NEWTON. 



187 



convex towards o x will be marked out, while if, on the other hand, 

 the motion be retarded, the curve will be concave. Now, P M in the 

 figure represents a certain position of the moving ordinate P M and 

 moving point P. When the ordinate has arrived at the position repre- 

 sented by N a, the point P has the position marked P', it has receded 

 from o x at a lessening velocity. If, while the ordinate was moving 

 from M to N, the moving point, 

 had continued to recede from 

 o x with the same velocity in 

 that direction which it had at 

 p, then it would not be in the 

 curve, but at some other point, 

 as a, and in its passage from p 

 to a it must necessarily have 

 described the straight line P a. 

 Suppose now that from p we 

 draw a line parallel to o x cut- 

 ting a N in b, then in Newton's 



FIG. 83. 



phraseology b a is \hzfluxion <f 

 the ordinate, that is, the ordi- 

 nate while passing from M to N would have (if uniform motion had con- 

 tinued) flowed on so as to increase its length by the amount b a. In a 

 similar way Newton would call p b the fluxion of the abscissa (remember 

 the movement along o x has been assumed to be uniform). The geo- 

 metrical truth enunciated on page 13 tells us that P#, which is \h.z fluxion 

 of the curve, has its square equal to the sum of the squares of ? b and ba. 

 It will be perceived that the principle of the rule for tangents is related 

 to this calculus; but we are not required to make any supposition 

 about curves being made up of very short straight lines, nor is it ne- 

 cessary to consider infinitely small quantities at all. From the equation 

 to any curve (page 150) it is always possible to deduce the ratios of 

 the two fluxions a b, P b; but the methods of doing this would be un- 

 intelligible to readers previously unacquainted with the subject. It- 

 must suffice to show the elementary conception of Newton's Fluxions ; 

 and these explanations are not out of place even if the non-mathema- 

 tical reader has gathered from them nothing more than a clear idea 

 of why Newton gave this name to his method. Further, we may state 

 that ^ Newton used certain symbols to indicate fluxionary quantities. 

 For instance, the b a and b p of Fig. 83 are particular cases of y and ,, 

 which respectively represent fluxions of the ordinate and abscissa. The 

 first application of the theory of fluxions is to find the tangents, and 

 Fig. 82 shows that j is the trigonometrical tangent of the angle which 

 the geometrical tangent to the curve at c forms with o x. 



It has already been mentioned that one of the first mathematical 

 works which Newton read was Dr. Wallis's " Arithmetic of Infinites." 

 Some theorems given in that work led Newton to the study of certain 



