Gottfried Leibniz · Mathematics
How Leibniz, independently of Newton, discovered the calculus - a method for handling the infinitely small, taming the twin problems of tangents and areas with a single symbolic machinery.
For two thousand years after Euclid, two problems resisted every method the geometers could bring to bear. The first was the problem of the tangent: given a curve, how steep is it at a single point? A straight line has one slope everywhere, but a curve bends, so its steepness changes from place to place - and asking for the slope ‘at a point’ seems to ask for the rise over the run when the run has shrunk to nothing, which threatens to become the meaningless ratio 0 divided by 0. The second was the problem of quadrature, or area: how much region lies under a curved line? The Greeks could find the area of a circle or a parabola by the heroic method of exhaustion, squeezing the answer between inscribed and circumscribed polygons, but each new curve demanded a fresh feat of ingenuity.
What Leibniz discovered in the 1670s - and what Newton had discovered a decade earlier by a different path - was that these two ancient problems are not separate at all. They are inverses of one another, two faces of a single operation, and both can be solved mechanically by one symbolic method. That method is the calculus. It was perhaps the single most consequential mathematical invention since the Greeks, and it remade physics, astronomy, and engineering in its image.
Leibniz’s method rested on a bold and slippery idea: the infinitesimal, a quantity smaller than any ordinary number you could name, yet not equal to zero. Imagine a curve, and on it two points so close together that the bit of curve between them is, to all intents, a straight line. Leibniz called the tiny horizontal gap between them dx (a ‘difference’ in x) and the tiny vertical gap dy. The slope of the curve at that point is then simply the ratio dy/dx - the rise over the run of an infinitely small triangle, what Leibniz called the ‘characteristic triangle’. Because dx and dy are vanishingly small, the slope they give is the slope at the point, not over a stretch.
For area, Leibniz imagined the region under a curve sliced into infinitely many infinitely thin vertical strips, each of width dx and height y, so each has area y·dx. The total area is the sum of all these slivers - written with his elongated S, ∫ y dx, the integral. Add up infinitely many infinitely thin pieces and you recover the whole. The infinitesimal let Leibniz turn the continuous - smooth curves, flowing areas - into something he could compute with, by treating the smooth as built from infinitely many infinitely small straight or flat pieces. It was logically dubious (what is a number smaller than every number yet not zero?), and it would take a century and a half to make rigorous - but it worked, with uncanny power.
Isaac Newton developed his version of the calculus - his ‘method of fluxions’ - around 1666, roughly a decade before Leibniz, but he did not publish it, circulating it only privately among friends. Leibniz developed his version independently in the mid-1670s and published it first, in 1684 and 1686. When the two systems became widely known to be equivalent, a bitter priority dispute erupted, pitting the Royal Society of London (championing Newton) against the Continental mathematicians (championing Leibniz). Newton’s partisans accused Leibniz of plagiarism, alleging he had seen Newton’s unpublished papers; the Royal Society, with Newton himself secretly drafting its supposedly impartial report, found against Leibniz. The quarrel poisoned the last years of Leibniz’s life and split European mathematics for a century.
The modern verdict is that the two men discovered the calculus independently. Newton came to it through physics - rates of change in time, the ‘fluxions’ of flowing quantities - while Leibniz came through geometry and the algebra of differences and sums. The deep irony is that Leibniz’s superior notation - the dy/dx and ∫ that we still use - proved so much clearer and more powerful than Newton’s dots-over-letters that British mathematics, clinging loyally to Newton’s inferior symbols, fell behind the Continent for a hundred years. In the contest of ideas Newton may have come first; in the contest of language, Leibniz won decisively, and it is his calculus we all learn today.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned how Leibniz invented the calculus by reasoning with infinitesimals - the differential dx and the integral ∫ . Explain what problems the calculus solved and what an infinitesimal is supposed to be.
Leads to Isaac Newton.
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