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+Project Gutenberg (https://www.gutenberg.org) public repository for
+eBook #54546 (https://www.gutenberg.org/ebooks/54546)
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-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***
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-<h1 class="pg">The Project Gutenberg eBook, Time and Clocks, by Sir Henry H. (Henry
-Hardinge) Cunynghame</h1>
-<p>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 <a
-href="http://www.gutenberg.org">www.gutenberg.org</a>.  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.</p>
-<p>Title: Time and Clocks</p>
-<p>       A Description of Ancient and Modern Methods of Measuring Time</p>
-<p>Author: Sir Henry H. (Henry Hardinge) Cunynghame</p>
-<p>Release Date: April 13, 2017  [eBook #54546]</p>
-<p>Language: English</p>
-<p>Character set encoding: UTF-8</p>
-<p>***START OF THE PROJECT GUTENBERG EBOOK TIME AND CLOCKS***</p>
-<p>&nbsp;</p>
-<h4>E-text prepared by deaurider, Charlie Howard,<br />
-    and the Online Distributed Proofreading Team<br />
-    (<a href="http://www.pgdp.net">http://www.pgdp.net</a>)<br />
-    from page images generously made available by<br />
-    Internet Archive<br />
-    (<a href="https://archive.org">https://archive.org</a>)</h4>
-<p>&nbsp;</p>
-<table border="0" style="background-color: #ccccff;margin: 0 auto;" cellpadding="10">
-  <tr>
-    <td valign="top">
-      Note:
-    </td>
-    <td>
-      Images of the original pages are available through
-      Internet Archive. See
-      <a href="https://archive.org/details/timeclocksdescri00cuny">
-      https://archive.org/details/timeclocksdescri00cuny</a>
-    </td>
-  </tr>
-</table>
-<p>&nbsp;</p>
-<hr class="full" />
-<p>&nbsp;</p>
-<p>&nbsp;</p>
-<div class="body">
-<h1 class="wspace">TIME AND CLOCKS.</h1>
-
-<div id="if_i_004" class="figcenter" style="max-width: 30em;">
-  <img src="images/i_004.jpg" width="296" height="634" alt="" />
-  <div class="captionr">[<i>Frontispiece.</i></div>
-  <div class="caption vspace wspace">
-    <p>NUREMBERG CLOCK. CONVERTED FROM A VERGE ESCAPEMENT<br />
-    TO A PENDULUM MOVEMENT.</p>
-  </div>
-</div>
-
-<hr />
-
-<p class="newpage p2 center xxlarge wspace bold">TIME AND CLOCKS:</p>
-
-<p class="p1 center vspace larger wspace"><i>A DESCRIPTION OF ANCIENT<br />
-AND MODERN METHODS OF<br />
-MEASURING TIME</i>.</p>
-
-<p class="p2 center vspace wspace larger"><span class="xsmall">BY</span><br />
-H. H. CUNYNGHAME M.A. C.B. M.I.E.E.</p>
-
-<p class="p2 center"><i>WITH MANY ILLUSTRATIONS.</i></p>
-
-<p class="p2 center larger vspace wspace"><span class="smaller">LONDON:</span><br />
-ARCHIBALD CONSTABLE &amp; CO. <span class="smcap">Ltd.</span><br />
-<span class="smaller">16 JAMES STREET HAYMARKET.<br />
-1906.</span>
-</p>
-
-<hr />
-
-<p class="newpage p4 center small">BRADBURY, AGNEW, &amp; CO. LD., PRINTERS,
-LONDON AND TONBRIDGE.
-</p>
-
-<hr />
-
-<div class="chapter">
-<h2><a id="CONTENTS"></a>CONTENTS.</h2>
-
-<div class="centertable">
-<table summary="Contents">
-  <tr class="small nopad">
-    <td class="tdl" colspan="2"> </td>
-    <td class="tdr nopad">PAGE</td></tr>
-  <tr>
-    <td class="tdl" colspan="2"><span class="smcap">Introduction</span></td>
-    <td class="tdr"><a href="#INTRODUCTION">1</a></td></tr>
-  <tr>
-    <td class="tdl"><span class="smcap">Chapter <span class="c1">I.</span></span></td>
-    <td> </td>
-    <td class="tdr"><a href="#CHAPTER_I">7</a></td></tr>
-  <tr>
-    <td class="tdl"><span class="smcap">Chapter <span class="c2">II.</span></span></td>
-    <td> </td>
-    <td class="tdr"><a href="#CHAPTER_II">50</a></td></tr>
-  <tr>
-    <td class="tdl"><span class="smcap">Chapter <span class="c3">III.</span></span></td>
-    <td> </td>
-    <td class="tdr"><a href="#CHAPTER_III">90</a></td></tr>
-  <tr>
-    <td class="tdl"><span class="smcap">Chapter <span class="c4">IV.</span></span></td>
-    <td> </td>
-    <td class="tdr"><a href="#CHAPTER_IV">123</a></td></tr>
-  <tr>
-    <td class="tdl" colspan="2"><span class="smcap">Appendix on the Shape of the Teeth of Wheels</span></td>
-    <td class="tdr"><a href="#APPENDIX">187</a></td></tr>
-  <tr>
-    <td class="tdl" colspan="2"><span class="smcap">Index</span></td>
-    <td class="tdr"><a href="#INDEX">199</a></td></tr>
-</table></div>
-</div>
-
-<hr />
-
-<p><span class="pagenum"><a id="Page_1">1</a></span></p>
-
-<div class="chapter">
-<h2><a id="TIME_AND_CLOCKS"></a><span class="large wspace">TIME AND CLOCKS.</span></h2>
-
-<hr class="narrow" />
-
-<h2 class="nobreak p1"><a id="INTRODUCTION"></a>INTRODUCTION.</h2>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_2">2</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_3">3</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>The one great requirement for the education<span class="pagenum"><a id="Page_4">4</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>This fatal practice of learning words without
-trying to obtain a clear comprehension of their
-meaning, causes many teachers to use mathematical<span class="pagenum"><a id="Page_5">5</a></span>
-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.”</p>
-
-<p>Another kindred error is to expect too much of
-science. For with all our efforts to obtain a further<span class="pagenum"><a id="Page_6">6</a></span>
-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 id="FNanchor_A" href="#Footnote_A" class="fnanchor">A</a> and all our analysis of
-incomprehensible things leads us only to things
-more incomprehensible still.</p>
-
-<div class="footnote">
-
-<p><a id="Footnote_A" href="#FNanchor_A" class="fnanchor">A</a><cite>Manfred</cite>, Act II., scene iv.</p></div>
-
-<p>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.</p>
-</div>
-
-<hr />
-
-<p><span class="pagenum"><a id="Page_7">7</a></span></p>
-
-<div class="chapter">
-<h2><a id="CHAPTER_I"></a>CHAPTER I.</h2>
-
-<p>All students of any subject are at first apt to be
-perplexed with the number and complication of the
-new ideas presented to them.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>The study of natural phenomena of the world
-around us is called the study of physics from the<span class="pagenum"><a id="Page_8">8</a></span>
-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).</p>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_9">9</a></span>
-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.</p>
-
-<p>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<span class="pagenum"><a id="Page_10">10</a></span>
-power of moving in two dimensions, still less could
-he jump over them, that would want three.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.<span class="pagenum"><a id="Page_11">11</a></span>
-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 <em>mass</em> of a certain
-amount of tea or the <em>weight</em> of that mass. For
-accurate thinking we must, of course, always deal
-with masses, not with weights. For so far as we
-can tell <em>mass</em> appears indestructible. A mass is a<span class="pagenum"><a id="Page_12">12</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>In physics it is usual to employ the letters of the
-alphabet as a sort of shorthand to represent words.<span class="pagenum"><a id="Page_13">13</a></span>
-So that the letter <i>m</i> stands for the mass of a body.
-So again <i>g</i> stands for the attractive pull of the
-earth at a given place. <i>w</i> 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, <i>w</i> = <i>m</i> × <i>g</i>; or <i>w</i> is equal to <i>m</i> multiplied
-by <i>g</i>; 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</p>
-
-<p class="center"><i>w</i> = <i>m</i><i>g</i>.</p>
-
-<p>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.</p>
-
-<p>In the same way, if when estimating the number
-of waggons, <i>w</i>, that would be wanted for an army of
-men, <i>n</i>, which consumed a number of pounds, <i>p</i>, of
-provisions a day, we might put</p>
-
-<p class="center"><i>w</i> = <i>n</i><i>p</i>.</p>
-
-<p>But this would not mean that we were multiplying
-soldiers into food to produce waggons, but only
-that we were performing a numerical calculation.</p>
-
-<p>Time is one of the most mysterious of our
-elementary ideas. It seems to exist or not to exist,<span class="pagenum"><a id="Page_14">14</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_15">15</a></span>
-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.</p>
-
-<p>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.</p>
-
-<div id="ip_15" class="figcenter" style="max-width: 29em;">
-  <img src="images/i_015.jpg" width="464" height="122" alt="" />
-  <div class="caption"><span class="smcap">Fig. 1.</span></div></div>
-
-<p>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?</p>
-
-<p>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 <i>A´</i>.
-In winter it would rise a little to the south of east,<span class="pagenum"><a id="Page_16">16</a></span>
-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.</p>
-
-<p>The moon also would rise at different places,
-varying between <i>A</i> and <i>B</i>, and set at places varying
-between <i>A´</i> and <i>B´</i>, but these would be independent
-of those at which the sun rose and set.</p>
-
-<p>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 <i>M´</i>.
-And in her path she would change her form from<span class="pagenum"><a id="Page_17">17</a></span>
-a crescent, as at <i>M</i>, 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 <i>A´</i>, 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.</p>
-
-<p>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,<span class="pagenum"><a id="Page_18">18</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_19">19</a></span>
-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.</p>
-
-<div id="ip_19" class="figcenter" style="max-width: 19.625em;">
-  <img src="images/i_019.jpg" width="314" height="134" alt="" />
-  <div class="caption"><span class="smcap">Fig. 2.</span></div></div>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_20">20</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p><span class="pagenum"><a id="Page_21">21</a></span></p>
-
-<div id="ip_21" class="figcenter" style="max-width: 19.8125em;">
-  <img src="images/i_021.jpg" width="317" height="242" alt="" />
-  <div class="caption"><span class="smcap">Fig. 3.</span></div></div>
-
-<p>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 <i>A</i> 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 <i>B</i> 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<span class="pagenum"><a id="Page_22">22</a></span>
-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<span class="pagenum"><a id="Page_23">23</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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
-“<i xml:lang="la" lang="la">primum mobile</i>,” 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<span class="pagenum"><a id="Page_24">24</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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 <span class="locked">top—</span></p>
-
-<div class="center"><div class="centerblock">
-<p class="in0">
-Saturn,<br />
-Jupiter,<br />
-Mars,<br />
-The Sun,<br />
-Venus,<br />
-Mercury,<br />
-The Moon.
