1 86 HISTORY OF SCIENCE. 



are, of course, extremely small. But, however .small these distances 

 may be, the proportion that one of them bears to the other is definite, 

 and may be very great. In illustration observe that a number so small 

 as 'the thousandth part of i may bear as large a proportion as you 

 please to some other small number ; for instance, the ratio of the small 

 number just named to the millionth part of i is 1,000. F D, then, 

 must have to B F a certain ratio ; and, as the triangle B D F is right- 

 angled, we give that proportion a name (see page 61), and say that 



T? T) ' 



is the (trigonometrical) tangent of the angle F D B. Observe here 



that the word tangent, as last used, has a sense quite other from that 

 of the geometrical tangent T B N, but it defines the position, of this 

 last by giving its inclination to o x, since angle N TX obviously equals 

 angle FED. The triangle B F D is the incremental triangle. 



JOHN WALLIS, who was Savillian professor of geometry at Oxford 

 during the last half of the seventeenth century, made further advances 

 towards the solution of problems which were now occupying the 

 attention of geometers. He pursued the investigation into methods 

 of greater generality for finding areas, than the comparatively few 

 cases in which geometers had been as yet able to assign the areas 

 bounded by curves. Descartes had given the highest degree of gene- 

 rality to the manner of constructing curves, and it was now sought to 

 assign by equally general methods, the areas of the spaces bounded by 

 these curves. This, of course, would depend upon the relation be- 

 tween the abscissa and the ordinate which the Cartesian equation 

 expresses, and Wallis found that he could exhibit the quadration or 

 area of all curves, when the one co-ordinate could be expressed in 

 terms of the other, involving no fractional or negative indices. Wallis 

 was the author of some ingenious researches on Series. Newton was 

 led to one of his great mathematical discoveries by the consideration 

 of certain series which, are discussed in Wallis's " Arithmetic of In- 

 finites." 



Newton presented the principles of the new calculus in a clear 

 manner, by basing his exposition on certain obvious notions regarding 

 motion. When a body moves uniformly it has at each instant the same 

 velocity; but when the case is otherwise, in a falling body for example, 

 the velocity at each instant is different, but at each instant the actual 

 velocity is measured by that uniform velocity which the body would 

 have if, at the instant considered, the body had ceased to be affected 

 by any force. Now, every curved line may be conceived as generated 

 by two motions, one consisting of a movement of the ordinate receding 

 from o Y (Fig. 83), and always parallel to it ; the other, that of a pbint 

 travelling along the moving ordinate and changing its distance from o x. 

 For simplicity, suppose the motion of the ordinate to be uniform, then, 

 if the motion of the point in the ordinate is also uniform, the point 

 mttst describe a straight line; but if the motion be accelerated, a curve 



