Hypsicles (Ancient Greek:Ὑψικλῆς; c. 190 – c. 120 BCE) was an ancientGreekmathematician andastronomer known for authoringOn Ascensions (Ἀναφορικός) and possibly the Book XIV ofEuclid'sElements. Hypsicles lived inAlexandria.[1]
Although little is known about the life of Hypsicles, it is believed that he authored the astronomical workOn Ascensions. The mathematicianDiophantus of Alexandria noted on a definition of polygonal numbers, due to Hypsicles:[2]
If there are as many numbers as we please beginning from 1 and increasing by the same common difference, then, when the common difference is 1, the sum of all the numbers is a triangular number; when 2 a square; when 3, a pentagonal number [and so on]. And the number of angles is called after the number which exceeds the common difference by 2, and the side after the number of terms including 1.
InOn Ascensions (Ἀναφορικός and sometimes translatedOn Rising Times), Hypsicles proves a number of propositions onarithmetical progressions and uses the results to calculate approximate values for the times required for thesigns of the zodiac to rise above thehorizon.[3] It is thought that this is the work from which the division of thecircle into 360parts may have been adopted[4] since it divides the day into 360 parts, a division possibly suggested byBabylonian astronomy,[5] although this is mere speculation and no actual evidence is found to support this.Heath 1921 notes, "The earliest extant Greek book in which the division of the circle into 360 degrees appears".[6]
This work by Hypsicles is believed to represent the earliest extant Greek text to use the Babylonian division of the zodiac into 12 signs of 30 degrees each.[7]
Hypsicles is more famously known for possibly writing the Book XIV of Euclid'sElements. The book may have been composed on the basis of a treatise byApollonius. The book continues Euclid's comparison ofregular solidsinscribed inspheres, with the chief result being that the ratio of the surfaces of thedodecahedron andicosahedron inscribed in the same sphere is the same as theratio of theirvolumes, the ratio being.[4]
Heath further notes, "Hypsicles says also that Aristaeus, in a work entitledComparison of the five figures, proved that the same circle circumscribes both the pentagon of the dodecahedron and the triangle of the icosahedron inscribed in the same sphere; whether this Aristaeus is the same as the Aristaeus of the Solid Loci, the elder (Aristaeus the Elder) contemporary of Euclid, we do not know."[6]
Hypsicles letter was a preface of the supplement taken from Euclid's Book XIV, part of the thirteen books ofEuclid's Elements, featuring a treatise.[1]
Basilides of Tyre, OProtarchus, when he came to Alexandria and met my father, spent the greater part of his sojourn with him on account of the bond between them due to their common interest in mathematics. And on one occasion, when looking into the tract written byApollonius (Apollonius of Perga) about the comparison of thedodecahedron andicosahedron inscribed in one and the same sphere, that is to say, on the question what ratio they bear to one another, they came to the conclusion that Apollonius' treatment of it in this book was not correct; accordingly, as I understood from my father, they proceeded to amend and rewrite it. But I myself afterwards came across another book published by Apollonius, containing a demonstration of the matter in question, and I was greatly attracted by his investigation of the problem. Now the book published by Apollonius is accessible to all; for it has a large circulation in a form which seems to have been the result of later careful elaboration.For my part, I determined to dedicate to you what I deem to be necessary by way of commentary, partly because you will be able, by reason of your proficiency in all mathematics and particularly in geometry, to pass an expert judgment upon what I am about to write, and partly because, on account of your intimacy with my father and your friendly feeling towards myself, you will lend a kindly ear to my disquisition. But it is time to have done with the preamble and to begin my treatise itself.
In ancient times it was not uncommon to attribute to a celebrated author works that were not by him; thus, some versions of Euclid'sElements 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, 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.)
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
