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The Method

The story

Archimedes wrote a book describing how he found his theorems — weighing imaginary slices of shapes on an imaginary balance until the truth tipped into view — and sent it to Eratosthenes at Alexandria. For two thousand years, that book was lost. Everyone knew it had existed; no one had read it since antiquity.

Then, in 1899, a scholar named Papadopulos Kerameus published a notice about a palimpsest — a parchment manuscript whose original writing had been scraped off and overwritten — of "mathematical content," quoting a few specimen lines. A Danish philologist, J. L. Heiberg, read those few lines and inferred that the manuscript must contain something by Archimedes. In 1906 he travelled to Constantinople, photographed the parchment, and read what remained beneath the overwriting.

Here is how the book survived. Sometime in the thirteenth century, a scribe needed parchment for a prayer book. He took a tenth-century copy of Archimedes, washed the mathematics off — and, in the words of Heath's 1912 supplement, the attempt was made "fortunately with only partial success." The prayers were written over the proofs. The book then sat for six hundred years as a Euchologion, a book of prayers, keeping the proofs silently beneath its devotions. Of 177 parchment leaves, only 29 were fully scrubbed clean; the rest held on, "tolerably legible with the aid of a magnifying glass."

And the recovered book was worth the wait. In it, Archimedes says plainly that his mechanical method of weighing slices did not furnish an actual demonstration — it made things clear first, and the geometry had to prove them afterwards. He credits Democritus, who first asserted the cone's volume a century and a half earlier "though he did not prove it," and gives him "no small share of the credit." And he bequeaths the method forward: some successor, he writes, "by means of the method when once established, will be able to discover other theorems... which have not yet occurred to me." They did. His balance-slices became Cavalieri's indivisibles, then the calculus, and eventually the machines that would read his own scraped pages.

The lesson

Discovery and proof are different acts, and they earn different credits. Archimedes files the finding and the showing separately, and gives credit to the man who only guessed. The guess without the proof still gets its share of the receipt. An honest record keeps both columns.

And the palimpsest is the archive's grace at its most radical: the proof survived because it was beneath a prayer book — a record kept for one purpose (devotion) preserving what no one meant to keep (the Method). Sometimes the safest place for a truth is under something more ordinary. The receipt precedes the record: Heiberg inferred the whole book from a few specimen lines in a catalogue notice.

The receipt