Thomas Little Heath Quotes

The method of exhaustion was not discovered all at once; we find traces of gropings after such a method before it was actually evolved. It was perhaps Antiphon. the sophist, of Athens, a contemporary of Socrates, who took the first step. He inscribed a square (or, according to another account, a triangle) in a circle, then bisected the arcs subtended by the sides, and so inscribed a polygon of double the number of sides; he then repeated the process, and maintained that, by continuing it, we should at last arrive at a polygon with sides so small as to make the polygon coincident with the circle. Thought this was formally incorrect, it nevertheless contained the germ of the method of exhaustion.

Eudoxus was perhaps the greatest of all Archimedes's predecessors, and it is his achievements, especially the discovery of the method of exhaustion, which interest us in connexion with Archimedes.

Theodorus of Cyrene and Theaetetus generalised the theory of irrationals, and we may safely conclude that a great part of the substance of Euclid's Book X. (on irrationals) was due to Theætetus. Theætetus also wrote on the five regular solids, and Euclid was therefore no doubt equally indebted to Theætetus for the contents of his Book XIII. In the matter of Book XII. Eudoxus was the pioneer. These facts are confirmed by the remark of Proclus that Euclid, in compiling his Elements, collected many of the theorems of Eudoxus, perfected many others by Theætetus, and brought to irrefragable demonstration the propositions which had only been somewhat loosely proved by his predecessors.

The actual writers of Elements of whom we hear were the following. Leon, a little younger than Eudoxus, was the author of a collection of propositions more numerous and more serviceable than those collected by Hippocrates. Theudius of Magnesia, a contemporary of Menæchmus and Dinostratus, "put together the elements admirably, making many partial or limited propositions more general". Theudius's book was no doubt the geometrical text-book of the Academy and that used by Aristotle.

The trisection of an angle was effected by means of a curve discovered by Hippias of Elis, the sophist, a contemporary of Hippocrates as well as of Democritus and Socrates. The curve was called the quadratrix because it also served (in the hands, as we are told, of Dinostratus, brother of Menæchmus, and of Nicomedes) for squaring the circle. It was theoretically constructed as the locus of the point of intersection of two straight lines moving at uniform speeds and in the same time, one motion being angular and the other rectilinear.

Menæchmus, a pupil of Eudoxus, and a contemporary of Plato, found the two mean proportionals by means of conic sections, in two ways, (α) by the intersection of two parabolas, the equations of which in Cartesian co-ordinates would be x²=ay, y²=bx, and (β) by the intersection of a parabola and a rectangular hyperbola, the corresponding equations being x²=ay, and xy=ab respectively. It would appear that it was in the effort to solve this problem that Menæchmus discovered the conic sections, which are called, in an epigram by Eratosthenes, "the triads of Menæchmus".

Archytas of Tarentum found the two mean proportionals by a very striking construction in three dimensions, which shows that solid geometry, in the hands of Archytas at least, was already well advanced. The construction was usually called mechanical, which it no doubt was in form, though in reality it was in the highest degree theoretical. It consisted in determining a point in space as the intersection of three surfaces: (a) a cylinder, (b) a cone, (c) an "anchor-ring" with internal radius = 0.

The problem of doubling the cube was henceforth tried exclusively in the form of the problem of the two mean proportionals.

Hippocrates also attacked the problem of doubling the cube. ...Hippocrates did not, indeed, solve the problem, but he succeeded in reducing it to another, namely, the problem of finding two mean proportionals in continued proportion between two given straight lines, i.e. finding x, y such that a:x=x:y=y:b, where a, b are the two given straight lines. It is easy to see that, if a:x=x:y=y:b, then b/a = (x/a)³, and, as a particular case, if b=2a, x³=2a³, so that the side of the cube which is double of the cube of side a is found.

Hippocrates himself is an example of the concurrent study of the two departments. On the one hand, he was the first of the Greeks who is known to have compiled a book of Elements. This book, we may be sure, contained in particular the most important propositions about the circle included in Euclid, Book III. But a much more important proposition is attributed to Hippocrates; he is said to have been the first to prove that circles are to one another as the squares on their diameters (= Eucl. XII., 2) with the deduction that similar segments of circles are to one another as the squares on their bases. These propositions were used by him in his tract on the squaring of lunes, which was intended to lead up to the squaring of the circle. The latter problem is one which must have exercised practical geometers from time immemorial. Anaxagoras for instance is said to have worked at the problem while in prison.

By the time of Hippocrates of Chios the scope of Greek geometry was no longer even limited to the Elements; certain special problems were also attacked which were beyond the power of the geometry of the straight line and circle, and which were destined to play a great part in determining the direction taken by Greek geometry in its highest flights. The main problems in question were three: (1) the doubling of the cube, (2) the trisection of any angle, (3) the squaring of the circle; and from the time of Hippocrates onwards the investigation of these problems proceeded pari passu with the completion of the body of the Elements.

The Pythagoreans discovered the existence of incommensurable lines, or of irrationals. This was, doubtless, first discovered with reference to the diagonal of a square which is incommensurable with the side, being in the ratio to it of √2 to 1. The Pythagorean proof of this particular case survives in Aristotle and in a proposition interpolated in Euclid's Book X.; it is by a reductio ad absurdum proving that, if the diagonal is commensurable with the side, the same number must be both odd and even. This discovery of the incommensurable... showed that the theory of proportion invented by Pythagoras was not of universal application and therefore that propositions proved by means of it were not really established. ...The fatal flaw thus revealed in the body of geometry was not removed till Eudoxus discovered the great theory of proportion (expounded in Euclid's Book V.), which is applicable to incommensurable as well as to commensurable magnitudes.

In geometry the following theorems are attributed to him [Thales]—and their character shows how the Greeks had to begin at the very beginning of the theory—(1) that a circle is bisected by any diameter (Eucl. I., Def. 17), (2) that the angles at the base of an isosceles triangle are equal (Eucl. I., 5), (3) that, if two straight lines cut one another, the vertically opposite angles are equal (Eucl. I., 15), (4) that, if two triangles have two angles and one side respectively equal, the triangles are equal in all respects (Eucl. I., 26). He is said (5) to have been the first to inscribe a right-angled triangle in a circle: which must mean that he was the first to discover that the angle in a semicircle is a right angle. He also solved two problems in practical geometry: (1) he showed how to measure the distance from the land of a ship at sea (for this he is said to have used the proposition numbered (4) above), and (2) he measured the heights of pyramids by means of the shadow thrown on the ground (this implies the use of similar triangles in the way that the Egyptians had used them in the construction of pyramids).

Almost the whole of Greek science and philosophy begins with Thales.

In illustration of his entire preoccupation with his studies, we are told that he would forget all about his food and such necessities of life, and would be drawing geometrical figures in the ashes of the fire, or, when anointing himself, in the oil on his body.