11. Taking this as the beginning of his discovery, it is said that he made two ma.s.ses of the same weight as the crown, one of gold and the other of silver. After making them, he filled a large vessel with water to the very brim, and dropped the ma.s.s of silver into it. As much water ran out as was equal in bulk to that of the silver sunk in the vessel.
Then, taking out the ma.s.s, he poured back the lost quant.i.ty of water, using a pint measure, until it was level with the brim as it had been before. Thus he found the weight of silver corresponding to a definite quant.i.ty of water.
12. After this experiment, he likewise dropped the ma.s.s of gold into the full vessel and, on taking it out and measuring as before, found that not so much water was lost, but a smaller quant.i.ty: namely, as much less as a ma.s.s of gold lacks in bulk compared to a ma.s.s of silver of the same weight. Finally, filling the vessel again and dropping the crown itself into the same quant.i.ty of water, he found that more water ran over for the crown than for the ma.s.s of gold of the same weight. Hence, reasoning from the fact that more water was lost in the case of the crown than in that of the ma.s.s, he detected the mixing of silver with the gold, and made the theft of the contractor perfectly clear.
13. Now let us turn our thoughts to the researches of Archytas of Tarentum and Eratosthenes of Cyrene. They made many discoveries from mathematics which are welcome to men, and so, though they deserve our thanks for other discoveries, they are particularly worthy of admiration for their ideas in that field. For example, each in a different way solved the problem enjoined upon Delos by Apollo in an oracle, the doubling of the number of cubic feet in his altars; this done, he said, the inhabitants of the island would be delivered from an offence against religion.
14. Archytas solved it by his figure of the semi-cylinders; Eratosthenes, by means of the instrument called the mesolabe.
Noting all these things with the great delight which learning gives, we cannot but be stirred by these discoveries when we reflect upon the influence of them one by one. I find also much for admiration in the books of Democritus on nature, and in his commentary ent.i.tled [Greek: Cheirokmeta], in which he made use of his ring to seal with soft wax the principles which he had himself put to the test.
15. These, then, were men whose researches are an everlasting possession, not only for the improvement of character but also for general utility. The fame of athletes, however, soon declines with their bodily powers. Neither when they are in the flower of their strength, nor afterwards with posterity, can they do for human life what is done by the researches of the learned.
16. But although honours are not bestowed upon authors for excellence of character and teaching, yet as their minds, naturally looking up to the higher regions of the air, are raised to the sky on the steps of history, it must needs be, that not merely their doctrines, but even their appearance, should be known to posterity through time eternal.
Hence, men whose souls are aroused by the delights of literature cannot but carry enshrined in their hearts the likeness of the poet Ennius, as they do those of the G.o.ds. Those who are devotedly attached to the poems of Accius seem to have before them not merely his vigorous language but even his very figure.
17. So, too, numbers born after our time will feel as if they were discussing nature face to face with Lucretius, or the art of rhetoric with Cicero; many of our posterity will confer with Varro on the Latin language; likewise, there will be numerous scholars who, as they weigh many points with the wise among the Greeks, will feel as if they were carrying on private conversations with them. In a word, the opinions of learned authors, though their bodily forms are absent, gain strength as time goes on, and, when taking part in councils and discussions, have greater weight than those of any living men.
18. Such, Caesar, are the authorities on whom I have depended, and applying their views and opinions I have written the present books, in the first seven treating of buildings and in the eighth of water. In this I shall set forth the rules for dialling, showing how they are found through the shadows cast by the gnomon from the sun"s rays in the firmament, and on what principles these shadows lengthen and shorten.
CHAPTER I
THE ZODIAC AND THE PLANETS
1. It is due to the divine intelligence and is a very great wonder to all who reflect upon it, that the shadow of a gnomon at the equinox is of one length in Athens, of another in Alexandria, of another in Rome, and not the same at Piacenza, or at other places in the world. Hence drawings for dials are very different from one another, corresponding to differences of situation. This is because the length of the shadow at the equinox is used in constructing the figure of the a.n.a.lemma, in accordance with which the hours are marked to conform to the situation and the shadow of the gnomon. The a.n.a.lemma is a basis for calculation deduced from the course of the sun, and found by observation of the shadow as it increases until the winter solstice. By means of this, through architectural principles and the employment of the compa.s.ses, we find out the operation of the sun in the universe.
2. The word "universe" means the general a.s.semblage of all nature, and it also means the heaven that is made up of the constellations and the courses of the stars. The heaven revolves steadily round earth and sea on the pivots at the ends of its axis. The architect at these points was the power of Nature, and she put the pivots there, to be, as it were, centres, one of them above the earth and sea at the very top of the firmament and even beyond the stars composing the Great Bear, the other on the opposite side under the earth in the regions of the south. Round these pivots (termed in Greek [Greek: poloi]) as centres, like those of a turning lathe, she formed the circles in which the heaven pa.s.ses on its everlasting way. In the midst thereof, the earth and sea naturally occupy the central point.
3. It follows from this natural arrangement that the central point in the north is high above the earth, while on the south, the region below, it is beneath the earth and consequently hidden by it.
Furthermore, across the middle, and obliquely inclined to the south, there is a broad circular belt composed of the twelve signs, whose stars, arranged in twelve equivalent divisions, represent each a shape which nature has depicted. And so with the firmament and the other constellations, they move round the earth and sea in glittering array, completing their orbits according to the spherical shape of the heaven.
4. They are all visible or invisible according to fixed times. While six of the signs are pa.s.sing along with the heaven above the earth, the other six are moving under the earth and hidden by its shadow. But there are always six of them making their way above the earth; for, corresponding to that part of the last sign which in the course of its revolution has to sink, pa.s.s under the earth, and become concealed, an equivalent part of the sign opposite to it is obliged by the law of their common revolution to pa.s.s up and, having completed its circuit, to emerge out of the darkness into the light of the open s.p.a.ce on the other side. This is because the rising and setting of both are subject to one and the same power and law.
5. While these signs, twelve in number and occupying each one twelfth part of the firmament, steadily revolve from east to west, the moon, Mercury, Venus, the sun, as well as Mars, Jupiter, and Saturn, differing from one another in the magnitude of their orbits as though their courses were at different points in a flight of steps, pa.s.s through those signs in just the opposite direction, from west to east in the firmament. The moon makes her circuit of the heaven in twenty-eight days plus about an hour, and with her return to the sign from which she set forth, completes a lunar month.
6. The sun takes a full month to move across the s.p.a.ce of one sign, that is, one twelfth of the firmament. Consequently, in twelve months he traverses the s.p.a.ces of the twelve signs, and, on returning to the sign from which he began, completes the period of a full year. Hence, the circuit made by the moon thirteen times in twelve months, is measured by the sun only once in the same number of months. But Mercury and Venus, their paths wreathing around the sun"s rays as their centre, retrograde and delay their movements, and so, from the nature of that circuit, sometimes wait at stopping-places within the s.p.a.ces of the signs.
7. This fact may best be recognized from Venus. When she is following the sun, she makes her appearance in the sky after his setting, and is then called the Evening Star, shining most brilliantly. At other times she precedes him, rising before day-break, and is named the Morning Star. Thus Mercury and Venus sometimes delay in one sign for a good many days, and at others advance pretty rapidly into another sign. They do not spend the same number of days in every sign, but the longer they have previously delayed, the more rapidly they accomplish their journeys after pa.s.sing into the next sign, and thus they complete their appointed course. Consequently, in spite of their delay in some of the signs, they nevertheless soon reach the proper place in their orbits after freeing themselves from their enforced delay.
8. Mercury, on his journey through the heavens, pa.s.ses through the s.p.a.ces of the signs in three hundred and sixty days, and so arrives at the sign from which he set out on his course at the beginning of his revolution. His average rate of movement is such that he has about thirty days in each sign.
9. Venus, on becoming free from the hindrance of the sun"s rays, crosses the s.p.a.ce of a sign in thirty days. Though she thus stays less than forty days in particular signs, she makes good the required amount by delaying in one sign when she comes to a pause. Therefore she completes her total revolution in heaven in four hundred and eighty-five days, and once more enters the sign from which she previously began to move.
10. Mars, after traversing the s.p.a.ces of the constellations for about six hundred and eighty-three days, arrives at the point from which he had before set out at the beginning of his course, and while he pa.s.ses through some of the signs more rapidly than others, he makes up the required number of days whenever he comes to a pause. Jupiter, climbing with gentler pace against the revolution of the firmament, travels through each sign in about three hundred and sixty days, and finishes in eleven years and three hundred and thirteen days, returning to the sign in which he had been twelve years before. Saturn, traversing the s.p.a.ce of one sign in twenty-nine months plus a few days, is restored after twenty-nine years and about one hundred and sixty days to that in which he had been thirty years before. He is, as it appears, slower, because the nearer he is to the outermost part of the firmament, the greater is the orbit through which he has to pa.s.s.
