1. The ancestors of the Greeks held the celebrated wrestlers who were victors in the Olympic, Pythian, Isthmian and Nemean games in such esteem, that, decorated with the palm and crown, they were not only publicly thanked, but were also, in their triumphant return to their respective homes, borne to their cities and countries in four horse chariots, and were allowed pensions for life from the public revenue. When I consider these circumstances, I cannot help thinking it strange that similar honours, or even greater, are not decreed to those authors who are of lasting service to mankind. Such certainly ought to be the case; for the wrestler, by training, merely hardens his own body for the conflict; a writer, however, not only cultivates his own mind, but affords every one else the same opportunity, by laying down precepts for acquiring knowledge, and exciting the talents of his reader.
2. What does it signify to mankind, that Milo of Crotona, and others of this class, should have been invincible, except that whilst living they were ennobled by their fellow countrymen? On the other hand the doctrines of Pythagoras, Democritus, Plato, Aristotle, and other sages, the result of their daily application, and undeviating industry, still continue to yield, not only to their own country, but to all nations, fresh and luscious fruit, and they, who from an early age are satiated therewith, acquire the knowledge of true science, civilize mankind, and introduce laws and justice, without which no state can long exist.
3. Since, therefore, individuals as well as the public are so indebted to these writers for the benefits they enjoy, I think them not only entitled to the honour of palms and crowns, but even to be numbered among the gods. I shall produce in illustration, some of their discoveries as examples, out of many, which are of utility to mankind, on the exhibition whereof it must be granted without hesitation that we are bound to render them our homage. The first I shall produce will be one of Plato, which will be found of the greatest importance, as demonstrated by him.
4. If there be an area or field, whose form is a square, and it is required to set out another field whose form is also to be a square, but double in area, as this cannot be accomplished by any numbers or multiplication, it may be found exactly by drawing lines for the purpose, and the demonstration is as follows. A square plot of ground ten feet long by ten feet wide, contains an hundred feet; if we have to double this, that is, to set out a plot also square, which shall contain two hundred, we must find the length of a side of this square, so that its area may be double, that is two hundred feet. By numbers this cannot be done;a for if the sides are made fourteen feet, these multiplied into each other give one hundred and ninety-six feet; if fifteen feet, they give a product of two hundred and twenty-five.
5. Since, therefore, we cannot find them by the aid of numbers, in the square of ten feet a diagonal is to be drawn from angle to angle, so that the square may thereby be divided into two equal triangles of fifty feet area each. On this diagonal another square being described, it will be found, that whereas in the first square there were two triangles, each containing fifty feet, so in the larger square formed on the diagonal there will be four triangles of equal size and number of feet to those in the larger square. In this way Plato shewed and demonstrated the method of doubling the square, as the figure appended explains.
6. Pythagoras demonstrated the method of forming a right triangle without the aid of the instruments of artificers: and that which they scarcely, even with great trouble, exactly obtain, may be performed by his rules with great facility. Let three rods be procured, one three feet, one four feet, and the other five feet long; and let them be so joined as to touch each other at their extremities; they will then form a triangle, one of whose angles will be a right angle. For if, on the length of each of the rods, squares be described, that whose length is three feet will have an area of nine feet; that of four, of sixteen feet; and that of five, of twenty-five feet:
7. so that the number of feet contained in the two areas of the square of three and four feet added together, are equal to those contained in the square, whose side is five feet. When Pythagoras discovered this property, convinced that the Muses had assisted him in the discovery, he evinced this gratitude to them by sacrifice. This proposition is serviceable on many occasions, particularly in measuring, no less than in setting out the staircases of buildings, so that each step may have its proper height.
8. For if the height from the pavement to the floor be divided into three parts, five of those parts will be the exact length of the inclined line which regulated the blocks of which the steps are formed. Four parts, each equal to one of the three into which the height from the pavement to the floor was divided, are set off from the perpendicular, for the position of the first or lower step. Thus the arrangement and ease of the flight of stairs will be obtained, as the figure will shew.
9. Though Archimedes discovered many curious matters which evince great intelligence, that which I am about to mention is the most extraordinary. Hiero, when he obtained the regal power in Syracuse, having, on the fortunate turn of his affairs, decreed a votive crown of gold to be placed in a certain temple to the immortal gods, commanded it to be made of great value, and assigned an appropriate weight of gold to the manufacturer. He, in due time, presented the work to the king, beautifully wrought, and the weight appeared to correspond with that of the gold which had been assigned for it.
10. But a report having been circulated, that some of the gold had been abstracted, and that the deficiency thus caused had been supplied with silver, Hiero was indignant at the fraud, and, unacquainted with the method by which the theft might be detected, requested Archimedes would undertake to give it his attention. Charged with this commission, he by chance went to a bath, and being in the vessel, perceived that, as his body became immersed, the water ran out of the vessel. Whence, catching at the method to be adopted for the solution of the proposition, he immediately followed it up, leapt out of the vessel in joy, and, returning home naked,º cried out with a loud voice that he had found that of which he was in search, for he continued exclaiming, in Greek, εὑρηκα, (I have found it out).
11. After this, he is said to have taken two masses, each of a weight equal to that of the crown, one of them of gold and the other of silver. Having prepared them, he filled a large vase with water up to the brim, wherein he placed the mass of silver, which caused as much water to run out as was equal to the bulk thereof. The mass being then taken out, he poured in by measure as much water as was required to fill the vase once more to the brim. By these means he found out what quantity of water was equal to a certain weight of silver.
12. He then placed the mass of gold in the vessel, and, on taking it out, found that the water which ran over was lessened, because, as the magnitude of the gold mass was smaller than that containing the same weight of silver. After again filling the vase by measure, he put the crown itself in, and discovered that more water ran over then than with the mass of gold that was equal to it in weight; and thus, from the superfluous quantity of water carried over the brim by the immersion of the crown, more than that displaced by the mass, he found, by calculation, the quantity of silver mixed with the gold, and made manifest the fraud of the manufacturer.b
13. Let us now consider the discoveries of Archytas the Tarentine, and Eratosthenes of Cyrene, who, by the aid of mathematics, invented many things useful to mankind; and though for other inventions they are remembered with respect, yet they are chiefly celebrated for their solution of the following problem. Each of these, by a different method, endeavoured to discover the way of satisfying the response of Apollo of Delos, which required an altar to be made similar to his, but to contain double the number of cube feet, on the accomplishment of which, the island was to be freed from the anger of the gods.
