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Elements","Platonists"],"text":"THE\n                    PHILOSOPHICAL and MATHEMATICAL\n                       COMMENTARIES OF PROCLUS,\n                                  ON\n                 THE FIRST BOOK OF EUCLID\u2019S ELEMENTS.\n\n                          TO WHICH ARE ADDED,\n          A History of the Restoration of PLATONIC THEOLOGY,\n                       BY THE LATTER PLATONISTS:\n                  And a Translation from the Greek of\n                    PROCLUS\u2019S THEOLOGICAL ELEMENTS.\n\n                            IN TWO VOLUMES.\n\n  [Illustration]\n\n\n                                VOL I.\n\n                    LONDON, PRINTED FOR THE AUTHOR:\n And Sold by T. PAYNE and SON; B. WHITE and SON; J. ROBSON; T. CADELL;\n   LEIGH and CO.; G. NICOL; R. FAULDER; and T. and J. EGERTON. 1792.\n\n                    [Price Two Guineas in Boards.]\n\n\n\n\n      _Extracts from Curiosities of Literature._ Second Edition.\n                     Printed for Murray. Page 385.\n\nMr. _T. TAYLOR_, the Platonic Philosopher and _the modern Plethon_,\nconsonant to that philosophy, professes Polytheism.[1]\n\n\n\n\nThe Reader is requested to correct the following Errors.\n\n\n   Page 4. of the Dissertation, Vol. I. line 8. for _admitted_,\n   read _omitted_. Page 16. line 8. for _from_, read _form_. Page\n   51. Vol. I. of the Commentaries, line 16. instead of _They are\n   surely not the_, &c. read _For surely it cannot be said that\n   there are_, &c. Line 17. for _but we_, read _but that we_. And\n   line 19. for _is by much prior to_, read _is by a much greater\n   priority_.\n\n   Vol. II. page 18. line 26. for _and one is_, read _and one part\n   is_. And line 27. for _another_, read _the other_. Page 114.\n   line 13. for _The angle_, read _Let the angle_; and instead\n   of _is bisected_, in the same line, read _be bisected_. Page\n   411. line 2. for _is filled with intellect_, read _fills\n   intellect_. And line 3. for _it also participates_, read _also\n   it participates_.\n\n\n\n\n                                  TO\n\n                              THE SACRED\n\n                                MAJESTY\n\n                                  OF\n\n                                TRUTH.\n\n\n\n\n                               PREFACE.\n\n\nThe design of the present work is to bring us acquainted with\nthe nature and end of Mathematics in general, and of Geometry in\nparticular: and in the execution of this design our Author has\ndisplayed an uncommon elegance of composition, and a most valuable\nstore of recondite learning. He is not content with every where\nunfolding the full, and most accurate meaning of Euclid; but he\ncontinually rises in his discourse, and leads us into the depths of the\nPythagoric and Platonic philosophy. We are surprised to find an use in\nGeometry, which at present it is by no means suspected to afford. For\nwho would conceive that it is the genuine passage to true theology, and\nthe vestibule of divinity? This, indeed, is by no means the case when\nit is studied for lucre, and applied to mechanical purposes; for then\nthe soul is neither elevated nor enlightened; but degraded and filled\nwith material darkness. Hence these Commentaries are alone valuable\nto the liberal part of mankind, who look beyond _sense_ for\ncertainty; and who prefer things desirable for their own sakes, before\nsuch as minister to the necessities of life.\n\nThe translation of this work is attended with great difficulty and\nlabour; not only from the sense of the philosopher, which is always\nprofound, and frequently obscure, but from the great incorrectness of\nthe Greek edition, in which, exclusive of numberless typographical\nerrors, entire sentences, essential to the connection, are frequently\nomitted; and in one place two pages of the Latin translation are\nwanting in the original, as will be shewn in our following notes.\nIndeed, the Latin translation of Francis Barocius the Venetian, (Patav.\n1560.) which was made from a variety of manuscripts, is inconceivably\nvaluable; for the diagrams, so necessary to a work of this kind, but\nwhich are omitted in the Greek, are here inserted; and the version\nis every where faithful, and sufficiently perspicuous to those who\nare conversant in the ancient philosophy. Barocius justly cautions\nthe reader not to compare his version with the printed Greek, which\nhe observes is rather lacerated than printed; as indeed, without his\ntranslation, it is impossible for any one to read the half of this\ninvaluable work, even though he should be as perfect in Greek as in\nhis native tongue. If I had not, therefore, fortunately acquired this\ntranslation, which is at present very rare, I would have by no means\nengaged in this arduous undertaking. Barocius, indeed, gives evident\nproofs of his possessing the philosophical genius, by the excellence of\nhis translation, and his preface to the reader; and it is greatly to\nbe lamented that he did not adorn his version with explanatory notes,\nwhich this profound work frequently demands, and which he was doubtless\nwell qualified to accomplish. This defect I have endeavoured, as far\nas I was able, to supply; and at the same time have been cautious\nneither to weary the reader by prolixity, nor by too much brevity to\nleave him destitute of proper information. In the distribution of\nthe first book of this work into chapters, I have followed the order\nof Barocius, because it is natural and obvious; and must beg leave\nto solicit the reader\u2019s indulgence for using the words _partible_\nand _impartible_, differently from their common signification. These\nwords I have generally employed to express the meaning of \u03bc\u03b5\u03c1\u03b9\u03c3\u03c4\u03cc\u03c2[2]\nand \u03b1\u03bc\u03b5\u03c1\u03b9\u03c3\u03c4\u03bf\u03c2[3] in the Greek, as I do not conceive that the words\n_divisible_ and _indivisible_ always convey their full signification.\nI have likewise used _quadrangle_ instead of square, and _quinquangle_\nfor the word pentagon. For if \u03c4\u03c1\u03af\u03b3\u03c9\u03bd\u03bf\u03c2 be rendered _triangle_, why\nshould not \u03c4\u03b5\u03c4\u03c1\u03b1\u03b3\u03ce\u03bd\u03bf\u03c2 be rendered _quadrangle_? And, as Barocius\nobserves, why, for a similar reason, should not \u03c0\u03b5\u03bd\u03c4\u03ac\u03b3\u03c9\u03bd\u03bf\u03c2 and\n\u1f11\u03be\u03ac\u03b3\u03c9\u03bd\u03bf\u03c2 be rendered _quinquangle_ and _sexangle_; and so of the rest?\nUniformity is always desirable when it can be obtained; and is no where\nso necessary as in scientifical disquisitions.\n\nIt is likewise necessary to inform the reader, that though I have\nalways endeavoured to give the faithful meaning of my Author, yet I\nhave occasionally paraphrased his sense, when most obscure, and added\nsuch elucidations of my own, as I either thought necessary to the\nfull comprehension of his matter; or which were naturally excited by\nthe fire and spirit of the Original. If it shall appear that I have\nsucceeded in the execution of this work, and rendered it intelligible\nto the lovers of truth, I shall rejoice in my success, and consider\nmy labours sufficiently rewarded. The applause of the multitude I am\nneither likely nor desirous to gain; but I am anxious to procure the\napprobation of the discerning few, who know that the age of philosophy\nis past; and who esteem the works of her ancient heroes as the most\nprecious treasures which have escaped the ravages of time.\n\n       *       *       *       *       *\n\nTime, indeed, is like a deep and rapid river; whatever is trifling and\nlight, is precipitately borne on its surface, and what is valuable and\nweighty, sinks to its bottom. Hence, the superficial observer collects\nnothing more than the rubbish, which it is forever devolving into the\nabyss of oblivion; while the profound and contemplative genius explores\nthe depths of the stream, and accounts himself happy if he can gather\nany of the pearl which its bottom contains. Thus the discoveries of\nexperimental philosophy, float like straws on the surface, while\nthe wisdom of Pythagoras and Plato lies concealed in the depths of\nthe river. I am well aware it will be said, that the reverse of this\nsimilitude is true; that the modern philosophy is the pearl, and\nthe ancient the stubble; and that the former will be celebrated by\nposterity, and increase in reputation when the latter shall scarcely\nbe known. But let us attentively examine the truth of this assertion,\nand shut our ears to the unsubstantial echoes of popular applause. Is\nit reasonable to suppose that men of such exalted abilities, as the\nPythagoric and Platonic philosophers possessed, even in the estimation\nof their opponents, accompanied with the greatest advantages of birth\nand fortune, and the most unwearied attention, have discovered nothing\nvaluable, and have left nothing behind them, but jargon and reveries?\nIs it to be supposed, that in an age when philosophy was almost\nadored; when it was esteemed by kings, cultivated by noblemen, and\neven reverenced by the vulgar; when empire was relinquished for its\npursuit, and every danger encountered for its possession: is it to be\nsupposed, that nothing but delusion was the offspring of so glorious a\nperiod, and nothing but folly the reward of such generous endurance?\nOr shall we say, that the discovery of truth was reserved for the age\nof experiment; and that she is alone to be apprehended in the infinite\nlabyrinth of particulars? That she is to be investigated with the\ncorporeal senses, and not with the powers of intellect; and that\nthe crucible, the alembic, and the air-pump, are the only mediums of\ndetection? If this be the case, truth is material, and may be calcined,\ndistilled, and rarefied, like any other corporeal substance. It is\nno longer eternal and immutable, but perishable and fluctuating; the\nphantastic subject of sensible inspection, and not the steady and real\nobject of the permanent energies of science. Shall we call this the\nage of philosophy, in which talents are prostituted for sustenance,\nand learning submits to the impudence of wealth? Shall we say that we\nhave strengthened the cause of philosophy, by demolishing her schools;\nand increased her independence, by enlarging the empire of commerce?\nWhere shall we find the man, who is at present reverenced for the\nprofession of teaching speculative truth, or indeed who teaches it\nat all? Or should we chance to meet with such an obsolete character,\nshall we find him supported by the profession? It is a well known fact,\nthat men formerly lived in the highest esteem by its propagation: it\nis equally as notorious, that a man at present would starve by such an\nattempt. Dare we assert, that the reason of this difference must be\nascribed to the greater liberality, and more philosophical spirit of\nthe present age? Shall we not rather say, that the period, in which\nthese ancient heroes lived, was the golden age of philosophy;--a\nperiod so different from the present, as to appear fabulous on the\ncomparison? For mark the distinguishing characteristics of our\ninferiority. The great object of ancient philosophy, was an accurate\nspeculation of principles and causes: but that of the modern, is a\nconfused investigation of effects. And if pursuits participate of the\nnature of their subjects, and causes are more noble than effects, the\nancient philosophy must undoubtedly be more elevated than the modern.\nAgain, the object of the Pythagorean and Platonic philosophy was to\nmake its possessors wise and virtuous; and to elevate them above the\ncommon frailties and imperfections of degraded humanity; and this end\nwas happily accomplished in its votaries, as their lives abundantly\nevince: but the object of modern philosophy, is a promotion of the\nconveniencies and refinements of life, by enlarging the boundaries\nof traffic; and the Mathematical Sciences are studied solely with a\nview to this enlargement. The design of the ancient philosophy was\nto remove the causes of wonder, by contemplating effects in their\ncauses: the grand object of the modern, is to increase admiration, by\nattempting to investigate causes through the infinity of particular\neffects. So that philosophy, as Mr. Harris justly observes, now\nends where it formerly began. For either there is no such thing as\nscience, or if its existence be admitted, it can never be obtained by\nexperimental enquiries; as these must be liable to all the inaccuracy\nand imperfection of their material subjects.\n\nIn short, the philosophy of Pythagoras and Plato will be found, when\nimpartially considered, to contain every thing which can enlighten the\nmind, improve the morals, and exalt the character of man. It is built\non the steady basis of truth, and will survive the wreck of ages. Its\nfoundation is deep, and its summit reaches the heavens. It is a mighty\nrock, which modern systems _may_ assail, like a raging sea; but,\nlike stormy waves, they will only be broken about its impenetrable\nsides. To war against wisdom is folly; for opposition in this case is\nthe destruction of its author. The moderns may, indeed, expect, because\ntheir merit is raised by the present age, above that of the ancients,\nto appear as giants in the eyes of posterity; but they will only verify\nthe elegant observation of the poet[4], that\n\n    Pygmies are Pygmies still, though perch\u2019d on Alps,\n    And Pyramids are Pyramids in vales.\n\n\n\n\n                                   A\n\n                             DISSERTATION\n\n                                ON THE\n\n                    PLATONIC DOCTRINE OF IDEAS, &c.\n\n\n\n\n                              SECTION I.\n\n\nThe Platonic doctrine of Ideas has been, in all ages, the derision of\nthe vulgar, and the admiration of the wise. Indeed, if we consider\nthat ideas are the most sublime objects of speculation, and that their\nnature is no less bright in itself, than difficult to investigate, this\nopposition in the conduct of mankind will be natural and necessary;\nfor, from our connection with a material nature, our intellectual eye,\nprevious to the irradiations of science, is as ill adapted to objects\nthe most splendid of all, \u201cas the eyes of bats to the light of day[5].\u201d\nAnd yet (as I presume, it will appear from the following discourse),\nunless the existence of these lucid beings is admitted, there can\nbe no such thing as science; nor, indeed, any genuine knowledge at\nall. Hence, an enquiry concerning their nature and reality, is highly\nproper, as an introduction to the ensuing Commentaries, in which they\nare considered as the stable pillars of all truth, and the prolific\nprinciples of the universe.\n\nBut previous to this enquiry, it is proper to observe, that Plato\nwas not the inventor, though he was a strenuous asserter, of ideas;\nfor, in the Sophista he affirms, that ideas were the discovery of men\nwho excelled in wisdom and piety, and who contended for an invisible\nessence. Diogenes La\u00ebrtius, indeed, asserts, that Plato received\nthe doctrine of ideas from Epicharmus. But Epicharmus was not their\ninventor, because Pythagoras, and others of still higher antiquity,\nwere well acquainted with ideas; so that it may be affirmed, with much\ngreater truth, that Plato was instructed in their nature by Philolaus\nhis preceptor, and the disciple of Pythagoras. For Pythagoras, after\nhis mysterious manner, signified ideas by numbers. But, prior to\nPythagoras, Orpheus was an asserter of ideas, and called Jupiter, or\nthe dimiurgus of the world, \u201cthe idea of all things.\u201d And, according\nto Syrianus, the mundane sphere, celebrated by Empedocles, is no other\nthan the ideal world; so that the doctrine of ideas is as ancient as\nthat of wisdom itself.\n\nBut to begin with our enquiry: in the first place, without universals\nthere can be no science; for the flowing and perishing nature of\nparticulars is perfectly foreign from that stability and duration which\nis requisite to objects of invariable truth. Neither is it possible,\nthat infinite individuals can exist without the subsistence of one\ncause endued with infinite power; for all multitude must necessarily\noriginate from one, and must resemble its cause in as great a degree of\nperfection as its nature can admit; by a diffused infinity, shadowing\nforth that infinite power which subsists in indivisible union. Hence,\nif this be the case, and if infinite men, horses, and a multitude\nof other univocals, are produced in an infinite time, an unity of\ninfinite power must be the source of each, according to which they are\ngenerated in a terminated manner to infinity in the universe. Again,\nall animals are transmuted from that which is in capacity (i. e. seed),\ninto energy. But if this be true, it is requisite there should be some\nanimal in the universe, subsisting in ever-vital energy, which may\ncall forth that which is concealed in dormant capacity, into perfect\nactuality. Thirdly, the celestial orbs would not perpetually revolve\nin the same spaces, and after the same manner, unless one and the same\nuniversal number, or idea, ruled in each. So, likewise, there is a\nnatural number in every animal; or those of the same species, would\nnot always (when perfect) be distinguished with the same invariable\norgans; nor would they be subject to puberty and old age, at the same\ntime, unless they were detained by the same measure of nature. Besides,\nthe participation of universals, is evident in every sensible object.\nThus, the rational nature is united with every individual man. Thus,\nanimal subsists in a lion and a horse, in a man and a dog. And thus\nthe pentad, or number five, is participated in the five fingers, and\nthe duad in the nostrils, eyes, hands, and feet. But since these do\nnot subsist without a cause, but are perfected by certain determinate\nnatures, it is necessary there should be an universal animal, in the\nwhole of nature, separate from sensibles, by means of which this\nsensible animal is generated. And that there should subsist in nature\na pentad, through which the hands are always adorned with that number\nof extremities; and a duad, from which the two eyes and nostrils are\nderived. But if nature does not possess these numbers from herself, as\nshe is not the first cause of all, but derives them from another cause,\nin the same manner as matter from nature, it is necessary there should\nbe universals and numbers prior to nature, subsisting in far greater\npurity and perfection.\n\nAgain, we may demonstrate the existence of ideas as follows: if the\nDeity, in fabricating the universe, operated essentially (and there\nis no other way in which we can conceive him to operate), he must\nfabricate the universe, an image of Himself. But, if this be the case,\nhe contains in himself, in the manner of an exemplar, the causes of\nthe universe; and these causes are no other than ideas. Besides, this\nconsideration is not to be omitted, that the perfect must necessarily\nantecede and preside over the imperfect; unity over multitude; the\nimpartible over the partible; and that which is perpetually the same,\nover that which admits of variation and change. From whence it may be\ninferred, that things do not originate from baser natures, but that\ntheir gradual processions end in these; and that they begin from the\nmost perfect, best, and most beautiful natures. But let us pursue this\nreasoning more minutely, as it affords the strongest arguments for the\nexistence of ideas.\n\nWhen the Deity fabricated the various species of animals, and bestowed\non them the different senses, it was doubtless with a view to the\nbenefit of their possessors, as he foresaw, that without these, the\nanimal could neither provide for its own support, nor defend itself\nfrom surrounding dangers. But may we not enquire from whence this\nprevious perception originated? For it is not to be supposed, that he\nfirst made animals destitute of senses, and so, being admonished by\ntheir sudden destruction, afterwards assigned them to their nature.\nShall we say, this foreknowledge was the result of a reasoning process?\nBut then, we again ask, What were the principles of this ratiocination?\nFor if they originated from other reasonings, it is necessary, at\nlength, to arrive at something prior to these discursive operations,\non which they ultimately depend; since all reasoning must be founded\non indemonstrable principles. Was sense, then, or intellect the\nprinciple of this previous perception? But, sense, in the present\ninstance, had not then a being, for it could not exist prior to the\nanimal nature: it was, therefore, intellect. But if intellect be the\nrepository of certain propositions, and the conclusion be science, it\nmust follow, that there could not then be a consultation of any thing\nsensible. For the principle and the conclusion must both depend on\nsomething intelligible. Besides, may we not ask, how such a habit of\nthought arose before the existence of a sensible nature! It is absurd\nin the extreme, to say from chance, and to resolve it into a sudden\nvolition of the Deity, is an assertion that may, indeed, satisfy vulgar\nminds, but can by no means quiet the restless spirit of philosophical\ninvestigation. Since, to suppose the cause of the universe, actuated by\nsudden volitions, is to place him on a level with the vilest natures,\nand subject him to the irrational impulses of the brute. Hence we\ninfer that the formation of animals, and by the same arguments of\nthe world, was not the result of any reasoning process. For, indeed,\nargument and foreknowledge cannot with propriety be attributed to\nthe Deity; but when they are ascribed to him, we must consider it as\nnothing more than an indication of his constituting particulars, in a\nmanner somewhat similar to the providence of a wise man, in inferior\nconcerns. For, in subordinate natures, whose operations cannot take\neffect prior to enquiry, reason is necessary, on account of the\ninferiority of that power which precedes the reasoning energy. In\nlike manner, foreknowledge is necessary, because a power is wanting\nto its possessor, which might render him superior to its use. For\nforeknowledge is directed to this end, that one particular circumstance\nmay take place in preference to another. But if it be requisite that\nevery energy in the Deity should be void of defect, and if it is not\nlawful that any thing should be present with him, which is not total\nand universal, it is necessary that all things should be contained in\nevery thing essential to the nature of the Deity. Hence, since even\nfuturity is with him present, there is nothing in him posterior; but\nwhat is present in him becomes posterior, by its participation in\nanother. If then futurity be present with the Deity, it is necessary it\nshould be so present, as if foreknown in a posterior nature; that is,\nin such a manner that nothing may be wanting to any being; and that is,\nlastly, so that every thing may be complete.\n\nBesides, reasoning cannot, by any means, belong to an eternal essence\nlike the deity; for if this be admitted, he must be forgetful of his\nformer operations. And if, in consequence of reasoning, he produces\nmore perfect natures afterwards, his works could not be perfectly\nbeautiful before: but if they were beautiful before, they must be\nco-existent with their cause, i.e. they must be eternally beautiful,\nantecedent to the reasoning energy. Again, if we suppose the supreme\nintellect, the demiurgus of the world, to operate by enquiry, his\nenergy could not be spontaneous, and truly his own; but his essence\nwould be similar to that of the artificer, who does not derive his\nproductions from himself, but procures them as something adventitious\nby learning and enquiry. But if the universe was not formed by\ndeliberation, it must be co-existent with its cause, and reside in\nhis essence; for if it be not co-existent there must have been some\nparticular time, in which its artificer determined on its production;\nand this determination must have been the result of a reasoning\nprocess, concluding that it would not be good to produce it before\nthat particular time, (from whence, by the way, we infer the eternity\nof the world.) And if the universe be co-existent with its author, it\nmust perpetually emanate from his nature, and be dependent on it, like\nthe shadow on its forming substance. But in this case, its archetype\nmust be contained in the essence of its author; for every cause is\nthat primarily, which its effect is secondarily. And hence we infer,\nthat if the sensible universe be replete with forms of every kind, the\nexemplars of those forms, must subsist in immaterial perfection, in the\nartificer of the world.\n\nIf this sensible world, then, be formed according to the exemplar of\nthat which is intelligible; may we not say, with the great Plotinus,\nthat it is requisite universal animal should there primarily subsist in\nperfect vital energy, containing all things in its omniform essence.\n\u201cHence (says he[6]) the heavens are there a divine animal, replete with\nideal stars. Earth too does not there subsist solitary, but is much\nmore vital than this corporeal earth, for it is full of intellectual\nlife. The sea too is there, and all water subsisting in life, and an\never-abiding stream. For how is it possible that any thing not vital,\ncan be the progeny of life itself? He, therefore, who enquires from\nwhence animals originate in the intelligible world, might as well\nenquire from whence all life, and soul, and universal intellect, arose.\nFor here there is nothing indigent nor defective, but every thing is\nperfect and exuberant. Here they all flow from one fountain, not as\nfrom a certain spirit, or heat, but as if from an universal quality,\npossessing and preserving in itself, all qualities; such as sweetness,\naccompanied with fragrance of smell, the vigour of wine, and the\nstrength of all juices, bright colours, and whatever is perceived by\nthe taste.\u201d\n\n3. Such then are the arguments which the Platonic philosophy affords\nin defence of ideas; the existence of which was so evident to Plato,\nthat, in the Sophista, he compares those who oppose the friends of\nideas to the giants of old, warring, as it were, on celestial souls,\nand such as are engaged in sublime investigations. Let us now consider\nto what universals these lucid beings are confined; since, according to\nthe Pythagoreans and Platonists, there are not ideas of all universal\nconceptions. \u201cFor, in the first place (says Syrianus[7]), there are no\nideas of things evil and base, because these subsist in nature rather\nby a privation and absence of ideas. And, on this account, they are\nsaid to exist contrary to nature. Nor, secondly, of negations, for\nthese are destructive of the bound and limitation which is attributed\nto every thing from the unifying and comprehending nature of ideas;\nand hence, separation is rather the result of material infinity than\nof that which is formal or ideal. Nor again, are there any ideas of\nthings which at different times receive a variety of conditions. For\nthese participate of transmutation from a moveable cause, but not from\nthe immoveable and stable illustration of ideas. Nor again of parts,\nsuch as the hand, head, fingers, and the like. For the causes of\nthings existing entire, produce whole species and forms; not divided\nabout the parts of these, like the reasons of nature. But neither did\nthese wise men place in intellect the determinate causes of accidents\nin bodies, such as sweetness and whiteness. For they considered that\nnatural reasons were sufficient for the production of accidents.\nNor again, of composites, as of a wise man. For since ideas are\nsimple, they preside over the simple essence of every thing. But the\ncomposition and division of things is the business of our intellect;\nideas, at the same time, and that intellection which is co-ordinate\nto ideas, being exempt from all these, on account of superlative\nsimplicity. Neither, therefore, must we establish ideas of things\ngenerated from dissimilars, such as mules; nor of fruit produced by\nengrafting from different trees. For all these have a posterior and\nadventitious generation, and are not the work of nature alone, nor\nof nature proceeding according to her own reasons, but, as it were,\ncompelled to labour contrary to her own determinations. Hence it is\nmanifest, that all art, which imitates nature, and alone ministers\nto the use of mortal life, is separated from the cause of ideas. But\nneither are the works which, depending on the purpose of the soul, are\nperfected by a concourse of many causes, and which we are accustomed\nto call the operations of fortune, to be conjoined to the cause of\nideas. For things which are there perfected, are eternal, and subsist\nperpetually the same, free from the nature of contingent events. It\nremains, therefore, that ideas must be confined to universal and\nperfect essences, and to whatever confers to their natural disposition;\nas for instance, to man, and every thing perfective of man, such as\nwisdom and virtue. For ideas existing as the generative and energetic\ncauses of the perfection of every thing, distribute being to essences,\nand convert them to the inexhaustible plenitude of their own omniform\nnatures.\u201d\n\n4. But let us now consider the nature of numbers; for as every form\nis a number, according to the Pythagoreans[8], a speculation of this\nkind must afford no small light to the arduous investigation of ideas.\nWill it not, therefore, be proper, in the first place, to enquire,\nwith the great Plotinus[9], whether multitude is not a departure and\ndistance from _one_, so that infinity itself is a separation\nfrom unity in the extreme, because it is no other than innumerable\nmultitude; that on this account it becomes evil; and that we contract a\nsimilar nature when departing from intellectual unity, we are divided\nby sensible multitude? For a being then properly becomes many, when no\nlonger able to remain collected in itself, the same, it is diffused\nabroad, and thus, being dispersed, is variously extended; so that\nwhen, by diffusion, it is absolutely deprived of unity, it becomes\nperfect multitude, destitute of that universal cement, which unites\none part with another. But whenever the conciliating one is present,\nthen that which was scattered and diffused, becoming permanent by its\nbounding power, passes into magnitude. But if any one should deny the\nsubsistence of unity, asserting that one is no where to be found,\nwhich is not some particular one; and should hence affirm, that what\nis called one abstractedly, is only a certain affection of the soul\ntowards any being; we ask, what prohibits the appellation of essence,\nfrom being nothing more than an affection of the soul, and consequently\nthe existence of _being_, a delusion? For we predicate unity of\nparticulars with as great propriety as being. I am well aware, that\nphilosophers of the present day will answer, that we have an evident\nproof of the reality of being, from its agitating the soul, and\nbecoming apparent in the phantasy: to which we reply, that in like\nmanner, the soul is agitated, and the imagination influenced about the\n_one_. For every individual as much excites the perception of one,\nas of being.\n\nBesides, it is necessary to enquire whether we behold this passion and\nconception of the soul, as one or multitude. And again, when we say\nnot one, we do not then possess one from the thing itself; for we say\nthat one is not contained in that individual. And hence we must possess\none in our own nature, and this must reside in the soul, separate\nfrom that which is denominated some particular one. But here it may\nbe objected, that the one we possess is received from externals, and\nis nothing more than a conception of the mind, produced by the thing\nitself. For it will be said, that as multitude is nothing besides a\nnumber of individuals, which are called many, so one is nothing besides\none thing; and is formed by thought separating that one particular from\nothers. To this we reply as follows:\n\nHow can it be consonant to reason to suppose that the conception of\n_one_ arises from the sensation of some one particular subject?\nFor one particular man, who is discerned by sense, is by no means the\nsame with one itself, since, if this were the case, thought could never\npredicate one of that which is not a man. Besides, as cogitation, on\nbeholding the different positions of things, affirms that this is here\nor there, so when it perceives an individual, pronounces one; for that\npassion is not vain, nor does it assert one of a non-entity. Nor must\nwe think it predicates ones, because this individual is different from\nanother; for when cogitation affirms such a thing is this, and not\nanother, it declares, in the mean time, that the other is _one_.\nLikewise when it affirms that any thing is this _alone_, it then\ndeclares, that what is alone is one: on which account, it predicates\none, prior to alone. Besides, if there be multitude, it is necessary\nthat one should antecede; since when it predicates _many_, it\npronounces more than one. And when it affirms that an army contains a\nmultitude of men, it conceives the soldiers reduced to one order.\n\nFor thought, indeed, does not permit multitude to remain perfect\nmultitude, destitute of the conciliating power of unity; in which\nvery circumstance, the subsistence of one is evinced; for acutely\nand swiftly perceiving the one which results from order, it reduces\nthe nature of the many into one. Besides, we affirm that a house and\nan army are each one, but that a house is more one than an army, on\naccount of the continuity of its parts. If therefore, one is contained\nmore in that which is continued than in that which is discrete, and\nstill more in what is perfectly indivisible, it is evident that _the\none_ is a certain nature, and has a real being. For it is impossible\nthat the more and the less should take place among things which have no\nsubsistence. If then it be not possible to understand any thing without\none or two, or some other number, it is by no means proper to deny\nexistence to that, without which we cannot comprehend the existence\nor properties of any being: but it is requisite _that_ nature\nshould antecede all discourse, and intelligence, which is every where\nnecessary to their existence.\n\nAgain, if unity has no real subsistence, and is nothing more than\na name or conception of the mind, it may be destroyed without the\ndestruction of its subject. The unity, therefore, of a house may be\ntaken away, without the ruin of a house. But if a house is nothing more\nthan certain materials, reduced into one form, this is impossible. And,\non the contrary, the alteration of that subject, of which unity is\npredicated, can make no real alteration in unity (on this hypothesis)\nany more than the death of a man can affect his name. When, therefore,\na body, of which _one_ was predicated, is divided into a multitude\nof parts, there is no real alteration made in the unity of the body,\nbecause unity is nothing more than a name.\n\nIt was in consequence of this reasoning, and perceiving that unity\nwas participated by every being, that the Pythagoreans placed a\nsuper-essential one at the top of the universe, intelligibly abstracted\nfrom all beings in simplicity and excellence of nature. For they\nconsidered, that unless there was a self-subsisting one in all things,\nthere could neither be universals nor particulars. Not the first,\nbecause they are by nature _one_ and _many_. But it is requisite that\nthe one itself, should preside over that which is not one alone.\nNor again, the second, because they are many and one, (that is,\nthey participate more of multitude than unity, and their nature is\ndetermined more by the _many_ than the _one_.) And because of things\nin participation, unless an unparticipated one is added, there can\nbe no cause of union to beings; in the same manner as the cause of\nessence to beings, is taken away by those who deny that being itself,\nis the principle of all essence. For as the good itself, is the one\nprinciple of good to the universe, and is nothing besides good; and\nas a self-motive nature, which is nothing besides self-motion, is the\ncause of motion to all things; so all things proceed from being itself,\nand all united natures receive their union from _the one_, abstracted\nfrom all things.\n\nHence (such is the absolute dominion of unity), continued quantities\nwould have no existence without its participation; for when they are\ndivided, so far as they lose unity, they change their being into some\nother form. Hence, the bodies of plants or animals, which are each of\nthem one, when they fly from unity, and are dissipated into multitude,\nimmediately lose the essence they formerly possessed, and become\nsomething else; which new state of being they likewise possess so far\nas they are one. Add too, that health then flourishes in the corporeal\nframe, when the body is conciliated into one; then beauty flourishes,\nwhen the power of one connects the members into proportion and consent;\nand then virtue reigns in the soul, when the soul is reduced into one\nsimilitude with that which is divine.\n\n5. But let us now investigate the nature of numbers. All number,\naccording to the Pythagoreans, originates from unity and the indefinite\nduad; the first having the relation of form, and the second, that\nof matter to all the orders of numbers. But they likewise divided\nnumber into two kinds, essential and monadic. The essential number\nthey considered as first subsisting in the intelligible world,\ntogether with being, and from thence distributed into all the various\ngradations of forms. But the monadic, or that which is composed from\ncertain units, they justly considered as nothing more than the image\nof essential number. And with respect to the numbers which the human\nsoul participates, these from its imperfect condition have a middle\nsubsistence; i. e. they exist in a vital, gnostic, and speculative,\nbut not in an operative manner. Hence, when receiving one thing with\nanother, we affirm, that they are two, as a dog and a man, or two men;\nor when we compute more than two, as ten, and say that there is a decad\nof men, this number is not essential to the two or ten individuals,\nnor is it to be conceived as subsisting in sensible natures; but it\nis purely quantity. But when we distribute this ten, into units,\nwe produce the principle of quantity, and generate a subject in\nopinion[10], capable of participating the essential decad of our soul.\nBut when, considering man in himself, we affirm that he is a certain\nnumber, as the duad, composed of animal and rational, we do not observe\none mode in this predication; but so far as by a discursive operation\nof the soul, we numerate, we effect a particular quantum; but so far as\nthe subjects are two, and at the same time both one (since one fills\nthe essence of both, and in both unity is contained), we pronounce\nanother, and an essential number: and this duad is not of a posterior\norigin, nor alone signifies a certain quantity, external to the\nsubject, but a duad subsisting in the essence of man, and containing\nhis nature. For here we do not produce a number by a discursive\noperation, while we pursue essential natures. But when we number any\nten things, which are not connected by any conciliating unity, like\na choir, or an army, then this decad, which we predicate of the ten\nparticulars, subsists alone in our numerating soul, which renders\nthe ten individuals in opinion, a definite quantum. But in a choir,\nor an army, essential number is participated exclusive of that which\nsubsists in our soul. And if it be enquired how number subsists in the\nhuman soul, we must say, that the soul, by her self-moving energies,\nprocreates number, while she numerates, and by this energy, causes the\nexistence of quantity; in the same manner as in walking, we give rise\nto a certain motion. Thus, monadic number, or a collection of units of\nvarious kinds, subsists in opinion, in a manner correspondent to that\nof geometrical figures; and by this means participates the essential\nnumber of the soul. For as a triangular figure in the phantasy, is the\nrecipient of a triangular nature, or of triangle itself; so every three\nunits in opinion, receive the essential triad of the soul, and, by this\nmeans, form a definite quantum.\n\nIn short, as in every being we may discern the resemblances of matter\nand form, so in the pentad, or any other number, the five units, which\nare the subject of participation, and the quantity of the number,\noriginate from the duad; but the form, that is the pentad itself, from\nunity. For every form is an unity, which unites its subject quantity,\nand connects it with its ideal species. It is, therefore, requisite\nto understand, that the two principles of mathematical numbers are\nresident in our souls, with which every mathematical number is\nco-existent; I mean unity, comprehending in itself all the forms of\nnumbers, and which corresponds to unity in intellectual natures; and\nthe duad, endued with a generative power, of a formless nature, and of\ninfinite virtue; and which is called boundless, on account of its being\nthe image of never-failing and intelligible duality. Hence, the unity\nof the soul, with a never-ceasing energy, continually distinguishes and\nforms all the orderly processions of her numbers, suffers no vacuum\nto intervene, and leaves no quantity formless and innumerable. Hence\ntoo, no essential number of the soul, as for instance, the pentad,\nis composed from substance and accident, as a white man; nor from\ngenus and difference, as man from animal and biped; nor again, from\nfive unities mutually touching each other, like a bundle of wood; nor\nfrom things mixt, like water and wine, nor from things subsisting\nby position, in the manner that stones compose a house; nor lastly,\ndoes it subsist like things numerable; for it is not because they are\ncomposed from indivisible units, that they possess any thing besides\nunits. For many points are indivisible, yet quantity is not produced\non this account; but because they participate of two natures, the one\ncorresponding to matter, and the other to form. Lastly, it is not\nproper to say, that the number seven (and so of any other number), is\ncomposed from the triad and the tetrad; for units, indeed, composed\nwith units, form a subject adapted to the reception of the heptad,\nor the ideal and essential number seven; but the definite numerical\nquantity seven, is formed from so many units, and the ideal heptad.\nHence, as the soul of the shipwright gives form to the timber, from her\ninherent art; so the numerative soul, from the unity endued with the\nrelation of a principle which she possesses, gives form and subsistence\nto all her inherent numbers. But there is this difference between the\ntwo, that the shipwright\u2019s art is not essential to our nature, and\nrequires manual operation, because it is conversant with sensible\nmatter; but the numerative art is essentially inherent in the soul,\nand is therefore present with all men, and possesses an intellectual\nmatter, which it easily forms without the assistance of time. And\nthis, perhaps, is what deceives many, who think that the heptad is\nnothing more than seven units. For the imagination of the vulgar,\nunless it first perceives a thing destitute of ornament, and afterwards\nthe operations of the adorning artificer supervening its nature; and\nlastly, beholds the thing perfect, and invested with form, cannot be\npersuaded that it possesses two natures, the one formless, but the\nother endued with an energetic and forming power.\n\nAnd here it is necessary to observe, that though unity is the form\nof all arithmetical forms, yet it is chiefly the form of the decad.\nFor what unity is simply to all the series of numbers, that the decad\nis to the following hundreds, thousands, and millions; from whence,\naccording to a secondary progression, it is denominated unity. As\nintellect, therefore, is the form of all things, but especially of the\nsoul, so unity, though it is the idea of all numbers, yet especially\nof the decad. But the reason why the Pythagoreans extended ideal\nnumbers no farther than ten, is because this number is the ultimate\nperfection of beings, containing all things in its omniform nature.\nFor all proportion subsists within the number ten; the arithmetical in\na natural progression of numbers from unity; the geometrical in the\nnumbers 1, 2, 4, and 1, 3, 9, and the harmonical in the numbers 2,\n3, 6, and 3, 4, 6. And since the causes of all things are contained\nin numbers, as far as to the decad[11], it is superfluous to suppose\nexemplars of the following numbers.\n\nIf it should be asked in what manner we must conceive number as\nsubsisting in the intelligible world, we answer, with the great\nPlotinus, that we must conceive it as subsisting in being itself,\nwith a power of impelling it to the production of multitude. \u201cHence\n(says he, Ennead vi. lib. vi.) number is either the essence or the\nenergy of being, and animal itself, and intellect is number. But,\nperhaps, we must call being, number united (\u1f00\u03c1\u03b9\u03b8\u03bc\u1f78\u03c2 \u03b7\u03bd\u03c9\u03bc\u03ad\u03bd\u03bf\u03c2),\nbut beings, number evolved, or unfolded; (\u1f10\u03be\u03b5\u03bb\u03b7\u03bb\u03b5\u03b3\u03bc\u03ad\u03bd\u03bf\u03c2 \u1f00\u03c1\u03b9\u03b8\u03bc\u1f78\u03c2)\nintellect, number moving in itself; (\u1f00\u03c1\u03b9\u03b8\u03bc\u1f78\u03c2 \u1f10\u03bd \u1f10\u03b1\u03c5\u03c4\u1ff6 \u03ba\u03b9\u03bd\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03c2)\nand lastly, animal, number comprehending (\u1f00\u03c1\u03b9\u03b8\u03bc\u1f78\u03c2 \u03c0\u03b5\u03c1\u03b9\u03ad\u03c7\u03c9\u03bd.\u201c) It\nwas in consequence of this reasoning, that the Pythagoreans called\nideas numbers; because the gradual evolution of these from ineffable\nunity, produced all the beautiful variety of forms. Their exalted\nconceptions of numbers, likewise, originated from the same sublime\ntheory. Hence, [12]Pythagoras, in the sacred discourse, calls number\n\u201c_the ruler of forms and ideas_.\u201d But [13]Philolaus, \u201c_the\ncommanding and self-begotten container of the eternal duration of\nmundane concerns_.\u201d And [14]Hippasus, and all those who were\ncalled \u1f00\u03ba\u03bf\u03c5\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03bf\u1f76 (or such as were yet under the probation of the\nquinquennial silence), \u201c_the first exemplar of the mundane fabric,\nand the judiciary instrument of its artificer_.\u201d\n\n6. And here I cannot but take notice, with regret, of the very\nunphilosophical mistake committed by that great mathematician Dr.\nBarrow[15]: I say, with regret, on account of the extraordinary\nobligations I am under to his writings, for my proficiency (whatever it\nmay be) in mathematical learning. But respect must yield to the truth.\n\u201cUnity, says he, is not indivisible. (For how ex. gr. can 2/6 added\nto 4/6 be equal to unity, if unity be indivisible and incomposed, and\nrepresent a point) but rather only unity is properly divisible, and\nnumbers arise from the division of unity.\u201d Here the Doctor evidently\nconfounds sensible units, which are the subjects of vulgar practical\narithmetic, with those units which are the objects of science.\nEvery individual sensible object, is indeed an unit, so far as it\nparticipates the connecting and conciliating power of an immaterial\n_one_: but the unity which stands at the top of speculative\narithmetic, is perfectly indivisible, or arithmetic would cease to be a\nscience. The truth of this is evident from Euclid\u2019s definition: \u201cUnity\n(says he) is that according to which each of the things which are, is\ncalled one.\u201d But if unity be a composite, the definition is false;\nsince a composite, or a certain multitude, can never be the cause of\nunity, but the contrary. And that this immaterial _one_ subsists\nin sensible natures, has, I hope, been sufficiently proved in the\npreceding part of this discourse. But the Platonic Theo[16] of Smyrna,\nfully establishes the indivisibility of unity, as follows: \u201cUnity is\nterminating quantity, the principle and element of numbers, which\nremains undiminished by the most immense multitude of subtractions,\nand being deprived of all number, continues firm and fixt, because\nit is impossible for division to proceed beyond the bound of unity.\nThus, if we separate any one corporeal substance into parts, the\n_one_ again becomes _many_; and by subtracting the several\nparts, we end in one part; and from this remaining part, again divided,\narises multitude; and by taking away every part, we again arrive at\n_one_. So that _one_, considered as _one_, is incapable\nof diminution, and perfectly indivisible. On the contrary, every\nnumber is diminished by division, and is separated into parts less\nthan itself; as the number 6 into 3 and 3, or into 4 and 2, or into 5\nand 1. But unity in sensible particulars, if divided, is diminished\nafter the manner of body, and by section is distributed into parts less\nthan itself: but it receives increase after the manner of number; for\ninstead of the one, multitude is produced. In this sense, therefore,\nis unity indivisible; for nothing is divided into parts greater than\nitself. But that which is cut into parts greater than the whole,\nand into parts equal to the whole, is divided as number. Thus, for\ninstance, if any one sensible body is divided into six parts, 1, 1, 1,\n1, 1, 1, these shall be equal to the whole; but by a section into 4\nand 2, it is divided into parts greater than the whole, considered as\n_one_; for 4 and 2 considered as numbers, exceed unity, and the\nbody was supposed to be one. Unity, therefore, as number is perfectly\nindivisible. But unity is called by the Greek word \u03bc\u03bf\u03bd\u03ac\u03c2, only, or\nalone, either because it remains immoveable, and does not desert\nitself, nor surpass the bounds of its nature (for it remains the same,\nhowever multiplied into itself, through an infinite progression) or\nbecause it is placed separate and apart from the multitude of other\nnumbers, it is denominated the _monad_, or _one_.\u201d\n\nIn consequence of this very mistaken hypothesis, which opposes not\nonly all the wisdom of antiquity, but the sublimest truths, the Doctor\nasserts, that an arithmetical cypher is the principle of numbers; and\nthat it is analogous to a point in geometry. Just as if a cypher, which\nis nothing more than a mark expressive by its position with numbers,\nof a certain quantity, had a real existence, and was productive of\nnumber: when, at the same time, any other arbitrary character would\nserve the same purposes, if applied in a similar manner. It must surely\nafflict every thinking mind, to see how dreadfully the mechanical\nsystem of philosophy, which has been so long in fashion, enslaves and\nperverts the minds of its votaries; for there cannot, I think, be a\nmore egregious instance of its fatal tendency, than the present, in\nwhich _nothing_ is considered as the foundation of that noble\nscience, arithmetic; which was deservedly placed by the ancients, in\nthe first rank of the mathematical disciplines. Such a foundation,\nindeed, _may be_ proper to the _mechanical philosophy_, but\nis very ill adapted to support the solid fabric of the arithmetical\nscience. But let us attend to the arguments of this most learned man,\nin defence of so strange an assertion, \u201cA cypher, or arithmetical\nnothing (says he) is really the bound of every number coming between\nit and the numbers next following, but not as a part. A cypher being\nadded to, or taken from a number, does neither increase nor diminish\nit; from it is taken the beginning of computation, while itself is not\ncomputed; and it bears a manifest relation to the principal properties\nof a geometrical point.\u201d But in what manner are we to conceive the\n_nothing_ which intervenes between any two numbers, to be their\nterm or boundary? For Euclid defines a term to be the extremity of\nany thing; implying by the extremity, something belonging to that of\nwhich it is the bound. But how can a cypher, or _nothing_, in any\nrespect belong to number, or _something_? For if _nothing_\nbe a boundary, merely from its intervention, a point existing between\nany two disjoined lines, though at the greatest distance from each,\nmust be their common boundary, which is evidently absurd. Besides, what\nrelation does it bear to a point, which is endued with a generative\npower, by its flux forming the simple extension of a line, and, at\nthe same time, every where limiting its progression, and subsisting\nin infinite capacity in its every part? Where are the real and divine\nproperties to be found in an arithmetical nothing, which Proclus,\nin the following Commentaries, exhibits in a point? And how can\ncomputation originate from a mere non-entity?\n\nBut a little consideration will convince us, that this Saracen, or\nIndian cypher, is nothing more than an arbitrary character, invented\nfor the purpose of facilitating computation. For, suppose the letter\n(_a_) to be placed in its stead, and to signify, when connected\nwith the mark for unity, ten, or ten multiplied by one; when connected\nwith the mark for two, ten multiplied by two, and so on. And again,\nwhen placed twice after unity, let it express the second power of ten,\nor one hundred, in this manner, _a a_; when thrice connected,\none thousand, or the third power of ten, and so on: shall we say, in\nconsequence of this, that (_a_) is the bound of numbers, and\nthe principle of arithmetic? Or, shall we not rather say, that it is\nan arbitrary symbol, like any other algebraic character, having no\nreal connection with numbers, and depending, for its existence and\napplication, entirely on the will of its inventor. But this opinion is\ntoo absurd to need any farther refutation.\n\n7. It may here, perhaps, be expected, that I should explain how, in the\nlanguage of Syrianus[17], \u201cdivine number proceeds from the immortal\nretreats of unity, until it arrives at the divine tetrad[18];\u201d and\nthat I should unfold the properties of the tetractys, according to\nthe Pythagoreans; but an undertaking of this kind, would not only far\nexceed the limits of this dissertation, but, perhaps, in the present\nage, might be justly deemed, by the lovers of wisdom, a prostitution\nand profanation of the most exalted truths. Enough, I hope, has been\nsaid to excite the curiosity, and rouse the attention of the thinking\nand liberal part of mankind; and those who understand what is here\nbriefly delivered, may apply themselves, with advantage, to Proclus on\nPlato\u2019s Theology, where they will find all the mysteries of numbers\nunravelled; and to the works of the great Plotinus, who will lead them\ninto the penetralia of the most recondite wisdom. But, in perusing\nthe works of these great men, the reader must not expect to find the\nsublimest truths explained in a familiar manner, and adapted, like\nmany modern publications, _to the meanest capacities_. For this,\nindeed, is impossible to be effected. \u201cMankind (says Petvin[19]),\nare not to be made any more truly knowing than happy by another\u2019s\nunderstanding.--There is no man can at once convey light in the higher\nsubjects, to another man\u2019s understanding. It must come into the mind\nfrom its own motions, within itself: and the grand art of philosophy,\n_is to set the mind a-going_; and, even when we think nothing\nof it, to assist it in its labour.\u201d After which he observes, that\n\u201cthe ancients never attempt to lead us into knowledge, by a continued\nchain of reasoning; on the contrary, they write in such a manner,\nas to force us to think for ourselves.\u201d And, previous to this, he\nremarks, \u201cthat there are certain truths acquired by a long exercise\nof reason, both in _particular_, and likewise in those subjects\nthat are most _general_, as much, perhaps, out of the reach of the\ngreatest mathematician, as Sir Isaac Newton\u2019s speculations are above\nthe capacity of some that are now called mathematicians.\u201d The truth of\nthis observation is sufficiently evinced, in Plato\u2019s definition of a\nphilosopher (in his Sophista), \u201cThe philosopher (says he) is the man\nwho sufficiently sees one idea every way extended through many, every\none of them lying apart; and many ideas different from one another,\nexternally comprehended under one.--And farther, one idea, throughout\nall manys, wrapt up in one; and many ideas, every way separate or\ndiscreet. This is to have the knowledge to discern how ideas, as they\nare general, agree and disagree.\u201d Now, he who thinks that a perception\nof this kind may be acquired by barely reading an accurate discourse\non the nature of ideas, composed in intelligible terms, without, at\nthe same time, employing a long course of profound meditation, and\npatient thought, knows but little the difficulty of the task, and\nuntil he changes his opinion will never be the wiser. But the folly\nand presumption of men, with respect to this sublime philosophy, is\nreally unpardonable; for there are very few who conceive that much\nprevious instruction is requisite to its acquisition; but almost\nevery man decides peremptorily on the most abstract speculations, and\nreckons himself sufficient for the most profound investigations. In the\nsciences and arts they are willing to proceed to perfection by gradual\nadvances; but they consider philosophy as easy, of instant access, and\nhastily approach to her embraces with an assured confidence of success.\nThough, like unhappy Ixion, through their presumption, instead of a\ngoddess, they grasp nothing but an empty cloud. Plato was so sensible\nof this truth, that, in his seventh epistle to Dion, he expressly\naffirms, that he neither has written, nor ever will write explicitly\nconcerning these sublime speculations; \u201cFor a thing of this kind (says\nhe) cannot be expressed by words, like other disciplines, but by a\nlasting familiarity, and conjunction of life, with this _divine\nobject_, a bright light[20] on a sudden, as it were leaping from\na fire, will illuminate the soul, and there preserve and nourish its\nsplendor. He adds, that a publication of such concerns, is alone\nuseful to a few of mankind, who from some small vestiges previously\ndemonstrated, are sufficiently sagacious to their invention. But it\nwill fill others partly with a base contempt, and partly with a rash\nand vain confidence, as if they had now learned some very excellent\nthings.\u201d He then subjoins the following instance of the difficulty\nattending such an undertaking: \u201cThere are three things (says he),\nfrom which science must necessarily be produced; but the fourth is\nscience itself. And it is requisite to establish the fifth as that\nwhich is the object of knowledge, and has a true existence. One of\nthese is the name of a thing; the second its definition; the third\nthe resemblance; the fourth science. Now take each of these, desiring\nto learn what we have lately asserted, and think concerning them all,\nin a similar manner. A circle is called something, whose _name_\nwe have just expressed. After this follows its definition, composed\nfrom nouns and verbs. For that which every where is equally distant\nfrom the extremes to the middle, is the definition of that which we\nsignify by the name of a round, and a circumference, and a circle. But\nthe third is the circle which may be painted, or blotted out, which\nmay be made by a wheel, or destroyed. None of which affections, the\ncircle itself, which each of these respects, suffers, as being of a\ndifferent nature. But the fourth is science, and intellect, and true\nopinion about these. And this again must be established as one whole,\nwhich neither subsists in voice, nor in corporeal figures, but in\nintellect and intelligence. It is therefore manifest, that this fourth\nis different from the nature itself of the circle, and again different\nfrom the three we have previously mentioned. But among the number of\nthese, intellect, by its relation and similitude, proximately adheres\nto the fifth, while the rest are more remote from its nature. The same\nmay likewise be affirmed of a straight and crooked figure, of colour,\nand of the good, the beautiful, and the just. And again, of every\nbody, whether fashioned by the hand, or the work of nature, whether\nfire or water, and the rest of this kind; likewise of every animal,\nand the manners of animals; and of all actions and passions. For\nunless, among these, some one, after a manner, receives that fourth,\nhe will never perfectly participate the science about the fifth.\u201d He\nthen proceeds to shew in what respect each of the preceding four are\ndifferent from the fifth. \u201cEvery circle (says he) which by the hands\nof men is either painted, or fashioned by a wheel, is plainly contrary\nto our fifth. For it every where participates of the right-line. But\nwe must affirm, that the circle itself has neither more nor less of\nany thing whatever; that is, it possesses in itself, nothing of a\ncontrary nature. Besides, none of these are endued with any stability\nof name. For nothing hinders our applying the appellation of straight\nto that which we now denominate round, and calling the straight by\nthe denomination of the round; nor will there be any less stability\nin these, when their names are changed into the contrary. The same\nreasoning is likewise true of definition, since it is composed from\nnouns and verbs, which possess no stability. And in a variety of ways,\nit may be proved, that no one of these four is certain and firm.\u201d Now,\nthis fifth division of Plato\u2019s entirely respects ideas, considered as\nflourishing in intellect; by a conjunction with which, we acquire true\nintelligence, and the perfection of human knowledge. The first three of\nthe preceding are obnoxious to various mutations; the fourth less; but\nthe last is perfectly stable and invariable. The three first are rather\nconversant about the qualities of things, about the image and shadow;\nthe fourth raises us to the participation of truth; but the fifth to\ntruth itself, and permanent essence. In the first degrees almost all\nare conversant; in the fourth a few; in the fifth, all the gods, but a\nvery small part of mankind, as it is asserted in the Tim\u00e6us. The four\nfirst may be known, indeed, without the fifth, confusedly; but from\nthe knowledge of the fifth they become perfectly manifest, as effects\nfrom the knowledge of their cause. But we cannot, by any means, attain\nto the apprehension of the fifth, unless we have been first accurately\nconversant with the rest; for from our imperfect condition we are\ncompelled to rise from difference to identity, from multitude to unity,\nand from shadow to substance. While we investigate the knowledge of\nthings, if we are alone desirous to apprehend their resemblance (which\nis the case with the multitude) we shall be placed in the third degree,\nand may easily acquire the object of our pursuit. But if we should\nfortunately possess the true philosophical genius, which is rare in the\nextreme, and aspiring to the fifth degree, should, by a happy event,\nattain to its conjunction, though such a contact is clearer and more\ncertain than all knowledge; yet it is difficult to express it in words,\nand to manifest it to others. And the reason of this is obvious: first,\nbecause words are wanting, which exactly correspond to the essence of a\nthing, since these are only the symbols of shadows. Secondly, because\nwe speak with those, who are alone conversant with shadows, and are on\nthis account derided by them, when they find that our fifth does not,\nby any means, accord with material resemblances, which they consider as\nthe only realities.\n\n8. And here a question very naturally presents itself for our solution,\nwhether the soul, while united with the body, is able to perceive\nideas, without the assistance of the phantasy, For it seems difficult\nto apprehend how the soul, thus depressed and weighed down with the\nterrene mass, should be able to raise herself to the supernal light\nof ideas, and become united with their refulgence. The opinion of the\nPeripatetics is well known, that some phantasm must always accompany\nintelligence; but this is denied by the Platonists, and I think with\ngreat reason. For the operations of intellect are not dependent on\nthe phantasy, though the perceptions of the latter proceed from the\nenergies of the former. Besides, as Plotinus beautifully observes, our\nmost vigorous energies are accompanied with the least animadversion;\nand there is no absurdity in supposing that by increasing the force of\nintellectual energy, we may speculate free from all imagination; since\nthe phantasms attending our conceptions, became weak in proportion\nas the intellectual sight increases in vigour. On this account, the\nPlatonists affirm, that the moral virtues free us from the vehemence\nof perturbations; but the contemplative from imagination, and the\nsenses. Hence too, the sciences may be called living waters; in which\nthe wings of the soul being dipt, her feathers, which were either\nseparated or broken by her lapse into body, are repaired, and restored\nto a resemblance of their former perfection. For the wings are the\npowers of the soul, leading to intelligibles: but the feathers are as\nwell the natural instincts to good and truth, as reasons inserted in\nthe soul; which either fall off, or are broken by her descent into\nbody, and conjunction with its ruinous bonds. But these are repaired\nand invigorated by the sciences, which, like living streams, flowing\nfrom the fountains of ideas, restore life and perfection to the soul.\nHence Plato, in the Ph\u00e6drus, asserts that these wings of the soul\nare increased by every thing which confers to supernal elevation; as\nbeauty, wisdom, and the like; and by a convenient metaphor, in the same\ndialogue, he considers the chariot of the souls lives, her charioteer,\nand the horses by which her car is drawn; and lastly, every thing which\ncontributes to the elevation of the soul, and her conjunction with\nintellect and ideas. We may therefore conclude, that this conjunction\nis possible to be effected, though it is rarely obtained; and that it\nis a flight too arduous and sacred for the groveling and sordid; a\nsplendor too bright for the sensible eye; and a contact too ineffable\nto be described by the unstable composition of words.\n\nBut I cannot conclude this section, without soliciting the reader\u2019s\nattention to a comparison of the difference between the ancient\nphilosophy, and that invented by Mr. Locke, and the moderns. According\nto Mr. Locke\u2019s system ideas are formed from sensible particulars, by a\nkind of mechanical operation; so that truth is something by its nature,\nposterior to sensation, and entirely dependent on it for existence.\nAccording to Plato, ideas are eternal and immaterial beings, the\noriginals of all sensible forms, and the fountains of all evidence\nand truth; so that on this system truth ranks among the first, and\nnot in the last of things; and would still retain its nature, though\nthe corporeal senses were no more. According to Mr. Locke, the soul\nis a mere _rasa tabula_, an empty recipient, a mechanical blank.\nAccording to Plato, she is an ever-written tablet, a plenitude of\nforms, a vital and intellectual energy. On the former system, she is\non a level with the most degraded natures, the receptacle of material\nspecies, and the spectator of delusion and non-entity[21]. Hence,\nher energies are nothing but somnolent perceptions, and encumbered\ncogitations; for all her knowledge terminates in sense, and her science\nin passion. Like a man between sleeping and waking, her visions are\nturbid and confused, and the phantoms of a material night, continually\nglide before her drowsy eye. But on the latter system, the soul is the\nconnecting medium of an intelligible and sensible nature, the bright\nrepository of all middle forms, and the vigilant eye of all cogitative\nreasons. Hence she is capable of rousing herself from the sleep of a\ncorporeal life, and emerging from this dark Cimmerian land, into the\nregions of light and reality. At first, indeed, before she is excited\nby science, she is oppressed with lethargy, and clouded with oblivion;\nbut in proportion as learning and enquiry stimulate her dormant powers,\nshe wakens from the dreams of ignorance, and opens her eye to the\nirradiations of wisdom. On Mr. Locke\u2019s system, the principles of\nscience and sense are the same, for the energies of both originate\nfrom material forms, on which they are continually employed. Hence,\nscience is subject to the flowing and perishable nature of particulars;\nand if body and its attributes were destroyed, would be nothing but\na name. But on the system of Plato, they differ as much as delusion\nand reality; for here the vital, permanent, and lucid nature of ideas\nis the fountain of science; and the inert, unstable, and obscure\nnature of sensible objects, the source of sensation. On Mr. Locke\u2019s\nsystem, body may be modified into thought, and become an intelligent\ncreature; it may be subtilized into life, and shrink, by its exility,\ninto intellect. On that of Plato, body can never alter its nature by\nmodification, however, it may be rarefied and refined, varied by the\ntransposition of its parts, or tortured by the hand of experiment. In\nshort, the two systems may be aptly represented by the two sections of\na line, in Plato\u2019s Republic. In the ancient, you have truth itself, and\nwhatever participates of the brightest evidence and reality: in the\nmodern, ignorance, and whatever belongs to obscurity and shadow. The\nformer fills the soul with intelligible light, breaks her lethargic\nfetters, and elevates her to the principle of things; the latter\nclouds the intellectual eye of the soul, by increasing her oblivion,\nstrengthens her corporeal bands, and hurries her downwards into the\ndark labyrinths of matter.\n\nNor is it wonderful there should be so great a difference between the\ntwo systems, and so much in favour of the ancients, if we consider the\ngreat advantages these ancients possessed over the moderns in every\nthing which contributes to the advancement of philosophy. For, in\nthe first place, they lived in an age when abstract investigations\nwere in the greatest request, and the professors of such pursuits in\nthe highest estimation. Besides this, they united the most exalted\nabilities with the most unwearied attention and obstinate perseverance;\nthey devoted their whole lives to the search of truth; and relinquished\nevery thing which might be an obstacle to its acquisition. We may\nadd, likewise, the advantages of a language extremely philosophical;\nand a freedom from the toil of learning any tongue but their own. Now\nthe reverse of all this is the portion of the moderns: for in the\npresent age, abstract speculations are ridiculed; and its professors\ndespised. The pursuit of truth is considered as perfectly consistent\nwith ordinary avocations, and is rather prosecuted as a relief from\nthe toils of business than as a thing desirable for its own sake,\nand of the greatest dignity and worth. Hence, a few years desultory\napplication at a college, where language is one of the first objects\nof attention, qualifies a modern for philosophy, raises him above\nPythagoras and Plato, and persuades him, with presumptuous confidence,\nto enter the lists against these venerable heroes. And lastly, all\nmodern languages are barbarous with respect to the Greek; falling far\nshort of its harmony and energy, its copiousness and propriety. If\nsuch then be the true state of the case, what judgment must we form\nof men who, with all these disadvantages, philosophized without the\nassistance of the ancients, despising their works, and being ignorant\nof their contents? Shall we call it prudence or presumption, wisdom\nor folly? Truth will certainly pronounce the latter; and the general\nvoice of posterity will confirm her decision. There are two egregious\ninstances in our own country of this daring presumption; I mean Bacon\nand Locke. The former of these is celebrated for having destroyed the\njargon of the schoolmen, and brought experimental enquiries into\nrepute; and for attempting to investigate causes through the immensity\nof particular effects. Hence, he fondly expected, by experiment piled\non experiment, to reach the principle of the universe; not considering\nthat his undertaking was as ridiculous as that of the giants of old,\nwho attempted to invade the heavens, by placing Ossa upon Pelion, and\nOlympus upon Ossa; and ignorant that\n\n    Heaven still, with laughter, the vain toil surveys,\n    And buries madmen in the heaps they raise.\n\nThe latter of these, Mr. Locke, is applauded for having, without\nassistance from the ancients, explained the nature, and exhibited\nthe genuine theory of human understanding. But that this applause\nis false, the preceding comparison between his and the ancient\nphilosophy, may evince; and the variety of other self-taught systems\nwhich, like nocturnal meteors, blaze for a while, and then vanish\nin obscurity, abundantly confirms. Had these men, indeed, when they\njustly derided the barbarous writings of the schoolmen, explored the\nworks of antiquity, penetrated the wisdom they contain, and enriched\ntheir native language with its illustration, they had doubtless been\ncelebrated by the latest posterity: but, desirous of becoming masters\nin philosophy by intuition, they disdained the instruction of the\nancients, and vainly attempted to soar on artificial wings to the\nvery summit of science. They are, however, destined, like Icarus, to\na precipitate fall; for the influence of time, which is continually\ndissolving the cement of their plumes, is likewise continually\nweakening their force, and will at last effect their final separation.\nAnd thus much concerning the doctrine of ideas, and numbers, according\nto Pythagoras and Plato.\n\n\n\n\n                            SECTION II.[22]\n\n\nBut let us now consider the properties of the demonstrative syllogism,\nand endeavour to unravel its intricate web; appointing Aristotle for\nour guide in this arduous investigation. For an enquiry of this kind\nis naturally connected with the doctrine of ideas, as it enables us to\ngain a glimpse of the universals participated in mathematical forms,\nand to rise to the principles of science. It brings us acquainted with\nthe laws which bind demonstration; and teaches us that objects of\nintellect are alone the objects of science, and the sources of truth.\n\nPrevious to the acquisition of all learning and ratiocinative\ndiscipline, it is necessary we should possess certain natural\nprinciples of knowledge, as subservient to our future progress and\nattainments. Thus, in every science there are some things which require\nan immediate assent as soon as proposed; whose certainty is too evident\nand illustrious to stand in need of any demonstrative proof deduced\nfrom that particular science which, like stately pillars, they equally\nsupport and adorn. Hence we are informed by the geometrician, that a\npoint is that which is destitute of all parts whatever; but we must\npreviously understand the meaning of the word _part_. Thus the\narithmetician defines an odd number, that which is divided according\nto unequal parts; but it is necessary we should antecedently know\nthe meaning of the word _unequal_. Thus, too, art as well as\nscience operates by antecedent knowledge; and hence the architect,\nthe statuary, and the shipwright, learn the names and the use of their\nrespective implements, previous to the exercise of the materials\nthemselves. This is particularly evident in the discursive arts of\nrhetoric and logic; thus the logician reasons by syllogism, the\nrhetorician by induction, and the sophist by digressions and examples;\nwhile each proceeds in an orderly progression from principles simple\nand evident, to the most remote and complicated conclusions.\n\n2. The antecedent knowledge of things may be divided into two parts:\nthe one a knowledge of their existence, or that they exist; the other\na knowledge of the terms expressive of their existence. Thus, previous\nto the enquiry why iron is attracted by the magnet, it is necessary\nwe should learn the reality of this attraction, and the general mode\nof its operation: thus too, in an enquiry concerning the nature of\nmotion and time, we must be previously convinced of their existence\nin the nature of things. The second division of antecedent knowledge\ntakes place in subjects whose very existence admits of a dispute: thus\nprevious to a solution of the questions, Whether there are any gods\nor not? Whether there is a providence or not? and the like, it is\nnecessary we should first understand the meaning of the terms; since\nwe in vain investigate the nature of any thing while we are ignorant\nof the meaning of its name; although, on the contrary, we may have a\nperfect conception of the meaning of some words, and yet be totally\nignorant whether the things they express have a real, or only an\nimaginary existence. Thus, the meaning of the word _centaur_ is\nwell understood by every one; but its existence is questioned by most.\n\n3. From hence it will easily appear, that no small difference subsists\nbetween _learning_ and _knowledge_. He who is about to understand the\ntruth of any proposition, may be said to possess a previous conception\nof its truth; while, on the contrary, it may happen that he who is in\nthe capacity of a learner, has no antecedent knowledge of the science\nhe is about to learn. Thus we attain to the distinct knowledge of a\nthing which we formerly knew in a general way; and frequently, things\nof which we were ignorant are learned and known in the same instant.\n\nOf this kind are the things contained under some general idea, of which\nwe possess a previous knowledge: thus, he who already knows that the\nthree interior angles of every triangle are equal to two right, and\nis as yet ignorant that some particular figure delineated on paper\nis a triangle, is no sooner convinced from inspection of its being\na triangle, than he immediately learns and knows: he learns it is a\ntriangle; he knows the equality of its angles to two right ones. That\nit is now a triangle he both sees and learns; but the equality of its\nangles he previously knew in that general and comprehensive idea, which\nembraces every particular triangle.\n\nIndeed, a definite knowledge of this triangle requires two conditions:\nthe one, that it is a triangle; and the other, that it has angles\nequal to two right. The first we receive from inspection; the second\nis the result of a syllogistic process; an operation too refined for\nthe energies of sense, and alone the province of _intellect_\nand _demonstration_. But demonstration without the knowledge of\nthat which is universal, cannot subsist; and since the proposition\nis universal, that in every triangle the angles are equal to two\nright; as soon as any figure is acknowledged to be a triangle, it must\nnecessarily possess this general property.\n\nHence we infer, that of the triangle delineated on paper, and\nconcealed, we are partly ignorant of this general property, the\nequality of its angles (because we are ignorant of its existence);\nand we partly understand it as included in that universal idea we\npreviously possessed. Hence too, it is evident that actual science\narises from a medium between absolute ignorance and perfect knowledge;\nand that he who possesses the principles of demonstration, possesses in\ncapacity the conclusions also, however complicated and remote; and that\nby an evocation of these principles from dormant power into energy, we\nadvance from general and abstracted knowledge to that which is sensible\nand particular.\n\n4. Two acceptations of knowledge may be admitted; the one common and\nwithout any restriction; the other limited and peculiar. Since all\nknowledge, whether arising from accidents, or supported by necessary\nprinciples, is called science. Knowledge, properly so called, arises\nfrom a possession of that cause from which a thing derives its\nexistence, and by which we infer the necessity of its existence; and\nthis constitutes simple and absolute science. Thus too, the definitions\nof those general conceptions and suppositions, which from their\nprimary nature are incapable of demonstration, are called science.\nBut the science which treats of the method of arriving at knowledge,\nis called demonstration; for every demonstration is a syllogism\nproducing science. Hence, if in every syllogism it is necessary that\nthe propositions should be the cause of the conclusion; and to know\nany thing properly, a knowledge of its cause is requisite; in the\npropositions of demonstration, both these conditions are required: that\nthey should be effective of the conclusion; and the causes of the thing\ndemonstrated.\n\nThus, from the ruins of a stately edifice, we may justly infer, that\nthe building was beautiful when entire; and from the smoke we may\ncollect the existence of the fire, though concealed: but the ruins of\nthe edifice are not the cause of its beauty; nor does fire originate\nfrom smoke, but, on the contrary, smoke is the natural result of fire:\nthe inference, therefore, is in neither case a demonstrative one.\nAgain, since every cause is both prior to, and more excellent than its\neffect, it is necessary that the propositions should be more peculiar,\nprimary, and excellent than the conclusions. And because we then know\na thing properly when we believe it to have a necessary existence,\nhence it is requisite that the propositions should be true; for if\nfalse, a false conclusion may ensue, such as, that the diameter of a\nsquare is commensurable with its side. But if every science arises\nfrom antecedent knowledge, demonstration must be founded on something\nprevious; and on this account it is requisite that the propositions\nshould be more known than the conclusions. The necessary properties,\nthen, of all demonstrative propositions, are these; that they exist\nas causes, are primary, more excellent, peculiar, true, and known,\nthan the conclusions. Indeed, every demonstration not only consists of\nprinciples prior to others, but of such as are eminently first; for\nif the assumed propositions may be demonstrated by other assumptions,\nsuch propositions may, indeed, appear prior to the conclusions, but\nare by no means entitled to the appellation of first. But others, on\nthe contrary, which require no demonstration, but are of themselves\nprobable or manifest, are deservedly esteemed the first, the truest,\nand the best. Such indemonstrable truths were called by the ancients,\naxioms, from their majesty and authority; as the assumptions which\nconstitute the best syllogisms derive all their force and efficacy from\nthese.\n\nAnd on this account, above all others, they merit the title of\nthe principles of demonstration. But here it is worth observing,\nthat these primary propositions are not the first in the order of\nour conceptions; but first to nature, or in the nature of things.\nTo us, that which is first is particular, and subject to sensible\ninspection; to nature, that which is universal, and far remote from\nthe apprehension of sense. Demonstration does not submit itself to the\nmeasure of our ingenuity, but, with invariable rectitude, tends to\ntruth as its ultimate aim; and without stopping to consider what our\nlimited powers can attain, it alone explores and traces out the nature\nof a thing, though to us unperceived and unknown.\n\nThis demonstrative syllogism differs not a little from others, by the\nabove property; the rest can as well educe a true conclusion from false\npremises, which is frequent among the rhetoricians, as that which is\nprior from that which is posterior; such as, Is every syllogism derived\nfrom conjecture?\n\nWith respect to the rest, as we have already confessed, they may be\nformed from principles that are true, but not from such as are proper\nand peculiar; as if a physician should endeavour to prove an orbicular\nwound the most difficult to coalesce and heal, because its figure is of\nall others, the most capacious; since the demonstration of this is not\nthe province of the physician, but of the geometrician _alone_.\n\n5. That proposition is called immediate, which has none superior to\nitself, and which no demonstration whatever can confirm: such as these\nare held together by the embraces of universals. There are some,\nindeed, united from that which is sensible and particular: thus, that\nthe garment is white, is an immediate proposition, but not of that\nkind whose principles require to be demonstrative ones; the cause\nof which we shall hereafter investigate. Of immediate propositions\nsubservient to the purposes of demonstration, some are of such a\nsuperior nature, that all men possess a knowledge of them without any\nprevious instruction; and these are called axioms, or general notions;\nfor without these all knowledge and enquiry is vain. Another species\nof immediate propositions is position; incapable of being strengthened\nby demonstration, yet not necessarily foreknown by the learner, but\nreceived from the teacher. With respect to the genus of position, one\nof its species is definition, and another hypothesis. Definition is an\noration, in which we neither speak of the existence, nor non-existence\nof a thing; but alone determine its nature and essence. It is common to\nevery hypothesis, not to be derived from nature, but to be the entire\nresult of the art of the preceptor.\n\nIt likewise always affirms the existence or non-existence of its\nsubject: such as, that motion is, and that from nothing nothing is\nproduced. Those which are not so perspicuous are called postulates, or\npetitions; as that a circle may be described from any centre, and with\nany radius; and such as these are properly hypotheses and postulates.\n\n6. We have now seen the privilege assigned to the principles of\ndemonstration:--whether or no our decision has been just, the\nensuing considerations will evince. We said that the assumptions in\ndemonstration were more known than the conclusions,--not indeed without\nreason, since through these our knowledge and belief of the conclusion\narises. For universally, that quality which is attributed to many\ndifferent things so as to be assigned to one through the medium of\nanother, abounds most in that medium by which it is transmitted to the\nrest.\n\nThus the sun, through the medium of the moon, illuminates the earth by\nnight; thus the father loves the preceptor through the medium of his\nchild. And in the first instance the moon is more lucid than any object\nit enlightens: in the second, the child possesses more of the father\u2019s\nregard than his preceptor. If then we assent to the conclusions through\nour belief of the principles alone, it is necessary that the principles\nshould be more known, and inherit a greater degree of our assent.\nHence, if it be true that the principles are more known than the\nconclusions, it follows, that either our knowledge of them is derived\nfrom demonstration, or that it is superior to any demonstrative proof;\nand after this manner we must conceive of those general self-evident\nnotions which, on account of their indemonstrable certainty, are\ndeservedly placed at the top of all human science.\n\nThese propositions not only possess greater credibility than their\nconclusions, they likewise inherit this property as an accession to\ntheir dignity and importance; that no contrary propositions deserve\ngreater belief; for if you give no more assent to any principle than\nto its contrary, neither can you give more credit to the conclusion\ndeduced from that principle than to its opposite. Were this the\ncase, the doctrine of these propositions would immediately lose its\ninvariable certainty.\n\n7. There are, indeed, some who, from erroneously applying what we\nhave rightly determined, endeavour to take away the possibility\nof demonstration. From the preceding doctrine it appears that the\nprinciples are more aptly known than the conclusions. This is not\nevident to some, who think nothing can be known by us without a\ndemonstrative process; and consequently believe that the most\nsimple principles must derive all their credit from the light of\ndemonstration.\n\nBut if it be necessary that all assumptions should be demonstrated by\nothers, and these again by others; either the enquiry must be continued\nto infinity, (but infinity can never be absolved), or if, wearied by\nthe immense process, you at length stop, you must doubtless leave those\npropositions unknown, whose demonstration was declined through the\nfatigue of investigation. But how can science be derived from unknown\nprinciples? For he who is ignorant of the principles, cannot understand\nthe conclusions which flow from these as their proper source, unless\nfrom an hypothesis or supposition of their reality.