# Hypsicles

Hypsicles (Ancient Greek: Ὑψικλῆς; ca. 190 BCE – ca. 120 BCE) was an ancient Greek mathematician and astronomer known for authoring On Ascensions (Ἀναφορικός) and the spurious Book XIV of Euclid's Elements.

## Life and work

Although little is known about the life of Hypsicles, it is believed that he authored the astronomical work On Ascensions (Ἀναφορικός and sometimes translated On Rising Times). In this work, Hypsicles proves a number of propositions on arithmetical progressions and uses the results to calculate approximate values for the times required for the signs of the zodiac to rise above the horizon.[1] It is thought that this is the work from which the division of the circle into 360 parts may have been adopted[2] since it divides the day into 360 parts, a division possibly suggested by Babylonian astronomy.,[3] although this is a mere speculation and no actual evidence is found to support this.

Hypsicles is more famously known for possibly writing the apocryphal Book XIV of Euclid's Elements. The spurious Book XIV may have been composed on the basis of a treatise by Apollonius. The book continues Euclid's comparison of regular solids inscribed in spheres, with the chief result being that the ratio of the surfaces of the dodecahedron and icosahedron inscribed in the same sphere is the same as the ratio of their volumes, the ratio being $\sqrt{\tfrac{10}{3(5-\sqrt{5})}}$.[2]

## Notes and references

1. ^ Evans, J., (1998), The History and Practice of Ancient Astronomy, page 90. Oxford University Press.
2. ^ a b Boyer (1991). "Euclid of Alexandria". pp. 118–119. "In ancient times it was not uncommon to attribute to a celebrated author works that were not by him; thus, some versions of Euclid's Elements include a fourteenth and even a fifteenth book, both shown by later scholars to be apocryphal. The so-called Book XIV continues Euclid's comparison of the regular solids inscribed in a sphere, the chief results being that the ratio of the surfaces of the dodecahedron and icosahedron inscribed in the same sphere is the same as the ratio of their volumes, the ratio being that of the edge of the cube to the edge of the icosahedron, that is, ${\scriptstyle\sqrt{\frac{10}{3(5-\sqrt{5})}}}.$ It is thought that this book may have been composed by Hypsicles on the basis of a treatise (now lost) by Apollonius comparing the dodecahedron and icosahedron. (Hypsicles, who probably lived in the second half of the second century B.C., is thought to be the author of an astronomical work, De ascensionibus, from which the division of the circle into 360 parts may have been adopted.)"
3. ^ Boyer (1991). "Greek Trigonometry and Mensuration". p. 162. "It is possible that he took over from Hypsicles, who earlier had divided the day into 360 parts, a subdivision that may have been suggested by Babylonian astronomy)"