-</p>
-</div></div>
-
-<p><span class="pagenum"><a id="Page_25">25</a></span>
-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<span class="pagenum"><a id="Page_26">26</a></span>
-corresponds to Venus, and thus the week is
-completed.</p>
-
-<div id="ip_26" class="figcenter" style="max-width: 24.6875em;">
-  <img src="images/i_025.jpg" width="395" height="162" alt="" />
-  <div class="caption"><span class="smcap">Fig. 4.</span></div></div>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>For the use of men engaged in practical astronomy,
-it is very convenient to consider the sun,<span class="pagenum"><a id="Page_27">27</a></span>
-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.</p>
-
-<p>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.</p>
-
-<div id="ip_27" class="figcenter" style="max-width: 29.25em;">
-  <img src="images/i_028.jpg" width="468" height="308" alt="" />
-  <div class="caption"><span class="smcap">Fig. 5.</span></div></div>
-
-<p>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<span class="pagenum"><a id="Page_28">28</a></span>
-the paper would serve to measure the hours, and
-the paper and pen would serve as a sort of sun-dial.</p>
-
-<p>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.</p>
-
-<div id="ip_28" class="figcenter" style="max-width: 25.5625em;">
-  <img src="images/i_029.jpg" width="409" height="424" alt="" />
-  <div class="caption"><span class="smcap">Fig. 6.</span></div></div>
-
-<p>Let us suppose that <i>N O S</i> is the axis of the<span class="pagenum"><a id="Page_29">29</a></span>
-heavens, and the lines <i>N A S</i>, <i>N B S</i>, <i>N C S</i>, are
-meridian lines drawn from one of the poles <i>N</i> of the
-heavens round on the surface of a celestial sphere
-whose centre is at <i>O</i>. Let <i>A B C</i> be a circle also
-on this sphere, passing through <i>O</i>, the centre of
-the sphere, in a plane at right angles to <i>N S</i>, the
-axis. Then <i>A B C</i> is called the equatorial. It is a
-circle in the heavens corresponding to the equator
-on the earth. At the vernal and autumnal equinox,<span class="pagenum"><a id="Page_30">30</a></span>
-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 <i>N</i>, as at <i>V</i>; in midwinter
-it is nearer to <i>S</i>, as at <i>W</i>. 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.</p>
-
-<p>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 <i>E</i>, due east; it traverses half the
-equatorial and sets in the ocean at a point <i>W</i>, due
-west. The day is twelve hours long, from 6 a.m.
-to 6 p.m.</p>
-
-<div id="ip_30" class="figcenter" style="max-width: 29.125em;">
-  <img src="images/i_031.jpg" width="466" height="421" alt="" />
-  <div class="caption"><span class="smcap">Fig. 7.</span></div></div>
-
-<p>In summer the sun is higher, and nearer to the
-pole <i>N</i>, say at a point <i>s</i>. It rises at a point <i>a</i> in
-the ocean more to the north than <i>E</i>, the eastern
-point, and sets at a point <i>b</i>, also more north than
-<i>W</i>, the western point, and traverses the path <i>a s b</i>.
-But to traverse this path it takes longer than
-twelve hours, for <i>a s b</i> is more than half the circle<span class="pagenum"><a id="Page_31">31</a></span>
-<i>a s b</i>. 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 <i>b</i> to <i>a</i>, 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 <i>C</i> in the ocean at
-7.30 a.m., and traverses the arc <i>c t d</i>, and sets at
-the point <i>d</i> at 4.30 p.m. So that the winter day is<span class="pagenum"><a id="Page_32">32</a></span>
-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
-<i>d r c</i> in the interval. It is therefore the obliquity
-of the poles <i>N S</i>, coupled with the fact that the
-sun’s position is nearer to one pole, <i>N</i>, in summer,
-and nearer to the other pole, <i>S</i>, 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 <i>a s b</i> in the summer, or <i>E S W</i> at the
-vernal or autumnal equinoxes, or <i>c t d</i> in the winter,
-in each of these cases, though the places of rising<span class="pagenum"><a id="Page_33">33</a></span>
-and setting in the ocean might vary in summer
-from <i>a</i> and <i>b</i> to <i>c</i> and <i>d</i> in winter, yet in each of
-these cases the path from <i>a</i> to <i>b</i>, <i>A</i> to <i>B</i>, and <i>c</i> to <i>d</i>
-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.</p>
-
-<div id="ip_33" class="figcenter" style="max-width: 28.25em;">
-  <img src="images/i_032.jpg" width="452" height="314" alt="" />
-  <div class="caption"><span class="smcap">Fig. 8.</span></div></div>
-
-<p>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 <i>s</i> nearer to the
-pole <i>N</i>. 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.</p>
-
-<div id="ip_33b" class="figcenter" style="max-width: 28.75em;">
-  <img src="images/i_034.jpg" width="460" height="502" alt="" />
-  <div class="caption"><span class="smcap">Fig. 9.</span></div></div>
-
-<p>Thus then all over the world the period of darkness
-and light is equivalent. At the tropics the<span class="pagenum"><a id="Page_34">34</a></span>
-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
-<i xml:lang="la" lang="la">vice versâ</i>.</p>
-
-<p><span class="pagenum"><a id="Page_35">35</a></span></p>
-
-<div id="ip_35" class="figright" style="max-width: 13.625em;">
-  <img src="images/i_035.jpg" width="218" height="161" alt="" />
-  <div class="caption"><span class="smcap">Fig. 10.</span></div></div>
-
-<p>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 (<a href="#ip_35">Fig. 10</a>).</p>
-
-<div id="ip_35b" class="figright" style="max-width: 13.1875em;">
-  <img src="images/i_035b.jpg" width="211" height="134" alt="" />
-  <div class="caption"><span class="smcap">Fig. 11.</span></div></div>
-
-<p>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
-<i>O A</i>, then when the sun was in the south, at midday,
-the shadow would lie along the same direction,
-<i>O B</i>, whether the sun
-were high in summer,
-as at <i>S</i>, or low in
-winter, as at <i>s</i>. But
-at other hours, such
-as nine o’clock in the
-morning, the shadow of
-the “style” <i>O A</i> would,
-when the sun was at its summer position <i>T</i>, lie
-along <i>O D</i>, whereas when the sun was at its winter
-position <i>t</i> the shadow would lie along <i>O C</i>. Thus
-the time would appear different in summer and in<span class="pagenum"><a id="Page_36">36</a></span>
-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.</p>
-
-<div id="ip_36" class="figright" style="max-width: 11.6875em;">
-  <img src="images/i_036.jpg" width="187" height="177" alt="" />
-  <div class="caption"><span class="smcap">Fig. 12.</span></div></div>
-
-<p>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.</p>
-
-<p>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
-<i>o a</i>, <i>o b</i>, <i>o c</i>, <i>o d</i>, of the inclined “style” <i>O N</i> 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<span class="pagenum"><a id="Page_37">37</a></span>
-is given at the end of this chapter in an
-<a href="#appendix1">appendix</a>.</p>
-
-<p>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.”</p>
-
-<div id="ip_37" class="figright" style="max-width: 8.375em;">
-  <img src="images/i_037.jpg" width="134" height="261" alt="" />
-  <div class="caption"><span class="smcap">Fig. 13.</span></div></div>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_38">38</a></span>
-of the water at the pipe, and this depends in its turn
-upon <i>P Q</i>, the head of water in the vessel. Whence
-it follows that the division <i>Q R</i>, due to say an hour’s
-run of the clock at <i>Q R</i>, will be more than <i>q r</i>,
-the division corresponding to an hour, at <i>q</i>, a point
-lower down between <i>P</i> and <i>Q</i>, 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.</p>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_39">39</a></span>
-such a system the hours in summer were longer
-than in winter.</p>
-
-<p>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 <i>A</i> is the receiving water
-vessel, <i>P</i> the pipe through which the water flows; <i>B</i>
-is the figure, <i>C</i> the rod; <i>D</i> is the drum, made to
-revolve by the cogwheel <i>E</i>, containing 365 teeth, of
-which one tooth was driven forward at the close of<span class="pagenum"><a id="Page_40">40</a></span>
-each day. A syphon <i>G</i> was fixed in the vessel <i>A</i>, so
-that when the figure had risen to the top and pushed
-forward the lever <i>F</i>, the syphon suddenly emptied
-the vessel through the pipe <i>H</i>, and the figure fell
-to the bottom of the vessel <i>A</i> and became ready to<span class="pagenum"><a id="Page_41">41</a></span>
-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.</p>
-
-<div id="ip_41" class="figcenter" style="max-width: 29.125em;">
-  <img src="images/i_040.jpg" width="466" height="544" alt="" />
-  <div class="caption"><span class="smcap">Fig. 14.</span></div></div>
-
-<p>Such water clocks as these were used by the
-ancient Romans.</p>
-
-<p>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<span class="pagenum"><a id="Page_42">42</a></span>
-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.</p>
-
-<p>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<span class="pagenum"><a id="Page_43">43</a></span>
-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.</p>
-
-<p>We use the expression “knots an hour” quite
-incorrectly. It should be “knots per half-minute,”
-or “nautical miles an hour.”</p>
-
-<p>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<span class="pagenum"><a id="Page_44">44</a></span>
-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.</p>
-
-<div id="ip_44" class="figcenter" style="max-width: 23.75em;">
-  <img src="images/i_044.jpg" width="380" height="309" alt="" />
-  <div class="caption"><span class="smcap">Fig. 15.</span></div></div>
-
-<p>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<span class="pagenum"><a id="Page_45">45</a></span>
-an egg comfortably. The extinguisher <i>E</i>, mounted
-on a hinge <i>Q</i>, is turned back, and the spirit lamp
-<i>L</i> is lit. As soon as the water boils, the tap <i>T</i> 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 <i>X</i> 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 <i>W</i> must have a weight <i>P</i> at
-one end of it, and the fulcrum must be
-nearer to that end, so that when empty it
-rests with the end <i>P</i> 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.</p>
-
-<div id="ip_45" class="figright" style="max-width: 1.9375em;">
-  <img src="images/i_045.jpg" width="31" height="223" alt="" />
-  <div class="caption"><span class="smcap">Fig. 16.</span></div></div>
-
-<p>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<span class="pagenum"><a id="Page_46">46</a></span>
-twelve hours for a bubble of mercury to descend in
-it. But the trouble of making so small a hole is
-considerable.</p>
-
-<div id="ip_46" class="figleft" style="max-width: 13.6875em;">
-  <img src="images/i_046.jpg" width="219" height="338" alt="" />
-  <div class="caption"><span class="smcap">Fig. 17.</span></div></div>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>Oil lamps were also used to mark the time.<span class="pagenum"><a id="Page_47">47</a></span>
-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.</p>
-
-<h3 id="appendix1"><span class="smcap">Appendix.</span></h3>
-
-<p>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 <i>A B C</i> with centre <i>O</i>. Make the angle
-<i>B O D</i> equal to the latitude of the place for London
-= 51° 30´´. From <i>A</i> draw <i>A E</i> parallel to <i>O B</i> to
-meet <i>O D</i> in <i>E</i>, and with radius <i>O E</i> describe<span class="pagenum"><a id="Page_48">48</a></span>
-another circle about <i>O</i>. Divide the inner circle
-<i>A B C</i> into twenty-four parts, and draw radii
-through them from <i>O</i> to meet the larger circle.
