## Eurocentric, but precursors include

• Zeno, Eudoxos, and Archimedes in ancient Greece
• Lui Hui and Zu Chongzhi in China
• Alhazen in the middle east
• and many others, known and unknown.

## 17th century Europe

• Fermat, Descartes, Pascal, and others developed tools for computing tangents and areas.
• First published proof of the Fundamental theorem of calculus was 1668 by James Gregory.
• This was later generalized by Isaac Barrow.
• But, this was all ad hoc with no overarching context.

## Newton and Leibniz

The co-founders of Calculus

### Newton (1643-1727)

• Key discoveries 1665-1666, though published 20 years later
• Developed the mathematical theory very clearly by the standards of the time.
• Established many applications, particularly the explanations of Kepler's observations which is really the birth of the scientific method.

### Leibniz (1646–1716)

• Key discoveries 1674-75, with his first publication in 1675
• Less interested in physics, more in pure math and intrigued by infinitesimals.
• Developed much of the notation we still use.

## Together, these two

• Placed the ad hoc results of those who came earlier in a broader context, and
• Developed algorithms to solve broad classes of problems

## The Analyst

• A book published in 1734 by Bishop George Berkeley
• Subtitle: A DISCOURSE Addressed to an Infidel MATHEMATICIAN. WHEREIN It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith
• Essentially, an attack on the theoretical foundations of calculus with the objective of showing that it was based in faith as much as religion was.

## Blind Faith with spectacular results

• The Bernoullis (Jakob born 1664 - Johann III died 1807)
• Euler (1707-1783)
• Lagrange (1736-1813)
• Laplace (1749-1827)
• Fourier (1768-1839)
• Many others

## Lagrange

• Promoted rigor in calculus
• As a teacher of mathematics, felt the need to understanding basic questions more completely.
• (Teaching was becoming more important after the decline of royal courts.)
• As a tremendously influential mathematician, his ideas persuaded much of Europe.

## Fourier

• Full solution to the heat problem in 1822, which led to the Fourier series
• Fourier series (or, more generally, trigonometric series) are much more challenging than Taylor series (or, more generally, power series).
• This truly pushed the limits of 18th century calculus.

## Example

The domain of convergence of a power series is an interval while a trigonometric series might converge on a much more complicated set.

## Another example

Term by term differentiation of a power series leads to another powers series which represents the same function on (almost) the same domain as the original power series. Term by term differentiation of a convergent trig series leads to a trig series which might very well diverge everywhere.

## Final example

It's relatively easy to understand where the Taylor series of a function actually converges to that function. The corresponding question for Fourier series is much more complicated.

## Cauchy (1789-1857)

Generally, considered to be one of the founders of analysis.

## Cauchy's definition of limit

When the successively attributed values of the same variable indefinitely approach a fixed value, so that finally they differ from it by as little as desired, the last is called the limit of the others.

## Newton's definition of limit

Doesn't look so terribly different:
Ultimate ratios are "limits to which the ratios of quantities decreasing without limit do always converge, and to which they approach nearer than by any given difference…"
• But, Cauchy wrote proofs which translated his verbal definition into inequalities involving $\varepsilon$s and $\delta$s.
• The beginnings of what we'll fondly come to know as epsilonics.

## Weierstrass (1815-1897)

• Generally credited to be the person who formulated analysis the way we see it today.
• Also, a teacher.

## Weierstrass's definition of limit

$\displaystyle \lim_{x\rightarrow a} f(x) = L$ if for every $\varepsilon < 0$ there is a $\delta < 0$ such that $\left|\,f(x)-L\right|<\varepsilon$ whenever $0<\left|x-a\right|<\delta$.