What idea of integration were Newton and Leibniz using?

Question

The integral that is taught in calculus courses is the Riemann Integral. Which presumably is named after Bernhard Riemann. But Riemann was born $99$ years after Newton died. So what kind of integration were Newton and Leibniz using?

I ask this because we are told that Newtons advisor discovered the Fundamental Theorem of Calculus. But that would require knowledge of integration. So what idea did Newton and Leibniz have of integration if Newton's advisor managed to discover the Fundamental Theorem of Calculus.

Answer 1

Things become clearer if you look at Barrow's original proof, which can be found in The Geometrical Lectures of Isaac Barrow (Proposition 11 of Lecture X).

According to the picture below, Barrow (= Newton's advisor) proved that, given two curves $ZGE$ and $AIF$ (with $AZ$, $PG$, $DE$, etc. increasing), if the area of the region $ADEZ$ is $DF\cdot R$ (for any $D$ and a given constant $R$) then $TF$ is tangent to $AIF$ at $F$, where $T$ is defined by the relation $\frac{DE}{DF}=\frac{R}{DT}$.

This is the Barrow's Fundamental Theorem of Calculus. In modern notation, if we take $R=1$, $ZGE$ is the graph of $g(x)$, $f(x)=-g(x)$, $AIF$ is the graph of $F(x)$, $x$ is the abscissa of $D$ and $a$ is abscissa of $A$, then Barrow's result implies that

$$\begin{aligned} \frac{d}{dx}\left[\int_a^x f(s)\,ds\right]&=\frac{d}{dx}\left[\text{area}(ADEZ)\right]&&\text{(modern meaning of integral)}\\ &=\frac{d}{dx}\left[F(x)\right]&&\text{(Barrow's hypothesis)}\\ &=\frac{DF}{DT}&&\text{(modern meaning of derivative)}\\ &=DE&&\text{(Barrow's hypothesis)}\\ &=f(x),&& \end{aligned}$$ which is one part of the modern Fundamental Theorem of Calculus.

The other part can be found in Proposition 19 of Lecture XI: if $FT$ is tangent, then $\text{area}(APGZ)=PI\cdot R$. In modern notation, taking $R=1$ and using the fact that $F(a)=0$ in the considered case and labeling $p$ the abscissa of $P$, we obtain the usual result $$\int_a^p f(x)\,dx=F(p)-F(a).$$

In a broad sense:

• From the presented statements, we can see that Barrow's result is not about integrals and derivative, but about areas and tangents.
• From the proofs, that I've omitted but can be found in the said book as well as here (with more details), we can see that the Barrow's integral (at least in this particular context of the FTC) is "sum of infinitesimal rectangles".
• Leibniz (and at least some of his contemporaries, as the Bernoullis) used a similar interpretation.

Answer 2

Leibniz didn't have a formal definition of an integral, much less the idea of a limit, but it was like a Riemann integral. Leibniz imagines the area under the graph of a function $f(x)$ to be an infinite collection of infinitely thin rectangles, each with height $f(x)$ and width $dx$, an infinitely small change in $x$. Thus each rectangle has area $f(x)dx$, and the expression

$$\int_a^b f(x)dx$$

means the sum of such rectangles which will comprise the area.

Answer 3

The following is more a hint than an elaborated answer. I would like to draw attention to an informative and exciting book by William Dunham. It is called The Calculus Gallery - Masterpieces from Newton to Lebesgue and the first two chapters are devoted to Newton and Leibniz.

In the chapter about Newton, the author treats integration in two sections.

Newton: Quadrature Rules from the De Analysi:

De Analysi written in 1669 but not published before 1711 was one of Newtons celebrated mathematical writings where he demonstrated calculations with fluxions to obtain the area beneath simple curves. He formulated three quadrature rules:

• Rule 1: The quadrature of simple curves: If $y=ax^{m/n}$ is $AD$, where $a$ is a constant and $m$ and $n$ are positive integers, then the area of region $ABD$ is $\frac{an}{m+n}x^{(m+n)/n}$ (see figure 1.1).

• Rule 2: The quadrature of curves compounded of simple ones: If the value of $y$ be made up of several such terms, the area likewise shall be made up of the areas which result from every one of the terms.

• Rule 3: The quadrature of all other curves: But if the value of $y$, or any of its terms be more compounded than the foregoing, it must be reduced into more simple terms ... and afterwards by the preceding rules you will discover the [area] of the curve sought.