\n\nThis argument of the sophists is, indeed, so far true, that he who does\nnot understand that which is first in the order of demonstration, must\nremain ignorant of that which is last:--But in this it fails, that\nall knowledge is demonstrative; since this is an assertion no less\nridiculous than to maintain that nothing can be known. For as it is\nmanifest that some things derive their credit and support from others,\nit is equally obvious that many, by their intrinsic excellence, possess\nindubitable certainty and truth; and command our immediate assent as\nsoon as proposed. They inherit, indeed, a higher degree of evidence\nthan those we assent to by the confirmation of others; and these are\nthe first principles of demonstration: propositions indisputable,\nimmediate, and perspicuous by that native lustre they always possess.\nBy means of these, we advance from proposition to proposition, and\nfrom syllogism to syllogism, till we arrive at the most complicated\nand important conclusions. Others, willing to decline this infinite\nprogression, defend the necessity of a circular or reciprocal\ndemonstration. But this is nothing more than to build error upon error,\nin order to attain the truth; an attempt no less ridiculous than that\nof the giants of old. For since, as we shall hereafter accurately\nprove, demonstration ought to consist from that which is first, and\nmost known; and since it is impossible that the same thing should be\nto itself both prior and posterior: hence we infer the absurdity of\ncircular demonstration; or those syllogisms in which the conclusions\nare alternately substituted as principles, and the principles as\nconclusions. It may, indeed, happen, that the same thing may be both\nprior and posterior to the same; but not at one and the same time, nor\naccording to the same mode of existence. Thus, what is prior in the\norder of our conceptions, is posterior in the order of nature; and what\nis first in the arrangement of things, is last in the progressions of\nhuman understanding. But demonstration always desires that _first_\nwhich is prior in the order and constitution of nature. But the folly\nof such a method will more plainly appear from considering its result:\nlet us suppose every _a_ is _b_, and every _b_ is _c_; hence we justly\ninfer, that every _a_ is _c_. In like manner, if we prove that every\n_a_ is _b_, and by a circular demonstration, that every _b_ is _a_, the\nconsequence from the preceding is no other than that every _a_ is _a_;\nand thus the conclusion terminates in that from which it first began; a\ndeduction equally useless and ridiculous. However, admitting that, in\nthe first figure, circular demonstration may be in some cases adopted,\nyet this can but seldom happen from the paucity of reciprocal terms.\n\nBut that reciprocal terms are very few, is plain from hence: let\nany species be assumed, as _man_; whatever is the predicate of\nman, is either constitutive of his essence, or expressive of some\naccident belonging to his nature. The superior genera and differences\ncompose his essence, among which no equal predicate can be assigned\nreciprocable with man, except the ultimate differences which cannot be\notherwise than one, i. e., risibility, which mutually reciprocates\nwith its subject; since every man is risible, and whatever is risible\nis man. Of accidents some are common, others peculiar; and the common\nare far more in number than the peculiar; consequently the predicates\nwhich reciprocate with man, are much fewer than those which do not\nreciprocate.\n\n8. It is now necessary to enumerate the questions pertaining to\ndemonstration; and for this purpose, we shall begin with propositions,\nsince from these, syllogisms are formed; and since every proposition\nconsists of a subject and predicate, the modes of predication must be\nconsidered, and these are three which I call _total_, _essential_,\nand _universal_; a _total_ predication takes place when that which is\naffirmed or denied of one individual is affirmed or denied of every\nindividual comprehended under the same common species.\n\nThus, animal is predicated of every man, and it has this farther\nproperty besides, that of whatever subject it is true to affirm man, it\nis at the same time true to affirm animal.\n\nThose things are said to be _essentially_ predicated; first,\nwhen the predicate is not only total, but constitutes the essence\nof the subject; instances of this kind are, animal of man; tree of\nthe plantain; a line of a triangle; for a triangle is that which is\ncontained under three right-lines. But here we must observe, that\nnot every total predicate is an essential one; thus, whiteness is\npredicated of every swan, because it is inherent in every swan, and at\nevery instant of time; but because whiteness does not constitute the\nessence of a swan, it is not essentially predicated; and this, first,\nis one of the modes of essential predication of the greatest importance\nin demonstration. The second mode is of accidents, in the definition\nof which their common subject is applied: thus, a line is essentially\ninherent in rectitude, because in its geometrical definition, a line\nis adopted; for rectitude is no other than a measure, equally extended\nbetween the points of a line. In the same manner, imparity is contained\nin number; for what is that which is odd, but a number divided into\nunequal parts? Thus, virtues are resident in the soul, because, in\ntheir definition, either some part of the soul, or some one of its\npowers, is always applied. The third mode of essential predicates\npertains to accidents which are inseparably contained in some\nparticular subject, so as to exclude a prior existence in any other\nsubject; such as colour in superficies. The fourth mode is of things\nneither contained in another, nor predicated of others; and such are\nall individuals, as Callias, Socrates, Plato. Causes are likewise said\nto exist substantially, which operate neither from accident nor\nfortune.\n\nThus, digging up the ground for the purposes of agriculture, may be the\ncause of discovering a treasure, but it is only an accidental one. But\nthe death of Socrates, in despite of vigilance, is not the result of a\nfortuitous cause, but of an essential one, viz. the operation of poison.\n\n9. These posterior significations of essential predicates are added\nmore for the sake of ornament than use; but the two former have a\nnecessary existence, since they cannot but exist in the definition of\nnames which predicate the essence of a thing, and in subjects which are\nso entirely the support of accidents, that they are always applied in\ntheir definition. But it is a doubt with some, whether those accidents\nare necessary, which cannot be defined independent of their common\nsubject? To this we answer, that no such accident can, from its nature,\nbe contained in every individual of any species; for curvature is not\ncontained in every line; nor imparity in every number; from whence we\ninfer, that neither is curvature necessarily existent in a line, nor\nparity in number. The truth of this is evident from considering these\naccidents abstracted from their subjects; for then we shall perceive\nthat a line may exist without curvature, and number without imparity.\n\nAgain, I call that an _universal_ predicate, which is predicated\nof a subject totally and essentially, and considered as primarily\nand inseparably inherent in that subject: for it does not follow\nthat a predicate, which is total, should be immediately universal;\nfor whiteness is affirmed of every swan, and blackness of every\ncrow, yet neither universally. In like manner, a substantial\npredicate is not consequently an universal one; for the third mode\nof essential predicates, and the two following (instanced before)\ncannot be universal. Thus, colour, although inherent in superficies\nessentially, is not inherent in every superficies, and consequently\nnot universally. Thus again, Socrates, Callias, and Plato, though they\nexist essentially, are not universals, but particulars; and thus,\nlastly, the drinking of poison was an essential cause of the death of\nSocrates, but not an universal one, because Socrates might have died\nby other means than poison. If then, we wish to render an accurate\ndefinition of an universal predicate, we must not only say it is total\nand essential, but that it is primarily present to its subject and no\nother. Thus, the possession of angles equal to two right, primarily\nbelongs to a triangle; for this assertion is essentially predicated of\ntriangle, and is inherent in every triangle. This property, therefore,\nis not universally in figure, because it is not the property of every\nfigure, not of a square, for instance; nor as universal in a scalene\ntriangle: for although it is contained in every scalene, and in every\nequilateral, and isosceles triangle, yet it is not primarily contained\nin them, but in triangle itself; because these several figures inherit\nthis property, not from the particular species to which they belong,\nbut from the common genus _triangle_. And thus much concerning\n_total_, _essential_, and _universal_ predicates.\n\n10. Concerning that which is universal, we are frequently liable to\nerr; often from a belief that our demonstration is universal, when it\nis only particular; and frequently from supposing it particular when it\nis, on the contrary, universal. There are three causes of this mistake;\nthe first, when we demonstrate any particular property of that which is\nsingular and individual, as the sun, the earth, or the world. For since\nthere is but one sun, one earth, and one world, when we demonstrate\nthat the orb of the earth possesses the middle place, or that the\nheavens revolve, we do not then appear to demonstrate that which is\nuniversal.\n\nTo this we answer: when we demonstrate an eclipse of the sun to arise\nfrom the opposition of the moon, we do not consider the sun as one\nparticular luminary, but we deduce this consequence as if many other\nsuns existed besides the present.\n\nJust as if there were but one species of triangles existed; for\ninstance, the _isosceles_; the equality of its angles at the base\nwould not be considered in the demonstration of the equality of all its\nangles to two right ones: but its triangularity would be essential,\nsupposing every species of triangles but the isosceles extinct, and no\nother the subject of this affection. So when we prove that the sun is\ngreater than the earth, our proof does not arise from considering it\nas this particular sun alone, but as sun in general; and by applying\nour reasoning to every sun, if thousands besides the present should\nenlighten the world. This will appear still more evident, if we\nconsider that such conclusions must be universal, as they are the\nresult of an induction of particulars: thus, he who demonstrates that\nan eclipse of the sun arises from the opposition of the moon between\nthe sun and earth, must previously collect, by induction, that when any\nluminous body is placed in a right-line with any two others opaque,\nthe lucid body shall be prevented, in a greater or less degree, from\nenlightening the last of these bodies, by the intervention of the\nsecond; and by extending this reasoning to the sun and earth, the\nsyllogism will run thus:\n\n  Every lucid body placed in a right-line with two others opaque,\n  will be eclipsed in respect of the last by the intervention of\n  the second;\n\n  The sun, or every sun, is a luminous body with these conditions;\n\n  And consequently the sun, and so every sun, will be eclipsed to\n  the earth by the opposition of the moon.\n\nHence, in cases of this kind, we must ever remember, that we\ndemonstrate no property of them as singulars, but as that universal\nconceived by the abstraction of the mind.\n\nAnother cause of deception arises, when many different species agree in\none ratio or analogy, yet that in which they agree is nameless. Thus\nnumber, magnitude, and time, differ by the diversities of species;\nbut agree in this, that as any four comparable numbers correspond\nin their proportions to each other, so that as the first is to the\nsecond, so is the third to the fourth; or alternately, as the first\nto the third; so is the second to the fourth: in a similar manner,\nfour magnitudes, or four times, accord in their mutual analogies and\nproportions. Hence, alternate proportion may be attributed to lines\nas they are lines, to numbers as they are numbers, and afterwards to\n_times_ and to bodies, as the demonstration of these is usually\nseparate and singular; when the same property might be proved of all\nthese by one comprehensive demonstration, if the common name of their\ngenus could be obtained: but since this is wanting, and the species\nare different, we are obliged to consider them separately and apart;\nand as we are now speaking of that universal demonstration which is\nproperly one, as arising from one first subject; hence none of these\nobtain an universal demonstration, because this affection of alternate\nproportion is not restricted to numbers or lines, considered in\nthemselves, but to that common something which is supposed to embrace\nall these, and is destitute of a proper name. Thus too we may happen\nto be deceived, should we attempt to prove the equality of three\nangles to two right, separately, of a scalene, an isosceles, and an\nequilateral triangle, only with this difference, that in the latter\ncase the deception is not so easy as in the former; since here the\nname triangle, expressive of their common genus, is assigned. A third\ncause of error arises from believing that to demonstrate any property\ninherent after some particular manner in the whole of a thing, is\nto demonstrate that property universally inherent. Thus, geometry\nproves[23] that if a right-line falling upon two right-lines makes\nthe outward angle with the one line a right-angle, and the inward\nand opposite angle with the other a right one, those two right-lines\nshall be parallel, or never meet, though infinitely extended. This\nproperty agrees to all lines which make right-angles: but they are not\nprimarily equidistant on this account, since, if they do not each make\na right-angle, but the two conjointly are equal to two right, they\nmay still be proved equidistant. This latter demonstration, then, is\nprimarily and universally conceived; the other, which always supposes\nthe opposite angles right ones, does not conclude universally; though\nit concludes totally of all lines with such conditions: the one may\nbe said to conclude of a greater _all_; the other of a lesser.\nIt is this greater _all_ which the mind embraces when it assents\nto any self-evident truth; or to any of the propositions of Euclid.\nBut by what method may we discover whether our demonstration is of\nthis greater or lesser all? We answer, that general affection which\nconstitutes universal demonstration is always present to that subject,\nwhich when taken away, the predicate is immediately destroyed, because\nthe first of all its inherent properties.\n\nThus, for instance, some particular sensible triangle possesses these\nproperties:--it consists of brass; it is scalene; it is a triangle. The\nquery is, by which of these we have just now enumerated, this affection\nof possessing angles equal to two right is predicated of the triangle?\nTake away the brass, do you by this means destroy the equality of its\nangles to two right ones? Certainly not:--take away its scalenity, yet\nthis general affection remains: lastly, take away its triangularity,\nand then you necessarily destroy the predicate; for no longer can this\nproperty remain, if it ceases to be a triangle.\n\nBut perhaps some may object from this reasoning, such a general\naffection extends to figure, superficies, and extremities, since, if\nany of these are taken away, the equality of its angles to two right\ncan no longer remain. It is true, indeed, that by a separation of\nfigure, superficies, and terms, from a body, you destroy all the modes\nand circumstances of its being; yet not because these are taken away,\nbut because the triangle, by the separation of these, is necessarily\ndestroyed; for if the triangle could still be preserved without figure,\nsuperficies, and terms, though these were taken away it would still\nretain angles equal to two right; but this is impossible. And if all\nthese remain, and triangle is taken away, this affection no longer\nremains. Hence the possession of this equality of three angles to two\nright, is primarily and universally inherent in triangle, since it is\nnot abolished by the abolition of the rest:----such as to consist of\nbrass; to be scalene, or the like. Neither does it derive its being\nfrom the existence of the rest alone; as figure, superficies, terms;\nsince it is not every figure which possesses this property, as is\nevident in such as are quadrangular, or multangular. And thus it is\npreserved by the preservation of triangle, it is destroyed by its\ndestruction.\n\n11. From the principles already established, it is plain that\ndemonstration must consist of such propositions as are universal and\nnecessary. That they must be universal, is evident from the preceding;\nand that they must be necessary, we gather probably from hence; that in\nthe subversion of any demonstration we use no other arguments than the\nwant of necessary existence in the principles.\n\nWe collect their necessity demonstratively, thus; he who does not\nknow a thing by the proper cause of its existence, cannot possess\nscience of that thing; but he who collects a necessary conclusion from\na medium not necessary, does not know it by the proper cause of its\nexistence, and therefore he has no proper science concerning it. Thus,\nif the necessary conclusion _c_ is _a_, be demonstrated by\nthe medium B, not necessary; such a medium is not the cause of the\nconclusion; for since the medium does not exist necessarily, it may\nbe supposed not to exist; and at the time when it no longer exists,\nthe conclusion remains in full force; because, since necessary,\nit is eternal. But an effect cannot exist without a cause of its\nexistence; and hence such a medium can never be the cause of such a\nconclusion. Again, since in all science there are three things, with\nwhose preservation the duration of knowledge is connected; and these\nare, first, he who possesses science; secondly, the thing known; and\nthirdly, the reason by which it is known; while these endure, science\ncan never be blotted from the mind, but on the contrary, if science be\never lost, it is necessary some of these three must be destroyed.\n\nIf then you infer that the science of a necessary conclusion may be\nobtained from a medium not necessary, suppose this medium, since\ncapable of extinction, to be destroyed; then the conclusion, since\nnecessary, shall remain; but will be no longer the object of knowledge,\nsince it is supposed to be known by that medium which is now extinct.\nHence, science is lost, though none of the preceding three are\ntaken away; but this is absurd, and contrary to the principles we\nhave just established. The thing known remains; for the conclusion,\nsince necessary, cannot be destroyed;--he who knows still remains,\nsince neither dead, nor forgetful of the conclusion:--lastly, the\ndemonstration by which it was known, still survives in the mind; and\nhence we collect, that if science be no more after the corruption\nof the medium, neither was it science by that medium before its\ncorruption; for if science was ever obtained through such a medium,\nit could not be lost while these three are preserved. The science,\ntherefore, of a necessary conclusion can never be obtained by a medium\nwhich is not necessary.\n\n12. From hence it is manifest, that demonstrations cannot emigrate from\none genus to another; or by such a translation be compared with one\nanother. Such as, for instance, the demonstrations of geometry with\nthose of arithmetic. To be convinced of this, we must rise a little\nhigher in our speculations, and attentively consider the properties\nof demonstration: one of which is, that predicate which is always\nfound in the conclusion, and which affirms or denies the existence of\nits subject: another is, those axioms or first principles by whose\nuniversal embrace demonstration is fortified; and from whose original\nlight it derives all its lustre. The third is the subject genus, and\nthat nature of which the affections and essential properties are\npredicated; such as magnitude and number. In these subjects we must\nexamine when, and in what manner a transition in demonstrations from\ngenus to genus may be allowed. First, it is evident, that when the\ngenera are altogether separate and discordant, as in arithmetic and\ngeometry, then the demonstrations of the one cannot be referred to\nthe other. Thus, it is impossible that arithmetical proofs can ever\nbe accommodated with propriety to the accidents of magnitudes: but\nwhen the genera, as it were, communicate, and the one is contained\nunder the other, then the one may transfer the principles of the other\nto its own convenience. Thus, optics unites in amicable compact with\ngeometry, which defines all its suppositions; such as lines that are\n_right_, angles acute, lines equilateral, and the like. The same\norder may be perceived between arithmetic and music: thus, the double,\nsesquialter, and the like, are transferred from arithmetic, from which\nthey take their rise, and are applied to the measures of harmony.\n\nThus, medicine frequently derives its proofs from nature, because\nthe human body, with which it is conversant, is comprehended under\nnatural body. From hence it follows, that the geometrician cannot, by\nany geometrical reasons demonstrate any truth, abstracted from lines,\nsuperficies, and solids; such as, that of contraries there is the same\nscience; or that contraries follow each other; nor yet such as have an\nexistence in lines and superficies, but not an essential one, in the\nsense previously explained.\n\nOf this kind is the question, whether a right-line is the most\nbeautiful of lines? or whether it is more opposed to a line perfectly\norbicular, or to an arch only. For the consideration of beauty, and the\nopposition of contraries, does not belong to geometry, but is alone the\nprovince of metaphysics, or the first philosophy.\n\nBut a question here occurs, If it be requisite that the propositions\nwhich constitute demonstration should be peculiar to the science they\nestablish, after what manner are we to admit in demonstration those\naxioms which are conceived in the most common and general terms; such\nas, if from equal things you take away equals, the remainders shall be\nequal:----as likewise, of every thing that exists, either affirmation\nor negation is true? The solution is this: such principles, though\ncommon, yet when applied to any particular science for the purposes\nof demonstration, must be used with a certain limitation. Thus the\ngeometrician applies that general principle, if from equal things, &c.\nnot simply, but with a restriction to magnitudes; and the arithmetician\nuniversally to numbers.\n\nThus too, that other general proposition:----of every thing,\naffirmation or negation is true; is subservient to every art, but not\nwithout accommodation to the particular science it is used by. Thus\nnumber _is_ or is _not_, and so of others. It is not then\nalone sufficient in demonstration that its propositions are true,\nnor that they are immediate, or such as inherit an evidence more\nillustrious than the certainty of proof; but, besides all these, it\nis necessary they should be made peculiar by a limitation of their\ncomprehensive nature to some particular subject. It is on this\naccount that no one esteems the quadrature of Bryso[24], a geometrical\ndemonstration, since he uses a principle which, although true,\nis entirely common. Previous to his demonstration he supposes two\nsquares described, the one circumscribing the circle, which will be\nconsequently greater; the other inscribed, which will be consequently\nless than the given circle. Hence, because the circle is a medium\nbetween the two given squares, let a mean square be found between\nthem, which is easily done from the principles of geometry; this mean\nsquare, Bryso affirms, shall be equal to the given circle. In order to\nprove this, he reasons after the following manner: those things which\ncompared with others without any respect, are either at the same time\ngreater, or at the same time less, are equal among themselves: the\ncircle and the mean square are, at the same time, greater than the\ninternal, and at the same time less than the external square; therefore\nthey are equal among themselves. This demonstration can never produce\nscience, because it is built only on one common principle, which may\nwith equal propriety be applied to numbers in arithmetic, and to times\nin natural science. It is defective, therefore, because it assumes no\nprinciple peculiar to the nature of the circle alone, but such a one as\nis common to quantity in general.\n\n13. It is likewise evident, that if the propositions be universal,\nfrom which the demonstrative syllogism consists, the conclusion must\nnecessarily be eternal. For necessary propositions are eternal; but\nfrom things necessary and eternal, necessary and eternal truth must\narise. There is no demonstration, therefore, of corruptible natures,\nnor any science absolutely, but only by accident; because it is not\nfounded on that which is universal. For what confirmation can there\nbe of a conclusion, whose subject is dissoluble, and whose predicate\nis neither always, nor simply, but only partially inherent? But as\nthere can be no demonstration, so likewise there can be no definition\nof corruptible natures; because definition is either the principle of\ndemonstration, or demonstration differing in the position of terms, or\nit is a certain conclusion of demonstration. It is the beginning of\ndemonstration, when it is either assumed for an immediate proposition,\nor for a term in the proposition; as if any one should prove that man\nis risible, because he is a rational animal. And it alone differs\nin position from demonstration, as often as the definition is such\nas contains the cause of its subjects existence. As the following:\nan eclipse of the sun is a concealment of its light, through the\ninterposition of the moon between that luminary and the earth. For\nthe order of this definition being a little changed, passes into a\ndemonstration; thus,\n\n  The moon is subjected and opposed to the sun:\n\n  That which is subjected and opposed, conceals:\n\n  The moon, therefore, being subjected and opposed, conceals the\n    sun.\n\nBut _that_ definition is the conclusion of demonstration, which\nextends to the material cause; as in the preceding instance, the\nconclusion affirming that the subjection and opposition of the moon\nconceals the sun, is a definition of an eclipse including the material\ncause.\n\nAgain, we have already proved that all demonstration consists of\nsuch principles as are prior in the nature of things; and from\nhence we infer, that it is the business of no science to prove its\nown principles, since they can no longer be called principles if\nthey require confirmation from any thing prior to themselves; for,\nadmitting this as necessary, an infinite series of proofs must ensue.\nOn the contrary, if this be not necessary, but things most known\nand evident are admitted, these must be constituted the principles\nof science. He who possesses a knowledge of these, and applies them\nas mediums of demonstration, is better skilled in science, than he\nwho knows only posterior or mediate propositions, and demonstrates\nfrom posterior principles. But here a doubt arises whether the first\nprinciples of geometry, arithmetic, music, and of other arts, can ever\nbe demonstrated? Or shall we allow they are capable of proof, not by\nthat particular science which applies them as principles or causes\nof its conclusions? If so, this will be the office of some superior\nscience,--which can be no other than the first philosophy, to whose\ncharge the task is committed; and whose universal embrace circumscribes\nthe whole circle of science, in the same manner as arithmetic\ncomprehends music, or geometry optics.--This is no other than that\ncelebrated _wisdom_ which merits the appellation of science in\na more simple, as well as in a more eminent degree than others: not,\nindeed, that all causes are within its reach, but such only as are the\nprincipal and the best, because no cause superior to them can ever be\nfound. Hence the difficulty of knowing whether we possess science or\nnot, from the difficulty of understanding whether it is founded on\npeculiar or common principles; since it is necessary that both these\nshould be applied in the constitution of all real knowledge and science.\n\n[25]Again, axioms differ from postulates in this:--they demand our\nassent without any previous solicitation, from the illustrious\ncertainty they possess. Their truth may, indeed, be denied by external\nspeech, but never from internal connection. He who denies that\nequal things shall remain from the subtraction of equal, dissents,\nas Euripides says, with his tongue, and not with his heart. But\ndemonstration depends not on external speech, but on intellectual and\ninternal conviction; and hence, axioms derive all their authority\nfrom intrinsic approbation, and not from public proclaim. For the\nprompt decisions of the tongue are frequently dissonant from the\nsentiments concealed in the secret recesses of the heart. Thus the\n[26]geometrician does not speculate those lines which are the objects\nof corporeal sight, but such as are exhibited by mental conception,\nand of which the delineations on paper, or in the dust, are no more\nthan imperfect copies, notes, and resemblances. Thus, when he draws\na pedal line which is not pedal, or an equilateral triangle which is\nnot equilateral, we must pay no regard to the designations of the pen,\nbut solely attend to the intellection of the mind; for the property\ndemonstrated of some particular line, is in the conclusion applied\nto one that is universal, and this true line could be no otherwise\nsignified to the learner than by a material description.\n\nThe certainty of axioms is, indeed, in a measure obvious to every one.\nFor what more evident than that nothing exists of which it is possible,\nat the same time, to affirm and deny any circumstance of being? Indeed,\nso illustrious and indubitable is the light of this axiom, that in any\ndemonstration we are ashamed to assign it the place of an assumption.\nIt would almost seem prolix and superfluous, since there is nothing\nmore manifest and certain; and yet there are cases in which it is\nnecessary to rank it among assumptions. And these take place whenever\nthe intention is to conclude the existence of something as true, and\nof its opposite as false. Thus, for instance, in the demonstration\nthat the world is finite, we assume this principle, and then reason as\nfollows:\n\n  Bound and infinite cannot be at the same time affirmed and denied\n    of any _body_:\n\n  The world is a body:\n\n  Therefore the world is not at the same time finite and infinite.\n\nAnd in this genus of demonstration, the major proposition ought always\nto assimilate with the conclusion. But the above axiom is not the\nonly one obvious, for the following possesses equal certainty; that\nof every thing which exists, either affirmation or negation is true.\nThis axiom is of great use in demonstrations leading to an absurdity;\nfor he who demonstrates the impossibility of any opposite assertion,\nnecessarily establishes his own. Hence it is we affirm that the\ndiameter of a square is either commensurable or incommensurable with\nits side; and this general principle is accommodated, and, as it were,\ndescends into its proper matter as often as that which it possesses of\nuniversal is contracted to a certain genus; for, as we have previously\nobserved, common principles are not admitted in demonstration without\nany restriction; but then only when their general nature is limited to\nsome particular subject, by which they become peculiar and apposite.\n\n14. [27]Wisdom, or the first philosophy and logic, agree in not using\naxioms after the same manner as other arts; but on the contrary,\nthey confirm and establish their certainty, though with this\ndifference, that the logician reasons only from probabilities, but the\nmetaphysician from the highest certainty and evidence. Besides, we do\nnot rank logic in the order of the sciences, because it is destitute\nof some determinate genus or subject, as it is neither conversant about\nlines, nor numbers, nor proportions. And its chief concern is about\napparent properties, and not such as are essential to a subject.\n\nHence, in logical disquisitions, we confidently employ interrogations,\nas equally subservient to the affirmation or negation of an opinion:--a\nmethod utterly impracticable, if we only employed those principles\nwhich are universally acknowledged; since it is impossible of the same\nthing to prove contrary properties,--as of the soul, that it is mortal\nand immortal; but he who demonstrates, assumes one definite part of a\nquestion, because his purpose is not to interrogate, but to trace out\nthe latent paths of truth. And hence, if any one affirms that the soul\nis moved, and immediately after denies it, he is no longer a subject\nworthy the exercise of our discursive and reasoning powers.\n\nAgain, it may so happen, that the same science at one time considers\n_why_ a thing is, at another only explains its existence, or that\nit exists, without considering the cause. Thus, the syllogism which\nconcludes by mediate propositions, demonstrates without assigning the\nproper cause: but that which determines by immediate ones, in a great\nmeasure explains the cause or reason of existence. Thus, he who infers\nthat trees do not breathe because they are not animals, reasons from a\nmediate and secondary cause, because there are many animals, such as\ninsects, which exist without breathing: but he who infers this from\ntheir want of lungs, demonstrates from the immediate and primary cause.\n\nThus, the following syllogism is a mediate one, or such as requires one\nor more mediums to establish its certainty:\n\n  Every thing that is not an animal does not breathe;\n\n  A tree is not an animal;\n\n  Therefore a tree does not breathe.\n\nHere the major proposition is evidently mediate, because we are still\nto seek why that which is not an animal does not breathe, which the\nfollowing immediate syllogism solves.\n\n  Every thing that is not endued with lungs does not breathe;\n\n  Every thing that is not an animal is not endued with lungs; ergo,\n\n  Every thing that is not an animal does not breathe.\n\nAgain, the same science may demonstrate the existence of a thing,\nor that it exists, and the cause of such existence as often as it\nassigns two immediate reasons; but the one from the proper cause,\nthe other only from a sign. Thus, he who demonstrates the increase\nof the moon, from the plenitude of her orb, infers the cause of such\nincrease; but on the contrary, he who collects the plenitude of her\norb from her increase, reasons only from a sign, and can alone declare\nits existence. And, indeed, it often happens that the _cause_\nand _sign_ reciprocate, so that as from the sign we advance to\nthe cause, demonstration from the cause frequently recurs to the\n_sign_. Thus, from the breadth and firmness of the basis, we\ncollect the permanent duration of the pyramid; and from its extended\nexistence we infer the strength of its support. Whenever, then, the\nargument originates from a sign, it gives evidence to the conclusion,\nas from something more known than its cause. When it begins from the\ncause, it proceeds from that which is first in the order of nature, to\nthat which is last, and reasons as from the proper principle of the\nthing.\n\nSometimes the _cause_ and _sign_ do not reciprocate. Thus,\nalthough wherever there is smoke, we infer the existence of fire; yet\nwe cannot infer, that wherever there is fire smoke exists. Thus, from\nthe palace and the picture we collect the existence of the architect\nand painter; but the last may exist without the first;--the living\narchitect without the actual palace; and the living painter without\nthe energies of his art. And thus it is that the cause is illustrated\nby its sign; but not always the sign by its cause.\n\nHence then, as all causes do not reciprocate with their effects; so\nneither is it always causes and effects which do reciprocate: because\na multitude of signs, mutually inferring each other, may accompany a\ncertain cause. Thus, the signs which attend the causes of a fever, are\na quick pulsation of the artery, and an intense heat: and these signs\nmutually assert each other; but no syllogism can be composed from\neither expressing the _why_, but only simply _that_ the other\nexists.\n\n15. We now propose to consider the mode in which the two preceding\ndemonstrations are distributed in different sciences. When sciences\nthen are so related, that the one is dependent on the other, as optics\non geometry, navigation on astronomy, and music composed by the\narbitration of the ear, on that which consists in the knowledge of\nmathematical proportions: in this case, the demonstration of simple\nexistence, or _that_ they _exist_, pertains to the science of\nsensibles; but the demonstration _why_ they _exist_ to the\nscience which is speculative and mathematical.\n\nThus the mathematician speculates the causes of a certain sensible\neffect, without considering its actual existence; for the contemplation\nof universals excludes the knowledge of particulars; and he whose\nintellectual eye is fixed on that which is general and comprehensive,\nwill think but little of that which is sensible and singular. Thus,\nby mathematics we may learn the responsive harmony of the _last\nchord_, and its consonance with the _mean_; but we cannot\nperceive this concord, if unaccustomed to the practice of the musical\nart. In fine, those sciences which are more of a mathematical nature,\nI mean such as are more amply conversant with the inspection of\nthings, considering their forms abstracted from every material subject,\nalways demonstrate the _why_; and such is geometry in respect\nof optics. Thus geometry considers only such things as are peculiar\nto right-lines, independent of every sensible connection. For the\ngeometrician does not investigate a right-line as contained in stone or\nbrass; but considers it as entirely detached and unconnected with any\nobject of sense.\n\nOn the contrary, optics receives a right-line just as it is perceived\nin a rule, or engraved in brass. And, indeed, in treating of some\nparticulars, natural science has the same relation to optics, as optics\nto geometry. Thus, in considering the reason of the appearance of\nthe rainbow, the natural philosopher defines the bow to be an image\nrefracted from a certain cloud against the sun; but why it is endued\nwith such a form, and seen with such a colour, must be assigned by him\nwho is skilled in optics. There are, again, sciences, one of which is\nnot subordinate to the other, because founded on principles totally\ndifferent; yet, in some particulars they agree with the preceding.\nThus, to know that an orbicular wound is the most difficult of cure,\nbelongs to the physician; but to know _why_, to the geometrician.\n\n16. Of all syllogistic figures, the first is the best adapted to\nscience, since the arithmetician, geometrician, and lastly all those\nwho demonstrate any effect from its proper cause, fabricate their\nreasonings according to this figure. For the middle figure is seldom\nused, because only adapted to a few occasions: and since the knowledge\nof the _why_ is of all others the most important, which is alone\nobtained by this figure: hence, in the pursuit of science, it is\nalways preferred before the rest. Besides, it is equally accommodated\nto the knowledge of final causes; to which it alone tends: for it\ncomposes definitions from words universal, and affirmative. In the\nsecond figure, a complex negative is conceived; and in the last, a\nparticular one. Add to this, that mediate propositions are no other\nways reducible to immediate ones than by this figure, in which the\nmediate proposition tends, by a continued series, to that which is\nimmediate. But the second does not conclude affirmatively, nor the\nlast universally; from whence it appears, that a mediate proposition\ncan never become immediate by these figures: not that all affirmative\npropositions are immediate ones, since some negatives are of this\nkind; for all propositions are equally immediate, which cannot be\nconfirmed by syllogism; and such are those negatives, of whose terms it\nis impossible any genus can be affirmed. Thus the proposition, _no\nsubstance is quality_, is an immediate negative of this kind, whose\nterms are two of the most universal genera of things.\n\nAgain, as we have frequently affirmed that he who demonstrates, always\nassumes such things as are essentially predicated; but that he who\nargues dialectically or topically, not always, but generally assumes\nsuch as are accidentally predicated, and which appear more probable and\nknown than such as are essentially inherent; it is proper we should\ndefine what is meant by accidental predication; or something predicated\nby means of another. Indeed, the term has a diffuse signification:\nfor, first, a body is said to be white by something else, because by\nits superficies; and in this manner vines are white, because their\nbranches are white. Thus, if accident be predicated of accident, it is\nby means of another; as when we say the musician is fair; for the being\na musician is an accident of man, and the being fair of the musician:\nand man is the subject of each. The predicate of substance is equally\naccidental, when not included in the number of things substantially\ninherent; as when we affirm of any particular man that he is red, or\nblack. But the predication is especially accidental, as often as, by\nperverting the order of nature, substance is predicated of accident; as\nwhen we say something white is an animal: for this assertion differs\nfrom that other, _animal is white_. In the latter, the subject\n_animal_ is neither inherent in another, nor subsists by another,\nbut has an essential existence. In the former, what is assumed as a\nsubject derives its existence from that of which it is the accident.\nIt is only dialectically, therefore, that we can argue from predicates\nas probable and known without any distinction: but in demonstration,\nall that are preposterous and accidental must be carefully avoided,\nexcepting such accidents as being essentially in a subject, admit of an\nessential predication; and some of these we have enumerated before.\n\n17. We are now entering on a disquisition neither ignoble nor useless:\nit is this, whether the number of things predicated essentially of a\nsubject is finite, or whether things in a continued series run on to\ninfinity. For instance, let us suppose some ultimate subject, which\nis not the predicate of any thing besides; and let _c_ represent such\na subject, of which _b_ is the first and immediate predicate; and in\nthe same manner _d_ of _b_, and _e_ of _d_: the query is, Whether or\nnot this extraction must necessarily stop, or will admit of an immense\nprogression, so that _f_ may be predicated of _e_, and _g_ of _f_,\nand so on infinitely; the power of the predicates, which supplies the\ncommon identity, still remaining inexhaustible and undiminished? The\nsecond query is this, Supposing some general subject, which we call\n_a_, of such a nature as to be no longer the subject of any farther\npredication, but to be itself the supreme and primary predicate; and\nsupposing that it is immediately inherent in _f_, and _f_ in _e_,\nand _e_ in _g_, whether or not the process must stop, or extend to\ninfinity, and no subject be found which is not directly predicable of\nanother? There is a remarkable difference in the two considerations;\nfor, in the former we enquire whether any ultimate subject can supply\nan infinite ascent of predicates; in the latter, whether any first\npredicate can exist in an infinite descending series of subjects. The\nthird question is, supposing two extremes constituted from a first\npredicate and last subject, whether it is possible an infinite number\nof mediums can intervene? And this is no other than to enquire whether\ndemonstrations admit of an infinite progression, so that whatever is\nassumed in proof of another, must be proved itself? Or whether it\nis not more agreeable to truth, that there should be some immediate\npropositions and ultimate terms, whose discovery may give respite to\nenquiry, and stay the elaborate process of demonstration? The same\nquestion occurs in negatives. But that some of these are immediate, the\ninstance lately alledged sufficiently evinces. The solution of this\nenquiry is not so difficult in subjects which mutually reciprocate;\nfor in these, when the ultimate subject is given, no one can doubt the\nexistence of their primary predicate; nor when the primary predicate\nis admitted, can there be any doubt of the existence of some ultimate\nsubject. For, in things which mutually reciprocate, whatever is\nenquired of the one, is immediately questioned of the other; and\nwherever there is a last subject, there must be a first predicate;\nfor by the conversion of the ultimate subject you effect the primary\npredicate.\n\nPrevious to the discussion of the first question, it is necessary to\nknow that infinite intermediates cannot intervene between two finite\nterms in an ascending and descending series of predications. I call the\nseries ascending which rises to universals; but descending, which, by\na contrary process, stops at particulars. Thus, if any one admits that\n_a_ is some first predicate, and _g_ some ultimate subject,\nand should contend, that between these terms there may be infinite\nmediums, he contradicts himself; since he who begins from _a_ in a\ndescending progression, will never, by this means, arrive at _g_;\nand he who departs from _g_ in an ascending series, can never\nfinally rise to _a_. So that the extremes can be no longer finite,\nas the hypothesis admitted. Indeed, the absurdity of such a supposition\nis the same as to contend that between one and ten, an infinity of\nnumbers may exist; which is evidently impossible, because the discrete\nnature of numbers excludes their actual existence in infinitum,\nbetween any finite limits; since they can only become infinite from\ntheir actual existence and precedence, and not from any dormant power\nor capacity they possess: for between any two given numbers there is\nnothing similar to number in capacity, which can ever become number\nin energy; as in quantity continuous between any two points there are\nalways parts in capacity, which, whenever a proper agent is at hand,\nbecome immediately actual. In like manner, he who admits the terms\nfinite, but believes that the mediums are infinite, asserts what is\nimpossible, since these logical predications are of the same discrete\nnature with numbers themselves. Thus all the predicates which can exist\nbetween Socrates and substance, must exist actually, or not at all; for\nsurely between these two terms, or periods, no predicate in capacity\ncan ever be supposed to subsist. If it be urged, that the capacity of\nreceiving these predicates exists between Socrates and substance, still\nwe reply, it is not that kind of capacity in which these predicates can\nretain the most shadowy existence; out of which they can ever be called\nforth into energy, as from some latent retreat; or into which they\ncan finally retire, when energy is no more. And hence we conclude it\nimpossible that infinite mediums can exist between any finite terms.\n\n18. It now remains that we prove, first, by probable arguments,\nand then by such as are demonstrative, that the extremes in any\nseries of predications are finite; and that an infinite progression\nis impossible, not only in substantial predicates, but in such as\nare accidental. For every thing predicated of another is either\nessentially or accidentally inherent; and is predicated in a natural\nor preposterous order. It is predicated according to nature, when\naccident is declared of substance; contrary to nature, when substance\nof accident. That essential predicates are finite, appears from hence,\nbecause a contrary hypothesis excludes the existence of definition,\nby admitting that all things are contained in some superior genus,\nand acknowledge some farther definition; since it is impossible that\nthe definitions of genus can ever be circumscribed, while there is a\ncontinual supply of other genera, which can never be known without\ndefinition; for thus we shall never obtain either a beginning or an\nend. But to define all things is not possible, because infinity can\nnever be absolved by the most unwearied progression. Predictions then,\nof this kind, are always circumscribed by a certain number of terms,\nwhich prevent their infinite process, and cause all the strength of\ndemonstration, and all the certainty of human knowledge. The same may\nbe proved in accidents; for such as are predicated of substance, are\neither predicated as qualities or quantities, as relatives, or as\nactions and passions; as expressive of some habit, or significant of\nsome place; or as connected with some time. Thus we say the wood is\nwhite, the triangle is scalene; whiteness being accidental to the\nwood, and scalenity to the triangle. It is therefore certain, that\nevery accident is predicated of substance; and it is no less certain\nthat the predicates of substance are finite, since they are all\nincluded in the ten universal genera of things.\n\n19. We have hitherto defended the impossibility of an infinite\nprogression of logical predicates and subjects, in a demonstrative\nprocess, by such arguments as are dialectical and common: it now\nremains that we adopt such as are peculiar and certain. Demonstrations,\nthen, are derived from affections essentially inherent in a subject;\nand these are either such as take place in definitions of a subject,\nas multitude and quantity, are essentially predicated of number; or,\nsecondly, accidents which are defined from their subjects, as imparity\nby number. But the predication cannot, in either case, be extended to\ninfinity. For it is not necessary that in the same manner that imparity\nis predicated of number, something else, suppose _c_, should be\npredicated of imparity; and so imparity be contained in its definition,\nsimilar to number in the definition of imparity. For in predications\nof this kind, the terms are always assumed more contracted than their\nsubject; and at length, by a continued procession, must terminate in an\nindivisible. Thus, as imparity is more contracted than number, _c_\nmust be more contracted than imparity. Hence, these predications either\nfinally stop, for the reasons we have assigned; or because whatever is\npredicated of imparity, is necessarily predicated of number; so that\none thing as number would be actually contained in the definition of an\ninfinity of things; and so actual infinity must ensue, which is absurd.\nLastly, whatever is said to reside in the terms, must be allowed to\nreside in the subject; so number must be applied in the definition\nof every affection; and an infinite number of properties will be\nessentially inherent in number; and number will inherit infinite\ndefinitions. But affections essentially resident in a subject cannot\nbe infinite, because it is necessary they should exist in energy.\nThus, imparity cannot exist potentially in number; nor reason in man;\nnor rotundity in a circle, because wherever these subjects have an\nactual being, it is necessary these essential attributes should be\nactually inherent. Again, in the definitions of a subject, an infinite\nprocess is impossible, because from such an hypothesis nothing could\never be defined; and thus it appears that neither can demonstrations\nbe infinitely extended, nor every thing admit of demonstration, an\nopinion we have already noticed in the beginning of this section: for\nif neither universally, nor in every proposition a middle term can\nbe assumed, but as soon as we arrive at immediate propositions, the\nlabour of investigation is finished, the possibility of demonstrating\nevery thing can no longer be defended; since it is proved above, that\nby limiting the extremes, an infinite number of mediums is necessarily\nexcluded.\n\nAnd thus, by taking away infinity from the reasoning art, we have\ngiven a support to science, which the most vigorous efforts of subtle\nsophistry can never finally subvert. We have set bounds to that\nrestless spirit of enquiry which wanders uncontrouled in the mind\nunenlightened by science, by every where circumscribing its progress\nwithin the limits of that which is most particular, and most universal,\na first predicate, and an ultimate subject: and finally, by asserting\nthat all the evidence of human knowledge results from the lustre\nof primary and immediate principles, we have held up a steady and\npermanent light, ever sufficient to direct our steps through the dark\nmazes of ignorance and error, into the bright paths of certainty and\ntruth.\n\n20. Let us next consider whether universal demonstration is preferable\nto particular, or not. And first, in favour of particulars we may\nsay that their evidence is more exquisite and certain than that of\nuniversals. Thus, the knowledge, from inspection, that Callias is a\nrational animal, is superior to that acquired by a reasoning process\nwhich infers his rationality, because every man is a rational animal.\nBy particular demonstration a thing is known as it is, by universal\nonly in common. Besides, particulars possess some solidity, universals\nnone: and the demonstration of things which have a real existence,\nis more excellent than that of things which have none. And there are\nno errors more frequent than those about universals; demonstration\nconsidering them as things entirely abstracted from singulars. On\nthe contrary, particulars are usurped by the sight, grasped, as it\nwere, by the hand, and the general subject of every sense; so that\nconcerning these, demonstration affirms nothing false or inconstant.\nBut these reasons, however plausible, are easily confused. And,\nfirst, the term essential is more closely connected with universals\nthan particulars. Thus the possession of three angles equal to two\nright, is an affection more essential to the triangle itself, than\nto one equilateral or scalene. Add too, that in the demonstration of\nuniversals we always infer some property of a subject from its simple\nexistence, or because it is such a subject. Again, many affections are\ncontained in singulars assumed from no particular nature, but from\nthat which is universal; as rationality in Socrates, which is not\ninferred from his existence as Socrates, but from his existence as man.\nFarther, that demonstration is the more excellent which is derived\nfrom the better cause: but an universal cause is more extended and\nexcellent than a particular one; since the arduous investigation of\nthe _why_ in any subject is stopt by the arrival at universals.\nThus, if we desire to know why the exterior angles of a triangle are\nequal to four right ones, and it is answered, because the triangle\nis isosceles; we again ask, But why because isosceles? And if it be\nreplied, because it is a triangle, we may again enquire, But why\nbecause a triangle? To which we finally answer, because a triangle is a\nright-lined figure; and here our enquiry rests at that universal idea\nwhich embraces every preceding particular one, and is contained in no\nother more general and comprehensive than itself. Add too, that the\ndemonstration of particulars is almost the demonstration of infinites;\nof universals, the demonstration of finites.--We add farther,\n_that_ demonstration is the best, which furnishes the mind with\nthe most ample knowledge; and this is alone the province of universals.\nAgain, the principles of science become immediate only in proportion\nas the demonstration becomes universal; and he who knows universals,\nknows particulars in capacity: but we cannot infer, that he who has\nthe best knowledge of particulars, knows any thing of universals.\nLastly, that which is universal, is the province of intellect and\nreason, particulars are the offspring of sense; and hence we conclude\nthat universal demonstration exceeds particular both in dignity and\nexcellence, and is first in the nature of things, although last in the\nprogressions of the reasoning power.\n\nAgain, That affirmative demonstration is superior to negative, appears\nfrom hence: the affirmative does not require the assistance of the\nnegative; but the negative cannot exist without the affirmative;\non which account, the demonstration composed from negatives alone,\nis incapable of producing real evidence and conviction. Besides,\naffirmation exceeds negation both in priority and simplicity of\nexistence.\n\nAgain, the demonstration which concludes _directly_, is better\nthan that which confirms a proposition by evincing the absurdity of\nits contrary. The first proceeding in a regular order, establishes,\nby a natural deduction, the truth which was first advanced. The\nsecond taking a wider circuit, yet with the same intentions produces\na conclusion quite opposite to its apparent design. The one may be\ncompared to the open attack of a valiant and skilful soldier, who\nexpects the conquest of his enemy from strength and courage alone: the\nprogress of the other resembles the same soldier, uniting force with\nstratagem, and advancing, by an irregular march, which his foe mistakes\nfor a retreat, but finds the secret cause of his destruction. The first\nis simple and impromiscuous, as composed from propositions alone: the\nsecond is compound and miscellaneous, calling in hypothesis to its\nassistance.\n\n21. One science is said to be prior to, and more certain than another\nin many respects;--when the one reasons from primary causes, but the\nother from such as are secondary:--when the one may be ranked in the\ngenera of intelligibles and universals; but the other in the genera of\nsensibles and particulars. And such is the relation of arithmetic to\nmusic; of geometry to optics; and lastly, of every superior to every\nsubordinate science. Again, this happens when the one reasons from\nsimple principles, the other from such as are complex and connected;\non which account arithmetic seems to possess greater certainty than\ngeometry. For the principle of arithmetic is unity; but of geometry\na point; and unity is without position, with which a point is always\nconnected. And in this manner geometry inherits greater evidence than\nastronomy; for the one considers body simply, the other as connected\nwith a circular motion. The science is called one which contemplates\nactions belonging to one genus: the genus is one which possesses the\nsame first principles; and hence geometry and stereometry form one\nscience. On the contrary, the sciences are called different which have\ndifferent principles, such as geometry and optics; the latter of which\ndoes not originate from the principles of the former.\n\nAgain, the same thing may admit of many demonstrations, and may be\nknown from many mediums: at one time from the application of such\nas are congenial: at another, from those of a different order or\ngenus. From congenials, as when we demonstrate that the plantain is a\nsubstance, first, by the medium of a tree, and then by the medium of a\nplant, thus:\n\n  Every tree is a substance;\n\n  The plantain is a tree:\n\n  Therefore the plantain is a substance. And again,\n\n  Every plant is a substance:\n\n  The plantain is a plant:\n\n  Therefore the plantain is a substance.\n\nWe demonstrate, from mediums, of a following order or genus, as when we\nprove man to be a substance, at one one time from his being rational,\nat another from his being a biped; and these mediums, in part, mutually\ncontain each other.\n\n22. Fortuitous events can never, in any science, become the subject\nof demonstration; since they are neither limited by necessity, nor\nadmit the arrangement of syllogism. Indeed, so far from obtaining\na necessary, they do not possess a frequent existence, but every\nsyllogism is composed from one or other of these.\n\nAgain, science is not the business of sense, since that which is\nuniversal is the object of perception in particulars themselves. For\nthe object of sight is colour in general, and not this particular\ncolour: the object of hearing is sound in general, and not any\nparticular sound; and, on this account we see or hear not only this\nor that colour or sound, but likewise every other which falls under\nthe cognizance of these senses. Hence, if it were possible for any\none to discern by his sight, the equality of the three angles of some\nparticular triangle to two right, he would not by this means possess a\ndemonstration of the conclusion which affirms this to be the property\nof every triangle; but his knowledge would extend no farther than the\ntriangle he inspects. Thus too, if we could perceive an eclipse of\nthe moon to arise from the interposition of the earth, we could not\nuniversally conclude that this is the cause of every eclipse, but\nonly of the particular one we behold. For the explication of causes\nextends to universals; and comprehends not only the knowledge of one\nparticular defect of the moon, but simply of every eclipse; since the\ninterposition of the earth is not so much the cause of any present\neclipse, as of all which can possibly exist in every age. Whenever,\nthen, the cause is universal, the knowledge of any effect deduced from\nsuch a cause is, in every respect, superior to the evidence arising\nfrom the perceptions of sense. It is likewise more excellent than\nthe apprehension which subsists independent of the proper cause; as\nif any one should give absolute credit to the proposition, that the\nthree angles of a triangle are equal to two right, without a previous\nconviction that the external angle of a triangle, is equal to the two\ninterior opposite ones; and without applying this last proposition as\nthe cause of the first. The comprehension, then, which is conjoined\nwith the proper cause, far exceeds the strongest evidence of sense.\n\nBut perhaps it may be said that science consists in sense, because the\nscience of any particular, fails from a defect of the sense by which\nit is apprehended. To this we reply, that science, indeed, is not\nacquired without the assistance of sense, but it does not follow from\nhence, that _to perceive_ is _to know_; because the object\nof science is that which is universal; but of sense, that which is\nparticular. Thus, if we could see light penetrating the pores of glass\n(on the atomical hypothesis) the cause why it illuminates would be\nmanifest from sensible inspection as the means, and from the universal\napprehension of science, by which we should understand this to be\nuniversally true.\n\nAgain, the principles of all sciences cannot be the same neither\nconsidered as remote or proximate. Not considered as proximate, because\nthe principles always correspond to the demonstrated conclusions; but\nthese are not the same, since they are often generically different;\nand consequently the propositions from which they result must be\nderived from discordant genera. But propositions consist of such\nthings as essentially exist; and hence we infer, that the principles\nof geometry are essentially distinguished from those of arithmetic,\nthat they cannot admit of reciprocal accommodation, so that the one may\nbe predicated, or become the subject of the other, and that the one\ncan never be subservient as a medium to the other. Again, common and\nfirst principles are not applied in every science; such as this, that\nevery thing must either be affirmed or denied. Nor can any thing be\nproved by their assistance alone, but as often as these are required\nin demonstration, other principles more proximate and peculiar to the\ngiven proposition, must always be adopted. Again, axioms universally\nconceived, cannot be assumed in syllogism, but they must be contracted,\nas it were, to some subject genus. Of this kind is that common\naxiom, that as often as any four quantities are proportionable, by\npermutation, or changing the order of the terms, the same ratio will\nresult. For the arts apply this axiom in a restricted sense; geometry,\nby considering the relatives as four magnitudes, and arithmetic as four\nnumbers; but the natural philosopher, by adapting the comparison to\nfour motions, or four times. Besides, if the principles of all sciences\nwere the same, it is necessary they should be comprehended by some\ncertain number, similar to the limitation of the elements: but every\nscience is capable of immense increase from the many different modes\nof amplification the conclusions will admit; and consequently it is\nrequisite to establish a correspondent number of proper principles;\nfor such as are common cannot be alone sufficient. Lastly, if the same\nprinciples accord with every science, it follows, that any thing may\nbe demonstrated from such principles: but the certainty of geometrical\nconclusions cannot be established from the principles of music; and\nfrom hence it follows, that although the principles of every science\nare not the same, they do not possess an entire diversity, nor yet an\nabsolute affinity of nature.\n\n23. There is a remarkable difference between science and opinion.\nWhatever is the subject of science must have a necessary existence; on\nthe contrary, opinion is conversant with things liable to mutation and\ndecay. Again, as science depends on necessary propositions for support;\nso opinion on such as possess only a possibility of existence; and so\nthere is one mode of approbation in subjects of opinion, and another\nin those of science. Hence science is distinguished from opinion by\ntwo discriminations, the one arising from their subjects, the other\nfrom the mode of approbation. That opinion is conversant with things\npossible or contingent, we may learn from hence; contingencies cannot\nbelong to science, because their existence is not necessary; nor to\nintellect, or that principle of science by which its terms are known;\nnor to the apprehension or belief of immediate propositions, called\nindemonstrable science. Hence, if every habit by which truth is known,\nis either science, or intellect, or opinion, it remains that opinion\nalone consists of things which are, indeed, true; but not necessary.\nIt is, therefore, inconstant and unstable, from the mutable nature of\nits subjects. Besides, no one thinks he possesses an opinion of things\nwhich he believes to have a necessary existence, so that they cannot be\notherwise than they are; but to such conviction he properly gives the\nname of knowledge, and to its contrary the name of opinion.\n\nAgain, the same thing from the same propositions may at one time\nbecome the subject of knowledge, at another, of opinion; and this\nhappens according to the different formation of the syllogism which\nthe propositions compose; whether reasoning from the proper cause it\nexplains the _why_, or only simply declares a thing exists. Hence\na doubt arises why opinions of this kind may not be called science,\nsince both the subjects and propositions are the same? The solution is\nobvious. If it is believed that the propositions cannot be otherwise\nthan they are, or that they have a necessary existence, such an assent\nof the mind is not opinion, but science; because things which inherit\nan essential existence are the ornaments of science alone. On the\ncontrary, if we are convinced that the propositions are true, but\nat the same time not necessary, such conviction is not science, but\nopinion. Hence, it is impossible that science and opinion can be the\nsame, since they vary in their definition and mode of approbation, and\nin a different manner demand our belief. Similar to this, although\nit may happen that of the same thing a true and a false opinion may\narise, it will not therefore follow, that true and false opinions\nare the same. For that which is firm and constant can never be the\nsame with that which is mutable and frail; and that which is always\ntrue must be essentially different from that which may be changed into\nfalse. By the power of habit indeed in different men, the same thing\nmay be comprehended by opinion and science. Thus it was opinion in\nEpicurus when he said that the sun was eclipsed by the moon passing\nunder its orb, because he thought it might otherwise happen, and that\nthe moon might be interposed without obscuring the light of the sun.\nIt was science in Hipparchus, because he knew it as a necessary event.\nBut in the same mind, at the same time, and of the same thing, it is\nimpossible that science and opinion can exist. And thus much concerning\nthe difference of the two.\n\n24. Lastly, sagacity is an acute and sudden apprehension of the medium,\nor proper cause of a certain effect: as if any one, beholding the\nmoon, should in a moment conjecture the cause of the part opposite to\nthe sun being lucid, and the other parts obscure, because she derives\nher splendor from the sun. Hence he is universally called acute and\nsagacious, who, from the aspect or hearing of the extremes, can readily\nperceive the medium which exists between them: as the term imports a\ncertain revolution of the conclusion into its first propositions, and,\nas it were, a swift comprehension and continuation of the medium.\n\n\n\n\n                             SECTION III.\n\n\nIn the ensuing Commentaries, the soul is considered as immaterial; and\nas possessing a middle nature between intelligibles and sensibles;\nbut that this important assertion may not remain without proof,\nthe following demonstrative arguments, derived from the Platonic\nphilosophy, are offered to the reader\u2019s consideration. And first, that\nthe soul is an unextended, and consequently immaterial essence, may\nbe thus proved. If, after the manner of magnitude, it consisted of\ncontinuous parts, it would be impossible that any one part could be\nsentient from the passion of another; but the soul, for instance, which\nis situated in the finger, would be sensible of passion, as if detached\nfrom soul in the other parts of the body, and existing by itself; for\nthe soul, from this hypothesis, would be sentient by parts, and not\nconsidered as a whole. Besides, there must be many souls governing\neach part of us, different among themselves, and endued with their\nown peculiar energies. For whatever may be said of continuity, is to\nno purpose, unless it conduces to unity of sensation; so that the\nhypothesis which supposes that the sensations gradually arrive at the\nprincipal part of the soul by a certain continued succession, is not\nto be admitted, since it may be reasonably asked, How is the principal\npart to be peculiarly distinguished? By what rule of quantity can the\nparts be discerned, by what difference are they to be distinguished,\nwhere the quantity is one, and the bulk continuous? Besides, is\nthe principal part alone, or are the other parts, sentient? If the\nprincipal part alone be perceptive, the soul can then alone be sentient\nwhen a sensible passion meets with this principal part, situated in\nits peculiar seat; but if a sensible passion falls upon any other part\nof the soul destitute of sense, it is impossible that such a part\nshould be able to transmit the same passion to the principal, or be at\nall sentient. For how can that which is void of sense, receive passion,\nand convey it entire to a sentient part? Besides, if passion accedes\nto the principal, it either falls on one of its parts, and so either\none part alone will be sentient, and the rest without sensation, and\nconsequently superfluous, or there must be innumerable and dissimilar\nsensations; for if the sensation of each of the parts singly, is the\nsame with the aggregate of them all, of what use is a multitude of\nparts? But if the sensations are various, a man may say, as it were, I\nam primarily sentient in this place, and secondarily in another; and\nevery sentient part besides the first, will be ignorant where sensation\nis most powerful: or perhaps (from such an hypothesis) every part of\nthe soul will be equally deceived, each part thinking the passion to\narise in the place where it is situated. But if not the principal\npart alone, but every part of the soul be sentient, a principal part\nis superfluous for the purpose of sensation; and if the soul be\ndivisible like magnitude, how is it able to recognize, as belonging\nto one subject, the qualities which flow, as it were, through many\nsenses, as through the eyes and ears? For that part of the soul which\nis perceptive through the eyes, distinguishes nothing but colours;\nthat which energises through the ears, nothing but sounds; and that\nwhich acts through the medium of the touch, nothing but the surfaces\nof bodies: what is it then which perceives all these properties of\nbodies united in one subject, or what is that which perceives any thing\nas a certain one? For unless the intentions of the senses, and of\nsensible objects, were collected together in one, the soul could never\nbe able to judge of the peculiar and different properties of bodies:\nHence it is necessary that the soul should be, as it were, a general\ncentre; that the several senses should be extended on all sides to\nthis, similar to lines verging from the circumference of a circle to\nthe centre; and that a power of this kind, comprehending all things;\nshould be truly one. For if the soul was any thing divisible, and the\nintentions of the senses reached the soul, and ended in its essence\nlike the extremities of various lines, they must either again concur\nin one and the same as a medium, or have different situations, in such\na manner that each sense may perceive different from one another:\nas if, for instance, the sense of sight should perceive the form of\nSocrates, and the sense of hearing recognize his voice; that essence\nwhich pronounces the whole to be one person, that of Socrates, must\nbe something different from each of the senses. Hence it is necessary\nthat the soul should be an indivisible essence; for if she possessed\nmagnitude, she must be divided along with every sensible object she\nperceives; so that one part of the soul would perceive a certain part\nof a sensible object, and we should possess no sentient power capable\nof perceiving the whole, or of pronouncing any thing one. Thus, in\nthe perception of a man, considered as one, how is it possible that\nthe soul can be divided, so that the perception of a part shall be\nthe same with the perception of the whole. But if we suppose the soul\ndivisible in all her perceptions, since it is impossible she should\nbe co-extended with every sentient object, in how many parts is the\ndivision to be made? Is the soul to be distributed into the same number\nof parts as the sensible object she perceives, so that every part of\nthe soul may perceive the same part of the object? Or shall we say that\nthe parts of the soul have no sensation of the parts of the object?\nBut this would be absurd and impossible. If every part then of the\nsoul perceived every part of a magnitude as a whole, since magnitude\nis divisible to infinity, and since, on this hypothesis, there must\nbe innumerable sensations of every sensible object, there must be\ninnumerable images, as it were, of the same thing in our principal\npart. Besides, if that which perceives is corporeal, it will not be\npossible for it to perceive in any other manner than as if certain\nimages were impressed from a seal in wax, or in brass, or in any other\nsensible substance. But if the images of sensation exist as in humid\nbodies (which is most probable), they will certainly be confounded\nlike images in water; nor can there be any memory, the image departing\nwith its forming substance. And if we suppose the figures to remain\nlike impressions in solid bodies, either it will not be possible for\nothers to succeed while the former endure; and thus, sensations of\nother things cannot take place; or if others succeed, the former images\nmust be immediately destroyed, and memory be no more. So that if we\nallow it possible to remember, and, besides this, to perceive other\nthings, without any hindrance from former impressions, it is impossible\nthat the soul should be corporeal. Since the soul, therefore, is an\nunextended, indivisible, and immaterial substance, it is consequently\nincorruptible and immortal; for every thing capable of dissolution and\ndispersion is either corporeal and composite, or exists in some subject\nfrom which it is inseparable. And indeed, whatever may be dissolved\nis corruptible, as being compounded from many. But whatever naturally\nsubsists in something different from itself, when it is separated\nfrom its subject, immediately vanishes into non-entity. But the soul,\nas we have proved, is incorporeal; it is likewise removed from every\nsubject, and naturally reverts to itself, and is therefore immortal and\nincorruptible.\n\n2. Let us now consider how, and on what account the soul is said to be\nof a middle nature, and to be the receptacle of all middle energies,\nboth vital and gnostic. Since, then, there is a long gradation of\nbeings, proceeding from the first being, even to formless matter,\nwhich is nothing more than the dark shadow of essence, it is requisite\nto enquire what the properties are of the first and last beings, and\nwhat the condition is of the middle orders; for thus we shall know\nwhere the essence of the soul ought to be placed. The properties of\nintelligible natures, therefore, are as follows: true being, eternal,\nindivisible, immoveable, total, perfect, full of essence, replete with\nlife, free, moving all things, similitude, presiding over all things,\nand at the same time separated from all; for each of these properties\nappears in intelligibles, according to the processions of being.\nBut the properties of sensible natures, different from these by the\ngreatest interval, are such as, not-true-being, temporal according to\nessence, partible, moveable, particular, indigent of another, always\nreplete with subsistence, living by participation, moved by another,\ndissimilitude, and occupying place by parts. But the middle properties\nof these are, not-true-being, an essence better than non-being, and\ninferior to true-being, according to essence eternal, but according to\nits energies extended with time, indivisible according to its divine\npart, but divisible according to the various processions of reasons,\nself-motive, governing things moved by another, but subordinate to\nsuch as are immoveable, bearing before itself a particular nature,\ntogether with its totality; (for, because it contains in itself all\nreasons, it is after a manner a whole, but because it is diminished and\nfallen, ends in parts, and suffers a transition of its energy, it must\nbe esteemed a particular nature): and again, perfecting itself, yet,\nnevertheless, perfected by natures prior to its own; filling itself\nwith power and strength, and at the same time filled by others: living\nfrom itself, and receiving life from others, being more divine, indeed,\nthan things which live only by participation, but inferior to things\nprimarily vital; moving other things, and itself moved by others; at\nthe same time similar and dissimilar; and separated, at the same time,\nfrom last natures, and co-ordinated with them. Such then, being the\nproperties of the first, middle, and last orders, let us consider where\nwe ought to place the soul, whether in the first order, or in those\nwhich retain the last place: but if we establish it among the first,\nit must be true being, every way eternal and immoveable, and it must\nconsequently possess every thing which we have attributed to the first\nbeings; add too, that on this hypothesis we can no longer attribute\nto the soul a power of self-motion, nor the discursive processions of\nreasoning, nor a variety of other particulars, which manifestly belong\nto the soul. But can we place it with propriety among the last of\nbeings? The least of all: for on this hypothesis we shall make it alone\nmoved by others, divisible, composite, and alone possessing perfection\nfrom others, the opposite of which is evident in all our souls; since\nthey move and perfect themselves, and are led wherever they please.\nSince then it is not possible to place the soul either in the first,\nor last order of beings, it is requisite to assign it a middle place,\nin imitation of its divine cause Rhea (according to the theologists),\nwho is the conciliating band of the two parents Saturn and Jupiter,\nand is reported, from her prolific bosom to produce the life of the\nsoul. But though the soul is thus the extremity of intelligible, and\nthe principle of sensible natures, we must not conceive it to be such\na principle or extremity as a point in a line, for it is not in both\nthe natures it terminates, like a point in both the sections of a\nline; but it is to be called the extremity of intelligibles, because it\nappears after an intelligible essence, and the principle of sensibles,\nas being abstracted from them, and the source of their motion. And\nthus it will preserve to us a certain proportion, that as the natures\nwhich are moved by others, are to those which are moved by themselves,\nso are these last to immoveable natures; and hence it will obtain the\ncondition of a bond, on account of its peculiar mediocrity, unfolding,\nindeed, united causes, but reducing the dissipated powers of sensibles\ninto _one_, and being contained by an immoveable and perpetually\nabiding cause; but containing itself the generation, which, moved by\nanother, is subject to continual mutation. It is likewise intelligible,\nif we regard generated natures; but generated, if we compare it with\nintelligibles; and thus it exhibits in its middle nature both extremes,\nimitating also, by this means, (according to the Greek theologists) its\ndivine cause, for it is said to be on both sides refulgent, \u1f00\u03bc\u03c6\u03b9\u03c6\u03b1\u1f74\u03c2,\nand to be endued with two faces \u1f00\u03bc\u03c6\u03b9\u03c0\u03c1\u03cc\u03c3\u03c9\u03c0\u03bf\u03c2, and to receive in its\nbosom the processions of intelligible natures. It is likewise said to\nbe replenished with intellectual life, and to be the fountain of the\never-running streams of corporeal life, and to contain in itself the\ncentre of the processions of all beings. On this account it is, with\ngreat propriety, affirmed to be generated, and at the same time without\ngeneration. For true being, according to the Platonists, is without\ngeneration, because it has an infinite power of being totally present\nat the same time: and body is said to be generated, because it always\npossesses in itself an infinite flowing power, which it cannot at\nonce totally receive. The soul, therefore, because it is incorporeal,\nabiding in itself, has an infinite power of being, and this total\nwith respect to its essence, and immortal without generation; but\naccording to parts it may be considered in infinite production. For it\nhas not the same total infinity ever present, or there would be the\nsame infinity of the whole and part, of the perfect and imperfect,\nof the contained and containing, which is impossible. But neither is\nit possible that the whole of its essence should be in the act of\nperpetual production, any more than that a part of it should be eternal\nbeing, lest the part should be more worthy and better than the whole.\nHence the \u1f51\u03c0\u03cc\u03db\u03b1\u03c3\u03b9\u03c2, or subsistence of the soul, is at the same time\nof infinite power, and is generated in infinitum; for by this means\nit participates of being, and obtains the first place among generated\nnatures; while body alone, both with respect to its whole and parts, is\nobnoxious to a perpetual generation.\n\n3. But let us now enquire from what genera Plato composes this\nnature of the soul, which contains in itself the bond of all beings;\nprevious to which it will be requisite to explain what these genera\nare, and from whence they originate. Of the species, then, existing\nin the intelligible world, or the divine intellect, which contains in\nitself the causes of all posterior natures, some are most general,\nextending themselves to the universality of things; but others are\nmore particular, like the most special species, and others subsisting\nbetween these, expand themselves, indeed, to a multitude of things,\nbut not to all, according to the division, of the Elean guest in the\nSophista. For man is produced from the ideal man, and horse from the\nideal horse, in the intelligible world; but the similitude which is\nfound in man and horse, and other animals, is produced from likeness\nitself, or the ideal similitude, as dissimilitude from unlikeness\nitself; but the sameness and difference which are found in all beings,\nproceed from the sameness and difference which subsist in ever-vital\nenergy and perfection, in the supreme intellect, or the ideal world.\nNow, as among the sciences some are especially universal, so in\nintelligible causes some are perfectly particular, presiding alone\nover the proper and peculiar number of one species; but others extend\nthemselves to a multitude, such as equality, likeness, totality, (for\nthe whole considered as a whole is not common to all things, since\nthe part is not a whole); but others, again, expand themselves to all\nthings, as all beings participate of these, considered as beings, and\nnot considered as vital or animated, or possessing any other property\nexclusive of the denomination of being. Because, therefore, _being_\nis the first, the causes of _being_ obtain the most universal order\namong genera; and these are five in number, as follows, _essence_,\n_sameness_, _difference_, _motion_, and _station_. For every being\nis endued with essence; is united to itself; is by itself, or its\nown sameness, separated from others; proceeds from itself, and its\nown state and principle, and no less appears to participate of a\ncertain _abiding_, in preserving its own proper species. All things,\ntherefore, whether intelligibles or sensibles, or subsisting between\nboth, depend on these genera for their existence. For without the\nbeing of _essence_, nothing could subsist; in like manner, without\n_sameness_ every whole would be dissipated, and divided from itself;\nand _difference_ being taken away, all things would be one alone, and\nmultitude be destroyed. But without _motion_ and _station_, all things\nwould either be inefficacious and dead, or, losing their proper state\nand stability, would end in non-entity.\n\n4. Such then being the middle nature of the soul, Plato, with great\npropriety, in the Ph\u00e6drus, and in his tenth book of laws, defines it to\nbe number moving itself; which definition he received from Philolaus,\nand Philolaus from Pythagoras. For since mathematical species have\na middle subsistence, as is proved in the following Commentaries,\nthey are of all things most accommodated to the nature of the soul.\nHence Plato, in imitation of Pythagoras, sometimes explains the\nsoul by number, as in the present instance; and sometimes by figure\nand magnitude, as in the Tim\u00e6us; while he considers in the soul the\nintersection of lines, and a twofold circle. For since mathematical\nforms are separated from the flux and inconstancy of matter, they\nparticipate of a certain, exact, sure, and exquisite condition, by\nmeans of which they eminently confer to the elevation of our ingenuity,\nand the explication of latent concerns; and, on this account, as they\npertain to numbers, we may say, preserving the analogy, that there are\nfive orders of numbers, _the divine_, _the essential_, _the\nanimative_, _the natural_, and _the mathematic_. The first\nof these is uniform, the second immoveable, the third self-motive, the\nfourth moved by another, and the last the image of the others, and\ntheir external measure. The divine number is considered eminently in\nthe deity, as in the principle of all things; the essential belongs to\nintellect through ideas, and is called _essence_, _unity_,\nand _the first being_; the animative number belongs to the soul,\nthrough the medium of her inherent reasons; the natural to physical\nconcerns, through the seeds of nature; and lastly, the mathematical\nbelongs to opinion, as it is nothing more than the image of essential\nnumber, formed by the energies of the rational soul. The soul,\ntherefore, is number, not limited by quantity, and mathematical,\nbut animative; it is number, not indeed numbering, but numbered,\ngenerating and converted into itself. Hence too, because harmony arises\nfrom number, the soul is called harmony; not, indeed, a harmony of\nthe parts of the body, nor the harmonic quantity which subsists in\nsound or in voice; but a harmony arising from its essential numbers,\nplaced in its inherent reasons, and in the genera which constitute its\nnature. It is this harmony which produces, as from its proper cause,\nthe harmony of the corporeal parts, the rhythm of motions, and the\nmelody of voices and sounds. It is this which produces that delight\nin the soul from sensible harmony, which sufficiently indicates it to\nbe something familiar and domestic to her nature. From hence it may\nbe inferred, that Plato is not inconsistent with himself when, in the\nPh\u00e6do, he denies, and in the Tim\u00e6us affirms, the soul to be harmony;\nfor he denies that it is a harmony of a definite quantity, or such as\narises from the parts of the body; but he asserts it to be a harmony\nin the manner already explained. It may likewise be inferred, that\nPlato is ignorantly accused by a many, for affirming that the soul is\nharmony, or number; for they only regard vulgar mathematical number,\nand sensible harmony; while Plato, far more elevated, discourses of\nintelligible numbers, and ideal harmony, subsisting in immaterial\nenergy and perfection.