HISTORY OF MATHEMATICS – LECTURE 17 – MONDAY 15
TH
NOVEMBER
The Age of Rigor
We have already looked at examples of how the mathematics of Newton and Euler, while brilliant in its
originality, lacked the rigor that characterizes the subject today. The concept of a formal proof was well
known for making geometric arguments, but for calculus it was not possible to achieve the same level of
precision until the concepts of a limit and convergence were better defined. In the meantime,
mathematicians had a hard time resolving the paradoxes that seemed to exist, and the ridicule that they
caused.
Ex.
The Taylor series for
ଵ
ଵି௫
is 1 + x + x
2
+ x
3
+ …
One of the most vocal critics of the methods of calculus was the bishop and philosopher George
Berkeley (1685
‐
1753), after whom the famous California university is named. Referring to the work of
Newton in his famously critical book
The Analyst,
Berkeley wrote
“And if the first [fluxions] are incomprehensible, what should we say of the second and third? He who
can digest a second or third fluxion…need not, methinks, be squeamish about any point in Divinity.”
“And what are these fluxions? The velocities of evanescent increments? And what are these same
evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet
nothing. May we not call them the ghosts of departed quantities?”
Augustin Louis Cauchy
The two mathematicians most responsible for providing clarity and rigor to the subject of calculus were
Augustin Louis Cauchy (1789
‐
1857) and Karl Weierstrass (1815
‐
1897).
Cauchy was originally an engineer, but was persuaded by Lagrange to devote his attention to
mathematics. He became a full professor in 1816, and won the grand prize of the Acad
é
mie des Sciences
the same year for his paper on wave propagation.
Due to the fragile political situation in France in the early 19th century, Cauchy spent time teaching in
Turin and Prague, but he communicated his ideas back to his homeland, and his output was so
prodigious that he established his own journal, the
Exercises des math
é
matiques
, and inundated the
journal of the Acad
é
mie with so many papers (589 of the 789 that he published in all) that they imposed
a rule limiting their length to four pages that is still in effect to this day.
Cauchy’s most famous work, the
Cours d’analyse
, was published in 1821, in which he developed the
notions of a limit, continuity, differentiability, and the definite integral which we see today in calculus.
For example, Cauchy stated that
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 Spring '08
 EVINSON
 Math, Calculus, The Land, Karl Weierstrass, Augustin Louis Cauchy, Académie des Sciences, Vladimir Kovalevsky

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