What Archimedes Did Two Thousand Years Before Newton

Sometime around 230 BC, in the Greek-Sicilian city of Syracuse, the mathematician Archimedes wrote a letter to his colleague Eratosthenes — the same Eratosthenes who at that moment was running the Library at Alexandria and who had recently calculated the circumference of the Earth from the angle of a shadow in a well. The letter, several thousand words long, was titled On the Method of Mechanical Theorems. It was an unusual document for Greek mathematics. It was also, in retrospect, the most candid piece of mathematical exposition produced in the ancient world.

The letter described, in technical detail, the actual process by which Archimedes discovered his most famous geometrical results — the volumes of spheres, paraboloids, hyperboloids, the centers of gravity of curved figures, the surface areas of three-dimensional bodies. The process, he confessed, was not the elegant axiomatic deduction that he had used to publish the results in his other treatises. The actual discovery process involved imagining the curved figures as sums of infinitesimally thin slices, mentally weighing those slices on a balance, finding the unknown volume by comparing it to a known volume that would balance it.

In other words, Archimedes had been using something very close to integral calculus, in his head, two thousand years before Newton and Leibniz independently formulated it as a coherent mathematical system. And the reason no one knew about it for those two thousand years was that The Method was lost.

It was rediscovered in 1906, by a Danish classical philologist named Johan Ludvig Heiberg, as the underwriting of a medieval Byzantine prayer book in a monastery library in Constantinople. The text was readable — barely — from the palimpsest. Modern multispectral imaging recovered the rest. The recovered text shows that Archimedes had anticipated the central technique of integral calculus, had used it routinely, and had — for reasons that the letter to Eratosthenes makes explicit — chosen not to publish it as a rigorous mathematical method.

What the Method actually says

The opening of the letter, as recovered from the palimpsest, contains a passage that has become one of the most-quoted in the history of mathematics:

“Certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the said method did not furnish an actual demonstration. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.”

The sentence is a confession. Archimedes is admitting, to Eratosthenes specifically and to posterity by way of the letter, that he discovered his most famous geometrical results not by rigorous proof but by physical-mechanical analogy. The classical Greek mathematical tradition, founded by Eudoxus of Cnidus a century earlier, had set down the strict standard that geometrical results had to be proven by exhaustion — by demonstrating that they could not be either greater or less than the asserted value, using a sequence of inscribed and circumscribed approximating figures. The exhaustion method was rigorous. It was also, in practice, almost impossible to use as a discovery method. To prove by exhaustion that the volume of a sphere equals two-thirds the volume of its circumscribing cylinder, you had to already know that the volume was two-thirds — the proof confirmed; it did not reveal.

How did Archimedes already know? The letter explains. He imagined the sphere and the cylinder, in his head, as composed of infinitely many infinitesimally thin parallel slices — a sphere as a stack of circular discs of varying radii, a cylinder as a stack of identical circular discs. He then imagined the slices placed on a mechanical balance — the kind of lever-and-fulcrum balance the ancient world used for weighing produce — with the disc-slices from one figure on one side and the disc-slices from another figure on the other, hanging at different distances from the fulcrum. By calculating the distances at which they balanced, he could derive the relative volumes of the figures.

This is, in modern mathematical language, an integration. The slices are differential elements; the balance is a weighted sum; the distance from the fulcrum is the moment-arm by which each element contributes. The mechanical-balance metaphor is exactly the integral-of-a-function-multiplied-by-a-position from elementary calculus.

The recovered text of The Method contains fifteen propositions in which Archimedes uses this technique to derive specific results. The most spectacular is Proposition 14, in which he derives the volume of a parabolic cylinder by balancing it against a triangular prism using infinitely thin parallel slices. The reconstruction is unambiguous. He is doing calculus.

He then converts each of these mechanical derivations into a rigorous exhaustion proof for publication in his other treatises. The published works contain the rigorous proofs but not the mechanical discoveries. The Method contains both — the mechanical discovery, followed by an indication of how the rigorous proof would proceed. The published works were copied and preserved across the centuries. The Method, which was the discovery-tool rather than the publication-quality treatise, was apparently considered less essential and was less widely copied. It survived in exactly one Byzantine manuscript, and that manuscript was scraped clean and reused as a prayer book in the early thirteenth century.

If the Byzantine scraper had pressed slightly harder with his pumice stone, calculus would have to wait for Newton and Leibniz to be reinvented.

Why Archimedes thought he had to convert

The convention of Greek mathematics around the time Archimedes was writing was that mechanical reasoning — reasoning about physical objects, levers, balances, motions — was not a valid form of mathematical proof. Mathematics dealt with abstract magnitudes; the physical world was where those magnitudes happened to be instantiated, but the validity of a proof had to be established without recourse to physical intuition. This was the Eudoxean tradition. It was rigorous. It was also restrictive.

Archimedes accepted the convention. The mechanical reasoning of the Method was, he wrote to Eratosthenes, “useful for the discovery” but “not for the demonstration.” The published proofs in his On the Sphere and Cylinder, On Conoids and Spheroids, On the Equilibrium of Planes, and other treatises were therefore exhaustion proofs that made no reference to the mechanical reasoning that had led him to the results. The mechanical reasoning was a private technique, communicated to one mathematical colleague in a long letter, treated as a kind of methodological footnote rather than as a central contribution.

This is, in retrospect, a significant cultural choice. Archimedes had in his hands a method of mathematical discovery that was capable of yielding results that exhaustion could not easily reach. He treated it as second-class. The Greek mathematical tradition that followed him agreed: the published exhaustion proofs were what was studied, taught, and transmitted; the mechanical discovery method was not.

Newton and Leibniz, in the 1670s and 1680s, would face exactly the same methodological problem in the early development of calculus — that their methods worked but did not yet have a rigorous logical foundation comparable to Euclidean geometry. They published anyway, treating the methodological worry as a problem to be solved later rather than a reason to suppress the technique. The strict logical foundations of calculus took another two centuries (Cauchy, Weierstrass, late nineteenth century) to fully work out. By then calculus had been used to do most of the work of the Industrial Revolution.

Had Archimedes published The Method as the central text of his mathematical work rather than as a private letter — had the Greek mathematical tradition accepted mechanical reasoning as a legitimate source of mathematical results — the development of calculus might have begun in the third century BC. Whether this would have meaningfully altered subsequent scientific history is a counterfactual question that no historian of mathematics has yet answered to anyone’s satisfaction.

The closest thing to a verdict on the question comes from Sir Thomas Heath, the British classicist who in 1897 produced the standard English edition of Archimedes’s works — before the Method had been rediscovered. In a passage written shortly after the 1906 rediscovery, Heath wrote that he had revised his estimate of Archimedes upward. “If his other works show him as a great mathematician,” Heath wrote, “the Method shows him as a great mathematician who knew where he was going, and why.”

The Method survives in one manuscript. The manuscript is in Baltimore. The text Archimedes wrote to Eratosthenes can be read, in full, by anyone with an internet connection. The audience he wrote it for is two thousand two hundred and fifty years dead.