Tangents before the calculus

When, around 1684, Newton and Leibniz gave the calculus its systematic form, the problem of drawing the tangent to a curve had already been tackled with surprisingly modern methods. Two protagonists of this “prehistory” of the derivative are Pierre de Fermat and Isaac Barrow.

Fermat’s adequality

As early as around 1636 Fermat had a method for finding maxima, minima and tangents. The idea, which he called adaequalitas (a word borrowed from the Greek of Diophantus), consists in comparing the value of the function at $x$ and at a nearby point $x+E$, treating the two values as “almost equal” and finally setting $E=0$.

To find the maximum of the area $f(x)=bx-x^2$ of a rectangle of fixed perimeter, Fermat sets up the adequality $f(x+E)\approx f(x),$ that is $b(x+E)-(x+E)^2 \approx bx-x^2 .$ Expanding and cancelling the equal terms gives $bE-2xE-E^2\approx 0$; dividing by $E$, $b-2x-E\approx 0,$ and finally, setting $E=0$, one finds $x=\dfrac{b}{2}$. This is exactly the result we obtain today by setting the derivative $f'(x)=b-2x$ to zero: the step “divide by $E$, then set $E=0$” is, in embryonic form, the limit of the difference quotient.

Barrow’s differential triangle

Isaac Barrow, the first holder of the Lucasian chair at Cambridge and Newton’s teacher, in his Lectiones Geometricae (1670) constructed tangents by means of a small “differential triangle” with sides $a$ and $e$: infinitesimal increments of the ordinate and of the abscissa whose ratio $a/e$ gives the slope of the tangent. It is the very figure of our ratio $\Delta f/\Delta x$, and it directly paves the way to the derivative as a limit.

These methods, however, remained tied to the individual geometric problem: the leap made by Newton and Leibniz was to recognise that behind tangents, velocities and areas lay a single algorithm — the calculus — with general rules and an efficient notation (Boyer, Katz).

Links

Topics: Derivatives
Concepts: Tangent line
People: Pierre de Fermat · Isaac Barrow · Diophantus of Alexandria