MATHEMATICS. 309 



The admiration of Plato for geometry is well known, from the in- Plato, 

 scription which he is said to have placed over the door of the place B * c ' 40 * 

 where he taught : " Let no one enter who is without geometry." The 

 acquisitions which are attributed to him and his school show how 

 rapidly the science advanced ; for the discoveries which we have now 

 to notice are no longer particular propositions, but general methods, 

 and long trains of investigation. We shall consider them in order. 



It appears by what has been just said of Menechmus, that the Conies. 

 conic sections had already been discovered. They are sometimes 

 ascribed to Plato himself, and many of their properties were known 

 soon after his time. 



Plato is said to have invented the geometrical analysis ; the method Analysis. 

 by which, assuming a problematical result to be true, we reason back- 

 ward to the other propositions which its truth presupposes, till we 

 arrive at something which is known to be true or to be false ; and 

 thus establish or overturn the proposition assumed. 



Another invention of this illustrious mathematical school was the Loci, 

 doctrine of geometrical loci. By this proceeding, when a required 

 point cannot be found by the intersections of straight lines and circles, 

 some new curve is constructed, consisting of the places which the 

 point might assume by leaving out one of the conditions ; and in this 

 curve the remaining condition enables us to determine the point de- 

 manded. The quadratrix (rrpaywWov<ra) of Deinostratus, a curve 

 so called from the scholar of Plato, who invented it, or discovered its 

 properties, and from its use in squaring the circle, was one of the first 

 of these loci. It may also be used in another celebrated problem, the 

 trisection of an angle. This problem, and that of the duplication of 

 the cube, gave rise also to the loci constructed by succeeding mathe- 

 maticians, and called the conchoid of Nicomedes, and the cissoid of 

 Diocles. Besides these, which were called loci at a line, similar con- 

 siderations led afterwards to the invention of loci at a surface, when j^d a t a 

 the possible positions of a point lay in a curved superficies. surface. 



To the active minds of Plato and his school we may attribute also solid 

 the prosecution of * Solid Geometry.' We have a treatise* by ZENO- g eometr y- 

 DOEUS, who is supposed to have lived somewhat about this time, in 

 which it is proved that the content of the sphere is greater than that 

 of any other solid of equal surface. This is preserved by Theon in 

 his commentary on the * Almagest,' and is the oldest work on geo- 

 metry extant. But the Platonists pursued this subject, and investi- 

 gated the properties of the five regular solids, called from that cause 

 the Platonic bodies. This branch could not previously have been 

 much attended to, for Plato (Rep. lib. 7) " notes it defective," to use 

 Lord Bacon's phrase in his * Survey of Human Learning.' 



In the passage of Plato just referred to, he divides mathematics Astronomy, 

 into the doctrine of planes, or plane geometry ; the doctrine of solids ; 

 and the doctrine of solids in motion. This last division is meant to 

 describe the mathematical part of ( Astronomy,' viz., the doctrine of 



