Mathematics 1684

A New Method for Maxima and Minima

Gottfried Wilhelm Leibniz

A handful of symbols — dx, dy — that turned change itself into algebra.

Choose your version

In depth · the introduction

Before Leibniz, finding the steepness of a curve was a puzzle solved one curve at a time. He turned it into a recipe almost anyone could follow.

The big idea

Calculus is the mathematics of change. One half of it asks a simple-sounding question: how steep is a curve at a single point? For a straight ramp the answer is easy — the rise divided by the run. But real curves bend, so the steepness is different at every point.

Leibniz's move was to zoom in. Look at a tiny enough piece of any smooth curve and it looks straight. Call the tiny step sideways dx and the tiny rise that goes with it dy; then the steepness at that point is just dy divided by dx. His real gift was a short list of rules for combining these tiny pieces — so you never have to zoom in by hand. You just calculate.

How it came about

Leibniz was a lawyer, diplomat and philosopher who taught himself mathematics late and brilliantly. In Paris in the 1670s he worked out both halves of the calculus and, just as importantly, the clean notation — dx, dy, and later the integral sign ∫ — that we still use. In October 1684 he published it: six dense pages in the German journal Acta Eruditorum.

Across the Channel, Isaac Newton had reached an equivalent method some twenty years earlier but had kept it mostly to himself. When Leibniz published first, a bitter argument broke out over who deserved the credit. A Royal Society panel — quietly steered by Newton — declared Leibniz a plagiarist. Historians now agree the two men discovered it independently. And it is Leibniz's tidy symbols, not Newton's dots, that schoolchildren learn today.

Why it mattered

Once you can compute how fast anything changes, you can describe motion, heat, growth, electricity, even money. Calculus became the shared language of physics and engineering: the laws of nature are mostly written as equations about rates of change, and those equations are written with Leibniz's d. Almost no bridge, engine, spacecraft or economic model exists without it.

A way to picture it

Imagine driving up a winding hill at night, headlights showing only a metre of road ahead. Over that one lit metre the road looks like a straight ramp, and its steepness is just how much you climbed divided by how far you went forward — that is dy over dx. The shorter the lit stretch, the more exactly it tells you the steepness at that precise spot. Leibniz's calculus is the rulebook for that ratio as the lit stretch shrinks to nothing.

Where it sits

Descartes (1637) had married algebra to geometry, and Fermat had a method for tangents and maxima — but it was Leibniz, and independently Newton, who forged the general tool. From here the river runs straight to Newton's Principia, to Euler, and to every differential equation in this Library: Fourier's heat, Maxwell's fields, Schrödinger's wave. Whenever something in modern science changes, it is described in the language Leibniz wrote down here.

The original document

Original source text

G. W. Leibniz · Nova Methodus pro Maximis et Minimis · Acta Eruditorum (Oct. 1684): 467–473

Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illis calculi genus

(“A new method for maxima and minima, as well as tangents, which is impeded neither by fractional nor by irrational quantities, and a remarkable type of calculus for them.”)

Leibniz takes a curve referred to an axis, with ordinates v, w, y, z, and tangents VB, WC, YD, ZE meeting the axis at B, C, D, E. He then introduces his central device: a fixed but arbitrary increment, dx, against which every other small difference is measured.

Now some right line taken arbitrarily may be called dx, and the right line which shall be to dx, as v (or w, y, z, resp.) is to VB (or WC, YD, ZE, respect.) may be called dv (or dw, dy, dz, resp.), or the differentials.

Because dv stands to dx as the ordinate stands to its subtangent, the ratio dy/dx is, by definition, the slope of the tangent — and the little right triangle with legs dx and dy, its hypotenuse lying along the tangent, becomes the engine of the whole method.

From this single definition Leibniz states the rules of the calculus, in a form every student still learns: the differential of a constant is zero; d(ax) = a·dx; differentials carry through sums and differences, so d(z − y + w + x) = dz − dy + dw + dx; for a product, d(xv) = x·dv + v·dx; for a quotient, d(v/y) = (y·dv − v·dy)/y²; and for powers and roots, d(xⁿ) = n·xⁿ⁻¹·dx — and, he stresses, these hold whether the exponent is whole, fractional, or irrational.

He then reads meaning into the signs. Where the ordinates stop increasing and turn to decrease (or the reverse), the curve reaches a maximum or a minimum, and there the differential dy is nothing — zero. Where the curve passes from concave to convex, the point of inflection, it is the differences of the differences that change sign.

[ … ]

To show the method's reach, Leibniz closes by solving a problem posed by Florimond de Beaune — to find a curve whose subtangent is everywhere constant — and arrives at the logarithmic curve, a transcendental curve that ordinary algebra cannot capture. The same calculus, he notes, opens problems that had resisted every earlier approach.

Acta Eruditorum · Leipzig · October 1684