**Hypsicles** (Greek : Ὑψικλῆς; c. 190 – c. 120 BCE) was an ancient Greek mathematician and astronomer known for authoring *On Ascensions* (Ἀναφορικός) and the Book XIV of Euclid's *Elements*. Hypsicles lived in Alexandria.^{ [1] }

Although little is known about the life of Hypsicles, it is believed that he authored the astronomical work *On Ascensions*. The mathematician Diophantus 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.

In *On Ascensions* (Ἀναφορικός and sometimes translated *On Rising Times*), 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.^{ [3] } It is thought that this is the work from which the division of the circle into 360 parts may have been adopted^{ [4] } since it divides the day into 360 parts, a division possibly suggested by Babylonian astronomy,^{ [5] } although this is a 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] }

Hypsicles is more famously known for possibly writing the Book XIV of Euclid's *Elements*. The book 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 .^{ [4] }

Heath further notes, "Hypsicles says also that Aristaeus, in a work entitled *Comparison 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 of Euclid's Elements, featuring a treatise.^{ [1] }

"Basilides of Tyre, O Protarchus, 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 by Apollonius (Apollonius of Perga) about the comparison of the dodecahedron and icosahedron 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."

- 1 2 Thomas Little Heath (1908). "The thirteen books of Euclid's Elements".
- ↑ Thomas Bulmer (1990). "Biography in Dictionary of Scientific Biography".Missing or empty
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(help) - ↑ Evans, J., (1998),
*The History and Practice of Ancient Astronomy*, page 90. Oxford University Press. - 1 2 Boyer (1991). "Euclid of Alexandria".
*A History of Mathematics*. pp. 130–131.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, 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.) - ↑ Boyer (1991). "Greek Trigonometry and Mensuration".
*A History of Mathematics*. 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

- 1 2 Thomas Little Heath (1921). "A history of Greek mathematics".

**Euclid**, sometimes called **Euclid of Alexandria** to distinguish him from Euclid of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry". He was active in Alexandria during the reign of Ptolemy I. His *Elements* is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics from the time of its publication until the late 19th or early 20th century. In the *Elements*, Euclid deduced the theorems of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and mathematical rigour.

In geometry, a **regular icosahedron** is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.

In geometry, an **icosidodecahedron** is a polyhedron with twenty (icosi) triangular faces and twelve (dodeca) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.

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In geometry, a **golden rectangle** is a rectangle whose side lengths are in the golden ratio, , which is , where is approximately 1.618.

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**Sir Thomas Little Heath** was a British civil servant, mathematician, classical scholar, historian of ancient Greek mathematics, translator, and mountaineer. He was educated at Clifton College. Heath translated works of Euclid of Alexandria, Apollonius of Perga, Aristarchus of Samos, and Archimedes of Syracuse into English.

**Apollonius of Perga** was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention of analytic geometry. His definitions of the terms ellipse, parabola, and hyperbola are the ones in use today.

**Theon of Alexandria** was a Greek scholar and mathematician who lived in Alexandria, Egypt. He edited and arranged Euclid's *Elements* and wrote commentaries on works by Euclid and Ptolemy. His daughter Hypatia also won fame as a mathematician.

A **regular polyhedron** is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.

**Pappus of Alexandria** was one of the last great Greek mathematicians of antiquity, known for his *Synagoge* (Συναγωγή) or *Collection*, and for Pappus's hexagon theorem in projective geometry. Nothing is known of his life, other than what can be found in his own writings: that he had a son named Hermodorus, and was a teacher in Alexandria.

In geometry, the **rhombic triacontahedron**, sometimes simply called the **triacontahedron** as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.

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A **rhombic enneacontahedron** is a polyhedron composed of 90 rhombic faces; with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim. The rhombic enneacontahedron is a zonohedron with a superficial resemblance to the rhombic triacontahedron.

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**Aristaeus the Elder** was a Greek mathematician who worked on conic sections. He was a contemporary of Euclid.

Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics and Babylonian mathematics. Trigonometry was also prevalent in Kushite mathematics. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata, who discovered the sine function. During the Middle Ages, the study of trigonometry continued in Islamic mathematics, by mathematicians such as Al-Khwarizmi and Abu al-Wafa. It became an independent discipline in the Islamic world, where all six trigonometric functions were known. Translations of Arabic and Greek texts led to trigonometry being adopted as a subject in the Latin West beginning in the Renaissance with Regiomontanus. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics and reaching its modern form with Leonhard Euler (1748).

A **regular dodecahedron** or **pentagonal dodecahedron** is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals. It is represented by the Schläfli symbol {5,3}.

**Basilides of Tyre** was a mathematician, mentioned by Hypsicles in his prefatory letter of Euclid's Elements, Book XIV. Barnes and Brunschwig suggested that Basilides of Tyre and Basilides the Epicurean could be the same Basilides.

- Boyer, Carl B. (1991).
*A History of Mathematics*(Second ed.). John Wiley & Sons, Inc. ISBN 0-471-54397-7. - Heath, Thomas Little (1981).
*A History of Greek Mathematics, Volume I*. Dover publications. ISBN 0-486-24073-8.

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