The Project Gutenberg eBook, Time and Clocks, by Sir Henry H. (Henry Hardinge) Cunynghame

TIME AND CLOCKS.

[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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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]).

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.

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.”

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Fig. 34.