NEWTON. 



185 



moving away from o Y, the diagonal dc would represent the direction of 

 the tangent E F. Roberval, however, had no general method for deter- 

 mining the ratios of these velocities, nnd, in fact, applied his method 

 only to a few particular curves, in which the ratio of the velocities 1 

 was readily discoverable. Indeed, he did not present the subject a , 

 we have here done, in reference to the relation between ordinates and 

 abscissa (see page 151), but deduced the ratios for the ellipse, etc., from 

 the properties of lines drawn from the foci. Nor did any considera- 

 tion of infinitely small quanties enter into his method. FERMAT, 

 another French mathematician, employed the conception of infinitely 

 small quantities to determine the tangent to curves, in a manner 

 which so nearly approached the method of the new calculus, that 

 some of Fermat's countrymen have even claimed for him the honour 

 of discovering the differential calculus. Dr. Barrow, Newton's pre- 

 decessor at Cambridge, simplified Fermat's method of tangents by his 

 conception of what has been called the * Incremental Triangle." This 

 notion, like that of indivisibles, is expressed in language which con- 

 tradicts the obvious truths of geometry, for it assumes that of any 

 curve a portion so small may be taken that it may be rendered a 

 straight line. It will perhaps assist a reader unaccustomed to this 

 language to a conception of its meaning, if, having actually drawn a 

 circle with a pair of compasses, he will draw on it a number of chords 

 progressively smaller and smaller. He may then observe that as the 

 chords become shorter the distances of their middle points from the 

 arcs they cut off not only become smaller in themselves, but bear a 

 continually lessening in proportion to the length of the chord itself. 

 The process of drawing shorter and shorter chords may theoretically 

 be carried so far that the chord differs from its arc by an amount less 

 than any assigned one. When the difficulty about the phraseology 

 used in discussing these infinitely small quantities has been sur- 

 mounted, "the incremental triangle" may easily be understood. Let 



B F G, Fig. 82, be any curve, o c and c B the co-ordinates of the point 

 B. Pass now from B to F, another point in the curve, so near to B that 

 the portion of the curve B F may be practically a straight one. The 

 abscissa of F is o E, its ordinate E F, greater than the abscissa and 

 ordinate of B by the distances BD and DF (BD is parallel to ox), which 



