To noble athletes who had conquered at the Olympic, Pythian, Isthmian, and Nemean games, the ancestors of the Greeks ordained such great honors that, not only standing in the assembly with palm and crown they should bear praises, but also, returning to their cities as triumphant victors, they should be driven into the walls and into their native lands in four-horse chariots, and, with fixed revenues from the commonwealth, enjoy a perpetual life. When therefore I consider this, I wonder why like or even greater honors have not been bestowed on writers, who furnish boundless utilities through all ages to all peoples. For this was more worthy to be instituted, because, while athletes make their own bodies stronger by exercise, writers prepare precepts for learning and sharpening minds, not only their own senses, but those of all. §1
For what use is it to men that Milo of Croton was unconquered, or the rest who were conquerors of that kind, unless that, while they lived, they themselves had renown among their fellow-citizens? But the precepts of Pythagoras, Democritus, Plato, Aristotle, and the rest of the wise, pursued daily with perpetual industries, put forth fresh and flourishing fruits not only for their own citizens but for all peoples. And those who from tender ages are sated with an abundance of learnings, having the best senses of wisdom, institute for cities the manners of civility, equal rights, and laws, without which no state can be safe. §2
Since then so great gifts have been prepared by the prudence of writers for men privately and publicly, I judge not only that palms and crowns should be awarded to them but that triumphs should be decreed, and they should be dedicated among the seats of the gods. Of their meditations useful to men for life, I shall put forth a few of many as examples, which when considered, men will necessarily confess that honors ought to be paid them. §3
First, of Plato, of his many most useful reasonings, I shall set forth one as he unfolded it. If there be a square place or field of equal sides, and it must be doubled, a thing which cannot be found by multiplication of numbers, it is found by exact descriptions of lines. This is the demonstration. A square place ten feet long and ten feet broad makes a hundred square feet of area. If then it must be doubled and, also of equal sides, of two hundred feet, it must be asked how great the side of that square be made, so that from it two hundred feet may answer to the doubling. No one can find this by number. For if we establish fourteen, multiplied it will be 196; if fifteen, 225. §4
Since therefore this is not to be explained by number, in that square ten feet long and ten feet wide, let a diagonal line from angle to angle be drawn, so that two triangles of equal size be divided, each of 50 square feet; and to the length of that diagonal line let a square with equal sides be described. Thus, as many triangles as were designated in the smaller square of 50 feet by the diagonal line, of the same size and same number of feet four shall be made in the larger. By this reasoning the doubling by lines, as by Plato the scheme is subscribed at the bottom of the page. §5
Likewise Pythagoras showed that the square, without the fabrications of artisans, was found; and what carpenters, with great labor making the square, scarcely can bring to truth, is corrected by reasonings and methods from his precepts. For if three rules be taken, of which one is three feet, another four, the third five, and these rules, composed with one another so as to touch each other at their extreme peaks, have the figure of a triangle, they will form a true square. And if to the lengths of these rules, each a square with equal sides be described, of the one whose side is three, the area will have nine feet; of the one of four, sixteen; of the one of five, twenty-five. §6
Thus as great a number of square feet as the two squares from three feet length of sides and four make, the same number is returned by one described from five. When Pythagoras had discovered this, not doubting that he had been prompted by the Muses in that invention, giving the greatest thanks is said to have immolated victims to them. This reasoning, as it is useful in many things and measures, is also, in the building of stairs, to have steps with balanced levelings, ready at hand. §7
For if the height of a floor from the top boarding to the lowest level has been divided into three parts, there shall be of these, five, in the stairs, as the just length of the strings (scapi), so that as great as the three parts of the height between the framing and the lowest level were, four shall recede from the perpendicular, and there shall the lower heels of the strings be placed. So, if they be thus, the placings of both the steps and of the stairs themselves will be well-proportioned. Of this matter too a figure will be appended. §8
Of Archimedes there are many wonderful and various inventions, but of all, with boundless dexterity, this which I shall set forth seems to have been expressed. Doubtless Hiero of Syracuse, when he had risen to royal power, his affairs being well managed, having vowed to place a gold crown in a certain shrine to the immortal gods, contracted it to be made by workmanship, and weighed out the gold to the contractor for the weight. In due time the contractor approved his work, subtly made by hand, to the king, and was seen to have furnished the weight of the crown according to the weigh. §9
Afterwards information was given that, gold being taken out, just so much silver had been mixed into that crown-work; indignant that he was held in contempt, nor finding by what reasoning he might catch that theft, Hiero asked Archimedes to take thought on his behalf. Then when he had this matter in mind, by chance he came to the bath, and there, when he went down into the tub, he observed that as much of his body as sat in it, so much water flowed out of the tub. When this had shown him the reasoning of the explanation, he did not delay, but leapt with joy out of the tub, and, going naked homewards, signified by a clear voice that he had found what he sought. For running, over and over he cried in Greek εὕρηκα εὕρηκα! §10
