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XIV. Researches in the Integral Calculus. — Part I. By H. F. Talbot, Esq. F.R.S. 
Received and read March 10, 1836. 
§ 1. Brief Historical Sketch of the Subject. 
The first inventors of the integral calculus observed that only a certain number of 
formulae were susceptible of exact integration, or could be reduced to a finite number 
of terms involving algebraic, circular, or logarithmic quantities. When this result 
could not be attained, they were accustomed to develop the integral in an infinite 
series. But this method, although useful when numerical values are to be computed, 
is entirely inadequate, in an analytical point of view, to supply the place of the exact 
integral ; for the progress of analysis has shown many instances of exact relation be- 
tween different integrals which cannot by any means be inferred from the infinite 
series in which they are developed. 
The first great improvement beyond this was made by Fagnani about the year 1714. 
This most acute and ingenious mathematician proposed the following question to the 
scientific world in an Italian journal*: “Given a biquadratic parabola whose equa- 
tion is x* — y, and an arc of it, to find another arc, so that their difference may be 
rectifiable.” 
No answer appearing, he published a solution of the problem in the year 1715-^, 
and extended it in a nearly similar manner to other curves whose equation is x n = y, 
viz. to those cases where n equals one of the numbers 3, -f-, x, •§-, -f, -f. 
In the year 17 18 and afterwards he published a variety of important theorems 
respecting the division into equal parts of the arcs of the lemniscate, and respecting 
the ellipse and hyperbola, in both of which he showed how two arcs may be deter- 
mined of which the difference is a known straight line. These discoveries justify us 
in regarding Fagnani as the founder of a new and very curious branch of analysis. 
Euler, who enriched almost every department of science with new discoveries, 
exhibited the complete algebraic integral of the equation 
d x . dy 
V « + j3 x + y 
x 2 + 8 x 3 + £ x 4 
+ 
= 0 
v'a + fiy + yy 2 + S «/ 3 + sy 4 
a remarkable theorem, which long continued to be the ne plus ultra of this branch of 
science, little success having attended the endeavours of mathematicians to arrive at 
results of greater generality. 
* Giomale de’Letterati d’ Italia, tom. xix. p. 438. 
2 A 
f tom. xxii. p. 229. 
MDCCCXXXVI. 
