316 



GREEK SCIENCE. 



ment of geometry ; for I well know his uncommon talents, and his 

 indefatigable industry in these studies." When Conon was dead, years 

 elapsed without any one attempting the proposed theorems. The 

 demonstrations were sent by Archimedes himself, at different times, 

 to Dositheus, an Athenian, whom he knew, as he tells him, to be 

 both a friend of Conon and a lover of mathematics ; and who, after 

 receiving a part, had pressed him much for the remaining portions. 

 These successive epistles form his treatises * On the Quadrature of the 

 Parabola,' 4 On the Sphere and Cylinder,' ' On Helices or Spirals,' and 

 ' On Spheroids and Conoids.' 



Quadrature The 'Treatise on the Quadrature of the Parabola,' was the first 

 of parabola. j ns t ance j n w hich a geometer had been able to determine the exact 

 space bounded by a curve line ; for though several before him had 

 pretended to assign the area of the circle and of portions of it, their 

 assumptions, as Archimedes asserts, were inadmissible : and their 

 conclusions must have been false, since the problem, as we have 

 already observed, is not soluble. The method which he employs is 

 most remarkable for its ingenuity and novelty. He divides the para- 

 bola into an endless series of decreasing terms ; and we may observe 

 in his process the tendency to that passage from finite to infinite, by 

 resolving a curve into its smallest portions, which, after assuming 

 various forms in the hands of Barrow, Cavallerius, Newton, &c., 

 produced at last the differential and integral calculus. And though 

 by means of these modern methods, a mere scholar in mathematics 

 may now obtain the answers to such questions as that of which we 

 are speaking, we cannot but regret, in the facilities of our technical 

 rules, the elegance and evidence of the ancient geometry. Difficult 

 as the problem appears in the way in which Archimedes has treated 

 it, his only axiom is, that of two unequal spaces, the excess of the 

 greater above the less, may be multiplied so as to exceed any given 

 space; and from this he proves, by the strictest reasoning, that a 

 parabola can be neither greater nor less than two-thirds of the paral- 

 lelogram described about it. 



The speculations respecting the sphere and cylinder are those with 

 which the author appears to have been most delighted, for he wished 

 to have his grave marked by these solids, as some more recent mathe- 

 maticians have had their discoveries engraved on their tomb-stones. 

 Indeed, all who have the perception of geometrical beauty, must be 

 struck both with his results and his methods. As he had been the 

 first to find the area of a plane curve, he here finds the. surface of a 

 curvilinear solid ; and determines the sphere to be two-thirds, both in 

 content and in surface, of the cylinder which circumscribes it; with 

 many other remarkable properties of these solids compared with each 

 other and with the cone. 



The subject of spiral lines, was also, so far as we know, altogether 

 new. In the one which he has examined he has discovered many 

 remarkable properties with respect to its area, tangent, &c. 



Sphere and 

 cylinder. 



Spirals. 