Then, from that approach to the invention, he is said to have made two masses of equal weight with which the crown had been, one of gold and the other of silver. When he had so done, he filled an ample vessel to the lips with water, in which he lowered the silver mass. As great a size as was sunk in the vessel, so much water flowed out. So with the mass taken out, by how much less it was done, he refilled with a sextarius measured, that in the same way as before it should be equalized to the lips. So from this he found how much weight of silver answered to a fixed measure of water. §11
When he had tried this, then he likewise lowered the gold mass into the full vessel, and, this being taken out, by the same reasoning measure being added, he found the water was lacking not as much, but less, by how much a mass of gold of the same weight is less in large body than of silver. Afterwards, the vessel being refilled, in the same water the crown itself being lowered, he found that more water flowed off in the crown than in the gold mass of the same weight; and so from this, that more water was lacking in the crown than in the mass, reasoning he detected the admixture of silver in the gold and the manifest theft of the contractor. §12
Let the mind be transferred to the meditations of Archytas of Tarentum and Eratosthenes of Cyrene. For these men discovered many pleasant things from mathematical things for men, and so, as in other inventions they were pleasing, in the meditations of this matter they are most to be regarded. For the one by one reasoning, the other by another unfolded what Apollo at Delos had commanded in his response: that, as many square feet as the altar had, that should be doubled, and so it would be that those in that island would then be freed from religious scruple. §13
Archytas, by the descriptions of the hemicylinder; Eratosthenes, by an organic reasoning of the mesolabe, unfolded the same. Since these things are noted with such great pleasantness of the disciplines, and we are naturally compelled, considering the effects, to be moved by the inventions of single things, I, attending to many things, also admire the volumes of Democritus on the nature of things, and his commentary which is inscribed χειροκμήτων, in which he even used a seal to mark with soft wax those things which he had tried. §14
Therefore the meditations of these men are not only prepared for correcting the manners but also for the utility of all, perpetually. The renowns of athletes, however, in a short space grow old with their bodies. So neither when they are most flourishing, nor to posterity, can they be of use, as the meditations of the wise are to men's lives. §15
But though neither by manners nor by institutions are honors paid to the writers, standing out, their very minds, looking at higher things of the air, lifted up to the sky by the steps of memories, compel, by the immortality of the age, not only their sentences but also their images to be known to posterity. So those who have minds tinged by the pleasantness of letters cannot but have dedicated in their breasts, as of the gods, so the image of the poet Ennius. And those who zealously delight in the verses of Accius seem to have not only the virtues of his words but also his figure present with them. §16
Likewise many born after our memory will seem, with Lucretius, as if face-to-face to debate of the nature of things; of rhetoric with Cicero; many of posterity will confer speech with Varro on the Latin language; no less many philologists, debating many things with the wise of the Greeks, will seem to have secret conversations with them; and, in sum, the sentences of the wise writers, their bodies absent, flourishing with age, when they lie among counsels and debates, have greater authorities than all of those present. §17
And so, Caesar, relying on these authors, with their senses and counsels employed, I have written these volumes, and in the first seven on buildings, in the eighth on waters, and in this on gnomonics I shall unfold how the rays of the sun in the world were discovered through the shadows of gnomons, and by what reasonings they are lengthened or shortened. §18
These things are prepared by the divine mind, and they have great wonder to those considering them, that the gnomon's equinoctial shadow is of one size at Athens, of another at Alexandria, of another at Rome, not the same at Placentia and in the rest of the places of the orb of the earth. Therefore the designs of sundials differ greatly with changes of place. For by the sizes of the equinoctial shadows are designated the forms of analemmata, from which are perfected, in proportion to the places and the gnomon's shadows, the descriptions of the hours. The analemma is a reasoning, sought out by the sun's course and the observation of the growing shadow to the winter solstice, from which, through architectural reasonings and the drawings of the compasses, the effect in the world has been found. §19
The world is the highest conception of all natural things, and the sky shaped by the stars and the courses of stars. It turns continuously about the earth and the sea at the utmost pivots of the axis. For in these places the natural power so contrived and placed the pivots as centers, one from the earth and sea at the top of the world, even behind the stars of the north; the other opposite, beneath the earth in the southern parts; and there, round these pivots, like little wheels around centers as on a lathe, perfected what the Greeks call πόλοι, through which the sky ever flies round. So the middle of the earth, with the sea, has been naturally placed at the place of the center. §20
With these so disposed by nature that in the northern part from the earth the pivot has a loftier height, but in the southern, in lower places, being below, it is obscured from the earth, then also through the middle, transverse and inclining to the south, a wide circular zone shaped of the twelve signs is formed, whose forms, by the stars disposed, equalized in twelve parts, express a figuration depicted by nature. Therefore, shining with the world and the rest of the stars, flying about the earth and the sea, they perfect their courses round the rotundity of the sky. §21
