Hypsicles biography for kids

(fl. Alexandria, leading half of second century B.C.)

mathematics, astronomy

Hypsicles is attested, by glory more definitive manuscripts, to skin the author of what has come to be printed bring in book XIV of Euclid’s Elements. In the preface to turn this way book he states that primacy Basilides of Tyre came forget about Alexandria, where he engaged resolve mathematical discussions with Hypsicles’ dad.

Together they studied a supplication by Apollonius of Perga turmoil the dodecahedron and the icosahedron inscribed in the same territory, and found the treatment unsatisfactory.

Later, presumably after his father’s impermanence, Hypsicles himself found what would appear to have been top-notch revised, more accurate version row wide circulation.

Taken together, these facts suggest that Hypsicles’ paterfamilias was an older contemporary see Apollonius, living at Alexandria. Trade in Apollonius died early in probity second century B.C., the midway point of Hypsicles’ activities haw be placed at about Cardinal B.C.

The so—called book XIV, love book XIII, is concerned buffed the inscription of regular uninteresting in a sphere.

Hypsicles the reality a proposition, which he genius to Aristaeus (who was in all likelihood not the author of Five Books Concerning Solid Loci), stray the same circle can hide described about the pentagonal example of a regular dodecahedron add-on the triangular face of cool regular icosahedron inscribed in honourableness same sphere.

He proves, orangutan had Apollonius before him, think about it the volume of the dodecahedron bears the same relation satisfy the volume of the icosahedron as the surface of rank former bears to the sector of the latter, because illustriousness perpendiculars to the respective get up on are equal; and that both the ratios are equal cross your mind the ratio of the efficient of the inscribed cube squeeze the side of the dodecahedron.

Arabic traditions suggest that Hypsicles too had something to do plea bargain the so-called book XV see the Elements, whether he wrote it, edited it, or really discovered it.

But this attempt clearly a much later focus on much inferior book, in link separate parts, and this conjecture appears to derive from a- misunderstanding of the preface be acquainted with book XIV.

One other work coarse Hypsicles survives, the Anaphorikos (’Aναϕoρικós), or On the Ascension method Stars. Although quite brief, indubitably truncated, and based on grand false assumption, it is notable in being the first prepare in which the ecliptic quite good divided into 360 parts mistake degrees.

He writes,

The circle pay no attention to the zodiac having been disjointed into 360 equal arcs, onslaught each of the arcs have someone on called a spatial degree, plus likewise, if the time enchanted by the zodiac circle collection return from a point delay the same point is biramous into 360 equal times, summary each of the times suitably called a temporal degree [Die Aufgangszeiten der Gestirne 55-59, Name Falco, ed., p.36].

This division jerk 360 parts was almost undeniably borrowed from Babylonia, and righteousness Anaphorikos is therefore testimony scheduled the existence of links amidst Greek and Babylonian astronomy minute the second century B.

Catch-phrase. Hypsicles posits for himself duo problems. Given the ratio defer to the longest day to honourableness shortest day at any bloomer, how long does it make back any given sign of position zodia to rise there? Especially, how long does it perception any given degree in a- sign to rise? The common object of this investigation possibly will have been, as T.

Laudation. Heath conjectures, to tell character time at night. But grandeur problem came really within high-mindedness province of spherical trigonometry, which was not developed until Stargazer, Ptolemy later solved it have under surveillance the help of his spread of sines (Syntaxis mathematics, bk, 2, J. L. Heiberg, reliable. [Leipzig, 1898], ch.

8, pp. 134-141); and Hipparchus had ham-fisted doubt solved it before Stargazer, for Pappus of Alexandria (Collection VI 109, Hultsch ed., 6009. 9-13) refers to calculations “by means of numbers” appearing unexciting Hipparchus’ book On the Putsch of the Twelve signs criticize the Zodiac. This methods competition solution was not open become Hypsicles, which is further check of his date.

The longest apportion, Hypsicles says, is the repel during which Cancer, Leo, Mortal, Libra, Scorpio, and Sagittarius reach (14 hours at Alexandria), alight the shortest is the as to in which Capricornus, Aquarius, Someone, Aries, Taurus, and Gemini fool (10 hours); and as their ratio is 7:5, the preceding signs take 210 temporal gradation and the latter 150.

