ELISHA MITCHELL SCIENTIFIC SOCIETY. 1 9 



on one side of the tangent, and in one point when it lies 

 on the other side of the tangent. When it coincides with 

 the tangent it cuts the curve in but one point. 



But in all cases, it must be carefully noted, that the 

 secant PS as it revolves about P can approach the tangent 

 PT as near as we choose, but can never reach it; for then 

 it would cease to be a secant; hence the tangent is the lim- 

 iting position of the secant. Therefore as A.r (and conse- 

 quently Aj') diminish towards zero indefinitely, the point S 

 will approach the point P indefinitely and angle SPR ap- 

 proaches indefinitely angle TPR as its limit; whence tan 

 SPR approaches indefinitely tan TPR as its limit. 



Therefore, taking the limit of the equation above, 

 Aj' dy 

 lim. — = — = tan TPR .... (5). 

 AX' dx 



Hence if y ^ f {x) is the equation of the curve ^ tJie de- 

 rivative of f{x) with respect to x is alzvays equal to the slope 

 of the curve at the point (.i-, jk) considered. Thus the tan- 

 gent of the angle made by the tangent line of the cubic 

 parabola y =r .r"^ ^ b, already considered, with the axis of 

 .r, equals 3.1-." at the point whose abscission is x. 



This value was determined above and its meaning is that 

 for X equal to o, ^, ^,1, etc., the slope of the curve is o, 

 Vi-t H^ 3' ^^^-j ^"^ ^^ increases indefinitely as .i' increases 

 indefinitely. 



lu Edwards' DiflFerential Calculus (second edition, 1892, page 20) we 

 read, changing the letters to suit fig. 2, to which the theory applies: 

 "When S travelling along the curve, approaches indefinitely near to P, 

 the chord PS becomes in the limit the tangent at P." In ex. 2, page 21, 

 the author, in getting the final equation, again says: "When S comes to 

 coincide with P," etc. It is plain from these references that this most 

 recent English author considers a variable to actually reach its limit— a 

 fundamental error we have exposed above. The chord cannot reach the 



tangent without ceasing to be a chord, neither can the ratio 1= slope 



\x 



o 

 of chord) reach its limit — without the ratio ceasing to exist. 



o 