All the things, visited and unvisited, are established by the necessity of time. Of these, six signs in number wander with the sky above the earth; the rest, passing under the earth, are obscured by its shadow. Six of them, however, are always above the earth. For as much part of the last sign, compressed by depression by turning, is hidden going under the earth, so much of the other, from its opposite conversion, overpassing by rotation turned about, in places opening out of the hidden places goes forth to the light. For one force and necessity together effects a rising and a setting on each side. §22
And since these signs are twelve in number, and each possesses the twelfth parts of the world and revolves continually from east to west, then through those signs, by a contrary course, the moon, the star of Mercury, Venus, the sun itself, likewise Mars, Jupiter, and Saturn, as by the ascent of steps running through, another with another of the size of the circuit, wander from west to east in the world. The moon, on the twenty-eighth day and about an hour more, running through the circuit of the sky, from whatever sign it began, by returning to that sign, completes the lunar month. §23
The sun, however, traverses in a month's end the space of a sign, which is the twelfth part of the world. Thus in twelve months wandering through the intervals of the twelve signs, when it returns to that sign whence it began, it completes the space of the returning year. Thus the circle which the moon runs thirteen times in twelve months, the sun measures once in the same months. The stars of Mercury and Venus, however, about the sun's rays, crowning the same center with their journeys, make retreats backwards and retardations; and thus by stations, on account of that circling, they delay in the spaces of the signs. §24
That this is so, is known chiefly from the star of Venus, which, following the sun, appearing after its setting in the sky and shining most clearly, is called Hesperus (evening star); but at other times, forerunning it and rising before light, is called Lucifer (light-bringer). For this reason sometimes they delay several days in one sign, at others enter more quickly into another. So because they do not equally go through a number of days in the single signs, as much as they have delayed, by leaping over before with quicker passages, they complete a just course. So it comes about that whatever they delay in some signs, nonetheless when they snatch themselves from the necessity of the delay, they quickly accomplish a just circuit. §25
The star of Mercury so flies in the world that, on the three hundred and sixtieth day, running through the spaces of the signs, it comes to that sign from which, in its previous circuit, it began to make its course; and so its journey is equalized, as to have, in the single signs, about thirty days of numerical reasoning. §26
Venus, however, when it is freed from the obstruction of the sun's rays, runs through the space of a sign in thirty days; by the fewer forty days in the single signs, when it has made a station, it restores that sum of numbers, lingering in one sign. Therefore, having measured the whole circuit in the sky on the four-hundred-and-eighty-fifth day, it again enters the sign from which it first began to make its journey. §27
Mars, wandering through the spaces of the stars about six hundred and eighty-third day, reaches where, having begun to make its course before, it was, and in those signs in which it runs more quickly, when it has made a station, completes the number of days reasonably. Jupiter, however, climbing with gentler steps against the world's turn, about 360 days measures out the single signs, and settles after the eleventh year and 313 days, and returns to that sign in which in the twelfth year before it was. Saturn, however, in twenty-nine months and a few days more, passes through the space of a sign, and in the twenty-ninth year and about 160 days is restored to that in which it was before the thirtieth year; and for that reason, by how much less it stands apart from the edge of the world, by so much more, running a greater circuit of its wheel, it appears to be slower. §28
Those which run their circuits above the sun's course, especially when they have been in a triangle that it enters, then do not advance, but, making retreats, linger until the same sun has made transition from that triangle into another sign. Some are pleased that this comes about for this reason, that they say the sun, when it is farther off by a certain distance, hinders, by the dimness of non-lucid journeys, the wandering stars through that, with the delays of obscurity. To us, however, it does not seem so. For the sun's splendor is perspicuous and open, without any obscurations, through all the world, so that also to us the stars may appear when they make retreats and delays. §29
Therefore if our sight at such intervals can observe this, why then with the divinities and splendors of the stars do we judge that obscurities can be cast before them? Therefore rather this reasoning will stand with us: that heat, as it evokes all things and draws them to itself (as we see fruits rising from the earth into the height through warmth, no less that the vapors of waters are stirred from springs to the clouds by risings), by the same reasoning the vehement onset of the sun, with rays extended in the form of a triangle, draws the following stars to itself, and those running before it, as it were reining in and holding back, does not suffer them to go forward, but compels them to return to itself until it has gone out into the sign of another triangle. §30