14. Archytas obtained a solution of the problem by the semicylinder, and Eratosthenes by means of a proportional instrument.º The pleasures derivable from scientific investigations, and the delight which inventions afford when we consider their effects, are such that I cannot help admiring the works of Democritus, on the nature of things, and his commentary, entitled Χειροτόνητον, wherein he sealed with a ring, on red wax, the account of those experiments he had tried.º
15. The discoveries, therefore, of these men are always at hand, not only to correct the morals of mankind, but also to be of perpetual advantage to them. But the glory of the wrestler and his body soon decay, so that neither whilst in vigour, nor afterwards by his instructions, is he of that service to society which the learned are by publication of their sentiments.
16. Since honours are not awarded for propriety of conduct, nor for the excellent precepts delivered by authors, their minds soaring higher, are raised to heaven in the estimation of posterity, they derive immortality from their works, and even leave their portraits to succeeding ages. For, those who are fond of literature, cannot help figuring to themselves the likeness of the poet Ennius, as they do that of any of the gods. So also those who are pleased with the verses of Accius, think they have himself, not less than the force of his expressions, always before them.
17. Many even in after ages will fancy themselves contending with Lucretius on the nature of things, as with Cicero on the art of rhetoric. Many of our posterity will think that they are in discourse with Varro when they read his work on the Latin language: nor will there be wanting a number of philologers, who, consulting in various cases the Greek philosophers, will imagine that they are actually talking with them. In short, the opinions of learned men who have flourished in all periods, though absent in body, have greater weight in our councils and discussions than were they even present.
18. Hence, O Cæsar, relying on these authorities, and using their judgment and opinions, I have written these books; the first seven related to buildings, the eighth to the conduct of water, and in this I propose treating on the rules of dialling, as deducible from the shadow produced by the rays of the sun from a gnomon, and I shall explain in what proportions it is lengthened and shortened.
1. It is clearly by a divine and surprising arrangement, that the equinoctial gnomons are of different lengths in Athens, Alexandria, Rome, Piacenza, and in other parts of the earth. Hence the construction of dials varies according to the places in which they are to be erected; for from the size of the equinoctial shadow, are formed analemmata, by means of which the shadows of gnomons are adjusted to the situation of the place and the lines which mark the hours. By an analemma is meant• a rule deduced from the sun's course, and founded on observation of the increase of the shadow from the winter solstice, by means of which, with mechanical operations and the use of compasses, we arrive at an accurate knowledge of the true shape of the world.
2. By the world is meant the whole system of nature together with the firmament and its stars. ºThis continually turns round the earth and sea on the extreme points of its axis, for in those points the natural power is so contrived that they must be considered as centres, one above the earth and sea at the extremity of the heavens by the north stars, the other opposite and below the earth towards the south; moreover in these central points as round the centres of wheels, which the Greeks call πόλοι, the heavens perpetually revolve. Thus the earth and sea occupy the central space.
3. Hence from the conclusion, the polar centre is raised above the earth in the northern part, whilst that in the southern part, which is underneath, is hidden from our view by the earth, and through the middle obliquely and inclined to the south, is a large band comprising the twelve signs, which, by the varied combinations of the stars being divided into twelve equal parts, contains that number of representations of figures. These are luminous, and with the firmament and the other stars and constellations, make their circuit round the earth and sea;
4. all these, visible as well as invisible, have their fixed seasons, six of the signs turning above the earth, the remaining six below it; which latter are hidden by the earth. Six of them, however, are always above the earth; for the portion of the last sign, which by the revolution is depressed below the earth and hidden by it, is on the opposite side equal to that of a fresh sign emerging from darkness by the force of the moving power; since it is the same power and motion which cause the rising and the setting at the same moment.
5. As these signs are twelve in number, each occupies a twelfth part of the heaven, and they move continually from east to west: and through them in a contrary course, the moon, Mercury, Venus, the sun itself, Mars, Jupiter and Saturn, as if ascending, pass through the heavens from west to east in different orbits. The moon making her circuit in twenty-eight days and about one hour, and thus returning to the sign from which she departed, completes the lunar month.c
6. The sun, in the course of a month, passes through the space of one sign which is a twelfth part of the heavens; hence in twelve months going through the twelve signs, when he has returned to that sign from which he set out, the period of a year is completed: but that circle which the moon passes through thirteen times in twelve months, the sun passes through only once in the same time. The planets Mercury and Venus nearest the rays of the sun, move round the sun as a centre,d and appear sometimes retrograde and sometimes progressive, seeming occasionally, from the nature of their circuit, stationary in the signs.e
7. This may be observed in the planet Venus, which when it follows the sun, and appears in the heavens with great lustre after his setting, is called the evening star; at other times preceding him in the morning before sunrise, it is called the morning star. Wherefore these planets at times appear as if they remained many days in one sign, whilst at other times they pass rapidly from one to another; but though they do not remain an equal number of days in each sign, the longer they are delayed in one the quicker they pass through the succeeding one,º and thus perform their appointed course: in this manner it happens that being delayed in some of the signs, when they escape from the retention, they quickly pass through the rest of their orbit.
8. Mercury revolves in the heavens in such a manner, that passing through the several signs in three hundred and sixty days,☿ he returns to that sign from which he set out, remaining about thirty days in each sign.
9. The planet Venus, as soon as she escapes from the influence of the suns' rays, runs through the space of one sign in forty days; and what she loses by stopping a long time in one sign, she makes up by her quick passage through others. She completes her circuit through the heavens in four hundred and eighty-five days;f by which time she has returned to the sign from whence she set out.
10. Mars, on about the six hundred and eighty-third day,♂ completes the circuit of the signs, and returns to his place; and if, in any sign, he move with a greater velocity, his stationary state in others equalizes the motion, so as to bring him round in the proper number of days. Jupiter moving also in contrary rotation, but with less velocity, takes three hundred and sixty days to pass through one sign; thus lengthening the duration of his circuit to eleven years and three hundred and twenty-three days♃ before he returns to the sign in which he was seen twelve years before his setting out. Lastly, Saturn, remaining thirty-one months and some days in each sign, returns to his point of departure at the end of twenty-nine years, and about one hundred and sixty days, or nearly thirty years.♄Hence, the nearer he is to the extremity of the universe, the larger does his circuit appear, as well as the slower his motion.