\n\n5. And here it is necessary to consider what number, in a particular\nmanner belongs to the soul; for various numbers, differently\nconsidered, accord with her self-motive nature. In the first place,\nunion and unity may be considered in the soul, as in her proper degree\nshe participates of divine unity; and likewise with relation to her\ntotality, for she is one certain whole. And because _a whole_ may\nbe considered in a triple respect, one before the parts, another rising\nfrom the aggregate of parts, and a third subsisting in the single\nparts; the soul is a whole in each of these respects. Thus she is a\nwhole prior to the parts, while she is considered as divisible into\nthem, in an incorporeal manner; she is a whole rising from parts, while\nassuming the parts in the first place, we consider how her nature is\nfabricated from their conjunction; and she is a whole in the single\nparts, since she is total in the whole, and in every part. Besides,\nthe duad belongs to the soul, because she contains in her nature bound\nand infinite, sameness and difference; and lastly, a conversion to\nintelligibles and sensibles. And, indeed, the duad conjoined with\nunity, very properly accords with the soul; for to intellect above\nsoul, unity particularly belongs; to body beneath soul, the infinite\nalone; and to soul situated in the middle, duality properly agrees,\nbeing, as it were, infinite, connected with unity. Again, the ternary\nnumber is attributed to the soul, as well on account of her beginning,\nmiddle, and end, as because she abides in herself, proceeds to\ninferiors, and returns to supernal natures. Besides, she flows from\n_the one_, recedes from him, and is reflected into the one when\nshe acquires her proper perfection. Lastly, as Proclus observes, the\nnature of the soul is divided into essence, power, and energy; so that\nshe may be said to rejoice in the ternary number, and to be replete\nwith its perfection. But the quaternary number belongs to the soul\nso far as she is connected with matter, which is tempered with four\nqualities, and four elements; and she is endued with four principal\nfaculties, nutrition, sensation, local motion, and intellection. But to\nomit other numbers, and their conformity with the soul, the quinary,\nand septenary numbers are especially attributed to the soul. The\nquinary, because the soul is composed from the five genera of things,\nwe have previously explained; and because five particulars merit a\nprincipal consideration in the soul; first, her essence; secondly,\nthe harmony of her reasons; thirdly, the species arising from the\nconcord of her parts or reasons; fourthly, her virtue; and, lastly, her\nenergies: and on this account, Proclus observes, the consideration\nof the soul ought to receive a quintuple distribution. Besides, as\nthe soul consists from a divisible and indivisible nature; so the\nquinary is composed from the first even and the first odd number.\nLastly, as the soul is the connecting medium of the universe; so the\nquinary obtains the middle situation in universal number, that is in\nthe decad. But the septenary number belongs to the soul, because, as\nPlato shews, in his Tim\u00e6us, all harmonical reasons are contained in\nthe seven numbers, 1, 2, 3, 4, 9, 8, 27; and from these the soul is\ncomposed. Hence Proclus elegantly observes, that the septenary number\nis dedicated to Apollo, the parent of all harmony; because in one, two,\nand four, from which the septenary results, the first bisdiapason is\nfound. Besides, the writers on harmony affirm that all the difference\nof voices proceeds as far as to the seventh degree. So that Plato uses,\nwith great propriety, the septenary number for the composition of the\nsoul. Again, in these numbers of the soul, every medium is found in\na convenient proportion; in these the geometric medium is contained,\ncorresponding to the right institution of laws; of which Plato, in his\nRepublic, says, that by this cities are properly governed: there is\nfound too, an harmonical medium, which is the similitude of justice:\nand lastly, we may discover an arithmetical medium, which is called\nthe illustrious symbol of peace. After this manner, therefore, Plato,\nwith a wonderful fecundity of significations, affirms that the soul is\ncomposed from numbers. He likewise considers the figures she contains,\nI mean the circle and triangle; because as the soul is the first\nnature which verges to body, so these are the first of all figures, as\nwell rectilinear as curvilinear. Besides, an orbiculation agrees to\nthe soul, through intellect; but progression and rectitude according\nto her own proper nature; and on this account she contains both a\nstraight and circular figure. I omit other correspondencies of a circle\nand triangle with the soul, as they are exhibited in the following\nCommentaries; and particularly by Proclus, in the third book of his\nCommentaries on the Tim\u00e6us: it is sufficient to the design of this\nDissertation, just to have mentioned this analogy, that the nature of\nthe numbers and figures may appear, which are considered by Plato in\nthe composition of the soul.\n\n6. Let us now pass from contemplating the nature of the soul, to\na survey of its various gradations of knowledge, and the means\nby which it acquires the illuminations of science; as this is a\nspeculation perfectly essential to a full comprehension of the ensuing\nCommentaries. According to Plato, then, in the sixth and seventh\nbooks of his Republic[28], there are four degrees of the internal\ncognitions of our soul; imagination, or assimilation; faith, cogitation\n(\u03b4\u03b9\u03ac\u03bd\u03bf\u03b9\u03b1); and lastly, science or wisdom. The two first degrees\nconjoined constitute opinion; but the two last equally joined produce\nintelligence in its large acceptation. I say in its large acceptation,\nbecause the word intelligence is considered by the Platonists in a\ntriple respect. First, as it rises from opinion and science; as Plato\nasserts, in the seventh book of his Republic. Secondly, as it passes\ninto the same with science; for thus, in the end of the sixth book,\nhe considers intelligence and science as the same. Lastly, as it is\ndistinguished from science, and intelligible from that which is the\nobject of cognition: thus, science regards the essential reasons\nof the soul; but intelligence elevates us to ideas, and this is\nintelligence in its proper acceptation. The distinction of these four\ndegrees, especially depends on the distinction of things with which\nthe soul is conversant. For these four degrees of things are, the\nimage of a sensible object, the sensible object itself, the image of\nan intelligible object, and the intelligible itself. Imagination or\nassimilation regards the image of the sensible object; which image is\nnothing more than the shadow or resemblance appearing in water, or\nother lucid and polished bodies. Faith is conversant with sensible\nobjects; and these are animals, plants, and every thing subject to\nthe energies of sense. From the junction of imagination and faith,\nopinion is produced. Cogitation is conversant with the image of an\nintelligible object; which is nothing more than a certain universal,\ncollected from sensibles, related to the reasons of things existing\nin the soul, and constituted by their assistance; but not elevated to\nideas, and resolved into their lucid nature. Those who are conversant\nwith this image of an intelligible object, use hypotheses; which do\nnot elevate us to principles, which are reasons and ideas, but bring\nus down to subordinate objects. Lastly, science, considered as the\nsame with intelligence, is conversant with that which is intelligible,\nor the essence of things; and of this kind are reasons pertaining to\nthe soul, and ideas to intellect. Hence, as _intelligible_ is to\n_sensible_ in splendor and truth, and form to its image; such is\nthe relation of the superior to the inferior degrees of cognition.\nAnd as these four degrees may be resolved into five, by separating\nintelligence from science; so they are reduced by Plato into two\nprincipal degrees; so far as the two first are conversant about\ngeneration; but the two last about essence. And thus much for the first\nparticular proposed.\n\nIt now remains that we investigate the mode in which science is\nproduced in the soul, according to the doctrine of Plato. For this\npurpose, I think it will be necessary to consider the rational soul,\nin whose nature all reasons corresponding to ideas, have been inserted\nfrom eternity, received into the human body, as into the plain of\noblivion; bordering on the river of negligence, that is placed near to\nthe flux of humours; which producing in its nature various kinds of\nperturbations, are the causes of its self-oblivion and neglect. Hence\nthe soul, thus constituted, as it were, sleeping and intoxicated;\nsleeping before it is roused; intoxicated before it is purified; begins\nfrom external sensible objects to be moved and excited, and to seek\nwith avidity, the knowledge congenial to its nature. The soul now,\nenquiring after truth and the sciences, is first conversant with the\nexternal images of things, in which the glimmering light of similitude\nto truth, presents itself to the view; then it distinguishes these\nimages among themselves by its reasoning power; and if they agree\nin any particular, collects them into one. Thus, being employed in\nseparating into many, things united, and reducing many into one, it\nadvances from shade to substance, and is elevated from similitude\nto truth itself; and thus apprehends the essence of a thing free\nfrom every foreign or contrary quality, shining in _reason_ and\n_idea_. Afterwards, from this contact, especially salutary to\nitself, it experiences an ineffable joy, as from a return into its\nproper nature, and best disposition; and so great is its exultation,\nthat it neglects and despises the shadows which it formerly pursued.\nThen the soul truly knows, that while a man regards corporeal natures,\nhe is employed in resemblances; and that though he may esteem himself\nknowing in many things, he knows nothing in reality; but is then alone\nelevated to the sublime degree of science, when he arrives at ideas.\nHence it appears, that there are four subordinate dispositions of\nmankind in order to science. For, in the first place, children, as\nnew guests of this world, are ignorant of every thing, without being\nconscious of their ignorance. But as they advance in years, they are\nemployed in the shadows and images of natural concerns; and not being\nyet converted to the essences of things, though they possess no real\nknowledge, yet they conceive themselves to abound in knowledge. But\nin the third place, being elevated to supernal natures, and judging\nthese to be alone true, they affirm themselves to be ignorant of all\nthey formerly imagined themselves to know; in which degree Socrates\nprofessed to find himself, when he said, _this one thing I know,\nthat I know nothing_; an ignorance preferable to all the knowledge\ngained by the most unwearied experimental enquiries. Indeed, this order\nis indicated by Socrates himself, in the Ph\u00e6do, when he says, that on\nhis first acquaintance with natural concerns, he thought he abounded\nin knowledge, but as he advanced in these pursuits, he perceived that\nhe knew nothing. Lastly, when, through the piercing sight of the soul,\nmen are elevated to ideas, they become illustrated with the splendors\nof true science, and pass into the regions of perfect reality. And\nhence it appears how true science, which is the same with intelligence\nand wisdom, produces piety and religion: for science elevates us to\nintellect and divinity; and copulates the soul with natures of the\ngreatest purity and perfection; so that an union of this kind cannot\ntake place without piety, sanctity, and religion; as dissimilars can\nnever be blended in amicable conjunction.\n\n\n\n\n                              SECTION IV.\n\n\nWe are informed by Proclus, in the ensuing Commentaries, that the end\nof geometry, and, indeed, of mathematics in general, is to be referred\nto the energies of intellect; and that it is degraded when made\nsubservient to the common utilities of a mere animal life. But as the\nvery opposite to this is the prevailing opinion of the present age,\nlet us examine the truth of this doctrine, and attend to the arguments\nwhich the Platonic philosophy affords in its defence. For if we can\nprove that this assertion of Proclus is supported by the strongest\nevidence, we shall vindicate the dignity of true geometry, restore it\nto its ancient esteem in the minds of the liberal, and shew how much it\nis perverted by applying it to contrary purposes.\n\nIn order to this, I shall endeavour to prove the following position,\nthat things valuable for their own sakes, are preferable to such as\nrefer to something else. Now, this may be demonstrated, by considering\nthat every natural production was made with reference to some end, as\nis evident from an induction of particulars; and if this be the case,\nit may be safely inferred, that every thing exists for the sake of the\nend. But that for the sake of which any being subsists is the best\nof all; and the end, according to nature, is that which is perfected\nthe last of all, from the birth of any being. Hence the human body\nreceives its end or perfection first, but the soul last. And hence the\nsoul is posterior to the body, in the accomplishment of its nature;\nand its ultimate perfection is wisdom. It is on this account that old\nage alone pursues and desires the goods of prudence and wisdom. Hence,\nwisdom is a certain end to us according to nature; and to be wise,\nis the extreme or final cause for which we were produced. It was,\ntherefore, beautifully said by Pythagoras, that man was constituted\nby divinity, that he might know, and contemplate. If then wisdom be\nthe end of our nature, to be endued with wisdom must be the best of\nall. So that other things are to be performed for the sake of the good\nwhich this contains. But to enquire in every science something besides\nthis, and to require that it should be useful, is alone the employment\nof one ignorant of the great difference between the most illustrious\ngoods, and things necessary. For they differ, indeed, widely; since\nthings are to be called necessaries, which are the objects of desire\nfor the sake of others, and without which it is impossible to live. But\nthose concerns alone are properly good, which are loved by themselves,\nthough nothing else should fall to the lot of their possessor; for one\nthing is not to be desired for the sake of another infinitely, but it\nis requisite to stop at some limited object of desire, of which it\nwould be ridiculous to require any utility abstracted from itself.\nBut you will ask, What is the emolument of contemplative wisdom, what\nthe good it confers on its possessor? What if we should say (for such\nis the truth of the case) that it transports us by intellect and\ncogitation, to regions similar to the fortunate islands; for utility\nand necessity are strangers to those happy and liberal realms. And\nif this be admitted, ought we not to blush, that having it in our\npower to become inhabitants of the fortunate islands, we neglect the\npursuit, through a sordid enquiry after what is useful and profitable\naccording to vulgar estimation? The rewards of science, therefore, are\nnot to be reprehended, nor is it a trifling good which results from\nits acquisition. Besides, as men travel to the mountain Olympus for\nits spectacle alone, preferring a view of its lofty summit to much\nwealth; and as many other spectacles are desired for their own sakes,\nand valued beyond gold, in like manner the speculation of the universe\nis to be prized above every thing which appears useful to the purposes\nof life: for it is surely shameful that we should eagerly frequent\nthe theatre, and the race, for the sake of the delight afforded to\nour corporeal sight, and should look for no farther utility in these\nthan the pleasure they produce; and yet should be so sordidly stupid\nas to think that the nature of things, and truth itself are not to be\nspeculated without some farther reward than the sincere delight their\ncontemplation affords.\n\nIt is on this account that the apprehension of truth is compared to\ncorporeal vision; for the sight is the most liberal of all the senses,\nas is confirmed by the general testimony of mankind. Hence, the sight\nof the sun and moon, and the glorious spectacle of the stars is desired\nby the most illiterate as well as the most knowing, for the delight\nsuch visions afford; while, on the contrary, the desires of the other\nsenses are for the most part directed to something farther than the\nmere objects of their energy. Thus, even the sense of hearing, which\nis the next in dignity to the sight, is not always desirable for its\nown sake; for light is the general object of sight, and sound that of\nhearing; but it is evident that light is more universally desired than\nsound, since all light, when not excessive, is always pleasing, but\nthis is by no means the case with every kind of sound. Hence it is,\nthat all contemplation is so delightful, and this in proportion as it\nbecomes abstracted from sensible objects; for the most beautiful forms\ndo not produce genuine delight, until they are strongly represented in\nthe phantasy, as is evident in the passion of love; since the fairest\nface then alone causes love when it presents itself clearly to the\ninward eye of thought, in the mirror of imagination, accompanied with\nliving elegance, and a resistless energy of form.\n\nIndeed, so liberal and so exalted an employment is contemplation,\nthat Plotinus, with his usual profundity, proves that the universe\nsubsists for its sake; that all the productions of nature originate\nfrom this; and that even actions themselves are undertaken with a\nview to the enjoyment of after-speculation. May we not, therefore,\nsay that the sportsman follows the chace for the sake of a subsequent\nreview of his favourite pursuit? That the glutton for this rejoices in\nthe meal; and even the miser in his wealth? And that conversation is\nalone sollicited, that it may recal past images to the soul? In short,\ncontemplation is the first spring of action, and its only end; since we\nare first incited to any external object by speculating its image in\nthe phantasy: and our subsequent conduct tends, without ceasing, to the\nenergy of reflection; for destroy prior and posterior contemplation,\nand action is no more.\n\nNow if this be the case, and if geometry is a speculative science (I\nmean the geometry of the ancients), it is both desirable for its own\nsake, and for still higher contemplations, the visions of intellect,\nto which it is ultimately subservient. For, when studied with this\nview, it opens the eye of the soul to spectacles of perfect reality,\nand purifies it from the darkness of material oblivion. Away then,\nye sordid vulgar, who are perpetually demanding the utility of\nabstract speculations, and who are impatient to bring down and debase\nthe noblest energies, to the most groveling purposes; ignorant of\nthat mighty principle of action, which influences every part of the\nuniverse, and through which even division and discord tend as much as\npossible to union and consent; ignorant that from the depravity of your\nnature, and the blindness of your inward eye, you are incapable of\nspeculating the substance of reality, and are therefore eagerly gazing\non its shadow: and lastly, unconscious that this is the point about\nwhich you are continually making excentric revolutions, mistaking the\ncircumference for the centre, motion for rest, and a departure from\ngood for a tendency to felicity.\n\nIt was for the sake of this most exalted and liberal contemplation that\nHeraclitus yielded his right of succession to a throne, to his brother;\nand that Anaxagoras neglected his patrimony, esteeming one drop of\ngenuine wisdom preferable to whole tuns of riches. Led by a desire of\nthis, as by some guiding star, Pythagoras travelled into Egypt, and\ncheerfully encountered the greatest difficulties, and maintained the\nmost obstinate perseverance, until at length he happily penetrated\nthe depths of Egyptian wisdom, and brought into Greece a treasury of\ntruth for future speculation. But these were happy days; this was the\nperiod destined to the reign of _true philosophy_, and to the\nadvancement of the human soul to the greatest perfection its union with\nthis terrene body can admit. For in our times, the voice of wisdom is\nno longer heard in the silence of sacred solitude; but _folly_\nusurping her place, has filled every quarter with the barbarous and\ndeafning clamours of despicable sectaries; while the brutal hand of\ncommerce has blinded the liberal eye of divine contemplation. For\nunfortunately, the circle of time, as it produces continual variations,\nat length reverses the objects of pursuit; and hence, that which was\nonce deservedly first, becomes at length, by a degraded revolution, the\nlast in the general esteem.\n\n2. If geometry, therefore, be both valuable for its own sake, and\nfor its subserviency to the most exalted contemplations, there can\nbe no doubt but that the great perfection to which this science was\nbrought by the Greeks, was entirely owing to their deep conviction\nof this important truth. Euclid, we are informed by Proclus, in\nthis work, was of the Platonic sect; and Archimedes is reported, by\nPlutarch, in his Life of Marcellus, to have possessed such elevated\nsentiments of the intrinsic dignity of geometry, that he considered\nit perverted and degraded, when subservient to mechanical operations;\nthough, at the request of king Hiero, he fabricated such admirable\nengines for the defence of Syracuse. From this source alone, the great\naccuracy and elegance of their demonstrations was derived, which have\nbeen so deservedly applauded by the greatest modern mathematicians,\nand the warmest advocates for the farrago of algebraic calculation.\nAlgebra, indeed, or as it is called, _specious analysis_, is the\nmodern substitute for the perfect method adopted by the ancients in\ngeometrical demonstrations; and this solely, because it is capable of\nbeing applied with greater facility to the common purposes of life.\nHence, hypotheses have been eagerly admitted in geometry, which the\nancients would have blushed to own: I mean the multiplications and\ndivisions of lines and spaces as if they were numbers, and considering\ngeometry and arithmetic as sciences perfectly the same. But we have\nfortunately the testimony of the first mathematicians among the moderns\nagainst the unlawfulness of this ungeometrical invasion. And to begin\nwith the great sir Isaac Newton, in his Universal Arithmetic[29]:\n\u201cEquations (says he) are expressions of arithmetical computation, and\nproperly have no place in geometry, except so far as quantities truly\ngeometrical (that is, lines, surfaces, solids, and proportions), may\nbe said to be some equal to others. _Multiplications, divisions,\nand such sort of computations, are newly received into geometry, and\nthat unwarily, and contrary to the first design of this science._\nFor whoever considers the construction of problems by a right-line and\na circle, found out by the first geometricians, will easily perceive\nthat geometry was invented that we might expeditiously avoid, by\ndrawing lines, the _tediousness of computation_. _Therefore,\nthese two sciences ought not to be confounded._ The ancients so\nindustriously distinguished them from one another, that they never\nintroduced arithmetical terms into geometry. _And the moderns, by\nconfounding both, have lost the simplicity in which all the elegancy\nof geometry consists._\u201d And in another part[30] of the same work he\nobserves, that \u201c_the modern geometers are too fond of the speculation\nof equations_.\u201d To this very high authority we may add that of Dr.\nHalley, in the preface to his translation of Apollonius de Sectione\nRationis; for which work he conceived so great an esteem, that he was\nat the pains to learn Arabic in order to accomplish its translation\ninto Latin[31]: \u201cThis method, says he, (of Apollonius) contends with\nspecious algebra in facility, but far excels it in evidence and\nelegance of demonstrations; as will be abundantly manifest if any one\ncompares this doctrine of Apollonius de Sectione Rationis, with the\nalgebraic analysis of the same problem, which the most illustrious\nWallis exhibits in the second volume of his mathematical works, cap.\nliv. p. 220.\u201d And in the conclusion of his preface, he observes[32],\n\u201cthat it is one thing to give the resolution of a problem some how\nor other, which may be accomplished by various ways, but another to\neffect this by the most elegant method; by an analysis the shortest,\nand at the same time perspicuous; by a synthesis elegant, and by\nno means operose.\u201d And Dr. Barrow, notwithstanding he was so great\nan advocate for the identity of arithmetic and geometry, expressly\nasserts[33], that _algebra is no science_. To these authorities\nwe may add Simson and Lawson, who, sensible of the superior skill\nof the ancients, both in analysis and synthesis, have made laudable\nattempts to restore the _Greek geometry_ to its pristine purity\nand perfection.\n\nAgain, the greatest men of the present times have been of opinion, that\nalgebra was not unknown to the ancients; and if this be true, their\nsilence respecting it is a sufficient proof of their disapprobation.\nIndeed, if we consider it when applied to geometry, as an art alone\nsubservient to the facility of practice, as conveying no evidence, and\npossessing no elegance of demonstration, we shall not wonder at its\nbeing unnoticed by the ancients, with whom practice was ever considered\nas subservient to speculation; and in whose writings elegance of theory\nand accuracy of reasoning are found perpetually united.\n\n3. But the lives of the first cultivators of this science (I mean the\nEgyptian priests) as well as of the Pythagoreans and Platonists, by\nwhom it afterwards received such improvements, sufficiently evince\nthat this science advanced to perfection from an intellectual theory\nas its source, and from being referred to contemplation as its end;\nand this will be evident, by attending to the following history of the\nEgyptian priests, as preserved to us by Porphyry, in his excellent\nwork on abstinence[34]; a translation of which will not, I presume, be\nunacceptable to the philosophical reader, \u201cCh\u00e6remon, the Stoic (says\nhe) explaining the rites of the Egyptian priests, who, he says, are\naccounted philosophers by the Egyptians, relates, that they choose a\nplace best adapted to the study and performance of sacred rites; so\nthat a desire of contemplation is excited by only frequenting those\nrecesses which are dedicated to their use, and which procure safety\nto the priests, on account of that reverence of the divinity, whose\nsacred mysteries they perform; so that all possible honour is paid to\nthese philosophers, in the same manner as to some sacred animals. But\nhe says they live entirely solitary, except at particular times, when\nthey mix with others in such assemblies as are usually held, and in\npublic feasts; and that on all other occasions they are scarcely to be\napproached. For he who desires to converse with them must first purify\nhimself, and abstain from a multitude of things after the manner of\nthese Egyptian priests. He adds, that these men, renouncing every other\noccupation, and all human affairs, give themselves entirety, through\nthe whole of life, to the contemplation of divine concerns, and to\nenquiring into the divine will: by the latter of these employments\nprocuring to themselves honour, security, and the estimation of\npiety; by contemplation, tracing out the latent paths of science; and\nby both these occupations united, accustoming themselves to manners\ntruly occult, and _worthy of antiquity_. For to dwell always on\ndivine knowledge, and be disposed for divine inspiration, removes a\nman beyond all immoderate desires, calms the passions of the soul, and\nraises her intellectual eye to the perception of that which is real\nand true. But they studied tenuity of aliment, and frugality in their\napparel, and cultivated temperance and patience, together with justice\nand equity, in all their concerns. Indeed, a solitary life rendered\nthem perfectly venerable; for during that period which they call the\ntime of purification, they scarcely mixed with the associates of their\nown order, or saw any one of them, except him who was conversant with\nthem in that exercise of purity, on account of necessary uses. But\nthey by no means concerned themselves with those who were unemployed\nin the business of purification. The remaining part of their time they\nconversed familiarly with those similar to themselves; but they lived\nseparate and apart from those who were estranged from their ceremonies\nand manner of living. He adds, they are always seen employed among the\nresemblances of the gods, either carrying their images, or preceding\nthem in their accustomed processions, and disposing them with gravity\nof deportment, and in a graceful order. In all which operations\nthey did not indicate any pride of disposition; but exhibited some\nparticular natural reason. But their gravity was conspicuous from their\nhabit; for when they walked, their pace was equable, and their aspect\nso perfectly steady, that they refrained from winking whenever they\npleased. Their risibility too, extended no farther than to a smile.\nBut their hands were always contained within their garments; and as\nthere were many orders of priests, every one carried about him some\nremarkable symbol of the order he was allotted in sacred concerns.\nTheir sustenance too was slender and simple; and with respect to wine,\nsome of them entirely refrained from it; and others drank it very\nsparingly, affirming that it hurt the nerves, was an impediment to the\ninvention of things, and an incentive to venereal desires. They also\nabstained from many other things, never using bread in exercises of\npurity; and if they ate it at other times, it was first cut in pieces,\nand mingled with hyssop. But they abstained, for the most part, from\noil, and when they used it mixt with olives, it was only in small\nquantities, and as much as was sufficient to mitigate the taste of the\nherbs.\n\nIn the mean time, it was not lawful for any one to taste of the\naliment, whether solid or fluid, which was brought into Egypt from\nforeign parts. They likewise abstained from the fish which Egypt\nproduced; and from all quadrupeds having solid or many fissured hoofs;\nfrom such as were without horns; and from all carniverous birds: but\nmany of them abstained entirely from animal food. And at those times\nwhen they all rendered themselves pure, they did not even eat an egg.\nBut when the time drew near in which they were to celebrate some sacred\nrites, or festival, they employed many days in previous preparation,\nsome of them setting apart forty-two days, others a greater length\nof time than this; and others again a shorter; but never less than\nseven days; abstaining, during this period, from all animals, and\nfrom all leguminous and oily nutriment, but especially from venereal\ncongress. Every day, they washed themselves three times in cold water;\nafter rising from bed, before dinner, and when they betook themselves\nto rest. And if they happened to be polluted in their sleep, they\nimmediately purified their bodies in a bath. They made cold water\ntoo subservient to the purposes of purification at other times, but\nnot so often as the bath. Their beds were composed from the branches\nof palm, which they called \u03b2\u03b1\u03af\u03c2, _bais_. A piece of wood, of a\nsemi-cylindrical form, and well planed, served them for a pillow.\nBut through the whole of life, they were exercised in the endurance\nof hunger and thirst, and accustomed to a paucity and simplicity of\nnutriment.\n\nBut as a testimony of their temperance, though they neither used the\nexercise of walking nor riding, yet they lived free from disease, and\nwere moderately strong. For, indeed, they endured great labour in\ntheir sacred ceremonies, and performed many services exceeding the\ncommon strength of men. They divided the night between observations of\nthe celestial bodies, and offices of purity; but the day was destined\nby them to the cultivation of the divinities, whom they worshiped\nwith hymns each day three or four times; in the morning and evening,\nwhen the sun is at his meridian, and when he is setting. But the rest\nof their time they were occupied in arithmetical and geometrical\nspeculations, always laborious and inventing, and continually employed\nin the investigation of things. In winter nights also, they were\ndiligent in the same employments, and were ever vigilant to literary\nstudies; since they were not solicitous about external concerns,\nand were freed from the base dominion of intemperate desires. Their\nunwearied and assiduous labour, therefore, is an argument of their\ngreat patience; and their continence is sufficiently indicated by their\nprivation of desire. Besides this, it was esteemed very impious to sail\nfrom Egypt, as they were particularly careful in abstaining from the\nmanners and luxuries of foreign nations; so that to leave Egypt was\nalone lawful to those who were compelled to it by state necessities.\nBut they discoursed much concerning a retention of their native\nmanners; and if any priest was judged to have transgressed the laws\nin the least particular, he was expelled from the college. Besides,\nthe true method of philosophizing was preserved in Commentaries and\nDiaries, by the prophets and ministers of sacred concerns: but the\nremaining multitude of priests, _Pastophori_, or priests of Isis\nand Osiris, governors of temples, and servants of the gods, studied\npurity, yet not so exactly, nor with so great continence as those we\nhave mentioned. And thus much is related of the Egyptians, by a man who\nis equally a lover of truth, and of accurate diligence, and who is\ndeeply skilled in the Stoic philosophy.\u201d\n\n4. But the lives of the Pythagoreans and Platonists, who carried this\ndivine science to its ultimate perfection, no less eminently evince\nthe truth of our position. For, as Porphyry informs us, in the same\ninvaluable treatise[35], \u201csome of the ancient Pythagoreans, and wise\nmen, inhabited the most desert places; and others retired into temples,\nfrom which the multitude and every tumult were expelled. But Plato was\nwilling to fix his academy in a place not only solitary, and remote\nfrom the city, but, as they report, insalubrious. Others, again, have\nnot spared their eyes, through a desire of more perfectly enjoying that\nblissful contemplation, from which they wished never to be separated.\u201d\nAfter this, he presents us with a description from Plato[36] of\nthose intellectual men, by whom the world has been enlightened with\nthe sublimest wisdom and truth: \u201cFor it was not falsly, or in vain\n(says he), that a certain philosopher, speaking of contemplative men,\naffirms, that such as these are ignorant, from their early youth,\nof the way which leads to the forum, or in what place the court or\nsenate-house is situated, or any public council of the state. They\nneither see nor hear the laws, whether decreed or promulgated, or\nwritten; and with respect to the factions and contentions of their\ncompanions for magistracy, for assemblies and splendid entertainments,\nluxurious eating and minstrels, they do not even think of these as in a\ndream. Such an one knows no more of the evil which has happened to some\none of his ancestors, whether male or female, or any thing belonging\nto them, than how many pitchers of water are contained in the sea. Nor\ndoes he abstain from things of this nature for the sake of acquiring\nfame; but in reality, his body alone abides in the city, and wanders\nabout from place to place, but his intellect esteeming all these as\nof small importance, or rather as non-entities, he despises them,\nand, according to Pindar, \u201cfrom these on every side he soars:\u201d by no\nmeans applying himself to things which are near him, and to sensible\nconcerns.\u201d\n\nIf such then were the lives of the men who brought this contemplative\nscience to its present perfection, and who are to this day our masters\nin geometry; if such were the exalted sentiments they entertained\nof its dignity and worth, what greater proof can we require of its\nbeing valuable for its own sake, and as subservient to the energies\nof intellect? We have ample evidence too, of its being degraded\nwhen brought down to the common purposes of life, in the example\nof those who, with this view, have disguised it with the dark and\nsordid involutions of algebraic calculation; for it was solely to\nfacilitate practice, that this barbarous invasion has been admitted by\nthe moderns. Let me then be permitted to persuade the few who study\ngeometry in its ancient purity, and who consider the ruins of Grecian\nliterature on this, as well as on every other science, the models\nof perfection, to enter with avidity on the study of the ensuing\nCommentaries, and endeavour to fathom the depth of our profound and\nelegant philosopher: for by this means they may happily obtain the end\nof all true science, the purification of the soul; and be able to draw\nthe light of perfect wisdom, from the undecaying and inexhaustible\nfountain of good.\n\nBut if it should be asked in what these energies of intellect\nconsist, to which all science ultimately refers? I answer, in the\ncontemplation of true being, or those ideal and divine forms, with\nwhich the intelligible world is replete. Now this great end is not\nto be accomplished without previous discipline, a long exercise of\nthe reasoning power, and a continued series of philosophic endurance.\nFor this end, when attained, is no other than the enjoyment of that\nfelicity congenial to the soul previous to her immersion in body. But,\nfor the further information of the liberal reader on this important\nsubject, the following paraphrases from Porphyry and Proclus are\nsubjoined; the former instructing us in the various purifications\nnecessary to this end; and the latter exhibiting the gradations by\nwhich we may rise to the speculation of reality, and (leaving all\n_multitude_ behind) ascend to the divinely solitary principle of\nthings, the ineffable _One_.\n\n5. \u201cIn the first place, then (says Porphyry[37]) my reasons are not\naddressed to those who are occupied in illiberal arts, nor to those\nengaged in corporeal exercises, neither to soldiers nor sailors,\nneither to rhetoricians nor to those who have undertaken the duties of\nan active life. But I write to the man continually employed in thinking\nwhat he is, from whence he comes, and whither he ought to tend: and\nwho, with respect to every thing pertaining to food, and other offices\nof life, is entirely changed from those who propose to themselves a\ndifferent manner of living; for to a man of this kind alone is my\npresent discourse addressed. Indeed, in this common state of existence,\none and the same mode of persuasion cannot be addressed to the sleeper,\nwho, if it was possible, would conciliate to himself perpetual sleep,\nand who, for this purpose, seeks on every side for soporiferous\nincentives, as to him who studies continually to drive away sleep,\nand to dispose every thing about him to vigilance and intellectual\nactivity. But to the former, it is necessary to advise intoxication,\nsurfeiting, and satiety, and to recommend a dark house; and, as the\npoets say, _a bed luxurious, broad, and soft_. Such a one should\nchuse whatever tends to produce stupor, and give birth to indolence\nand oblivion, whether consisting of odours, ointments, or medicaments\nwhich are accustomed to be eat or drank. But it is necessary that the\nintellectual man should use sober drink, unmixed with the lethargic\nfumes of wine; nutriment slender, and almost approaching to fasting;\na lucid house, receiving a subtle air and wind; that he should be\ncontinually agitated with cares and griefs; and lastly, that he\nprepares for himself a small and hard bed, while thus employed in\npurifying his soul from the stains contracted by corporeal involution.\nBut whether we are born for this exalted purpose, I mean for vigilant\nintellectual energies, allowing as small a part of our life as possible\nto sleep; (since we do not exist in a place where souls perpetually\nvigilant abide), or whether we are destined to a contrary purpose,\nI mean, to sleep and oblivion, would be foreign from our design to\nexplain; and would require a longer demonstration than the limits of\nour work will admit.\n\nBut whoever once cautiously surmizes the delusions of our life in the\npresent world, and the inchantments of this material house in which we\nare employed, and who perceives himself naturally adapted to vigilant\nenergies; lastly, who apprehends the soporiferous nature of the place\nin which he acts, to such a one we would prescribe a diet congruous\nto his suspicion of this fallacious abode, and to the knowledge he\npossesses of himself; in the mean time, advising him to bid a long\nfarewel to the sleeper, stretched on his couch, as on the lap of\noblivion. Nevertheless, we should be careful lest, as those who behold\nthe bleer-eyed, contract a similar defect, and as we gape when present\nwith those who are gaping, so we should be filled with drowsiness\nand sleep, when the place in which we reside is cold, and adapted to\nfill the eyes with watery humours, from its abounding with marshes and\nvapours, which incline their inhabitants to heaviness and sleep. If\nthen, legislators had composed the laws with a view to the utility of\nthe state, and had referred these to a contemplative and intellectual\nlife as their end, we ought to submit to their institutions, and\nacquiesce in the diet they have prescribed for our subsistence. But if\nthey, only regarding that life which is according to nature, and is\ncalled of the middle kind, ordain such things as the vulgar admit, who\nonly estimate good and evil as they respect the body, why should any\none, adducing these laws, weary himself in endeavouring to subvert a\nlife which is far more excellent than every law written and composed\nfor the sake of the vulgar, and which follows a law not written, but\ndivinely delivered? For such is the truth of the case.\n\n_That_ contemplation which procures us felicity, is not a mass\nof discourses, and a multitude of disciplines; or, as some may\nthink, consisting from hence; nor does it receive any increase from\na quantity of words. For if this was the case, nothing could hinder\nthose from being happy, who comprehend all disciplines, and _are\naccurately skilled in a variety of languages_. But the whole\ncircle of the sciences cannot by any means accomplish this blissful\ncontemplation, nor even those disciplines which are conversant with\ntrue and substantial being, unless there is also a conformation of\nour nature and life to this divine end. For since there are, as they\nsay, three ends of living, if we regard the particular objects to\nwhich mankind tend, the end with us is to follow the contemplation of\n_true being_, promoting, as much as possible, by an acquisition\nof this kind, an intimate union of the contemplating individual with\nthe object of contemplation. For, in nothing else besides _true\nbeing_, is it possible for the soul to return to its pristine\nfelicity; nor can this be effected by any other conjunction. But\nintellect is _true being itself_: so that the proper end is to\nlive according to intellect. And on this account, exoteric discourses\nand disciplines, retarding the purgation of the soul, are far from\nfilling up the measure of our felicity. If then, felicity was defined\nby the comprehension of words or sciences, they who do not pay a proper\nattention to the kind and quantity of their food, nor to any thing\nelse pertaining to their present existence, might obtain this end: but\nsince it is requisite to change our life, and to be pure both in speech\nand action, let us consider what discourses and what works may render\nus partakers of this most necessary means of acquiring substantial\nfelicity.\n\nAre, then, those things which separate us from sensible objects,\nand from the affections which they excite, and which lead to a life\nintellectual, and void of imagination and passion, are these the\nmeans we are in pursuit of? So that every thing contrary is foreign\nfrom our purpose, and worthy to be rejected? And in such proportion\nas it draws us aside from intellect? Indeed, I think it is consonant\nto truth, that we should eagerly contend where intellect leads; for\nin this material abode, we are similar to those who enter or depart\nfrom a foreign region, not only in casting aside our native manners\nand customs, but from the long use of a strange country, we are imbued\nwith affections, manners, and laws foreign from our natural and true\nregion, and with a strong propensity to these unnatural habits. Such\nan one, therefore, should not only think earnestly of the way, however\nlong and laborious, by which he may return to his own, but that he may\nmeet with a more favourable reception from his proper kindred, should\nalso meditate by what means he may divest himself of every thing alien\nfrom his true country, which he has contracted; and in what manner he\nmay best recal to his memory, those habits and dispositions without\nwhich he cannot be admitted by his own, and which, from long disuse,\nhave departed from his soul. In like manner, it is requisite, if we\nwish to return to such things as are truly our own, and proper to man\nconsidered as a rational soul, to lay aside whatever we have associated\nto ourselves from a mortal nature, together with all that propensity\nto material connections, by which the soul is allured, and descends\ninto the obscure regions of sense; but to be mindful of that blessed\nand eternal essence intellect, our true father, and hastening our\nreturn to the contemplation of the uncoloured light of _good_, to\ntake especial care of these two things; one, that we divest ourselves\n(as of foreign garments) of every thing mortal and material; the\nother how we may return with safety, since thus, ascending to our\nnative land, we are different from ourselves before we descended into\nmortality. For we were formerly intellectual natures; and even now we\nare essences purified from every stain contracted by sense, and from\nthat part which is destitute of reason: but we are complicated with\nsensible connections, on account of our impotence and infirmity, which\nis the cause that we cannot always be conversant with intellectual\nconcerns; but with mundane affairs we can be present with frequency\nand ease; for all our energetic powers are stupified and clouded\nwith oblivion, through body and sense; the soul not remaining in an\nintellectual state; (as the earth when badly affected, though good\nfruit is deposited in its bosom, produces nothing but weeds); and this,\nthrough the improbity of the soul, which does not, indeed, destroy\nits essence, while it acquires brutality; but by such an accession it\nbecomes complicated with a perishing nature, is bound in the dark folds\nof matter, and is drawn aside from its proper state, into one that is\nforeign and base.\n\nSo that it is highly requisite to study, if we are solicitous of\nreturning to our pristine state of felicity, how to depart from sense\nand imagination, and her attendant brutality, and from those passions\nwhich are raised by her phantastic eye, as much as the necessity of our\nnature will permit. For the intellect must be accurately composed; and\nit is proper it should obtain a peace and tranquility free from the\ncontentions of that part which is destitute of reason, that we may not\nonly hear with attention concerning intellect and intelligible objects,\nbut to the utmost of our ability, may enjoy their contemplation; and\nthus, being reduced into an incorporeal nature, may truly lead an\nintellectual life, and not in a false delusive manner, like these\nwho are at the same time entangled with corporeal concerns. We must,\ntherefore, divest ourselves of the various garments of mortality by\nwhich our vigour is impeded; as well this visible and fleshly garment,\nas that more interior one with which we are invested contiguous to\nthe skin. We must enter the place of contest naked, and without the\nincumbrance of dress, striving for the most glorious of all prizes, the\nOlympiad of the soul. But the first requisite, and without which it is\nnot lawful to contend, is, that we strip off our garments. And since\nour vestments are some of them exterior, and some interior, so with\nrespect to the denudation of the soul, one process is by things more\nopen, another by such as are more occult. For instance, not to eat, or\nnot to accept what is offered, is among things obvious and open; but\nnot to desire is more obscure; so that it is here requisite not only\nto abstain from things improper in deeds, but likewise in desire. For\nwhat does it profit to abstain in actions from what is base, in the\nmean time adhering to the causes which produce such actions, as if\nbound in indissoluble chains?\n\nBut this receding from material affections is brought about partly\nby force, and partly by persuasion; and by the assistance of reason\nthe affections languish, and are, as it were, buried in oblivion, or\nin a certain philosophical death; which is, indeed, the best mode\nof desertion, without oppressing the terrene bandage from which the\nsoul departs. For in things which are the objects of sense, a violent\ndevulsion cannot take place without either a laceration of some part,\nor at least a vestige of separation. But vice steals in upon the\nsoul through continual negligence: and carelessness is produced by\nnot sufficiently attending to intelligible objects; the affections\nin the mean time being excited by the drowsy perceptions of sense,\namong which must be also reckoned the sensations arising from food.\nWe must therefore abstain, not less than from other things, from such\nfood as usually excites the passions of our soul. Let us then in this\nparticular enquire a little farther.\n\nThere are two fountains, whose noxious streams detain the soul in\nmatter, and with which, as if saturated with lethargic potions, she\nforgets her own proper speculations: I mean pleasure and grief, the\nartificer of which is sense and its perceptions, together with the\noperations attendant on the senses, imaginations, opinions, and memory.\nThe passions, roused by the energies of these, and the irrational part,\nnow fattened with noxious nutriment, draw down the soul, and avert\nher inclinations from her native love of true being. It is requisite,\ntherefore, that we revolt from these to the utmost of our ability.\nBut true defections can alone take place by avoiding the passions and\nrash motions produced by the senses. But, sensation respects whatever\nmoves the sight, or the hearing, or the taste, or the smell. And sense\nis, as it were, the metropolis of that foreign colony of passions which\nreside in the soul, and which must be expelled by him who wishes, while\nconnected with body, to become an inhabitant of the royal regions of\nintellect. Let us then enquire how much fuel of the passions enters\ninto us through each of the senses; and this either when we behold the\nspectacles of horses in the race, and the labours of the athletic, or\nthe contests of those who twist and bend their bodies in leaping, or\nwhen we survey beautiful women. For all these insnare us, unconscious\nof the danger, and subject to their dominion the irrational appetite,\nby proffered inchantments of every kind.\n\nFor by all such inchantments the soul, as if driven into fury, compels\nthe compound man to leap rashly, and without reason, and full of the\nbrutal nature to bellow and exclaim. In the mean time, the perturbation\nappearing from without, being inflamed by the internal, which was\nfirst of all roused by sense. But the vehement motions excited by the\nhearing, arise from certain noises and sounds, from base discourse,\nand mixed assemblies; so that some, exiled from reason, behave as if\nstruck mad; and others, enervated by effeminate softness, agitate\nthemselves by a multitude of trifling gesticulations. And who is\nignorant how much the soul is fattened, and infested with material\ngrossness, by the ointments and perfumes which commend lovers to each\nother? But why is it necessary to speak of the passions originating\nfrom the taste: in this respect especially, binding the soul in a\ndouble band; one of which is thickened by the passions excited by the\ntaste; the other becomes strong and powerful by the different bodies\nwhich we receive in food. For as a certain physician observed, those\nare not the only poisons which are prepared by the medical art, but\nsuch things as we daily receive for food, as well liquid as solid, are\nto be reckoned among this number; and much greater danger arises to\nour life from these, than to our bodies from poisons. But the touch\ndoes all but transmute the soul into body, and excites in it, as in a\ndissonant body, certain broken and enervated sounds. The remembrance,\nimagination, and cogitation of all these raise a collected swarm of\npassions, i. e. of fear, desire, anger, love, emulation, cares, and\ngriefs, they fill the soul with perturbations of this kind, cloud its\nintellectual eye with oblivion, and bury its divine light in material\ndarkness.\n\nOn which account it is a great undertaking to be purified from all this\nrout of pollutions; and to bestow much labour in meditating day and\nnight, what measures we shall adopt to be freed from these bonds, and\nthis because we are complicated with sense, from a certain necessity.\nFrom whence, as much as our ability will permit, we ought to recede\nfrom those places in which we may (perhaps unwillingly), meet with\nthis hostile rout; and it is requisite we should be solicitous not to\nengage in combat with these dangerous foes, lest, through too great a\nconfidence of victory and success, instead of vigorous contention, we\nproduce only unskilfulness and indolence.\u201d\n\nAnd in the conclusion of the first book, he adds, \u201cFor, indeed, if\nit be lawful to speak freely, and without fear, we can by no other\nmeans obtain the true end of a contemplative, intellectual life, but\nby adhering to the Deity (if I may be allowed the expression), as if\nfastened by a nail, at the same time being torn away and separated\nfrom body and corporeal delights; having procured safety from our\ndeeds, and not from the mere attention to words. But if friendship\nis not to be conciliated with a divinity, who is only the governor\nof some particular region, with any kind of food, or by the use of\nanimal nutriment, much less can a gross diet effect an union with\nthat God who is exalted above all things, and who is superior to a\nnature simply incorporeal; but after every mode of purgation, and the\ngreatest chastity of body, and purity of soul, we shall scarcely be\nthought worthy to obtain the vision of his ineffable beauty; though\nthis is sometimes permitted to him whose soul is well disposed, and\nwho has passed through life with the greatest sanctity and purity of\nmanners. So that, by how much the Father of all exceeds every nature\nin simplicity, purity, and self-sufficiency, as being infinitely\nremote from all suspicion of material contagion, by so much the more\nought he who approaches to the Deity, to be entirely pure and holy,\nfirst in his body, and afterwards in the most secret recesses of his\nsoul; having distributed a purgation adapted to every part, and being\ncompletely invested with purity, as with a transparent garment, fit\nfor the intimate reception of divine illumination.\u201d Thus far Porphyry,\nwhose excellent sentiments on this subject are a lasting monument of\nthe elevation and purity of soul which the Platonic philosophy affords;\nand at the same time sufficiently prove the arrogance and ignorance of\nthose who depreciate the wisdom of the ancients, and consider their\ngreatest philosophers as involved in mental darkness and delusion.\nBut presumption of this kind is continually increased by indolence,\nand strengthened by interest; and it is common to find _scribblers\nof every kind_, laughing at Plato and his philosophy, who are too\nmean for criticism, and even too insignificant for contempt. Let us,\ntherefore, leave such in their native inanity, and listen to the\ninstructions of the divinely elegant Proclus, by which we may ascend to\nthe contemplation of true being, and the ineffable principle of things.\n\n6. [38]\u201cPythagoras and Plato command us to fly from the multitude,\nthat we may pursue the most simple truth, and apply ourselves wholly\nto the contemplation of real being. From the multitude of exterior\npeople drawing us aside in various ways, and deceiving us by fallacious\nappearances. But much more to shun the multitude of interior people;\nfor this much more distracts and deceives. We must, therefore, fly from\nthe various multitude of affections, the obscure informations of sense,\nthe shadowy objects of imagination, and the dusky light of opinion. For\nevery multitude of this kind is so different in itself, that its parts\nare contrary to one another; from whence it is necessary to betake\nourselves to the sciences, in which multitude has no contrariety.\nFor though affections are contrary to affections, one perception\nof sense to another, imaginations to imaginations, and opinions to\nopinions, yet no one science is found contrary to another. In this\nmultitude, therefore, of propositions and notions, we may collect\ninto one the number of sciences binding them in one according bond.\nFor they are so remote from contrariety to each other, that notion is\nsubservient to notion, and inferior sciences minister to superior,\ndepending on them for their origin. Above all, it is here necessary,\nfrom many sciences which pre-suppose one, to betake ourselves to one\nscience itself, no longer supposing another, and in an orderly series\nto refer them all to this original one. But after science, and its\nstudy, it will be necessary to lay aside compositions, divisions, and\nmultiform discourses, and from thence to ascend to intellectual life,\nto its simple vision, and intimate perception. For science is not the\nsummit of knowledge, but beyond it is intellect; not that intellect\nonly which is separated from soul, but the illustration infused from\nthence into the soul, which Aristotle affirms to be the intellect by\nwhich we acknowledge the principles of science; and Tim\u00e6us says, that\nthis exists in no place but the soul. Ascending, therefore, to this\nintellect, we must contemplate together with it intelligible essence,\nby indivisible and simple perceptions, speculating the simple genera\nof beings. But after venerable intellect itself, it will be proper to\ncontemplate that summit of the soul, by which we are one, and under\nwhose influence our multitude is united. For as by our intellect\nwe touch the divine intellect; so by our unity, and as it were the\n_flower_ of our essence, it will be lawful to touch that first\none, from whom all subordinate unities proceed. And by this our\n_one_, we are especially conjoined with divinity. For similitude\nmay be every where comprehended by that which is similar; the objects\nof knowledge by science; things intelligible by intellect; and the most\nunifying measures of being, by the unity of the soul. But this unity\nand its energy is the summit of our actions; for by this we become\ndivine, when, flying from all multitude, we retire into the depths of\nour unity, and, being collected into one, uniformly energize. Thus far\nwe admonish to shun the multitude, by steps proceeding from the order\nof knowledge: in the next place, we shall proceed in the same design\nby the series of knowable objects. Fly then every sensible species,\nfor they are heaped together, are divisible, and perfectly mutable,\nand incapable of affording sincere and genuine knowledge. From these\ndark informations, therefore, betake yourself to incorporeal essence;\nsince every sensible object possesses adventitious unity, is by itself\nscattered and confused, and full of formless infinity. Hence its good\nis divisible, and adventitious, distant and separated from itself, and\nresiding in a foreign seat. When you have ascended thither, and are\nplaced among incorporeal beings, you will behold above the fluctuating\nempire of bodies, the sublime animal order, self-moving, spontaneously\nenergizing in itself, and from itself possessing its own essence, yet\nmultiplied, and anticipating in itself a certain apparition or image\nof the essence divisible about the unstable order of bodies. You will\nthere perceive many habitudes of reasons, various proportions, and\naccording bonds. Likewise the whole and parts, vivid circles, and\na multiform variety of powers; together with a perfection of souls\nnot-eternal, not subsisting, together as a whole, but, unfolded by\ntime, gradually departing from their integrity, and conversant with\ncontinual circulations. For such is the nature of the soul.\n\nBut after the multitude belonging to souls, betake yourself to\nintellect, and the intellectual kingdoms, that you may possess the\nunity of things. There remain in contemplation of a nature ever abiding\nin eternity, of life ever flourishing, intelligence ever vigilant, to\nwhich no perfection of being is wanting, and which does not desire\nthe chariot of time, for the full energy of its essence. When you\nhave beheld natures of this exalted kind, and have seen by how great\nan interval they are superior to souls; in the next place enquire\nwhether any multitude is there, and if intellect, since it is one, is\nalso universal; and again, since it is uniform, if not also multiform:\nfor you will find it subsists after this manner. When, therefore, you\nhave intimately beheld this intellectual multitude, though profoundly\nindivisible and united, transport yourself again to another principle,\nand having considered, as in a more exalted rank, the _unities_\nof intellectual essences, in the last place proceed to unity perfectly\nseparate and free from all things. And when advanced thus far, lay\naside all multitude, and you will at length arrive at the ineffable\nfountain of _good_. And since it appears, from these various\ngradations, that the soul then properly obtains perfection, when she\nflies from all external and internal multitude, and the boundless\nvariety of the universe, we may likewise conclude from hence, that\nour souls do not alone collect their knowledge from the obscure\nobjects of sense, nor from things particular and divisible discover a\nperfect whole, and a perfect one, but draw forth science from their\ninmost recesses, and produce accuracy and perfection from whatever\nin appearances is inaccurate and imperfect. For it is not proper to\nsuppose that things false and obscure, should be the principal sources\nof knowledge to the soul; and that things discordant among themselves,\nwhich require the reasonings and arguments of the soul, and which are\nambiguous and confused, should precede science which is immutable; nor\nthat things variously changed, should generate reasons abiding in one;\nnor that indeterminate beings should exist as the causes of determinate\nintelligence. It is not, therefore, fit to receive the truth of\neternal entities from boundless multitude; nor from sensible objects\nthe judgment of universals; nor from things destitute of reason,\naccurate discrimination of that which is good: but it is proper that\nthe soul, retiring into her immortal essence, should there scrutinize\nthe _good_ and the _true_, and the immutable reasons of all\nthings: for the essence of the soul is full of these, though they are\nclouded by oblivion. The soul, therefore, beholding exteriors, enquires\nafter truth, in the mean time possessing it in the depths of her\nessence, and deserting herself, explores the good in the dark regions\nof matter. Hence, every one in the pursuit of reality ought to begin\nwith the knowledge of himself. For, if we constantly extend our views\namong the multitude of men, we shall never discern the one species man,\nobscured by the multitude, and distracted by the division and discord,\nand the various mutations of those who participate the species. But\nif we turn our eye inwards, there, remote from perturbation, we shall\nbehold one reason and nature of men; since multitude is an impediment\nto the conversion of the soul into herself. For here variety darkens\nunity, difference obscures identity, and dissimilitude clouds\nsimilitude; since species are confused in the folds of matter; and\nevery where that which is excellent is mixed with the base.\u201d Thus far\nProclus; and thus much for our intended Dissertation.\n\n\n\n\n                                 LIFE\n                                  AND\n                             COMMENTARIES\n                                  OF\n                               PROCLUS.\n\n\n\n\n                                  THE\n                           LIFE OF PROCLUS,\n                            BY MARINUS[39];\n                                  OR,\n                         CONCERNING FELICITY.\n\n\nWhen I consider the magnitude of mind, and dignity of character\nbelonging to Proclus, a philosopher of our time, and attend to those\nrequisites, and that power of composition which those ought to possess\nwho undertake a description of his life; and lastly, when I regard\nmy own poverty of diction, I am inclined to believe it more proper\nto refrain from such an undertaking, not to leap over the fosse\n(according to the proverb), and to decline a discourse involved in so\nmuch difficulty and danger. But my scruples are something diminished\nwhen I consider, on the other hand, that even in temples, those who\napproach to the altars do not all sacrifice alike; but that some are\nsolicitously employed in preparing bulls, goats, and other things of a\nsimilar kind, as not unworthy the beneficence of the Gods to whom those\naltars belong: likewise that they compose hymns, some of which are more\nelegant in verse, but others in prose; while some, who are destitute\nof all such gifts, and sacrifice with nothing more perhaps than a cake\nand a small quantity of bread, with frankincense, and who finish their\ninvocations with a short address to the particular divinity they adore,\nare not less heard than others. While I thus think with myself, I am\nafraid, according to Ibycus[40], lest I should not offend against the\nGods (for these are his words) but against a wise man, and thus obtain\nthe praise of men.\n\nFor I do not think it lawful, that I who was one of his familiars,\nshould be silent concerning his life; and should not, according to my\nutmost ability, relate such particulars concerning him as are true, and\nwhich perhaps ought to be published in preference to others. And indeed\nby such a neglect I shall not perhaps obtain the esteem and honour\nof mankind, who will not entirely ascribe my conduct to the desire\nof avoiding ostentation, but will suppose I avoided such a design\nfrom indolence, or some, more dreadful disease of the soul. Incited,\ntherefore, by all these considerations, I have taken upon me to relate\nsome illustrious particulars of this philosopher, since they are almost\ninfinite, and may be depended on for their undoubted reality.\n\nI shall begin therefore not according to the usual manner of writers,\nwho are accustomed to distribute their discourse into chapters; but\nI consider that the felicity of this blessed man ought, with the\ngreatest propriety, to be placed as the foundation of this treatise.\nFor I regard him as the most happy of those men who were celebrated\nin former ages; I do not say happy only from the felicity of wisdom,\nthough he possessed this in the highest degree of all men; nor because\nhe abundantly enjoyed the goods of an animal life; nor again on account\nof his fortune, though this belonged to him in a most eminent degree,\nfor he was supplied with a great abundance of all such things as are\ncalled external goods: but I call him happy, because his felicity\nwas perfect, complete in all parts, and composed from each of the\npreceding particulars. Having then in the first place distributed[41]\nvirtues according to their kinds into natural, moral, and political,\nand also into those of a sublimer rank, which are wholly conversant\nwith purification and contemplation, and are therefore called Cathartic\nand Theoretic, and also such as are denominated Theurgic, by which we\nacquire a similitude with some particular divinity; but omitting such\nas are superior to these, as beyond the reach of man, we shall begin\nfrom such as are more natural, and which are first in the progressions\nof the human soul, though not first in the nature of things.\n\nThis blessed man, then, whose praise is the subject of this treatise,\nnaturally possessed, from the hour of his birth, all those physical\nvirtues which fall to the lot of mankind; the traces of which were\nmanifest in the latest period of his life, and appeared to surround and\ninvest his body after the manner of a tenacious shell. In the first\nplace, he was endued with a singular perfection of sensation, which\nthey denominate corporeal prudence; and this was particularly evident\nin the nobler senses of seeing and hearing, which are indeed given by\nthe gods to men for the purpose of philosophizing, and for the greater\nconvenience of the animal life; and which remained entire to this\ndivine man through the whole of his life. Secondly, he possessed a\nstrength of body which was not affected by cold, and which was neither\nweakened nor disturbed by any vicious or negligent diet, nor by any\nendurance of labours, though it was exhausted day and night, while he\nwas employed in prayer, in perusing the works of others, in writing\nbooks himself, and in conversing with his familiars; all which he\nperformed with such expedition, that he appeared to study but one thing\nalone. But a power of this kind may with propriety be called fortitude\nof body, from the singular strength employed in such exertions. The\nthird corporeal virtue with which he was endued was beauty, which, when\ncompared with temperance, the authors of these appellations have very\nproperly considered as possessing a similitude of nature. For as we\nconsider temperance as consisting in a certain symphony and consent of\nthe powers of the soul, so corporeal beauty is understood to consist\nin a certain agreement of the organical parts. He was indeed of a most\npleasing aspect, not only because he was endued with this excellent\nproportion of body, but because the flourishing condition of his soul\nbeamed through his corporeal frame like a living light, with splendors\ntoo wonderful for language to explain. And indeed he was so beautiful\nthat no painter could accurately describe his resemblance; and all the\npictures of him which were circulated, although very beautiful, were\nfar short of the true beauty of the original. But the fourth corporeal\nvirtue which he possessed was health, which they affirm corresponds\nto justice in the soul; and that this is a certain justice in the\ndisposition of the corporeal parts, as the other in those of the soul.\nFor justice is nothing more than a certain habit, containing the parts\nof the soul in their proper duty. Hence, that is called health by\nphysicians, which conciliates the jarring elements of the body into\nunion and consent; and which Proclus possessed in such perfection, that\nhe affirmed he was not ill above twice or thrice, in the course of so\nlong a life as seventy-five years. But a sufficient proof of this is\nevident from hence, that, in his last illness, he was entirely ignorant\nwhat the disorders were which invaded his body, on account of the great\nrarity of their incursions.\n\nSuch then were the corporeal goods which Proclus possessed, and which\nmay be called the forerunners, and as it were messengers, of those\nforms into which we have distributed perfect virtue. But the first\npowers and progeny of his soul, which he naturally possessed, previous\nto instruction, and those parts of virtue with which he was adorned,\nand which Plato reckons the elements of a philosophic nature[42], must\nexcite the wonder of any one who considers their excellent quality. For\nhe was remarkable for his memory and ingenuity; he was of a disposition\nmagnificent, gentle, and friendly; and a companion, as it were, of\ntruth, justice, fortitude and temperance; and his love of truth was\nso great, that he never admitted any prudent dissimulation, but hated\nfalsehood vehemently. Indeed it is necessary that he who prosecutes\ntruth, with so much earnestness and sincerity, should be extremely\ndesirous of it from his infancy, since truth is the source of every\ngood, both to gods and men. But that he despised corporeal pleasures,\nand was an eminent lover of temperance, is sufficiently evident from\nhis great propensity to disciplines, and his desire of every kind\nof studies; for dispositions of this kind never suffer beastly and\nilliberal pleasure to dwell it the mind, but are able to excite in the\nsoul, from her own internal operations, sincere pleasure and delight.\nBut it is impossible to say how foreign he was from avarice, so that\nwhen a boy he despised the wealth of his parents, though very rich, on\naccount of his incredible love towards philosophy. Hence he was far\nremoved from illiberality, and from the care of lesser concerns, as he\nwas most studious of the universe, and of every thing divine and human.\nBut from such a disposition of the rational soul, having acquired true\nmagnanimity, he considered human life as of no account, and, unlike the\nmultitude, viewed nothing dreadful in death. So that he by no means\nfeared all that rout of molestations which appear terrible to others,\nand this in consequence of that natural affection which it is proper to\ncall by no other name than that of Fortitude alone. But, from all these\nvirtues, I think it must be evident to those who have not experienced\nhis best of dispositions, that he loved equity from a boy; that he was\njust and mild, and by no means difficult or unjust in his associations\nor contracts. To us indeed he certainly appeared modest and elegant,\nneither avaritious nor illiberal, neither arrogant nor timid.\n\nBut will it not be superfluous to mention the goodness and fertility of\nhis ingenuity? Especially among those who know and who have heard, that\nhe was full of the most beautiful disciplines, and who are acquainted\nwith the multitude he produced and published to the world, so that\nhe alone seemed to have drank nothing of the cup of oblivion, as he\nwas endued with a power of memory which was never disturbed, and that\nwhich belongs to the oblivious, never happened to him. Besides, he\nnever neglected fresh acquisitions, as if possessing a sufficiency of\ndisciplines; and as one who is merely delighted with their study. But\nhe was most remote from a nature rustic and horrid, and averse from the\nMuses, and particularly propense to more cultivated endowments: for on\naccount of his singular urbanity and festivity (without transgressing\nthe bounds of true honesty) in his common associations, sacred feasts,\nand other actions, he allured and charmed his familiars, and always\ndismissed them more cheerful and pleased.\n\nHis mother, therefore, Marcella, lawfully united to his father\nPatricius, both of the Lycian nation, and excelling in birth and\nvirtue, produced our philosopher, thus endued from the beginning with\nall these, and other gifts of nature. And[43] Minerva, the tutelar\ngoddess of Byzantium, received him when born, and took care of him as a\nmidwife, she being the cause of his birth in that city: but afterwards\nshe provided for his well-being, when he was numbered among boys and\nyoung men; for she appeared once to him in a dream, and exhorted him\nto the study of philosophy, from whence arose his great propensity\nto this goddess, as he particularly performed her sacred rites, and\ncultivated with a greater fury (as I may say) her institutions. Lastly,\nhis parents brought him, when born, into their native country Zanthus,\nconsecrated to Apollo: and I cannot but think that this country\nhappened to him by a certain divine providence; as it was requisite\nthat he, who was to be the prince of all sciences, should be educated\nunder the presiding deity of the Muses. Here, being instituted in the\nmost elegant manners, he pursued moral virtues, and was accustomed\nto right conduct, and to a declination of its contrary, that which\nis base.\n\nBut at that time the love of the gods, who had attended him from his\nnativity, manifestly appeared; for being once detained by some disease\nof body, and it appearing very difficult, and scarcely possible to\ncure him, there stood at his bed a youth of a more than ordinary\nappearance, so that even previous to the declaration of his name,\nhe might be considered as[44] Telesphorus or Apollo: but the god\nproclaiming who he was, and pronouncing his name, touched the head of\nProclus (for he stood reclining his head on Proclus\u2019 pillow) and having\nimmediately restored him to health, vanished from his sight. And such\nwas the[45] divine vision, and the divine benevolence at that time\nexhibited to our youth.\n\nBut having, for a short space of time, in Lycia, applied himself\nto grammar, he went to Alexandria in Egypt, bringing with him very\nsingular moral virtues, by which he excited towards himself the love\nof the masters resident in that place. Hence Leonas the rhetorician,\nwho derived (as I think) his lineage from Isaurus, and was illustrious\namong many of that profession, who were then at Alexandria, not only\nmade him a partaker of his studies, but thought him worthy to become\nhis domestic, and ordered that he should be supplied with food,\ntogether with his wife and children, no otherwise than if he had been\nhis[46] true son. He likewise took care to procure him the notice of\nthe principal men in Egypt, who being wonderfully delighted with the\nacumen of the youth\u2019s ingenuity, and with the elegance and integrity\nof his morals, reckoned him among their greatest friends. But he was\nalso instructed by Orion the grammarian, whose ancestors discharged\nthe sacerdotal office among the Egyptians, and who had made such a\nprogress in the knowledge of his art, that he composed elaborate books\non this subject, which he left not without advantage to posterity. He\nalso went to the schools of the Roman preceptors, and made a great\nprogress in that language; for he was at first led to the study of his\nfather\u2019s profession, in which he was illustrious, his employment being\nthe study of law in the royal city. But when it appeared how vehemently\nthe young man was delighted with the study of rhetoric, as he had not\nyet touched the writings of the philosophers, he both acquired great\nglory from his acquisitions, and became the admiration of his fellow\npupils and masters, on account of the elegance of his discourse, and\nhis celerity in perceiving; and from his exhibiting more the habit and\nindustry of the master, than that of the scholar.\n\nBut while he yet frequented the rhetorical school, the sophist\nLeonas, made him the companion of his journey to Byzantium: which he\nundertook for the purpose of gratifying his friend Theodorus, who was\nat that time pr\u00e6fect of Alexandria, and who was a man both polite and\nmagnificent, and a lover of philosophy. But Proclus, though a youth,\nfollowed his master more cheerfully in this journey, lest he should\nbe compelled to interrupt his studies. However, that I may speak\nmore truly, a certain good fortune brought him back to the source of\nhis nativity. For, on his return, his tutelar goddess exhorted him\nto philosophy, and to visit the Athenian schools. But having first\nreturned to Alexandria, and bid farewel to rhetoric, and the other arts\nwhich he had formerly studied, he gave himself up to the discourses of\nthe philosophers then resident at Alexandria. But he frequented[47]\nOlympiodorus, the most illustrious of philosophers, for the sake of\nimbibing the doctrine of Aristotle; and in mathematical disciplines\ngave himself to Hero[48], a religious man, and one who was eminently\nskilful in the proper methods of institution. But these men were so\ndelighted with the manners of the youth, that Olympiodorus wished\nhim to espouse his daughter, whom he had taken care to instruct in\nphilosophy, and Hero committed to him all his religion, and made\nhim his constant companion. But having, on a certain time, heard\nOlympiodorus, a man who was endued with a great power of speaking,\nand on account of the celerity of his speech, and the gravity of\nhis subjects, was understood by very few of his auditors, as he was\ndeparting with the dismissed multitude, he repeated to his companions\nall that was said, and almost verbatim, though the discourse was\ncopious; as Ulpianus Gaz\u00e6us, one of his fellow-disciples informed\nme, who also consumed not the least part of his life in the study of\nphilosophy. But he likewise learned, with great facility, the writings\nof Aristotle, pertaining to rational philosophy[49], though the bare\nreading of them is difficult to those who are engaged in the attempt.\n\nHaving therefore, at Alexandria, applied himself to these masters,\nand enjoyed their confidence in such instruction as they were able to\nafford, when upon reading together with them a certain author, they\nappeared to him not to interpret the mind of the philosopher as they\nought; conceiving a contempt for these schools, and at the same time\nbeing mindful of the exhortation which had been divinely sent to him\nat Byzantium, he went to Athens, attended by the presiding deities of\neloquence and philosophy, and by beneficent d\u00e6mons. For that he might\npreserve the genuine and entire succession[50] of Plato, he was brought\nby the gods to the guardian city of philosophy, as the circumstances\nwhich happened on his first entrance into the city, and all the divine\nexcitations manifestly evince: for they openly presaged, that this\ngift was sent from the father Apollo, and was a future suffrage of his\nsuccession confirmed by divine events. For when his vessel drove to\nthe Pyr\u00e6um, and it was told to the citizens, Nicolaus, who afterwards\nflourished in the rhetorical art, but at that time studied under the\nmasters of Athens, descended to the shore as if to an acquaintance,\nreceived him for his guest as if he had been a citizen, and brought\nhim to the city; for Nicolaus was also a Lycian. But Proclus, who\nperceived himself weary from his journey, sat down at the temple of\nSocrates, though he did not yet know, nor had heard that Socrates was\nworshipped in that place, and requested Nicolaus that he would stay\nthere for a short time, and, if possible, procure him some water,\nas he said that he was exceeding thirsty. Immediately Nicolaus,\nfrom that very consecrated place, brought him some water; for the\nfountain belonging to the statue of Socrates was not far distant. But\nwhile he was drinking, Nicolaus, for the first time, considering the\ncircumstance: This is an omen, says he, because you have sate in the\ntemple of Socrates, and have there first drank the Attic water. Then\nProclus rising, and having paid due reverence to the place, proceeded\nto the city. But when he came to the tower, the porter who was present\nat his entrance, and was about to close the gates with bars, said to\nhim, (that I may repeat the words of the man,) \u201cCertainly unless you\nhad come, I should have closed the gates.\u201d And what omen could be more\nmanifest, or could less require the interpretations of[51] Polletes or\nMelampodes, or such like diviners?\n\nBut Proclus, despising the schools of the rhetoricians, though they\nvery much desired his association, as if he had come for that very\npurpose, met with the prince of philosophers Syrianus[52], the son of\nPhiloxenus. At that time too, Lachares was in the same company, a man\nmuch conversant in philosophy, and the companion of Syrianus in his\nstudy, but in eloquence he was in as great admiration as Homer in the\npoetic art. He then was, as I have said, present at the same time. But\nit was now the evening twilight, and while they were engaged in mutual\nconverse, the sun sate, and the moon made her first appearance after\nher change: wherefore, having saluted the stranger, they endeavoured to\ndismiss him, as being a young man, from their company, that they might\nadore the goddess apart. But he not having proceeded far, beheld also\nthe moon appearing from the same house, and laying aside his sandals,\nin their presence saluted the goddess. Here Lachares, admiring the\nconfidence of the youth, said, speaking to Syrianus, \u201cThis is what\nPlato[53] divinely affirms of great geniuses; that they either produce\ngreat good, or its contrary.\u201d And such, that I may mention a few out\nof many, were the signatures of divine original, which happened to our\nphilosopher, on his first arrival at Athens.\n\nBut Syrianus brought him to the great Plutarch[54], the son of Nestor,\nwho, when he saw the young man, not yet twenty years old, and heard\nof his love and desire of a philosophic life, he was much delighted,\nand immediately made him a partaker of his philosophic study, though\nhis age almost forbade such an attempt; for he was then very old. He\ntherefore read to Proclus his commentary on Aristotle\u2019s books on the\nsoul, and on the Ph\u00e6do of Plato: and this great man exhorted him to\ncommit to writing what he heard, employing the ambition of the youth\nas an instrument, by telling him, that if he completed those scholia,\nthey would be reported as the commentaries of Proclus. And as he\nloved the youth very much on account of his inclination to the best\nstudies, he called him his son, and caused him to reside with him as a\ndomestic. But after he saw his temperance, with regard to animal food,\nhe exhorted him not to abstain from animals entirely, but to use them\nso far as was necessary to the vital energies of the corporeal part. He\nlikewise gave the same advice to Syrianus, concerning the youth\u2019s diet.\nBut he answered the old man, as that divine head (Proclus) informed\nus: \u201cSuffer him with that frugality to learn what I wish, and then, if\nhe pleases, he may die.\u201d Such was the care of his masters respecting\nhim, in every concern. But the old man lived about two years after the\narrival of Proclus; and dying, commended the youth to his successor\nSyrianus, as also his grandson Archiadas. But Syrianus, when he had\nreceived Proclus as his pupil, not only much assisted him in learning,\nbut made him his domestic as to other concerns, and the companion of\nhis philosophic life, having found him such an auditor and successor\nas he had for a long time sought for; and one who was adapted for the\nreception of a multitude of disciplines, and divine dogmata.\n\nIn a shorter space, therefore, than two years, he read, together\nwith Syrianus, all the works of Aristotle, i. e. his logic, ethics,\npolitics, physics, and theological science. But being sufficiently\ninstructed in these, as in certain[55] proteleia, and small mysteries,\nSyrianus led him to the sacred discipline of Plato, and this by an\norderly progression, and not[56] according to the oracle, with a\ntranscendent foot. And he was careful that he might survey with him\ntrue mysteries, with the eyes of his soul, free from material darkness,\nand with a speculation of intellect refined and pure. Hence Proclus\nwas employed night and day in vigilant energies, and in writing\ncompendiously what he had heard, employing his own judgment in the\nselection and order. And in consequence of this unwearied assiduity,\nhe made so great a progress in a short time, that by then he was\ntwenty-eight years of age, he composed a multitude of works; and among\nthe rest his very learned and elegant commentaries on the Tim\u00e6us. But\nfrom an institution of this kind, his manners also received a greater\nornament, since as he advanced in science he accumulated virtue.