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

Life and work

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.

On Ascensions

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 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]

Euclid's Elements

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 ${\sqrt {\tfrac {10}{3(5-{\sqrt {5}})}}}$ .^[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

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.

Notes

^ ^a ^b Thomas Little Heath (1908). "The thirteen books of Euclid's Elements". Cambridge, The University Press.
^ Thomas Bulmer (1990), Biography in Dictionary of Scientific Biography
^ Evans, J., (1998), The History and Practice of Ancient Astronomy, page 90. Oxford University Press.
^ ^a ^b 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, ${\scriptstyle {\sqrt {\frac {10}{3(5-{\sqrt {5}})}}}}.$ 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.)
^ 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
^ ^a ^b Thomas Little Heath (1921). "A history of Greek mathematics". Oxford, The Clarendon Press.
^ Montelle, Clemency (2016), "The Anaphoricus of Hypsicles of Alexandria", in Steele, John M., The Circulation of Astronomical Knowledge in the Ancient World, Time, Astronomy, and Calendars: Texts and Studies, 6, Leiden: Brill, pp. 287–315, ISBN 978-90-0431561-7

References

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.

External links

The mac-tutor biography of Hypsicles
Hypsicles, from Smith, Dictionary of Greek and Roman Biography and Mythology
Scan of Manitius' edition of On Ascensions at wilbourhall.org (German introduction, Greek text and Latin translation)

[Heath2-1] Thomas Little Heath (1908). "The thirteen books of Euclid's Elements". Cambridge, The University Press.

[2] Thomas Bulmer (1990), Biography in Dictionary of Scientific Biography

[3] Evans, J., (1998), The History and Practice of Ancient Astronomy, page 90. Oxford University Press.

[Boyer_Apocrypha-4] 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, ${\scriptstyle {\sqrt {\frac {10}{3(5-{\sqrt {5}})}}}}.$ 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.)

[5] 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

[Heath-6] Thomas Little Heath (1921). "A history of Greek mathematics". Oxford, The Clarendon Press.

[7] Montelle, Clemency (2016), "The Anaphoricus of Hypsicles of Alexandria", in Steele, John M., The Circulation of Astronomical Knowledge in the Ancient World, Time, Astronomy, and Calendars: Texts and Studies, 6, Leiden: Brill, pp. 287–315, ISBN 978-90-0431561-7

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[2]

[3]

[4]

[5]

[6]

[7]