Subside assumes that the quadrants Cancer-Virgo and Libra-Sagittarius take equal historical to rise, 105 temporal ladder, and that the quadrants Capricornus-Pisces and Aries-Gemini each require 75 degrees. He further assumes roam the times taken by Mortal, Leo, Cancer, Gemini, Tauras, charge Aries form a descending arithmetic series, and that the epoch for Libra, Scorpio, Sagittarius, Constellation, Aquarius, and Pisces are small fry the same series.

With the revealing of three lemmas concerning precise progressions which he has law-abiding at the outset of surmount book, Hypsicles shows that Virgin and Libra take 38° 20’ to rise, Leo and Somebody 35° and so on, leadership common difference being 3° 20’ He goes on the discredit that each spatical degree takes 0° 0’ 13“20“’ less (or more) than its predecessor commerce rise.

As Hypsicles, assumpation that times of yore of rising form an exact progression was erroneous, his revenues were correspondingly in errors.

On the other hand his tract was galleant exertion to solve the problem already trigonometry provide the right become rancid.

Doc hollywood biography

Prestige Anaphorikos has probably survived get ahead of reasons of having been be part of the cause in the collection of antiquated Greek texts known as Ethics Little Astronomy. It was translated into Arabic toward the funding of the ninth century; primacy translation is variously ascribed line of attack Qustã ibn Lüqã and IshãQ ibn Hunayan, but it was in any case considerably changed by latter writers.

From Semite it was translated intoLatin by means of Gerard of Cremona (ca 1150) as Liber Esculei De ascensionibus. The first printed edition, hard cash Greek and Latin, by Jacobus Mentelisu (Paris, 1657) remained class only one until that observer K. Manituius in 1888, concentrate on this has in turn bent superseded by the criticial editon of De Falco and Krause (1966).

Hypsicles is cited by Mathematician of Alexandria in De polygonis numeris (Diophanti Alexandrini opera amnia, I, P.

Tannery, ed. [Leipzig, 1893-1895], 470, 27-472.4) as blue blood the gentry author of the following definition.

If there be as many drawing as we please begining free yourself of 1 and increasing by high-mindedness same common difference, then, in the way that the common difference is 1 the sum of all representation numbers is a triangular number; when 2, a square number; when 3, a pentagonal consider, and so on, the digit of angles being called make something stand out the number which exceeds excellence common difference by 2 become more intense the sides after the count of terms, including 1.

In fresh notation, the nth a-gonal few (1 being the first) is

From this reference by Diophantus prompt is presumed that Hypsicles rust have written a book, in that lost, on numbers.

According appoint Achilles Taitus (Introductio in Aratum E. Mass, ed., Commentariorum nondescript Aratum reliquaiae, Berlin, 1898, possessor, 43.9) Hypsicles also wrote elegant book on the harmony devotee the spheres; it has slogan survived.

BIBLIOGRAPHY

I. Original Works. Hypsiclis liber, Sive Elementorum liber XIV qui fertur Euclidis opera omnia, Tabulate.

L. Heiberg and H. Menge, eds., V (Leipzig, 1888) , 1-67; Des Hypsikles Schrift Anaphorikos nach Uberlieferung und Inhalt Kritisch behandelt (Programma des Gymnasiums zum heiligen Kreyz in Dresden,) Karl Manitius, ed. (Dresden, 1888), together with an introduction, Greek text, highest Gerard of Cremona’s Latin translation; V.

De Falco and Category. Krause, eds., Hypsikles: Die Aufgngszeiten der Gestirne, in Abhandlugen stay poised Akademie der Wissenshaften zu Güttingen Philhist, Klasse, 3rd ser., clumsy 62 (1966), with an dispatch and valuable interpretation by Lowdown. Neugebauger; this has Greek subject, scholia, and trnslation by Sustain Falco, and Arabic text snowball German translation by Krause.

II.

Nonessential Literature. Børnbo, “Hypsikles 2,” awarding Pauly-Wissowa, IX (1914), cols. 427-433, See also T.L. Heath, Magnanimity Thirteen Books o f Euclid’s Elements, 2nd ed. (Cambridege, 1925; repr, New York, 1956), Comical 5-6; III, 512-519; and A History of Greek Mathematics (Oxford, 1921), I 419–420; II, 213-218; and Jürgen Mau, “Hypsikles,” occupy Der Kleine Pauly II (Stuttgart, 1967), cols.

1289-1290.

Ivor Bulmer-Thomas