diff options
Diffstat (limited to 'old/54546-0.txt')
| -rw-r--r-- | old/54546-0.txt | 4860 |
1 files changed, 0 insertions, 4860 deletions
diff --git a/old/54546-0.txt b/old/54546-0.txt deleted file mode 100644 index 4d1b7d8..0000000 --- a/old/54546-0.txt +++ /dev/null @@ -1,4860 +0,0 @@ -The Project Gutenberg eBook, Time and Clocks, by Sir Henry H. (Henry -Hardinge) Cunynghame - - -This eBook is for the use of anyone anywhere in the United States and most -other parts of the world at no cost and with almost no restrictions -whatsoever. You may copy it, give it away or re-use it under the terms of -the Project Gutenberg License included with this eBook or online at -www.gutenberg.org. If you are not located in the United States, you'll have -to check the laws of the country where you are located before using this ebook. - - - - -Title: Time and Clocks - A Description of Ancient and Modern Methods of Measuring Time - - -Author: Sir Henry H. (Henry Hardinge) Cunynghame - - - -Release Date: April 13, 2017 [eBook #54546] - -Language: English - -Character set encoding: UTF-8 - - -***START OF THE PROJECT GUTENBERG EBOOK TIME AND CLOCKS*** - - -E-text prepared by deaurider, Charlie Howard, and the Online Distributed -Proofreading Team (http://www.pgdp.net) from page images generously made -available by Internet Archive (https://archive.org) - - - -Note: Project Gutenberg also has an HTML version of this - file which includes the original illustrations. - See 54546-h.htm or 54546-h.zip: - (http://www.gutenberg.org/files/54546/54546-h/54546-h.htm) - or - (http://www.gutenberg.org/files/54546/54546-h.zip) - - - Images of the original pages are available through - Internet Archive. See - https://archive.org/details/timeclocksdescri00cuny - - -Transcriber’s note: - - Text enclosed by underscores is in italics (_italics_). - However, to improve readability, italics notation has been - omitted from equations. If the variables are in italics in - the surrounding text, they were in italics in the equations. - - - - - -TIME AND CLOCKS. - - -[Illustration: - - [_Frontispiece._ - -NUREMBERG CLOCK. CONVERTED FROM A VERGE ESCAPEMENT TO A PENDULUM -MOVEMENT.] - - -TIME AND CLOCKS: - -A Description of Ancient and Modern Methods of -Measuring Time. - -by - -H. H. CUNYNGHAME M.A. C.B. M.I.E.E. - -With Many Illustrations. - - - - - - -London: -Archibald Constable & Co. Ltd. -16 James Street Haymarket. -1906. - -Bradbury, Agnew, & Co. Ld., Printers, -London and Tonbridge. - - - - -CONTENTS. - - - PAGE - INTRODUCTION 1 - - CHAPTER I. 7 - - CHAPTER II. 50 - - CHAPTER III. 90 - - CHAPTER IV. 123 - - APPENDIX ON THE SHAPE OF THE TEETH OF WHEELS 187 - - INDEX 199 - - - - -TIME AND CLOCKS. - - - - -INTRODUCTION. - - -When we read the works of Homer, or Virgil, or Plato, or turn to the -later productions of Dante, of Shakespeare, of Milton, and the host -of writers and poets who have done so much to instruct and amuse us, -and to make our lives good and agreeable, we are apt to look with some -disappointment upon present times. And when we turn to the field of art -and compare Greek statues and Gothic or Renaissance architecture with -our modern efforts, we must feel bound to admit our inferiority to our -ancestors. And this leads us perhaps to question whether our age is the -equal of those which have gone before, or whether the human intellect -is not on the decline. - -This feeling, however, proceeds from a failure to remember that each -age of the world has its peculiar points of strength, as well as of -weakness. During one period that self-denying patriotism and zeal -for the common good will be developing, which is necessary for the -formation of society. During another, the study of the principles of -morality and religion will be in the ascendant. During another the arts -will take the lead; during another, poetry, tragedy, and lyric poetry -and prose will be cultivated; during another, music will take its turn, -and out of rude peasant songs will evolve the harmony of the opera. - -To our age is reserved the glory of being easily the foremost in -scientific discovery. Future ages may despise our literature, -surpass us in poetry, complain that in philosophy we have done -nothing, and even deride and forget our music; but they will only -be able to look back with admiration on the band of scientific -thinkers who in the seventeenth century reduced to a system the -laws that govern the motions of worlds no less than those of atoms, -and who in the eighteenth and nineteenth founded the sciences of -chemistry, electricity, sound, heat, light, and who gave to mankind -the steam-engine, the telegraph, railways, the methods of making -huge structures of iron, the dynamo, the telephone, and the thousand -applications of science to the service of man. - -And future students of history who shall be familiar with the -conditions of our life will, I think, be also struck with surprise at -our estimate of our own peculiar capabilities and faculties. They will -note with astonishment that a gentleman of the nineteenth century, an -age mighty in science, and by no means pre-eminent in art, literature -and philosophy, should have considered it disgraceful to be ignorant -of the accent with which a Greek or a Roman thought fit to pronounce -a word, should have been ashamed to be unable to construe a Latin -aphorism, and yet should have considered it no shame at all not to -know how a telephone was made and why it worked. They will smile when -they observe that our highest university degrees, our most lucrative -rewards, were given for the study of dead languages or archæological -investigations, and that science, our glory and that for which we have -shown real ability, should only have occupied a secondary place in our -education. - -They will smile when they learn that we considered that a knowledge -of public affairs could only be acquired by a grounding in Greek -particles, or that it could ever have been thought that men could not -command an army without a study of the tactics employed at the battle -of Marathon. - -But the battle between classical and scientific education is not in -reality so much a dispute regarding subjects to be taught, as between -methods of teaching. It is possible to teach classics so that they -become a mental training of the highest value. It is possible to teach -science so that it becomes a mere enslaving routine. - -The one great requirement for the education of the future is firmly to -grasp the fact that a study of words is not a study of things, and that -a man cannot become a carpenter merely by learning the names of his -tools. - -It was the mistake of the teachers of the Middle Ages to believe that -the first step in knowledge was to get a correct set of concepts of -all things, and then to deduce or bring out all knowledge from them. -Admirable plan if you can get your concepts! But unfortunately concepts -do not exist ready made—they must be grown; and as your knowledge -increases, so do your concepts change. A concept of a thing is not a -mere definition, it is a complete history of it. And you must build up -your edifice of scientific knowledge from the earth, brick by brick and -stone by stone. There is no magic process by which it can with a word -be conjured into existence like a palace in the Arabian Nights. - -For nothing is more fatal than a juggle with words such as force, -weight, attraction, mass, time, space, capacity, or gravity. Words are -like purses, they contain only as much money as you put into them. You -may jingle your bag of pennies till they sound like sovereigns, but -when you come to pay your bills the difference is soon discovered. - -This fatal practice of learning words without trying to obtain a -clear comprehension of their meaning, causes many teachers to use -mathematical formulæ not as mere steps in a logical chain, but like -magical chaldrons into which they put the premises as the witches put -herbs and babies’ thumbs into their pots, and expect the answers to -rise like apparitions by some occult process that they cannot explain. -This tendency is encouraged by foolish parents who like to see their -infant prodigies appear to understand things too hard for themselves, -and look on at their children’s lessons in mathematics like rustics -gaping at a fair. They forget that for the practical purposes of -life one thing well understood is worth a whole book-full of muddled -ill-digested formulæ. Unfortunately it is possible to cram boys up -and run them through the examination sieves with the appearance of -knowledge without its reality. If it were cricket or golf that were -being tested how soon would the fraud be discovered. No humbug would be -permitted in those interesting and absorbing subjects. And really, when -one reflects how easy it is to present the appearance of book knowledge -without the reality, one can hardly blame those who select men for -service in India and Egypt a good deal for their proficiency in sports -and games. Better a good cricketer than a silly pedant stuffed full of -learning that “lies like marl upon a barren soil encumbering what is -not in its power to fertilize.” - -Another kindred error is to expect too much of science. For with all -our efforts to obtain a further knowledge of the mysteries of nature, -we are only like travellers in a forest. The deeper we penetrate it, -the darker becomes the shade. For science is “but an exchange of -ignorance for that which is another kind of ignorance”[A] and all -our analysis of incomprehensible things leads us only to things more -incomprehensible still. - - [A] _Manfred_, Act II., scene iv. - -It is, therefore, by the firm resolution never to juggle with words or -ideas, or to try and persuade ourselves or others that we understand -what we do not understand, that any scientific advance can be made. - - - - -CHAPTER I. - - -All students of any subject are at first apt to be perplexed with the -number and complication of the new ideas presented to them. - -The need of comprehending these ideas is felt, and yet they are -difficult to grasp and to define. Thus, for instance, we are all apt -to think we know what is meant when force, weight, length, capacity, -motion, rest, size, are spoken of. And yet when we come to examine -these ideas more closely, we find that we know very little about -them. Indeed, the more elementary they are, the less we are able to -understand them. - -The most primordial of our ideas seem to be those of number and -quantity; we can count things, and we can measure them, or compare -them with one another. Arithmetic is the science which deals with the -numbers of things and enables us to multiply and divide them. The -estimation of quantities is made by the application of our faculty of -comparison to different subjects. The ideas of number and quantity -appear to pervade all our conceptions. - -The study of natural phenomena of the world around us is called the -study of physics from the Greek word φυσίς or “inanimate nature,” -the term physics is usually confined to such part of nature as is not -alive. The study of living things is usually termed biology (from βια, -life). - -In the study of natural phenomena there are, however, three ideas which -occupy a peculiar and important position, because they may be used -as the means of measuring or estimating all the rest. In this sense -they seem to be the most primitive and fundamental that we possess. -We are not entitled to say that all other ideas are formed from and -compounded of these ideas, but we are entitled to say that our correct -understanding of physics, that is of the study of nature, depends -in no slight degree upon our clear understanding of them. The three -fundamental ideas are those of space, time and mass. - -Space appears to be the universal accompaniment of all our impressions -of the world around us. Try as we may, we cannot think of material -bodies except in space, and occupying space. Though we can imagine -space as empty we cannot conceive it as destroyed. And this space has -three dimensions, length, breadth measured across or at right angles to -length, and thickness measured at right angles to length and breadth. -More dimensions than this we cannot have. For some inscrutable reason -it has been arranged that space shall present these three dimensions -and no more. A fourth dimension is to us unimaginable—I will not say -inconceivable—we can conceive that a world might be with space in four -dimensions, but we cannot imagine it to ourselves or think what things -would be like in it. - -With difficulty we can perhaps imagine a world with space of only two -dimensions, a “flat land,” where flat beings of different shapes, -like figures cut out of paper, slide or float about on a flat table. -They could not hop over one another, for they would only have length -and breadth; to hop up you would want to be able to move in a third -dimension, but having two dimensions only you could only slide forward -and sideways in a plane. To such beings a ring would be a box. You -would have to break the ring to get anything out of it, for if you -tried to slide out you would be met by a wall in every direction. You -could not jump out of it like a sheep would jump out of a pen over the -hurdles, for to jump would require a third dimension, which you have -not got. Beings in a world with one dimension only would be in a worse -plight still. Like beads on a string they could slide about in one -direction as far as the others would let them. They could not pass one -another. To such a being two other beings would be a box one on each -side of him, for if thus imprisoned, he could not get away. Like a -waggon on a railway, he could not walk round another waggon. That would -want power of moving in two dimensions, still less could he jump over -them, that would want three. - -We have not the smallest idea why our world has been thus limited. Some -philosophers think that the limitation is in us, not in the world, and -that perhaps when our minds are free from the limitations imposed by -their sojourn in our bodies, and death has set us free, we may see not -only what is the length and breadth and height, but a great deal more -also of which we can now form no conception. But these speculations -lead us out of science into the shadowy land of metaphysics, of which -we long to know something, but are condemned to know so little. Area -is got by multiplying length by breadth. Cubic content is got by -multiplying length by breadth and by height. Of all the conceptions -respecting space, that of a line is the simplest. It has direction, and -length. - -The idea of mass is more difficult to grasp than that of space. It -means quantity of matter. But what is matter? That we do not know. It -is not weight, though it is true that all matter has weight. Yet matter -would still have mass even if its property of weight were taken away. - -For consider such a thing as a pound packet of tea. It has size, it -occupies space, it has length, breadth, and thickness. It has also -weight. But what gives it weight? The attraction of the earth. Suppose -you double the size of the earth. The earth being bigger would attract -the package of tea more strongly. The weight of the tea, that is, the -attraction of the earth on the package of tea, would be increased—the -tea would weigh more than before. Take the package of tea to the planet -Jupiter, which, being very large, has an attraction at the surface 2½ -times that of the earth. Its size would be the same, but it would feel -to carry like a package of sand. Yet there would be the same “mass” -of tea. You could make no more cups of tea out of it in Jupiter than -on earth. Take it to the moon, and it would weigh a little over two -ounces, but still it would be a pound of tea. We are in the habit of -estimating mass by its weight, and we do so rightly, for at any place -on the earth, as London, the weights of masses are always proportioned -to the masses, and if you want to find out what mass of tea you have -got, you weigh it, and you know for certain. Hence in our minds we -confuse mass with weight. And even in our Acts of Parliament we have -done the same thing, so that it is difficult in the statutes respecting -standard weights to know what was meant by those who drew them up, and -whether a pound of tea means the _mass_ of a certain amount of tea or -the _weight_ of that mass. For accurate thinking we must, of course, -always deal with masses, not with weights. For so far as we can tell -_mass_ appears indestructible. A mass is a mass wherever it is, and -for all time, whereas its weight varies with the attractive force of -the planet upon which it happens to be, and with its distance from that -planet’s centre. A flea on this earth can skip perhaps eight inches -high; put that flea on the moon, and with the expenditure of the same -energy he could skip four feet high. Put him on the planet Jupiter and -he could only skip 3⅕ inches high. A man in a street in the moon could -jump up into a window on the first floor of a house. One pound of tea -taken to the sun would be as heavy as twenty-eight pounds of it at the -earth’s surface; and weight varies at different parts of the earth. -Hence the true measure of quantity of matter is mass, not weight. - -The mass of bodies varies according to their size; if you have the same -nature of material, then for a double size you have a double mass. Some -bodies are more concentrated than others, that is to say, more dense; -it is as though they were more tightly squeezed together. Thus a ball -of lead of an inch in diameter contains forty-eight times as much mass -as a ball of cork an inch in diameter. In order to know the weight of a -certain mass of matter, we should have to multiply the mass by a figure -representing the attractive force or pull of the earth. - -In physics it is usual to employ the letters of the alphabet as a sort -of shorthand to represent words. So that the letter _m_ stands for -the mass of a body. So again _g_ stands for the attractive pull of the -earth at a given place. _w_ stands for the weight of the body. Hence -then, since the weight of a body depends on its mass and also on the -attractive pull of the earth, we express this in short language by -saying, w = m × g; or _w_ is equal to _m_ multiplied by _g_; the symbol -= being used for equality, and × the sign of multiplication. In common -use × is usually omitted, and when letters are put together they are -intended to be understood as multiplied. So that this is written - - w = mg. - -Of course by this equation we do not mean that weight is mass -multiplied into the force of gravity, we only mean that the number of -units of weight is to be found by multiplying the number of units of -mass into the number of units of the earth’s force of gravity. - -In the same way, if when estimating the number of waggons, _w_, that -would be wanted for an army of men, _n_, which consumed a number of -pounds, _p_, of provisions a day, we might put - - w = np. - -But this would not mean that we were multiplying soldiers into food -to produce waggons, but only that we were performing a numerical -calculation. - -Time is one of the most mysterious of our elementary ideas. It seems to -exist or not to exist, according as we are thinking or not thinking. -It seems to run or stand still and to go fast or slowly. How it drags -through a wearisome lesson; how it flies during a game of cricket; how -it seems to stop in sleep. If we measured time by our own thoughts it -would be a very uncertain quantity. But other considerations seem to -show us that Nature knows no such uncertainty as regards time, that she -produces her phenomena in a uniform manner in uniform times, and that -time has an existence independent of our thoughts and wills. - -The idea of a state of things in which time existed no more was quite -familiar to mediæval thinkers, and was regarded by many of them as the -condition that would exist after the Day of Judgment. In recent times -Kant propounded the theory that time was only a necessary condition of -our thoughts, and had no existence apart from thinking beings—in fact, -that it was our way of looking at things. - -Scientifically, however, we are warranted in treating time as perfectly -real and capable of the most exact measurement. For example, if we -arrange a stream of sand to run out of an orifice, and observe how -much will run out while an egg is being boiled hard, we find as a fact -that if the same quantity of sand runs out, the state of the egg is -uniform. If we walk for an hour by a watch, we find that we can go -half the distance that we should if we walked two hours. It is the -correspondence of these various experiments that gives us faith in the -treatment of time as a thing existing independently of ourselves, or, -at all events, independent of our transient moods. - -The ideas of time acquired by the races of men that first evolved from -a state of barbarism were no doubt derived from the observation of day -and night, the month and the year. - -[Illustration: FIG. 1.] - -For, suppose that a shepherd were on the plains of Chaldea, or perhaps -on those mountains of India known as the roof of the world, which -according to some archæologists was the site of the garden of Eden and -the early home of the European race, what would he see? - -He would see the sun rise in the east, slowly mount the heavens till it -stood over the south at middle day, then it would sink towards the west -and disappear. In summer the rising point of the sun would be more to -the northward than in winter, and so also would be its point of setting -_A´_. In winter it would rise a little to the south of east, and set -a little to the south of west, and not rise so high in the heavens at -midday, so that the summer day would be longer than the winter day. -If the day were always divided into twelve hours, whether it were long -or short, then in summer the hours of the day would be long; in winter -they would be short. This mode of dividing the day was that used by -the Greeks. The Egyptians, on the other hand, averaged their day by -dividing the whole round of the sun into twenty-four hours, so that the -summer day contained more hours than the winter day. Hence, for the -Egyptians, sun-rise did not always take place at six o’clock. For in -winter it took place after six, and in summer before six; and this is -the system that has descended to us. - -The moon also would rise at different places, varying between _A_ and -_B_, and set at places varying between _A´_ and _B´_, but these would -be independent of those at which the sun rose and set. - -Moreover, the moon each day would appear to get further and further -away from the sun in the direction of the arrow, as shown in the -sketch. If the moon rose an hour after the sun on one day, the next day -it would rise more than two hours after the sun, and so on. This delay -in rising of the moon would go on day by day till at last she came -right round to the sun again, as shown at _M´_. And in her path she -would change her form from a crescent, as at _M_, up to a full moon, -when she would be half way round from the sun, that is, when she would -rise twelve hours after him, or just be rising as the sun set. This -delay and accompanying change of form would go on, till after three -weeks she would have got round to a position _A´_, when she would rise -eighteen hours after the sun, and have become a crescent with her back -to the sun; in fact, she would always turn her convex side to the sun. -At length, when twenty-eight days had passed, she would be round again -about opposite to the sun, and consequently her pale light would be -extinguished in his beams, and she would gradually reappear as a new -moon on the other side of him. This series of changes of the moon takes -place once every twenty-eight days, and is called a lunar or “moon” -month, and was used as a division of time by very early nations. The -changes of the seasons recurred with the changes in the times of rising -of the sun, and took a year to bring about. And there were nearly -thirteen moon changes in the year. - -It was also observed that during its cycle of changes, the sun was -slowly moving round backwards among the stars in the same direction as -the moon, only it made its retrograde cycle in a year, and thus arose -the division of time into months and years. The stars turned round in -the heavens once in the complete day. The sun, therefore, appeared to -move back among them, passing successively through groups of stars, so -as to make the circuit of them all in a year. The stars through which -he passed in a year, and through which the moon travelled in a month, -were divided by the ancients into groups called constellations, and its -yearly path in the heavens was called the zodiac. There were twelve of -these constellations in the zodiac called the signs. Hence, then, the -sun passed through a sign in every month, making the tour of them all -in the year. To these signs fanciful names were given, such as “the -Ram,” “the Water-bearer,” “the Virgin,” “the Scorpion,” and so on, and -the sun and moon were then said to pass through the signs of the zodiac. - -Hence, since the path of the sun marked the year, you could tell the -seasons by knowing what sign of the zodiac the sun was in. The age of -the moon was easily known by her form. - -When the winter was over, then, just as the sun set the dog star would -be rising in the east, and this would show that the spring was at hand. -Then the peasants prepared to till the earth and sow the seed and lead -the oxen out to pasture, and celebrated with joyful mirth the glad -advent of the spring, corresponding to our Easter, when the sun had -run through three constellations of the zodiac. Then came the summer -heat, and with many a mystic rite they celebrated Midsummer’s Day. In -autumn, after three more signs of the zodiac have been traversed by the -sun, the sun again rises exactly in the east and sets in the west, and -the days and nights are equal. This is the autumnal equinox, and was -once celebrated by the feast which we now know as Michaelmas Day, and -the goose is the remnant of the ancient festival. - -[Illustration: FIG. 2.] - -And the great winter feast of the ancients is now known to us as -Christmas, and chosen to celebrate the birth of our Lord; for when -Christianity came into the world and the heathens were converted, the -old feast days were deliberately changed into Christian festivals. - -To us, therefore, the whole heavens, and the fixed stars with them, -appear to turn from east to west, or from left to right, as we look -towards the south, as shown by the big arrow. But the moon and sun, -though apparently placed in the heavens, move backwards among the fixed -stars, as shown by the small arrows. The sun moves at such a rate that -he goes round the circle of the heavens in a year of three hundred and -sixty-five days. The moon goes round the circle in twenty-eight and a -half days, or a lunar month. Of course, in reality the sun is at rest, -and it is the earth that moves round the sun and spins on its axis -as it moves. But it will presently be shown that the appearance to a -person on the earth is the same whether the earth goes round the sun or -the sun round the earth. - -From the works of Greek writers we know a good deal about the ideas of -the world that were entertained by the ancients. The most early notions -were, of course, connected with the worship of the gods. The sun was -considered as a huge light carried in a chariot, driven by Apollo, with -four spirited steeds. It descended to the ocean when the day declined, -and then the horses were unyoked by the nymphs of the ocean and led -round to the east, so as to be ready for the journey of the following -day. The Egyptians figured the sun as placed in a boat which sailed -over the heavens. At night the sun god descended into the infernal -regions, carrying with him the souls of those who had died during the -day. There they passed through different regions of hell, with portals -guarded by hideous monsters. Those who had well learned the ritual of -the dead knew the words of power wherewith to appease the demons. Those -unprovided with the watchwords were subjected to terrible dangers. -Then the soul appeared before Minos, and was weighed and dealt with -according to its deserts. - -[Illustration: FIG. 3.] - -The earth was considered as a huge island in the midst of a circular -sea. Gradually, however, astronomical ideas became subjected to -science. One of the first truths that dawned on astronomers was the -fact that the earth was a sphere. For they noticed that as people -went further and further to the north, the elevation of the sun at -midday above the horizon became smaller and smaller. This can easily -be seen from the diagram. When an observer is at _A_ the sun appears -at an altitude above the horizon equal to the angle α, but as he goes -along the curved surface of the earth to a point _B_ nearer to the -north pole, the sun appears to be lower and only to have an altitude -β. From this it was easy for men so skilled in geometry as the Greeks -to calculate how big the earth was. They did so, and it appeared to -have the enormous diameter of 8,000 miles. They only knew quite a -small portion of it. They thought that the rest was ocean. But they -had, of course, a clear idea of the “antipodes” or up-side-down side -of it, and they believed that if men were on the other side of it -that their feet must all point towards its centre. From this they -got the idea of the centre of the earth as a point of attraction for -all things that had an earth-seeking or earthy nature. Fire appeared -always to desire to go upwards, so they thought that fire had an -earth-repellent, heaven-seeking character. Water they thought partly -earth-seeking, partly heaven-seeking, for it appeared in the ocean or -floated as clouds. Air they thought to be indifferent. And out of the -four elements fire, water, earth, and air they believed the world was -made. The earth they thought must be at rest; for if it was in motion -things would fly off from it. They saw that either the sun must be -moving round the earth, or else the earth must be turning on its axis. -They chose the former hypothesis, because they argued that if the earth -were twisting round once in twenty-four hours then such a country as -Greece must be flying round like a spot on the surface of a top, at the -rate of about 18,000 miles in twenty-four hours, that is, at the rate -of about 180 yards in a second, or faster than an arrow from a bow. -But if that was the case then a bird that flew up from the earth would -be left far behind. If a ball were thrown up it would fall hundreds -of yards behind the person who threw it. They could not conceive how -it was possible for a ball thrown up by someone standing on a moving -object not to fall behind the thrower. - -This decided them in their error. The mistaken astronomy of the Greeks -was also much forwarded by Aristotle, the tutor of Alexander the Great. -This great genius in politics and philosophy was only in the second -rank as a man of science, and, as I think, hardly equal to Archimedes -or Hipparchus, or even to Ptolemy. Aristotle wrote a book concerning -the heavens which bristles with the most wantonly erroneous scientific -ideas, such as, for instance, that the motion of the heavenly bodies -must be circular because the most perfect curve is a circle, and -similar assumptions as to the course of nature. - -The earth, then, being fixed, they thought that the moon, the sun, -and the seven planets were carried round it, fixed each of them in -an enormous crystal spherical shell. These spheres, like coats of -an onion, slid round one upon another, each carrying his celestial -luminary. The moon was the nearest, then Mercury, then Venus, then the -sun, then Mars, Jupiter and Saturn. Outside them was the sphere of the -stars, and outside all the “_primum mobile_,” or great Prime Mover of -the universe. When one of the celestial bodies, such as the moon, got -in front of another, such as the sun, there was an eclipse. They soon -observed that the moon derived