prThe 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.
2What 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.
3Since, 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.
4If 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 feet, 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; 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.
5Since, 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.
6Pythagoras demonstrated the method of forming a right angle 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:
7so 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 his 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.
8For if the height from the pavement to the floor above be divided into three parts, five of those parts will be the exact length of the inclined line which regulates 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.
9Though 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.
10But 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, 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).
11After 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 what quantity of water was equal to a certain weight of silver.
12He 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.
13Let 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.
14Archytas obtained a solution of the problem by the semicylinder, and Eratosthenes by means of a proportional instrument. 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.
15The 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.
16Since 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.
17Many 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.
18Hence, 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.