# How did Isaac Newton write the integral symbol?

Isaac Newton is known as the discoverer of the FTC (Fundamental Theorem of Calculus), so maybe he wrote the integral symbol and derivative symbol. I know he wrote the derivative symbol as $$\dot y$$ but I cannot find the integral symbol he wrote. How did Isaac Newton write the integral symbol?

• Mar 30, 2023 at 15:14
• As far as I think by Wikipedia, Newton's notation is like $\fbox{$x$}=\frac12x^2+\mathit{Const.}$. Mar 30, 2023 at 15:32
• @MIKANkankitsu I would like to know the answer to this also. The tall S $\int$ is agreed to have been invented by Leibniz, so Newton almost certainly didn't use it. Mar 30, 2023 at 15:48

Newton used both vertical bars ($$\overset{|}{x}$$) and rectangles ($$\boxed{x}$$) to denote integrals in his Quadratura curvarum published in 1704.

Here, the bar notation is used on the bottom of page 9 in the linked digital scan (multiple bars would represent multiple integrals):

This source states that the notation did not become popular because the bar could be misinterpreted as a "prime", and a rectangle was difficult to print at the time.

Additional information from post linked by @KurtG. in the comments:

• Newton used $$\overset{|}{x}$$ to indicate the quantity whos fluxion (a fluent was a changing value and a fluxion was its instantaneous rate of change) is $$x$$.

In fact, it seems this Wikipedia article sums everything up very clearly. Given $$y=f(t)$$, $$\overset{|}{y}=\square y=\boxed{y}=\int y\ \mathrm{d}t.$$

• See also Mauro Allegranza's comments to this post . Maybe you want to compile this all into a more complete answer? I cannot type formulas right now. Mar 30, 2023 at 18:47

The section II.4 Integral Calculus in Analysis by Its History by E. Hairer and E. Wanner begins with quotations, one of them from a letter by Newton:

Newton, letter to Keill, April 20, 1714: And whereas $$\mathrm{M^r}$$ Leibnits praefixes the letter $$\int$$ to the Ordinate of a curve to denote the Summ of the Ordinates or area of the Curve. I did some years before represent the same thing by inscribing the Ordinate in a square ...

My symbols therefore ... are the oldest in the kind.