{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_54546/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_54546","id":"gutenberg_54546","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/54546","summary":null,"tags":["Category: Engineering & Technology","Category: History - Ancient","Category: History - Modern (1750+)","Category: Philosophy & Ethics","Category: Science - Physics","Clocks and watches","Time"],"text":"Note: Project Gutenberg also has an HTML version of this\n      file which includes the original illustrations.\n      See 54546-h.htm or 54546-h.zip:\n      (https://www.gutenberg.org/cache/epub/54546/pg54546-images.html)\n      or\n      (https://www.gutenberg.org/files/54546/54546-h.zip)\n\n\n      Images of the original pages are available through\n      Internet Archive. See\n      https://archive.org/details/timeclocksdescri00cuny\n\n\nTranscriber\u2019s note:\n\n      Text enclosed by underscores is in italics (_italics_).\n      However, to improve readability, italics notation has been\n      omitted from equations. If the variables are in italics in\n      the surrounding text, they were in italics in the equations.\n\n\n\n\n\nTIME AND CLOCKS.\n\n\n[Illustration:\n\n  [_Frontispiece._\n\nNUREMBERG CLOCK. CONVERTED FROM A VERGE ESCAPEMENT TO A PENDULUM\nMOVEMENT.]\n\n\nTIME AND CLOCKS:\n\nA Description of Ancient and Modern Methods of\nMeasuring Time.\n\nby\n\nH. H. CUNYNGHAME M.A. C.B. M.I.E.E.\n\nWith Many Illustrations.\n\n\n\n\n\n\nLondon:\nArchibald Constable & Co. Ltd.\n16 James Street Haymarket.\n1906.\n\nBradbury, Agnew, & Co. Ld., Printers,\nLondon and Tonbridge.\n\n\n\n\nCONTENTS.\n\n\n                                                   PAGE\n  INTRODUCTION                                        1\n\n  CHAPTER   I.                                        7\n\n  CHAPTER  II.                                       50\n\n  CHAPTER III.                                       90\n\n  CHAPTER  IV.                                      123\n\n  APPENDIX ON THE SHAPE OF THE TEETH OF WHEELS      187\n\n  INDEX                                             199\n\n\n\n\nTIME AND CLOCKS.\n\n\n\n\nINTRODUCTION.\n\n\nWhen we read the works of Homer, or Virgil, or Plato, or turn to the\nlater productions of Dante, of Shakespeare, of Milton, and the host\nof writers and poets who have done so much to instruct and amuse us,\nand to make our lives good and agreeable, we are apt to look with some\ndisappointment upon present times. And when we turn to the field of art\nand compare Greek statues and Gothic or Renaissance architecture with\nour modern efforts, we must feel bound to admit our inferiority to our\nancestors. And this leads us perhaps to question whether our age is the\nequal of those which have gone before, or whether the human intellect\nis not on the decline.\n\nThis feeling, however, proceeds from a failure to remember that each\nage of the world has its peculiar points of strength, as well as of\nweakness. During one period that self-denying patriotism and zeal\nfor the common good will be developing, which is necessary for the\nformation of society. During another, the study of the principles of\nmorality and religion will be in the ascendant. During another the arts\nwill take the lead; during another, poetry, tragedy, and lyric poetry\nand prose will be cultivated; during another, music will take its turn,\nand out of rude peasant songs will evolve the harmony of the opera.\n\nTo our age is reserved the glory of being easily the foremost in\nscientific discovery. Future ages may despise our literature,\nsurpass us in poetry, complain that in philosophy we have done\nnothing, and even deride and forget our music; but they will only\nbe able to look back with admiration on the band of scientific\nthinkers who in the seventeenth century reduced to a system the\nlaws that govern the motions of worlds no less than those of atoms,\nand who in the eighteenth and nineteenth founded the sciences of\nchemistry, electricity, sound, heat, light, and who gave to mankind\nthe steam-engine, the telegraph, railways, the methods of making\nhuge structures of iron, the dynamo, the telephone, and the thousand\napplications of science to the service of man.\n\nAnd future students of history who shall be familiar with the\nconditions of our life will, I think, be also struck with surprise at\nour estimate of our own peculiar capabilities and faculties. They will\nnote with astonishment that a gentleman of the nineteenth century, an\nage mighty in science, and by no means pre-eminent in art, literature\nand philosophy, should have considered it disgraceful to be ignorant\nof the accent with which a Greek or a Roman thought fit to pronounce\na word, should have been ashamed to be unable to construe a Latin\naphorism, and yet should have considered it no shame at all not to\nknow how a telephone was made and why it worked. They will smile when\nthey observe that our highest university degrees, our most lucrative\nrewards, were given for the study of dead languages or arch\u00e6ological\ninvestigations, and that science, our glory and that for which we have\nshown real ability, should only have occupied a secondary place in our\neducation.\n\nThey will smile when they learn that we considered that a knowledge\nof public affairs could only be acquired by a grounding in Greek\nparticles, or that it could ever have been thought that men could not\ncommand an army without a study of the tactics employed at the battle\nof Marathon.\n\nBut the battle between classical and scientific education is not in\nreality so much a dispute regarding subjects to be taught, as between\nmethods of teaching. It is possible to teach classics so that they\nbecome a mental training of the highest value. It is possible to teach\nscience so that it becomes a mere enslaving routine.\n\nThe one great requirement for the education of the future is firmly to\ngrasp the fact that a study of words is not a study of things, and that\na man cannot become a carpenter merely by learning the names of his\ntools.\n\nIt was the mistake of the teachers of the Middle Ages to believe that\nthe first step in knowledge was to get a correct set of concepts of\nall things, and then to deduce or bring out all knowledge from them.\nAdmirable plan if you can get your concepts! But unfortunately concepts\ndo not exist ready made\u2014they must be grown; and as your knowledge\nincreases, so do your concepts change. A concept of a thing is not a\nmere definition, it is a complete history of it. And you must build up\nyour edifice of scientific knowledge from the earth, brick by brick and\nstone by stone. There is no magic process by which it can with a word\nbe conjured into existence like a palace in the Arabian Nights.\n\nFor nothing is more fatal than a juggle with words such as force,\nweight, attraction, mass, time, space, capacity, or gravity. Words are\nlike purses, they contain only as much money as you put into them. You\nmay jingle your bag of pennies till they sound like sovereigns, but\nwhen you come to pay your bills the difference is soon discovered.\n\nThis fatal practice of learning words without trying to obtain a\nclear comprehension of their meaning, causes many teachers to use\nmathematical formul\u00e6 not as mere steps in a logical chain, but like\nmagical chaldrons into which they put the premises as the witches put\nherbs and babies\u2019 thumbs into their pots, and expect the answers to\nrise like apparitions by some occult process that they cannot explain.\nThis tendency is encouraged by foolish parents who like to see their\ninfant prodigies appear to understand things too hard for themselves,\nand look on at their children\u2019s lessons in mathematics like rustics\ngaping at a fair. They forget that for the practical purposes of\nlife one thing well understood is worth a whole book-full of muddled\nill-digested formul\u00e6. Unfortunately it is possible to cram boys up\nand run them through the examination sieves with the appearance of\nknowledge without its reality. If it were cricket or golf that were\nbeing tested how soon would the fraud be discovered. No humbug would be\npermitted in those interesting and absorbing subjects. And really, when\none reflects how easy it is to present the appearance of book knowledge\nwithout the reality, one can hardly blame those who select men for\nservice in India and Egypt a good deal for their proficiency in sports\nand games. Better a good cricketer than a silly pedant stuffed full of\nlearning that \u201clies like marl upon a barren soil encumbering what is\nnot in its power to fertilize.\u201d\n\nAnother kindred error is to expect too much of science. For with all\nour efforts to obtain a further knowledge of the mysteries of nature,\nwe are only like travellers in a forest. The deeper we penetrate it,\nthe darker becomes the shade. For science is \u201cbut an exchange of\nignorance for that which is another kind of ignorance\u201d[A] and all\nour analysis of incomprehensible things leads us only to things more\nincomprehensible still.\n\n    [A] _Manfred_, Act II., scene iv.\n\nIt is, therefore, by the firm resolution never to juggle with words or\nideas, or to try and persuade ourselves or others that we understand\nwhat we do not understand, that any scientific advance can be made.\n\n\n\n\nCHAPTER I.\n\n\nAll students of any subject are at first apt to be perplexed with the\nnumber and complication of the new ideas presented to them.\n\nThe need of comprehending these ideas is felt, and yet they are\ndifficult to grasp and to define. Thus, for instance, we are all apt\nto think we know what is meant when force, weight, length, capacity,\nmotion, rest, size, are spoken of. And yet when we come to examine\nthese ideas more closely, we find that we know very little about\nthem. Indeed, the more elementary they are, the less we are able to\nunderstand them.\n\nThe most primordial of our ideas seem to be those of number and\nquantity; we can count things, and we can measure them, or compare\nthem with one another. Arithmetic is the science which deals with the\nnumbers of things and enables us to multiply and divide them. The\nestimation of quantities is made by the application of our faculty of\ncomparison to different subjects. The ideas of number and quantity\nappear to pervade all our conceptions.\n\nThe study of natural phenomena of the world around us is called the\nstudy of physics from the Greek word \u03c6\u03c5\u03c3\u03af\u03c2 or \u201cinanimate nature,\u201d\nthe term physics is usually confined to such part of nature as is not\nalive. The study of living things is usually termed biology (from \u03b2\u03b9\u03b1,\nlife).\n\nIn the study of natural phenomena there are, however, three ideas which\noccupy a peculiar and important position, because they may be used\nas the means of measuring or estimating all the rest. In this sense\nthey seem to be the most primitive and fundamental that we possess.\nWe are not entitled to say that all other ideas are formed from and\ncompounded of these ideas, but we are entitled to say that our correct\nunderstanding of physics, that is of the study of nature, depends\nin no slight degree upon our clear understanding of them. The three\nfundamental ideas are those of space, time and mass.\n\nSpace appears to be the universal accompaniment of all our impressions\nof the world around us. Try as we may, we cannot think of material\nbodies except in space, and occupying space. Though we can imagine\nspace as empty we cannot conceive it as destroyed. And this space has\nthree dimensions, length, breadth measured across or at right angles to\nlength, and thickness measured at right angles to length and breadth.\nMore dimensions than this we cannot have. For some inscrutable reason\nit has been arranged that space shall present these three dimensions\nand no more. A fourth dimension is to us unimaginable\u2014I will not say\ninconceivable\u2014we can conceive that a world might be with space in four\ndimensions, but we cannot imagine it to ourselves or think what things\nwould be like in it.\n\nWith difficulty we can perhaps imagine a world with space of only two\ndimensions, a \u201cflat land,\u201d where flat beings of different shapes,\nlike figures cut out of paper, slide or float about on a flat table.\nThey could not hop over one another, for they would only have length\nand breadth; to hop up you would want to be able to move in a third\ndimension, but having two dimensions only you could only slide forward\nand sideways in a plane. To such beings a ring would be a box. You\nwould have to break the ring to get anything out of it, for if you\ntried to slide out you would be met by a wall in every direction. You\ncould not jump out of it like a sheep would jump out of a pen over the\nhurdles, for to jump would require a third dimension, which you have\nnot got. Beings in a world with one dimension only would be in a worse\nplight still. Like beads on a string they could slide about in one\ndirection as far as the others would let them. They could not pass one\nanother. To such a being two other beings would be a box one on each\nside of him, for if thus imprisoned, he could not get away. Like a\nwaggon on a railway, he could not walk round another waggon. That would\nwant power of moving in two dimensions, still less could he jump over\nthem, that would want three.\n\nWe have not the smallest idea why our world has been thus limited. Some\nphilosophers think that the limitation is in us, not in the world, and\nthat perhaps when our minds are free from the limitations imposed by\ntheir sojourn in our bodies, and death has set us free, we may see not\nonly what is the length and breadth and height, but a great deal more\nalso of which we can now form no conception. But these speculations\nlead us out of science into the shadowy land of metaphysics, of which\nwe long to know something, but are condemned to know so little. Area\nis got by multiplying length by breadth. Cubic content is got by\nmultiplying length by breadth and by height. Of all the conceptions\nrespecting space, that of a line is the simplest. It has direction, and\nlength.\n\nThe idea of mass is more difficult to grasp than that of space. It\nmeans quantity of matter. But what is matter? That we do not know. It\nis not weight, though it is true that all matter has weight. Yet matter\nwould still have mass even if its property of weight were taken away.\n\nFor consider such a thing as a pound packet of tea. It has size, it\noccupies space, it has length, breadth, and thickness. It has also\nweight. But what gives it weight? The attraction of the earth. Suppose\nyou double the size of the earth. The earth being bigger would attract\nthe package of tea more strongly. The weight of the tea, that is, the\nattraction of the earth on the package of tea, would be increased\u2014the\ntea would weigh more than before. Take the package of tea to the planet\nJupiter, which, being very large, has an attraction at the surface 2\u00bd\ntimes that of the earth. Its size would be the same, but it would feel\nto carry like a package of sand. Yet there would be the same \u201cmass\u201d\nof tea. You could make no more cups of tea out of it in Jupiter than\non earth. Take it to the moon, and it would weigh a little over two\nounces, but still it would be a pound of tea. We are in the habit of\nestimating mass by its weight, and we do so rightly, for at any place\non the earth, as London, the weights of masses are always proportioned\nto the masses, and if you want to find out what mass of tea you have\ngot, you weigh it, and you know for certain. Hence in our minds we\nconfuse mass with weight. And even in our Acts of Parliament we have\ndone the same thing, so that it is difficult in the statutes respecting\nstandard weights to know what was meant by those who drew them up, and\nwhether a pound of tea means the _mass_ of a certain amount of tea or\nthe _weight_ of that mass. For accurate thinking we must, of course,\nalways deal with masses, not with weights. For so far as we can tell\n_mass_ appears indestructible. A mass is a mass wherever it is, and\nfor all time, whereas its weight varies with the attractive force of\nthe planet upon which it happens to be, and with its distance from that\nplanet\u2019s centre. A flea on this earth can skip perhaps eight inches\nhigh; put that flea on the moon, and with the expenditure of the same\nenergy he could skip four feet high. Put him on the planet Jupiter and\nhe could only skip 3\u2155 inches high. A man in a street in the moon could\njump up into a window on the first floor of a house. One pound of tea\ntaken to the sun would be as heavy as twenty-eight pounds of it at the\nearth\u2019s surface; and weight varies at different parts of the earth.\nHence the true measure of quantity of matter is mass, not weight.\n\nThe mass of bodies varies according to their size; if you have the same\nnature of material, then for a double size you have a double mass. Some\nbodies are more concentrated than others, that is to say, more dense;\nit is as though they were more tightly squeezed together. Thus a ball\nof lead of an inch in diameter contains forty-eight times as much mass\nas a ball of cork an inch in diameter. In order to know the weight of a\ncertain mass of matter, we should have to multiply the mass by a figure\nrepresenting the attractive force or pull of the earth.\n\nIn physics it is usual to employ the letters of the alphabet as a sort\nof shorthand to represent words. So that the letter _m_ stands for\nthe mass of a body. So again _g_ stands for the attractive pull of the\nearth at a given place. _w_ stands for the weight of the body. Hence\nthen, since the weight of a body depends on its mass and also on the\nattractive pull of the earth, we express this in short language by\nsaying, w = m \u00d7 g; or _w_ is equal to _m_ multiplied by _g_; the symbol\n= being used for equality, and \u00d7 the sign of multiplication. In common\nuse \u00d7 is usually omitted, and when letters are put together they are\nintended to be understood as multiplied. So that this is written\n\n                                w = mg.\n\nOf course by this equation we do not mean that weight is mass\nmultiplied into the force of gravity, we only mean that the number of\nunits of weight is to be found by multiplying the number of units of\nmass into the number of units of the earth\u2019s force of gravity.\n\nIn the same way, if when estimating the number of waggons, _w_, that\nwould be wanted for an army of men, _n_, which consumed a number of\npounds, _p_, of provisions a day, we might put\n\n                                w = np.\n\nBut this would not mean that we were multiplying soldiers into food\nto produce waggons, but only that we were performing a numerical\ncalculation.\n\nTime is one of the most mysterious of our elementary ideas. It seems to\nexist or not to exist, according as we are thinking or not thinking.\nIt seems to run or stand still and to go fast or slowly. How it drags\nthrough a wearisome lesson; how it flies during a game of cricket; how\nit seems to stop in sleep. If we measured time by our own thoughts it\nwould be a very uncertain quantity. But other considerations seem to\nshow us that Nature knows no such uncertainty as regards time, that she\nproduces her phenomena in a uniform manner in uniform times, and that\ntime has an existence independent of our thoughts and wills.\n\nThe idea of a state of things in which time existed no more was quite\nfamiliar to medi\u00e6val thinkers, and was regarded by many of them as the\ncondition that would exist after the Day of Judgment. In recent times\nKant propounded the theory that time was only a necessary condition of\nour thoughts, and had no existence apart from thinking beings\u2014in fact,\nthat it was our way of looking at things.\n\nScientifically, however, we are warranted in treating time as perfectly\nreal and capable of the most exact measurement. For example, if we\narrange a stream of sand to run out of an orifice, and observe how\nmuch will run out while an egg is being boiled hard, we find as a fact\nthat if the same quantity of sand runs out, the state of the egg is\nuniform. If we walk for an hour by a watch, we find that we can go\nhalf the distance that we should if we walked two hours. It is the\ncorrespondence of these various experiments that gives us faith in the\ntreatment of time as a thing existing independently of ourselves, or,\nat all events, independent of our transient moods.\n\nThe ideas of time acquired by the races of men that first evolved from\na state of barbarism were no doubt derived from the observation of day\nand night, the month and the year.\n\n[Illustration: FIG. 1.]\n\nFor, suppose that a shepherd were on the plains of Chaldea, or perhaps\non those mountains of India known as the roof of the world, which\naccording to some arch\u00e6ologists was the site of the garden of Eden and\nthe early home of the European race, what would he see?\n\nHe would see the sun rise in the east, slowly mount the heavens till it\nstood over the south at middle day, then it would sink towards the west\nand disappear. In summer the rising point of the sun would be more to\nthe northward than in winter, and so also would be its point of setting\n_A\u00b4_. In winter it would rise a little to the south of east, and set\na little to the south of west, and not rise so high in the heavens at\nmidday, so that the summer day would be longer than the winter day.\nIf the day were always divided into twelve hours, whether it were long\nor short, then in summer the hours of the day would be long; in winter\nthey would be short. This mode of dividing the day was that used by\nthe Greeks. The Egyptians, on the other hand, averaged their day by\ndividing the whole round of the sun into twenty-four hours, so that the\nsummer day contained more hours than the winter day. Hence, for the\nEgyptians, sun-rise did not always take place at six o\u2019clock. For in\nwinter it took place after six, and in summer before six; and this is\nthe system that has descended to us.\n\nThe moon also would rise at different places, varying between _A_ and\n_B_, and set at places varying between _A\u00b4_ and _B\u00b4_, but these would\nbe independent of those at which the sun rose and set.\n\nMoreover, the moon each day would appear to get further and further\naway from the sun in the direction of the arrow, as shown in the\nsketch. If the moon rose an hour after the sun on one day, the next day\nit would rise more than two hours after the sun, and so on. This delay\nin rising of the moon would go on day by day till at last she came\nright round to the sun again, as shown at _M\u00b4_. And in her path she\nwould change her form from a crescent, as at _M_, up to a full moon,\nwhen she would be half way round from the sun, that is, when she would\nrise twelve hours after him, or just be rising as the sun set. This\ndelay and accompanying change of form would go on, till after three\nweeks she would have got round to a position _A\u00b4_, when she would rise\neighteen hours after the sun, and have become a crescent with her back\nto the sun; in fact, she would always turn her convex side to the sun.\nAt length, when twenty-eight days had passed, she would be round again\nabout opposite to the sun, and consequently her pale light would be\nextinguished in his beams, and she would gradually reappear as a new\nmoon on the other side of him. This series of changes of the moon takes\nplace once every twenty-eight days, and is called a lunar or \u201cmoon\u201d\nmonth, and was used as a division of time by very early nations. The\nchanges of the seasons recurred with the changes in the times of rising\nof the sun, and took a year to bring about. And there were nearly\nthirteen moon changes in the year.\n\nIt was also observed that during its cycle of changes, the sun was\nslowly moving round backwards among the stars in the same direction as\nthe moon, only it made its retrograde cycle in a year, and thus arose\nthe division of time into months and years. The stars turned round in\nthe heavens once in the complete day. The sun, therefore, appeared to\nmove back among them, passing successively through groups of stars, so\nas to make the circuit of them all in a year. The stars through which\nhe passed in a year, and through which the moon travelled in a month,\nwere divided by the ancients into groups called constellations, and its\nyearly path in the heavens was called the zodiac. There were twelve of\nthese constellations in the zodiac called the signs. Hence, then, the\nsun passed through a sign in every month, making the tour of them all\nin the year. To these signs fanciful names were given, such as \u201cthe\nRam,\u201d \u201cthe Water-bearer,\u201d \u201cthe Virgin,\u201d \u201cthe Scorpion,\u201d and so on, and\nthe sun and moon were then said to pass through the signs of the zodiac.\n\nHence, since the path of the sun marked the year, you could tell the\nseasons by knowing what sign of the zodiac the sun was in. The age of\nthe moon was easily known by her form.\n\nWhen the winter was over, then, just as the sun set the dog star would\nbe rising in the east, and this would show that the spring was at hand.\nThen the peasants prepared to till the earth and sow the seed and lead\nthe oxen out to pasture, and celebrated with joyful mirth the glad\nadvent of the spring, corresponding to our Easter, when the sun had\nrun through three constellations of the zodiac. Then came the summer\nheat, and with many a mystic rite they celebrated Midsummer\u2019s Day. In\nautumn, after three more signs of the zodiac have been traversed by the\nsun, the sun again rises exactly in the east and sets in the west, and\nthe days and nights are equal. This is the autumnal equinox, and was\nonce celebrated by the feast which we now know as Michaelmas Day, and\nthe goose is the remnant of the ancient festival.\n\n[Illustration: FIG. 2.]\n\nAnd the great winter feast of the ancients is now known to us as\nChristmas, and chosen to celebrate the birth of our Lord; for when\nChristianity came into the world and the heathens were converted, the\nold feast days were deliberately changed into Christian festivals.\n\nTo us, therefore, the whole heavens, and the fixed stars with them,\nappear to turn from east to west, or from left to right, as we look\ntowards the south, as shown by the big arrow. But the moon and sun,\nthough apparently placed in the heavens, move backwards among the fixed\nstars, as shown by the small arrows. The sun moves at such a rate that\nhe goes round the circle of the heavens in a year of three hundred and\nsixty-five days. The moon goes round the circle in twenty-eight and a\nhalf days, or a lunar month. Of course, in reality the sun is at rest,\nand it is the earth that moves round the sun and spins on its axis\nas it moves. But it will presently be shown that the appearance to a\nperson on the earth is the same whether the earth goes round the sun or\nthe sun round the earth.\n\nFrom the works of Greek writers we know a good deal about the ideas of\nthe world that were entertained by the ancients. The most early notions\nwere, of course, connected with the worship of the gods. The sun was\nconsidered as a huge light carried in a chariot, driven by Apollo, with\nfour spirited steeds. It descended to the ocean when the day declined,\nand then the horses were unyoked by the nymphs of the ocean and led\nround to the east, so as to be ready for the journey of the following\nday. The Egyptians figured the sun as placed in a boat which sailed\nover the heavens. At night the sun god descended into the infernal\nregions, carrying with him the souls of those who had died during the\nday. There they passed through different regions of hell, with portals\nguarded by hideous monsters. Those who had well learned the ritual of\nthe dead knew the words of power wherewith to appease the demons. Those\nunprovided with the watchwords were subjected to terrible dangers.\nThen the soul appeared before Minos, and was weighed and dealt with\naccording to its deserts.\n\n[Illustration: FIG. 3.]\n\nThe earth was considered as a huge island in the midst of a circular\nsea. Gradually, however, astronomical ideas became subjected to\nscience. One of the first truths that dawned on astronomers was the\nfact that the earth was a sphere. For they noticed that as people\nwent further and further to the north, the elevation of the sun at\nmidday above the horizon became smaller and smaller. This can easily\nbe seen from the diagram. When an observer is at _A_ the sun appears\nat an altitude above the horizon equal to the angle \u03b1, but as he goes\nalong the curved surface of the earth to a point _B_ nearer to the\nnorth pole, the sun appears to be lower and only to have an altitude\n\u03b2. From this it was easy for men so skilled in geometry as the Greeks\nto calculate how big the earth was. They did so, and it appeared to\nhave the enormous diameter of 8,000 miles. They only knew quite a\nsmall portion of it. They thought that the rest was ocean. But they\nhad, of course, a clear idea of the \u201cantipodes\u201d or up-side-down side\nof it, and they believed that if men were on the other side of it\nthat their feet must all point towards its centre. From this they\ngot the idea of the centre of the earth as a point of attraction for\nall things that had an earth-seeking or earthy nature. Fire appeared\nalways to desire to go upwards, so they thought that fire had an\nearth-repellent, heaven-seeking character. Water they thought partly\nearth-seeking, partly heaven-seeking, for it appeared in the ocean or\nfloated as clouds. Air they thought to be indifferent. And out of the\nfour elements fire, water, earth, and air they believed the world was\nmade. The earth they thought must be at rest; for if it was in motion\nthings would fly off from it. They saw that either the sun must be\nmoving round the earth, or else the earth must be turning on its axis.