OF THE HIGHER CALCULUS. 519 



portional to that function (Constructs, sect. 26). In his first logarithms, also, 

 he supposes, that at the outset the two velocities are alike. From these pre- 

 mises, he constructed his three auxiliary tables which are really what we now 

 call antilogarithmic. After the computation of these tables of radicates— after 

 the compilation of his Construct™ ; and, probably, even after the completion of 

 the Canon Mirificus, he began to view the matter from the other side, and to 

 think of his system as applicable to calculation in general ; that is, he viewed the 

 Naturalis as the argument, the Artificially to which he now gives the expressive 

 name Logarithm, as the function. Inverting the relations of primary and 

 function, he states in his appendix, article Habitudines, sect. 2, that the velocity 

 of the increase of the logarithm is inversely proportional to the number (sinus). 

 Thus, the genesis and computation of logarithms given by Nepair, is a perfect 



example of the transition from the equation -s- = <p x , which belongs to the cal- 



Rt 1 

 cuius of primaries, to the new equation ~- = ^- , which belongs to the integral 



calculus. 



The great problem in mechanics, " having given the law of attraction to 

 compute the motion of a body," belongs to the second chapter of our calculus, for 

 the attraction is the second derivative of the position regarded as a function 

 of the time, and thus all mechanical problems of this class are typified by 



the general formula -^ =(/>x, or 2t x — <px, in which the law connecting the func- 

 tion with its second derivative being given, that which connects it with the pri- 

 mary is sought. In this case also, the problem can be brought back to the cal- 

 culus of integrals, the artifice being to multiply each member of the equation by 

 the first derivative, and then to obtain by integration the square of the velocity 

 in terms of the position. 



The calculus of primaries has thus been unfortunate in that its first two 

 chapters have been absorbed by the Integral Calculus. But this absorption ceases 

 when we have to do with derivatives of a higher order, for then the resources of 

 integration fail us. As yet, no artifice has been discovered whereby the equation 



8 3 x 



p = ##, or st x = (px, can be rendered integrable, and problems involving such 



equations must be resolved by methods special to themselves ; or, as not unfre- 

 quently happens, must be let alone. 



There are many problems connected with the geometry of motion and with 

 mechanics, which resist all the powers of the calculus ; thus, although we know 

 the law connecting the curvature of an elastic plate with its angular tension, that 

 is, although we be able to write down an equation of the shape in differentials, 

 we are unable thence to arrive at that shape ; in other words, we are unable to 

 integrate. And thus our progress in mechanical science, even in matters of 



VOL. XXIV. PART III. 7 B 