11. All those which make their circuit above that of the sun,º especially when they are in trine aspect, do not advance, but, on the contrary, are retrograde, and seem to stop till the sun passes from the trinal sign into another. Some are of opinion, that this happens on account of their great distance from the sun, on which account their paths not being sufficiently lighted, they are retarded by the darkness. But I am not of that opinion, since the brightness of the sun is perceptible, evident and unobscured throughout the system, just as it appears to us, as well when the planets are retrograde as when they are stationary. If, then, our vision extends to such a distance, how can we imagine it possible to obscure the glorious splendour of the planets?
12. It appears more probable, that it is the heat which draws and attracts all things towards itself: we, in fact, see the heat raise the fruits of the earth to a considerable height, and the spray of waters from fountains ascend to the clouds by the rainbow: in the same manner the excessive power of the sun spreading his rays in a triangular form, attracts the planets which follow him, and, as it were, stops and restrains those which precede him, preventing them from leaving him, and, indeed, forcing them to return to him, and to remain in the other trinal sign.
13. One may perhaps ask, whence it happens that the sun, by its heat, causes a detention in the fifth sign from itself, rather than in the second or third, which are nearer.º This may be thus explained. Its rays diverge through the heavens in lines which form a triangle whose sides are equal. Those sides fall exactly in the fifth sign. For if the rays fell circularly throughout the system, and were not bounded by a triangular figure, the nearer places would be absolutely burnt. This seems to have struck the Greek poet, Euripides; for he observes, that those places more distant from the sun are more intensely heated than those temperate ones that are nearer to him: hence, in the tragedy of Phaëthon, he says, Καίει τὰ πόῤῥω, τὰ δ’ ἐγγὺς εὔκατ’ ἔχει. (The distant places burn, those that are near are temperate.)
14. If, therefore, experience, reason, and the testimony of an antientº poet, prove it, I do not see how it can be otherwise than I have above shewn. Jupiter performs his circuit between those of Mars and Saturn: thus it is greater than that of Mars, but less than that of Saturn. In short, all the planets, the more distant they are from the extremity of the heaven, and the nearer their orbit is to the earth, seem to move swifter; for those which have a smaller orbit, often pass those above them.
15. Thus, on a wheel similar to those in use among potters, if seven ants be placed in as many channels round the centre, which are necessarily greater in proportion to their distance therefrom, and the ants are forced to make their circuits in these channels, whilst the wheel moves round in an opposite direction, they will assuredly complete their circuit, notwithstanding the contrary motion of the wheel; and, moreover, that nearest the centre will perform his journey sooner than he who is travelling in the outer channel of the wheel, who, though he move with great velocity, yet, from the greater extent of its circuit, will require a longer time for its completion. It is even so with the planets, which, each in its particular orbit, revolve in a direction contrary to the motion of the heavens, although, in their diurnal motion, they are carried backwards by its rotation.
16. The reason why some planets are temperate, some hot, and others cold, appears to be this; that all fire has a flame, whose tendency is upward. Hence the sun warms, by his rays, the air above him, wherein Mars moves, and that planet is therefore heated thereby. Saturn, on the contrary, who is near the extremity of the universe, and comes in contact with the frozen regions of the heavens, is exceedingly cold. Jupiter, however, whose orbit lies between those of the two just mentioned, is tempered by the cold and heat, and has an agreeable and moderate temperature. Of the band comprising the twelve signs, of the seven planets, and their contrary motions and orbits, also of the manner and time in which they pass from one sign into another, and complete their circuits, I have set forth all that I have learnt from authors. I will now speak of the moon's increase and wane, as taught by the antients.º
1. Berosus, who travelled into Asia from the state or country of the Chaldeans, teaching his doctrines, maintained that the moon was a ball, half whereof was luminous, and the remaining half of a blue colour; and that when, in its course, it approached the sun; attracted by the rays and the force of the heat, it turned its bright side in that direction, from the sympathy existing between light and light; whence, when the sun is above it, the lower part, which is not luminous, is not visible, from the similarity of its colour to the air. When thus perpendicular to the sun's rays, all the light is confined to its upper surface, and it is then called the new moon.
2. When it passes towards the east, the sun begins to have less effect upon it, and a thin line on the edge of its bright side emits its splendour towards the earth. This is on the second day: and thus, from day to day, advancing in its circuit, the third and fourth days are numbered: but, on the seventh day, when the sun is in the west, the moon is in the middle, between the east and the west; and being distant from the sun half the space of the heavens, the luminous half side will be towards the earth. Lastly; when the sun and the moon are the whole distance of the heavens from each other, and the former, passing towards the west, shines full on the moon behind it in the east, being the fourteenth day, it is then at the greatest distance from its rays, and the complete circle of the whole orb emits its light. In the remaining days it gradually decreases till the completion of the lunar month, and then returns to re-pass under the sun; its monthly rays being determined by the number of days.
3. I shall now subjoin what Aristarchus, the Samian mathematician, learnedly wrote on this subject, though of a different nature. He asserted, that the moon possesses no light of its own, but is similar to a speculum,º which receives its splendour from the sun's rays.g Of the planets, the moon makes the smallest circuit, and is nearest to the earth; whence, on the first day of its monthly course, hiding itself under the sun, it is invisible; and when thus in conjunction with the sun, it is called the new moon. The following day, which is called the second, removing a little from the sun, it receives a small portion of light on its disc. When it is three days distant from him, it has increased, and become more illuminated; thus daily elongating from him, on the seventh day, being half the heavens distant from the western sun, one half of it shines, namely, that half which is lighted by the sun.
4. On the fourteenth day, being diametrically opposite to the sun, and the whole of the heavens distant from him, it becomes full, and rises as the sun sets; and its distance being the whole extent of the heavens, it is exactly opposite to, and its whole orb receives, the light of the sun. On the seventeenth day, when the sun rises, it inclines towards the west; on the twenty-first day, when the sun rises, the moon is about mid-heaven, and the side next the sun is enlightened, whilst the other is in shadow. Thus advancing every day, about the twenty-eighth day it again returns under the rays of the sun, and completes its monthly rotation. I will now explain how the sun, in his passage through a sign every month, causes the days and hours to increase and diminish.
