44 INTRODUCTION. 



SECTION IV. OF THE PROPERTIES OF CURVES. 



189. Definition. Any parallel right lines, inter- 

 cepted between a curve and a given right line, are called 

 ordinates; and each part of that line, intercepted between 

 an ordinate and a given point, is the absciss corresponding 

 to that ordinate. 



" 190." [141,B.] Theorem. The fluxion of the area 

 of any figure is equal to the parallelogram contained by 

 the ordinate and the fluxion of the absciss. 



Let AB be the absciss, and BC the ordinate 

 through C draw DCE||AB, and take DCzi 

 DEzihalf the increment of AB, then the simul- 

 taneous increment of the figure ABC will ulti- 

 mately coincide with the figure I'CGEB, since 

 the curve ultimately coincides with its tangent 

 (141), but the triangles CDF, CEG, are equal, therefore the paral- 

 lelogram DBE is ultimately equal to the increment of ABC. And 

 if any other line^ than DE represent the fluxion of AB, as DE is to 

 this line, so is the parallelogram DBE to the parallelogram contained 

 by BC and this line ; therefore that parallelogram is the fluxion of 

 ABC (46). 



Scholium. Those, who prefer the geometrical mode of represen- 

 tation, may deduce from this proposition a demonstration of the 

 theorem for determining the fluxion of the product of two quantities 

 (48) ; for every rectangle may be diagonally divided into two such 

 figures as are here considered, and the sum of their fluxions, accord- 

 ing to this proposition, will be the same with the fluxion of the rect- 

 angle determined by that theorem. It is obvious that this theorem 

 ought not to have followed article 180. 



191. Definition. A flexible line being supposed to 

 be applied to any curve, and to be gradually unbent, the 

 curve, described by its extremity, is called the involute of 

 the first curve, and that curve the evolute of the second. 



192. Definition. The radius of curvature of the in- 