\nThey chose the former hypothesis, because they argued that if the earth\nwere twisting round once in twenty-four hours then such a country as\nGreece must be flying round like a spot on the surface of a top, at the\nrate of about 18,000 miles in twenty-four hours, that is, at the rate\nof about 180 yards in a second, or faster than an arrow from a bow.\nBut if that was the case then a bird that flew up from the earth would\nbe left far behind. If a ball were thrown up it would fall hundreds\nof yards behind the person who threw it. They could not conceive how\nit was possible for a ball thrown up by someone standing on a moving\nobject not to fall behind the thrower.\n\nThis decided them in their error. The mistaken astronomy of the Greeks\nwas also much forwarded by Aristotle, the tutor of Alexander the Great.\nThis great genius in politics and philosophy was only in the second\nrank as a man of science, and, as I think, hardly equal to Archimedes\nor Hipparchus, or even to Ptolemy. Aristotle wrote a book concerning\nthe heavens which bristles with the most wantonly erroneous scientific\nideas, such as, for instance, that the motion of the heavenly bodies\nmust be circular because the most perfect curve is a circle, and\nsimilar assumptions as to the course of nature.\n\nThe earth, then, being fixed, they thought that the moon, the sun,\nand the seven planets were carried round it, fixed each of them in\nan enormous crystal spherical shell. These spheres, like coats of\nan onion, slid round one upon another, each carrying his celestial\nluminary. The moon was the nearest, then Mercury, then Venus, then the\nsun, then Mars, Jupiter and Saturn. Outside them was the sphere of the\nstars, and outside all the \u201c_primum mobile_,\u201d or great Prime Mover of\nthe universe. When one of the celestial bodies, such as the moon, got\nin front of another, such as the sun, there was an eclipse. They soon\nobserved that the moon derived its light from the sun. As they knew\nthe size of the earth, by comparison they got some vague idea of the\nhuge distances that the heavenly bodies must be from us. In fact, they\nmeasured the distance of the moon with approximate accuracy, making it\n240,000 miles, or about thirty times the earth\u2019s diameter.\n\nThis, of course, gave them the moon\u2019s diameter, for they were easily\nable to calculate how big an object must be, that looks as big as the\nmoon and is 240,000 miles away.\n\nThis large size of the moon gave them some idea of the distance of the\nsun, but they failed to realise how big and far away he really is.\n\nSeveral ancient nations used weeks as means of measuring time. They\nmade four weeks to the lunar month. The order of the days was rather\ncuriously arranged. For, assuming that the earth is the centre of the\nplanetary system, put the planets in a column, putting the nearest (the\nmoon) at the bottom and the furthest off at the top\u2014\n\n  Saturn,\n  Jupiter,\n  Mars,\n  The Sun,\n  Venus,\n  Mercury,\n  The Moon.\n\nThen divide the day into three watches of eight hours each, and let\neach watch be presided over by one of the planet-gods: begin with\nSaturn. We then have Saturn as the first god ruling Saturday, and\nJupiter and Mars, the two other gods, for that day. The first watch for\nSunday will be the sun; Venus and Mercury will preside over the next\ntwo watches of that day. The planet that will preside over the first\nwatch of the next day will be the moon, and the day will, therefore, be\ncalled Monday; Saturn and Jupiter will be the other gods for Monday.\nThe first watch of the next day will be presided over by Mars, and the\nday will, therefore, be called Mars-day or Mardi, or, in the Teutonic\nlanguages, Tuesday, after Tuesco, a Scandinavian god of war. Mercury\nwill give a name to Mercredi, or to Wednesday, or Wodin\u2019s-day. Jupiter\nto Jeudi, or \u201cThurs\u201d day. Venus to Vendredi, or in the Scandinavian,\nFriday, the day of the Scandinavian goddess Freya, the goddess of love\nand beauty, who corresponds to Venus, and thus the week is completed.\n\n[Illustration: FIG. 4.]\n\nThis weekly scheme came probably from the Chaldean astronomers. It\nappears probable that the great tower of Babel, the ruins of which\nexist to this day, consisted of seven stages, one over the other, the\ntop one painted white, or perhaps purple, to represent the Moon, the\nnext lower blue for Mercury, then green for Venus, yellow for the Sun,\nred for Mars, orange for Jupiter, and black for Saturn. Unfortunately,\nof the colours no trace now remains.\n\nBut nightly on the long terraces the Babylonian priests observed\neclipses and other celestial phenomena. Their records were afterwards\ntaken to Alexandria and kept in the great library that was subsequently\nburned by the Turks. In that library they were seen by the astronomer\nPtolemy, who used them in the writing of his work on astronomy called\nthe \u201cGreat Syntaxis\u201d or \u201cCollection.\u201d The original work perished, but\nit had been translated into Arabic by the Arab astronomers, who called\nit \u201cAl Magest,\u201d the Great Book. It was translated from Arabic into\nLatin, and remained the textbook for astronomers in Europe quite down\nto the time of Queen Elizabeth, when a better system took its place.\n\nFor the use of men engaged in practical astronomy, it is very\nconvenient to consider the sun, moon, stars, and planets as going\nround the earth at rest. For instance, seamen use the heavenly bodies\nas in a way hands of a huge clock from which they can know the time and\ntheir position on the earth. \u201cThe Nautical Almanac,\u201d which is printed\nyearly, gives the true position of these heavenly bodies for every\nhour, minute, and second of the year, and I will presently show how\nuseful this is to sailors.\n\nWe will deal with the sun first. From the motions of the sun we can\nobserve the time. This is done in every garden by means of sun-dials,\nand I will now describe how they are constructed. If a light, such as\nthe light of a candle, be moved round in a circle at a uniform pace so\nas to go round once in some given period, such as twenty-four hours, it\nis obvious that it would serve to measure time. Thus, for example, if a\nsheet of white paper be placed upon the table, and a pencil be stuck on\nto it upright with some sealing wax, or a pen propped up in an ink-pot,\nthen a candle held by anyone will cast the shadow of the pen on the\npaper.\n\n[Illustration: FIG. 5.]\n\nIf the person holding the candle walk round the table at a uniform\nspeed, the shadow will go round like the hand of a clock, and might be\nmade to mark the time. If the candle took twenty-four hours to go round\nthe table, as the sun takes twenty-four hours to go round the earth,\nthen marks placed on the paper would serve to measure the hours, and\nthe paper and pen would serve as a sort of sun-dial.\n\nBut the sun does not go round the earth as the candle round the\ntable. Its path is an inclined one, like that shown by the dotted\nline. Sometimes it is above the level of the table, sometimes below\nit. And, moreover, its winter path is different from its summer path.\nWhence then it follows that the hour-marks on the paper cannot be\nput equidistant like the hours on the dial of a clock, and that some\narrangement must be made so that the line as shown by the summer sun\nshall correspond with the time as shown by the winter sun.\n\n[Illustration: FIG. 6.]\n\nLet us suppose that _N O S_ is the axis of the heavens, and the lines\n_N A S_, _N B S_, _N C S_, are meridian lines drawn from one of the\npoles _N_ of the heavens round on the surface of a celestial sphere\nwhose centre is at _O_. Let _A B C_ be a circle also on this sphere,\npassing through _O_, the centre of the sphere, in a plane at right\nangles to _N S_, the axis. Then _A B C_ is called the equatorial. It\nis a circle in the heavens corresponding to the equator on the earth.\nAt the vernal and autumnal equinox, namely on March 25 and September\n25, the sun is in the equatorial. In midsummer and midwinter it is on\nopposite sides of the equatorial. In midsummer it is nearer to _N_, as\nat _V_; in midwinter it is nearer to _S_, as at _W_. Suppose we were\non an island in the midst of a surrounding ocean, we should only have\na limited range of view. If the highest point on the island were 100\nfeet, then from that altitude we should be able to see about thirteen\nmiles to the horizon. More than that could not be seen on account of\nthe rotundity of the earth.\n\nLet us suppose then such an island surrounded for thirteen miles\ndistant on every side by an ocean, and let us consider what would be\nthe apparent motions of the sun when seen from such an island. At the\nvernal and autumnal equinoxes, when the sun is on the equatorial, it\nwould appear to rise out of the ocean at a point _E_, due east; it\ntraverses half the equatorial and sets in the ocean at a point _W_, due\nwest. The day is twelve hours long, from 6 a.m. to 6 p.m.\n\n[Illustration: FIG. 7.]\n\nIn summer the sun is higher, and nearer to the pole _N_, say at a point\n_s_. It rises at a point _a_ in the ocean more to the north than _E_,\nthe eastern point, and sets at a point _b_, also more north than _W_,\nthe western point, and traverses the path _a s b_. But to traverse\nthis path it takes longer than twelve hours, for _a s b_ is more than\nhalf the circle _a s b_. Hence then it rises say at 4.30 a.m. and\nsets at 7.30 p.m. The night, during which the sun moves round the path\nfrom _b_ to _a_, is correspondingly short, being only nine hours in\nlength, from 7.30 p.m. till 4.30 a.m. So you have a long summer day and\na short summer night. But in winter, when the sun gets nearer to the\nsouth pole of the heavens, it rises at a point _C_ in the ocean at 7.30\na.m., and traverses the arc _c t d_, and sets at the point _d_ at 4.30\np.m. So that the winter day is only nine hours long. But the winter\nnight lasts from 4.30 p.m. till 7.30 a.m., and is therefore fifteen\nhours long, the sun going round the path _d r c_ in the interval. It is\ntherefore the obliquity of the poles _N S_, coupled with the fact that\nthe sun\u2019s position is nearer to one pole, _N_, in summer, and nearer to\nthe other pole, _S_, in winter, that produces the inequality of days\nand nights in our latitudes. Suppose our island were on the equator.\nThe north pole and the south pole would appear to be on the horizon,\nand then whether the sun moved in the circle _a s b_ in the summer, or\n_E S W_ at the vernal or autumnal equinoxes, or _c t d_ in the winter,\nin each of these cases, though the places of rising and setting in the\nocean might vary in summer from _a_ and _b_ to _c_ and _d_ in winter,\nyet in each of these cases the path from _a_ to _b_, _A_ to _B_, and\n_c_ to _d_ would still always be a half-circle and occupy twelve hours.\nHence at the equator the days and nights never vary in length, but the\nsun always rises at six and sets at six. And, besides, it always rises\nstraight up from the ocean and plunges down vertically into it, so that\nthere is but little twilight and dawn.\n\n[Illustration: FIG. 8.]\n\nBut now let us suppose we were living at the north pole. In this case\nthe north pole would be directly overhead, the south pole directly\nunder our feet. At the vernal and autumnal equinoxes the sun would\nappear with half its disc above the ocean, and go round the ocean\nhorizon, always appearing with half its disc above the sea. In summer\nit would appear at a point _s_ nearer to the pole _N_. It would go\nround in the heavens, always appearing above the horizon, and would\nnever set at all. As the summer waned the sun would become lower and\nlower, still, however, going round and round without setting till at\nthe autumn equinox it reached the horizon. So that for six months it\nwould never have set. But when it did set, there would then be six\nmonths without a sun at all.\n\n[Illustration: FIG. 9.]\n\nThus then all over the world the period of darkness and light is\nequivalent. At the tropics the days and nights are always equal. At\nthe poles light for six months is followed by darkness for six months.\nIn the intermediate temperate regions nights of varying lengths follow\ndays of varying lengths, a short night following a long day and _vice\nvers\u00e2_.\n\n[Illustration: FIG. 10.]\n\nIt is evident that for a person living on the north pole a sun-dial\nwould be an easy thing to make. All that would be needful would be to\nput a post vertically in the ground, and observe its shadow as the sun\nwent round (Fig.\u00a010).\n\n[Illustration: FIG. 11.]\n\nIn latitudes such as that of England, where the pole of the earth is\ninclined at an angle to the horizon, it is necessary that the rod, or\n\u201cstyle\u201d as it is called, of the sun-dial should be inclined to the\nhorizontal. For if we used an upright \u201cstyle,\u201d as _O A_, then when the\nsun was in the south, at midday, the shadow would lie along the same\ndirection, _O B_, whether the sun were high in summer, as at _S_, or\nlow in winter, as at _s_. But at other hours, such as nine o\u2019clock in\nthe morning, the shadow of the \u201cstyle\u201d _O A_ would, when the sun was\nat its summer position _T_, lie along _O D_, whereas when the sun was\nat its winter position _t_ the shadow would lie along _O C_. Thus the\ntime would appear different in summer and in winter; and the dial\nwould lead to errors. But if the \u201cstyle\u201d is inclined in the direction\nof the poles, then, however, the sun moves from or towards the pole.\nAs its position varies in winter and summer, the shadow still remains\nunchanged for any particular hour, and it is only the circular motion\nof the sun round in its daily path that affects the position of the\nshadows.\n\n[Illustration: FIG. 12.]\n\nTherefore the first condition of making a sun-dial is that the \u201cstyle\u201d\nwhich casts the shadow should be parallel to the earth\u2019s axis, that\nis to say should point to the polar star. This is the case whether\nthe sun-dial is horizontal or is vertical, and whether it stands on a\npillar in the garden or is attached to the wall of a house.\n\nTo divide the dial, we have only to imagine it surrounded by a sort\nof cage formed of twenty-four arcs drawn from the north pole to the\nsouth pole, and equidistant from one another. In its course the sun\nwould cross one of them every hour. Hence the points to which the\nshadows _o a_, _o b_, _o c_, _o d_, of the inclined \u201cstyle\u201d _O N_ would\npoint are the points where these arcs meet the horizontal circle. This\nconsideration leads to a simple method of constructing a sun-dial,\nwhich is given at the end of this chapter in an appendix.\n\nSun-dials were largely in use in ancient times. It is thought that the\ncircular rows of stones used by the Druids were used to mark the sun\u2019s\npath, and indicate the times and seasons. Obelisks are also supposed\nto have been used to cast sun-shadows. The Greeks were perfectly\nacquainted with the method of making sun-dials with inclined \u201cstyles,\u201d\nor \u201cgnomons.\u201d\n\n[Illustration: FIG. 13.]\n\nSmall portable sun-dials were once much used before the introduction of\nwatches, and were provided with compasses by which they could be turned\nround, so that the \u201cstyle\u201d pointed to the north.\n\nSun-dials were only available during the hours of the day when the\nsun was shining. The desire to mark the hours of the night led to\nthe adoption of water clocks, which measured time by the amount of\nwater which escaped from a small hole in a level of water. Some care,\nhowever, is required to secure correct registration. For suppose that\nwe have a vessel with a small pipe leading out near the bottom, then\nthe amount of water which will run out of the pipe in a given time\ndepends upon the pressure of the water at the pipe, and this depends\nin its turn upon _P Q_, the head of water in the vessel. Whence it\nfollows that the division _Q R_, due to say an hour\u2019s run of the clock\nat _Q R_, will be more than _q r_, the division corresponding to an\nhour, at _q_, a point lower down between _P_ and _Q_, and hence the\ndivisions marked on the vessel to show the hours by means of the level\nof the water would be uneven, becoming smaller and smaller as the water\nfell in the vessel.\n\nTo avoid the inconvenience of unequal divisions, the water to be\nmeasured was allowed to escape into an empty vessel from a vessel in\nwhich its surface was always kept at a constant level. Inasmuch as the\npressure on the pipe or orifice in the vessel in which the water was\nalways kept at a constant level was always constant, it followed that\nequal volumes of water indicated equal times, and the vessel into which\nthe water fell needed only to be equally divided.\n\nAs a measure of hours of the day in countries such as Egypt, where the\nhours were always equal, and thus where the longer days contained more\nhours, the water clock was very suitable; but in Greece and Rome, where\nthe day, whatever its length, was always divided into twelve hours, the\nsimple water clock was as unsuitable as a modern clock would be, for it\nalways divided the hours equally, and took no account of the fact that\nby such a system the hours in summer were longer than in winter.\n\nIn order, therefore, to make the water clock available in Greece and\nItaly, it became necessary to make the hours unequal, and to arrange\nthem to correspond with unequal hours of the Greek day. This plan was\naccomplished as follows. Upon the water which was poured into the\nvessel that measured the hours was placed a float; and on the float\nstood a figure made of thin copper, with a wand in its hand. This wand\npointed to an unequally divided scale. A separate scale was provided\nfor every day in the year, and these scales were mounted on a drum\nwhich revolved so as to turn round once in the year. Thus as the figure\nrose each day by means of a cogwheel it moved the drum round one\ndivision, or one three hundred and sixty-fifth part of a revolution.\nBy this means the scale corresponding to any particular day of winter\nor summer was brought opposite the wand of the figure, and thus the\nscale of hours was kept true. In fact, the water clock, which kept\ntrue time, was made by artificial means to keep untrue time, in order\nto correspond with the unequal hours of the Greek days. In the picture\n_A_ is the receiving water vessel, _P_ the pipe through which the\nwater flows; _B_ is the figure, _C_ the rod; _D_ is the drum, made to\nrevolve by the cogwheel _E_, containing 365 teeth, of which one tooth\nwas driven forward at the close of each day. A syphon _G_ was fixed in\nthe vessel _A_, so that when the figure had risen to the top and pushed\nforward the lever _F_, the syphon suddenly emptied the vessel through\nthe pipe _H_, and the figure fell to the bottom of the vessel _A_ and\nbecame ready to rise and register another day. The divisions on the\ndrum are, of course, uneven. On one side, corresponding to the summer,\nthe day hours would reckon about seventy minutes each, the night hours\nwould be only about fifty minutes each, so that the day divisions on\nthe scale would be long, and the night divisions short. The reverse\nwould be the case in winter. And, therefore, the lines round the drum\nwould go in an uneven wavy form.\n\n[Illustration: FIG. 14.]\n\nSuch water clocks as these were used by the ancient Romans.\n\nSand was also used to measure time. As soon as the art of blowing glass\nhad been perfected by the people of Byzantium, from whom the art passed\nto the Venetians, sand-glasses were made. These glasses were used for\nall sorts of purposes, for speeches and for cooking, but their most\nimportant use was at sea. For it was very important in the early days\nof navigation to know the speed at which the vessel was proceeding in\norder that one\u2019s place at sea might be calculated. The earliest method\nwas to throw over a heavy piece of wood of a shape that resisted being\ndragged through the water, and with a string tied to it. The block of\nwood was called the log, and the string had knots in it. The knots\nwere so arranged that when one of them ran through one\u2019s fingers in\na half-minute measured by a sand-glass it indicated that the vessel\nwas going at the speed of one nautical mile in an hour. The nautical\nmile was taken so that sixty of them constituted one degree, that is\none three hundred and sixtieth part of a great circle of the earth.\nEach nautical mile has, therefore, 6,080 feet. This is bigger than an\nordinary mile on land, which has only 5,280 feet. The knots, therefore,\nhave to be arranged so that when the ship is going one nautical\nmile\u2014that is to say, 6,080 feet\u2014in an hour, a knot shall run out during\nthe half-minute run of the minute glass. This is attained by putting\nthe knots 1/120 \u00d7 6,080 = 50 feet 7 inches apart. As one sailor heaved\nthe log over he gave a stamp on the deck and allowed the cord to run\nout through his fingers. Another sailor then turned the sand-glass.\nWhen the sand had all run out, showing that half a minute had passed,\nthe man who was letting the cord run through his fingers gripped it\nfast, and observed how many knots or parts of knots of string had run\nout, and thus was able to tell how many \u201cknots\u201d per half-minute the\nvessel was going, that is to say, how many nautical miles an hour.\n\nThe modern plan of observing the speed of vessels is different. Now we\nuse a patent log, consisting of a miniature screw propeller tied to a\nstring and dragged through the water after the vessel. As it is pulled\nthrough the water it revolves, and the number of revolutions it makes\nshows how much water it has passed through, and thus what distance\nit has gone. The number of revolutions is measured by a counting\nmechanism, and can be read off when the log is pulled in. Or sometimes\nthe screw is attached to a stiff wire, and the counting mechanism is\nkept on board the ship.\n\nWe use the expression \u201cknots an hour\u201d quite incorrectly. It should be\n\u201cknots per half-minute,\u201d or \u201cnautical miles an hour.\u201d\n\nIt is easy to use the flow of sand for all sorts of purposes to measure\ntime. Thus, if sand be allowed to flow from a hopper through a fine\nhole into a bucket, the bucket may be arranged so that when a given\ntime has elapsed, and a given weight of sand has therefore fallen, the\nbucket shall tip over, and release a catch, which shall then allow\na weight to fall and any mechanical operation to be done that is\nrequired. Thus, for example, we might put an egg in a small holder tied\nto a string and lower it into a saucepan of boiling water. The string\nmight have a counter-weight attached to it, acting over a pulley and\nthus always trying to pull it up out of the water. But this might be\nprevented by a pin passing through a loop in the string and preventing\nit moving. A hopper or funnel might be filled with sand which was\nallowed gradually to escape into a small tip-waggon or other similar\ndevice, so that when a given amount of sand had entered the tip-waggon\nwould tip over, lurch the pin out of the loop, and thus release the\nweight, which in its turn would pull the egg up out of the water in\nthree minutes or any desired time after it had been put in, or a hole\ncould be made in the saucepan, furnished with a little tap, and the\nwater that ran out might be made to fall into a tip-waggon and tip it\nover, and thus when it had run out to put an extinguisher on to the\nspirit lamp that was heating the saucepan, and at the same time make\na contact and ring an electric bell. By this means the egg would be\nalways exactly cooked to the right amount, would be kept warm after it\nwas cooked, and a signal given when it was ready.\n\n[Illustration: FIG. 15.]\n\nThe sketch shows such an arrangement. The saucepan is about three\ninches in diameter and two inches high. When filled with water it will\nhold an egg comfortably. The extinguisher _E_, mounted on a hinge _Q_,\nis turned back, and the spirit lamp _L_ is lit. As soon as the water\nboils, the tap _T_ is turned, and the water gradually trickles away\ninto the tip-waggon. As soon as it is full it tips over and strikes the\narm _X_ of the extinguisher, and turns the lamp out. The little hot\nwater left in the saucepan will keep the egg warm for some time. The\nwaggon _W_ must have a weight _P_ at one end of it, and the fulcrum\nmust be nearer to that end, so that when empty it rests with the end\n_P_ down, but when full it tips over on the fulcrum, when the waggon\nhas received the right quantity of water. I leave to the ingenious\nreader the task of working out the details of such a machine, which, if\nmade properly, will act very well and may be made for a number of eggs\nand worked with very little trouble.\n\n[Illustration: FIG. 16.]\n\nMercury has been used also as an hour-glass. The orifice must be\nexceedingly fine. Or a bubble of mercury may be put into a tube which\ncontains air, and made gradually as it falls to drive the air out\nthrough a minute hole. The difficulty is to get the hole fine enough.\nAll that can be done is to draw out a fine tube in the blow-lamp, break\nit off, and put the broken point in the blow-lamp until it is almost\ncompletely closed up. A tube may thus be made about twelve inches long\nthat will take twelve hours for a bubble of mercury to descend in it.\nBut the trouble of making so small a hole is considerable.\n\n[Illustration: FIG. 17.]\n\nKing Alfred is said to have used candles made of wax to mark the time.\nAs they blew about with the draughts, he put them in lanterns of horn.\nThey had no glass windows in those days, but only openings closed\nwith heavy wooden shutters. These large shutters were for use in fine\nweather. Smaller shutters were made in them, so as to let a little\nlight in in rainy weather without letting in too much wind and rain.\n\nRooms must then have been very draughty, so that people required to\nwear caps and gowns, and beds had thick curtains drawn round them.\nWhen glass was first invented it was only used by kings and princes,\nand glass casements were carried about with them to be fixed into\nthe windows of the houses to which they came, and removed at their\ndeparture.\n\nOil lamps were also used to mark the time. Some of them certainly as\nearly as the fifteenth century were made like bird-bottles; that is\nto say, they consisted of a reservoir closed at the top with a pipe\nleading out of the bottom. When full, the pressure of the external\natmosphere keeps the oil in the bottle, and the oil stands in the neck\nand feeds the wick. As the oil is consumed bubbles of air pass back\nalong the neck and rise up to the top of the oil, the level of which\ngradually sinks. Of course the time shown by the lamp varies with the\nrate of burning of the oil, and hence with the size of the wick, so\nthat the method of measuring time is a very rough one.\n\n\nAPPENDIX.\n\nTo make a sun-dial, procure a circular piece of zinc, about \u215b inch\nthick, and say twelve inches in diameter. Have a \u201cstyle\u201d or \u201cgnomon\u201d\ncast such that the angle of its edge equals the latitude of the place\nwhere the sun-dial is to be set up. This for London will be equal to\n51\u00b0 30\u00b4\u00b4. A pattern may be made for this in wood; it should then be\ncast in gun-metal, which is much better for out-of-door exposure than\nbrass. On a sheet of paper draw a circle _A B C_ with centre _O_. Make\nthe angle _B O D_ equal to the latitude of the place for London = 51\u00b0\n30\u00b4\u00b4. From _A_ draw _A E_ parallel to _O B_ to meet _O D_ in _E_, and\nwith radius _O E_ describe another circle about _O_. Divide the inner\ncircle _A B C_ into twenty-four parts, and draw radii through them\nfrom _O_ to meet the larger circle. Through any divisions (say that\ncorresponding to two o\u2019clock) draw lines parallel to _O B_, _O C_,\nrespectively to meet in _a_. Then the line _O a_ is the shadow line\nof the gnomon at two o\u2019clock. The lines thus drawn on paper may be\ntransferred to the dial and engraved on it, or else eaten in with acid\nin the manner in which etchings are done.\n\n[Illustration: FIG. 18.]\n\nThe centre _O_ need not be in the centre of the zinc disc, but may\nbe on one side of it, so as to give better room for the hours, etc.\nA motto may be etched upon the dial, such as \u201cHoras non numero nisi\nserenas,\u201d or \u201cQual \u2019hom senza Dio, son senza sol io,\u201d or any suitable\ninscription, and the dial is ready for use. It is best put up by\nturning it till the hour is shown truly as compared with a correctly\ntimed watch. It must be levelled with a spirit level. It must be\nremembered that the sun does not move quite uniformly in his yearly\npath among the fixed stars. This is because he moves not in a circle,\nbut in an ellipse of which the earth is in one of the foci. Hence the\nhours shown on the dial are slightly irregular, the sun being sometimes\nin advance of the clock, sometimes behind it. The difference is never\nmore than a quarter of an hour. There is no difference at midsummer and\nmidwinter.\n\n[Illustration: FIG. 19.]\n\nCivil time is solar time averaged, so as to make the hours and days all\nequal. The difference between civil time and apparent solar time is\ncalled the equation of time, and is the amount by which the sun-dial is\nin advance of or in retard of the clock. In setting a dial by means of\na watch, of course allowance must be made for the equation of time.\n\n\n\n\nCHAPTER II.\n\n\nIn the last chapter a short description has been given of the ideas of\nthe ancients as to the nature of the earth and heavens. Before we pass\nto the changes introduced by modern science, it will be well to devote\na short space to an examination of ancient scientific ideas.\n\nAll science is really based upon a combination of two methods,\ncalled respectively inductive and deductive reasoning. The first of\nthese consists in gathering together the results of observation and\nexperiment, and, having put them all together, in the formulation of\nuniversal laws. Having, for example, long observed that all heavy\nthings tended to go towards the centre of the earth, we might conclude\nthat, since the stars remain up in the sky, they can have no weight.