-Through any divisions (say that corresponding to
-two o’clock) draw lines parallel to <i>O B</i>, <i>O C</i>, respectively
-to meet in <i>a</i>. Then the line <i>O a</i> 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.</p>
-
-<div id="ip_48" class="figcenter" style="max-width: 14.25em;">
-  <img src="images/i_048.jpg" width="228" height="228" alt="" />
-  <div class="caption"><span class="smcap">Fig. 18.</span></div></div>
-
-<p>The centre <i>O</i> 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<span class="pagenum"><a id="Page_49">49</a></span>
-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.</p>
-
-<div id="ip_49" class="figcenter" style="max-width: 16.6875em;">
-  <img src="images/i_049.jpg" width="267" height="273" alt="" />
-  <div class="caption"><span class="smcap">Fig. 19.</span></div></div>
-
-<p>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.</p>
-</div>
-
-<hr />
-
-<p><span class="pagenum"><a id="Page_50">50</a></span></p>
-
-<div class="chapter">
-<h2><a id="CHAPTER_II"></a>CHAPTER II.</h2>
-
-<p>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.</p>
-
-<p>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 <i>tend</i> 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<span class="pagenum"><a id="Page_51">51</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>The only difference between the old methods and
-the new is that in modern times men have learned<span class="pagenum"><a id="Page_52">52</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>His errors lived after him and dominated the
-thought of the middle ages, and for a long time
-delayed the progress of science.</p>
-
-<p>The other great writer on astronomy of ancient
-times was Ptolemy of Alexandria.</p>
-
-<p><span class="pagenum"><a id="Page_53">53</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>A system which, like that of Aristotle and
-Ptolemy, was based on deductions from concepts,<span class="pagenum"><a id="Page_54">54</a></span>
-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.</p>
-
-<div id="ip_54" class="figleft" style="max-width: 11.375em;">
-  <img src="images/i_054.jpg" width="182" height="245" alt="" />
-  <div class="caption"><span class="smcap">Fig. 20.</span></div></div>
-
-<p>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 <i>J</i>. Round this island was an ocean <i>O</i>. 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 <i>C</i>. Heaven was at the opposite
-side <i>H</i> of the earth to Jerusalem, and was<span class="pagenum"><a id="Page_55">55</a></span>
-beyond the circles of the planets, in the <i xml:lang="la" lang="la">primum
-mobile</i>. 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 <i xml:lang="la" lang="la">primum mobile</i> presided over the celestial
-hosts.</p>
-
-<p>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<span class="pagenum"><a id="Page_56">56</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_57">57</a></span>
-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?</p>
-
-<p>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<span class="pagenum"><a id="Page_58">58</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_59">59</a></span>
-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.</p>
-
-<p>“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.”</p>
-
-<p>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 <em>it is called</em> 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.”</p>
-
-<p>Too often are we still inclined to put the name
-for the thing, and to think when we use big words<span class="pagenum"><a id="Page_60">60</a></span>
-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.</p>
-
-<p>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.</p>
-
-<div id="ip_60" class="figcenter" style="max-width: 22.5625em;">
-  <img src="images/i_061.jpg" width="361" height="534" alt="" />
-  <div class="caption"><span class="smcap">Fig. 21.</span></div></div>
-
-<p>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,<span class="pagenum"><a id="Page_61">61</a></span>
-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<span class="pagenum"><a id="Page_62">62</a></span>
-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 <i>P</i> the objects will be released together.
-This can be done by making a loop at the end of
-another piece of string, <i>A</i>, 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.</p>
-
-<div id="ip_62" class="figleft" style="max-width: 12.3125em;">
-  <img src="images/i_062.jpg" width="197" height="173" alt="" />
-  <div class="caption"><span class="smcap">Fig. 22.</span></div></div>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_63">63</a></span>
-chair at Pisa. He then took a professorship at
-Padua.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_64">64</a></span>
-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.</p>
-
-<p>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<span class="pagenum"><a id="Page_65">65</a></span>
-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.”</p>
-
-<p>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<span class="pagenum"><a id="Page_66">66</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>Let us now go on to examine the second fallacy<span class="pagenum"><a id="Page_67">67</a></span>
-that was derived from the Aristotelians, and that so
-long impeded the advance of science, namely, that
-the earth must be at rest.</p>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_68">68</a></span>
-behind the point from which it was dropped, but
-just underneath it.</p>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_69">69</a></span>
-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
-<em>so long as the change is going on</em>. 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.</p>
-
-<div id="ip_69" class="figright" style="max-width: 8em;">
-  <img src="images/i_069.jpg" width="128" height="142" alt="" />
-  <div class="caption"><span class="smcap">Fig. 23.</span></div></div>
-
-<p>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<span class="pagenum"><a id="Page_70">70</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_71">71</a></span>
-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.</p>
-
-<div id="ip_71" class="figright" style="max-width: 10.5625em;">
-  <img src="images/i_071.jpg" width="169" height="200" alt="" />
-  <div class="caption"><span class="smcap">Fig. 24.</span></div></div>
-
-<p>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
-<em>pressures</em>. 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<span class="pagenum"><a id="Page_72">72</a></span>
-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 <em>acts</em>. 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<span class="pagenum"><a id="Page_73">73</a></span>
-the mass is free to move. This is to be our definition
-of force.</p>
-
-<p>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.</p>
-
-<p>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 <em>moving</em> does not tell us that force is acting
-on it. All we know is that, if it is moving, force <em>has</em>
-acted on it. It is only when we see it changing its
-speed or direction, that is changing its motion, that
-we say <em>force</em> 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.</p>
-
-<p><span class="pagenum"><a id="Page_74">74</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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. <em>Weight</em> is the action of<span class="pagenum"><a id="Page_75">75</a></span>
-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 <em>mass</em> 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. <em>Weights</em>, that is earth attractions on masses,
-are proportional to the masses at the same place.
-This, as we have seen, is known by experiment.</p>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_76">76</a></span>
-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.”</p>
-
-<p>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.</p>
-
-<p>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 <em>force</em>, acting
-on <em>mass</em> during a <em>time</em>, produces acceleration.</p>
-
-<p>From this, then, it follows that if a <em>force</em> 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<span class="pagenum"><a id="Page_77">77</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p><span class="pagenum"><a id="Page_78">78</a></span>
-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.</p>
-
-<p>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 <em>numbers only</em>.</p>
-
-<p>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<span class="pagenum"><a id="Page_79">79</a></span>
-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.</p>
-
-<p>Now let us see what the speed will be if the force
-is gravity, that is the attraction of the earth.</p>
-
-<p>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.</p>
-
-<p>What I want to know then is this: how far will a
-body under the action of gravity fall in a second of
-time?</p>
-
-<p>This, of course, is a matter for measurement. If
-we can get a machine to measure seconds, we shall<span class="pagenum"><a id="Page_80">80</a></span>
-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.</p>
-
-<div id="ip_80" class="figleft" style="max-width: 16.3125em;">
-  <img src="images/i_080.jpg" width="261" height="290" alt="" />
-  <div class="caption"><span class="smcap">Fig. 25.</span></div></div>
-
-<p>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.</p>
-
-<p>Can any law be discovered tending to show what
-the force is with which gravity tends to drag a mass
-down an incline?</p>
-
-<p>There is a simple one, and before Galileo’s time<span class="pagenum"><a id="Page_81">81</a></span>
-it had been discovered by Stevinus, an engineer.
-Stevinus’ solution was as follows. Suppose that
-<i>A B C</i> 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 <i>C</i> and no friction anywhere.
-Then the chain will hang at rest. But the
-lower part, from <i>A</i> to <i>B</i>, 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
-<i>A</i> to <i>D</i> will balance the other half from <i>B</i> to <i>D</i>.
-Therefore, like weights in a scale, you may remove
-both, and then the force of gravity acting down the
-plane on the part <i>A C</i> will balance the force of
-gravity acting vertically on the part <i>C B</i>. 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 <i>C A</i> is equal to the gravitational
-force vertically of a piece whose weight
-equals <i>C B</i>. In other words, the force of gravity
-acting down a plane is diminished in the ratio of
-<i>C B</i> to <i>C A</i>.</p>
-
-<p>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,<span class="pagenum"><a id="Page_82">82</a></span>
-instead of G we must put as the accelerating force</p>
-
-<p class="center">G × (vertical height of the end of the plane)/(length of the plane).</p>
-
-<p class="in0">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.</p>
-
-<p>But the slope of the plane is one thirty-sixth to
-the vertical. Therefore the vertical acceleration
-of gravity, <i>i.e.</i>, 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>i.e.</i>,</p>
-
-<p>36 × (36 × 2)/(9)²; and this equals 32 feet per second.</p>
-
-<p>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<span class="pagenum"><a id="Page_83">83</a></span>
-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.</p>
-
-<div id="ip_83" class="figright" style="max-width: 6.5em;">
-  <img src="images/i_083.jpg" width="104" height="173" alt="" />
-  <div class="caption"><span class="smcap">Fig. 26.</span></div></div>
-
-<p>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 <i>W</i>, equal say
-to a hundredth of the combined weight of the weights
-<i>A</i> and <i>B</i> and <i>W</i>, were suddenly put on <i>A</i>, then it
-would descend under an accelerating force equal to
-a hundredth part of ordinary gravity. We should
-then have</p>
-
-<p class="center">S (the space moved through by the weights) = 1/2 × G/100 × <i>t</i>².</p>
-
-<p>With such a system, he found that in 7½ seconds<span class="pagenum"><a id="Page_84">84</a></span>
-the weights moved through 9 feet. Whence he got</p>
-
-<p class="center">9 = 1/2 G/100 × (7½)².</p>
-
-<p class="in0">From which</p>
-
-<p class="center">G = (2 × 9 × 100)/(7½)² = 32 feet per second nearly.</p>
-
-<p>Thus by letting gravity only act on a hundredth
-part of the total weight moved, namely <i>A</i>, <i>B</i>, and
-<i>W</i>, 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.</p>
-
-<p>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<span class="pagenum"><a id="Page_85">85</a></span>
-straight line, which, however, by the earth’s attraction
-is at once deflected into a curve, and altered
-by the resistance of the air.</p>
-
-<div id="ip_85" class="figright" style="max-width: 16.375em;">
-  <img src="images/i_085.jpg" width="262" height="100" alt="" />
-  <div class="caption"><span class="smcap">Fig. 27.</span></div></div>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<div id="ip_85b" class="figright" style="max-width: 16.4375em;">
-  <img src="images/i_086.jpg" width="263" height="98" alt="" />
-  <div class="caption"><span class="smcap">Fig. 28.</span></div></div>
-
-<p>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<span class="pagenum"><a id="Page_86">86</a></span>
-uniform. In the direction in which the accelerating
-force is acting, it will move faster and faster. Thus
-along <i>A B</i> it will proceed uniformly. If it proceeded
-uniformly also along <i>A C</i> (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 <i>A D</i>, the
-obliquity being determined by the relative magnitude
-of the forces acting on it. But how if it went
-uniformly along <i>A B</i>, but at an accelerated pace
-along <i>A C</i>? Then while in equal times the distances
-along <i>A B</i> would be uniform the distances in the
-same times along
-<i>A C</i> would be
-getting bigger
-and bigger. It
-<em>would not describe
-a straight line; it
-would go in a curve</em>. 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 <i>A B</i>
-will be as the squares of the times. Here, then,<span class="pagenum"><a id="Page_87">87</a></span>
-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.</p>
-
-<div id="ip_87" class="figright" style="max-width: 14.875em;">
-  <img src="images/i_087.jpg" width="238" height="368" alt="" />
-  <div class="caption"><span class="smcap">Fig. 29.</span></div></div>
-
-<p>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;<span class="pagenum"><a id="Page_88">88</a></span>
-if I weaken down the force of gravity by making
-the plane nearly horizontal, then it is wide and
-flat.</p>
-
-<p>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.</p>
-
-<p>Instead of peas we can use water. A stream of
-it rushing horizontally out of an orifice will soon
-bend down into a parabola.</p>
-
-<p>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<span class="pagenum"><a id="Page_89">89</a></span>
-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.</p>
-
-<p>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.</p>
-</div>
-
-<hr />
-
-<p><span class="pagenum"><a id="Page_90">90</a></span></p>
-
-<div class="chapter">
-<h2><a id="CHAPTER_III"></a>CHAPTER III.</h2>
-
-<p>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.</p>
-
-<div id="ip_90" class="figleft" style="max-width: 4.375em;">
-  <img src="images/i_090.jpg" width="70" height="390" alt="" />
-  <div class="caption"><span class="smcap">Fig.</span> 30.</div></div>
-
-<p>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, “<span xml:lang="la" lang="la">Ut tensio, sic
-vis</span>”: “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<span class="pagenum"><a id="Page_91">91</a></span>
-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,<span class="pagenum"><a id="Page_92">92</a></span>
-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.</p>
-
-<div id="ip_92" class="figcenter" style="max-width: 16.75em;">
-  <img src="images/i_091.jpg" width="268" height="162" alt="" />
-  <div class="caption"><span class="smcap">Fig. 31.</span></div></div>
-
-<p>Let a heavy ball (<i>A</i>) 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
-<i>B</i> and <i>C</i>. Then when the ball is pulled aside a
-little, say to a position <i>D</i>, it will tend to fly back to<span class="pagenum"><a id="Page_93">93</a></span>
-<i>A</i> with a force proportioned to the distance <i>A D</i>.
