HISTORY OF SCIENCE. 



FIG. 80. 



of infinitely thin solids. For example, we may suppose a number o{ 

 parallelograms described in the figure ABC, Fig. 8cx ' These fall short 

 of the figure by the triangular spaces at their ends ; but if their number 

 were doubled, we should obviously have only about half the space left 

 between their boundaries and that of the figure, and as this space 

 would be again halved by doubling the num- 

 ber, we may suppose that the number of pa- 

 rallelograms has been so multiplied that their 

 boundaries coincide as nearly as may be de- 

 sired with the curve. As the parallelograms 

 increase in number, they become thinner and 

 thinner, and they approach more and more 

 nearly to lines. Cavalieri deduced some impor- 

 tant truths by his method, and succeeded in 

 solving by it many problems which had defied 

 the efforts of preceding mathematicians. While 

 Cavalieri was applying his new conceptions 

 to these problems, the French geometers were studying many of the 

 higher curves, seeking methods for describing their tangents, etc. 

 Great rivalry existed among the mathematicians of the seventeenth 

 century for the honour of making first discoveries, and many contro- 

 versies arose on claims of priority. It was customary for mathema- 

 ticians to challenge the skill of their competitors for scientific fame by 

 publicly proposing some difficult problem for solution. The proper- 

 ties of the cycloid were discovered in this learned rivalry of Roberval, 

 Torricelli, Pascal, and others. 



A method for the tangents of curves was devised by Roberval about 

 1636, founded upon a notion much akin to Newton's principle of 

 Fluxions. He conceived a curve to be formed by a composition of 

 motions. To fix our ideas, suppose that in Fig. 8 1 o Y and o x being 



fixed lines, a line sets out from o Y, 

 remaining always parallel to o Y, and 

 moving at a rate which for the present 

 we shall suppose uniform. Let a 

 point simultaneously move along the 

 moving line away from o x, at a rate 

 such that the squares of its distances 

 from o x increase uniformly in equal 

 times, then the curve o c D, described 

 by the point, will be a parabola. The 

 tangent to a curve is the direction in 

 which the moving point generating the 

 curve is going at the point of contact. 



Now, Roberval conceived that if a parallelogram be constructed the 

 side of which, a c, is proportional to the velocity of the generating point 

 moving away from o x, and the side a d to the velocity of the line A c 



FIG. 81. 



