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.

Plot or Argument

• David Eugene Smith's introduction

  Smith describes the 1906 discovery in Constantinople as fitting, since the West first received Archimedes' other works from that city, and praises Heiberg as the leading editor of Greek mathematics. He draws from the greeting new evidence of a friendship between Archimedes and Eratosthenes, stresses that Archimedes was primarily a discoverer, summarizes the propositions, and lists four ways the work adds to knowledge of Archimedes.

• The letter to Eratosthenes

  Archimedes restates the two theorems on the cylinder slice and the intersection of two cylinders that he had sent without proofs. He explains that, because Eratosthenes values a mathematical method of investigation, he will set down a mechanical method of discovery, notes that results found this way still need geometrical proof, credits Democritus and Eudoxus for the cone and pyramid, and hopes later investigators will use the method to find new propositions.

• The lemmas

  Eleven numbered assumptions follow, mostly on centres of gravity, drawn from Archimedes' own On the Equilibrium of Planes: the centre of gravity of a line, a triangle, a parallelogram, a circle, a cylinder, a prism and a cone, and a rule on sums of proportional magnitudes from On Conoids and Spheroids.

• Proposition I: the parabolic segment

  The segment of a parabola and a related triangle are imagined as filled with straight lines and hung from a lever so that each line of one balances the corresponding line of the other. The balance yields the result that the segment is four thirds of the triangle with the same base and height, which Archimedes says indicates the truth without proving it.

• Proposition II: sphere, cone and cylinder

  By weighing circular sections of a sphere and a cone against sections of a cylinder, Archimedes finds that the sphere is four times the cone whose base is its great circle and whose height is its radius, and that the cylinder circumscribing the sphere is one and a half times the sphere. Reasoning by analogy from the circle, he then perceives that the surface of the sphere is four times its great circle.

• Propositions III–X: other solids and centres of gravity

  The same method gives the volume of a spheroid and of a segment of a paraboloid, the centre of gravity of a paraboloid segment, of a hemisphere and of segments of a sphere and an ellipsoid, and a ratio for a segment of a hyperboloid. Archimedes leaves further cases aside because the method has been made clear.

• Propositions XI–XIV: the cylinder slice

  The final propositions treat the solid cut from a cylinder by a plane through the centre of one base and a side of the opposite square. Archimedes first finds by balancing that it equals one sixth of the prism, then begins a strict geometrical proof that inscribes and circumscribes figures made of prisms and argues by contradiction. The manuscript breaks off in Proposition XIV, and the proof of the two-cylinder theorem is lost.

Characters

• Archimedes, author

  Syracusan mathematician writing late in life to explain his method of discovery to a respected colleague.

• Eratosthenes, addressee

  Alexandrian scholar, geographer and head of the Library, whom Archimedes calls a capable scholar and teacher of philosophy.

• Democritus, earlier discoverer

  Atomist philosopher credited with first stating, without proof, that a cone is one third of a cylinder with the same base and height.

• Eudoxus, earlier prover

  Mathematician of Cnidus who first gave the rigorous demonstrations of Democritus' results on the cone and pyramid.

• J. L. Heiberg, discoverer and editor

  Danish philologist who identified and deciphered the palimpsest in Constantinople in 1906 and translated it into German.

• David Eugene Smith, introducer

  American historian of mathematics whose introduction precedes the English translation.

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.

  • Why does Archimedes say his mechanical argument indicates the result without proving it?
  • How does knowing the answer in advance make a proof easier to find?
  • Can you think of a case in your own studies where you were sure of an answer before you could show why it was true?

• 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.

  • Why does Archimedes mention Democritus and Eudoxus at all?
  • What does his hope that future investigators will find new propositions reveal about how he saw mathematics?
  • How should credit be shared between the person who first sees a truth and the person who first proves it?

• 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.

  • How does Archimedes use the idea of a balance to compare the size of two figures?
  • Does The Method support or complicate Plutarch's picture of Archimedes?
  • Why might physical intuition help in abstract reasoning?

• 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.