-What will be the time it will take to do this? If
-the distance <i>A D</i> 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 <i>A D</i>
-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.</p>
-
-<div id="ip_93" class="figcenter" style="max-width: 22.0625em;">
-  <img src="images/i_092.jpg" width="353" height="211" alt="" />
-  <div class="caption"><span class="smcap">Fig. 32.</span></div></div>
-
-<p>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<span class="pagenum"><a id="Page_94">94</a></span>
-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.</p>
-
-<div id="ip_94" class="figleft" style="max-width: 7.5625em;">
-  <img src="images/i_094.jpg" width="121" height="331" alt="" />
-  <div class="caption"><span class="smcap">Fig. 33.</span></div></div>
-
-<p>Next take a short, stiff spring of
-steel. One of the kind known as
-tuning forks may be employed.</p>
-
-<p>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<span class="pagenum"><a id="Page_95">95</a></span>
-gives the taps to the air by which the note is
-produced.</p>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_96">96</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_97">97</a></span>
-therefore remains invariable, whatever the amplitude
-of the vibration may be.</p>
-
-<p>Upon this law depends the correct going of both
-clocks and watches.</p>
-
-<p>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<span class="pagenum"><a id="Page_98">98</a></span>
-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.</p>
-
-<p>But before we get to that we have yet to go back
-a little.</p>
-
-<p>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.</p>
-
-<p>This greatly interested him, and at home he
-began to try some experiments. As he got older<span class="pagenum"><a id="Page_99">99</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>We will follow these investigations, which will
-largely depend on what we have already learned.</p>
-
-<p>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.</p>
-
-<p>Suppose that we have a flat piece of wood of any
-shape like <a href="#ip_100">Fig. 34</a>, and that we put a screw<span class="pagenum"><a id="Page_100">100</a></span>
-through any spot <i>A</i> in it, no matter where, and
-screw it to a wall, so that it can turn round the
-screw as round a pivot.</p>
-
-<div id="ip_100" class="figcenter" style="max-width: 28.5625em;">
-  <img src="images/i_100.jpg" width="457" height="446" alt="" />
-  <div class="caption"><span class="smcap">Fig. 34.</span></div></div>
-
-<div id="ip_101" class="figright" style="max-width: 14.4375em;">
-  <img src="images/i_101.jpg" width="231" height="291" alt="" />
-  <div class="caption"><span class="smcap">Fig. 35.</span></div></div>
-
-<p>Next we will knock a tintack into any point <i>B</i>,
-and tie a string on to <i>B</i>. Then if I pull at the
-string in any direction <i>B C</i> the board tends to
-twist round the screw at <i>A</i>. What will the strength
-of the twisting force be? It will depend on the<span class="pagenum"><a id="Page_101">101</a></span>
-strength of the pull, and on the “leverage,” or
-distance of the line <i>C B</i> from <i>A</i>. We might imagine
-the string, instead of being attached at <i>B</i>, to be
-attached at <i>D</i>; then, if I put <i>P</i> as the strength
-of the pull, the twisting power would be represented
-by <i>P</i> × <i>A D</i>. This is called the “moment”
-of the force <i>P</i> round
-the centre <i>A</i>. It
-would be the same
-as if I had simply
-an arm <i>A D</i>, and
-pulled upon it with
-the force <i>P</i>. 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.</p>
-
-<p>Now suppose <i>A B</i> is a pendulum, with a bob
-<i>B</i> of 10 lbs. weight, and suppose it has been
-drawn aside out of the vertical so that the bob is
-in the position <i>B</i>. Then the weight of the bob will
-act vertically downwards along the line <i>B C</i>. The
-moment, or twisting power, of the weight will be<span class="pagenum"><a id="Page_102">102</a></span>
-equal to 10 lbs. multiplied by <i>A D</i>, <i>A D</i> being a
-line perpendicular to <i>B C</i>.</p>
-
-<div id="ip_102" class="figcenter" style="max-width: 19.625em;">
-  <img src="images/i_102.jpg" width="314" height="297" alt="" />
-  <div class="caption"><span class="smcap">Fig. 36.</span></div></div>
-
-<p>Now suppose that another string were tied to the
-bob <i>B</i>, and pulled in a direction at right angles to
-<i>A B</i>, with a force <i>P</i> just enough to hold the bob
-back in the position <i>B</i>. The pull along <i>D B</i> × <i>A B</i>
-would be the moment of that pull round the point
-<i>A</i>. But, because this moment just holds the
-pendulum up, it follows that the moment of the
-weight of the pendulum round <i>A</i> is equal to the
-moment of the pull of the string <i>B D</i> round <i>A</i>.</p>
-
-<p class="center">Whence <i>P</i> × <i>A B</i> = 10 lbs. × <i>A D</i>.</p>
-
-<p class="center"><span class="in5">Whence <i>P</i> = 10 lbs. × (<i>A D</i>)/(<i>A B</i>).</span></p>
-
-<p><span class="pagenum"><a id="Page_103">103</a></span>
-But <i>A B</i> 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 <i>E B</i> (which is always
-equal to <i>A D</i>), then the force tending to bring it
-back to <i>E</i> is always proportional to <i>E B</i>. But if
-the pendulum be fairly long, say 39-1/7 inches, and
-the displacement <i>E B</i> 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
-<i>F</i>, its position of rest, is not very different from the
-force tending to bring it back to <i>E</i>. But <i>F B</i> 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 <i>C</i> is a small pulley, and <i>B C</i> a string attached to
-a pendulum <i>A B</i> whose bob is <i>B</i>. Then if a weight
-<i>D</i> be tied to the string and passed over a pulley <i>C</i>,
-the amount <i>F B</i> by which the weight <i>D</i> will deflect
-the bob <i>B</i> is almost exactly proportional to <i>D</i>, so
-long as we only make the deflection <i>E B</i> small,
-that is two or three inches, where say 39-1/7 inches is
-the length <i>A B</i> of the pendulum.</p>
-
-<p>If <i>F B</i> is made too big, then the line <i>B F</i> can no
-longer be considered nearly equal to the arc of
-deflection <i>E B</i>, and the proposition is no longer true.</p>
-
-<p><span class="pagenum"><a id="Page_104">104</a></span>
-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.</p>
-
-<p>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 <i>E</i>, acts
-just as an elastic
-band would act,
-that is to say pulls
-more strongly in
-proportion as the
-distance <i>F B</i> is
-bigger. In fact, if
-we could remove
-the force of gravity
-still leaving the
-mass <i>B</i> 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.</p>
-
-<div id="ip_104" class="figleft" style="max-width: 15.625em;">
-  <img src="images/i_104.jpg" width="250" height="344" alt="" />
-  <div class="caption"><span class="smcap">Fig. 37.</span></div></div>
-
-<p>But the motion of the bob, under the influence<span class="pagenum"><a id="Page_105">105</a></span>
-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.</p>
-
-<p>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).</p>
-
-<p>Now fix a similar pendulum <i>A B</i> 39-1/7 inches long,
-horizontally, with a weight <i>B</i> of 10 lbs. on it. Fasten
-it to a vertical shaft <i>C D</i>, with a tie rod of wire or
-string <i>A B</i> so as to keep it up, and attach to each
-side of the rod <i>A B</i> elastic threads <i>E F</i> and <i>E G</i>.<span class="pagenum"><a id="Page_106">106</a></span>
-Let these threads be tied on at such a point that
-when <i>B</i> 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 <i>A B</i>
-vertically and let it swing on the pivots <i>C</i> and <i>D</i>
-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.</p>
-
-<div id="ip_106" class="figcenter" style="max-width: 26.625em;">
-  <img src="images/i_106.jpg" width="426" height="267" alt="" />
-  <div class="caption"><span class="smcap">Fig. 38.</span></div></div>
-
-<p>It is this curious property of the circle that
-makes the vertical force of gravity on a pendulum<span class="pagenum"><a id="Page_107">107</a></span>
-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.”</p>
-
-<p>But it must be remembered that this equal
-swinging is only approximate, and only true when
-the arc of vibration is small.</p>
-
-<p>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.</p>
-
-<p>The next question that arises is whether the
-weight of the pendulum has any influence upon the
-time of its vibration.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>For inasmuch as the fall of a pendulum is due<span class="pagenum"><a id="Page_108">108</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>Hence, then, you cannot increase the speed of
-the vibration of a pendulum by adding weight
-to the bob.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p><span class="pagenum"><a id="Page_109">109</a></span></p>
-
-<div id="ip_109" class="figright" style="max-width: 16.875em;">
-  <img src="images/i_109.jpg" width="270" height="89" alt="" />
-  <div class="caption"><span class="smcap">Fig. 39.</span></div></div>
-
-<p>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 <i>F B</i>
-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.</p>
-
-<p><span class="pagenum"><a id="Page_110">110</a></span>
-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 <span class="locked">this:—</span></p>
-
-<p>(1) A harmonic motion is one in which the
-accelerating force increases with the distance of
-the body from some fixed point.</p>
-
-<p>(2) Bodies moving harmonically make their
-swings about this point in equal times.</p>
-
-<p>(3) A spring of any sort or shape always has a
-restitutional force proportional to the displacement.</p>
-
-<p>(4) And therefore masses attached to springs
-vibrate in equal times however large the vibration
-may be.</p>
-
-<p>(5) The bob of a pendulum, oscillating backwards
-and forwards, acts like a weight under the
-influence of a spring, and is therefore isochronous.</p>
-
-<p>(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.</p>
-
-<p>We have now to deal with the application of
-these principles to clocks and watches.</p>
-
-<p>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<span class="pagenum"><a id="Page_111">111</a></span>
-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.</p>
-
-<p>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?</p>
-
-<p>First, the machine must have a reservoir of
-force. I can’t get a machine to do work unless I<span class="pagenum"><a id="Page_112">112</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>Such an arrangement, if it can be contrived
-mechanically, is called an “escapement.”</p>
-
-<p>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
-<i>A B</i> has an arm <i>A H</i> attached to it, and there is a
-ratchet <i>C D</i> which engages with the pins. The
-ratchet has a projecting arm <i>E F</i>.</p>
-
-<div id="ip_112" class="figcenter" style="max-width: 21.5625em;">
-  <img src="images/i_113.jpg" width="345" height="401" alt="" />
-  <div class="caption"><span class="smcap">Fig. 40.</span></div></div>
-
-<p>When the pendulum comes back towards the
-end of its beat, the arm <i>A H</i> strikes the arm <i>E F</i>,
-and raises the ratchet <i>C D</i>. 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 <i>G</i> strikes upon the arm
-<i>A H</i>, and thus on its return stroke gives an impetus<span class="pagenum"><a id="Page_113">113</a></span>
-to the pendulum. As the pin <i>G</i> moves forward it
-slides on the arm <i>A H</i> till it slips over the point <i>H</i>.