\nThe conclusion would be wrong in this case, not because the method\nis wrong, but because it is wrongly applied. It is true that all\nheavy things _tend_ to go to the centre of the earth, but if they are\nbeing whirled round like a stone in a sling the centrifugal force\nwill counteract this tendency. The first part of the reasoning would\nbe inductive, the second deductive. All this reasoning consists,\ntherefore, in forming as complete an idea as possible respecting the\nnature of a thing, and then concluding from that idea what the thing\nwill do or what its other properties will be. In fact, you form correct\nideas, or \u201cconcepts,\u201d as they are called, and reason from them.\n\nBut the danger arises when you begin to reason before you are sure of\nthe nature of your concepts, and this has been the great source of\nerror, and it was this error that all men of science so commonly fell\ninto all through ancient and modern times up to the seventeenth century.\n\nOf course, if it were possible by mere observation to derive a\ncomplete knowledge of any objects, it would be the simplest method.\nAll that would be necessary to do would be to reason correctly from\nthis knowledge. Unfortunately, however, it is not possible to obtain\nknowledge of this kind in any branch of science.\n\nThe ancient method resembled the action of one who should contend that\nby observing and talking to a man you could acquire such a knowledge of\nhis character as would infallibly enable you to understand and predict\nall his actions, and to take little trouble to see whether what he did\nverified your predictions.\n\nThe only difference between the old methods and the new is that in\nmodern times men have learned to give far more care to the formation\nof correct ideas to start with, are much more cautious in arguing\nfrom them, and keep testing them again and again on every possible\nopportunity.\n\nThe constant insistence on the formation of clear ideas and the\npractice of, as Lord Bacon called it, \u201cputting nature to the torture,\u201d\nis the main cause of the advance of physical science in modern times,\nand the want of application of these principles explains why so little\nprogress is being made in the so-called \u201chumanitarian\u201d studies, such as\nphilosophy, ethics, and politics.\n\nThe works of Aristotle are full of the fallacious method of the old\nsystem. In his work on the heavens he repeatedly argues that the\nheavenly bodies must move in circles, because the circle is the most\nperfect figure. He affects a perplexity as to how a circle can at the\nsame time be convex and also its opposite, concave, and repeatedly\nentangles his readers in similar mere word confusion.\n\nRegarded as a man of science, he must be placed, I think, in spite\nof his great genius, below Archimedes, Hipparchus, and several other\nancient astronomers and physicists.\n\nHis errors lived after him and dominated the thought of the middle\nages, and for a long time delayed the progress of science.\n\nThe other great writer on astronomy of ancient times was Ptolemy of\nAlexandria.\n\nHis work was called the \u201cGreat Collection,\u201d and was what we should\nnow term a compendium of astronomy. Although based on a fundamental\nerror, it is a thoroughly scientific work. There is none of the false\nphilosophy in it that so much disfigures the work of Aristotle. The\nreasons for believing that the earth is at rest are interesting.\nPtolemy argues that if the earth were moving round on its axis once in\ntwenty-four hours a bird that flew up from it would be left behind.\nAt first sight this argument seems very convincing, for it appears\nimpossible to conceive a body spinning at the rate at which the earth\nis alleged to move, and yet not leaving behind any bodies that become\ndetached from it.\n\nOn the other hand, the system which taught that the sun and planets\nmoved round the earth, and which had been adopted largely on account\nof its supposed simplicity, proved, on further examination, to be\nexceedingly complicated. Each planet, instead of moving simply and\nuniformly round the earth in a circle, had to be supposed to move\nuniformly in a circle round another point that moved round the earth in\na circle. This secondary circle, in which the planet moved, was called\nan epicycle. And even this more complicated view failed to explain the\nfacts.\n\nA system which, like that of Aristotle and Ptolemy, was based on\ndeductions from concepts, and which consisted rather of drawing\nconclusions than of examining premises, was very well adapted to\nmedi\u00e6val thought, and formed the foundation of astronomy and geography\nas taught by the schoolmen.\n\n[Illustration: FIG. 20.]\n\nThe poem of Dante accurately represents the best scientific knowledge\nof his day. According to his views, the centre of the earth was a fixed\npoint, such that all things of a heavy nature tended towards it. Thus\nthe earth and water collected round it in the form of a ball. He had no\nidea of the attraction of one particle of matter for another particle.\nThe only conception he had of gravity was of a force drawing all heavy\nthings to a certain point, which thus became the point round which the\nworld was formed. The habitable part of the earth was an island, with\nJerusalem in the middle of it _J_. Round this island was an ocean _O_.\nUnder the island, in the form of a hollow cone, was hell, with its\nseven circles of torment, each circle becoming smaller and smaller,\ntill it got down into the centre _C_. Heaven was at the opposite side\n_H_ of the earth to Jerusalem, and was beyond the circles of the\nplanets, in the _primum mobile_. When Lucifer was expelled from heaven\nafter his rebellion against God, having become of a nature to be\nattracted to the centre of the earth, and no longer drawn heavenwards,\nhe fell from heaven, and impinged upon the earth just at the antipodes\nof Jerusalem, with such violence that he plunged right through it to\nthe centre, throwing up behind him a hill. On the summit of this hill\nwas the Garden of Eden, where our first parents lived, and down the\nsides of the hill was a spiral winding way which constituted purgatory.\nDante, having descended into hell, and passed the centre, found his\nhead immediately turned round so as to point the other way up, and,\nhaving ascended a tortuous path, came out upon the hill of Purgatory.\nHaving seen this, he was conducted to the various spheres of the\nplanets, and in each sphere he became put into spiritual communion with\nthe spirits of the blessed who were of the character represented by\nthat sphere, and he supposes that he was thus allowed to proceed from\nsphere to sphere until he was permitted to come into the presence of\nthe Almighty, who in the _primum mobile_ presided over the celestial\nhosts.\n\nThe astronomical descriptions given by Dante of the rising and setting\nof the sun and moon and planets are quite accurate, according to the\nsystem of the world as conceived by him, and show not only that he was\na competent astronomer, but that he probably possessed an astrolabe and\nsome tables of the motions of the heavenly bodies.\n\nOur own poet Chaucer may also be credited with accurate knowledge of\nthe astronomy of his day. His poems often mention the constellations,\nand one of them is devoted to a description of the astrolabe, an\ninstrument somewhat like the celestial globe which used to be employed\nin schools.\n\nBut with the revival of learning in Europe and the rise of freedom of\nthought, the old theories were questioned in more than one quarter.\n\nIt occurred to Copernicus, an ecclesiastic who lived in the sixteenth\ncentury, to re-examine the theory that had been started in ancient\ntimes, and to consider what explanation of the appearance of the\nheavenly bodies could be given on the hypothesis put forward by\nPythagoras, that the earth moved round on its own axis, and also round\nthe sun.\n\nIt may appear rather curious that two theories so different, one that\nthe sun goes round the earth and the other that the earth goes round\nthe sun, should each be capable of explaining the observed appearances\nof those bodies. But it must be remembered that motion is relative. If\nin a waltz the gentleman goes round the lady, the lady also goes round\nthe gentleman. If you take away the room in which they are turning,\nand consider them as spinning round like two insects in space, who is\nto say which of them is at rest and which in motion? For motion is\nrelative. I can consider motion in a train from London to York. As I\nleave London I get nearer to York, and I move with respect to London\nand York. But if both London and York were annihilated how should I\nknow that I was in motion at all? Or, again, if, while I was at rest\nin the train at a station on the way, instead of the train moving the\nwhole earth began to move in a southward direction, and the train in\nsome way were left stationary, then, though the earth was moving, and\nthe train was at rest, yet, so far as I was concerned, the train would\nappear to have started again on its journey to York, at which place it\nwould appear to arrive in due time. The trees and hedges would fly by\nat the proper rate, and who was to say whether the train was in motion\nor the earth?\n\nThe theory of Copernicus, however, remained but a theory. It was\nopposed to the evidence of the senses, which certainly leads us to\nthink that the earth is at rest, and it was opposed also to the ideas\nof some among the theologians who thought that the Bible taught us that\nthe earth was so fast that it could not be moved. Therefore the theory\nfound but little favour. It was in fact necessary before the question\ncould be properly considered on its merits that more should be known\nabout the laws of motion, and this was the principal work of Galileo.\n\nThe merit of Galileo is not only to have placed on a firm basis the\nstudy of mechanics, but to have set himself definitely and consciously\nto reverse the ancient methods of learning.\n\nHe discarded authority, basing all knowledge upon reason, and protested\nagainst the theory that the study of words could be any substitute for\nthe study of things.\n\nAlluding to the mathematicians of his day, \u201cThis sort of men,\u201d says\nGalileo in a letter to the astronomer Kepler, \u201cfancied that philosophy\nwas to be studied like the \u2018\u00c6neid\u2019 or \u2018Odyssey,\u2019 and that the true\nreading of nature was to be detected by the collating of texts.\u201d And\nmost of his life was spent in fighting against preconceived ideas. It\nwas maintained that there could only be seven planets, because God\nhad ordered all things in nature by sevens (\u201cDianoia Astronomica,\u201d\n1610); and even the discoveries of the spots on the sun and the\nmountains in the moon were discredited on the ground that celestial\nbodies could have no blemishes. \u201cHow great and common an error,\u201d\nwrites Galileo, \u201cappears to me the mistake of those who persist in\nmaking their knowledge and apprehension the measure of the knowledge\nand apprehension of God, as if that alone were perfect which they\nunderstand to be so. But ... nature has other scales of perfection,\nwhich we, being unable to comprehend, class among imperfections.\n\n\u201cIf one of our most celebrated architects had had to distribute the\nvast multitude of fixed stars over the great vault of heaven, I believe\nhe would have disposed them with beautiful arrangements of squares,\nhexagons, and octagons; he would have dispersed the larger ones among\nthe middle-sized or lesser, so as to correspond exactly with each\nother; and then he would think he had contrived admirable proportions;\nbut God, on the contrary, has shaken them out from His hand as if by\nchance, and we, forsooth, must think that He has scattered them up\nyonder without any regularity, symmetry, or elegance.\u201d\n\nIn one of Galileo\u2019s \u201cDialogues\u201d Simplicio says, \u201cThat the cause that\nthe parts of the earth move downwards is notorious, and everyone knows\nthat it is gravity.\u201d Salviati replies, \u201cYou are out, Master Simplicio:\nyou should say that everyone knows that _it is called_ gravity; I do\nnot ask you for the name, but for the nature, of the thing of which\nnature neither you nor I know anything.\u201d\n\nToo often are we still inclined to put the name for the thing, and to\nthink when we use big words such as art, empire, liberty, and the\nrights of man, that we explain matters instead of obscuring them. Not\none man in a thousand who uses them knows what he means; no two men\nagree as to their signification.\n\nThe relativity of motion mentioned above was very elegantly illustrated\nby Galileo. He called attention to the fact that if an artist were\nmaking a drawing with a pen while in a ship that was in rapid passage\nthrough the water, the true line drawn by the pen with regard to the\nsurface of the earth would be a long straight line with some small\ndents or variations in it. Yet the very same line traced by the pen\nupon a paper carried along in the ship made up a drawing. Whether you\nsaw a long uneven line or a drawing in the path that the pen had traced\ndepended altogether on the point of view with which you regarded its\nmotion.\n\n[Illustration: FIG. 21.]\n\nBut the first great step in science which Galileo made when quite a\nyoung professor at Pisa was the refutation of Aristotle\u2019s opinion that\nheavy bodies fell to the earth faster than light ones. In the presence\nof a number of professors he dropped two balls, a large and a small\none, from the parapet of the leaning tower of Pisa. They fell to the\nground almost exactly in the same time. This experiment is quite an\neasy one to try. One of the simplest ways is as follows: Into any beam\n(the lintel of a door will do), and about four inches apart, drive\nthree smooth pins so as to project each about a quarter of an inch;\nthey must not have any heads. Take two unequal weights, say of 1 lb.\nand 3 lbs. Anything will do, say a boot for one and pocket-knife for\nthe other; fasten loops of fine string to them, put the loops over the\ncentre peg of the three, and pass the strings one over each of the side\npegs. Now of course if you hitch the loops off the centre peg _P_ the\nobjects will be released together. This can be done by making a loop\nat the end of another piece of string, _A_, and putting it on to the\ncentre peg behind the other loops. If the string be pulled of course\nthe loop on it pulls the other two loops off the central peg, and\nallows the boot and the knife to drop. The boot and the knife should be\nhung so as to be at the same height. They will then fall to the ground\ntogether. The same experiment can be tried by dropping two objects from\nan upper window, holding one in each hand, and taking care to let them\ngo together.\n\n[Illustration: FIG. 22.]\n\nThis result is very puzzling; one does not understand it. It appears as\nthough two unequal forces produced the same effect. It is as though a\nstrong horse could run no faster than a weaker one.\n\nThe professors were so irritated at the result of this experiment, and\nindeed at the general character of young Professor Galileo\u2019s attacks on\nthe time-honoured ideas of Aristotle, that they never rested till they\nworried him out of his very poorly paid chair at Pisa. He then took a\nprofessorship at Padua.\n\nLet us now examine this result and see why it is that the ideas we\nshould at first naturally form are wrong, and that the heavy body will\nfall in exactly the same time as the light one.\n\nWe may reason the matter in this way. The heavy body has more force\npulling on it; that is true, but then, on the other hand there is more\nmatter which has got to be moved. If a crowd of persons are rushing out\nof a building, the total force of the crowd will be greater than the\nforce of one man, but the speed at which they can get out will not be\ngreater than the speed of one man; in fact, each man in the crowd has\nonly force enough to move his own mass. And so it is with the weights:\neach part of the body is occupied in moving itself. If you add more to\nthe body you only add another part which has itself to move. A hundred\nmen by taking hands cannot run faster than one man.\n\nBut, you will say, cannot a man run faster than a child? Yes, because\nhis impelling power is greater in proportion to his weight than that of\na child.\n\nIf it were the fact that the attraction of gravity due to the earth\nacted on some bodies with forces greater in proportion to their\nmasses than the forces that acted on other bodies, then it is true\nthat those different bodies would fall in unequal time. But it is\nan experimental fact that the attractive force of gravity is always\nexactly proportional to the mass of a body, and the resistance to\nmotion is also proportional to mass, hence the force with which a\nbody is moved by the earth\u2019s attraction is always proportional to the\ndifficulty of moving the body. This would not be the case with other\nmethods of setting a body in motion. If I kick a small ball with all\nmy might, I shall send it further than a kick of equal strength would\nsend a heavier ball. Why? Because the impulse is the same in each case,\nbut the masses are different. But if those balls are pulled by gravity,\nthen, by the very nature of the earth\u2019s attraction (the reason of which\nwe cannot explain), the small ball receives a little pull, and the big\nball receives a big pull, the earth exactly apportioning its pull in\neach case to the mass of the body on which it has to act. It is to this\nfact, that the earth pulls bodies with a strength always in each case\nexactly proportional to their masses, that is due the result that they\nfall in equal times, each body having a pull given to it proportional\nto its needs.\n\nThe error of the view of Aristotle was not only demonstrated by\nGalileo by experiment, but was also demonstrated by argument. In this\nargument Galileo imitated the abstract methods of the Aristotelians,\nand turned those methods against themselves. For he said, \u201cYou\u201d (the\nAristotelians) \u201csay that a lighter body will fall more slowly than a\nheavy one. Well, then, if you bind a light body on to a heavy one by\nmeans of a string, and let them fall together, the light body ought\nto hang behind, and impede the heavy body, and thus the two bodies\ntogether ought to fall more slowly than the heavy body alone; this\nfollows from your view: but see the contradiction. For the two bodies\ntied together constitute a heavier body than the heavy body alone, and\nthus, on your own theory, ought to fall more quickly than the heavy\nbody alone. Your theory, therefore, contradicts itself.\u201d\n\nThe truth is that each body is occupied in moving itself without\ntroubling about moving its neighbour, so that if you put any number of\nmarbles into a bag and let them drop they all go down individually, as\nit were, and all in the time which a single marble would take to fall.\nFor any other result would be a contradiction. If you cut a piece of\nbread in two, and put the two halves together, and tie them together\nwith a thread, will the mere fact that they are two pieces make each of\nthem fall more slowly than if they were one? Yet that is what you would\nbe bound to assert on the Aristotelian theory. Hold an egg in your\nopen hand and jump down from a chair. The egg is not left behind; it\nfalls with you. Yet you are the heavier of the two, and on Aristotelian\nprinciples you ought to leave the egg behind you. It is true that when\nyou jump down a bank your straw hat will often come off, but that is\nbecause the air offers more resistance to it than the air offers to\nyour body. It is the downward rush through the air that causes your hat\nto be left behind, just as wind will blow your hat off without blowing\nyou away. For since motion is relative, it is all one whether you jump\ndown through the air, or the air rushes past you, as in a wind. If\nthere were no air, the hat would fall as fast as your body.\n\nThis is easy to see if we have an airpump and are thus enabled to\npump out almost all the air from a glass vessel. In that vessel so\nexhausted, a feather and a coin will fall in equal times. If we have\nnot an airpump, we can try the experiment in a more simple way. For\nlet us put a feather into a metal egg-cup and drop them together. The\ncup will keep the air from the feather, and the feather will not come\nout of the cup. Both will fall to the ground together. But if the\nlighter body fall more slowly, the feather ought to be left behind. If,\nhowever, you tie some strings across a napkin ring so as to make a sort\nof rough sieve, and put a feather in it, and then drop the ring, then\nas the ring falls the air can get through the bottom of the ring and\nact on the feather, which will be left floating as the ring falls.\n\nLet us now go on to examine the second fallacy that was derived from\nthe Aristotelians, and that so long impeded the advance of science,\nnamely, that the earth must be at rest.\n\nThe principal reason given for this was that if bodies were thrown\nup from the earth they ought, if the earth were in motion, to remain\nbehind. Now, if this were so, then it would follow that if a person\nin a train which was moving rapidly threw a ball vertically, that is\nperpendicularly, up into the air, the ball, instead of coming back into\nhis hand, ought to hit the side of the carriage behind him. The next\ntime any of my readers travel by train he can easily satisfy himself\nthat this is not so. But there are other ways of proving it. For\ninstance, if a little waggon running on rails has a spring gun fixed in\nit in a perpendicular position, so arranged that when the waggon comes\nto a particular point on the rails a catch releases the trigger and\nshoots a ball perpendicularly upwards, it will be found that the ball,\ninstead of going upwards in a vertical line, is carried along over the\nwaggon, and the ball as it ascends and descends keeps always above the\nwaggon, just as a hawk might hover over a running mouse, and finally\nfalls not behind the waggon, but into it.\n\nSo, again, if an article is dropped out of the window of a train, it\nwill not simply be left behind as it falls, but while it falls it will\nalso partake of the motion of the train, and touch the ground, not\nbehind the point from which it was dropped, but just underneath it.\n\nThe reason is, that when the ball is dropped or thrown it acquires\nnot only the motion given to it by the throw, or by gravity, but it\ntakes also the motion of the train from which it is thrown. If a ball\nis thrown from the hand, it derives its motion from the motion of the\nhand, and if at the time of throwing the person who does so is moving\nrapidly along in a train, his hand has not only the outward motion\nof the throw, but also the onward motion of the train, and the ball\ntherefore acquires both motions simultaneously. Hence then it is not\ncorrect reasoning to say, because a ball thrown up vertically falls\nvertically back to the spot from which it was thrown, that therefore\nthe earth must be at rest; the same result will happen whether the\nearth is at rest or in motion. You can no more tell whether the earth\nis at rest or in motion from the behaviour of falling bodies than you\ncan tell whether a ship on the ocean is at rest or in motion from the\nbehaviour of bodies on it.\n\nBut you will say. Then why do we feel sea-sick on a ship? The answer\nis, that that is because the motion of the ship is not uniform. If the\nearth, instead of turning round uniformly, were to rock to and fro,\neverything on it would be flung about in the wildest fashion. For as\nsoon as the earth had communicated its motion to a body which then\nmoved with the earth, if the earth\u2019s motion were reversed, the body\nwould go on like a passenger in a train on which the break is quickly\napplied, and he would be shot up against the side of the room. Nay,\nmore, the houses would be shaken off their foundations. Changes of\nmotion are perceptible _so long as the change is going on_. We are\ntherefore justified in inferring from the behaviour of bodies on the\nearth, not that the earth is at rest, but that it is either at rest, or\nelse, if it is in motion, that its motion is uniform and not in jerks\nor variable.\n\n[Illustration: FIG. 23.]\n\nFor if it were not so, consider what would be happening around us. The\nearth is about 8,000 miles in diameter, and a parallel of latitude\nthrough London is therefore about 19,000 miles long, and this space\nis travelled in twenty-four hours. So that London is spinning through\nspace at the rate of over 1,000 feet a second, due to the earth\u2019s\nrotary motion alone, not to speak of the motion due to the earth\u2019s\npath round the sun. If a boy jumped up two and a half feet into the\nair, he would take about half a second to go up and come down, but if\nin jumping he did not partake of the earth\u2019s motion, he would land\nmore than 500 feet to the westward of the point from which he jumped\nup, and if he did it in a room, he would be dashed against the wall\nwith a force greater than he would experience from a drop down from\nthe top of Mont Blanc. He would be not only killed, but dashed into an\nindistinguishable mass. If the earth suddenly stood still, everything\non it would be shaken to pieces. It is bad enough to have the\nconcussion of a train going thirty miles an hour when dashed against\nsome obstacle. But the concussion due to the earth\u2019s stoppage would\nbe as of a train going about 800 miles an hour, which would smash up\neverything and everybody.\n\nThus, then, the first effect of the new ideas formulated by Galileo was\nto show that the Copernican theory that the earth moved round on its\naxis, and round the sun, was in agreement with the laws of motion. In\nfact, he introduced quite new ideas of force, and these ideas I must\nnow endeavour to explain.\n\nLet us consider what is meant by the word \u201cforce.\u201d If I press my\nhand against the table, I exert force. The harder I press, the more\nforce there is. If I put a weight on a stand, the weight presses the\nstand down with a force. If I squeeze a spring, the spring tries to\nrecover itself and exerts a certain force. In all these cases force\nis considered as a pressure. And I can measure the force by seeing\nhow much it will press things. If I take a spring, and press it in an\ninch, it takes perhaps a force of 1 lb. It will take a force of 2 lbs.\nto press it in another inch. Or again, if I pull it out an inch, it\ntakes a force of 1 lb. If I pull it out another inch, it takes a force\nof 2 lbs. We thus always get into the habit of conceiving forces as\nproducing pressures and being measured by pressures.\n\n[Illustration: FIG. 24.]\n\nThis is a perfectly legitimate way of looking at the matter, just as\nthe cook\u2019s method of employing a spring balance to weigh masses of meat\nis a perfectly legitimate way of estimating the forces acting upon\nbodies at rest. But when you come to consider the laws of the pendulums\nof clocks, to which all that I am saying is a preparation, then you\nhave to deal with bodies in motion. And for this purpose a new idea of\nforce altogether is requisite. We shall no longer speak of forces as\nproducing _pressures_. We shall treat them quite independently of their\npressing power. The sun exerts a force of attraction on the earth, but\nit does not press upon it. It exerts its force at a distance. Hence\nthen we want a new idea of \u201cforce.\u201d This idea is to be the following.\nWe will consider that when a force acts upon a body it endeavours to\ncause it to move; in fact, it tries to impart motion to the body. We\nmay treat this motion as a sort of thing or property. The longer the\nforce acts on the body, the more motion it imparts to it, provided the\nbody is free to receive that motion. So that we may say that the test\nof the strength of the force is how much motion it can give to a body\nof a given mass in a given time. It does not matter how the force acts.\nIt may act by means of a string and pull it; it may act by means of a\nstick and push it; it may act by attraction and draw it; it may act\nby repulsion and repel it; it may act as a sort of little spirit and\nfly away with it. In all these cases it _acts_. The more it acts, the\nmore effect it has. In double the time it produces double the motion.\nIf the mass is big, it takes more force to make the mass move; if the\nmass of the body is small, it is moved more easily. Therefore when we\nwant to measure a force in this way we do not press it against springs\nto see how much it will press them in. What we do is to cause it to act\non bodies that are free to move and see what motions it will produce\nin them. Of course we can only do this with things that are free to\nmove. You cannot treat force in this way if you have only a pair of\nscales; in that case you would have to be content with simply measuring\npressures. It is important clearly to grasp this idea. If a body has\na certain mass, then the force acting on it is measured by the amount\nof motion that will in a given time be imparted to that mass, provided\nthat the mass is free to move. This is to be our definition of force.\n\nTherefore, by the action of an attraction or any other force on a body\nfree to move; motion is continually being imparted to the body. Motion\nis, as it were, poured into it, and therefore the body continually\nmoves faster and faster.\n\nHere is a ball flying through the air. Let us suppose that forces are\nacting on it. How can we measure them? We cannot feel what pressures\nare being exerted on it. The only thing we can do is to watch its\nmotions, and see how it flies. If it goes more and more quickly, we\nsay, \u201cThere is propelling force acting on it\u201d; if it begins to stop,\nwe say again, \u201cThere is retarding force acting on it.