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

Chapter 15
Elliptic integrals and functions:
Chronology and topics
After the calculus was invented in the seventeenth century, it was quickly applied
to classical problems concerning curves. One of the main achievements of the new
tool was the ability to treat curves which had previously been outside the reach of
geometry. For instance, the quadrature of the hyperbola provided an analytical way
of describing and treating logarithmic functions. After the calculus had conquered
such basic curves as the logarithmic and trigonometric ones, the determination of the
length of an ellipse became a major obstacle on the path to generality.
In the eighteenth century, L. EULER’S (1707–1783) new vision of the calculus as
founded upon functions also transformed the way in which curves were approached. 1
The way in which the elliptic transcendentals enter into the realm of analysis touches
upon a number of points which will be described below and — primarily — connected
to N. H. ABEL’S (1802–1829) work:
1. During the eighteenth century, a need came to be felt to include higher transcen-
dentals (i.e. functions different from the algebraic, logarithmic, and trigonomet-
ric ones) into analysis on a par with the well established elementary functions. In
order to accept these new objects into analysis, they had to undergo a process of
becoming known. This process manifested itself in various ways, e.g. in the search
for acceptable analytical representations of the new objects.
2. Because the new functions were in some senses generalizations of the elementary
functions, their study opened possibilities of generalization of existing results.
In the process, insights into the new objects were obtained which also helped
making them known.
3. Ultimately, the consensus on how to introduce particular new higher transcen-
dentals — as primitive functions of algebraic differentials — led to a research pro-
1 For EULER’S approach to analysis, see section 10.1.
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