its light from the sun. As they knew -the size of the earth, by comparison they got some vague idea of the -huge distances that the heavenly bodies must be from us. In fact, they -measured the distance of the moon with approximate accuracy, making it -240,000 miles, or about thirty times the earth’s diameter. - -This, of course, gave them the moon’s diameter, for they were easily -able to calculate how big an object must be, that looks as big as the -moon and is 240,000 miles away. - -This large size of the moon gave them some idea of the distance of the -sun, but they failed to realise how big and far away he really is. - -Several ancient nations used weeks as means of measuring time. They -made four weeks to the lunar month. The order of the days was rather -curiously arranged. For, assuming that the earth is the centre of the -planetary system, put the planets in a column, putting the nearest (the -moon) at the bottom and the furthest off at the top— - - Saturn, - Jupiter, - Mars, - The Sun, - Venus, - Mercury, - The Moon. - -Then divide the day into three watches of eight hours each, and let -each watch be presided over by one of the planet-gods: begin with -Saturn. We then have Saturn as the first god ruling Saturday, and -Jupiter and Mars, the two other gods, for that day. The first watch for -Sunday will be the sun; Venus and Mercury will preside over the next -two watches of that day. The planet that will preside over the first -watch of the next day will be the moon, and the day will, therefore, be -called Monday; Saturn and Jupiter will be the other gods for Monday. -The first watch of the next day will be presided over by Mars, and the -day will, therefore, be called Mars-day or Mardi, or, in the Teutonic -languages, Tuesday, after Tuesco, a Scandinavian god of war. Mercury -will give a name to Mercredi, or to Wednesday, or Wodin’s-day. Jupiter -to Jeudi, or “Thurs” day. Venus to Vendredi, or in the Scandinavian, -Friday, the day of the Scandinavian goddess Freya, the goddess of love -and beauty, who corresponds to Venus, and thus the week is completed. - -[Illustration: FIG. 4.] - -This weekly scheme came probably from the Chaldean astronomers. It -appears probable that the great tower of Babel, the ruins of which -exist to this day, consisted of seven stages, one over the other, the -top one painted white, or perhaps purple, to represent the Moon, the -next lower blue for Mercury, then green for Venus, yellow for the Sun, -red for Mars, orange for Jupiter, and black for Saturn. Unfortunately, -of the colours no trace now remains. - -But nightly on the long terraces the Babylonian priests observed -eclipses and other celestial phenomena. Their records were afterwards -taken to Alexandria and kept in the great library that was subsequently -burned by the Turks. In that library they were seen by the astronomer -Ptolemy, who used them in the writing of his work on astronomy called -the “Great Syntaxis” or “Collection.” The original work perished, but -it had been translated into Arabic by the Arab astronomers, who called -it “Al Magest,” the Great Book. It was translated from Arabic into -Latin, and remained the textbook for astronomers in Europe quite down -to the time of Queen Elizabeth, when a better system took its place. - -For the use of men engaged in practical astronomy, it is very -convenient to consider the sun, moon, stars, and planets as going -round the earth at rest. For instance, seamen use the heavenly bodies -as in a way hands of a huge clock from which they can know the time and -their position on the earth. “The Nautical Almanac,” which is printed -yearly, gives the true position of these heavenly bodies for every -hour, minute, and second of the year, and I will presently show how -useful this is to sailors. - -We will deal with the sun first. From the motions of the sun we can -observe the time. This is done in every garden by means of sun-dials, -and I will now describe how they are constructed. If a light, such as -the light of a candle, be moved round in a circle at a uniform pace so -as to go round once in some given period, such as twenty-four hours, it -is obvious that it would serve to measure time. Thus, for example, if a -sheet of white paper be placed upon the table, and a pencil be stuck on -to it upright with some sealing wax, or a pen propped up in an ink-pot, -then a candle held by anyone will cast the shadow of the pen on the -paper. - -[Illustration: FIG. 5.] - -If the person holding the candle walk round the table at a uniform -speed, the shadow will go round like the hand of a clock, and might be -made to mark the time. If the candle took twenty-four hours to go round -the table, as the sun takes twenty-four hours to go round the earth, -then marks placed on the paper would serve to measure the hours, and -the paper and pen would serve as a sort of sun-dial. - -But the sun does not go round the earth as the candle round the -table. Its path is an inclined one, like that shown by the dotted -line. Sometimes it is above the level of the table, sometimes below -it. And, moreover, its winter path is different from its summer path. -Whence then it follows that the hour-marks on the paper cannot be -put equidistant like the hours on the dial of a clock, and that some -arrangement must be made so that the line as shown by the summer sun -shall correspond with the time as shown by the winter sun. - -[Illustration: FIG. 6.] - -Let us suppose that _N O S_ is the axis of the heavens, and the lines -_N A S_, _N B S_, _N C S_, are meridian lines drawn from one of the -poles _N_ of the heavens round on the surface of a celestial sphere -whose centre is at _O_. Let _A B C_ be a circle also on this sphere, -passing through _O_, the centre of the sphere, in a plane at right -angles to _N S_, the axis. Then _A B C_ is called the equatorial. It -is a circle in the heavens corresponding to the equator on the earth. -At the vernal and autumnal equinox, namely on March 25 and September -25, the sun is in the equatorial. In midsummer and midwinter it is on -opposite sides of the equatorial. In midsummer it is nearer to _N_, as -at _V_; in midwinter it is nearer to _S_, as at _W_. Suppose we were -on an island in the midst of a surrounding ocean, we should only have -a limited range of view. If the highest point on the island were 100 -feet, then from that altitude we should be able to see about thirteen -miles to the horizon. More than that could not be seen on account of -the rotundity of the earth. - -Let us suppose then such an island surrounded for thirteen miles -distant on every side by an ocean, and let us consider what would be -the apparent motions of the sun when seen from such an island. At the -vernal and autumnal equinoxes, when the sun is on the equatorial, it -would appear to rise out of the ocean at a point _E_, due east; it -traverses half the equatorial and sets in the ocean at a point _W_, due -west. The day is twelve hours long, from 6 a.m. to 6 p.m. - -[Illustration: FIG. 7.] - -In summer the sun is higher, and nearer to the pole _N_, say at a point -_s_. It rises at a point _a_ in the ocean more to the north than _E_, -the eastern point, and sets at a point _b_, also more north than _W_, -the western point, and traverses the path _a s b_. But to traverse -this path it takes longer than twelve hours, for _a s b_ is more than -half the circle _a s b_. Hence then it rises say at 4.30 a.m. and -sets at 7.30 p.m. The night, during which the sun moves round the path -from _b_ to _a_, is correspondingly short, being only nine hours in -length, from 7.30 p.m. till 4.30 a.m. So you have a long summer day and -a short summer night. But in winter, when the sun gets nearer to the -south pole of the heavens, it rises at a point _C_ in the ocean at 7.30 -a.m., and traverses the arc _c t d_, and sets at the point _d_ at 4.30 -p.m. So that the winter day is only nine hours long. But the winter -night lasts from 4.30 p.m. till 7.30 a.m., and is therefore fifteen -hours long, the sun going round the path _d r c_ in the interval. It is -therefore the obliquity of the poles _N S_, coupled with the fact that -the sun’s position is nearer to one pole, _N_, in summer, and nearer to -the other pole, _S_, in winter, that produces the inequality of days -and nights in our latitudes. Suppose our island were on the equator. -The north pole and the south pole would appear to be on the horizon, -and then whether the sun moved in the circle _a s b_ in the summer, or -_E S W_ at the vernal or autumnal equinoxes, or _c t d_ in the winter, -in each of these cases, though the places of rising and setting in the -ocean might vary in summer from _a_ and _b_ to _c_ and _d_ in winter, -yet in each of these cases the path from _a_ to _b_, _A_ to _B_, and -_c_ to _d_ would still always be a half-circle and occupy twelve hours. -Hence at the equator the days and nights never vary in length, but the -sun always rises at six and sets at six. And, besides, it always rises -straight up from the ocean and plunges down vertically into it, so that -there is but little twilight and dawn. - -[Illustration: FIG. 8.] - -But now let us suppose we were living at the north pole. In this case -the north pole would be directly overhead, the south pole directly -under our feet. At the vernal and autumnal equinoxes the sun would -appear with half its disc above the ocean, and go round the ocean -horizon, always appearing with half its disc above the sea. In summer -it would appear at a point _s_ nearer to the pole _N_. It would go -round in the heavens, always appearing above the horizon, and would -never set at all. As the summer waned the sun would become lower and -lower, still, however, going round and round without setting till at -the autumn equinox it reached the horizon. So that for six months it -would never have set. But when it did set, there would then be six -months without a sun at all. - -[Illustration: FIG. 9.] - -Thus then all over the world the period of darkness and light is -equivalent. At the tropics the days and nights are always equal. At -the poles light for six months is followed by darkness for six months. -In the intermediate temperate regions nights of varying lengths follow -days of varying lengths, a short night following a long day and _vice -versâ_. - -[Illustration: FIG. 10.] - -It is evident that for a person living on the north pole a sun-dial -would be an easy thing to make. All that would be needful would be to -put a post vertically in the ground, and observe its shadow as the sun -went round (Fig. 10). - -[Illustration: FIG. 11.] - -In latitudes such as that of England, where the pole of the earth is -inclined at an angle to the horizon, it is necessary that the rod, or -“style” as it is called, of the sun-dial should be inclined to the -horizontal. For if we used an upright “style,” as _O A_, then when the -sun was in the south, at midday, the shadow would lie along the same -direction, _O B_, whether the sun were high in summer, as at _S_, or -low in winter, as at _s_. But at other hours, such as nine o’clock in -the morning, the shadow of the “style” _O A_ would, when the sun was -at its summer position _T_, lie along _O D_, whereas when the sun was -at its winter position _t_ the shadow would lie along _O C_. Thus the -time would appear different in summer and in winter; and the dial -would lead to errors. But if the “style” is inclined in the direction -of the poles, then, however, the sun moves from or towards the pole. -As its position varies in winter and summer, the shadow still remains -unchanged for any particular hour, and it is only the circular motion -of the sun round in its daily path that affects the position of the -shadows. - -[Illustration: FIG. 12.] - -Therefore the first condition of making a sun-dial is that the “style” -which casts the shadow should be parallel to the earth’s axis, that -is to say should point to the polar star. This is the case whether -the sun-dial is horizontal or is vertical, and whether it stands on a -pillar in the garden or is attached to the wall of a house. - -To divide the dial, we have only to imagine it surrounded by a sort -of cage formed of twenty-four arcs drawn from the north pole to the -south pole, and equidistant from one another. In its course the sun -would cross one of them every hour. Hence the points to which the -shadows _o a_, _o b_, _o c_, _o d_, of the inclined “style” _O N_ would -point are the points where these arcs meet the horizontal circle. This -consideration leads to a simple method of constructing a sun-dial, -which is given at the end of this chapter in an appendix. - -Sun-dials were largely in use in ancient times. It is thought that the -circular rows of stones used by the Druids were used to mark the sun’s -path, and indicate the times and seasons. Obelisks are also supposed -to have been used to cast sun-shadows. The Greeks were perfectly -acquainted with the method of making sun-dials with inclined “styles,” -or “gnomons.” - -[Illustration: FIG. 13.] - -Small portable sun-dials were once much used before the introduction of -watches, and were provided with compasses by which they could be turned -round, so that the “style” pointed to the north. - -Sun-dials were only available during the hours of the day when the -sun was shining. The desire to mark the hours of the night led to -the adoption of water clocks, which measured time by the amount of -water which escaped from a small hole in a level of water. Some care, -however, is required to secure correct registration. For suppose that -we have a vessel with a small pipe leading out near the bottom, then -the amount of water which will run out of the pipe in a given time -depends upon the pressure of the water at the pipe, and this depends -in its turn upon _P Q_, the head of water in the vessel. Whence it -follows that the division _Q R_, due to say an hour’s run of the clock -at _Q R_, will be more than _q r_, the division corresponding to an -hour, at _q_, a point lower down between _P_ and _Q_, and hence the -divisions marked on the vessel to show the hours by means of the level -of the water would be uneven, becoming smaller and smaller as the water -fell in the vessel. - -To avoid the inconvenience of unequal divisions, the water to be -measured was allowed to escape into an empty vessel from a vessel in -which its surface was always kept at a constant level. Inasmuch as the -pressure on the pipe or orifice in the vessel in which the water was -always kept at a constant level was always constant, it followed that -equal volumes of water indicated equal times, and the vessel into which -the water fell needed only to be equally divided. - -As a measure of hours of the day in countries such as Egypt, where the -hours were always equal, and thus where the longer days contained more -hours, the water clock was very suitable; but in Greece and Rome, where -the day, whatever its length, was always divided into twelve hours, the -simple water clock was as unsuitable as a modern clock would be, for it -always divided the hours equally, and took no account of the fact that -by such a system the hours in summer were longer than in winter. - -In order, therefore, to make the water clock available in Greece and -Italy, it became necessary to make the hours unequal, and to arrange -them to correspond with unequal hours of the Greek day. This plan was -accomplished as follows. Upon the water which was poured into the -vessel that measured the hours was placed a float; and on the float -stood a figure made of thin copper, with a wand in its hand. This wand -pointed to an unequally divided scale. A separate scale was provided -for every day in the year, and these scales were mounted on a drum -which revolved so as to turn round once in the year. Thus as the figure -rose each day by means of a cogwheel it moved the drum round one -division, or one three hundred and sixty-fifth part of a revolution. -By this means the scale corresponding to any particular day of winter -or summer was brought opposite the wand of the figure, and thus the -scale of hours was kept true. In fact, the water clock, which kept -true time, was made by artificial means to keep untrue time, in order -to correspond with the unequal hours of the Greek days. In the picture -_A_ is the receiving water vessel, _P_ the pipe through which the -water flows; _B_ is the figure, _C_ the rod; _D_ is the drum, made to -revolve by the cogwheel _E_, containing 365 teeth, of which one tooth -was driven forward at the close of each day. A syphon _G_ was fixed in -the vessel _A_, so that when the figure had risen to the top and pushed -forward the lever _F_, the syphon suddenly emptied the vessel through -the pipe _H_, and the figure fell to the bottom of the vessel _A_ and -became ready to rise and register another day. The divisions on the -drum are, of course, uneven. On one side, corresponding to the summer, -the day hours would reckon about seventy minutes each, the night hours -would be only about fifty minutes each, so that the day divisions on -the scale would be long, and the night divisions short. The reverse -would be the case in winter. And, therefore, the lines round the drum -would go in an uneven wavy form. - -[Illustration: FIG. 14.] - -Such water clocks as these were used by the ancient Romans. - -Sand was also used to measure time. As soon as the art of blowing glass -had been perfected by the people of Byzantium, from whom the art passed -to the Venetians, sand-glasses were made. These glasses were used for -all sorts of purposes, for speeches and for cooking, but their most -important use was at sea. For it was very important in the early days -of navigation to know the speed at which the vessel was proceeding in -order that one’s place at sea might be calculated. The earliest method -was to throw over a heavy piece of wood of a shape that resisted being -dragged through the water, and with a string tied to it. The block of -wood was called the log, and the string had knots in it. The knots -were so arranged that when one of them ran through one’s fingers in -a half-minute measured by a sand-glass it indicated that the vessel -was going at the speed of one nautical mile in an hour. The nautical -mile was taken so that sixty of them constituted one degree, that is -one three hundred and sixtieth part of a great circle of the earth. -Each nautical mile has, therefore, 6,080 feet. This is bigger than an -ordinary mile on land, which has only 5,280 feet. The knots, therefore, -have to be arranged so that when the ship is going one nautical -mile—that is to say, 6,080 feet—in an hour, a knot shall run out during -the half-minute run of the minute glass. This is attained by putting -the knots 1/120 × 6,080 = 50 feet 7 inches apart. As one sailor heaved -the log over he gave a stamp on the deck and allowed the cord to run -out through his fingers. Another sailor then turned the sand-glass. -When the sand had all run out, showing that half a minute had passed, -the man who was letting the cord run through his fingers gripped it -fast, and observed how many knots or parts of knots of string had run -out, and thus was able to tell how many “knots” per half-minute the -vessel was going, that is to say, how many nautical miles an hour. - -The modern plan of observing the speed of vessels is different. Now we -use a patent log, consisting of a miniature screw propeller tied to a -string and dragged through the water after the vessel. As it is pulled -through the water it revolves, and the number of revolutions it makes -shows how much water it has passed through, and thus what distance -it has gone. The number of revolutions is measured by a counting -mechanism, and can be read off when the log is pulled in. Or sometimes -the screw is attached to a stiff wire, and the counting mechanism is -kept on board the ship. - -We use the expression “knots an hour” quite incorrectly. It should be -“knots per half-minute,” or “nautical miles an hour.” - -It is easy to use the flow of sand for all sorts of purposes to measure -time. Thus, if sand be allowed to flow from a hopper through a fine -hole into a bucket, the bucket may be arranged so that when a given -time has elapsed, and a given weight of sand has therefore fallen, the -bucket shall tip over, and release a catch, which shall then allow -a weight to fall and any mechanical operation to be done that is -required. Thus, for example, we might put an egg in a small holder tied -to a string and lower it into a saucepan of boiling water. The string -might have a counter-weight attached to it, acting over a pulley and -thus always trying to pull it up out of the water. But this might be -prevented by a pin passing through a loop in the string and preventing -it moving. A hopper or funnel might be filled with sand which was -allowed gradually to escape into a small tip-waggon or other similar -device, so that when a given amount of sand had entered the tip-waggon -would tip over, lurch the pin out of the loop, and thus release the -weight, which in its turn would pull the egg up out of the water in -three minutes or any desired time after it had been put in, or a hole -could be made in the saucepan, furnished with a little tap, and the -water that ran out might be made to fall into a tip-waggon and tip it -over, and thus when it had run out to put an extinguisher on to the -spirit lamp that was heating the saucepan, and at the same time make -a contact and ring an electric bell. By this means the egg would be -always exactly cooked to the right amount, would be kept warm after it -was cooked, and a signal given when it was ready. - -[Illustration: FIG. 15.] - -The sketch shows such an arrangement. The saucepan is about three -inches in diameter and two inches high. When filled with water it will -hold an egg comfortably. The extinguisher _E_, mounted on a hinge _Q_, -is turned back, and the spirit lamp _L_ is lit. As soon as the water -boils, the tap _T_ is turned, and the water gradually trickles away -into the tip-waggon. As soon as it is full it tips over and strikes the -arm _X_ of the extinguisher, and turns the lamp out. The little hot -water left in the saucepan will keep the egg warm for some time. The -waggon _W_ must have a weight _P_ at one end of it, and the fulcrum -must be nearer to that end, so that when empty it rests with the end -_P_ down, but when full it tips over on the fulcrum, when the waggon -has received the right quantity of water. I leave to the ingenious -reader the task of working out the details of such a machine, which, if -made properly, will act very well and may be made for a number of eggs -and worked with very little trouble. - -[Illustration: FIG. 16.] - -Mercury has been used also as an hour-glass. The orifice must be -exceedingly fine. Or a bubble of mercury may be put into a tube which -contains air, and made gradually as it falls to drive the air out -through a minute hole. The difficulty is to get the hole fine enough. -All that can be done is to draw out a fine tube in the blow-lamp, break -it off, and put the broken point in the blow-lamp until it is almost -completely closed up. A tube may thus be made about twelve inches long -that will take twelve hours for a bubble of mercury to descend in it. -But the trouble of making so small a hole is considerable. - -[Illustration: FIG. 17.] - -King Alfred is said to have used candles made of wax to mark the time. -As they blew about with the draughts, he put them in lanterns of horn. -They had no glass windows in those days, but only openings closed -with heavy wooden shutters. These large shutters were for use in fine -weather. Smaller shutters were made in them, so as to let a little -light in in rainy weather without letting in too much wind and rain. - -Rooms must then have been very draughty, so that people required to -wear caps and gowns, and beds had thick curtains drawn round them. -When glass was first invented it was only used by kings and princes, -and glass casements were carried about with them to be fixed into -the windows of the houses to which they came, and removed at their -departure. - -Oil lamps were also used to mark the time. Some of them certainly as -early as the fifteenth century were made like bird-bottles; that is -to say, they consisted of a reservoir closed at the top with a pipe -leading out of the bottom. When full, the pressure of the external -atmosphere keeps the oil in the bottle, and the oil stands in the neck -and feeds the wick. As the oil is consumed bubbles of air pass back -along the neck and rise up to the top of the oil, the level of which -gradually sinks. Of course the time shown by the lamp varies with the -rate of burning of the oil, and hence with the size of the wick, so -that the method of measuring time is a very rough one. - - -APPENDIX. - -To make a sun-dial, procure a circular piece of zinc, about ⅛ inch -thick, and say twelve inches in diameter. Have a “style” or “gnomon” -cast such that the angle of its edge equals the latitude of the place -where the sun-dial is to be set up. This for London will be equal to -51° 30´´. A pattern may be made for this in wood; it should then be -cast in gun-metal, which is much better for out-of-door exposure than -brass. On a sheet of paper draw a circle _A B C_ with centre _O_. Make -the angle _B O D_ equal to the latitude of the place for London = 51° -30´´. From _A_ draw _A E_ parallel to _O B_ to meet _O D_ in _E_, and -with radius _O E_ describe another circle about _O_. Divide the inner -circle _A B C_ into twenty-four parts, and draw radii through them -from _O_ to meet the larger circle. Through any divisions (say that -corresponding to two o’clock) draw lines parallel to _O B_, _O C_, -respectively to meet in _a_. Then the line _O a_ is the shadow line -of the gnomon at two o’clock. The lines thus drawn on paper may be -transferred to the dial and engraved on it, or else eaten in with acid -in the manner in which etchings are done. - -[Illustration: FIG. 18.] - -The centre _O_ need not be in the centre of the zinc disc, but may -be on one side of it, so as to give better room for the hours, etc. -A motto may be etched upon the dial, such as “Horas non numero nisi -serenas,” or “Qual ’hom senza Dio, son senza sol io,” or any suitable -inscription, and the dial is ready for use. It is best put up by -turning it till the hour is shown truly as compared with a correctly -timed watch. It must be levelled with a spirit level. It must be -remembered that the sun does not move quite uniformly in his yearly -path among the fixed stars. This is because he moves not in a circle, -but in an ellipse of which the earth is in one of the foci. Hence the -hours shown on the dial are slightly irregular, the sun being sometimes -in advance of the clock, sometimes behind it. The difference is never -more than a quarter of an hour. There is no difference at midsummer and -midwinter. - -[Illustration: FIG. 19.] - -Civil time is solar time averaged, so as to make the hours and days all -equal. The difference between civil time and apparent solar time is -called the equation of time, and is the amount by which the sun-dial is -in advance of or in retard of the clock. In setting a dial by means of -a watch, of course allowance must be made for the equation of time. - - - - -CHAPTER II. - - -In the last chapter a short description has been given of the ideas of -the ancients as to the nature of the earth and heavens. Before we pass -to the changes introduced by modern science, it will be well to devote -a short space to an examination of ancient scientific ideas. - -All science is really based upon a combination of two methods, -called respectively inductive and deductive reasoning. The first of -these consists in gathering together the results of observation and -experiment, and, having put them all together, in the formulation of -universal laws. Having, for example, long observed that all heavy -things tended to go towards the centre of the earth, we might conclude -that, since the stars remain up in the sky, they can have no weight. -The conclusion would be wrong in this case, not because the method -is wrong, but because it is wrongly applied. It is true that all -heavy things _tend_ to go to the centre of the earth, but if they are -being whirled round like a stone in a sling the centrifugal force -will counteract this tendency. The first part of the reasoning would -be inductive, the second deductive. All this reasoning consists, -therefore, in forming as complete an idea as possible respecting the -nature of a thing, and then concluding from that idea what the thing -will do or what its other properties will be. In fact, you form correct -ideas, or “concepts,” as they are called, and reason from them. - -But the danger arises when you begin to reason before you are sure of -the nature of your concepts, and this has been the great source of -error, and it was this error that all men of science so commonly fell -into all through ancient and modern times up to the seventeenth century. - -Of course, if it were possible by mere observation to derive a -complete knowledge of any objects, it would be the simplest method. -All that would be necessary to do would be to reason correctly from -this knowledge. Unfortunately, however, it is not possible to obtain -knowledge of this kind in any branch of science. - -The ancient method resembled the action of one who should contend that -by observing and talking to a man you could acquire such a knowledge of -his character as would infallibly enable you to understand and predict -all his actions, and to take little trouble to see whether what he did -verified your predictions. - -The only difference between the old methods and the new is that in -modern times men have learned to give far more care to the formation -of correct ideas to start with, are much more cautious in arguing -from them, and keep testing them again and again on every possible -opportunity. - -The constant insistence on the formation of clear ideas and the -practice of, as Lord Bacon called it, “putting nature to the torture,” -is the main cause of the advance of physical science in modern times, -and the want of application of these principles explains why so little -progress is being made in the so-called “humanitarian” studies, such as -philosophy, ethics, and politics. - -The works of Aristotle are full of the fallacious method of the old -system. In his work on the heavens he repeatedly argues that the -heavenly bodies must move in circles, because the circle is the most -perfect figure. He affects a perplexity as to how a circle can at the -same time be convex and also its opposite, concave, and repeatedly -entangles his readers in similar mere word confusion. - -Regarded as a man of science, he must be placed, I think, in spite -of his great genius, below Archimedes, Hipparchus, and several other -ancient astronomers and physicists. - -His errors lived after him and dominated the thought of the middle -ages, and for a long time delayed the progress of science. - -The other great writer on astronomy of ancient times was Ptolemy of -Alexandria. - -His work was called the “Great Collection,” and was what we should -now term a compendium of astronomy. Although based on a fundamental -error, it is a thoroughly scientific work. There is none of the false -philosophy in it that so much disfigures the work of Aristotle. The -reasons for believing that the earth is at rest are interesting. -Ptolemy argues that if the earth were moving round on its axis once in -twenty-four hours a bird that flew up from it would be left behind. -At first sight this argument seems very convincing, for it appears -impossible to conceive a body spinning at the rate at which the earth -is alleged to move, and yet not leaving behind any bodies that become -detached from it. - -On the other hand, the system which taught that the sun and planets -moved round the earth, and which had been adopted largely on account -of its supposed simplicity, proved, on further examination, to be -exceedingly complicated. Each planet, instead of moving simply and -uniformly round the earth in a circle, had to be supposed to move -uniformly in a circle