1. When the sun has entered the sign of Aries, and run through about an eighth part of it,h it is the vernal equinox. When he has arrived at the tail of Taurusº and the Pleiades, for which the fore part of the Bull is conspicuous, he has advanced in the heavens more than half his course towards the north. From Taurus, he enters into Gemini, at the time when the Pleiades rise, and being more over the earth increases the length of the days. From Gemini entering into Cancer, which occupies the smallest space in the heavens, and coming to the eighth division of it he determines the solstice, and moving forward arrives at the head and breast of Leo, which are parts properly within the division assigned to Cancer.i
2. From the breast of Leo and the boundaries of Cancer, the sun moving through the other parts of Leo, has by that time diminished the length of the day, as well as of his circuit, and resumes the equal motion he had when in Gemini. Hence from Leo passing to Virgo and proceeding to the indented part of her garment, he contracts his circuit, which is now equal to that which it had in Taurus. Proceeding then from Virgo through the indentation which includes the beginning of Libra, in the eighth part of that sign, the autumnal equinox is completed; the circuit being then equal to that in the sign Aries.
3. When the sun enters into Scorpio at the setting of the Pleiades, he diminishes, in passing to the southern parts, the length of the days; and from Scorpio passing to a point near the thighs of Sagittarius, he makes a shorter diurnal circuit. Then beginning from the thighs of Sagittarius, which are in Capricornus, at the eighth part of the latter he makes the shortest course in the heavens. This time from the shortness of the days, is called Bruma (winter) and the days Brumales. From Capricornus passing into Aquarius, the length of days is increased to that of those when he was in Sagittarius. From Aquarius he passes into Pisces at the time that the west wind blows; and his course is equal to that he made in Scorpio. Thus the sun travelling through these signs at stated times, increases and diminishes the duration of the days and hours. I shall now treat of the other constellations on the right and left side of the zodiac, as well those on the south as on the north side of the heavens.
1. The Great Bear, which the Greeks call ἄρκτος, and also ἑλίκη, has his keeper behind him. Not far distant is the constellation of the Virgin, on whose right shoulder is a very brilliant star, called by us Provindemia Major, and by the Greeks προτρύγετος, which shines with extraordinary lustre and colour.j Opposite to it is another star, between the knees of the Keeper of the Bear, which bears the name of Arcturus.
2. Opposite the head of the Bear, across the feet of the Twins, is Auriga (the charioteer) standing on the point of the horns of the Bull, and one side, above the left horn towards the feet of Auriga, there is a star called the hand of Auriga; on the other side the Goat's Kids and the Goat over the left shoulder. Above both the Bull and the Ram stands Perseus, which on the right extends under the bottom of the Pleiades, on the left towards the head of the Ram; his right hand rests on the head of Cassiopeia, the left holding the Gorgon's head by its top over the Bull, and laying it at the feet of Andromeda.
3. Above Andromeda are the Fishes, one under her belly, and the other above the back of the Horse; the brilliant star in the belly of the Horse is also in the head of Andromeda.k The right hand of Andromeda is placed on the figure of Cassiopeia, the left upon the north eastern fish. Aquarius stands on the head of the horse; the ears of the horse turn towards the knees of Aquarius, and the middle star of Aquarius is also common to Capricornus. Above on high is the Eagle and the Dolphin, and near them Sagitta. On the side is the Swan, the right wing of which is turned towards the hand and sceptre of Cepheus, the left leans on Cassiopeia, and under the tail of Avis the feet of the horse are hidden.
4. Above Sagittarius, Scorpio, and Libra, comes the Serpent, the point of whose snout touches the Crown; in the middle of the Serpent is Ophiuchus, who holds the Serpent in his hands, and with his left foot treads on the head of the Scorpion. Near the middle of the head of Ophiuchus is the head of the Kneeler; their heads are easily distinguished from being marked with luminous stars.
5. The foot of the Kneeler is placed on the temple of the Serpent, which is entwined between the two northern bears, called Septentriones. The Dolphin is a short distance from them. Opposite the bill of the Swan is the Lyre. The Crown lies between the shoulders of the Keeper and the Kneeler. In the northern circle are two Bears, with their shoulders and breasts in opposite directions; of these the Less is called κυνοσούρα, and the Larger ἑλίκη by the Greeks. Their heads are turned downwards, and each of their tails is towards the other's head, for both their tails are raised,
6. and that which is called the pole-star, is that near the tail of the Little Bear. Between these tails, as we have before stated, extends the Serpent, who turns round the head of that nearest to him, whence he takes a folding direction round the head of the smaller bear, and then spreading under his feet, and rising up, returns and folds from the head of the Less to the Greater Bear, with his snout opposite and shewing the right temple of his head. The feet of Cepheus are also on the tail of the Small Bear; towards which part more above our heads, are the stars which form the equilateral triangle above Aries. There are many stars common to the Lesser Bear and Cepheus. I have enumerated the constellations which are in the heavens to the right of the east between the zodiac and the north. I shall now describe those which are distributed on the southern side to the left of the east.
1. First, under Capricornus is the southern Fish looking towards the tail of the whale. Between it and Sagittarius is a vacant space. The Altar is under the sting of Scorpio. The fore parts of the Centaur are near Libra and Scorpio, and he holds in his hand that constellation which astronomers call the Beast. Near Virgo, Leo, and Cancer, the Snake stretches through a range of stars, and with its foldings encircles the region of Cancer, raising its snout towards Leo and on the middle of its body supporting the cup; its tail extends towards the hand of Virgo, and upon that is the Crow; the stars on its back are all equally luminous.
2. Under its belly, at the tail, is the Centaur. Near the cup and Leo is the ship Argo, whose prow is hidden, but the mast and parts about the steerage are clearly seen. The Ship and its poop touch the tip of the Dog's tail.º The smaller Dog is behind the Twins at the head of the Snake, and the larger follows the smaller Dog. Orion lies transversely under, pressed on by the hoof of the Bull, holding a shield in his left hand and with the club in his right hand raised towards Gemini;
3. near his feet is the Dog at a short distance following the Hare. Below Aries and Pisces is the Whale, from whose top to the two Fishes a small train of stars, which the Greeks call Ἑρμηδόνη, regularly extends, and this ligature of the Fishes twisting considerably inwards, at one part touches the top of the Whale. A river of stars, in the shape of the river Po,l begins from the left foot of Orion. The water that runs from Aquarius takes its course between the head of the southern Fish and the tail of the Whale.