\n\nIn the original \u03ba\u03b1\u03c4\u1f70 \u03c4\u1f78 \u03bb\u03cc\u03b3\u03b9\u03bf\u03bd, which I wonder Fabricius should\ntranslate, _quod aiunt_, as it is usual with the Platonists, to\ncite the Zoroastrian oracles exactly in these words, instances of which\nmay be found in Proclus on Plato\u2019s theology; and the very words prove\nthemselves to be a part of an oracle, when attentively considered.]\n\nBut he likewise pursued the civil virtues from Aristotle\u2019s political\nwritings, and Plato\u2019s books concerning laws and a republic. However,\nlest he should be conversant only in the contemplation of these, and\nshould attempt nothing actual, since he was hindered from engaging\nin public affairs himself, from his being employed in more important\nconcerns, he exhorted Archiadas, who was a religious man, to a\npolitical life, at the same time instructing him in its nature and\nvirtues, and explaining to him the methods proper to be observed in\nthe discharge of its duties. And in the same manner as they incite\nthose who run in the race, so he exhorted him to preside in common\nover the whole of his city, and to confer benefits privately upon each\naccording to all the virtues, but particularly in conformity with the\nlaws of justice. But he excited in reality the emulation of himself in\nArchiadas, when he exhibited to him his own liberality and magnificence\nwith respect to riches, by bestowing gifts at one time on his friends,\nand at another time on his kindred, guests, and citizens; proving\nhimself by this means not in the least affected with avarice, and\nconferring not a little to the good of public concerns. But when dying,\nhe bequeathed that part of his possessions which he did not leave to\nArchiadas, to his own country, and to Athens. But Archiadas became so\ngreat a lover of truth, as well from the company of his own associates\nas from the friendship of Proclus, that he is never mentioned by\nour acquaintance, without being at the same time celebrated as the\nreligious Archiadas.\n\nBut the philosopher also employed himself in civil consultations\namong public assemblies, where things pertaining to the republic\nwere discussed; giving the most prudent counsels, conferring with\ngovernors concerning equity, and not only exhorting them to an\nimpartial distribution of justice, but in a manner compelling them\nby philosophical authority. For he had a certain public care of the\nmorals of princes, and not only instructed them in the art of temperate\ngovernment by his discourse, but also by his own example through the\nwhole of his life; since he was, as it were, the exemplar of temperance\nto the rest. But he gave a specimen of civil fortitude perfectly\nHerculean: for since at that time there was, as it were, a sea of\ntroubles upon him, and mighty waves of stormy employments were roused\nby adverse winds against his upright life, he conducted himself, though\nin danger, with gravity, and an unshaken constancy. And when he was\nonce very much molested by the improbity of some violent men, which\nwas both pernicious and dangerous to himself, he undertook a journey\ninto Asia, which contributed greatly to his own advantage: for as he\nwas not unskilful in the more ancient rites of that place, which he\nyet preserved, a divine power afforded him this occasion of departure.\nHence, as he well knew the whole of their concerns, he taught them\nmore accurately in things pertaining to the gods, if they happened to\nhave neglected any thing through a long interval of time. And while\nhe was engaged in all these employments, and lived in a correspondent\nmanner, he was so concealed from the multitude that he even excelled\nthe Pythagoreans, who keep with unshaken constancy this precept of\ntheir founder, \u03bb\u03ac\u03b8\u03b5 \u03b2\u03b9\u03ce\u03c3\u03b1\u03c2, _live concealed_. But having passed\na year only in the parts of Lydia, he returned to Athens, under the\nprovidential protection of the presiding goddess of philosophy. And\nthus fortitude was perfected in our philosopher, first by nature,\nthen by custom, and afterwards by science and the consideration of\ncauses. Besides this, he exhibited in another manner his politic habit\npractically, by composing letters for noblemen; and by this means\nprocuring good to entire cities. But of this I have a sufficient\ntestimony from those on whom they were bestowed, as well Athenians as\nArgives, and others of different nations.\n\nBut he likewise much promoted and increased literary studies, demanding\nof princes rewards for the preceptors, according to their several\ndeserts. Nor did he undertake this rashly, nor with any interested\nviews, but he compelled them (as he considered it a matter of great\nmoment) to be diligent in their profession, interrogating and\ndiscoursing with them respecting every particular: for he was a judge\nsufficiently instructed in the employments of them all. And if he ever\nfound any one negligent in his profession, he sharply reproved him;\nso that he appeared very vehement and ambitious, because he was both\nwilling and able to give a just determination on every subject: and\nhe was indeed a lover of glory. But this was not a fault in him, as\nin most, because it alone regarded virtue and goodness. And, perhaps,\nwithout an energy of this kind, nothing great and excellent would ever\nsubsist in the human mind. But he was in this respect vehement: this I\nwill not deny. Yet, at the same time, he was gentle; for he was easily\npleased, and demonstrated in a moment that his anger was as pliable\nas wax. For, almost at the same time, he was (as I may say) wholly\ntransported in reprehension, and with a desire of becoming subservient\nto their interest, and that he might intercede with princes in their\nnames; being moved with a certain natural conjunction of soul, and, as\nit were, sympathy of grief.\n\nAnd here, I very opportunely recollect a peculiar example of his\nnatural sympathy of soul with others: nor do I think the like was\never related of any other man. For, notwithstanding he was unmarried,\nand had no children (because he was not desirous of such connections,\nbut remained free from them all, though many noble and rich alliances\nwere offered him), yet his care of all his familiars and friends,\nand of their wives and children, was as great as if he had been\nsome common father, and the cause of their birth; for he bestowed a\nsingular attention on the life of each. And whenever any one of them\nwas detained by any disease, he first earnestly supplicated the gods\non their behalf with sacrifices and hymns; afterwards he gave a prompt\nattendance on the sick person himself, convened the physicians, and\nurged them to make an immediate trial, if they knew of any thing in\ntheir art advantageous to the condition of the diseased; and sometimes\nhe produced some singular advice himself, among the physicians; and\nthus delivered many from imminent dangers. And the greatness of this\nblessed man\u2019s humanity towards his servants, may be understood by\nthose who desire it, from his will. But of all his familiars, he loved\nArchiadas and his kindred the most; because, in the first place, their\nsuccession was derived from the genus of Plutarch the philosopher;\nand afterwards on account of that[57] Pythagoric friendship which he\nmaintained with Archiadas, as he was both the companion and preceptor\nof his studies. And this other kind of friendship, differing from\nthe two already mentioned, appears to have been the most firm and\nexcellent, among these illustrious men. For Archiadas desired nothing,\nwhich was not also the wish of Proclus; and on the contrary, the\ndesires of Proclus were the constant wishes of Archiadas.\n\nBut having now brought the political virtues, which are inferior to\nthe true ones, to an end, and terminating them in[58] friendship, as\ntheir proper bound, we shall now pass to the Cathartic differing from\nthe politic virtues. Indeed, the employment of these last consists in\npurifying the soul, that so being liberated from the body as much as\nthey are able to effect, it may regard human concerns, and possess a\ncertain similitude with divinity; which is the soul\u2019s best and most\nexalted end. Yet they do not all liberate after the same manner,\nbut some more, and others less. Since there are certain political\npurgations which adorn their possessors, even while connected with\nbody, and reduce them to a better condition; bringing under the\ndominion of reason, anger and desire, and entirely destroying passion\nand every false opinion: but the Cathartic virtues, which are superior\nto these, separate entirely from this truly leaden weight of body, and\nprocure an easy flight from mundane concerns. And in these, indeed, our\nphilosopher was studiously employed during the whole of his life, which\nwas devoted to philosophy; since he both taught by his discourses what\nthey were, and after what manner they were preparatory to felicity,\nand in a particular manner conformed his life to their institutions;\nperforming every thing which could contribute to the separation of\nhis soul, using both night and day prayers, lustrations, and other\npurifications, as well according to the Orphic as the Chaldaic\ninstitutions: and every month he descended, with great diligence, to\nthe sea; and this sometimes twice or thrice. But he was exercised in\nthese, not only in the vigour of his age, but also towards the close of\nhis life; and these customs he observed perpetually, as if they were\ncertain invariable statutes.\n\nBut he used meat and drink, and other necessary pleasures, only so\nfar as was necessary to avoid the molestations of disease; for he was\nin these by much the most frugal, and particularly loved abstinence\nfrom animal food. And if at any time he was invited to eat it more\nvehemently, he was so cautious in its use, that he ate it merely after\nthe manner of a taster. But he purified himself every month by the\nsacred rites, in honour of the mother of the gods, celebrated by the\nRomans, and prior to them by the Phrygians: he likewise more diligently\nobserved the unfortunate days of the Egyptians than they themselves;\nand, besides this, fasted, on certain days in a peculiar manner, on\naccount of the lunar appearances[59]. He likewise instituted a fast\non the last day of the month, not having supped the day before. But\nin what a splendid manner, and with what piety, he celebrated the new\nmoon, and properly observed, with sacrifices, the more illustrious\nfeasts of almost all nations, according to the manner of each country;\nand how from these he did not, according to the custom of many, take\noccasion of becoming idle and intemperate, but employed himself in\ncontinual prayers, hymns, and the like, his hymns sufficiently evince,\nwhich not only celebrate the divinities of the Greeks, but likewise\nMarna Gaz\u00e6us, Esculapius Leonteuchus, Ascalonites, and Theandrites,\nanother god much venerated by the Arabians; together with Isis,\nworshipped by the Philians; and lastly, all the rest which were the\nsubjects of his devotion. For this sentiment was very familiar to\nthis most religious man, that it was proper a philosopher should not\nbe careful in the observance of the rites and institutions of one\nparticular city, nor of certain nations only, but that he should be the\ngeneral priest of the universe. And thus was he pure and holy, so far\nas pertains to the virtue of temperance.\n\nBut he declined, as much as possible, pain: and if it ever happened\nto him, he bore it with gentleness, and diminished it with this view,\nthat his best part might not at the same time be affected with its\nmolestations. And the fortitude of his soul in this respect, was\nsufficiently evinced in his last illness; for when, at that time,\nhe was oppressed and tormented with the most excruciating pains, he\nendeavoured to the utmost to mitigate and expel their afflictive\ninvasions. Hence, on such occasions, he often commanded us to repeat\ncertain hymns, which when repeated procured him a remission and\ncessation of pain. And what is more wonderful, he remembered what he\nheard of these, though forgetful of almost all human concerns, from the\ndissolution of his corporeal part continually increasing. For when we\nbegan to repeat, he supplied what was unfinished of the hymns, together\nwith many of the Orphic verses; for it was these we were then reciting.\nNor was he only thus constant in enduring corporeal evils, but much\nmore so in external unfortunate events, and such as appeared to happen\ncontrary to expectation. So that he would say, concerning particulars\nof this kind, _So it is, such things are usual_; which seemed to\nme, or rather at that time appeared to be, worthy of remembrance, and\nan evident argument of our philosopher\u2019s magnanimity. But besides this,\nhe restrained anger as much as possible, so that it might either remain\nfree from all excitation, or that at least reason might not consent to\nits indulgence, but the irrational faculty alone, contrary to his will,\nmight be moderately and lightly excited. And with respect to venereal\nconcerns, he used them in the natural way, but so as that he might not\nproceed beyond a very moderate and light phantasy, in their indulgence.\n\nAnd thus the soul of this blessed man, having collected itself from\nall parts, and retiring into the depths of its essence, departed\nafter a manner from body; while it yet appeared to be contained in\nits dark receptacle. For he possessed a prudence, not like that of a\ncivil nature, which is conversant in the administration of fluctuating\nparticulars, but prudence itself, by itself sincere, which is engaged\nin contemplating, and converting itself into itself, without any longer\nconsenting to a corporeal nature. He likewise possessed a temperance\nfree from evil; and which is not even moderately influenced by\nperturbations, but is abstracted from all affections. And lastly, he\nacquired a fortitude, which does not fear a departure from body. But\nreason and intellect having obtained in him a perfect dominion, and the\ninferior powers of his soul no longer opposing themselves to purifying\njustice, his whole life was adorned with the divine irradiations of\ngenuine virtue.\n\nOur philosopher, therefore, having most happily absolved this form of\nvirtues, advancing now, as it were, by the highest and most mystical\nstep, he ascended to the greatest and most consummate or telestic\nvirtues; employing for this purpose, the felicity of his nature, and\na sciential institution. Hence, being now purified, and the victor of\nhis nativity, and despising the vain Thyrsus-bearers, and boasters of\nwisdom, he happily penetrated into her profound recesses, and enjoyed\nthe contemplation of the truly blessed spectacles she contains. No\nlonger requiring prolix dissertations, or demonstrations, for the\npurpose of collecting the science of these, but, with a simple vision\nand energy of intellect, beholding the exemplar of the divine mind,\nhe obtained a virtue which cannot with sufficient propriety be called\nprudence, but is more properly denominated wisdom, or something,\nif possible, still more venerable and divine. But the philosopher\nenergizing, according to this virtue, easily comprehended all the\ntheology of the Greeks and Barbarians, and whatever is shadowed over\nby the figments of fables, and placed it in a clear light, for the use\nof those who are willing and able to pursue its latent signification.\nBut having interpreted divinely every thing of this kind, and shewing\nthe symphony between them all; at the same time, investigating all the\nwritings of the ancients, whatever he found in them of genuine wisdom,\nand approved by general consent, this he judiciously applied to use;\nbut if he found any thing of a different and dissonant nature, this\nhe entirely rejected, as vicious and false. And whatever he met with\ncontrary to wisdom, though endued with a friendly appearance, this he\nvigorously subverted by a diligent examination. Nor did he employ less\nforce and perspicuity in his association with other men. For he was a\nman laborious to a miracle; as he often, in one day absolved five, and\nsometimes more lectures; and writ besides, many verses, often to the\nnumber of seven hundred. Besides this, he went to other philosophers,\nand frequented their company; and again celebrated with them an\nevening association, ceasing from the labour of writing. And all\nthese employments he executed in such a manner, as not to neglect his\nnocturnal and vigilant piety to the gods, and assiduously supplicating\nthe sun when rising, when at his meridian, and when he sets.\n\nBut he was the parent of many dogmata, which were never known before,\nboth in physics, and in intellectual and more divine concerns. For he\nfirst taught, that there was a kind of souls[60], endued with the power\nof contemplating many forms at once, which he placed, not without great\nprobability, between intellect, collectively, and as it were with one\nintuition comprehending every thing, and souls, which are alone able to\ndirect their vision to one particular form. And those who are willing\nto peruse his works will meet with a great variety of dogmata, peculiar\nto him alone; the relation of which I shall omit, lest I should give a\ntoo great extent to my discourse. But he who evolves his writings, will\neasily perceive that all we have above related of him is most true, and\nmuch more if he happens to have known him, to have seen his face, and\nto have heard him interpreting in the most excellent dissertations, and\ndelivering the Platonic and Socratic dogmata in his yearly schools.\nNor did he seem destitute of divine inspiration; for he produced from\nhis wise mouth, words similar to the most white and thick falling\nsnow[61]; so that his eyes emitted a bright radiance, and the rest of\nhis countenance was resplendent with a divine light. Hence, when on a\ncertain time, one Rufinus, a man of a great name in the republic, who\nwas studious of truth, and in other respects worthy of veneration, came\nto him when he was teaching and interpreting, he perceived that the\nhead of Proclus was surrounded with a light; and when the philosopher\nhad finished his interpretation, Rufinus rising, adored him; and\noffered to give a public testimony, by oath, of the divine vision which\nhe had observed. And much gold was offered to him, by the same Rufinus,\non his return from Asia, having escaped the danger of the war. But\nProclus likewise rejected this gift, and was by no means willing to\nreceive it.\n\nBut that we may return to our first design, having now discoursed\nconcerning the contemplative wisdom of the philosopher, though\nin a manner but little suited to its dignity, it remains that we\nnow speak of the justice pertaining to this kind of virtues. For\nthis, unlike that of which we spoke before, is not conversant in\ndistribution, or proportion; but must be equally removed from the kind\nof self-energizing justice, by which all things are alone directed to\nthe rational soul. For to that, concerning which we now treat, it is\nalone proper to refer every energy to intellect and the deity, which\nour philosopher performed in the most exalted manner. For he scarcely\nrested from his diurnal labours, or refreshed his body with sleep,\nand perhaps even then was not free from meditation and contemplation.\nThis is certain, that having very speedily roused himself from sleep,\nas from a certain torpor of the soul, he aspired after the morning,\nthe time of prayer; and lest the greater part of the night, should\nglide from him without advantage, as he was lying alone in his bed, he\neither composed hymns, or examined and fortified those dogmata which\nafterwards, in the day time, he committed to writing.\n\nAfter a similar manner he pursued that temperance which has an affinity\nwith these virtues, and which consists in a conversion of the soul to\nintellect, so as not to suffer itself to be touched, nor moved with any\nother concerns. Lastly, he joined fortitude in alliance with these, by\na certain perfect method, zealously aspiring after that liberty which\nis ignorant of all passion, and which he perceived was natural to the\ndivine object of his contemplation. And thus, through the whole of his\nconduct, he did not lead the life of a man merely good, to which, as\nPlotinus says, the political virtues may lead, but leaving this far\nbehind him, he endeavoured to change it for one much more perfect and\ndivine, the life of the gods themselves; since, to become similar to\nthese, and not to virtuous men, was the great object of his contention.\n\nAnd thus he had rendered virtues of this kind familiar to himself,\nwhile he frequented the philosopher Syrianus, and evolved and studied\nthe commentaries of the ancients. But he received from the mouth of\nhis preceptor certain small seeds, as it were, of the Orphic and\nChaldaic theology; because he was prevented from hearing the complete\ninterpretation of his master on the Orphic verses. For Syrianus left to\nthe choice of Proclus and one Domninus[62], a philosopher of the Syrian\nnation, and who afterwards succeeded Syrianus, the exposition of the\nOrphic writings, or the oracles. But they were by no means unanimous\nin their choice; for Domninus preferred the interpretation of the\nOrphic verses, and Proclus that of the oracles. But our philosopher did\nnot perfect his undertaking, because the death of the great Syrianus\nhappened not long after. Having therefore, as I have said, received\nthe outlines from the mouth of his master, he applied himself with\nthe greatest diligence to the written commentaries of Syrianus upon\nOrpheus; and being assiduously nourished with the copious lucubrations\nof Porphyry and Jamblichus on the oracles, and similar writings of\nthe Chaldeans, he arrived, as much as is possible to man, to the top\nof those highest virtues, which the divine Jamblichus was accustomed\nto call after a truly divine manner, theurgic. He laboured therefore,\nnot without exquisite judgment, in collecting the expositions of\nphilosophers prior to his time; and contracted into one, other\nChaldaic hypotheses, and the most excellent of the commentaries on\nthe divine oracles, completing this great work in the space of five\nyears; concerning which this divine vision appeared to him in his\nsleep. For he saw the great Plutarch approach to him, affirming that\nhe should live so many years as he had composed tetrads or quaternions\non the oracles. Afterwards, having collected the number of these, he\nfound they amounted to seventy. But that this was a divine dream,\nwas sufficiently evinced by the last part of his life; for though\nhe lived seventy-five years, as we have mentioned above, yet he had\nnot the perfect use of his powers, in the five last. But his body,\nthough naturally of the best constitution, being debilitated by that\nhard and scarcely tolerable kind of diet which he used, and by so\nmany frequent labours and fastings, began to languish exactly at his\nseventieth year, so that he then became much more remiss in all his\nenergies than before. Yet, even at this period, and thus affected, he\ncomposed orations and hymns: he also writ some things, and conversed\nwith his friends; but his ancient vigour was wanting in each. Hence\nthe memory of the dream excited his wonder, and he every where said,\nthat he had only lived seventy years. But while he laboured under this\ninfirmity of body, a certain youth, named Hegias, rendered him more\nalert in the business of interpretation. This young man, who already\nexhibited from his tender years such egregious signs of all the virtues\nof his ancestors, was one of that golden chain of philosophers, who\nhad formerly appeared to men; and adhered most diligently to Proclus\ndelivering the Platonic and other theologies. But Proclus, at this\nadvanced period, was not moderately rejoiced in communicating with\nthe young man, his own writings, after he understood that he made\ncubital advances in every kind of disciplines. And thus we have briefly\ndiscoursed concerning the apparatus of the philosopher in the Chaldean\noracles.\n\nBut I, on a certain time, having read with him the Orphic verses, and\nheard, among his interpretations, not only the recondite theology\nwhich is to be found in Jamblichus and Syrianus, but also, among many\nother divine men, I requested the philosopher that he would not leave\nthese divine verses also without his explanation: but his answer was,\nThat he had often thought of writing commentaries on Orpheus, but that\nhe had been strongly prohibited in more than one dream. For Syrianus\nappearing to him in his sleep, had deterred him with threats from the\ndesign. Having therefore employed other machines, I intreated that at\nleast he would mark what he principally approved of in the books of\nhis master; which when this best of men had performed, in consequence\nof my persuasions, and had noted some things in the front of each of\nSyrianus\u2019s commentaries, we obtained a collection of all these, and\nby this means scholia, and commentaries of no small bulk; though to\naccomplish this on the whole of that divine poetry, and on all the\nOrphic rhapsiodies, was not the intention of Proclus.\n\nBut since, as we have said, his incredible study of these concerns,\nprocured him a greater and more perfect degree of theurgic virtue,\nhence he no longer remained in the contemplative order, nor contented\nwith either of the twofold properties in divine concerns, exercised\nintellect and thought alone on the more excellent objects of\nspeculation: but he was careful to obtain another kind of virtues more\ndivine than the former, and separated from the politic mode; for he\nused the Chaldean assemblies and conferences, and their divine and\nineffable concealments. And having comprehended these, he learned the\nmanner of pronouncing and expressing them, with their remaining use,\nfrom Asclepigenia, the daughter of Plutarch: for she alone, at that\ntime, preserved the knowledge of the great Orgies, and of the whole\ntheurgic discipline, delivered to her by her father, who received it\nfrom Nestor. Besides this, our philosopher also being purified in an\norderly manner in the Chaldean lustrations, was an inspector of the\nlucid hecatic phasmai (or visions) of which he himself makes mention in\none of his commentaries. But by moving a certain hecatic sph\u00e6rula[63],\nhe very opportunely brought down showers of rain, and freed Athens\nfrom an unseasonable heat. Besides this, by certain phylacteria, or\ncharms, he stopt an earthquake, and had thoroughly tried the energies\nof a divining tripod, having learned from certain verses concerning its\ndefect. For when he was in his fortieth year, he thought in a dream,\nthat he repeated to himself, the following verses:\n\n    High above \u00e6ther there with radiance bright,\n    A pure immortal splendor wings its flight;\n    Whose beams divine with vivid force aspire,\n    And leap resounding from a fount of fire.\n\nAnd in the beginning of his forty-second year, he appeared to himself\nto pronounce these verses, with a loud voice:\n\n    Lo! on my soul a sacred fire descends,\n    Whose vivid pow\u2019r the intellect extends;\n    From whence far-beaming through dull bodies night,\n    It soars to \u00e6ther deck\u2019d with starry light;\n    And with soft murmurs through the azure round,\n    The lucid regions of the gods resound.\n\nBesides, he clearly perceived that he belonged to the Mercurial series;\nand was persuaded from a dream that he possessed the soul of[64]\nNicomachus the Pythagorean.\n\nIndeed, if I were willing to be prolix, I could relate many theurgical\noperations of this blessed man; from the infinite number of which,\nI will only relate one in this place of a very wonderful nature.\nAsclepigenia, the daughter of Archiades and Plutarch, but the wife of\nTheagenes, from whom I have received many benefits, while she was yet a\ngirl, and was nourished by her parents, fell into a disease, grievous,\nand incurable by the physicians. And Archiades, who had reposed in her\nalone the hope of his race, was deeply afflicted with her condition.\nBut the physicians despairing of her recovery, the father fled to his\nlast anchor, Proclus the philosopher, which he was accustomed to do in\naffairs of the greatest moment, considering him as his best preserver,\nand earnestly intreated him to intercede by his prayers with the god\nfor his daughter, whose condition was well known to our philosopher.\nProclus therefore, taking with him Pericles[65] the Lydian, a man\nwell deserving the name of a philosopher, went to the temple of\nEsculapius[66], that he might pray to the god for the health of the\nsick virgin: for the city at that time happily possessed this divinity,\nand rejoiced in the temple of the saviour god, which was as yet free\nfrom the destruction of the Christians. As soon, therefore, as Proclus\nhad prayed after the ancient manner, the girl immediately perceived a\ngreat change and alleviation of her disease; for the preserving god had\neasily restored her to health. The sacred rites being finished, Proclus\nwent to Asclepigenia, and found her delivered from the molestations of\ndisease, and in a healthy condition. But this affair, with many others,\nwas performed privately, so that no traces of it remained to succeeding\ninvestigators; and the house in which he resided greatly assisted him\nin this design. For besides his other fortunes, he enjoyed a most\nconvenient dwelling, which his father Syrianus and his grand-father\nPlutarch (for so he did not hesitate to call him) had once inhabited;\nand this was situated near the temple of Esculapius, together with\nthat of the celebrated Sophocles, and of Bacchus, which is next to the\ntheatre, and looks towards, or is at least seen from, the lofty towers\nof Minerva.\n\nBut how much Proclus was loved by the philosophic goddess is\nabundantly evinced by his philosophic life, which he chose through\nher persuasions, and that with the great success we have hitherto\ndescribed. But she clearly demonstrated her affection to Proclus, by\nthe following circumstance. When her image, which had been so long\ndedicated in the parthenon, or temple, was taken away by those[67] who,\nwithout any hesitation, moved out of their places things the most holy,\nand which ought to be immoveable, there appeared to the philosopher\nin a dream, a woman of a graceful form, who admonished him to build a\ntemple with great expedition, for, says she, _it pleases Minerva, the\npresiding deity of philosophy, to dwell with you_. And how familiar\nhe was with Esculapius, besides what we have mentioned above, may be\nevinced from his perceiving the presence of the same god in his last\nillness. For while he was between sleeping and waking, he perceived a\ndragon creeping on his head, which vision occasioned a remission of\nhis disease, and a mitigation of his pain; so that it is probable he\nwould have been restored to perfect health, if his desire and vehement\nexpectation of death had not prevented his recovery, or his no longer\nbestowing a diligent attention on his body.\n\nBut he also related the following circumstances, (worthy of being\nremembered) but not without tears, through the sympathy of his mind.\nFor, when a young man, he was afraid lest he should be infested\nwith the gout, which was the disease of his father, and which loves\nto descend from parents to their children. Nor was he afraid, as\nit seems, without reason; for prior to that which we have already\nrelated concerning him, he was tormented with pains of this kind, when\nanother extraordinary circumstance happened to this blessed man. In\nconsequence of a certain person\u2019s prescription, whom he had consulted,\nhe applied a plaster to his tormented foot, which an unexpected bird\nflew away with as he was lying on his bed. And this was certainly a\ndivine and salutary symbol to Proclus, and might have persuaded him\nnot to be afraid of this calamity in future. But he, as I have said,\nwas notwithstanding afflicted with the dread of this disease; he\ntherefore supplicated the healing god concerning this, and intreated\nhim to afford him a more evident token of his will: and after this he\nsaw in his sleep (it is bold indeed to conceive such a circumstance in\nthe mind, but we must dare, nevertheless, nor dread to bring truth\nto light) Esculapius approaching to him, and accurately contemplating\nhis leg, nor disdaining, through his great philanthropy to embrace his\nknee. Hence, in consequence of this vision, Proclus was, through the\nwhole of his life, free from the apprehensions of this disease, and was\nnever more tortured with this kind of pains.\n\nAgain, the god who is worshipped by the Adrotteni, did not less\nconspicuously evince his great familiarity with this friend of the\ngods; for, more than once, the god benevolently presented himself to\nthe sight of Proclus, frequenting his temple. And when he was doubtful,\nand wished to know what god or gods resided and were worshipped in\nthat place, because the inhabitants were of different opinions in\nthis affair, some believing that the temple of Esculapius was there,\npersuaded of this by various signs, as voices are certainly said to\nbe heard in that place, and a table is consecrated to the gods, and\noracles are given concerning the recovery of health, and those who\napproach are miraculously delivered from the most grievous dangers\nof life. Others, on the contrary, think that the Dioscuri reside in\nthat place, because there have been those who saw in the way tending\ntowards Adrotta, two young men of the most beautiful form, and riding\nwith great celerity on horseback; at the same time declaring, that\nthey hastened to the temple. They add besides, that the countenance of\nthese was indeed human, but that they immediately gave evidence of a\nmore divine presence; for when the men were arrived at the temple, the\nabove mentioned youths appeared to them, making no enquiry concerning\nthe affair, and occupied in the sacred concerns; but presently after,\nwithdrew themselves from their sight. As the philosopher, therefore,\nwas in doubt concerning these divinities, and did not discredit the\nrelations, having requested the gods who inhabited that place, that\nthey would condescend to manifest who they were; a god appeared to him\nin a dream, and clearly spoke to him as follows: _What! Hast thou not\nheard of Jamblichus teaching concerning, and celebrating those two,\nMachaon and Podalirius?_ And besides this, the god condescended to\nafford so great an instance of benevolence to this blessed man, that\nhe stood after the manner of those who bestow encomiums on others in\nthe theatres, and with a clear voice, and composed habit, extending his\nright hand, did not hesitate to exclaim, (that I may relate the words\nof the god,) _Proclus the ornament of the city_. And what could\npossibly be a greater testimony of this blessed man\u2019s friendship with\ndivinity? But he, indeed, in consequence of a most remarkable sympathy,\nby which he was united with the gods, could never refrain from tears\nwhen he mentioned this affair to us, and related the divine encomium of\nhimself.\n\nIndeed, if I were willing to pursue every particular in like manner,\nand to relate his familiarity with the Hermetic Pan, together with\nthe great benevolence and manifold assistance which that divinity\ncondescended to afford Proclus at Athens, and of that perfectly\nsingular kind of felicity which he obtained from the mother of the\ngods, and in which he was accustomed especially to triumph and rejoice;\nI should perhaps seem to many readers, to be rash in my assertions,\nand to others, the author of things perfectly incredible. For many and\ngreat were the daily instances of this goddess\u2019s benevolence towards\nhim, in words and actions, which are both innumerable and unheard\nof, and concerning which, I do not at present retain an accurate\nremembrance. But if any one is desirous of knowing how great he was in\nthese, he must evolve his book concerning the mother of the gods, from\nwhich he will understand, that it was not without divine assistance\nhe produced all the theology of that goddess into light; as likewise,\nwhatever is acted or affirmed concerning the same in the fables of\nAthis, which he has explained after a philosophical manner, that\nvulgar ears might afterwards cease to be disturbed, on perceiving the\nlamentations and other obscure ceremonies with which her mysteries are\ncelebrated.\n\nAfter having, therefore, run through the theurgic virtue of Proclus,\nand its energies, and the happy circumstances which befell him,\nrespecting its operations, and having shewn, that he did not less excel\nin every kind of virtue, and that he was a man such as mortals have\nnot beheld for a long period of time, it remains that we now bring our\ndiscourse concerning him to a conclusion: for we are not now at the\nbeginning of our narration, nor does the half of the whole remain to\nus, according to the proverb, but the whole itself is now perfect and\ncomplete. Since having begun from the felicity of the philosopher,\nand proceeded in its exemplification, our discourse now returns to\nit again. For we have explained the great goods, and providential\nexertions, which were granted to this most excellent man from the\ngods, and have shewn their prompt attention to his prayers, the divine\nvisions which he enjoyed, and the help and solicitude which the gods\ntestified towards him. We have likewise explained his prosperous fate,\nand propitious fortune, his country, parents, the good habit of his\nbody, his masters, friends, and other external advantages; each of\nwhich we have shewn to have been far greater and more splendid in\nhim than in other men; and have likewise diligently enumerated such\nthings as cannot be reckoned amongst outward allurements, but entirely\ndepended on his will, such as are the upright and illustrious deeds of\nhis soul, according to universal virtue. And thus we have summarily\ndemonstrated that his soul arrived in reality at the summit of the most\nconsummate virtue, and was happily established in a perfect life, by\nhuman and divine goods of every kind.\n\nBut that the lovers of more elegant studies way be able to conjecture\nfrom the position of the stars under which he was born, that the\ncondition of his life, was by no means among the last or middling\nclasses, but ranked among the highest and most happy orders, we have\nthought fit to expose in this place, the following scheme of his\nnativity[68].\n\n            \u2609      16\u00b0    26\u2019    \u2652\ufe0e\n            \u263d      17     29     \u264a\ufe0e\n            \u2644      24     23     \u2649\ufe0e\n            \u2643      24     41     \u2649\ufe0e\n            \u2642       29     50     \u2650\ufe0e\n            \u2640       --     23     \u2653\ufe0e\n            \u263f       4     42     \u2652\ufe0e\n       Horos.        8     19     \u2648\ufe0e\n    Mid.    }\n    Heaven. }        4     42     \u2651\ufe0e\n            \u260b or the head of the dragon.\n                    24     33     \u264f\ufe0e\n    The new moon preceding his birth.\n                     8     51     \u2652\ufe0e\n\nBut Proclus departed from this corporeal life, in the one hundred and\ntwenty-fourth year after the government of Julian, on the seventeenth\nday of the Attic Munichion, or the April of the Romans. Nicagoras,\nthe junior, being at that time the Athenian archon. The dead body was\ntaken care of according to the Athenian rites, as he himself while\nliving had directed. And if that diligent attention which is due to\nthe deceased, was ever paid by any one, it was certainly bestowed by\nthis most blessed man. For he did not neglect any particular of the\naccustomed ceremonies, but every year, on certain days, visited the\nsepulchres of the Athenian heroes and philosophers, and besides this\nsacrificed himself, and not through the medium of another, to the\nmanes of his friends and familiars. And having thus exhibited to each\na proper reverence and honour, proceeding into the Academy, he there,\nin a certain place, pleased one by one the manes of his ancestors and\nkindred; and shortly after, in another part of the same Academy,\nhe supplicated in common the souls of all philosophers. And these\nceremonies being ended, this most excellent man, having chosen a third\nplace, performed sacred rites to all the souls of deceased mankind. The\ndead body of Proclus, therefore, being disposed of according to his\nappointment, as we have said, was carried by his friends and buried\nin the eastern part of the suburbs[69], near to Lycabetus, where also\nthe body of his preceptor Syrianus reposes, who, when he was alive,\nhad requested this of our philosopher, and, in consequence of this,\nhad taken care to procure a twofold receptacle in his sepulchre. But\nwhen this most holy man was consulted how he would wish his funeral to\nbe conducted, lest there should be any thing indecent, or without a\nproper decorum, he desired flutes, with which he was threatened in a\ndream, and nothing besides. An epigram, consisting of four verses, is\ninscribed on his tomb, which he composed himself, as follows:\n\n    I Proclus, here the debt of nature paid,\n    (My country Lycia) in the dust am laid;\n    Great Syrianus form\u2019d my early youth,\n    And left me his successor in _the truth_.\n    One common tomb, our earthly part contains,\n    One place our kindred souls,--th\u2019 \u00e6therial plains.\n\nNor were prognostics wanting in the year prior to his decease, such\nas an eclipse[70] of the sun, so great that night was produced in the\nday-time: for the darkness was so profound, that the stars became\nvisible. And this happened when the sun was in Capricorn, in the\neastern centre. Besides, the writers of Ephemeride observe that there\nwill be another eclipse of the sun, at the conclusion of the next year.\nBut such like affections of the heavenly bodies are said to portend\nevents on the earth: to us indeed, the eclipse perspicuously signified\na privation and, as it were, defection of the illustrious luminary of\nphilosophy.\n\nAnd thus much may suffice for the life of the philosopher. It now\nremains for those who are willing to undertake the labour, to write\nconcerning his disciples and friends. For it appears that many resorted\nto him from all parts; among which some were only his auditors, but\nothers firm adherents to his doctrine; and, on account of philosophy,\nhis familiars. I could likewise wish that some one, who is more\nlaborious than myself, would give a particular account of his writings.\nFor I have alone delivered these particulars of his life, that I might\nsatisfy my conscience, and might testify that I religiously reverence\nthe soul of Proclus, and his good d\u00e6mon. I shall, therefore, add\nnothing concerning the philosopher\u2019s writings, except that I have\nalways heard him prefer his commentaries on the Tim\u00e6us of Plato beyond\nall his works. He likewise vehemently approved of his commentaries on\nthe The\u00e6tetus. And he used frequently to say, that if he was endued\nwith despotic power, he would alone preserve, of all the writings of\nthe ancients, the Oracles and the Tim\u00e6us. _For_, said he, _I\nwould abolish all the rest, and remove them from the present age,\nbecause it happens that many are offended, who undertake to read them\nrashly, and without proper institution_.\n\n\n\n\n             Concerning the Published Writings of PROCLUS.\n\n\nMarinus having neglected to give us an account of the writings of\nProclus, I thought it would not be unacceptable to the reader, to\npresent him with a catalogue of his works which are still extant;\nand the most perfect relation I am able of such as are unfortunately\nlost. In the execution of this design, I shall follow, for the most\npart, the accurate Fabricius in the order, and critical account of his\nworks; but shall not hesitate to dissent from him in deciding on their\nphilosophical merit: for it is very rare that philology and philosophy\nare united in the same person, and coalesce in amicable conjunction.\n\n       *       *       *       *       *\n\n_1. Four Hymns._ In the preceding life we are informed, by\nMarinus, that Proclus composed many hymns, in which the divinities,\nboth of Greeks and Barbarians, were celebrated; but unfortunately there\nare only four preserved, the first of which is to the Sun, the second\nand third to Venus (which may be found in the Florilegium of Stobus, p.\n249. edit. Grotii,) and the fourth to the Muses. They are collected by\nFabricius, in the eighth volume of his Bibliotheca Gr\u00e6ca; and are most\nhappy specimens of philosophical poetry. Indeed, they bear most evident\nmarks of a mind full of divine light, and agitated with the fury of\nthe Muses; and possess all that elegance of composition for which the\nwritings of Proclus are so remarkable. So that it is very strange\nGyraldus should ascribe them to a Hierophant of Laodic\u00e6a, of that name;\nsince, as Fabricius observes, Suidas mentions other writings of this\npriest, but does not speak of him as the author of any hymns. And if he\nhad, these hymns breathe too much of the spirit and manner of Proclus,\nto be the production of any other.\n\n_2. Two Books concerning the useful Parts of Learning, contained\nin Grammar._ Three books on this subject are mentioned by Suidas,\nand four by Photius; but there are alone extant fragments of two,\nin the Bibliotheca of Photius; which were published separate by\nAndrew Schottus, at the end of the syntax of Apollonius Alexandrinus,\nFrancof. 1590. Hanov. 1615. quarto. It appears from these extracts,\nthat Proclus, in this work, not only treats of every kind of verse, but\nalso of the more celebrated poets. And hence Fabricius conjectures,\nthat the short life of Homer, which Leo Allatius published under the\nname of Proclus, was taken from the first book of this Chrestomathia.\n\n_3. Eighteen Arguments against the Christians._ The learned Cave,\nwas egregiously mistaken, in supposing that this work, as well as the\nanswer to it by Philoponus, is lost; not considering, probably, that\nthese arguments in defence of the world\u2019s eternity are (as Fabricius\nwell observes) directly against one of the Christian dogmata, the\ncreation of the world. These arguments (except the first, which is\nlost,) are preserved in the answer of Philoponus; the Greek edition of\nwhich was published at Venice, in 1535, folio; and in Latin from the\nversion of Joh. Mahatius, at Lyons, 1557. folio. Simplicius, in his\ncommentaries on the first book of Aristotle de C\u0153lo, and elsewhere,\nopposes this work of Philoponus; at the same time, suppressing\nhis name. The arguments are, in my opinion, extremely subtle, and\nunanswerable; and Philoponus, in his refutation, every where discovers,\nthat he has by no means fathomed the depth of our philosopher.\n\n_4. A Commentary on the Tim\u00e6us of Plato, in five Books._ A\nmost admirable work, composed in the flower of his age, when he was\ntwenty-eight years old, according to Marinus. This invaluable treasure,\nmay be said to contain all the wisdom of the highest antiquity:\nfor Proclus every where cites the most celebrated interpreters on\nthe Tim\u00e6us, such as Adrastus the Peripatetic; \u00c6lianus and Albinus,\nPlatonists; Aristocles, Clearchus, Crantor, Dercyllides; and especially\nJamblichus; likewise Longinus, whom he often refutes; Numenius the\nPythagorean, and Origen, (different from Adamantius,) together with\nPlotinus, Porphyry the Platonic, Ptolemy, Severus, and Theodorus\nAsin\u00e6us; but he always prefers the explications of his master Syrianus,\nbefore all the rest. Add too, that he every where conciliates the\nOrphic writings and the Chaldean oracles with the theology of Plato.\nAnd with respect to Orpheus in particular, we are indebted to these\nsublime commentaries for the greatest part of the fragments of that\ndivine poet. He is perpetually on the wing, and rises gradually in\nelevated conceptions, until his mind, like Homer\u2019s discord, reaches the\nheavens. His style is at the same time nervous and diffuse, accurate\nand elegant. We are astonished with the magnificence of his metaphors,\ndelighted with the copious variety of his matter, and filled with\na divine light, by the sacred truths he unfolds. This great work,\nhowever, unfortunately, scarcely explains a third part of the Tim\u00e6us;\nfrom whence it is probable, as Fabricius well observes, that several\nbooks have been lost through the injury of time. It was published in\nGreek, at Basil, in the year 1556; and is full of mistakes, as is\ngenerally the case with the Basil editions of books, so that a more\ncorrect edition is greatly to be desired, though at present not much\nto be expected. On concluding my account of this inestimable work, I\nfind my indignation roused by the following words of Dr. Cudworth,\nin his Intellectual System, p. 306. \u201cProclus (says he) had some\npeculiar fancies and whims of his own, and was indeed a confounder\nof the Platonic theology, and a mingler of much unintelligible stuff\nwith it.\u201d I must confess, (and I am neither afraid, nor ashamed of\nthe declaration,) that I never found any thing in Proclus, but what\nby patient thought, accompanied with a sincere and vehement thirst\nafter truth, I have been able to fathom. Had Dr. Cudworth been endued\nwith these requisites, he would doubtless have had equal success;\nbut without them, the sublimest truths will certainly appear to be\n_unintelligible stuff_. Besides this consideration is not to be\nomitted, that a modern priest makes a bad philosopher.\n\n_5. On Plato\u2019s Republic._ These commentaries, or rather fragments\nof commentaries, are extant in Greek, at the end of Proclus on the\nTim\u00e6us. Suidas mentions four books of Proclus on Plato\u2019s politics; and\nsome of Proclus\u2019s dissertations on these books, were found (according\nto Fabricius) in the library of Lucas Holstenius. The chief design of\nthis work seems to be the unveiling the theological mysteries concealed\nunder the fables of Homer, and other divine poets; which Proclus has\naccomplished (in my opinion) in a most wonderful manner. That Homer,\nindeed, every where abounds with Egyptian learning, is obvious to\nevery one; but few are acquainted with the profound wisdom which his\nfables conceal. The latent meaning of most of these is unfolded in\nthe present invaluable, though imperfect work; and he whose mind is\nsufficiently enlightened by the ancient philosophy, to comprehend the\nbeauty of these illustrations, will receive an additional delight from\nthe study of Homer, which it is impossible to express. An epitome of\nthis work was published in Latin, by the learned Gesner, 1542. 8vo.\nunder the following title: Apologi\u00e6 qu\u00e6dam pro Homero, et Arte Poetica,\nFabularumque aliquot Enarrationes ex commentariis Procli Lycii Diadochi\nphilosophi Platonici in libros Platonis de Rep. in quibus plurim\u00e6 de\nDiis Fabul\u00e6 non juxta grammaticorum vulgus historic\u00e8, physic\u00e8 aut\nethic\u00e8 tractantur, sed Theologicis, ut Gentiles loquntur, ex prima\nPhilosophia rationibus explanantur.\n\n_6. On the first Alcibiades of Plato._ Ficinus translated parts\nof this work into Latin, and published them under the title of Procli\nde Anima ac d\u00e6mone, de Sacrificio et Magia, Venice 1497. and 1516.\nfol. by Aldus; and in a more simple form at Lyons. Fabricius informs\nus, that the manuscript commentary of Proclus in Greek, but scarcely\nexplaining the half part of the Alcibiades, is to be found in various\nlibraries of France, England, and Italy. Also at Lyons, among the books\nof Isaac Vossius; and at Hamburgh in the Johannean library. From the\nspecimen given of this work by Ficinus, it appears, like all Proclus\u2019s\nphilosophical writings, to be an invaluable treasury of wisdom; and\nnothing certainly, reflects greater disgrace on a nation than suffering\nsuch monuments of ancient learning and wisdom to lie concealed in\ncolleges, covered with dust, and never consulted.\n\n_7. Six Books on Plato\u2019s Theology._ A most divine work, in which\nthe philosopher collects into a system the theology dispersed in the\nwritings of Plato, and establishes it by invincible demonstrations. He\ndeduces, in a beautiful and connected series, all the divine orders,\nfrom the retreats of ineffable unity; every where connects them by\nproper mediums, and, after leading us through the long gradation of\nprinciples, brings us back again to the original from whence they\nflowed, and to which they constantly tend. The whole is uncommonly\nprofound and abstruse; and it was not before the third reading, that\nI could fathom the depth it contains. Fabricius observes, \u201cthat it\nis a subtle and learned work, but from which, you will sooner learn\nthe opinion of Syrianus and Proclus, concerning the deity and divine\nconcerns, than that of Plato. He adds, that it is usual with the\nPlatonists, even from Plotinus, to unite to the doctrine of Plato, a\nthousand dogmata, foreign from his philosophy, as if Plato, though he\ndid not perceive after this manner, ought certainly so to perceive.\u201d\nWhen men mistake their abilities, they always act absurdly, and often\ndangerously. As a laborious and accurate critic on philological\nmatters, Fabricius merits the highest commendation such attainments\ncan deserve; but when he leaves the beaten road in which nature\ndesigned him to walk, and attempts the tractless paths of philosophy,\nhe perpetually stumbles, and often falls on the ground. The wings of\nphilology, like those of the swallow, were never destined for a lofty\nflight:--it must be the eagle wing of genius, which can alone soar to\nthe sun of philosophy. The Greek and Latin edition of this valuable\nwork, was published at Hamburgh, by \u00c6milius Portus, 1618, folio.\n\n_8. Theological Institutions_; or, as it may be called, _the\nElements of Theology_. This admirable work contains two hundred and\nten propositions, disposed in a scientific order, and supported by the\nfirmest demonstrations. They begin from super-essential unity, and\nproceed gradually through all the beautiful and wonderful progressions\nof divine causes, ending in the self-moving energies of soul. They\npossess all the accuracy of Euclid, and all the subtilty and sublimity\nnecessary to a knowledge of the most profound theology; and may be\nconsidered as bearing the same relation to the Pythagoric and Platonic\nwisdom, as Euclid\u2019s Elements, to the most abstruse geometry. Patricius,\nthe first Latin translator of this divine work, seems to have been very\nsensible of the truth of this observation: for he every where carefully\ndistinguishes the propositions from their demonstrations; and adds\nthe word corollary to such consequencies as merit that appellation.\nHis edition was published at Ferraria, 1583. quarto, under the title\nof Theological Elements. The Greek and Latin edition, is subjoined to\nProclus\u2019s six books on Plato\u2019s Theology, Hamburgh 1618. folio.\n\n_9. Two Books concerning Motion._ This useful work, collected, as\nFabricius observes, from the third and following books of Aristotle\u2019s\nphysics, was published in Greek at Basil, 1531, and with the Latin\nversion of one Justus Velsius, a physician, Basil, 1545. octavo. It\nwas likewise translated by Patricius, and is annexed to his version of\nthe Theological Institutions.\n\n_10. An Hypotyposis, or Information concerning Astronomical\nHypotheses._ This work, which Fabricius observes is a compendium of\nPtolemy\u2019s Almagest, was published in Greek, at Basil, 1540. quarto;\nand in Latin by George Valla, folio, 1541. A part of this work, which\ntreats of the use of the astrolabe, Fabricius informs us, is extant in\nmanuscript, in various libraries. The same accurate critic likewise\nobserves, that a small treatise, inscribed Uranodromus, is extant,\nunder the name of Proclus, in some libraries, as in that of Vindobona,\nand of Oxford, among the Barrocian volumes. The comprehensive variety\nof Proclus\u2019s genius equally demands our admiration and applause.\n\n_11. A small Treatise concerning the Sphere, or Celestial\nCircles._ This little work is an accurate and elegant introduction\nto astronomy; and is almost wholly taken from the Isagoge of Geminus\nRhodius, on the ph\u00e6nomena. The best editions are the Greek and Latin\none published at Paris in 1553, quarto; and that of Bainbridge,\nprofessor of astronomy at Oxford, London 1620. quarto.\n\n_12. A Paraphrase in four Books, on the Quadripartite of Ptolemy._\nThis elegant work must, I should imagine, be an invaluable treasure to\nthe lovers of astrology. It was first published in Greek by Melancthon;\nand afterwards in Greek and Latin by Leo Allatius, at Lyons Bat. 1654.\noctavo.\n\n_13. Four Books, on the first Book of Euclid\u2019s Elements._ For an\naccount of this work, see the introduction, and the following sheets,\nin which it speaks for itself, in an English dress.\n\n_14. A Commentary on Hesiod\u2019s Works and Days._ This work contains\na valuable moral explanation of this great poet\u2019s meaning; and\nFabricius justly observes, that he is often assaulted without occasion,\nby the petulant jeers of that vain man Joh. Tzetzes. The best edition\nof this work is that of Daniel Heinsius, Lugd. Bat. 1603. quarto.\n\n_15._ Fabricius informs us, that in some manuscripts, as in the\nVindobonensian and Barrocian, a small treatise is usually ascribed\nto Proclus, entitled _Epistolic Characters_; and is prefixed to\nthe Epistles of Phalaris and Brutis, and published under the name of\nLibanius, in Greek, with the version of Casp. Stibilinus, Commelin.\n1597, octavo. But it is doubtful whether Proclus is the genuine author\nof this work: from the title, I should suppose the contrary. And thus\nmuch for an account of those writings of Proclus which have escaped\nthe ravages of time, and have been fortunately exposed to public\ninspection: it now remains that we relate such inestimable works of\nthis philosopher, as are yet preserved in shameful concealment; or are\nutterly lost in the ruins of antiquity.\n\n\n            Concerning the Unpublished Writings of PROCLUS.\n\n16. _On the Alcibiades of Plato._ See num. 6.\n\n17. _On Plato\u2019s Politics._ See num. 5.\n\n18. _On Plato\u2019s Parmenides._ A commentary, in seven books; the\nlast of which was not completed by Proclus, but by Damascius. From\noccasional fragments, which have been published of this commentary, it\nappears to be a most divine work; and indeed it cannot be otherwise, if\nwe consider it as the production of one of the greatest philosophers,\non the most sublime and profound of all Plato\u2019s Dialogues. It is\ndedicated to Asclepiodotus, a physician and philosopher, and is\nnot only extant in Greek MS. in the library of the German emperor,\naccording to Lambecius, lib. vii. p. 41. but also in Latin, from the\nunpublished version of one Antonius Hermannus Gogava, as the same\nLambecius informs us, p. 41. Four books of this work are extant in\nGreek, in the Bodleian library at Oxford; and it is much to be lamented\nthat Thomson did not publish these, instead of his trifling edition of\nthe Parmenides. Fabricius likewise informs us, that Livius Galantes\nmentions his having found six of these books in some of the Italian\nlibraries. They are also extant in the Medic\u00e6an library of the great\nEtruscan commander.\n\n_19. On the Cratylus of Plato._ We have already observed, in the\ndissertation on the Orphic theology, p. 105. what a great treasure of\nancient mythology, must be contained in this work; but there is little\nhope of its ever emerging from the obscurity of public libraries. It\nis extant in Greek, not only in the Italian libraries, but also among\nthe manuscript books of Isaac Vossius.\n\n_20. Uranodromus._ See above, num. 10.\n\n_21. Ten Doubts concerning Providence, in one Book._ Philoponus\nmentions this work, in his second book against Proclus on the eternity\nof the world; and a Latin version of it is extant by one William de\nMorbeka, in the Johannean library of Hamburgh. Extracts from this\ntranslation are preserved by Fabricius, in his Greek Library; and they\nare in every respect worthy of the genius of Proclus.\n\n_22. Concerning Providence and Fate, and that which is in our Power,\none Book._ This work is dedicated to one Theodorus, a mechanist; and\nis extant in the Latin translation of the same Morbeka, in the Greek\nLibrary of Fabricius. The translation is for the most part barbarous,\nbut is, however, sufficiently legible to discover that it is a most\nvaluable treatise, replete with the usual elegance, subtilty, and\nsublimity of our philosopher.\n\n_23. Concerning the Hypostasis, or Subsistence of Evil._ This\nbook is extant in Latin, in the Johannean library; and fragments of\nit are preserved by Fabricius, in his Greek library. It is to be\nregretted, that Fabricius did not preserve the whole in that excellent\nphilological work.\n\n\n               Concerning the Lost Writings of PROCLUS.\n\n_24. On the Speech of Diotima, in Plato\u2019s Banquet, concerning the\nSubsistence of the Beautiful._ Fabricius informs us, that this\nwork is distributed into many books; and Holstenius observes, that it\nis mentioned in a certain scholium of the Medicean copy of Proclus\u2019s\ncommentaries on Plato\u2019s politics; but it is unfortunately no where\nextant.\n\n_25. On the Philebus of Plato_; as may be inferred from the\nnarration of Damascius in Photius, p. 550; and Suidas in Marinus.\nFor Damascius relates, that Marinus having composed a commentary on\nthis dialogue, on shewing it to Isidorus for his approbation, that\nphilosopher observed, _that those of his master were sufficient_;\nwhich words Fabricius, with great propriety, applies to the\ncommentaries of Proclus on the Philebus.\n\n_26. On the The\u00e6tetus of Plato._ This work is praised by Marinus,\nin the last chapter of the preceding life; and no doubt with great\npropriety: for this abstruse and sublime dialogue would naturally call\nforth all the divine fire and elegance of our philosopher.\n\n_27. Commentaries on the Enneads of Plotinus._ This work is\nmentioned by Gyraldus, in his second dialogue on ancient poets; by\nFicinus on Plotinus; by Philip Labbeus, in his account of MS. books,\np. 286; and in the notes of Bullialdus to Theo of Smyrna, p. 224. But\nalso in a certain note prefixed to an ancient manuscript of Jamblichus,\non the Egyptian mysteries, to this effect: \u201cThe philosopher Proclus,\ncommenting on the Enneads of the great Plotinus, says, that it is the\ndivine Jamblichus who answers the epistle of Porphyry.\u201d This note is\nin Greek, in the original, and is (in my opinion) of itself sufficient\nto prove that such a work was once extant, though now unfortunately\nlost. How much the want of these commentaries is to be regretted, must\nbe deeply felt by every lover of the Platonic philosophy. For the\nunequalled profundity, and divine mysteries, contained in the writings\nof Plotinus, could never be more happily illustrated than by the\nirradiations of such a genius as Proclus.\n\n_28. Lectures on Aristotle\u2019s Book_ \u03a0\u03b5\u03c1\u1f76 \u0395\u03c1\u03bc\u03b7\u03bd\u03b5\u03af\u03b1\u03c2, _or concerning\nInterpretation_. This work, it seems, was never published; but Ammonius\nHermeas, the disciple of Proclus, has inserted in his valuable\ncommentary on this book all that he could retain in his memory of\nProclus\u2019s lectures.\n\n_29. Hymns_, not a few, see num. 1.\n\n_30. Chrestomathea._ See num. 3.\n\n_31. On the Mother of the Gods_, one book, mentioned by Marinus,\nin the preceding Life.\n\n_32. On the Theology of Orpheus._ This work is mentioned by\nMarinus, in the preceding Life, and by Suidas; and its loss must be\nparticularly regretted by all the lovers of recondite theology.\n\n_33. Ten Books, on the Chaldean Oracles._ This most valuable work\nis mentioned by Marinus, in the preceding Life, and by Proclus himself\non Plato\u2019s Politics, p. 359. It was doubtless not extant at the time\nwhen Psellus and Pletho undertook the illustration of a few of these\noracles: at least the inconsiderable merit of their commentaries,\nstrongly favours this supposition.\n\n_34. A Commentary on the whole of Homer._ Suidas. A specimen\nof the great value of this work may be seen in our philosopher\u2019s\ncommentaries on Plato\u2019s republic. The works of Homer are not only the\ngreat fountain of poetry, but likewise of philosophy; and are no less\nadmirable for inspiring the fury of the Muses than for containing the\nmysteries of the most recondite theology.\n\n_35. Concerning the Gods, according to Homer._ Had this work been\npreserved, we should doubtless have been furnished with a defence of\nthe heathen religion, which would have silenced the ignorant clamours\nof its opponents.\n\n_36. The Symphony or Concord of Orpheus, Pythagoras, and Plato._\nSuidas. Proclus, in his published writings, is every where studious of\nreconciling the doctrines of these great men, and is always successful\nin this undertaking. Indeed, the same divine genius seems to have\nirradiated and inspired these wonderful heroes, but in different\nways: in Orpheus it was accompanied with the fire of the Muses; in\nPythagoras it shone through the mysterious veil of numbers; and in\nPlato, combining the preceding modes, it was seen enshrined in awful\nmajesty of thought, clothed with the graces of poetical diction, and\nresplendent with ineffable light.\n\n_37. Two Books on the Theurgic Discipline._ Suidas. How much\nProclus excelled in this art, may be seen in the preceding Life.\n\n_38. Concerning the Oppositions of Aristotle to Plato\u2019s Tim\u00e6us._\nThis work is mentioned by Proclus in the 3d book of his commentary\non the Tim\u00e6us, p. 226. and seems to have escaped the notice of the\naccurate Fabricius. Aristotle may, no doubt, in many particulars\nbe reconciled with Plato; but it is also certain, that in some he\nis perfectly dissonant. And thus much for the Life and Writings of\nProclus.\n\n\n\n\n                             COMMENTARIES\n\n                                  OF\n\n                               PROCLUS.\n\n\n\n\n                                BOOK I.\n\n\n\n\n                               CHAP. I.\n\n          _On the Middle Nature of the Mathematical Essence._\n\n\nIt is necessary that the mathematical essence should neither be\nseparated from the first nor last genera of things, nor from that\nwhich obtains a simplicity of essence; but that it should obtain\na middle situation between substances destitute of parts, simple,\nincomposite and indivisible, and such as are subject to partition, and\nare terminated in manifold compositions and various divisions. For\nsince that which subsists in its inherent reasons remains perpetually\nthe same, is firm and durable, and cannot be confuted, it evidently\ndeclares it is superior to the forms existing in matter. But that\npower of progression which apprehends, and which besides uses the\ndimensions of subjects, and prepares different conclusions from\ndifferent principles, gives it an order inferior to that nature which\nis allotted an indivisible essence, perfectly constituted in itself.\nHence (as it appears to me)[71] Plato also divides the knowledge\nof things which are, into first, middle, and last substances. And\nto indivisible natures, indeed, he attributes an intelligence,\nwhich, in a collective manner, and by a certain simple power, divides\nthe objects of intellectual perception; so that being divested of\nmatter, and endued with the greatest purity, it apprehends things\nthemselves, by a certain unifying perception, and excels the other\nkinds of knowledge. But to divisible essences, and such as are\nallotted the lowest nature, and to all sensible beings, he attributes\nopinion, which obtains an obscure and imperfect truth. But to middle\nessences (and such are mathematical forms), and to things inferior\nto an indivisible and superior to a divisible nature, he attributes\ncogitation. For this, indeed, is inferior to intellect, and the supreme\nscience dialectic; but is more perfect than opinion, and more certain\nand pure. For it advances by a discursive procession, expands the\nindivisibility of intellect, and unfolds that which was involved in\nthe unity of intellectual apprehension: but it collects things which\nare divided, and brings them back to mind. Hence, as knowledges differ\namong themselves, so the objects of knowledge are distinguished by\nnature. So that intelligible essences having an uniform subsistence,\nevidently excel all others. But sensibles are entirely excelled by\nprimary essences: and mathematical natures, and whatever falls under\ncogitation, are allotted a middle order: for they are excelled by the\ndivision of intelligibles; but because destitute of matter, they are\nsuperior to sensible natures; and by a certain simple power, they are\nexcelled by the first; but by a certain reason are more exalted than\nthe last. Hence they possess notions of an intellectual essence, which\nare more manifest than sensibles, but which are, at the same time,\nonly the images of an intellectual nature; and they imitate divisibly\nthe indivisible, and, in a multiform manner, the uniform exemplars\nof things. And, that I may sum up the whole in a few words, they are\nplaced in the vestibules or entrances of primary forms, and disclose\ntheir indivisible and prolific subsistence collected into one, but\nthey do not yet excel the division and composition of reasons, and an\nessence accommodated to the obscurity of images; nor are they capable\nof passing beyond the various notions of the soul, endued with a\ndiscursive power, and of adhering to intellections perfectly simple,\nand purified from all material imperfection. After this manner then, is\nthe middle nature of mathematical genera and forms to be understood; as\nfilling up the medium between essences entirely indivisible, and such\nas are divisible about matter.\n\n\n\n\n                               CHAP. II.\n\n _Concerning the common Principles of Beings, and of the Mathematical\n                   Essence,[72] bound and infinite._\n\n\nBut it is necessary that, considering the principles of the whole\nmathematical essence, we should return to those general principles,\nwhich pervade through and produce all things from themselves, I mean\n_bound_ and _infinite_. For from these two after that cause of _one_,\nwhich can neither be explained, nor entirely comprehended, every other\nthing, as well as the nature of the mathematical disciplines, is\nconstituted. In the former, indeed, producing all things collectively\nand separately; but in these proceeding in a convenient measure, and\nreceiving a progression in a becoming order; and in some, subsisting\namong primary, but in others among middle, and in others again among\nposterior natures. For intelligible genera, by their simplicity of\npower, are the first participants of _bound_ and _infinite_: because,\non account of their union and identity, and their firm and stable\nexistence, they are perfected by bound: but on account of their\ndivision into multitude, their copious power of generation, and their\ndivine diversity and progression, they obtain the nature of infinite.\nBut mathematical genera originate, indeed, from bound and infinite,\nyet not from primary, intelligible, and occult principles only; but\nalso from those principles which proceed from the first to a secondary\norder, and which are sufficient to produce the middle ornaments of\nbeings, and the variety which is alternately found in their natures.\nHence, in these also, the reasons and proportions advance to infinity,\nbut are restrained and confined by that which is the cause of bound.\nFor number rising from the retreats of unity, receives an incessant\nincrease, but that which is received as it stops in its progression,\nis always finite. Magnitude also suffers an infinite division, yet\nall the parts which are divided are bounded, and the particles of the\nwhole exist finite in energy. So that without the being of infinity,\nall magnitudes would be commensurable, and no one would be found but\nwhat might either be explained by words, or comprehended by reason (in\nwhich indeed geometrical subjects appear to differ from such as are\narithmetical;) and numbers would be very little able to evince the\nprolific power of unity, and all the multiplex and super-particular\nproportions which they contain. For every number changes its\nproportion, looking back upon, and diligently enquiring after unity,\nand a reason prior to itself. But _bound_ being taken away, the\ncommensurability and communication of reasons, and one and the same\nperpetual essence of forms, together with equality, and whatever\nregards a better co-ordination, would never appear in mathematical\nanticipations: nor would there be any science of these; nor any firm\nand certain comprehensions. Hence then, as all other genera of beings\nrequire these two principles, so likewise the mathematical essences.\nBut such things as are last in the order of beings, which subsist in\nmatter, and are formed by the plastic hand of nature, are manifestly\nseen to enjoy these two principles essentially. Infinite as the subject\nseat of their forms; but bound as that which invests them with reasons,\nfigures, and forms. And hence it is manifest that mathematical essences\nhave the same pre-existent principles with all the other genera of\nbeings.\n\n\n\n\n                              CHAP. III.\n\n     _What the common Theorems are of the Mathematical Essences._\n\n\nBut as we have contemplated the common principles of things, which are\ndiffused through all the mathematical genera, after the same manner\nwe must consider those common and simple theorems, originating from\none science, which contains all mathematical knowledge in one. And we\nmust investigate how they are capable of according with all numbers,\nmagnitudes and motions. But of this kind are all considerations\nrespecting proportions, compositions, divisions, conversions, and\nalternate changes: also the speculation of every kind of reasons,\nmultiplex, super-particular, super-partient, and the opposite to these:\ntogether with the common and universal considerations respecting equal\nand unequal, not as conversant in figures, or numbers, or motions,\nbut so far as each of these possesses a common nature essentially,\nand affords a more simple knowledge of itself. But beauty and order\nare also common to all the mathematical disciplines, together with\na passage from things more known, to such as are sought for, and\na transition from these to those which are called resolutions and\ncompositions. Besides, a similitude and dissimilitude of reasons are\nby no means absent from the mathematical genera: for we call some\nfigures similar, and others dissimilar; and the same with respect\nto numbers. And again, all the considerations which regard powers,\nagree in like manner to all the mathematical disciplines, as well the\npowers themselves, as things subject to their dominion: which, indeed,\nSocrates, in the Republic, dedicates to the Muses, speaking things\narduous and sublime, because he had embraced things common to all\nmathematical reasons, in terminated limits, and had determined them in\ngiven numbers, in which the measures both of abundance and sterility\nappear.\n\n\n\n\n                               CHAP. IV.\n\n  _How these Common Properties subsist, and by what Science they are\n                             considered._\n\n\nBut it is requisite to believe, that these common properties do not\nprimarily subsist in many and divided forms, nor originate from things\nmany and the last: but we ought to place them as things preceding\nin a certain simplicity and excellence. For the knowledge of these\nantecedes many knowledges, and supplies them with principles; and\nthe multitude of sciences subsist about this, and are referred to\nit as their source. Thus the geometrician affirms, that when four\nmagnitudes are proportional, they shall be alternately proportional;\nand he demonstrates this from principles peculiar to his science, and\nwhich the arithmetician never uses. In like manner, the arithmetician\naffirms, that when four numbers are proportional, they shall be so\nalternately: and this he evinces from the proper principles of his\nscience. For who is he that knows alternate ratio considered by\nitself, whether it subsists in magnitudes or in numbers? And the\ndivision of composite magnitudes or numbers, and in like manner, the\ncomposition of such as are divided? For surely it cannot be said that\nthere are sciences and cognitions of things divisible: but that we\nhave no science of things destitute of matter, and which are assigned\na more intellectual contemplation; for the knowledge of these is by\na much greater priority science, and from these the common reasons\nof many sciences are derived. And there is a gradual ascent in\ncognitions from things more particular to more universal, till we\nrevert to the science of that which _is_, considered _as it\nis_, abstracted from all secondary properties. For this sublime\nscience does not think it suitable to its dignity, to contemplate\nthe common properties which are essentially inherent in numbers, and\nare common to all quantities; but it contemplates the one, and firm\nessence of all the things which _are_. Hence, it is the most\ncapacious of all sciences, and from this all the rest assume their\nown peculiar principles. For the superior sciences always afford the\nfirst suppositions of demonstrations to such as are subordinate. But\nthat which is the most perfect of all the sciences, distributes from\nitself principles to all the rest, to some indeed, such as are more\nuniversal, but to others, such as are more particular. Hence, Socrates,\nin the The\u00e6tetus, mingling the jocose with the serious, compares the\nsciences which reside in us to doves: but he says they fly away, some\nin flocks, but others separate from one another. For such, indeed,\nas are more common and more capacious, comprehend in themselves many\nsuch as are more particular: but such as being distributed into forms,\ntouch things subject to knowledge, are distant from one another, and\ncan by no means be copulated together, since they are excited by\ndifferent primary principles. One science, therefore, precedes all\nsciences and disciplines, since it knows the common properties which\npervade through all the genera of beings, and supplies principles to\nall the mathematical sciences. And thus far our doctrine concerning\ndialectic[73] is terminated.\n\n\n\n\n                               CHAP. V.\n\n _What the Instrument is, which judges of the Mathematical Genera and\n                               Species._\n\n\nLet us now consider what that instrument is[74], adapted to the\njudgment of mathematical concerns; and let us appoint Plato as our\nguide in this affair, who, in his Republic, divides cognitions\nseparately from such things as are the objects of knowledge; and\ndistributes cognitions in conjunction with things subject to knowledge.\nFor of the things which are, some he ranks among intelligibles, and\nothers among sensibles. And of intelligibles, some are again pure\nintelligibles, and others subject to cogitation. And of sensibles,\nsome are purely sensibles, but others conjectural. To intelligibles,\nindeed, which are the first of the four genera, he assigns an\nintelligible knowledge; but to those which are subject to cogitation,\nhe attributes thought: to sensibles, faith; but to conjecturals, a\nconjectural or assimilatory power. And he shews, that the assimilatory\npower has the same proportion to sense as thought to intelligence. For\nthe conjectural power knows the spectres of sensible forms, while they\nare beheld in water and other bodies, which perspicuously represent\ntheir image: since, by their situation in water, they are after a\nmanner, allotted the last seat in the gradations of forms, and truly\nbecome the resemblances of resemblances. In like manner, thought\nbeholds the images of intelligibles in a degraded state, fallen from\nprimary simple and indivisible forms, into multitude and division.\nHence, a knowledge of this kind, depends on other more ancient\nhypotheses; but intelligence arrives at that principle which is no\nlonger supposed. If then, mathematical concerns are neither allotted an\nessence separate from all division and variety, nor that nature which\nis apprehended by sense, which is obnoxious to many mutations, and is\nin every proportion divisible, it must be manifest to every one, that\nthey are essentially subject to cogitation: but cogitation presides\nover these as an instrument adapted to judgment, in the same manner as\nsense to sensibles, and the assimilatory power to conjecturals. From\nwhence, indeed, Socrates determines that the knowledge of these is more\nobscure than the first science, but is more evident than the impulsive\napprehension of opinion. For in this the mathematical sciences are\ninferior to intelligence, because they contemplate that which is\nevolved, and is endued with a power of progression; but they are\nsuperior to opinion, by that stability of reasons which they contain,\nand which cannot be confuted. And they originate from supposition,\nthrough a diminution of the first science; but they contain forms\nindependent of matter, from their possessing a knowledge more perfect\nthan that of sensibles. We have therefore determined an instrument\nadapted to the judgment of all mathematical concerns, i. e. cogitation,\naccording to the mind of Plato; which places itself indeed above\nopinion, but is excelled by intelligence.\n\n\n\n\n                               CHAP. VI.\n\n   _Concerning the Essence of Mathematical Genera and Species[75]._\n\n\nIt now remains, that we consider what subsistence or essence ought\nto be assigned to mathematical genera and species? Whether we must\ndeduce their origin and subsistence from sensible objects, or from\nabstraction, or from a collection of such things as are dispersed by\nparts into one common definition; or must allow them an existence\nprior to that of sensibles, as Plato affirms, and as the progression\nof universal being demonstrates? First then, if we affirm that\nmathematical species are composed from sensibles; whilst the soul\nfrom material triangles or circles, forms in herself the trigonic, or\ncircular species, by a kind of secondary generation; I would ask from\nwhence is derived the great certainty and accuracy of definitions?\nFor it must either proceed from sensibles, or from the soul herself.\nBut from sensibles is impossible, for these, in a continual flow of\ngeneration and decay, do not for a moment retain an exact sameness\nof being; and consequently fall far short of the exactness contained\nin the definitions themselves. It must therefore proceed from the\nsoul, which, by her immaterial nature, procures perfection from the\nimperfect, accurate subtilty from that which is neither accurate nor\nsubtle, and rekindles the light of ideas from the obscure and unreal\nobjects of sense.\n\nFor where shall we find, amongst sensible objects, an indivisible\nnature, such as that of a point, or a line without the dimension\nof breadth, or a superficies without depth, or the ever constant\nproportion of sides, and exact rectitude of angles? For my part, I\ncannot see where, since all divisible natures are thus mixed and\nconfused together, nothing sincere, nothing free from its contrary,\nbut things every where yielding to separation, as well such as are\nremoved by distance of place, as those which are united together.\nHow then shall we obtain this durable essence for these immoveable\nnatures from the ever fluctuating forms of sense? For whatever derives\nits existence from moveable beings, must of necessity be mutable and\nfrail. And how shall we gain this perfect accuracy for the stable\nspecies, from the inaccurate and imperfect? For whatever is the cause\nof a conception, always immutable, is itself much more stable than its\neffect. We must therefore admit the soul to be the generator of these\nmathematical species and reasons. But if she contains them in herself,\nas first exemplars, she gives them an essential being, so that the\ngenerations are nothing else than propagations of species, which had\na prior subsistence in herself: and thus we shall speak agreeably to\nthe sentiments of Plato, and discover the true essence of mathematical\nentities. But if the soul, though she neither possesses nor received\nthe mathematical reasons prior to the energies of sense, yet fabricates\nthis admirable immaterial building, and generates this fair series of\nspeculations; how can she discern whether her productions are stable\nand constant, or things which the winds may dissipate, and phantoms\nrather than realities? What standard can she apply as the measure of\ntheir truth? Or how, since she is destitute of their essence, can she\ngenerate such a variety of reasons? For from such an hypothesis, we\nmake their subsistence fortuitous, not tending to any scientific bound.\nMathematical species are therefore the genuine offspring of the soul:\nnor does she derive from sensible objects the definitions she frames,\nbut rather the first are propagated from the second; they are the\nenergies of soul, which, as it were, pregnant with forms, delivers her\nimmaterial progeny into the dark and fluctuating regions of matter, as\nevidences of the permanent duration of her species.\n\nAgain, if we collect mathematical reasons from externals, why\nare not demonstrations composed from sensibles, better than the\ndemonstrations of universal and simple species? For we say, in order\nto the investigation of any thing sought, that the principles and\npropositions, should be allied to the conclusions. If then, particulars\nare the causes of universals, and sensibles the sources of reasoning,\nwhy does the boundary of demonstration always refer to that which is\nmore universal, and not to that which is partial and particular? And\nhow can we prove that the essence of intelligibles is more allied to\ndemonstration than the essence of sensibles? For thus they speak[76]:\nhis knowledge is not legitimate, who demonstrates that the isosceles,\nthe equilateral, or the scalene triangle, have angles equal to two\nright; but he possesses science, properly so called, who demonstrates\nthis of every triangle simply, or of triangle itself. And again,\nthat universals, for the purpose of demonstration, are superior to\nparticulars; that demonstrations concern things more universal; but\nthat the principles from which demonstrations are composed, have a\npriority of existence, and a precedency in nature to singulars, and\nare the causes of the propositions they prove. It is very remote,\ntherefore, from the nature of Apodictical sciences, that from converse\nwith things of posterior origin, and from the dark perceptions of\nsense, they should falsely collect their indubitable propositions.\nI add farther, that they who affirm this, make the soul of a baser\nnature than the material species themselves. For if matter derives from\nnature beings essential, and participating a high degree of entity and\nevidence; but the soul, by a posterior energy, receives these from\nsensible objects, and fashions in herself resemblances and images\nof posterior origin, contemplating vile essences, and abstracting\nfrom matter, the forms inseparable from its nature; do they not make\nthe soul more obscure and indigent than matter itself? For matter\nis the receptacle of forms materialized, as the soul is of species\nimmaterialized. But in this case, matter would be the place of primary\nbeings, and the soul of such as are secondary and subordinate: matter\nand its forms obtaining the lead in being, and existing as the sources\nof the subsistence of immaterial forms. Lastly, the material forms\nwould have an essential existence, the others only an intentional\ndenomination. How then can the soul, which is the first participant of\nintellect, and an intellective essence, and which derives from thence\nconsummate knowledge, and a plenitude of life, become the receptacle\nof the most obscure species, the lowest in the order of things, and\nparticipating the most imperfect existence. But this opinion, which has\nbeen sufficiently exploded by others, needs no farther confutation.