-The wheel now being free, would fly round were
-it not that when the pendulum returned, and the
-arm <i>A H</i> was lowered, the ratchet had got into
-position again and its point <i>D</i> 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<span class="pagenum"><a id="Page_114">114</a></span>
-moves the pendulum makes the “tick-tock” sound
-with which we are all familiar.</p>
-
-<p>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.</p>
-
-<p>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 <i>G</i>. 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 <i>G</i> at the end
-of the stroke of the pendulum is disadvantageous,
-as it interferes with the free play of the pendulum.</p>
-
-<p><span class="pagenum"><a id="Page_115">115</a></span>
-From all these causes the <a href="#ip_112">above escapement</a> is
-imperfect in character, and would not do where
-precision was required.</p>
-
-<div id="ip_115" class="figright" style="max-width: 12.875em;">
-  <img src="images/i_115.jpg" width="206" height="148" alt="" />
-  <div class="caption"><span class="smcap">Fig. 41.</span></div></div>
-
-<p>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” <i>e f</i>. Across the axis a verge or rod <i>e f</i>
-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<span class="pagenum"><a id="Page_116">116</a></span>
-the balls to a standstill, and ultimately to impart a
-motion in a contrary direction to them.</p>
-
-<p>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 <i>A</i> to <i>B</i> and <i>B</i> to <i>A</i>. 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.</p>
-
-<div id="ip_116" class="figleft" style="max-width: 10.1875em;">
-  <img src="images/i_116.jpg" width="163" height="246" alt="" />
-  <div class="caption"><span class="smcap">Fig. 42.</span></div></div>
-
-<p>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.</p>
-
-<p>Nevertheless it was on this principle that clocks
-were made during the thirteenth, fourteenth and
-fifteenth centuries. They were mostly made for<span class="pagenum"><a id="Page_117">117</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_118">118</a></span>
-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.</p>
-
-<p>In this manner small
-table clocks were made
-which kept very tolerable
-time.</p>
-
-<div id="ip_118" class="figleft" style="max-width: 12.375em;">
-  <img src="images/i_118.jpg" width="198" height="302" alt="" />
-  <div class="caption"><span class="smcap">Fig. 43.</span></div></div>
-
-<p>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.</p>
-
-<p><span class="pagenum"><a id="Page_119">119</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_120">120</a></span>
-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.</p>
-
-<div id="ip_120" class="figcenter" style="max-width: 16.0625em;">
-  <img src="images/i_120.jpg" width="257" height="238" alt="" />
-  <div class="caption"><span class="smcap">Fig. 44.</span></div></div>
-
-<p><span class="pagenum"><a id="Page_121">121</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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<i>s.</i> The wheels<span class="pagenum"><a id="Page_122">122</a></span>
-are well cut, and the internal mechanism very
-good.</p>
-
-<p>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.</p>
-</div>
-
-<hr />
-
-<p><span class="pagenum"><a id="Page_123">123</a></span></p>
-
-<div class="chapter">
-<h2><a id="CHAPTER_IV"></a>CHAPTER IV.</h2>
-
-<p>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.</p>
-
-<div id="ip_123" class="figcenter" style="max-width: 14.6875em;">
-  <img src="images/i_123.jpg" width="235" height="426" alt="" />
-  <div class="caption"><span class="smcap">Fig. 45.</span></div></div>
-
-<p>In the first place, we have to consider the axle
-which carries the minute hand, and which turns<span class="pagenum"><a id="Page_124">124</a></span>
-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.</p>
-
-<p>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 <i>S T</i> the clock faces, and <i>C</i>, the arbor
-of the minute hand.</p>
-
-<p>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.</p>
-
-<div id="ip_124" class="figleft" style="max-width: 15.75em;">
-  <img src="images/i_124.jpg" width="252" height="71" alt="" />
-  <div class="caption"><span class="smcap">Fig. 46.</span></div></div>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>It is then clear that if the arbor <i>A</i> turns round<span class="pagenum"><a id="Page_125">125</a></span>
-once in an hour, the arbor <i>B</i> will turn round
-eight times in an hour, and <i>C</i> will turn round
-(60 × 64)/(8 × 8) = 60 times in an hour, or once in each
-minute.</p>
-
-<p>By having 480 teeth on the cogwheel on <i>A</i>, you
-could, of course, make <i>C</i> go round once in a minute
-without the use of any intermediate arbor such as <i>B</i>.</p>
-
-<div id="ip_125" class="figcenter" style="max-width: 21.0625em;">
-  <img src="images/i_125.jpg" width="337" height="324" alt="" />
-  <div class="caption"><span class="smcap">Fig. 47.</span></div></div>
-
-<p>But this would not be a very convenient plan.
-For as the wheel on <i>A</i> 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<span class="pagenum"><a id="Page_126">126</a></span>
-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 <i>A</i> 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.</p>
-
-<div id="ip_126" class="figleft" style="max-width: 3.9375em;">
-  <img src="images/i_126.jpg" width="63" height="223" alt="" />
-  <div class="caption"><span class="smcap">Fig. 48.</span></div></div>
-
-<p>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,<span class="pagenum"><a id="Page_127">127</a></span>
-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.</p>
-
-<div id="ip_127" class="figright" style="max-width: 15.3125em;">
-  <img src="images/i_127.jpg" width="245" height="409" alt="" />
-  <div class="caption"><span class="smcap">Fig. 49.</span></div></div>
-
-<p>If we put a pinion
-of eight leaves on
-the minute hand
-arbor <i>c</i>, and engage
-it with a wheel of
-sixty-four teeth on
-another arbor <i>b</i>,
-then <i>b</i> 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 <i>b</i> 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<span class="pagenum"><a id="Page_128">128</a></span>
-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.</p>
-
-<div id="ip_128" class="figright" style="max-width: 12.0625em;">
-  <img src="images/i_129.jpg" width="193" height="381" alt="" />
-  <div class="caption"><span class="smcap">Fig. 50.</span></div></div>
-
-<p>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<span class="pagenum"><a id="Page_129">129</a></span>
-it, as shown in the figure <i>A</i>. A pall is a piece of
-metal, mounted on a pin, and kept pressed up
-against the ratchet wheel by a spring <i>C</i>. It is
-obvious that if I turn the wheel <i>A</i> round, and
-thus wind up a weight, fastened to a cord wound
-round the drum <i>D</i>, that
-the pall <i>B</i> 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 <i>E</i> will
-pull on the drum <i>D</i>, and
-try and turn the ratchet
-wheel back the opposite
-way to that in which I
-twisted it at first. If
-the pall <i>B</i> is held fast,
-it is impossible to move
-it, but if the pall is fixed
-to a cogwheel <i>F</i>, which rides loose on the arbor
-of the drum <i>D</i>, then the pull of the weight <i>E</i>
-will tend to twist the cogwheel <i>F</i> 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 <i>E</i><span class="pagenum"><a id="Page_130">130</a></span>
-gradually runs down, and, at last all the string
-is unwound from the drum <i>D</i>. 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 <i>D</i>, 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.</p>
-
-<div id="ip_130" class="figleft" style="max-width: 14.875em;">
-  <img src="images/i_130.jpg" width="238" height="396" alt="" />
-  <div class="caption"><span class="smcap">Fig. 51.</span></div></div>
-
-<p>I have thus described
-the winding
-mechanism. It now
-remains to describe the escapement.</p>
-
-<p>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<span class="pagenum"><a id="Page_131">131</a></span>
-to go one-sixtieth part of its journey round in each
-second. In fact, we need an “escapement” and a
-pendulum.</p>
-
-<p>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.</p>
-
-<p>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 (<a href="#ip_120">Fig. 44</a>). 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<span class="pagenum"><a id="Page_132">132</a></span>
-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.</p>
-
-<p>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<span class="pagenum"><a id="Page_133">133</a></span>
-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.</p>
-
-<p>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.</p>
-
-<div id="ip_135" class="figright" style="max-width: 9.75em;">
-  <img src="images/i_135.jpg" width="156" height="175" alt="" />
-  <div class="caption"><span class="smcap">Fig. 52.</span></div></div>
-
-<p>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<span class="pagenum"><a id="Page_134">134</a></span>
-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 <em>freely</em> 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<span class="pagenum"><a id="Page_135">135</a></span>
-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.</p>
-
-<p>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 <i>a b</i>
-and a sliding face <i>b c</i>.</p>
-
-<div id="ip_135b" class="figcenter" style="max-width: 17.1875em;">
-  <img src="images/i_135b.jpg" width="275" height="63" alt="" /></div>
-
-<p>When the tooth <i>b</i> 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,<span class="pagenum"><a id="Page_136">136</a></span>
-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 <i>a´ b´</i>. As soon, however, as the tooth leaves
-<i>b´</i>, another tooth on the other side at once engages
-the sliding face <i>b c</i> of the other pallet, and so the
-motion goes on.</p>
-
-<p>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.</p>
-
-<p>There are several other ways even better than
-this for securing a free pendulum movement. We
-have now to return to our clock.</p>
-
-<p>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 <i>N</i> 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 <a href="#ip_123">Fig. 45</a>). 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 <i>n</i> has thirty teeth.<span class="pagenum"><a id="Page_137">137</a></span>
-This cogwheel engages another cogwheel <i>o</i> with thirty
-teeth, on a separate arbor, which carries a third
-cogwheel, <i>p</i>, with six teeth, and this again engages a
-fourth cogwheel, <i>q</i>, with seventy-two teeth, mounted
-on a tube which slips over the tube to which the
-cogwheel <i>a</i> is attached. It is now easy to see that
-for each turn of the minute hand arbor the
-arbor <i>p</i> makes one turn, and for each turn of the
-arbor <i>p</i> the cogwheel <i>d</i>, makes one-twelfth of a turn.
-From which it follows that for each turn of the
-minute hand arbor the cogwheel <i>d</i> 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.</p>
-
-<p>The minute hand is attached to the tube or slieve
-which carries the cogwheel <i>N</i>. The hour hand is
-attached to the tube or slieve which carries the
-cogwheel <i>Q</i>, and one goes twelve times as slowly
-as the other.</p>
-
-<p>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<span class="pagenum"><a id="Page_138">138</a></span>
-lynch-pin driven through a hole in the minute
-hand arbor.</p>
-
-<p>It remains to explain a few subsidiary arrangements,
-not always found upon all clocks, but which
-are useful.</p>
-
-<p>In order to prevent the overwinding of the clock
-(see <a href="#ip_118">Fig. 43</a>), 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.</p>
-
-<p>Another arrangement is the “maintaining spring.”</p>
-
-<p>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<span class="pagenum"><a id="Page_139">139</a></span>
-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.</p>
-
-<p>The arrangement shown in <a href="#ip_139">Fig. 53</a> will explain it.</p>
-
-<div id="ip_139" class="figcenter" style="max-width: 19.375em;">
-  <img src="images/i_139.jpg" width="310" height="331" alt="" />
-  <div class="caption"><span class="smcap">Fig. 53.</span></div></div>
-
-<p>The cogwheel <i>a</i> and the ratchet wheel are both
-mounted loosely on the arbor carrying the drum.
-<i>a</i> is linked to <i>b</i> by a spring <i>c</i>. The ratchet wheel
-<i>b</i> 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 <i>b</i>, and this acts on the spring <i>c</i> and<span class="pagenum"><a id="Page_140">140</a></span>
-pulls it out a little. As soon as the spring <i>c</i> is
-pulled out as far as its elasticity permits, a pull is
-communicated to the cogwheel <i>a</i>, and the clock is
-driven round. When the clock is wound the
-pressure of the weight is removed, and therefore
-the ratchet wheel <i>e</i> no longer presses on the pall,
-and thus no pressure is communicated to the ratchet
-wheel <i>b</i>, or through it to the clock. But here the
-spring <i>c</i> comes into play. For since the ratchet
-wheel <i>b</i> is held fast by the pall <i>d</i>, the spring <i>c</i>
-pulls at the wheel <i>a</i>, 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.</p>
-
-<p>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.</p>
-
-<p>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.<span class="pagenum"><a id="Page_141">141</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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 =</p>
-
-<p class="center">9/(2 × 24 × 60 × 60 × 8) feet or 1/12800th inch.</p>
-
-<p class="in0">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.</p>
-
-<p><span class="pagenum"><a id="Page_142">142</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<div id="ip_142" class="figright" style="max-width: 14.75em;">
-  <img src="images/i_143.jpg" width="236" height="274" alt="" />
-  <div class="caption"><span class="smcap">Fig. 54.</span></div></div>
-
-<p>As has been previously said, the power of the<span class="pagenum"><a id="Page_143">143</a></span>
-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 <a href="#appendix4">appendix</a> 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<span class="pagenum"><a id="Page_144">144</a></span>
-which you must measure in order to get its length.
