6 NEAVTON S PKINCIPIA. 



portion of the curve a maximum or a minimum value in 

 some respect. Thus, instead of inquiring at what value 

 of x (the abscissa) in a known equation between x and 

 the ordinate y, y became a minimum, or the curve ap 

 proached the nearest to its axis, the question was what 

 relation x must have to y (or what must be the equation 

 as yet unknown) in order to make the whole curve, for 

 example, of the shortest length between two given points, 

 or inclose with two given lines the largest space, or 

 (having some property given) inclose within itself the 

 largest space, or be traversed in the shortest possible time 

 by a body impelled by a given force between two given 

 points. Here the ordinary resources of the differential 

 calculus failed us, because that calculus only enabled us, 

 by substituting in the differential equation the value of one 

 co-ordinate in terms of the other, to make the whole equal 

 to nothing, as it must be at the maximum or minimum 

 point where there is no further increase or decrease. But 

 here no means were afforded of making this substitution, 

 and the problem seemed, as far as this method went, 

 indeterminate. Various very ingenious resources were 

 employed by Sir Isaac Newton, who in the Principia 

 seems to have first solved a problem of the Isoperimetrical 

 class that is, finding the solid of least resistance; and 

 soon after by the Bernouillis and other continental ma 

 thematicians, who worked by skilful constructions and 

 suppositions consistent with the data. The calculus called 

 that of Variations has since been invented for the general 

 solution of these and other similar problems. It con 

 sists in treating the relations of quantities, or of their 

 functions, as themselves varying, but varying according 

 to prescribed rules, just as the differential calculus regards 

 the quantities themselves, or their functions, as varying 

 according to prescribed rules. It bears to the differential 