4. I have explained the constellations displayed and formed in the heavens by nature with a divine intelligence, according to the system of the philosopher Democritus, confining myself to those whose rising and setting are visible. Some, however, such as the two Bears turning round the pole, never set nor pass under the earth. So also, the constellations about the south pole, which from the obliquity of the heavens is under the earth, are always hidden, and their revolution never brings them above the horizon. Whence the interposition of the earth prevents a knowledge of their forms. The constellation Canopus proves this, which is unknown in these countries, though well known to merchants who have travelled to the extremity of Egypt and other boundaries of the earth.
1. I have described the true circuit of the heavens about the earth, the arrangement of the twelve signs, also that of the northern and southern constellations, because therefrom, from the opposite course of the sun through the signs, and from the shadows of gnomons at the equinoxes, are formed the diagrams of analemmata.
2. The rest which relates to astrology, and the effects produced upon human life by the twelve signs, the five planets, the sun and the moon, must be left to the discussions of the Chaldeans, whose profession it is to cast nativities, and by means of the configurations of the stars to explain the past and the future. The talent, the ingenuity, and reputation of those who come from the country of the Chaldeans, is manifest from the discoveries they have left us in writing. Berosus was the first of them. He settled in the island and state of Cos, and there established a school. Afterwards came Antipater and Achinapolus, which latter not only gave rules for predicting a man's fate by a knowledge of the time of his birth, but even by that of the moment wherein he was conceived.
3. In respect of natural philosophy Thales the Milesian, Anaxagoras of Clazomenæ, Pythagoras the Samian, Xenophanes of Colophon Democritus the Abderite, have published systems which explain the mode in which Nature is regulated, and how every effect is produced. Eudoxus, Endæmon, Callippus, Melo, Philip, Hipparchus, Aratus, and others, following in the steps of the preceding, found, by the use of instruments, the rising and setting of the stars and the changes of the seasons, and left treatises thereon for the use of posterity. Their learning will be admired by mankind, because, added to the above, they appear as if by divine inspiration to have foretold the weather at particular seasons of the year. For a knowledge of these matters reference must therefore be made to their labours and investigation.
1. From the doctrines of the philosophers above mentioned, are extracted the principles of dialling, and the explanation of the increase and decrease of the days in the different months. The sun at the times of the equinoxes, that is when he is in Aries or Libra, casts a shadow in the latitude of Rome equal to eight ninths of the length of the gnomon.m At Athens the length of the shadow is three fourths of that of the gnomon. At Rhodes five sevenths; at Tarentum nine elevenths; at Alexandria three fifths; and thus at all other places the shadow of the gnomon at the equinoxes naturally differs.
2. Hence in whatever place a dial is to be erected, we must first obtain the equinoctial shadow. If, at Rome, the shadow be eight ninths of the gnomon, let a line be drawn on a plane surface, in the center whereof is raised a perpendicular thereto; this is called the gnomon, and from the line on the plane in the direction of the gnomon, let nine equal parts be measured. Let the end of the ninth part A, be considered as a centre, and extending the compasses from that centre to the extremity B of the said line, let a circle be described. This is called the meridian.
3. Then of those nine parts between the plane and the point of the gnomon, let eight be allotted to the line on the plane, whose extremity is marked C. This will be the equinoctial shadow of the gnomon. From the point C through the centre A let a line be drawn, and it will be a ray of the sun at the equinoxes. Then extend the compasses from the centre to the line on the plane, and mark on the left an equidistant point E, and on the right another, lettered I, and join them by a line through the centre, which will divide the circle into two semicircles. This line by mathematicians is called the horizon.
4. A fifteenth part of the whole circumference is to be then taken, and placing the point of the compasses in that point of the circumference F, where the equinoctial ray is cut, mark with it to the right and left the points G and H. From these, through the centre, draw lines to the plane where the letters T and R are placed, thus one ray of the sun is obtained for the winter, and the other for the summer. Opposite the point E, will be found the point I, in which a line drawn through the centre, cuts the circumference; and opposite to G and the points K and L, and opposite to C, F, and A, will be the point N.
5. Diameters are then to be drawn from G to L, and from H to K. The lower one will determine the summer, and the upper the winter portion. These diameters are to be equally divided in the middle at the points M and O, and the points being thus marked, through them and the centre A a line must be drawn to the circumference, where the letters P and Q are placed. This line will be perpendicular to the equinoctial ray, and is called in mathematical disquisitions, the Axon. From the last obtained points as centres (M and O) extending the compasses to the extremity of the diameter, two semicircles are to be described, one of which will be for summer, the other for winter.
6. In respect of those points where the two parallels cut that line which is called the horizon; on the right hand is placed the letter S, and on the left the letter V, and at the extremity of the semicircle, lettered G, a line parallel to the Axon is drawn to the extremity on the left, lettered H. This parallel line is called Lacotomus. Finally, let the point of the compasses be placed in that point where this line is cut by the equinoctial ray, and letter the point X, and let the other point be extended to that where the summer ray cuts the circumference, and be lettered H. Then with a distance equal to that from the summer interval on the equinoctial point, as a centre, describe the circle of the months, which is called Manacus. Thus will the analemma be completed.
7. Having proceeded with the diagram and its formation, then our lines may be projected on the analemma according to the place, either by winter lines, or summer lines, or equinoctial lines, or lines of the months, and as many varieties and species of dials as can be desired, may be constructed by this ingenious method. In all figures and diagrams the effect will be the same, that is to say, the equinoctial as well as the solstitial days, will always be divided into twelve equal parts. These matters, however, I pass over, not from indolence, but to avoid prolixity. I will merely add, by whom the different species and figures of dials were invented; for I have not been able to invent a new sort, neither will I pass off the inventions of others as my own. I shall therefore mention those of which I have any information, and by whom they were invented.
1. Berosus the Chaldean, was the inventor of the semicircle, hollowed in a square, and inclined according to the climate. Aristarchus the Samian, of the Scaphe or Hemisphere, as also of the discus on a plane. The Arachne was the invention of Eudoxus the astrologer, although some attribute it to Apollonius. The Plinthium or Lacunar, an example of which is to be seen in the Circus Flaminius, was invented by Scopas the Syracusan. The sort called Πρὸς τὰ ἱστορούμενα, by Parmenio. That called Πρὸς πᾶν κλίμα,º by Theodosius and Andrias. The Pelicinon by Patrocles. The Cone by Dionysodorus. The Quiver by Apollonius. The persons above mentioned not only invented other sorts; but the inventions of others have come down to us, such as the Gonarche, the Engonatos, and the Antiboreus.• Many also have left instructions for constructing the portable pendulous dials.º
2. Ctesibius Alexandrinus was the first who found out the properties of the wind, and of pneumatic power, the origin of which inventions is worthy of being known. Ctesibius, whose father was a barber, was born at Alexandria. Endowed with extraordinary talent and industry, he acquired great reputation by his taste for his mechanical contrivances. Wishing to suspend a mirror in his father's shop, in such a way that it might be easily raised and lowered by means of a concealed cord, he used the following expedient.
