NEWTON S PRINCIPIA. 5 



required. The same process was attempted with the 

 lengths of the curves, considering them as polygons whose 

 sides diminished while their numbers increased indefinitely. 

 In this way Cavalleri, Fermat, and \Tallis, and still more 

 Harriot and Roberval, appear to have come exceedingly 

 near the discovery of the general rule for performing these 

 operations before Newton and Leibnitz, unknown to each 

 other, made the great step. Roberval especially had 

 solved many problems of quadrature and of drawing 

 tangents, by methods extremely similar to the Newtonian. 

 Nor were the ancient methods of Exhaustion and Indi 

 visibles so far distant as to let us doubt that, had the 

 old geometers been possessed of the great instrument of 

 algebra, and bethought them of its truly felicitous ap 

 plication according to the idea of Descartes, long before 

 our times they would have anticipated the discoveries 

 which form the great glory of modern science.* 



The discovery of the Calculus of Variations affords 

 a similar example of gradual progress. When the 

 differential calculus had enabled us to ascertain the 

 maxima and minima of quantities, for example the value 

 of one co-ordinate to a curve, at which the other becomes 

 a maximum or a minimum, or, which is the same thing, 

 the point of greatest and least distance between the 

 curve and a given right line, or, which is the same 

 thing, when the general relation of the co-ordinates 

 being given we were enabled by means of the calculus to 

 examine what that particular value was at which a 

 maximum property belonged to one of them then 

 geometricians next inquired into the maxima and minima 

 of different curves, that is to say, into the general re 

 lation between the co-ordinates which gave to every 



* Among other marvels in Galileo s history he seems to have made 

 a near approach to the calculus. See M. Libri s most able and learned 

 work, Hist, de Math, en Italic, torn iv. 



B 3 