\n\nIf then, mathematical species do not subsist by material abstraction,\nnor by a collection of those common properties inherent in individuals;\nnor are at all, in their origin, posterior to sensibles, nor derived\nin any manner from them: it is necessary that the soul should either\ndeduce them from herself, or from intellect; or lastly, from herself\nand intellect united. But if from herself alone, Whence do the\nimages of intellectual species arise; whence do they derive their\nmiddle nature, linking, as it were, the divisible and indivisible\nessence together; if they do not participate the fullness of entity\nfrom primary essences? Lastly, how, upon this hypothesis, are the\nfirst exemplars, paradigms, or ideas, which subsist in intellect,\nthe principles of universals? But if they are derived from intellect\nalone into the soul, how can the soul remain self-operative, and\nself-motive, if her inherent reasons flow from an external source, and\nare regulated by its operations? And in what respect does the soul\ndiffer from matter, which is all things in mere dormant capacity,\nbut generates nothing appertaining to material species? It remains,\ntherefore, that the soul deduces these species from herself, and\nintellect; and that she is the absolute consummation of the forms which\noriginate from intellectual exemplars, but which are allotted from\nthemselves a transition to permanent being. The soul, therefore, is by\nno means to be compared to a smooth tablet, void of all reasons; but\nshe is an ever-written tablet, herself inscribing the characters in\nherself, of which she derives an eternal plenitude from intellect.\nFor soul is a certain subordinate intellect, revolving round an\nintellect prior to herself, formed to its image, and participating its\ndivine irradiations. If then, this superior intellect is all things\nintellectually, soul will all things animally; if the first exists as\nthe exemplar, soul will be as its image; if as contracted and united\nin itself, soul as divisible and expanded. And this is what Plato\nunderstood, when in his Tim\u00e6us, he composes the soul of the world\nfrom all things, dividing her according to harmonical reasons, and\nanalogies; assigning to her the first principles effective of figures,\nI mean the right and circular line, and giving an intellectual motion\nto her inherent circles. All mathematical species, therefore, have a\nprimary subsistence in the soul: so that, before sensible numbers,\nthere are to be found in her inmost recesses, self-moving numbers;\nvital figures, prior to the apparent, ideal proportions of harmony\nprevious to concordant sounds; and invisible orbs, prior to the bodies\nwhich revolve in a circle. So that soul is the prolific abundance of\nall these, and is another ornament producing herself, and produced\nfrom a proper principle, filling herself with life, and at the same\ntime filled from the demiurgus of the universe, is an incorporeal\nand indistant manner. When, therefore, she produces and unfolds\nher latent reasons, she then detects every science and virtue. The\nessence of soul then consists in these species, nor must we suppose\nher inherent numbers to be a multitude of units, nor her archytipal\nideas of divisible forms to be corporeal: but we must conceive all\nthese as subsisting ever vitally, and intellectually, as the exemplars\nof apparent numbers, figures, reasons and motions. And here we must\nfollow the doctrine of Tim\u00e6us, who derives the origin, and consummates\nthe fabric of the soul, from mathematical forms, and reposes in her\nnature the causes of every thing which exists. For the seven bounding\nterms[77], comprehending the principles of all numbers, lines,\nplanes and solids, pre-exist in soul according to cause. And again,\nthe principles of figures are placed in her essence, according to a\ndemiurgical power. And lastly, the first of all motions, which embraces\nevery other motion in its comprehensive ambit, is co-existent with\nsoul. For the principle of every thing which is moved is a circle, and\nthe circular motion. The mathematical reasons, therefore, which fully\nconsummate the soul, are essential, and self-moving: and the soul, by\nher cogitative power, diffusing, propagating, and evolving these, from\nher profound recesses, constitutes all the fair variety of mathematical\nsciences. Nor will she ever cease to generate, and waken into energy,\nsucceeding species, while she divests her indivisible reasons of\ntheir intellectual simplicity. For she previously received all\nthings, after a primary manner; and according to her infinite power,\nfrom pre-existent principles, deduces a beautiful series of various\nspeculations.\n\n\n\n\n                              CHAP. VII.\n\n   _What the Employments and Powers are of the Mathematical Science,\n        and how far they extend themselves in their Energies._\n\n\nBut, after contemplating the essence of mathematical forms, it is\nnecessary we should recur to that one master-science of these, which\nwe have shewn is prior to a multitude of others, and that we should\ncontemplate what its employment is, what are its powers, and how far\nit advances in its energies. The employment, therefore, of the whole\nmathematical science, possessing, as we have before said, the power\nof cogitation, must not be placed so high as that of intelligence;\nwhich is firmly seated in its own stable essence, is perfect, is\ncontained by itself, and in itself continually verges. Nor must it be\nsituated so low as that of opinion and sense, since these cognitions\ndwell upon external concerns, energize upon them, and do not possess\nthe causes of the objects of their knowledge. But the mathematical\nscience, receives its commencement, indeed, extrinsically from[78]\nrecollection, but ends in the most intimate reasons, residing in the\ndepths of the soul; and is excited, indeed, from things posterior,\nbut arrives by gradual advances at the principal essence of forms.\nNor is its energy immoveable, like that of intelligence, nor is it\naffected with local motion and alteration, like sense, but it revolves\nwith a vital energy, and runs through the ornament of incorporeal\nreasons, sometimes advancing from principles to such things as are\nperfected by principles, but at other times yielding in a retrograde\nprogression from conclusions to their forming principles: and sometimes\nproceeding from things previously known, to such as are the subject\nof investigation: but at other times, from things placed in the\nquestion, to such as precede in cognition. Besides, it does not excel\nall inquisition, as if it were perfect from itself, like intellect,\nnor is it perfected from others, like sense, but it proceeds by\nenquiry to invention, and ascends from the imperfect to perfection.\nBut it likewise possesses twofold powers, one kind of these deducing\nprinciples into multitude, and generating the different paths of\ncontemplation: but the other endued with a power of collecting many\ntransitions into proper suppositions. For since it proposes to itself\nas principles, as well unity, and multitude, as bound and infinite,\nand such things as are subject to its comprehension, are allotted a\nmiddle order, between forms indivisible and every way divisible; with\ngreat propriety (I think) the gnostic powers of the whole science of\nthese are essentially twofold. One species indeed, hastens to union,\nand contracts the expansion of multitude: but the other possesses\na power of distinguishing things simple into such as are various,\nmore universals into more particulars, and reasons digested in their\nprinciple, into things secondary and multifariously multiplied from\nthese principles. For rising higher from its commencement it penetrates\neven to such things as are the perfections of sensible concerns,\nis joined with nature, and demonstrates many things together with\nnatural science. Since ascending from inferiors, it accedes in a\ncertain respect proximate to intellectual knowledge, and touches the\ncontemplation of things primary and divine. And hence, in the limits\nwhich flow from its essence, it produces the whole mechanic, optic,\nand catoptric speculation, together with many other sciences which are\ninwoven and entangled with sensible concerns, and which operate through\ntheir assistance. Besides, in its ascensions from corporeal natures,\nit derives intelligences indivisible and destitute of matter: and with\nthese it perfects its divisible apprehensions, those cognitions which\nsubsist in progressions, and its own genera and forms: it likewise\nindicates the truth respecting the gods themselves, and in its peculiar\ntreatises exhibits a contemplation of the things which _are_. And\nthus much concerning the employment and powers of the Mathematical\nScience.\n\n\n\n\n                              CHAP. VIII.\n\n         _Concerning the Utility of the Mathematical Science._\n\n\nBut let us now consider the utility of this Science, which extends\nitself from the most principal to the last cognitions. Tim\u00e6us,\ntherefore, calls the knowledge of the mathematical disciplines the\npath of erudition, because, indeed, it has the same proportion to\nuniversal science, and the first philosophy, which learning has to\nvirtue. For this last frames our soul to a perfect life, by the\npossession of worthy manners; but the former prepares our cogitation,\nand the divine eye of our soul to an elevation from the obscurity of\nsensible information. Hence, Socrates in the Republic, says, \u201cThat\nthe eye of the soul, which is darkened and buried by other studies,\ncan by the mathematical disciplines alone be invigorated, and again\nexcited to the contemplation of that which _is_, and transferred\nfrom resemblances to real beings, from an obscure light to that\nlight which has the power of intelligence, and from a cave, and\nthose bonds which exist in it as the authors of generation, and from\nmaterial impediments be able to rise to an incorporeal and indivisible\nessence. For the beauty and order of mathematical reasons, and the\nfirmness and stability of the contemplations they afford, conjoins\nus with intelligible objects, and perfectly determines us in their\nessences; which perpetually remain the same, ever shining with divine\nbeauty, and preserving a mutual order without end. But Socrates, in\nthe Ph\u00e6drus, delivers to us three characters who are elevated from\nsense, because they fill up and accomplish the primary life of the\nsoul, i. e. the philosopher, the lover, and the musician. But the\nbeginning and path of elevation to the lover, is a progression from\napparent beauty, using as excitations the middle forms of beautiful\nobjects. But to the musician, who is allotted the third seat, the\nway consists in a transition from sensible to invisible harmonies,\nand to the reasons existing in these. So that to the one, sight is\nthe instrument of reminiscence, and to the other, hearing. But to\nhim who is by nature a philosopher, from whence and by what means is\nreminiscence the prelude of intellectual knowledge, and an excitation\nto that which truly is, and to truth itself? For this character also,\non account of its imperfection, requires a proper principle: for it is\nallotted a natural virtue, an imperfect eye, and a degraded manner.\nIt must therefore be excited from itself; and he who is of such a\nnature, rejoices in that which _is_. But to the philosopher,\nsays Plotinus, the mathematical disciplines must be exhibited, that\nthey may accustom him to an incorporeal nature, and that afterwards\nusing these as figures, he may be led to dialectic reasons, and to\nthe contemplation of all the things which _are_. And thus it is\nmanifest, from hence, that the mathematics are of the greatest utility\nto philosophy. But it is requisite that we should be more explicit, and\nmention the several particulars to which they conduce, and evince that\nthey prepare the intellectual apprehensions of theology. For whatever\nto imperfect natures appears difficult and arduous in obtaining the\ntrue knowledge of the gods, the mathematical reasons render, by their\nimages, credible, manifest, and certain. Thus, in numbers, they\nindicate the significations of super-essential properties, but they\nevince the powers of intellectual figures, in those figures which\nfall under cogitation. Hence it is, that Plato, by mathematical forms\nteaches us many and admirable sentences concerning the gods, and the\nphilosophy of the Pythagoreans, using these as veils, conceals from\nvulgar inspection the discipline of divine sentences. For such is the\nwhole of the _Sacred and Divine Discourse_[79], that of Philolaus\nin his Bacchics, and the universal method of the Pythagoric narration\nconcerning the Gods. But it especially refers to the contemplation of\nnature, since it discloses the order of those reasons by which the\nuniverse is fabricated, and that proportion which binds, as Tim\u00e6us\nsays, whatever the world contains, in union and consent; besides, it\nconciliates in amity things mutually opposing each other, and gives\nconvenience and consent to things mutually disagreeing, and exhibits\nto our view simple and primary elements, from which the universe is\ncomposed, on every side comprehended by commensurability and equality,\nbecause it receives convenient figures in its proportions, and numbers\nproper to every production, and finds out their revolutions and\nrenovations, by which we are enabled to reason concerning the best\norigin, and the contrary dissolution of particulars. In consequence\nof this, as it appears to me, Tim\u00e6us discloses the contemplation\nconcerning the nature of the universe, by mathematical names, adorns\nthe origin of the elements with numbers and figures, referring to\nthese their powers, passions, and energies; and esteeming as well\nthe acuteness as the obtuseness of angles, the levity of sides, or\ncontrary powers, and their multitude and paucity to be the cause of the\nall-various mutation of the elements. But why may we not say, that it\nprofits much, and in an admirable manner, to that philosophy which is\ncalled Politic, as well by measuring the times of actions as affording\nthe various revolutions of the universe, and numbers convenient to\nthings rising into being; I mean the assimilating, and authors of\ndissimilitude, the prolific too and the perfect, and the contraries\nto these; together with orderly and elegant ministers of life, and\ninelegance; and finally, such numbers as procure fertility and\nsterility. Which, indeed, the speech of the Muses in the Republic[80]\nevinces, placing the universal Geometric Number as the author of better\nand more debased generations, and as the cause of the indissoluble\nperseverance of good manners, and of the mutation of the best Republics\ninto such as are remote from reason, and are given to affections. For\nit is sufficiently evident, that it belongs to the whole mathematical\ndiscipline to deliver the science of this number which is called\ngeometrical, and not to one particular science, such as arithmetic, or\ngeometry: since the reasons or proportions of abundance and sterility,\npermeate through all the mathematical disciplines. Again, it is the\nmeans of our institution in moral philosophy which it brings to its\nultimate perfection, and gives order and an elegant life to our\nmanners. Besides this, it delivers to us figures, and modulations\nand motions convenient to virtue, by which the Athenian guest wishes\nthose to be instituted and perfected, who are destined to pursue\nmoral virtue from their early youth. Add too, that it places before\nour view the reasons of virtues, in one manner, indeed, in numbers,\nin another in figures, but differently in musical symphonies; and\nlastly, it indicates the excess and defect of vices, by which we are\nenabled to moderate and adorn our manners. Hence it is, that Socrates,\nin the Gorgias, accusing Calicles of an inordinate and intemperate\nlife, says to him, \u201cYou neglect geometry and geometric equality:\u201d but,\nin the Republic, he finds out the proportion of tyrannic pleasure to\na royal interval, according to a plane and solid generation. But we\nshall learn what great utility is derived to other sciences and arts\nfrom the mathematical science, when we consider that it adds order and\nperfection to contemplative arts; I mean rhetoric, and all such as\nconsist in discourse. But it proposes to the poetic arts, the reasons\nof poems in the place of an example, because it presides over the\nmeasures existing in these. But to the active arts it determines action\nand motion, by its own abiding and immoveable forms. For all arts, as\nSocrates says, in the Philebus, require arithmetic, mensuration, and\nstatics, either in all, or in some of their operations. But all these\nare contained in the discourses of the mathematical science, and are\nterminated according to their diversity. For from this science the\ndivisions of numbers, and the variety of dimensions, and the difference\nof weights are known. The utility, therefore, of the whole mathematical\nscience to philosophy itself, and to other sciences and arts, may be\nfrom hence known to intelligent hearers.\u201d\n\n\n\n\n                               CHAP. IX.\n\n _A Solution of an Objection raised by some against the Utility of the\n                        Mathematical Sciences._\n\n\nBut some, who are prone to contradiction through those who wish to\nsubvert geometry, endeavour to destroy the dignity of this science.\nOne part, indeed, depriving it of ornament and good, because it\ndoes not discourse on these. But another part[81] affirming that\nsensible experiments are more useful than the universal objects of its\nspeculation; I mean, that Geodesia (for instance,) or the mensuration\nof the earth, is preferable to geometry, and vulgar arithmetic to\nthat arithmetic which is conversant with theorems alone: and that\nnautical astrology is more useful than that which teaches universally,\nabstracted from any application to sensible concerns. For we are not,\nsay they, made rich by our knowledge of riches, but by using them;\nnor are we happy by the merely understanding felicity, but by living\nhappily. Hence we must confess that those mathematical sciences,\nwhich are conversant with cognition, do not profit human life, and\nconfer to action, but those only which are engaged in exercise. For\nthose who are ignorant of the reasons of things, but are exercised\nin particular and sensible experiments, are in every respect more\nexcellent, for the purposes of human life, than those who are employed\nin contemplation alone. Against objections then, of this kind, we\nshall reply, by shewing the beauty of the mathematical disciplines\nfrom those arguments by which Aristotle endeavours to persuade us. We\nmust therefore confess that there are three things which especially\ncause beauty, both in bodies and souls; I mean, order, convenience, and\ndetermination. Since corporeal baseness, indeed, arises from material\ninordination, deformity, and inconvenience, and from the dominion\nof the indefinite in the composite body. But the baseness of the\nsoul originates from its irrationality, and inordinate motion, and\nfrom its being in a state of discord with reason, and not receiving\nfrom thence its proper limitation. Hence, beauty exists even in\ncontraries, by means of order, convenience and determination. But we\nmay behold these in a more eminent degree in the mathematical science;\norder, indeed, in the perpetual exhibition of things posterior and\nmore various, from such as are primary and more simple; for things\nsubsequent are always annexed to their precedents, the latter ranking\nas principles, and the former as the first suppositions of things\nconsequent: but convenience is evinced in the mutual consonance of\nthings demonstrated, and in the relation of all of them to intellect,\nsince intellect is the common measure of all science, from which it\nreceives its principles, and to which it converts the learner: but\ndetermination is perceived in its perpetually abiding and immoveable\nreasons, for the objects of its knowledge are not, at times, subject\nto variation, like those of opinion and sense, but present themselves\nfor ever the same, and are bounded by intellectual forms. If such\nthen, are the principal requisites of beauty, it is evident, that\nin these sciences that illustrious ornament and gracefulness is\nfound. For how is it possible this should not be the case with a\nscience receiving a supernal illumination from intellect, to which\nit continually advances, hastening to transfer us from the obscure\nlight of sensible information? With respect to the second objection,\nwe think it proper to judge of its utility, without regarding the\nconveniencies and necessities of human life. For otherwise, we must\nconfess that contemplative virtue is also useless, which separates\nitself from human concerns, which it is very little desirous to look\ndown upon and understand. Indeed Socrates, in the The\u00e6tetus, affirming\nthis concerning noblemen endued with the prophetic power, says, \u201cthat\nit withdraws them from all regard to human life, and raises their\nthoughts, properly liberated, from all necessity and use, to the very\nsummit of all true being.\u201d The mathematical science, therefore, must\nbe considered as desirable for its own sake, and for the contemplation\nit affords, and not on account of the utility it administers to human\nconcerns. But if it is necessary to refer the utility it produces to\nsomething different from itself, it must be referred to intellectual\nknowledge. For it leads us to this, and prepares the eye of the\nsoul for the knowledge of universals, removing and obliterating the\nimpediments arising from the senses, and from corporeal involution.\nAs therefore we call the whole of purgative virtue useful, or the\ncontrary, not regarding the use of the sensible life, but of that\nwhich is contemplative, so indeed it is requisite to refer the end\nof mathematics to intellect, and universal wisdom. Hence its energy\nis worthy our study, both on its own account, and on account of an\nintellectual life. But it appears, as Aristotle[82] says, that this\nscience is desirable of itself to its votaries, because though no\nreward is proposed to its enquirers, yet the mathematical contemplation\nreceives, in a small time, an abundant increase. Besides, this is\nfarther evident from hence, that all men are willingly employed in\nits pursuit, and wish to dwell on its speculations, omitting every\nother concern; even those who have, with their lips, as it were, but\njust touched its utility. And hence it follows, that they who despise\nthe knowledge of the mathematical disciplines, have very little\ntasted of the pleasures they contain. The mathematics, therefore,\nare not to be despised because their speculative parts do not\nimmediately confer to human utility, (for the ultimate limits of its\nprogressions, and whatever operates with matter, consider a use of\nthis kind;) but on the contrary we should admire its immateriality,\nand the good which it contains, considered by itself alone. For when\nmankind were entirely disengaged from the care of necessary concerns,\nthey converted themselves to the investigation of the mathematical\ndisciplines; and this, indeed, with the greatest propriety. Since\naffairs familiar to human life in its most imperfect state, and which\nare immediately connected with its origin, first of all employed the\nstudies of mankind: but, in the second place, those concerns succeeded\nwhich separate the soul from generation, and restore its memory of\nthat which _IS_. After this manner, then, we are engaged in\nnecessaries, before things honourable for their own sakes, on account\nof their intrinsic dignity and worth; and in things related to sense,\nbefore such as are apprehended by the nobler energies of mind. For\nevery origin and life of the soul which is converted into herself, is\nnaturally adapted to proceed from the imperfect to the perfect. And\nthus much against those who despise the mathematical science.\u201d\n\n\n\n\n                               CHAP. X.\n\n  _A Solution of another Objection of certain Platonists, against the\n                Utility of the Mathematical Sciences._\n\n\nBut, perhaps, some of our own family will here rise up against us,\nand, proposing Plato as a witness, will endeavour to provoke ruder\nunderstandings into a contemptuous disregard of the mathematical\ndisciplines. For they will say, that this philosopher entirely\nexcludes (in his Republic) the mathematical knowledge from the choir\nof the sciences, and that he accuses it as being ignorant of its own\nprinciples, that its very principle is to itself unknown, and its ends\nand mediums composed from things of which it is ignorant. To these\nobjections they may likewise add whatever other reproaches are there\nurged by Socrates against this contemplation. In answer then, to the\nobjections of our friends, we shall recall into their memory, that\nPlato himself perspicuously asserts the mathematical science to be the\npurgation of the soul, and that it is endued with a power of leading\nit on high; because, like the Homeric Minerva, it removes the darkness\nof a sensible nature from the intellectual light of thought, which\nis better worth saving than ten thousand corporeal eyes, and which\nnot only participates of a mercurial gift, (preserving us from the\nincantations and delusions of this material abode, which is similar\nto the fascinating realms of Circe,) but also of the more divine arts\nof Minerva. He likewise every where calls it by the name of science,\nand asserts that it is the cause of the greatest felicity to those\nwho are exercised in its contemplation. But I will briefly explain\nwhy, in the Republic he takes from it the surname of science: for my\npresent discourse is addressed to the learned. Plato, indeed, in most\nplaces, calls all the knowledge (as I may say) of universals by the\nname of science, opposing it in a division to sense which apprehends\nonly particulars, whether such a mode of cognition is accomplished\nby art or experience. And in this sense, as it appears to me in\nthe _Civil_ Dialogue, and in the Sophista, he seems to use the\nname of science; placing likewise the illustrious Sophistic science,\nwhich Socrates in the Gorgias, says, is a certain experience: also,\nthe adulatory, and many others, which are experiences, but not true\nsciences. But again, dividing this knowledge of universals into\nthat which knows causes, and into that which understands without a\ncause, he thinks that the one should be called science, but the other\nexperience. And hence, to arts he sometimes attributes the name of\nscience, but to experience never. For how (says he in the Banquet)\ncan a thing which possesses no reason be science? All knowledge,\ntherefore, which contains the reason and cause of the things known,\nis a certain science. Again, therefore, he divides this science which\nis endued with a power from the cause of knowing, by the peculiarity\nof its subjects, and he places one, conjectural of things divisible;\nbut the other of such as subsist by themselves, and are ever knowable\nafter the same manner. And according to this division he separates from\nscience, medicine, and every faculty which is conversant with material\nconcerns. But he calls mathematical knowledge, and whatever possesses\na power of contemplating eternal objects, by the name of science.\nLastly, dividing this science, which we distinguished from arts, he\nconsiders one part as void of supposition; but the other as flowing\nfrom supposition. And that the one which is void of supposition, has\na power of knowing universals: that it rises to good, and the supreme\ncause of all; and that it considers good as the end of its elevation:\nbut that the other, which previously fabricates for itself definite\nand determinate principles, from which it evinces things consequent to\nsuch principles, does not tend to the principle, but to the conclusion.\nAnd hence he asserts, that mathematical knowledge, because it makes\nuse of supposition, falls short of that science which is without\nsupposition, and is perfect. For there is one true science, by means\nof which we are disposed to know all the things _which are_, and\nfrom which also principles emerge to all sciences; to some, indeed,\nconstituted more proximately, but to others more remotely. We must\nnot say, therefore, that Plato expels mathematical knowledge from the\nnumber of the sciences, but that he asserts it to be the second from\nthat one science, which possesses the supreme seat of all: nor must we\naffirm, that he accuses it as ignorant of its own principles, but that\nreceiving these from the master science dialectic, and possessing them\nwithout any demonstration, it demonstrates from these its consequent\npropositions. For, indeed, he sometimes allows the soul, which is\nconstituted from mathematical reasons, to be the principle of motion:\nand sometimes he affirms, that it receives its motion from genera\nwhich are subject to intelligence. And these variations accord among\nthemselves. For to such things as are moved by another, the soul is a\ncertain cause of motion, but it is not the cause of every motion. After\nthe same manner, the mathematical science is indeed the second from the\nfirst of all sciences, and, with reference to it, imperfect: but it\nis, nevertheless, a science, not as being free from supposition, but\nas knowing the peculiar reasons resident in the soul, and as bringing\nthe causes of conclusions, and containing the reason of such things as\nare subject to its knowledge. And thus much for the opinion of Plato\nrespecting mathematics.\n\n\n\n\n                               CHAP. XI.\n\n\nBut let us now consider what are the things which may be required of\na mathematician, and how any one may rightly judge concerning his\ndistinguishing peculiarities. For[83] Aristotle indeed, says, that\nhe who is simply learned in all disciplines, is adapted to judge of\nall: but that he who is alone skilled in the mathematical sciences,\ncan alone determine concerning the magnitude of reasons inherent in\nthese. It is requisite, therefore, that we should previously assume\nthe terms of judging, and that we should know, in the first place,\nin what things it is proper to demonstrate generally, and in what to\nregard the peculiarities of singulars. For many of the same properties\nreside in things differing in species, as two right angles in all\ntriangles: but many have indeed the same predicament, yet differ\nin their individuals in a common species, as similitude in figures\nand numbers. But one demonstration is not to be sought for by the\nmathematician in these, for the principles of figures and numbers are\nnot the same, but differ in their subject genus. And if the essential\naccident is one, the demonstration will also be one[84]: for the\npossession of two right angles is the same in all triangles, and that\ngeneral something to which this pertains is the same in all, I mean\ntriangle, and a triangular reason. In the same manner, likewise, the\npossession of external angles to four right ones, not only pertains to\ntriangles, but also to all right-lined figures; and the demonstration,\nso far as they are right-lined, agrees in all. For every reason brings\nwith it, at the same time, a certain property and passion, of which all\nparticipate through that reason, whether triangular, or rectilinear, or\nuniversally figure. But the second limit by which a mathematician is\nto be judged, is, if he demonstrates according to his subject-matter,\nand renders necessary reasons, and such as cannot be confuted, but are\nat the same time neither probable, nor replenished with a similitude\nof truth. For, says Aristotle, it is just the same to require\ndemonstrations from a rhetorician, and to assent to a mathematician\ndisputing probably; since every one, endued with science and art, ought\nto render reasons adapted to the subjects of his investigation. In\nlike manner also, Plato in the Tim\u00e6us, requires credible reasons of\nthe natural philosopher, as one who is employed in the resemblances\nof truth: but of him who discourses concerning intelligibles, and a\nstable essence, he demands reasons which can neither be confuted nor\nmoved. For subjects every where cause a difference in sciences and\narts, since, if some of them are immoveable, others are conversant\nwith motion; and some are more simple, but others more composite;\nand some are intelligibles, but others sensibles. Hence we must\nnot require the same certainty from every part of the mathematical\nscience. For if one part, after a manner, borders upon sensibles, but\nanother part is the knowledge of intelligible subjects, they cannot\nboth be equally certain, but one must inherit a higher degree of\nevidence than the other. And hence it is, that we call arithmetic more\ncertain than the science of harmony. Nor must we think it just that\nmathematics and other sciences should use the same demonstrations;\nfor their subjects afford them no small variety. In the third place,\nwe must affirm, that he who rightly judges mathematical reasons, must\nconsider sameness and difference, what subsists by itself, and what is\naccidental, what proportion is, and every consideration of a similar\nkind. For almost all errors of this sort happen to those who think\nthey demonstrate mathematically, when at the same time they by no\nmeans demonstrate, since they either demonstrate the same thing as if\ndifferent in each species, or that which is different as if it were\nthe same: or when they regard that which is accidental, as if it were\nan essential property; or that which subsists by itself, as if it were\naccidental. For instance, when they endeavour to demonstrate that the\ncircumference of a circle is more beautiful than a right line, or an\nequilateral than an isosceles triangle. For the determination of these\ndoes not belong to the mathematician, but to the first philosopher\nalone. Lastly, in the fourth place, we must affirm, that since the\nmathematical science obtains a middle situation between intelligibles\nand sensibles, and exhibits in itself many images of divine concerns,\nand many exemplars of natural reasons, we may behold in it three kinds\nof demonstration[85], one approaching nearer to intellect, the second\nmore accommodated to cogitation, and the third bordering on opinion.\nFor it is requisite that demonstrations should differ according to the\nvarieties of problems, and receive a division correspondent to the\ngenera of beings, since the mathematical science is connected with all\nthese, and adapts its reasons to the universality of things. And thus\nmuch for a discussion of the subject proposed.\n\n\n\n\n                              CHAP. XII.\n\n _What and how many the Species of the whole Mathematical Science are,\n            according to the Opinion of the Pythagoreans._\n\n\nBut after these considerations, it is requisite to determine concerning\nthe parts of the mathematical science, what, and how many they are.\nFor it is just, after speculating its whole and entire genus, to\nconsider the differences of its more particular sciences, according\nto their species. The Pythagoreans[86], therefore, thought that the\nwhole mathematical science should receive a fourfold distribution,\nattributing one of its parts to the how-many, but the other to the\nhow-much; and they assigned to each of these parts a twofold division.\nFor they said, that discrete quantity, or the how-many, either subsists\nby itself, or must be considered with relation to some other; but that\ncontinued quantity, or the how-much, is either stable or in motion.\nHence they affirmed, that arithmetic contemplates that discrete\nquantity which subsists by itself, but music that which is related\nto another; and that geometry considers continued quantity so far as\nit is immoveable; but spherics contemplates continued quantity as\nmoving from itself, in consequence of its union with a self-motive\nnature. They affirmed besides, that these two sciences, discrete and\ncontinued quantity, did not consider either magnitude or multitude\nabsolutely, but that alone which in each of these is definite from the\nparticipation of bound. For sciences alone speculate the definite,\nrejecting as vain the comprehension of infinite quantity. But when\nthese wise men assigned this distribution, we must not suppose they\nunderstood that discrete quantity which is found in sensible natures,\nnor that continued quantity which subsists about the fluctuating\norder of bodies. For, I think, the contemplation of these pertains\nto the natural and not to the mathematical science. But because the\ndemiurgus of the universe, employed the union, division, and identity\nof general natures, together with difference, station, and motion, for\nthe purpose of completing the essence of the soul, and composed it from\nthese genera, as Tim\u00e6us informs us, we must affirm, that cogitation,\nabiding according to its diversity, its division of reasons, and its\nmultitude, and understanding itself to be both one and many, proposes\nindeed to itself, and produces numbers, together with an arithmetical\nknowledge of these: but it provides for itself music according to an\nunion of its multitude, and a communication and junction with itself;\nand hence it is that arithmetic excels music in antiquity; since,\naccording to the narration of Plato, the demiurgus first divided the\nsoul, and afterwards collected it in harmonical proportions. Again,\nthought establishing its energy according to the stability which it\ncontains, draws from its inmost retreats geometry, together with one\nessential figure, and the demiurgical principles of all figures[87]:\nbut, according to its inherent motion, it produces the spherical\nscience. For it is moved also by circles, but abides perpetually the\nsame from the causes of circles. Hence, likewise, geometry precedes\nspherics, in the same manner as station is prior to motion. But because\ncogitation itself produces these sciences, not by looking back upon\nits convolution of forms, endued with an infinite power, but upon the\ninclosure of _bound_ according to its definite genera; hence they\nsay, that the mathematical sciences take away infinite from multitude\nand magnitude, and are only conversant about finite quantity. Indeed,\nintellect has placed in cogitation all the principles both of multitude\nand magnitude. For since it wholly consists, with reference to itself,\nof similar parts, and is one and indivisible, and again divisible,\neducing the ornament of forms, it participates of _bound_ and\n_infinite_, from intelligible essences themselves. But it\nunderstands, indeed, from its participation of bound, and generates\nvital energies, and various reasons from the nature of infinite.\nThe intellections, therefore, of thought, constitute these sciences\naccording to the _bound_ which they contain, and not according to\nan infinity of life; since they bring with them an image of intellect,\nbut not of life. Such then is the opinion of the Pythagoreans, and the\ndivision of the four mathematical sciences.\n\n\n\n\n                              CHAP. XIII.\n\n      _Another Division of the Mathematical Science, according to\n                               Geminus._\n\n\nAgain, some think (among whom is Geminus) that the mathematical science\nis to be divided in a different manner from the preceding. And they\nconsider that one of its parts is conversant with intelligibles only,\nbut the other with sensibles, upon which it borders; denominating as\nintelligibles whatever inspections the soul rouses into energy by\nherself, when separating herself from material forms. And of that which\nis conversant with intelligibles they establish two, by far the first\nand most principal parts, arithmetic and geometry: but of that which\nunfolds its office and employment in sensibles, they appoint six parts,\nmechanics, astrology, optics, geod\u00e6sia, canonics, and logistics, or\nthe art of reckoning. But they do not think that the military art, or\ntactics, should be called any one part of mathematics, according to\nthe opinion of some[88]; but they consider it as using at one time\nthe art of reckoning, as in the numbering of legions; but at another\ntime geod\u00e6sia, as in dividing and measuring the spaces filled by a\nfield of camps. As, say they, neither the art of writing, nor the\nart of healing, are any part of mathematics, though frequently both\nthe historian and physician use mathematical theorems. This is the\ncase with historians indeed, when relating the situation of climates,\nor collecting the magnitudes and dimensions of cities, or their\ncompass and circuit: but with physicians, when elucidating by ways of\nthis kind, many things in their art. For Hippocrates himself shews\nthe utility derived to medicine from astrology, and almost all who\nspeak of opportune times and places. By the same reason he also, who\naccommodates his work to tactics, uses indeed mathematical theorems,\nyet is not on this account a mathematician, although he is sometimes\nwilling that a numerous camp should exhibit a very small multitude,\nand forms his army according to a circular figure; but sometimes in a\nquadrangular, quinquangular, or some other multangular figure, when\nhe desires it to appear numerous. But since these are the species of\nthe whole mathematical science, geometry is again divided into the\ncontemplation of planes, and the dimension of solids, which is called\nstereometry. For there is not any peculiar treatise about points and\nlines, because no figure can be produced from these without planes\nor solids. For geometry treats of nothing else in every one of its\nparts, than that it may constitute either planes or solids: or that\nwhen constituted, it may compare and divide them among themselves.\nIn like manner, arithmetic is distributed into the contemplation of\nlinear, plane, and solid numbers. For it considers the species of\nnumbers separate from sensible connections, proceeding from unity,\nand the origin of plane numbers; I mean of the similar, dissimilar,\nand solid, even to the third increase. But geod\u00e6sia, and the art\nof reckoning, are divided similarly to arithmetic and geometry, as\nthey do not discourse concerning intelligible numbers or figures,\nbut of such as are sensible alone. For neither is it the office of\ngeodesia to measure the cylinder or the cone, but material masses\nas if they were cones, and wells as if they were cylinders. Neither\ndoes it accomplish this purpose by intelligible right lines, but by\nsuch as are sensible, sometimes indeed by a more certain means, as\nby the solar rays: but at other times by grosser ones, as by a line\nand perpendicular. In like manner, the reckoner does not survey the\npassions of numbers by themselves, but as they are resident in sensible\nobjects. From whence he also imposes a name upon these derived from\nthe things which he reckons, calling them \u03bc\u03b7\u03bb\u03af\u03b1\u03b9, & \u03c6\u03b9\u03b1\u03bb\u03af\u03c4\u03b1\u03b9. Besides\nthis, he does not, admit of any least, like the arithmetician, who\nreceives that minimum, as a genus of relation. For some one man is\nconsidered by him as the measure of the whole multitude of men, as\nunity also is the common measure of all numbers. Again, optics and\ncanonics are produced from geometry and arithmetic. And optics uses\nthe visual rays which are constituted by the rays of the eyes, as\nlines and angles. But it is divided into that which is properly called\noptics (because it renders the cause of these appearances, which are\naccustomed to present themselves to us different from their reality, on\naccount of the different situations and distances of visible objects,\nas the coincidence of parallel lines, or the appearance of quadrangles\nas if they were circles); and into universal catoptrics, which is\nconversant about various and manifold refractions, and is connected\nwith imaginative or conjectural knowledge: as also into that which\nis called sciography[89], or the delineation of shadows, which shews\nhow appearances in images may seem neither inelegant nor deformed, on\naccount of the distances and altitudes of the things designed. But\ncanonics (music) or the regular art, considers the apparent reasons\nof harmonies, finding out the sections of rules, every where using\nthe assistance of sense, and, as Plato says, seeming to prefer the\ntestimony of the ears to intellect itself. But to the parts we have\nhitherto enumerated, mechanics must he added, as it is a certain part\nof the whole science, and of the knowledge of sensible objects, and\nof things united with matter. But under this exists the art effective\nof instruments, which is called (\u1f40\u03c1\u03b3\u03b1\u03bdo\u03c0\u03bf\u03b9\u03b7\u03c4\u03b9\u03ba\u1f74) I mean of those\ninstruments proper for the purposes of war: such, indeed, as Archimedes\nis reported to have constructed, resisting the besiegers of sea and\nland; and that which is effective of miracles, and which is called\n(\u03b8\u03b1\u03c5\u03bc\u03b1\u03c4\u03bf\u03c0\u03bf\u03b9\u03b7\u03c4\u03b9\u03ba\u1f74.) One part of this constructs with the greatest\nartifice pneumatic engines, such as Ctesibius and Heron fabricated: but\nanother operates with weights, the motion of which is reckoned to be\nthe cause of inequilibrity; but their station of equilibrity, as Tim\u00e6us\nalso has determined: and again, another part imitates animate foldings\nand motions by strings and ropes. Again, under mechanics is placed\nthe knowledge of equilibriums, and of such instruments as are called\ncentroponderants: also (\u03c3\u03c6\u03b1\u03b9\u03c1\u03bf\u03c0\u03bf\u03b9\u03af\u03b1) or the art effective of spheres,\nimitating the celestial revolutions, such as Archimedes fabricated;\nand lastly, every thing endued with a power of moving matter. But the\nlast of all is astrology, which treats of the mundane motions, of the\nmagnitudes of the celestial bodies, their figures and illuminations,\ntheir distances from the earth, and every thing of this kind; assuming\nmany things indeed to itself from sense, but communicating much with\nthe natural speculation. One part of this is gnomonics, which is\nexercised in settling the dimension of hororary gnomons: but the other\nis metheoroscopics, which finds out the differences of elevations, and\nthe distances of the stars, and also teaches many other and various\nastrological theorems. The third part is dioptrics, which ascertains by\ndioptric instruments of this kind the distances of the sun and moon,\nand of the five other stars. And such is the account of the parts of\nthe mathematical science, delivered by the ancients, and transmitted to\nour memory by the informing hand of time.\n\n\n\n\n                              CHAP. XIV.\n\n   _How Dialectic is the Top of the Mathematical Sciences, and what\n              their Conjunction is, according to Plato._\n\n\nLet us again consider after what manner Plato, in his Republic, calls\ndialectic the top of the mathematical disciplines; and what their\nconjunction is, according to the tradition of the author of the\nEpinomis[90]. And in order to this we must assert, that as intellect\nis superior to cogitation, supplying it with supernal principles,\nand from itself giving perfection to cogitation; in the same manner\ndialectic also, being the purest part of philosophy, excels in\nsimplicity the mathematical disciplines, to which it is proximate,\nand with which it is conjoined. Indeed it embraces the complete\ncircle of these sciences, to which it elevates from itself various\nenergies, endued with a power of causing perfection, judgment, and\nintelligence. And these energies consist in resolving, dividing,\ndefining, and demonstrating; by which mathematics itself, receiving\nassistance and perfection, invents some things by resolution, but\nothers by composition: and some things it explains by division, others\nby definition: but collects other subjects of its investigation by\ndemonstration; accommodating, indeed, these ways to its subjects, but\nusing each of them for the purpose of beholding its middle enquiries.\nFrom whence indeed, both the resolutions, definitions, divisions, and\ndemonstrations which it contains, are peculiar, and adapted to its\nnature, and revolve according to the mode of mathematical cognition.\nNot undeservedly, therefore, is dialectic the vertex as it were,\nand summit of mathematics. Since it perfects all which mathematics\ncontains of intelligence; renders its certainty free from reprehension,\npreserves the stability of its immovable essence, and refers what it\ncontains destitute of matter and pure to the simplicity of intellect,\nand a nature separated from material connections. Besides, it\ndistinguishes the first principles of these sciences, by definitions:\nexhibits the separations of genera and forms contained under the\ngenera themselves: and besides this, teaches the compositions, which,\nfrom principles, produce things consequent to principles: and the\nresolutions which rise and mount up to things first, and to principles\nthemselves. But with respect to what remains, proportion itself is not\nto be considered (as Eratosthenes thought it was) as the conjunction\nof the mathematical disciplines. Since proportion is said to be, and\nindeed is one of those things common to the mathematics. But in short,\nmany other things besides proportion regard all the mathematical\ndisciplines, which are essentially inherent in the common nature of\nthe mathematics. But as it appears to me, we should say, that there\nis one proximate conjunction of these, and of the whole mathematical\nscience, which especially embraces in itself, in a more simple manner,\nthe principles of all sciences; which considers their community and\ndifference; teaches whatever is found in these the same; together\nwith what things are inherent in a many, and what in a few. So\nthat to those who aptly learn there is a reversion from many other\nsciences to this alone[91]. But, dialectic is a conjunction of the\nmathematical disciplines superior to the preceding; which Plato, as I\nhave already observed, calls in his Republic their vertex: for, indeed,\nit perfects the whole of mathematics, brings it back to intellect by\nits powers, shews it to be a true science, and causes it to be certain\nand obnoxious to no reproof. But, intellect obtains the third order\nbetween these conjunctions, which comprehends in itself uniformly\nall the dialectic powers, contracts their variety by its simplicity,\ntheir partition by its indivisible knowledge, and their multitude\nby its occult union. Hence, intellect itself congregates indeed the\ninvolutions and deviations of the dialectic paths, into an intelligible\nessence, but it collects supernally all the progression of mathematical\ndiscourses: and it is the best end both of the elevating power of the\nsoul, and of the energy consisting in cognition. And such are the\nsentiments declared by me on the present enquiry.\n\n\n\n\n                               CHAP. XV.\n\n            _From whence the Name Mathematics originated._\n\n\nAgain, from whence shall we say this name of mathematics, and\nmathematical disciplines, was assigned by the ancients, and what\napt reason can we render of its position? Indeed, it appears to me,\nthat such an appellation of a science which respects cogitative\nreasons, was not, like most names, invented by indifferent persons,\nbut (as the truth of the case is, and according to report) by the\nPythagoreans alone. And this, when they perceived, that whatever is\ncalled mathesis or discipline, is nothing more than reminiscence;\nwhich does not approach the soul extrinsically, like the images which\nrising from sensible objects are formed in the phantasy: nor is it\nadventitious and foreign, like the knowledge consisting in opinion,\nbut it is excited, indeed, from apparent objects, and is perfected\nwithin, by thought intimately converted to itself. And when they\nlikewise perceived that though reminiscence might be shewn from many\nparticulars, yet it was evinced in a more eminent manner (as Plato\nalso says[92]) from the mathematical disciplines. For if any one, says\nhe, is led into the descriptions, he will there easily prove that\ndiscipline is reminiscence. From whence Socrates also, in the Meno,\nshews by this method of arguing, that learning is nothing else than the\nsoul\u2019s recollection of her inherent reasons. And this, because that\nwhich recollects, is alone the cogitative part of the soul; but this\nperfects her essence in the reasons of the mathematical disciplines,\nthe sciences of which she previously received into herself, though she\ndoes not always energize on their fair variety. Indeed, she contains\nthem all essentially and occultly; but she produces each of them when\nshe is freed from the impediments originating from sense. For sense\nunites her with divisible objects: the phantasy fills her with forming\nmotions, and appetite bends her to an indulgent and luxurious life.\nBut every thing divisible is an obstacle to our self-conversion. And\nwhatever invests with form, disturbs and offends that knowledge which\nis destitute of form. And whatever is obnoxious to perturbations is\nan impediment to that energy which is unimpaired by affections. When,\ntherefore, we have moved all these from the cogitative power, then\nshall we be able to understand by thought itself, the reasons which\nthought contains: then shall we become scientific in energy; and\nunfold our essential knowledge. But whilst we are captive and bound,\nand winking with the eye of the soul, we cannot by any means attain\nto a perfection convenient to our nature. Such then is mathesis or\ndiscipline: a reminiscence of the eternal reasons contained in the\nsoul. And the mathematical or disciplinative science is on this account\nparticularly denominated that knowledge which especially confers to our\nreminiscence of these essential reasons. Hence, the business and office\nof this science[93], is apparent from its name. For its duty is to\nmove the inherent knowledge of the soul; to awaken its intelligence;\nto purify its cogitation; to call forth its essential forms from their\ndormant retreats; to remove that oblivion and ignorance, which are\ncongenial with our birth; and to dissolve the bonds arising from our\nunion with an irrational nature. It plainly leads us to a similitude\nof that divinity who presides over this science, who manifests\nintellectual gifts, and fills the universe with divine reasons; who\nelevates souls to intellect, wakens them as from a profound sleep,\nconverts them by enquiry to themselves; and by a certain obstetric art,\nand invention of pure intellect, brings them to a blessed life. To whom\nindeed, dedicating the present work, we here conclude our contemplation\nof the mathematical science.\n\n\n\n\n                               BOOK II.\n\n\n\n\n                               CHAP. I.\n\n   _What Part Geometry is of Mathematics, and what the Matter is of\n                          which it consists._\n\n\nIn the preceding discourses we have considered those common properties\nwhich respect the whole of the mathematical science; and this we have\ndone agreeable to the doctrine of Plato; at the same time collecting\nsuch particulars as pertain to our present design. But consequent to\nthis it is requisite that we should discourse on geometry itself, and\non the proposed institution of the elements, for the sake of which we\nhave undertaken the whole of the present work. That geometry then, is\na part of the whole of mathematics, and that it obtains the second\nplace after arithmetic, since it is perfected and bounded by this,\n(for whatever in geometry may be expressed and known, is determined\nby arithmetical reasons) has been asserted by the ancients, and\nrequires no long discussion in the present enquiry. But we also may\nbe able to relate our opinion on this particular, if we consider what\nplace, and what essence its subject matter[94] is allotted among the\nuniversality of things. For from a proper survey of this, the power\nof the science which knows this subject matter, the utility arising\nfrom it, and the good acquired by its learners, will immediately\nappear. Indeed, some one may doubt in what genus of things he ought\nto place geometrical matter, so as not to deviate from the truth it\ncontains. For if the figures concerning which geometry discourses,\nexist in sensible natures, and cannot be separated from the dark\nreceptacle of matter; how can we assert that geometry frees us from\nsensible objects, that it brings us to an incorporeal essence, that it\naccustoms us to an inspection of intelligibles, and prepares us for\nintellectual energy? Where shall we ever survey among sensible objects\na point without parts, or a line destitute of breadth, or a superficies\nwithout profundity, or the equality of lines from the centre to the\ncircumference; or the multangles, and all the figures of many bases,\nconcerning which geometry informs us? Lastly, after what manner can the\nreasons of such a science remain free from all possible confutation;\nsince, indeed, sensible forms and figures are susceptive of the more\nand the less, are all moveable and mutable, and are full of material\nvariety; among which equality subsists mixt and confused with its\ncontrary inequality, and into which things without parts have proceeded\ninto partition, and interval, darkened with the shades of matter, and\nlost in its infinite folds? But if the subjects of geometry are removed\nfrom matter, are pure forms, and are separated from sensible objects:\nthey will be all of them, without doubt, void of parts, incorporeal,\nand destitute of magnitude. For extension, tumor, and interval,\napproach to forms, on account of the material receptacle in which they\nare involved, and which receives things destitute of parts, distributed\ninto parts; things void of dimension, extended into dimension; and\nimmoveable natures accompanied with motion. How then, if this is the\ncase, shall we cut a right line, triangle, and circle? How can we speak\nof the diversities of angles, and the increments and decrements of\ntriangular and quadrangular figures? Or how exhibit the contacts of\ncircles or right lines? For all these evince that the geometric matter\nconsists of parts, and does not reside among indivisible reasons.\nSuch then are the doubts concerning the matter of geometry, to which\nwe may add, that Plato considers the forms of geometry as placed in\ncogitation; and grants, that we advance from sensibles to forms of\nthis kind, and that we rise from sensibles to intellect, though (as\nwe have previously observed) the reasons subsisting in cogitation\nare indivisible, are separated by no interval, and subsist according\nto the peculiarity of the soul. But if reasons are to be rendered\nagreeable to things themselves, and to the doctrine of Plato, the\nfollowing division must be adopted. [95]Every universal, and one\nthing containing many, is either naturally disposed to be thought of\nin particulars, or to appear such, because it possesses its existence\nin these; is inseparable from them; is disposed and distributed in\nthem; and together with these is either moved, or firmly and immoveably\nabides. Or it is adapted to subsist prior to many, and to possess a\npower of generating multitude, affording to many things images from\nitself, being furnished with a nature destitute of parts, from the\nessences which it participates, and raising various participations to\nsecondary natures: or it is disposed to be formed by thought, from the\nmany, to possess a generating existence, and to reside in the last\nplace in the many. For, according to these three modes of subsistence,\nwe shall find, I think, that some subsist before the many, others in\nthe many, and others from the relation and predication which they\npossess to these. But, that I may absolve all in one word, universal\nforms being threefold, we shall consider the differences of that form\nwhich many participate, which exists in many, and fills particular\nnatures according to its subject matter. Besides this, establishing\na twofold order of participants, one subsisting in sensible objects,\nbut the other in the phantasy, (since matter is twofold; one indeed,\nof things united with sense, but the other of such as fall under the\ninspection of phantasy, as Aristotle asserts, in a certain place[96])\nwe must allow that the universal, which is distributed in the many,\nis likewise twofold. The one, indeed, sensible, as being that which\nsensible objects participate; but the other imaginative, as that\nwhich subsists in the _many_ of the phantasy. For the phantasy,\non account of its forming motion, and because it subsists with, and\nin body, always receives impressions which are both divided and\nfigured. So that whatever is known by it, is allotted a correspondent\nexistence: on which account, Aristotle[97] does not hesitate to call\nit passive intellect. But if it is intellect, why is it not impassive,\nand destitute of matter? And if it operates with passion, how can it\nwith propriety be called intellect? For impassivity, indeed, properly\nbelongs to intellect and an intelligent nature: but passivity is very\nremote from such an essence. But (unless I am deceived) Aristotle\nbeing willing to explain its middle nature between cognitions the most\nprimary, and such as are the last, calls it at the same time intellect,\nbecause similar to primary cognitions, and passive from that alliance\nwhich it possesses with such as are posterior. For first cognitions are\nindeed destitute of figures and forms; comprehending in themselves,\nintelligible natures, energizing about themselves, united with the\nobjects of knowledge, and free from all extrinsical impression and\npassion. But last cognitions exercise themselves through the medium\nof instruments, are rather passions than energies, admit extrinsical\nknowledge, and move themselves together with their various subjects.\nFor such (says Plato) are the sensations which arise from violent\npassions. But the phantasy, obtaining a middle centre in the order of\ncognitions, is excited, indeed, by itself, and produces that which\nfalls under cogitation: but because it is not separate from body,\nit deduces into partition, interval, and figure, the objects of its\nknowledge, from the indivisibility of an intellectual life. Hence,\nwhatever it knows, is a certain impression and form of intelligence.\nFor it understands the circle, together with its interval, void,\nindeed, of external matter, but possessing intelligible matter. On\nthis account, like sensible matter, it does not contain one circle\nonly: for we behold in its receptacle, distance, together with the\nmore and the less, and a multitude of circles and triangles. If then\nan universal nature is distributed in sensible circles, since each of\nthese completes a circular figure, and they are all mutually similar,\nsubsisting in one reason, but differing in magnitudes or subjects:\nin like manner, there is a common something in the circles, which\nsubsist in the receptacle of the phantasy, of which all its circles\nparticipate, and according to which they all possess the same form;\nbut in the phantasy they possess but one difference only, that of\nmagnitude. For when you imagine many circles about the same centre,\nthey all of them exist in one immaterial subject and life, which is\ninseparable from a simple body, which, by the possession of interval,\nexceeds an essence destitute of parts; but they differ in magnitude\nand parvitude, and because they are contained and contain. Hence,\nthat universal is twofold, which is understood as subsisting in the\nmany: one, indeed, in sensible forms; but the other in such as are\nimaginative. And the reason of a circular and triangular figure, and of\nfigure universal, is twofold. The one subsisting in intelligible, but\nthe other in sensible matter. But prior to these is the reason which\nresides in cogitation, and that which is seated in nature herself.\nThe former being the author of imaginative circles, and of the one\nform which they contain; but the other, of such as are sensible.\nFor there are circles existing in the heavens, and universally\nthose produced by nature, the reason of which does not fall under a\ncogitative distribution. For in incorporeal causes, things possessing\ninterval, are distinguished by no intervals: such as are endued with\nparts, subsist without parts: and magnitudes without the diffusion of\nmagnitude, as on the contrary in corporeal causes, things without parts\nsubsist divisibly, and such as are void of magnitude with the extension\nof magnitude. Hence, the circle resident in cogitation, is one, simple\nand free from interval: and magnitude itself is there destitute of\nmagnitude; and figure expressed by no figure: for such are reasons\nseparate from matter. But the circle subsisting in the phantasy, is\ndivisible, figured, endued with interval, not one only, but one and\nmany, nor form alone, but distributed with form. And the circle, in\nsensible objects, is composite, distant with magnitude, diminished by\na certain reason, full of ineptitude, and very remote from the purity\nof immaterial natures. We must therefore say, that geometry, when it\nasserts any thing of circle and diameter, and of the passions and\naffections which regard the circle; as of contacts, divisions, and the\nlike: neither teaches nor discourses concerning sensible forms, (since\nit endeavours to separate us from these), nor yet concerning the form\nresident in cogitation, (for here the circle is one, but geometry\ndiscourses of many, proposing something of each, and contemplating the\nsame of all: and here it is indivisible, but the geometric circle is\ndivisible); but we must confess, that it considers universal itself;\nyet as distributed in imaginative circles. And that it beholds, indeed,\none circle[98]: and by the medium of another, contemplates the circle\nresident in the depths of cogitation: but by another, different from\nthe preceding, fabricates the fair variety of its demonstrations.\nFor since cogitation is endued with reasons, but cannot behold them\ncontractedly, separated from material figure; it distributes and\nremoves them, and draws them forth seated in the shadowy bosom of the\nphantasy, and placed in the vestibules of primary forms; revolving\nin it, or together with it, the knowledge of these: loving, indeed,\na separation from sensibles, but finding imaginative matter proper\nfor the reception of its universal forms. Hence, its intellection\ndoes not subsist without the phantasy. And the compositions and\ndivisions of figures are imaginative; and their knowledge is the way\nwhich leads us to that essence pursued by cogitation: but cogitation\nitself, does not yet arrive at this stable essence, while it looks\nabroad to externals, contemplates its internal forms according to\nthese, uses the impressions of reasons, and is moved from itself to\nexternal and material forms. But if it should ever be able to return\nto itself, when it has contracted intervals and impressions, and\nbeholds multitude without impression, and subsisting uniformly; then\nit will excellently perceive geometrical reasons, void of division and\ninterval, essential and vital, of which there is a copious variety.\nAnd this energy will be the best end of the geometric study; and truly\nthe employment of a Mercurial gift, bringing it back as from a certain\nCalypso, and her detaining charms, to a more intellectual knowledge;\nand freeing it from those _forming_ apprehensions with which the\nmirror of the phantasy is replete. Indeed, it is requisite that a\ntrue geometrician should be employed in this meditation, and should\nestablish, as his proper end, the excitation and transition from the\nphantasy to cogitation alone; and that he should accomplish this by\nseparating himself from intervals, and the passive intellect to that\nenergy which cogitation contains. For by this means he will perceive\nall things without an interval, the circle and diameter without a part,\nthe polygons in the circle, all in all, and yet every one separate and\napart. Since, on this account, we exhibit also in the phantasy, both\ncircles inscribed in polygons, and polygons in circles; imitating the\nalternate exhibition of reasons destitute of parts. Hence, therefore,\nwe describe the constitutions, the origin, divisions, positions, and\napplications of figures: because we use the phantasy, and distances\nof this kind proceeding from its material nature; since form itself\nis immoveable, without generation, indivisible, and free from every\nsubject. But whatever form contains occultly, and in an indistant\nmanner, is produced into the phantasy subsisting with intervals,\ndivisibly and expanded. And that which, indeed, produces the forms of\ngeometric speculation, is cogitation: but that from which they are\nproduced, is the form resident in cogitation: and that in which the\nproduced figure resides is what is called the passive intellect. Which\nfolds itself about the impartibility of true intellect, separates from\nitself the power of pure intelligence free from interval; conforms\nitself according to all formless species, and becomes perfectly\nevery thing from which cogitation itself, and our indivisible reason\nconsists. And thus much concerning the geometric matter, as we are\nnot ignorant of whatever Porphyry the Philosopher has observed in his\nmiscellanies, and whatever many of the Platonists describe. But we\nthink that the present discussions are more agreeable to geometric\ndissertations, and to Plato himself, who subjects to geometry the\nobjects of cogitation. For these mutually agree among themselves;\nbecause the causes, indeed, of geometrical forms, by which cogitation\nproduces demonstrations, pre-exist in demonstration itself: but the\nparticular figures which are divided and compounded, are situated in\nthe receptacle of the phantasy.\n\n\n\n\n                               CHAP. II.\n\n                  _What kind of Science Geometry is._\n\n\nBut let us now speak of that science which possesses a power of\ncontemplating the universal forms participated by imaginative matter.\nGeometry, therefore, is endued with the knowledge of magnitudes and\nfigures, and of the terms and reasons subsisting in these; together\nwith the passions, various positions and motions which are contingent\nabout these. For it proceeds, indeed, from an impartible point, but\ndescends even to solids, and finds out their multiform diversities.\nAnd again, runs back from things more composite, to things more\nsimple, and to the principles of these: since it uses compositions\nand resolutions, always beginning from suppositions, and assuming its\nprinciples from a previous science; but employing all the dialectic\nways. In principles, by the divisions of forms from their genera, and\nby defining its orations. But in things posterior to principles, by\ndemonstrations and resolutions. As likewise, it exhibits things more\nvarious, proceeding from such as are more simple, and returning to\nthem again. Besides this, it separately discourses of its subjects;\nseparately of its axioms; from which it rises to demonstrations;\nand separately of essential accidents, which it shews likewise are\nresident in its subjects. For every science has, indeed, a genus, about\nwhich it is conversant, and whose passions it proposes to consider:\nand besides this, principles, which it uses in demonstrations; and\nessential accidents. Axioms, indeed, are common to all sciences\n(though each employs them in its peculiar subject matter), but genus\nand essential accident vary according to the sciential variety. The\nsubjects of geometry are therefore, indeed, triangles, quadrangles,\ncircles, and universally figures and magnitudes, and the boundaries of\nthese. But its essential accidents are divisions, ratios, contacts,\nequalities, applications, excesses, defects, and the like. But its\npetitions and axioms, by which it demonstrates every particular are,\nthis, to draw a right line from any point to any point; and that,\nif from equals you take away equals, the remainders will be equal;\ntogether with the petitions and axioms consequent to these. Hence,\nnot every problem nor thing sought is geometrical, but such only as\nflow from geometric principles. And he who is reproved and convicted\nfrom these, is convinced as a geometrician. But whoever is convinced\nfrom principles different from these, is not a geometrician, but is\nforeign from the geometric contemplation. But the objects of the\nnon-geometric investigation, are of two kinds. For the thing sought\nfor, is either from entirely different principles, as we say that a\nmusical enquiry is foreign from geometry, because it emanates from\nother suppositions, and not from the principles of geometry: or it is\nsuch as uses, indeed, geometrical principles, but at the same time\nperversely, as if any one should say, that parallels coincide. And on\nthis account, geometry also exhibits to us instruments of judging, by\nwhich we may know what things are consequent to its principles, and\nwhat those are which fall from the truth of its principles: for some\nthings attend geometrical, but others arithmetical principles. And why\nshould we speak of others, since they are far distant from these? For\none science is more certain than another (as Aristotle says[99]) that,\nindeed, which emanates from more simple suppositions, than that which\nuses more various principles; and that which tells the _why_,\nthan that which knows only the simple existence of a thing; and that\nwhich is conversant about intelligibles, than that which touches and\nis employed about sensibles. And according to these definitions of\ncertainty, arithmetic is, indeed, more certain than geometry, since its\nprinciples excel by their simplicity. For unity is void of position,\nwith which a point is endued. And a point, indeed, when it receives\nposition, is the principle of geometry: but unity, of arithmetic. But\ngeometry is more certain than spherics; and arithmetic, than music.\nFor these render universally the causes of those theorems, which are\ncontained under them. Again, geometry is more certain than mechanics,\noptics, and catoptrics. Because these discourse only on sensible\nobjects. The principles, therefore, of geometry and arithmetic, differ,\nindeed, from the principles of other sciences; but the hypotheses of\nthese two, alternately differ and agree according to the difference we\nhave already described. Hence, also, with respect to the theorems which\nare demonstrated in these sciences, some are, indeed, common to them,\nbut others peculiar. For the theorem which says, _every proportion\nmay be expressed_, alone belongs to arithmetic; but by no means\nto geometry: since this last science contains things which cannot be\nexpressed[100]. That theorem also, which affirms, that _the gnomons\nof quadrangles are terminated according to the least_[101], is the\nproperty of arithmetic: for in geometry, a minimum cannot be given.\nBut those things are peculiar to geometry, which are conversant about\npositions; for numbers have no position: which respect contacts; for\ncontact is found in continued quantities: and which are conversant\nabout ineffable proportions; for where division proceeds to infinity,\nthere also that which is ineffable is found[102]. But things common to\nboth these sciences, are such as respect divisions, which Euclid treats\nof in the second book; except that proposition which divides a right\nline into extreme and mean proportion[103]. Again, of these common\ntheorems, some, indeed, are transferred from geometry into arithmetic;\nbut others, on the contrary, from arithmetic into geometry: and others\nsimilarly accord with both, which are derived into them from the whole\nmathematical science. For the permutation, indeed, conversions,\ncompositions, and divisions of ratios are, after this manner, common\nto both. But such things as are commensurable, arithmetic first\nbeholds; but afterwards geometry, imitating arithmetic. From whence,\nalso, it determines such things to be commensurables of this kind,\nwhich have the same mutual ratio to one another, as number to number;\nbecause commensurability principally subsists in numbers. For where\nnumber is, there also that which is commensurable is found; and where\ncommensurable is, there also number. Lastly, geometry first inspects\ntriangles and quadrangles: but, arithmetic, receiving these from\ngeometry, considers them according to proportion. For in numbers,\nfigures reside in a causal manner. Being excited, therefore, from\neffects, we pass to their causes, which are contained in numbers. And\nat one time, we indifferently behold the same accidents, as when every\npolygon is resolved by us into triangles[104]: but, at another time,\nwe are content with what is nearest to the truth, as when we find in\ngeometry one quadrangle the double of another, but not finding this\nin numbers, we say that one square is double of another, except by a\ndeficience of unity. As for instance, the square from 7, is double the\nsquare from 5, wanting one. But we have produced our discussion to this\nlength, for the purpose of evincing the communion and difference in the\nprinciples of these two sciences. Since it belongs to a geometrician\nto survey from what common principles common theorems are divided;\nand from what principles such as are peculiar proceed; and thus to\ndistinguish between the geometrical, and non-geometrical, referring\neach of them to different sciences.\n\n\n\n\n                              CHAP. III.\n\n  _From whence the whole of Geometry originated, how far it proceeds,\n                  and in what its Utility consists._\n\n\nBut, beginning still higher, let us contemplate the whole of geometry,\nfrom whence it originated, and how far it proceeds in its energies:\nfor thus we shall properly perceive the ornament which it contains.\nIndeed, it is necessary to understand that it is extended through the\nuniversality of things: that it accommodates its animadversions[105]\nto all beings; and contains in itself the forms of all things: that,\naccording to its supreme part, and which is endued with the highest\npower of intelligence, it surveys true beings; and teaches by images\nthe properties of divine ornaments, and the powers of intellectual\nforms: for it contains the reasons of these also in its peculiar\ncontemplations. And it exhibits what figures are convenient to the\ngod, to primary essences, and to the natures of souls. But, according\nto its middle cognitions, it evolves cogitative reasons; explains and\nbeholds the variety which they contain; exhibits their existence, and\ninherent passions; as also, their communities and diversities. From\nwhich, indeed, it comprehends, in terminated bounds, the imaginative\nformations of figures, and reduces them to the essential substance\nof reasons. But, according to the third propagations of cogitative\nintelligence, it considers nature, and delivers the manner in which\nthe forms of sensible elements, and the powers which they contain, are\npreviously received according to cause, in the reasons themselves. For\nit possesses, indeed, the images of universal intelligible genera;\nbut the exemplars of such as are sensible: and completes its own\nessence, according to such things as are subject to cogitation. And\nthrough these, as through proper mediums, it ascends and descends to\nthose universals which truly are, and to sensible forms which are in a\nstate of perpetual formation. But always geometrically philosophising\nconcerning the things which are, it comprehends in all the proportions\nof virtues, the images of intellectual, animal, and natural concerns.\nAnd it delivers, in an orderly manner, all the ornaments of republics:\nand exhibits in itself their various mutations. Such then are its\nenergies arising from a certain immaterial power of cognition: but\nwhen it touches upon matter, it produces from itself a multitude of\nsciences; such as geod\u00e6sia, mechanics, and perspective: by which it\nprocures the greatest benefit to the life of mortals. For it constructs\nby these sciences, war-instruments, and the bulwarks of cities; and\nmakes known the circuits of mountains, and the situations of places.\nLastly, it instructs us in measures: at one time of the diversified\nways of the earth; and at another, of the restless paths of the deep.\nAdd too, that it constructs balances and scales, by which it renders\nto cities a sure equality according to the invariable standard of\nnumber. Likewise, it clearly expresses, by images, the order of the\nwhole orb of the earth; and by these, manifests many things incredible\nto mankind, and renders them credible to all. Such, indeed, as Hiero\nof Syracuse is reported to have said of Archimedes[106], when he had\nfabricated a ship furnished with three sails, which he had prepared to\nsend to Ptolemy king of Egypt. For when all the Syracusians together,\nwere unable to draw this ship, Archimedes enabled Hiero to draw it\nhimself, without any assistance from others. But he, being astonished,\nsaid, From this day, Archimedes shall be believed in whatever he shall\naffirm. They also report, that Gelo said the same, when Archimedes\ndiscovered the weight of the several materials from which his crown was\ncomposed, without dissolving their union. And such are the narrations\nwhich many of the ancients have delivered to our memory, who were\nwilling to speak in praise of the mathematics: and, on this account, we\nhave placed before the reader, for the present, a few out of the many,\nas not foreign from our design of exhibiting the knowledge and utility\nof geometry.\n\n\n\n\n                               CHAP. IV.\n\n            _On the Origin of Geometry, and its Inventors._\n\n\nBut let us now explain the origin of geometry, as existing in the\npresent age of the world. For the demoniacal Aristotle[107] observes,\nthat the same opinions often subsist among men, according to certain\norderly revolutions of the world: and that sciences did not receive\ntheir first constitution in our times, nor in those periods which are\nknown to us from historical tradition, but have appeared and vanished\nagain in other revolutions of the universe; nor is it possible to say\nhow often this has happened in past ages, and will again take place\nin the future circulations of time. But, because the origin of arts\nand sciences is to be considered according to the present revolution\nof the universe, we must affirm, in conformity with the most general\ntradition, that geometry was first invented by the Egyptians, deriving\nits origin from the mensuration of their fields: since this, indeed,\nwas necessary to them, on account of the inundation of the Nile washing\naway the boundaries of land belonging to each. Nor ought it to seem\nwonderful that the invention of this as well as of other sciences,\nshould receive its commencement from convenience and opportunity.\nSince whatever is carried in the circle of generation, proceeds\nfrom the imperfect to the perfect. A transition, therefore, is not\nundeservedly made from sense to consideration, and from this to the\nnobler energies of intellect[108]. Hence, as the certain knowledge\nof numbers received its origin among the Ph\u0153nicians, on account of\nmerchandise and commerce, so geometry was found out among the Egyptians\nfrom the distribution of land. When Thales, therefore, first went into\nEgypt, he transferred this knowledge from thence into Greece: and\nhe invented many things himself, and communicated to his successors\nthe principles of many. Some of which were, indeed, more universal,\nbut others extended to sensibles. After him Ameristus, the brother\nof Stesichorus the poet, is celebrated as one who touched upon, and\ntasted the study of geometry, and who is mentioned by Hippias the\nElean, as restoring the glory of geometry. But after these, Pythagoras\nchanged that philosophy, which is conversant about geometry itself,\ninto the form of a liberal doctrine, considering its principles in a\nmore exalted manner; and investigating its theorems immaterially and\nintellectually; who likewise invented a treatise of such things as\ncannot be explained[109] in geometry, and discovered the constitution\nof the mundane figures. After him, Anaxagoras the Clazomenian\nsucceeded, who undertook many things pertaining to geometry. And\nOenopides the Chian, was somewhat junior to Anaxagoras, and whom Plato\nmentions in his Rivals, as one who obtained mathematical glory. To\nthese, succeeded Hippocrates, the Chian, who invented the quadrature\nof the lunula[110], and Theodorus the Cyrenean, both of them eminent\nin geometrical knowledge. For the first of these, Hippocrates composed\ngeometrical elements: but Plato, who was posterior to these, caused\nas well geometry itself, as the other mathematical disciplines, to\nreceive a remarkable addition, on account of the great study he\nbestowed in their investigation. This he himself manifests, and his\nbooks, replete with mathematical discourses, evince: to which we\nmay add, that he every where excites whatever in them is wonderful,\nand extends to philosophy. But in his time also lived Leodamas the\nThasian, Archytas the Tarantine, and The\u00e6tetus the Athenian; by whom\ntheorems were increased, and advanced to a more skilful constitution.\nBut Neoclides was junior to Leodamas, and his disciple was Leon; who\nadded many things to those thought of by former geometricians. So that\nLeon also constructed elements more accurate, both on account of their\nmultitude, and on account of the use which they exhibit: and besides\nthis, he discovered a method of determining when a problem, whose\ninvestigation is sought for, is possible, and when it is impossible.\nBut Eudoxus the Cnidian, who was somewhat junior to Leon, and the\ncompanion of Plato, first of all rendered the multitude of those\ntheorems which are called universals more abundant; and to three\nproportions added three others; and things relative to a section,\nwhich received their commencement from Plato, he diffused into a\nricher multitude, employing also resolutions in the prosecution of\nthese. Again, Amyclas the Heracleotean, one of Plato\u2019s familiars, and\nMen\u00e6chmus, the disciple, indeed, of Eudoxus, but conversant with Plato,\nand his brother Dinostratus, rendered the whole of geometry as yet\nmore perfect. But Theudius, the Magnian, appears to have excelled, as\nwell in mathematical disciplines, as in the rest of philosophy. For he\nconstructed elements egregiously, and rendered many particulars more\nuniversal. Besides, Cyzicinus the Athenian, flourished at the same\nperiod, and became illustrious in other mathematical disciplines, but\nespecially in geometry. These, therefore, resorted by turns to the\nAcademy, and employed themselves in proposing common questions. But\nHermotimus, the Colophonian, rendered more abundant what was formerly\npublished by Eudoxus and The\u00e6tetus, and invented a multitude of\nelements, and wrote concerning some geometrical places. But Philippus\nthe Mend\u00e6an[111], a disciple of Plato, and by him inflamed in the\nmathematical disciplines, both composed questions, according to the\ninstitutions of Plato, and proposed as the object of his enquiry\nwhatever he thought conduced to the Platonic philosophy. And thus far\nhistorians produce the perfection of this science. But Euclid was not\nmuch junior to these, who collected elements, and constructed many\nof those things which were invented by Eudoxus; and perfected many\nwhich were discovered by The\u00e6tetus. Besides, he reduced to invincible\ndemonstrations, such things as were exhibited by others with a weaker\narm. But he lived in the times of the first Ptolemy: for Archimedes\nmentions Euclid, in his first book, and also in others. Besides, they\nrelate that Euclid was asked by Ptolemy, whether there was any shorter\nway to the attainment of geometry than by his elementary institution,\nand that he answered, there was no other royal path which led to\ngeometry. Euclid, therefore, was junior to the familiars of Plato, but\nmore ancient than Eratosthenes and Archimedes (for these lived at one\nand the same time, according to the tradition of Eratosthenes) but he\nwas of the Platonic sect, and familiar with its philosophy: and from\nhence he appointed the constitution of those figures which are called\nPlatonic[112], as the end of his elementary institutions.\n\n\n\n\n                               CHAP. V.\n\n             _What Mathematical Volumes Euclid composed._\n\n\nThere are, therefore, many other mathematical volumes of this man,\nfull of admirable diligence, and skilful consideration: for such are\nhis Optics[113], and Catoptrics: and such also, are his elementary\ninstitutions, which conduce to the attainment of music[114]; and his\nbook concerning divisions[115]. But his geometrical institution of the\nElements is especially admirable, on account of the order and election\nof those theorems and problems, which are distributed through the\nElements. For he does not assume all which might be said, but that\nonly which could be delivered in an elementary order. Besides this, he\nexhibits modes of syllogisms of every kind; some, indeed, receiving\ncredibility from causes, but others proceeding from certain signs; but\nall of them invincible and sure, and accommodated to science. But,\nbesides these, he employs all the dialectic ways, dividing, indeed, in\nthe inventions of forms; but defining in essential reasons: and again,\ndemonstrating in the progressions from principles to things sought,\nbut resolving in the reversions from things sought to principles.