# The Calculus Gallery Masterpieces from Newton to Lebesgue

# How much do I want to read more? 7/10

I love this kind of book so that I can engage in those formulas and check the examples for myself. It's like going back in time, and really understand those strikes of genius within the right context.

# INTRODUCTION

"The calculus," wrote John von Neumann (1903-1957), "was the first achievement of modern mathematics, and it is difficult to overesti- mate its importance"

calculus contin- ues to warrant such praise.

from the finite to the infinite, from the discrete to the continuous, from the superficial to the profound.

Like any great intellectual pursuit, the calculus has a rich history and a nch prehistory.

- Archimedes of Syracuse (ca. 287-212 BCE) found certain areas, volumes, and surfaces with a technique we now recognize as proto- integration.
- Pierre de Fermat (1601-1665) determined slopes of tangents and areas under curves in a remarkably modern fashion.
- Isaac Newton 0642-1727) and Gottfried Wilhelm Leibniz (1646-1716) gave birth to the calculus.
- Bernoulli brothers, Jakob 0654-1705) and Johann 0667-1748), and the incomparable Leonhard Euler 0707- 1783), whose research filled many thousands of pages with mathematics of the highest quality. limits, derivatives, integrals, infinite sequences, infinite series…
- Augustin-Louis Cauchy (1789-1857), Georg Friedrich Bernhard Riemann (1826-1866), Joseph Liouville (1809-1882), and Karl Weier- strass (1815-1897). These individuals worked with unprecedented care, taking pains to define their terms exactly and to prove results that had hitherto been accepted uncritically.
- Georg Cantor (1845-1918), Vito Volterra (1860-1940), Rene Baire (1874-1932), and Henri Lebesgue (1875-1941). pushed analysis ever further toward generality and abstraction.

I intend that the book be accessible to those who have majored or minored in college mathematics and who are not put off by an integral here or an epSilon there.

my foremost motivation is Simple: to share some favorite results from the rich history of analysis.

Students of lit- erature read Shakespeare; students of music listen to Bach. In mathematics it is what I tried to write in this book. not the paintings of Rembrandt or Van Gogh but the theorems of Euler or Riemann.

# Chapter 1 - Newton (1642-1727)

he had absorbed the work of such predecessors as Rene Descartes (1596--1650), John Wallis (1616-1703), and Trinitys own Isaac Barrow (1630-1677).

his generalized binomial expansion for turning certain expressions into infinite series, his technique for finding inverses of such series, and his quadrature rule for determining areas under curves. We conclude with a spectacular consequence of these: the series expanSion for the sine of an angle.

Although this chapter is restricted to Newton's early work, we note that "early" Newton tends to surpass the mature work of just about anyone else.

## GENERALIZED BINOMIAL EXPANSION

Newton had found a simple way to expand-his word was "reduce"-binomial expressions into series.

such reductions would be a means of recasting binomials in alternate form as well as an entryway into the method of fluxions.

The reduction, as Newton explained to Leibniz, obeyed the rule:

Example:

To identify A, B, C, and the rest, we recall that each is the immediately preceding term.

Working 8e 16e from left to right in this fashion, Newton arrived at:

Obviously, the technique has a recursive flavor.

Although the modern reader is probably accustomed to a "direct" statement of the binomial theorem, Newton's recursion has an un- deniable appeal, for it streamlines the arithmetic when calculating a nu- merical coefficient from its predecessor.

For the record, it is a simple matter to replace A, B, C, … by their eqUivalent expressions in terms of P and Q, then factor, and so arrive at the result found in today's

texts:

Newton likened such reductions to the conversion of square roots into infinite decimals:

"It is a convenience attending infinite series, that all kinds of complicated terms . . . may be reduced to the class of simple quantities, i.e., to an infinite series of fractions whose numerators and denominators are simple terms, which will thus be freed from those difficulties that in their original form seem'd almost insuperable."

Another example:

Newton would "check" by squaring the series and examining the answer. where all of the coefficients miraculously tum out to be 1 (try it!).

The re- sulting product, of course, is an infinite geometric series with common ratio x2 which, by the well-known formula, sums to 1/(1-x2)

He asserted that the "common analysis performed by means of equations of a finite number of terms" may be extended to such infinite ex- pressions "albeit we mortals whose reasoning powers are confined within narrow limits, can neither express nor so conceive all the terms of these equations, as to know exactly from thence the quantities we want".