Perhaps it will be asked, why the sun makes in these heats these retentions at the fifth sign from it rather than at the second or third which are nearer? Therefore I will set forth how this seems to take place. Its rays in the world are extended by their directions in the form of a triangle of equal sides; and this is neither more nor less than to the fifth sign from it. Therefore if the rays, diffused through all the world, wandered by circles and were not projected in the form of a triangle, the nearer things would burn. And Euripides, the Greek poet, seems to have observed this. For he says that things farther from the sun are hotter, but nearer it has tempered. So he writes in the play Phaethon: καίει τὰ πόρρω, τἀγγύθεν δ᾽ εὔκρατ᾽ ἔχει. §31
If therefore the fact and the reasoning and the testimony of the ancient poet show this, I do not think it ought to be judged otherwise than as we have written above about this matter. Jupiter, running between the circuits of Mars and Saturn, flies through a greater course than Mars, less than Saturn. Likewise the other stars, by how much greater space they are distant from the outermost sky, and by how much nearer to the earth they have their circling, the quicker they seem to move, because whichever of them runs a lesser circuit oftener enters going under and passes the higher. §32
As on the wheel which potters use, if seven ants were placed, and as many channels in the wheel were made around the center, growing from the lowest to the outermost, in which these ants were compelled to make a circuit, and the wheel were turned in one direction, it would be necessary for them to complete their journeys against the wheel's turn no less in the opposite direction, and for that which had its way nearest the center to wander more swiftly, and that which ran the outermost orbit, even though it walked equally swiftly, on account of the magnitude of its circuit, to complete its course much more slowly. Similarly, stars straining against the course of the world perfect their circuits by their own journeys, but by the sky's turn are carried back by overflowings in the daily circumambulation of time. §33
That there are some stars tempered, others fervid, others also cold, this seems to be the cause: that all fire has a flame climbing into higher places. Therefore the sun, by burning the ether which is above itself with its rays, makes it glowing, in which places the star Mars has its course; so Mars becomes fervid from the burning of the sun. But Saturn, because it is nearest the edge of the world, and touches the frozen regions of the sky, is vehemently cold. Jupiter, since it has its course between the circuits of both, is seen, by the middle cooling and heat of those, to have agreeing and most temperate effects. Of the zone of the twelve signs and the contrary work and course of the seven stars, by what reasonings and numbers they pass from sign to sign, and of their circuit, as I received from my teachers, I have set forth. Now of the waxing light of the moon and its diminution, as it was handed down to us by our forebears, I shall speak. §34
Berosus, who from the city or nation of the Chaldaeans advanced into Asia, and also disclosed the Chaldean discipline, declared that the moon is a ball, half of which is glowing, the rest of a blue color; but that, running its course, when it goes under the orb of the sun, it is then seized by the rays and the onset of heat, and is turned glowing on account of its own property of light to light; but when, called by the orb of the sun, it looks at the upper parts, its lower part is seen dark because, for the likeness of air, it is not glowing. When it stands perpendicular to the sun's rays, the whole light is retained to its upper aspect, and it is then called the first (new). §35
When, going past, it goes to the eastern parts of the sky, it is relaxed from the onset of the sun, and the extreme part of its brightness, on a most thin line toward the earth, sends its splendor, and so from this it is called the second. By daily remission of its turning, the third and fourth are daily counted. On the seventh day, when the sun is in the west, but the moon holds the middle regions of the sky between east and west, because it stands off by half of the sky's space from the sun, it has half of the brightness turned to the earth. But when the whole space of the world lies between the sun and moon, and the moon rising, the sun is opposite in the west, it, the farther it is from the rays, on the fourteenth day, by the full wheel of its whole orb sends splendor; and on the remaining days, by daily decreases, to the completion of the lunar month, by turns and courses called back by the sun, it goes under the wheel and its rays, and so makes the monthly reasonings of the days. §36
But as Aristarchus of Samos the mathematician, with great vigor by the disciplines of variety, left reasonings on the same matter, I will set them forth. For it is not hidden that the moon has no light of its own and proper, but is like a mirror, and receives splendor from the sun's onset. For the moon, of the seven stars, wanders the circle nearest the earth, least in its courses. So every month, under the wheel of the sun and its rays, one day before it passes by, being hidden it is obscured, and, when with the sun, is called new. On the next day, which is counted as the second, passing by the sun, it makes a visit of a thin outermost curvature. When it has receded three days from the sun, it grows and is more lighted. Daily departing, when it has come to the seventh day, standing apart from the setting sun about half the sky's region, it shines half; and the part of it which looks to the sun is lighted. §37
But on the fourteenth day, when in diameter's space of the whole world it is distant from the sun, it is made full and rises when the sun is in the west; because standing apart the whole space of the world, it meets opposite, and, from the onset of the sun, the whole orb receives into itself splendor. On the seventeenth day, when the sun rises, the moon is pressed into the west. On the twenty-second, when the sun has risen, the moon holds about the middle regions of the sky, and so what looks to the sun, that it has lighted, the rest being dark. Likewise, running a daily course, about the twenty-eighth day it goes under the rays of the sun, and so makes the monthly reasonings. Now how in the single months the sun, going through the signs, increases and diminishes the spaces of the days and hours, I will say. §38
