a 
ON THE RECENT PROGRESS OF ANALYSIS. 57 
peared *, his method is, in a historical point of view, of considerable in- 
_ terest. 
18. In the second volume of Crelle’s Journal, p. 101, we find Abel’s first 
memoir on elliptic functions. It was published in the spring of 1827, and 
therefore before M. Jacobi’s announcement in No. 123 of Schumacher’s 
Journal. But it contains nothing which interferes with M. Jacobi’s disco- 
very of the general theory of transformation. Abel’s researches on this part 
of the subject appeared in the third volume of Crelle’s Journal, p. 160. 
This second communication is dated, as we are informed by an editorial 
note, the 12th of February, 1828, and though it is announced as a continu- 
ation of the former memoir, it is yet in effect distinct from it, as its contents 
are not mentioned in the general summary prefixed to the first communica- 
‘tion. 
These details may not be without interest, though it is not often that ques- 
tions of priority deserve the importance sometimes given to them. There 
is no doubt that Abel’s researches were wholly independent of those of 
M. Jacobi; and though the coincidence of some of their results is therefore 
interesting, yet the general view which they respectively took of the theory of 
elliptic functions is essentially different, as different as the style and manner 
of their writings. 
With M. Jacobi the problem of transformation occupied the first place ; 
with Abel that of the division of elliptic integrals. Both introduced a nota- 
tion inverse to that which had previously been used, and as an immediate 
consequence recognised the double periodicity of elliptic functions. Ex- 
pressions of these functions in continued products and series were given by 
both, but those of Abel were deduced by considering the limiting case of 
the 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 continued 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 
_ ean be more distinct. In M. Jacobi’s we meet perpetually with the traces 
_ of patient and philosophical induction ; we observe a frequent reference to 
4 particular 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 
is it may require. The principle which he has laid down in a remarkable pas- 
| sage 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 +. 
: 
___ * The fundamental formula of his transformation is incidentally mentioned in Legendre’s 
_ second work (Traité des Fonct., i. 61). 
___t For instance, Is it possible to trisect an angle by the rule and compass? The ques- 
tion 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 
_ 1s to be answered in the negative. But if the last clause be omitted or neglected, we can 
Dowd proceed, as many persons have done, tentatively, 7. e. by attempting actually to solve 
oo 
: e problem. If we fail, the question remains unanswered; if we succeed, we do answer 
it, but as it were only by accident. 
