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.