ON THE RECENT PROGRESS OF ANALYSIS. 35 
Bacon, that knowledge, after it has been systematized, is less likely to increase 
than before, seem less applicable to mathematical than to natural science. 
Nevertheless, almost immediately after the publication of Legendre’s work, 
the earlier researches of Abel and Jacobi became known, and it was at once 
seen that what had been already accomplished formed but a part, and not a 
_ large one, of the whole subject. 
__ To say this is not to derogate from the merit of Legendre. He created 
_ the theory of elliptic functions; and it is impossible not to admire the per- 
_ severance with which he devoted himself to it. The attention of mathema- 
- ticians was given to other things, and though the practical importance of his 
labours was probably acknowledged, yet scarcely any one seems to have 
_ entered on similar researches*. This kind of indifference was doubtless dis- 
_ couraging, but not long before his death he had the satisfaction of knowing 
_ that there were some by whom that which he had done would not willingly 
_ be let die. 
__ The considerations here suggested have led me to select the theory of the 
integrals of algebraical functions as the subject of the report which I have the 
honour to lay before the Association. 
__ 2. The theory of the comparison of transcendental functions appears to 
have originated with Fagnani. In 1714, he proposed, in the ‘Giornale de 
Litterati d’ Italia,’ the following problem: To assign an arc of the parabola 
_ whose equation is 
, y= 
such that its difference from a given are shall be rectifiable. 
_ Of this problem he gave a solution in the twentieth volume of the same 
rnal. 
___ The principle of the solution consists in the transformation of a certain 
differential expression by means of an algebraical and rational assumption 
which introduces a new variable. The transformed expression -is of the same 
form as the original one, but is affected with a negative sign. By integrating 
both we are enabled to compare two integrals, neither of which can be as- 
| Signed in a finite form. It is difficult, however, to perceive how Fagnani was 
to make the assumption in question: a remark which applies more or 
to his subsequent researches on‘similar subjects. 
he theorem which has made his name familiar to all mathematicians, ap» 
ed in the twenty-sixth volume of the ‘Giornale.’ In its application to 
comparison of hyperbolic ares we find some indications of a more general 
lethod. We have here a symmetrical relation between two variables, x and 
» such that the differential expression J(#)dx may be written in the form 
z. It follows at once that f(z) dz =< dz, and consequently that 
WhiOLE +f f(z) dz =f {xdz+zdzx}=x2+ on 
3 remarkable manner in which the idea of symmetry here presents itself, 
gested to Mr. Fox Talbot his ‘ Researches in the Integral Calculus.’ 
In applying bis methods to the division of the are of the lemniscate, Fag- 
i obtained some very curious results, and has accordingly taken for the 
hette of his collected Works a figure of this curve with the singular motto, 
eo veritatis gloria.” 
3. In MacLaurin’s Fluxions, and in the writings of D’Alembert, instances 
3 to be found where the solution of a problem is made to depend on the 
ose of M. Gauss, which would doubtless have been exceedingly valuable, have not, I 
» been published. They are mentioned in a letter from M. Crelle to Abel. Vide the 
eduction to the collected works of the latter, ps Vii. 
‘ D2 