\u201d So long as it\ndoes not change its speed or direction, we say, \u201cThere is no force\nacting on it.\u201d By this method, therefore, we tell whether a body is\nbeing acted on by force, simply by observing its speed or its change of\nspeed. Merely to say a body is _moving_ does not tell us that force is\nacting on it. All we know is that, if it is moving, force _has_ acted\non it. It is only when we see it changing its speed or direction, that\nis changing its motion, that we say _force_ is acting. Every change of\nmotion, either in direction or speed, must be the result of force, and\nmust be proportional to that force. This is what we mean when we say\nmotion is the test and measure of force.\n\nThis most interesting way of looking at the matter lies at the root\nof the whole theory of mechanics. It is the foundation of the system\nwhich the stupendous genius of Newton conceived in order to explain the\nmotion of the sun, moon, and stars.\n\nForces were treated by him as proportional to the motions, and the\nmotions proportional to the forces, and with this idea he solved a part\nof the riddle of the universe. Galileo had partly seen the same thing,\nbut he never saw it so clearly as Newton. Great discoveries are only\nmade by seeing things clearly. What required the force of a genius in\none age to see in the next may be understood by a child.\n\nHence then we say a force is that which in a given time produces a\ngiven motion in a given mass which is free to move.\n\nYou must have time for a force to act in; for however great the force,\nin no time there can be no motion. You must have mass for a force to\nact on; no mass, no effect. You must have free space for the mass to\nmove in; no freedom to move, no movement.\n\nBut what is this \u201cmass\u201d? We do not know; it is a mystery. We call it\n\u201cquantity of matter.\u201d In uniform substances it varies with size. Double\nthe volume, double the mass. Cut a cake in half, each half has the\nsame \u201cmass.\u201d But then is mass \u201cweight\u201d? No, it is not. _Weight_ is the\naction of the earth\u2019s attraction on matter. No earth to attract, and\nyou would have no weight, but you would still have \u201cmass.\u201d What then\nis matter? Of that we have no idea. The greatest minds are now at work\nupon it. But _mass_ is quantity of matter. Knock a brick against your\nhead, and you will know what mass is. It is not the weight of the brick\nthat gives you a bump; it is the mass. Try to throw a ball of lead, and\nyou will know what mass is. Try to push a heavy waggon, and you will\nknow what mass is. _Weights_, that is earth attractions on masses, are\nproportional to the masses at the same place. This, as we have seen, is\nknown by experiment.\n\nTherefore, when a force acts for a certain time on a mass that is free\nto move, however small the force and however small the time, that body\nwill move. When a baby in a temper stamps upon the earth it makes the\nearth move\u2014not much, it is true, but still it moves; nay, more, in\ntheory, not a fly can jump into the air without moving the earth and\nthe whole solar system. Only, as you may imagine they do not show it\nappreciably. Still, in theory the motion is there.\n\nHence then there are two different ways of considering and estimating\nforces, one suitable for observations on bodies at rest, the other\nsuitable for observations of bodies that are free to move. The force\nof course always tends to produce motion. If, however, motion is\nimpossible, then it develops pressures which we can measure, and\ncalculate, and observe. If the body is free to move, then the force\nproduces motions which we can also measure, calculate, and observe.\nAnd we can compare these two sets of effects. We can say, \u201cA force\nwhich, acting on a ball of a mass of one pound, would produce such and\nsuch motions, would if it acted on a certain spring produce so much\ncompression.\u201d\n\nThe attraction of the earth on masses of matter that are not free to\nmove gives rise to forces which are called weights. Thus the attraction\nof gravitation on a mass of one pound produces a pressure equal to a\nweight of one pound. Unfortunately the same word \u201cpound\u201d is used to\nexpress both the mass and the weight, and has come down to us from days\nwhen the nature of mass was not very well appreciated. But great care\nmust be taken not to confuse these two meanings.\n\nBut the earth\u2019s attractions and all other forces acting upon matter\nwhich is free to move give rise to changes of motion. The word used for\na change of motion is \u201cacceleration\u201d or a quickening. \u201cHe accelerated\nhis pace,\u201d we say. That is, he quickened it; he added to his motion. So\nthat _force_, acting on _mass_ during a _time_, produces acceleration.\n\nFrom this, then, it follows that if a _force_ continues to act on a\nbody the body keeps moving quicker and quicker. When the force stops\nacting, the motion already acquired goes on, but the acceleration\nstops. That is to say, the body goes on moving in a straight line\nuniformly at the pace it had when the force stopped.\n\nIf, then, a body is exposed to the action of a force, and held tight,\nwhat will happen? It will, of course, remain fixed. Now let it go\u2014it\nwill then, being a free body, begin to move. As long as the force\nacts, the force keeps putting more and more motion into the body, like\npouring water into a jug, the longer you pour the faster the motion\nbecomes. The body keeps all the motion it had, and keeps adding all the\nmotion it gains. It is like a boy saving up his weekly pocket-money:\nhe has what he had, and he keeps adding to that. So if in one second\na motion is imparted of one foot a second, then in another second a\nmotion of one foot a second more will be added, making together a\nmotion of two feet a second; in another second of force action the\nmotion will have been increased or \u201caccelerated\u201d by another foot per\nsecond, and so on. The speed will thus be always proportional to\nthe force and the time. If we write the letter V to represent the\nmotion, or speed, or velocity; F to represent the acceleration or gain\nof motion; and T to represent the time, then V = FT. Here V is the\nvelocity the body will have acquired at the end of the time T, if free\nto move and submitted to a force capable of producing an acceleration\nof F feet per second in a unit of time.\n\nV is the final velocity. The average velocity will be 1/2 V, for it\nbegan with no velocity and increased uniformly. How far will the body\nhave fallen in the interval? Manifestly we get that by multiplying the\ntime by the average velocity, that is S = 1/2 VT, where V, as I said,\nis the final velocity, but we found that V = FT. Hence by substitution\nS = 1/2 FT \u00d7 T = 1/2 FT\u00b2.\n\nIt is to be carefully borne in mind that these letters V, S, and T\ndo not represent velocities, spaces, and times, but merely represent\narithmetical numbers of units of velocities, spaces, and times. Thus\nV represents V feet per second, S represents S feet, and T represents\nT seconds. And when we use the equation V = FT we do not mean that\nby multiplying a force by a time you can produce a velocity. If, for\ninstance, it be true that you can obtain the number of inhabitants (H)\nin London by multiplying the average number of persons (P) who live\nin a house by the number of houses (N), this may be expressed by the\nequation H = PN. But this does not mean that by multiplying people into\nhouses you can produce inhabitants. H, P, and N are numbers of units,\nand they are _numbers only_.\n\nTherefore when a body is being acted on by an accelerating force\nit tends to go faster and faster as it proceeds, and therefore its\nvelocity increases with the time. But the space passed through\nincreases faster still, for as the time runs on not only does the\nspace passed through increase, but the rate of passing also gets\nbigger. It goes on increasing at an increasing rate. It is like a man\nwho has an increasing income and always goes on saving it. His total\nmounts up not merely in proportion to the time, but the very rate of\nincrease also increases with the time, so that the total increase is\nin proportion to the time multiplied into the time, in other words to\nthe square of the time. So then, if I let a body drop from rest under\nthe action of any force capable of producing an acceleration, the space\npassed through will be as the square of the time.\n\nNow let us see what the speed will be if the force is gravity, that is\nthe attraction of the earth.\n\nTurning back to what was said about Galileo, it will be remembered that\nhe showed that all bodies, big and small, light and heavy, fell to the\nearth at the same speeds. What is that speed? Let us denominate by G\nthe number of feet per second of increase of motion produced in a body\nby the earth\u2019s action during one second. Then the velocity at the end\nof that second will be V = GT. The space fallen through will be S = 1/2\nGT\u00b2.\n\nWhat I want to know then is this: how far will a body under the action\nof gravity fall in a second of time?\n\nThis, of course, is a matter for measurement. If we can get a machine\nto measure seconds, we shall be able to do it; but inasmuch as falling\nbodies begin by falling sixteen feet in the first second and afterwards\ngo on falling quicker and quicker, the measurements are difficult.\nGalileo wanted to see if he could make it easier to observe. He said\nto himself, \u201cIf I can only water down the force of gravity and make\nit weaker, so that the body will move very slowly under its action,\nthen the time of falling will be easier to observe.\u201d But how to do it?\nThis is one of those things the discovery of which at once marks the\ninventor.\n\n[Illustration: FIG. 25.]\n\nThe idea of Galileo was, instead of letting the body drop vertically,\nto make it roll slowly down an incline, for a body put upon an incline\nis not urged down the incline with the same force which tends to make\nit fall vertically.\n\nCan any law be discovered tending to show what the force is with which\ngravity tends to drag a mass down an incline?\n\nThere is a simple one, and before Galileo\u2019s time it had been\ndiscovered by Stevinus, an engineer. Stevinus\u2019 solution was as follows.\nSuppose that _A B C_ is a wedge-shaped block of wood. Let a loop of\nheavy chain be hung over it, and suppose that there is a little pulley\nat _C_ and no friction anywhere. Then the chain will hang at rest. But\nthe lower part, from _A_ to _B_, is symmetrical; that is to say, it\nis even in shape on both sides. Hence, so far as any pull it exerts\nis concerned, the half from _A_ to _D_ will balance the other half\nfrom _B_ to _D_. Therefore, like weights in a scale, you may remove\nboth, and then the force of gravity acting down the plane on the part\n_A C_ will balance the force of gravity acting vertically on the part\n_C B_. Now the weight of any part of the chain, since it is uniform,\nis proportional to its length. Hence, then, the gravitational force\ndown the plane of a piece whose weight equals _C A_ is equal to the\ngravitational force vertically of a piece whose weight equals _C B_. In\nother words, the force of gravity acting down a plane is diminished in\nthe ratio of _C B_ to _C A_.\n\nBut when a body falls vertically, then, as we have seen, S = 1/2 GT\u00b2,\nwhere S is the space it will fall through, G the number of feet per\nsecond of velocity that gravity, acting vertically on a body, will\nproduce in it in a second, and T the number of seconds of time. If\nthen, instead of falling vertically, the body is to fall obliquely down\na plane, instead of G we must put as the accelerating force\n\n  G \u00d7 (vertical height of the end of the plane)/(length of the plane).\n\nTo try the experiment, he took a beam of wood thirty-six feet long with\na groove in it. He inclined it so that one end was one foot higher than\nthe other. Hence the acceleration down the plane was 1/36 G, where G is\nthe vertical acceleration due to gravity which he wanted to discover.\nThen he measured the time a brass ball took to run down the plane\nthirty-six feet long, and found it to be nine seconds. Whence from\nthe equation given above 36 feet = 1/2 acceleration of gravity down\nthe plane \u00d7 (9 seconds)\u00b2. Whence it follows that the acceleration of\ngravity down the plane is (36 \u00d7 2)/(9)\u00b2 feet per second.\n\nBut the slope of the plane is one thirty-sixth to the vertical.\nTherefore the vertical acceleration of gravity, _i.e._, the velocity\nwhich gravity would induce in a vertical direction in a second, is\nequal to thirty-six times that which it exercises down the plane,\n_i.e._,\n\n36 \u00d7 (36 \u00d7 2)/(9)\u00b2; and this equals 32 feet per second.\n\nThough this method is ingenious, it possesses two defects. One is the\nerror produced by friction, the other from failure to observe that\nthe force of gravity on the ball is not only exerted in getting it\ndown the plane, but also in rotating it, and for this no allowance has\nbeen made. The allowance to be made for rotation is complicated, and\ninvolves more knowledge than Galileo possessed. Still the result is\napproximately true.\n\n[Illustration: FIG. 26.]\n\nThe next attempt to measure G, that is the velocity that gravity will\nproduce on a body in a second of time, was made by Attwood, a Cambridge\nprofessor. His idea was to weaken the force of gravity and thus make\nthe action slow, not by making it act obliquely, but by allowing it to\nact, not on the whole, but only on a portion of the mass to be moved.\nFor this purpose he hung two equal weights over a very delicately\nconstructed pulley. Gravity, of course, could not act on these, for\nany effect it produced on one would be negatived by its effect on\nthe other. The weights would therefore remain at rest. If, however,\na small weight _W_, equal say to a hundredth of the combined weight\nof the weights _A_ and _B_ and _W_, were suddenly put on _A_, then it\nwould descend under an accelerating force equal to a hundredth part of\nordinary gravity. We should then have\n\n    S (the space moved through by the weights) = 1/2 \u00d7 G/100 \u00d7 t\u00b2.\n\nWith such a system, he found that in 7\u00bd seconds the weights moved\nthrough 9 feet. Whence he got\n\n                        9 = 1/2 G/100 \u00d7 (7\u00bd)\u00b2.\n\nFrom which\n\n         G = (2 \u00d7 9 \u00d7 100)/(7\u00bd)\u00b2 = 32 feet per second nearly.\n\nThus by letting gravity only act on a hundredth part of the total\nweight moved, namely _A_, _B_, and _W_, he weakened its action 100\ntimes, and thus made the time of falling and the space fallen through\nsufficiently large to be capable of measurement. To sum up, when a body\nfree to move is acted upon by the force of gravity, its speed will\nincrease in proportion to the time it has been acted upon, and the\nspace it will pass through from rest is proportional to the square of\nthe time during which the accelerating force has acted on it.\n\nGravity is, of course, not the only accelerating force with which\nwe are acquainted. If a spring be suddenly allowed to act on a body\nand pull it, the body begins to move, and its action is gradually\naccelerated, just as though it were attracted, and the acceleration\nof its motion will be proportional to the time during which the\naccelerating force acts. Similarly, if gunpowder be exploded in a\ngun-barrel, and the force thus produced be allowed to act on a bullet,\nthe motion of the bullet is accelerated so long as it is in the barrel.\nWhen the bullet leaves the barrel it goes on with a uniform pace in a\nstraight line, which, however, by the earth\u2019s attraction is at once\ndeflected into a curve, and altered by the resistance of the air.\n\n[Illustration: FIG. 27.]\n\nIt has been already stated that motions may be considered independently\none of another, so that if a body be exposed to two different forces\nthe action of these forces can be considered and calculated each\nindependently of the other. Let us take an example of this law. We have\nseen if a body is propelled forwards, and then the force acting on it\nceases, that it proceeds on with uniform unchanging velocity, and if\nnothing impeded it, or influenced it, it would go on in a straight line\nat a uniform speed.\n\nWe have also seen that if a body is exposed to the action of an\naccelerating force such as gravity it constantly keeps being\naccelerated, it constantly keeps gaining motion, and its speed becomes\nquicker and quicker.\n\n[Illustration: FIG. 28.]\n\nLet us suppose a body exposed to both of these forces at the same time.\nShoot it out of a cannon, and let an accelerating force act on it, not\nin the direction it is going, but in some other direction, say at right\nangles. What will happen? In the direction in which it is going, its\nspeed will remain uniform. In the direction in which the accelerating\nforce is acting, it will move faster and faster. Thus along _A B_ it\nwill proceed uniformly. If it proceeded uniformly also along _A C_ (as\nit would do if a simple force acted on it and then ceased to act), then\nas a result it would go in the oblique line _A D_, the obliquity being\ndetermined by the relative magnitude of the forces acting on it. But\nhow if it went uniformly along _A B_, but at an accelerated pace along\n_A C_? Then while in equal times the distances along _A B_ would be\nuniform the distances in the same times along _A C_ would be getting\nbigger and bigger. It _would not describe a straight line; it would go\nin a curve_. This is very interesting. Let us take an example of it.\nSuppose we give a ball a blow horizontally; as soon as it quits the bat\nit would of course go on horizontally in a straight line at a uniform\nspeed; but now if I at the same instant expose it to the accelerating\nforce of gravity, then, of course, while its horizontal movement will\ngo on uniformly, its downward drop will keep increasing at a speed\nvarying as the time. And while the total distances horizontally will\nbe uniform in equal times, the total downward drop from _A B_ will\nbe as the squares of the times. Here, then, you have a point moving\nuniformly in a horizontal direction, but as the squares of the times in\na vertical direction. It describes a curve. What curve? Why, one whose\ndistances go uniformly one way, but increase as the squares the other\nway.\n\n[Illustration: FIG. 29.]\n\nThis interesting curve is called a parabola. With a ball simply hit by\na bat, the motion is so very fast that we cannot see it well. Cannot\nwe make it go slowly? Let us remember what Galileo did. He used an\ninclined plane to water down his force of gravity. Let us do the same.\nLet us take an inclined plane and throw on it a ball horizontally.\nIt will go in a curve. Its speed is uniform horizontally, but is\naccelerated downwards. If we desire to trace the curve it is easy to\ndo. We coat the ball with cloth and then dip it in the inkpot. It will\nthen describe a visible parabola. If I tilt up the plane and make the\nforce of gravity big, the parabola is long and thin; if I weaken down\nthe force of gravity by making the plane nearly horizontal, then it is\nwide and flat.\n\nOne can also show this by a stream of peas or shot. The little bullets\ngo each with a uniform velocity horizontally, and an accelerated force\ndownwards.\n\nInstead of peas we can use water. A stream of it rushing horizontally\nout of an orifice will soon bend down into a parabola.\n\nThus then I have tried to show what force is and how it is measured. I\nrepeat again, when a body is free to move, then, if no further force\nacts on it, it will go on in a straight line at a uniform speed, but\nif a force continues to act on it in any direction, then that force\nproduces in each unit of time a unit of acceleration in the direction\nin which the force acts, and the result is that the body goes on moving\ntowards the direction of acceleration at a constantly increasing speed,\nand hence passing over spaces that are greater and greater as the speed\nincreases. This is the notion of a \u201cforce.\u201d In all that has been said\nabove it has been assumed that the attraction of gravity on a body\ndoes not increase as that body gets nearer to the earth. This is not\nstrictly true; in reality the attractive force of gravity increases as\nthe earth\u2019s centre is approached. But small distances through which\nthe weights in Attwood\u2019s machine fall make no appreciable difference,\nbeing as nothing compared to the radius of earth. For practical\npurposes, therefore, the force may be considered uniform on bodies that\nare being moved within a few feet of the earth\u2019s surface. It is only\nwhen we have to consider the motions of the planets that considerations\nof the change of attractive force due to distance have to be considered.\n\nI am glad to say that the most tiresome, or rather the most difficult,\npart of our inquiry is now over. With the help of the notions already\nacquired, we are now ready to get to the pendulum, and to show how it\ncame about that a boy who once in church amused himself by watching the\nswinging of the great lamps instead of attending to the service laid\nthe foundation of our modern methods of measuring time.\n\n\n\n\nCHAPTER III.\n\n\nWe have examined the action of a body under the accelerating or\nspeed-quickening force due to gravity, the attractive force of which on\nany body is always proportional to the mass of that body. Let us now\nconsider another form of acceleration.\n\n[Illustration: FIG. 30.]\n\nTake the case of a strip of indiarubber. If pulled it resists and tends\nto spring back. The more I pull it out the harder is the pull I have\nto exert. This is true of all springs. It is true of spiral springs,\nwhether they are pulled out or pushed in, and in each case the amount\nby which the spring is pulled out or pushed in is proportional to the\npressure. This law is called Hooke\u2019s law. It was expressed by him in\nLatin, \u201cUt tensio, sic vis\u201d: \u201cAs the extension, so the force.\u201d It is\ntrue of all elastic bodies, and it is true whether they are pulled out\nor pushed in or bent aside. The common spring balance is devised on\nthis principle. The body to be weighed is hung on a hook suspended from\na spring. The amount by which the spring is pulled out is a measure\nof the weight of the body. If you take a fishing rod and put the butt\nend of it on a table and secure it by putting something heavy on the\nend, then the tip will bend down on account of its own weight. Mark the\npoint to which it goes. Now, if you hang a weight on the tip, the tip\nwill bend down a little further. If you put double the weight the tip\nwill go down double the distance, and so on until the fishing rod is\nconsiderably bent, so that its form is altered and a new law of flexure\ncomes into play. Suppose I use a spring as an accelerating force. For\nexample, suppose I suspend a heavy ball by a string and then attach a\nspiral spring to it and pull the spring aside. The ball will be drawn\nafter the spring. If then I let the ball go, it will begin to move. The\nforce of the spring will act upon it as an accelerating force, and the\nball will go on moving quicker and quicker. But the acceleration will\nnot be like that of gravity. There will be two differences. The pull of\nthe spring will in no way depend on the mass of the ball, and the pull\nof the spring, instead of being constant, like the pull of gravity,\nwill become weaker and weaker as the ball yields to it. Consequently\nthe equations above given which determine the relations between this\nspace passed through, the velocity, and the time which were determined\nin the case of gravity are no longer true, and a different set of\nrelations has to be determined. This can be easily done by mathematics.\nBut I do not propose to go into it. I prefer to offer a rough and ready\nexplanation, which, though it does not amount to a proof, yet enables\nus to accept the truth that can be established both by experiment and\nby calculation.\n\n[Illustration: FIG. 31.]\n\nLet a heavy ball (_A_) be suspended by a long string, so that the\naction of gravity sideways on the ball is very small and may be\nneglected, and to each side attach an indiarubber thread fastened\nat _B_ and _C_. Then when the ball is pulled aside a little, say\nto a position _D_, it will tend to fly back to _A_ with a force\nproportioned to the distance _A D_. What will be the time it will take\nto do this? If the distance _A D_ is small, the ball has only a small\ndistance to go, but then, on the other hand, it has only small forces\nacting on it. If the distance _A D_ is bigger, then it has a longer\ndistance to go, but larger forces to urge it. These counteract one\nanother, so that the time in each case will be the same.\n\n[Illustration: FIG. 32.]\n\nThe question is this:\u2014Will you go a long distance with a powerful\nhorse, or a small distance with a weak horse? If the distance in each\ncase is proportioned to the power of the horse, then the amount of the\ndistance does not matter. The powerful horse goes the long distance in\nthe same time that the weak horse goes the short distance. And so it\nis here. However far you pull out the spring, the accelerative pull on\nthe ball is proportioned to the distance. But the time of pulling the\nball in depends on the distance. So that each neutralises the other.\nWhence then we have this most important fact, that springs are all\nisochronous; that is to say, any body attached to any spring whatever,\nwhether it is big or small, straight or curly, long or short, has a\ntime of vibration quite independent of the bigness of the vibration.\nThe experiment is easy to try with a ball mounted on a long arm that\ncan swing horizontally. It is attached on each side to an elastic\nthread. If pulled aside, it vibrates, but observe, the vibration is\nexactly the same whether the bigness of the vibration is great or\nsmall. If the pull aside is big, the force of restitution is big; if\nthe pull is small, the force of restitution is small. In one case the\nball has a longer distance to go, but then at all points of its path\nit has a proportionally stronger force to pull it; if the ball has a\nsmaller distance to go, then at all the corresponding points of its\npath it has a proportionally weaker force to pull it. Thus the time\nremains the same whether you have the powerful horse for the long\njourney or the weaker horse for the smaller journey.\n\n[Illustration: FIG. 33.]\n\nNext take a short, stiff spring of steel. One of the kind known as\ntuning forks may be employed.\n\nThe reader is probably aware that sounds are produced by very rapid\npulsations of the air. Any series of taps becomes a continuous sound\nif it is only rapid enough. For example, if I tap a card at the rate\nof 264 times in a second, I should get a continuous sound such as that\ngiven by the middle C note of the piano. That, in fact, is the rate\nat which the piano string is vibrating when C is struck, and that\nvibration it is that gives the taps to the air by which the note is\nproduced.\n\nThis can be very easily proved. For if you lift up the end of a bicycle\nand cause the driving wheel to spin pretty rapidly by turning the pedal\nwith the hand, then the wheel will rotate perhaps about three times in\na second. If a visiting card be held so as to be flipped by the spokes\nas they fly by, since there are about thirty-six of them, we should\nget a series of taps at the rate of about 108 a second. This on trial\nwill be found to nearly correspond to the note A, the lowest space on\nthe bass clef of music. As the speed of rotation is lowered, the tone\nof the note becomes lower; if the speed is made greater, the pitch of\nthe note becomes higher, and the note more shrill. However far or near\nthe card is held from the centre of the wheel makes no difference, for\nthe number of taps per second remains the same. So, again, if a bit of\nwatch-spring be rapidly drawn over a file, you hear a musical note. The\nfiner the file, and the more rapid the action, the higher the note. The\naction of a tuning fork and of a vibrating string in producing a note\ndepends simply on the beating of the air. The hum of insects is also\nsimilarly produced by the rapid flapping of their wings.\n\nIt is an experimental fact that when a piano note is struck, as the\nvibration gradually ceases the sound dies away, but the pitch of the\nnote remains unchanged. A tune played softly, so that the strings\nvibrate but little, remains the same tune still, and with the same\npitch for the notes.\n\nA \u201csiren\u201d is an ingenious apparatus for producing a series of very\nrapid puffs of air. It consists of a small wheel with oblique holes in\nit, mounted so as to revolve in close proximity to a fixed wheel with\nsimilar holes in it. If air be forced through the wheels, by reason\nof the obliquity of the orifices in the movable wheel it is caused to\nrotate. As it does so, the air is alternately interrupted and allowed\nto pass, so that a series of very rapid puffs is produced. As the\nair is forced in, the wheel turns faster and faster. The rapidity of\nsuccession of the puffs increases so that the note produced by them\ngradually increases in pitch till it rises to a sort of scream. For\nsteamers these \u201csirens\u201d are worked by steam, and make a very loud noise.\n\nIt is, however, impossible to make a tuning fork or a stretched piano\nspring alter the pitch of its note without altering the elastic force\nof the spring by altering its tension, or without putting weights on\nthe arms of the tuning fork to make it go more slowly. And this is\nbecause the tuning fork and the piano spring, being elastic, obey\nHooke\u2019s law, \u201cAs the deflection, so the force\u201d; and therefore the time\nof back spring is in each case invariable, and the pitch of the note\nproduced therefore remains invariable, whatever the amplitude of the\nvibration may be.\n\nUpon this law depends the correct going of both clocks and watches.