11. The three that complete their circuits above the sun"s course do not make progress while they are in the triangle which he has entered, but retrograde and pause until the sun has crossed from that triangle into another sign. Some hold that this takes place because, as they say, when the sun is a great distance off, the paths on which these stars wander are without light on account of that distance, and so the darkness r.e.t.a.r.ds and hinders them. But I do not think that this is so. The splendour of the sun is clearly to be seen, and manifest without any kind of obscurity, throughout the whole firmament, so that those very retrograde movements and pauses of the stars are visible even to us.
12. If then, at this great distance, our human vision can discern that sight, why, pray, are we to think that the divine splendour of the stars can be cast into darkness? Rather will the following way of accounting for it prove to be correct. Heat summons and attracts everything towards itself; for instance, we see the fruits of the earth growing up high under the influence of heat, and that spring water is vapourised and drawn up to the clouds at sunrise. On the same principle, the mighty influence of the sun, with his rays diverging in the form of a triangle, attracts the stars which follow him, and, as it were, curbs and restrains those that precede, not allowing them to make progress, but obliging them to retrograde towards himself until he pa.s.ses out into the sign that belongs to a different triangle.
13. Perhaps the question will be raised, why the sun by his great heat causes these detentions in the fifth sign from himself rather than in the second or third, which are nearer. I will therefore set forth what seems to be the reason. His rays diverge through the firmament in straight lines as though forming an equilateral triangle, that is, to the fifth sign from the sun, no more, no less. If his rays were diffused in circuits spreading all over the firmament, instead of in straight lines diverging so as to form a triangle, they would burn up all the nearer objects. This is a fact which the Greek poet Euripides seems to have remarked; for he says that places at a greater distance from the sun are in a violent heat, and that those which are nearer he keeps temperate. Thus in the play of Phaethon, the poet writes: [Greek: kaiei ta porro, tangythen d eukrat echei].
14. If then, fact and reason and the evidence of an ancient poet point to this explanation, I do not see why we should decide otherwise than as I have written above on this subject.
Jupiter, whose orbit is between those of Mars and Saturn, traverses a longer course than Mars, and a shorter than Saturn. Likewise with the rest of these stars: the farther they are from the outermost limits of the heaven, and the nearer their orbits to the earth, the sooner they are seen to finish their courses; for those of them that have a smaller orbit often pa.s.s those that are higher, going under them.
15. For example, place seven ants on a wheel such as potters use, having made seven channels on the wheel about the centre, increasing successively in circ.u.mference; and suppose those ants obliged to make a circuit in these channels while the wheel is turned in the opposite direction. In spite of having to move in a direction contrary to that of the wheel, the ants must necessarily complete their journeys in the opposite direction, and that ant which is nearest the centre must finish its circuit sooner, while the ant that is going round at the outer edge of the disc of the wheel must, on account of the size of its circuit, be much slower in completing its course, even though it is moving just as quickly as the other. In the same way, these stars, which struggle on against the course of the firmament, are accomplishing an orbit on paths of their own; but, owing to the revolution of the heaven, they are swept back as it goes round every day.
16. The reason why some of these stars are temperate, others hot, and others cold, appears to be this: that the flame of every kind of fire rises to higher places. Consequently, the burning rays of the sun make the ether above him white hot, in the regions of the course of Mars, and so the heat of the sun makes him hot. Saturn, on the contrary, being nearest to the outermost limit of the firmament and bordering on the quarters of the heaven which are frozen, is excessively cold. Hence, Jupiter, whose course is between the orbits of these two, appears to have a moderate and very temperate influence, intermediate between their cold and heat.
I have now described, as I have received them from my teacher, the belt of the twelve signs and the seven stars that work and move in the opposite direction, with the laws and numerical relations under which they pa.s.s from sign to sign, and how they complete their orbits. I shall next speak of the waxing and waning of the moon, according to the accounts of my predecessors.
CHAPTER II
THE PHASES OF THE MOON
1. According to the teaching of Berosus, who came from the state, or rather nation, of the Chaldees, and was the pioneer of Chaldean learning in Asia, the moon is a ball, one half luminous and the rest of a blue colour. When, in the course of her orbit, she has pa.s.sed below the disc of the sun, she is attracted by his rays and great heat, and turns thither her luminous side, on account of the sympathy between light and light. Being thus summoned by the sun"s disc and facing upward, her lower half, as it is not luminous, is invisible on account of its likeness to the air. When she is perpendicular to the sun"s rays, all her light is confined to her upper surface, and she is then called the new moon.