@@ Line 1: / Line 1: @@
+{{Short description|Greek mathematician and astronomer}}
-'''Hypsicles''' ({{lang-grc-gre|Ὑψικλῆς}}; c. 190 – c. 120 BCE) was an ancient [[Greeks|Greek]] [[mathematician]] and [[astronomer]] known for authoring ''On Ascensions'' (Ἀναφορικός) and the Book XIV of [[Euclid's Elements|Euclid's ''Elements'']].
+'''Hypsicles''' ({{langx|grc|Ὑψικλῆς}}; c. 190 – c. 120 BCE) was an ancient [[Greeks|Greek]] [[mathematician]] and [[astronomer]] known for authoring ''On Ascensions'' (Ἀναφορικός) and possibly the Book XIV of [[Euclid's Elements|Euclid's ''Elements'']]. Hypsicles lived in [[Alexandria]].<ref name="Heath2">{{cite web|title=The thirteen books of Euclid's Elements|author=Thomas Little Heath|url=https://archive.org/stream/thirteenbookseu03heibgoog#page/n521/mode/2up/search/Antiphon|year=1908|publisher=Cambridge, The University Press }}</ref>
 == 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''). The mathematician [[Diophantus]] of Alexandria noted on a definition of polygonal numbers, due to Hypsicles:<ref>{{cite web|author=Thomas Bulmer|title=Biography in Dictionary of Scientific Biography|year=1990}}</ref>
+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:<ref>{{citation|author=Thomas Bulmer|title=Biography in Dictionary of Scientific Biography|year=1990}}</ref>
 {{Quote|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.}}
 ===On Ascensions===
-In this work, Hypsicles proves a number of propositions on [[arithmetic progression|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]].<ref>Evans, J., (1998), ''The History and Practice of Ancient Astronomy'', page 90. Oxford University Press.</ref> It is thought that this is the work from which the division of the [[circle]] into 360 [[Degree (angle)|parts]] may have been adopted<ref name="Boyer Apocrypha"/> since it divides the day into 360 parts, a division possibly suggested by [[History of astronomy|Babylonian]] astronomy,<ref>{{cite book|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|date=1991|chapter=Greek Trigonometry and Mensuration|pages=162|quote=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}}</ref> although this is a mere speculation and no actual evidence is found to support this. [[Thomas Little Heath|Heath]] 1921 notes, "The earliest extant Greek book in which the division of the circle into 360 degrees appears".<ref name="Heath">{{cite web|url=https://archive.org/stream/historyofgreekma01heat#page/418/mode/2up/search/Hypsicles|title=A history of Greek mathematics|year=1921|author=Thomas Little Heath}}</ref>
+In ''On Ascensions'' (Ἀναφορικός and sometimes translated ''On Rising Times''), Hypsicles proves a number of propositions on [[arithmetic progression|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]].<ref>Evans, J., (1998), ''The History and Practice of Ancient Astronomy'', page 90. Oxford University Press.</ref> It is thought that this is the work from which the division of the [[circle]] into 360 [[Degree (angle)|parts]] may have been adopted<ref name="Boyer Apocrypha"/> since it divides the day into 360 parts, a division possibly suggested by [[History of astronomy|Babylonian]] astronomy,<ref>{{cite book|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|date=1991|chapter=Greek Trigonometry and Mensuration|pages=162|quote=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}}</ref> although this is mere speculation and no actual evidence is found to support this. [[Thomas Little Heath|Heath]] 1921 notes, "The earliest extant Greek book in which the division of the circle into 360 degrees appears".<ref name="Heath">{{cite web|url=https://archive.org/stream/historyofgreekma01heat#page/418/mode/2up/search/Hypsicles|title=A history of Greek mathematics|year=1921|author=Thomas Little Heath|publisher=Oxford, The Clarendon Press }}</ref>
+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.<ref>Montelle, Clemency (2016), "The Anaphoricus of Hypsicles of Alexandria", in Steele, John M., The Circulation of Astronomical Knowledge in the Ancient World, Time, Astronomy, and Calendars: Texts and Studies, 6, Leiden: Brill, pp. 287–315, {{ISBN|978-90-0431561-7}}</ref>
-===Euclid's Elements===
-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 of Perga|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 [[volume]]s, the ratio being <math>\sqrt{\tfrac{10}{3(5-\sqrt{5})}}</math>.<ref name = "Boyer Apocrypha">{{cite book|url=https://archive.org/stream/AHistoryOfMathematics/Boyer-AHistoryOfMathematics#page/n145/mode/2up/search/Hypsicles|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|date=1991|chapter=Euclid of Alexandria|pages=130–131|quote=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, <math>{\scriptstyle\sqrt{\frac{10}{3(5-\sqrt{5})}}}.</math> 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.)}}</ref>
+===Euclid's ''Elements''===
-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 contemporary of Euclid, we do not know."<ref name="Heath"/>
+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 of Perga|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 [[volume]]s, the ratio being <math>\sqrt{\tfrac{10}{3(5-\sqrt{5})}}</math>.<ref name = "Boyer Apocrypha">{{cite book|chapter-url=https://archive.org/stream/AHistoryOfMathematics/Boyer-AHistoryOfMathematics#page/n145/mode/2up/search/Hypsicles|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|date=1991|chapter=Euclid of Alexandria|pages=130–131|quote=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, <math>{\scriptstyle\sqrt{\frac{10}{3(5-\sqrt{5})}}}.</math> 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.)}}</ref>
+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."<ref name="Heath"/>
-==Hypsicles letter==
-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.<ref>{{cite web|title=The thirteen books of Euclid's Elements|author=Thomas Little Heath|url=https://archive.org/stream/thirteenbookseu03heibgoog#page/n521/mode/2up/search/Antiphon|year=1908}}</ref>
-{{Quote|"[[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 of Perga|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."
+==Hypsicles letter==
-"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."}}
+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.<ref name="Heath2"/>
+{{Quote|[[Basilides of Tyre]], O [[Protarchos (mathematician)|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 of Perga|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.}}
-==See also==
-*[[Hypsicles of Alexandria]]
 == Notes ==
-{{reflist}}
+{{reflist|4}}
 == References ==
@@ Line 32: / Line 32: @@
  | title=A History of Mathematics
  | edition=Second
- | publisher=John Wiley & Sons, Inc.
+ | publisher=[[John Wiley & Sons, Inc.]]
  | date=1991
  | isbn=0-471-54397-7
@@ Line 48: / Line 48: @@
 == External links ==
 *[http://www-history.mcs.st-andrews.ac.uk/Biographies/Hypsicles.html The mac-tutor biography of Hypsicles]
-*[http://ancientlibrary.com/smith-bio/1649.html Hypsicles, from  Smith, Dictionary of Greek and Roman Biography and Mythology]
+*[https://web.archive.org/web/20121017003524/http://www.ancientlibrary.com/smith-bio/1649.html Hypsicles, from  Smith, Dictionary of Greek and Roman Biography and Mythology]
+* [https://wilbourhall.org/index.html#hypsicles Scan of Manitius' edition of ''On Ascensions''] at wilbourhall.org (German introduction, Greek text and Latin translation)
 {{Ancient Greek astronomy}}
@@ Line 60: / Line 62: @@
 [[Category:190s BC births]]
 [[Category:120s BC deaths]]
+[[Category:2nd-century BC mathematicians]]

v t e Ancient Greek astronomy
Astronomers	Aglaonice Agrippa Anaximander Andronicus Apollonius Aratus Aristarchus Aristyllus Attalus Autolycus Bion Callippus Cleomedes Cleostratus Conon Eratosthenes Euctemon Eudoxus Geminus Heraclides Hicetas Hipparchus Hippocrates of Chios Hypsicles Menelaus Meton Oenopides Philip of Opus Philolaus Posidonius Ptolemy Pytheas Seleucus Sosigenes of Alexandria Sosigenes the Peripatetic Strabo Thales Theodosius Theon of Alexandria Theon of Smyrna Timocharis
Works	Almagest (Ptolemy) Catasterismi (Eratosthenes) On Sizes and Distances (Hipparchus) On the Sizes and Distances (Aristarchus) On the Heavens (Aristotle)
Instruments	Antikythera mechanism Armillary sphere Astrolabe Dioptra Equatorial ring Gnomon Mural instrument Triquetrum
Concepts	Callippic cycle Celestial spheres Circle of latitude Counter-Earth Deferent and epicycle Equant Geocentrism Heliocentrism Hipparchic cycle Inferior and superior planets Metonic cycle Octaeteris Solstice Spherical Earth Sublunary sphere Zodiac
Influences	Babylonian astronomy Egyptian astronomy
Influenced	Medieval European science Indian astronomy Medieval Islamic astronomy

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