The figure 1.1 used to illustrate rule 1 is of similar kind as the picture provided by @pedro. The calculation was done by Newton using fluxions with an application of the binomial series expansion and the theorem becomes in modern notation: \begin{align*} \int_{0}^xat^{m/n}\,dt=\frac{ax^{(m/n)+1}}{(m/n)+1}=\frac{an}{m+n}x^{(m+n)/n}\tag{1} \end{align*}

Here we should keep in mind, that Newton was just $27$ years old at that time and the binomial series expansion and the concept of fluxions were creative inventions done by Newton himself.

The second rule addresses the linearity of the integral operator, whereas the third rule was another great idea, basis for more mathematical highlights. In order to calculate the area beneath a curve which has a compound form, the idea was to reduce it at first, which means to make a series expansion, then integrate the series term by term and finally sum up all the terms. This technique led to another highlight nicely shown by W. Dunham in the next section:

Newton's Derivation of the Sine Series:

Here Newton started with a figure showing a circle centered at the origin with radius $1$. The figure is again of similar complexity as the figure given by @pedro using similarities of triangles and infinitely small triangles.

In a next step he exploited the circular relationship $y=\sqrt{1-x^2}$ and obtained \begin{align*} dz &= \frac{dx}{y}=\frac{dx}{\sqrt{1-x^2}}\\ &=\left(1+\frac{1}{2}x^2+\frac{3}{8}x^4+\frac{5}{16}x^6+\cdots\right)\,dx \end{align*} Finding the quadratures of these indiviaual powers and summing the results by rule 3, Newton derived the arcsine series \begin{align*} \arcsin x=z=x+\frac{1}{6}x^3+\frac{3}{40}x^5+\frac{5}{112}x^7+\cdots \end{align*} Newton had at that time already another trick at hand which he could masterfully apply. It was an inversion technique based on some clever substitution of a series term by term to derive the inverted series. This way he was able to derive the celebrated series expansion for the sine.

\begin{align*} \color{blue}{\sin z}&\color{blue}{=z-\frac{1}{6}z^3+\frac{1}{120} z^5-\frac{1}{5\,040} z^7+\frac{1}{362\,880}z^9-\cdots}\\ &\color{blue}{=\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)!}z^{2k+1}} \end{align*}

The following chapter about Leibniz is also exciting and informative, again containing two sections about integration. These sections cover theorems which were developed by Leibniz in the years 1673 - 1674. Leibniz named his technique to derive the quadrature of a curve:

Leibniz: The Transmutation Theorem:

Seeking the area beneath a curve was a hot topic at that time and Leibniz called his discovery to calculate the area transmutation theorem. Working with figures containing similarities of triangles and infinitely small triangles he was able to calculate the area beneath curves similarly as shown in figures from that of Newton.

• As symbol for this process he took $\int$, an elongated S for summa. Earlier he had already introduced the symbol $dx$ for the differential of $x$ and he introduced the notation \begin{align*} \color{blue}{\int x\,dx} \end{align*} which is now familiar to us and which was called later by Pascal a very happy notation.

• In fact Leibniz, as is nicely shown by figures reproduced in Dunhams book could not only derive formulas to calculate the area beneath curves, he also found by insightful application of geometric relations the scheme for integration by parts \begin{align*} \color{blue}{\int_{a}^bf(x)g^{\prime}(x)\,dx=f(x)g(x)\Big|_{a}^b-\int_a^bg(x)f^{\prime}(x)\,dx} \end{align*} which was a great technique to simplify certain integrals and to solve further interesting problems this way.

Leibniz succeded in finding even more great relations. He applied the transmutation theorem and found a series which nowadays carries his name and is the main theme of the next section.

The Leibniz Series:

Using derivations as outstanding as those of Newton he was able to derive starting from a figure showing \begin{align*} (x-1)^2+y^2=1\qquad\qquad resp.\qquad\qquad x^2+y^2=2x \end{align*} and calculating the area beneath $y=\sqrt{2x-x^2}$ together with a quadratix $z=\sqrt{\frac{x}{2-x}}$, a function with a helper area beneath it for the wanted quadrature the famous Leibniz series. \begin{align*} \color{blue}{\frac{\pi}{4}}&\color{blue}{=1-\int_{0}^1\frac{z^2}{1+z^2}}\\ &=1-\int_{0}^1\left(z^2-z^4+z^6-z^8+\cdots\right)\,dz\\ &\color{blue}{=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots} \end{align*}

Notes:

• Of course to better appreciate all these derivations the figures presented in Dunhams book are highly useful to really see what's going on.

• One of the main sources about Newton was The Mathematical Works of Isaac Newton, Vol. 1 and Mathematical Papers of Isaac Newtwon, Vol. 2 by Derek Whiteside.

• One of the main sources about Leibniz was The Early Mathematical Manuscripts of Leibniz by J. M. Child.