\n\nWonderful nature, that causes the uniformity of sounds of a piano, or a\nviolin, to depend on the same laws that govern the uniform going of a\nwatch! Nay, more, all creation is vibrating. The surge of the sea upon\nthe coast that swishes in at regular intervals, the colours of light,\nwhich consist of ripples made in an elastic ether, which springs back\nwith a restitutional force proportioned to its displacement, all depend\nupon the same law. This grand law by which so many phenomena of nature\nare governed has a very beautiful name, which I hope you will remember.\nIt is called \u201charmonic motion,\u201d by which is meant that when the atoms\nof nature vibrate they vibrate, like piano strings, according to the\nlaws of harmony. The ancient Pythagorean philosophers thought that all\nnature moved to music, and that dying souls could begin to hear the\ntones to which the stars moved in their orbits. They called it, as you\nknow, the music of the spheres. But could they have seen what science\nhas revealed to man\u2019s patient efforts, they would have seen a vision\nof harmony in which not a ray of light, not a string of a musical\ninstrument, not a pipe of an organ, not an undulation of all-pervading\nelectricity, not a wing of a fly, but vibrates according to the law\nof harmony, the simple easy law of which a boy\u2019s catapult is the type,\nand which, as we have seen, teaches us that when an elastic body is\ndisplaced the force of restitution, in other words, the force tending\nto restore it to its old position, is proportional to the displacement,\nand the time of vibration is uniform. The last is the important thing\nfor us; we seem to get a gleam of a notion of how the clock and watch\nproblem is going to be solved.\n\nBut before we get to that we have yet to go back a little.\n\nAbout the year 1580 an inattentive youth (it was our friend Galileo\nagain) watched the swing of one of the great chandeliers in the\ncathedral church at Pisa. The chandeliers have been renewed since his\nday, it was one of the old lamps that he watched. It had been lit, and\nallowed to swing through a considerable space. He expected that as it\ngradually came to rest it would swing in a quicker and quicker time,\nbut it seemed to be uniform. This was curious. He wanted to measure the\ntime of its swing. For this purpose he counted his pulse-beats. So far\nas he could judge, there were exactly the same number in each pendulum\nswing.\n\nThis greatly interested him, and at home he began to try some\nexperiments. As he got older his attention was repeatedly turned to\nthat subject, and he finally established in a satisfactory way the law\nthat, if a weight is hung to the end of a string and caused to vibrate,\nit is isochronous, or equal-timed, no matter what the extent of the arc\nof vibration.\n\nThe first use of this that he made was to make a little machine with\na string of which you could vary the length, for use by doctors. For\nthe doctors of that day had no gold watch to pull out while with\nsolemn face they watched the ticks. They were delighted with the new\ninvention, and for years doctors used to take out the little string and\nweight, and put one hand on the patient\u2019s pulse while they adjusted the\nstring till the pendulum beat in unison with the pulse. By observing\nthe length of the string, they were then able to tell how many beats\nthe pulse made in a minute. But Galileo did not stop there. He\nproceeded to examine the laws which govern the pendulum.\n\nWe will follow these investigations, which will largely depend on what\nwe have already learned.\n\nBefore, however, it is possible to understand the laws which govern\nthe pendulum, there are one or two simple matters connected with the\nbalance and operation of forces which have to be grasped.\n\nSuppose that we have a flat piece of wood of any shape like Fig.\u00a034,\nand that we put a screw through any spot _A_ in it, no matter where,\nand screw it to a wall, so that it can turn round the screw as round a\npivot.\n\n[Illustration: FIG. 34.]\n\n[Illustration: FIG. 35.]\n\nNext we will knock a tintack into any point _B_, and tie a string on\nto _B_. Then if I pull at the string in any direction _B C_ the board\ntends to twist round the screw at _A_. What will the strength of the\ntwisting force be? It will depend on the strength of the pull, and\non the \u201cleverage,\u201d or distance of the line _C B_ from _A_. We might\nimagine the string, instead of being attached at _B_, to be attached\nat _D_; then, if I put _P_ as the strength of the pull, the twisting\npower would be represented by _P_ \u00d7 _A D_. This is called the \u201cmoment\u201d\nof the force _P_ round the centre _A_. It would be the same as if I\nhad simply an arm _A D_, and pulled upon it with the force _P_. It\nis an experimental truth, known to the old Greek philosophers, that\nmoments, or twisting powers, are equal when in each case the result of\nmultiplying the arm by the power acting at right angles to it is equal.\n\nNow suppose _A B_ is a pendulum, with a bob _B_ of 10 lbs. weight, and\nsuppose it has been drawn aside out of the vertical so that the bob is\nin the position _B_. Then the weight of the bob will act vertically\ndownwards along the line _B C_. The moment, or twisting power, of the\nweight will be equal to 10 lbs. multiplied by _A D_, _A D_ being a\nline perpendicular to _B C_.\n\n[Illustration: FIG. 36.]\n\nNow suppose that another string were tied to the bob _B_, and pulled\nin a direction at right angles to _A B_, with a force _P_ just enough\nto hold the bob back in the position _B_. The pull along _D B_ \u00d7 _A B_\nwould be the moment of that pull round the point _A_. But, because this\nmoment just holds the pendulum up, it follows that the moment of the\nweight of the pendulum round _A_ is equal to the moment of the pull of\nthe string _B D_ round _A_.\n\n                 Whence P \u00d7 A B = 10 lbs. \u00d7 A D.\n\n                       Whence P = 10 lbs. \u00d7 (A D)/(A B).\n\nBut _A B_ is always the same, whatever the side deflection or\ndisplacement of the pendulum may be. Whence then we see that when\na pendulum is pulled aside a distance _E B_ (which is always equal\nto _A D_), then the force tending to bring it back to _E_ is always\nproportional to _E B_. But if the pendulum be fairly long, say 39-1/7\ninches, and the displacement _E B_ be small,\u2014in other words, if we do\nnot drag it much out of the vertical,\u2014then we may say that the force\ntending to bring it back to _F_, its position of rest, is not very\ndifferent from the force tending to bring it back to _E_. But _F B_\nis the \u201cdisplacement\u201d of the pendulum, and, therefore, we find that\nwhen a pendulum is displaced, or deflected, or pulled aside a little,\nthe amount of the deflection is always very nearly proportional to the\nforce which was used to produce the deflection. This important law\ncan be verified by experiment. If _C_ is a small pulley, and _B C_ a\nstring attached to a pendulum _A B_ whose bob is _B_. Then if a weight\n_D_ be tied to the string and passed over a pulley _C_, the amount _F\nB_ by which the weight _D_ will deflect the bob _B_ is almost exactly\nproportional to _D_, so long as we only make the deflection _E B_\nsmall, that is two or three inches, where say 39-1/7 inches is the\nlength _A B_ of the pendulum.\n\nIf _F B_ is made too big, then the line _B F_ can no longer be\nconsidered nearly equal to the arc of deflection _E B_, and the\nproposition is no longer true.\n\nHence then, both by experiment and on theory, we find that for small\ndistances the displacement of a pendulum bob is approximately equal to\nthe force by which that displacement is produced.\n\nBut if so, then from what has gone before, we have an example of\nharmonic motion. The weight of the bob, tending to pull the bob back to\n_E_, acts just as an elastic band would act, that is to say pulls more\nstrongly in proportion as the distance _F B_ is bigger. In fact, if we\ncould remove the force of gravity still leaving the mass _B_ of the\npendulum bob, the force of an elastic band acting so as to tend to pull\nthe bob back to rest might be used to replace it. It would be all one\nwhether the bob were brought back to rest by the downward force of its\nown gravity, or by the horizontal force of a properly arranged elastic\nband of suitable length.\n\n[Illustration: FIG. 37.]\n\nBut the motion of the bob, under the influence of the pull of\nan elastic band where the strain was always proportional to the\ndisplacement, would, as we have seen, be harmonic motion, and performed\nin equal times whatever the extent of the swing. Whence then we\nconclude that if the swings of a pendulum are not too big, say not\nexceeding two and a half inches each way, the motion may be considered\nharmonic motion, and the swings will be made in equal times whether\nthey are large or small ones. In other words, a clock with a 39-1/7\ninch pendulum and side swing on each side if not over two inches will\nkeep time, whatever the arc of swing may be.\n\nThis may be verified experimentally. Take a pendulum of wood 39-1/7\ninches long, and affix to its end a bob of 10 lbs. weight. The pendulum\nwill swing once in each second. To pull it aside two inches we should\nwant a weight such that its moment about the point of support was equal\nto the moment of the force of gravity acting on the bob, about the\npoint of support. In other words, the weight required \u00d7 39-1/7 inches =\n10 lbs. \u00d7 2 inches. Whence the weight required = 1/2 lb. (nearly).\n\nNow fix a similar pendulum _A B_ 39-1/7 inches long, horizontally,\nwith a weight _B_ of 10 lbs. on it. Fasten it to a vertical shaft _C\nD_, with a tie rod of wire or string _A B_ so as to keep it up, and\nattach to each side of the rod _A B_ elastic threads _E F_ and _E G_.\nLet these threads be tied on at such a point that when _B_ is pulled\naside two inches the force tending to bring it back to rest is half a\npound. Then if set vibrating the rod will swing backwards and forwards\nin equal times, no matter how big, the arc of vibration (provided the\narc is kept small), and the time of oscillation will be that of a\npendulum, namely, one swing in a second. In fact, whether you put _A B_\nvertically and let it swing on the pivots _C_ and _D_ by the force of\ngravity, or put it horizontally, and thus prevent gravity acting on it,\nbut make it swing under the accelerating influence of a pair of elastic\nbands so arranged as to be equivalent to gravity, in each case it will\nswing in seconds.\n\n[Illustration: FIG. 38.]\n\nIt is this curious property of the circle that makes the vertical\nforce of gravity on a pendulum pull it as though it were a\nhorizontally acting elastic band; that is the reason why a pendulum is\nequal-time-swinging, or, as it is called, isochronous, from two Greek\nwords that mean \u201cthe same\u201d and \u201ctime.\u201d\n\nBut it must be remembered that this equal swinging is only approximate,\nand only true when the arc of vibration is small.\n\nHere then we have a proof which shows us that the pendulum of a clock\nand the balance wheel of a watch depend on exactly the same principles.\nThey are each an example of harmonic motion.\n\nThe next question that arises is whether the weight of the pendulum has\nany influence upon the time of its vibration.\n\nA little reflection will soon convince us that it has none. For we know\nthat the time that bodies take to fall to the ground under the action\nof gravity is independent of the weight. A falling 2 lb. weight is only\nequivalent to two pound-weights falling side by side.\n\nIn the same way and by the same reasoning we might take two pendulums\nof equal length, and each with a bob weighing 1 lb. They would, if put\nside by side close together swing in equal times. But the time would be\nthe same if they were fastened together, and made into one pendulum.\n\nFor inasmuch as the fall of a pendulum is due to gravity, and the\naction of gravity upon a body is proportional to its mass, it follows\nthat in a pendulum the part of the gravitational force that acts upon\neach part of the mass is occupied in moving that mass, and the whole\npendulum may be considered as a bundle of pendulums tied together and\nvibrating together.\n\nThe same would be the case with a pendulum vibrating under the\ninfluence of a spring. If you have two bobs and two springs, they will\nvibrate in the same time as one bob accelerated by one spring. In\nthis case, however, the force of the one spring must be equal to the\ncombined force of the two springs. In other words, the springs must be\nmade proportional in strength to the masses.\n\nHence, then, you cannot increase the speed of the vibration of a\npendulum by adding weight to the bob.\n\nOn the other hand, if you have a bob vibrating under the influence of a\nspring, like the balance wheel of a watch, then if you increase the bob\nwithout increasing the spring, since the mass to be moved has increased\nwithout a corresponding increase in the accelerating force acting on\nit, the time of swing will alter accordingly.\n\nBut in the case of gravity, by altering the mass, you thereby\nproportionally alter the attraction on it, and therefore the time of\nswing is unaltered.\n\n[Illustration: FIG. 39.]\n\nThe explanation which has been given above of the reasons why a\npendulum swings backwards and forwards in a given time independently\nof the length of the arc through which it swings, that is to say of\nthe amount by which it sways from side to side, is only approximate,\nbecause in the proof we assumed that the arc of swing and the line _F\nB_ were equal, which is not really and exactly true. Galileo never got\nat the real solution, though he tried hard. It was reserved for another\nthan he to find the true path of an isochronous pendulum and completely\nto determine its laws. Huygens, a Dutch mathematician, found that the\ntrue path in which a pendulum ought to swing if it is to be really\nisochronous is a curve called a cycloid, that is to say the curve which\nis traced out by a pencil fixed on the rim of a hoop when the hoop is\nrolled along a straight ruler. It is the curve which a nail sticking\nout of the rim of a waggon wheel would scratch upon a wall. I will\nnot go into the mathematical proof of this. Clocks are not made with\ncycloidal pendulums, because when the arc of a pendulum is small the\nswing is so very near a cycloid as to make no appreciable difference in\ntime-keeping.\n\nI am now glad to be able to say that I have dealt with all the\nmathematics that is necessary to enable the mechanism of a clock to be\nunderstood. It all leads up to this:\u2014\n\n(1) A harmonic motion is one in which the accelerating force increases\nwith the distance of the body from some fixed point.\n\n(2) Bodies moving harmonically make their swings about this point in\nequal times.\n\n(3) A spring of any sort or shape always has a restitutional force\nproportional to the displacement.\n\n(4) And therefore masses attached to springs vibrate in equal times\nhowever large the vibration may be.\n\n(5) The bob of a pendulum, oscillating backwards and forwards, acts\nlike a weight under the influence of a spring, and is therefore\nisochronous.\n\n(6) The time of vibration of a pendulum is uninfluenced by changes in\nthe weight of the bob, but is influenced by changes in the length of\nthe pendulum rod. The time of vibration of a mass attached to a spring\nis influenced by changes in the mass.\n\nWe have now to deal with the application of these principles to clocks\nand watches.\n\nClocks had been known before the time of Galileo, and before the\ninvention of the pendulum. They had what is known as balance, or verge\nescapements. Strictly in order of time I ought to explain them here.\nBut I will not do so. I will go on to describe the pendulum clock, and\nthen I will go back and explain the verge escapement, which, we shall\nsee, is really a sort of huge watch of a very imperfect character.\n\nAs soon as Galileo had discovered that pendulums were isochronous,\nthat is, equi-time-swinging, he set to work to see whether he could\nnot contrive to make a timepiece by means of them. This would be easy\nif only he could keep a pendulum swinging. When a pendulum is set\nswinging, it soon comes to rest. What brings it to rest? The resistance\nof the air and the friction of the pivots. Therefore what is obviously\nwanted is something to give it a kick now and then, but the thing must\nkick with discretion. If it kicked at the wrong time, it might actually\nstop the pendulum instead of keeping it going. You want something that,\njust as the pendulum is at one end and has begun to move, will give it\na further push. Suppose that I have a swing and that I put a boy in it,\nand I swing him to and fro. I time my pushes. As he comes back against\nmy hand I let him push it back, and then just as the swing turns I give\nit a further push. But I cannot stand doing that all day. I must make a\nmachine to do it. Now what sort of a machine?\n\nFirst, the machine must have a reservoir of force. I can\u2019t get a\nmachine to do work unless I wind it up, nor a man to do work unless I\nfeed him, which is his way of being wound up. But then what do I want\nhim to do? I want him, when I give him a push, to push me back harder.\nI want a reservoir of force such that when a pendulum comes back and\ntouches it, the touch, like the pressure of the trigger of a gun, shall\nallow some pent-up power to escape and to drive the pendulum forward.\n\nThis is the case in a swing. Each time that the swing returns to my\nhands I give it a push, which serves to sustain the motion that would\notherwise be destroyed by friction and the resistance of the air.\n\nSuch an arrangement, if it can be contrived mechanically, is called an\n\u201cescapement.\u201d\n\nAn arrangement of this kind was contrived by Galileo. He provided a\nwheel, as is here shown, with a number of pins round it. The pendulum\n_A B_ has an arm _A H_ attached to it, and there is a ratchet _C D_\nwhich engages with the pins. The ratchet has a projecting arm _E F_.\n\n[Illustration: FIG. 40.]\n\nWhen the pendulum comes back towards the end of its beat, the arm _A H_\nstrikes the arm _E F_, and raises the ratchet _C D_. This releases the\nwheel, which has a weight wound up upon it, and therefore at once tries\nto go round. The consequence is, that the pin _G_ strikes upon the arm\n_A H_, and thus on its return stroke gives an impetus to the pendulum.\nAs the pin _G_ moves forward it slides on the arm _A H_ till it slips\nover the point _H_. The wheel now being free, would fly round were it\nnot that when the pendulum returned, and the arm _A H_ was lowered,\nthe ratchet had got into position again and its point _D_ was ready to\nmeet and stop the next pin that was coming on against it. At each blow\nof the pins against the pendulum a \u201ctick\u201d is made, at each blow of a\npin against the ratchet a \u201ctock\u201d is sounded, so that as it moves the\npendulum makes the \u201ctick-tock\u201d sound with which we are all familiar.\n\nHence then a clock consists of a wheel, or train of wheels, urged by\na weight or spring, which strives continually to spin round, but its\nrotation is controlled by an escapement and pendulum, so contrived as\nonly to allow it to go a step forward at regular equal intervals of\ntime.\n\nBut this would make only a poor sort of escapement. For the mode of\ndriving the pendulum adds a complication to the swing of the pendulum.\nInstead of the pendulum being simply under the accelerative force of\ngravity, it is also subjected to the acceleration of the pin _G_. This\nacceleration is not of the \u201charmonic\u201d order. Hence so far as it goes it\ndoes not tend to assist in giving a harmonic motion to the pendulum,\nbut, on the contrary, disturbs that harmonic motion. Besides this, the\nimpulse of the pin is in practice not always uniform. For if the wheel\nis at the end of a train of wheels driven by a weight, though the force\nacting on it is constant, yet, as that force is transmitted through a\ntrain of wheels, it is much affected by the friction of the oil. And\non colder days the oil becomes more coagulated, and offers greater\nresistance. Moreover, as will be explained more in detail afterwards,\nthe fact that the impulse is administered by _G_ at the end of the\nstroke of the pendulum is disadvantageous, as it interferes with the\nfree play of the pendulum.\n\nFrom all these causes the above escapement is imperfect in character,\nand would not do where precision was required.\n\n[Illustration: FIG. 41.]\n\nIt is now time to return to the old-fashioned escapements which were\nin use before the time of Galileo. These consisted of a wheel called\na crown wheel, with triangular teeth. On one side of this wheel a\nvertical axis was fitted, with projecting \u201cpallets\u201d _e f_. Across\nthe axis a verge or rod _e f_ was placed, fitted with a ball at each\nend. When the crown wheel attempted to move on, one of its teeth came\nin contact with a pallet. This urged the pallet forward, and thereby\ncaused an impulse to be given to the axis, on which was mounted the\nverge, carrying the balls. These of course began to move under the\nacceleration of the force thus impressed upon the pallet. Meantime,\nhowever, the other pallet was moving in the opposite direction, and by\nthe time the first pallet had been pushed so far that it escaped or\nslid past the tooth of the crown wheel, which was pressing upon it, the\nother pallet had come into contact with the tooth on the other side\nof the crown wheel. This tended to arrest the motion of the verge, to\nbring the balls to a standstill, and ultimately to impart a motion in\na contrary direction to them.\n\nThus then the arrangement was that of a pendulum not acted on by\ngravity, for the balls neutralised one another. The pendulum was,\nhowever, not subjected to a harmonic acceleration, but alternately\nto a nearly uniform acceleration from _A_ to _B_ and _B_ to _A_. As\na result, therefore, the time of oscillation was not independent of\nthe arc of swing, but varied according to it, as also according to\nthe driving power of the crown wheel. At each stroke there was a\nconsiderable \u201crecoil.\u201d For as each tooth of the wheel came into play it\nwas unable at first to overcome and drive back the pallet against which\nit was pressing, but, on the contrary, was for a time itself driven\nback by the pallet.\n\n[Illustration: FIG. 42.]\n\nOf course, so long as the motions of the wheel and verge were exactly\nuniform, fair time was kept. But the least inequality of manufacture\nproduced differences.\n\nNevertheless it was on this principle that clocks were made during the\nthirteenth, fourteenth and fifteenth centuries. They were mostly made\nfor cathedrals and monasteries. One was put up at Westminster, erected\nout of money paid as a fine upon one of the few English judges who have\nbeen convicted of taking bribes.\n\nThe time of swing of these clocks depended entirely upon the ratio of\nthe mass of the balls at the end of the verge as compared with the\nstrength of the driving force by which the acceleration on the pallets\nwas produced. They were very commonly driven by a spring instead of a\nweight. The spring consisted of a long strip of rather poor quality\nsteel coiled up on a drum. As it unwound it became weaker, and thus the\nacceleration on the verge became weaker, and the clock went slower.\n\nIn order, therefore, to keep the time true, it became necessary to\ndevise some arrangement by which the driving force on the crown wheel\nshould be kept more constant.\n\nThis gave rise to the invention of the fusee. The spring was put inside\na drum or cylindrical box. One end of the spring was fastened to an\naxis, which was kept fixed while the clock was going; the other was\nfastened to the inside of the drum. Round the drum a cord was wound,\nwhich, as the drum was moved by the spring, tended to be wound up on\nthe surface of the drum. Owing to the unequal pull of the spring, this\ncord was pulled by the drum strongly at first, and afterwards more\nfeebly. To compensate its action a conical wheel was provided, with a\nspiral path cut in it in such a way and of such a size and proportion\nthat as the wheel was turned round by the pull of the drum the cord was\non different parts of it, so that the leverage or turning power on it\nvaried, becoming greater as the pull of the cord became weaker, and in\nsuch a ratio that one just compensated the other, and the turning power\nof the axle was kept uniform.\n\nIn this manner small table clocks were made which kept very tolerable\ntime.\n\n[Illustration: FIG. 43.]\n\nHuygens converted these clocks into pendulum clocks in a very simple\nmanner. He removed one of the balls, lengthened the verge, and slightly\nincreased the weight of the other ball. By this means, while the crown\nwheel still continued to drive the verge and remaining ball, the\nacceleration on that ball now no longer depended entirely on the force\nof the crown wheel. The acceleration and retardation were now almost\nentirely governed by the force of gravity on the remaining ball, and\nthis acceleration was harmonic.\n\nThe clock, therefore, was immensely improved as a time-keeper. Still,\nhowever, the acceleration remained partly due to the driving power, and\nthis was partly non-harmonic and introduced errors.\n\nMost of the old clocks were converted shortly after the time of\nHuygens. As there was in general no room for the pendulum inside the\nclock-case, they usually brought the axle on which the pallets were\nmounted outside the clock and made it vibrate in front of the face.\n\nMany old clocks exist, of which the engraving in the frontispiece is\nan example, that have been thus converted. A true old verge escapement\nclock is now a rarity.\n\nThe type of escapement invented by Galileo never came into vogue for\nclocks, on account of its imperfections, except till after a long\ninterval, when, with certain modifications, it became the basis of a\nnew improvement at the hands of Sir George Airey.\n\nThe crown wheel fell into disuse and was replaced by the anchor\nescapement, which was employed in that popular and excellent timepiece\nused throughout the eighteenth and the early part of the nineteenth\ncentury, and is now known as \u201cThe Grandfather\u2019s Clock.\u201d It was after\nall the crown wheel in another shape. The wheel, however, was flattened\nout, the teeth being put in the same plane. This made it much easier to\nconstruct. The pallets were fixed on an axis, and were a little altered\nso as to suit the changed arrangement of the teeth. The pendulum was\nno longer hung on the axis which carried the pallets. A cause of a good\ndeal of friction and loss of power was thus removed. The pendulum was\nhung from a strip of thin steel spring, which allowed it to oscillate,\nand which supported it without friction. This excellent manner of\nsuspending pendulums is now universal. It enabled the pendulum to be\nmade very heavy. The bob was usually some eight or nine pounds weight.\nBy this means the acceleration on the pendulum was due almost entirely\nto gravity acting on the bob, and thus the motion of the pendulum\nbecame almost wholly harmonic. Whence it followed that variations in\nthe pendulum swing became of secondary importance, and did not greatly\nalter the going of the clock.\n\n[Illustration: FIG. 44.]\n\nTherefore when the wheels became worn, and the pivots choked with old\noil and dust, the old clock still went on. If it showed a tendency to\nstop for want of power, a little more was added to the driving weight,\nand the clock kept as good time as ever.\n\nThe swing of the pendulum was by this escapement enabled to be made\nsmall, so that the arc of swing of the bob differed but little from a\ncycloid.\n\nThe secret of the time-keeping qualities of these old \u201cGrandfather\u201d\nclocks is the length of pendulum. This renders it possible to have\nbut a small arc of oscillation, and therefore the motion is kept very\nnearly harmonic. For practical purposes nothing will even now beat\nthese old clocks, of which one should be in every house. At present\nthe tendency is to abolish them and to substitute American clocks with\nvery short pendulums, which never can keep good time. They are made\nof stamped metal. When they get out of order no one thinks of having\nthem mended. They are thrown into the ash-pit and a new one bought. In\nreality this is not economy.\n\nGood \u201cGrandfather\u201d clocks are not now often made. The last place I\nremember to have seen them being manufactured is at Morez, in the\ndistrict of the Jura. An excellent clock, enclosed in a dust-tight iron\ncase, with a tall painted case of quaint old design, can be bought for\nabout 55_s._ The wheels are well cut, and the internal mechanism very\ngood.\n\nI visited the town of Morez in the year 1893. The clock industry was\ndeclining. The farmers of France seemed to prefer small clocks of\nhideous appearance, made in Germany and in America, to the excellent\nwork of their own country. Probably by now the old clockmaking industry\nis extinct. One I purchased at that time has gone well ever since.\n\n\n\n\nCHAPTER IV.\n\n\nIt is now time to give a description of the various parts of an\nordinary pendulum clock. We will take the \u201cGrandfather\u201d clock as an\nexample. We shall want an hour hand and a minute hand in the centre of\nthe face, and a seconds hand to show seconds a little above them. There\nwill be a seconds pendulum 39\u00b714 inches long, and the centre of the\nface of the clock will be about seven feet above the ground, so as to\ngive practically about five feet of fall for the weight.\n\n[Illustration: FIG. 45.]\n\nIn the first place, we have to consider the axle which carries the\nminute hand, and which turns round once in each hour. This is usually\nmade of a piece of steel about one-sixth of an inch in diameter.\nClockmakers usually call an axle an \u201carbor,\u201d or \u201ctree,\u201d whence our word\naxletree.