280 A SHORT HISTORY OF SCIENCE 



as it was to criticism, were solved simple area problems involving 

 the parabola and the hyperbola. 



The principle of comparing areas by comparing lengths of a 

 system of parallel lines crossing them is easily illustrated in the 

 case of the ellipse by comparing it with the circle having as its 

 diameter the (horizontal) major axis of the ellipse. If a and b 



are the semi-axes of the ellipse the two 

 curves are known to be so related that every 

 vertical chord of the circle is in a fixed ratio 

 a : b to the part of it lying within the ellipse. 

 The area of the circle must bear the same 

 relation to the area of the ellipse. The 

 transition from length to area while not 

 DE _ CB _ _a_ rigorously worked out by Cavalieri does not 



necessarily involve the false assumption that 

 area consists of the sum of parallel lines. A similar method is 

 evidently applicable to volumes. Thus was anticipated one of the 

 most interesting and important processes of modern mathematics, 

 integration as a summation. 



Similarly Cavalieri determined volumes by a consideration of 

 the thin sections or elements into which they may be resolved by 

 parallel planes. The principle that "two bodies have the same 

 volume if sections at the same level have the same area" is still 

 known by his name. 



Descartes's work with tangents seems not to have led him to 

 develop the fundamental ideas of the differential calculus, and it 

 appeared that the integral calculus would be evolved first from the 

 work of Cavalieri. 



PROJECTTVE GEOMETRY : DESARGUES. Hardly less interesting 

 than the new ideas of Descartes and Cavalieri are those of their 

 contemporary Desargues (1593-1662), an engineer and architect 

 of Lyons, who made important researches in geometry. But for 

 the still more brilliant geometrical achievements of Descartes, 

 these might have led to the immediate development of projective 

 geometry, the elements of which are contained in Desargues's 

 work. In general this geometry instead of dealing with definite 