  • What is a palimpsest, and why were manuscripts reused?
  • What did Heiberg's discovery add to what was known about Archimedes and Eratosthenes?
  • What responsibilities do libraries and scholars have toward texts from the past?

Context

Archimedes worked in the third century BCE, when Greek mathematics had been organized by Euclid's Elements and Alexandria, under the Ptolemies, was the centre of learning. Greek geometers presented results in a strict synthetic form: definitions, postulates and proofs, often using the method of exhaustion associated with Eudoxus, which proves equalities of areas and volumes by showing that either inequality leads to a contradiction. Such proofs conceal how the results were found. The Method is the rare ancient text in which a mathematician explains his heuristic, a mechanical procedure of balancing slices of figures.

Composition

Smith's introduction argues from the cross-references that the treatise is one of Archimedes' late works. It survives only in the Archimedes Palimpsest, a tenth-century Byzantine copy of several of his treatises that was erased and overwritten with a liturgical text in the thirteenth century. The manuscript was listed in a Constantinople library catalogue in the 1890s, and Heiberg examined it there in 1906, publishing a German translation in the journal Hermes in 1907 and the Greek text in his edition of Archimedes. Lydia G. Robinson's English version of Heiberg's German appeared in The Monist in April 1909 and as a booklet from Open Court, and T. L. Heath published his own English version in 1912 as a supplement to The Works of Archimedes.

Reception

The discovery attracted wide attention, because the method of summing slices of a figure resembles the method of indivisibles used by Cavalieri in the seventeenth century and anticipates ideas later developed in the integral calculus. Smith's introduction already draws this comparison. The palimpsest disappeared from view in the twentieth century, reappeared at auction in 1998, and was conserved and imaged with modern techniques at the Walters Art Museum in Baltimore, which allowed scholars to read passages Heiberg could not. Scholars differ on how far Archimedes' reasoning involved infinitely many parts: some, including Reviel Netz, have argued that the recovered passages show him treating infinite collections directly, while others read the method as a heuristic that stays within Greek finitist practice.

Comparisons

• euclid-elements-book-i

  Euclid shows the synthetic style of Greek proof, from definitions and postulates to demonstrated propositions, that Archimedes followed in his published works; The Method lets readers see the exploratory reasoning that such finished proofs do not record.

• galileo-selected-writings

  Galileo admired Archimedes and, like him, joined mathematics with the study of weights, balances and motion; reading the two together shows how Archimedean mechanics informed the new physics of the seventeenth century.

• plutarch-lives

  Plutarch's Life of Marcellus describes Archimedes' war machines at Syracuse, his reported contempt for practical arts and his death; The Method, which uses mechanics to serve geometry, adds nuance to that portrait.

• aristotle-posterior-analytics-and-physics

  Aristotle's account of demonstrative science requires proof from first principles; Archimedes accepts that standard for proof while openly defending a separate, non-demonstrative path of discovery.

• discourse-on-method

  Descartes also wrote about method and wished that ancient geometers had revealed how they found their results; The Method is one ancient answer to his complaint.

Quotes

For Students

A reading companion for classical Christian high school students. See the student guide.

Reading Guide

Watch for

• The opening greeting and what it shows about Archimedes and Eratosthenes.
• Every place where Archimedes says a result has been found but not yet proved.
• How the image of a lever or balance is used to compare two figures.
• The credit given to Democritus and Eudoxus.
• The gaps in Propositions XI–XIV, which mark damage to the only surviving manuscript.

Key terms and names

Palimpsest

A manuscript whose original writing was scraped off so the parchment could be reused, with traces of the older text still visible beneath.

Centre of gravity

The point at which the whole weight of a body may be considered to act, so that the body balances when supported there.

Method of exhaustion

The Greek technique for proving areas and volumes by approximating a figure with inscribed and circumscribed figures and showing that any other answer leads to a contradiction.

Heuristic

A method that helps one discover a result without itself establishing that the result is true.

Parabolic segment

The region bounded by a parabola and a straight line cutting it.

Lemma

A preliminary proposition assumed or proved in order to be used in proving others.

