(1) Archytas of Tarentum (who flourished in first half of fourth century B.C.) 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. (2) Menaechmus, a pupil of Eudoxus, and a contemporary of Plato, found the two mean proportionals by means of conic sections, in two ways, ([alpha]) by the intersection of two parabolas, the equations of which in Cartesian co-ordinates would be x^2 = ay, y^2 = bx, and ([beta]) by the intersection of a parabola and a rectangular hyperbola, the corresponding equations being x^2 = ay, and xy = ab respectively. It would appear that it was in the effort to solve this problem that Menaechmus discovered the conic sections, which are called, in an epigram by Eratosthenes, "the triads of Menaechmus". 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 (470-399 B.C.). The curve was called the _quadratrix_ because it also served (in the hands, as we are told, of Dinostratus, brother of Menaechmus, 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. Suppose OA, OB are two radii of a circle at right angles to one another. Tangents to the circle at A and B, meeting at C, form with the two radii the square OACB. The radius OA is made to move uniformly about O, the centre, so as to describe the angle AOB in a certain time. Simultaneously AC moves parallel to itself at uniform speed such that A just describes the line AO in the same length of time. The intersection of the moving radius and AC in their various positions traces out the _quadratrix_. The rest of the geometry which concerns us was mostly the work of a few men, Democritus of Abdera, Theodorus of Cyrene (the mathematical teacher of Plato), Theaetetus, Eudoxus, and Euclid. The actual writers of Elements of whom we hear were the following. Leon, a little younger than Eudoxus (408-355 B.C.), was the author of a collection of propositions more numerous and more serviceable than those collected by Hippocrates. Theudius of Magnesia, a contemporary of Menaechmus 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.