RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

15.1. Elliptic transcendentals before the nineteenth century 287
of points on the curve, its rectification and quadrature and the determination of its
tangents, centres of curvature, involutes, and evolutes. A number of these properties
together constituted the knowledge required for a curve to be known and it was one
of the greatest achievements of the calculus to devise a method for obtaining most of
them from a single defining equation. 2
A variety of different curves was treated in the first decades of the calculus, and
many problems were reduced to “simpler” problems, primarily to the construction of
points on algebraic curves, the rectification of the circle (inverse trigonometric func-
tions) and the quadrature of the hyperbola (logarithmic functions). A classification of
construction problems was established based on the simpler problems to which the
solution of the given problem could be reduced.
Despite the efforts of the mathematicians, certain problems defied the accepted
known means of solution; for instance, when asked to compute the arc lengths of
ellipses and some other curves, mathematicians found that the known simple integrals
did not suffice.
In the year 1675, G. W. LEIBNIZ (1646–1716) directed two enquiries concerning the
rectification of the ellipse to the British mathematicians J. GREGORY (1638–1675) and
NEWTON. LEIBNIZ received the answer that the British mathematicians could only
compute the length of an ellipse by approximation, i.e. with the help of infinite series,
and did not possess any closed expression for the length. LEIBNIZ, himself, at the time
believed that he could reduce this problem (and the rectification of the hyperbola) to
the quadratures of the circle and the hyperbola. Later, LEIBNIZ realized that he had
been misled by a computational error. 3
As can be seen from box 6, the rectification of the ellipse involved the computation
of an integral of the form � R(x) dx
√ P4(x) in which R and P4 were polynomials such that
deg P4 ≤ 4. Such integrals (with the relaxed assumption that R be only a rational function)
were soon called elliptic integrals by G. C. FAGNANO DEI TOSCHI (1682–1766). 4
LEIBNIZ’ question reflects the search for simpler, finite representations of elliptic inte-
grals.
In a paper written in 1732 but not published until six years later, 5 EULER deduced
a series representation of a quarter of the circumference of an ellipse. Based on a figure
(see figure 15.1) in which M represented a point on the ellipse with center C and semi-
axes CA = a and CB = b, EULER expressed the differential of the arc-length �AM
as
ds = b2√ b 2 + t 2 + nt 2 dt
(b 2 + t 2 ) 3 2
2 For information on these aspects of curves, see e.g. (Loria, 1902). For a general discussion on the
conceptions of curves before the advent of the calculus, see (H. J. M. Bos, 2001).
3 (Hofmann, 1949, 75,118).
4 (Natucci, 1971, 516). For more on FAGNANO DEI TOSCHI’S work on elliptic integrals, see below.
5 (L. Euler, 1732a).