Seminar Questions

1. Comprehension What two theorems does Archimedes say he had earlier sent Eratosthenes without proof?
2. Comprehension In Proposition II, what relation does Archimedes find between the sphere, the cone and the cylinder?
3. Comprehension How did The Method survive, and how was it rediscovered?
4. Interpretation Why does Archimedes refuse to treat his mechanical arguments as proofs, even when he is confident of the results?
5. Interpretation What does the letter suggest about Archimedes' respect for Eratosthenes and for his readers?
6. Interpretation Smith compares Archimedes to Cavalieri and Newton. What features of the method justify the comparison, and where might it be misleading?
7. Big Question Is the way we come to know something part of the knowledge itself, or only a ladder to be thrown away once the proof is found?

Faith Connections

• Wisdom of Solomon 11:20 (Scripture)

  'Thou hast ordered all things in measure and number and weight' was a favourite text for medieval and early modern Christians who studied geometry and mechanics; Archimedes' joining of weight and measure illustrates the order it praises.

• Proverbs 25:2 (Scripture)

  'It is the glory of God to conceal a thing: but the honour of kings is to search out a matter' fits a work devoted to the search for hidden mathematical truths, and a text that itself lay hidden beneath a prayer book.

• Augustine, On Christian Doctrine II.38 (Church Fathers)

  Augustine teaches that the truths of number are not invented by men but discovered, and that Christians may study them with profit; The Method speaks of mathematical results as discovered rather than made.

• Plutarch, Life of Marcellus 14–19 (Great Conversation)

  Plutarch's portrait of Archimedes' absorption in geometry and his death at Syracuse is the classic ancient account of the man.

Logic and Rhetoric

Central argument: Some geometrical truths can be found by imagining figures as composed of their parallel slices and balancing them on a lever against figures already known. Results found this way are not yet proved, because the mechanical argument is not a geometrical demonstration. Yet it is easier to prove what one already knows to be true. Therefore the mechanical method is a valuable aid to discovery, worth publishing so that others may find new results and then prove them rigorously.

Rhetorical device: argument from analogy. At the end of Proposition II Archimedes reasons that just as a circle equals a triangle whose base is its circumference and whose height is its radius, so a sphere equals a cone whose base is the sphere's surface and whose height is its radius, and from this he perceives the surface area (see quote 5).

Debate: Resolved: That a method of discovery deserves as much honour as a rigorous proof.

Memory Work

it is easier to establish a proof if one has in this way previously obtained a conception of the questions, than for him to seek it without such a preliminary notion.

It is true that this is not proved by what we have said here; but it indicates that the result is correct.

Writing Prompts

Essay: Explain Archimedes' mechanical method using Proposition I or II, and argue whether his distinction between discovering and proving still matters in science and mathematics.

Comparison with Elements, Book I: Compare a proposition from Euclid's Book I with Archimedes' Proposition I. What does each text show, and what does each hide, about how mathematical knowledge is reached?

Medical Lens

A treatise on geometry and mechanics with no medical content.

Creative Prompts

• Classroom

  Cut a cardboard parabolic segment and a triangle with the same base and height, weigh or balance them, and compare the result with Archimedes' four-thirds ratio. Then discuss why the experiment, like his method, falls short of a proof.

• Fiction

  Write a scene in Alexandria in which Eratosthenes opens the letter from Syracuse and tries to explain the method to a student at the Library.

• Essay

  Archimedes separates the discovery of a result from its proof. Using Propositions I and II, explain his method and argue whether a method of discovery deserves a place in a finished mathematical work.

• Dialogue

  Write a conversation between J. L. Heiberg in 1906 and the thirteenth-century scribe who reused the parchment, each explaining why the book mattered to him.

Further Reading

• The Works of Archimedes, T. L. Heath (ed. and trans.)

  The standard older English version; its 1912 supplement contains Heath's translation of The Method.

• Archimedes, E. J. Dijksterhuis

  A thorough study of Archimedes' mathematics, including the mechanical method.

• The Archimedes Codex, Reviel Netz and William Noel

  An account of the palimpsest's history and the modern project to read it.

• A History of Greek Mathematics, T. L. Heath