3. Fixing a wooden tube under the beam, he attached pulleys to it upon which the cord passed and made an angle in descending into the wood which he had hollowed out: there he placed small tubes, within which a leaden ball attached to the cord was made to descend. It happened that the weight, in passing through the narrow parts of the tube, pressed on the inclosed air, and violently driving out at its mouth the quantity of air compressed in the tubes, produced by obstruction and contact a distinct sound.
4. Ctesibius having thus observed that by the compression and concussion of the air, sounds might be produced, he made use of the discovery in his application of it to hydraulic machines, to those automata which act by the power of inclosed water, to lever and turning engines, and to many other entertaining devices, but principally to water dials. First he made a perforation in a piece of gold or a smooth gem, because these materials are not liable to be worn by the action of the water, nor to collect filth, by which the passage of the water might be obstructed:
5. the water flowing through the hole equably, raises an inverted bowl, called by the workmen phellos, or the tympanum, with which are connected a rule and revolving drum wheels with perfectly equal teeth, which teeth, acting on one another, produce revolutions and measured motion. There are other rules and other wheels, toothed in a similar manner, which acted upon by the same force in their revolutions, produce different species of motion, by which figures are made to move, cones are turned round, stones or oviform bodies are ejected, trumpets sounded, and similar conceits effected.
6. On these also, either on columns or pillars, the hours are marked, to which a figure, holding a wand and rising from the lower part, points throughout the day, the increase and decrease whereof is daily and monthly adjusted, by adding or taking away certain wedges.º To regulate the flow of the water, stoppers are thus formed. Two cones are prepared, one convex, the other concave, and rounded so as to fit exactly into each other. A rod, by elongating these, or bringing them together, increases or diminishes the flow of water into the vessel. In this manner, and according to the principles of this machine, water-dials for winterº are constructed.
7. If the addition or removal of the wedges should not be attended by a correspondent increase or decrease in the days, for the wedges are frequently imperfect, it is thus to be remedied.º Let the hours from the analemma be placed on the column transversely, and let the lines of the months be also marked thereon. The column is to turn round, so that, in its continual revolution, the wand of the figure, as it rises, points to the hours, and, according to the respective months, makes the hours long or short.
8. Other kinds of winter-dials are made, which are called Anaporica. They are constructed as follows. With the aid of the analemma the hours are marked by brazen rods on their face, beginning from the centre, whereon circles are drawn, shewing the limits of the months. Behind these rods a wheel is placed, on which are measured and painted the heavens and the zodiac with the figures of the twelve celestial signs, by drawing lines from the centre, which mark the greater and smaller spaces of each sign. On the back part of the middle of the wheel is fixed a revolving axis, round which a pliable brass chain is coiled, at one of whose ends a phellos or tympanum hangs, which is raised by the water, and at the other end a counterpoise of sand equal to the weight of the phellos.
9. Thus as the phellos ascends by the action of the water, the counterpoise of sand descends and turns the axis, as does that the wheel, whose rotation causes at times the greater part of the circle of the zodiac to be in motion, and at other times the smaller; thus adjusting the hours to the seasons. Moreover in the sign of each month are as many holes as there are days in it, and the index which in dials is generally a representation of the sun, shews the spaces of the hours; and whilst passing from one hole to another, it completes the period of the month.
10. Wherefore, as the sun passing through the signs, lengthens and shortens the days and hours, so the index of the dial, entering by the points opposite the centre round which the wheel turns, by its daily motions, sometimes in greater, at other times in less periods, will pass through the limits of the months and days. The management of the water, and its equable flow, is thus regulated.
11. Inside, behind the face of the dial, a cistern is placed, into which the water is conveyed by a pipe. In its bottom is a hole, at whose side is fixed a brazen tympanum, with a hole in it, through which the water in the cistern may pass into it. Within this is inclosed a lesser tympanum attached to the greater, with male and female joints rounded, so that the lesser tympanum turning within the greater, similar to a stopple, fits closely, though it moves easily. Moreover, on the lip of the greater tympanum are three hundred and sixty-five points, at equal distances. On the circumference of the smaller tympanum a tongue is fixed, whose tip points to the marks. In this smaller tympanum a proportionable hole is made, through which the water passes into the tympanum, and serves the work.
12. On the lip of the large tympanum, which is fixed, are the figures of the celestial signs; above, is the figure of Cancer, and opposite to it, below, that of Capricornus. On the right of the spectator is Libra, on his left Aries.ºAll the other signs are arranged in the spaces between these, as they are seen in the heavens.
13. Thus, when the sun is in the portion of the circle occupied by Capricornus, the tongue stands in that part of the larger tympanum where Capricornus is placed, touching a different point every day: and as it then vertically bears the great weight of the running water, this passes with great velocity through the hole into the vase, which, receiving it, and being soon filled, diminishes and contracts the lengths of the days and hours. When, by the diurnal revolution of the lesser tympanum, the tongue enters Aquarius, all the holes fall perpendicular, and the flow of water being thus lessened, it runs off more slowly; whence the vase receiving the water with less velocity, the length of the hours is increased.
14. Thus, going gradually through the points of Aquarius and Pisces, as soon as the hole of the small tympanum touches the eighth part of Aries, the water flows more gently, and produces the equinoctial hours. From Aries, through the spaces of Taurus and Gemini, advancing to the upper points where the Crab is placed, the hole or tympanum touching it at its eighth division, and arriving at the summit, the power is lessened; and hence running more slowly, its stay is lengthened, and the solstitial hours are thereby formed. When it descends from Cancer, and passes through Leo and Virgo, returning to the point of the eighth part of Libra, its stay is shortened by degrees, and the hours diminished, till, arriving at the same point of Libra, it again indicates the equinoctial hours.