-So that if <i>B</i> were a circular bob, and the rod of the
-pendulum were very light, the distance <i>A B</i> 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 <em>roll</em> 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 <i>C</i> 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 <i>C</i> were always uppermost, as, for instance,
-if the pendulum consisted of two parallel rods, <i>A B</i>
-and <i>C D</i>, suspended from <i>A</i> and <i>C</i>, then we might
-consider the bob as that of a pendulum suspended
-from <i>E</i>, and the pendulum would swing once in a<span class="pagenum"><a id="Page_145">145</a></span>
-second if <i>A B</i> = <i>C D</i> = <i>E F</i> 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 <i>E F</i>, the
-rotation must be taken into account.</p>
-
-<div id="ip_145" class="figleft" style="max-width: 12.25em;">
-  <img src="images/i_144.jpg" width="196" height="260" alt="" />
-  <div class="caption"><span class="smcap">Fig. 55.</span></div></div>
-
-<p>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.</p>
-
-<div id="ip_145b" class="figright" style="max-width: 9.5em;">
-  <img src="images/i_145.jpg" width="152" height="259" alt="" />
-  <div class="caption"><span class="smcap">Fig. 56.</span></div></div>
-
-<p>The mode of suspension of
-a pendulum of the best class
-is that shown in <a href="#ip_145b">Fig. 56</a>,
-which allows the pendulum
-to fall into its true position without strain. <i>A</i> 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 <i>t</i> 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<span class="pagenum"><a id="Page_146">146</a></span>
-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.</p>
-
-<div id="ip_146" class="figcenter" style="max-width: 16.8125em;">
-  <img src="images/i_146.jpg" width="269" height="147" alt="" />
-  <div class="caption"><span class="smcap">Fig. 57.</span></div></div>
-
-<p>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.<span class="pagenum"><a id="Page_147">147</a></span>
-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.</p>
-
-<div id="ip_147" class="figleft" style="max-width: 15.3125em;">
-  <img src="images/i_147.jpg" width="245" height="272" alt="" />
-  <div class="caption"><span class="smcap">Fig. 58.</span></div></div>
-
-<p>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<span class="pagenum"><a id="Page_148">148</a></span>
-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.</p>
-
-<div id="ip_149" class="figright" style="max-width: 4.6875em;">
-  <img src="images/i_149.jpg" width="75" height="403" alt="" />
-  <div class="caption"><span class="smcap">Fig. 59.</span></div></div>
-
-<p>Hence a zinc tube, about twenty inches long
-(shown shaded in <a href="#ip_149">Fig. 59</a>), 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 <i>A</i>, from which is suspended an iron
-tube <i>A</i>, which carries the bob <i>B</i>. 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.</p>
-
-<p><span class="pagenum"><a id="Page_149">149</a></span>
-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.</p>
-
-<p>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<span class="pagenum"><a id="Page_150">150</a></span>
-increase of swing makes the pendulum go slower
-by about a second a day.</p>
-
-<p>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.</p>
-
-<h3 id="appendix4"><span class="smcap">Appendix to Chapter IV.</span></h3>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_151">151</a></span>
-as it flies round equals V. Let <i>a</i> be the body whirled
-round by a string <i>o a</i> from a centre at <i>O</i>. 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 <i>O</i> 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.</p>
-
-<div id="ip_151" class="figright" style="max-width: 11.625em;">
-  <img src="images/i_151.jpg" width="186" height="217" alt="" />
-  <div class="caption"><span class="smcap">Fig. 60.</span></div></div>
-
-<p>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.</p>
-
-<p>Suppose that at some given instant of time the
-body is at a point <i>a</i>. At that instant its <em>direction</em>
-will be along <i>a b</i>, tangential to the circle at <i>a</i>, and
-that is the path it would take if the centripetal or
-attractive force ceased to act just as the body got
-to <i>a</i>. In that case the body would be whirled off<span class="pagenum"><a id="Page_152">152</a></span>
-like a stone from a sling along the line <i>a b</i>, and would
-at the end of a given time, let us suppose a second,
-arrive at <i>b</i>. But it is not so whirled off; it is attracted
-towards <i>O</i> and pulled inwards, and comes to <i>c</i>. Hence,
-then, the attractive force acting during one second
-must have been sufficient to pull the mass in from
-<i>b</i> to <i>c</i>. 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.</p>
-
-<p>Hence, then, the space <i>b c</i>, by which the body
-has been pulled in, is represented by half F, but
-<i>a b</i>, 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 <i>b c</i> travelled in a second is very
-small, then the triangles <i>a b d</i> and <i>a b c</i> are approximately
-similar, and the smaller <i>a b</i> is the more
-nearly similar they are. Whence then (<i>a b</i>)/(<i>b c</i>) = (<i>a d</i>)/(<i>a b</i>),
-that is to say (<i>a b</i>)² = <i>a d</i> × <i>b c</i>.</p>
-
-<p>But <i>a b</i> 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; <i>a d</i> is
-the diameter of the circle, and hence equals 2 R;
-<i>b c</i> is the space through which the body has been
-drawn in the second by the attractive force F, and
-therefore equals half F.</p>
-
-<p>Whence then V² = 2 R × half F = R F.</p>
-
-<p><span class="pagenum"><a id="Page_153">153</a></span>
-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.</p>
-
-<p>The time of rotation, is of course = length of the
-path ÷ velocity</p>
-
-<p class="center">= (2πR)/V = (2πR)/√(R F) = 2π√(R/F).</p>
-
-<p>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<span class="pagenum"><a id="Page_154">154</a></span>
-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.</p>
-
-<p>It also follows that if a body hung by a string
-of length <i>l</i>, under the action of gravity, be travelling
-in a circle round and round, then, <em>if the circle is a
-small one compared with the length of the string</em>, the
-inward acceleration <i>f</i> towards the centre will be
-approximately proportional to the radius <i>r</i> of the
-circle, and the time of rotation will be</p>
-
-<p class="center"><i>t</i> = 2π√(<i>r</i>/<i>f</i>).</p>
-
-<p class="in0">But in this case <i>f</i>, the inward acceleration, is to
-<i>g</i> the acceleration downwards of gravity as A B:A P
-or</p>
-
-<p class="center"><i>f</i>/<i>g</i> = (A B)/(A P) = (A P)/(O P) = <i>r</i>/<i>l</i>.</p>
-
-<div class="center"><div class="container">
-<div id="ip_154" class="figleft fig61" style="max-width: 8.9375em;">
-  <img src="images/i_154.jpg" width="143" height="181" alt="" />
-  <div class="caption"><span class="smcap">Fig. 61.</span></div></div>
-
-<div id="ip_154b" class="figright" style="max-width: 6.4375em;">
-  <img src="images/i_154b.jpg" width="103" height="217" alt="" />
-  <div class="caption"><span class="smcap">Fig. 62.</span></div></div>
-</div></div>
-
-<p>Whence then the time of rotation of this body
-would be if the circle of rotation was small</p>
-
-<p><span class="pagenum"><a id="Page_155">155</a></span></p>
-
-<p class="center">= 2π√(<i>l</i>/<i>g</i>).</p>
-
-<p class="in0">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
-<em>small</em> circle. Now <i>g</i> 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√(<i>l</i>/<i>g</i>) and its
-motion harmonic.</p>
-
-<p>The length of a seconds pendulum, that is a
-pendulum that makes its double swing in two
-seconds, will therefore be</p>
-
-<p class="center"><span class="in1"><i>l</i> = 4/((2π)²) × <i>g</i> feet</span></p>
-
-<p class="center"><span class="in2">= (<i>g</i> × 12)/π² inches</span></p>
-
-<p class="center"> = 39·14 inches.</p>
-</div>
-
-<hr />
-
-<p><span class="pagenum"><a id="Page_156">156</a></span></p>
-
-<div class="chapter">
-<h2><a id="CHAPTER_V"></a>CHAPTER V.</h2>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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?</p>
-
-<p>The answer to this is, that it is under the<span class="pagenum"><a id="Page_157">157</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>For watches, however, a different plan was contrived.
-One end of a slender spiral spring was<span class="pagenum"><a id="Page_158">158</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>But the escapement of the early watches left<span class="pagenum"><a id="Page_159">159</a></span>
-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.</p>
-
-<p>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, <em>but not otherwise</em>,
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>Now about this time—I am speaking of the<span class="pagenum"><a id="Page_160">160</a></span>
-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.</p>
-
-<p>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
-<em>direction</em> he was sailing in, but it was quite a
-problem to determine where he was on the surface
-of the earth.</p>
-
-<div id="ip_160" class="figcenter" style="max-width: 25em;">
-  <img src="images/i_161.jpg" width="400" height="405" alt="" />
-  <div class="caption"><span class="smcap">Fig. 63.</span></div></div>
-
-<p>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<span class="pagenum"><a id="Page_161">161</a></span>
-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, <i>a</i> to <i>b</i>, <i>b</i> to <i>c</i>, etc., from
-pole to pole. This will divide the earth up like a<span class="pagenum"><a id="Page_162">162</a></span>
-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 <i>p q</i> 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 <i>q s</i>
-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.</p>
-
-<div id="ip_162" class="figleft" style="max-width: 6.5em;">
-  <img src="images/i_162.jpg" width="104" height="176" alt="" />
-  <div class="caption"><span class="smcap">Fig. 64.</span></div></div>
-
-<p>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
-<i>p q</i> = 37 nautical miles and <i>q s</i> = 60 nautical
-miles.</p>
-
-<p>Now of course we can see at once that it is easy
-at any place on the earth’s surface to find your
-<em>latitude</em> 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<span class="pagenum"><a id="Page_163">163</a></span>
-nautical almanac the angle he is above the equator,
-you can soon determine your place <i>A</i> on the globe.
-Or at night, if you measure the angular distance
-that the polar star <i>P</i> is from the zenith, or point
-exactly over your head—that is, the angle <i>P O Z</i>—you
-can subtract it from a right angle and get your
-latitude, <i>A O E</i>, at once.</p>
-
-<div id="ip_163" class="figcenter" style="max-width: 26.75em;">
-  <img src="images/i_163.jpg" width="428" height="402" alt="" />
-  <div class="caption"><span class="smcap">Fig. 65.</span></div></div>
-
-<p>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,<span class="pagenum"><a id="Page_164">164</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_165">165</a></span>
-£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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>The peculiarity of the chronometer is its escapement.</p>
-
-<p>Let <i>A B</i> be the scape wheel, and <i>C D</i> a small
-lever attached to <i>C</i>, the pivot on which the balance
-wheel and spring is fastened. Let <i>E G</i> be a lever,
-with a tooth <i>F</i> which engages the teeth of the
-scape wheel and prevents it moving round. Let
-<i>H</i> be a spring holding the lever <i>E G</i> up to its
-work.</p>
-
-<p><span class="pagenum"><a id="Page_166">166</a></span></p>
-
-<div id="ip_166" class="figcenter" style="max-width: 17.125em;">
-  <img src="images/i_166.jpg" width="274" height="421" alt="" />
-  <div class="caption"><span class="smcap">Fig. 66.</span></div></div>
-
-<p>The lever has a spring <i>K E</i> fastened to it at
-the point <i>K</i>. This spring is very delicate. If the
-lever <i>C D</i> is turned so that the little projection
-<i>M</i> on it strikes the spring <i>E</i> 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 <i>C D</i> is
-just opposite the tooth <i>D</i> of the scape wheel, so
-that the scape wheel, instead of running away,<span class="pagenum"><a id="Page_167">167</a></span>
-leaps with its tooth <i>D</i> on to the lever <i>C D</i> 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
-<i>M</i> passes the spring <i>E</i>, then the lever <i>G E</i> 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 <i>M</i> merely flips
-aside the spring <i>E</i> and the balance wheel goes
-back, till again it is brought to rest and returns.
