TANGENT TO A CURVE. 23 



The new value of the ordinate is represented in the figure 



A?/ 

 by NQ, and NR represents Ay. The fraction -^ represents 



i\X 



the ratio of the increase of the function to the increase of 

 the variable, and is equal to the trigonometrical tangent 

 of the angle NME formed by the secant MN with the axis 

 of x. 



37. It is evident that this fraction is a natural measure of 

 the degree of rapidity with which the function y increases 

 when the independent variable x increases; for the greater 

 this fraction is, the greater will be the increase of the func- 

 tion y corresponding to the given increase Ace of the variable. 



A?/ 

 But it is important to observe that the value of will 



depend not only on the value given to x, but also on the 

 magnitude of the increment Ace, except in the case in which 

 the curve becomes a straight line. 



If then we left this increment arbitrary, it would be im- 



Ay 

 possible to assign to the fraction ~- any definite value, and 



QG 



it is thus necessary to adopt some convention which will 

 remove this uncertainty. 



38. Suppose that after giving to Ace a certain value, to 

 which will correspond a certain value for A?/ and a certain 

 direction for the secant MN, we make the value of Ace 

 gradually diminish and become ultimately zero. The value 

 of Ay will also gradually diminish and become ultimately 

 zero. The point ^will move along the curve towards M, and 

 we shall find in every example we consider, that the straight line 

 MN will approach towards some limiting position MT. This 

 is in fact equivalent to the assertion made in Art. 24, that 

 by examining every case in detail we could shew that every 

 function has a differential coefficient. The limiting position 

 which the secant assumes when N coincides with M is called 



cLij 

 the tangent to the curve at the point M, and thus is the 



trigonometrical tangent of the inclination to the axis of x 

 of the tangent line to the curve. 



