# Archimedes

‘Do not disturb my circles’

The famous and fitting last words of Archimedes of Syracuse.

Edward Worth’s copy of the Opera of Archimedes was published by the famous Venetian Aldine press in 1558 and contains two books. The first contains the work of Archimedes of Syracuse (c.287 BC-c.212 BC), in particular his works in geometry, which is split into five separate pieces of writing conveying propositions on various shapes and other mathematical ideas. The second part of the publication is the commentary on the work of Archimedes by Federico Commandino (1506-1575) who also translated the text (O’Connor and Robertson, 2002).

Archimedes, Opera non nulla à Federico Commandino Vrbinate nuper in Latinum conuersa, et commentariis illustrata (Venice, 1558), title page.

Archimedes was better known historically for his work in engineering, for his inventions such as his screw (O’Connor and Robertson, 1999) or for his anecdotal story concerning the discovery of his principle (BBC, 2014). Unlike other Greek mathematicians who preceded him, most of the written work of Archimedes did not survive. Though this publication was written in Latin, the works of Archimedes would have been originally translated from the Doric Greek, the dialect of ancient Syracuse. Most translations of texts into Latin would have taken place during and after the medieval period when Latin became a universal language due to the spread of Christianity. Although it mainly contains text, the writing of Archimedes does contain some very familiar illustrations.

Archimedes, Opera non nulla à Federico Commandino Vrbinate nuper in Latinum conuersa, et commentariis illustrata (Venice, 1558), p. 1.

Roughly translating as ‘dimensions of the circle’, Circvli Dimensio contains only three propositions. Within the writing of Archimedes there are attempts to give an approximate value for π, a number in which the Greeks had a particular interest. Proposition I states that ‘The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference, of the circle.’

Archimedes proves this using the method of exhaustion. Proposition II simply states ‘The area of a circle is to the square on its diameter as 11 (‘XI’) to 14 (‘XIIII’).’ In other words, the area of a circle is equal to  where Archimedes estimates  . The proof relies heavily on the outcome of proposition III which uses approximations of the square root of three to pinpoint the value of π to be between 3.1408 and 3.1429 (Shuttleworth, 2010), a very accurate approximation considering the time in which it was calculated.

The second piece of writing in this collection is believed to have been written around 225 BC and contains 28 propositions concerning spirals.

Archimedes, Opera non nulla à Federico Commandino Vrbinate nuper in Latinum conuersa, et commentariis illustrata (Venice, 1558), p. 10.

In the subsequent books, Archimedes goes on to discuss the quadrature of the parabola, conoids and spheroids. In the final text, Archimedes poses an interesting question: if you were to fill the space of the universe with sand what would be the number of grains? He references Aristarchus, a Greek astronomer who considered the universe to be a set of bodies orbiting around the sun (O’Connor and Robertson, 1999). This is despite the proliferation at the time of the belief that the planetary bodies orbited the Earth (the Ptolemaic system), a belief shared by lauded astronomers such as Ptolemy and Aristotle.

Sources

BBC, History of Archimedes (2014).
O’Connor, J.J. and E.F. Robertson, Archimedes, MacTutor History of Mathematics (University of St Andrews, 1999).
O’Connor, J.J. and E.F. Robertson, Frederico Commandino, MacTutor History of Mathematics (University of St Andrews, 2002).
Shuttleworth, M., 2010, Archimedes, explorable.com.

Text: Claire Maguire and Fionnán Howard