15. The hole being lowered through the space of Scorpio and Sagittarius, in its revolution it returns to the eighth division of Capricornus, and, by the velocity of the water, the winter hours are produced. To the best of my ability I have explained the construction and proportions of dials, so that they may be easily set up. It now remains for me to speak of machines, and the principles which govern them. These will be found in the following book, and will complete this Treatise on Architecture.
a "By numbers this cannot be done": the unhappy results of a poor system of mathematical notation in a nutshell. The usual interpretation of this is that the square root of two is an irrational number, i.e., a number not reducible to a fraction with a whole numerator and a whole denominator, and thus not calculatable in ancient arithmetic: and that, of course, is quite true; and the essence of the problem.
Let us suppose for a moment, however, that I do not know how to extract a square root: what would I do today? The answer suggests itself immediately: I would zoom in on the answer by trial and error, squaring 14.1, then maybe 14.15, and so on. In about five minutes I would have an answer precise enough for any architectural work.
The contortions involved in multiplying non-integers in Antiquity made this practically impossible, however. What to us is 14.1, to a Roman might become 14 + 1⁄ + 1⁄, although Jove knows how that would be written: the nearest writable approximation seems to be 14 + 1⁄ + 1⁄ + 5⁄, an irregular continued fraction of sorts, the beginning of which would be written XIIII𐆑 ℈IIII. The problem is twofold: not that they had no decimal base, but that they had no consistent base at all from which any number might be expressed in terms of a power series; then the absence of a zero to serve as a placeholder: so that while you can add and subtract and multiply on an abacus• with relative ease, division is fiendish and fractions will kill you.
In the example above, I've cheated a bit: the non-decimal Romans would have done their trial and error with 14 + 1⁄ (XIIII 𐆑) and 14 + ⅙ (XIIII 𐆐) but the principle, or rather absence thereof, remains the same; anyone to whom this is a revelation is invited to look at how Frontinus measures the diameter, the circumference and the cross-section of a pipe.
Calculation was obviously not impossible, though — as is proved by those aqueducts and countless feats of astonishing Roman engineering. An interesting attempt to reconstruct how a "scientific calculator" in Antiquity might have worked (based in part on a known artifact, the Salamis Tablet) may be found in this series of pages by S. K. Stephenson.
c There are two ways of looking at the orbit of the moon around the earth: but neither one yields this result now, and it's exceedingly unlikely that it did 2000 years ago.
The Ptolemaic system, in which all the heavenly bodies revolve around the earth, just happens to be right with respect to the moon (actually only 54% right, but that's good enough), so that's not the problem.
She takes roughly 27d 7h 43m to circle the earth and return to the same place, so that an earth observer might see her at the same point of the celestial sphere, say, in conjunction with the same star: this is her sidereal period (and appears to be the period meant in the text).
Since in that time both of us have traveled quite a ways around the sun, she won't be in the same phase, and needs additional time. If she is full now, she will be full again in 29d 12h 44m: this is her synodic period. It's an average, the actual time oscillating around that by a few hours, but certainly not by a day and a half.
I'm at a loss to explain this discrepancy: the moon's orbit was among the earliest and best-known astronomical phenomena (Pliny for example gets the sidereal figure right, II.44), and I haven't yet figured out how 27⅓ days could turn into 28 days and an hour thru manuscript corruption. Mystery.
d Tantalizingly close to the system we call Copernican, but no cigar. Vitruvius does not mean that Mercury and Venus revolve around the sun. He is referring to the fact that, seen from earth and in terms of the celestial backdrop, they are never very far from the sun: the maximum angular distance from the sun, or elongation, of Mercury is 28° and that of Venus 48°. They thus appear to oscillate around the sun, which is the statistical center of their apparent path. This is an effect of their being, in our terms, "inferior planets" (planets closer to the sun than us).
e seeming occasionally, from the nature of their circuit, stationary in the signs: This is a straightforward and relatively correct translation. Vitruvius is discussing the 2 points in a planet's orbit in which, seen from the earth, the planet appears to be stationary: an effect due to the relative motions of the earth and the planet cancelling each other out briefly. These "stations" thus occur at places along the orbit determined each time by the relative positions of the earth and the planet: which are in each cycle at different places along the planet's orbit, as measured against the backdrop of the fixed stars.
The translation in the Loeb edition, "thus because of their orbit they delay at the nodes" is thus a mistake, and its footnote (although it gives a perfectly correct definition of a node) is quite irrelevant. The nodes of a planet's orbit around the sun are determined solely by the angle of that orbit with the ecliptic, are at an (almost) fixed point against the heavens, and have nothing to do with what we see from earth.
f Venus completes her circuit through the heavens in 485 days: Vitruvius says that every 485 days Venus will be, as seen from earth, in the same position against the zodiac. Not only this isn't true, it doesn't make any sense at all, no simple change in the number can fix it (i.e., it is probably not a manuscript problem), and ancient astronomers knew it wasn't true: we must be dealing with some kind of error on Vitruvius' part.
First, as to the truth of the statement. Even a casual perusal of geocentric ephemerides will show it is false. Reading a few successive passages of Venus at 0 Aries, from Bryant Tuckerman, Planetary, Lunar, and Solar Positions 601 B.C. to A.D. 1 At Five-day and Ten-day Intervals (The American Philosophical Society, Philadelphia, 1962):
|Year||Date of Passage at 0 Ariº||Days Interval|
|-19 (= 20 B.C.)º|
Venus travels with the sun, more or less, but may be ahead of it or behind it: there is no constant period between its geocentric returns (successive passes at any given degree). I picked 0 Aries arbitrarily; but any other degree produces similar results. Statistically, of course, there is a mean period: 365.25 days, precisely because Venus, viewed from the earth, oscillates around the sun. Mercury has the same average period, for the same reason.
Now there does exist a fairly constant period of revolution of any planet, as viewed from the earth: the synodic period, or time between two successive similar alignments of that planet and the earth, which varies only slightly depending on the eccentricity of the two orbits. That synodic period which can be easily calculated today from the solar orbital periods of the planet and earth, is 584 days for Venus: but the position of Venus will not be in the same degree of the ecliptic by any means.
The case of the outer planets, Mars, Jupiter and Saturn, is somewhat different. The farther out a planet is, the closer its mean synodic period will be to 365.25 days, and the closer its geocentric period (the interval between two successive passages at any fixed degree of the ecliptic) will be to its tropical year (its period of revolution around the sun). The figures given by Vitruvius for the outer planets are thus increasingly good approximations of their tropical years:
Vitruvius' Geocentric Period
approx. 10 752
g Aristarchus did take the moon to be a mere reflector of the sun's light (Sizes and Distances of the Sun and Moon, Hypotheses, §1), but this had been the commonly accepted scientific doctrine for about two hundred years before him, due to Anaxagoras, Parmenides, or maybe even Pythagoras or Thales: (T. L. Heath, Aristarchus of Samos, pp75‑76).
h A modern reader might be surprised at this definition of the vernal equinox, since modern astronomy, modern astrology and its popular bastard all agree in placing it at 0 Aries.
