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 f?s?? 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 ?a, 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 3? 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.

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.

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.

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 a, 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 . 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—

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.”

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.

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.

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.

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.

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 ? 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.

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.

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.