Archimedes

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Archimedes : biography

circa 287 BC – circa 212 BC
  • The Sand Reckoner
In this treatise, Archimedes counts the number of grains of sand that will fit inside the universe. This book mentions the heliocentric theory of the solar system proposed by Aristarchus of Samos, as well as contemporary ideas about the size of the Earth and the distance between various celestial bodies. By using a system of numbers based on powers of the myriad, Archimedes concludes that the number of grains of sand required to fill the universe is 8 in modern notation. The introductory letter states that Archimedes’ father was an astronomer named Phidias. The Sand Reckoner or Psammites is the only surviving work in which Archimedes discusses his views on astronomy.
  • The Method of Mechanical Theorems
This treatise was thought lost until the discovery of the Archimedes Palimpsest in 1906. In this work Archimedes uses infinitesimals, and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. Archimedes may have considered this method lacking in formal rigor, so he also used the method of exhaustion to derive the results. As with The Cattle Problem, The Method of Mechanical Theorems was written in the form of a letter to Eratosthenes in Alexandria.

Apocryphal works

Archimedes’ Book of Lemmas or Liber Assumptorum is a treatise with fifteen propositions on the nature of circles. The earliest known copy of the text is in Arabic. The scholars T. L. Heath and Marshall Clagett argued that it cannot have been written by Archimedes in its current form, since it quotes Archimedes, suggesting modification by another author. The Lemmas may be based on an earlier work by Archimedes that is now lost.

It has also been claimed that Heron’s formula for calculating the area of a triangle from the length of its sides was known to Archimedes. However, the first reliable reference to the formula is given by Heron of Alexandria in the 1st century AD.

Notes

a. In the preface to On Spirals addressed to Dositheus of Pelusium, Archimedes says that "many years have elapsed since Conon’s death." Conon of Samos lived , suggesting that Archimedes may have been an older man when writing some of his works.

b. The treatises by Archimedes known to exist only through references in the works of other authors are: On Sphere-Making and a work on polyhedra mentioned by Pappus of Alexandria; Catoptrica, a work on optics mentioned by Theon of Alexandria; Principles, addressed to Zeuxippus and explaining the number system used in The Sand Reckoner; On Balances and Levers; On Centers of Gravity; On the Calendar. Of the surviving works by Archimedes, T. L. Heath offers the following suggestion as to the order in which they were written: On the Equilibrium of Planes I, The Quadrature of the Parabola, On the Equilibrium of Planes II, On the Sphere and the Cylinder I, II, On Spirals, On Conoids and Spheroids, On Floating Bodies I, II, On the Measurement of a Circle, The Sand Reckoner.

c. Boyer, Carl Benjamin A History of Mathematics (1991) ISBN 0-471-54397-7 "Arabic scholars inform us that the familiar area formula for a triangle in terms of its three sides, usually known as Heron’s formula — k = √(s(s − a)(s − b)(s − c)), where s is the semiperimeter — was known to Archimedes several centuries before Heron lived. Arabic scholars also attribute to Archimedes the ‘theorem on the broken chord’ … Archimedes is reported by the Arabs to have given several proofs of the theorem."

d. "It was usual to smear the seams or even the whole hull with pitch or with pitch and wax". In Νεκρικοὶ Διάλογοι (Dialogues of the Dead), Lucian refers to coating the seams of a skiff with wax, a reference to pitch (tar) or wax.