For when it enters the sign of Aries and wanders through the eighth part, it perfects the vernal equinox. When it advances to the tail of Taurus, and to the stars of the Pleiades, from which the forward half of Taurus rises, it runs into a greater space of the world than half, advancing toward the northern part. When from Taurus it enters into Gemini, at the rising of the Pleiades, it grows more above the earth, and increases the spaces of days. Then, from Gemini, when it enters Cancer, in which it holds the longest space of sky, when it has come to the eighth part, it perfects the solstitial time, and going on it reaches the head and breast of the Lion, for those parts are assigned to Cancer. §39
From the breast of the Lion and the boundaries of Cancer, the sun, going through the remaining parts of the Lion as it departs, diminishes the daily magnitude and the circle, and returns to an equal course with Gemini. Then crossing from Leo into Virgo, and going on to the fold of her vestment, it contracts the circle, and equalizes to that which Taurus has as the reasoning of course. From Virgo, going on through her bosom (which bosom holds the first parts of Libra), in the eighth part of Libra it perfects the autumn equinox. This course equals that circle which was in the sign of Aries. §40
When the sun has entered Scorpio, at the setting of the Pleiades, advancing it lessens toward the southern parts the lengths of days. From Scorpio, running on, it enters Sagittarius at his thighs, and flies a more contracted daily course. But when it begins from the thighs of Sagittarius, which part is assigned to Capricorn, up to the eighth part, it runs the shortest space of the sky. From this brevity of day, the bruma (winter solstice) and the brumal days are named. From Capricorn, crossing into Aquarius, it augments the space of the day from the length equal to Sagittarius. From Aquarius, when it has entered into Pisces, with Favonius blowing, it prepares a course equal to Scorpio. Thus the sun, wandering round those signs at fixed times, augments or diminishes the spaces of days and hours. Now of the other stars, which are disposed and figured on the right and left of the zone of signs, on the southern and northern part of the world, I will speak. §41
Behold Septentrio, which the Greeks call ἄρκτον or ἑλίκην, has placed behind it the Watcher (Bootes). Not far from it is shaped the Virgin, above whose right shoulder a most shining star shines forth, which ours call Provindemiator, the Greeks προτρυγητήν. Shining more is her Spica. Placed likewise another, opposite the middle of the knees of the Guardian of the Bear, is the star which is called Arcturus, there dedicated. §42
In the region of the head of Septentrio, transversely to the feet of Gemini, stands the Charioteer at the top of the horn of Taurus; and so at the top of the left horn of Taurus and of the Charioteer's right foot, one star holds the place. And joined to the hand of the Charioteer are the Kids; the She-goat at the left shoulder. Above Taurus and Aries, Perseus, having on the right below the sole of his running the Pleiades, on the left the head of Aries, and leaning with his right hand upon the image of Cassiopeia, holding at the left, above Aries, the Gorgon's head, tossing it beneath the feet of Andromeda. §43
Likewise the Pisces are above Andromeda, and the Horse's belly and the bristles which are above the Horse's spine, of whose belly the most shining star finishes the Horse's belly and the head of Andromeda. Andromeda's right hand is placed above the image of Cassiopeia, the left above the northern Fish. Likewise Aquarius is above the Horse's head. The Horse's hooves touch the knees of Aquarius. Cassiopeia is placed in the middle. Above Capricorn, in the heights, are the Eagle and the Dolphin. After these is the Arrow. From it the Bird (Swan), whose right wing touches the hand and scepter of Cepheus, the left above Cassiopeia leans. Beneath the tail of the bird are set the Horse's feet. §44
Thence, above Sagittarius, Scorpio, and Libra, the Serpent touches the Crown with the tip of its snout. From it, Ophiuchus holds the middle Serpent in his hands, trampling with his left foot upon the middle of Scorpio's forehead. From the part of Ophiuchus's head, not far placed, is the head of him who is called 'striving on his knees' (Hercules). Of these, the tops of the heads are easier for recognition, because they are shaped with not dim stars. §45
The foot of the Kneeler is propped on the temple of the head of the serpent, in which the Bears, which are called Septentriones, are entwined. In front of the Horse, little Dolphin bends. Opposite the Bird's snout, the Lyre is set. Between the shoulders of the Guardian and the Kneeler, the Crown is ordered. In the northern circle, the two Bears are placed, composed of the bones of shoulder between themselves and averted at the breasts; of which the less is called by the Greeks κυνόσουρα, the greater ἑλίκη; and their heads, looking away from each other, are placed; their tails, opposite to the heads of the others, are figured opposite. For both by outdoing stand out at the top. §46
Through their tails, too, the Serpent is said to be stretched, and where the star which is called the pole shines, around the head of the greater Septentrio. For where is nearest, about its head it is involved; but the serpent is thrown about, once, Cynosura's head, and extended next to her feet. Twisting here and folding back, it raises itself, reflected from the head of the less to the greater round the snout and the right temple. Likewise above the tail of the less are the feet of Cepheus, and there at the topmost peak are stars making an equilateral triangle, above the sign of Aries. Of the less Septentrio and of Cepheus's image are several stars mingled. What are on the right of the rising, between the zone of signs and the constellations of the north, placed in the sky, I have said. Now I will unfold what are on the left of the rising and in the southern parts distributed by nature. §47