Let's start with a minor matter: Gwilt either has a poor text here or mistranslates. Teubner has partem octavam, which is in fact, strictly speaking, ambiguous (literally, "eighth part": either one-eighth, or a part eighth in number in some succession of parts); but a few lines after this, Gwilt renders a second identical partem octavam by "the eighth division of it", and from then on makes no further mistakes: and 8 degrees is in fact meant, or in modern terms 7° since we start numbering degrees at 0, but ancient astronomers at 1.
Still, an apparent oddity remains: that Vitruvius places the vernal equinox within the seventh degree of Aries when modern astronomers place it at 0° Aries. The essential explanation follows:
Vitruvius is orienting himself by the visible constellations of the zodiac, and in his time the point of the spring equinox, or vernal point, was in the 7th degree of the constellation Aries.
Gravitational effects involving the oblateness and tilt of the earth with respect to its solar orbit cause this vernal point to move very slowly backwards thru the signs, carrying with it equally of course the whole circle of sun-referenced points, of which the salient ones are the equinoctial and solstitial points. This precession of the equinoxes is a regular motion taking about 25,800 years to make a full circle: the rate of precession is thus 50.23 seconds of arc per year.
Modern astronomers orient themselves in terms of the vernal point, regarded as zero wherever it is. They have also firmly delimited each of the constellations: the vernal point is currently (1999) in the constellation Pisces, about 28 degrees east, along the ecliptic, of the modern boundary of Aries, and about 10 degrees west of the boundary of Aquarius. (The modern Pisces is a large constellation, and eats up 38° of ecliptic, more than some of the others.)
Thus, assuming a date of roughly 20 B.C. for the de Architectura (further difficulties, see my note elsewhere), the vernal point has now precessed by
Taking the middle of Vitruvius' "eighth degree", or 7°30, as his vernal point and assuming equal division of the zodiacal constellations along the ecliptic (of which there is one clear indication in this very Book, see next note), in his terms that point is now at 9°19 (which he'd call the tenth degree) of Pisces, give or take 30 minutes.
It may be noted in passing that when Vitruvius places the vernal point in a given degree of Aries, he is writing within a precisely defined framework of astronomical coördinates, constellation boundaries and star positions: over three hundred years old, in fact.
A couple of related comments:
a. If, as noted, there is nothing magical about the beginning of Aries, why is something called "0 Aries" (and even astronomers use ♈, the symbol for a Ram, to denote the point) taken as a fixed point? I would answer: convenience; but it is pleasant to see, and to some extent true, that embedded in both astrology and scientific astronomy, a specific historical time will remain marked as long as those disciplines retain their present form: roughly A.D. 517, following my earlier assumptions. (For any NewAgers out there, the vernal point will pass out of the equal-sign Pisces in roughly the year 2667; the so‑called age of Aquarius is hardly with us yet.)
b. Astrologers, like everyone else, need to be very careful in reading the works of Antiquity. I mentioned that the ancients numbered degrees within a sign starting at 1, not at 0 like us (which the most cursory check of Ptolemy will instantly reveal: he often speaks of the 30th degree of a given house or of a given zodiacal constellation, and never of a 0th degree). So if you are using 3 Taurus as the degree of the Moon's exaltation, or 15 Virgo for Mercury's etc. — you are one degree off. Yes, the ancient texts are unanimous in referring to the third degree of Taurus, etc.: meaning the space of one degree after two have passed, or in modern astrological terms 2 Tau, 14 Vir. . . . For a somewhat more detailed view of the problem, see my note to the article Astrologia in Smith's Dictionary.
i Vitruvius has just said, and he's right, that the constellation Cancer is smaller than the others; here he is saying that part of the constellation Leo is assigned to the sign Cancer. Elsewhere in this Book, he refers to the zodiac as a "large band comprising the twelve signs, which, by the varied combinations of the stars being divided into twelve equal parts. . ." It is clear that the signs are artificially bounded twelfths of the zodiac, and do not necessarily correspond exactly to the constellations to which they owe their names.
j This translation is based on an apparent garble in a manuscript, and conflates two stars. Provindemia, modern name Vindemiatrix, is only the fifth-brightest star in Virgo (ε Virginis), and of no extraordinary lustre: yet the brightest, which, furthermore, is one of the brightest in the entire sky, appears not be mentioned. By emending species to Spica, that star is included, and the passage makes much better sense.
The footnote in the Loeb edition, "Spica, however, is said to be pure white", is best ignored: Spica, spectral class B2, is prominently visible to the naked eye, and, to an observer not beset by light pollution, of a rather striking blue, which with its brilliance has traditionally made it considered by astrologers to be the most benefic fixed star in the heavens.
k Modern astronomy has delimited the constellations so that there is no overlap and no vacant space: a star always belongs to one, and only one, constellation. The star here is α Andromedae, variously known now also as Sirrah or Alpherat(z), both derivatives of its Arabic name as-Surrat al‑Faras, the Horse's Navel. Ptolemy calls it Andromeda's Head, and Aratus refers to it as a ξῦνος ἀστήρ: shared star.
l In the Latin text of modern editions (though not in Gwilt's), what has been translated here as "Po" is given as Eridanus. But the identification of the Eridanus is in fact very problematic, or maybe just pointless. The Po (Padus) and the Rhône (Rhodanus) were just two of several ancient candidates, most of which are discussed in Allen's Star Names, s.v. Eridanus.
m At either equinox, if G is the length of a gnomon and S is the length of the shadow it casts,
S = G tan λ
where λ is the latitude of the place. The figures for the places mentioned in the text are:
|Place||Latitude N||tan λ||Vitruvius' Value||% Error|
|Rhodes (at Lindos)|
Guess where the astronomers worked. . . . (Actually, the fact that the errors are smallest for Rome and Alexandria might be in part something of a coincidence, since Vitruvius' values are conditioned by the limited choice of fractions available to him in which to express them; we're back to the arithmetical problems covered in my earlier note. On the other hand, a much better value for Athens could have been expressed by 7⁄, the error for Tarentum would have pretty much been eliminated with 6⁄, etc.)
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