\n\nThis \u201carbor\u201d is turned in the lathe, so as to have pivots on each end,\nfitted into holes in the clock plates, that is to say, the flat pieces\nof brass that serve as the body of the clock. The adjoining diagram\nshows _S T_ the clock faces, and _C_, the arbor of the minute hand.\n\nInasmuch as the seconds hand is to turn round sixty times while the\nminute hand turns round once, it is obvious that the arbor of the\nminute hand must be connected to the arbor of the seconds hand by a\ntrain of cogwheels so arranged as to multiply by sixty. This of course\ninvolves us in having large and small cogwheels.\n\n[Illustration: FIG. 46.]\n\nThe small cogwheels usually have eight teeth, and are for convenience\nof manufacture, as also to stand prolonged wear, cut out of the solid\nsteel of the arbor. They are nicely polished.\n\nThe easiest pair of wheels to use will be two pinions of eight teeth,\nor \u201cleaves,\u201d as they are called, and two cogwheels, one of sixty-four\nteeth, the other of sixty teeth.\n\nIt is then clear that if the arbor _A_ turns round once in an hour,\nthe arbor _B_ will turn round eight times in an hour, and _C_ will turn\nround (60 \u00d7 64)/(8 \u00d7 8) = 60 times in an hour, or once in each minute.\n\nBy having 480 teeth on the cogwheel on _A_, you could, of course, make\n_C_ go round once in a minute without the use of any intermediate arbor\nsuch as _B_.\n\n[Illustration: FIG. 47.]\n\nBut this would not be a very convenient plan. For as the wheel on _A_\nis usually about two and a quarter inches in diameter, to cut 480 teeth\non so small a wheel would involve us in cutting about sixty teeth to\nthe inch. The teeth would thus be microscopically small, and would\nhave to be set so fine that the least dirt would clog them. Moreover,\nthe pinion of eight leaves would have to be microscopic. For these\nreasons, therefore, it is usual in clocks not to use wheels with teeth\nmore than sixty or sixty-four in number, and to diminish the motion\ngradually by means, where needful, of intermediate arbors. We have next\nto consider how the weight is to be arranged so as to turn the arbor\n_A_ once round in an hour. We know that we have five feet of space for\nthe weights to fall in. If we arrange to have what is called a double\nfall, as shown in the sketch, then, allowing room for pulley wheels, we\nshall find that our string may be practically about nine feet in length.\n\n[Illustration: FIG. 48.]\n\nThe clock will be wanted to go for a week without winding, and as\npeople may forget to wind it at the proper hour of the day, we will\ngive it a day extra, and make an \u201ceight-day\u201d clock of it. Hence then,\nwhile nine feet of cord is being pulled out by a weight which falls\nfour and a half feet, the minute hand is to be turned round as many\ntimes as there are hours in eight days, viz., 192 times. This could be\naccomplished, of course, by winding the cord round the arbor of the\nminute hand. But this would require 192 turns. If our cord is to be\nordinary whipcord, or catgut, say one-twelfth of an inch in diameter,\nin order that the cord could be wound upon it, the arbor would have\nto be 192/12 inches long = 14\u2153 inches long. This would make the clock\ncase unnecessarily deep. We must therefore again have recourse to an\nintermediate wheel.\n\n[Illustration: FIG. 49.]\n\nIf we put a pinion of eight leaves on the minute hand arbor _c_, and\nengage it with a wheel of sixty-four teeth on another arbor _b_, then\n_b_ will obviously turn round once in eight hours, that is to say,\ntwenty-four times in the period of eight days. And, if we fix on _b_\na \u201cdrum\u201d or cylinder two inches long, the twenty-four turns of our\ncord will just fit upon it, since, as has been said, our cord is to be\none-twelfth of an inch in diameter. The diameter of the drum must be\nsuch that a cord nine feet long can be wound twenty-four times round\nit. That is to say, each lap must take (9 \u00d7 12)/24 = 4\u00bd inches of cord.\nFrom this it is easy to calculate that the diameter of the drum must be\nrather less than one and a half inches. From this then it results that\nwe want for a \u201cGrandfather\u2019s\u201d clock a drum two inches long and one and\na half inches diameter, on this a cogwheel of sixty-four teeth working\ninto a minute hand arbor, with a pinion wheel with eight leaves, and\na cogwheel of sixty-four teeth, an intermediate or idle wheel with\nan eight-leaved pinion, and a cogwheel of sixty teeth, engaging with\na seconds hand arbor with a pinion of eight leaves. This is called\nthe \u201ctrain of wheels.\u201d With it a weight such as can be arranged in an\nordinary \u201cGrandfather\u2019s\u201d clock case will cause by its fall during eight\ndays the second hand arbor to turn round once in each minute during the\nwhole time, and the minute hand arbor to turn round once in each hour.\n\n[Illustration: FIG. 50.]\n\nWe must next provide an arrangement for winding the clock up. It\nis obvious that we cannot do so by twisting the hands back. It is\ntrue that this could be done, but it would take about five minutes\nto do each time and be wearisome. In order to save this trouble, an\narrangement called a ratchet wheel and pall must be provided. A ratchet\nwheel consists of a wheel with a series of notches cut in it, as\nshown in the figure _A_. A pall is a piece of metal, mounted on a pin,\nand kept pressed up against the ratchet wheel by a spring _C_. It is\nobvious that if I turn the wheel _A_ round, and thus wind up a weight,\nfastened to a cord wound round the drum _D_, that the pall _B_ will go\nclick-click-click as the ratchet wheel goes round, but that the pall\nwill hold it from slipping back again. When, however, I take my hands\naway, and let the ratchet wheel alone, then the weight _E_ will pull on\nthe drum _D_, and try and turn the ratchet wheel back the opposite way\nto that in which I twisted it at first. If the pall _B_ is held fast,\nit is impossible to move it, but if the pall is fixed to a cogwheel\n_F_, which rides loose on the arbor of the drum _D_, then the pull of\nthe weight _E_ will tend to twist the cogwheel _F_ round, and this, if\nengaged with a pinion wheel on the minute hand arbor, will therefore\ndrive the clock. As the clock arbors move, of course the weight _E_\ngradually runs down, and, at last all the string is unwound from the\ndrum _D_. The clock is said then to have \u201crun down,\u201d but if I take a\nclock key, and by means of it wind the string up upon the drum _D_,\nthen the pall lets the drum and ratchet slip; the clock hands are not\naffected. When I have given twenty-four turns to the arbor, the nine\nfeet of cord will then be wound upon the drum again, and the clock will\nbe ready to go for eight more days, and will begin to move as soon as I\ncease to press upon the clock key.\n\n[Illustration: FIG. 51.]\n\nI have thus described the winding mechanism. It now remains to describe\nthe escapement.\n\nIt is of course obvious that, if the weight and train of wheels were\nsimply let go, the weight would rush down, and the seconds-hand\nwheel would fly round at a tremendous pace; but we want it to be so\nrestrained as only to be allowed to go one-sixtieth part of its\njourney round in each second. In fact, we need an \u201cescapement\u201d and a\npendulum.\n\nThe escapement usually employed in \u201cGrandfather\u201d clocks is the anchor\nescapement above described. It is not by any means the best sort of\nescapement, but it is the easiest to make; and hence its popularity in\nthe days sometimes called the \u201cdear, good old days,\u201d when people had to\nfile everything out by hand, and had to take a day to do badly what can\nnow be done well in five minutes.\n\nThe escape wheel of an anchor escapement has thirty sharp angular teeth\non its rim. The wheel is made as light as possible, so that the shock\nof stoppage at each tick of the clock may be as slight as possible, for\na heavy blow of course wastes power and gradually wears out the clock.\nThe anchor consists of two arms of the shape shown in the illustration\n(Fig.\u00a044). As the escape wheel goes round in the direction of the\narrow, the anchor, mounted on its arbor, rocks to and fro. The wheel\ncannot run away, because the act of pushing one arm or \u201cpallet,\u201d as it\nis called, outwards, and thus freeing the tooth pulls the other pallet\nin, and this stops the motion of the tooth opposite to it, but when the\nanchor rocks back again, so as to disengage the pallet from the tooth\nthat holds it, then the opposite tooth is free to fly forward against\nthe other pallet. This tends to rock the anchor the other way, but\nat that instant the pallet just engages the next tooth of the wheel,\nand so the action goes on. The anchor rocks from side to side; the\npallets alternately engage the teeth of the wheel, making at each rock\nof the anchor the tick-tock sound with which we are so familiar. If\nthe anchor were free to rock at any speed it could, the ticking of the\nclock would be very quick; so, to restrain the vivacity of the anchor,\nwe have a pendulum. The pendulum might be simply hung on to the anchor.\nBut the disadvantage of doing this would be that the heavy bob of the\npendulum would cause such a pressure on the arbor of the anchor that\nthere would be great friction, and the arbor would soon be worn out,\nand the accurate going of the clock disturbed. The pendulum therefore\nis hung on a piece of steel spring on a separate hook, which lets it\ngo backwards and forwards and carries the weight easily, while a rod\nprojecting from the anchor has a pin, which works in a slot on the\npendulum. The pendulum is therefore able to control and regulate the\nmovements of the escapement, and thus the time of the clock.\n\nOf course it is clear that the heavier the driving weight put on the\ndrum of the clock, and the better the cut and finish of the wheels, and\nthe greater the cleanliness and oil, the more will be the pressure\ntending to drive round the escape wheel, and the harder the pressure\non the pallets, and hence the bigger the impulses on the pendulum, and\ntherefore the larger the amplitude of its swing.\n\nIf the amplitude of the pendulum\u2019s swing affected the time of its\nswing, then the time kept by the clock would vary with the weight, and\nthe dirt and friction, and the drying up of the oil. But here precisely\nis where the value of the beautiful law governing the harmonic motion\nof the pendulum comes in. The time of the pendulum is (for small arcs)\nindependent of the length of swing, and therefore of the driving force\nof the clock, and hence within limits the clock, even though roughly\nmade and foul with the dirt of years, continues to keep good time.\nBut the anchor escapement has imperfections. The only way in which\na pendulum can be relied on to keep accurate time is by leaving it\nunimpeded. But the pressure of the teeth on the pallets in an anchor\nescapement constantly interferes with this.\n\n[Illustration: FIG. 52.]\n\nA little consideration will easily show that there are some times\nduring the swing of a pendulum at which interference is far more\nfatal to its time-keeping than at others. Thus the bob of a pendulum\nmay be regarded as a weight shot outwards from its position of rest\nagainst the influence of a retarding force varying as its distance\nfrom rest\u2014in fact, shot out against a spring. The time of going out\nand coming in again will be quite independent of the force exerted to\nthrow it out, quite independent of its original velocity. Therefore\na variation in the impulse given to the bob is of no consequence,\nprovided that impulse is given when the bob is near the position of\nrest. This follows from the nature of the motion. If a ball be attached\nto a piece of elastic thread, and thrown from the hand, so that it\nflies out, and then stops and is brought back by the elastic force of\nthe thread, the time of the outward motion and the return is the same\nwhatever be the force of the throw. And so if a pendulum be impelled\noutwards from a position of rest, the time of the swing out and back\nis the same, however big (within limits) is the impelling force and\nthe consequent length of the swing. The use of a pendulum as a measure\nof time is to impel it outwards, and then let it fly _freely_ out and\nback. But if its motion is not free, if forces other than gravity act\nupon it while on its path, then its time of swing will be disturbed.\nIt does not matter with what force you originally impel it, but what\ndoes matter is, that when it once starts it should be allowed to travel\nunimpeded and uninfluenced. Now that is what an anchor escapement does\nnot do. The impulse is given the whole way out on one of the pallets,\nand then when it is at its extreme of swing, and ought to be left\ntranquil, the other pallet fastens on it. But a perfect escapement\nought to give its impulse at the middle point of the swing, when the\npendulum is at the lowest, and then cease, and allow the pendulum to\nadapt its swing to the impulse it has received, and thus therefore to\nkeep its time constant. This is done by an escapement called the dead\nbeat escapement, which, though in an imperfect way, realises these\nconditions.\n\nThe alteration is made in the shape of the pallets of the anchor. The\nwheel is much the same. Each pallet consists of two faces: a driving\nface _a b_ and a sliding face _b c_.\n\n[Illustration]\n\nWhen the tooth _b_ has done its work by pressing on the driving face,\nand thus driving the anchor over, say, to the left, then the tooth on\nthe opposite side falls on the sliding face of the other pallet. This\nbeing an arc of a circle, has no effect in driving the anchor one way\nor the other; hence the pendulum is free to swing to the left as far as\nit likes and return when it feels inclined, always with the exception\nof a little friction of the tooth on the faces of the pallets, but\nwhen it returns and begins to move towards the right, the tooth slides\nback along the face of the pallet till the pendulum is almost at the\nmiddle of its swing; then an impulse is given by the pressure of the\ntooth upon the inclined plane _a\u00b4 b\u00b4_. As soon, however, as the tooth\nleaves _b\u00b4_, another tooth on the other side at once engages the\nsliding face _b c_ of the other pallet, and so the motion goes on.\n\nThis beautiful escapement is at present used for astronomical clocks;\nthe pallets are made of agate or sapphire, and therefore do not grind\naway the teeth of the wheel perceptibly, and the loss by friction on\nthe sliding surfaces is exceedingly small.\n\nThere are several other ways even better than this for securing a free\npendulum movement. We have now to return to our clock.\n\nThe centre arbor moves round once in an hour, and carries the minute\nhand. In order to provide an hour hand, which shall turn round once in\ntwelve hours, we fasten a cogwheel and tube _N_ on to the minute hand\narbor by means of a small spring, which keeps it rather tight, but\nallows it to slip if turned round hard (see Fig.\u00a045). This spring is\na little bent plate slipped in behind the cogwheel on which its ends\nrest; its centre presses on a shoulder on the minute hand arbor; it is\na sort of small carriage spring. The cogwheel _n_ has thirty teeth.\nThis cogwheel engages another cogwheel _o_ with thirty teeth, on a\nseparate arbor, which carries a third cogwheel, _p_, with six teeth,\nand this again engages a fourth cogwheel, _q_, with seventy-two teeth,\nmounted on a tube which slips over the tube to which the cogwheel _a_\nis attached. It is now easy to see that for each turn of the minute\nhand arbor the arbor _p_ makes one turn, and for each turn of the\narbor _p_ the cogwheel _d_, makes one-twelfth of a turn. From which\nit follows that for each turn of the minute hand arbor the cogwheel\n_d_ with its tube, or, as it is sometimes called, its \u201cslieve,\u201d makes\none-twelfth of a turn, and thus makes a hand fastened to it show one\nhour for every complete turn of the minute hand.\n\nThe minute hand is attached to the tube or slieve which carries the\ncogwheel _N_. The hour hand is attached to the tube or slieve which\ncarries the cogwheel _Q_, and one goes twelve times as slowly as the\nother.\n\nBut if you want to set the clock it is easy to do so by reason of the\nfact that the minute hand is not fixed to the arbor, but only to the\nslieve on the cogwheel that fits on the arbor, and is held somewhat\ntight to the arbor by means of the spring. The hands can thus be\nturned, but they are a little stiff. A washer on the minute hand arbor\nkeeps the slieve on the cogwheel pressed tight against the spring,\nbeing secured in its turn by a very small lynch-pin driven through a\nhole in the minute hand arbor.\n\nIt remains to explain a few subsidiary arrangements, not always found\nupon all clocks, but which are useful.\n\nIn order to prevent the overwinding of the clock (see Fig.\u00a043), which\nwould cause the cord to overrun the drum, an arm is provided, fitted\nwith a spring. As the weight is wound up the free part of the cord\ntravels along the drum or the fusee; and the cord, when it is near the\nend of the winding, comes up against the arm and pushes it a little\naside. This causes the end of the arm to be pushed against a stop on\nthe axis of the fusee, and thus prevents the clock being further wound\nup. The stop, being ratchet-shaped, does not prevent the weight from\npulling the ratchet wheel round the other way, and thus driving the\nclock; it only prevents the rotation of that wheel when the string is\nnear it, and the winding is finished.\n\nAnother arrangement is the \u201cmaintaining spring.\u201d\n\nIt will be remembered that during the process of winding the clock the\nhand twisting the key takes the pressure of the ratchet wheel off the\npall, so that during that operation no force is at work to drive the\nclock. In consequence the pendulum receives no impulse, but swings\nsimply by virtue of its former motion. If the process of winding were\ndone slowly enough the clock might even stop. To avoid this, a very\ningenious arrangement is made to keep the cogwheel mounted on the\nwinding shaft going during the winding-up process. This is called a\nmaintaining spring.\n\nThe arrangement shown in Fig.\u00a053 will explain it.\n\n[Illustration: FIG. 53.]\n\nThe cogwheel _a_ and the ratchet wheel are both mounted loosely on the\narbor carrying the drum. _a_ is linked to _b_ by a spring _c_. The\nratchet wheel _b_ is engaged by a pall fixed to some convenient place\non the body of the clock frame. When the weight pulls on the drum the\npull is communicated to the ratchet wheel _b_, and this acts on the\nspring _c_ and pulls it out a little. As soon as the spring _c_ is\npulled out as far as its elasticity permits, a pull is communicated\nto the cogwheel _a_, and the clock is driven round. When the clock is\nwound the pressure of the weight is removed, and therefore the ratchet\nwheel _e_ no longer presses on the pall, and thus no pressure is\ncommunicated to the ratchet wheel _b_, or through it to the clock. But\nhere the spring _c_ comes into play. For since the ratchet wheel _b_ is\nheld fast by the pall _d_, the spring _c_ pulls at the wheel _a_, and\nthus for a minute or so will continue to drive the clock. This driving\nforce, it is true, is less than that caused by the weight, but it is\njust enough to keep the pendulum going for a short time, so that the\ngoing of the clock is not interfered with.\n\nIf the reader can get possession of a clock, preferably one that does\nnot strike, and, with the aid of a small pair of pincers and one or two\nscrewdrivers, will take it to pieces and put it together again, the\nmechanism above described will soon become familiar to him. Not every\nclock is provided with maintaining spring and overwinding preventer.\n\nThe cause of stoppage of a clock generally is dirt. Where possible,\nclocks should always be put under glass cases. \u201cGrandfather\u201d clocks\nwill go much better if brown paper covers are fitted over the works\nunder the cases. In this way a quantity of dust may be avoided. To get\na good oil is very important. It will be noticed that pivot-holes in\nclocks are usually provided with little cup-like depressions. This is\nto aid in keeping in the oil. The best clock oil is that which does not\neasily solidify or evaporate. Ordinary machine oil, such as used for\nsewing machines, is good as a lubricant, but rapidly evaporates. Olive\noil corrodes the brass.\n\nIt is best to procure a little clock oil, or else the oil used for gun\nlocks, sold by the gunsmiths. The holes should be cleaned out with the\nend of a wooden lucifer match, cut to a tapering point. The pivots\nshould be well rubbed with a rag dipped in spirits of wine. If the\npivots are worn they should be repolished in the lathe. If the cogs of\nthe wheels are worn, there is no remedy but to get new ones. Old clocks\nsometimes want a little addition to the driving weight to make them go.\n\nThe weight necessary to drive the clock depends on its goodness of\nconstruction, and on the weight of the pendulum. If the clock is driven\nfor eight days with a cord of nine feet in length with a double fall,\nthen during each beat of the pendulum that weight will descend by an\namount =\n\n             9/(2 \u00d7 24 \u00d7 60 \u00d7 60 \u00d7 8) feet or 1/12800th inch.\n\nWhence, if the clock weight is 10 lbs., the impulse received by the\nclock at each beat is equivalent to a weight of 10 lbs. falling through\n1/12800th of an inch, or to the fall of six grains through an inch.\n\nThe power thus expended goes in friction of the wheels and hands, and\nin maintaining the pendulum in spite of the friction of the air.\n\nThe work therefore that is put into the clock by the operation of\nwinding is gradually expended during the week in movement against\nfriction. The work is indestructible. The friction of the parts of the\nclock develops heat, which is dissipated over the room and gradually\nabsorbed in nature. But this heat is only another form of work. Amounts\nof work are estimated in pressures acting through distances. Thus, if\nI draw up a weight of 1 lb. against the accelerative force of gravity\nthrough a distance of one foot, I am said to do a foot-pound of work.\n\nOne pound of coal consumed in a perfect engine would do eight millions\nfoot-pounds of work. Hence, if the energy in a pound of coal could be\nutilized, it would keep about 100,000 grandfather\u2019s clocks going for a\nweek. As it is consumed in an ordinary steam engine it will do about\nhalf a million foot-pounds of work. One pound of bread contains about\nthree million foot-pounds of energy. A man can eat about three pounds\nof bread in a day, and, as he is a very good engine, he can turn this\ninto about three-quarters of a million foot-pounds of work. The rest of\nthe work contained in the bread goes off in the form of heat.\n\n[Illustration: FIG. 54.]\n\nAs has been previously said, the power of the action of gravity in\ndrawing back a pendulum that has been pushed aside from its position of\nrest becomes less in proportion as the pendulum is longer, and hence\nas the pendulum is longer the time of vibrations increases. In the\nappendix to this chapter a short proof will be given showing that the\nlength of a pendulum varies as the square of the time of its vibration.\nA pendulum which is 39\u00b714 inches in length vibrates at London once in\neach second. Of course at other parts of the earth, where the force of\ngravity is slightly different, the time of vibration will be different,\nbut, since the earth is nearly a globe in shape, the force of gravity\nat different parts of it does not vary much, and therefore the time of\nvibration of the same pendulum in different parts of the earth does not\nvary very much. The length of a pendulum is measured from its point of\nsuspension down to a point in the bob or weight. At first sight one\nwould be inclined to think that the centre of gravity of the pendulum\nwould be the point to which you must measure in order to get its\nlength. So that if _B_ were a circular bob, and the rod of the pendulum\nwere very light, the distance _A B_ to the centre of the bob would be\nthe length of the pendulum. But if we were to fly to this conclusion,\nwe should be making the same error that Galileo made when he allowed a\nball to _roll_ down an inclined plane. He forgot that the motion was\nnot a simple one of a body down a plane, but was also a rolling motion.\nThe pendulum does not vibrate so as always to keep the bob immovable\nwith the top side _C_ always uppermost. On the contrary, at each beat\nthe bob rotates on its centre and makes, as it were, some swings of\nits own. Therefore in the total motions of the pendulum this rotation\nof the bob has to be taken into account. Of course, if the pendulum\nwere so arranged that the bob did not rotate, and the point _C_ were\nalways uppermost, as, for instance, if the pendulum consisted of two\nparallel rods, _A B_ and _C D_, suspended from _A_ and _C_, then we\nmight consider the bob as that of a pendulum suspended from _E_, and\nthe pendulum would swing once in a second if _A B_ = _C D_ = _E F_\nwere equal to 39\u00b714 inches, for by this arrangement there would be no\nrotation of the bob. But as pendulums are generally made with the bob\nrigidly fixed to the rod _E F_, the rotation must be taken into account.\n\n[Illustration: FIG. 55.]\n\nIt wants some rather advanced mathematical knowledge to do this. But\nin practice clockmakers take no account of it. The correction is not a\nlarge one, so they make the rod as nearly true as they can, arrange a\nscrew on the bob to allow of adjustment, and then screw the bob up and\ndown until in practice the time of oscillation is found to be correct.\n\n[Illustration: FIG. 56.]\n\nThe mode of suspension of a pendulum of the best class is that shown\nin Fig.\u00a056, which allows the pendulum to fall into its true position\nwithout strain. _A_ is a tempered steel spring, which bends to and\nfro at each oscillation. It is wonderful how long these springs can\nbe bent to and fro without breaking. Inasmuch as lengthening the\npendulum increases the time, so that the time of vibration _t_ varies\nas the square of the length of the pendulum, a very small lengthening\nof the pendulum causes a difference in the time. In practice, for\neach thousandth of an inch that we lengthen the pendulum we make a\ndifference of about one second a day in the going of the clock. If we\ncut a screw with eighteen threads to the inch on the bottom of the\npendulum rod, and put a circular nut on it, with the rim divided into\nsixty parts, then each turn through one division will raise or lower\nthe bob by 1/1080th of an inch, and this first causes an alteration\nof time of the clock by one second in the day. This is a convenient\narrangement in practice, for it affords an easy means of adjusting the\npendulum. We need only observe how many seconds the clock loses or\ngains in the day, and then turn the nut through a corresponding number\nof divisions in order to rectify the pendulum.\n\n[Illustration: FIG. 57.]\n\nAnother needful correction of the pendulum is that due to changes in\ntemperature. If the rod of the pendulum be made of thoroughly dried\nmahogany, soaked in a weak solution of shellac in spirits of wine,\nand then dried, there will not be much variation either from heat or\nmoisture. But for clocks required to have great precision the pendulum\nrod is usually made of metal. A rod of iron expands about 1/160000th\nof its length for each degree Fahrenheit; and therefore for each degree\nFahrenheit a pendulum rod of 39\u00b714 inches will expand about 1/4000\nthousandths of an inch, and thus make a difference in the going of the\nclock of about one-fourth of a second per day. The expansion will, of\ncourse, make the clock go slower. It would be possible to correct this\nexpansion if some arrangement could be made, whenever it occurred, to\nlift up the bob of the pendulum by an amount corresponding to it, as,\nfor instance, to make the bob of some material which expanded very much\nmore by heat than the material of which the pendulum rod was made.\n\n[Illustration: FIG. 58.]\n\nThus if we hang on to the end of a pendulum of iron a bottle of iron\nabout seven inches long, and almost fill it with mercury, then, as\nsoon as the heat increases, the iron of the rod and of the bottle\nexpands, and the centre of oscillation of the pendulum is lowered.\nBut as the linear expansion of mercury contained in a bottle is about\nfive times that of iron, the mercury rises in the bottle, and thus the\nexpansion downwards of the pendulum rod is compensated by the expansion\nupwards of the mercury in the bottle. The rod may be fastened to the\nmouth of the bottle by a screw, so that as the bottle is turned round\nit may be raised or lowered on the rod, and thus the length of the\npendulum may be adjusted. The bottle is made of steel tube, screwed\ninto a thin turned iron top and bottom. Of course no solder must be\nused to unite the iron, for mercury dissolves solder. A little oil and\nwhite-lead will make the screwed joints tight. This is an excellent\nform of pendulum. Another plan is to use zinc as the metal which is to\ncounteract the expansion of the iron. The expansion of zinc is about\nthree times that of iron.\n\n[Illustration: FIG. 59.]\n\nHence a zinc tube, about twenty inches long (shown shaded in Fig.\u00a059),\nis made to rest upon a disc fastened to the lower part of the iron\npendulum rod. On the top of the zinc rests a flat ring _A_, from which\nis suspended an iron tube _A_, which carries the bob _B_. The expansion\nof the zinc tube is large enough to compensate the expansion both of\nthe rod and the tube, and the bob consequently remains at the same\ndepth below the point of suspension, whatever be the temperature.\n\nThere is, however, a new method which is far superior to all these, and\nthis is due to the discovery by M. Guilliaume, of Paris, of a compound\nof nickel and steel which expands so little that it can be compensated\nby a bob of lead instead of by a bob of mercury. This material is sold\nin England under the name of \u201cinvar.