First, beneath Capricorn is placed the Southern Fish, looking toward the tail of the Whale. From it to Sagittarius is an empty place. The Altar is under the sting of Scorpio. The fore-parts of the Centaur are nearest Libra and Scorpio. He holds in his hands that image which the skilled in stars have named the Beast. Pointing, the Snake stretches a train of stars, entwined surrounding the Virgin, the Lion, and the Cancer; at the region of Cancer raising the snout, sustaining the Cup at the Lion and the middle of its body, laying the tail beneath the hand of the Virgin, in which is the Raven. Those that are above the shoulders are of equal brightness. §48
At the lower belly of the Snake, under the tail, is placed the Centaur. Near the Cup and the Lion is the Ship which is called Argo, whose prow is obscured, but the mast and those things that are around the rudder are seen standing out; and the very small boat's stern, at the top of the tail, is joined to the Dog. The lesser Dog also follows the Twins, opposite the head of the Snake. The greater likewise follows the less. Orion, indeed, is set transverse, pressed by the hoof of the Bull, holding in his left hand a club, the other raising to the Twins. §49
At his base, the Dog follows the Hare at a small distance. To the Ram and the Fishes, the Whale is set; from whose crest, arranged to each of the Fishes, a thin stretch of stars is disposed (which the Greeks call ἁρπεδόναι); and at a great interval, drawn inwards, the knot of the clinging reaches the topmost crest of the Whale. Through the form of the stars a river flows, its beginning taking from the left foot of Orion. The water which is said to be poured from Aquarius flows between the Southern Fish's head and the Whale's tail. §50
The figures and images of the stars in the world, designed, as Democritus the natural philosopher held, by nature and the divine mind, I have set out; but only those whose risings and settings we can observe and behold with our eyes. For as the Septentriones, turning about the axis' pole, do not set nor go under the earth, so around the southern pole, which, on account of the inclination of the world, is subject to the earth, the stars are unseen and hidden, and do not rise above the earth. Therefore their figures, on account of the obstruction of the earth, are not known. The index of this is the star Canopus, which is unknown in these regions, but merchants have announced it, who have reached the farthest regions of Egypt and the boundaries next to the outermost bounds of the earth. §51
Of the flying of the world around the earth, and of the disposition of the twelve signs and of the constellations from the northern and the southern part, that it may be perspicuous, I have taught. For from that world-turning, and from the contrary course of the sun through the signs, and from the equinoctial shadows of gnomons, the descriptions of the analemmata are found. §52
As for what effects the twelve signs, the five stars, the sun, and the moon have on the life of man from astrology, this is to be conceded to the reasonings of the Chaldaeans, since it is proper to their reasoning of genethlialogy to be able to unfold past and future events from the reasonings of the stars. Their inventions they have left, and of keenness and acumen they were great, who flowed from the very nation of the Chaldaeans; and first Berosus settled in the island and city of Cos, and there opened the discipline; after, Antipater devoting himself to it, and then Athenodorus, who even from conception, not from birth, left reasonings of genethlialogy unfolded. §53
Of natural things Thales of Miletus, Anaxagoras of Clazomenae, Pythagoras of Samos, Xenophanes of Colophon, Democritus of Abdera left reasonings by which those matters are governed by the nature of things, however effects they may have. Following their inventions, the risings and settings of the stars and the significances of weather, Eudoxus, Euctemon, Callippus, Meton, Philippus, Hipparchus, Aratus, and the rest, found by the disciplines of parapegmata from astrology, and left these unfolded for posterity. Their sciences are to be looked up to by men, because they were so careful that they seem, by divine mind, to announce signs of weather to come. Therefore these things are to be conceded to their cares and studies. §54
From these we must separate our reasonings, and unfold the monthly brevities of the day, and likewise dilations. For the sun, turning at equinoctial time in Aries and Libra, of the parts of the gnomon which it has nine, makes eight of the shadow in the inclination of the sky which is at Rome. Likewise at Athens, as great as four parts of the gnomon are, the shadows are three; at Rhodes, to seven, five; at Tarentum, to eleven, nine; to five at Alexandria, three; and in all other places, the equinoctial shadows of gnomons are found, one in one fashion, another in another, distributed by the nature of things. §55
So in whatsoever places sundials are to be described, in that place the equinoctial shadow is to be taken; and if they are, as at Rome, nine parts of the gnomon, shadow of eight, let a line be described on a plane and, from the middle, let there be raised at right angles πρὸς ὀρθάς what is called the gnomon; and from the line which is the plane, on the line of the gnomon, with compasses, nine spaces be measured; and where the sign of the ninth part is, let the center be established where the letter A will be; and drawing the compasses from that center to the line of the plane where the letter B will be, let a circle be drawn, which is called the meridian. §56
Then of the nine parts which are from the plane to the gnomon's center, let eight be taken and marked on the line which is on the plane, where the letter C will be. This will be the equinoctial shadow of the gnomon. And from that sign and the letter C, through the center where is the letter A, let a line be drawn where will be the solar equinoctial ray. Then from the center, with the compasses drawn to the line of the plane, let an equal breadth be marked where the letter E will be, on the left side, and I on the right, at the ends of the line of the circumference; and through the center let a line be drawn, that two equal hemicycles may be divided. This line is called by the mathematicians the horizon. §57
