ON THE RECENT PROGRESS OF ANALYSIS. 239 



not to admire the perseverance with which he devoted himself 

 to it The attention of mathematicians 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 discouraging, 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 o algebraical functions as the subject 

 of the report which I have the honour to lay before the Associa- 

 tion. 



2. The theory of the comparison of transcendental func- 

 tions appears to have originated with Fagnani. In 1714, he 

 proposed, in the ' Griornale de Litterati d'ltalia,' the follow- 

 ing problem: To assign an arc of the parabola whose equa- 

 tion is 



y=x\ 



such that its difference from a given arc shall be rectifiable. 



Of this problem he gave a solution in the twentieth volume 

 of the same journal. 



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 

 assigned in a finite form. It is difficult, however, to perceive 

 how Fagnani was led to make the assumption in question: a 

 remark which applies more or less to his subsequent researches 

 on similar subjects. 



The theorem which has made his name familiar to all 

 mathematicians, appeared in the twenty-sixth volume of the 

 1 Giornale.' In its application to the comparison of hyperbolic 

 arcs we find some indications of a more general method. We 

 have here a symmetrical relation between two variables, x and 



* Those of M. Gauss, which would doubtless have been exceedingly valuable, 

 have not, I believe, been published. They are mentioned in a letter from M. 

 Crelle to Abel. Vide the introduction to the collected works of the latter, p. vii. 



