Abstract
We consider several appearances of the notion of convexity in Greek antiquity, more specifically in mathematics and optics, in the writings of Aristotle, and in art.
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Notes
- 1.
Timaeus, [23], 53C-55C.
- 2.
The Conics, Book IV, [3, p. 172].
- 3.
The Conics, Book IV, [3, p. 178].
- 4.
The Conics, Book IV, [3, p. 183].
- 5.
The quotations of Archimedes are from Heath’s translation [13].
- 6.
“Terminated surfaces”, in Heath’s translation.
- 7.
In this and the next quotes, since we follow Heath’s translation, we are using the word “extremities”, although the word “boundary” would have been closer to what we intend in modern geometry.
- 8.
We are using modern terminology.
- 9.
Ptolemy’s proof with the reference to Heiberg’s edition is quoted in Heath’s edition of Euclid [14, Vol. 2 p. 225].
- 10.
Actually, in the cave passage ([22], Book VII, 514a–521d), not only images are distorted because the walls are not planar, but also one sees only shadows, apparent contours. Thom, in his Esquisse d’une sémiophysique ([34, p. 218] of the English translation) sees there the mathematical problem of reconstructing figures from their apparent contours.
- 11.
The Latin word focus means fireplace, which led to the expression “burning mirror.”
- 12.
The name refers to Ibn al-Haytham, the Arab scholar from the Middle Ages known in the Latin world as Alhazen, a deformation of the name “Al-Haytham.” Ibn al-Haytham is especially famous for his treatise on Optics (Kitāb al-manāzir), in seven books (about 1400 pages long), which was translated into Latin at the beginning of the thirteenth century, and which was influential on Johannes Kepler, Galileo Galilei, Christiaan Huygens and René Descartes, among others. An important part of what survives from his work in geometry and optics was translated and edited by Rashed [27, 28]. Ibn al-Haytham is the author of an “intromission” theory of vision saying that it is the result of light rays penetrating our eyes, contradicting the theories held by Euclid and Ptolemy who considered, on the contrary, that vision is the result of light rays emanating from the eye (“extramission” theory). It is possible though that Euclid, as a mathematician, adhered to the theory where visual perception is caused by light rays traveling along straight lines emitted from the eye that strike the objects seen, in order to develop his mathematical theory of optics as an application of Euclidean geometry. This also explains the fact that Euclid’s optics does not include any physiological theory of vision, nor any physical theory of colors, etc. Needless to say, besides this rough classification into an intromission theory and an extramission theory of light, there is a large amount of highly sophisticated and complex theories of vision and of light that were developed by Greek authors, which were related to the various philosophical schools of thought, and at the same time to the mathematical theories that were being developed.
- 13.
See Footnote 12.
- 14.
Au commencement de 1669 nous voyons Huygens absorbé par la mathématique. Il s’occupa du problème d’Alhazen. C’est là un des problèmes dont il a toujours eu l’ambition de trouver, par les sections coniques, la solution la plus élégante.
- 15.
Vitellion is the name of a thirteenth-century mathematician who edited works of Alhazen on optics.
- 16.
Nicomachean Ethics [6], 1102a-30.
- 17.
Meteorology [5], 350a10.
- 18.
Physics [8], 217a30-b5.
- 19.
Physics [8], 222b1.
- 20.
Mechanical problems [9], 847b.
- 21.
Mechanical problems [9], 848a.
- 22.
On the Gait of Animals [4], 704a15.
- 23.
On the Gait of Animals [4], 712a.
- 24.
Nel mezzo del cammin di nostra vita
mi ritrovai per una selva oscura
ché la diritta via era smarrita.
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Papadopoulos, A. (2019). Convexity in Greek Antiquity. In: Dani, S.G., Papadopoulos, A. (eds) Geometry in History. Springer, Cham. https://doi.org/10.1007/978-3-030-13609-3_4
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