The infinitely small
In the late 1600s a revolution took place in mathematics. Scholars had long attempted to understand the relationships between numbers and geometric objects like lines, plane figures, and bodies. It should be possible to measure lengths, areas, and volumes. But there always has to be a unit to measure with. Sometimes this is not possible. It is said that one of Pythagoras' students who discovered in the 5th century B.C. that the length of a diagonal in a square is not measurable was drowned in the sea. Such a horrifying discovery could not be allowed to be passed on.
It was difficult to understand infinity, or rather the infinitely small. It was possible to divide a length a finite number of times, any number of times, but not an infinite number of times, making it infinitely small. Newton and Leibniz succeeded in constructing theories for the infinitely small, the so-called infinitesimal.
Newton's theory of fluxions
Newton regarded curves as paths of a moving point. This motion is dependent on time. Newton called a quantity that varies when moving (for example x) a fluent. He called the velocity at which the fluent was moving the fluxion. This is marked with a dot, as ẋ. In the infinitely brief time o, the point thus moves ẋo. We can get an idea of how Newton conceived of his calculations in his Method of Fluxions, which was first published nine years after his death. Pay particular attention to paragraphs 13–18 in the copy below.
Leibniz's differential calculus
Leibniz regarded curves as they were. The differentials dx or dy are infinitely small changes in the values of the variables x and y.
Equalities that were puzzling were x + dx = x and y + dy = y. The equals sign apparently assumes a new meaning together with differentials.
Guillaume de l’Hospital's Analyse des Infiniment Petits (published in 1696) was the first textbook on differential calculus. The book's first illustration shows Ap = x, PM = pR = y, Pp = MR = dx and mR = dy. It was highly criticized. Can we really see something that is infinitely tiny?