Then of the whole circumference, a fifteenth part is to be taken, and the center of the compasses placed on the line of the circumference where that line is cut by the equinoctial ray, where will be the letter F, and marked on the right and the left where are letters G and H. Then from these let lines be drawn up to the line of the plane, where will be letters T and R. So there will be one solar ray of winter, the other of summer; and opposite the letter E will be I, where that line through the center, where is the letter A, cuts the circumference; and opposite G and H will be the letters L and K, and opposite C and F and A will be the letter N. §58
Then diameters must be drawn from G to L and from H to K. The upper will be of the summer part, the lower of the winter. These diameters are to be divided equally in the middle, where will be the letters M and O; and there centers are to be marked, and through these marks and the center A a line is to be drawn to the extremes of the line of the circumference where will be the letters P Q. This will be a line πρὸς ὀρθάς to the equinoctial ray; it will be called by the mathematical reasonings the axon. And from the same centers, the compasses being drawn to the extremes of the diameters, let hemicircles be described, of which one will be of summer, the other of winter. §59
Then in the places where the parallel lines cut the line that is called horizon, on the right side will be the letter S, on the left V, and from the extremity of the hemicircle where is the letter G, let a line be drawn parallel to the axon to the left hemicircle where is the letter H. This parallel line is called logotomus. And then the center of the compasses must be placed in that place where the equinoctial ray cuts that line, where the letter D will be, and drawn to that place where the summer ray cuts the circumference, where the letter H is. From the equinoctial center, with the summer interval, let the circle of the monthly (menaeus) be drawn, which is called menaeus. §60
Thus will be had the shaping of the analemma. When this has been so described and unfolded, whether by winter lines, by summer lines, by equinoctial, or even by monthly, the reasonings of hours in the projections are to be described from the analemma; and many varieties and kinds of sundials are sub-projected in it and are described by these artistic reasonings. Of all the figures and descriptions, the effect is one: that the equinoctial day, and the brumal, and likewise the solstitial, be divided equally into twelve parts. Wherefore, not deterred by sloth, I have not passed it over; but, lest by writing many things I offend, I will set forth by whom the kinds and descriptions of sundials have been discovered. For neither can I now invent new kinds, nor does it seem right to claim others' as mine. So, what have been handed down to us, and by whom they were invented, I will say. §61
The hemicycle, hollowed out of a square and cut back to the inclination, Berosus the Chaldean is said to have invented. The scaphe or hemisphere, Aristarchus of Samos, who also made the disc on the plane. The spider, Eudoxus the astrologer (some say Apollonius). The plinthion or coffer-dial (which is also placed in the Flaminian Circus), Scopinas of Syracuse. The πρὸς τὰ ἱστορούμενα, Parmenion; the πρὸς πᾶν κλίμα, Theodosius and Andreas; Patrocles the pelekinon; Dionysodorus the cone; Apollonius the quiver; and other kinds, and those men above written, and many more, left inventions: as conarachne, conic plinthion, antiboreum. Likewise many left written that hanging traveler's dials should be made from these kinds. From whose books, if anyone wishes, he can find projections, provided he knows the descriptions of analemmata. §62
Likewise there are reasonings of water-clocks, found out by the same writers, and first by Ctesibius of Alexandria, who also discovered the force of natural breath and pneumatic things. But how these were sought out, it is worth while for students to know. Ctesibius had been born at Alexandria, his father a barber. He, excelling beyond the rest in genius and great industry, was said to delight himself in artistic things. For when he wished in his father's shop to hang a mirror so that when drawn up and drawn back the hidden counter-weight of a cord should draw it, he so placed his mechanism. §63
He fixed a wooden channel under the beam, and there placed pulleys. Through the channel he led a cord to a corner, and there he built pipes. Into these he caused a leaden ball to be lowered by the cord. So the weight, when, running down, it pressed into the narrows of the pipes the density of air, by a vehement descent through the throats, thrust out into the open air the compressed air, solidified by compression, and, upon the blow of the touch, expressed a clearness of sound. §64
Therefore Ctesibius, when he had observed that from the touch of air and the expressions of breath, sounds and voices are born, using these principles, first instituted hydraulic machines. Likewise he unfolded the expressions of waters, and self-moving machines, and many kinds of delights, and among them the constructions of water-clocks. And first he set up an orifice perfected of gold, or of pierced gem. For these neither are worn by the running of the water, nor take in filth so as to be stopped up. §65
For water flowing equally through that orifice lifts up a little inverted boat (which by the craftsmen is called phellos or tympanum); in which is placed a rule, perfected by a turning drum with equal teeth. These teeth, pushing one another, make measured turnings and motions. Likewise other rules and other drums, toothed in the same way, compelled by one motion, by turning make effects and varieties of motions, in which figures are moved, cones are turned, little stones or eggs are thrown forth, trumpets sound, and the rest of the side-shows. §66