round another point that moved round the earth in -a circle. This secondary circle, in which the planet moved, was called -an epicycle. And even this more complicated view failed to explain the -facts. - -A system which, like that of Aristotle and Ptolemy, was based on -deductions from concepts, and which consisted rather of drawing -conclusions than of examining premises, was very well adapted to -mediæval thought, and formed the foundation of astronomy and geography -as taught by the schoolmen. - -[Illustration: FIG. 20.] - -The poem of Dante accurately represents the best scientific knowledge -of his day. According to his views, the centre of the earth was a fixed -point, such that all things of a heavy nature tended towards it. Thus -the earth and water collected round it in the form of a ball. He had no -idea of the attraction of one particle of matter for another particle. -The only conception he had of gravity was of a force drawing all heavy -things to a certain point, which thus became the point round which the -world was formed. The habitable part of the earth was an island, with -Jerusalem in the middle of it _J_. Round this island was an ocean _O_. -Under the island, in the form of a hollow cone, was hell, with its -seven circles of torment, each circle becoming smaller and smaller, -till it got down into the centre _C_. Heaven was at the opposite side -_H_ of the earth to Jerusalem, and was beyond the circles of the -planets, in the _primum mobile_. When Lucifer was expelled from heaven -after his rebellion against God, having become of a nature to be -attracted to the centre of the earth, and no longer drawn heavenwards, -he fell from heaven, and impinged upon the earth just at the antipodes -of Jerusalem, with such violence that he plunged right through it to -the centre, throwing up behind him a hill. On the summit of this hill -was the Garden of Eden, where our first parents lived, and down the -sides of the hill was a spiral winding way which constituted purgatory. -Dante, having descended into hell, and passed the centre, found his -head immediately turned round so as to point the other way up, and, -having ascended a tortuous path, came out upon the hill of Purgatory. -Having seen this, he was conducted to the various spheres of the -planets, and in each sphere he became put into spiritual communion with -the spirits of the blessed who were of the character represented by -that sphere, and he supposes that he was thus allowed to proceed from -sphere to sphere until he was permitted to come into the presence of -the Almighty, who in the _primum mobile_ presided over the celestial -hosts. - -The astronomical descriptions given by Dante of the rising and setting -of the sun and moon and planets are quite accurate, according to the -system of the world as conceived by him, and show not only that he was -a competent astronomer, but that he probably possessed an astrolabe and -some tables of the motions of the heavenly bodies. - -Our own poet Chaucer may also be credited with accurate knowledge of -the astronomy of his day. His poems often mention the constellations, -and one of them is devoted to a description of the astrolabe, an -instrument somewhat like the celestial globe which used to be employed -in schools. - -But with the revival of learning in Europe and the rise of freedom of -thought, the old theories were questioned in more than one quarter. - -It occurred to Copernicus, an ecclesiastic who lived in the sixteenth -century, to re-examine the theory that had been started in ancient -times, and to consider what explanation of the appearance of the -heavenly bodies could be given on the hypothesis put forward by -Pythagoras, that the earth moved round on its own axis, and also round -the sun. - -It may appear rather curious that two theories so different, one that -the sun goes round the earth and the other that the earth goes round -the sun, should each be capable of explaining the observed appearances -of those bodies. But it must be remembered that motion is relative. If -in a waltz the gentleman goes round the lady, the lady also goes round -the gentleman. If you take away the room in which they are turning, -and consider them as spinning round like two insects in space, who is -to say which of them is at rest and which in motion? For motion is -relative. I can consider motion in a train from London to York. As I -leave London I get nearer to York, and I move with respect to London -and York. But if both London and York were annihilated how should I -know that I was in motion at all? Or, again, if, while I was at rest -in the train at a station on the way, instead of the train moving the -whole earth began to move in a southward direction, and the train in -some way were left stationary, then, though the earth was moving, and -the train was at rest, yet, so far as I was concerned, the train would -appear to have started again on its journey to York, at which place it -would appear to arrive in due time. The trees and hedges would fly by -at the proper rate, and who was to say whether the train was in motion -or the earth? - -The theory of Copernicus, however, remained but a theory. It was -opposed to the evidence of the senses, which certainly leads us to -think that the earth is at rest, and it was opposed also to the ideas -of some among the theologians who thought that the Bible taught us that -the earth was so fast that it could not be moved. Therefore the theory -found but little favour. It was in fact necessary before the question -could be properly considered on its merits that more should be known -about the laws of motion, and this was the principal work of Galileo. - -The merit of Galileo is not only to have placed on a firm basis the -study of mechanics, but to have set himself definitely and consciously -to reverse the ancient methods of learning. - -He discarded authority, basing all knowledge upon reason, and protested -against the theory that the study of words could be any substitute for -the study of things. - -Alluding to the mathematicians of his day, “This sort of men,” says -Galileo in a letter to the astronomer Kepler, “fancied that philosophy -was to be studied like the ‘Æneid’ or ‘Odyssey,’ and that the true -reading of nature was to be detected by the collating of texts.” And -most of his life was spent in fighting against preconceived ideas. It -was maintained that there could only be seven planets, because God -had ordered all things in nature by sevens (“Dianoia Astronomica,” -1610); and even the discoveries of the spots on the sun and the -mountains in the moon were discredited on the ground that celestial -bodies could have no blemishes. “How great and common an error,” -writes Galileo, “appears to me the mistake of those who persist in -making their knowledge and apprehension the measure of the knowledge -and apprehension of God, as if that alone were perfect which they -understand to be so. But ... nature has other scales of perfection, -which we, being unable to comprehend, class among imperfections. - -“If one of our most celebrated architects had had to distribute the -vast multitude of fixed stars over the great vault of heaven, I believe -he would have disposed them with beautiful arrangements of squares, -hexagons, and octagons; he would have dispersed the larger ones among -the middle-sized or lesser, so as to correspond exactly with each -other; and then he would think he had contrived admirable proportions; -but God, on the contrary, has shaken them out from His hand as if by -chance, and we, forsooth, must think that He has scattered them up -yonder without any regularity, symmetry, or elegance.” - -In one of Galileo’s “Dialogues” Simplicio says, “That the cause that -the parts of the earth move downwards is notorious, and everyone knows -that it is gravity.” Salviati replies, “You are out, Master Simplicio: -you should say that everyone knows that _it is called_ gravity; I do -not ask you for the name, but for the nature, of the thing of which -nature neither you nor I know anything.” - -Too often are we still inclined to put the name for the thing, and to -think when we use big words such as art, empire, liberty, and the -rights of man, that we explain matters instead of obscuring them. Not -one man in a thousand who uses them knows what he means; no two men -agree as to their signification. - -The relativity of motion mentioned above was very elegantly illustrated -by Galileo. He called attention to the fact that if an artist were -making a drawing with a pen while in a ship that was in rapid passage -through the water, the true line drawn by the pen with regard to the -surface of the earth would be a long straight line with some small -dents or variations in it. Yet the very same line traced by the pen -upon a paper carried along in the ship made up a drawing. Whether you -saw a long uneven line or a drawing in the path that the pen had traced -depended altogether on the point of view with which you regarded its -motion. - -[Illustration: FIG. 21.] - -But the first great step in science which Galileo made when quite a -young professor at Pisa was the refutation of Aristotle’s opinion that -heavy bodies fell to the earth faster than light ones. In the presence -of a number of professors he dropped two balls, a large and a small -one, from the parapet of the leaning tower of Pisa. They fell to the -ground almost exactly in the same time. This experiment is quite an -easy one to try. One of the simplest ways is as follows: Into any beam -(the lintel of a door will do), and about four inches apart, drive -three smooth pins so as to project each about a quarter of an inch; -they must not have any heads. Take two unequal weights, say of 1 lb. -and 3 lbs. Anything will do, say a boot for one and pocket-knife for -the other; fasten loops of fine string to them, put the loops over the -centre peg of the three, and pass the strings one over each of the side -pegs. Now of course if you hitch the loops off the centre peg _P_ the -objects will be released together. This can be done by making a loop -at the end of another piece of string, _A_, and putting it on to the -centre peg behind the other loops. If the string be pulled of course -the loop on it pulls the other two loops off the central peg, and -allows the boot and the knife to drop. The boot and the knife should be -hung so as to be at the same height. They will then fall to the ground -together. The same experiment can be tried by dropping two objects from -an upper window, holding one in each hand, and taking care to let them -go together. - -[Illustration: FIG. 22.] - -This result is very puzzling; one does not understand it. It appears as -though two unequal forces produced the same effect. It is as though a -strong horse could run no faster than a weaker one. - -The professors were so irritated at the result of this experiment, and -indeed at the general character of young Professor Galileo’s attacks on -the time-honoured ideas of Aristotle, that they never rested till they -worried him out of his very poorly paid chair at Pisa. He then took a -professorship at Padua. - -Let us now examine this result and see why it is that the ideas we -should at first naturally form are wrong, and that the heavy body will -fall in exactly the same time as the light one. - -We may reason the matter in this way. The heavy body has more force -pulling on it; that is true, but then, on the other hand there is more -matter which has got to be moved. If a crowd of persons are rushing out -of a building, the total force of the crowd will be greater than the -force of one man, but the speed at which they can get out will not be -greater than the speed of one man; in fact, each man in the crowd has -only force enough to move his own mass. And so it is with the weights: -each part of the body is occupied in moving itself. If you add more to -the body you only add another part which has itself to move. A hundred -men by taking hands cannot run faster than one man. - -But, you will say, cannot a man run faster than a child? Yes, because -his impelling power is greater in proportion to his weight than that of -a child. - -If it were the fact that the attraction of gravity due to the earth -acted on some bodies with forces greater in proportion to their -masses than the forces that acted on other bodies, then it is true -that those different bodies would fall in unequal time. But it is -an experimental fact that the attractive force of gravity is always -exactly proportional to the mass of a body, and the resistance to -motion is also proportional to mass, hence the force with which a -body is moved by the earth’s attraction is always proportional to the -difficulty of moving the body. This would not be the case with other -methods of setting a body in motion. If I kick a small ball with all -my might, I shall send it further than a kick of equal strength would -send a heavier ball. Why? Because the impulse is the same in each case, -but the masses are different. But if those balls are pulled by gravity, -then, by the very nature of the earth’s attraction (the reason of which -we cannot explain), the small ball receives a little pull, and the big -ball receives a big pull, the earth exactly apportioning its pull in -each case to the mass of the body on which it has to act. It is to this -fact, that the earth pulls bodies with a strength always in each case -exactly proportional to their masses, that is due the result that they -fall in equal times, each body having a pull given to it proportional -to its needs. - -The error of the view of Aristotle was not only demonstrated by -Galileo by experiment, but was also demonstrated by argument. In this -argument Galileo imitated the abstract methods of the Aristotelians, -and turned those methods against themselves. For he said, “You” (the -Aristotelians) “say that a lighter body will fall more slowly than a -heavy one. Well, then, if you bind a light body on to a heavy one by -means of a string, and let them fall together, the light body ought -to hang behind, and impede the heavy body, and thus the two bodies -together ought to fall more slowly than the heavy body alone; this -follows from your view: but see the contradiction. For the two bodies -tied together constitute a heavier body than the heavy body alone, and -thus, on your own theory, ought to fall more quickly than the heavy -body alone. Your theory, therefore, contradicts itself.” - -The truth is that each body is occupied in moving itself without -troubling about moving its neighbour, so that if you put any number of -marbles into a bag and let them drop they all go down individually, as -it were, and all in the time which a single marble would take to fall. -For any other result would be a contradiction. If you cut a piece of -bread in two, and put the two halves together, and tie them together -with a thread, will the mere fact that they are two pieces make each of -them fall more slowly than if they were one? Yet that is what you would -be bound to assert on the Aristotelian theory. Hold an egg in your -open hand and jump down from a chair. The egg is not left behind; it -falls with you. Yet you are the heavier of the two, and on Aristotelian -principles you ought to leave the egg behind you. It is true that when -you jump down a bank your straw hat will often come off, but that is -because the air offers more resistance to it than the air offers to -your body. It is the downward rush through the air that causes your hat -to be left behind, just as wind will blow your hat off without blowing -you away. For since motion is relative, it is all one whether you jump -down through the air, or the air rushes past you, as in a wind. If -there were no air, the hat would fall as fast as your body. - -This is easy to see if we have an airpump and are thus enabled to -pump out almost all the air from a glass vessel. In that vessel so -exhausted, a feather and a coin will fall in equal times. If we have -not an airpump, we can try the experiment in a more simple way. For -let us put a feather into a metal egg-cup and drop them together. The -cup will keep the air from the feather, and the feather will not come -out of the cup. Both will fall to the ground together. But if the -lighter body fall more slowly, the feather ought to be left behind. If, -however, you tie some strings across a napkin ring so as to make a sort -of rough sieve, and put a feather in it, and then drop the ring, then -as the ring falls the air can get through the bottom of the ring and -act on the feather, which will be left floating as the ring falls. - -Let us now go on to examine the second fallacy that was derived from -the Aristotelians, and that so long impeded the advance of science, -namely, that the earth must be at rest. - -The principal reason given for this was that if bodies were thrown -up from the earth they ought, if the earth were in motion, to remain -behind. Now, if this were so, then it would follow that if a person -in a train which was moving rapidly threw a ball vertically, that is -perpendicularly, up into the air, the ball, instead of coming back into -his hand, ought to hit the side of the carriage behind him. The next -time any of my readers travel by train he can easily satisfy himself -that this is not so. But there are other ways of proving it. For -instance, if a little waggon running on rails has a spring gun fixed in -it in a perpendicular position, so arranged that when the waggon comes -to a particular point on the rails a catch releases the trigger and -shoots a ball perpendicularly upwards, it will be found that the ball, -instead of going upwards in a vertical line, is carried along over the -waggon, and the ball as it ascends and descends keeps always above the -waggon, just as a hawk might hover over a running mouse, and finally -falls not behind the waggon, but into it. - -So, again, if an article is dropped out of the window of a train, it -will not simply be left behind as it falls, but while it falls it will -also partake of the motion of the train, and touch the ground, not -behind the point from which it was dropped, but just underneath it. - -The reason is, that when the ball is dropped or thrown it acquires -not only the motion given to it by the throw, or by gravity, but it -takes also the motion of the train from which it is thrown. If a ball -is thrown from the hand, it derives its motion from the motion of the -hand, and if at the time of throwing the person who does so is moving -rapidly along in a train, his hand has not only the outward motion -of the throw, but also the onward motion of the train, and the ball -therefore acquires both motions simultaneously. Hence then it is not -correct reasoning to say, because a ball thrown up vertically falls -vertically back to the spot from which it was thrown, that therefore -the earth must be at rest; the same result will happen whether the -earth is at rest or in motion. You can no more tell whether the earth -is at rest or in motion from the behaviour of falling bodies than you -can tell whether a ship on the ocean is at rest or in motion from the -behaviour of bodies on it. - -But you will say. Then why do we feel sea-sick on a ship? The answer -is, that that is because the motion of the ship is not uniform. If the -earth, instead of turning round uniformly, were to rock to and fro, -everything on it would be flung about in the wildest fashion. For as -soon as the earth had communicated its motion to a body which then -moved with the earth, if the earth’s motion were reversed, the body -would go on like a passenger in a train on which the break is quickly -applied, and he would be shot up against the side of the room. Nay, -more, the houses would be shaken off their foundations. Changes of -motion are perceptible _so long as the change is going on_. We are -therefore justified in inferring from the behaviour of bodies on the -earth, not that the earth is at rest, but that it is either at rest, or -else, if it is in motion, that its motion is uniform and not in jerks -or variable. - -[Illustration: FIG. 23.] - -For if it were not so, consider what would be happening around us. The -earth is about 8,000 miles in diameter, and a parallel of latitude -through London is therefore about 19,000 miles long, and this space -is travelled in twenty-four hours. So that London is spinning through -space at the rate of over 1,000 feet a second, due to the earth’s -rotary motion alone, not to speak of the motion due to the earth’s -path round the sun. If a boy jumped up two and a half feet into the -air, he would take about half a second to go up and come down, but if -in jumping he did not partake of the earth’s motion, he would land -more than 500 feet to the westward of the point from which he jumped -up, and if he did it in a room, he would be dashed against the wall -with a force greater than he would experience from a drop down from -the top of Mont Blanc. He would be not only killed, but dashed into an -indistinguishable mass. If the earth suddenly stood still, everything -on it would be shaken to pieces. It is bad enough to have the -concussion of a train going thirty miles an hour when dashed against -some obstacle. But the concussion due to the earth’s stoppage would -be as of a train going about 800 miles an hour, which would smash up -everything and everybody. - -Thus, then, the first effect of the new ideas formulated by Galileo was -to show that the Copernican theory that the earth moved round on its -axis, and round the sun, was in agreement with the laws of motion. In -fact, he introduced quite new ideas of force, and these ideas I must -now endeavour to explain. - -Let us consider what is meant by the word “force.” If I press my -hand against the table, I exert force. The harder I press, the more -force there is. If I put a weight on a stand, the weight presses the -stand down with a force. If I squeeze a spring, the spring tries to -recover itself and exerts a certain force. In all these cases force -is considered as a pressure. And I can measure the force by seeing -how much it will press things. If I take a spring, and press it in an -inch, it takes perhaps a force of 1 lb. It will take a force of 2 lbs. -to press it in another inch. Or again, if I pull it out an inch, it -takes a force of 1 lb. If I pull it out another inch, it takes a force -of 2 lbs. We thus always get into the habit of conceiving forces as -producing pressures and being measured by pressures. - -[Illustration: FIG. 24.] - -This is a perfectly legitimate way of looking at the matter, just as -the cook’s method of employing a spring balance to weigh masses of meat -is a perfectly legitimate way of estimating the forces acting upon -bodies at rest. But when you come to consider the laws of the pendulums -of clocks, to which all that I am saying is a preparation, then you -have to deal with bodies in motion. And for this purpose a new idea of -force altogether is requisite. We shall no longer speak of forces as -producing _pressures_. We shall treat them quite independently of their -pressing power. The sun exerts a force of attraction on the earth, but -it does not press upon it. It exerts its force at a distance. Hence -then we want a new idea of “force.” This idea is to be the following. -We will consider that when a force acts upon a body it endeavours to -cause it to move; in fact, it tries to impart motion to the body. We -may treat this motion as a sort of thing or property. The longer the -force acts on the body, the more motion it imparts to it, provided the -body is free to receive that motion. So that we may say that the test -of the strength of the force is how much motion it can give to a body -of a given mass in a given time. It does not matter how the force acts. -It may act by means of a string and pull it; it may act by means of a -stick and push it; it may act by attraction and draw it; it may act -by repulsion and repel it; it may act as a sort of little spirit and -fly away with it. In all these cases it _acts_. The more it acts, the -more effect it has. In double the time it produces double the motion. -If the mass is big, it takes more force to make the mass move; if the -mass of the body is small, it is moved more easily. Therefore when we -want to measure a force in this way we do not press it against springs -to see how much it will press them in. What we do is to cause it to act -on bodies that are free to move and see what motions it will produce -in them. Of course we can only do this with things that are free to -move. You cannot treat force in this way if you have only a pair of -scales; in that case you would have to be content with simply measuring -pressures. It is important clearly to grasp this idea. If a body has -a certain mass, then the force acting on it is measured by the amount -of motion that will in a given time be imparted to that mass, provided -that the mass is free to move. This is to be our definition of force. - -Therefore, by the action of an attraction or any other force on a body -free to move; motion is continually being imparted to the body. Motion -is, as it were, poured into it, and therefore the body continually -moves faster and faster. - -Here is a ball flying through the air. Let us suppose that forces are -acting on it. How can we measure them? We cannot feel what pressures -are being exerted on it. The only thing we can do is to watch its -motions, and see how it flies. If it goes more and more quickly, we -say, “There is propelling force acting on it”; if it begins to stop, -we say again, “There is retarding force acting on it.” So long as it -does not change its speed or direction, we say, “There is no force -acting on it.” By this method, therefore, we tell whether a body is -being acted on by force, simply by observing its speed or its change of -speed. Merely to say a body is _moving_ does not tell us that force is -acting on it. All we know is that, if it is moving, force _has_ acted -on it. It is only when we see it changing its speed or direction, that -is changing its motion, that we say _force_ is acting. Every change of -motion, either in direction or speed, must be the result of force, and -must be proportional to that force. This is what we mean when we say -motion is the test and measure of force. - -This most interesting way of looking at the matter lies at the root -of the whole theory of mechanics. It is the foundation of the system -which the stupendous genius of Newton conceived in order to explain the -motion of the sun, moon, and stars. - -Forces were treated by him as proportional to the motions, and the -motions proportional to the forces, and with this idea he solved a part -of the riddle of the universe. Galileo had partly seen the same thing, -but he never saw it so clearly as Newton. Great discoveries are only -made by seeing things clearly. What required the force of a genius in -one age to see in the next may be understood by a child. - -Hence then we say a force is that which in a given time produces a -given motion in a given mass which is free to move. - -You must have time for a force to act in; for however great the force, -in no time there can be no motion. You must have mass for a force to -act on; no mass, no effect. You must have free space for the mass to -move in; no freedom to move, no movement. - -But what is this “mass”? We do not know; it is a mystery. We call it -“quantity of matter.” In uniform substances it varies with size. Double -the volume, double the mass. Cut a cake in half, each half has the -same “mass.” But then is mass “weight”? No, it is not. _Weight_ is the -action of the earth’s attraction on matter. No earth to attract, and -you would have no weight, but you would still have “mass.” What then -is matter? Of that we have no idea. The greatest minds are now at work -upon it. But _mass_ is quantity of matter. Knock a brick against your -head, and you will know what mass is. It is not the weight of the brick -that gives you a bump; it is the mass. Try to throw a ball of lead, and -you will know what mass is. Try to push a heavy waggon, and you will -know what mass is. _Weights_, that is earth attractions on masses, are -proportional to the masses at the same place. This, as we have seen, is -known by experiment. - -Therefore, when a force acts for a certain time on a mass that is free -to move, however small the force and however small the time, that body -will move. When a baby in a temper stamps upon the earth it makes the -earth move—not much, it is true, but still it moves; nay, more, in -theory, not a fly can jump into the air without moving the earth and -the whole solar system. Only, as you may imagine they do not show it -appreciably. Still, in theory the motion is there. - -Hence then there are two different ways of considering and estimating -forces, one suitable for observations on bodies at rest, the other -suitable for observations of bodies that are free to move. The force -of course always tends to produce motion. If, however, motion is -impossible, then it develops pressures which we can measure, and -calculate, and observe. If the body is free to move, then the force -produces motions which we can also measure, calculate, and observe. -And we can compare these two sets of effects. We can say, “A force -which, acting on a ball of a mass of one pound, would produce such and -such motions, would if it acted on a certain spring produce so much -compression.” - -The attraction of the earth on masses of matter that are not free to -move gives rise to forces which are called weights. Thus the attraction -of gravitation on a mass of one pound produces a pressure equal to a -weight of one pound. Unfortunately the same word “pound” is used to -express both the mass and the weight, and has come down to us from days -when the nature of mass was not very well appreciated. But great care -must be taken not to confuse these two meanings. - -But the earth’s attractions and all other forces acting upon matter -which is free to move give rise to changes of motion. The word used for -a change of motion is “acceleration” or a quickening. “He accelerated -his pace,” we say. That is, he quickened it; he added to his motion. So -that _force_, acting on _mass_ during a _time_, produces acceleration. - -From this, then, it follows that if a _force_ continues to act on a -body the body keeps moving quicker and quicker. When the force stops -acting, the motion already acquired goes on, but the acceleration -stops. That is to say, the body goes on moving in a straight line -uniformly at the pace it had when the force stopped. - -If, then, a body is exposed to the action of a force, and held tight, -what will happen? It will, of course, remain fixed. Now let it go—it -will then, being a free body, begin to move. As long as the force -acts, the force keeps putting more and more motion into the body, like -pouring water into a jug, the longer you pour the faster the motion -becomes. The body keeps all the motion it had, and keeps adding all the -motion it gains. It is like a boy saving up his weekly pocket-money: -he has what he had, and he keeps adding to that. So if in one second -a motion is imparted of one foot a second, then in another second a -motion of one foot a second more will be added, making together a -motion of two feet a second; in another second of force action the -motion will have been increased or “accelerated” by another foot per -second, and so on. The speed will thus be always proportional to -the force and the time. If we write the letter V to represent the -motion, or speed, or velocity; F to represent the acceleration or gain -of motion; and T to represent the time, then V = FT. Here V is the -velocity the body will have acquired at the end of the time T, if free -to move and submitted to a force capable of producing an acceleration -of F feet per second in a unit of time. - -V is the final velocity. The average velocity will be 1/2 V, for it -began with no velocity and increased uniformly. How far will the body -have fallen in the interval? Manifestly we get that by multiplying the -time by the average velocity, that is S = 1/2 VT, where V, as I said, -is the final