\nBesides this, we may view in his geometrical elements, the various\nspecies of conversions, as well of such as are simple as of such as\nare more composite. And again, what wholes may be converted with\nwholes: what wholes with parts; and on the other hand, what as parts\nwith parts[116]. Besides this, we must say, that in the continuation\nof inventions, the dispositions and order of things preceding and\nfollowing, and in the power with which he treats every particular,\nhe is not deceived, as if falling from science, and carried to its\ncontrary, falsehood and ignorance. But because we may imagine many\nthings as adhering to truth, and which are consequent to principles\nproducing science, which nevertheless tend to that error which flows\nfrom the principles, and which deceives ruder minds, he has also\ndelivered methods of the perspicacious prudence belonging to these.\nFrom the possession of which, we may exercise those in the invention of\nfallacies, who undertake this inspection, and may preserve ourselves\nfrom all deception. And this book, by which he procures us this\npreparation, is inscribed \u03c8\u03b5\u03c5\u03b4\u03b1\u03c1\u03af\u03bf\u03c2, or, concerning fallacies[117].\nBecause he enumerates in order their various modes, and in each\nexercises our cogitation with various theorems. And he compares truth\nwith falsehood, and adapts the confutation of deception to experience\nitself. This book, therefore, contains a purgative and exercising\npower. But the institution of his elementary, skilful contemplation of\ngeometrical concerns, possesses an invincible and perfect narration.\n\n\n\n\n                               CHAP. VI.\n\n                 _Concerning the Purport of Geometry._\n\n\nBut, perhaps, some one may enquire in what the design of this treatise\nconsists? To this I answer, that its design is to be distinguished as\nwell according to the objects of enquiry, as according to the learner.\nAnd, indeed, regarding the subject, we must affirm, that all the\ndiscourse of geometry is concerning the mundane figures. Because it\nbegins from such things as are simple, but ends in the variety of their\nconstitution. And, indeed, it constitutes each of them separately,\nbut at the same time delivers their inscriptions in a sphere, and the\nproportions which they contain. On which account some have thought,\nthat the design of each of the books is to be referred to the world;\nand they have delivered to our memory, the utility which they afford\nus in the contemplation of the universe. But distinguishing the\ndesign with respect to the learner, we must affirm, that its purpose\nis the institution of elements; and the perfection of the learners\ncogitative powers in universal geometry. For beginning from these,\nwe are enabled to understand the other parts of this science, and to\ncomprehend the variety which they contain. And, indeed, without these,\nthe discipline of the rest, is to us impossible and incomprehensible.\nFor such theorems as are most principal and simple, and are most allied\nto first suppositions, are here collected in a becoming order. And the\ndemonstrations of other mathematicians, use these as most known, and\nadvance from these in their most complicated progressions. For thus\nArchimedes, in what he has writ concerning the sphere and cylinder, and\nApollonius, and the rest of mathematicians, use, as evident principles,\nthe things exhibited in this treatise. Its purpose, therefore, is the\ninstitution of learners in the whole geometric science, and to deliver\nthe determinate constitutions of the mundane figures.\n\n\n\n\n                              CHAP. VII.\n\n  _From whence the Name of Elementary Institution originated, and why\n             Euclid is called the Institutor of Elements._\n\n\nBut what gave rise to the name of elementary institution, and of\nelement itself, from which elementary institution was derived?\nTo this we shall reply, by observing, that of theorems some are\nusually called elements, but others elementary, and others again are\ndetermined beyond the power of these. Hence, an element is that whose\nconsideration passes to the science of other things, and from which we\nderive a solution of the doubts incident to the particular science we\ninvestigate. For as there are certain first principles of speech, most\nsimple and indivisible, which we denominate elements, and from which\nall discourse is composed; so there are certain principal theorems of\nthe whole of geometry, denominated elements, which have the respect of\nprinciples to the following theorems; which regard all the subsequent\npropositions, and afford the demonstrations of many accidents essential\nto the subjects of geometric speculation. But things elementary\nare such as extend themselves to a multitude of propositions, and\npossess a certain simplicity and sweetness, yet are not of the same\ndignity with elements; because their contemplation is not common to\nall the science to which they belong, as is the case in the following\ntheorem, that in triangles, perpendiculars, drawn from their angles\nto their sides, coincide in one point[118]. Lastly, whatever neither\npossesses a knowledge extended into multitude, nor exhibits any thing\nskilful and elegant, falls beyond the elementary power. Again, an\nelement, as Men\u00e6chmus says, may have a twofold definition. For that\nwhich confirms, is an element of that which is confirmed; as the first\nproposition of Euclid with respect to the second, and the fourth with\nregard to the fifth. And thus, indeed, many things may be mutually\ncalled elements one of another; for they are mutually confirmed.\nThus, because the external angles of right-lined figures, are equal\nto four right angles, the multitude of internal ones equal to right\nangles; and, on the contrary, that from this is exhibited[119].\nBesides, an element is otherwise called that into which, because it\nis more simple, a composite is dissolved. But it must be observed,\nthat every element cannot be called the element of every thing: but\nsuch as are more principal are the elements of such as are constituted\nin the reason of the thing effected; as petitions are the elements\nof theorems. And, according to this signification of an element,\nEuclid\u2019s elements are constructed. Some, indeed, of that geometry\nwhich is conversant about planes; but others of stereometry. In the\nsame manner, likewise, in arithmetic and astronomy, many have composed\nelementary institutions. But it is difficult, in each science, to chuse\nand conveniently ordain elements, from which all the peculiarities of\nthat science originate, and into which they may be resolved. And among\nthose who have undertaken this employment, some have been able to\ncollect more, but others fewer elements. And some, indeed, have used\nshorter demonstrations; but others have extended their treatise to an\ninfinite length. And some have omitted the method by an impossibility;\nbut others that by proportion; and others, again, have attempted\npreparations against arguments destroying principles. So that many\nmethods of elementary institution have been invented by particular\nwriters on this subject. But it is requisite that this treatise should\nentirely remove every thing superfluous, because it is an impediment to\nscience. But every thing should be chosen, which contains and concludes\nthe thing proposed; for this is most convenient and useful in science.\nThe greatest care, likewise, should be paid to clearness and brevity;\nfor the contraries to these, disturb our cogitation. Lastly, it should\nvindicate to itself, the universal comprehension of theorems, in their\nproper bounds: for such things as divide learning into particular\nfragments, produce an incomprehensible knowledge. But in all these\nmodes, any one may easily find, that the elementary institution of\nEuclid excels the institutions of others. For its utility, indeed,\nespecially confers to the contemplation of primary figures: but the\ntransition from things more simple to such as are more various, and\nalso that perception, which from axioms possesses the beginning of\nknowledge, produces clearness, and an orderly tradition: and the\nmigration from first and principal theorems to the objects of enquiry,\neffects the universality of demonstration. For whatever he seems to\nomit, may either be known by the same ways, as the construction of a\nscalene and isosceles triangle[120]: or because they are difficult,\nand capable of infinite variety, they are far remote from the election\nof elements, such as the doctrine of perturbate proportions, which\nApollonius has copiously handled: or, lastly, because they may be\neasily constructed from the things delivered, as from causes, such as\nmany species of angles and lines. For these, indeed, were omitted by\nEuclid, and are largely discoursed of by others, and are known from\nsimple propositions. And thus much concerning the universal elementary\ninstitution of geometry.\n\n\n\n\n                              CHAP. VIII.\n\n           _Concerning the Order of Geometrical Discourses._\n\n\nBut let us now explain the universal order of the discourses contained\nin geometry. Because then, we assert that this science consists from\nhypothesis[121], and demonstrates its consequent propositions from\ndefinite principles (for one science only, I mean the first philosophy,\nis without supposition, but all the rest assume their principles\nfrom this) it is necessary that he who constructs the geometrical\ninstitution of elements, should separately deliver the principles of\nthe science, and separately the conclusions which flow from those\nprinciples; and that he should render no reason concerning the nature\nor truth of the principles, but should confirm by reasons, the things\nconsequent to these geometric principles. For no science demonstrates\nits own principles, nor discourses concerning them; but procures to\nitself a belief of their reality, and they become more evident to the\nparticular science to which they belong than the things derived from\nthem as their source. And these, indeed, science knows by themselves;\nbut their consequents, through the medium of these. For thus, also, the\nnatural philosopher propagates his reasons from a definite principle,\nsupposing the existence of motion. Thus too, the physician, and he\nwho is skilled in any of the other sciences and arts. For if any one\nmingles principles, and things flowing from principles into one and\nthe same, he disturbs the whole order of knowledge, and conglutinates\nthings which can never mutually agree; since a principle, and its\nemanating consequent, are naturally distinct from each other. In the\nfirst place, therefore (as I have said), principles in the geometric\ninstitution are to be distinguished from their consequents, which is\nperformed by Euclid in each of his books; who, before every treatise,\nexhibits the common principles of this science; and afterwards divides\nthese common principles into hypotheses, petitions, and axioms. For all\nthese mutually differ; nor is an axiom, petition, and hypothesis the\nsame, according to the demoniacal Aristotle; but when that which is\nassumed in the order of a principle, is indeed known to the learner,\nand credible by itself, it is an axiom: such as, that things equal\nto the same, are mutually equal to each other. But when any one,\nhearing another speak concerning that of which he has no self-evident\nknowledge, gives this assent to its assumption, this is hypothesis. For\nthat a circle is a figure of such a particular kind, we presume (not\naccording to any common conception) without any preceding doctrine. But\nwhen, again, that which is asserted was neither known, nor admitted by\nthe learner, yet is assumed, then (says he) we call it petition; as\nthe assumption that all right angles are equal. But the truth of this\nis evinced by those who study to treat of some petition, as of that\nwhich cannot by itself be admitted by any one. And thus, according to\nthe doctrine of Aristotle[122], are axiom, petition, and supposition\ndistinguished. But oftentimes, some denominate all these hypotheses,\nin the same manner as the Stoics call every simple enunciation an\naxiom. So that, according to their opinion, hypotheses also will\nbe axioms; but, according to the opinion of others, axioms will be\ncalled suppositions. Again, such things as flow from principles are\ndivided into problems and theorems. The first, indeed, containing the\norigin, sections, ablations, or additions of figures, and all the\naffections with which they are conversant; but the other exhibiting\nthe accidents essential to each figure. For, as things effective of\nscience, participate of contemplation, in the same manner things\ncontemplative previously assume problems in the place of operations.\nBut formerly some of the ancient mathematicians thought that all\ngeometrical propositions should be called theorems, as the followers of\nSpeusippus and Amphinomus, believing, that to contemplative sciences,\nthe appellation of theorems is more proper than that of problems;\nespecially since they discourse concerning eternal and immutable\nobjects. For origin does not subsist among things eternal: on which\naccount, problems cannot have any place in these sciences; since they\nenunciate origin, and the production of that which formerly had no\nexistence, as the construction of an equilateral triangle, or the\ndescription of a square on a given right line, or the position of a\nright line at a given point. It is better, therefore (say they), to\nassert that all propositions are of the speculative kind; but that we\nperceive their origin, not by production, but by knowledge, receiving\nthings eternal as if they were generated; and on this account we ought\nto conceive all those theorematically, but not problematically. But\nothers, on the contrary, think that all should be called problems;\nas those mathematicians who have followed Men\u00e6chmus. But that the\noffice of problems is twofold, sometimes, indeed, to procure the thing\nsought; but at other times when they have received the determinate\nobject of enquiry, to see, either what it is, or of what kind it is,\nor what affection it possesses, or what its relation is to another.\nAnd, indeed, the assertions of each are right; for the followers of\nSpeusippus well perceive. Since the problems of geometry are not of\nthe same kind, with such as are mechanical. For these are sensibles,\nand are endued with origin, and mutation of every kind. And, on the\nother hand, those who follow Men\u00e6chmus do not dissent from truth: since\nthe inventions of theorems cannot by any means take place without an\napproach into matter; I mean intelligible matter. Reasons, therefore,\nproceeding into this, and giving form to its formless nature, are not\nundeservedly said to be assimilated to generations. For we say that the\nmotion of our cogitation, and the production of its inherent reasons,\nis the origin of the figures situated in the phantasy, and of the\naffections with which they are conversant: for there constructions and\nsections, positions and applications, additions and ablations, exist:\nbut every thing resident in cogitation, subsists without origin and\nmutation. There are, therefore, both geometrical problems and theorems.\nBut, because contemplation abounds in geometry, as production in\nmechanics, all problems participate of contemplation; but every thing\ncontemplative is not problematical. For demonstrations are entirely\nthe work of contemplation; but every thing in geometry posterior to\nthe principles, is assumed by demonstration. Hence, a theorem is more\ncommon: but all theorems do not require problems; for there are some\nwhich possess from themselves the demonstration of the thing sought.\nBut others, distinguishing a theorem from a problem, say, that indeed\nevery problem receives whatever is predicated of its matter, together\nwith its own opposite: but that every theorem receives, indeed, its\nsymptom predicate, but not its opposite. But I call the matter of\nthese, that genus which is the subject of enquiry; as for instance,\na triangle, quadrangle, or a circle: but the symptom predicate, that\nwhich is denominated an essential accident, as equality, or section,\nor position, or some other affection of this kind. When, therefore,\nany one proposes to inscribe an equilateral triangle in a circle, he\nproposes a problem: for it is possible to inscribe one that is not\nequilateral. But when any one asserts that the angles at the base\nof an isosceles triangle are equal, we must affirm that he proposes\na theorem; for it is not possible that the angles at the base of an\nisosceles triangle should be unequal to each other. On which account,\nif any one forming problematically, should say that he wishes to\ninscribe a right angle in a semi-circle, he must be considered as\nignorant of geometry; since every angle in a semi-circle is necessarily\na right one. Hence, propositions which have an universal symptom,\nattending the whole matter, must be called theorems; but those in\nwhich the symptom is not universal, and does not attend its subject,\nmust be considered as problems. As to bisect a given terminated right\nline, or to cut it into equal parts: for it is possible to cut it\ninto unequal parts. To bisect every rectilinear angle, or divide it\ninto equal parts; for a division may be given into unequal parts. On\na given right line to describe a quadrangle; for a figure that is not\nquadrangular may be described. And, in short, all of this kind belong\nto the problematical order. But the followers of Zenodotus, who was\nfamiliar with the doctrine of Oenopides, but the disciple of Andron,\ndistinguish a theorem from a problem, so far as a theorem enquires what\nthe symptom is which is predicated of the matter it contains; but a\nproblem enquires what that is, the existence of which is granted. From\nwhence the followers of Possidonius define a theorem a proposition,\nby which it is enquired whether a thing exists or not; but a problem,\na proposition, in which it is enquired what a thing is, or the manner\nof its existence. And they say that we ought to form the contemplating\nproposition by enunciating, as that every triangle has two sides\ngreater than the remaining one, and that the angles at the base of\nevery isosceles triangle are equal: but we must form the problematical\nproposition, as if enquiring whether a triangle is to be constructed\nupon this right line. For there is a difference, say they, absolutely\nand indefinitely, to enquire whether the thing proposed is from a given\npoint to erect a right line at right angles to a given line, and to\nbehold what the perpendicular is. And thus, from what has been said, it\nis manifest there is some difference between a problem and a theorem.\nBut that the elementary institution of Euclid, also, consists partly of\nproblems, and partly of theorems, will be manifest from considering the\nseveral propositions. Since, in the conclusion of his demonstrations,\nhe sometimes adds (which was to be shewn) sometimes (which was to be\ndone) the latter sentence being the mark or symbol of problems, and\nthe former of theorems. For although, as we have said, demonstration\ntakes place in problems, yet it is often for the sake of generation;\nfor we assume demonstration in order to shew, that what was commanded\nis accomplished: but sometimes it is worthy by itself, since the nature\nof the thing sought after may be brought into the midst. But you will\nfind Euclid sometimes combining theorems with problems, and using them\nalternately, as in the first book; but sometimes abounding with the one\nand not the other. For the fourth book is wholly problematical; but the\nfifth is entirely composed from theorems. And thus much concerning the\norder of geometrical propositions.\n\n\n\n\n                               CHAP. IX.\n\n    _Concerning the Design of the first Book,--its Division,--and a\n                  previous Admonition to the Reader._\n\n\nBut, after these considerations, when we have determined the design\nof the first book, and have exhibited its division, we shall enter\nupon the treatise of the definitions. The design, then, of this book,\nis to deliver the principles of the contemplation of right lines. For\nthough a circle, and its consideration, is more excellent than the\nessence and knowledge of right lines, yet the doctrine concerning these\nis more adapted to us, who are hastening to transfer our cogitation\nfrom more imperfect and sensible natures, to such as are intelligible.\nFor, indeed, right lined figures are proper to sensibles, but a\ncircle to intelligibles. Because that which is simple, uniform, and\ndefinite, is proper to the nature of the things which are: but that\nwhich is various, and which increases indefinitely from the number\nof its containing sides, regards the fluctuating essence of sensible\nparticulars. Hence, in this book, the first and most principal of right\nlined figures are delivered; I mean the triangle and parallelogram.\nFor in these, as under their proper genus, the causes of the elements\nare contained: viz. the isosceles and scalene, and those which are\nformed from these, the equilateral triangle, and the quadrangle,\nfrom which the four figures of the elements are composed. We shall\nfind, therefore, as well the origin of the equilateral triangle as\nof the quadrangle; of the last, indeed, upon, but of the first from\na given right line. [[123]An equilateral triangle, therefore, is the\nproximate cause of the three elements, fire, air, and water: but a\nquadrangle is annexed to earth.] And lastly, the design of the first\nbook is adapted to the whole treatise, and confers to the universal\nknowledge of the mundane elements. Besides, it instructs learners in\nthe science concerning right-lined figures; since it rightly invents,\nand accurately collects, the first principles of these.\n\nBut this book is divided into three greatest parts, of which the first\ndeclares the origin and properties of triangles, as well according\nto angles, as also according to sides. Besides, it makes mutual\ncomparisons of these, and beholds every one by itself. For receiving\none triangle, sometimes it considers the angles from the sides; but\nsometimes the sides from the angles: and this according to equality\nand inequality. And supposing two triangles, it discovers the same\nproperty again, by various methods. But the second part combines the\ncontemplation of parallelograms, describing their properties and\ngenerations. And the third part shews the communication of triangles\nand parallelograms, both in symptoms and mutual comparisons. For it\nshews that triangles and parallelograms constituted on the same and on\nequal bases, are affected with the same passions; and by complication,\nwhen both stand upon one base: and again, after what manner a\nparallelogram may be made equal to a triangle; and lastly, concerning\nthe proportion which in right angled triangles, the square made from\nthe side subtending, has to the squares containing the right angle. And\nsuch is the division of the first Book.\n\nBut, previous to our enquiry into each of these parts, we think it\nrequisite to admonish the reader, that he must not require of us,\nthose small assumptions, and cases, and whatever else there may be of\nthat kind, which has been divulged by our predecessors. For we are\nsatiated with these, and shall, therefore, but rarely adopt them in our\ndiscourse. But whatever has a more difficult contemplation, and regards\nuniversal philosophy, of this we shall make a particular relation:\nimitating the Pythagoreans, with whom this \u00e6nigma was common, \u201ca[124]\nfigure and a step: but not a figure and three oboli,\u201d shewing by this,\nthat it is requisite to pursue that philosophy which ascends every\ntheorem by a step, and raises the soul on high; but does not suffer it\nto remain among sensibles, to fill up the use attendant on mortals,\nand, consulting for this, to neglect the elevation which rises from\nhence to an intelligible essence.\n\n\n\n\n                             DEFINITIONS.\n\n\n\n\n                             DEFINITION I.\n\n                  A POINT is that which has no PARTS.\n\n\nThat geometry, according to the transition which takes place from\nthings more composite to such as are more simple, runs from body, which\nis diffused into distance by three dimensions, to a superficies by\nwhich it is bounded; but from superficies to a line, the boundary of\nsuperficies; and from a line to a point destitute of all dimension,\nhas been often said, and is perfectly manifest. But because these\nterms, in many places, on account of their simplicity, appear to be\nmore excellent than the nature of composites; but in many, as when they\nsubsist in things which they terminate, they are similar to accidents,\nit is necessary to determine in what genera of beings each of these\nmay be beheld[125]. I say then, that such things as are destitute of\nmatter, which subsist in separate reasons, and in those forms which\nare placed under themselves, are always allotted a subsistence of\nmore simple essences, superior to the subsistence of such as are more\ncomposite. On this account, both in intellect, and in the ornaments,\nas well of the middle kind as among those peculiar to the soul, and\nin natures themselves, the terms which proximately vivify bodies,\nexcel according to essence the things which are terminated; and are\nmore impartible, more uniform, and more primary than these. For in\nimmaterial forms, unity is more perfect than multitude; that which is\nimpartible, than that which is endued with unbounded progression; and\nthat which terminates, than that which receives bound from another.\nBut such things as are indigent of matter, and abide in others, and\ndegenerate from the perfection of their essence, which are scattered\nabout subjects, and have an unnatural union, are allotted more\ncomposite reasons, prior to such as are more simple. Hence, things\nwhich appear in the phantasy invested with form, and the matter of\nthe figures which the phantasy contains, and whatever in sensibles is\ngenerated by nature, have, in a preceding order, the reasons of the\nthings terminated; but the reasons which terminate, in a following\nand adventitious rank[126].] For lest that which is distributed into\nthree dimensions, should be extended into infinite magnitude, either\naccording to intelligence or sense, it was every way terminated by\nsuperficies. And lest a plane superficies should conceal itself in an\ninfinite progression, a line approaching opposed its diffusion, and\ngave bound to its indefinite extension. And, in like manner, a point\nlimited the progressions of a line; composite natures deriving their\nsubsistence from such as are simple. For this also is again manifest,\nthat in separate forms the reasons of terms subsist in themselves,\nbut not in those which are terminated; and abiding such as they are\nin reality, possess a power of constituting secondary natures. But,\nin inseparable forms they give themselves up to things which are\nterminated, reside in them, become, as it were, their parts, and\nare replenished with baser natures. On which account, that which is\nimpartible is there endued with a partible essence, and that which is\nvoid of latitude is diffused into breadth. And terms are no longer\nable to preserve their simplicity and purity. For since they abide in\nanother, they necessarily change their own nature into the matter of\ntheir containing subject. Matter, indeed, disturbs the perfection of\nthese, and causes the reason of a plane to become a profound plane;\nbut obscuring the one dimension of a line, causes it to be every way\npartible; and gives corporeity to the indivisibility of a point, and\nseparates it together with the natures which it terminates. For all\nthese reasons falling into matter, the one kind from cogitation into\nintelligible matter, but the other from nature into that which is\nsensible, are replenished with their containing subjects; and depart\nfrom their own simplicity, into foreign compositions and intervals. But\nhere a doubt arises how all these, existing in intellect and soul in\nan impartible manner, and without any dimension, are distributed into\nmatter, some indeed, principally, but others on account of its nature?\nShall we say that there is a certain order in immaterial forms, so\nthat some are allotted the first, some the middle, and others the last\nplace; and that of forms some are more uniform, but that others are\nmore multiplied; and that some have their powers collected together,\nbut others tending into interval; and that some, again, border upon\nbound, but that others are proximate to infinity? For though all\nparticipate of these two principles, yet some originate from bound,\nbut others from infinity, of which they more largely participate.\nHence, a point is entirely impartible, since it subsists according\nto bound, yet it occultly contains an infinite power, by which it\nproduces every interval, and the progression of all intervals, unfolds\nits infinite power. But body, and the reason of body, participates\nmore of an infinite nature; on which account it is among the number of\nthings terminated by another, and divisible in infinitum, according\nto all dimensions. But the mediums between these, according to the\ndistance of the extremes, are either among the number of things which\nhave an abundance of bound; or among such as have an affluence of\ninfinity: on which account they both terminate and are terminated.\nFor, indeed, so far as they consist from bound, they are able to\nterminate others; but so far as they participate of infinity, they are\nindigent of termination from others, Hence, since a point is also a\nbound, it preserves its proper power in participation: but since it\nlikewise contains infinity occultly, and is compelled to be every where\npresent with the natures which it terminates, it resides with them\ninfinitely. And, because among immaterial forms there was a certain\ninfinite power capable of producing things distant from each other by\nintervals, a point is present with its participants in capacity. For\ninfinity in intelligibles is the primary cause and prolific power of\nthe universe; but in material natures it is imperfect, and is alone\nall things in dormant capacity. And in short, those forms which, on\naccount of their simplicity and impartibility, hold a superior rank\namong principles, preserve, indeed, (in conformity to their nature,)\ntheir own property in their participations, but become worse than\nmore composite reasons. For matter is able to participate these more\nclearly, and to be prepared for their reception, rather than that of\nthe most simple causes of beings. On which account, the vestigies of\nseparate principles descend into matter; but the participations of\nthose in a second and third order, become more conspicuous. Hence,\nmatter participates more of the cause of body, than of a plane; and of\nthis more than the form of a line; and of this still more than that of\na point, which contains all these, and is the boundary of them all.\nFor the reason of a point presides over this whole series, unites\nand contains all partible natures, terminates their progressions,\nproduces them all by its infinite power, and comprehends them in its\nindivisible bound. On which account also, in the images of immaterial\nforms, some are the boundaries of others; but a point is the limit\nof them all. But that we must not think with the Stoics, that these\nboundaries of bodies alone subsist from cogitation; but that there are\ncertain natures of this kind among beings, which previously contain\nthe demiurgical reasons of things, we shall be enabled to remember, if\nwe regard the whole world, the convolutions of its parts, the centres\nof those convolutions, and the axes which penetrate through the whole\nof these revolving circles. For the centres subsist in energy, since\nthey contain the spheres, preserve them in their proper state, unite\ntheir intervals, and bind and establish to themselves the powers which\nthey possess. But the axes themselves being in an immoveable position,\nevolve the spheres, give them a circular motion, and a revolution round\ntheir own abiding nature. And the poles of the spheres, which both\nterminate the axes, and bind in themselves the other convolutions, do\nthey not perspicuously evince, that points are endued with demiurgical\nand capacious powers, that they are perfective of every thing distant\nby intervals, and are the sources of union, and an unceasing motion?\nFrom whence, indeed, Plato[127] also says, that they have an adamantine\nsubsistence; shewing by this, the immutable, eternal, and stable\npower of their essence, ever preserving itself in the same uniform\nmode of existence. He adds too, that the whole spindle of the Fates,\nis turned about these, and leaps round their coercive union. But other\nmore recondite and abstruse discourses affirm, that the demiurgus\npresides over the world, seated in the poles, and, by his divine love,\nconverting the universe to himself. But the Pythagoreans thought\nthat the pole should be called the Seal of Rhea[128]; because the\nzoogonic, or vivific goddess, pours through these into the universe,\nan inexplicable and efficacious power. And the centre they called the\nprison of Jupiter; because, since Jupiter has placed a demiurgical\nguard in the bosom of the world, he has firmly established it in the\nmidst. For, indeed, the centre abiding, the universe possesses its\nimmoveable ornament, and unceasing convolution: and the gods who\npreside over the poles, obtain a power collective of divisible natures,\nand unific of such as are multiplied: and those who are allotted the\ngovernment of the axes, restrain and eternally evolve their perpetual\nconvolutions. And, if it is lawful to offer our own opinion on this\nsubject, the centres and poles of all the spheres are the symbols of\nthe conciliating gods, shadowing forth their imperceptible and unifying\ncomposition. But the axes express the coherencies of the universal\nornaments; and are endued with a power of comprehending the mundane\nintegrities and periods, in the same manner as their presiding deities,\nof such as are intellectual. But the spheres themselves are images\nof the gods, called perfectors of works, copulating the principle\nwith the end, and excelling all figures in simplicity, similitude,\nand perfection. But we have been thus prolix, that we might evince\nthe power of impartibles, and of the terms which the world contains,\nand that so far as they bear an image of primary and most principal\ncauses, they are allotted the most excellent order in the universe.\nFor centres and poles are not of the same kind with things which\nare terminated; but they subsist in energy, and possess an essence,\nand perfect power, which pervades through all partible natures. But\nmany beholding those terms which imperfectly subsist in terminated\nessences, consider them as endued with a slender subsistence; and\nsome indeed say, that they are alone separated from sensibles by\nthought; but others, that they have an essence no where but in our\nthoughts. However, since the forms of all these are found both in\nthe nature of intellect, in the ornaments of soul, in the nature of\nthings, and in inferior bodies, let us consider how, according to the\norder they contain, they subsist in the genera of beings. And indeed,\nall of them pre-exist in intellect, but in an impartible and uniform\nmanner: so that they all subsist according to one form, the reason of\na point, which exists occultly and impartibly. But they all subsist\nin soul according to the form of a line: on which account Tim\u00e6us also\ncomposes the soul from right and circular lines: for every circle is\na line alone[129]. But they all subsist in natures, according to the\nreason of a plane; and on this account, Plato commands us to manifest\nthose natural reasons, which are endued with a power of constituting\nbodies by a plane. And the resolution of bodies into planes leads us\nto the proximate cause of appearances. Lastly, they all subsist in\nbodies, but in a corporeal manner; since all forms have their being\nin these, according to the partible nature of bodies. Hence, all of\nthem appear every where, and each according to its proper order; and\ndiversity arises from pre-dominating power. The point, indeed, is every\nwhere impartible, and when that which is divisible into parts, excels\naccording to the diminution of beings, it vindicates to itself, an\nillustrious subsistence of partible natures. And sometimes the point is\nentirely superior, according to the excellence of cause; but sometimes\nit is connected with divisibles, and sometimes it is allotted in them\nan adventitious existence; and, as if swallowed up by the partition of\nthe lowest natures, loses its own proper impartibility. As, therefore,\nwith respect to the monad, one[130] is the mother of number, but the\nother is as matter spread under, and the receptacle of numbers; and\neach of them a principle, (yet neither of them is number), but in a\ndifferent respect: in the same manner a point also, is partly the\nparent and author of magnitudes; but is partly a principle in another\nrespect, and not according to a generative cause. But is a point, then,\nthe only impartible? Or may we affirm this of the now in time, and of\nunity in numbers? Shall we not say, that to the philosopher, indeed,\ndiscoursing concerning the universality of things, it is proper to\nbehold every thing, however falling under distribution; but that to\nhim who is endued with the science of particulars, who produces his\ncontemplation from certain definite principles, and runs back even to\nthese, but very little scrutinizes the progressions of beings, it is\nrequisite to attempt, consider, and treat concerning that impartible\nnature alone, which regards his first principles; and to behold that\nsimplicity which presides over all the particular subjects of his\nknowledge? In consequence of this reasoning, therefore, a point alone,\naccording to the geometric matter, is destitute of partition; but unity\naccording to that which is arithmetical. And the reason of a point,\nhowever in some other respects it may be imperfect, yet is perfect in\nthe present science. For, indeed, the physician also says, that the\nelements of bodies are fire and water, and things similar to these; and\nas far as to these the resolution of bodies proceeds. But the natural\nphilosopher passes on to more simple elements; and the one defines an\nelement simple as to sense, but the other simple as to reason; and both\nof them properly as to their peculiar science. We must not, therefore,\nthink that the definition of a point is faulty, nor determine it as\nimperfect; for so far as pertains to the geometric matter, and its\nprinciples, it is sufficiently delivered. This alone, indeed, is\nwanting to its completion, that the definition does not clearly say,\n_that which is impartible with me is a point; and my principle, and\nthat which I contain as most simple, is nothing else than this_.\nAnd after this manner it is proper to hear the geometrician addressing\nus. Euclid, therefore, from a negation of parts, declares to us a\nprinciple, leading to the theory of its whole subject nature. For\nnegative discourses are proper to principles, as Parmenides teaches\nus, who delivers the doctrine concerning the first and last cause,\nby negations alone. Since every principle consists of an essence\ndifferent from its flowing consequents; and the negations of these\nexhibit to us the property of their source. For that it is, indeed, the\ncause of these, yet at the same time has nothing in common with these,\nbecomes perspicuous from a doctrine of this kind. But here a doubt may\narise, how, since the phantasy receives all things invested with forms,\nand in a partible manner, the geometrician beholds in it the point\ndestitute of parts? For it is not because they are reasons existing in\ncogitation, but the phantasy receives the resemblances of intellectual\nand divine forms according to its own proper nature, exhibiting in its\nshadowy bosom the forms of formless natures, and clothing with figure\nthings entirely free from the affections of figure. To this ambiguity\nwe must say, that the species of imaginative motion is neither alone\npartible, nor impartible; but that it proceeds from the impartible to\nthe partible, and from the formless nature to that which is expressed\nby form. For if it was partible alone, it could not preserve in itself\nmany impressions of forms, since the subsequent would obscure the\npre-existent figures: for no body can contain at once, and according\nto the same situation, a multitude of figures; but the former will\nbe blotted out by the succession of the latter. But if it was alone\nimpartible, it would not be inferior to cogitation, and to soul, which\nsurveys all things in an impartible manner. Hence, it is necessary that\nit should indeed begin from an impartible according to its motion, and\nfrom thence draw forth the folded and scattered form of every thing\nfalling under cogitation, and penetrating to its shadowy receptacle:\nbut, that it should at length end in form, figure, and interval. And\nif it be allotted a nature of this kind, it will, after a certain\nmanner, contain an impartible essence: and a point, according to this,\nmust be said to have its principal subsistence: for the form of a\nline is contracted in the phantasy according to this. Hence, because\nit possesses a twofold power, impartible and partible, it will indeed\ncontain a point in an impartible, and intervals in a partible manner.\nBut as the Pythagoreans define a point to be unity having position, let\nus consider what they mean. That numbers, indeed, are more immaterial\nand more pure than magnitudes, and that the principle of numbers is\nmore simple than the principle of magnitudes, is manifest to every one:\nbut when they say that a point is unity endued with position, they\nappear to me to evince that unity and number subsist in opinion: I mean\nmonadic number[131]. On which account, every number, as the pentad and\nthe heptad, is one in every soul, and not many; and they are destitute\nof figure and adventitious form. But a point openly presents itself in\nthe phantasy, subsists, as it were, in place, and is material according\nto intelligible matter. Unity, therefore, has no position, so far as it\nis immaterial, and free from all interval and place: but a point has\nposition, so far as it appears seated in the bosom of the phantasy,\nand has a material subsistence. But unity is still more simple than a\npoint, on account of the community of principles. Since a point exceeds\nunity according to position; but appositions in incorporeals produce\ndiminutions of those natures, by which the appositions are received.\n\n\n\n\n                            DEFINITION II.\n\n                  A Line is a Length without Breadth.\n\n\nA Line obtains the second place in the Definitions, as it is by far\nthe first and most simple interval, which the geometrician calls a\nlength, adding also without breadth; since a line, in respect of a\nsuperficies, ranks as a principle. For he defines a point, as it is the\nprinciple of all magnitudes, by negation alone; but a line, as well by\naffirmation as by negation. Hence it is a length, and by this exceeds\nthe impartibility of a point; but it is without breadth, because it\nis separated from other dimensions. For, indeed, every thing which\nis void of breadth, is also destitute of bulk, but the contrary is\nnot true, that every thing void of bulk is also destitute of breadth.\nSince, therefore, he has removed breadth from a line, he has also\nremoved at the same time bulk. On which account he does not add, that\na line also has no thickness, because this property is consequent\nto the notion of being without breadth. But it is defined by others\nin various ways: for some call it the flux of a point, but others a\nmagnitude contained by one interval. And this definition, indeed; is\nperfect, and sufficiently explains the essence of a line; but that\nwhich calls it the flux of a point, appears to manifest its nature from\nits producing cause; and does not express every line, but alone that\nwhich is immaterial. For this is produced by a point, which though\nimpartible itself, is the cause of being to partible natures. But the\nflux of a point, shews its progression and prolific power, approaching\nto every interval, receiving no detriment, perpetually abiding the\nsame, and affording essence to all partible magnitudes. However, these\nobservations are known, and manifest to every one. But we shall recall\ninto our memory, discourses more Pythagorical, which determine a point\nas analogous to unity, a line to the duad, a superficies to the triad,\nand body to the tetrad. [[132]Yet when we compare those which receive\ninterval together, we shall find a line monadic; but a superficies\ndyadic, and a solid body triadic.] From whence also, Aristotle[133]\nsays; that body is perfected by the ternary number. And, indeed, this\nis not wonderful, that a point, on account of its impartibility,\nshould be assimilated to unity; but that things subsequent to a point,\nshould subsist according to numbers proceeding from unity, and should\npreserve the same proportion to a point, as numbers to unity; and that\nevery one should participate of its proximate superior, and have the\nsame proportion to its kindred, and following degree, as the superior\nto this, which is the immediate consequent. [[134]For example, that\na line has the order of the duad with respect to the point, but of\nunity to a superficies; and that this last has the relation of a triad\nto the point, but of the duad to a solid.] And on this account, body\nis tetradic, with respect to a point, but triadic as to a line. Each\norder, therefore, has its proportion; but the order of the Pythagoreans\nis the more principal, which receives its commencement from an exalted\nsource, and follows the nature of beings. For a point is indeed\ntwofold; since it either subsists by itself, or in a line; in which\nlast respect also, since as a boundary it is alone and one, neither\nhaving a whole nor parts, it imitates the supreme nature of beings.\nOn which account too, it was placed in a correspondent proportion to\nunity. [135]For as the oracle says, _Unity is there first, where the\npaternal unity abides_. But a line is the first endued with parts\nand a whole, and it is monadic because it is distant by one interval\nonly; and dyadic on account of its progression: for if it be infinite,\nit participates of the indefinite duad; but if finite, it requires two\nterms, from whence and to what place; since, on account of these it\nimitates totality, and is allotted an order among totals. For unity,\naccording to the oracle, is extended[136], and generates two; and\nthis produces a progression into longitude, together with that which\nis distant extendedly, and with one interval, and the matter of the\nduad. But superficies, since it is both a triad and duad, as also the\nreceptacle of the primary figures, and that which receives the first\nform and species, is in a certain respect similar to the triadic\nnature, which first terminates beings; and to the duad, by which they\nare divided and dispersed. But a solid, since it has a triple distance,\nand is distinguished by the tetrad, which is endued with a power of\ncomprehending all reasons, is reduced to that order in which the\ndistinction of corporeal ornaments appears; as also the division of the\nuniverse into three parts, together with the tetradic property, which\nis generative and female. And these observations, indeed, might be more\nlargely discussed, but for the present, must be omitted. Again, the\ndiscourse of the Pythagoreans, not undeservedly, calls a line, which is\nthe second in order, and is constituted according to the first motion\nfrom an impartible nature, dyadic. And that a point is posterior to\nunity, a line to the duad, and a superficies to the triad, Parmenides\nhimself shews, by first of all taking away multitude from one by\nnegation, and afterwards the whole. Because, if multitude is before\nthat which is a whole, number also will be prior to that which is\ncontinuous, and the duad to the line, and unity to the point: since the\nepithet _not many_, belongs to unity which generates multitude,\nbut to the point, the term _not a whole_, is proper, because it\nproduces a whole; for this is said to have no part. And these things\nare affirmed of a line, while we more accurately contemplate its\nnature. But we should also admit the followers of Apollonius, who say,\nthat we obtain a notion of a line, when we are ordered to measure the\nlengths alone, either of ways or walls; for we do not then subjoin\neither breadth or bulk, but only make one distance the object of our\nconsideration. In the same manner we perceive superficies, when we\nmeasure fields; and a solid, when we take the dimensions of wells. For\nthen, collecting all the distances together, we say, that the space of\nthe well is so much, according to length, breadth, and depth. But a\nline may become the object of our sensation, if we behold the divisions\nof lucid places from those which are dark, and survey the moon when\ndichotomized: for this medium has no distance with respect to latitude;\nbut is endued with longitude, which is extended together with the light\nand shadow.\n\n\n\n\n                            DEFINITION III.\n\n               But the Extremities of a Line are Points.\n\n\nEvery composite receives its bound from that which is simple, and every\nthing partible from that which is impartible; and the images of these\nopenly present themselves in mathematical principles. For when it is\nsaid that a line is terminated by points, it seems manifestly to make\nit of itself infinite, because, on account of its proper progression,\nit has no extremity. As, therefore, the duad is terminated by unity,\nand reduces its own intolerable boldness under bound, when it is\nrestrained in its comprehensive embrace: so a line also is limited by\nthe points which it contains. For, since it is similar to the duad,\nit participates of a point having the relation of unity, according to\nthe nature of the duad. Indeed, in imaginative, as well as in sensible\nforms, the points themselves terminate the lines in which they reside.\nBut in immaterial forms, the reason of the impartible point pre-exists\nseparate and apart; but when proceeding from thence by far the first of\nall, by determining itself with interval, moving itself, and flowing\nin infinite progression, and imitating the indefinite duad, it is\nrestrained indeed, by its proper principle, is united by its power,\nand on every side seized by its coercive bound. Hence it is, at the\nsame time, both infinite and finite: infinite, indeed, according to its\nprogression; but finite according to its participation of a terminating\ncause. So that, when it approaches to this cause, it is detained in its\ncomprehension, and is terminated according to its union. Hence too, in\nthe images of incorporeal forms, a point is said to terminate a line,\nby occupying its beginning and end. Bound, therefore, in immaterials,\nis separated from that which is bounded: but here it is twofold; for\nit subsists in that which is terminated. And this affords a wonderful\nsymptom, that forms; indeed, abiding in themselves, precede their\nparticipants according to cause; but when giving themselves up to their\nsubordinate natures, subsist according to their diversified properties:\nsince they are multiplied and distributed together with these, and\nreceive the division of their subjects. Besides, this also must be\npreviously received concerning a line, that our geometrician uses it in\na threefold acceptation. As terminated on both sides, and finite; as\nin the problem[137] which says, Upon a given terminated right line to\nconstruct an equilateral triangle. And as partly infinite and partly\nfinite; as in the problem which commands us from three right lines,\nwhich are equal to three given right lines, to construct a triangle;\nfor in the construction of the problem, he says, Let there be placed a\ncertain right line, on one part finite, but on the other part infinite.\nAnd again, a line is received by Euclid as on both sides infinite; as\nin the problem which says, Upon a given infinite right line, from a\ngiven point, which is not in that line, to let fall a perpendicular.\nBut, besides this, the following doubts, since they are worthy of\nsolution, must not be omitted. How are points called the extremities\nof a line? and of what line, since they can neither be the bounds of\none that is infinite, nor of every finite? For there is a certain line,\nwhich is both finite, and has not points for its extremities. And such\nis a circular line, which returns into itself, and is not bounded by\npoints, like a right line. And such also is the ellipsis, or line like\na shield. Is it therefore requisite to behold a line, considered as a\nline? for we must receive a certain circumference, which is terminated\nby points, and a part of the elliptic line; having, in like manner,\nits extremities bounded by points. But every circular and elliptic\nline, assumes to itself another certain property, by which it is not\nline alone, but is also endued with a power of perfecting figure[138].\nLines, themselves, therefore, have their extremities terminated by\npoints; but those which are effective of such like figures, return into\nthemselves. And, indeed, if you conceive them to be described, you\nwill also find how they are bounded by points; but if you receive them\nalready described, and connect the end with the beginning, you can no\nlonger behold their extremes.\n\n\n\n\n                            DEFINITION IV.\n\n      A Right Line, is that which is equally situated between its\n                          _bounding_ Points.\n\n\nPlato establishing two most simple and principal species of lines,\nthe right and the circular, composes all the rest from the mixture of\nthese; I mean such as are called curve lines, some of which are formed\nfrom planes; but others subsist about solids; and whatever species\nof curve lines are produced by the sections of solids. And it seems,\nindeed, that a point (if it be lawful so to speak) bears an image of\nthe one itself, according to Plato: for unity has no part, as he also\nshews in the Parmenides. But, because after unity itself there are\nthree hypostases, or substances, bound, infinite, and that which is\nmixed from these, the species of lines, angles, and figures, which\nsubsist in the nature of things originate from thence. And, indeed, a\ncircumference and a circular angle, and a circle among plane figures,\nand a sphere among solids, are analogous to _bound_. But a right\nline corresponds to _infinity_, according to all these; for it\nproperly belongs to all, if it is beheld as existing in each. But that\nwhich is mixed in all these, is analogous to the mixt which subsists\namong intelligibles. For lines are mixed, as those which are called\nspirals. And angles, as the semi-circular and cornicular[139]. And\nplane figures, as segments and apsides; but solids, as cones and\ncylinders, and others of that kind. Bound, therefore, infinite, and\nthat which is mixed, are participated by all these. But Aristotle[140]\nlikewise assents to Plato; for every species of lines, says he, is\neither right or circular, or mixed from these two. From whence also\nthere are three motions, one according to a right line; the other\ncircular; and the third mixed. But some oppose this division, and\nsay that there are not two simple lines alone, but that there is a\ncertain third line given, i. e. a helix or spiral, which is described\nabout a cylinder[141], when, whilst a right line is moved round the\nsuperficies of the cylinder, a point in the line is carried along with\nan equal celerity. For by this means, a helix, or circumvolute line,\nis produced, which adapts all the parts of itself to all, according to\na similitude of parts, as Apollonius shews in his book concerning the\nCochlea; which passion, among all spirals, agrees to this alone. For\nthe parts of a plane helix are dissimilar among themselves; as also of\nthose which are described about a cone and sphere. But the cylindric\nspiral alone, consists of similar parts in the same manner as a right\nand circular line. Are there, then, three simple lines, and not two\nonly? To which doubt we reply, that a helix of this kind is, indeed, of\nsimilar parts, as Apollonius teaches, but is by no means simple; since\namong natural productions, gold and silver are composed of similar\nparts, but are not simple bodies. But the generation of the cylindric\nhelix evinces that its mixture is from things simple; for it originates\nwhile a right line is circularly moved round the axis of the cylinder,\na point at the same time flowing along in the right line. Two simple\nmotions, therefore, compose its nature; and, on this account, it is\namong the number of mixt lines, and not among such as are simple: for\nthat which is composed from dissimilars is not simple, but mixt. Hence,\nGeminus, with great propriety, when he admits that some simple lines\nmay be produced from many motions, does not grant that every such line\nis mixt; but that alone, which arises from dissimilar motions. For if\nyou conceive a square, and two motions which are performed with an\nequal celerity, one according to the length, but the other according\nto the breadth, a right line or the diameter will be produced; but\nthe right line will not, on this account, be mixed: for no other line\nprecedes it, formed by a simple motion, as we asserted of the cylindric\nhelix. Nor yet, if you suppose a right line, moving in a right angle,\nand by a bisection to describe a circle[142], is the circular line, on\nthis account, produced with mixture: for the extremities of that which\nis moved after this manner, since they are equally moved, will describe\na right line; and the bisection, since it is unequally devolved, will\ndelineate a circle; but the other points will describe an ellipsis. On\nwhich account, the generation of a circular line is the consequence\nof that inequality of lation arising from the bisection; because a\nright line was supposed to be moved in a right angle, but not in a\nnatural manner. And thus much concerning the generation of lines. But\nit seems, that of the two simple lines, the right and the circular,\nthe right line is the more simple; for in this, dissimilitude cannot\nbe conceived, even in opinion. But in the circular line, the concave\nand the convex, indicate dissimilitude. And a right line, indeed, does\nnot infer a circumference according to thought; but a circumference\nbrings with it a right line, though not according to its generation,\nyet with respect to its centre. But what if it should be said that a\ncircumference requires a right line to its construction! For if either\nextreme of a right line remains fixt, but the other is moved, it will\ndoubtless describe a circle, whose centre will be the abiding extreme\nof the right line. Shall we say that the generator of the circle is\nthe point which is carried about the abiding point, but not the right\nline itself? For the line only determines the distance, but the point\ncomposes the circular line, while it is moved in a circular manner:\nbut of this enough. Again, a circumference appears to be proximate to\nbound, and to have the same proportion to other lines, as bound to the\nuniversality of things. For it is finite, and is alone among simple\nlines perfective of figure. But a right line is proximate to infinity;\nfor its capacity of infinite extension never fails: and as all the\nrest are produced from bound and infinite, in the same manner from the\ncircular and right line, every mixt genus of lines is composed, as\nwell of planes as of those which consist in solid bodies. And on this\naccount, the soul also[143] previously assumed into herself the right\nand circular according to her essence, that she might moderate all\nthe co-ordination of infinite, and all the nature of bound, which the\nworld contains. By a right line, indeed, constituting the progression\nof these principles into the universe; but by a circular line, their\nreturn to their original source: and by the one, producing all things\ninto multitude; but by the other, collecting them into one. And not\nonly the soul, but he also who produced the soul, and endued her\nwith these powers, contains in himself both these primary causes.\nFor when he previously assumed the beginning, middle, and end of all\nthings, he terminated right lines (says Plato[144]), by a circular\nprogression according to nature. And proceeding to all things by\nprovident energies, and returning to himself, he established himself,\nsays Tim\u00e6us, after his own peculiar manner. But a right line is the\nmark or symbol of a providence, indeclinable, incapable of perversion,\nimmaculate, never-failing, omnipotent, and present to all beings, and\nto every part of the universe. But a circumference, and that which\nenvirons, is the symbol of an energy retiring into union with itself,\nand which rules over all things according to one intellectual bound.\nWhen, therefore, the demiurgus of the universe had established in\nhimself these two principles, the right and the circular line, and had\ngiven them dominion, he produced from himself two unities; the one,\nindeed, energizing according to the circular line, and being effective\nof intellectual essences; but the other according to the right line,\nand affording an origin to sensible natures. But because the soul is\nallotted a middle situation between intellectuals and sensibles, so\nfar, indeed, as she adheres to an intellectual nature, she energizes\naccording to the circle; but so far as she presides over sensibles, she\nprovides for their welfare according to the right line: and thus much\nconcerning the similitude of these forms to the universality of things.\nBut Euclid, indeed, has properly delivered the present definition of\na line; by which he shews that a right line alone occupies a space\nequal to that which is situated between its points: for as much as is\nthe distance of one point from another, so great is the magnitude of\nthe lines terminated by the points. And this is the meaning of being\nequally situated between its extremes. For if you take two points in a\ncircumference, or in any other certain line, the space of line which\nis included between these, exceeds their distance from each other;\nand every line, besides a right one, appears to suffer this property.\nHence, according to a common conception, the vulgar also say, that he\nwho walks by a right line, performs only a necessary journey: but that\nthey necessarily wander much, who do not proceed in a right line. But\nPlato thus defines it; a right line is that whose middle parts darken\nits extremes. For this passion necessarily attends things which have\na direct position; but it is not necessary that things situated in the\ncircumference of a circle, or in another interval, should be endued\nwith this property. Hence, the astrologers also say, that the sun then\nsuffers an eclipse when that luminary, the moon, and our eye are in one\nright line; for it is then darkened through the middle position of the\nmoon between us and its orb. And perhaps, the passion of a right line\nwill evince, that in the things which are, according to processions\nemanating from causes, the mediums are endued with a power of dividing\nthe distance of the extremes, and their mutual communication with each\nother. As also, according to regressions, such things as are distant\nfrom the extremes, are converted by mediums to their primary causes.\nBut Archimedes defines a right line the least of things having the\nsame bounds. For since, according to Euclid, a right line is equally\nsituated between its points, it is on this account, the least of things\nhaving the same bounds: for if a less line could be given, it would\nnot lie equally between its extremes: but all the other definitions of\na right line, fall into the same conclusions; as for instance, that\nit is constituted in its extremities, and that one part of it is not\nin its subject plane, but another, in one more sublime: and that all\nits parts similarly agree to all: and that its extremes abiding, it\nalso abides. Lastly, that it does not perfect figure, with one line\nsimilar in species to itself: for all these definitions express the\nproperty of a right line, which it possesses from the simplicity of its\nessence, and from its having one progression the shortest of all from\none extremity to another. And thus much concerning the definitions of a\nright line. But again, Geminus divides a line first into an incomposite\nand composite; calling a composite, that which is refracted, and forms\nan angle; but all the rest of them, he denominates incomposites.\nAfterwards, he divides a composite line into that which produces\nfigure, and that which may be infinitely extended. And he calls that\nwhich produces figure, a circular line, and the line of a shield[145],\nand that which is similar to an ivy leaf[146]; but that which is not\neffective of figure, the section of a rectangular and obtuse angular\ncone, the line similar to a shell[147], the right line, and all of\nthat kind. And again, after another manner, of the incomposite line,\none sort is simple, but the other mixt. And of the simple, one produces\nfigure, as the circular; but the other is indefinite, as the right\nline. But of the mixt, one subsists in planes, but the other in solids.\nAnd of that which is in planes, one coincides in itself, as the figure\nof the ivy leaf, which is called the cissoid; but the other may be\nproduced in infinitum, as the helix. But of that which is in solids,\none may be considered in the sections of solids; but the other as\nconsisting about the solids themselves. For the helix, indeed, which\nis described about a sphere or a cone, consists about solids; but\nconic, or spirical sections are generated from a particular section of\nsolids. But, with respect to these sections, the conic were invented by\nM\u00e6nechmus, which also Eratosthenes relating, says,\n\n    \u201cNor in a cone M\u00e6nechmian ternaries divide.\u201d\n\nBut the spiric by Perseus, who composed an epigram on their invention,\nto this purpose, \u201cWhen Perseus had invented three spiral lines in five\nsections, he sacrificed to the gods on the occasion.\u201d And the three\nsections of a cone, are the parabola, hyperbola, and ellipsis: but of\nspiral sections, one kind is twisted and involved, like the fetlock of\na horse; but another is dilated in the middle, and deficient in each\nextremity: and another which is oblong, has less space in the middle,\nbut is dilated on each side. But the multitude of the other mixt lines\nis infinite. For there is an innumerable multitude of solid figures,\nfrom which there are constituted multiform sections. For a right line,\nwhile it is circularly moved[148], does not make a certain determinate\nsuperficies, nor yet conical, nor conchoidal lines, nor circumferences\nthemselves. Hence, if these solids are multifariously cut, they will\nexhibit various species of lines. Lastly, of those lines which consist\nabout solids, some are of similar parts, as the helixes about a\ncylinder; but others of dissimilar parts, as all the rest. From these\ndivisions, therefore, we may collect, that there are only three lines\nof similar parts, the right, the circular, and the cylindric helix. The\ntwo simple ones, indeed, existing in a plane, but the one mixt, about\na solid. And this Geminus evidently demonstrates, when he shews, that\nif two right lines are extended from one point, to a line of similar\nparts, so as to make equal angles upon that line, they shall be equal\nto each other. And the demonstrations of this may be received by the\nstudious, from his volumes; since in these he delivers the origin of\nspiral, conchoidal, and cissoidal lines. But we have barely related\nthe names and divisions of these lines, for the purpose of exciting\nthe ingenious to their investigation; as we think, that an accurate\nenquiry after the method of detecting the properties of each, would be\nsuperfluous in the present undertaking: since the geometrician only\nunfolds to us in this work, simple and primary lines, i.e. the right\nline, in the present definition; but a circular line, in the tradition\nof a circle. For he then says, that the line terminating the circle,\nis the circumference. But he makes no mention of mixt lines, though\nhe was well acquainted with mixt angles, I mean, the semi-circular\nand cornicular: as also with plane mixt figures, i.e. segments and\nsectors; and with solids, viz. cones and cylinders. Of each of the\nrest, therefore, he delivers three species; but of lines only two, i.\ne. the right and circular: for he thought it requisite in discourses\nconcerning things simple, to assume simple species; and all the rest\nare more composite than lines. Hence, in imitation of the geometrician,\nwe also shall terminate their explanation with simple lines.\n\n\n\n\n                             DEFINITION V.\n\n       A SUPERFICIES is that which has only Length and Breadth.\n\n\nAfter a point and a line, a superficies is placed, which is distant\nby a twofold interval, length and breadth. But this also remaining\ndestitute of thickness or bulk, possesses a nature more simple than\nbody, which is distant by a triple dimension. On which account the\ngeometrician adds to the two intervals the particle _only_,\nbecause the third interval does not exist in superficies. And this is\nequivalent to a negation of bulk, as here also he shews the excellency\nof superficies compared to a solid with respect to simplicity, by\nnegation, or by an addition equivalent to negation: but the diminution\nwhich it possesses, if compared with the preceding terms, by the\naffirmations themselves. But others define a superficies to be the\nboundary of body, which is almost affirming the same as the definition\nof Euclid; since that which terminates is exceeded in one dimension,\nby that which is terminated. And others, a magnitude different by\ntwo intervals. Lastly, others declaring the same affection, form its\nassignation in a somewhat different manner. But they say we have a\nknowledge of superficies when we measure fields, and distinguish their\nextremities according to length and breadth; but that we receive a\ncertain sensation of it, when we behold shadows. For as they are\nwithout bulk, because they cannot penetrate into the interior part of\nthe earth, they have only length and breadth. But the Pythagoreans\nsay, that it is assimilated to the triad; because the ternary is by\nfar the first cause to all the figures; which a superficies contains.\nFor a circle, which is the principle of orbicular figures, occultly\npossesses the ternary, by its centre, interval, and circumference. But\na triangle, which ranks as the first among all right-lined figures, on\nevery side evinces that it is enclosed by the triad, and receives its\nform from its perfect nature.\n\n\n\n\n                            DEFINITION VI.\n\n              The Extremities of a SUPERFICIES are Lines.\n\n\nFrom these also, as images, we may understand, that things more simple\nprocure bound and an end to every one of their proximate natures: for\nsoul perfects and determines the operations of nature; and nature\nthe motion of bodies. And prior to these, intellect measures the\nconvolutions of soul; and unity the life of intellect; for that is\nthe measure of all. Just as in these also, a solid is terminated by a\nsuperficies; but a superficies by a line; and a line by a point; for\nthat is the boundary of them all. Hence, the line existing uniformly\nin immaterial forms and impartible reasons, terminates and restrains\nthe various motion of a superficies in its progression, and proximately\nunites its infinity. But in the images of these, when that which\nbounds supervenes that which is bounded, it causes, by this means,\nits limitation and bound. But if it should be enquired how lines are\nthe extremities of every superficies, since they are not the extremes\nof every finite figure; for the superficies of a sphere is terminated\nindeed, yet not by lines, but by itself? In answer to this, we must\nsay, that by receiving a superficies so far as it is distant by a\ntwofold interval, we shall find it terminated by lines according to\nlength and breadth. But if we behold a spherical superficies, we\nmust receive it as that which is endued with figure; which possesses\nanother quality, and conjoins the end with the beginning; and loses\nits two extremities in the comprehensive embraces of one: and this one\nextremity subsists in capacity only, and not in energy.\n\n\n\n\n                            DEFINITION VII.\n\n     A Plane SUPERFICIES is that which is equally situated between\n                          its bounding Lines.\n\n\nIt was not agreeable to the ancient philosophers to establish a plane\nspecies of superficies; but they considered superficies in general,\nas the representative of magnitude, which is distant by a twofold\ninterval. For thus the divine Plato[149] says, that geometry is\ncontemplative of planes, opposing it in division to stereometry, as\nif a plane and a superficies were the same. And this was likewise the\nopinion of the demoniacal Aristotle[150]. But Euclid and his followers\nconsider superficies as a genus, but a plane as its species, in the\nsame manner as rectitude of a line. And on this account he defines a\nplane separate from a superficies, after the similitude of a right\nline. For he defines this last as equal to the space, placed between\nits points. And in like manner, he says, that two right lines being\ngiven, a plane superficies occupies a place equal to the space situated\nbetween those two lines. For this is equally situated between its\nlines; and others also explaining the same boundary, assert that it\nis constituted in its extremities. But others define it as that to\nall the parts of which a right line may be adapted[151]. But perhaps\nothers will say, that it is the shortest of superficies, having the\nsame boundaries; and that its middle parts darken its extremities; and\nthat all the definitions of a right line may be transferred into a\nplane superficies, by only changing the genus: since a right, circular,\nand mixt line, commencing from lines, arrive even at solids, as we\nhave asserted above; for they are proportionally, both in superficies\nand solids. Hence also, Parmenides says, that every figure is either\nright, or circular, or mixt. But if you wish to consider the right in\nsuperficies, take a plane, to which a right line agrees in various\nways; but if a circular receive a spherical superficies; and if a\nmixt, a conic or cylindric, or some one of that genus. But it is\nrequisite (says Geminus) since a line, and also a superficies is\ncalled mixt, to know the measure of mixture, because it is various.\nFor mixture in lines, is neither by composition, nor by temperament\nonly: since, indeed, a helix is mixed, yet one part of it is not\nstraight, and another part circular, like those things which are\nmixed by composition: nor if a helix is cut after any manner, does\nit exhibit an image of things simple, such as those which are mixed\nthrough temperament; but in these the extremes are, at the same time,\ncorrupted and confused. Hence, Theodorus the mathematician, does not\nrightly perceive, in thinking that this mixture is in lines. But\nmixture in superficies, is neither by composition, nor by confusion;\nbut subsists rather by a certain temperament. For conceiving a circle\nin a subject plane, and a point on high, and producing a right line\nfrom the point to the circumference of the circle, the revolution of\nthis line will produce a conical superficies which is mixt. And we\nagain resolve it into its simple elements, by a parallel section: for\nby drawing a section between the vertex and the base, which shall\ncut the plane of the generative right line, we effect a circular\nline. But the idea of lines, shews that the mode of mixture is not by\ntemperament; for neither does it send us back to the simple nature of\nelements: on the contrary, when superficies are cut, they immediately\nexhibit to us their producing lines. The mode of mixture, therefore, is\nnot the same in lines and superficies. But as among lines there were\nsome simple, that is, the right and circular, of which the vulgar also\npossess an anticipated knowledge without any previous instruction;\nbut the species of mixt lines require a more artificial apprehension:\nso among superficies, we possess an innate notion of those which are\nespecially elementary, the plane and spherical; but science and its\nreason investigates the variety of those which are composed through\nmixture. But this is an admirable property of superficies, that their\nmixture in generation is oftentimes produced from a circular line;\nand this also happens to a spiral superficies. For this is understood\nby the revolution of a circle remaining erect, and turning itself\nabout the same point which is not its centre. And on this account, a\nspiral also is threefold; for its centre is either in a circumference,\nor within, or external to a circumference. If the centre is in\nthe circumference, a continued spiral is produced: if within the\ncircumference, an intangled one; if without, a divided one. And there\nare three spiral sections corresponding to these three differences.\nBut every spiral line is mixt, although the motion from which it is\nproduced is one and circular. And mixt superficies are produced as\nwell from simple lines, (as we have said,) while they are moved with a\nmotion of this kind, as from mixt lines. Since, therefore, there are\nthree conic lines, they produce four mixt superficies, which they call\nconoids. For a rectangular conoid, is produced from the revolution of\nthe parabola about its axis: but that which is formed by the ellipsis,\nis called a spheroid; and is the revolution is made about the greater\naxis, it is an oblong; but if about the lesser a broad spheroid.\nLastly, an obtuse-angled conoid is generated from the revolution of the\nhyperbola. But it is requisite to know, that sometimes we arrive at the\nknowledge of superficies from lines, and sometimes the contrary; for\nfrom conical and spiral superficies, we apprehend conical and spiral\nlines. Besides, this also must be previously received concerning the\ndifference of lines and superficies, that there are three lines of\nsimilar parts (as we have already observed), but only two superficies,\nthe plane and the spherical. For this is not true of the cylindric,\nsince all parts of the cylindric superficies cannot agree to all. And\nthus much concerning the differences of superficies, one of which\nthe geometrician having chosen (I mean the plane), this also he has\ndefined; and in this, as a subject, he contemplates figures, and their\nattendant passions: for his discourse is more copious in this than in\nother superficies: since, indeed, we may understand right lines, and\ncircles, and helixes in a plane; also the sections of circles and right\nlines, contacts, and applications, and the constructions of angles\nof every kind. But in other superficies, all these cannot be beheld.\nFor how in one that is spherical, can we apprehend a right line, or a\nright-lined angle? How, lastly, in a conic or cylindric superficies,\ncan we behold sections of circles or right lines? Not undeservedly,\ntherefore, does he both define this superficies, and discuss his\ngeometrical concerns, by exhibiting every thing in this as in a\nsubject; for from hence he calls the present treatise plane. And, after\nthis manner, it is requisite to understand that which is plane, as\nprojected and constituted before the eyes: but cogitation as describing\nall things in this, the phantasy corresponding to a plane mirror, and\nthe reasons resident in cogitation as dropping their images[152] into\nits shadowy receptacle.\n\n\n\n\n                           DEFINITION VIII.\n\n   [153]A PLANE ANGLE, is the inclination of two Lines to each\n   other in a Plane, which meet together, but are not in the same\n   direction.\n\nSome of the ancient philosophers, placing an angle in the predicament\nof relation, have said, that it is the mutual inclination of lines or\nplanes to each other. But others, including this in quality, as well\nas rectitude and obliquity, say, that it is a certain passion of a\nsuperficies or a solid. And others, referring it to quantity, confess\nthat it is superficies or a solid. For the angle which subsists in\nsuperficies is divided by a line; but that which is in solids, by a\nsuperficies. But (say they) that which is divided by these, is no other\nthan magnitude, and this is not linear, since a line is divided by a\npoint; and therefore it follows that it must be either a superficies\nor a solid. But if it is magnitude, and all finite magnitudes of the\nsame kind have a mutual proportion; all angles of the same kind, i.\ne. which subsist in superficies, will have a mutual proportion. And\nhence, the cornicular will be proportionable to a right-lined angle.\nBut things which have a mutual proportion, may, by multiplication,\nexceed each other; and therefore it may be possible for the cornicular\nto exceed a right-lined angle, which, it is well known, is impossible,\nsince it is shewn to be less than every right-lined angle. But if\nit is quality alone, like heat and cold, how is it divisible into\nequal parts? For equality, inequality, and divisibility, are not less\nresident in angles than in magnitudes; but they are, in like manner,\nessential. But if the things in which these are essentially inherent,\nare quantities, and not qualities, it is manifest that angles also are\nnot qualities. Since the more and the less are the proper passions\nof quality[154], but not equal and unequal. On this hypothesis,\ntherefore, angles ought not to be called unequal, and this greater,\nbut the other less; but they ought to be denominated dissimilars, and\none more an angle, but the other less. But that these appellations\nare foreign from the essence of mathematical concerns, is obvious to\nevery one: for every angle receives the same definition, nor is this\nmore an angle, but that less. Thirdly, if an angle is inclination, and\nbelongs to the category of relation, it must follow, that from the\nexistence of one inclination, there will also be one angle, and not\nmore than one. For if it is nothing else than the relation of lines\nor planes, how is it possible there can be one relation of lines or\nplanes, but many angles? If, therefore, we conceive a cone cut by a\ntriangle from the vertex to the base, we shall behold one inclination\nof the triangular lines in the semicone to the vertex; but two distinct\nangles: one of which is plane, I mean that of the triangle; but the\nother subsists in the mixt superficies of the cone, and both are\ncomprehended by the two triangular lines. The relation, therefore,\nof these, do not make the angle. Again, if is necessary to call an\nangle either quality or quantity, or relation; for figures, indeed,\nare qualities, but their mutual proportions belong to relation. It is\nnecessary, therefore, that an angle should be reduced under one of\nthese three genera. Such doubts, then, arising concerning an angle,\nand Euclid calling it inclination, but Apollonius the collection of\na superficies, or a solid in one point, under a refracted line or\nsuperficies (for he seems to define every angle universally), we\nshall affirm, agreeable to the sentiments of our preceptor Syrianus,\nthat an angle is of itself none of the aforesaid; but is constituted\nfrom the concurrence of them all. And that, on this account, a doubt\narises among those who regard one category alone. But this is not\npeculiar to an angle, but is likewise the property of a triangle. For\nthis, too, participates of quantity, and is called equal and unequal;\nbecause it has to quantity the proportion of matter. But quality also,\nis present with this, in consequence of its figure (since triangles\nare called as well similar as equal); but it possesses this from one\ncategory, and that from another. Hence, an angle is perfectly indigent\nof quantity, the subject of magnitude. But it is also indigent of\nquality, by which it possesses, as it were, its proper form and figure,\nLastly, it is indigent of the relation of lines terminating, or of\nsuperficies comprehending its form. So that an angle consists from\nall these, yet is not any one of them in particular. And it is indeed\ndivisible, and capable of receiving equality and inequality, according\nto the quantity which it contains. But it is not compelled to admit\nthe proportion of magnitudes of the same kind, since it has also a\npeculiar quantity, by which angles are also incapable of a comparison\nwith each other. Nor can one inclination perfect one angle: since the\nquantity also, which is placed between the inclined lines, completes\nits essence. If then we regard these distinctions, we shall dissolve\nall absurdities, and discover that the property of an angle is not the\ncollection of a superficies or solid, according to Apollonius (since\nthese also complete its essence,) but that it is nothing else than\na superficies itself, collected into one point, and comprehended by\ninclined lines, or by one line inclined to itself: and that a solid\nangle is the collection of superficies mutually inclined to each\nother. Hence, we shall find that a formed quantum, constituted in a\ncertain relation, supplies its perfect definition. And thus much we\nhave thought requisite to assert concerning the substance of angles,\npreviously contemplating the common essence of every triangle, before\nwe divide it into species. But since there are three opinions of an\nangle, Eudemus the Peripatetic, who composed a book concerning an\nangle, affirms that it is quality. For, considering the origin of an\nangle, he says that it is nothing else than the fraction of lines:\nbecause, if rectitude is quality, fraction also will be quality. And\nhence, since its generation is in quality, an angle will be entirely\nquality. But Euclid, and those who call it inclination, place it in the\ncategory of relation. But they call it quantity, who say that it is the\nfirst interval under a point, that is immediately subsisting after a\npoint. In the number of which is Plutarch, who constrains Apollonius\nalso into the same opinion. For it is requisite (says he) there should\nbe some first interval, under the inclination of containing lines\nor superficies. But since the interval, which is under a point, is\ncontinuous, it is not possible that a first interval can be assumed;\nsince every interval is divisible in infinitum. Besides, if we any\nhow distinguish a first interval, and through it draw a right line, a\ntriangle is produced, and not one angle. But Carpus Antiochenus says,\nthat an angle is quantity, and is the distance of its comprehending\nlines, or superficies; and that this is distant by one interval, and\nyet an angle is not on that account a line: since it is not true that\nevery thing which is distant by only one interval, is a line. But this\nsurely is the most absurd of all, that there should be any magnitude\nexcept a line, which is distant only by one interval. And thus much\nconcerning the nature of an angle. But with respect to the division\nof angles, some consist in superficies, but others in solids. And of\nthose which are in superficies, some are in simple ones, but others in\nsuch as are mixt. For an angle may be produced in a cylindric, conic,\nspherical, and plane superficies. But of those which consist in simple\nsuperficies, some are constituted in the spherical; but others in the\nplane. For the zodiac itself forms angles, dividing the equinoctial\nin two parts, at the vertex of the cutting superficies. And angles\nof this kind subsist in a spherical superficies. But of those which\nare in planes, some are comprehended by simple lines, others by mixt\nones; and others, again, by both. For in the shield-like figure[155],\nan angle is comprehended by the axis, and the line of the shield: but\none of these lines is mixt, and the other simple. But if a circle\ncuts the shield, the angle will be comprehended by the circumference,\nand the ellipsis. And when cissoids, or lines similar to an ivy leaf,\nclosing in one point like the leaves of ivy (from whence they derive\ntheir appellation) make an angle, such an angle is comprehended by\nmixt lines. Also, when the hippopede, or line familiar to the foot\nof a mare, which is one of the spirals, inclining to another line,\nforms an angle, it is comprehended by mixt lines. Lastly, the angles\ncontained by a circumference and a right line, are comprehended by\nsimple lines. But of these again, some are contained by such as are\nsimilar in species, but others by such as are dissimilar. For two\ncircumferences, mutually cutting, or touching each other, produce\nangles: and these triple, for they are either on both sides convex,\nwhen the convexities of the circumferences are external: or on both\nsides concave, when both the concavities are external; which they call\nsistroides; or mixt from convex and concave lines, as the lines called\nlunulas. But besides this, angles are contained in a twofold manner,\nby a right line and a circumference: for they are either contained by\na right line, and a concave circumference, as the semi-circular angle;\nor by a right line and a convex circumference, as the cornicular angle.