Zenodorus (mathematician)

Zenodorus ( ancient Greek Ζηνόδωρος ) was an ancient mathematician . He lived in the 2nd century BC. And wrote a treatise on the isoperimetric problem .

Origin and time of life

Nothing certain is known about the origin of Zenodorus. Analyzes of the frequency of the name in antiquity showed that in the Greek-speaking world it was only used in Palestine and Syria , occasionally found in Cyrenaica and Ptolemaic Egypt, and was otherwise extremely rare with the exception of Attica .

In a biography of the philosopher Philonides from Laodikeia , found on a papyrus from Herculaneum , a Zenodorus known to Philonides is mentioned in connection with visits to Athens . The identity with the mathematician in question is assumed and his lifetime is assumed to be the first half of the second century BC. It is certain that Zenodorus was younger than Archimedes , since he used his evidence.

Due to the circumstances mentioned, Gerald J. Toomer prefers the possibility that the mathematician Zenodoros was Athenian, and draws particular attention to a family from the Demos Lamptrai in which the name Zenodoros was hereditary.

Mentioned by ancient authors

In addition to the biography of Philonides, Zenodorus is named by four other ancient authors: Theon of Alexandria quotes in detail his - the original lost - treatise on isoperimetric figures, figures of the same size; Proclus writes that Zenodorus called squares with an overstretched angle (concave at one corner) as hollow-angled ones (Greek κοιλοδωνια); Simplikios mentions Zenodoros' mathematical explanations on the area of ​​figures of the same size and the volume of bodies of the same surface. In the work On Burning Mirror of Diocles , of which only excerpts were known for a long time and an Arabic translation only since the 1970s, there is talk of a Zenodorus known to the author. There he is, however, referred to as an astronomer; moreover, the reading of the name in the present Arabic transcription is a question of interpretation. An acquaintance of Diocles with the mathematician Zenodorus would fit in with his widely assumed lifetime.

Zenodorus' Treatise on Isoperimetric Figures

The only work by Zenodorus that has survived is the treatise On isoperimetric figures ( Περὶ ἰσοπεριμέτρων σχημάτων ); isoperimetric figures are those of the same size. The topic is the question of which geometric figure of all the same circumference comprises the largest area and its spatial correspondence - also known as the isoperimetric problem. The treatise of Zenodorus is the oldest known argument on this question.

The evidence lost in the original can be found in three ancient writings. The most faithful is the rendering in Book I of Theons of Alexandria commentary on the Almagest ; only Theon also mentions Zenodorus as the author. Similar versions can be found in the synagogue of Pappos and in an introduction to the Almagest by an anonymous author, so that it can be assumed that all three are based on the same source.

Importance of the subject in ancient times

In ancient Greece it was known long before Zenodoros that the size of an area could not be determined by its perimeter, for example the size of an island could not be determined by the time required to circumnavigate it. The problem of dido is also about this thematic context . The fact that of all figures of the same circumference the circle has the largest area, the sphere with the same area the largest volume of all bodies, is obvious from a simple view and is also addressed by ancient authors. This special property of the circle and sphere underlines the special position of the two forms, which was also established in other ways in antiquity.

Proven theorems and recourse to Archimedes

The most important theorems for the plane , proved by Zenodoros, are:

• Of the straight, regular , i.e. equilateral and equiangular , polygons with the same circumference, the one with the larger number of corners is the larger.
• With the same circumference, the circle is larger than a straight, regular polygon.
• Of rectilinear polygons with the same number of sides and the same circumference, the largest is equilateral and equiangular.

In addition, the treatise contains an argument for three-dimensional space, i.e. for the fact that the sphere is larger than all bodies with the same surface.

In his argument, Zenodorus uses Archimedes' theorem that a rectangle with the circumference and radius of a circle is twice as large as the circle.

In several respects, Zenodorus' theorems are insufficient to prove the maximum property of the circle of equal circumference with regard to the surface. On the one hand, it is assumed that the geometrical figures and bodies compared are convex . For the solids, only solids with a number of areas divisible by four are used. Final proof, including the existence of a solution for the maximum area of ​​isoperimetric figures, was not given until the 19th century; methods of integral geometry were used.

Later taking up the evidence

The evidence of Zenodorus is preserved in several medieval manuscripts. In his book Geometria speculativa , Thomas Bradwardine presents his own argumentation for the maximum property of the isoperimetric circle and the spherical surface with the same surface, but sticks to Zenodorus' treatise in the sequence of steps. Jakob Steiner performed simple proofs of the isoperimetric main theorems in the 19th century ; in one of his proof methods, which has become known as the symmetrization process, he, like Zenodorus, follows the approach using polygons as comparative figures.

literature

• Menso Folkerts : Zenodoros 1. In: The New Pauly (DNP). Volume 12/2, Metzler, Stuttgart 2002, ISBN 3-476-01487-8 , Sp. 736.
• Helmuth Gericke : On the history of the isoperimetric problem . In: Mathematical semester reports. To maintain the connection between school and university. Volume XXIX (1982), ISSN  0720-728X , pp. 160-187 (with evidence and modern history of the isoperimetric problem).
• Wilhelm Müller: The isoperimetric problem in antiquity - with a translation of the treatise of Zenodorus after Theon of Alexandria . In: Sudhoff's archive for the history of medicine and the natural sciences. Volume 37 (1953), ISSN 0931-9425, pp. 39-71 (with evidence and additional mathematical remarks).
• Anton Nokk ( arr .): Zenodorus' treatise on the isoperimetric figures, based on the excerpts that the Alexandrians Theon and Pappus have given us from the same, edited in German by Dr. Nokk . In: Program of the Grand Ducal Lyceum in Freiburg im Breisgau - as an invitation to the public exams , 1860, supplement, pp. 1–35. Digitized , resource of the Bayerische Staatsbibliothek (probably the most faithful reproduction of the evidence).
• Gerald J. Toomer : The Mathematician Zenodorus . In: Greek, Roman, and Byzantine Studies , Volume 13 (1972), No. 2, pp. 177-192. Digitized , PDF document from Duke University Libraries (on origin and dating).