velocity, but we found that V = FT. Hence by substitution -S = 1/2 FT × T = 1/2 FT². - -It is to be carefully borne in mind that these letters V, S, and T -do not represent velocities, spaces, and times, but merely represent -arithmetical numbers of units of velocities, spaces, and times. Thus -V represents V feet per second, S represents S feet, and T represents -T seconds. And when we use the equation V = FT we do not mean that -by multiplying a force by a time you can produce a velocity. If, for -instance, it be true that you can obtain the number of inhabitants (H) -in London by multiplying the average number of persons (P) who live -in a house by the number of houses (N), this may be expressed by the -equation H = PN. But this does not mean that by multiplying people into -houses you can produce inhabitants. H, P, and N are numbers of units, -and they are _numbers only_. - -Therefore when a body is being acted on by an accelerating force -it tends to go faster and faster as it proceeds, and therefore its -velocity increases with the time. But the space passed through -increases faster still, for as the time runs on not only does the -space passed through increase, but the rate of passing also gets -bigger. It goes on increasing at an increasing rate. It is like a man -who has an increasing income and always goes on saving it. His total -mounts up not merely in proportion to the time, but the very rate of -increase also increases with the time, so that the total increase is -in proportion to the time multiplied into the time, in other words to -the square of the time. So then, if I let a body drop from rest under -the action of any force capable of producing an acceleration, the space -passed through will be as the square of the time. - -Now let us see what the speed will be if the force is gravity, that is -the attraction of the earth. - -Turning back to what was said about Galileo, it will be remembered that -he showed that all bodies, big and small, light and heavy, fell to the -earth at the same speeds. What is that speed? Let us denominate by G -the number of feet per second of increase of motion produced in a body -by the earth’s action during one second. Then the velocity at the end -of that second will be V = GT. The space fallen through will be S = 1/2 -GT². - -What I want to know then is this: how far will a body under the action -of gravity fall in a second of time? - -This, of course, is a matter for measurement. If we can get a machine -to measure seconds, we shall be able to do it; but inasmuch as falling -bodies begin by falling sixteen feet in the first second and afterwards -go on falling quicker and quicker, the measurements are difficult. -Galileo wanted to see if he could make it easier to observe. He said -to himself, “If I can only water down the force of gravity and make -it weaker, so that the body will move very slowly under its action, -then the time of falling will be easier to observe.” But how to do it? -This is one of those things the discovery of which at once marks the -inventor. - -[Illustration: FIG. 25.] - -The idea of Galileo was, instead of letting the body drop vertically, -to make it roll slowly down an incline, for a body put upon an incline -is not urged down the incline with the same force which tends to make -it fall vertically. - -Can any law be discovered tending to show what the force is with which -gravity tends to drag a mass down an incline? - -There is a simple one, and before Galileo’s time it had been -discovered by Stevinus, an engineer. Stevinus’ solution was as follows. -Suppose that _A B C_ is a wedge-shaped block of wood. Let a loop of -heavy chain be hung over it, and suppose that there is a little pulley -at _C_ and no friction anywhere. Then the chain will hang at rest. But -the lower part, from _A_ to _B_, is symmetrical; that is to say, it -is even in shape on both sides. Hence, so far as any pull it exerts -is concerned, the half from _A_ to _D_ will balance the other half -from _B_ to _D_. Therefore, like weights in a scale, you may remove -both, and then the force of gravity acting down the plane on the part -_A C_ will balance the force of gravity acting vertically on the part -_C B_. Now the weight of any part of the chain, since it is uniform, -is proportional to its length. Hence, then, the gravitational force -down the plane of a piece whose weight equals _C A_ is equal to the -gravitational force vertically of a piece whose weight equals _C B_. In -other words, the force of gravity acting down a plane is diminished in -the ratio of _C B_ to _C A_. - -But when a body falls vertically, then, as we have seen, S = 1/2 GT², -where S is the space it will fall through, G the number of feet per -second of velocity that gravity, acting vertically on a body, will -produce in it in a second, and T the number of seconds of time. If -then, instead of falling vertically, the body is to fall obliquely down -a plane, instead of G we must put as the accelerating force - - G × (vertical height of the end of the plane)/(length of the plane). - -To try the experiment, he took a beam of wood thirty-six feet long with -a groove in it. He inclined it so that one end was one foot higher than -the other. Hence the acceleration down the plane was 1/36 G, where G is -the vertical acceleration due to gravity which he wanted to discover. -Then he measured the time a brass ball took to run down the plane -thirty-six feet long, and found it to be nine seconds. Whence from -the equation given above 36 feet = 1/2 acceleration of gravity down -the plane × (9 seconds)². Whence it follows that the acceleration of -gravity down the plane is (36 × 2)/(9)² feet per second. - -But the slope of the plane is one thirty-sixth to the vertical. -Therefore the vertical acceleration of gravity, _i.e._, the velocity -which gravity would induce in a vertical direction in a second, is -equal to thirty-six times that which it exercises down the plane, -_i.e._, - -36 × (36 × 2)/(9)²; and this equals 32 feet per second. - -Though this method is ingenious, it possesses two defects. One is the -error produced by friction, the other from failure to observe that -the force of gravity on the ball is not only exerted in getting it -down the plane, but also in rotating it, and for this no allowance has -been made. The allowance to be made for rotation is complicated, and -involves more knowledge than Galileo possessed. Still the result is -approximately true. - -[Illustration: FIG. 26.] - -The next attempt to measure G, that is the velocity that gravity will -produce on a body in a second of time, was made by Attwood, a Cambridge -professor. His idea was to weaken the force of gravity and thus make -the action slow, not by making it act obliquely, but by allowing it to -act, not on the whole, but only on a portion of the mass to be moved. -For this purpose he hung two equal weights over a very delicately -constructed pulley. Gravity, of course, could not act on these, for -any effect it produced on one would be negatived by its effect on -the other. The weights would therefore remain at rest. If, however, -a small weight _W_, equal say to a hundredth of the combined weight -of the weights _A_ and _B_ and _W_, were suddenly put on _A_, then it -would descend under an accelerating force equal to a hundredth part of -ordinary gravity. We should then have - - S (the space moved through by the weights) = 1/2 × G/100 × t². - -With such a system, he found that in 7½ seconds the weights moved -through 9 feet. Whence he got - - 9 = 1/2 G/100 × (7½)². - -From which - - G = (2 × 9 × 100)/(7½)² = 32 feet per second nearly. - -Thus by letting gravity only act on a hundredth part of the total -weight moved, namely _A_, _B_, and _W_, he weakened its action 100 -times, and thus made the time of falling and the space fallen through -sufficiently large to be capable of measurement. To sum up, when a body -free to move is acted upon by the force of gravity, its speed will -increase in proportion to the time it has been acted upon, and the -space it will pass through from rest is proportional to the square of -the time during which the accelerating force has acted on it. - -Gravity is, of course, not the only accelerating force with which -we are acquainted. If a spring be suddenly allowed to act on a body -and pull it, the body begins to move, and its action is gradually -accelerated, just as though it were attracted, and the acceleration -of its motion will be proportional to the time during which the -accelerating force acts. Similarly, if gunpowder be exploded in a -gun-barrel, and the force thus produced be allowed to act on a bullet, -the motion of the bullet is accelerated so long as it is in the barrel. -When the bullet leaves the barrel it goes on with a uniform pace in a -straight line, which, however, by the earth’s attraction is at once -deflected into a curve, and altered by the resistance of the air. - -[Illustration: FIG. 27.] - -It has been already stated that motions may be considered independently -one of another, so that if a body be exposed to two different forces -the action of these forces can be considered and calculated each -independently of the other. Let us take an example of this law. We have -seen if a body is propelled forwards, and then the force acting on it -ceases, that it proceeds on with uniform unchanging velocity, and if -nothing impeded it, or influenced it, it would go on in a straight line -at a uniform speed. - -We have also seen that if a body is exposed to the action of an -accelerating force such as gravity it constantly keeps being -accelerated, it constantly keeps gaining motion, and its speed becomes -quicker and quicker. - -[Illustration: FIG. 28.] - -Let us suppose a body exposed to both of these forces at the same time. -Shoot it out of a cannon, and let an accelerating force act on it, not -in the direction it is going, but in some other direction, say at right -angles. What will happen? In the direction in which it is going, its -speed will remain uniform. In the direction in which the accelerating -force is acting, it will move faster and faster. Thus along _A B_ it -will proceed uniformly. If it proceeded uniformly also along _A C_ (as -it would do if a simple force acted on it and then ceased to act), then -as a result it would go in the oblique line _A D_, the obliquity being -determined by the relative magnitude of the forces acting on it. But -how if it went uniformly along _A B_, but at an accelerated pace along -_A C_? Then while in equal times the distances along _A B_ would be -uniform the distances in the same times along _A C_ would be getting -bigger and bigger. It _would not describe a straight line; it would go -in a curve_. This is very interesting. Let us take an example of it. -Suppose we give a ball a blow horizontally; as soon as it quits the bat -it would of course go on horizontally in a straight line at a uniform -speed; but now if I at the same instant expose it to the accelerating -force of gravity, then, of course, while its horizontal movement will -go on uniformly, its downward drop will keep increasing at a speed -varying as the time. And while the total distances horizontally will -be uniform in equal times, the total downward drop from _A B_ will -be as the squares of the times. Here, then, you have a point moving -uniformly in a horizontal direction, but as the squares of the times in -a vertical direction. It describes a curve. What curve? Why, one whose -distances go uniformly one way, but increase as the squares the other -way. - -[Illustration: FIG. 29.] - -This interesting curve is called a parabola. With a ball simply hit by -a bat, the motion is so very fast that we cannot see it well. Cannot -we make it go slowly? Let us remember what Galileo did. He used an -inclined plane to water down his force of gravity. Let us do the same. -Let us take an inclined plane and throw on it a ball horizontally. -It will go in a curve. Its speed is uniform horizontally, but is -accelerated downwards. If we desire to trace the curve it is easy to -do. We coat the ball with cloth and then dip it in the inkpot. It will -then describe a visible parabola. If I tilt up the plane and make the -force of gravity big, the parabola is long and thin; if I weaken down -the force of gravity by making the plane nearly horizontal, then it is -wide and flat. - -One can also show this by a stream of peas or shot. The little bullets -go each with a uniform velocity horizontally, and an accelerated force -downwards. - -Instead of peas we can use water. A stream of it rushing horizontally -out of an orifice will soon bend down into a parabola. - -Thus then I have tried to show what force is and how it is measured. I -repeat again, when a body is free to move, then, if no further force -acts on it, it will go on in a straight line at a uniform speed, but -if a force continues to act on it in any direction, then that force -produces in each unit of time a unit of acceleration in the direction -in which the force acts, and the result is that the body goes on moving -towards the direction of acceleration at a constantly increasing speed, -and hence passing over spaces that are greater and greater as the speed -increases. This is the notion of a “force.” In all that has been said -above it has been assumed that the attraction of gravity on a body -does not increase as that body gets nearer to the earth. This is not -strictly true; in reality the attractive force of gravity increases as -the earth’s centre is approached. But small distances through which -the weights in Attwood’s machine fall make no appreciable difference, -being as nothing compared to the radius of earth. For practical -purposes, therefore, the force may be considered uniform on bodies that -are being moved within a few feet of the earth’s surface. It is only -when we have to consider the motions of the planets that considerations -of the change of attractive force due to distance have to be considered. - -I am glad to say that the most tiresome, or rather the most difficult, -part of our inquiry is now over. With the help of the notions already -acquired, we are now ready to get to the pendulum, and to show how it -came about that a boy who once in church amused himself by watching the -swinging of the great lamps instead of attending to the service laid -the foundation of our modern methods of measuring time. - - - - -CHAPTER III. - - -We have examined the action of a body under the accelerating or -speed-quickening force due to gravity, the attractive force of which on -any body is always proportional to the mass of that body. Let us now -consider another form of acceleration. - -[Illustration: FIG. 30.] - -Take the case of a strip of indiarubber. If pulled it resists and tends -to spring back. The more I pull it out the harder is the pull I have -to exert. This is true of all springs. It is true of spiral springs, -whether they are pulled out or pushed in, and in each case the amount -by which the spring is pulled out or pushed in is proportional to the -pressure. This law is called Hooke’s law. It was expressed by him in -Latin, “Ut tensio, sic vis”: “As the extension, so the force.” It is -true of all elastic bodies, and it is true whether they are pulled out -or pushed in or bent aside. The common spring balance is devised on -this principle. The body to be weighed is hung on a hook suspended from -a spring. The amount by which the spring is pulled out is a measure -of the weight of the body. If you take a fishing rod and put the butt -end of it on a table and secure it by putting something heavy on the -end, then the tip will bend down on account of its own weight. Mark the -point to which it goes. Now, if you hang a weight on the tip, the tip -will bend down a little further. If you put double the weight the tip -will go down double the distance, and so on until the fishing rod is -considerably bent, so that its form is altered and a new law of flexure -comes into play. Suppose I use a spring as an accelerating force. For -example, suppose I suspend a heavy ball by a string and then attach a -spiral spring to it and pull the spring aside. The ball will be drawn -after the spring. If then I let the ball go, it will begin to move. The -force of the spring will act upon it as an accelerating force, and the -ball will go on moving quicker and quicker. But the acceleration will -not be like that of gravity. There will be two differences. The pull of -the spring will in no way depend on the mass of the ball, and the pull -of the spring, instead of being constant, like the pull of gravity, -will become weaker and weaker as the ball yields to it. Consequently -the equations above given which determine the relations between this -space passed through, the velocity, and the time which were determined -in the case of gravity are no longer true, and a different set of -relations has to be determined. This can be easily done by mathematics. -But I do not propose to go into it. I prefer to offer a rough and ready -explanation, which, though it does not amount to a proof, yet enables -us to accept the truth that can be established both by experiment and -by calculation. - -[Illustration: FIG. 31.] - -Let a heavy ball (_A_) be suspended by a long string, so that the -action of gravity sideways on the ball is very small and may be -neglected, and to each side attach an indiarubber thread fastened -at _B_ and _C_. Then when the ball is pulled aside a little, say -to a position _D_, it will tend to fly back to _A_ with a force -proportioned to the distance _A D_. What will be the time it will take -to do this? If the distance _A D_ is small, the ball has only a small -distance to go, but then, on the other hand, it has only small forces -acting on it. If the distance _A D_ is bigger, then it has a longer -distance to go, but larger forces to urge it. These counteract one -another, so that the time in each case will be the same. - -[Illustration: FIG. 32.] - -The question is this:—Will you go a long distance with a powerful -horse, or a small distance with a weak horse? If the distance in each -case is proportioned to the power of the horse, then the amount of the -distance does not matter. The powerful horse goes the long distance in -the same time that the weak horse goes the short distance. And so it -is here. However far you pull out the spring, the accelerative pull on -the ball is proportioned to the distance. But the time of pulling the -ball in depends on the distance. So that each neutralises the other. -Whence then we have this most important fact, that springs are all -isochronous; that is to say, any body attached to any spring whatever, -whether it is big or small, straight or curly, long or short, has a -time of vibration quite independent of the bigness of the vibration. -The experiment is easy to try with a ball mounted on a long arm that -can swing horizontally. It is attached on each side to an elastic -thread. If pulled aside, it vibrates, but observe, the vibration is -exactly the same whether the bigness of the vibration is great or -small. If the pull aside is big, the force of restitution is big; if -the pull is small, the force of restitution is small. In one case the -ball has a longer distance to go, but then at all points of its path -it has a proportionally stronger force to pull it; if the ball has a -smaller distance to go, then at all the corresponding points of its -path it has a proportionally weaker force to pull it. Thus the time -remains the same whether you have the powerful horse for the long -journey or the weaker horse for the smaller journey. - -[Illustration: FIG. 33.] - -Next take a short, stiff spring of steel. One of the kind known as -tuning forks may be employed. - -The reader is probably aware that sounds are produced by very rapid -pulsations of the air. Any series of taps becomes a continuous sound -if it is only rapid enough. For example, if I tap a card at the rate -of 264 times in a second, I should get a continuous sound such as that -given by the middle C note of the piano. That, in fact, is the rate -at which the piano string is vibrating when C is struck, and that -vibration it is that gives the taps to the air by which the note is -produced. - -This can be very easily proved. For if you lift up the end of a bicycle -and cause the driving wheel to spin pretty rapidly by turning the pedal -with the hand, then the wheel will rotate perhaps about three times in -a second. If a visiting card be held so as to be flipped by the spokes -as they fly by, since there are about thirty-six of them, we should -get a series of taps at the rate of about 108 a second. This on trial -will be found to nearly correspond to the note A, the lowest space on -the bass clef of music. As the speed of rotation is lowered, the tone -of the note becomes lower; if the speed is made greater, the pitch of -the note becomes higher, and the note more shrill. However far or near -the card is held from the centre of the wheel makes no difference, for -the number of taps per second remains the same. So, again, if a bit of -watch-spring be rapidly drawn over a file, you hear a musical note. The -finer the file, and the more rapid the action, the higher the note. The -action of a tuning fork and of a vibrating string in producing a note -depends simply on the beating of the air. The hum of insects is also -similarly produced by the rapid flapping of their wings. - -It is an experimental fact that when a piano note is struck, as the -vibration gradually ceases the sound dies away, but the pitch of the -note remains unchanged. A tune played softly, so that the strings -vibrate but little, remains the same tune still, and with the same -pitch for the notes. - -A “siren” is an ingenious apparatus for producing a series of very -rapid puffs of air. It consists of a small wheel with oblique holes in -it, mounted so as to revolve in close proximity to a fixed wheel with -similar holes in it. If air be forced through the wheels, by reason -of the obliquity of the orifices in the movable wheel it is caused to -rotate. As it does so, the air is alternately interrupted and allowed -to pass, so that a series of very rapid puffs is produced. As the -air is forced in, the wheel turns faster and faster. The rapidity of -succession of the puffs increases so that the note produced by them -gradually increases in pitch till it rises to a sort of scream. For -steamers these “sirens” are worked by steam, and make a very loud noise. - -It is, however, impossible to make a tuning fork or a stretched piano -spring alter the pitch of its note without altering the elastic force -of the spring by altering its tension, or without putting weights on -the arms of the tuning fork to make it go more slowly. And this is -because the tuning fork and the piano spring, being elastic, obey -Hooke’s law, “As the deflection, so the force”; and therefore the time -of back spring is in each case invariable, and the pitch of the note -produced therefore remains invariable, whatever the amplitude of the -vibration may be. - -Upon this law depends the correct going of both clocks and watches. - -Wonderful nature, that causes the uniformity of sounds of a piano, or a -violin, to depend on the same laws that govern the uniform going of a -watch! Nay, more, all creation is vibrating. The surge of the sea upon -the coast that swishes in at regular intervals, the colours of light, -which consist of ripples made in an elastic ether, which springs back -with a restitutional force proportioned to its displacement, all depend -upon the same law. This grand law by which so many phenomena of nature -are governed has a very beautiful name, which I hope you will remember. -It is called “harmonic motion,” by which is meant that when the atoms -of nature vibrate they vibrate, like piano strings, according to the -laws of harmony. The ancient Pythagorean philosophers thought that all -nature moved to music, and that dying souls could begin to hear the -tones to which the stars moved in their orbits. They called it, as you -know, the music of the spheres. But could they have seen what science -has revealed to man’s patient efforts, they would have seen a vision -of harmony in which not a ray of light, not a string of a musical -instrument, not a pipe of an organ, not an undulation of all-pervading -electricity, not a wing of a fly, but vibrates according to the law -of harmony, the simple easy law of which a boy’s catapult is the type, -and which, as we have seen, teaches us that when an elastic body is -displaced the force of restitution, in other words, the force tending -to restore it to its old position, is proportional to the displacement, -and the time of vibration is uniform. The last is the important thing -for us; we seem to get a gleam of a notion of how the clock and watch -problem is going to be solved. - -But before we get to that we have yet to go back a little. - -About the year 1580 an inattentive youth (it was our friend Galileo -again) watched the swing of one of the great chandeliers in the -cathedral church at Pisa. The chandeliers have been renewed since his -day, it was one of the old lamps that he watched. It had been lit, and -allowed to swing through a considerable space. He expected that as it -gradually came to rest it would swing in a quicker and quicker time, -but it seemed to be uniform. This was curious. He wanted to measure the -time of its swing. For this purpose he counted his pulse-beats. So far -as he could judge, there were exactly the same number in each pendulum -swing. - -This greatly interested him, and at home he began to try some -experiments. As he got older his attention was repeatedly turned to -that subject, and he finally established in a satisfactory way the law -that, if a weight is hung to the end of a string and caused to vibrate, -it is isochronous, or equal-timed, no matter what the extent of the arc -of vibration. - -The first use of this that he made was to make a little machine with -a string of which you could vary the length, for use by doctors. For -the doctors of that day had no gold watch to pull out while with -solemn face they watched the ticks. They were delighted with the new -invention, and for years doctors used to take out the little string and -weight, and put one hand on the patient’s pulse while they adjusted the -string till the pendulum beat in unison with the pulse. By observing -the length of the string, they were then able to tell how many beats -the pulse made in a minute. But Galileo did not stop there. He -proceeded to examine the laws which govern the pendulum. - -We will follow these investigations, which will largely depend on what -we have already learned. - -Before, however, it is possible to understand the laws which govern -the pendulum, there are one or two simple matters connected with the -balance and operation of forces which have to be grasped. - -Suppose that we have a flat piece of wood of any shape like Fig. 34, -and that we put a screw through any spot _A_ in it, no matter where, -and screw it to a wall, so that it can turn round the screw as round a -pivot. - -[Illustration: FIG. 34.] - -[Illustration: FIG. 35.] - -Next we will knock a tintack into any point _B_, and tie a string on -to _B_. Then if I pull at the string in any direction _B C_ the board -tends to twist round the screw at _A_. What will the strength of the -twisting force be? It will depend on the strength of the pull, and -on the “leverage,” or distance of the line _C B_ from _A_. We might -imagine the string, instead of being attached at _B_, to be attached -at _D_; then, if I put _P_ as the strength of the pull, the twisting -power would be represented by _P_ × _A D_. This is called the “moment” -of the force _P_ round the centre _A_. It would be the same as if I -had simply an arm _A D_, and pulled upon it with the force _P_. It -is an experimental truth, known to the old Greek philosophers, that -moments, or twisting powers, are equal when in each case the result of -multiplying the arm by the power acting at right angles to it is equal. - -Now suppose _A B_ is a pendulum, with a bob _B_ of 10 lbs. weight, and -suppose it has been drawn aside out of the vertical so that the bob is -in the position _B_. Then the weight of the bob will act vertically -downwards along the line _B C_. The moment, or twisting power, of the -weight will be equal to 10 lbs. multiplied by _A D_, _A D_ being a -line perpendicular to _B C_. - -[Illustration: FIG. 36.] - -Now suppose that another string were tied to the bob _B_, and pulled -in a direction at right angles to _A B_, with a force _P_ just enough -to hold the bob back in the position _B_. The pull along _D B_ × _A B_ -would be the moment of that pull round the point _A_. But, because this -moment just holds the pendulum up, it follows that the moment of the -weight of the pendulum round _A_ is equal to the moment of the pull of -the string _B D_ round _A_. - - Whence P × A B = 10 lbs. × A D. - - Whence P = 10 lbs. × (A D)/(A B). - -But _A B_ is always the same, whatever the side deflection or -displacement of the pendulum may be. Whence then we see that when -a pendulum is pulled aside a distance _E B_ (which is always equal -to _A D_), then the force tending to bring it back to _E_ is always -proportional to _E B_. But if the pendulum be fairly long, say 39-1/7 -inches, and the displacement _E B_ be small,—in other words, if we do -not drag it much out of the vertical,—then we may say that the force -tending to bring it back to _F_, its position of rest, is not very -different from the force tending to bring it back to _E_. But _F B_ -is the “displacement” of the pendulum, and, therefore, we find that -when a pendulum is displaced, or deflected, or pulled aside a little, -the amount of the deflection is always very nearly proportional to the -force which was used to produce the deflection. This important law -can be verified by experiment. If _C_ is a small pulley, and _B C_ a -string attached to a pendulum _A B_ whose bob is _B_. Then if a weight -_D_ be tied to the string and passed over a pulley _C_, the amount _F -B_ by which the weight _D_ will deflect the bob _B_ is almost exactly -proportional to _D_, so long as we only make the deflection _E B_ -small, that is two or three inches, where say 39-1/7 inches is the -length _A B_ of the pendulum. - -If _F B_ is made too big, then the line _B F_ can no longer be -considered nearly equal to the arc of deflection _E B_, and the -proposition is no longer true. - -Hence then, both by experiment and on theory, we find that for small -distances the displacement of a pendulum bob is approximately equal to -the force by which that displacement is produced. - -But if so, then from what has gone before, we have an example of -harmonic motion. The weight of the bob, tending to pull the bob back to -_E_, acts just as an elastic band would act, that is to say pulls more -strongly in proportion as the distance _F B_ is bigger. In fact, if we -could remove the force of gravity still leaving the mass _B_ of the -pendulum bob, the force of an elastic band acting so as to tend to pull -the bob back to rest might be used to replace it. It would be all one -whether the bob were brought back to rest by the downward force of its -own gravity, or by the horizontal force of a properly arranged elastic -band of suitable length. - -[Illustration: FIG. 37.] - -But the motion of the bob, under the influence of the pull of -an elastic band where the strain was always proportional to the -displacement, would, as we have seen, be harmonic motion, and performed -in equal times whatever the extent of the swing. Whence then we -conclude that if the swings of a pendulum are not too big, say not -exceeding two and a half inches each way, the motion may be considered -harmonic motion, and the swings will be made in equal times whether -they are large or small ones. In other words, a clock with a 39-1/7 -inch pendulum and side swing on each side if not over two inches will -keep time, whatever the arc of swing may be. - -This may be verified experimentally. Take a pendulum of wood 39-1/7 -inches long, and affix to its end a bob of 10 lbs. weight. The