in 



able geometrician, the method of investigating infinite 

 sericscs took its rise from his Arithmetic of Infinites, 

 published in 1656 ; and he himself acknowledges, 

 that both of them are founded on tho method of ex- 

 haustions used by the ancients. He farther says, that 

 the method of indivisibles introduced by Cavallieri, is 

 no other than an abridgment of that of exhaustions, 

 though somewhat more obscure. He observes, that 

 the lines and surfaces, whose proportion and contents 

 are enquired into, and ascertained by Cavallieri, differ 

 in nothing from the inscribed and circumscribed trian- 

 gles, whose approaches Archimedes brought so near, 

 that the difference of the space enclosed between them, 

 and that which they approached, and about which they 

 were drawn, to vvit, the contents of the circle might 

 become less than any assignable quantity : and this he 

 proves afterwards, by an analytic exposition of both. 

 I may however remark, that from the time of Dio- 

 phantes, algebra made but small progress, till that of 

 Vietus, who restored and perfected it, and was the 

 first who marked the known quantities of the letters 

 of the alphabet. 



7. Besides the discoveries made in astronomy by 

 the ancients, which we have been reading, there are 

 a great many others, which I cannot bring into view, 

 in that full manner they deserve. Yet 1 cannot omit 

 mentioning here one important observation of Aris- 

 tarchus. lie was the first who suggested u a method 

 of measuring the distance of the suri from the earth, 

 by means of the half section of the moon's disk," or 

 that phasis of it wherein it appears to us when it is in 

 its quadratures. 



8. Hipparchus was the first who calculated tables of 

 the motion of the sun and moon, and composed a cata- 

 logue of the fixed stars. He was also the first, who, 

 from the observation of eclipses, determined the longi- 

 tude of places upon earth ; but, what above all does 

 immortal honour to his genius is ? that he laid the first 