-As soon as the lever comes opposite <i>D</i> the projection
-<i>M</i> then again hits the spring <i>E</i>, and releases
-the catch at <i>F</i>, and another tooth of the scape
-wheel goes by.</p>
-
-<p>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.</p>
-
-<p>There is but little remaining in a watch that is
-not in a clock, for the wheel-trains and general
-arrangements are very similar.</p>
-
-<p>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:</p>
-
-<p><i>A</i> is a block of metal fitted to the bottom of the
-pendulum, <i>B</i> a light lever pivoted on it. <i>C</i> is the<span class="pagenum"><a id="Page_168">168</a></span>
-scape wheel, with four teeth; <i>D</i> a tooth of the
-scape wheel, which hops on to the projection of
-the pendulum the moment that the impact of the
-point <i>E</i> of the lever <i>B E</i> has pushed aside the
-lever <i>G F</i>, 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.</p>
-
-<div id="ip_168" class="figleft" style="max-width: 16.375em;">
-  <img src="images/i_168.jpg" width="262" height="288" alt="" />
-  <div class="caption"><span class="smcap">Fig. 67.</span></div></div>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p><span class="pagenum"><a id="Page_169">169</a></span>
-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 <cite>Encyclopædia Britannica</cite>,
-written by Lord Grimthorpe himself.</p>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_170">170</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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<span class="pagenum"><a id="Page_171">171</a></span>
-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.</p>
-
-<p>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<span class="pagenum"><a id="Page_172">172</a></span>
-which occasionally occurs of a “Grandfather” clock
-striking the whole of the hours for the week
-without stopping.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p><span class="pagenum"><a id="Page_173">173</a></span>
-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.</p>
-
-<p>Electric clocks of
-many kinds have
-been invented. The
-principle of an electric
-escapement is
-similar to that of
-an ordinary escapement.</p>
-
-<div id="ip_173" class="figright" style="max-width: 15.1875em;">
-  <img src="images/i_173.jpg" width="243" height="347" alt="" />
-  <div class="caption"><span class="smcap">Fig. 68.</span></div></div>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>If put at a proper position it would at each time<span class="pagenum"><a id="Page_174">174</a></span>
-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 <i>P</i>
-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 <i>R</i> strikes
-the obstruction. But if that side <i>R</i> 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.</p>
-
-<p>Such pendulums do not act very well, because it
-is difficult to keep metallic surfaces like <i>Q</i> clean,
-and therefore misses often occur. Besides, the
-strength of the current varies with the goodness
-of the contact and with other things.</p>
-
-<p>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.</p>
-
-<p>The advantage of electricity is, that by means of
-the current that actuates the clock, or winds it up,<span class="pagenum"><a id="Page_175">175</a></span>
-you can at regular intervals set the hands in motion
-of a great number of clocks.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p><span class="pagenum"><a id="Page_176">176</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p><span class="pagenum"><a id="Page_177">177</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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<i>s.</i>
-than could have been got for £30 twenty years ago.</p>
-
-<p>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<i>s.</i> for the<span class="pagenum"><a id="Page_178">178</a></span>
-watch, but you have £28 10<i>s.</i> left to spend on
-pictures.</p>
-
-<p>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 <em>ornament</em> be
-hand work, your utility machine work.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>The tuning fork consists of a piece of steel bent
-into a U shape. The arms are set vibrating so as<span class="pagenum"><a id="Page_179">179</a></span>
-alternately to approach and recede from one
-another.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>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?</p>
-
-<div id="ip_180" class="figcenter" style="max-width: 20.5em;">
-  <img src="images/i_180.jpg" width="328" height="443" alt="" />
-  <div class="caption"><span class="smcap">Fig. 69.</span></div></div>
-
-<p>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,<span class="pagenum"><a id="Page_180">180</a></span>
-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.</p>
-
-<div id="ip_181" class="figright" style="max-width: 5.125em;">
-  <img src="images/i_181.jpg" width="82" height="309" alt="" />
-  <div class="caption"><span class="smcap">Fig. 70.</span></div></div>
-
-<p><span class="pagenum"><a id="Page_181">181</a></span>
-If an arm <i>a b</i> be fitted to move sideways when
-a little string <i>c d</i> 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 <i>a</i> is the trace of the tuning
-fork, <i>b</i> that of the indicating arm. The
-time which has elapsed between the
-jerk which produced the indentation <i>c</i>
-and that which produced the indentation
-<i>d</i> 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.</p>
-
-<p>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<span class="pagenum"><a id="Page_182">182</a></span>
-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.</p>
-
-<div id="ip_182" class="figcenter" style="max-width: 22.6875em;">
-  <img src="images/i_182.jpg" width="363" height="162" alt="" />
-  <div class="caption"><span class="smcap">Fig. 71.</span></div></div>
-
-<p>Such an apparatus is used in modern gunnery
-experiments. It is an elaborate one, but is based
-on the principle above described.</p>
-
-<p>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
-<i>E</i> 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 <i>Q</i>, from which a<span class="pagenum"><a id="Page_183">183</a></span>
-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 <i>R</i>. When the
-fork vibrates the arm, being bent outwards, makes
-the wire <i>R</i> touch <i>Q</i>. 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 <a href="#ip_182">Fig. 71</a>
-with <a href="#ip_173">Fig. 68</a>.</p>
-
-<p>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,</p>
-
-<p class="center">S = 1/2 <i>g</i> <i>t</i>².</p>
-
-<p class="in0">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.</p>
-
-<p>An interesting adaptation of this method can
-serve as a basis of a curious toy.</p>
-
-<p>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 <i>a</i> on a wooden wall. On the spot <i>a</i> 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 <i>a</i> be the cat,<span class="pagenum"><a id="Page_184">184</a></span>
-<i>b</i> the loop by which it is hung over the nail <i>c</i>, that is
-fixed to another piece of iron furnished with a hinge
-at <i>c</i>, so that when the electric current is turned on
-the nail <i>c</i> 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.</p>
-
-<p>Now here you have an apparatus such that exactly
-as the bolt leaves the crossbow, the cat drops. Now
-what will happen?</p>
-
-<div id="ip_184" class="figcenter" style="max-width: 36.5625em;">
-  <img src="images/i_184.jpg" width="585" height="160" alt="" />
-  <div class="caption"><span class="smcap">Fig. 72.</span></div></div>
-
-<p>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 <i>b a</i> from the bow to the
-target.</p>
-
-<p>But it is also subject to another force, namely
-that of gravity, which acts on it vertically, and
-deflects it <em>in a vertical direction</em> 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<span class="pagenum"><a id="Page_185">185</a></span>
-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 <i>a</i> to <i>c</i>, the cat will also have fallen vertically
-from <i>a</i> to <i>c</i>, 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 <i>a b</i> is
-horizontal or pointing upwards
-or downwards.</p>
-
-<div id="ip_185" class="figright" style="max-width: 9.25em;">
-  <img src="images/i_185.jpg" width="148" height="241" alt="" />
-  <div class="caption"><span class="smcap">Fig. 73.</span></div></div>
-
-<p>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.</p>
-
-<p>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)/<i>g</i>) = √(6/<i>g</i>) = ·43 seconds. But the bolt in<span class="pagenum"><a id="Page_186">186</a></span>
-this time went thirty feet; hence its velocity was
-thirty feet in ·43 seconds, or seventy feet per second.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-
-<p>May I end with a word of advice to parents?</p>
-
-<p>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 <span class="locked">mind:—</span></p>
-
-<p><i>First</i>, that almost everything a boy wants can be
-made with wood, and metal, and wire, and string,
-<em>if</em> 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.</p>
-
-<p><i>Second</i>, that he wants not toy tools, but good
-tools. If an expert wants a good tool, how much
-more a beginner.</p>
-
-<p><i>Third</i>, 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.</p>
-
-<p><span class="pagenum"><a id="Page_187">187</a></span>
-<i>Fourth</i>, 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.</p>
-
-<p><i>Fifth</i>, that the making of apparatus to show
-scientific facts is more useful than making bootjacks
-for his father or workboxes for his mother.</p>
-
-<p>And, <i>lastly</i>, 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, <span class="smcap">The Examiner</span>, he will often thank his
-stars that he learned in play the fundamental
-formula S = 1/2 <i>g</i> <i>t</i>², and that he knows the nature of
-“harmonic motion,” the two most important principles
-in the measurement of time.</p>
-
-<p class="p2 center">THE END.</p>
-</div>
-
-<hr />
-<div class="chapter">
-<h2><a id="APPENDIX"></a><span class="smcap">Appendix on the Shape of the Teeth of Wheels.</span></h2>
-
-<div id="ip_187" class="figcenter" style="max-width: 26.6875em;">
-  <img src="images/i_188.jpg" width="427" height="235" alt="" />
-  <div class="caption"><span class="smcap">Fig. 74.</span></div></div>
-
-<p>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<span class="pagenum"><a id="Page_188">188</a></span>
-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 <i>a c</i>
-turned round, driving before it the arm <i>b d</i>, the
-point <i>c</i> 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, <i>a c</i>
-and <i>d b</i>, mounted on pivots at <i>a</i> and <i>d</i>. The answer
-to this question is, that when a lever such as <i>a c</i>
-presses with its end against another, <i>d b</i>, the power<span class="pagenum"><a id="Page_189">189</a></span>
-is exercised in a direction <i>c e</i> at right angles to <i>d b</i>.