\nBut all those which are comprehended by two right lines, are called\nrectilinear angles, which have likewise a triple difference[156]. The\ngeometrician, therefore, in the present hypothesis, defines all those\nangles which are constituted in plane superficies, and gives them the\ncommon name of a plane angle. And the genus of these he denominates\ninclination: but the place, the plane itself, for angles have position:\nbut their origin such, that it is requisite there should be two lines\nat least, and not three as in a solid. And that these should touch\neach other, and by touching, must not lie in a right line, as an angle\nis the inclination and comprehension of lines: but is not distance\nonly, according to one interval. But if we examine this definition,\nin the first place it appears that it does not admit, an angle can be\nperfected by one line; though a cissoid, which is but one, perfects\nan angle. And, in like manner, the hippopede. For we call the whole\na cissoid, and not its portions (lest any one should say, that the\nconjunction of these forms an angle) and the whole a spiral, but not\nits parts. Each, therefore, since it is one, forms an angle to itself,\nand not to another. But after this, he is faulty, in defining an angle\nto be inclination. For how, on this hypothesis, will there be two\nangles, from one inclination? How can we call angles equal and unequal?\nAnd whatever else is usually objected against this opinion. Thirdly,\nand lastly, that part of the definition, which says, _and not placed\nin a right line_, is superfluous in certain angles, as in those\nwhich are formed from orbicular lines. For without the assistance of\nthis part, the definition is perfect; since the inclination of one of\nthe lines to the other, forms the angle. And it is not possible that\norbicular angles should be placed in a right line. And thus much we\nhave thought proper to say concerning the definition of Euclid; partly,\nindeed, interpreting, and partly doubting its truth.\n\n\n\n\n                            DEFINITION IX.\n\n        But when the Lines containing the Angle, are right, the\n                     Angle is called RECTILINEAR.\n\n\nAn angle is the symbol and image of the connection and compression,\nwhich subsists in the divine genera, and of that order which collects\ndivisibles into one, particles into an impartible nature, and the many\ninto conciliating community. For it is the bond of a multitude of lines\nand superficies, the collector of magnitude into the impartibility of\npoints, and the comprehender of every figure which is composed by its\nconfining nature. On which account, the oracles[157] call the angular\njunctions of figures, knots, so far as they bring with them an image of\nconnecting union, and divine conjunctions, by which discrete natures\nmutually cohere with each other. The angles, therefore, subsisting in\nsuperficies, express the more immaterial, simple, and perfect unions\nwhich superficies contain: but those which are in solids, represent\nthe unions, which proceed even to inferiors, and supply a community\nto things disjunct, and a construction of the same nature, to things\nwhich on every side receive a perfect partition. But of the angles in\nsuperficies, some shadow forth primary and unmixt unions; but others,\nsuch as comprehend in themselves, an infinity of progressions. And\nsome, indeed, are the sources of union to intellectual forms; but\nothers, to sensible reasons; and others, again, are copulative of those\nforms which obtain between these, a middle situation. Hence, the angles\nwhich are made from circumferences, imitate those causes which envelop\nintellectual variety in coercive union; for circumferences, hastening\nto coalesce with each other, are images of intellect, and intellectual\nforms. On the contrary, rectilineal angles, are the symbols of those\nunions which preside over sensibles, and afford a conjunction of the\nreasons subsisting in these: but mixt angles represent the preservers\nof the communion, as well of sensible, as of intellectual forms,\naccording to one immoveable union. It is requisite, therefore, by\nregarding these paradigms, or exemplars, to render the causes of each.\nFor among the Pythagoreans we shall find various angles dedicated to\nvarious gods. Thus, Philolaus, consecrates to some a triangular, but to\nothers a quadrangular angle; and to others, again, different angles.\nLikewise, he permits the same to many gods, and many to the same god,\naccording to the different powers which they contain. And with a view\nto this, and to the demiurgic triangle, which is the primary cause of\nall the ornament of the elements, it appears to me, that Theodorus\nAsin\u00e6us the philosopher, constitutes some of the gods, according to\nsides; but others, according to angles. The first, indeed, supplying\nprogression and power; but the second, the conjunction of the universe,\nand the collection of progressive natures again into one. But these,\nindeed, direct us to the knowledge of the things which are. And we must\nnot wonder that lines are here said to contain an angle. For the one\nand impartible nature which is found in these, is adventitious: but\nin the gods themselves, and in true beings, the whole, and impartible\ngood, precedes things many, and divided.\n\n\n\n\n                             DEFINITION X.\n\n  When a Right Line standing on a Right Line, makes the successive\n  Angles on each side equal to one another, each of the equal\n  Angles, is a Right Angle; and the insisting Right Line, is called\n  a PERPENDICULAR to that upon which it stands.\n\n                            DEFINITION XI.\n\n         An OBTUSE ANGLE is that which is greater than a RIGHT\n                                ANGLE.\n\n                            DEFINITION XII.\n\n           But an ACUTE ANGLE, is that which is less than a\n                             RIGHT ANGLE.\n\n\nThese are the triple species of angles, which Socrates speaks of in\nthe Republic, and which are received by geometricians from hypothesis;\na right-line constituting these angles, according to a division into\nspecies; I mean, the right, the obtuse, and the acute. The first of\nthese being defined by equality, identity and similitude; but the\nothers being composed through the nature of the greater and the lesser;\nand lastly, through inequality and diversity, and through the more and\nthe less, indeterminately assumed. But many geometricians, are unable\nto render a reason of this division, and use the assertion, that there\nare three angles, as an hypothesis[158]. So that, when we interrogate\nthem concerning its cause, they answer, this is not to be required\nof them as geometricians. However, the Pythagoreans, referring the\nsolution of this triple distribution to principles, are not wanting in\nrendering the causes of this difference of right-lined angles. For,\nsince one of the principles subsists according to bound, and is the\ncause of limitation, identity, and equality, and lastly, of the whole\nof a better co-ordination: but the other is of an infinite nature,\nand confers on its progeny, a progression to infinity, increase,\nand decrease, inequality, and diversity of every kind, and entirely\npresides over the worse series; hence, with great propriety, since\nthe principles of a right-lined angle are constituted by these, the\nreason proceeding from bound, produces a right angle, one, with respect\nto the equality of every right angle, endued with similitude, always\nfinite and determinate, ever abiding the same, and neither receiving\nincrement nor decrease. But the reason proceeding from infinity,\nsince it is the second in order, and of a dyadic nature, produces\ntwofold angles about the right angle, distinguished by inequality,\naccording to the nature of the greater and the lesser, and possessing\nan infinite motion, according to the more and the less, since the one\nbecomes more or less obtuse; but the other more or less acute. Hence,\nin consequence of this reason, they ascribe right angles to the pure\nand immaculate gods of the divine ornaments, and divine powers which\nproceed into the universe, as the authors of the invariable providence\nof inferiors; for rectitude, and an inflexibility and immutability to\nsubordinate natures accords with these gods: but they affirm, that the\nobtuse and acute angles should be ascribed to the gods, who afford\nprogression, and motion, and a variety of powers. Since obtuseness is\nthe image of an expanded progression of forms; but acuteness possesses\na similitude to the cause dividing and moving the universe. But\nlikewise, among the things which are, rectitude is, indeed, similar to\nessence, preserving the same bound of its being; but the obtuse and\nacute, shadow forth the nature of accidents. For these receive the more\nand the less, and are indefinitely changed without ceasing. Hence,\nwith great propriety, they exhort the soul to make her descent into\ngeneration, according to this invariable species of the right angle,\nby not verging to this part more than to that; and by not affecting\nsome things more, and others less. For the distribution of a certain\nconvenience and sympathy of nature, draws it down into material error,\nand indefinite variety[159]. A perpendicular line is, therefore, the\nsymbol of inflexibility, purity, immaculate, and invariable power,\nand every thing of this kind. But it is likewise the symbol of divine\nand intellectual measure: since we measure the altitudes of figures\nby a perpendicular, and define other rectilineal angles by their\nrelation to a right angle, as by themselves they are indefinite and\nindeterminate. For they are beheld subsisting in excess and defect,\neach of which is, by itself, indefinite. Hence they say, that virtue\nalso stands according to rectitude; but that vice subsists according\nto the infinity of the obtuse and acute, that it produces excesses and\ndefects, and that the more and the less exhibit its immoderation, and\ninordinate nature. Of rectilineal angles, therefore, we must establish\nthe right angle, as the image of perfection, and invariable energy,\nof limitation, intellectual bound, and the like; but the obtuse and\nacute, as shadowing forth infinite motion, unceasing progression,\ndivision, partition and infinity. And thus much for the theological\nspeculation of angles. But here we must take notice, that the genus\nis to be added to the definitions of an obtuse and acute angle; for\neach is right-lined, and the one is greater, but the other less than a\nright-angle. But it is not absolutely true, that every angle which is\nless than a right one, is acute. For the cornicular is less than every\nright-angle, because less than an acute one, yet is not on this account\nan acute angle. Also, a semi-circular is less than any right-angle, yet\nis not acute. And the cause of this property is because they are mixt,\nand not rectilineal angles. Besides, many curve-lined angles appear\ngreater than right-lined angles, yet are not on this account obtuse;\nbecause it is requisite that an obtuse should be a right-lined angle.\nSecondly, as it was the intention of Euclid, to define a right-angle,\nhe considers a right-line standing upon another right-line, and making\nthe angles on each side equal. But he defines an obtuse and acute\nangle, not from the inclination of a right line to either part, but\nfrom their relation to a right-angle. For this is the measure of\nangles deviating from the right, in the same manner as equality of\nthings unequal. But lines inclined to either part, are innumerable,\nand not one alone, like a perpendicular. But after this, when he says,\n(_the angles equal to one another_) he exhibits to us a specimen\nof the greatest geometrical diligence; since it is possible that angles\nmay be equal to others, without being right. But when they are equal\nto one another, it is necessary they should be right. Besides, the\nword _successive_ appears to me not to be added superfluously, as\nsome have improperly considered it; since it exhibits the reason of\nrectitude. For it is on this account that each of the angles is right;\nbecause, when they are _successive_, they are equal. And, indeed,\nthe insisting right-line, on account of its inflexibility to either\npart, is the cause of equality to both, and of rectitude to each.\nThe cause, therefore, of the rectitude of angles, is not absolutely\nmutual equality, but position in a consequent order, together with\nequality. But, besides all this, I think it here necessary to call to\nmind, the purpose of our author; I mean, that he discourses in this\nplace, concerning the angles consisting in one plane. And hence, this\ndefinition is not of every perpendicular; but of that which is in one\nand the same plane. For it is not his present design to define a solid\nangle. As, therefore, he defines, in this place, a plane angle, so\nlikewise a perpendicular of this kind. Because a solid perpendicular\nought not to make right angles to one right-line only; but to all which\ntouch it, and are contained in its subject plane: for this is its\nnecessary peculiarity.\n\n\n\n\n                           DEFINITION XIII.\n\n       A BOUND is that which is the Extremity of any thing[160].\n\n\nA Bound, in this place, is not to be referred to all magnitudes, for\nthere is a bound and extremity of a line; but to the spaces which are\ncontained in superficies, and to solid bodies. For he now calls a\nbound, the ambit which terminates and distinguishes every space. And\na bound of this kind, he defines to be an extremity: but not after\nthe manner in which a point is called the extremity of a line, but\naccording to its property of including and excluding from circumjacent\nfigures. But this name is proper to geometry in its infant state, by\nwhich they measured fields, and preserved their boundaries distinct and\nwithout confusion, and from which they arrived at the knowledge of the\npresent science. Since, therefore, Euclid calls the external ambit,\na bound, it is not without propriety that he, by this means, defines\nthe extremity of spaces. For by this, every thing comprehended is\ncircumscribed. I say, for example, in a circle, its bound and extremity\nis the circumference; but itself, a certain plane space: and so of the\nrest.\n\n\n\n\n                            DEFINITION XIV.\n\n         A FIGURE is that which is comprehended by one or more\n                              Boundaries.\n\n\nBecause figure is predicated in various ways, and is divided into\ndifferent species, it is requisite, in the first place, to behold\nits differences; and afterwards to discourse concerning that figure\nwhich is proposed in this Definition. There is, then a certain figure\nwhich is constituted by mutation, and is produced from passion,\nwhile the recipients of the figure are disturbed, divided, or taken\naway; while they receive additions, or are altered, or suffer other\nvarious affections. There is also a figure, which is produced by the\npotter\u2019s, or statuary\u2019s art, according to the pre-existent reason,\nwhich art itself contains: art, indeed, producing the form, but matter\nreceiving from thence, form, and beauty, and elegance. But there are\nstill more noble and more illustrious figures than these, the skilful\noperations of nature. Some, indeed, existing in the elements under\nthe moon[161], and having a power of comprehending the reasons those\nelements contain: but others are situated in the celestial regions,\ndistinguishing their powers, and endless revolutions. For the heavenly\nbodies, both when considered by themselves, and with relation to each\nother, exhibit an abundant and admirable variety of figures; and at\ndifferent times they present to our view different forms, bringing with\nthem a splendid image of intellectual species; and, by their elegant\nand harmonious revolutions, describing the incorporeal and immaterial\npowers of figures. But there are, again, besides all these, most pure\nand perfect beauties, the figures of souls, which, because they are\nfull of life, and self-motive, have an existence prior to things moved\nby another; and which, because they subsist immaterially, and without\nany dimension, excel the forms which are endued with dimension and\nmatter. In the nature of which we are instructed by Tim\u00e6us, who has\nexplained to us the demiurgic, and essential figure of souls[162].\nBut again, the figures of intellects are by far more divine than the\nfigures of souls; for these, on every side, excel partible essences;\nare every where resplendent with impartible and intellectual light;\nare prolific, effective, and perfective of the universe; are equally\npresent, and firmly abide in all things; and procure union to the\nfigures of souls; but recall the mutation of sensible figures to the\nlimitation of their proper bound. Lastly, there are, separate from\nall these, those perfect, uniform, unknown, and ineffable figures of\nthe gods, which are resident, indeed, in the figures of intellects;\nbut jointly terminate all figures, and comprehend all things in their\nunifying boundaries. The properties of which the theurgic art, also\nexpressing, surrounds various resemblances of the gods, with various\nfigures. And some, indeed, it fashions by characters, in an ineffable\nmanner; for characters of this kind, manifest the unknown powers of the\ngods: but others it imitates by forms and images; fashioning some of\nthem erect, and others fitting; and some similar to a heart, but others\nspherical, and others expressed by different figures. And again, some\nit fabricates of a simple form; but others it composes from a multitude\nof forms; and some are sacred and venerable; but others are domestic,\nexhibiting the peculiar gentleness of the gods. And some it constructs\nof a severe aspect; and lastly, attributes to others, different\nsymbols, according to the similitude and sympathy pertaining to the\ngods[163]. Since, therefore, figure derives its origin from the gods\nthemselves, it arrives, by a gradual progression, even to inferiors,\nin these also appearing from primary causes. Since it is requisite to\nsuppose the perfect before the imperfect, and things situated in the\nstability of their own essence, prior to those which subsist in others,\nand previous to things full of their own privation, such as preserve\ntheir proper nature sincere. Such figures, therefore, as are material,\nparticipate of material inelegance, and do not possess a purity\nconvenient to their nature. But the celestial figures are divisible,\nand subsist in others. And the figures of souls are endued with\ndivision, and variety, and involution of every kind; but the figures\nof intellects, together with immaterial union, possess a progression\ninto multitude. And lastly, the figures of the gods are free, uniform,\nsimple, and generative; they subsist before all things, containing all\nperfection in themselves, and extending from themselves to all things,\nthe completion of forms. We must not, therefore, listen to, and endure\nthe opinions of many, who affirm, that certain additions, ablations,\nand alterations, produce sensible figures, (for motions, since they are\nimperfect, cannot possess the principle and primary cause of effects;\nnor could the same figures often be produced from contrary motions; for\nthe same form is sometimes generated from addition and detraction,)\nbut we must consider operations of this kind as subservient to other\npurposes in generation, and derive the perfection of figure from other\nprimogenial causes. Nor must we subscribe to their opinion, who assert\nthat figures destitute of matter can have no subsistence; but those\nonly which appear in matter. Nor to theirs, who acknowledge, indeed,\nthat they are external to matter, but consider them as subsisting\nalone, according to thought and abstraction. For where shall we\npreserve in safety, the certainty, beauty, and order of figures, among\nthings which subsist by abstraction? For, since they are of the same\nkind with sensibles, they are far distant from indubitable and pure\ncertainty. But from whence do they derive the certainty, order, and\nperfection which they receive? For they either derive it from sensibles\n(but they have no subsistence in these), or from intelligibles (but in\nthese they are more perfect), since, to say from that which is not, is\nthe most absurd of all. For nature does not produce imperfect figures,\nand leave the perfect without any subsistence. Nor is it lawful, that\nour soul should fabricate more certain, perfect, and orderly figures,\nthan intellect and the gods themselves. There are, therefore, prior\nto sensible figures, self-moving, intellectual, and divine reasons of\nfigures. And we are excited, indeed, from the obscurity of sensible\nforms, but we produce internal reasons, which are the lucid images\nof others. And we possess a knowledge of sensible figures, by their\nexemplars resident in soul (\u03c0\u03b1\u03c1\u03b1\u03b4\u03b5\u03b9\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u1ff6\u03c2), but we comprehend by\nimages (\u03b5\u1f30\u03ba\u03bf\u03bd\u03b9\u03ba\u1ff6\u03c2) such as are intellectual and divine. For the reasons\nwe contain, emerging from the dark night of oblivion, and propagating\nthemselves in sciential variety, exhibit the forms of the gods, and\nthe uniform bounds of the universe, by which they ineffably convert\nall things into themselves. In the gods, therefore, there is both an\negregious knowledge of universal figures, and a power of generating and\nconstituting all inferiors. But in natures, figures are endued with\na power generative of apparent forms; but are destitute of cognition\nand intellectual perception. And, in particular souls, there is,\nindeed, an immaterial intellection, and a self-energizing knowledge;\nbut there is wanting a prolific, and efficacious cause. As, therefore,\nnature, by her forming power presides over sensible figures, in the\nsame manner, soul, by her gnostic energy, drops in the phantasy as in\na mirror, the reasons of figures. But the phantasy receiving these in\nher shadowy forms, and possessing images of the inherent reasons of\nthe soul, affords by these the means of inward conversion to the soul,\nand of an energy directed to herself, from the spectres of imagination.\nJust as if any one beholding his image in a mirror, and admiring the\npower of nature, and his own beauty, should desire to see himself in\nperfection, and should receive a power of becoming, at the same time,\nthe perceiver, and the thing perceived. For the soul, after this\nmanner, looking abroad into the bright mirror of the phantasy, and\nsurveying the shadowy figures it contains, and admiring their beauty\nand order, pursues, in consequence of her admiration, the reasons from\nwhich these images proceed; and being wonderfully delighted, dismisses\ntheir beauty, as conversant about spectres alone; but afterwards\nseeks her own purer beauty, and desires to pass into her own profound\nretreats, and there to perceive the circle and the triangle, and all\nthings subsisting together, in an impartible manner, and to insert\nherself in the objects, to contract her multitude into one; and lastly,\nto behold the occult and ineffable figures of the gods, seated in the\nmost sacred and divine recesses of her nature. She is likewise desirous\nof bringing into light, from its awful concealment, the solitary\nbeauty, of the gods, and of perceiving the circle, subsisting in its\ntrue perfection, more impartible than any centre, and the triangle\nwithout interval; and lastly, by ascending into an union with herself,\nof surveying every object which is subject to the power of cognition.\nThe figure, therefore, which is self-motive, precedes that which is\nmoved by another; and the impartible that which is self-motive: but\nthat which is the same with _one_, precedes the impartible itself.\nFor all things are bounded, when they return to the unities of their\nnature; since all things pass through these as a divine entrance into\nbeing. And thus much for this long digression, which we have delivered\naccording to the sentiments of the Pythagoreans. But the geometrician,\ncontemplating that figure which is seated in the phantasy, and defining\nthis, in the first place, (since this definition agrees with sensibles,\nin the second place) says, that figure is that which is comprehended by\none or more boundaries. For, since he receives it together with matter,\nand conceives of it as distant with intervals, he does not improperly\ncall it finite and terminated[164]. [Since every thing which contains\neither intelligible or feasible matter, is allotted an adventitious\nbound; and is not itself bound, but that which is bounded.] Nor is it\nthe bound of itself; but one of its powers is terminating, and the\nother terminated. Nor does it subsist in bound itself, but is contained\nby bound. For figure is joined to quantity, and subsists together\nwith it; and, at the same time, quantity is subjected to figure; but\nthe reason and aspect of that quantity is nothing else than figure\nand form. Since, indeed, reason terminates quantity, and adds to it\na particular character and bound, either simple or composite. For,\nsince this also exhibits the twofold progression of _bound_ and\n_infinite_ in its proper forms, (in the same manner as the reason\nof an angle,) it invests the objects of its comprehension with one\nboundary and simple form, according to _bound_, but with many,\naccording to infinity[165]. Hence, every thing figured, vindicates to\nitself either one boundary, or a many. Euclid, therefore, denominating\nthat which is figured and material, and annexed to quantity figure,\ndoes not improperly say, that it is contained by one or more terms.\nBut Possidonius defines figure to be concluding bound, separating\nthe reason of figure from quantity; and considering it as the cause\nof terminating, defining, and comprehending quantity. For that which\nencloses, is different from that which is enclosed; and bound from that\nwhich is bounded. And Possidonius, indeed, seems to regard the external\nsurrounding bound; but Euclid, the whole subject. Hence, the one calls\na circle a figure, with relation to its whole plane, and exterior\nambit; but the other with relation to its circumference only. And the\none defines that which is figured, and which is beheld together with\nits subject: but the other desires to define the reason of the circle;\nI mean that which terminates and concludes its quantity. But if any\nlogician, and captious person, should blame the definition of Euclid,\nbecause he defines genus from species (for things contained by one or\nmore terms, are the species of figure,) we shall assert, in opposition\nto such an objection, that genera also pre-occupy in themselves the\npowers of species. And when men of ancient authority, were willing to\nmanifest genera themselves, from those powers which genera contain,\nthey appeared, indeed, to enter on their design from species, but, in\nreality, they explained genera from themselves, and from the powers\nwhich they contain. The reason of figure, therefore, since it is one,\ncomprehends the differences of many figures, according to the bound\nand infinity residing in its nature. And he who defined this reason,\nwas not void of understanding, whilst he comprehended in a definition,\nthe differences of the powers it contained. But you will ask, From\nwhence does the reason of figure originate, and by what causes is\nit perfected? I answer, that it first arises from _bound_ and\n_infinite_, and that which is mixed from these. Hence it produces\nsome species from _bound_, others from _infinite_, and others\nfrom the _mixt_. And this it accomplishes by bringing the form of\nbound to circles; but that of infinite, to right lines: and that of\nthe mixt to figures composed from right and circular lines. But, in\nthe second place, this reason is perfected from that totality, which\nis separated into dissimilar parts. From whence, indeed, it occasions\na whole to every form, and each figure is cut into different species.\nFor a circle, and every right-lined figure may be divided, by reason or\nproportion, into dissimilar figures; which is the business of Euclid\nin his book of divisions, where he divides one figure into figures\nsimilar to each as are given; but another into such as are dissimilar.\nIn the third place, it is invigorated from accumulated multitude, and,\non account of this, extends forms of every kind, and produces the\nmultiform reasons of figures. Hence, in propagating itself, it does\nnot cease till it arrives at something last, and has unfolded all the\nvariety of forms. And, as in the intelligible world, _one_ is\nshewn to abide in that which _is_; and, at the same time, that\nwhich _is_ in _one_, so likewise, reason exhibits circular in\nright-lined figures; and on the contrary, rectilinear comprehended in\ncircular figures. And it peculiarly manifests its whole nature in each,\nand all these in all. Since the whole subsists in all collectively, and\nin each separate and apart. From that order, therefore, it is endued\nwith this power. In the fourth place, it receives from the first of\nnumbers[166], the measures of the progression of forms. From whence it\nconstitutes all figures according to numbers; some, indeed, according\nto the more simple, but others according to the more composite. For\ntriangles, quadrangles, quinquangles, and all multangles, proceed in\ninfinitum, together with the mutations of numbers. But the cause of\nthis is, indeed, unknown to the vulgar, though, to those who understand\nwhere number and figure subsist, the reason is manifest. Fifthly,\nit is replete with that division of forms, which divides forms into\nother similar forms, from another second totality, which is also\ndistributed into similar parts. And by this, a triangular reason is\ndivided into triangles, and a quadrangular reason into quadrangles.\nAnd hence, exercising our inward powers, we effect what I have said\nin images, since it pre-existed by far the first in its principles.\nBut by regarding these distributions, we may render many causes of\nfigures, reducing them to their first principles. And the more common,\nor geometrical figure, is allotted an order of this kind, and from so\nmany causes, receives the perfection of its nature. But, from hence\nit advances to the genera of the gods, and is variously attributed\naccording to its various forms, and energizes differently in different\ngods. To some, indeed, affording more simple figures; but to others,\nsuch as are more composite. And to some, again, assigning primary\nfigures, and those which are produced in superficies; but to others\n(entering the tumor of solid bodies) such figures, as in solids are\nconvenient to themselves. For all figures, indeed, subsist in all,\nsince the forms of the gods are accumulated, and full of universal\npowers: but, by their peculiarity, they produce one thing according\nto another. For one possesses all things circularly, another in a\ntriangular manner, but another according to a quadrangular reason. And\nin a similar manner in solids.\n\n                            DEFINITION XV.\n\n   A CIRCLE is a Plane Figure, comprehended by one Line, which is\n   called the Circumference, to which all Right Lines falling from\n   a certain Point within the Figure, are equal to each other.\n\n                            DEFINITION XVI.\n\n          And that Point is called the CENTRE of the CIRCLE.\n\n\nA Circle is the first, most simple, and most perfect of figures. For\nit excels all solids, because it exists in a more simple place; but\nit is superior to the figures subsisting in planes, on account of\nits similitude and identity. And it has a corresponding proportion\nto bound, and unity, and a better co-ordination of being. Hence, in\na distribution of mundane and super-mundane figures, you will always\nfind that the circle is of a diviner nature. For if you make a division\ninto the heavens, and the universal regions of generation, you must\nassign to the heavens a circular form; but to generation, that of a\nright line, For whatever among generable natures is circular, descends\nfrom the heavens; since generation revolves into itself, through\ntheir circumvolutions, and reduces its unstable mutation to a regular\nand orderly continuance. But if you distribute incorporeal natures\ninto soul and intellect, you will say, that the circle belongs to\nintellect, and the right line to the soul. And on this account, the\nsoul, by its conversion to intellect, is said to be circularly moved;\nand it possesses the same proportion to intellect, as generation to\nthe heavens. For it is circularly moved, (says Socrates[167],) because\nit imitates intellect. But the generation and progression of soul is\nmade according to a right-line. For it is the property, of the soul\nto apply herself at different times to different forms. But if you\nwish to divide into body and soul, you must constitute every thing\ncorporeal, according to the right line; but you must assign to every\nanimal a participation of the identity and similitude of the circle.\nFor body is a composite, and is endued with various powers, similar to\nright-lined figures: but soul is simple and intelligent; self-motive,\nand self-operative; converted into, and energizing in herself. From\nwhence, indeed, Tim\u00e6us also, when he had composed the elements of\nthe universe from right-lined figures, assigned to them a circular\nmotion and formation, from that divine soul which is seated in the\nbosom of the world. And thus, that the circle every where holds the\nfirst rank, in respect of other figures, is sufficiently evident from\nthe preceding observations. But it is requisite to survey its whole\nseries, beginning supernally, ending in inferiors, and perfecting all\nthings, according to the aptitude of the natures which receive its\nalliance. To the gods, therefore, it affords a conversion to their\ncauses, and ineffable union: it occasions their abiding in themselves,\nprevents their departing from their own beatitude, strengthens their\nhighest unions, as centres desirable to inferior natures; and stably\nplaces about these the multitude of the powers which the gods possess,\ncontaining them in the simplicity of their essences. But the circle\naffords to intellectual natures, a perpetual energy in themselves, is\nthe cause of their being filled with knowledge from themselves, and\nof possessing in their essences, intelligibles contractedly; and of\nperfecting intellections in themselves. For every intellect, proposes\nto itself that which is intelligible; and this is as a centre to\nintellect, about which it continually revolves: for intellect folds\nitself, and operates about this, and is united within itself on all\nsides, by universal intellectual energies. But it extends to souls by\nillumination, a self-vital, and self-motive power, and an ability of\nturning, and leaping round intellect, and of returning according to\nproper convolutions, unfolding the impartibility of intellect. Again,\nthe intellectual orders excel souls after the manner of centres, but\nsouls energize circularly about their nature. For every soul, according\nto its intellectual part, and the supreme _one_, which is the\nvery flower of its essence, receives a centre: but, according to its\nmultitude, it has a circular revolution, desiring, by this means, to\nembrace the intellect which it participates. But, to the celestial\nbodies, the circle affords an assimilation to intellect, equality, a\ncomprehension of the universe, in proper limits, revolutions which\ntake place in determinate measures, a perpetual subsistence, a nature\nwithout beginning and end, and every thing of this kind. And to the\nelements under the concave of the moon\u2019s orb, it is the cause of a\nperiod, conversant with mutations; an assimilation to the heavens; that\nwhich is without generation, in generated natures; that which abides in\nthings which are moved; and whatever is bounded in partible essences.\nFor all things are perpetual, through the circle of generation; and\nequability is every where preserved on account of the reciprocation\nof corruption. Since, if generation did not return, in a circular\nrevolution, in a short space of time, the order, and all the ornament\nof the elements would vanish. But again, the circle procures to animals\nand plants, that similitude which is found in generations; for these\nare produced from seeds, and seeds from these. Hence, generation here,\nand a circumvolution, alternately takes place, from the imperfect to\nthe perfect, and the contrary; so that corruption subsists together\nwith generation. But, besides this, to unnatural productions it\nimposes order, and reduces their indeterminate variety to the\nlimitation of bound; and, through this, nature herself is gracefully\nornamented in the last vestiges of her powers. Hence, things contrary\nto nature have a revolution according to determinate numbers, and not\nonly fertility, but also sterility, subsists according to the alternate\nconvolutions of circles (as the discourse of the Muses evinces), and\nall evils though they are dismissed from the presence of the gods,\ninto the place of mortals, yet these roll round, says Socrates, and to\nthese there is present a circular revolution, and a circular order;\nso that nothing immoderate and evil is deserted by the gods; but that\nprovidence, which is perfective of the universe, reduces also the\ninfinite variety of evils, to bound, and an order convenient to their\nnature. The circle, therefore, is the cause of ornament to all things,\neven to the last participations, and leaves nothing destitute of\nitself, since it supplies beauty, similitude, formation, and perfection\nto the universe. Hence too, in numbers it contains the middle centres\nof the whole progression of numbers, which revolves from unity to the\ndecad (or ten). For five and six exhibit a circular power, because, in\nthe progressions from themselves, they return again into themselves,\nas is evident in the multiplication of these numbers. Multiplication,\ntherefore, is an image of progression, since it is extended into\nmultitude; but an ending in the same species, is an image of regression\ninto themselves. But a circular power affords each of these, exciting,\nindeed, as from an abiding centre, those causes which are productive\nof multitude; but converting multitude after the productions to their\ncauses. Two numbers, therefore, having the properties of a circle,\npossess the middle place between all numbers: of which one, indeed,\nprecedes every convertible genus of males and an odd nature; but the\nother, recalls every thing feminine and even, and all prolific series,\nto their proper principles, according to a circular power. And thus\nmuch concerning the perfection of the circle. Let us now contemplate\nthe mathematical definition of the circle, which is every way perfect.\nIn the first place, therefore, he defines it a figure, because, indeed,\nit is finite, and every where comprehended by one limit, and is not of\nan infinite nature, but associated to bound. Likewise plane, because,\nsince figures are either beheld in superficies, or in solid bodies, a\ncircle is the first of plane figures, excelling solids in simplicity,\nbut possessing the proportion of unity to planes. But comprehended by\none line, because it is similar to one, by which it is defined, and\nbecause it does not extrinsically receive a variety of surrounding\nterms. And again, that this line makes all the lines drawn to it from\na certain point within equal, because of the figures which are bounded\nby one line, some have all the lines proceeding from the middle equal;\nbut others not at all. For the ellipsis is comprehended by one line,\nyet all the lines issuing from the centre, and bounded by is curvature,\nare not equal, but only two. Also the plane, which is included by the\nline called a cissoid, has one containing line, yet it does not contain\na centre, from which all the lines are equal. But, because the centre\nin a circle is entirely one point (for there are not many centres of\none circle), on this account, the geometrician adds, that lines falling\nfrom one point to the bound of the circle, are equal. For there are\ninfinite points within it, but of all these, one only has the power of\na centre. And because this one point, from which all the lines drawn to\nthe circumference of the circle are equal, is either within the circle,\nor without (for every circle has a pole, from which all the lines drawn\nto its circumference are equal), on this account he adds, _of the\npoints within the figure_, because, here he receives the centre\nalone, and not the pole. For he wishes to behold all its properties in\none plane, but the pole is more elevated than the subject plane. Hence,\nhe necessarily adds, in the end of the definition, that this point,\nwhich is placed within the circle, and to which all right lines drawn\nfrom it to the circumference, are equal, is the centre of the circle.\nFor there are only two points of this kind, the pole and the centre.\nBut the former is without, and the other within the plane. Thus, for\ninstance, if you conceive a perpendicular standing on the centre of a\ncircle, its superior extremity is the pole: for all lines drawn from\nit to the circumference of the circle, are demonstrated to be equal.\nAnd, in like manner, in a cone, the vertex of the whole cone, is the\npole of the circle at the base. And thus far we have determined what a\ncircle is, and its centre, and what the nature is of its circumference,\nand the whole circular figure. Again, therefore, from these, let us\nreturn to the speculation of their exemplars, contemplating in them\nthe centre, according to one impartible and stable excellence. But the\ndistances from the centre, according to the progressions which are made\nfrom one, to multitude infinite in capacity. And the circumference of\nthe circle, according to the regression of the progressions to the\ncentre, by means of which the multitude of powers are rolled round\ntheir union, and all of them hasten to its comprehension, and desire\nto energize about its indivisible embrace. And, as in the circle\nitself, all things subsist together, the centre, intervals, and\nexternal circumference; so in these which are its image, one thing\nhas not an essence pre-existent, and another consequent in time;\nbut all things are, indeed, together, permanency, progression, and\nregression. But these differ from those, because the former subsist\nindivisibly, and without any dimension; but the latter with dimension;\nand in a divisible manner; the centre existing in one place, the lines\nemanating from the centre, in another; and the external circumference\nterminating the circle, having a still different situation. But there\nall things abide in one: for if you regard that which performs the\noffice of a centre, you will find it the receptacle of all things. If\nthe progression distant from the centre, in this, likewise, you will\nfind all things contained. And, in a similar manner, if you regard\nits regression. When, therefore, you are able to perceive all things\nsubsisting together, and have taken away the defect proceeding from\ndimension, and have removed from your inward vision, the position about\nwhich partition subsists, you will find the true circle, advancing to\nitself, bounding, and energizing in itself, existing both one and many,\nand abiding, proceeding, and returning; likewise firmly establishing\nthat part of its essence which is most impartible, and especially\nsingular; but advancing from this according to rectitude, and the\ninfinity which it contains; and rolling itself from itself to one, and\nexciting itself by similitude and identity to the impartible centre\nof its nature, and to the occult power of the one which it contains.\nBut this one, which the circle contains, and environs in its bosom, it\nemulates according to the multitude of its own nature. For that which\nis convolved, imitates that which abides, and the periphery is as a\ncentre which is distant with interval, and nods to itself, hastening\nto receive, and to become one with the centre, and to terminate its\nregress where it received the principle of its progression. For the\ncentre is every where in the place of that which is lovely, and the\nobject of desire, presiding over all things which subsist about its\nnature, and existing as the beginning and author of all progressions.\nAnd this the mathematical centre also expresses, by terminating all the\nlines falling from itself to the circumference, and by affording to\nthem equality, as an image of proper union. But the oracles likewise\ndefine the centre, after this manner: _The centre is that from which\nand[168] to which all the lines to the circumference are equal_.\nIndicating the beginning of the distance of the lines, by the particle\n_from which_; but the middle of the circumference by the particle\n_to which_: for this, in every part, is joined with the centre.\nBut if it be necessary to declare the first cause, through which a\ncircular figure appears and receives its perfection, I affirm, that\nit is the supreme order of intelligibles. For the centre, indeed, is\nassimilated to the cause of bound; but the lines emanating from this,\nand which are infinite, with respect to themselves, both in multitude\nand magnitude, represent infinity; and the line which terminates their\nextension, and conjoins the circular figure with the centre, is similar\nto that occult ornament, consisting from the intelligible orders;\nwhich Orpheus also says, is circularly borne, in the following words,\n_But it is carried with an unwearied energy, according to an infinite\ncircle_. For, since it is moved intelligibly, about that which is\nintelligible, having it for the centre of its motion, it is, with great\npropriety, said to energize in a circular manner. Hence, from these\nalso, the triadic god[169] proceeds, who contains in himself the cause\nof the progression of right-lined figures. For on this account, wise\nmen, and the most mystic of theologists, have fabricated his name.\n[[170]Hence too, it is manifest, that a circle is the first of all\nfigures:] but a triangle is the first of such as are right-lined.\nFigures, therefore, appear first in the regular ornaments of the gods;\nbut they have a latent subsistence, according to pre-existent causes,\nin intelligible essences.\n\n\n\n\n                           DEFINITION XVII.\n\n   A DIAMETER of a Circle is a certain straight Line, drawn through\n   the Centre, which is terminated both ways by the Circumference\n   of the Circle, and, divides the Circle into two equal Parts.\n\n\nEuclid here perspicuously shews, that he does not define every\ndiameter, but that which belongs to a circle only. Because there\nis a diameter of quadrangles and all parallelograms, and likewise\nof a sphere among solid figures. But in the first of these, it is\ndenominated a diagonal: but in a sphere, the axis; and in circles\nthe diameter only. Indeed, we are accustomed to speak of the axis of\nan ellipsis, cylinder, and cone; but of a circle, with propriety,\nthe diameter. This, therefore, in its genus, is a right-line; but as\nthere are many right-lines in a circle, as likewise infinite points,\none of which is a centre, so this only is called a diameter, which\npasses through the centre, and neither falls within the circumference,\nnor transcends its boundary; but is both ways terminated by its\ncomprehensive bound. And these observations exhibit its origin. But\nthat which is added in the end, that it also divides the circle into\ntwo equal parts, indicates its proper energy in the circle, exclusive\nof all other lines drawn through the centre, which are not terminated\nboth ways by the circumference. But they report, that Thales first\ndemonstrated, that the circle was bisected by the diameter. And the\ncause of this bisection, is the indeclineable transit of the right\nline, through the centre. For, since it is drawn through the middle,\nand always preserves the same inflexible motion, according to all its\nparts, it cuts off equal portions on both sides to the circumference\nof the circle. But if you desire to exhibit the same mathematically,\nconceive the diameter drawn, and one part of the circle placed on\nthe other[171]. Then, if it is not equal, it either falls within, or\nwithout; but the consequence either of these ways must be, that a less\nright-line will be equal to a greater. Since all lines from the centre\nto the circumference are equal. The line, therefore, which tends to\nthe exterior circumference, will be equal to that which tends to the\ninterior. But this is impossible. These parts of the circle, then,\nagree, and are on this account equal. But here a doubt arises, if two\nsemi-circles are produced by one diameter, and infinite diameters may\nbe drawn through the centre, a double of infinities will take place,\naccording to number. For this is objected[172] by some against the\nsection of magnitudes to infinity. But this we may solve by affirming,\nthat magnitude may, indeed, be divided infinitely, but not into\ninfinites. For this latter mode produces infinites in energy, but the\nformer in capacity only. And the one affords essence to infinite,\nbut the other is the source of its origin alone. Two semi-circles,\ntherefore, subsist together with one diameter, yet there will never be\ninfinite diameters, although they may be infinitely assumed. Hence,\nthere can never be doubles of infinites; but the doubles which are\ncontinually produced, are the doubles of finites; for the diameters\nwhich are always assumed, are finite in number. And what reason can be\nassigned why every magnitude should not have finite divisions, since\nnumber is prior to magnitudes, defines all their sections, pre-occupies\ninfinity, and always determines the parts which rise into energy, from\ndormant capacity?\n\n\n\n\n                           DEFINITION XVIII.\n\n   A SEMI-CIRCLE is the Figure contained by the Diameter, and that\n   Part of the Circumference which is cut off by the Diameter.\n\n                         DEFINITION XIX.[173]\n\n      But the CENTRE of the Semi-circle, is the same with that of\n                              the Circle.\n\n\nFrom the definition of a circle Euclid finds out the nature of the\ncentre, differing from all the other points which the circle contains.\nBut from the centre he defines the diameter, and separates it from the\nother right lines, which are described within the circle. And from\nthe diameter, he teaches the nature of the semi-circle; and informs\nus, that it is contained by two terms, always differing from each\nother, viz. a right-line and a circumference: and that this right-line\nis not any one indifferently, but the diameter of the circle. For\nboth a less and a greater segment of a circle, are contained by a\nright-line and circumference; yet these are not semi-circles, because\nthe division of the circle is not made through the centre. All these\nfigures, therefore, are biformed, as a circle was monadic, and are\ncomposed from dissimilars. For every figure which is comprehended by\ntwo terms, is either contained by two circumferences, as the lunular:\nor by a right-line and circumference, as the above mentioned figures;\nor by two mixt lines, as if two ellipses intersect each other (since\nthey enclose a figure, which is intercepted between them), or by a\nmixt line and circumference, as when a circle cuts an ellipsis; or by\na mixt and right-line, as the half of an ellipsis. But a semi-circle\nis composed from dissimilar lines, yet such as are, at the same time,\nsimple, and touching each other by apposition. Hence, before he defines\ntriadic figures, he, with great propriety, passes from the circle to\na biformed figure. For two right-lines can, indeed, never comprehend\nspace. But this may be effected by a right-line and circumference.\nLikewise by two circumferences, either making angles, as in the\nlunular figure; or forming a figure without angles, as that which is\ncomprehended by concentric circles. For the middle space intercepted\nbetween both, is comprehended by two circumferences; one interior,\nbut the other exterior, and no angle is produced. For they do not\nmutually intersect, as in the lunular figure, and that which is on\nboth sides convex. But that the centre of the semi-circle is the same\nwith that of the circle, is manifest. For the diameter, containing in\nitself the centre, completes the semi-circle, and from this all lines\ndrawn to the semi-circumference are equal. For this is a part of the\ncircumference of the circle. But equal right lines proceed from the\ncentre to all parts of the circumference. The centre, therefore, of the\ncircle and semi-circle is one and the same. And it must be observed,\nthat among all figures, this alone contains the centre in its own\nperimeter, I say, among all plane figures. Hence you may collect, that\nthe centre has three places. For it is either within a figure, as in\nthe circle; or in its perimeter, as in the semi-circle; or without\nthe figure, as in certain conic lines[174]. What then is indicated by\nthe semi-circles, having the same centre with the circle, or of what\nthings does it bear an image, unless that all figures which do not\nentirely depart from such as are first, but participate them after\na manner, may be concentric with them, and participate of the same\ncauses? For the semi-circle communicates with the circle doubly, as\nwell according to the diameter, as according to the circumference.\nOn this account, they possess a centre also in common. And perhaps,\nafter the most simple principles, the semi-circle is assimilated to\nthe second co-ordinations, which participate those principles; and by\ntheir relation to them, although imperfectly, and by halves, they are,\nnevertheless, reduced to that which is, and to their first original\ncause.\n\n                            DEFINITION XX.\n\n        RECTILINEAR FIGURES are those which are comprehended by\n                            Straight Lines.\n\n                            DEFINITION XXI.\n\n          TRILATERAL FIGURES, or TRIANGLES, by three Straight\n                                Lines.\n\n                           DEFINITION XXII.\n\n                QUADRILATERAL, by four Straight Lines.\n\n                           DEFINITION XXIII.\n\n         MULTILATERAL FIGURES, or POLYGONS, by more than four\n                            Straight Lines.\n\n\nAfter the monadic figure having the relation of a principle to all\nfigures, and the biformed semi-circle, the progression of right-lined\nfigures in infinitum, according to numbers, is delivered. For on this\naccount also, mention was made of the semi-circle, as communicating\naccording to terms or boundaries; partly, indeed, with the circle,\nbut partly with right-lines: just as the duad is the medium between\nunity and number. For unity, by composition, produces more than by\nmultiplication; but number, on the contrary, is more increased by\nmultiplication than composition: and the duad, whether multiplied into,\nor compounded with itself, produces an equal quantity. As, therefore,\nthe duad is the middle of unity and number, so likewise, a semi-circle\ncommunicates, according to its base, with right-lines; but according\nto its circumference, with the circle. But right-lined figures proceed\norderly to infinity, attended by number and its bounding power,\nwhich begins from the triad. On this account, Euclid also begins\nfrom hence[175]. For he says, trilateral and quadrilateral, and the\nfollowing figures, called by the common name of multilateral: since\ntrilateral figures are also multilaterals; but they have likewise\na proper, besides a common denomination. But, as we are but little\nable to pursue the rest, on account of the infinite progression of\nnumbers, we must be content with a common denomination. But he only\nmakes mention of trilaterals and quadrilaterals, because the triad and\ntetrad are the first in the order of numbers; the former being a pure\nodd among the odd; but the latter, an entire even among even numbers.\nEuclid, therefore, assumes both in the origin of right-lined figures,\nfor the purpose of exhibiting their subsistence, according to all even\nand odd numbers. Besides, since he is about to teach concerning these\nin the first book, as especially elementary (I mean triangles and\nparallelograms) he does not undeservedly, as far as to these, establish\na proper enumeration: but he embraces all other right-lined figures by\na common name, calling them multilaterals: but of these enough. Again,\nassuming a more elevated exordium, we must say, that of plane figures,\nsome are contained by simple lines, others by such as are mixt, but\nothers again by both. And of those which are comprehended by simple\nlines, some are contained by similars in species, as right-lines; but\nothers by dissimilars in species, as semi-circles, and segments, and\napsides, which are less than semi-circles. Likewise of those which\nare contained by similars in species, some are comprehended by a\ncircular line; but others by a right-line. And of those comprehended\nby a circular line, some are contained by one, others by two, but\nothers by more than two. By one, indeed, the circle itself. But by\ntwo, some without angles, as the crowns[176] terminated by concentric\ncircles; but others angular (\u03b3\u03b5\u03b3\u03c9\u03bd\u03b9\u03c9\u03bc\u03ad\u03bd\u03b1) as the lunula. And of those\ncomprehended by more than two, there is an infinite procession. For\nthere are certain figures contained by three and four and succeeding\ncircumferences. Thus, if three circles touch each other, they will\nintercept a certain trilateral space; but if four, one terminated by\nfour circumferences, and in like manner, by a successive progression.\nBut of those contained by right lines, some are comprehended by three,\nothers by four, and others by a multitude of lines. For neither is\nspace comprehended by two right-lines, nor much more by one right-line.\nHence, every space comprehended by one boundary, or by two, is either\nmixt or circular. And it is mixt in a twofold manner, either because\nthe mixt lines comprehend it, as the space intercepted by the cissoidal\nline; or because it is contained by lines dissimilar in species,\nas the apsis: since mingling is twofold, either by apposition or\nconfusion. Every right-lined figure, therefore, is either trilateral,\nor quadrilateral, or gradually multilateral; but every trilateral, or\nquadrilateral, or multilateral figure, is not right-lined; since so\ngreat a number of sides is also produced from circumferences. And thus\nmuch concerning the division of plane figures. But we have already\nasserted[177], that rectitude of progression is both a symbol of motion\nand infinity, and that it is peculiar to the generative co-ordinations\nof the gods, and to the producers of difference, and to the authors\nof mutation and motion. Right-lined figures, therefore, are peculiar\nto these gods, who are the principles of the prolific energy of the\nwhole progression of forms. On which account, generation also, was\nprincipally adorned by these figures, and is allotted its essence from\nthese, so far as it subsists in continual motion and mutation without\nend.\n\n\n\n\n                           DEFINITION XXIV.\n\n          Of three-sided FIGURES: an EQUILATERAL TRIANGLE is\n                   that which has three equal Sides.\n\n                            DEFINITION XXV.\n\n        An ISOSCELES TRIANGLE, is that which has only two Sides\n                                equal.\n\n                           DEFINITION XXVI.\n\n          A SCALENE TRIANGLE, is that which has three unequal\n                                Sides.\n\n                           DEFINITION XXVII.\n\n           A RIGHT-ANGLED TRIANGLE is that which has a Right\n                                Angle.\n\n                          DEFINITION XXVIII.\n\n            An OBTUSE-ANGLED TRIANGLE is that which has an\n                             Obtuse Angle.\n\n                           DEFINITION XXIX.\n\n           An ACUTE-ANGLED TRIANGLE is that which has three\n                             Acute Angles.\n\n\nThe division of triangles sometimes commences from angles, but\nsometimes from sides. And that, indeed, which originates from\nsides, precedes as known; but that from angles follows as a proper\ndistribution. For these three angles alone belong to right-lined\nfigures, viz. the right, the obtuse, and the acute: but the equality\nand inequality of sides subsist also in non-rectilinear figures.\nEuclid says, therefore, that of triangles, some are equilateral,\nothers isosceles, and others scalene: for they have either all their\nsides equal, or all unequal, or only two equal. And again, that of\ntriangles some are right-angled, others obtuse-angled, and others\nacute-angled. And he defines a right-angled triangle, that which has\none right angle, as likewise an obtuse-angled triangle, that which has\none obtuse angle: for it is impossible that a triangle can have more\nthan one right, or obtuse angle[178]. But he defines an acute-angled\ntriangle, that which has all its angles acute. For here it is not\nsufficient that it should have only one acute; since, in this case,\nall triangles would be acute-angled, as every triangle has necessarily\ntwo acute angles[179]. But, to possess three acute angles, is the\nproperty of an acute-angled triangle alone. But Euclid appears to me to\nhave made a separate division into angles and sides, from considering\nthis alone, that every triangle is not also trilateral. For there are\nquadrilateral triangles, which are called by mathematicians themselves\n(\u1f00\u03ba\u03b9\u03b4\u03bf\u03b5\u03b9\u03b4\u1fc6) that is, similar to the point of a spear[180]: but by\nZenodorus (\u03ba\u03bf\u03b9\u03bb\u03bf\u03b3\u03ce\u03bd\u03b9\u03b1) that is, having an hollow angle. For on one of\nthe sides of a trilateral figure, constitute two right-lines inwardly;\nby this means a certain space will be enclosed, which is comprehended\nby external and internal right-lines, and which has three angles;\none, indeed, contained by the external lines; but two comprehended by\nthese and the internal lines, at the extremities in which these lines\nare conjoined. A figure of this kind, therefore, is a quadrilateral\ntriangle. And hence, it does not immediately follow, that because a\nfigure has three angles (whether they are all acute, or one right,\nor one obtuse), we shall find it trilateral; for it may be, perhaps,\nquadrilateral. In like manner, you may also find quadrangles having\nmore than four sides. And therefore, we must not rashly determine the\nnumber of sides from the multitude of angles. But of this enough. But\nthe Pythagoreans affirm that the triangle is simply the principle\nof generation, and of the formation of generable natures. On which\naccount, Tim\u00e6us says, that natural reasons, as well as those of the\nconstruction of the elements, are triangular. For they are distant by a\ntriple interval, are on all sides collective of partible, and variously\nmutable natures, are replete with material infinity, and bear before\nthemselves the conjunctions of material bodies, loosened and free: as,\nindeed, triangles also are comprehended by three right-lines, but they\npossess angles which collect the multitude of lines, and afford to them\nan adventitious angle and conjunction. With great propriety, therefore,\nPhilolaus has consecrated the angle of a triangle to four gods,\nSaturn, Pluto, Mars, and Bacchus, comprehending in these the whole\nquadripartite ornament of the elements descending from the heavens,\nor from the four segments of the zodiac. For Saturn constitutes an\nessence wholly humid and frigid; but Mars a nature totally fiery; and\nPluto contains the whole terrestrial life; but Bacchus governs a humid\nand hot generation; of which wine also is a symbol, for this is humid\nand hot. Hence, all these gods differ according to their operations\nin inferior concerns: but they are mutually united according to their\nproper natures. And on this account, Philolaus collects their union\naccording to one angle. But if the differences of triangles contribute\nto generation, we shall very properly confess that a triangle is\nthe principle and author of the constitution of sublunary natures.\nFor a right angle, indeed, affords them essence, and determines the\nmeasure of being; and the reason of a right-angled triangle produces\nthe essence of the elements of generable natures; but an obtuse angle\nassigns to them universal distance; and the reason of an obtuse-angled\ntriangle increases material forms in magnitude, and in mutation of\nevery kind. But an acute angle effects their divisible nature; and the\nreason of an acute-angled triangle prepares them to receive infinite\ndivision. But simply, a triangular reason constitutes the essence of\nmaterial bodies distant with interval, and on all sides divisible.\nAnd thus much should we speculate concerning the nature of triangles.\nBut from these divisions you may understand, that all the species of\ntriangles are neither more nor less than seven. For the equilateral\ntriangle is one, since it is acute-angled only; but each of the rest\nis triple. For the isosceles is either right-angled, or obtuse-angled,\nor acute-angled; and, in like manner, the scalene triangle possesses\nthis triple difference. If then, these have a triple distinction, but\nthe equilateral has but one mode of existence, all the species of\ntriangles will be seven. But again, you will understand the proportion\nof triangles to the things which _are_, according to the division\nof sides; for the equilateral, entirely excelling in equality and\nsimplicity, is allied to divine souls; since it is the measure and\nequality of things unequal, in the same manner as divinity of all\ninferior concerns. But the isosceles triangle is allied to the better\ngenera, which govern a material nature, the greater part of which\ngenera is held by the limitation of measure; but their extremes extend\nto inequality and material immoderation; for the two sides of an\nisosceles triangle are equal, but the base is unequal. But a scalene\ntriangle symbolizes with partible lives, which are on all sides lame\nand defective, which prepare themselves for generation, and are replete\nwith matter and material imperfection.\n\n\n\n\n                            DEFINITION XXX.\n\n   Of Quadrilateral Figures, a QUADRANGLE or SQUARE is that which\n   has all its Sides equal, and all its Angles Right Angles.\n\n                           DEFINITION XXXI.\n\n     An OBLONG is that which has all its Angles right Angles, but\n                     has not all its Sides equal.\n\n                           DEFINITION XXXII.\n\n       A RHOMBUS, is that which has all its Sides equal, but its\n                     Angles are not right Angles.\n\n                          DEFINITION XXXIII.\n\n   A RHOMBOID is that which has its opposite Sides equal to one\n   another, but all its Sides are not equal, nor its Angles Right\n   Angles.\n\n                           DEFINITION XXXIV.\n\n       All other Quadrilateral Figures besides these, are called\n                              TRAPEZIUMS.\n\n\nIt is requisite that the first division of quadrilateral figures should\ntake place in two numbers; and that some of them should be called\nparallelograms, but others non-parallelograms. But of parallelograms\nsome are rectangular and equilateral, as quadrangles; but others\nneither of these, as rhomboids: others again, are rectangular, but not\nequilateral, as oblongs: but others, on the contrary, are equilateral,\nbut not rectangular, as the rhombuses. For it is requisite either\nto possess both, viz. equality of sides and rectitude of angles, or\nneither; or one of these, and this in a twofold respect. Hence a\nparallelogram has a quadruple subsistence. But of non-parallelograms,\nsome have only two parallel sides, and not the rest; but others have\nnone of their sides parallel. And those are called Trapeziums, but\nthese Trapezoids. But of Trapeziums, some, indeed, have the sides\nequal, by which the parallel sides of this kind are conjoined;\nbut others unequal; and the former of these are called isosceles\ntrapeziums; but the latter scalene trapeziums. A quadrilateral figure,\ntherefore, is constituted by us according to a seven-fold distribution.\nFor one is a quadrangle; but the other an oblong; the third a rhombus;\nthe fourth a rhomboides; the fifth an isosceles trapezium; the sixth\na scalene trapezium; the seventh a trapezoid. But Possidonius makes\na perfect division of right-lined quadrilateral figures into so many\nmembers; for he establishes seven species of these; as likewise\nof triangles. But Euclid could not divide into parallelograms and\nnon parallelograms, because he neither mentions parallels, nor\nteaches us concerning the parallelogram itself. But trapeziums, and\nall trapezoids, he calls by a common name, describing trapeziums\nthemselves, according to the difference of those four figures[181], in\nwhich the property of parallelograms is verified. And this is to have\nthe opposite sides and angles equal. For a quadrangle and an oblong,\nand a rhombus, have their opposite sides and angles equal. But in a\nrhomboides he only adds this, _that its opposite sides are equal_,\nlest he should define it by negations alone, since he neither calls it\nequilateral, nor rectangular. For where we want proper appellations,\nit is necessary to use such as are common. But we should hear Euclid\nshewing that this is common to all parallelograms. But a rhombus\nappears to be a quadrangle having its sides moved, and a rhomboides\na moved oblong. Hence, according to sides, these do not differ from\nthose; but they vary only according to the obtuseness and acuteness\nof angles; since the quadrangle and the oblong are rectangular. For\nif you conceive a quadrangle or an oblong, having its sides drawn in\nsuch a manner, that while two of its opposite angles are dilated, the\nother two are contracted; then the dilated angles will appear obtuse,\nand the contracted, acute. And the appellation of rhombus[182] seems\nto have been imposed from motion. For if you conceive a quadrangle\nmoving after the manner of a rhombus, it will appear to you changed in\norder, according to its angles: just as if a circle is moved after the\nmanner of a sling, it will immediately exhibit the appearance of an\nellipsis. But here you may perhaps enquire concerning the quadrangle,\nwhy it has this denomination? and why the appellation of quadrangle\nmay not be applied to other quadrilateral figures, as the name of\ntriangle is common to all those which are neither equiangular nor\nequilateral, and in like manner of quinquangles or pentagons; for\nthe geometrician, in these, adds only the particle _an equilateral\ntriangle_, or a _quinquangle_, _which is equilateral and\nequiangular_, as if these could not be otherwise than such as they\nare? But when he mentions a quadrangle, he immediately indicates that\nit must be equilateral and rectangular. But the reason of this is as\nfollows: a quadrangle alone has the best space, both according to\nits sides and angles. For each of the latter is right, intercepting\na measure of angles, which neither receives intention nor remission.\nAs it excels, therefore, in both respects, it deservedly obtains a\ncommon appellation. But a triangle, though it may have equal sides,\nyet will in this case have all its angles acute, and a quinquangle all\nits angles obtuse. Since, therefore, of all quadrilateral figures,\na quadrangle alone is replete with equality of sides, and rectitude\nof angles, it was not undeservedly allotted this appellation: for,\nto excellent forms, we often dedicate the name of the whole. But it\nappeared also to the Pythagoreans, that this property of quadrilateral\nfigures, principally conveyed an image of a divine essence. For they\nparticularly signified by this, a pure and immaculate order. Since\nrectitude imitates inflexibility, but equality a firm and permanent\npower: for motion emanates from inequality, but quiet from equality\nitself. The gods, therefore, who are the authors to all things\nof stable disposition, of pure and uncontaminated order, and of\nindeclinable power, are deservedly manifested as from an image, by a\nquadrangular figure. But, besides these, Philolaus also, according to\nanother apprehension, calls a quadrangular angle, the angle of Rhea,\nCeres and Vesta. For, since a quadrangle constitutes the earth, and is\nits proximate element, as we learn from Tim\u00e6us, but the earth herself\nreceives from all these divinities, genital seeds, and prolific powers,\nhe does not unjustly consecrate the angle of a quadrangle to these\ngoddesses, the bestowers of life. For some call both the earth and\nCeres, Vesta[183], and they say that Rhea totally participates her\nnature, and that all generative causes are contained in her essence.\nPhilolaus, therefore, says that a quadrangular angle comprehends, by\na certain terrestrial power, one union of the divine genera. But some\nassimilate a quadrangle to universal virtue, so far as every quadrangle\nfrom its perfection has four right angles. Just as we say that each of\nthe virtues is perfect, content with itself, the measure and bound of\nlife, and the middle of every thing which, in morals, corresponds to\nthe obtuse and acute. But it is by no means proper to conceal, that\nPhilolaus attributes a triangular angle to four, but a quadrangular\nangle to three gods, exhibiting their alternate transition, and the\ncommunity of all things in all, of odd natures in the even, and of even\nin the odd. Hence, the tetradic ternary, and the triadic quaternary,\nparticipating of prolific and efficacious goods, contain the whole\nornament of generable natures, and preserve them in their proper\nstate. From which the duodenary, or the number twelve, is excited\nto a singular unity, viz. the government of Jupiter. For Philolaus\nsays, that the angle of a dodecagon (or twelve-sided figure) belongs\nto Jove, so far as Jupiter contains and preserves, by his singular\nunion, the whole number of the duodenary. For also, according to Plato,\nJupiter presides over the duodenary[184], and governs and moderates\nthe universe with absolute sway. And thus much we have thought proper\nto discourse concerning quadrilateral figures, as well declaring the\nsense of our author, as likewise affording an occasion of more profound\ninspections to such as desire the knowledge of intelligible and occult\nessences.\n\n\n\n\n                           DEFINITION XXXV.\n\n   PARALLEL RIGHT LINES are such as being in the same Plane, and\n   produced both ways infinitely, will in no part mutually coincide.\n\n\nWhat the elements of parallels are, and by what accidents in these\nthey may be known, we shall afterwards learn: but what parallel right\nlines are, he defines in these words: \u201cIt is requisite, therefore (says\nhe), that they should be in one plane, and while they are produced\nboth ways have no coincidence, but be extended in infinitum.\u201d For\nnon-parallel lines also, if they are produced to a certain distance,\nwill not coincide. But to be produced infinitely, without coincidence,\nexpresses the property of parallels. Nor yet this absolutely, but to be\nextended both ways infinitely, and not coincide. For it is possible\nthat non-parallel lines may also be produced one way infinitely, but\nnot the other; since, verging in this part, they are far distant from\nmutual coincidence in the other. But the reason of this is, because two\nright-lines cannot comprehend space; for if they verge to each other\nboth ways, this cannot happen. Besides this, he very properly considers\nthe right-lines as subsisting in the same plane. For if the one should\nbe in a subject plane, but the other in one elevated, they will not\nmutually coincide according to every position, yet they are not on this\naccount parallel. The plane, therefore, should be one, and they should\nbe produced both ways infinitely, and not coincide in either part. For\nwith these conditions, the right-lines will be parallel. And agreeable\nto this, Euclid defines parallel right-lines. But Posidonius says,\nparallel lines are such as neither incline nor diverge in one plane;\nbut have all the perpendiculars equal which are drawn from the points\nof the one to the other. But such lines as make their perpendiculars\nalways greater and less, will some time or other coincide, because\nthey mutually verge to each other. For a perpendicular is capable of\nbounding the altitudes of spaces, and the distances of lines. On which\naccount, when the perpendiculars are equal, the distances of the right\nlines are also equal; but when they are greater and less, the distance\nalso becomes greater and less, and they mutually verge in those parts,\nin which the lesser perpendiculars are found. But it is requisite to\nknow, that non-coincidence does not entirely form parallel lines. For\nthe circumferences of concentric circles do not coincide: but it is\nlikewise requisite that they should be infinitely produced. But this\nproperty is not only inherent in right, but also in other lines: for\nit is possible to conceive spirals described in order about right\nlines, which if produced infinitely together with the right lines,\nwill never coincide[185]. Geminus, therefore, makes a very proper\ndivision in this place, affirming from the beginning, that of lines\nsome are bounded, and contain figure, as the circle and ellipsis,\nlikewise the cissoid, and many others; but others are indeterminate,\nwhich may be produced infinitely, as the right-line, and the section\nof a right-angled, and obtuse angled cone; likewise the conchoid\nitself. But again, of those which may be produced in infinitum, some\ncomprehend no figure, as the right-line and the conic sections; but\nothers, returning into themselves, and forming figure, may afterwards\nbe infinitely produced. And of these some will not hereafter coincide,\nwhich resist coincidence, how far soever they may be produced; but\nothers are coincident, which will some time or other coincide. But\nof non-coincident lines, some are mutually in one plane; and others\nnot. And of non-coincidents subsisting in one plane, some are always\nmutually distant by an equal interval; but others always diminish the\ninterval, as an hyperbola in its inclination to a right-line, and\nlikewise the conchoid[186]. For these, though they always diminish\nthe interval, never coincide. And they mutually converge, indeed,\nbut never perfectly nod to each other; which is indeed a theorem in\ngeometry especially admirable, exhibiting certain lines endued with a\nnon-assenting nod. But the right-lines, which are always distant by an\nequal interval, and which never diminish the space placed between them\nin one plane, are parallel lines. And thus much we have extracted from\nthe studies of the elegant Geminus, for the purpose of explaining the\npresent definition.\n\n\n                       END OF THE FIRST VOLUME.\n\n\nFOOTNOTES:\n\n[1] The Grecian literature of this writer will now prove of real\nutility; and the graces and the sublimities of PLATO will\nsoon be familiarised to the English reader, by a hand that I am\npersuaded will not appear inferior to his great original. Let me also\nbe permitted to recommend his version of PLOTINUS on THE\nBEAUTIFUL.\n\n[2] i.e. Capable of parts.\n\n[3] i.e. Not capable of parts.\n\n[4] Dr. Young, in his Night Thoughts.\n\n[5] See book the second, of Aristotle\u2019s Metaphysics.\n\n[6] Ennead vi. lib. vii.\n\n[7] In his commentary on the 2d, 12th, and 13th books of Aristotle\u2019s\nMetaphysics, page 60. A Latin translation only of this invaluable work\nis extant; but I have fortunately a copy in my possession, with the\nversion every where corrected by the learned Thomas Gale, and with\nlarge extracts from the Greek.\n\n[8] See Proclus on Plato\u2019s Theology, p. 226.\n\n[9] Ennead vi. lib. 6.\n\n[10] In giving monadic number a subsistence in opinion, I have followed\nthe distribution of Proclus, in the conclusion of his comment on a\npoint; and, I think, not without sufficient reason. For since monadic\nnumbers are more immaterial than geometrical lines and figures, they\nmust have a more immaterial subsistence. But as they are correspondent\nto matter, they cannot reside in the essential reasons of the soul;\nnor can they subsist in the phantasy, because they are superior to\ngeometrical figures. It remains, therefore, that we must place them\nbetween \u03b4\u03b9\u03ac\u03bd\u03bf\u03b9\u03b1 or cogitation, and the phantasy; and this middle\nsituation is that of opinion. For cogitation, which Plato defines,\nin his Sophista, to be an inward discourse, without voice, is an\nenergy of the rational soul, extending itself from propositions to\nconclusions. And, according to Plato, in the same place, opinion is\nthe silent affirmation, or negation of \u03b4\u03b9\u03ac\u03bd\u03bf\u03b9\u03b1, or thought. Hence,\nsays he, \u201copinion is the conclusion of cogitation; but imagination,\nthe mutual mixture of sense and opinion.\u201d So that opinion may, with\ngreat propriety, be said to contain monadic number, to which it bears\nthe proportion of matter. And hence the reason is obvious, why the\nPythagoreans called the duad opinion.\n\n[11]\n\n     \u1f0c\u03c4\u03c1\u03bf\u03c0\u03bf\u03bd, \u1f00\u03ba\u03b1\u03bc\u03ac\u03c4\u03bf\u03bd \u0394\u03b5\u03ba\u03ac\u03b4\u03b1 \u03ba\u03bb\u03b5\u03af\u03bf\u03c5\u03c3\u03b9\u03bd \u03bc\u03b9\u03bd \u1f01\u03b3\u03b9\u1f74\u03bd,\n    \u1f08\u03b8\u03ac\u03bd\u03b1\u03c4\u03bf\u03af \u03c4\u03b5 \u03b8\u03b5\u03bf\u1f76 \u03ba\u03b1\u1f76 \u03b3\u03b7\u03b3\u03b5\u03bd\u03ad\u03b5\u03b9\u03c2 \u1f03\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03b9.\n\n    Syrian. in Meta. Aristot. p. 113. Gr.\n\ni.e. (According to the Pythagoreans) \u201cthe immortal gods and earth-born\nmen, call the venerable decad, immutable and unwearied.\u201d\n\n[12] \u0391\u03c5\u03c4\u1f78\u03c2 \u03bc\u1f72\u03bd \u03a0\u03c5\u03b8\u03b1\u03b3\u03cc\u03c1\u03b1\u03c2 \u1f10\u03bd \u03c4\u1ff7 \u1f31\u03b5\u03c1\u1ff7 \u03bb\u03cc\u03b3\u1ff3 \u03b4\u03b9\u03b1\u1fe4\u1fe5\u03b7\u03b4\u03b7\u03bd \u03bc\u03bf\u03c1\u03c6\u1ff6\u03bd \u03ba\u03b1\u1f76 \u1f30\u03b4\u03b5\u1ff6\u03bd\n\u03ba\u03c1\u03ac\u03bd\u03c4\u03bf\u03c1\u03b1 \u03c4\u1f78\u03bd \u1f00\u03c1\u03b9\u03b8\u03bc\u03cc\u03bd \u1f14\u03bb\u03b5\u03b3\u03b5\u03bd \u03b5\u1f36\u03bd\u03b1\u03b9.\n\n    Vid. Syrian. in Arist. Meta. p. 85. Gr.\n\n\n[13] \u03a6\u03b9\u03bb\u03cc\u03bb\u03b1\u03bf\u03c2 \u03b4\u03ad, \u03c4\u1fc6\u03c2 \u03c4\u1ff6\u03bd \u03ba\u03bf\u03c3\u03bc\u03b9\u03ba\u1f7c\u03bd \u03b1\u1f30\u03c9\u03bd\u03af\u03b1\u03c2 \u03b4\u03b9\u03b1\u03bc\u03bf\u03bd\u1fc6\u03c2 \u03c4\u1f74\u03bd \u03ba\u03c1\u03b1\u03c4\u03b9\u03c3\u03c4\u03b5\u03cd\u03bf\u03c5\u03c3\u03b1\u03bd\n\u03ba\u03b1\u1f76 \u03b1\u1f50\u03c4\u03bf\u03b3\u03b5\u03b9\u1fc6 \u03c3\u03c5\u03bd\u03bf\u03c7\u1f74\u03bd \u03b5\u1f36\u03bd\u03b1\u03b9 \u1f00\u03c0\u03b5\u03c6\u03ae\u03bd\u03b1\u03c4\u03bf \u03c4\u1f78\u03bd \u1f00\u03c1\u03b9\u03b8\u03bc\u03cc\u03bd.\n\n    Syrian. in eodem loco.\n\n\n[14] \u039f\u1f31 \u03b4\u1f72 \u03c0\u03b5\u03c1\u1f76 \u1f3c\u03c0\u03c0\u03b1\u03c3\u03bf\u03bd \u1f00\u03ba\u03bf\u03c5\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03bf\u1f76, \u1f00\u03c1\u03b9\u03b8\u03bc\u03cc\u03bd \u03b5\u1f36\u03c0\u03bf\u03bd \u03c0\u03b1\u03c1\u03ac\u03b4\u03b5\u03b9\u03b3\u03bc\u03b1 \u03c0\u03c1\u1ff6\u03c4\u03bf\u03bd\n\u03ba\u03bf\u03c3\u03bc\u03bf\u03c0\u03bf\u03b9\u03af\u03b1\u03c2. \u039a\u03b1\u1f76 \u03c0\u03ac\u03bb\u03b9\u03bd \u03ba\u03c1\u03b9\u03c4\u03b9\u03ba\u1f78\u03bd \u03ba\u03bf\u03c3\u03bc\u03bf\u03c5\u03c1\u03b3\u03bf\u1fe6 \u03b8\u03b5\u03bf\u1fe6 \u1f44\u03c1\u03b3\u03b1\u03bd\u03bf\u03bd.\n\n    Jamb. in Nicomach. Arith. p. 11.\n\n\n[15] In his Mathematical Lectures, page 48.\n\n[16] In Arithmet. p. 23.\n\n[17] In Aristot. Meta. p. 113. Gr. vel 59. b. Lat.\n\n[18] For the tetrad contains all numbers within its nature, in the\nmanner of an exemplar; and hence it is, that in monadic numbers, 1, 2,\n3, 4, are equal to ten.\n\n[19] Notes to Letters on Mind, page 83.\n\n[20] This bright light is no other than that of ideas themselves;\nwhich, when it is once enkindled, or rather re-kindled in the soul,\nbecomes the general standard, and criterion of truth. He who possesses\nthis, is no longer the slave of opinion; puzzled with doubts, and lost\nin the uncertainties of conjecture. Here the fountain of evidence is\nalone to be found.--This is the true light, whose splendors can alone\ndispel the darkness of ignorance, and procure for the soul undecaying\ngood, and substantial felicity. Of this I am certain, from my own\nexperience; and happy is he who acquires this invaluable treasure. But\nlet the reader beware of mixing the extravagancies of modern enthusiasm\nwith this exalted illumination. For this light is alone brought into\nthe mind by science, patient reflection, and unwearied meditation:\nit is not produced by any violent agitation of spirits, or extasy of\nimagination; for it is far superior to the energies of these: but it is\ntranquil and steady, intellectual and divine. Avicenna, the Arabian,\nwas well acquainted with this light, as is evident from the beautiful\ndescription he gives of it, in the elegant introduction of Ebn Tophail,\nto the Life of Hai Ebn Yokdhan. \u201cWhen a man\u2019s desires (says he) are\nconsiderably elevated, and he is competently well exercised in these\nspeculations, there will appear to him some small glimmerings of the\ntruth, as it were flashes of lightning, very delightful, which just\nshine upon him, and then become extinct. Then the more he exercises\nhimself, the oftener will he perceive them, till at last he will\nbecome so well acquainted with them, that they will occur to him\nspontaneously, without any exercise at all; and then as soon as he\nperceives any thing, he applies himself to the divine essence, so as to\nretain some impression of it; then something occurs to him on a sudden,\nwhereby he begins to discern the _truth_ in every thing; till\nthrough frequent exercise he at last attains to a perfect tranquillity;\nand that which used to appear to him only by fits and starts, becomes\nhabitual, and that which was only a glimmering before, a constant\nlight; and he obtains a constant and steady knowledge.\u201d He who desires\nto know more concerning this, and a still brighter light, that arising\nfrom an union with the supreme, must consult the eighth book of\nPlotinus\u2019 fifth Ennead, and the 7th and 9th of the sixth, and his book\non the Beautiful, of which I have published a translation.\n\n[21] Lest the superficial reader should think this is nothing more than\ndeclamation, let him attend to the following argument. If the soul\npossesses another eye different from that of sense (and that she does\nso, the sciences sufficiently evince), there must be, in the nature\nof things, species accommodated to her perception, different from\nfeasible forms. For if our intellect speculates things which have no\nreal subsistence, such as Mr. Locke\u2019s ideas, its condition must be much\nmore unhappy than that of the sensitive eye, since this is co-ordinated\nto beings; but intellect would speculate nothing but illusions. Now, if\nthis be absurd, and if we possess an intellectual eye, which is endued\nwith a visive power, there must be forms correspondent and conjoined\nwith its vision; forms immoveable, indeed, by a corporeal motion, but\nmoved by an intellectual energy.\n\n[22] The present section contains an illustration of almost all the\nfirst book of Aristotle\u2019s last Analytics. I have for the most part\nfollowed the accurate and elegant paraphrase of Themistius, in the\nexecution of this design, as the learned reader will perceive: but I\nhave likewise everywhere added elucidations of my own, and endeavoured\nto render this valuable work intelligible to the thinking mathematical\nreader.\n\n[23] See the twenty-eighth proposition of the first book of Euclid\u2019s\nElements.