# Evolution of Analysis

By: Abraham Schroeder and Tongzhou Wang

### Early Calculus

In the past few weeks of class we have been rigorously defining and proving all of the rules of calculus. The proofs that we have been learning in class are rooted in the definitions for functional limits and continuity. Strangely enough these proofs and definitions for calculus were not developed until the early 19th century even though calculus was created in the mid 17th century. For over a hundred years there were no formal proofs for most of the conclusions of calculus. We will discuss why there were no rigorous proofs for calculus and what occurred to motivate the mathematics community to create formal proofs for calculus after over a century of having none.

### Development

#### Fermat

Initially the derivative was used to answer geometric questions, such as how to find a maximum and the tangent to a line. Pierre de Fermat most notably used derivatives to find maxima and tangent lines but he was not fully aware of the underlying mathematics that he used in his methods. In order to find the slope of a tangent line at a point $x$ Fermat would consider a curve and a point of $E$ distance away from $x$ as shown below.

Using a property of similar triangles Fermat said that $\frac{s}{s+E} = \frac{f(x)}{f(x+E)}$ where $s$ is the distance from $0$ to $x$. Then by isolating $s$ Fermat found that $s = \frac{f(x)}{(f(x+E)-f(x))/E}$. Using this method Fermat unwittingly uses a limit and with no justification. We will demonstrate this by the following example.

In order to find the slope of the tangent line, we must find $\frac{f(x)}{s}$. Firstly we must find $s$. Let’s consider using the function $f(x) = x^2$. Using Fermat’s method we say $s = \frac{f(x)}{(f(x+E)-f(x))/(E)} = \frac{x^2}{(2xE+E^2)/E} = \frac{x^2}{2x+E}$. Fermat then with little justification let $E$ be $0$ by claiming that $s = \frac{x}{2}$. Then using $f'(x) = \frac{f(x)}{s}$, we find $f'(x) = 2x$. Fermat did this without realizing that he had used $E$ as a limit. In addition to this Fermat treated $E$ as a non-zero number by dividing it out but immediately after this, he treated it as zero without any justification(Dan Ginsburg, Brian Groose, Josh Taylor, The History of the Calculus and the Development of Computer Algebra Systems, Section 2). Despite this lack of proof, Fermat’s results were correct. Fermat unknowingly used limits and the derivative again to find the maximum of a function.

Here in the early stages of the development of calculus it is clear that calculus was used as a tool to solve geometric problems and the lack of proper justification was ignored by the fact that the solutions were correct and the application was useful.

#### Newton and Leibniz

Although derivatives and integrals were initially used by mathematicians such as Fermat and Descartes, it was not until Newton and Leibniz that the existing methods to find tangents, maxima, and areas were simplified to derivatives and integrals. Newton and Leibniz both developed the fundamental theorem of calculus which shows that the derivative and the integral are inverse operations of each other. The most controversial part of Newton and Leibniz’s work is the way in which they chose to define derivatives. Newton called limit as fluxion which means a rate of flux or change while Leibniz saw the derivative as a ratio of infinitesimal differences , $\frac{dy}{dx}$, and called it the differential quotient.(Judith V. Grabiner, The Changing Concepty of Change:The Derivative from Fermat to Weierstrasshttps://www.maa.org/sites/default/files/0025570×04690.di021131.02p02223.pdf)

Both of these methods had similar problems to Fermat’s use of the derivative where they treated a very small number as a non- zero number and then as a zero number. In addition to this Leibniz’s notation was the cause for further controversy since his difference quotient violated the Archimedian’s axiom of ratios which states that given any magnitudes of ratios one can find a multiple of either ratio that will be larger than the other (WolframMathWorld, Archimede’s Axiom, http://mathworld.wolfram.com/ArchimedesAxiom.html).

Since the ratio of infinitesimals has no well defined value, it violates this axiom. However, this notation is what eventually became the more widely used notation. Despite these shortcomings, the work of Leibniz and Newton was generally accepted since calculus was such a powerful tool that answered many questions in both mathematics and physics. Calculus was instrumental in physics since it allowed Newton to define velocity as a derivative of position and acceleration as the derivative of velocity. In addition to this, calculus allowed Euler to properly model the motion of a vibrating string. In mathematics taylor series were also developed so that any function could be approximated with polynomials, however this was developed without a rigorous understanding of the convergence of a series.

#### Cauchy

The shift in the mathematics community that sparked an interest in proving calculus was when in 1797 Lagrange defined the derivative to be a function. With the notation of Newton the derivative was a rate of change and with Leibniz the derivative was a ratio of infinitesimals, however Lagrange changed this. He stated that any function $f(x+h)$ had the following expansion, $f(x+h) = f(x)+f'(x)h+f''(x)h^2+f'''(x)h^3....$. This was the first time that the derivative was thought of as a function in which Lagrange referred to $f'(x)$ as the “first derived function” of $f(x)$ and $f''(x)$ as the first derived function of $f'(x)$.

With the derivative newly defined as a function Cauchy then was able to define the derivative as a fixed value that is approached by successively attributed value of one variable. Although this definition may not be as solid as the $\epsilon-\delta$ version, it helped Cauchy to prove his famous theorems. With the help of Weierstrass, Cauchy and other mathematicians were able to define continuity, convergence and many other underlying mathematical properties in Calculus.

### Reason to change

While the derivative was first introduced during 1600s, the methods used to determine it were not that definite due to the underlying nature of limit was not understood.

However, those early mathematicians did not care too much about the rigor of their methods because those non-rigorous methods helped them to apply calculus in physics.

For example, Fermat applied his principle of determining maximum to study optics.

Early mathematicians did not feel the need to justify their methods until 1780s, explanations of the nature of calculus were found importantly in introductions of calculus. They realized in order to introduce this new subject to the public, rigorous proofs were desperately needed. Besides people who were interested in learning math, scientific professionals also had lots of potential readers of calculus books. Therefore, mathematicians started to prove the nature of calculus(Grabiner, Origins of Cauchy’s Rigorous Calculus, 23).

Furthermore, though calculus conclusions were widely accepted, there were people who questioned the correctness of math theories such as Bishop George Berkeley who said. “Scientists attack religion for being unreasonable; well, let them improve their own reasoning first”(Judith V. Grabiner, The Changing Concepty of Change:The Derivative from Fermat to Weierstrasshttps://www.maa.org/sites/default/files/0025570×04690.di021131.02p02223.pdf). Berkeley’s attack pointed out real deficiencies and mathematicians moved to try to answer it(Grabiner, Origins of Cauchy’s Rigorous Calculus, 27). Thus after Lagrange published his paper on the power series in 1797, Cauchy and others started to rigorously prove and define the rules of calculus.

### Reference

Judith V. Grabiner, The Origins of Cauchy’s Rigorous Calculus. (http://users.uoa.gr/~spapast/TomeasDidaktikhs/Caychy/GrabinerOriginsofCauchysRigorousCalculus.pdf)

Judith V. Grabiner, The Changing Concepty of Change:The Derivative from Fermat to Weierstrass (https://www.maa.org/sites/default/files/0025570×04690.di021131.02p02223.pdf)

Dan Ginsburg, Brian Groose, Josh Taylor, The History of the Calculus and the Development of Computer Algebra Systems, Section 2 (http://www.math.wpi.edu/IQP/BVCalcHist/calc2.html)

WolframMathWorld, Archimede’s Axiom (http://mathworld.wolfram.com/ArchimedesAxiom.html)