In these also, whether on a column or on a pilaster, the hours are described, which a little figure, emerging from the bottom, by its rod signals through the whole day. The diminishings or increases of these, the addition or removal of wedges compels to be perfected on single days and months. The regulations of waters for tempering are so established. Two cones are made, one solid, one hollow, so perfected on the lathe that one can fit into the other; and by the same rule, their loosening or tightening makes either a vehement or gentle flow of water flowing into those vessels. So by these reasonings and by this mechanism, the arrangements of water-clocks for winter use are composed of water. §67
But if the contractings of the days or their increases be not approved to be made by additions and subtractions of wedges, because the wedges very often make defects, thus must it be unfolded. Let the hours be described on a little column transversely from the analemma, and let the monthly lines be marked on the little column. And let that column be made to turn, so that, with reference to the figure and its rod (by which rod, issuing, the figure shows the hours), the column, turning continuously, may make the brevities and increases of the hours of each its own month. §68
There are made also, by another kind, winter sundials, which are called anaphoric, and are perfected by these reasonings. Hours are disposed by bronze rods, from the analemma's description, arranged from the center, on the front. In it, circles are drawn around, bounding the monthly spaces. Behind these rods is a drum, on which is described and painted the world and the zodiacal circle; and the description from the twelve heavenly signs makes a figure, of which from the center is shaped one greater, another less. In the rear part, a turning axle is enclosed in the middle of the drum, and in that axle a flexible bronze chain is wound, from which hangs on one side a float which is raised by the water, on the other an equal weight of ballast-lead. §69
So as much as the float is lifted by the water, so much the weight of ballast drawing below turns the axle, and the axle the drum. The turning of this drum makes now a greater part of the zodiacal circle, now a lesser, in its turnings, designate the proprieties of the hours for their times. For in the single signs are perfected holes for the number of days of each its own month, by which the knob, which in sundials seems to hold the image of the sun, signifies the spaces of the hours. This knob, transferred from hole to hole, completes the course of the running month. §70
So as the sun, going through the spaces of the stars, dilates and contracts the days and hours, so the knob in the sundials entering through the points, against the drum's central turning, daily, as it is transferred at different times through wider, at other times through narrower spaces, by monthly limits makes the images of hours and days. Of the administration of the water, as it may be tempered to the reasoning, thus must it be done. §71
Behind the front of the clock, within let a castellum be placed, and into it through a pipe let water flow; and let it have at the bottom a cavity. Let there be fixed to it a bronze drum having an opening through which, from the castellum, water may flow into it. In that one, moreover, let a smaller drum be included, with hinges made on the lathe, male and female, pressed together, so that the smaller drum, as an epitonium, may turn closely and gently in the larger. §72
Let the lip of the larger drum have 365 points marked at equal intervals; the smaller little orb at the outside of the circumference have a tongue fixed, whose point directs to the regions of the points, and in that little orb let a tempered opening be, because through it the water flows into the drum and preserves the administration. Let there be shaped on the lip of the larger drum the shapings of the heavenly signs, and let that be immovable; let it have at the top the sign of Cancer shaped, perpendicular to it at the bottom Capricorn, on the right of the beholder Libra, on the left Aries, and the rest designated between these spaces as they are seen in the sky. §73
When therefore the sun is in Capricorn, the tongue of the little orb, in the part of Capricorn of the larger drum, daily touching single points, perpendicularly holding the water's running vehement weight, swiftly through the little orb's opening thrusts it into a vessel which, receiving it, is filled in a short space, and thus snatches and contracts the spaces of days and hours. But when, by daily turning of the smaller drum, the tongue enters the points of Aquarius, the opening will depart from the perpendicular, and the water will be compelled from vehement course to send forth its stream more slowly. So by how much less with swift course the vessel receives the water, by so much it dilates the spaces of the hours. §74
The points of Aquarius and Pisces, as by steps climbing, the little orb's opening touching Aries in the eighth part, the water moderately flowing, gives the equinoctial hours. From Aries through the spaces of Taurus and Gemini, to the topmost points of the eighth part of Cancer, the opening, moving by the turnings of the drum, and going back into the height, is exhausted of forces, and so by flowing more slowly it dilates by delaying the spaces, and makes the hours solstitial in the sign of Cancer. From Cancer, when it inclines and runs through Leo and Virgo to the eighth part of Libra, returning and gradually snatching the spaces, it contracts the hours, and so arriving at the points of Libra restores again the equinoctial hours. §75
Through the spaces of Scorpio and Sagittarius, sloping more steeply, pressing itself down, and returning by turning to the eighth part of Capricorn, is restored, by the swiftness of the flow, to the winter hours' brevities. What reasonings and preparations in the descriptions of sundials, that they may be more ready for use, as aptly as I could, I have written. It remains now to reason of machines and their principles. Therefore of these, that a corrected body of architecture may be perfected, in the next volume I will begin to write. §76