pendulum -will swing once in each second. To pull it aside two inches we should -want a weight such that its moment about the point of support was equal -to the moment of the force of gravity acting on the bob, about the -point of support. In other words, the weight required × 39-1/7 inches = -10 lbs. × 2 inches. Whence the weight required = 1/2 lb. (nearly). - -Now fix a similar pendulum _A B_ 39-1/7 inches long, horizontally, -with a weight _B_ of 10 lbs. on it. Fasten it to a vertical shaft _C -D_, with a tie rod of wire or string _A B_ so as to keep it up, and -attach to each side of the rod _A B_ elastic threads _E F_ and _E G_. -Let these threads be tied on at such a point that when _B_ is pulled -aside two inches the force tending to bring it back to rest is half a -pound. Then if set vibrating the rod will swing backwards and forwards -in equal times, no matter how big, the arc of vibration (provided the -arc is kept small), and the time of oscillation will be that of a -pendulum, namely, one swing in a second. In fact, whether you put _A B_ -vertically and let it swing on the pivots _C_ and _D_ by the force of -gravity, or put it horizontally, and thus prevent gravity acting on it, -but make it swing under the accelerating influence of a pair of elastic -bands so arranged as to be equivalent to gravity, in each case it will -swing in seconds. - -[Illustration: FIG. 38.] - -It is this curious property of the circle that makes the vertical -force of gravity on a pendulum pull it as though it were a -horizontally acting elastic band; that is the reason why a pendulum is -equal-time-swinging, or, as it is called, isochronous, from two Greek -words that mean “the same” and “time.” - -But it must be remembered that this equal swinging is only approximate, -and only true when the arc of vibration is small. - -Here then we have a proof which shows us that the pendulum of a clock -and the balance wheel of a watch depend on exactly the same principles. -They are each an example of harmonic motion. - -The next question that arises is whether the weight of the pendulum has -any influence upon the time of its vibration. - -A little reflection will soon convince us that it has none. For we know -that the time that bodies take to fall to the ground under the action -of gravity is independent of the weight. A falling 2 lb. weight is only -equivalent to two pound-weights falling side by side. - -In the same way and by the same reasoning we might take two pendulums -of equal length, and each with a bob weighing 1 lb. They would, if put -side by side close together swing in equal times. But the time would be -the same if they were fastened together, and made into one pendulum. - -For inasmuch as the fall of a pendulum is due to gravity, and the -action of gravity upon a body is proportional to its mass, it follows -that in a pendulum the part of the gravitational force that acts upon -each part of the mass is occupied in moving that mass, and the whole -pendulum may be considered as a bundle of pendulums tied together and -vibrating together. - -The same would be the case with a pendulum vibrating under the -influence of a spring. If you have two bobs and two springs, they will -vibrate in the same time as one bob accelerated by one spring. In -this case, however, the force of the one spring must be equal to the -combined force of the two springs. In other words, the springs must be -made proportional in strength to the masses. - -Hence, then, you cannot increase the speed of the vibration of a -pendulum by adding weight to the bob. - -On the other hand, if you have a bob vibrating under the influence of a -spring, like the balance wheel of a watch, then if you increase the bob -without increasing the spring, since the mass to be moved has increased -without a corresponding increase in the accelerating force acting on -it, the time of swing will alter accordingly. - -But in the case of gravity, by altering the mass, you thereby -proportionally alter the attraction on it, and therefore the time of -swing is unaltered. - -[Illustration: FIG. 39.] - -The explanation which has been given above of the reasons why a -pendulum swings backwards and forwards in a given time independently -of the length of the arc through which it swings, that is to say of -the amount by which it sways from side to side, is only approximate, -because in the proof we assumed that the arc of swing and the line _F -B_ were equal, which is not really and exactly true. Galileo never got -at the real solution, though he tried hard. It was reserved for another -than he to find the true path of an isochronous pendulum and completely -to determine its laws. Huygens, a Dutch mathematician, found that the -true path in which a pendulum ought to swing if it is to be really -isochronous is a curve called a cycloid, that is to say the curve which -is traced out by a pencil fixed on the rim of a hoop when the hoop is -rolled along a straight ruler. It is the curve which a nail sticking -out of the rim of a waggon wheel would scratch upon a wall. I will -not go into the mathematical proof of this. Clocks are not made with -cycloidal pendulums, because when the arc of a pendulum is small the -swing is so very near a cycloid as to make no appreciable difference in -time-keeping. - -I am now glad to be able to say that I have dealt with all the -mathematics that is necessary to enable the mechanism of a clock to be -understood. It all leads up to this:— - -(1) A harmonic motion is one in which the accelerating force increases -with the distance of the body from some fixed point. - -(2) Bodies moving harmonically make their swings about this point in -equal times. - -(3) A spring of any sort or shape always has a restitutional force -proportional to the displacement. - -(4) And therefore masses attached to springs vibrate in equal times -however large the vibration may be. - -(5) The bob of a pendulum, oscillating backwards and forwards, acts -like a weight under the influence of a spring, and is therefore -isochronous. - -(6) The time of vibration of a pendulum is uninfluenced by changes in -the weight of the bob, but is influenced by changes in the length of -the pendulum rod. The time of vibration of a mass attached to a spring -is influenced by changes in the mass. - -We have now to deal with the application of these principles to clocks -and watches. - -Clocks had been known before the time of Galileo, and before the -invention of the pendulum. They had what is known as balance, or verge -escapements. Strictly in order of time I ought to explain them here. -But I will not do so. I will go on to describe the pendulum clock, and -then I will go back and explain the verge escapement, which, we shall -see, is really a sort of huge watch of a very imperfect character. - -As soon as Galileo had discovered that pendulums were isochronous, -that is, equi-time-swinging, he set to work to see whether he could -not contrive to make a timepiece by means of them. This would be easy -if only he could keep a pendulum swinging. When a pendulum is set -swinging, it soon comes to rest. What brings it to rest? The resistance -of the air and the friction of the pivots. Therefore what is obviously -wanted is something to give it a kick now and then, but the thing must -kick with discretion. If it kicked at the wrong time, it might actually -stop the pendulum instead of keeping it going. You want something that, -just as the pendulum is at one end and has begun to move, will give it -a further push. Suppose that I have a swing and that I put a boy in it, -and I swing him to and fro. I time my pushes. As he comes back against -my hand I let him push it back, and then just as the swing turns I give -it a further push. But I cannot stand doing that all day. I must make a -machine to do it. Now what sort of a machine? - -First, the machine must have a reservoir of force. I can’t get a -machine to do work unless I wind it up, nor a man to do work unless I -feed him, which is his way of being wound up. But then what do I want -him to do? I want him, when I give him a push, to push me back harder. -I want a reservoir of force such that when a pendulum comes back and -touches it, the touch, like the pressure of the trigger of a gun, shall -allow some pent-up power to escape and to drive the pendulum forward. - -This is the case in a swing. Each time that the swing returns to my -hands I give it a push, which serves to sustain the motion that would -otherwise be destroyed by friction and the resistance of the air. - -Such an arrangement, if it can be contrived mechanically, is called an -“escapement.” - -An arrangement of this kind was contrived by Galileo. He provided a -wheel, as is here shown, with a number of pins round it. The pendulum -_A B_ has an arm _A H_ attached to it, and there is a ratchet _C D_ -which engages with the pins. The ratchet has a projecting arm _E F_. - -[Illustration: FIG. 40.] - -When the pendulum comes back towards the end of its beat, the arm _A H_ -strikes the arm _E F_, and raises the ratchet _C D_. This releases the -wheel, which has a weight wound up upon it, and therefore at once tries -to go round. The consequence is, that the pin _G_ strikes upon the arm -_A H_, and thus on its return stroke gives an impetus to the pendulum. -As the pin _G_ moves forward it slides on the arm _A H_ till it slips -over the point _H_. The wheel now being free, would fly round were it -not that when the pendulum returned, and the arm _A H_ was lowered, -the ratchet had got into position again and its point _D_ was ready to -meet and stop the next pin that was coming on against it. At each blow -of the pins against the pendulum a “tick” is made, at each blow of a -pin against the ratchet a “tock” is sounded, so that as it moves the -pendulum makes the “tick-tock” sound with which we are all familiar. - -Hence then a clock consists of a wheel, or train of wheels, urged by -a weight or spring, which strives continually to spin round, but its -rotation is controlled by an escapement and pendulum, so contrived as -only to allow it to go a step forward at regular equal intervals of -time. - -But this would make only a poor sort of escapement. For the mode of -driving the pendulum adds a complication to the swing of the pendulum. -Instead of the pendulum being simply under the accelerative force of -gravity, it is also subjected to the acceleration of the pin _G_. This -acceleration is not of the “harmonic” order. Hence so far as it goes it -does not tend to assist in giving a harmonic motion to the pendulum, -but, on the contrary, disturbs that harmonic motion. Besides this, the -impulse of the pin is in practice not always uniform. For if the wheel -is at the end of a train of wheels driven by a weight, though the force -acting on it is constant, yet, as that force is transmitted through a -train of wheels, it is much affected by the friction of the oil. And -on colder days the oil becomes more coagulated, and offers greater -resistance. Moreover, as will be explained more in detail afterwards, -the fact that the impulse is administered by _G_ at the end of the -stroke of the pendulum is disadvantageous, as it interferes with the -free play of the pendulum. - -From all these causes the above escapement is imperfect in character, -and would not do where precision was required. - -[Illustration: FIG. 41.] - -It is now time to return to the old-fashioned escapements which were -in use before the time of Galileo. These consisted of a wheel called -a crown wheel, with triangular teeth. On one side of this wheel a -vertical axis was fitted, with projecting “pallets” _e f_. Across -the axis a verge or rod _e f_ was placed, fitted with a ball at each -end. When the crown wheel attempted to move on, one of its teeth came -in contact with a pallet. This urged the pallet forward, and thereby -caused an impulse to be given to the axis, on which was mounted the -verge, carrying the balls. These of course began to move under the -acceleration of the force thus impressed upon the pallet. Meantime, -however, the other pallet was moving in the opposite direction, and by -the time the first pallet had been pushed so far that it escaped or -slid past the tooth of the crown wheel, which was pressing upon it, the -other pallet had come into contact with the tooth on the other side -of the crown wheel. This tended to arrest the motion of the verge, to -bring the balls to a standstill, and ultimately to impart a motion in -a contrary direction to them. - -Thus then the arrangement was that of a pendulum not acted on by -gravity, for the balls neutralised one another. The pendulum was, -however, not subjected to a harmonic acceleration, but alternately -to a nearly uniform acceleration from _A_ to _B_ and _B_ to _A_. As -a result, therefore, the time of oscillation was not independent of -the arc of swing, but varied according to it, as also according to -the driving power of the crown wheel. At each stroke there was a -considerable “recoil.” For as each tooth of the wheel came into play it -was unable at first to overcome and drive back the pallet against which -it was pressing, but, on the contrary, was for a time itself driven -back by the pallet. - -[Illustration: FIG. 42.] - -Of course, so long as the motions of the wheel and verge were exactly -uniform, fair time was kept. But the least inequality of manufacture -produced differences. - -Nevertheless it was on this principle that clocks were made during the -thirteenth, fourteenth and fifteenth centuries. They were mostly made -for cathedrals and monasteries. One was put up at Westminster, erected -out of money paid as a fine upon one of the few English judges who have -been convicted of taking bribes. - -The time of swing of these clocks depended entirely upon the ratio of -the mass of the balls at the end of the verge as compared with the -strength of the driving force by which the acceleration on the pallets -was produced. They were very commonly driven by a spring instead of a -weight. The spring consisted of a long strip of rather poor quality -steel coiled up on a drum. As it unwound it became weaker, and thus the -acceleration on the verge became weaker, and the clock went slower. - -In order, therefore, to keep the time true, it became necessary to -devise some arrangement by which the driving force on the crown wheel -should be kept more constant. - -This gave rise to the invention of the fusee. The spring was put inside -a drum or cylindrical box. One end of the spring was fastened to an -axis, which was kept fixed while the clock was going; the other was -fastened to the inside of the drum. Round the drum a cord was wound, -which, as the drum was moved by the spring, tended to be wound up on -the surface of the drum. Owing to the unequal pull of the spring, this -cord was pulled by the drum strongly at first, and afterwards more -feebly. To compensate its action a conical wheel was provided, with a -spiral path cut in it in such a way and of such a size and proportion -that as the wheel was turned round by the pull of the drum the cord was -on different parts of it, so that the leverage or turning power on it -varied, becoming greater as the pull of the cord became weaker, and in -such a ratio that one just compensated the other, and the turning power -of the axle was kept uniform. - -In this manner small table clocks were made which kept very tolerable -time. - -[Illustration: FIG. 43.] - -Huygens converted these clocks into pendulum clocks in a very simple -manner. He removed one of the balls, lengthened the verge, and slightly -increased the weight of the other ball. By this means, while the crown -wheel still continued to drive the verge and remaining ball, the -acceleration on that ball now no longer depended entirely on the force -of the crown wheel. The acceleration and retardation were now almost -entirely governed by the force of gravity on the remaining ball, and -this acceleration was harmonic. - -The clock, therefore, was immensely improved as a time-keeper. Still, -however, the acceleration remained partly due to the driving power, and -this was partly non-harmonic and introduced errors. - -Most of the old clocks were converted shortly after the time of -Huygens. As there was in general no room for the pendulum inside the -clock-case, they usually brought the axle on which the pallets were -mounted outside the clock and made it vibrate in front of the face. - -Many old clocks exist, of which the engraving in the frontispiece is -an example, that have been thus converted. A true old verge escapement -clock is now a rarity. - -The type of escapement invented by Galileo never came into vogue for -clocks, on account of its imperfections, except till after a long -interval, when, with certain modifications, it became the basis of a -new improvement at the hands of Sir George Airey. - -The crown wheel fell into disuse and was replaced by the anchor -escapement, which was employed in that popular and excellent timepiece -used throughout the eighteenth and the early part of the nineteenth -century, and is now known as “The Grandfather’s Clock.” It was after -all the crown wheel in another shape. The wheel, however, was flattened -out, the teeth being put in the same plane. This made it much easier to -construct. The pallets were fixed on an axis, and were a little altered -so as to suit the changed arrangement of the teeth. The pendulum was -no longer hung on the axis which carried the pallets. A cause of a good -deal of friction and loss of power was thus removed. The pendulum was -hung from a strip of thin steel spring, which allowed it to oscillate, -and which supported it without friction. This excellent manner of -suspending pendulums is now universal. It enabled the pendulum to be -made very heavy. The bob was usually some eight or nine pounds weight. -By this means the acceleration on the pendulum was due almost entirely -to gravity acting on the bob, and thus the motion of the pendulum -became almost wholly harmonic. Whence it followed that variations in -the pendulum swing became of secondary importance, and did not greatly -alter the going of the clock. - -[Illustration: FIG. 44.] - -Therefore when the wheels became worn, and the pivots choked with old -oil and dust, the old clock still went on. If it showed a tendency to -stop for want of power, a little more was added to the driving weight, -and the clock kept as good time as ever. - -The swing of the pendulum was by this escapement enabled to be made -small, so that the arc of swing of the bob differed but little from a -cycloid. - -The secret of the time-keeping qualities of these old “Grandfather” -clocks is the length of pendulum. This renders it possible to have -but a small arc of oscillation, and therefore the motion is kept very -nearly harmonic. For practical purposes nothing will even now beat -these old clocks, of which one should be in every house. At present -the tendency is to abolish them and to substitute American clocks with -very short pendulums, which never can keep good time. They are made -of stamped metal. When they get out of order no one thinks of having -them mended. They are thrown into the ash-pit and a new one bought. In -reality this is not economy. - -Good “Grandfather” clocks are not now often made. The last place I -remember to have seen them being manufactured is at Morez, in the -district of the Jura. An excellent clock, enclosed in a dust-tight iron -case, with a tall painted case of quaint old design, can be bought for -about 55_s._ The wheels are well cut, and the internal mechanism very -good. - -I visited the town of Morez in the year 1893. The clock industry was -declining. The farmers of France seemed to prefer small clocks of -hideous appearance, made in Germany and in America, to the excellent -work of their own country. Probably by now the old clockmaking industry -is extinct. One I purchased at that time has gone well ever since. - - - - -CHAPTER IV. - - -It is now time to give a description of the various parts of an -ordinary pendulum clock. We will take the “Grandfather” clock as an -example. We shall want an hour hand and a minute hand in the centre of -the face, and a seconds hand to show seconds a little above them. There -will be a seconds pendulum 39·14 inches long, and the centre of the -face of the clock will be about seven feet above the ground, so as to -give practically about five feet of fall for the weight. - -[Illustration: FIG. 45.] - -In the first place, we have to consider the axle which carries the -minute hand, and which turns round once in each hour. This is usually -made of a piece of steel about one-sixth of an inch in diameter. -Clockmakers usually call an axle an “arbor,” or “tree,” whence our word -axletree. - -This “arbor” is turned in the lathe, so as to have pivots on each end, -fitted into holes in the clock plates, that is to say, the flat pieces -of brass that serve as the body of the clock. The adjoining diagram -shows _S T_ the clock faces, and _C_, the arbor of the minute hand. - -Inasmuch as the seconds hand is to turn round sixty times while the -minute hand turns round once, it is obvious that the arbor of the -minute hand must be connected to the arbor of the seconds hand by a -train of cogwheels so arranged as to multiply by sixty. This of course -involves us in having large and small cogwheels. - -[Illustration: FIG. 46.] - -The small cogwheels usually have eight teeth, and are for convenience -of manufacture, as also to stand prolonged wear, cut out of the solid -steel of the arbor. They are nicely polished. - -The easiest pair of wheels to use will be two pinions of eight teeth, -or “leaves,” as they are called, and two cogwheels, one of sixty-four -teeth, the other of sixty teeth. - -It is then clear that if the arbor _A_ turns round once in an hour, -the arbor _B_ will turn round eight times in an hour, and _C_ will turn -round (60 × 64)/(8 × 8) = 60 times in an hour, or once in each minute. - -By having 480 teeth on the cogwheel on _A_, you could, of course, make -_C_ go round once in a minute without the use of any intermediate arbor -such as _B_. - -[Illustration: FIG. 47.] - -But this would not be a very convenient plan. For as the wheel on _A_ -is usually about two and a quarter inches in diameter, to cut 480 teeth -on so small a wheel would involve us in cutting about sixty teeth to -the inch. The teeth would thus be microscopically small, and would -have to be set so fine that the least dirt would clog them. Moreover, -the pinion of eight leaves would have to be microscopic. For these -reasons, therefore, it is usual in clocks not to use wheels with teeth -more than sixty or sixty-four in number, and to diminish the motion -gradually by means, where needful, of intermediate arbors. We have next -to consider how the weight is to be arranged so as to turn the arbor -_A_ once round in an hour. We know that we have five feet of space for -the weights to fall in. If we arrange to have what is called a double -fall, as shown in the sketch, then, allowing room for pulley wheels, we -shall find that our string may be practically about nine feet in length. - -[Illustration: FIG. 48.] - -The clock will be wanted to go for a week without winding, and as -people may forget to wind it at the proper hour of the day, we will -give it a day extra, and make an “eight-day” clock of it. Hence then, -while nine feet of cord is being pulled out by a weight which falls -four and a half feet, the minute hand is to be turned round as many -times as there are hours in eight days, viz., 192 times. This could be -accomplished, of course, by winding the cord round the arbor of the -minute hand. But this would require 192 turns. If our cord is to be -ordinary whipcord, or catgut, say one-twelfth of an inch in diameter, -in order that the cord could be wound upon it, the arbor would have -to be 192/12 inches long = 14⅓ inches long. This would make the clock -case unnecessarily deep. We must therefore again have recourse to an -intermediate wheel. - -[Illustration: FIG. 49.] - -If we put a pinion of eight leaves on the minute hand arbor _c_, and -engage it with a wheel of sixty-four teeth on another arbor _b_, then -_b_ will obviously turn round once in eight hours, that is to say, -twenty-four times in the period of eight days. And, if we fix on _b_ -a “drum” or cylinder two inches long, the twenty-four turns of our -cord will just fit upon it, since, as has been said, our cord is to be -one-twelfth of an inch in diameter. The diameter of the drum must be -such that a cord nine feet long can be wound twenty-four times round -it. That is to say, each lap must take (9 × 12)/24 = 4½ inches of cord. -From this it is easy to calculate that the diameter of the drum must be -rather less than one and a half inches. From this then it results that -we want for a “Grandfather’s” clock a drum two inches long and one and -a half inches diameter, on this a cogwheel of sixty-four teeth working -into a minute hand arbor, with a pinion wheel with eight leaves, and -a cogwheel of sixty-four teeth, an intermediate or idle wheel with -an eight-leaved pinion, and a cogwheel of sixty teeth, engaging with -a seconds hand arbor with a pinion of eight leaves. This is called -the “train of wheels.” With it a weight such as can be arranged in an -ordinary “Grandfather’s” clock case will cause by its fall during eight -days the second hand arbor to turn round once in each minute during the -whole time, and the minute hand arbor to turn round once in each hour. - -[Illustration: FIG. 50.] - -We must next provide an arrangement for winding the clock up. It -is obvious that we cannot do so by twisting the hands back. It is -true that this could be done, but it would take about five minutes -to do each time and be wearisome. In order to save this trouble, an -arrangement called a ratchet wheel and pall must be provided. A ratchet -wheel consists of a wheel with a series of notches cut in it, as -shown in the figure _A_. A pall is a piece of metal, mounted on a pin, -and kept pressed up against the ratchet wheel by a spring _C_. It is -obvious that if I turn the wheel _A_ round, and thus wind up a weight, -fastened to a cord wound round the drum _D_, that the pall _B_ will go -click-click-click as the ratchet wheel goes round, but that the pall -will hold it from slipping back again. When, however, I take my hands -away, and let the ratchet wheel alone, then the weight _E_ will pull on -the drum _D_, and try and turn the ratchet wheel back the opposite way -to that in which I twisted it at first. If the pall _B_ is held fast, -it is impossible to move it, but if the pall is fixed to a cogwheel -_F_, which rides loose on the arbor of the drum _D_, then the pull of -the weight _E_ will tend to twist the cogwheel _F_ round, and this, if -engaged with a pinion wheel on the minute hand arbor, will therefore -drive the clock. As the clock arbors move, of course the weight _E_ -gradually runs down, and, at last all the string is unwound from the -drum _D_. The clock is said then to have “run down,” but if I take a -clock key, and by means of it wind the string up upon the drum _D_, -then the pall lets the drum and ratchet slip; the clock hands are not -affected. When I have given twenty-four turns to the arbor, the nine -feet of cord will then be wound upon the drum again, and the clock will -be ready to go for eight more days, and will begin to move as soon as I -cease to press upon the clock key. - -[Illustration: FIG. 51.] - -I have thus described the winding mechanism. It now remains to describe -the escapement. - -It is of course obvious that, if the weight and train of wheels were -simply let go, the weight would rush down, and the seconds-hand -wheel would fly round at a tremendous pace; but we want it to be so -restrained as only to be allowed to go one-sixtieth part of its -journey round in each second. In fact, we need an “escapement” and a -pendulum. - -The escapement usually employed in “Grandfather” clocks is the anchor -escapement above described. It is not by any means the best sort of -escapement, but it is the easiest to make; and hence its popularity in -the days sometimes called the “dear, good old days,” when people had to -file everything out by hand, and had to take a day to do badly what can -now be done well in five minutes. - -The escape wheel of an anchor escapement has thirty sharp angular teeth -on its rim. The wheel is made as light as possible, so that the shock -of stoppage at each tick of the clock may be as slight as possible, for -a heavy blow of course wastes power and gradually wears out the clock. -The anchor consists of two arms of the shape shown in the illustration -(Fig. 44). As the escape wheel goes round in the direction of the -arrow, the anchor, mounted on its arbor, rocks to and fro. The wheel -cannot run away, because the act of pushing one arm or “pallet,” as it -is called, outwards, and thus freeing the tooth pulls the other pallet -in, and this stops the motion of the tooth opposite to it, but when the -anchor rocks back again, so as to disengage the pallet from the tooth -that holds it, then the opposite tooth is free to fly forward against -the other pallet. This tends to rock the anchor the other way, but -at that instant the pallet just engages the next tooth of the wheel, -and so the action goes on. The anchor rocks from side to side; the -pallets alternately engage the teeth of the wheel, making at each rock -of the anchor the tick-tock sound with which we are so familiar. If -the anchor were free to rock at any speed it could, the ticking of the -clock would be very quick; so, to restrain the vivacity of the anchor, -we have a pendulum. The pendulum might be simply hung on to the anchor. -But the disadvantage of doing this would be that the heavy bob of the -pendulum would cause such a pressure on the arbor of the anchor that -there would be great friction, and the arbor would soon be worn out, -and the accurate going of the clock disturbed. The pendulum therefore -is hung on a piece of steel spring on a separate hook, which lets it -go backwards and forwards and carries the weight easily, while a rod -projecting from the anchor has a pin, which works in a slot on the -pendulum. The pendulum is therefore able to control and regulate the -movements of the escapement, and thus the time of the clock. - -Of course it is clear that the heavier the driving weight put on the -drum of the clock, and the better the cut and finish of the wheels, and -the greater the cleanliness and oil, the more will be the pressure -tending to drive round the escape wheel, and the harder the pressure -on the pallets, and hence the bigger the impulses on the pendulum, and -therefore the larger the amplitude of its swing. - -If the amplitude of the pendulum’s swing affected the time of its -swing, then the time kept by the clock would vary with the weight, and -the dirt and friction, and the drying up of the oil. But here precisely -is where the value of the beautiful law governing the harmonic motion -of the pendulum comes in. The time of the pendulum is (for small arcs) -independent of the length of swing, and therefore of the driving force -of the clock, and hence within limits the clock, even though roughly -made and foul with the dirt of years, continues to keep good time. -But the anchor escapement has imperfections. The only way in which -a pendulum can be relied on to keep accurate time is by leaving it -unimpeded. But the pressure of the teeth on the pallets in an anchor -escapement constantly interferes with this. - -[Illustration: FIG. 52.] - -A little consideration will easily show that there are some times -during the swing of a pendulum at which interference is far more -fatal to its time-keeping than at others. Thus the bob of a pendulum -may be regarded as a weight shot outwards from its position of rest -against the influence of a retarding force varying as its distance -from rest—in fact, shot out against a spring. The time of going out -and coming in again will be quite independent of the force exerted to -throw it out, quite independent of its original velocity. Therefore -a variation in the impulse given to the bob is