\u201d An invar rod with a properly\nproportioned lead bob makes an almost perfect pendulum, the expansion\nof the invar and the lead going on together. The exact expansion of the\ninvar is given by the makers, who also supply information as to the\nsize and suspension of the bob proper to use with it.\n\nIt has been already shown that the uniformity of time of swing of a\npendulum is only true when the arc through which it swings is very\nsmall. If the total swing from one side to another is not more than\nabout two inches very little difference in time-keeping is made by\nputting a little more driving weight on the clock, and thus increasing\nits arc of swing; but when the arc of swing becomes say three inches,\nor one and a half inches on each side of the pendulum, then the time of\nvibration is affected. At this distance each tenth of an inch increase\nof swing makes the pendulum go slower by about a second a day.\n\nThe resistance of the air, of course, has a great influence on a\npendulum, and is one of the chief causes that bring it ultimately to\nrest. Even the variations of pressure of the atmosphere which the\nbarometer shows as the weather varies have an effect on the going of\na clock. Attempts have been made by fixing barometers on to pendulums\nwith an ingenious system of counter balancing to counteract this, but\nthese refinements are not in common use, and are too complicated to be\nsusceptible of effective regulation.\n\n\nAPPENDIX TO CHAPTER IV.\n\nIt may be useful to give a simple form of proof of the law which\ngoverns the time of oscillation of a pendulum whose length is given.\n\nUnfortunately, it is impossible to give one so simple as to be\ncomprehended by those who know nothing whatever of mathematics. It\nis, however, possible to give a proof that requires very little\nmathematical knowledge.\n\nWe know that when a mass of matter is whirled round at the end of a\nstring it tends to fly outwards and puts a strain on the string. The\nfaster the speed at which the mass is whirled, the stronger will be the\nstrain on the string. Suppose that the length of the string equals R,\nthe velocity of the mass as it flies round equals V. Let _a_ be the\nbody whirled round by a string _o a_ from a centre at _O_. The body\nalways, of course, tends to fly on in a straight line from the point at\nwhich it is at any instant. But that tendency is frustrated by the pull\nof the string which constrains it to take a circular path. It is, of\ncourse, all one whether the force that tends to pull the body inwards\ntowards _O_ is a string or an attractive force of any kind acting\nthrough a distance without any string at all. Evidently if the body\nkeeps its place in the circle it must be because the centrifugal force\ntending to whirl it out is equal to the centripetal or attractive force\ntending to pull it in.\n\n[Illustration: FIG. 60.]\n\nThe strain on the body, due to the force tending to pull it inwards, we\nshall designate by F, meaning by F the number of feet of velocity that\nwould in one second be imparted to the body by the attractive force.\n\nSuppose that at some given instant of time the body is at a point _a_.\nAt that instant its _direction_ will be along _a b_, tangential to the\ncircle at _a_, and that is the path it would take if the centripetal\nor attractive force ceased to act just as the body got to _a_. In that\ncase the body would be whirled off like a stone from a sling along\nthe line _a b_, and would at the end of a given time, let us suppose a\nsecond, arrive at _b_. But it is not so whirled off; it is attracted\ntowards _O_ and pulled inwards, and comes to _c_. Hence, then, the\nattractive force acting during one second must have been sufficient to\npull the mass in from _b_ to _c_. But we know that if an accelerating\nforce (F) acts on a body for a second it produces a final velocity\nequal to F at the end of the second, and an average velocity half F\nduring the second.\n\nHence, then, the space _b c_, by which the body has been pulled in,\nis represented by half F, but _a b_, the space which the body would\nhave travelled forwards, will be represented by V, the velocity of the\nbody in a second; but if the motion be such that the distance _b c_\ntravelled in a second is very small, then the triangles _a b d_ and\n_a b c_ are approximately similar, and the smaller _a b_ is the more\nnearly similar they are. Whence then (a b)/(b c) = (a d)/(a b), that is\nto say (a b)\u00b2 = a d \u00d7 b c.\n\nBut _a b_ represents the space which would have been traversed by the\nbody in one second at the rate it was going, and hence is equal to V;\n_a d_ is the diameter of the circle, and hence equals 2 R; _b c_ is\nthe space through which the body has been drawn in the second by the\nattractive force F, and therefore equals half F.\n\nWhence then V\u00b2 = 2 R \u00d7 half F = R F.\n\nWe took a second as the limit of time during which the motion was to be\nconsidered. Of course any other time could have been taken. Now what\nis true of the motion of a body during a very short time is also true\nof the body during the whole of its path, assuming that the path is a\ncircle, and that F remains constant, as it obviously will if the path\nis a circle, and the velocity is uniform. Whence then we may generally\nsay that if a body is being whirled round at the end of a string the\nstrain F on the string is directly proportional to the square of the\nvelocity, and is inversely proportional to the length of the string.\n\nThe time of rotation, is of course = length of the path \u00f7 velocity\n\n                  = (2\u03c0R)/V = (2\u03c0R)/\u221a(R F) = 2\u03c0\u221a(R/F).\n\nWhence then we see that for motion in a circle of a mass under the\nattraction of a centripetal force, or pull of a string, the time of\nrotation will be uniform, provided that the centripetal force always\nvaries as the radius of the path. From this it is evident that a body\nfixed on to an elastic thread where the pull varies as the extension\nwould make its rotations always in equal times. If your sling consists\nof elastic, whirl as you will, you can only whirl the body round so\nmany times in a second, and no more. Any increase in your efforts only\nmakes the string stretch, and the circle get bigger. The velocity of\nthe body in its path of course increases, but the time it takes to go\nonce round is invariable.\n\nIt also follows that if a body hung by a string of length _l_, under\nthe action of gravity, be travelling in a circle round and round, then,\n_if the circle is a small one compared with the length of the string_,\nthe inward acceleration _f_ towards the centre will be approximately\nproportional to the radius _r_ of the circle, and the time of rotation\nwill be\n\n                             t = 2\u03c0\u221a(r/f).\n\nBut in this case _f_, the inward acceleration, is to _g_ the\nacceleration downwards of gravity as A B:A P or\n\n                 f/g = (A B)/(A P) = (A P)/(O P) = r/l.\n\n[Illustration: FIG. 61.]\n\n[Illustration: FIG. 62.]\n\nWhence then the time of rotation of this body would be if the circle of\nrotation was small\n\n                              = 2\u03c0\u221a(l/g).\n\nAnd if you try you will find that this is so. For instance, take a\nthread 39-1/7 inches long, that is 3\u00b725 feet. Hang anything heavy from\none end of it, and cause it to swing round and round in a _small_\ncircle. Now _g_ the acceleration of gravity = 32\u00b72 feet per second.\n\u03c0 the ratio of the circumference of a circle to its diameter = 3\u00b714.\nFrom which it follows that the time of rotation = 2 \u00d7 3\u00b714\u221a(3\u00b725/32\u00b72)\nseconds = 2 seconds. But if we look at the rotating body sideways, it\nappears to act as a pendulum; it matters nothing whether we swing it\nround and round or to and fro. For in any case the accelerative force\ntending to bring it back to a position of rest is always proportional\nto the distance of displacement, and, therefore, its time of motion\nmust always be 2\u03c0\u221a(l/g) and its motion harmonic.\n\nThe length of a seconds pendulum, that is a pendulum that makes its\ndouble swing in two seconds, will therefore be\n\n                        l = 4/((2\u03c0)\u00b2) \u00d7 g feet\n\n                          = (g \u00d7 12)/\u03c0\u00b2 inches\n\n                          = 39\u00b714 inches.\n\n\n\n\nCHAPTER V.\n\n\nI have thus described the principal features of ordinary clocks. For\nthe details many treatises must be studied, and knowledge acquired\nwhich is not in any books at all.\n\nI now, however, pass to watches. It will be remembered that a verge\nescapement consists of a crown wheel with teeth, engaging two pallets\nfixed upon a verge, furnished with balls at its extremities.\n\nAs the crown wheel was urged forwards each pallet in succession was\npushed till it slipped over the tooth which was engaging it. Then\na tooth on the other side came into sharp collision with the other\npallet, and drove the verge the other way, and so on.\n\nNow here we have a driving force, and a sort of pendulum. But how did\nthe verge act as a pendulum to measure time? It is not a body rocking\nunder the action of gravity, nor under the acceleration of a spring.\nHow then can it act as a regulator of time, and what is the period of\nits swing?\n\nThe answer to this is, that it is under the acceleration of gravity,\nbut that gravity does not act freely on the bobs or weights, but only\nthrough the driving weight and teeth. The impulse that drives the\nverge is really also the accelerating force upon it, and the only\naccelerating force upon it.\n\nAnd the worst feature about the movement is, that as the teeth and\npallets move, the leverage of the teeth on the pallets alters, and\nthus the bobs on the verge are under the influence not of a uniform or\nduly regulated force, but of a constantly varying one, and one that\nvaries in a very complicated and erratic way. It would be hopeless to\nexpect much time-keeping from such a contrivance. The most that could\nbe expected would be by putting on a very big weight to reduce to\ncomparative insignificance the friction, and then hope that the swings\nwould be uniform, so that whatever went on in one swing would go on in\nthe next, and thus the time-keeping be regular.\n\nBut any course tending to diminish the driving force, such as the\nthickening of the oil, would greatly affect the going. It was for this\nreason that Huygens turned the verge into a pendulum by removing one of\nthe bobs, and letting gravity thus act on the other.\n\nFor watches, however, a different plan was contrived. One end of a\nslender spiral spring was affixed to the verge. The other end of the\nspring was made fast to the clock frame. The verge was now, therefore,\nchiefly under the action of the acceleration of the spring. To make the\nacceleration of the teeth of the \u2019scape wheel less embarrassing, the\nteeth were so shaped as only to give a short push at stated intervals,\nand not interfere with the free swing of the verge under the alternate\nto-and-fro accelerations and retardations of the spring. By this means\nthe verge became in every way an excellent pendulum, not dependent on\ngravity, and permitting the watch to be held in any position.\n\nThe verge thus fitted was turned into a wheel, and became a \u201cbalance\nwheel.\u201d It was compensated for heat expansion by a cunning use of the\nunequal expansion of brass and steel, in a manner analogous to the way\nthis unequal expansion of metals had been employed to compensate the\npendulum, and became the beautiful and accurate time-measurer that we\nsee to-day, with its pivots mounted in jewels to diminish friction, and\nwith screws round the rims of the balance wheel to enable the centre of\ngravity to be exactly adjusted to its centre of rotation, and with a\ndelicate hair-spring of tempered steel that is a marvel of microscopic\nwork.\n\nBut the escapement of the early watches left much to be desired. In\norder to make it clear how imperfect that early escapement was, we\nhave to turn back and remember what has been said about the dead beat\nescapement.\n\nIt will then be remembered that it was shown that for small arcs the\npendulum would keep good time provided you let it have as much swing as\nit wanted to use up the force which the escapement had applied to it,\n_but not otherwise_, so the pendulums only acted really well when the\nimpulse was given about the middle of the swing, and they were free to\ngo on and stop when they pleased, and turn back at the end of it.\n\nThis essential condition was fairly approximated to in the dead beat\nescapement of clocks which left them at the end of their swing with\nonly a very slight friction to impede their free motion.\n\nBut when you come to deal with a watch the case is quite different.\nHere the escapement is of a great size compared with the balance wheel,\nand the friction even of the most dead beat watch escapement that could\nbe contrived was so big compared with the forces acting on the balance\nwheel as seriously to derange its motion, and render it far from a\nperfect time-keeper.\n\nNow about this time\u2014I am speaking of the early part of the eighteenth\ncentury\u2014a demand of a very exceptional character arose for a really\nperfect watch. The demand did not arise from gentlemen who wanted\nto keep appointments to play at ombre at their clubs, or even\nfrom merchants to time their counting house hours. For these the\nold-fashioned watch did very well. The demand came from mariners.\nBut the seamen did not want to know the time merely to arrange the\nhours for meals on the ship or to determine when the watch was to be\nrelieved, but for a far more important purpose, namely, to find out by\nobservation of the heavens their place upon the ocean when far out of\nsight of the land. It will be very interesting to see how this problem\narose, and how the patient industry and ingenuity of man has solved it.\n\nThe ancient navigators never went very far from the shore, for, once\nout of sight of land, a ship was out of all means of knowing where she\nwas. On clear days and nights the compass, and the sun and stars would\ntell the mariner the _direction_ he was sailing in, but it was quite a\nproblem to determine where he was on the surface of the earth.\n\n[Illustration: FIG. 63.]\n\nLet us consider the problem. Suppose for convenience that the earth is\ndivided up into \u201csquares,\u201d as nearly, at least, as you can consider a\nglobe to be so marked out. Let us suppose that it has been agreed to\ndraw on it from pole to pole 360 lines of longitude, commencing with\none through say Greenwich Observatory as a starting-point, and going\nright round the earth till you come back to Greenwich again. Also\nsuppose that there have been drawn a series of circles parallel to the\nequator, but going up at equal distances apart towards the poles. Let\nus have 179 of these circles, so as to leave 180 spaces, _a_ to _b_,\n_b_ to _c_, etc., from pole to pole. This will divide the earth up like\na bird-cage into squares, as if we had robed it in a well-fitting\nScotch plaid. The length measured along the equator of the side _p q_\nof each square at the equator is taken as exactly sixty nautical miles\n(apart from a small error of measurement, which makes it in actual\npractice 59\u00b796). This is equal to sixty-nine and a quarter English\nstatute miles. The side of the square leading towards the poles _q s_\nwould also be sixty nautical miles were it not that the earth is not\ntruly spherical, which introduces a slight error. We may, however,\nroughly say that at the equator each square measures sixty nautical\nmiles each way.\n\n[Illustration: FIG. 64.]\n\nAs we get towards the poles the squares become rectangular figures,\nwith the heights of latitude still sixty nautical miles, but the widths\nbecoming smaller. Thus in England our squares measure _p q_ = 37\nnautical miles and _q s_ = 60 nautical miles.\n\nNow of course we can see at once that it is easy at any place on the\nearth\u2019s surface to find your _latitude_ by a simple observation of the\nsun at noon, if you know the day of the year, and have got a nautical\nalmanac. For by an instrument called a sextant you can measure the\nangle he appears to be above the horizon, and then, as you know from\na nautical almanac the angle he is above the equator, you can soon\ndetermine your place _A_ on the globe. Or at night, if you measure the\nangular distance that the polar star _P_ is from the zenith, or point\nexactly over your head\u2014that is, the angle _P O Z_\u2014you can subtract it\nfrom a right angle and get your latitude, _A O E_, at once.\n\n[Illustration: FIG. 65.]\n\nBut how are you to determine your longitude? The pole-star, or sun,\nor any other star won\u2019t help you, for as the earth is moving they\nkeep shifting, and at one time or another appear exactly in the same\nposition to everyone on the same parallel of latitude, as it is easy to\nsee. The fact is that you are on a ball turning round. You know easily\nwhat latitude you are on, but you cannot tell your longitude unless\nyou can tell how many hours and minutes you get to a position before\nGreenwich gets to the same position. If when a particular star got to\nGreenwich a gong were sounded which could be heard all over the earth,\nthen of course, by seeing what stars were overhead, everyone would\nknow their longitude at once. Perhaps by means of the new electric\nwaves this will before long be done, and the Greenwich hours will be\nsounded all over the world for the use of mariners. But till this is\naccomplished all that can be done is to keep an accurate clock on\nboard, so as always to give you Greenwich time.\n\nEarly attempts were made to take a pendulum clock to sea, suspending it\nso as to avoid disturbance to its motion by the rocking of the ship.\nThese proved vain.\n\nIt therefore became desirable that a watch with a balance wheel\nshould be contrived to go with a degree of accuracy in some respects\ncomparable with the accuracy of a pendulum clock. To encourage\ninventors an Act of Parliament was passed in the thirteenth year of\nQueen Anne\u2019s reign (chapter xv.) (1713) promising a reward of \u00a320,000\nto anyone who would invent a method of finding the longitude at sea\ntrue to half a degree\u2014that is, true to thirty geographical miles.\n\nIf the finding of the longitude were to be accomplished by the\ninvention of an accurate watch, then this involved the use of a watch\nthat should not, in several months\u2019 going, have an error of more than\ntwo minutes, which is the time which the earth takes to turn through\nhalf a degree of longitude.\n\nThis was the problem which John Harrison, a carpenter, of Yorkshire,\nmade it his life business to solve. His efforts lasted over forty\nyears, but at the end he succeeded in winning the prize.\n\nThese instruments have been much improved by subsequent inventors, and\nhave resulted in the construction of the modern ship\u2019s chronometer,\na large watch about six inches in diameter, mounted on axles, in a\nmahogany box. Several of these are taken to sea by every ship.\n\nThe peculiarity of the chronometer is its escapement.\n\nLet _A B_ be the scape wheel, and _C D_ a small lever attached to _C_,\nthe pivot on which the balance wheel and spring is fastened. Let _E G_\nbe a lever, with a tooth _F_ which engages the teeth of the scape wheel\nand prevents it moving round. Let _H_ be a spring holding the lever _E\nG_ up to its work.\n\n[Illustration: FIG. 66.]\n\nThe lever has a spring _K E_ fastened to it at the point _K_. This\nspring is very delicate. If the lever _C D_ is turned so that the\nlittle projection _M_ on it strikes the spring _E_ from left to right,\nthen, as the spring rests on the lever, the whole lever is pushed over,\nand the teeth of the scape wheel set free. At that instant, however,\nthe escapement is so arranged that the arm _C D_ is just opposite\nthe tooth _D_ of the scape wheel, so that the scape wheel, instead\nof running away, leaps with its tooth _D_ on to the lever _C D_ and\nswings the balance wheel round. The balance wheel is free to twist as\nmuch as it pleases, but the moment it has twisted so much that the\nprojection _M_ passes the spring _E_, then the lever _G E_ flies back\nto its place, and the scape wheel is again checked. Meanwhile the\nbalance wheel flies round till at last it is brought to rest by the\nbalance spring. It then recoils and sets out on its return path. This\ntime, however, the projection _M_ merely flips aside the spring _E_\nand the balance wheel goes back, till again it is brought to rest and\nreturns. As soon as the lever comes opposite _D_ the projection _M_\nthen again hits the spring _E_, and releases the catch at _F_, and\nanother tooth of the scape wheel goes by.\n\nThere then you have a completely free escapement, and consequently an\naccurate one. Many watches are made with these escapements, but they\nare more expensive than those in common use.\n\nThere is but little remaining in a watch that is not in a clock, for\nthe wheel-trains and general arrangements are very similar.\n\nIt is possible to apply the chronometer\u2019s detached escapement to a\nclock. This was done by several clock-makers in the eighteenth and\nearly part of the nineteenth century. One method of doing it is as\nfollows:\n\n_A_ is a block of metal fitted to the bottom of the pendulum, _B_ a\nlight lever pivoted on it. _C_ is the scape wheel, with four teeth;\n_D_ a tooth of the scape wheel, which hops on to the projection of the\npendulum the moment that the impact of the point _E_ of the lever _B E_\nhas pushed aside the lever _G F_, and thus released the scape wheel.\nThe advantage is that it is a very easy escapement to make. But it is\nin reality a detached (that is to say, a completely free) chronometer\nescapement, as can easily be seen.\n\n[Illustration: FIG. 67.]\n\nTurret-clocks are open to considerable disadvantages, for the wind\nblowing on the hands gives rise to considerable pressure, so that the\nclocks are sometimes urging the hands against the wind, sometimes are\nbeing helped by the wind. And this inequality of driving force makes\nthe pendulum at some times make a bigger arc of swing than at others.\n\nBut we saw above that though difference of arc of swing ought to make\nno difference in the time of swing of the pendulum, yet this was only\nstrictly true if the arc of swing were a cycloid.\n\nBut as for practical convenience we are obliged to make it a circle,\nit follows, as we saw, that for every tenth of an inch of increase of\nswing of an ordinary seconds pendulum about a second a day of error is\nintroduced. To remove this difficulty a gravity escapement was invented\nby Mudge in the eighteenth century, improved by Bloxam, a barrister,\nand perfected by the late Lord Grimthorpe. The idea was to make the\nscape wheel, instead of directly driving the pendulum, lift a weight,\nwhich, being subsequently released, drove the pendulum. The consequence\nwas, that inequalities in wind pressure, which affected the driving\nforce of the scape wheel, would not act on the pendulum, which would\nbe always driven by the uniform fall of a fixed and definite weight. A\nmovement of this kind has been fixed in the great clock at Westminster,\nand has gone admirably. A description of its details will be found in\nthe _Encyclop\u00e6dia Britannica_, written by Lord Grimthorpe himself.\n\nAll sorts of eccentric clocks and watches have been proposed. For\ninstance, it seems wonderful to see a pair of hands fitted to the\ncentre of a transparent sheet of glass go round and keep time with\napparently nothing to drive them.\n\nBut the mystery is simple. The seeming sheet of glass is not one sheet,\nbut four. The two centre sheets move round invisibly, carrying the hour\nhand and minute hand with them, being urged by little rollers below\non which they rest. When you touch the glass the outside sheets appear\nat rest, and you do not suspect that it is other than a single sheet.\nBut beware of dust, for if dust gets on the inner plate you detect the\ntrick. In this way a mechanical hand was made that wrote down answers\nto questions. This plan can be applied to all sorts of tricks.\n\nSir William Congreve, an ingenious inventor, proposed to make a clock\nthat measured time by letting a ball roll down an incline. When it got\nto the bottom it hit a lever, which released a spring and tipped the\nplane up again, so that the ball now ran down the other way. It is a\npoor time-keeper, and the idea was not original, for a ball had been\npreviously designed for the same purpose.\n\nSometimes clocks are constructed by attaching pendulums to bronze\nfigures, which have so small a movement that the eye is unable to\ndetect it. The figure appears to be at rest, but is in reality slowly\nrocking to and fro. It is necessary to make the movement as small as\nabout one four hundredth of an inch in half a second, if the movement\nis to escape human observation. For a movement of one two hundredth\nof an inch per second is about the largest that will certainly remain\nunperceived.\n\nIn medi\u00e6val times clocks were constructed with all sorts of queer\ndevices. The people of the upper town at Basle having quarrelled with\nthose of the lower town, fought and beat them. To commemorate this\nvictory they put on the old bridge at the upper town a clock provided\nwith an iron head, that slowly put out and drew in a long tongue of\nderision. This clock may still be seen in the museum. It is as though\nthe council of the city of London put a clock of derision at Temple Bar\nto put out its tongue at the County Council.\n\nI do not propose here to describe the striking mechanism of clocks.\nThere are several different ways of arranging it. They are rather\ncomplicated to follow out, but they all resolve themselves into a\nfew simple principles. As the hour hand revolves it carries a cam so\narranged as to be deeper cut away for the twelfth hour, less for the\neleventh, and so on. When the minute hand comes to the hour it releases\nthe striking mechanism, which, urged by a weight, begins to revolve,\nand, driving an arm carrying a pin, raises a hammer, which goes on\nstriking away as the arm revolves. This would continue for ever if\nit were not that at the same moment an arm is liberated which falls\nagainst the cam. At each stroke the arm is (by the striking apparatus)\nraised a bit back into position. When it comes back into position it\nstops the striking. It thus acts as a counter, or reckoner of the blows\ngiven, stopping the movement when the clock has struck sufficiently.\nIf the counting mechanism fails to act, we have the phenomenon which\noccasionally occurs of a \u201cGrandfather\u201d clock striking the whole of the\nhours for the week without stopping.\n\nA chiming clock is simpler still. For here we have a barrel covered\nwith pins, like the barrel in a musical box. As the pins go round they\nraise hammers which fall against bells. The barrel is wound up and\ndriven by a spring or weight. When the clock comes to the hour, the\nbarrel is released, and rotating, plays the tune.\n\nIf you want to make a clock wake you up in the morning it can be\ndone by making the striking arrangement hammer away with no counting\nmechanism to stop it until the weight has run down. If, not content\nwith that, you want the sheets pulled off the bed or the bed tilted\nup, or a can of water emptied over the person who will not rise, a\nmechanical device known as a relay must be used. It is very simple.\nWhat is wanted is that, after the lapse of a time which a clock\nmust measure, a considerable force must be exerted to pull off the\nbedclothes. It would be absurd to make the clock exercise this pull.\nIt is obviously better to attach the clothes by a hook to a rope which\npasses over a pulley, and from which hangs a weight. A pin secures the\nweight from falling, the pin being withdrawn by the clock. The work is\nthus done by the weight when released by the clock.\n\nIn like manner, if you have a telegraph designed to print messages at\na distance, you do not send along the wires the whole force necessary\nfor doing the printing. You only send impulses, which, like triggers,\nrelease the forces by which the letters are to be stamped.\n\nElectric clocks of many kinds have been invented. The principle of an\nelectric escapement is similar to that of an ordinary escapement.\n\n[Illustration: FIG. 68.]\n\nThe reader no doubt knows that, when a circuit of wire is joined or\ncompleted leading to a source of electricity, electricity flows through\nthe wire.\n\nIf the wire is wound round a piece of iron, then, whenever the circuit\nis joined, a current is set in motion, and the iron becomes an\nelectro-magnet. When the circuit is severed the iron ceases to be a\nmagnet.\n\nIf put at a proper position it would at each time an iron pendulum\napproached give it a small impulse provided that at that instant\nthe current is turned on. This can easily be made to be done by the\npendulum itself. For just as the pendulum is coming back to the\ncentral position a flipper _P_ attached to the rod can be caused\nto make contact with a piece of metal fixed on its path. Then the\nelectro-magnet, becoming magnetised, exerts a pull on the iron\npendulum. On the return beat of the pendulum the other side of the\nflipper _R_ strikes the obstruction. But if that side _R_ is covered\nwith ebonite or some non-conducting material no current will be set\nin motion, and the electro-magnet will not (as it would otherwise do)\nretard the pendulum. Such a pendulum has therefore an impulse given to\nit every second beat.\n\nSuch pendulums do not act very well, because it is difficult to keep\nmetallic surfaces like _Q_ clean, and therefore misses often occur.\nBesides, the strength of the current varies with the goodness of the\ncontact and with other things.