-NEWTON S PRINCIPIA. 3 



opened the way to the grand discovery of New ton and 

 Leibnitz, the foundation of modern science, mathematicians 

 had been intent upon the resolution of problems connected 

 with the rectification and quadrature of curves, and the 

 determination of points that possess properties of maxima 

 and minima, as well as the finding of normals, tangents, 

 and osculating circles. These inquiries had led them to 

 consider the laws by which the relations between the or- 

 dinates and abscissae referred to any given axis are go 

 verned at different points of that axis ; for in truth that 

 implies the nature of the curvature itself, and includes 

 the manner in which the length of the curve line increases 

 or diminishes, as well as the space which it incloses. They 

 were thus led to examine the generation of those curve 

 lines and curvilinear spaces, whether that is conceived 

 to be effected by the movement in the one case of points, 

 and in the other of straight lines, or is supposed to be 

 produced by the constant juxtaposition of indefinitely 

 small straight lines inclined to each other according to 

 a given law, in the one case, and indefinitely small rect 

 angles in the other. The latter is perhaps the more 

 natural supposition of the two, and not the less easy. 

 For if any one is set to measure the area of a field 

 bounded by a curvilinear outline, as he can at once 

 measure a space inclosed within straight lines, his course 

 will be to divide the given space into rectangles, and 

 then to divide each of the smaller curvilinear spaces 

 into other rectangles, and so on till he has exhausted 

 the whole by a series of rectangles, always decreasing in 

 size as they increase in number, and the last of which 

 seem to coincide nearly or sensibly with the area of the 

 outer or curved line of boundary. Thus he would proceed 

 by trial and actual measurement of the space ; and thus 

 do land-measurers (the lineal descendants of the first 



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