of no consequence, -provided that impulse is given when the bob is near the position of -rest. This follows from the nature of the motion. If a ball be attached -to a piece of elastic thread, and thrown from the hand, so that it -flies out, and then stops and is brought back by the elastic force of -the thread, the time of the outward motion and the return is the same -whatever be the force of the throw. And so if a pendulum be impelled -outwards from a position of rest, the time of the swing out and back -is the same, however big (within limits) is the impelling force and -the consequent length of the swing. The use of a pendulum as a measure -of time is to impel it outwards, and then let it fly _freely_ out and -back. But if its motion is not free, if forces other than gravity act -upon it while on its path, then its time of swing will be disturbed. -It does not matter with what force you originally impel it, but what -does matter is, that when it once starts it should be allowed to travel -unimpeded and uninfluenced. Now that is what an anchor escapement does -not do. The impulse is given the whole way out on one of the pallets, -and then when it is at its extreme of swing, and ought to be left -tranquil, the other pallet fastens on it. But a perfect escapement -ought to give its impulse at the middle point of the swing, when the -pendulum is at the lowest, and then cease, and allow the pendulum to -adapt its swing to the impulse it has received, and thus therefore to -keep its time constant. This is done by an escapement called the dead -beat escapement, which, though in an imperfect way, realises these -conditions. - -The alteration is made in the shape of the pallets of the anchor. The -wheel is much the same. Each pallet consists of two faces: a driving -face _a b_ and a sliding face _b c_. - -[Illustration] - -When the tooth _b_ has done its work by pressing on the driving face, -and thus driving the anchor over, say, to the left, then the tooth on -the opposite side falls on the sliding face of the other pallet. This -being an arc of a circle, has no effect in driving the anchor one way -or the other; hence the pendulum is free to swing to the left as far as -it likes and return when it feels inclined, always with the exception -of a little friction of the tooth on the faces of the pallets, but -when it returns and begins to move towards the right, the tooth slides -back along the face of the pallet till the pendulum is almost at the -middle of its swing; then an impulse is given by the pressure of the -tooth upon the inclined plane _a´ b´_. As soon, however, as the tooth -leaves _b´_, another tooth on the other side at once engages the -sliding face _b c_ of the other pallet, and so the motion goes on. - -This beautiful escapement is at present used for astronomical clocks; -the pallets are made of agate or sapphire, and therefore do not grind -away the teeth of the wheel perceptibly, and the loss by friction on -the sliding surfaces is exceedingly small. - -There are several other ways even better than this for securing a free -pendulum movement. We have now to return to our clock. - -The centre arbor moves round once in an hour, and carries the minute -hand. In order to provide an hour hand, which shall turn round once in -twelve hours, we fasten a cogwheel and tube _N_ on to the minute hand -arbor by means of a small spring, which keeps it rather tight, but -allows it to slip if turned round hard (see Fig. 45). This spring is -a little bent plate slipped in behind the cogwheel on which its ends -rest; its centre presses on a shoulder on the minute hand arbor; it is -a sort of small carriage spring. The cogwheel _n_ has thirty teeth. -This cogwheel engages another cogwheel _o_ with thirty teeth, on a -separate arbor, which carries a third cogwheel, _p_, with six teeth, -and this again engages a fourth cogwheel, _q_, with seventy-two teeth, -mounted on a tube which slips over the tube to which the cogwheel _a_ -is attached. It is now easy to see that for each turn of the minute -hand arbor the arbor _p_ makes one turn, and for each turn of the -arbor _p_ the cogwheel _d_, makes one-twelfth of a turn. From which -it follows that for each turn of the minute hand arbor the cogwheel -_d_ with its tube, or, as it is sometimes called, its “slieve,” makes -one-twelfth of a turn, and thus makes a hand fastened to it show one -hour for every complete turn of the minute hand. - -The minute hand is attached to the tube or slieve which carries the -cogwheel _N_. The hour hand is attached to the tube or slieve which -carries the cogwheel _Q_, and one goes twelve times as slowly as the -other. - -But if you want to set the clock it is easy to do so by reason of the -fact that the minute hand is not fixed to the arbor, but only to the -slieve on the cogwheel that fits on the arbor, and is held somewhat -tight to the arbor by means of the spring. The hands can thus be -turned, but they are a little stiff. A washer on the minute hand arbor -keeps the slieve on the cogwheel pressed tight against the spring, -being secured in its turn by a very small lynch-pin driven through a -hole in the minute hand arbor. - -It remains to explain a few subsidiary arrangements, not always found -upon all clocks, but which are useful. - -In order to prevent the overwinding of the clock (see Fig. 43), which -would cause the cord to overrun the drum, an arm is provided, fitted -with a spring. As the weight is wound up the free part of the cord -travels along the drum or the fusee; and the cord, when it is near the -end of the winding, comes up against the arm and pushes it a little -aside. This causes the end of the arm to be pushed against a stop on -the axis of the fusee, and thus prevents the clock being further wound -up. The stop, being ratchet-shaped, does not prevent the weight from -pulling the ratchet wheel round the other way, and thus driving the -clock; it only prevents the rotation of that wheel when the string is -near it, and the winding is finished. - -Another arrangement is the “maintaining spring.” - -It will be remembered that during the process of winding the clock the -hand twisting the key takes the pressure of the ratchet wheel off the -pall, so that during that operation no force is at work to drive the -clock. In consequence the pendulum receives no impulse, but swings -simply by virtue of its former motion. If the process of winding were -done slowly enough the clock might even stop. To avoid this, a very -ingenious arrangement is made to keep the cogwheel mounted on the -winding shaft going during the winding-up process. This is called a -maintaining spring. - -The arrangement shown in Fig. 53 will explain it. - -[Illustration: FIG. 53.] - -The cogwheel _a_ and the ratchet wheel are both mounted loosely on the -arbor carrying the drum. _a_ is linked to _b_ by a spring _c_. The -ratchet wheel _b_ is engaged by a pall fixed to some convenient place -on the body of the clock frame. When the weight pulls on the drum the -pull is communicated to the ratchet wheel _b_, and this acts on the -spring _c_ and pulls it out a little. As soon as the spring _c_ is -pulled out as far as its elasticity permits, a pull is communicated -to the cogwheel _a_, and the clock is driven round. When the clock is -wound the pressure of the weight is removed, and therefore the ratchet -wheel _e_ no longer presses on the pall, and thus no pressure is -communicated to the ratchet wheel _b_, or through it to the clock. But -here the spring _c_ comes into play. For since the ratchet wheel _b_ is -held fast by the pall _d_, the spring _c_ pulls at the wheel _a_, and -thus for a minute or so will continue to drive the clock. This driving -force, it is true, is less than that caused by the weight, but it is -just enough to keep the pendulum going for a short time, so that the -going of the clock is not interfered with. - -If the reader can get possession of a clock, preferably one that does -not strike, and, with the aid of a small pair of pincers and one or two -screwdrivers, will take it to pieces and put it together again, the -mechanism above described will soon become familiar to him. Not every -clock is provided with maintaining spring and overwinding preventer. - -The cause of stoppage of a clock generally is dirt. Where possible, -clocks should always be put under glass cases. “Grandfather” clocks -will go much better if brown paper covers are fitted over the works -under the cases. In this way a quantity of dust may be avoided. To get -a good oil is very important. It will be noticed that pivot-holes in -clocks are usually provided with little cup-like depressions. This is -to aid in keeping in the oil. The best clock oil is that which does not -easily solidify or evaporate. Ordinary machine oil, such as used for -sewing machines, is good as a lubricant, but rapidly evaporates. Olive -oil corrodes the brass. - -It is best to procure a little clock oil, or else the oil used for gun -locks, sold by the gunsmiths. The holes should be cleaned out with the -end of a wooden lucifer match, cut to a tapering point. The pivots -should be well rubbed with a rag dipped in spirits of wine. If the -pivots are worn they should be repolished in the lathe. If the cogs of -the wheels are worn, there is no remedy but to get new ones. Old clocks -sometimes want a little addition to the driving weight to make them go. - -The weight necessary to drive the clock depends on its goodness of -construction, and on the weight of the pendulum. If the clock is driven -for eight days with a cord of nine feet in length with a double fall, -then during each beat of the pendulum that weight will descend by an -amount = - - 9/(2 × 24 × 60 × 60 × 8) feet or 1/12800th inch. - -Whence, if the clock weight is 10 lbs., the impulse received by the -clock at each beat is equivalent to a weight of 10 lbs. falling through -1/12800th of an inch, or to the fall of six grains through an inch. - -The power thus expended goes in friction of the wheels and hands, and -in maintaining the pendulum in spite of the friction of the air. - -The work therefore that is put into the clock by the operation of -winding is gradually expended during the week in movement against -friction. The work is indestructible. The friction of the parts of the -clock develops heat, which is dissipated over the room and gradually -absorbed in nature. But this heat is only another form of work. Amounts -of work are estimated in pressures acting through distances. Thus, if -I draw up a weight of 1 lb. against the accelerative force of gravity -through a distance of one foot, I am said to do a foot-pound of work. - -One pound of coal consumed in a perfect engine would do eight millions -foot-pounds of work. Hence, if the energy in a pound of coal could be -utilized, it would keep about 100,000 grandfather’s clocks going for a -week. As it is consumed in an ordinary steam engine it will do about -half a million foot-pounds of work. One pound of bread contains about -three million foot-pounds of energy. A man can eat about three pounds -of bread in a day, and, as he is a very good engine, he can turn this -into about three-quarters of a million foot-pounds of work. The rest of -the work contained in the bread goes off in the form of heat. - -[Illustration: FIG. 54.] - -As has been previously said, the power of the action of gravity in -drawing back a pendulum that has been pushed aside from its position of -rest becomes less in proportion as the pendulum is longer, and hence -as the pendulum is longer the time of vibrations increases. In the -appendix to this chapter a short proof will be given showing that the -length of a pendulum varies as the square of the time of its vibration. -A pendulum which is 39·14 inches in length vibrates at London once in -each second. Of course at other parts of the earth, where the force of -gravity is slightly different, the time of vibration will be different, -but, since the earth is nearly a globe in shape, the force of gravity -at different parts of it does not vary much, and therefore the time of -vibration of the same pendulum in different parts of the earth does not -vary very much. The length of a pendulum is measured from its point of -suspension down to a point in the bob or weight. At first sight one -would be inclined to think that the centre of gravity of the pendulum -would be the point to which you must measure in order to get its -length. So that if _B_ were a circular bob, and the rod of the pendulum -were very light, the distance _A B_ to the centre of the bob would be -the length of the pendulum. But if we were to fly to this conclusion, -we should be making the same error that Galileo made when he allowed a -ball to _roll_ down an inclined plane. He forgot that the motion was -not a simple one of a body down a plane, but was also a rolling motion. -The pendulum does not vibrate so as always to keep the bob immovable -with the top side _C_ always uppermost. On the contrary, at each beat -the bob rotates on its centre and makes, as it were, some swings of -its own. Therefore in the total motions of the pendulum this rotation -of the bob has to be taken into account. Of course, if the pendulum -were so arranged that the bob did not rotate, and the point _C_ were -always uppermost, as, for instance, if the pendulum consisted of two -parallel rods, _A B_ and _C D_, suspended from _A_ and _C_, then we -might consider the bob as that of a pendulum suspended from _E_, and -the pendulum would swing once in a second if _A B_ = _C D_ = _E F_ -were equal to 39·14 inches, for by this arrangement there would be no -rotation of the bob. But as pendulums are generally made with the bob -rigidly fixed to the rod _E F_, the rotation must be taken into account. - -[Illustration: FIG. 55.] - -It wants some rather advanced mathematical knowledge to do this. But -in practice clockmakers take no account of it. The correction is not a -large one, so they make the rod as nearly true as they can, arrange a -screw on the bob to allow of adjustment, and then screw the bob up and -down until in practice the time of oscillation is found to be correct. - -[Illustration: FIG. 56.] - -The mode of suspension of a pendulum of the best class is that shown -in Fig. 56, which allows the pendulum to fall into its true position -without strain. _A_ is a tempered steel spring, which bends to and -fro at each oscillation. It is wonderful how long these springs can -be bent to and fro without breaking. Inasmuch as lengthening the -pendulum increases the time, so that the time of vibration _t_ varies -as the square of the length of the pendulum, a very small lengthening -of the pendulum causes a difference in the time. In practice, for -each thousandth of an inch that we lengthen the pendulum we make a -difference of about one second a day in the going of the clock. If we -cut a screw with eighteen threads to the inch on the bottom of the -pendulum rod, and put a circular nut on it, with the rim divided into -sixty parts, then each turn through one division will raise or lower -the bob by 1/1080th of an inch, and this first causes an alteration -of time of the clock by one second in the day. This is a convenient -arrangement in practice, for it affords an easy means of adjusting the -pendulum. We need only observe how many seconds the clock loses or -gains in the day, and then turn the nut through a corresponding number -of divisions in order to rectify the pendulum. - -[Illustration: FIG. 57.] - -Another needful correction of the pendulum is that due to changes in -temperature. If the rod of the pendulum be made of thoroughly dried -mahogany, soaked in a weak solution of shellac in spirits of wine, -and then dried, there will not be much variation either from heat or -moisture. But for clocks required to have great precision the pendulum -rod is usually made of metal. A rod of iron expands about 1/160000th -of its length for each degree Fahrenheit; and therefore for each degree -Fahrenheit a pendulum rod of 39·14 inches will expand about 1/4000 -thousandths of an inch, and thus make a difference in the going of the -clock of about one-fourth of a second per day. The expansion will, of -course, make the clock go slower. It would be possible to correct this -expansion if some arrangement could be made, whenever it occurred, to -lift up the bob of the pendulum by an amount corresponding to it, as, -for instance, to make the bob of some material which expanded very much -more by heat than the material of which the pendulum rod was made. - -[Illustration: FIG. 58.] - -Thus if we hang on to the end of a pendulum of iron a bottle of iron -about seven inches long, and almost fill it with mercury, then, as -soon as the heat increases, the iron of the rod and of the bottle -expands, and the centre of oscillation of the pendulum is lowered. -But as the linear expansion of mercury contained in a bottle is about -five times that of iron, the mercury rises in the bottle, and thus the -expansion downwards of the pendulum rod is compensated by the expansion -upwards of the mercury in the bottle. The rod may be fastened to the -mouth of the bottle by a screw, so that as the bottle is turned round -it may be raised or lowered on the rod, and thus the length of the -pendulum may be adjusted. The bottle is made of steel tube, screwed -into a thin turned iron top and bottom. Of course no solder must be -used to unite the iron, for mercury dissolves solder. A little oil and -white-lead will make the screwed joints tight. This is an excellent -form of pendulum. Another plan is to use zinc as the metal which is to -counteract the expansion of the iron. The expansion of zinc is about -three times that of iron. - -[Illustration: FIG. 59.] - -Hence a zinc tube, about twenty inches long (shown shaded in Fig. 59), -is made to rest upon a disc fastened to the lower part of the iron -pendulum rod. On the top of the zinc rests a flat ring _A_, from which -is suspended an iron tube _A_, which carries the bob _B_. The expansion -of the zinc tube is large enough to compensate the expansion both of -the rod and the tube, and the bob consequently remains at the same -depth below the point of suspension, whatever be the temperature. - -There is, however, a new method which is far superior to all these, and -this is due to the discovery by M. Guilliaume, of Paris, of a compound -of nickel and steel which expands so little that it can be compensated -by a bob of lead instead of by a bob of mercury. This material is sold -in England under the name of “invar.” An invar rod with a properly -proportioned lead bob makes an almost perfect pendulum, the expansion -of the invar and the lead going on together. The exact expansion of the -invar is given by the makers, who also supply information as to the -size and suspension of the bob proper to use with it. - -It has been already shown that the uniformity of time of swing of a -pendulum is only true when the arc through which it swings is very -small. If the total swing from one side to another is not more than -about two inches very little difference in time-keeping is made by -putting a little more driving weight on the clock, and thus increasing -its arc of swing; but when the arc of swing becomes say three inches, -or one and a half inches on each side of the pendulum, then the time of -vibration is affected. At this distance each tenth of an inch increase -of swing makes the pendulum go slower by about a second a day. - -The resistance of the air, of course, has a great influence on a -pendulum, and is one of the chief causes that bring it ultimately to -rest. Even the variations of pressure of the atmosphere which the -barometer shows as the weather varies have an effect on the going of -a clock. Attempts have been made by fixing barometers on to pendulums -with an ingenious system of counter balancing to counteract this, but -these refinements are not in common use, and are too complicated to be -susceptible of effective regulation. - - -APPENDIX TO CHAPTER IV. - -It may be useful to give a simple form of proof of the law which -governs the time of oscillation of a pendulum whose length is given. - -Unfortunately, it is impossible to give one so simple as to be -comprehended by those who know nothing whatever of mathematics. It -is, however, possible to give a proof that requires very little -mathematical knowledge. - -We know that when a mass of matter is whirled round at the end of a -string it tends to fly outwards and puts a strain on the string. The -faster the speed at which the mass is whirled, the stronger will be the -strain on the string. Suppose that the length of the string equals R, -the velocity of the mass as it flies round equals V. Let _a_ be the -body whirled round by a string _o a_ from a centre at _O_. The body -always, of course, tends to fly on in a straight line from the point at -which it is at any instant. But that tendency is frustrated by the pull -of the string which constrains it to take a circular path. It is, of -course, all one whether the force that tends to pull the body inwards -towards _O_ is a string or an attractive force of any kind acting -through a distance without any string at all. Evidently if the body -keeps its place in the circle it must be because the centrifugal force -tending to whirl it out is equal to the centripetal or attractive force -tending to pull it in. - -[Illustration: FIG. 60.] - -The strain on the body, due to the force tending to pull it inwards, we -shall designate by F, meaning by F the number of feet of velocity that -would in one second be imparted to the body by the attractive force. - -Suppose that at some given instant of time the body is at a point _a_. -At that instant its _direction_ will be along _a b_, tangential to the -circle at _a_, and that is the path it would take if the centripetal -or attractive force ceased to act just as the body got to _a_. In that -case the body would be whirled off like a stone from a sling along -the line _a b_, and would at the end of a given time, let us suppose a -second, arrive at _b_. But it is not so whirled off; it is attracted -towards _O_ and pulled inwards, and comes to _c_. Hence, then, the -attractive force acting during one second must have been sufficient to -pull the mass in from _b_ to _c_. But we know that if an accelerating -force (F) acts on a body for a second it produces a final velocity -equal to F at the end of the second, and an average velocity half F -during the second. - -Hence, then, the space _b c_, by which the body has been pulled in, -is represented by half F, but _a b_, the space which the body would -have travelled forwards, will be represented by V, the velocity of the -body in a second; but if the motion be such that the distance _b c_ -travelled in a second is very small, then the triangles _a b d_ and -_a b c_ are approximately similar, and the smaller _a b_ is the more -nearly similar they are. Whence then (a b)/(b c) = (a d)/(a b), that is -to say (a b)² = a d × b c. - -But _a b_ represents the space which would have been traversed by the -body in one second at the rate it was going, and hence is equal to V; -_a d_ is the diameter of the circle, and hence equals 2 R; _b c_ is -the space through which the body has been drawn in the second by the -attractive force F, and therefore equals half F. - -Whence then V² = 2 R × half F = R F. - -We took a second as the limit of time during which the motion was to be -considered. Of course any other time could have been taken. Now what -is true of the motion of a body during a very short time is also true -of the body during the whole of its path, assuming that the path is a -circle, and that F remains constant, as it obviously will if the path -is a circle, and the velocity is uniform. Whence then we may generally -say that if a body is being whirled round at the end of a string the -strain F on the string is directly proportional to the square of the -velocity, and is inversely proportional to the length of the string. - -The time of rotation, is of course = length of the path ÷ velocity - - = (2πR)/V = (2πR)/√(R F) = 2π√(R/F). - -Whence then we see that for motion in a circle of a mass under the -attraction of a centripetal force, or pull of a string, the time of -rotation will be uniform, provided that the centripetal force always -varies as the radius of the path. From this it is evident that a body -fixed on to an elastic thread where the pull varies as the extension -would make its rotations always in equal times. If your sling consists -of elastic, whirl as you will, you can only whirl the body round so -many times in a second, and no more. Any increase in your efforts only -makes the string stretch, and the circle get bigger. The velocity of -the body in its path of course increases, but the time it takes to go -once round is invariable. - -It also follows that if a body hung by a string of length _l_, under -the action of gravity, be travelling in a circle round and round, then, -_if the circle is a small one compared with the length of the string_, -the inward acceleration _f_ towards the centre will be approximately -proportional to the radius _r_ of the circle, and the time of rotation -will be - - t = 2π√(r/f). - -But in this case _f_, the inward acceleration, is to _g_ the -acceleration downwards of gravity as A B:A P or - - f/g = (A B)/(A P) = (A P)/(O P) = r/l. - -[Illustration: FIG. 61.] - -[Illustration: FIG. 62.] - -Whence then the time of rotation of this body would be if the circle of -rotation was small - - = 2π√(l/g). - -And if you try you will find that this is so. For instance, take a -thread 39-1/7 inches long, that is 3·25 feet. Hang anything heavy from -one end of it, and cause it to swing round and round in a _small_ -circle. Now _g_ the acceleration of gravity = 32·2 feet per second. -π the ratio of the circumference of a circle to its diameter = 3·14. -From which it follows that the time of rotation = 2 × 3·14√(3·25/32·2) -seconds = 2 seconds. But if we look at the rotating body sideways, it -appears to act as a pendulum; it matters nothing whether we swing it -round and round or to and fro. For in any case the accelerative force -tending to bring it back to a position of rest is always proportional -to the distance of displacement, and, therefore, its time of motion -must always be 2π√(l/g) and its motion harmonic. - -The length of a seconds pendulum, that is a pendulum that makes its -double swing in two seconds, will therefore be - - l = 4/((2π)²) × g feet - - = (g × 12)/π² inches - - = 39·14 inches. - - - - -CHAPTER V. - - -I have thus described the principal features of ordinary clocks. For -the details many treatises must be studied, and knowledge acquired -which is not in any books at all. - -I now, however, pass to watches. It will be remembered that a verge -escapement consists of a crown wheel with teeth, engaging two pallets -fixed upon a verge, furnished with balls at its extremities. - -As the crown wheel was urged forwards each pallet in succession was -pushed till it slipped over the tooth which was engaging it. Then -a tooth on the other side came into sharp collision with the other -pallet, and drove the verge the other way, and so on. - -Now here we have a driving force, and a sort of pendulum. But how did -the verge act as a pendulum to measure time? It is not a body rocking -under the action of gravity, nor under the acceleration of a spring. -How then can it act as a regulator of time, and what is the period of -its swing? - -The answer to this is, that it is under the acceleration of gravity, -but that gravity does not act freely on the bobs or weights, but only -through the driving weight and teeth. The impulse that drives the -verge is really also the accelerating force upon it, and the only -accelerating force upon it. - -And the worst feature about the movement is, that as the teeth and -pallets move, the leverage of the teeth on the pallets alters, and -thus the bobs on the verge are under the influence not of a uniform or -duly regulated force, but of a constantly varying one, and one that -varies in a very complicated and erratic way. It would be hopeless to -expect much time-keeping from such a contrivance. The most that could -be expected would be by putting on a very big weight to reduce to -comparative insignificance the friction, and then hope that the swings -would be uniform, so that whatever went on in one swing would go on in -the next, and thus the time-keeping be regular. - -But any course tending to diminish the driving force, such as the -thickening of the oil, would greatly affect the going. It was for this -reason that Huygens turned the verge into a pendulum by removing one of -the bobs, and letting gravity thus act on the other. - -For watches, however, a different plan was contrived. One end of a -slender spiral spring was affixed to the verge. The other end of the -spring was made fast to the clock frame. The verge was now, therefore, -chiefly under the action of the acceleration of the spring. To make the -acceleration of the teeth of the ’scape wheel less embarrassing, the -teeth were so shaped as only to give a short push at stated intervals, -and not interfere with the free swing of the verge under the alternate -to-and-fro accelerations and retardations of the spring. By this means -the verge became in every way an excellent pendulum, not dependent on -gravity, and permitting the watch to be held in any position. - -The verge thus fitted was turned into a wheel, and became a “balance -wheel.” It was compensated for heat expansion by a cunning use of the -unequal expansion of brass and steel, in a manner analogous to the way -this unequal expansion of metals had been employed to compensate the -pendulum, and became the beautiful and accurate time-measurer that we -see to-day, with its pivots mounted in jewels to diminish friction, and -with screws round the rims of the balance wheel to enable the centre of -gravity to be exactly adjusted to its centre of rotation, and with a -delicate hair-spring of tempered steel that is a marvel of microscopic -work. - -But the escapement of the early watches left much to be desired. In -order to make it clear how imperfect that early escapement was, we -have to turn back and remember what has been said about the dead beat -escapement. - -It will then be remembered that it was shown that for small arcs the -pendulum would keep good time provided you let it have as much swing as -it wanted to use up the force which the escapement had applied to it, -_but not otherwise_, so the pendulums only acted really well when the -impulse was given about the middle of the swing, and they were free to -go on and stop when they pleased, and turn back at the end of it. - -This essential condition was fairly approximated to in the dead beat -escapement of clocks which left them at the end of their swing with -only a very slight friction to impede their free motion. - -But when you come to deal with a watch the case is quite different. -Here the escapement is of a great size compared with the balance wheel, -and the friction even of the most dead beat watch escapement that could -be contrived was so big compared with the forces acting on the balance -wheel as seriously to derange its motion, and render it far from a -perfect time-keeper. - -Now about this time—I am speaking of the early part of the eighteenth -century—a demand of a very exceptional character arose for a really -perfect watch. The demand did not arise from gentlemen who wanted -to keep appointments to play at ombre at their clubs, or even -from merchants to time their counting house hours. For these the -old-fashioned watch did very well. The demand came from mariners. -But the seamen did not want to know the time merely to arrange the -hours for meals on the ship or to determine when the watch was to be -relieved, but for a far more important purpose, namely, to find out by -observation of the heavens their place upon the ocean when far out of -sight of the land. It will be very interesting to see how this problem -arose, and how the patient industry and ingenuity of man has solved it. - -The ancient navigators never went very far from the shore, for, once -out