\n\nWhat is now preferred is to make an arrangement by which an electric\ncurrent winds the clock up every minute or so. By this means the\nimpulse which drives the clock is not a varying electric one, but is\na steady weight. The most successful clocks have been made on these\nprinciples.\n\nThe advantage of electricity is, that by means of the current that\nactuates the clock, or winds it up, you can at regular intervals set\nthe hands in motion of a great number of clocks.\n\nSo that only one going clock with a pendulum is needed. The other\nclocks distributed over the building have only faces and hands, and\na very few simple wheels, to which a slight push is given by an\nelectro-magnet, say, every minute or so. The system is therefore well\nadapted for offices and hotels.\n\nIn America, by means of electric contacts, clocks have been arranged to\nput gramophones into action. You will remember that it was pointed out\nthat if a wire were dragged over a file a sound would be produced due\nto the little taps made as the wire clicked against the rough cuts on\nthe file, and that the tone of the note depended on the fineness of the\ncuts, and hence the rapidity of the little taps. You can imagine that,\nif the roughnesses were properly arranged, we might get the tones to\nvary, and thus imitate speech. This is the principle of the gramophone.\nThe roughnesses are produced by a tool, which, vibrating under the\ninfluence of human speech, makes small cuts in a soft material. This\nis hardened, and then, when another wire is dragged over the cuts, the\nvoice is reproduced.\n\nIn this way clocks are made to speak and tell the children when dinner\nis ready and when to go to bed. On this simple plan, too, dolls can be\nmade to speak.\n\nThe modern methods of clock and watch-making are very different from\nthose in use in olden days. In former times the pivots were turned up\nby hand on small lathes, and even the teeth of the wheels were filed\nout. Each hole in the clock or watch frame was drilled out separately,\nand each wheel separately fitted in, so that the watch was gradually\nbuilt up as one would build a house. Each wheel, of course, only fitted\nits own watch, and the parts of watches were not interchangeable.\n\nThis has now all been altered. By means of elaborate machinery the\nwhole of the work of cutting out every wheel and the making of every\nsingle part is done by tools moved independently of the will of the\nworkman, whose only duty is to sit still and see the things made. He\nis, as it were, the slave of the machine, watching it and answering to\nits calls. Or shall we rather say that he is the machine\u2019s employer and\nmaster? He has here a servant who never tires nor ever disobeys him.\nAll the machine requires is that its cutting edges should be exactly\ntrue and sharp and microscopically perfect; then it will cut away and\nmake wheel after wheel. It oils itself. It only wants the man to act as\nsuperintendent, and stop it if any cutting edge gets unduly worn. For\nthis purpose he measures the work it is doing from time to time with a\nmicroscope to see that it is good and true and exact.\n\nWhen all the parts have thus been made you have perhaps a hundred\nboxes, each with a thousand watch parts in it, each part exactly like\nits fellows. You take one wheel or bit from each box indiscriminately,\nand you then have the materials for a watch, screws, fittings, pins,\nand all. All you have now got to do is simply to screw them all\ntogether, like putting together a puzzle. Everything fits; there is no\nsnipping or filing.\n\nIn such a watch if a bit gets broken you simply send for another bit of\nthe same kind and fit it into its place.\n\nMotor cars, bicycles, and many other machines are, or ought to be, made\nin this manner, so that if a driver at York breaks a part of the car he\nsimply sends to London for another. It comes and fits into its place at\nonce. But for this sort of plan you must do work true to much less than\na thousandth of an inch, and, of course, no one must want to indulge\nhis individual fancy as to the shape or appearance of the watch. The\nwhole advantage consists in dead uniformity. But the cheapness is\nsurprising. You can have a better watch now for 30_s._ than could have\nbeen got for \u00a330 twenty years ago.\n\nArtistic people are in the habit of condemning this uniformity as\nthough it were inartistic and degrading. In truth, it is not degrading\nto get a machine to do what you want at the expense of as little labour\nas possible. You pay 30_s._ for the watch, but you have \u00a328 10_s._\nleft to spend on pictures.\n\nOnly one ought not to confuse industry with art. Watches made in this\nway have no pretence to be artistic products. They are simply useful.\nTo rule them all over with machine lines or to put hideous machine\nornament on them is purely and simply base and degrading. Let your\n_ornament_ be hand work, your utility machine work.\n\nThus then I have endeavoured to give a very brief sketch of the modes\nof measuring time, and incidentally to introduce my readers to those\nlaws of motion which are the foundation of so large a part of modern\nscience.\n\nIt only remains that I should shortly describe modern apparatus by\nmeans of which it is possible to measure with accuracy periods of time\nso short as to appear impossible. But when you see how it is done the\nmethod seems easy enough. It is still by means of a pendulum, only a\npendulum beating time not once, but hundreds and even thousands of\ntimes in a second.\n\nAnd such pendulums, instead of being difficult to make, are remarkably\nsimple, and present no difficulty whatever. For we have only to use the\ntuning fork which has been previously described.\n\nThe tuning fork consists of a piece of steel bent into a U shape. The\narms are set vibrating so as alternately to approach and recede from\none another.\n\nThe reason why there are two arms is that, if they come together and\nrecede, they balance, and hence the instrument as a whole does not\nshake on its base. This balance of moving parts of a rapidly moving\nmachine is very important. Some motor cars are arranged so that\nthe engines are \u201cbalanced,\u201d and the moving parts come in and out\nsimultaneously, leaving the centre of gravity unchanged whatever be the\nposition of the motion. This makes the vibration of the car very small.\n\nThe tuning fork is therefore balanced. Being elastic, it obeys Hook\u2019s\nlaw, \u201cAs the force, so the deflection.\u201d And therefore, as we have seen,\nthe vibrations of the fork are isochronous.\n\nA fork with arms about six or seven inches long will make about fifty\nor sixty vibrations in a second. How are we to record those vibrations,\nand how keep the tuning fork vibrating?\n\n[Illustration: FIG. 69.]\n\nA train of wheels is almost an impossibility, not perhaps so impossible\nas might be supposed, but still very difficult. So a different method\nis adopted. A little wire projects from one tuning fork arm. A piece of\nglazed paper is gently smoked by means of a wax taper, and is stretched\nround a well-made brass drum. The tuning fork is then put so that the\nlittle wire just touches the paper. The tuning fork is then made to\nvibrate by a blow, and while it is vibrating the drum is revolved.\nThus a wavy line is formed on the drum by the wire on the tuning fork.\nIf the tuning fork made fifty complete vibrations to and fro in a\nsecond there would be one hundred such indentations, fifty to the right\nand fifty to the left, and by these the time can be measured as you\nwould measure a length upon a rule.\n\n[Illustration: FIG. 70.]\n\nIf an arm _a b_ be fitted to move sideways when a little string _c\nd_ is pulled, and be also provided with a small wire, so as to touch\nthe drum, then it also will trace a straight line on the drum as the\nwire lightly scratches away the thin coating of smoke. Now, if it is\nsuddenly jerked and flips back, then a little indentation will be\nmade in the line, and if when we are to measure a rapid lapse of time\na jerk is given at the beginning, and another jerk at the end of it,\nwe should get a diagram like that in the adjoining figure, where _a_\nis the trace of the tuning fork, _b_ that of the indicating arm. The\ntime which has elapsed between the jerk which produced the indentation\n_c_ and that which produced the indentation _d_ will be about three\nand three-quarter double indentations of the tuning fork line, thus\nindicating three and three-quarter fiftieths of a second. It is easy\nto see how delicate this means of measurement can be made. With small\ntuning forks we can easily measure times to a thousandth part of a\nsecond, and much less if desired.\n\nThe jerk may be given by electricity if it is wished. When the current\nis joined a little electro-magnet pulls a bit of iron and gives a pull\nto the string. So extremely rapid is the flight of electricity that no\nappreciable time is lost in its transit through the wires, so that the\nimpulse may be given from a distance. Thus we may arrange that when a\ncannon ball leaves a gun an electric impulse shall be given. When it\nreaches and hits a target another electric impulse is given. These make\nnicks in the tracing line on the drum from which we can easily compute\nthe time that has elapsed between the leaving of the mouth of the gun\nand the arrival of the shot at its destination.\n\n[Illustration: FIG. 71.]\n\nSuch an apparatus is used in modern gunnery experiments. It is an\nelaborate one, but is based on the principle above described.\n\nDrums are sometimes driven by clockwork, and tuning forks are also\noften kept vibrating by electricity, thus constituting very rapidly\nmoving electric clocks. The arrangement is simple. An electro-magnet\n_E_ is put in the vicinity of the arm of the tuning fork. A small piece\nof wire from the arm is in contact with a piece of metal _Q_, from\nwhich a wire runs to the electro-magnet, thence to a battery, and from\nthe battery to the tuning fork, through which the current runs to the\nwire _R_. When the fork vibrates the arm, being bent outwards, makes\nthe wire _R_ touch _Q_. This at once causes the electro-magnet to give\na small pull to the steel arm of the tuning fork, and thus assists the\nswing of the arm. The whole arrangement is exactly analogous to an\nelectric clock, as may be seen by comparing Fig.\u00a071 with Fig.\u00a068.\n\nThere is another method of measuring rapid intervals of time which also\nmerits attention. It is to let a body drop at the commencement of the\nperiod of time to be measured, and mark how far it falls in the time,\nand then find the time from the equation given previously,\n\n                             S = 1/2 g t\u00b2.\n\nIt is practically done by letting a piece of smoked glass fall and\nmaking a small pointer make two dots upon it, one at the beginning,\nanother at the end, of the time to be measured.\n\nAn interesting adaptation of this method can serve as a basis of a\ncurious toy.\n\nTake a crossbow, with a bolt with a spike on it; fix it firmly in a\nvice so that the barrel points at a spot _a_ on a wooden wall. On the\nspot _a_ hang a cardboard figure of a cat on to a nail so contrived\nthat when an electro-magnet acts the nail is pulled aside, and the\ncat drops. Thus let _a_ be the cat, _b_ the loop by which it is hung\nover the nail _c_, that is fixed to another piece of iron furnished\nwith a hinge at _c_, so that when the electric current is turned on\nthe nail _c_ is withdrawn and the cat drops. Carry the wires from the\nelectro-magnet and battery to the crossbow, and so arrange them that\nwhen the bolt leaves the muzzle one is pressed against the other, and\ncontact made.\n\nNow here you have an apparatus such that exactly as the bolt leaves the\ncrossbow, the cat drops. Now what will happen?\n\n[Illustration: FIG. 72.]\n\nWhen the bolt leaves the bow it is subject to two motions, one a motion\nof projection at a uniform pace in the direction of _b a_ from the bow\nto the target.\n\nBut it is also subject to another force, namely that of gravity, which\nacts on it vertically, and deflects it _in a vertical direction_\nexactly as much and as fast as a body would do if dropped from rest at\nthe same instant as the bolt leaves the bow. But the cat is such a\nbody. Hence, then, since by the electric arrangement they are both let\ngo together, they will both drop simultaneously, and thus will always\nbe on the same level, and when the bolt reaches the wooden wall and\nhas fallen vertically from _a_ to _c_, the cat will also have fallen\nvertically from _a_ to _c_, and the bolt will pin him to the wall. It\ndoes not matter how far you take the bow from the wall, nor how strong\nthe bow is, nor how heavy the bolt is, nor how heavy the cat is, nor\nwhether _a b_ is horizontal or pointing upwards or downwards.\n\n[Illustration: FIG. 73.]\n\nIn every case, if only the barrel is pointed directly at the cat, then\nthe bolt and cat fall simultaneously and at the same rate, and the bolt\nwill pin the cat to the wall.\n\nIn trying the experiment the bolt should be pretty heavy, say half a\npound, and have a good spike; but if carefully done the experiment will\nsucceed every time. It enables you also to measure the speed of flight\nof the bolt. For if the distance of the bow from the wall be thirty\nfeet, and the cat have fallen three feet when it is struck, then the\ntime of fall is T\u00b2 = \u221a((2S)/g) = \u221a(6/g) = \u00b743 seconds. But the bolt\nin this time went thirty feet; hence its velocity was thirty feet in\n\u00b743 seconds, or seventy feet per second.\n\nOf course if you make the bolt heavier the velocity of projection will\nbecome slower, the time longer, and hence the cat will fall further\nbefore it is transfixed by the bolt.\n\nMy task is now at a close. I have endeavoured not merely to give a\ndescription of clocks and various apparatus for measuring time, but to\nexplain the fundamental principles of mechanics which lie at the root\nof the subject.\n\nMay I end with a word of advice to parents?\n\nThere is a certain number of boys, but only a certain number, who have\na real love for mechanical science. Such boys should be encouraged\nin every way by the possession of tools and apparatus, but in the\nselection of this apparatus the following principles should be borne in\nmind:\u2014\n\n_First_, that almost everything a boy wants can be made with wood, and\nmetal, and wire, and string, _if_ he has someone to give him a little\ninstruction how to do it. A bent bit of steel jammed in a vice makes an\nexcellent tuning fork.\n\n_Second_, that he wants not toy tools, but good tools. If an expert\nwants a good tool, how much more a beginner.\n\n_Third_, that he ought to have a reasonably dry and comfortable place\nto work in, and the help and advice of the village carpenter or\nblacksmith.\n\n_Fourth_, that he ought not to be allowed to potter with his tools, but\nto make something really sensible and useful, and not begin a dozen\nthings and finish none.\n\n_Fifth_, that the making of apparatus to show scientific facts is more\nuseful than making bootjacks for his father or workboxes for his mother.\n\nAnd, _lastly_, that a little money spent in this way will keep many a\nyoung rascal from worrying his sisters and stoning the cat; and when\nthe inevitable time comes at which he must face the young man\u2019s first\ntrial, THE EXAMINER, he will often thank his stars that he learned in\nplay the fundamental formula S = 1/2 g t\u00b2, and that he knows the\nnature of \u201charmonic motion,\u201d the two most important principles in the\nmeasurement of time.\n\n\nTHE END.\n\n\n\n\nAPPENDIX ON THE SHAPE OF THE TEETH OF WHEELS.\n\n\n[Illustration: FIG. 74.]\n\nThe teeth of wheels for watches and clocks need particular care in\nshaping, and it may be of interest if I describe briefly the principles\nupon which these wheels are made. What is required is that the motion\nshall not be communicated by jerks as the teeth successively engage one\nanother, but that the motion shall be perfectly smooth. The problem\ntherefore becomes this: How are we to arrange the teeth of the wheels\nso that as one of them turns and drives the other round the leverage\nor turning power exercised by the driving wheel on the driven wheel\nshall always be uniform? Now if the teeth were simple spikes one can\neasily see that this would not be the case. For instance, as the arm _a\nc_ turned round, driving before it the arm _b d_, the point _c_ would\nscrape along, and the leverage between the two teeth would constantly\nalter. Evidently some other construction must be adopted. Before we can\ndetermine what it is to be, we must inquire what the leverage would\nbe between two rods, _a c_ and _d b_, mounted on pivots at _a_ and\n_d_. The answer to this question is, that when a lever such as _a c_\npresses with its end against another, _d b_, the power is exercised in\na direction _c e_ at right angles to _d b_. Hence the leverage between\nthe two arms is in the ratio of _a e_ to _d c_. The system is just as\nif we had a lever _a e_ united to a lever _d c_ by a rigid rod _e c_ at\nright angles to both of them.\n\n[Illustration: FIG. 75.]\n\nWhence then the ratio of the power is as _a e_ is to _d c_.\n\n[Illustration: FIG. 76.]\n\nBut since the triangles _a e f_, _d c f_, are similar, _a e_ is to _d\nc_ as _a f_ to _f d_. Whence then we get this general proposition: If\none body mounted on an axis is pressing upon another body mounted\non an axis, the pressure exerted between them is always exercised in\na direction, shown by the dotted line, at right angles to the two\nsurfaces in contact; and the ratio of the leverage is found by drawing\na line from one axis to the other, so as to cut the line of direction\nof pressure in _f_. The leverage of one on the other is then as _a f_\nto _f d_. Our problem has now become the following: Given a rod _b d_,\nsuppose that it is pressed upon by a curved surface mounted on an axis\nat _a_. Then the direction of the pressure that the curved surface\n(called in engineering language a cam) will exercise on the rod _b d_\nis shown by the dotted line; and the ratio of the driving power to the\ndriven power is as _d f_ to _a f_. Now how can we shape the cam so that\nas it moves round, and different parts of its surface come successively\ninto contact with _b c_, the ratio of the leverage is always the\nsame; that is to say, the ratio of _a f_ to _f d_ shall always be\nconstant; that is to say, the line drawn through the point of contact\nperpendicular to the curve at that point, shall always pass through the\npoint _f_?\n\n[Illustration: FIG. 77.]\n\n[Illustration: FIG. 78.]\n\nEvidently, if this is to be so, the point _d_ must be on a semicircle,\nwhose diameter is _f b_, for in that case the angle _f d b_ will always\nbe a right angle.\n\n[Illustration: FIG. 79.]\n\nThe surface must then be so arranged that, whatever be the position of\nthe cam and of the rod _b d_, the point of contact between them must\nalways be on the semicircle _f c d_; that is to say, as the cam moves\nround the axis _a_ its shape must be such that a line drawn from _f_ to\nthe point where it cuts the circle _f d b_ is always perpendicular to\nthe curve.\n\nNow suppose that we move a circle whose centre is at _a_, and radius _a\nf_, so as to roll the circle _f d b_ by simple surface friction round\nits centre _o_, then any point _d_ on it would mark out a curve on a\npiece of paper attached to the moving circle whose centre is at _a_,\nand the direction of motion of the curve would always be such that the\npoint _d_ on it would at any instant be describing a circle round _f_,\nand the direction of the curve would thus at any point always be at\nright angles to the line _d f_ for the time being.\n\n[Illustration: FIG. 80.]\n\nThis curve, caused by the rolling of one circle on another, is called\nan epicycloid. Hence, then, for a clock, if we make the pinion wheel\nwith straight spokes and the driving wheel with its teeth cut in the\nform of epicycloids, caused by rolling a circle with a diameter equal\nto the radius of the pinion upon the driving wheel, we shall get a\nuniform ratio of leverage one upon the other.\n\nThe circles with radii _a f_, _b f_, are called the \u201cpitch circles,\u201d\nand these radii are in the ratio of the movement that is required for\nthe wheels, usually six to one or eight to one, as the case may be. The\nsides of the teeth of the pinion wheels are straight lines radiating\nfrom the centre, and rounded off at the ends so as to avoid accidental\njambing. The teeth of the cogwheel have epicycloidal sides. The tips\nare cut off so as to be out of the way, and spaces are left between\nthem for the width of the leaves of the pinion wheel.\n\n[Illustration: FIG. 81.]\n\nBoth pinion wheels and cogwheels are cut by cutters rotating at a\nhigh speed, about 3,500 times in a minute, the cutters being carefully\nshaped for the pinion wheels with straight edges, for the cogwheels\nin epicycloids. It is a pretty thing to see a wheel-cutting engine at\nwork, the cutter flying round with a hum, cutting the rim of a brass\nwheel into teeth, the brass coming off in flakes thinner than fine\nhairs and falling in fine dust. When a tooth is cut, the wheel is moved\nround one division of an apparatus called a \u201cdividing plate,\u201d so as to\npresent a new part of the wheel to the cutter. Of course, the cutter\nand wheels must all be properly proportioned. Cutters are sold in sets\nduly shaped for the work they have to do. Wheel-cutting is a special\nbranch of the clockmaking industry. The reason the speed of cutting is\nso high is because it is desired to take off small portions of metal\nat a time, and thus not strain the wheel and the cutting machinery. If\nbigger cuts were made, then the machine would have to go slower, for\nit is a principle in the construction of cutting machinery that the\nspeed of the cut must always be proportioned to the depth of it. If\nyou want to take deep cuts you must move the cutting edge slowly, and\n_vice vers\u00e2_. The most modern method of making cogwheels of brass, and\nthe best, is to stamp them out of solid sheet metal at a single punch\nof a punching machine, and cheap watches are always made in this way.\nIn fact, the whole method of watch and clock-making is undergoing a\ntransformation.\n\nBefore the time of the great engineering development which took place\ntowards the end of the eighteenth century, the making of machines was\na sort of fine art. Cogwheels were cut by hand. The circumference\nwas marked out by means of compasses. Then holes were drilled round\nthe rim, and teeth cut out leading into them, and shaped by means of\nspecial files constructed for the purpose (Fig.\u00a082). Big machinery was\nall shaped out at the forge and by the file. The engineers complained\nthat you could not get big work made true even to the eighth of an\ninch. But watches and clocks were beautifully made, though only at the\ncost of hours of patient measuring and filing. The taste for ornament\nstill existed. The wheels and backs of watches were chased over with\nthe most beautiful patterns; the frames of the clocks were wrought\ninto beautiful figures and forms. Even astronomical instruments were\nembellished.\n\n[Illustration: FIG. 82.]\n\nThen came the era of severe accuracy. Men of science took the\ngovernment of machine-making whose feelings were repugnant to art in\nany form. They hated to see any effort expended in ornament. With\nseverely utilitarian aims, they banished all appearance of beauty from\ninstruments and tools of all sorts, so that our modern machines, from a\nsteam engine down to a watch, are now models of precise but perfectly\nunornamented workmanship. They are the fitting implements of a nation\nthat wears trousers and tall hats. One has only to compare an old\nvessel of war, with its sculptured prow and streamers, with a modern\nironclad to take note of the difference. The art of ornamentation\nis now little more than a spasmodic imitation of the past, with a\nhistorical interest only. As a living entity it has almost ceased to\nexist.\n\nBut in precision of manufacture the present age is without a rival in\nthe history of the world. People believe no longer in the old methods\nof scraping and filing, and hand-work directly exercised on metal is\nrapidly falling into desuetude. It is possible, of course, with a file\nand scraper and days of labour to get two flat surfaces of metal so\nperfect that when put together one will lift the other like a sucker\non a stone, but it is waste labour. A small machine will do it as well\nin a few minutes. No longer is a watch built up as one would build a\nhouse, fitting part to part. By expensive machines thousands of watch\nparts are stamped and cut out to pattern, and then a watch is made by\ntaking one of each indiscriminately and just putting them into their\nplaces. Comparatively unskilled workmen can do this. Where the skill\nis wanted is to design and make the machinery and watch the cutters,\nmeasuring them with microscopic gauges from time to time, and at once\nremedying them if an edge is found to be a ten-thousandth part of an\ninch out of place. So that the labour of man is becoming more and more\na labour of design and of supervision. Machines are the slaves that do\nthe work, for in a good machine we have an eye and an arm that never\ntires, and only needs to be kept in working order. But machines are not\nartistic, and thus art is lost while precision is gained. At present\na desperate attempt is being made to supply by means of machinery\nthe craving of the human mind for art. But it is destined to failure.\nArt of this kind is generally produced by the same brain that designs\nmachines, and therefore presents an appearance of rigid accuracy and\nuniformity, which, while essential to an engine, is out of place in an\nartistic product.\n\nThe great manufacturers of our Midlands do not seem to understand that\nthere is no object in making a towel-horse as geometrically accurate as\na turning lathe. It will apparently be years before they learn to put\nart and precision each in the place where it is wanted\u2014precision in the\nworks of the watch, art in the face and the case of it; machine work in\nthe inside of a watch, hand work on the outside. When the public taste\nis educated so as to see the odious character of the jumble of Gothic,\nEgyptian, and meaningless ornament on such an article as the case of\nan American organ, one step will have been made towards the revival of\nartistic taste.\n\nBut to propose as a means of reviving art that we should discontinue\nthe use of machinery or abandon our modern cutters of precision to go\nback to a hack-saw and file is ridiculous, and could only be suggested\nby men quite destitute of scientific ideas. Art and precision each has\nits place: there is room for both; let neither intrude on the domain of\nthe other.\n\n\n\n\nINDEX.\n\n\n  Acceleration, 73, 77\n\n  Almagest, 53\n\n  Anchor escapement, 120\n\n  Ancient science, 50\n\n  Aristotle\u2019s ideas, 23, 52\n\n  Attwood\u2019s machine, 83\n\n\n  Babylon, temple of, 24\n\n  Balance wheel, 159\n\n\n  Candles to measure time, 46\n\n  Chaldean day, 15\n\n  Chaucer, 56\n\n  Chronographs, 179\n\n  Chronometer, 165\n\n  Chronometer escapement, 166\n\n  Clock movement, 123\n\n  Copernicus, 56\n\n  Crossbow experiment, 183\n\n  Crown wheel, 115\n\n  Cycloid, the, 109\n\n\n  Dante\u2019s Inferno, 54\n\n  Day, length of, 29\n\n  Dead beat escapement, 135\n\n  Density, 12\n\n  Driving weight, 127, 141\n\n\n  Earth, a sphere, 21\n\n  Earth\u2019s motion, 57, 69\n\n  Earth not at rest, 67\n\n  Egg-boiler, 43\n\n  Electric clocks, 179\n\n  Epicycloidal wheels, 191\n\n  Escapements, anchor, 120\n    crown, 115\n    chronometer, 166\n    dead beat, 135\n    gravity, 169\n\n\n  Falling bodies, laws of, 62\n\n  Force, 76\n\n  Forces, revolution of, 89\n\n  Fusee, the, 117\n\n\n  Galileo\u2019s \u201cDialogues,\u201d 58\n    clock, 111\n\n  Grandfather\u2019s clock, 119\n\n  Gravity, action of, 13, 65\n\n  Gravity escapement, 169\n\n  Greek day, 16\n\n\n  Harmonic motion, 97\n\n  Hooke\u2019s law, 71\n\n\n  Isochronism of springs, 93\n\n\n  Lamps to measure time, 46\n\n  Latitude and longitude, 161\n    finding, 163\n\n\n  Mass, nature of, 10\n\n  Mercury clock, 45\n\n  Modern methods, 177, 197\n\n  Moments, 101\n\n  Moon\u2019s appearance, 17\n\n  Motion, reliability of, 57\n\n  Musical notes, 95\n\n\n  North pole, days at, 33\n\n\n  Oscillations, law of, 151\n\n\n  Parabola, the, 87\n\n  Pendulum, the, 103, 145\n    suspension, 145\n    mercury, 147\n    gridiron, 149\n    theory of, 155\n    free, 133\n\n  Pisa, leaning tower of, 61\n\n  Planets, names of, 11\n\n  Pulse measurer, 99\n\n\n  Ratchet wheels, 129\n\n  Roman clocks, 40\n\n\n  Sand-glasses, 41\n\n  Space, nature of, 8\n\n  Speed of falling bodies, 79\n\n  Spring balance, 107\n\n  Stevinus\u2019 theory, 81\n\n  Style of sun-dials, 35\n\n  Sun-dials, 27\n    to make, 48\n\n  Synchronous clocks, 175\n\n\n  Time, 13\n\n  Toothed wheels, 125, 137\n\n  Tuning fork, 94, 181\n\n\n  Velocities, composition of, 85\n\n\n  Water pressure, 37\n\n  Water clocks, 39\n\n  Watches, 156\n\n  Week days, names of, 24\n\n  Wheels, shape of teeth, 190\n\n  Wheel-cutting machines, 193\n\n  Winding drum, 131\n\n  Winter sun, 31\n\n\n  Zodiac, 18\n\n\nBRADBURY, AGNEW, & CO. LD., PRINTERS, LONDON AND TONBRIDGE.\n\n\n\n\n      *      *      *      *      *      *\n\n\n\n\nTranscriber\u2019s note:\n\nPunctuation, hyphenation, and spelling were made consistent when a\npredominant preference was found in this book; otherwise they were not\nchanged.\n\nSimple typographical errors were corrected; occasional unpaired\nquotation marks were retained.\n\nAmbiguous hyphens at the ends of lines were retained.\n\nSquare roots are represented as \u221a(values).\n\nIndex not checked for proper alphabetization or correct page references.\n\nPage 16: \u201csix o\u2019clock\u201d was printed as \u201csix clock\u201d; changed here.","title":"Time and Clocks: A Description of Ancient and Modern Methods of Measuring Time","tradition":"philosophy-ethics","translated_text":null,"word_count":43942}