# Zenodorus

### Born: about 200 BC in Athens, Greece

Died: about 140 BC in Greece

We know little of **Zenodorus**'s life but he is mentioned in the Arabic translation of Diocles' *On burning mirrors* where it is stated [3]:-

*And when Zenodorus the astronomer came down to Arcadia and was introduced to us, he asked us how to find a mirror surface such that when it is placed facing the sun the rays reflected from it meet a point and thus cause burning.*

Toomer notes that his translation of *'when Zenodorus the astronomer came down to Arcadia and was introduced to us'* could, perhaps, be translated *'when Zenodorus the astronomer came down to Arcadia and was appointed to a teaching position there'*.

Before the discovery of the Arabic text of Diocles' *On burning mirrors*, Zenodorus was known to us mainly because of references to his treatise *On isometric figures* which is lost. There is another interesting source of information however. When Vesuvius erupted in 79 AD, Herculaneum together with Pompeii and Stabiae, was destroyed. Herculaneum was buried by a compact mass of material about 16 metres deep which preserved the city until excavations began in the 18^{th} century. Special conditions of humidity of the ground conserved wood, cloth, food, and in particular many papyri.
The papyri contain remarkable information and in particular there is a biography of the philosopher Philonides. This biography speaks of Zenodorus as a friend of Philonides and, although complete certainty is impossible, we can be confident that this reference to Zenodorus is to the mathematician described in this article. Two visits by Zenodorus to Athens are described in the biography.

Despite the loss of Zenodorus's treatise *On isometric figures*, we do know something of the results which it contained since Theon of Alexandria quotes a number of propositions from Zenodorus's work when he is giving his commentary on Ptolemy's *Syntaxis*. Pappus also made use of Zenodorus's *On isometric figures* in Book V of his own work and in fact a comparison with what Theon of Alexandria has presented shows that Pappus followed Zenodorus's presentation rather closely.

In *On isometric figures* Zenodorus himself follows the style of Euclid and Archimedes quite closely and he refers to results of Archimedes from his treatise *Measurement of a circle*.

Zenodorus studied the area of a figure with a fixed perimeter and the volume of a solid figure with fixed surface. For example he showed that among polygons with equal perimeter and an equal number of sides, the regular polygon has the greatest area.

He also showed that a circle is greater than any regular polygon of the same perimeter. To do this Zenodorus makes use of Archimedes result that the area of a circle is equal to that of a right-angled triangle of perpendicular side equal to the radius of the circle and base equal to the length of the circumference of the circle.

The treatise contains three-dimensional geometry results as well as two-dimensional. In particular he proved that the sphere was the solid figure of least surface area for a given volume.

**Article by:** *J J O'Connor* and *E F Robertson*

**April 1999**

**MacTutor History of Mathematics**

[http://www-history.mcs.st-andrews.ac.uk/Biographies/Zenodorus.html]