308 GREEK SCIENCE. 



mathematical subject ; the properties of numbers were pursued with 

 an inquisitiveness which led to a curious spirit of mysticism ; and the 

 doctrine of the sphere was applied to the explanation of astronomical 

 phenomena. 



Under these circumstances geometry and its related sciences soon 

 became of considerable extent. We have the titles of several treatises 



Democritus. upon a variety of its branches by DEMOCRITUS and others of the times 

 before Pericles; and at the period of the Peloponnesian war, 

 geometers had not only travelled over most elementary problems, 

 but had, in some instances, struck against those limits which they 

 have been ever since vainly struggling to pass. According to 



Anaxagoras. Plutarch, ANAXAGORAS, the friend of Pericles, employed himself 



B.C. 530. i . '. . .' . , , /> 7 7 



Squaring the m ms P nson m investigations on the quadrature of the circle; and 



circle. steps of the same problem were also attempted by Antiphon and 



Bryson, whose reasonings Aristotle calls paralogisms, though it would 



Hippocrates, seem undeservedly with respect to the former. HIPPOCRATES, who 

 was originally a merchant of Chio, and became a geometer at Athens, 

 whither he had gone in consequence of pecuniary misfortunes, entered 

 upon a train of research, which at first seemed to promise success, in 

 measuring the circle. He went so far as to find the area of a space 



Limes. comprehended between two circular areas, and called a lune, from its 

 resemblance to the horned moon ; but it was found impossible to 

 extend this to a whole circle. Another problem, now also known to 

 be impracticable by plane geometry, namely, the discovery of two 

 mean proportionals, excited much interest about this time. It is 



Doubling the identical with the problem of doubling the cube, said to have been 



cube. proposed by the oracle at Delos ; though this story is probably only 



one of those fictions in which mathematicians used often to present 

 their questions. However that may be, it is certain that we have 

 several solutions of this problem, purporting to be of the time of 

 Plato, given by Eutocius in his commentary on Archimedes. 



Archytas. ARCHYTAS, a Pythagorean, the master of Plato, solved it by a some- 

 what complicated construction, in which a conical and cylindrical 



Menechmus. surface are made to intersect. MENECHMUS, a scholar of Plato, 

 obtained the result by the intersection of two conic sections. 



Eudox US; EUDOXUS, another of Plato's scholars, is said to have applied to it 

 B. c. 370. curve ij nes invented by himself. Plato himself devised a kind of 

 parallel ruler, by means of which it might easily be mechanically 

 executed. Indeed, the Greek geometry seems sometimes to have had 

 a rather curious tendency to solve its problems by mechanical con- 

 trivances: of which practice, according to Plutarch, in his account of 

 Archimedes, Plato strongly disapproved ; notwithstanding the instance 

 we have just given of his adoption of it. 1 



1 Plutarch, in. Marcello. Plutarch obviously confounds, as it was easy for a 

 writer to do who was not a mathematician, the solution of problems by mechanical 

 contrivances (opydvtKi)'), with the application of mathematics to problems concerning 



