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JOURNAL OF THE 



the great Newton, have assumed that a variable can reach its limit, so 

 that (4) above should "ultimately become" t,-'^-. 



That Newton failed to establish a true theory of limits is shown in 

 Bledsoe's Philosophy of Mathematics. As it was, he made a great advance 

 over previous methods; but now that a correct theory of limits is so uni- 

 versally known, there can be no excuse for later writers in perpetuating 

 the same errors that seemed inevitable in ihe dawn of the infinitesimal 

 method. The French writers (following the lead of Duhamel), and also 

 some American writers, have been more logical in their development of 

 the infinitesimal calculus. 



The problem of tangents is one which gave rise to the 

 differential calcnhis and needs to be carefully considered. 



In fig. 2, let y z= /(x) be the equation of the curve DPS 

 referred to the rectangular axes x and y. Suppose we wish 

 to find the tangent of the angle PEX made by a tangent at 

 a point P of the curve with the X axis or l/ie slope of the 

 curve at P wdiose co-ordinates are x and y. The co-ordi- 

 nates of a point S to the right of P are y + aj^, x + ^x so 

 that, PQ = A,r, SQ = a>' and, 



AJ 



— = tan SPR. 

 A a; 



This equation gives 

 the slope of the secant 

 PS which varies with 

 the values of A.^■ and 

 Aj'. If PS regarded 

 as a line simply, but 

 not a secant, is re- 

 volved around P, in 

 one position only, it 

 coincides with the tangent, where it touches the curve in 

 but one point. If it is revolved further, it cuts the curve 

 on the other side of P whether the curve in the vicinity of 

 P is convex to the X axis as drawn or concave. If the 

 curve is convex on one side of P and concave on the other, 

 the line PS will cut the curve in three points when it lies 



