240 ON THE RECENT PROGRESS OF ANALYSIS. 



z, such that the differential expression f(x) dx may be written 

 in the form z dx. It follows at once that f(z) dz xdz^ and 

 consequently that 



ff(x) dx + jf(z) dz = j{x dz + z dx} = xz+C. 



The remarkable manner in which the idea of symmetry here 

 presents itself, suggested to Mr Fox Talbot his ' Kesearches in 

 the Integral Calculus.' 



In applying his methods to the division of the arc of the 

 lemniscate, Fagnani obtained some very curious results, and has 

 accordingly taken for the vignette of his collected works a 

 figure of this curve with the singular motto, ' Deo veritatis 

 gloria.' 



3. In MacLaurin's Fluxions, and in the writings of 

 D'Alembert, instances are to be found where the solution of a 

 problem is made to depend on the rectification of elliptic arcs, 

 or, as we should now express it, is reduced to elliptic integrals. 

 But of these instances Legendre has remarked that they are 

 isolated results, and form no connected theory. MacLaurin is 

 charged, in a letter appended to the works of Fagnani, with 

 taking from the latter without acknowledgement, a portion of 

 his discoveries with respect to the lemniscate and the elastic 

 curve. 



4. In 1761, Euler, in the 'Novi Commentarii Petropolitani ' 

 for 1758 and 1759, published his memorable discovery of the 

 algebraical integral of the equation 



mdx _ ndy 



(A + Bx + Cx z + Dx 3 + Ex 4 ) k ~ (A + By + Gf + Dy* + % 4 )* ' 



m and n being any rational numbers. 



He says he had been led to this result by no regular method, 

 ' sed id potius tentando, vel divinando elicui,' and recommends 

 the discovery of a direct method to the attention of analysts. 

 In effect his investigations resemble those of Fagnani : he begins 

 by assuming a symmetrical algebraical relation between the 

 variables, and hence finds a differential equation which it satis- 

 fies. In this differential equation the variables are separated, 

 so that each term may be considered as the differential of some 