of sight of land, a ship was out of all means of knowing where she -was. On clear days and nights the compass, and the sun and stars would -tell the mariner the _direction_ he was sailing in, but it was quite a -problem to determine where he was on the surface of the earth. - -[Illustration: FIG. 63.] - -Let us consider the problem. Suppose for convenience that the earth is -divided up into “squares,” as nearly, at least, as you can consider a -globe to be so marked out. Let us suppose that it has been agreed to -draw on it from pole to pole 360 lines of longitude, commencing with -one through say Greenwich Observatory as a starting-point, and going -right round the earth till you come back to Greenwich again. Also -suppose that there have been drawn a series of circles parallel to the -equator, but going up at equal distances apart towards the poles. Let -us have 179 of these circles, so as to leave 180 spaces, _a_ to _b_, -_b_ to _c_, etc., from pole to pole. This will divide the earth up like -a bird-cage into squares, as if we had robed it in a well-fitting -Scotch plaid. The length measured along the equator of the side _p q_ -of each square at the equator is taken as exactly sixty nautical miles -(apart from a small error of measurement, which makes it in actual -practice 59·96). This is equal to sixty-nine and a quarter English -statute miles. The side of the square leading towards the poles _q s_ -would also be sixty nautical miles were it not that the earth is not -truly spherical, which introduces a slight error. We may, however, -roughly say that at the equator each square measures sixty nautical -miles each way. - -[Illustration: FIG. 64.] - -As we get towards the poles the squares become rectangular figures, -with the heights of latitude still sixty nautical miles, but the widths -becoming smaller. Thus in England our squares measure _p q_ = 37 -nautical miles and _q s_ = 60 nautical miles. - -Now of course we can see at once that it is easy at any place on the -earth’s surface to find your _latitude_ by a simple observation of the -sun at noon, if you know the day of the year, and have got a nautical -almanac. For by an instrument called a sextant you can measure the -angle he appears to be above the horizon, and then, as you know from -a nautical almanac the angle he is above the equator, you can soon -determine your place _A_ on the globe. Or at night, if you measure the -angular distance that the polar star _P_ is from the zenith, or point -exactly over your head—that is, the angle _P O Z_—you can subtract it -from a right angle and get your latitude, _A O E_, at once. - -[Illustration: FIG. 65.] - -But how are you to determine your longitude? The pole-star, or sun, -or any other star won’t help you, for as the earth is moving they -keep shifting, and at one time or another appear exactly in the same -position to everyone on the same parallel of latitude, as it is easy to -see. The fact is that you are on a ball turning round. You know easily -what latitude you are on, but you cannot tell your longitude unless -you can tell how many hours and minutes you get to a position before -Greenwich gets to the same position. If when a particular star got to -Greenwich a gong were sounded which could be heard all over the earth, -then of course, by seeing what stars were overhead, everyone would -know their longitude at once. Perhaps by means of the new electric -waves this will before long be done, and the Greenwich hours will be -sounded all over the world for the use of mariners. But till this is -accomplished all that can be done is to keep an accurate clock on -board, so as always to give you Greenwich time. - -Early attempts were made to take a pendulum clock to sea, suspending it -so as to avoid disturbance to its motion by the rocking of the ship. -These proved vain. - -It therefore became desirable that a watch with a balance wheel -should be contrived to go with a degree of accuracy in some respects -comparable with the accuracy of a pendulum clock. To encourage -inventors an Act of Parliament was passed in the thirteenth year of -Queen Anne’s reign (chapter xv.) (1713) promising a reward of £20,000 -to anyone who would invent a method of finding the longitude at sea -true to half a degree—that is, true to thirty geographical miles. - -If the finding of the longitude were to be accomplished by the -invention of an accurate watch, then this involved the use of a watch -that should not, in several months’ going, have an error of more than -two minutes, which is the time which the earth takes to turn through -half a degree of longitude. - -This was the problem which John Harrison, a carpenter, of Yorkshire, -made it his life business to solve. His efforts lasted over forty -years, but at the end he succeeded in winning the prize. - -These instruments have been much improved by subsequent inventors, and -have resulted in the construction of the modern ship’s chronometer, -a large watch about six inches in diameter, mounted on axles, in a -mahogany box. Several of these are taken to sea by every ship. - -The peculiarity of the chronometer is its escapement. - -Let _A B_ be the scape wheel, and _C D_ a small lever attached to _C_, -the pivot on which the balance wheel and spring is fastened. Let _E G_ -be a lever, with a tooth _F_ which engages the teeth of the scape wheel -and prevents it moving round. Let _H_ be a spring holding the lever _E -G_ up to its work. - -[Illustration: FIG. 66.] - -The lever has a spring _K E_ fastened to it at the point _K_. This -spring is very delicate. If the lever _C D_ is turned so that the -little projection _M_ on it strikes the spring _E_ from left to right, -then, as the spring rests on the lever, the whole lever is pushed over, -and the teeth of the scape wheel set free. At that instant, however, -the escapement is so arranged that the arm _C D_ is just opposite -the tooth _D_ of the scape wheel, so that the scape wheel, instead -of running away, leaps with its tooth _D_ on to the lever _C D_ and -swings the balance wheel round. The balance wheel is free to twist as -much as it pleases, but the moment it has twisted so much that the -projection _M_ passes the spring _E_, then the lever _G E_ flies back -to its place, and the scape wheel is again checked. Meanwhile the -balance wheel flies round till at last it is brought to rest by the -balance spring. It then recoils and sets out on its return path. This -time, however, the projection _M_ merely flips aside the spring _E_ -and the balance wheel goes back, till again it is brought to rest and -returns. As soon as the lever comes opposite _D_ the projection _M_ -then again hits the spring _E_, and releases the catch at _F_, and -another tooth of the scape wheel goes by. - -There then you have a completely free escapement, and consequently an -accurate one. Many watches are made with these escapements, but they -are more expensive than those in common use. - -There is but little remaining in a watch that is not in a clock, for -the wheel-trains and general arrangements are very similar. - -It is possible to apply the chronometer’s detached escapement to a -clock. This was done by several clock-makers in the eighteenth and -early part of the nineteenth century. One method of doing it is as -follows: - -_A_ is a block of metal fitted to the bottom of the pendulum, _B_ a -light lever pivoted on it. _C_ is the scape wheel, with four teeth; -_D_ a tooth of the scape wheel, which hops on to the projection of the -pendulum the moment that the impact of the point _E_ of the lever _B E_ -has pushed aside the lever _G F_, and thus released the scape wheel. -The advantage is that it is a very easy escapement to make. But it is -in reality a detached (that is to say, a completely free) chronometer -escapement, as can easily be seen. - -[Illustration: FIG. 67.] - -Turret-clocks are open to considerable disadvantages, for the wind -blowing on the hands gives rise to considerable pressure, so that the -clocks are sometimes urging the hands against the wind, sometimes are -being helped by the wind. And this inequality of driving force makes -the pendulum at some times make a bigger arc of swing than at others. - -But we saw above that though difference of arc of swing ought to make -no difference in the time of swing of the pendulum, yet this was only -strictly true if the arc of swing were a cycloid. - -But as for practical convenience we are obliged to make it a circle, -it follows, as we saw, that for every tenth of an inch of increase of -swing of an ordinary seconds pendulum about a second a day of error is -introduced. To remove this difficulty a gravity escapement was invented -by Mudge in the eighteenth century, improved by Bloxam, a barrister, -and perfected by the late Lord Grimthorpe. The idea was to make the -scape wheel, instead of directly driving the pendulum, lift a weight, -which, being subsequently released, drove the pendulum. The consequence -was, that inequalities in wind pressure, which affected the driving -force of the scape wheel, would not act on the pendulum, which would -be always driven by the uniform fall of a fixed and definite weight. A -movement of this kind has been fixed in the great clock at Westminster, -and has gone admirably. A description of its details will be found in -the _Encyclopædia Britannica_, written by Lord Grimthorpe himself. - -All sorts of eccentric clocks and watches have been proposed. For -instance, it seems wonderful to see a pair of hands fitted to the -centre of a transparent sheet of glass go round and keep time with -apparently nothing to drive them. - -But the mystery is simple. The seeming sheet of glass is not one sheet, -but four. The two centre sheets move round invisibly, carrying the hour -hand and minute hand with them, being urged by little rollers below -on which they rest. When you touch the glass the outside sheets appear -at rest, and you do not suspect that it is other than a single sheet. -But beware of dust, for if dust gets on the inner plate you detect the -trick. In this way a mechanical hand was made that wrote down answers -to questions. This plan can be applied to all sorts of tricks. - -Sir William Congreve, an ingenious inventor, proposed to make a clock -that measured time by letting a ball roll down an incline. When it got -to the bottom it hit a lever, which released a spring and tipped the -plane up again, so that the ball now ran down the other way. It is a -poor time-keeper, and the idea was not original, for a ball had been -previously designed for the same purpose. - -Sometimes clocks are constructed by attaching pendulums to bronze -figures, which have so small a movement that the eye is unable to -detect it. The figure appears to be at rest, but is in reality slowly -rocking to and fro. It is necessary to make the movement as small as -about one four hundredth of an inch in half a second, if the movement -is to escape human observation. For a movement of one two hundredth -of an inch per second is about the largest that will certainly remain -unperceived. - -In mediæval times clocks were constructed with all sorts of queer -devices. The people of the upper town at Basle having quarrelled with -those of the lower town, fought and beat them. To commemorate this -victory they put on the old bridge at the upper town a clock provided -with an iron head, that slowly put out and drew in a long tongue of -derision. This clock may still be seen in the museum. It is as though -the council of the city of London put a clock of derision at Temple Bar -to put out its tongue at the County Council. - -I do not propose here to describe the striking mechanism of clocks. -There are several different ways of arranging it. They are rather -complicated to follow out, but they all resolve themselves into a -few simple principles. As the hour hand revolves it carries a cam so -arranged as to be deeper cut away for the twelfth hour, less for the -eleventh, and so on. When the minute hand comes to the hour it releases -the striking mechanism, which, urged by a weight, begins to revolve, -and, driving an arm carrying a pin, raises a hammer, which goes on -striking away as the arm revolves. This would continue for ever if -it were not that at the same moment an arm is liberated which falls -against the cam. At each stroke the arm is (by the striking apparatus) -raised a bit back into position. When it comes back into position it -stops the striking. It thus acts as a counter, or reckoner of the blows -given, stopping the movement when the clock has struck sufficiently. -If the counting mechanism fails to act, we have the phenomenon which -occasionally occurs of a “Grandfather” clock striking the whole of the -hours for the week without stopping. - -A chiming clock is simpler still. For here we have a barrel covered -with pins, like the barrel in a musical box. As the pins go round they -raise hammers which fall against bells. The barrel is wound up and -driven by a spring or weight. When the clock comes to the hour, the -barrel is released, and rotating, plays the tune. - -If you want to make a clock wake you up in the morning it can be -done by making the striking arrangement hammer away with no counting -mechanism to stop it until the weight has run down. If, not content -with that, you want the sheets pulled off the bed or the bed tilted -up, or a can of water emptied over the person who will not rise, a -mechanical device known as a relay must be used. It is very simple. -What is wanted is that, after the lapse of a time which a clock -must measure, a considerable force must be exerted to pull off the -bedclothes. It would be absurd to make the clock exercise this pull. -It is obviously better to attach the clothes by a hook to a rope which -passes over a pulley, and from which hangs a weight. A pin secures the -weight from falling, the pin being withdrawn by the clock. The work is -thus done by the weight when released by the clock. - -In like manner, if you have a telegraph designed to print messages at -a distance, you do not send along the wires the whole force necessary -for doing the printing. You only send impulses, which, like triggers, -release the forces by which the letters are to be stamped. - -Electric clocks of many kinds have been invented. The principle of an -electric escapement is similar to that of an ordinary escapement. - -[Illustration: FIG. 68.] - -The reader no doubt knows that, when a circuit of wire is joined or -completed leading to a source of electricity, electricity flows through -the wire. - -If the wire is wound round a piece of iron, then, whenever the circuit -is joined, a current is set in motion, and the iron becomes an -electro-magnet. When the circuit is severed the iron ceases to be a -magnet. - -If put at a proper position it would at each time an iron pendulum -approached give it a small impulse provided that at that instant -the current is turned on. This can easily be made to be done by the -pendulum itself. For just as the pendulum is coming back to the -central position a flipper _P_ attached to the rod can be caused -to make contact with a piece of metal fixed on its path. Then the -electro-magnet, becoming magnetised, exerts a pull on the iron -pendulum. On the return beat of the pendulum the other side of the -flipper _R_ strikes the obstruction. But if that side _R_ is covered -with ebonite or some non-conducting material no current will be set -in motion, and the electro-magnet will not (as it would otherwise do) -retard the pendulum. Such a pendulum has therefore an impulse given to -it every second beat. - -Such pendulums do not act very well, because it is difficult to keep -metallic surfaces like _Q_ clean, and therefore misses often occur. -Besides, the strength of the current varies with the goodness of the -contact and with other things. - -What is now preferred is to make an arrangement by which an electric -current winds the clock up every minute or so. By this means the -impulse which drives the clock is not a varying electric one, but is -a steady weight. The most successful clocks have been made on these -principles. - -The advantage of electricity is, that by means of the current that -actuates the clock, or winds it up, you can at regular intervals set -the hands in motion of a great number of clocks. - -So that only one going clock with a pendulum is needed. The other -clocks distributed over the building have only faces and hands, and -a very few simple wheels, to which a slight push is given by an -electro-magnet, say, every minute or so. The system is therefore well -adapted for offices and hotels. - -In America, by means of electric contacts, clocks have been arranged to -put gramophones into action. You will remember that it was pointed out -that if a wire were dragged over a file a sound would be produced due -to the little taps made as the wire clicked against the rough cuts on -the file, and that the tone of the note depended on the fineness of the -cuts, and hence the rapidity of the little taps. You can imagine that, -if the roughnesses were properly arranged, we might get the tones to -vary, and thus imitate speech. This is the principle of the gramophone. -The roughnesses are produced by a tool, which, vibrating under the -influence of human speech, makes small cuts in a soft material. This -is hardened, and then, when another wire is dragged over the cuts, the -voice is reproduced. - -In this way clocks are made to speak and tell the children when dinner -is ready and when to go to bed. On this simple plan, too, dolls can be -made to speak. - -The modern methods of clock and watch-making are very different from -those in use in olden days. In former times the pivots were turned up -by hand on small lathes, and even the teeth of the wheels were filed -out. Each hole in the clock or watch frame was drilled out separately, -and each wheel separately fitted in, so that the watch was gradually -built up as one would build a house. Each wheel, of course, only fitted -its own watch, and the parts of watches were not interchangeable. - -This has now all been altered. By means of elaborate machinery the -whole of the work of cutting out every wheel and the making of every -single part is done by tools moved independently of the will of the -workman, whose only duty is to sit still and see the things made. He -is, as it were, the slave of the machine, watching it and answering to -its calls. Or shall we rather say that he is the machine’s employer and -master? He has here a servant who never tires nor ever disobeys him. -All the machine requires is that its cutting edges should be exactly -true and sharp and microscopically perfect; then it will cut away and -make wheel after wheel. It oils itself. It only wants the man to act as -superintendent, and stop it if any cutting edge gets unduly worn. For -this purpose he measures the work it is doing from time to time with a -microscope to see that it is good and true and exact. - -When all the parts have thus been made you have perhaps a hundred -boxes, each with a thousand watch parts in it, each part exactly like -its fellows. You take one wheel or bit from each box indiscriminately, -and you then have the materials for a watch, screws, fittings, pins, -and all. All you have now got to do is simply to screw them all -together, like putting together a puzzle. Everything fits; there is no -snipping or filing. - -In such a watch if a bit gets broken you simply send for another bit of -the same kind and fit it into its place. - -Motor cars, bicycles, and many other machines are, or ought to be, made -in this manner, so that if a driver at York breaks a part of the car he -simply sends to London for another. It comes and fits into its place at -once. But for this sort of plan you must do work true to much less than -a thousandth of an inch, and, of course, no one must want to indulge -his individual fancy as to the shape or appearance of the watch. The -whole advantage consists in dead uniformity. But the cheapness is -surprising. You can have a better watch now for 30_s._ than could have -been got for £30 twenty years ago. - -Artistic people are in the habit of condemning this uniformity as -though it were inartistic and degrading. In truth, it is not degrading -to get a machine to do what you want at the expense of as little labour -as possible. You pay 30_s._ for the watch, but you have £28 10_s._ -left to spend on pictures. - -Only one ought not to confuse industry with art. Watches made in this -way have no pretence to be artistic products. They are simply useful. -To rule them all over with machine lines or to put hideous machine -ornament on them is purely and simply base and degrading. Let your -_ornament_ be hand work, your utility machine work. - -Thus then I have endeavoured to give a very brief sketch of the modes -of measuring time, and incidentally to introduce my readers to those -laws of motion which are the foundation of so large a part of modern -science. - -It only remains that I should shortly describe modern apparatus by -means of which it is possible to measure with accuracy periods of time -so short as to appear impossible. But when you see how it is done the -method seems easy enough. It is still by means of a pendulum, only a -pendulum beating time not once, but hundreds and even thousands of -times in a second. - -And such pendulums, instead of being difficult to make, are remarkably -simple, and present no difficulty whatever. For we have only to use the -tuning fork which has been previously described. - -The tuning fork consists of a piece of steel bent into a U shape. The -arms are set vibrating so as alternately to approach and recede from -one another. - -The reason why there are two arms is that, if they come together and -recede, they balance, and hence the instrument as a whole does not -shake on its base. This balance of moving parts of a rapidly moving -machine is very important. Some motor cars are arranged so that -the engines are “balanced,” and the moving parts come in and out -simultaneously, leaving the centre of gravity unchanged whatever be the -position of the motion. This makes the vibration of the car very small. - -The tuning fork is therefore balanced. Being elastic, it obeys Hook’s -law, “As the force, so the deflection.” And therefore, as we have seen, -the vibrations of the fork are isochronous. - -A fork with arms about six or seven inches long will make about fifty -or sixty vibrations in a second. How are we to record those vibrations, -and how keep the tuning fork vibrating? - -[Illustration: FIG. 69.] - -A train of wheels is almost an impossibility, not perhaps so impossible -as might be supposed, but still very difficult. So a different method -is adopted. A little wire projects from one tuning fork arm. A piece of -glazed paper is gently smoked by means of a wax taper, and is stretched -round a well-made brass drum. The tuning fork is then put so that the -little wire just touches the paper. The tuning fork is then made to -vibrate by a blow, and while it is vibrating the drum is revolved. -Thus a wavy line is formed on the drum by the wire on the tuning fork. -If the tuning fork made fifty complete vibrations to and fro in a -second there would be one hundred such indentations, fifty to the right -and fifty to the left, and by these the time can be measured as you -would measure a length upon a rule. - -[Illustration: FIG. 70.] - -If an arm _a b_ be fitted to move sideways when a little string _c -d_ is pulled, and be also provided with a small wire, so as to touch -the drum, then it also will trace a straight line on the drum as the -wire lightly scratches away the thin coating of smoke. Now, if it is -suddenly jerked and flips back, then a little indentation will be -made in the line, and if when we are to measure a rapid lapse of time -a jerk is given at the beginning, and another jerk at the end of it, -we should get a diagram like that in the adjoining figure, where _a_ -is the trace of the tuning fork, _b_ that of the indicating arm. The -time which has elapsed between the jerk which produced the indentation -_c_ and that which produced the indentation _d_ will be about three -and three-quarter double indentations of the tuning fork line, thus -indicating three and three-quarter fiftieths of a second. It is easy -to see how delicate this means of measurement can be made. With small -tuning forks we can easily measure times to a thousandth part of a -second, and much less if desired. - -The jerk may be given by electricity if it is wished. When the current -is joined a little electro-magnet pulls a bit of iron and gives a pull -to the string. So extremely rapid is the flight of electricity that no -appreciable time is lost in its transit through the wires, so that the -impulse may be given from a distance. Thus we may arrange that when a -cannon ball leaves a gun an electric impulse shall be given. When it -reaches and hits a target another electric impulse is given. These make -nicks in the tracing line on the drum from which we can easily compute -the time that has elapsed between the leaving of the mouth of the gun -and the arrival of the shot at its destination. - -[Illustration: FIG. 71.] - -Such an apparatus is used in modern gunnery experiments. It is an -elaborate one, but is based on the principle above described. - -Drums are sometimes driven by clockwork, and tuning forks are also -often kept vibrating by electricity, thus constituting very rapidly -moving electric clocks. The arrangement is simple. An electro-magnet -_E_ is put in the vicinity of the arm of the tuning fork. A small piece -of wire from the arm is in contact with a piece of metal _Q_, from -which a wire runs to the electro-magnet, thence to a battery, and from -the battery to the tuning fork, through which the current runs to the -wire _R_. When the fork vibrates the arm, being bent outwards, makes -the wire _R_ touch _Q_. This at once causes the electro-magnet to give -a small pull to the steel arm of the tuning fork, and thus assists the -swing of the arm. The whole arrangement is exactly analogous to an -electric clock, as may be seen by comparing Fig. 71 with Fig. 68. - -There is another method of measuring rapid intervals of time which also -merits attention. It is to let a body drop at the commencement of the -period of time to be measured, and mark how far it falls in the time, -and then find the time from the equation given previously, - - S = 1/2 g t². - -It is practically done by letting a piece of smoked glass fall and -making a small pointer make two dots upon it, one at the beginning, -another at the end, of the time to be measured. - -An interesting adaptation of this method can serve as a basis of a -curious toy. - -Take a crossbow, with a bolt with a spike on it; fix it firmly in a -vice so that the barrel points at a spot _a_ on a wooden wall. On the -spot _a_ hang a cardboard figure of a cat on to a nail so contrived -that when an electro-magnet acts the nail is pulled aside, and the -cat drops. Thus let _a_ be the cat, _b_ the loop by which it is hung -over the nail _c_, that is fixed to another piece of iron furnished -with a hinge at _c_, so that when the electric current is turned on -the nail _c_ is withdrawn and the cat drops. Carry the wires from the -electro-magnet and battery to the crossbow, and so arrange them that -when the bolt leaves the muzzle one is pressed against the other, and -contact made. - -Now here you have an apparatus such that exactly as the bolt leaves the -crossbow, the cat drops. Now what will happen? - -[Illustration: FIG. 72.] - -When the bolt leaves the bow it is subject to two motions, one a motion -of projection at a uniform pace in the direction of _b a_ from the bow -to the target. - -But it is also subject to another force, namely that of gravity, which -acts on it vertically, and deflects it _in a vertical direction_ -exactly as much and as fast as a body would do if dropped from rest at -the same instant as the bolt leaves the bow. But the cat is such a -body. Hence, then, since by the electric arrangement they are both let -go together, they will both drop simultaneously, and thus will always -be on the same level, and when the bolt reaches the wooden wall and -has fallen vertically from _a_ to _c_, the cat will also have fallen -vertically from _a_ to _c_, and the bolt will pin him to the wall. It -does not matter how far you take the bow from the wall, nor how strong -the bow is, nor how heavy the bolt is, nor how heavy the cat is, nor -whether _a b_ is horizontal or pointing upwards or downwards. - -[Illustration: FIG. 73.] - -In every case, if only the barrel is pointed directly at the cat, then -the bolt and cat fall simultaneously and at the same rate, and the bolt -will pin the cat to the wall. - -In trying the experiment the bolt should be pretty heavy, say half a -pound, and have a good spike; but if carefully done the experiment will -succeed every time. It enables you also to measure the speed of flight -of the bolt. For if the distance of the bow from the wall be thirty -feet, and the cat have fallen three feet when it is struck, then the -time of fall is T² = √((2S)/g) = √(6/g) = ·43 seconds. But the bolt -in this time went thirty feet; hence its velocity was thirty feet in -·43 seconds, or seventy feet per second. - -Of course if you make the bolt heavier the velocity of projection will -become slower, the time longer, and hence the cat will fall further -before it is transfixed by the bolt. - -My task is now at a close. I have endeavoured not merely to give a -description of clocks and various apparatus for measuring time, but to -explain the fundamental principles of mechanics which lie at the root -of the subject. - -May I end with a word of advice to parents? - -There is a certain number of boys, but only a certain number, who have -a real love for mechanical science. Such boys should be encouraged -in every way by the possession of tools and apparatus, but in the -selection of this apparatus the following principles should be borne in -mind:— - -_First_, that almost everything a boy wants can be made with wood, and -metal, and wire, and string, _if_ he has someone to give him a little -instruction how to do it. A bent bit of steel jammed in a vice makes an -excellent tuning fork. - -_Second_, that he wants not toy tools, but good tools. If an expert -wants a good tool, how much more a beginner. - -_Third_, that he ought to have a reasonably dry and comfortable place -to work in, and the help and advice of the village carpenter or -blacksmith. - -_Fourth_, that he ought not to be allowed to potter with his tools, but -to make something really sensible and useful, and not begin a dozen -things and finish none. - -_Fifth_, that the making of apparatus to show scientific facts is more -useful than making bootjacks for his father or workboxes for his mother. - -And, _lastly_, that a little money spent in this way will keep many a -young rascal from worrying his sisters and stoning the cat; and when -the inevitable time comes at which he must face the young man’s first -trial, THE EXAMINER, he will often thank his stars that he learned in -play the fundamental formula S = 1/2 g t², and that he knows the -nature of “harmonic motion,” the two most important principles in the -measurement of time. - - -THE END. - - - - -APPENDIX ON THE SHAPE OF THE TEETH OF WHEELS. - - -[Illustration: FIG. 74.] - -The teeth of wheels for watches and clocks need particular care in -shaping, and it may be of interest if I describe briefly the principles -upon which these wheels are made. What is required is that the motion -shall not be communicated by jerks as the teeth successively engage one -another, but that the motion shall be perfectly smooth. The problem -therefore becomes this: How are we to arrange the teeth of the wheels -so that as one of them turns and drives the other round the leverage -or turning power exercised by the driving wheel on the driven wheel -shall always be uniform? Now if the teeth were simple spikes one can -easily see that this would not be the case. For instance, as the arm _a -c_ turned round, driving before it the arm _b d_, the point _c_ would -scrape along, and the leverage between the two teeth would constantly -alter. Evidently some other construction must be adopted. Before we can -determine what it is to be, we must inquire what the leverage would -be between two rods, _a c_ and _d b_, mounted on pivots at _a_ and -_d_. The answer to this question is, that when a lever such as _a c_ -presses with its end against another, _d b_, the power is exercised in -a direction _c e_ at right angles to _d b_. Hence the leverage between -the two arms is in the ratio of _a e_ to _d c_. The system is just as -if we had a lever _a e_ united to a lever _d c_ by a rigid rod _e c_ at -right angles to both of them. - -[Illustration: FIG. 75.] - -Whence then the ratio of the power is as _a e_ is to _d c_. - -[Illustration: FIG. 76.] - -But since the triangles _a e f_, _d c f_, are similar, _a e_ is to _d -c_ as _a f_ to _f d_. Whence then we get this general proposition: If -one body mounted on an axis is pressing upon another body mounted -on an axis, the pressure exerted between them is always exercised in -a direction, shown by the dotted line, at right angles to the two -surfaces in contact; and the ratio of the leverage is found by drawing -a line from one axis to the other, so as to cut the line of direction -of pressure in _f_. The leverage of one on the other is then as _a f_ -to _f d_. Our problem has now become the following: Given a rod _b d_, -suppose that it is pressed upon by a curved surface mounted on an axis -at _a_. Then the direction of the pressure that the curved surface -(called in engineering language a cam) will exercise on the rod _b d_ -is shown by the dotted line; and the ratio of the driving power to the -driven power is as _d f_ to _a f_. Now how can we shape the cam so that -as it moves round, and different parts of its surface come successively -into contact with _b c_, the ratio of the leverage is always the -same; that is to say, the ratio of _a f_ to _f d_ shall always be -constant; that is to say, the line drawn through the point of contact -perpendicular to the curve at that point, shall always pass through the -point _f_? - -[Illustration: FIG. 77.] - -[Illustration: FIG. 78.] - -Evidently, if this is to be so, the point _d_ must be on a semicircle, -whose diameter is _f b_, for in that case the angle _f d b_ will always -be a right angle. - -[Illustration: FIG. 79.] - -The surface must then be so arranged that, whatever be the position of -the cam and of the rod _b d_, the point of contact between them must -always be on the semicircle _f c d_; that is to say, as the cam moves -round the axis _a_ its shape must be such that a line drawn from _f_ to -the point where it cuts the circle _f d b_ is always perpendicular to -the curve. - -Now suppose that we move a circle whose centre is at _a_, and radius _a -f_, so as to roll the circle _f d b_ by simple surface friction round -its centre _o_, then any point _d_ on it would mark out a curve on a -piece of paper attached to the moving circle whose centre is at _a_, -and the direction of motion of the curve would always be such that the -point _d_ on it would at any instant be describing a circle round _f_, -and the direction of the curve would thus at any point always be at -right angles to the line _d f_ for the time being. - -[Illustration: FIG. 80.] - -This curve, caused by the rolling of one circle on another, is called -an epicycloid. Hence, then, for a clock, if we make the pinion wheel -with straight spokes and the driving wheel with its teeth cut in the -form of epicycloids, caused by rolling a circle with a diameter equal -to the radius of the pinion upon the driving wheel, we shall get a -uniform ratio of leverage one upon the other. - -The circles with radii _a f_, _b f_, are called the “pitch circles,” -and these radii are in the ratio of the movement that is required for -the wheels, usually six to one or eight to one, as the case may be. The -sides of the teeth of the pinion wheels are straight lines radiating -from the centre, and rounded off at the ends so as to avoid accidental -jambing. The teeth of the cogwheel have epicycloidal sides. The tips -are cut off so as to be out of the way, and spaces are left between -them for the width of the leaves of the pinion wheel. - -[Illustration: FIG. 81.] - -Both pinion wheels and cogwheels are cut by cutters rotating at a -high speed, about 3,500 times in a minute, the cutters being carefully -shaped for the pinion wheels with straight edges, for the cogwheels -in epicycloids. It is a pretty thing to see a wheel-cutting engine at -work, the cutter flying round with a hum, cutting the rim of a brass -wheel into teeth, the brass coming off in flakes thinner than fine -hairs and falling in fine dust. When a tooth is cut, the wheel is moved -round one division of an apparatus called a “dividing plate,” so as to -present a new part of the wheel to the cutter. Of course, the cutter -and wheels must all be properly proportioned. Cutters are sold in sets -duly shaped for the work they have to do. Wheel-cutting is a special -branch of the clockmaking industry. The reason the speed of cutting is -so high is because it is desired to take off small portions of metal -at a time, and thus not strain the wheel and the cutting machinery. If -bigger cuts were made, then the machine would have to go slower, for -it is a principle in the construction of cutting machinery that the -speed of the cut must always be proportioned to the depth of it. If -you want to take deep cuts you must move the cutting edge slowly, and -_vice versâ_. The most modern method of making cogwheels of brass, and -the best, is to stamp them out of solid sheet metal at a single punch -of a punching machine, and cheap watches are always made in this way. -In fact, the whole method of watch and clock-making is undergoing a -transformation. - -Before the time of the great engineering development which took place -towards the end of the eighteenth century, the making of machines was -a sort of fine art. Cogwheels were cut by hand. The circumference -was marked out by means of compasses. Then holes were drilled round -the rim, and teeth cut out leading into them, and shaped by means of -special files constructed for the purpose (Fig. 82). Big machinery was -all shaped out at the forge and by the file. The engineers complained -that you could not get big work made true even to the eighth of an -inch. But watches and clocks were beautifully made, though only at the -cost of hours of patient measuring and filing. The taste for ornament -still existed. The wheels and backs of watches were chased over with -the most beautiful patterns; the frames of the clocks were wrought -into beautiful figures and forms. Even astronomical instruments were -embellished. - -[Illustration: FIG. 82.] - -Then came the era of severe accuracy. Men of science took the -government of machine-making whose feelings were repugnant to art in -any form. They hated to see any effort expended in ornament. With -severely utilitarian aims, they banished all appearance of beauty from -instruments and tools of all sorts, so that our modern machines, from a -steam engine down to a watch, are now models of precise but perfectly -unornamented workmanship. They are the fitting implements of a nation -that wears trousers and tall hats. One has only to compare an old -vessel of war, with its sculptured prow and streamers, with a modern -ironclad to take note of the difference. The art of ornamentation -is now little more than a spasmodic imitation of the past, with a -historical interest only. As a living entity it has almost ceased to -exist. - -But in precision of manufacture the present age is without a rival in -the history of the world. People believe no longer in the old methods -of scraping and filing, and hand-work directly exercised on metal is -rapidly falling into desuetude. It is possible, of course, with a file -and scraper and days of labour to get two flat surfaces of metal so -perfect that when put together one will lift the other like a sucker -on a stone, but it is waste labour. A small machine will do it as well -in a few minutes. No longer is a watch built up as one would build a -house, fitting part to part. By expensive machines thousands of watch -parts are stamped and cut out to pattern, and then a watch is made by -taking one of each indiscriminately and just putting them into their -places. Comparatively unskilled workmen can do this. Where the skill -is wanted is to design and make the machinery and watch the cutters, -measuring them with microscopic gauges from time to time, and at once -remedying them if an edge is found to be a ten-thousandth part of an -inch out of place. So that the labour of man is becoming more and more -a labour of design and of supervision. Machines are the slaves that do -the work, for in a good machine we have an eye and an arm that never -tires, and only needs to be kept in working order. But machines are not -artistic, and thus art is lost while precision is gained. At present -a desperate attempt is being made to supply by means of machinery -the craving of the human mind for art. But it is destined to failure. -Art of this kind is generally produced by the same brain that designs -machines, and therefore presents an appearance of rigid accuracy and -uniformity, which, while essential to an engine, is out of place in an -artistic product. - -The great manufacturers of our Midlands do not seem to understand that -there is no object in making a towel-horse as geometrically accurate as -a turning lathe. It will apparently be years before they learn to put -art and precision each in the place where it is wanted—precision in the -works of the watch, art in the face and the case of it; machine work in -the inside of a watch, hand work on the outside. When the public taste -is educated so as to see the odious character of the jumble of Gothic, -Egyptian, and meaningless ornament on such an article as the case of -an American organ, one step will have been made towards the revival of -artistic taste. - -But to propose as a means of reviving art that we should discontinue -the use of machinery or abandon our modern cutters of precision to go -back to a hack-saw and file is ridiculous, and could only be suggested -by men quite destitute of scientific ideas. Art and precision each has -its place: there is room for both; let neither intrude on the domain of -the other. - - - - -INDEX. - - - Acceleration, 73, 77 - - Almagest, 53 - - Anchor escapement, 120 - - Ancient science, 50 - - Aristotle’s ideas, 23, 52 - - Attwood’s machine, 83 - - - Babylon, temple of, 24 - - Balance wheel, 159 - - - Candles to measure time, 46 - - Chaldean day, 15 - - Chaucer, 56 - - Chronographs, 179 - - Chronometer, 165 - - Chronometer escapement, 166 - - Clock movement, 123 - - Copernicus, 56 - - Crossbow experiment, 183 - - Crown wheel, 115 - - Cycloid, the, 109 - - - Dante’s Inferno, 54 - - Day, length of, 29 - - Dead beat escapement, 135 - - Density, 12 - - Driving weight, 127, 141 - - - Earth, a sphere, 21 - - Earth’s motion, 57, 69 - - Earth not at rest, 67 - - Egg-boiler, 43 - - Electric clocks, 179 - - Epicycloidal wheels, 191 - - Escapements, anchor, 120 - crown, 115 - chronometer, 166 - dead beat, 135 - gravity, 169 - - - Falling bodies, laws of, 62 - - Force, 76 - - Forces, revolution of, 89 - - Fusee, the, 117 - - - Galileo’s “Dialogues,” 58 - clock, 111 - - Grandfather’s clock, 119 - - Gravity, action of, 13, 65 - - Gravity escapement, 169 - - Greek day, 16 - - - Harmonic motion, 97 - - Hooke’s law, 71 - - - Isochronism of springs, 93 - - - Lamps to measure time, 46 - - Latitude and longitude, 161 - finding, 163 - - - Mass, nature of, 10 - - Mercury clock, 45 - - Modern methods, 177, 197 - - Moments, 101 - - Moon’s appearance, 17 - - Motion, reliability of, 57 - - Musical notes, 95 - - - North pole, days at, 33 - - - Oscillations, law of, 151 - - - Parabola, the, 87 - - Pendulum, the, 103, 145 - suspension, 145 - mercury, 147 - gridiron, 149 - theory of, 155 - free, 133 - - Pisa, leaning tower of, 61 - - Planets, names of, 11 - - Pulse measurer, 99 - - - Ratchet wheels, 129 - - Roman clocks, 40 - - - Sand-glasses, 41 - - Space, nature of, 8 - - Speed of falling bodies, 79 - - Spring balance, 107 - - Stevinus’ theory, 81 - - Style of sun-dials, 35 - - Sun-dials, 27 - to make, 48 - - Synchronous clocks, 175 - - - Time, 13 - - Toothed wheels, 125, 137 - - Tuning fork, 94, 181 - - - Velocities, composition of, 85 - - - Water pressure, 37 - - Water clocks, 39 - - Watches, 156 - - Week days, names of, 24 - - Wheels, shape of teeth, 190 - - Wheel-cutting machines, 193 - - Winding drum, 131 - - Winter sun, 31 - - - Zodiac, 18 - - -BRADBURY, AGNEW, & CO. LD., PRINTERS, LONDON AND TONBRIDGE. - - - - - * * * * * * - - - - -Transcriber’s note: - -Punctuation, hyphenation, and spelling were made consistent when a -predominant preference was found in this book; otherwise they were not -changed. - -Simple typographical errors were corrected; occasional unpaired -quotation marks were retained. - -Ambiguous hyphens at the ends of lines were retained. - -Square roots are represented as √(values). - -Index not checked for proper alphabetization or correct page references. - -Page 16: “six o’clock” was printed as “six clock”; changed here. - - - -***END OF THE PROJECT GUTENBERG EBOOK TIME AND CLOCKS*** - - -******* This file should be named 54546-0.txt or 54546-0.zip ******* - - -This and all associated files of various formats will be found in: -http://www.gutenberg.org/dirs/5/4/5/4/54546 - - -Updated editions will replace the previous one--the old editions will -be renamed. - -Creating the works from print editions not protected by U.S. copyright -law means that no one owns a United States copyright in these works, -so the Foundation (and you!) can copy and distribute it in the United -States without permission and without paying copyright -royalties. Special rules, set forth in the General Terms of Use part -of this license, apply to copying and distributing Project -Gutenberg-tm electronic works to protect the PROJECT GUTENBERG-tm -concept and trademark. Project Gutenberg is a registered trademark, -and may not be used if you charge for the eBooks, unless you receive -specific permission. If you do not charge anything for copies of this -eBook, complying with the rules is very easy. You may use this eBook -for nearly any purpose such as creation of derivative works, reports, -performances and research. They may be modified and printed and given -away--you may do practically ANYTHING in the United States with eBooks -not protected by U.S. copyright law. Redistribution is subject to the -trademark license, especially commercial redistribution. - -START: FULL LICENSE - -THE FULL PROJECT GUTENBERG LICENSE -PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK - -To protect the Project Gutenberg-tm mission of promoting the free -distribution of electronic works, by using or distributing this work -(or any other work associated in any way with the phrase "Project -Gutenberg"), you agree to comply with all the terms of the Full -Project Gutenberg-tm License available with this file or online at -www.gutenberg.org/license. - -Section 1. General Terms of Use and Redistributing Project -Gutenberg-tm electronic works - -1.A. By reading or using any part of this Project Gutenberg-tm -electronic work, you indicate that you have read, understand, agree to -and accept all the terms of this license and intellectual property -(trademark/copyright) agreement. If you do not agree to abide by all -the terms of this agreement, you must cease using and return or -destroy all copies of Project Gutenberg-tm electronic works in your -possession. If you paid a fee for obtaining a copy of or access to a -Project Gutenberg-tm electronic work and you do not agree to be bound -by the terms of this agreement, you may obtain a refund from the -person or entity to whom you paid the fee as set forth in paragraph -1.E.8. - -1.B. "Project Gutenberg" is a registered trademark. It may only be -used on or associated in any way with an electronic work by people who -agree to be bound by the terms of this agreement. There are a few -things that you can do with most Project Gutenberg-tm electronic works -even without complying with the full terms of this agreement. See -paragraph 1.C below. There are a lot of things you can do with Project -Gutenberg-tm electronic works if you follow the terms of this -agreement and help preserve free future access to Project Gutenberg-tm -electronic works. See paragraph 1.E below. - -1.C. The Project Gutenberg Literary Archive Foundation ("the -Foundation" or PGLAF), owns a compilation copyright in the collection -of Project Gutenberg-tm electronic works. Nearly all the individual -works in the collection are in the public domain in the United -States. If an individual work is unprotected by copyright law in the -United States and you are located in the United States, we do not -claim a right to prevent you from copying, distributing, performing, -displaying or creating derivative works based on the work as long as -all references to Project Gutenberg are removed. Of course, we hope -that you will support the Project Gutenberg-tm mission of promoting -free access to electronic works by freely sharing Project Gutenberg-tm -works in compliance with the terms of this agreement for keeping the -Project Gutenberg-tm name associated with the work. You can easily -comply with the terms of this agreement by keeping this work in the -same format with its attached full Project Gutenberg-tm License when -you share it without charge with others. - -1.D. The copyright laws of the place where you are located also govern -what you can do with this work. Copyright laws in most countries are -in a constant state of change. If you are outside the United States, -check the laws of your country in addition to the terms of this -agreement before downloading, copying, displaying, performing, -distributing or creating derivative works based on this work or any -other Project Gutenberg-tm work. The Foundation makes no -representations concerning the copyright status of any work in any -country outside the United States. - -1.E. Unless you have removed all references to Project Gutenberg: - -1.E.1. The following sentence, with active links to, or other -immediate access to, the full Project Gutenberg-tm License must appear -prominently whenever any copy of a Project Gutenberg-tm work (any work -on which the phrase "Project Gutenberg" appears, or with which the -phrase "Project Gutenberg" is associated) is accessed, displayed, -performed, viewed, copied or distributed: - - This eBook is for the use of anyone anywhere in the United States and - most other parts of the world at no cost and with almost no - restrictions whatsoever. You may copy it, give it away or re-use it - under the terms of the Project Gutenberg License included with this - eBook or online at www.gutenberg.org. If you are not located in the - United States, you'll have to check the laws of the country where you - are located before using this ebook. - -1.E.2. If an individual Project Gutenberg-tm electronic work is -derived from texts not protected by U.S. copyright law (does not -contain a notice indicating that it is posted with permission of the -copyright holder), the work can be copied and distributed to anyone in -the United States without paying any fees or charges. If you are -redistributing or providing access to a work with the phrase "Project -Gutenberg" associated with or appearing on the work, you must comply -either with the requirements of paragraphs 1.E.1 through 1.E.7 or -obtain permission for the use of the work and the Project Gutenberg-tm -trademark as set forth in paragraphs 1.E.8 or 1.E.9. - -1.E.3. If an individual Project Gutenberg-tm electronic work is posted -with the permission of the copyright holder, your use and distribution -must comply with both paragraphs 1.E.1 through 1.E.7 and any -additional terms imposed by the copyright holder. Additional terms -will be linked to the Project Gutenberg-tm License for all works -posted with the permission of the copyright holder found at the -beginning of this work. - -1.E.4. Do not unlink or detach or remove the full Project Gutenberg-tm -License terms from this work, or any files containing a part of this -work or any other work associated with Project Gutenberg-tm. - -1.E.5. Do not copy, display, perform, distribute or redistribute this -electronic work, or any part of this electronic work, without -prominently displaying the sentence set forth in paragraph 1.E.1 with -active links or immediate access to the full terms of the Project -Gutenberg-tm License. - -1.E.6. You may convert to and distribute this work in any binary, -compressed, marked up, nonproprietary or proprietary form, including -any word processing or hypertext form. However, if you provide access -to or distribute copies of a Project Gutenberg-tm work in a format -other than "Plain Vanilla ASCII" or other format used in the official -version posted on the official Project Gutenberg-tm web site -(www.gutenberg.org), you must, at no additional cost, fee or expense -to the user, provide a copy, a means of exporting a copy, or a means -of obtaining a copy upon request, of the work in its original "Plain -Vanilla ASCII" or other form. Any alternate format must include the -full Project Gutenberg-tm License as specified in paragraph 1.E.1. - -1.E.7. Do not charge a fee for access to, viewing, displaying, -performing, copying or distributing any Project Gutenberg-tm works -unless you comply with paragraph 1.E.8 or 1.E.9. - -1.E.8. You may charge a reasonable fee for copies of or providing -access to or distributing Project Gutenberg-tm electronic works -provided that - -* You pay a royalty fee of 20% of the gross profits you derive from - the use of Project Gutenberg-tm works calculated using the method - you already use to calculate your applicable taxes. The fee is owed - to the owner of the Project Gutenberg-tm trademark, but he has - agreed to donate royalties under this paragraph to the Project - Gutenberg Literary Archive Foundation. Royalty payments must be paid - within 60 days following each date on which you prepare (or are - legally required to prepare) your periodic tax returns. Royalty - payments should be clearly marked as such and sent to the Project - Gutenberg Literary Archive Foundation at the address specified in - Section 4, "Information about donations to the Project Gutenberg - Literary Archive Foundation." - -* You provide a full refund of any money paid by a user who notifies - you in writing (or by e-mail) within 30 days of receipt that s/he - does not agree to the terms of the full Project Gutenberg-tm - License. You must require such a user to return or destroy all - copies of the works possessed in a physical medium and discontinue - all use of and all access to other copies of Project Gutenberg-tm - works. - -* You provide, in accordance with paragraph 1.F.3, a full refund of - any money paid for a work or a replacement copy, if a defect in the - electronic work is discovered and reported to you within 90 days of - receipt of the work. - -* You comply with all other terms of this agreement for free - distribution of Project Gutenberg-tm works. - -1.E.9. If you wish to charge a fee or distribute a Project -Gutenberg-tm electronic work or group of works on different terms than -are set forth in this agreement, you must obtain permission in writing -from both the Project Gutenberg Literary Archive Foundation and The -Project Gutenberg Trademark LLC, the owner of the Project Gutenberg-tm -trademark. Contact the Foundation as set forth in Section 3 below. - -1.F. - -1.F.1. Project Gutenberg volunteers and employees expend considerable -effort to identify, do copyright research on, transcribe and proofread -works not protected by U.S. copyright law in creating the Project -Gutenberg-tm collection. Despite these efforts, Project Gutenberg-tm -electronic works, and the medium on which they may be stored, may -contain "Defects," such as, but not limited to, incomplete, inaccurate -or corrupt data, transcription errors, a copyright or other -intellectual property infringement, a defective or damaged disk or -other medium, a computer virus, or computer codes that damage or -cannot be read by your equipment. - -1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the "Right -of Replacement or Refund" described in paragraph 1.F.3, the Project -Gutenberg Literary Archive Foundation, the owner of the Project -Gutenberg-tm trademark, and any other party distributing a Project -Gutenberg-tm electronic work under this agreement, disclaim all -liability to you for damages, costs and expenses, including legal -fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT -LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE -PROVIDED IN PARAGRAPH 1.F.3. YOU AGREE THAT THE FOUNDATION, THE -TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE -LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR -INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH -DAMAGE. - -1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a -defect in this electronic work within 90 days of receiving it, you can -receive a refund of the money (if any) you paid for it by sending a -written explanation to the person you received the work from. If you -received the work on a physical medium, you must return the medium -with your written explanation. The person or entity that provided you -with the defective work may elect to provide a replacement copy in -lieu of a refund. If you received the work electronically, the person -or entity providing it to you may choose to give you a second -opportunity to receive the work electronically in lieu of a refund. If -the second copy is also defective, you may demand a refund in writing -without further opportunities to fix the problem. - -1.F.4. Except for the limited right of replacement or refund set forth -in paragraph 1.F.3, this work is provided to you 'AS-IS', WITH NO -OTHER WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT -LIMITED TO WARRANTIES OF MERCHANTABILITY OR FITNESS FOR ANY PURPOSE. - -1.F.5. Some states do not allow disclaimers of certain implied -warranties or the exclusion or limitation of certain types of -damages. If any disclaimer or limitation set forth in this agreement -violates the law of the state applicable to this agreement, the -agreement shall be interpreted to make the maximum disclaimer or -limitation permitted by the applicable state law. The invalidity or -unenforceability of any provision of this agreement shall not void the -remaining provisions. - -1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the -trademark owner, any agent or employee of the Foundation, anyone -providing copies of Project Gutenberg-tm electronic works in -accordance with this agreement, and any volunteers associated with the -production, promotion and distribution of Project Gutenberg-tm -electronic works, harmless from all liability, costs and expenses, -including legal fees, that arise directly or indirectly from any of -the following which you do or cause to occur: (a) distribution of this -or any Project Gutenberg-tm work, (b) alteration, modification, or -additions or deletions to any Project Gutenberg-tm work, and (c) any -Defect you cause. - -Section 2. Information about the Mission of Project Gutenberg-tm - -Project Gutenberg-tm is synonymous with the free distribution of -electronic works in formats readable by the widest variety of -computers including obsolete, old, middle-aged and new computers. It -exists because of the efforts of hundreds of volunteers and donations -from people in all walks of life. - -Volunteers and financial support to provide volunteers with the -assistance they need are critical to reaching Project Gutenberg-tm's -goals and ensuring that the Project Gutenberg-tm collection will -remain freely available for generations to come. In 2001, the Project -Gutenberg Literary Archive Foundation was created to provide a secure -and permanent future for Project Gutenberg-tm and future -generations. To learn more about the Project Gutenberg Literary -Archive Foundation and how your efforts and donations can help, see -Sections 3 and 4 and the Foundation information page at -www.gutenberg.org - -Section 3. Information about the Project Gutenberg Literary -Archive Foundation - -The Project Gutenberg Literary Archive Foundation is a non profit -501(c)(3) educational corporation organized under the laws of the -state of Mississippi and granted tax exempt status by the Internal -Revenue Service. The Foundation's EIN or federal tax identification -number is 64-6221541. Contributions to the Project Gutenberg Literary -Archive Foundation are tax deductible to the full extent permitted by -U.S. federal laws and your state's laws. - -The Foundation's principal office is in Fairbanks, Alaska, with the -mailing address: PO Box 750175, Fairbanks, AK 99775, but its -volunteers and employees are scattered throughout numerous -locations. Its business office is located at 809 North 1500 West, Salt -Lake City, UT 84116, (801) 596-1887. Email contact links and up to -date contact information can be found at the Foundation's web site and -official page at www.gutenberg.org/contact - -For additional contact information: - - Dr. Gregory B. Newby - Chief Executive and Director - gbnewby@pglaf.org - -Section 4. Information about Donations to the Project Gutenberg -Literary Archive Foundation - -Project Gutenberg-tm depends upon and cannot survive without wide -spread public support and donations to carry out its mission of -increasing the number of public domain and licensed works that can be -freely distributed in machine readable form accessible by the widest -array of equipment including outdated equipment. Many small donations -($1 to $5,000) are particularly important to maintaining tax exempt -status with the IRS. - -The Foundation is committed to complying with the laws regulating -charities and charitable donations in all 50 states of the United -States. Compliance requirements are not uniform and it takes a -considerable effort, much paperwork and many fees to meet and keep up -with these requirements. We do not solicit donations in locations -where we have not received written confirmation of compliance. To SEND -DONATIONS or determine the status of compliance for any particular -state visit www.gutenberg.org/donate - -While we cannot and do not solicit contributions from states where we -have not met the solicitation requirements, we know of no prohibition -against accepting unsolicited donations from donors in such states who -approach us with offers to donate. - -International donations are gratefully accepted, but we cannot make -any statements concerning tax treatment of donations received from -outside the United States. U.S. laws alone swamp our small staff. - -Please check the Project Gutenberg Web pages for current donation -methods and addresses. Donations are accepted in a number of other -ways including checks, online payments and credit card donations. To -donate, please visit: www.gutenberg.org/donate - -Section 5. General Information About Project Gutenberg-tm electronic works. - -Professor Michael S. Hart was the originator of the Project -Gutenberg-tm concept of a library of electronic works that could be -freely shared with anyone. For forty years, he produced and -distributed Project Gutenberg-tm eBooks with only a loose network of -volunteer support. - -Project Gutenberg-tm eBooks are often created from several printed -editions, all of which are confirmed as not protected by copyright in -the U.S. unless a copyright notice is included. Thus, we do not -necessarily keep eBooks in compliance with any particular paper -edition. - -Most people start at our Web site which has the main PG search -facility: www.gutenberg.org - -This Web site includes information about Project Gutenberg-tm, -including how to make donations to the Project Gutenberg Literary -Archive Foundation, how to help produce our new eBooks, and how to -subscribe to our email newsletter to hear about new eBooks. - |