The Method (Geometrical Solutions Derived from Mechanics)

Summary

The Method, known in Greek as the Ephodos and translated in the library text as Geometrical Solutions Derived from Mechanics, is a letter from Archimedes of Syracuse to Eratosthenes, the head of the Library at Alexandria. In it Archimedes does something almost unique in ancient mathematics: he explains how he found his results before he proved them. The work was lost for centuries and survives only in a single tenth-century Byzantine manuscript that was scraped and overwritten as a prayer book in the thirteenth century. The Danish philologist J. L. Heiberg examined this palimpsest in Constantinople in 1906, read most of the faded text with a magnifying glass, and published it with a German translation. The library text is Lydia G. Robinson's English version of Heiberg's German, printed in The Monist and by Open Court in Chicago in 1909, with an introduction by David Eugene Smith of Teachers College, Columbia University. The letter opens by recalling two theorems Archimedes had earlier sent Eratosthenes without proofs: that a certain slice cut from a cylinder inscribed in a square prism is one sixth of the prism, and that the solid common to two cylinders inscribed in a cube is two thirds of the cube. He then announces a peculiar method by which certain mathematical questions may be investigated by means of mechanics. The method imagines a figure as made up of all its parallel slices and balances these slices, as on a lever, against the slices of a figure whose size and centre of gravity are known. Archimedes insists that this is a means of discovery, not a proof, but argues that it is easier to find a proof once one already knows what is true, and gives credit to Democritus for stating the volume of the cone before Eudoxus proved it. After a list of lemmas on centres of gravity, Proposition I shows by weighing that a segment of a parabola is four thirds of the inscribed triangle, followed by Archimedes' own warning that this has been conjectured rather than proved. Proposition II finds that a sphere is four times a cone on its great circle, and that the circumscribing cylinder is one and a half times the sphere, the result Archimedes asked to have carved on his tomb; he adds how this led him to the surface of the sphere. Propositions III to X treat spheroids, paraboloids and hyperboloids and their centres of gravity. Propositions XI to XIV return to the cylinder slice, giving a mechanical argument and then parts of a rigorous geometrical demonstration, broken off by gaps in the manuscript. The treatise is brief, and the stored text, prepared from Gutenberg's typeset source, includes geometrical notation and Smith's introductory essay.

Top Lessons

• Discovery and proof are different tasks

  Archimedes distinguishes the way a truth is found from the way it is established. Weighing imaginary slices showed him that a parabolic segment is four thirds of a triangle, but he says plainly that this has been conjectured rather than proved, and he supplies a geometrical demonstration elsewhere. Good inquiry uses both: a fruitful way of seeing and a strict way of checking.

• Giving credit to predecessors

  Archimedes says that no little credit is due to Democritus, who stated the volume of the cone without proof, even though Eudoxus was the first to demonstrate it. He treats insight and rigour as both worthy of honour, and he writes so that later mathematicians can build on his method.

• Mathematics and the physical world

  Plutarch reports that Archimedes despised practical arts, yet The Method uses the lever and the centre of gravity, ideas from mechanics, to discover truths of pure geometry. Smith's introduction draws attention to this link between pure and applied mathematics.

• A lost book recovered

  The only copy of The Method survived because a thirteenth-century scribe reused the parchment for a prayer book. Heiberg recognized the older writing underneath and read it in 1906, so a work lost since the Middle Ages returned to mathematics. Knowledge depends on copying, preservation and the patient labour of scholars.

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Medical Lens

A treatise on geometry and mechanics with no medical content.