MATHEMATICS. 313 



which we have on this science. The hypothesis of vision which is 

 there attempted to be proved is, that it does not take place by images 

 coming from the objects and entering the eye, but by rays proceeding 

 from the eye to the different points of the objects. It is evident that 

 the mathematical results would be the same on either supposition. 



The subjects which we have mentioned, geometry, plane and solid, 

 and arithmetic, in the department of pure mathematics ; astronomy, 

 music, mechanics, and optics, in that of their application ; formed the 

 exact sciences cultivated by the ancients. To this division some of 

 their authors added logistics and geodesy ; the former indicating the 

 application of arithmetic to questions respecting material objects ; 

 the latter, the mensuration of land by geometry : for which addition, 

 however, there seems no necessity. At the time of Plato, some 

 portion of these sciences seems to have formed a common part of a 

 liberal education : see the dialogues ' Meno,' ' Erastse,' and ' Thea?tetus.' 

 It is not probable, however, that any extensive information on such 

 subjects was popularly diffused. Thucydides seems to have been 

 ignorant of the cause of a solar eclipse ; and Aristophanes ridicules the 

 geometricians and natural philosophers, under the character of Socrates 

 in ' The Clouds,' and of Me ton in ' The Birds.' 



The preceding sketch shows how nourishing was the condition of 

 the mathematics among the Greeks at the time when almost every 

 other department of literature and art was at its greatest splendour. 

 Their poetry, eloquence, and sculpture soon began to decline ; but in 

 the usual progress of the human mind the sciences continue to advance 

 after these aits have become retrograde. In most countries a short 

 period only of original excellence has been allowed to the literature 

 which depends upon the imagination. The exercise of that faculty, 

 like the liberty of a turbulent republic, seems to lead, after a few 

 generations, to its slavery; but the reason, a better-governed kingdom, 

 goes on making acquisitions which are imperishable and perpetually 

 accumulating. The science and literature, of which Athens had been 

 the metropolis, were transferred to the other coasts of the Mediterra- 

 nean, and particularly fostered by the successors of Alexander. The 

 encouragement of the Ptolemies produced no poets who are now 

 considered as great; but royal patronage may be more successfully 

 extended to men of science ; and the mathematical school of Alex- school of 

 andria exhibited an extraordinary succession of remarkable men. The A1 e*andria. 

 cloistered walks, and public halls, and ample libraries of this Egyptian 

 college, were for nearly a thousand years the resort of the most 

 eminent of the men of science among the ancients ; in whose hands 

 the exact sciences, though often stationary, were sometimes advancing 

 and never going back. 



One of the principal founders of this school was EUCLID, whose Euclid. 

 * Elements ' form a groundwork of geometry, which the mathematical B " c * 300 ' 

 world has hardly yet been able to improve upon. Of his history we 

 know little. Pappus, contrasting his character with that of Apol- 