-Hence the leverage between the two arms is in the
-ratio of <i>a e</i> to <i>d c</i>. The system is just as if we had
-a lever <i>a e</i> united to a lever <i>d c</i> by a rigid rod <i>e c</i> at
-right angles to both of them.</p>
-
-<div id="ip_189" class="figcenter" style="max-width: 25.5em;">
-  <img src="images/i_189.jpg" width="408" height="189" alt="" />
-  <div class="caption"><span class="smcap">Fig. 75.</span></div></div>
-
-<p>Whence then the ratio of the power is as <i>a e</i> is to <i>d c</i>.</p>
-
-<div id="ip_189b" class="figcenter" style="max-width: 24.8125em;">
-  <img src="images/i_189b.jpg" width="397" height="186" alt="" />
-  <div class="caption"><span class="smcap">Fig. 76.</span></div></div>
-
-<p>But since the triangles <i>a e f</i>, <i>d c f</i>, are similar,
-<i>a e</i> is to <i>d c</i> as <i>a f</i> to <i>f d</i>. Whence then we get this
-general proposition: If one body mounted on an<span class="pagenum"><a id="Page_190">190</a></span>
-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 <i>f</i>. The leverage of one on
-the other is then as <i>a f</i> to <i>f d</i>. Our problem has
-now become the following: Given a rod <i>b d</i>, suppose
-that it is pressed upon by a curved surface mounted
-on an axis at <i>a</i>. Then the direction of the pressure<span class="pagenum"><a id="Page_191">191</a></span>
-that the curved surface (called in engineering language
-a cam) will exercise on the rod <i>b d</i> is shown by
-the dotted line; and the ratio of the driving power to
-the driven power is as <i>d f</i> to <i>a f</i>. Now how can we
-shape the cam so that as it moves round, and different
-parts of its surface come successively into contact
-with <i>b c</i>, the ratio of the leverage is always the
-same; that is to say, the ratio of <i>a f</i> to <i>f d</i> 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 <i>f</i>?</p>
-
-<div id="ip_191" class="figcenter" style="max-width: 28.625em;">
-  <img src="images/i_190.jpg" width="458" height="191" alt="" />
-  <div class="caption"><span class="smcap">Fig. 77.</span></div></div>
-
-<div id="ip_191b" class="figcenter" style="max-width: 25.8125em;">
-  <img src="images/i_190b.jpg" width="413" height="160" alt="" />
-  <div class="caption"><span class="smcap">Fig. 78.</span></div></div>
-
-<p>Evidently, if this is to be so, the point <i>d</i> must be
-on a semicircle, whose diameter is <i>f b</i>, for in that
-case the angle <i>f d b</i> will always be a right angle.</p>
-
-<div id="ip_191c" class="figcenter" style="max-width: 21.25em;">
-  <img src="images/i_191.jpg" width="340" height="279" alt="" />
-  <div class="caption"><span class="smcap">Fig. 79.</span></div></div>
-
-<p>The surface must then be so arranged that, whatever
-be the position of the cam and of the rod <i>b d</i>,<span class="pagenum"><a id="Page_192">192</a></span>
-the point of contact between them must always be
-on the semicircle <i>f c d</i>; that is to say, as the cam
-moves round the axis <i>a</i> its shape must be such that
-a line drawn from <i>f</i> to the point where it cuts the
-circle <i>f d b</i> is always perpendicular to the curve.</p>
-
-<p>Now suppose that we move a circle whose centre
-is at <i>a</i>, and radius <i>a f</i>, so as to roll the circle <i>f d b</i> by
-simple surface friction round its centre <i>o</i>, then any
-point <i>d</i> on it would mark out a curve on a piece of
-paper attached to
-the moving circle
-whose centre is at
-<i>a</i>, and the direction
-of motion of the
-curve would always
-be such that the
-point <i>d</i> on it would
-at any instant be
-describing a circle round <i>f</i>, and the direction of the
-curve would thus at any point always be at right
-angles to the line <i>d f</i> for the time being.</p>
-
-<div id="ip_192" class="figleft" style="max-width: 14.8125em;">
-  <img src="images/i_192.jpg" width="237" height="147" alt="" />
-  <div class="caption"><span class="smcap">Fig. 80.</span></div></div>
-
-<p>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.</p>
-
-<p><span class="pagenum"><a id="Page_193">193</a></span>
-The circles with radii <i>a f</i>, <i>b f</i>, 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.</p>
-
-<div id="ip_193" class="figright" style="max-width: 11.75em;">
-  <img src="images/i_193.jpg" width="188" height="232" alt="" />
-  <div class="caption"><span class="smcap">Fig. 81.</span></div></div>
-
-<p>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 cogwheel 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,<span class="pagenum"><a id="Page_194">194</a></span>
-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
-<i xml:lang="la" lang="la">vice versâ</i>. 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.</p>
-
-<p>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 (<a href="#ip_195">Fig. 82</a>). Big machinery was all shaped out<span class="pagenum"><a id="Page_195">195</a></span>
-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.</p>
-
-<div id="ip_195" class="figright" style="max-width: 9.75em;">
-  <img src="images/i_195.jpg" width="156" height="61" alt="" />
-  <div class="caption"><span class="smcap">Fig. 82.</span></div></div>
-
-<p>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.</p>
-
-<p><span class="pagenum"><a id="Page_196">196</a></span>
-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<span class="pagenum"><a id="Page_197">197</a></span>
-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.</p>
-
-<p>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.</p>
-
-<p>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.</p>
-</div>
-
-<hr />
-
-<p><span class="pagenum"><a id="Page_199">199</a></span></p>
-
-<div class="chapter"><div class="index">
-<h2 class="nobreak p1"><a id="INDEX"></a>INDEX.</h2>
-
-<ul class="index">
-<li class="ifrst">Acceleration, <a href="#Page_73">73</a>, <a href="#Page_77">77</a></li>
-
-<li class="indx">Almagest, <a href="#Page_53">53</a></li>
-
-<li class="indx">Anchor escapement, <a href="#Page_120">120</a></li>
-
-<li class="indx">Ancient science, <a href="#Page_50">50</a></li>
-
-<li class="indx">Aristotle’s ideas, <a href="#Page_23">23</a>, <a href="#Page_52">52</a></li>
-
-<li class="indx">Attwood’s machine, <a href="#Page_83">83</a></li>
-
-<li class="ifrst">Babylon, temple of, <a href="#Page_24">24</a></li>
-
-<li class="indx">Balance wheel, <a href="#Page_159">159</a></li>
-
-<li class="ifrst">Candles to measure time, <a href="#Page_46">46</a></li>
-
-<li class="indx">Chaldean day, <a href="#Page_15">15</a></li>
-
-<li class="indx">Chaucer, <a href="#Page_56">56</a></li>
-
-<li class="indx">Chronographs, <a href="#Page_179">179</a></li>
-
-<li class="indx">Chronometer, <a href="#Page_165">165</a></li>
-
-<li class="indx">Chronometer escapement, <a href="#Page_166">166</a></li>
-
-<li class="indx">Clock movement, <a href="#Page_123">123</a></li>
-
-<li class="indx">Copernicus, <a href="#Page_56">56</a></li>
-
-<li class="indx">Crossbow experiment, <a href="#Page_183">183</a></li>
-
-<li class="indx">Crown wheel, <a href="#Page_115">115</a></li>
-
-<li class="indx">Cycloid, the, <a href="#Page_109">109</a></li>
-
-<li class="ifrst">Dante’s Inferno, <a href="#Page_54">54</a></li>
-
-<li class="indx">Day, length of, <a href="#Page_29">29</a></li>
-
-<li class="indx">Dead beat escapement, <a href="#Page_135">135</a></li>
-
-<li class="indx">Density, <a href="#Page_12">12</a></li>
-
-<li class="indx">Driving weight, <a href="#Page_127">127</a>, <a href="#Page_141">141</a></li>
-
-<li class="ifrst">Earth, a sphere, <a href="#Page_21">21</a></li>
-
-<li class="indx">Earth’s motion, <a href="#Page_57">57</a>, <a href="#Page_69">69</a></li>
-
-<li class="indx">Earth not at rest, <a href="#Page_67">67</a></li>
-
-<li class="indx">Egg-boiler, <a href="#Page_43">43</a></li>
-
-<li class="indx">Electric clocks, <a href="#Page_179">179</a></li>
-
-<li class="indx">Epicycloidal wheels, <a href="#Page_191">191</a></li>
-
-<li class="indx">Escapements, anchor, <a href="#Page_120">120</a></li>
-<li class="isub1">crown, <a href="#Page_115">115</a></li>
-<li class="isub1">chronometer, <a href="#Page_166">166</a></li>
-<li class="isub1">dead beat, <a href="#Page_135">135</a></li>
-<li class="isub1">gravity, <a href="#Page_169">169</a></li>
-
-<li class="ifrst">Falling bodies, laws of, <a href="#Page_62">62</a></li>
-
-<li class="indx">Force, <a href="#Page_76">76</a></li>
-
-<li class="indx">Forces, revolution of, <a href="#Page_89">89</a></li>
-
-<li class="indx">Fusee, the, <a href="#Page_117">117</a></li>
-
-<li class="ifrst">Galileo’s “Dialogues,” <a href="#Page_58">58</a></li>
-<li class="isub1">clock, <a href="#Page_111">111</a></li>
-
-<li class="indx">Grandfather’s clock, <a href="#Page_119">119</a></li>
-
-<li class="indx">Gravity, action of, <a href="#Page_13">13</a>, <a href="#Page_65">65</a></li>
-
-<li class="indx">Gravity escapement, <a href="#Page_169">169</a></li>
-
-<li class="indx">Greek day, <a href="#Page_16">16</a></li>
-
-<li class="ifrst">Harmonic motion, <a href="#Page_97">97</a></li>
-
-<li class="indx">Hooke’s law, <a href="#Page_71">71</a></li>
-
-<li class="ifrst">Isochronism of springs, <a href="#Page_93">93</a></li>
-
-<li class="ifrst">Lamps to measure time, <a href="#Page_46">46</a></li>
-
-<li class="indx">Latitude and longitude, <a href="#Page_161">161</a></li>
-<li class="isub1">finding, <a href="#Page_163">163</a></li>
-
-<li class="ifrst">Mass, nature of, <a href="#Page_10">10</a></li>
-
-<li class="indx">Mercury clock, <a href="#Page_45">45</a></li>
-
-<li class="indx">Modern methods, <a href="#Page_177">177</a>, <a href="#Page_197">197</a></li>
-
-<li class="indx">Moments, <a href="#Page_101">101</a><span class="pagenum"><a id="Page_200">200</a></span></li>
-
-<li class="indx">Moon’s appearance, <a href="#Page_17">17</a></li>
-
-<li class="indx">Motion, reliability of, <a href="#Page_57">57</a></li>
-
-<li class="indx">Musical notes, <a href="#Page_95">95</a></li>
-
-<li class="ifrst">North pole, days at, <a href="#Page_33">33</a></li>
-
-<li class="ifrst">Oscillations, law of, <a href="#Page_151">151</a></li>
-
-<li class="ifrst">Parabola, the, <a href="#Page_87">87</a></li>
-
-<li class="indx">Pendulum, the, <a href="#Page_103">103</a>, <a href="#Page_145">145</a></li>
-<li class="isub1">suspension, <a href="#Page_145">145</a></li>
-<li class="isub1">mercury, <a href="#Page_147">147</a></li>
-<li class="isub1">gridiron, <a href="#Page_149">149</a></li>
-<li class="isub1">theory of, <a href="#Page_155">155</a></li>
-<li class="isub1">free, <a href="#Page_133">133</a></li>
-
-<li class="indx">Pisa, leaning tower of, <a href="#Page_61">61</a></li>
-
-<li class="indx">Planets, names of, <a href="#Page_11">11</a></li>
-
-<li class="indx">Pulse measurer, <a href="#Page_99">99</a></li>
-
-<li class="ifrst">Ratchet wheels, <a href="#Page_129">129</a></li>
-
-<li class="indx">Roman clocks, <a href="#Page_40">40</a></li>
-
-<li class="ifrst">Sand-glasses, <a href="#Page_41">41</a></li>
-
-<li class="indx">Space, nature of, <a href="#Page_8">8</a></li>
-
-<li class="indx">Speed of falling bodies, <a href="#Page_79">79</a></li>
-
-<li class="indx">Spring balance, <a href="#Page_107">107</a></li>
-
-<li class="indx">Stevinus’ theory, <a href="#Page_81">81</a></li>
-
-<li class="indx">Style of sun-dials, <a href="#Page_35">35</a></li>
-
-<li class="indx">Sun-dials, <a href="#Page_27">27</a></li>
-<li class="isub1">to make, <a href="#Page_48">48</a></li>
-
-<li class="indx">Synchronous clocks, <a href="#Page_175">175</a></li>
-
-<li class="ifrst">Time, <a href="#Page_13">13</a></li>
-
-<li class="indx">Toothed wheels, <a href="#Page_125">125</a>, <a href="#Page_137">137</a></li>
-
-<li class="indx">Tuning fork, <a href="#Page_94">94</a>, <a href="#Page_181">181</a></li>
-
-<li class="ifrst">Velocities, composition of, <a href="#Page_85">85</a></li>
-
-<li class="ifrst">Water pressure, <a href="#Page_37">37</a></li>
-
-<li class="indx">Water clocks, <a href="#Page_39">39</a></li>
-
-<li class="indx">Watches, <a href="#Page_156">156</a></li>
-
-<li class="indx">Week days, names of, <a href="#Page_24">24</a></li>
-
-<li class="indx">Wheels, shape of teeth, <a href="#Page_190">190</a></li>
-
-<li class="indx">Wheel-cutting machines, <a href="#Page_193">193</a></li>
-
-<li class="indx">Winding drum, <a href="#Page_131">131</a></li>
-
-<li class="indx">Winter sun, <a href="#Page_31">31</a></li>
-
-<li class="ifrst">Zodiac, <a href="#Page_18">18</a></li>
-</ul>
-</div>
-
-<p class="p2 center small wspace">BRADBURY, AGNEW, &amp; CO. LD., PRINTERS, LONDON AND TONBRIDGE.</p>
-</div>
-
-<div class="chapter"><div class="transnote">
-<h2 class="nobreak p1"><a id="Transcribers_Notes"></a>Transcriber’s Note</h2>
-
-<p>Punctuation, hyphenation, and spelling were made consistent when a predominant
-preference was found in this book; otherwise they were not changed.</p>
-
-<p>Simple typographical errors were corrected; occasional unpaired
-quotation marks were retained.</p>
-
-<p>Ambiguous hyphens at the ends of lines were retained.</p>
-
-<p>Square roots are represented as √(values).</p>
-
-<p>Index not checked for proper alphabetization or correct page references.</p>
-
-<p>Page <a href="#Page_16">16</a>: “six o’clock” was printed as “six clock”; changed here.</p>
-</div></div>
-</div>
-
-<p>&nbsp;</p>
-<p>&nbsp;</p>
-<hr class="full" />
-<p>***END OF THE PROJECT GUTENBERG EBOOK TIME AND CLOCKS***</p>
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