i ON THE RECENT PROGRESS OF ANALYSIS. 37 
isolated fragment, is the foundation of our knowledge of the higher trans- 
-cendents. We may therefore conveniently divide the subject of this report 
_ into two portions, viz. the general theory of the comparison of algebraical 
integrals, and the investigations which are founded on it. Mathematicians 
have been led, by comparing different transcendents, to introduce new func- 
tions into analysis, and the theory of these functions has become an important 
subject of research. 
The second portion may again be divided into two, viz. the theory of 
elliptic functions, and that of the higher transeendents. 
This classification, though not perhaps unexceptionable, will, I think, be 
found convenient. 
_ 7. About sixteen years after the publication of Lagrange’s earlier researches 
on the comparison of algebraical integrals, he gave, in the New Turin Me- 
_ moirs for 1784 and 1785, a method of approximating to the value of any 
a where P is.a rational function of z and R the 
- integral of the form 
_ square root of a polynomial of the fourth degree. I shall consider this im- 
_ portant contribution to the theory of elliptic functions in connexion with the 
_ writings of Legendre. At present, in order to give a connected view of the 
first division of my subject, it will be necessary to go on at once to the works 
of Abel, and to those of subsequent writers. In the history of any branch 
of science the chronological order must be subordinate to that which is 
_ founded on the natural connexion of different parts of the subject. 
__ Ishall merely mention in passing, that in 1775, Landen published in the 
Philosophical Transactions a very remarkable theorem with respect to the 
_ ares of a hyperbola. He showed that any arc of a hyperbola is equal to the 
_ difference of two elliptic ares together with an algebraical quantity. In 1780 
he published his researches on this subject in the first volume of his ‘ Mathe- 
matical Memoirs,’ p. 23. This theorem, as Legendre has remarked, might 
have led him to more important results. It contains the germ of the general 
theory of transformation, the eccentricities of the two ellipses being con- 
nected by the modular equation of transformations of the second order*. It 
is on this account that in a report on M. Jacobi’s ‘ Fundamenta Nova,’ con- 
lined in the tenth volume of the Memoirs of the Institute, Poisson speaks 
f Landen’s theorem as the first step made in the comparison 6f dissimilar 
elliptic integrals. Several writers have accordingly given Landen’s name to 
_ the transformation commonly known as Lagrange’s. 
_ 8. We have seen that even Lagrange failed in obtaining a result more 
eneral than that which had been made known by Euler, and yet, as we now 
now, Euler’s theorem is but a particular case of a far more general proposi- 
. But in order to further progress, it was necessary to introduce a wholly 
ew idea. The resources of the integral calculus were apparently exhausted ; 
el, however, was enabled to pass on into new fields of research, by bring- 
ing it into intimate connexion with another branch of analysis, namely, the 
theory of equations. The manner in which this was done shows that he was 
not unworthy to follow in the path of Euler and of Lagrange. 
wi pel attempt to state in a few words the fundamental idea of Abel’s 
thod. 
Let us suppose that the variable z is a root of the algebraical equation 
0, and that the coefficients of this equation are rational functions of 
in quantities a, b, ...¢, which we shall henceforth consider independent 
ables. Let us suppose also that in virtue of this equation we can express 
* Vide infra, pp. 50 and 67. 