\n\n[24] We are informed by Simplicius, in his Commentary on Aristotle\u2019s\nthird Category of Relation, \u201cthat though the quadrature of the\ncircle seems to have been unknown to Aristotle, yet, according to\nJamblichus, it was known to the Pythagoreans, as appears from the\nsayings and demonstrations of Sextus Pythagoricus, who received (says\nhe) by succession, the art of demonstration; and after him Archimedes\nsucceeded, who invented the quadrature by a line, which is called\nthe line of Nicomedes. Likewise, Nicomedes attempted to square the\ncircle by a line, which is properly called \u03c4\u03b5\u03c4\u03b1\u03c1\u03c4\u03b7\u03bc\u03cc\u03c1\u03b9\u03bf\u03bd, or _the\nquadrature_. And Apollonius, by a certain line, which he calls the\nsister of the curve line, similar to a cockle, or tortoise, and which\nis the same with the _quadratix_ of Nicomedes. Also Carpus wished\nto square the circle, by a certain line, which he calls simply formed\nfrom a twofold motion. And many others, according to Jamblichus, have\naccomplished this undertaking in various ways.\u201d Thus far Simplicius.\nIn like manner, Boethius, in his Commentary on the same part of\nAristotle\u2019s Categories (p. 166.) observes, that the quadrature of\nthe circle was not discovered in Aristotle\u2019s time, but was found out\nafterwards; the demonstration of which (says he) because it is long,\nmust be omitted in this place. From hence it seems very probable,\nthat the ancient mathematicians applied themselves solely to squaring\nthe circle geometrically, without attempting to accomplish this by an\narithmetical calculation. Indeed, nothing can be more ungeometrical\nthan to expect, that if ever the circle be squared, the square to which\nit is equal must be commensurable with other known rectilineal spaces;\nfor those who are skilled in geometry know that many lines and spaces\nmay be exhibited with the greatest accuracy, geometrically, though they\nare incapable of being expressed arithmetically, without an infinite\nseries. Agreeable to this, Tacquet well observes (in lib. ii. Geom.\nPract. p. 87.) \u201cDenique admonendi hic sunt, qui geometri\u00e6, non satis\nperiti, sibi persuadent ad quadraturam necessarium esse, ut ratio line\u00e6\ncircularis ad rectam, aut circuli ad quadratum in numeris exhibeatur.\nIs sane error valde crassus est, et indignus geometr\u00e2, quamvis enim\nirrationalis esset ea proportio, modo in rectis lineis exhibeatur,\nreperta erat quadratura.\u201d And that this quadrature is possible\ngeometrically, was not only the opinion of the above mentioned learned\nand acute geometrician, but likewise of Wallis and Barrow; as may be\nseen in the Mechanics of the former, p. 517 and in the Mathematical\nLectures of the latter, p. 194. But the following discovery will, I\nhope, convince the liberal geometrical reader, that the quadrature of\nthe circle may be obtained by means of a circle and right-line only,\nwhich we have no method of accomplishing by any invention of the\nancients or moderns. At least this method, if known to the ancients, is\nnow lost, and though it has been attempted by many of the moderns, it\nhas not been attended with success.\n\nIn the circle _g o e f_, let _g o_ be the quadrantal arch, and the\nright-line _g x_ its tangent. Then conceive that the central point\n_a_ flows uniformly along the radius _a e_, infinitely produced; and\nthat it is endued with an uniform impulsive power. Let it likewise be\nsupposed, that during its flux, radii emanate from it on all sides,\nwhich enlarge themselves in proportion to the distance of the point\n_a_ from its first situation. This being admitted, conceive that the\npoint _a_ by its impulsive power, through the radii _a n_, _a m_, &c.\nacting every where equally on the arch _g o_, impells it into its equal\ntangent arch _g r_. And when, by its uniform motion along the infinite\nline _a_ \u03c6, it has at the same time arrived at _b_, the centre of the\narch _g r_, let it impel in a similar manner the arch _g r_, into its\nequal tangent arch _g s_, by acting every where equally through radii\nequal to _b r_. Now, if this be conceived to take place infinitely\n(since a circular line is capable of infinite remission) the arch _g\no_ will at length be unbent into its equal, the tangent line _g x_;\nand the extreme point _o_, will describe by such a motion of unbending\na circular line _o x_. For since the same cause, acting every where\nsimilarly and equally, produces every where similar and equal effects;\nand the arch _g o_, is every where equally remitted or unbent, it\nwill describe a line similar in every part. Now, on account of the\nsimplicity of the impulsive motion, such a line must either be straight\nor circular; for there are only three lines every where similar, i. e.\nthe right and circular line, and the cylindric helix; but this last,\nas Proclus well observes in his following Commentary on the fourth\ndefinition, is not a simple line, because it is generated by two simple\nmotions, the rectilineal and circular. But the line which bounds more\nthan two equal tangent arches cannot be a right line, as is well known\nto all geometricians; it is therefore a circular line. It is likewise\nevident, that this arch _o x_ is concave towards the point _g_: for if\nnot, it would pass beyond the chord _o x_, which is absurd. And again,\nno arch greater than the quadrant can be unbent by this motion: for any\none of the radii, as _a p_ beyond _g o_, has a tendency from, and not\nto the tangent _g x_, which last is necessary to our hypothesis. Now if\nwe conceive another quadrantal arch of the circle _g o e f_, that is\n_g y_, touching the former in _g_ to be unbent in the same manner, the\narch _x y_ shall be a continuation of the arch _x o_; for if _\u03b3 x \u03ba_ be\ndrawn perpendicular to _x g_, as in the figure, it shall be a tangent\nin _x_ to the equal arches _y x_, _x o_; because it cannot fall within\neither, without making the sine of some one of the equal arches, equal\nto the right-line _x g_, which would be absurd. And hence we may easily\ninfer, that the centre of the arch _y x o_, is in the tangent line _x\ng_. Hence too, we have an easy method of finding a tangent right-line\nequal to a quadrantal arch: for having the points _y_, _o_ given, it is\neasy to find a third point, as _s_; and then the circle passing through\nthe three points _o_, _s_, _y_, shall cut off the tangent _x g_,\nequal to the quadrantal arch _g o_. And the point _s_ may be speedily\nobtained, by describing the arch _g s_ with a radius, having to the\nradius _a g_ the proportion of 6 to 4; for then _g s_ is the sixth part\nof its whole circle, and is equal to the arch _g o_. And thus, from\nthis hypothesis, which, I presume, may be as readily admitted as the\nincrements and decrements of lines in fluxions, the quadrature of the\ncircle may be geometrically obtained; for this is easily found, when a\nright-line is discovered equal to the periphery of a circle. I am well\naware the algebraists will consider it as useless, because it cannot be\naccommodated to the farrago of an arithmetical calculation; but I hope\nthe lovers of the ancient geometry will deem it deserving an accurate\ninvestigation; and if they can find no paralogism in the reasoning,\nwill consider it as a legitimate demonstration.\n\n  [Illustration]\n\n  [Illustration]\n\n[25] Axioms have a subsistence prior to that of magnitudes and\nmathematical numbers, but subordinate to that of ideas; or, in other\nwords, they have a middle situation between essential and mathematical\nmagnitude. For of the reasons subsisting in soul, some are more simple\nand universal, and have a greater ambit than others, and on this\naccount approach nearer to intellect, and are more manifest and known\nthan such as are more particular. But others are destitute of all\nthese, and receive their completion from more ancient reasons. Hence it\nis necessary (since conceptions are then true, when they are consonant\nwith things themselves) that there should be some reason, in which the\naxiom asserting, _if from equals you take away equals_, _&c._\nis primarily inherent; and which is neither the reason of magnitude,\nnor number, nor time, but contains all these, and every thing in which\nthis axiom is naturally inherent. Vide Syrian. in Arith. Meta. p. 48.\n\n[26] Geometry, indeed, wishes to speculate the impartible reasons\nof the soul, but since she cannot use intellections destitute of\nimagination, she extends her discourses to imaginative forms, and\nto figures endued with dimension, and by this means speculates\nimmaterial reasons in these; and when imagination is not sufficient\nfor this purpose, she proceeds even to external matter, in which she\ndescribes the fair variety of her propositions. But, indeed, even then\nthe principal design of geometry is not to apprehend sensible and\nexternal form, but that interior vital one, resident in the mirror of\nimagination, which the exterior inanimate form imitates, as far as its\nimperfect nature will admit. Nor yet is it her principal design to\nbe conversant with the imaginative form; but when, on account of the\nimbecility of her intellection, she cannot receive a form destitute of\nimagination, she speculates the immaterial reason in the purer form of\nthe phantasy; so that her principal employment is about universal and\nimmaterial forms. Syrian. in Arist. Meta. p. 49.\n\n[27] Syrianus, in his excellent Commentary on Aristotle\u2019s Metaphysics,\n(which does not so much explain Aristotle, as defend the doctrine of\nideas, according to Plato, from the apparent if not real opposition of\nAristotle to their existence), informs us that it is the business of\nwisdom, properly so called, to consider immaterial forms or essences,\nand their essential accidents. By the method of resolution receiving\nthe principles of being; by a divisive and and definitive method,\nconsidering the essences of all things; but by a demonstrative process,\nconcluding concerning the essential properties which substances\ncontain. Hence (says he) because intelligible essences are of the\nmost simple nature, they are neither capable of definition nor\ndemonstration, but are perceived by a simple vision and energy of\nintellect alone. But middle essences, which are demonstrable, exist\naccording to their inherent properties: since, in the most simple\nbeings, nothing is inherent besides their being. On which account we\ncannot say that _this_ is their essence, and _that_ something\nelse; and hence they are better than definition and demonstration.\nBut in universal reasons, considered by themselves, and adorning a\nsensible nature, essential accidents supervene; and hence demonstration\nis conversant with these. But in material species, individuals, and\nsensibles, such things as are properly accidents are perceived by the\nimagination, and are present and absent without the corruption of their\nsubjects. And these again being worse than demonstrable accidents, are\napprehended by signs, not indeed by a wise man, considered as wise, but\nperhaps by physicians, natural philosophers, and all of this kind.\n\n[28] See Note to Chap. i. Book i. of the ensuing Commentaries.\n\n[29] Page 227.\n\n[30] Page 250.\n\n[31] Methodus h\u00e6c cum algebr\u00e2 specios\u00e2 facilitate contendit, evidenti\u00e2\nvero et demonstrationum eleganti\u00e2 eam longe superare videtur: ut abunde\nconstabit, si quis conferat hanc Apollonii doctrinam _de Sectione\nRationis_ cum ejusdem Problematis Analysi Algebraic\u00e2, quam exhibuit\nclarissimus Wallisius, tom. ii. Operum Math. cap. liv. p. 220.\n\n[32] Verum perpendendum est, aliud esse problema aliqualiter resolutum\ndare, quod modis variis, plerumque fieri potest, aliud methodo\nelegantissim\u00e2 ipsum efficere; Analysi brevissim\u00e2 et simul perspicu\u00e2,\nSynthesi concinn\u00e2 et minime operos\u00e2.\n\n[33] In his Mathematical Lectures, p. 44.\n\n[34] Lib. iv.\n\n[35] Lib. i. p. 30.\n\n[36] In The\u00e6teto.\n\n[37] In his most excellent work on Abstinence, lib. i. p. 22, &c.\n\n[38] See the Excerpta of Ficinus from Proclus, on the first Alcibiades\nof Plato; his Latin version only of which is extant. Ficini Opera, tom.\nii.\n\n[39] Marinus, the author of the ensuing life, was the disciple of\nProclus; and his successor in the Athenian school. His philosophical\nwritings were not very numerous, and have not been preserved. A\ncommentary ascribed to him, on Euclid\u2019s data, is still extant; but\nhis most celebrated work, appears to have been, the present life of\nhis master. It is indeed in the original elegant and concise; and may\nbe considered as a very happy specimen of philosophical biography.\nEvery liberal mind must be charmed and elevated with the grandeur and\nsublimity of character, with which Proclus is presented to our view. If\ncompared with modern philosophical heroes, he appears to be a being of\na superior order; and we look back with regret on the glorious period,\nso well calculated for the growth of the philosophical genius, and the\nencouragement of exalted merit. We find in his life, no traces of the\ncommon frailties of depraved humanity; no instances of meanness, or\ninstability of conduct: but he is uniformly magnificent, and constantly\ngood. I am well aware that this account of him will be considered by\nmany as highly exaggerated; as the result of weak enthusiasm, blind\nsuperstition, or gross deception: but this will never be the persuasion\nof those, who know by experience what elevation of mind and purity of\nlife the Platonic philosophy is capable of procuring; and who truly\nunderstand the divine truths contained in his works. And the testimony\nof the multitude, who measure the merit of other men\u2019s characters by\nthe baseness of their own, is surely not to be regarded. I only add,\nthat our Philosopher flourished 412 years after Christ, according to\nthe accurate chronology of Fabricius; and I would recommend those who\ndesire a variety of critical information concerning Proclus, to the\nProlegomena prefixed by that most learned man to his excellent Greek\nand Latin edition of this work, printed at London in 1703.\n\n[40] Plato in Ph\u00e6dro. Meminit et Plutarch. VIII. Sympos. Suidas in\n\u03bc\u03ae\u03c4\u03bf\u03b9. Fabricius.\n\n[41] For a full account of the distribution of the virtues according to\nthe Platonists, consult the sentences of Porphyry, and the Prolegomena\nof Fabricius to this work.\n\n[42] See the sixth book of his Republic, and the Epinomis.\n\n[43] We are informed by Fabricius, that the Platonic Olympiodorus in\nhis MS. Commentary on the Alcibiades of Plato, divides the orders of\nthe Gods, into \u1f51\u03c0\u03b5\u03c1\u03ba\u03cc\u03c3\u03bc\u03b9\u03bf\u03b9, or super-mundane, which are separate from\nall connection with body; and into \u1f10\u03b3\u03ba\u03cc\u03c3\u03bc\u03b9\u03bf\u03b9, or mundane. And that\nof these, some are \u03bf\u1f50\u03c1\u03ac\u03bd\u03b9\u03bf\u03b9, or celestial, others \u03b1\u1f30\u03b8\u03ad\u03c1\u03b9\u03bf\u03b9, or, or\netherial, or \u03c0\u03cd\u03c1\u03b9\u03bf\u03b9, fiery, others \u1f00\u03ad\u03c1\u03b9\u03bf\u03b9, or aerial, others \u1f14\u03bd\u03c5\u03b4\u03c1\u03bf\u03b9,\nor watry, others \u03c7\u03b8\u03cc\u03bd\u03b9\u03bf\u03b9, or earthly; and others \u1f51\u03c0\u03bf\u03c4\u03b1\u03c1\u03c4\u03ac\u03c1\u03b9\u03bf\u03b9, or\nsubterranean. But among the terrestrial, some are \u03ba\u03bb\u03b9\u03bc\u03b1\u03c4\u03ac\u03c1\u03c7\u03b1\u03b9, or\ngovernors of climates, others \u03c0\u03bf\u03bb\u03b9\u03bf\u1fe6\u03c7\u03bf\u03b9, or rulers over cities, and\nothers lastly \u03ba\u03b1\u03c4\u03bf\u03b9\u03ba\u03af\u03b4\u03b9\u03bf\u03b9, or governors of houses.\n\n[44] This epithet is likewise ascribed by Onomacritus to the Moon, as\nmay be seen in his hymn to that deity; and the reason of which we have\ngiven in our notes to that hymn.\n\n[45] Divine visions, and extraordinary circumstances, may be fairly\nallowed to happen to such exalted geniuses as Proclus; but deserve\nridicule when ascribed to the vulgar.\n\n[46] What glorious times! when it was considered as an extraordinary\ncircumstance for a teacher of rhetoric to treat a noble and wealthy\npupil as his domestic. When we compare them with the present, we can\nonly exclaim, _O tempora! O mores!_ Philosophy sunk in the ruins\nof ancient Greece and Rome.\n\n[47] Fabricius rightly observes, that this Olympiodorus is not the\nsame with the Philosopher of that name, whose learned commentaries, on\ncertain books of Plato, are extant in manuscript, in various libraries.\nAs in these, not only Proclus himself, but Damascius, who flourished\nlong after Proclus, is celebrated.\n\n[48] Concerning the various mathematicians of this name, see Fabricius\nin Bibliotheca Gr\u00e6ca.\n\n[49] The word in the original is \u03bb\u03bf\u03b3\u03b9\u03ba\u1f70, which Fabricius rightly\nconjectures has in this place a more extensive signification than\neither Logic, or Rhetoric: but I must beg leave to differ from that\ngreat critic, in not translating it simply _philosophical_, as I\nshould rather imagine, Marinus intended to confine it to that part of\nAristotle\u2019s works, which comprehends only logic, rhetoric, and poetry.\nFor the verb \u1f10\u03be\u03b5\u03bc\u03ac\u03bd\u03b8\u03b1\u03bd\u03c9, or _to learn_, which Marinus uses on\nthis occasion, cannot with propriety be applied to the more abstruse\nwritings of Aristotle.\n\n[50] Hence Proclus was called, by way of eminence, \u03b4\u03b9\u03ac\u03b4\u03bf\u03c7\u03bf\u03c2 \u03a0\u03bb\u03b1\u03c4\u03c9\u03bd\u03b9\u03ba\u03cc\u03c2,\nor the Platonic Successor.\n\n[51] Concerning Polletes, see Suidas; and for Melampodes, consult\nFabricius in Bibliotheca Gr\u00e6ca.\n\n[52] This Syrianus was indeed a most excellent philosopher, as we may\nbe convinced from his commentary on the metaphysics of Aristotle,\na Latin translation only of which, by one Hieronimus Bagolinus,\nwas published at Venice in 1558. The Greek is extant, according to\nFabricius, in many of the Italian libraries, and in the Johannean\nlibrary at Hamburg. According to Suidas, he writ a commentary on the\nwhole of Homer in six books; on Plato\u2019s politics, in four books; and\non the consent of Orpheus, Pythagoras, and Plato, with the Chaldean\nOracles, in ten books. All these are unfortunately lost; and the\nliberal few, are by this means deprived of treasures of wisdom, which\nanother philosophical age, in some distant revolution, is alone likely\nto produce.\n\n[53] Socrates, in the 6th book of Plato\u2019s Republic, says, that from\ngreat geniuses nothing of a middle kind must be expected; but either\ngreat good, or great evil.\n\n[54] The reader will please to take notice, that this great man is not\nthe same with Plutarch the biographer, whose works are so well known;\nbut an Athenian philosopher of a much later period.\n\n[55] Aristotle\u2019s philosophy, when compared with the discipline of\nPlato is, I think, deservedly considered in this place as bearing the\nrelation of the proteleia to the epopteia in sacred mysteries. Now the\nproteleia, or things previous to perfection, belong to the initiated,\nand the mystics; the former of whom were introduced into some lighter\nceremonies only: but the mystics, were permitted to be present with\ncertain preliminary and lesser sacred concerns. On the other hand, the\nepopt\u00e6 were admitted into the sanctuary of the greater sacred rites;\nand became spectators of the symbols, and more interior ceremonies.\nAristotle indeed appears to be every where an enemy to the doctrine of\nideas, as understood by Plato; though they are doubtless the leading\nstars of all true philosophy. However, the great excellence of his\nworks, considered as an introduction to the divine theology of Plato,\ndeserves the most unbounded commendation. Agreeable to this, Damascius\ninforms us that Isidorus the philosopher, \u201cwhen he applied himself to\nthe more holy philosophy of Aristotle, and saw that he trusted more to\nnecessary reasons than to his own proper sense, yet did not entirely\nemploy a divine intellection, was but little solicitous about his\ndoctrine: but that when he had tasted of Plato\u2019s conceptions, he no\nlonger deigned to behold him in the language of Pindar. But hoping he\nshould obtain his desired end, if he could penetrate into the sanctuary\nof Plato\u2019s mind, he directed to this purpose the whole course of his\napplication.\u201d Photii Bibliotheca. p. 1034.\n\n[56] _according to the oracle._\n\n[57] Nothing is more celebrated by the ancients than that strict\nfriendship which subsisted among the Pythagoreans; to the exercise of\nwhich they were accustomed to admonish each other, _not to divide\nthe god which they contained_, as Jamblichus relates, lib. i. c.\n33. De Vita Pythagor\u00e6. Indeed, true friendship can alone subsist in\nsouls, properly enlightened with genuine wisdom and virtue; for it then\nbecomes an union of intellects, and must consequently be immortal and\ndivine.\n\n[58] Pythagoras, according to Damascius, said, that friendship was the\nmother of all the political virtues.\n\n[59] A genuine modern will doubtless consider the whole of Proclus\u2019\nreligious conduct as ridiculously superstitious. And so, indeed,\nat first sight, it appears; but he who has penetrated the depths\nof ancient wisdom, will find in it more than meets the vulgar ear.\nThe religion of the Heathens, has indeed, for many centuries, been\nthe object of ridicule and contempt: yet the author of the present\nwork is not ashamed to own, that he is a perfect convert to it in\nevery particular, so far as it was understood and illustrated by\nthe Pythagoric and Platonic philosophers. Indeed the theology of\nthe ancient, as well as of the modern vulgar, was no doubt full of\nabsurdity; but that of the ancient philosophers, appears to be worthy\nof the highest commendations, and the most assiduous cultivation.\nHowever, the present prevailing opinions, forbid the defence of such\na system; for this must be the business of a more enlightened and\nphilosophic age. Besides, the author is not forgetful of Porphyry\u2019s\ndestiny, whose polemical writings were suppressed by the decrees of\nemperors; and whose arguments in defence of his religion were so\nvery futile and easy of solution, that, as St. Hierom informs us,\nin his preface on Daniel, Eusebius answered him in twenty-five, and\nApollinaris in thirty volumes!\n\n[60] See Proclus on Plato\u2019s Politics, p. 399. Instit. Theolog. num.\n196; and the extracts of Ficinus from Proclus\u2019s commentary on the first\nAlcibiades, p. 246. &c.\n\n[61] Alluding to the beautiful description given of Ulysses, in the 3d\nbook of the Iliad, v. 222.\n\n    \u039a\u03b1\u03af \u1f14\u03c0\u03b5\u03b1 \u03bd\u03b9\u03c6\u03ac\u03b4\u03b5\u03c3\u03b9\u03bd \u1f10\u03bf\u03b9\u03ba\u03cc\u03c4\u03b1 \u03c7\u03b5\u03b9\u03bc\u03b5\u03c1\u03af\u03b7\u03c3\u03b9\u03bd.\n\nWhich is thus elegantly paraphrased by Mr. Pope.\n\n    But when he speaks, what elocution flows!\n    Soft as the fleeces of descending snows\n    The copious accents fall, with easy art;\n    Melting they fall, and sink into the heart! &c.\n\n\n[62] Concerning Domninus, see Photius and Suidas from Damascius in his\nLife of Isidorus.\n\n[63] Nicephorus, in his commentary on Synesius de Insomniis, p. 562.\ninforms us, that the hecatic orb, is a golden sphere, which has a\nsapphire stone included in its middle part, and through its whole\nextremity, characters and various figures. He adds, that turning this\nsphere round, they perform invocations, which they call Jyng\u00e6. Thus\ntoo, according to Suidas, the magician Julian of Chaldea, and Arnuphis\nthe Egyptian, brought down showers of rain, by a magical power. And by\nan artifice of this kind, Empedocles was accustomed to restrain the\nfury of the winds; on which account he was called \u1f00\u03bb\u03b5\u03be\u03ac\u03bd\u03b5\u03bc\u03bf\u03c2, or a\nchaser of winds.\n\n[64] No opinion is more celebrated, than that of the metempsychosis\nof Pythagoras: but perhaps, no doctrine is more generally mistaken.\nBy most of the present day it is exploded as ridiculous; and the few\nwho retain some veneration for its founder, endeavour to destroy the\nliteral, and to confine it to an allegorical meaning. By some of the\nancients this mutation was limited to similar bodies: so that they\nconceived the human soul might transmigrate into various human bodies,\nbut not into those of brutes; and this was the opinion of Hierocles, as\nmay be seen in his comment on the Golden Verses. But why may not the\nhuman soul become connected with subordinate as well as with superior\nlives, by a tendency of inclination? Do not similars love to be united;\nand is there not in all kinds of life, something similar and common?\nHence, when the affections of the soul verge to a baser nature, while\nconnected with a human body, these affections, on the dissolution of\nsuch a body, become enveloped as it were, in a brutal nature, and the\nrational eye, in this case, clouded with perturbations, is oppressed by\nthe irrational energies of the brute, and surveys nothing but the dark\nphantasms of a degraded imagination. But this doctrine is vindicated\nby Proclus with his usual subtilty, in his admirable commentary on the\nTim\u00e6us, lib. v. p. 329, as follows, \u201cIt is usual, says he, to enquire\nhow souls can descend into brute animals. And some, indeed, think\nthat there are certain similitudes of men to brutes, which they call\nsavage lives: for they by no means think it possible that the rational\nessence can become the soul of a savage animal. On the contrary,\nothers allow it may be sent into brutes, because all souls are of\none and the same kind; so that they may become wolves and panthers,\nand ichneumons. But true reason, indeed, asserts that the human soul\nway be lodged in brutes, yet in such a manner, as that it may obtain\nits own proper life, and that the degraded soul may, as it were, be\ncarried above it, and be bound to the baser nature, by a propensity and\nsimilitude of affection. And that this is the only mode of insinuation,\nwe have proved by a multitude of reasons, in our commentaries on the\nPh\u00e6drus. But if it is requisite to take notice, that this is the\nopinion of Plato, we add, that in his politics, he says, that the soul\nof Thersites assumed an ape, but not the body of an ape: and in the\nPh\u00e6drus, that the soul descends into a savage life, but not into a\nsavage body; for life is conjoined with its proper soul. And in this\nplace he says it is changed into a brutal nature: for a brutal nature\nis not a brutal body, but a brutal life.\u201d\n\n[65] Pericles Lydus, a Stoic philosopher.\n\n[66] Vide Pausan. lib. i. Atticorum, cap. 21. et 20.\n\n[67] He means the Christians.\n\n[68] Proclus was born in the year of Christ 412, on the 6th of the Ides\nof February. But, for the sake of the astrologers, I have subjoined\nthe following figure from the Prolegomena of Fabricius to this life:\nand though I am not skilled in the art myself, I am persuaded, from\nthe arguments of Plotinus, that it contains many general truths; but\nwhen made subservient to particulars, is liable to great inaccuracy\nand error. In short, its evidence is wholly of a physiognomic nature;\nfor such is the admirable order and connection of things, that\nthroughout the universe, one thing is signified by another, and wholes\nare after a manner contained in their parts. So that the language of\nthe obscure and profound Heraclitus is perfectly just, when he says,\n\u201cYou must connect the perfect and the imperfect, the agreeing and\nthe disagreeing, the consonant and the dissonant, and out of one all\nthings, and out of all things one.\u201d\n\n  Illustration: A Scheme of the situation of the Stars, such as it\n  was at Byzantium, when the philosopher Proclus was born.\n\n[69] It was formerly the custom of almost all nations, to have their\nburial places in the suburbs, and not in the city itself.\n\n[70] This eclipse happened, according to Fabricius, in A. C. 484. 19\nCal. Feb. at sun-rise.\n\n[71] All the ancient theologists, among whom Plato holds a\ndistinguished rank, affirmed that the soul was of a certain middle\nnature and condition between intelligibles and sensibles: agreeable\nto which doctrine, Plotinus divinely asserts that she is placed in\nthe horizon, or in the boundary and isthmus, as it were, of eternal\nand mortal natures; and hence, according to the Magi, she is similar\nto the moon, one of whose parts is lucid, but the other dark. Now the\nsoul, in consequence of this middle condition, must necessarily be the\nreceptacle of all middle energies, both vital and gnostic; so that her\nknowledge is inferior to the indivisible simplicity of intellectual\ncomprehension, but superior to the impulsive perceptions of sense.\nHence the mathematical genera and species reside in her essence, as\nin their proper and natural region; for they are entirely of a middle\nnature, as Proclus proves in this and the sixth following chapter.\nBut this doctrine of Plato\u2019s, originally derived from Brontinus and\nArchytas, is thus elegantly explained by that philosopher, in the\nconcluding part of the sixth book of his Republic. \u201cSocrates, know\nthen, they are, as we say, two (the Good itself, and the Sun,) and\nthat the one reigns over the intelligible world, but the other over\nthe visible, not to say the heavens, lest I should deceive you by the\nname. You comprehend then, these two orders of things, I mean the\nvisible and the intelligible?--Glauco. I do.--Socrates. Continue this\ndivision then, as if it were a line divided into two unequal segments;\nand each part again, i. e. the sensible and intelligible, divided\nafter a similar manner, and you will have evidence and obscurity\nplaced by each other. In the visible segment, indeed, one part will\ncontain images. But I call images, in the first place, shadows;\nafterwards, the resemblances of things appearing in water, and in\ndense, smooth, and lucid bodies, and every thing of this kind, if you\napprehend me?--Glauco. I apprehend you.--Socrates. Now conceive that\nthe other section comprehends the things, of which these images are\nnothing more than similitudes, such as the animals around us, together\nwith plants, and whatever is the work of nature and art.--Glauco. I\nconceive it.--Socrates. Do you consider this section then, as divided\ninto true and false? And that the hypothesis of opinion is to the\nknowledge of science, as a resemblance to its original?--Glauco. I\ndo, very readily.--Socrates. Now then, consider how the section of\nthe intelligible is to be divided.--Glauco. How?--Socrates. Thus:\none segment is that which the soul enquires after, using the former\ndivisions as images, and compelled to proceed from hypotheses, not\nto the principle, but to the conclusion. The other is that which\nemploys the cogitative power of the soul, as she proceeds from\nan hypothesis to a principle no longer supposed, and, neglecting\nimages, advances through their _obscurity_ into _the light of_ ideas\nthemselves.--Glauco. I do not, in this, sufficiently understand\nyou.--Socrates. But again, for you will more easily understand me from\nwhat has been already premised. I think you are not ignorant, that\nthose who are conversant in geometry, arithmetic, and the like, suppose\neven and odd, together with various figures, and the three species of\nangles, and other things similar to these, according to each method of\nproceeding. Now, having established these, as hypotheses sufficiently\nknown, they conceive that no reason is to be required for their\nposition: but beginning from these, they descend through the rest, and\narrive at last, at the object of their investigation.--Glauco. This\nI know perfectly well.--Socrates. This also you know, that they use\nvisible forms, and make them the subject of their discourse, at the\nsame time not directing their intellect to the perception of these,\nbut to the _originals_ they resemble; I mean the square itself, and\nthe diameter itself; and not to the figures they delineate. And thus,\nother forms, which are represented by shadows and images in water,\nare employed by them, merely as resemblances, while they strive to\nbehold that which can be seen by cogitation alone.--Glauco. You speak\nthe truth.--Socrates. This is what I called above a species of the\nintelligible, in the investigation of which, the soul was compelled to\nuse hypotheses; not ascending to the principle, as incapable of rising\nabove hypotheses, but using the images formed from inferior objects,\nto a similitude of such as are superior, and which are so conceived\nand distinguished by opinion, as if they perspicuously contributed to\nthe knowledge of things themselves.--Glauco. I understand indeed, that\nyou are speaking of the circumstances which take place in geometry,\nand her kindred arts.--Socrates. Understand now, that by the other\nsection of the intelligible, I mean that which reason herself reaches,\nby her power of demonstrating, when no longer esteeming hypotheses for\nprinciples, but receiving them in reality for hypotheses, she uses\nthem as so many steps and handles in her ascent, until she arrives at\nthat which is no longer hypothetical, the principle of the universe;\nand afterwards descending, holding by ideas which adhere to the\nprinciple, she arrives at the conclusion, employing nothing sensible\nin her progress, but proceeding through ideas, and in these at last\nterminating her descent.--Glauco. I understand you, but not so well\nas I desire: for you seem to me to propose a great undertaking. You\nendeavour, indeed, to determine that the portion of true being and\nintelligible, which we speculate by the science of demonstration, is\nmore evident than the discoveries made by the sciences called arts;\nbecause in the first hypotheses are principles, and their masters are\ncompelled to employ the eye of cogitation, and not the perceptions\nof the senses. Yet, because they do not ascend to the principle,\nbut investigate from hypotheses, they seem to you not to have\nintelligence concerning these, though they are intelligible, through\nthe light of the principle. But you seem to me to call the habit of\nreasoning on geometrical and the like concerns, cogitation, rather\nthan intelligence, as if cogitation held the middle situation between\nopinion and intellect.--Socrates. You understand me sufficiently well.\nAnd again: with these four proportions take these four corresponding\naffections of the soul: with the highest intelligence; with the second\ncogitation; against the third set opinion; and against the fourth\nassimilation, or imagination. Besides this, establish them in the\norder of alternate proportion, so that they may partake of evidence,\nin the same manner as their corresponding objects participate of\nreality.\u201d I have taken the liberty of translating this fine passage\ndifferently from both Petvin and Spens; because they have neglected\nto give the proper meaning of the word \u03b4\u03b9\u03ac\u03bd\u03bf\u03b9\u03b1, or cogitation, the\nformer translating it _mind_, and the _eye of the mind_, and by\nthis means confounding it with intellect; and the latter calling it\n_understanding_. But it is certain that Plato, in this place, ranks\nintellect as the first, on account of the superior evidence of its\nperceptions; in the next place, cogitation; in the third, opinion;\nand in the fourth, imagination. However, the reader will please to\nremember, that by \u03b4\u03b9\u03ac\u03bd\u03bf\u03b9\u03b1, or cogitation, in the present work, is\nunderstood that power of the soul which reasons from premises to\nconclusions, and whose syllogistic energy, on active subjects, is\ncalled prudence; and on such as are speculative, science. But for\nfarther information concerning its nature, see the dissertation\nprefixed to this work, and the following fifth chapter.\n\n[72] These two principles, _bound_ and _infinite_, will\ndoubtless be considered by the unthinking part of mankind, as\nnothing more than general terms, and not as the most real of beings.\nHowever, an accurate contemplation of the universe, will convince\nevery _truly_ philosophic mind of their reality. For the heavens\nthemselves, by the coherence and order of their parts, evince their\nparticipation of bound. But by their prolific powers, and the\nunceasing revolutions of the orbs they contain, they demonstrate their\nparticipation of _infinity_. And the finite and perpetually\nabiding forms with which the world is replete, bear a similitude _to\nbound_: while, on the contrary, the variety of particulars, their\nnever-ceasing mutation, and the connection of more and less in the\ncommunion of forms, represents an image of _infinity_. Add too,\nthat every natural species, by its form is similar to _bound_;\nbut by its matter, to _infinity_. For these two, form and matter,\ndepend on _bound_ and _infinity_, and are their ultimate\nprogressions. And each of these, indeed, participates of unity; but\nform is the measure and bound of matter, and is more _one_. But\nmatter is in capacity all things, because it subsists by an emanation\nfrom the first capacity, or the _infinite itself_.\n\n[73] Of human disciplines, those alone deserve to be called sciences\nwhich use no hypotheses, which resolve things into their principles,\nwhich are conversant with true being, and elevate us to ideas\nthemselves. Dialectic is wholly of this kind (I mean the dialectic of\nPlato); for this alone uses no suppositions, but, neglecting shadows\nand images, raises us, by a sublime investigation, to the principle\nof the universe; and on this account, deserves to be called the\nvery apex of disciplines. But we must not imagine, that by the word\ndialectic here, is meant logic, or any part of logic, or that method\nof disputation, by which we fabricate probable reasons; but we must\nconceive it as signifying a discipline, endued with the greatest\nacuteness; neglecting all hypotheses, truly soaring to primary causes,\nand ultimately reposing in their contemplation. Plotinus has given us\nmost happy specimens of this method, in his books _on the genera of\nbeing_.\n\n[74] See note to the first chapter.\n\n[75] I would particularly recommend this chapter to modern\nmathematicians, most of whom, I am afraid, have never considered\nwhether or not the subjects of their speculation have any real\nsubsistence: though it is surely an enquiry worthy the earnest\nattention of every liberal mind. For if the objects of mathematical\ninvestigation are merely imaginary, I mean the point without parts,\nthe line without breadth, &c. the science, founded on these false\nprinciples, must of course be entirely delusive. Indeed, an absolutely\ntrue conclusion, can never flow from an erroneous principle, as from\nits cause: as the stream must always participate of its source. I\nmean such a conclusion as is demonstrated by the proper cause, \u03c0\u03bb\u1f74\u03bd\n\u03bf\u1f50 \u03b4\u03b9\u03cc\u03c4\u03b9, \u1f00\u03bb\u03bb\u2019 \u1f45\u03c4\u03b9, says Aristotle, in his first Analytics; that is,\na syllogism from false principles will not prove the _why_, but\nonly simply _that it is_: indeed it can only simply prove _that\nit is_, to him who admits the false propositions; because he who\nallows the premises, cannot deny the conclusion, when the syllogism is\nproperly constructed. Thus we way syllogize in the first figure,\n\n               Every thing white, is an animal:\n               Every bird is white:\n    Therefore, Every bird is an animal.\n\nAnd the conclusion will be true, though the major and minor terms are\nfalse; but then these terms are not the causes of the conclusion, and\nwe have an inference without a proof. In like manner, if mathematical\nspecies are delusive and fictitious, the conclusions deduced from them\nas principles, are merely hypothetical, and not demonstrative.\n\n[76] Aristotle, in his last Analytics. The reader will please to\nobserve, that the whole force of this nervous, accurate, and elegant\nreasoning, is directed against Aristotle; who seems unfortunately to\nhave considered, with the moderns, that mathematical species subsist\nin the soul, by an abstraction from sensibles. See the preceding\nDissertation.\n\n[77] Viz. 1, 2, 4, 8, 3, 9, 27. Concerning which, see lib. iii. of\nProclus\u2019s excellent Commentary on the Tim\u00e6us.\n\n[78] Plato frequently, both in the Meno and elsewhere, shews that\nscience is Reminiscence; and I think not without the strongest\nreason. For since the soul is immaterial, as we have demonstrated\nin the dissertation to this work, she must be truly immortal, i. e.\nboth _a parte ante_, & _a parte post_. That she must be\neternal, indeed, with respect to futurity, if immaterial, is admitted\nby all; and we may prove, with Aristotle, in his first book de C\u0153lo,\nthat she is immortal, likewise _a parte ante_, as follows.\nEvery thing without generation, is incorruptible, and every thing\nincorruptible, is without generation: for that which is without\ngeneration, has a necessity of existing infinitely _a parte ante_\n(from the hypothesis); and therefore, if it possesses a capacity of\nbeing destroyed, since there is no greater reason why it should be\ncorrupted now, rather than in some former period, it is endued with\na capacity of being destroyed and ceasing to be, in every instant of\ninfinite time, in which it necessarily is. In like manner, that which\nis incorruptible, has a necessity of existing infinitely _a parte\npost_; therefore, if it possesses a capacity of being generated,\nsince there is no greater reason why it should be generated now rather\nthan afterwards, it possesses a capacity of being generated, in every\ninstant of time, in which it necessarily is. If then the soul is\nessentially immortal, with respect to the past and future circulations\nof time; and if she is replete with forms or ideas of every kind, as\nwe have proved in the dissertation, she must, from her circulating\nnature, have been for ever conversant in alternately possessing and\nlosing the knowledge of these. Now, the recovery of this knowledge by\nscience, is called by Plato, reminiscence; and is nothing more than\na renewed contemplation of those divine forms, so familiar to the\nsoul, before she became involved in the dark vestment of an earthly\nbody. So that we may say, with the elegant Maximus Tyrus, (Disser.\n28.) \u201cReminiscence is similar to that which happens to the corporeal\neye, which, though always endued with a power of vision, yet darkness\nsometimes obstructs its passage, and averts it from the perception of\nthings. Art therefore, approaches, which though it does not give to the\neye the power of vision, yet removes its impediments, and affords a\nfree egress to its rays. Conceive now, that our rational soul is such a\npower of perceiving, which sees and knows the nature of beings. To this\nthe common calamity of bodies happens, that darkness spreading round\nit, hurries away its aspect, blunts its sharpness, and extinguishes its\nproper light. Afterwards, the art of reason approaches, which, like\na physician, does not bring or afford it a new science, but rouses\nthat which it possesses, though very slender, confused, and unsteady.\u201d\nHence, since the soul, by her immersion in body, is in a dormant state,\nuntil she is roused by science to an exertion of her latent energies;\nand yet even previous to this awakening, since she contains the\nvivid sparks, as it were, of all knowledge, which only require to be\nventilated by the wings of learning, in order to rekindle the light of\nideas, she may be said in this case to know all things as in a dream,\nand to be ignorant of them with respect to vigilant perceptions. Hence\ntoo, we may infer that time does not antecede our essential knowledge\nof forms, because we possess it from eternity: but it precedes our\nknowledge with respect to a production of these reasons into perfect\nenergy. I only add, that I would recommend the liberal English reader,\nto Mr. Sydenham\u2019s excellent translation of Plato\u2019s Meno, where he\nwill find a familiar and elegant demonstration of the doctrine of\nReminiscence.\n\n[79] Concerning this valuable work, entitled \u0399\u0395\u03a1\u039f\u2018\u03a3 \u039b\u039f\u0393\u039f\u2019\u03a3, see the\nBibliotheca Gr\u00e6ca of Fabricius, vol. i. p. 118 and 462, and in the\ncommentary of Syrianus on Aristotle\u2019s metaphysics, p. 7, 71, 83, and\n108, the reader will find some curious extracts from this celebrated\ndiscourse; particularly in p. 83. Syrianus informs us, \u201cthat he who\nconsults this work will find all the orders both of Monads and Numbers,\nwithout neglecting one, fully celebrated (\u1f50\u03bc\u03bd\u03bf\u03c5\u03bc\u03ad\u03bd\u03b1\u03c2.)\u201d There is no\ndoubt, but that Pythagoras and his disciples concealed the sublimest\ntruths, under the symbols of numbers; of which he who reads and\nunderstands the writings of the Platonists will be fully convinced.\nHence Proclus, in the third book of his excellent commentary on the\nTim\u00e6us, observes, \u201cthat Plato employed mathematical terms for the sake\nof mystery and concealment, as certain veils, by which the penetralia\nof truth might be secluded from vulgar inspection, just as the\ntheologists made fables, but the Pythagoreans symbols, subservient to\nthe same purpose: for in images we may speculate their exemplars, and\nthe former afford us the means of access to the latter.\u201d\n\n[80] Concerning this Geometric Number, in the 8th book of Plato\u2019s\nRepublic, than which Cicero affirms there is nothing more obscure, see\nthe notes of Bullialdus to Theo. p. 292.\n\n[81] I am sorry to say, that this part of the enemies to pure geometry\nand arithmetic, are at the present time very numerous; conceptions\nof utility in these sciences, extending no farther than the sordid\npurposes of a mere animal life. But surely, if intellect is a part of\nour composition, and the noblest part too, there must be an object of\nits contemplation; and this, which is no other than truth in the most\nexalted sense, must be the most noble and useful subject of speculation\nto every rational being.\n\n[82] In the 13th book of his Metaphysics, cap. iii.\n\n[83] In. I. De Partib. Animalium, et in primo Ethic. cap. iii.\n\n[84] See more concerning this in the Dissertation.\n\n[85] Since number is prior to magnitude, the demonstrations of\narithmetic must be more intellectual, but those of geometry more\naccommodated to the rational power. And when either arithmetic or\ngeometry is applied to sensible concerns, the demonstrations, from the\nnature of the subjects, must participate of the obscurity of opinion.\nIf this is the case, a true mathematician will value those parts of his\nscience most, which participate most of evidence; and will consider\nthem as degraded, when applied to the common purposes of life.\n\n[86] This division of the mathematical science, according to the\nPythagoreans, which is nearly coincident with that of Plato, is blamed\nby Dr. Barrow in his Mathematical Lectures, p. 15. as being confined\nwithin too narrow limits: and the reason he assigns for so partial a\ndivision, is, \u201cbecause, in Plato\u2019s time, others were either not yet\ninvented, or not sufficiently cultivated, or at least were not yet\nreceived into the number of the mathematical sciences.\u201d But I must\nbeg leave to differ from this most illustrious mathematician in this\naffair; and to assert that the reason of so confined a distribution\n(as it is conceived by the moderns) arose from the exalted conceptions\nthese wise men entertained of the mathematical sciences, which they\nconsidered as so many preludes to the knowledge of divinity, when\nproperly pursued; but they reckoned them degraded and perverted, when\nthey became mixed with sensible objects, and were applied to the common\npurposes of life.\n\n[87] That is, a right and circular line.\n\n[88] I am afraid there are few in the present day, who do not consider\ntactics as one of the most principal parts of mathematics; and who\nwould not fail to cite, in defence of their opinions, that great\nreformer of philosophy, _as he is called_, Lord Bacon, commending\npursuits which come home to men\u2019s businesses and bosoms. Indeed, if\nwhat is lowest in the true order of things, and best administers to the\nvilest part of human nature, is to have the preference, their opinion\nis right, and Lord Bacon is a _philosopher_!\n\n[89] By this is to be understood the art new called Perspective: from\nwhence it is evident that this art was not unknown to the ancients,\nthough it is questioned by the moderns.\n\n[90] From hence it appears, that it is doubtful whether Plato is the\nauthor of the dialogue called Epinomis; and I think it may with great\npropriety be questioned. For though it bears evident marks of high\nantiquity, and is replete with genuine wisdom, it does not seem to be\nperfectly after Plato\u2019s manner; nor to contain that great depth of\nthought with which the writings of this philosopher abound. Fabricius\n(in his Bibliotheca Gr\u00e6ca, lib. i. p. 27.) wonders that Suidas should\nascribe this work to a philosopher who distributed Plato\u2019s laws into\ntwelve books, because it was an usual opinion; from whence it seems,\nthat accurate critic had not attended to the present passage.\n\n[91] This proximate conjunction of the mathematical sciences, which\nProclus considers as subordinate to dialectic, seems to differ from\nthat vertex of science in this, that the former merely embraces the\nprinciples of all science, but the latter comprehends the universal\ngenera of being, and speculates the principle of all.\n\n[92] In the Meno.\n\n[93] This is certainly the true or philosophical employment of the\nmathematical science; for by this means we shall be enabled to\nascend from sense to intellect, and rekindle in the soul that divine\nlight of truth, which, previous to such an energy, was buried in the\nobscurity of a corporeal nature. But by a contrary process, I mean,\nby applying mathematical speculations, to experimental purposes, we\nshall blind the liberal eye of the soul, and leave nothing in its stead\nbut the darkness of corporeal vision, and the phantoms of a degraded\nimagination.\n\n[94] The design of the present chapter is to prove that the figures\nwhich are the subjects of geometric speculation, do not subsist in\nexternal and sensible matter, but in the receptacle of imagination,\nor the matter of the phantasy. And this our philosopher proves with\nhis usual elegance, subtilty, and depth. Indeed, it must be evident\nto every attentive observer, that sensible figures fall far short\nof that accuracy and perfection which are required in geometrical\ndefinitions: for there is no sensible circle perfectly round, since\nthe point from which it is described is not without parts; and, as\nVossius well observes, (de Mathem. p. 4.) there is not any sphere in\nthe nature of things, that only touches in a point, for with some\npart of its superficies it always touches the subjected plane in a\nline, as Aristotle shews Protagoras to have objected against the\ngeometricians. Nor must we say, with that great mathematician Dr.\nBarrow, in his Mathematical Lectures, page 76, \u201cthat all imaginable\ngeometrical figures, are really inherent in every particle of matter,\nin the utmost perfection, though not apparent to sense; just as the\neffigies of C\u00e6sar lies hid in the unhewn marble, and is no new thing\nmade by the statuary, but only is discovered and brought to sight by\nhis workmanship, i. e. by removing the parts of matter by which it is\novershadowed and involved. Which made Michael Angelus, the most famous\ncarver, say, _that sculpture was nothing but a purgation from things\nsuperfluous. For take all that is superfluous_, (says he) _from\nthe wood or stone, and the rest will be the figure you intend_. So,\nif the hand of an angel (at least the power of God) should think fit to\npolish any particle of matter, without vacuity, a spherical superficies\nwould appear to the eyes, of a figure exactly round; not as created\nanew, but as unveiled and laid open from the disguises and covers of\nits circumjacent matter.\u201d For this would be giving a perfection to\nsensible matter, which it is naturally incapable of receiving: since\nexternal body is essentially full of pores and irregularities, which\nmust eternally prevent its receiving the accuracy of geometrical body,\nthough polished by the hand of an angel. Besides, what polishing\nwould ever produce a point without parts, and a line without breadth?\nFor though body may be reduced to the greatest exility, it will not\nby this means ever pass into an incorporeal nature, and desert its\ntriple dimension. Since external matter, therefore, is by no means the\nreceptacle of geometrical figures, they must necessarily reside in the\ncatoptric matter of the phantasy, where they subsist with an accuracy\nsufficient for the energies of this science. It is true, indeed, that\neven in the purer matter of imagination, the point does not appear\nperfectly impartible, nor the line without latitude: but then the\nmagnitude of the point, and the breadth of the line is indefinite, and\nthey are, at the same time, unattended with the qualities of body, and\nexhibit to the eye of thought, magnitude alone. Hence, the figures in\nthe phantasy, are the proper recipients of that universal, which is\nthe object of geometrical speculation, and represent, as in a mirror,\nthe participated subsistence of those vital and immaterial forms which\nessentially reside in the soul.\n\n[95] This division is elegantly explained by Ammonius, (in Porphyr. p.\n12.) as follows, \u201cConceive a seal-ring, which has the image of some\nparticular person, for instance, of Achilles, engraved in its seal, and\nlet there be many portions of wax, which are impressed by the ring.\nAfterwards conceive that some one approaches, and perceives all the\nportions of wax, stamped with the impression of this one ring, and\nkeeps the impression of the ring in his mind: the seal engraved in the\nring, represents the universal, prior to the many: the impression in\nthe portions of wax, the universal in the many: but that which remains\nin the intelligence of the beholder, may be called the universal,\nafter and posterior to the many. The same must we conceive in genera\nand species. For that best and most excellent artificer of the world,\npossesses within himself the forms and exemplars of all things: so that\nin the fabrication of man, he looks back upon the form of man resident\nin his essence, and fashions all the rest according to its exemplar.\nBut if any one should oppose this doctrine, and assert that the forms\nof things do not reside with their artificer, let him attend to the\nfollowing arguments. The artificer either knows, or is ignorant of that\nwhich he produces: but he who is ignorant will never produce any thing.\nFor who will attempt to do that, which he is ignorant how to perform?\nsince he cannot act from an irrational power like nature, whose\noperations are not attended with animadversion. But if he produces\nany thing by a certain reason, he must possess a knowledge of every\nthing which he produces. If, therefore, it is not impious to assert,\nthat the operations of the Deity, like those of men, are attended with\nknowledge, it is evident that the forms of things must reside in his\nessence: but forms are in the demiurgus, like the seal in the ring;\nand these forms are said to be prior to the many, and separated from\nmatter. But the species man, is contained in each particular man, like\nthe impression of the seal in the wax, and is said to subsist in the\nmany, without a separation from matter. And when we behold particular\nmen, and perceive the same form and effigy in each, that form seared\nin our soul, is said to be after the many, and to have a posterior\ngeneration: just as we observed in him, who beheld many seals impressed\nin the wax from one and the same ring. And this one, posterior to the\nmany, may be separated from body, when it is conceived as not inherent\nin body, but in the soul: but is incapable of a real separation from\nits subject.\u201d We must here, however, observe, that when Ammonius speaks\nof the knowledge of the Deity, it must be conceived as far superior\nto ours. For he possesses a nature more true than all essence, and a\nperception clearer than all knowledge. And as he produced all things\nby his unity, so by an ineffable unity of apprehension, he knows the\nuniversality of things.\n\n[96] In lib. vii. Metaphys. 35 & 39.\n\n[97] In lib. iii. de Anima, tex. 20.\n\n[98] That is, geometry first speculates the circle delineated on\npaper, or in the dust: but by the medium of the circular figure in\nthe phantasy, contemplates the circle resident in cogitation; and by\nthat universal, or circular reason, participated in the circle of the\nphantasy, frames its demonstrations.\n\n[99] In his first Analytics, t. 42. See the Dissertation to this work.\n\n[100] Such as the proportion of the diagonal of a square to its side;\nand that of the diameter of a circle, to the periphery.\n\n[101] The gnomons, from which square numbers are produced, are odd\nnumbers in a natural series from unity, i. e. 1, 3, 5, 7, 9, 11, &c.\nfor these, added to each other continually, produce square numbers\n_ad infinitum_. But these gnomons continually decrease from the\nhighest, and are at length terminated by indivisible unity.\n\n[102] This doctrine of ineffable quantities, or such whose proportion\ncannot be expressed, is largely and accurately discussed by Euclid, in\nthe tenth book of his Elements: but its study is neglected by modern\nmathematicians, _because it is of no use_, that is, because it\ncontributes to nothing mechanical.\n\n[103] This proposition is the 11th of the second book: at least,\nthe method of dividing a line into extreme and mean proportion, is\nimmediately deduced from it; which is done by Euclid, in the 30th, of\nthe sixth book. Thus, Euclid shews (11. 2.) how to divide the line (A\nG B)[diagram] A B, so that the rectangle under the whole A B, and\nthe segment G B, may be equal to the square made from A G: for when\nthis is done, it follows, that as A B is to A G, so is A G to G B; as\nis well known. But this proposition, as Dr. Barrow observes, cannot be\nexplained by numbers; because there is not any number which can be so\ndivided, that the product from the whole into one part, may be equal to\nthe square from the other part.\n\n[104] All polygonous figures, may, it is well known, be resolved\ninto triangles; and this is no less true of polygonous numbers,\nas the following observations evince. All number originates from\nindivisible unity, which corresponds to a point: and it is either\nlinear, corresponding to a line; or superficial, which corresponds to\na superficies; or solid, which imitates a geometrical solid. After\nunity, therefore, the first of linear numbers is the duad; just as\nevery finite line is allotted two extremities. The triad is the first\nof superficial numbers; as the triangle of geometrical figures. And\nthe tetrad, is the first of solids; because a triangular pyramid, is\nthe first among solid numbers, as well as among solid figures. As,\ntherefore, the monad is assimilated to the point, so the duad to the\nline, the triad to the superficies, and the tetrad to the solid. Now,\nof superficial numbers, some are triangles, others squares, others\npentagons, hexagons, heptagons, &c. Triangular numbers are generated\nfrom the continual addition of numbers in a natural series, beginning\nfrom unity. Thus, if the numbers 1, 2, 3, 4, 5, &c. be added to each\nother continually, they will produce the triangular numbers 1, 3, 6,\n10, 15, &c. and if every triangular number be added to its preceding\nnumber, it will produce a square number. Thus 3 added to 1 makes 4; 6\nadded to 3 is equals 9; 10 added to 6 is equal to 16; and so of the\nrest. Pentagons, are produced from the junction of triangular and\nsquare numbers, as follows. Let there be a series of triangular numbers\n1, 3, 6, 10, 15, &c.\n\nAnd of squares 1, 4, 9, 16, 25, &c.\n\nThen the second square number, added to the first triangle, will\nproduce the first pentagon from unity, i.e. 5. The third square added\nto the second triangle, will produce the second pentagon, i.e. 12;\nand so of the rest, by a similar addition. In like manner, the second\npentagon, added to the first triangle, will form the first hexagon from\nunity; the third pentagon and the second triangle, will form the second\nhexagon, &c. And, by a similar proceeding, all the other polygons may\nbe obtained.\n\n[105] Intellections are universally correspondent to their objects,\nand participate of evidence or the contrary, in proportion as their\nsubjects are lucid or obscure. Hence, Porphyry, in his sentences,\njustly observes, that \u201cwe do not understand in a similar manner with\nall the powers of the soul, but according to the particular essence\nof each. For with the intellect we understand intellectually; and\nwith the soul, rationally: our knowledge of plants is according to a\nseminal conception; our understanding of bodies is imaginative; and our\nintellection of the divinely solitary principle of the universe, who is\nabove all things, is in a manner superior to intellectual perception,\nand by a super-essential energy.\u201d \u1f08\u03c6\u03bf\u03c1\u03bc\u03b1\u1f76 \u03c0\u03c1\u1f78\u03c2 \u03c4\u1f70 \u039d\u03bf\u03b7\u03c4\u1f70, (10.) So that,\nin consequence of this reasoning, the speculations of geometry are then\nmost true, when most abstracted from sensible and material natures.\n\n[106] See Plutarch, in the life of Marcellus.\n\n[107] In lib. i. de C\u00e6lo, tex. 22. et lib. i. Meteo. cap. 3. Aristotle\nwas called demoniacal by the Platonic philosophers, in consequence of\nthe encomium bestowed on him by his master, Plato, \u201cThat he was the\nd\u00e6mon of nature.\u201d Indeed, his great knowledge in things subject to\nthe dominion of nature, well deserved this encomium; and the epithet\n_divine_, has been universally ascribed to Plato, from his\nprofound knowledge of the intelligible world.\n\n[108] \u0395\u1f30\u03c2 \u03bd\u03bf\u1fe6\u03bd, is wanting in the original, but is supplied by the\nexcellent translation of Barocius.\n\n[109] \u1f08\u03bb\u03cc\u03b3\u03c9\u03bd, in the printed Greek, which Fabricius, in his Bibliotheca\nGr\u00e6ca, vol. i. page 385, is of opinion, should be read \u1f00\u03bd\u03b1\u03bb\u03cc\u03b3\u03c9\u03bd; but\nI have rendered the word according to the translation of Barocius,\nwho is likely to have obtained the true reading, from the variety of\nmanuscripts which he consulted.\n\n[110] The quadrature of the Lunula is as follows.\n\n  [Illustration]\n\nLet A B C be a right-angled triangle, and B A C a semi-circle on the\ndiameter B C: B N A a semi-circle described on the diameter A B; A M\nC a semi-circle described on the diameter A C. Then the semi-circle B\nA C is equal to the semi-circle B N A, and A M C together: (because\ncircles are to each other as the squares of their diameters, 31, 6.)\nIf, therefore, you take away the two spaces B A, A C common on both\nsides, there will remain the two lunulas B N A, A M C, bounded on both\nsides with circular lines, equal to the right-angled triangle B A C.\nAnd if the line B A, be equal to the line A C, and you let fall a\nperpendicular to the hypotenuse B C, the triangle B A O will be equal\nto the lunular space B N A, and the triangle C O A will be equal to\nthe lunula C M A. Those who are curious, may see a long account of an\nattempt of Hippocrates to square the circle, by the invention of the\nlunulas, in Simplicius on Aristotle\u2019s Physics, lib. i.\n\n[111] So Barocius reads, but Fabricius \u039c\u03b5\u03b4\u03bc\u1fb6\u03b9\u03bf\u03c2.\n\n[112] i. e. The five regular bodies, the pyramid, cube, octaedron,\ndodecaedron and icosaedron; concerning which, and their application to\nthe theory of the universe, see Kepler\u2019s admirable work, De Harmonia\nMundi.\n\n[113] It may be doubted whether the optics and catoptrics, ascribed to\nEuclid in the editions of his works are genuine: for Savil, and Dr.\nGregory, think them scarcely worthy so great a man.\n\n[114] There are two excellent editions of this work, one by Meibomius,\nin his collection of ancient authors on harmony; and the other by Dr.\nGregory, in his collection of Euclid\u2019s works.\n\n[115] This work is most probably lost. See Dr. Gregory\u2019s Euclid.\n\n[116] All this is shewn by Proclus in the following Commentaries; and\nis surely most admirable and worthy the investigation of every liberal\nmind; but I am afraid modern mathematicians very little regard such\nknowledge, because it cannot be applied to practical and mechanical\npurposes.\n\n[117] This work is unfortunately lost.\n\n[118] Because this is true only in isosceles and equilateral triangles.\n\n[119] This follows from the 32d proposition of the first book of\nEuclid; and is demonstrated by Dr. Barrow, in his scholium to that\nproposition.\n\n[120] The method of constructing these is shewn by our philosopher,\nin his comment on the first proposition, as will appear in the second\nvolume of this work.\n\n[121] The reader will please to observe, that the definitions are,\nindeed, hypotheses, according to the doctrine of Plato, as may be seen\nin the note to chap, i. book I. of this work.\n\n[122] In his last Analytics. See the preceding Dissertation.\n\n[123] That part of this work enclosed within the brackets, is wanting\nin the original; which I have restored from the excellent version of\nBarocius. The philosophical reader, therefore, of the original, who may\nnot have Barocius in his possession, will, I hope, be pleased, to see\nso great a vacancy supplied; especially, as it contains the beginning\nof the commentary on the definition of a point.\n\n[124] I do not find this \u00e6nigma among the Pythagoric symbols which are\nextant; so that it is probably no where mentioned but in the present\nwork. And I am sorry to add, that a _figure and three oboli_, in\ntoo much the general cry of the present times.\n\n[125] The present Comment, and indeed most of the following, eminently\nevinces the truth of Kepler\u2019s observation, in his excellent work,\n_De Harmonia Mundi_, p. 118. For, speaking of our author\u2019s\ncomposition in the present work, which he every where admires and\ndefends, he remarks as follows, \u201coratio fluit ipsi torrentis instar,\nripas inundans, et c\u00e6ca dubitationum vada gurgitesque occultans, dum\nmens plena majestatis tantarum rerum, luctatur in angustiis lingu\u00e6, et\nconclusio nunquam sibi ipsi verborum copi\u00e2 satisfaciens, propositionum\nsimplicitatem excedit.\u201d But Kepler was skilled in the Platonic\nphilosophy, and appears to have been no less acquainted with the great\ndepth of our author\u2019s mind than with the magnificence and sublimity of\nhis language. Perhaps Kepler is the only instance among the moderns,\nof the philosophical and mathematical genius being united in the same\nperson.\n\n[126] That is, the reason of a triangular figure (for instance) in the\nphantasy, or triangle itself, is superior to the triangular nature\nparticipated in that figure.\n\n[127] In the tenth book of his Republic.\n\n[128] See the Hymn to the Mother of the Gods, in my translation of the\nOrphic Initiations.\n\n[129] The philosopher here seems to contradict what he asserts in the\nend of his comment on the 13th Definition: for there he asserts, that\nthe circle is a certain plane space. Perhaps he may be reconciled, by\nconsidering, that as the circle subsists most according to bound, when\nwe speculate its essence in this respect we may define it according\nto the circumference, which is the cause of its bound. But when we\nconsider it as participating of infinity also, though not in so eminent\na degree, and view it from its emanations from the centre as well as in\nits regressions, we may define it a plane space.\n\n[130] That is, the essential one of the soul is the mother of number;\nbut that which subsists in opinion is nothing more than the receptacle\nof the former; just as matter is the seat of all forms. For a farther\naccount of the subsistence of numbers, see the first section of the\npreceding Dissertation.\n\n[131] That is, number composed from units.\n\n[132] This sentence within the brackets, is wholly omitted in the\nprinted Greek.\n\n[133] In i. De C\u00e6lo.\n\n[134] This sentence within the brackets, which is very imperfect in the\nGreek, I have supplied from the excellent translation of Barocius. In\nthe Greek there is nothing more than \u03bb\u1f72\u03b3\u03c9 \u03b4\u1f72 \u1f11\u03bd\u1f78\u03bd \u03c4\u1fc2\u03bd \u03b3\u03c1\u03b1\u03bc\u03bc\u1f74\u03bd \u03b4\u03c5\u03b1\u03b4\u03cc\u03c2\n\u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78 \u03c3\u03c4\u03b5\u03c1\u03b5\u03cc\u03bd.\n\n[135] In the Greek, \u03b3\u1f70\u03c1 \u1f21 \u03bc\u03bf\u03bd\u1f70\u03c2 \u1f10\u03ba\u03b5\u1fd6 \u03c0\u03c1\u1ff6\u03c4\u03bf\u03bd, \u1f45\u03c0\u03bf\u03c5 \u03c0\u03b1\u03c4\u03c1\u03b9\u03ba\u1f74 \u03bc\u03bf\u03bd\u03ac\u03c2 \u1f10\u03c3\u03c4\u03b9\n\u03c6\u03b7\u03c3\u1f76 \u03c4\u1f78 \u03bb\u03cc\u03b3\u03b9\u03bf\u03bd. The latter part only of this oracle, is to be found\nin all the printed editions of the Zoroastrian oracles; though it is\nwonderful how this omission could escape the notice of so may able\ncritics, and learned men. It seems probable, from hence, that it is\nonly to be found perfect in the present work.\n\n[136] The word \u03c4\u03b1\u03bd\u03ac\u03b7, is omitted in the Greek.\n\n[137] This and the following problems, are the 1st, 22d, and 12th\npropositions of the first book. But in the two last, instead of the\nword \u1f04\u03c0\u03b5\u03b9\u03c1\u03bf\u03c2 or infinite, which is the term employed by Euclid, Mr.\nSimson, in his edition of the Elements, uses the word unlimited. But\nit is no unusual thing with this great geometrician, to alter the\nwords of Euclid, when they convey a philosophical meaning; as we shall\nplainly evince in the course of these Commentaries. He certainly\ndeserves the greatest praise for his zealous attachment to the ancient\ngeometry: but he would (in my opinion) have deserved still more, had\nhe been acquainted with the Greek philosophy; and fathomed the depth\nof Proclus; for then he would never have attempted to restore Euclid\u2019s\nElements, by depriving them of some very considerable beauties.\n\n[138] This is doubtless the reason why the proportion between a right\nand circular line, cannot be exactly obtained in numbers; for on this\nhypothesis, they must be incommensurable quantities; because the one\ncontains property essentially different from the other.\n\n[139]\n\n  [Illustration]\n\nThe cornicular angle is that which is made from the periphery of a\ncircle and its tangent; that is, the angle comprehended by the arch\nL A, and the right line F A, which Euclid in (16. 3.) proves to be\nless than any right-lined angle. And from this admirable proposition\nit follows, by a legitimate consequence, that any quantity may\nbe continually and infinitely increased, but another infinitely\ndiminished; and yet the augment of the first, how great soever it\nmay be, shall always be less than the decrement of the second: which\nCardan demonstrates as follows. Let there be proposed an angle of\ncontact B A E, and an acute angle H G I. Now if there be other lesser\ncircles described A C, A D, the angle of contact will be evidently\nincreased. And if between the right lines G H, G I, there fall other\nright lines G K, G L, the acute angle shall be continually diminished:\nyet the angle of contact, however increased, is always less than the\nacute angle, however diminished. Sir Isaac Newton likewise observes,\nin his Treatise on Fluxions, that there are angles of contact made\nby other curve lines, and their tangents infinitely less than those\nmade by a circle and right line; all which is demonstrably certain:\nyet, such is the force of prejudice, that Mr. Simson is of opinion,\nwith Vieta, that this part of the 16th proposition is adulterated; and\nthat the space made by a circular line and its tangent, is no angle.\nAt least his words, in the note upon this proposition, will bear such\na construction. Peletarius was likewise of the same opinion; but is\nelaborately confuted by the excellent Clavius, as may be seen in his\ncomment on this proposition. But all the difficulties and paradoxes in\nthis affair, may be easily solved and admitted, if we consider, with\nour philosopher, that the essence of an angle does not subsist in ether\nquantity, quality, or inclination, taken singly, but in the aggregate\nof them all. For if we regard the inclination of a circular line to\nits tangent, we shall find it possess the property, by which Euclid\ndefines an angle: if we respect its participation of quantity, we shall\nfind it capable of being augmented and diminished; and if we regard\nit as possessing a peculiar quality, we shall account for its being\nincommensurable with every right-lined angle. See the Comment on the\n8th Definition.\n\n[140] In i. De C\u00e6lo.\n\n[141] It is from this cylindric spiral that the screw is formed.\n\n[142]\n\n  [Illustration]\n\nThe present very obscure passage, may be explained by the following\nfigure. Let A B C, be a right angle, and D E the line to be moved,\nwhich is bisected in G. Now, conceive it to be moved along the lines A\nB, B C, in such a manner, that the point D may always remain in A B,\nand the point E in B C. Then, when the line D E, is in the situations\n_d e_, _\u03b4 \u03b5_, the point G, shall be in _g, \u03b3_, and these\npoints G, _g, \u03b3_, shall be in a circle. And any other point F in\nthe line D E, will, at the same time, describe an ellipsis; the greater\naxis being in the line A B, when the point F is between D and G; and in\nthe line B C, when the point F is between G and E.\n\n[143] That is, the soul of the world.\n\n[144] In Tim\u00e6o.\n\n[145] The ellipsis.\n\n[146] The cissoid. For the properties of this curve, see Dr. Wallis\u2019s\ntreatise on the cycloid, p. 81.\n\n[147] The conchoid.\n\n[148] Thus, a right line, when considered as the side of a\nparallelogram, moving circularly, generates a cylindrical superficies:\nwhen moving circularly, as the side of a triangle, a conical surface;\nand so in other lines, the produced superficies varying according to\nthe different positions of their generative lines.\n\n[149] Inv ii. De Rep.\n\n[150] In multis locis.\n\n[151] This definition is the same with that which Mr. Simson has\nadopted instead of Euclid\u2019s, expressed in different words: for he says,\n\u201ca plane superficies is that in which any two points being taken, the\nstraight line between them lies wholly in that superficies.\u201d But he\ndoes not mention to whom he was indebted for the definition; and this,\ndoubtless, because he considered _it was not worth while to relate\nthe trifles of Proclus at full length_: for these are his own words,\nin his note to proposition 7, book i. Nor has he informed us in what\nrespect Euclid\u2019s definition is _indistinct_.\n\n[152] In the Greek \u1f10\u03bd\u03bd\u03bf\u03b9\u1f70\u03c2, but it should doubtless be read \u03b5\u1f30\u03ba\u03cc\u03bd\u03b1\u03c2,\n_images_, as in the translation of Barocius.\n\n[153] Mr. Simson, in his note on this definition, supposes it to\nbe the addition of some less skilful editor; on which account, and\nbecause it is quite useless (in his opinion) he distinguishes it from\nthe rest by inverted double commas. But it is surely strange that the\ndefinition of angle in general should be accounted useless, and the\nwork of an unskilful geometrician. Such an assertion may, indeed, be\nvery suitable to a professor of experimental philosophy, who considers\nthe useful as inseparable from practice; but is by no means becoming\na restorer of the liberal geometry of the ancients. Besides, Mr.\nSimson seems continually to forget that Euclid was of the Platonic\nsect; and consequently was a philosopher as well as a mathematician.\nI only add, that the commentary on the present definition is, in my\nopinion, remarkably subtle and accurate, and well deserves the profound\nattention of the greatest geometricians.\n\n[154] For a philosophical discussion of the nature of quality and\nquantity, consult the Commentaries of Ammonius, and Simplicius on\nAristotle\u2019s Categories, Plotinus on the genera of beings, and Mr.\nHarris\u2019s Philosophical Arrangements.\n\n[155] That is, the ellipsis.\n\n[156] That is, they are either right, acute, or obtuse.\n\n[157] This oracle is not mentioned by any of the collectors of the\nZoroastrian oracles.\n\n[158] This, indeed, must always be the case with those geometricians,\nwho are not at the same time, philosophers; a conjunction no less\nvaluable than rare. Hence, from their ignorance of principles and\nintellectual concerns, when any contemplative enquiry is proposed,\nthey immediately ask, in what its utility consists; considering every\nthing as superfluous, which does not contribute to the solution of some\npractical problem.\n\n[159] Concerning the soul\u2019s descent into body, see lib. ix., Ennead\niv. of Plotinus; and for the method by which she may again return to\nher pristine felicity, study the first book of Porphyry\u2019s Treatise on\nAbstinence.\n\n[160] This Definition too, is marked by Mr. Simson with inverted\ncommas, as a symbol of its being interpolated. But for what reason\nI know not, unless because it is useless, that is, because it is\n_philosophical_!\n\n[161] That is, the various species of forms, with which the four\nelements are replete.\n\n[162] That is, the circle.\n\n[163] An admirer of the moderns, and their pursuits, will doubtless\nconsider all this as the relics of heathen superstition and ignorance;\nand will think, perhaps, he makes a great concession in admitting\nthe existence of one supreme god, without acknowledging a multitude\nof deities subordinate to the first. For what the ancients can urge\nin defence of this obsolete opinion, I must beg leave to refer the\nreader to the dissertation prefixed to my translation of Orpheus; in\naddition to which let him attend to the following considerations. Is it\npossible that the machinery of the gods in Homer could be so beautiful,\nif such beings had no existence? Or can any thing be beautiful which\nis destitute of all reality? Do not things universally please in\nproportion as they resemble reality? Perhaps it will be answered,\nthat the reverse of this is true, and that fiction more generally\npleases than truth, as is evident from the great avidity with which\nromances are perused. To this I reply, that fiction itself ceases to be\npleasing, when it supposes absolute impossibilities: for the existence\nof genii and fairies cannot be proved impossible; and these compose all\nthe marvellous of romance. This observation is verified in Spencer\u2019s\nFairy Queen: for his allegories, in which the passions are personified,\nare tedious and unpleasant, because they are not disguised under the\nappearance of reality: while the magic of Circe, the bower of Calypso,\nthe rocks of Scylla and Charybdis, and the melody of the Syrens, in the\nOdyssey of Homer, though nothing but allegories, universally enchant\nand delight, because they are covered with the semblance of truth.\nIt is on this account that Mikon\u2019s battles in heaven are barbarous\nand ridiculous in the extreme; for every one sees the impossibility\nof supposing gun-powder and cannons in the celestial regions: the\nmachinery is forced and unnatural, contains no elegance of fancy, and\nis not replete with any mystical information. On the contrary, Homer\u2019s\nmachinery is natural and possible, is full of dignity and elegance, and\nis pregnant with the sublimest truths; it delights and enobles the mind\nof the reader, astonishes him with its magnificence and propriety, and\nanimates him with the fury of poetic inspiration. And this, because it\nis possible and true.\n\n[164] The sentence within the brackets is omitted in the Greek.\n\n[165] That is, the circular form proceeds from _bound_, but\nright-lined figures from _infinity_.\n\n[166] That is, the number three.\n\n[167] In Tim\u00e6o.\n\n[168] \u03c0\u03c1\u1f78\u03c2 \u1f43, or, _to which_, is wanting in the original, and in\nall the published collections of the Zoroastrian oracles.\n\n[169] That is Jupiter, who is called triadic, because he proceeds from\nSaturn and Rhea; and because his government is participated by Neptune\nand Pluto, for each of these is called Jupiter by Orpheus.\n\n[170] This sentence, within the brackets, is omitted in the printed\nGreek.\n\n[171]\n\n  [Illustration: Fig. I.]\n\n  [Illustration: Fig. II.]\n\nThus let a part A E B cut off by the diameter A B (fig. I.) of the\ncircle A E B D be placed on the other part A D B, as in fig. II.. Then,\nif it is not equal to the other part, either A E B will fall within A D\nB, or A D B within A E B: but in either case, C E will be equal to C D,\nwhich is absurd.\n\n[172] This objection is urged by Philoponus, in his book against\nProclus on the eternity of the world; but not, in my opinion, with\nany success. See also Simplicius, in his third digression against\nPhiloponus, in his commentary on the 8th book of Aristotle\u2019s Physics.\n\n[173] This definition is no where extant but in the commentaries of\nProclus. Instead of it, in almost all the printed editions of Euclid,\nthe following is substituted. _A segment of a circle is the figure\ncontained by a diameter, and the part of the circumference cut off by\nthe diameter._ This Mr. Simson has marked with commas, as a symbol\nof its being interpolated: but he has taken no notice of the different\nreading in the commentaries of Proclus. And what is still more\nremarkable, this variation is not noticed by any editor of Euclid\u2019s\nElements, either ancient or modern.\n\n[174] As in every hyperbola.\n\n[175] The Platonic reader must doubtless be pleased to find that Euclid\nwas deeply skilled in the philosophy of Plato, as Proclus every where\nevinces. Indeed, the great accuracy, and elegant distribution of these\nElements, sufficiently prove the truth of this assertion. And it is no\ninconsiderable testimony in favour of the Platonic philosophy, that its\nassistance enabled Euclid to produce such an admirable work.\n\n[176] Concerning these crowns, or annular spaces, consult the great\nwork of that very subtle and elegant mathematician Tacquet, entitled\n_Cylindrica et Annularia_.\n\n[177] In the preceding tenth commentary.\n\n[178] This in consequence of every triangle possessing angles alone\nequal to two right.\n\n[179] This too, follows from the same cause as above.\n\n[180] Thus the following figure A B D C has four sides, and but three\nangles.\n\n  [Illustration]\n\n[181] The Greek in this place is very erroneous, which I have restored\nfrom the version of Barocius.\n\n[182] For the Greek word \u1fe5\u03cc\u03bc\u03b2\u03bf\u03c2 is derived from the verb \u1fe5\u03ad\u03bc\u03b2\u03c9, which\nsignifies to have a circumvolute motion.\n\n[183] See the Orphic Hymns of Onomacritus to these deities; my\ntranslation of which I must recommend to the English reader, because\nthere is no other.\n\n[184] These twelve divinities, of which Jupiter is the head, are,\n_Jupiter_, _Neptune_, _Vulcan_, _Vesta_, _Minerva_, _Mars_, _Ceres_,\n_Juno_, _Diana_, _Mercury_, _Venus_, and _Apollo_. The first triad of\nthese is demiurgic, the second comprehends guardian deities, the third\nis vivific, or zoogonic, and the fourth contains elevating gods. But,\nfor a particular theological account of these divinities, study Proclus\non Plato\u2019s Theology, and you will find their nature unfolded, in page\n403, of that admirable work.\n\n[185] For it is easy to conceive a cylindric spiral described about a\nright-line, so as to preserve an equal distance from it in every part;\nand in this case the spiral and right-line will never coincide though\ninfinitely produced.\n\n[186]\n\n  [Illustration]\n\nAs the conchoid is a curve but little known, I have subjoined the\nfollowing account of its generation and principal property. In\nany given right line A P, call P the pole, A the vertex, and any\nintermediate point C the centre of the conchoid: likewise, conceive\nan infinite right line C H, which is called a rule, perpendicular to\nA P. Then, if the right line A _p_ continued at _p_ as much\nas is necessary, is conceived to be so turned about the abiding pole\n_p_, that the point C may perpetually remain in the right line C\nH, the point A will describe the curve A _o_, which the ancients\ncalled a conchoid.\n\nIn this curve it is manifest (on account of the right line P O, cutting\nthe rule in H that the point _o_ will never arrive at rule C H;\nbut because _h_ O is perpetually equal to C A, and the angle of\nsection is continually more acute, the distance of the point O from C\nH will at length be less than any given distance, and consequently the\nright line C H will be an asymptote to the curve A O.\n\nWhen the pole is at P, so that P C is equal to C A, the conchoid A O\ndescribed by the revolution of P A, is called a primary conchoid, and\nthose described from the poles _p_, and _\u03c0_, or the curves\nA _o_, A _\u03c9_, secondary conchoids; and these are either\ncontracted or protracted, as the eccentricity P C, is greater or less\nthan the generative radius C A, which is called the altitude of the\ncurve.\n\nNow, from the nature of the conchoid, it may be easily inferred, that\nnot only the exterior conchoid A _\u03c9_ will never coincide with\nthe right line C H, but this is likewise true of the conchoids A O, A\n_o_; and by infinitely extending the right-line A _\u03c0_, an\ninfinite number of conchoids may be described between the exterior\nconchoid A _\u03c9_, and the line C H, no one of which shall ever\ncoincide with the asymptote C H. And this paradoxical property of\nthe conchoid which has not been observed by any mathematician, is a\nlegitimate consequence of the infinite divisibility of quantity. Not,\nindeed, that quantity admits of an actual division in infinitum, for\nthis is absurd and impossible; but it is endued with an unwearied\ncapacity of division, and a power of being diffused into multitude,\nwhich can never be exhausted. And this infinite capacity which it\npossesses arises from its participation of the indefinite duad; the\nsource of boundless diffusion, and innumerable multitude.\n\n  [Illustration]\n\nBut this singular property is not confined to the conchoid, but is\nfound in the following curve. Conceive that the right line A C which is\nperpendicular to the indefinite line X Y, is equal to the quadrantal\narch H D, described from the centre C, with the radius C D: then from\nthe same centre C, with the several distances C E, C F, C G, describe\nthe arches E _l_, F _n_, G _p_, each of which must be conceived equal\nto the first arch H D, and so on infinitely. Now, if the points H, _k_,\n_l_, _n_, _p_, be joined, they will form a curve line, approaching\ncontinually nearer to the right-line A B (parallel to C Y) but never\neffecting a perfect coincidence. This will be evident from considering\nthat each of the sines of the arches H D, _l_ E, _n_ F, &c. being less\nthan its respective arch, must also be less than the right-line A C,\nand consequently can never coincide with the right-line A B.\n\nBut if other arches D _i_, E _m_, F _o_, &c. each of them equal to\nthe right-line A C, and described from one centre, tangents to the\nformer arches H D, _l_ E, _n_ F, &c. be supposed; it is evident that\nthe points H, _i_, _m_, _o_, &c. being joined, will form a curve line,\nwhich shall pass beyond the former curve, and converge still nearer\nto the line A B, without a possibility of ever becoming coincident:\nfor since the arches D _i_, E _m_, F _o_, &c. have less curvature than\nthe former arches, but are equal to them in length, it is evident that\nthey will be subtended by longer lines, and yet can never touch the\nright-line A B. In like manner, if other tangent arches be drawn to the\nformer, and so on infinitely, with the same conditions, an infinite\nnumber of curve-lines will be formed, each of them passing between\nH _p_ and A B, and continually diverging from the latter, without a\npossibility of ever coinciding with the former. This curve, which I\ninvented some years since, I suspect to be a parabola; but I have not\nyet had opportunity to determine it with certainty.\n\n\nTranscriber\u2019s Notes:\n\n1. Obvious printers\u2019, punctuation and spelling errors have been\ncorrected silently.\n\n2. Where hyphenation is in doubt, it has been retained as in the\noriginal.\n\n3. Some hyphenated and non-hyphenated versions of the same words have\nbeen retained as in the original.\n\n4. The errata have been soilently corrected.\n\n5. Italics are shown as _xxx_.","title":"The philosophical and mathematical commentaries of Proclus on the first book of Euclid's elements (Vol. 1 of 2) : $b To which are added, A history of the restoration of Platonic theology, by the latter Platonists: And a translation from the Greek of Proclus's Theological elements","tradition":"philosophy-ethics","translated_text":null,"word_count":115861}