2. As she moves on, pa.s.sing by to the east, the effect of the sun upon her relaxes, and the outer edge of the luminous side sheds its light upon the earth in an exceedingly thin line. This is called the second day of the moon. Day by day she is further relieved and turns, and thus are numbered the third, fourth, and following days. On the seventh day, the sun being in the west and the moon in the middle of the firmament between the east and west, she is half the extent of the firmament distant from the sun, and therefore half of the luminous side is turned toward the earth. But when the sun and moon are separated by the entire extent of the firmament, and the moon is in the east with the sun over against her in the west, she is completely relieved by her still greater distance from his rays, and so, on the fourteenth day, she is at the full, and her entire disc emits its light. On the succeeding days, up to the end of the month, she wanes daily as she turns in her course, being recalled by the sun until she comes under his disc and rays, thus completing the count of the days of the month.
3. But Aristarchus of Samos, a mathematician of great powers, has left a different explanation in his teaching on this subject, as I shall now set forth. It is no secret that the moon has no light of her own, but is, as it were, a mirror, receiving brightness from the influence of the sun. Of all the seven stars, the moon traverses the shortest orbit, and her course is nearest to the earth. Hence in every month, on the day before she gets past the sun, she is under his disc and rays, and is consequently hidden and invisible. When she is thus in conjunction with the sun, she is called the new moon. On the next day, reckoned as her second, she gets past the sun and shows the thin edge of her sphere.
Three days away from the sun, she waxes and grows brighter. Removing further every day till she reaches the seventh, when her distance from the sun at his setting is about one half the extent of the firmament, one half of her is luminous: that is, the half which faces toward the sun is lighted up by him.
4. On the fourteenth day, being diametrically across the whole extent of the firmament from the sun, she is at her full and rises when the sun is setting. For, as she takes her place over against him and distant the whole extent of the firmament, she thus receives the light from the sun throughout her entire orb. On the seventeenth day, at sunrise, she is inclining to the west. On the twenty-second day, after sunrise, the moon is about mid-heaven; hence, the side exposed to the sun is bright and the rest dark. Continuing thus her daily course, she pa.s.ses under the rays of the sun on about the twenty-eighth day, and so completes the account of the month.
I will next explain how the sun, pa.s.sing through a different sign each month, causes the days and hours to increase and diminish in length.
CHAPTER III
THE COURSE OF THE SUN THROUGH THE TWELVE SIGNS
1. The sun, after entering the sign Aries and pa.s.sing through one eighth of it, determines the vernal equinox. On reaching the tail of Taurus and the constellation of the Pleiades, from which the front half of Taurus projects, he advances into a s.p.a.ce greater than half the firmament, moving toward the north. From Taurus he enters Gemini at the time of the rising of the Pleiades, and, getting higher above the earth, he increases the length of the days. Next, coming from Gemini into Cancer, which occupies the shortest s.p.a.ce in heaven, and after traversing one eighth of it, he determines the summer solstice. Continuing on, he reaches the head and breast of Leo, portions which are reckoned as belonging to Cancer.
2. After leaving the breast of Leo and the boundaries of Cancer, the sun, traversing the rest of Leo, makes the days shorter, diminishing the size of his circuit, and returning to the same course that he had in Gemini. Next, crossing from Leo into Virgo, and advancing as far as the bosom of her garment, he still further shortens his circuit, making his course equal to what it was in Taurus. Advancing from Virgo by way of the bosom of her garment, which forms the first part of Libra, he determines the autumn equinox at the end of one eighth of Libra. Here his course is equal to what his circuit was in the sign Aries.
3. When the sun has entered Scorpio, at the time of the setting of the Pleiades, he begins to make the days shorter as he advances toward the south. From Scorpio he enters Sagittarius and, on reaching the thighs, his daily course is still further diminished. From the thighs of Sagittarius, which are reckoned as part of Capricornus, he reaches the end of the first eighth of the latter, where his course in heaven is shortest. Consequently, this season, from the shortness of the day, is called bruma or dies brumales. Crossing from Capricornus into Aquarius, he causes the days to increase to the length which they had when he was in Sagittarius. From Aquarius he enters Pisces at the time when Favonius begins to blow, and here his course is the same as in Scorpio. In this way the sun pa.s.ses round through the signs, lengthening or shortening the days and hours at definite seasons.
I shall next speak of the other constellations formed by arrangements of stars, and lying to the right and left of the belt of the signs, in the southern and northern portions of the firmament.