- BRANCHES AND ROOTS OF THE CALCULUS 483
a minimum assuming that all the relevant quantities are differentiable a sufficient
number of times. Obviously, if a functional can be extended to a wider class of
functions in a natural way, the minimum reached may be smaller, or the maximum
larger. To make the restrictions as weak as possible, Weierstrass imposed the
condition that the partial derivatives of the integrand should be continuous at
corners. An extremal among all functions satisfying these less restrictive hypotheses
was called a strong extremal. The corner condition was also found by G. Erdmann,
a teacher at the Gymnasium in Konigsberg, who proved that Jacobi's sufficient
condition for a weak extremal was also necessary.
3.4. Foundations of the calculus. The British and Continental mathematicians
both found the power of the calculus so attractive that they applied and developed
it (sending forth new branches), all the while struggling to be clear about the
principles they were using (extending its roots). The branches grew more or less
continuously from the beginning. The development of the roots was slower and more
sporadic. A satisfactory consensus was achieved only late in the nineteenth century,
with the full development of real analysis, which is discussed in the Chapter 17.
The source of all the difficulty was the introduction of the infinite into analysis,
in the form of infinitesimal reasoning. Leibniz believed in actual infinitesimals,
levels of magnitude that were real, not zero, but so small that no accumulation
of them could ever exceed any finite quantity. His dx was such an infinitesimal,
and a product of two, such as dxdy or dx^2 , was a higher-order infinitesimal, so
small that no accumulation of such could ever exceed any infinitesimal of the first
order. On this view, even though theorems established using calculus were not
absolutely accurate, the errors were below the threshold of human perception and
therefore could not matter in practice. Newton was probably alluding to Leibniz
when in his discussion of the quadrature of curves, he wrote, "Errores quam minimi
in rebus mathematicis non sunt contemnendi" ("Errors, no matter how small, are
not to be considered in mathematics"). Newton knew that his arguments could
have been phrased using the Eudoxan method of exhaustion. In his Principia he
wrote that he used his method of first and last ratios "to avoid the tediousness
of deducing involved demonstrations ad absurdum, according to the method of the
ancient geometers."
There seemed to be three approaches that would allow the operation that we
now know as integration to be performed by antidifferentiation of tangents. One is
the infinitesimal approach of Leibniz, characterized by Mancosu (1989) as "static."
That is, a tangent is a state or position of a line, namely that of passing through
two infinitely near points. The second is Newton's "dynamic" approach, in which
a fluxion is the velocity of a moving object. The third is the ancient method of
exhaustion. In principle, a reduction of calculus to the Eudoxan theory of propor-
tion is possible. Psychologically, it would involve not only a great deal of tedium,
as Newton noted, but also a great deal of unnecessary confusion, which he did not
point out. If mathematicians had been shackled by the requirements of this kind of
rigor, the amount of geometry and analysis created would have been incomparably
less than it was. Still, Newton felt the objection and tried to phrase his exposition
of the method of first and last ratios in such a way as not to outrage anyone's logical
scruples. He said: