Zenodorus (mathematician)

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

Origin and lifetime

About the origin of Zenodorus nothing definite is known. Analysis of the frequency of the name in ancient times showed that he used in the Greek-speaking world only in Palestine and Syria, occasionally encountered in Cyrenaica and in Ptolemaic Egypt and was otherwise with the exception of Attica extremely rare.

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

Due to the circumstances mentioned Gerald J. Toomer preferred the possibility that the mathematician was Zenodorus Athenians effect, giving special attention to a family of the deme Lamptrai, in the name Zenodorus was hereditary.

Mentioned by ancient authors

Except in the biography of the Philonides Zenodorus is called with four other ancient authors by name: Theon of Alexandria quotes extensively from his - lost in the original - Treatise on isoperimetric figures, figures of equal size; Proclus writes Zenodorus have quadrangles with a hyperextended angle ( at a corner concave) as Hohlwinklige (Greek κοιλοδωνια ) refers; Simplicius mentions Zenodorus ' mathematical models to the area of figures of equal size and the volume of bodies of equal surface. In the treatise On Burning Mirrors of Diocles, from a long time and only excerpts were known only since the 1970s, an Arabic translation, is the author of a well-known Zenodorus the speech. He is, however, referred to there as an astronomer; Moreover, the reading of the name in the present Arab transcription is a matter of interpretation. An acquaintance with the mathematician Diocles Zenodorus would suit its widely assumed lifetime.

Zenodorus ' treatise on isoperimetric figures

As a single work of Zenodorus the treatise On isoperimetric figures ( Περὶ ἰσοπεριμέτρων σχημάτων ) is preserved; isoperimetric figures are those of equal size. Issue is the question of what geometric figure of all same scope covers the largest area and its equivalent in the spatial - also called isoperimetrisches problem. The treatise of the Zenodorus is the oldest known evidence on this issue.

In three ancient writings we find the lost original evidence. As most faithful reproduction applies in Book I of the commentary Theon of Alexandria on the Almagest; only when Theon also Zenodorus is named as the author. In the synagogue of Pappus and an introduction to the Almagest an anonymous authors find similar versions, so that it can be assumed that all three are based on the same source.

Importance of the issue in antiquity

In ancient Greece, was long before Zenodorus known that the size of an area can not be determined over its circumference, about the size of an island does not have the needed time in their circumnavigation. To these subjects context, it is also the problem of Dido. That same scope of all the figures the circle has the largest area, the ball at the same surface, the largest volume of all bodies, is simple intuition close and is also mentioned by ancient authors. This special property of circles and spheres in ancient times also underscores the otherwise justified special position of the two forms.

Proven records and recourse to Archimedes

The main proven by Zenodorus rates are for the layer:

• From the straight, regular, ie equilateral and equiangular polygons with the same scope that is with the larger number of corners of the larger ones.
• At the same circumference of the circle is greater than a rectilinear regular polygon.
• From rectilinear polygons with the same number of pages and the same amount is the largest equilateral and equiangular.

In addition, the paper contains an argument for the three-dimensional space, so that the ball is bigger than any body with the same surface.

Zenodorus used in his proof the theorem of Archimedes, that a rectangle of the circumference and the radius of a circle is twice as large as the circle.

Sets of Zenodorus are not sufficiently in several ways to prove the maximum property of the extensive circle equal in terms of area. On the one hand, it is assumed that the geometric figures and compared body are convex. When the bodies are used only body with divisible by four space index. A final evidence, including the existence of a solution for the maximum area isoperimetric figures was performed only in the 19th century; there were methods of integral geometry are used.

Later picking up the evidence Ganges

The evidence of Zenodorus are preserved in several medieval manuscripts. In his book Geometria speculativa Thomas Bradwardine shall submit a separate argument for the maximum property of the isoperimetric circle and the sphere with the same surface, but holds up upon the sequence of steps to the treatise of Zenodorus. Jakob Steiner led in the 19th century, Simple proofs of the isoperimetric main clauses; during one of his methods of proof, which become known as the Symmetrisierungsverfahren he pursued as Zenodorus the approach on polygons as comparative figures.