746 PEOrESSOE BOOLE ON THE COMPAEISON OF TEANSCENDENTS, ^TH 
peculiarity to which it seems necessary to direct attention, is the introduction of a 
symbol, diifering in interpretation only by the addition of one element, from that which 
Cauchy has employed in his ‘ Calculus of Eesidues.’ Of the nature of this connexion I 
was not aware until my researches were nearly completed; and I should then have 
abandoned my own symbol and adopted the other, already associated by the labours of 
its distinguished inventor with so many important discoveries in the higher departments 
of the integral calculus, if it had not appeared to me that the several elements combined 
in the interpretation of the former symbol were so allied that no one of them could 
without a manifest defect of completeness be omitted. It seemed to me also that many 
of Cauchy’s own applications of his symbol would gain in simplicity and in generality 
of expression, by the adoption of the more enlarged interpretation. 
2. Beside the above special objects, in the attainment of which whatever claim to 
originality this paper may possess will consist, I have proposed to myself, as a general 
object, the simplification of a branch of analysis which possesses some practical and 
much speculative importance. To this object the introduction of the symbol above 
referred to, contributes in a very important degree. The necessity’ of simplification 
will, I think, be admitted by all who are acquainted with the literature of the subject. 
As presented in the writings of Abel and of those who immediately followed in his 
steps, the doctrine of the comparison of transcendents is repulsive fr’om the complexity 
of the formulae in which its general conclusions are embodied. The particular result 
known as Abel’s theorem, the only one of its class which has been adopted into English 
works of education, will at once suggest itself in confirmation of this remark. Perhaps 
this complexity will not be thought surprising if we consider the nature of the problems 
involved, — the discovery of finite relations among integrals which derive their very name 
from the circumstance that individually their finite expression transcends the powers of 
analysis. On the other hand, and this is a juster ground of mference, the theory upon 
which such applications rest is far from being difficult or recondite, and, considered a 
priori, should be capable of a simpler analytical development than it has yet received. 
I hope that I shall be able to show that this anticipation is confirmed by the results of 
the present inquiry. Simplicity, though it is not to be gained at the expense of that 
which is the chief object of scientific methods, the discovery of truth, is nevertheless a 
highly valuable quality. And so far from being inconsistent Avith generality in the 
processes and the results of analysis, it is sometimes an indication of the measure of our 
approach to completeness and unity. I think that this is more especially the case 
where, as through the labours of Abel in the present instance, the subject matter of 
investigation has been clearly defined, and the entire series of methods and results 
foreshown to be the evolution of some one general principle or idea. 
3. It will be proper, before entering upon the special investigation, to giA’e some 
general account of the doctrine of the comparison of transcendents. In doing this, I 
cannot but refer to the able report of Mr. Leslie Ellis on the Progress of Analysis, 
published in the Eeport of the British Association for 1846. It contains a most A aluable 
