**Hypsicles of Alexandria** wrote a treatise on regular polyhedra. He is the author of what has been called Book XIV of Euclid's *Elements*, a work which deals with inscribing regular solids in a sphere.

What little is known of Hypsicles' life is related by him in the preface to the so-called Book XIV. He writes that Basilides of Tyre came to Alexandria and there he discussed mathematics with Hypsicles' father. Hypsicles relates that his father and Basilides studied a treatise by Apollonius on a dodecahedron and an icosahedron in the same sphere and decided that Apollonius's treatment was not satisfactory.

In the so-called Book XIV Hypsicles proves some results due to Apollonius. He had clearly studied Apollonius's tract on inscribing a dodecahedron and an icosahedron in the same sphere and clearly had, as his father and Basilides before him, found it poorly presented and Hypsicles attempts to improve on Apollonius's treatment.

Arab writers also claim that Hypsicles was involved with the so-called Book XV of the *Elements.* Bulmer-Thomas writes in [1] that various aspects are ascribed to him, claiming that either:-

... he wrote it, edited it, or merely discovered it. But this is clearly a much later and much inferior book, in three separate parts, and this speculation appears to derive from a misunderstanding of the preface to Book XIV.

Diophantus quotes a definition of polygonal number due to Hypsicles (see either [1] or [2]):-

If there are as many numbers as we please beginning from1and increasing by the same common difference, then, when the common difference is1,the sum of all the numbers is a triangular number; when2a square; when3,a pentagonal number[and so on]. And the number of angles is called after the number which exceeds the common difference by2,and the side after the number of terms including 1.

This says that, in modern notation, the *n*th *m*-agonal number is

n[2 + (n- 1) (m- 2)]/2.

We do not know for certain that Hypsicles wrote a text on polygonal numbers, but it is fairly certain that he did write such a text which has been lost. This work on polygonal numbers is related to the ideas on arithmetic progressions that appear in another work by Hypsicles, making it more likely that indeed Hypsicles had indeed done original work on this topic.

The work which involves arithmetic progressions is Hypsicles' *On the Ascension of Stars*. In this work he was the first to divide the Zodiac into 360°. He says (see [1] or [2]):-

The circle of the zodiac having been divided into360equal arcs, let each of the arcs be called a spatial degree, and likewise, if the time taken by the zodiac circle to return from a point to the same point is divided into360equal times, let each of the times be called a temporal degree.

Hypsicles considers two problems in this work [2]:-.

(i)

Given the ratio of the longest to the shortest day at any place, how long does it take any given sign of the zodiac to rise there?

(ii)How long does it take any given degree in a sign to rise?

Hypsicles makes a false assumption involving arithmetic progressions so that his results are wrong. Heath writes [2]:-

True, the treatise(if it really be by Hypsicles, and not a clumsy effort by a beginner working from an original by Hypsicles)does no credit to its author; but it is in some respects interesting...

The mistake which Hypsicles makes is to assume that the rising times form an arithmetical progression. Having made this assumption his results are correct and Neugebauer [4] certainly values this work much more highly than Heath does. In fact without the aid of the sine function and trigonometry it is hard to see how Hypsicles could have done better.

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

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