274 ON THE RECENT PROGRESS OF ANALYSIS. 



multiplication of elliptic integrals, those of M. Jacobi, as we 

 have seen, from the limiting case of their transformation. Hence 

 Abel's fundamental expressions depend on doubly infinite con- 

 tinued products, corresponding to the double periodicity of elliptic 

 functions. On the other hand, M. Jacobi's continued products 

 are all singly infinite. 



Other differences might of course be pointed out, but the 

 most remarkable is that which we find in the character and 

 style of their writings. Nothing can be more distinct. In 

 M. Jacobi's we meet perpetually with the traces of patient and 

 philosophical induction ; we observe a frequent reference to par- 

 ticular cases and a most just and accurate perception of analogy. 

 Abel's are distinguished by great facility of manner, which 

 seems to result from his power of bringing different classes of 

 mathematical ideas into relation with each other, and by the 

 scientific character of his method. We meet in his works with 

 nothing tentative, with but little even that seems like artifice. 

 He delights in setting out with the most general conception of 

 a problem, and in introducing successively the various conditions 

 and limitations which it may require. The principle which he 

 has laid down in a remarkable passage of an unfinished essay 

 on equations seems always to have guided him that a question 

 should be so stated that it may be possible to answer it. When 

 so stated it contains, he remarks, the germ of its solution*. 



I do not presume to compare the merits of these two mathe- 

 maticians. The writings of both are admirable, and may serve 

 to show that if ever the modern method of analysis seems to be 

 an fjL7T6ipLa rather than a re^vrj, it does so, either because it 

 has not been rightly used, or because it is not duly understood. 



To obtain a general view of Abel's writings it may be re- 

 marked, that his earliest researches related to the theory of equa- 

 tions. Of the ideas with which he was then conversant he has 

 made two principal applications. The one is to the comparison 



For instance, Is it possible to trisect an angle by the rule and compass ? 

 The question thus stated leads us to consider the general character of all problems 

 soluble by the methods of elementary geometry ; and following the suggestion thus 

 given, we find that it is to be answered in the negative. But if the last clause 

 be omitted or neglected, we can only proceed, as many persons have done, 

 tentatively, i.e. by attempting actually to solve the problem. If we fail, the 

 question remains unanswered; if we succeed, we do answer it, but as it were 

 only by accident. 



