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of a surface by passing to the limit with n → ∞ using the recur-
rence formulas


,
etc. (by letter 1636). Dissatisfied with this method, he
discovered (about 1629) that the area

of the parabola

p could be found in general form by a fusion of the
Euclidean determination of a pyramid (see I, p. 24) with the
geometric Archimedean quadrature of the parabola (see I, p.
25) in which inscribed and circumscribed figures are constructed
in steps so that the abscissas of their points of division form a
geometrical progression on the X-axis (first term a, second term
ta; in the computation of the area of the entire infinite series
of these step-figures, t → a). A short time later he extended the
method to the parabolas

(p, q integral, prime
to each other and positive), to the spirals

and

(mentioned in Mersenne's writings 1637-
1644) and to the hyperbolas

= 1. He knew that
equal areas lay in this "logarithmic" division of the base line in
the case of the hyperbola xy = ab before passing to the limit.
From 1636, there was added to this, the cubature and determina-
tion of the centroid for solids of revolution with parabolic merid-
ian sections and the calculation of hyperbolic areas extending to
infinity by the use of improper integrals.

The impetus for the extreme value rule ( 1629) came from
Pappus VII (see I, p. 35): if in


the coefficient
of the lowest power of h is set equal to zero, then a value of x will
be found for which f(x) will assume an extreme value (necessary

-13-

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Publication Information: Book Title: Classical Mathematics: A Concise History of the Classical Era in Mathematics. Contributors: Joseph Ehrenfried Hofmann - author. Publisher: Philosophical Library. Place of Publication: New York. Publication Year: 1959. Page Number: 13.
    
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