the first who found out, that in right lines cutting one 

 another, the opposite angles are equal. In short he 

 taught a great many other valuable truths, too long to 

 be narrated. We owe to A naxiinander, the successor 

 of Thales, the invention of the armillary sphere, 

 and of sua. dials; he 'was likewise the first who drew 

 a geographical map. 



3. Pythagoras has already afforded to us many in. 

 stances of his profound knowledge in all the sciences. 

 There are few philosophers, even among the ancients 

 who had so much sagacity and depth of genius. He 

 was the first who gave sure and fundamental precepts 

 with respect to music, which he fixed upon by a reach 

 of discernment that was extraordinary. Struck by the 

 difference of sounds which issued from the hammers 

 of a forge, but came into unison at the fourth, and 

 fifth, and eighth percussions ; he concluded that this 

 must proceed from the difference of weight in the 

 hammers : he had them weighed, and found that he had 

 conjectured right. Upon this he wound up some 

 musical strings, in number equal to the hammers, and 

 of a length proportioned to their weight ; and found 

 that at the same intervals, they corresponded with the 

 hammers in sound. It was upon the same principle 

 that h? -.le vised the mouochord; an instrument consist, 

 ing of one string, yet capable of easily determining 

 the various relations of sound. He also made many 

 fine discoveries in geometry, among others that pro- 

 perty of a. right-angled triangle, ' that the square of 

 thehypothenuse, or side subtending the right angle, 

 is equal to the squares of the two other sides.' And 

 he gave the fim sketch of the doctrine of isoperimeters 

 in demonstrating that of all plain figures, the circle is 

 the largest ; .and of ail solids, the sphere. 



4. Plato likewise applied himself to the study of 

 mathematics,; ami. we owe <o him many fine discove- 

 ries in tiiat science. He-it was, who first introdu- 

 ced u tbe analytic method, or that geometric analysis, 



