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Division of angles and circles

(923 words)

Author(s): Folkerts, Menso (Munich)
[German version] I. Ancient Orient see  Mathematics I Folkerts, Menso (Munich) II. Classical Antiquity [German version] A. Division of circles The division of circles, i.e. the division of the circumference of a circle into any number of arcs of equal length, is directly correlated to the regular polygons: if a regular n-gon is inscribed in a circle, the circumference of the circle is divided into n sections and the angle at the centre belonging to the side of the n-gon has the value 360°/ n . The Pythagoreans ( Pythagoras [2]) were already interested in the regular polygons a…


(1,146 words)

Author(s): Nesselrath, Heinz-Günther (Göttingen) | Strothmann, Meret (Bochum) | Folkerts, Menso (Munich)
(Διοφαντός; Diophantós). [German version] [1] Writer of comedies Author of comedies, dates unknown; one fragment and the title of one play (Μετοικιζόμενος) have been preserved. Nesselrath, Heinz-Günther (Göttingen) Bibliography 1 PCG V, 42. [German version] [2] Commander under Mithridates VI Eupator From Sinope, son of Asclepiodotus, commander to Mithridates VI Eupator. In 110 BC he provided skilful military and diplomatic support to the inhabitants of the city of Chersonesus and thus enabled them to withstand the Scythians (Str. 7,3,1…


(3,100 words)

Author(s): Folkerts, Menso (Munich) | Frede, Michael (Oxford) | Matthaios, Stephanos (Cologne) | Berger, Albrecht (Berlin) | Groß-Albenhausen, Kirsten (Frankfurt/Main) | Et al.
[German version] I Greek (Θεοδόσιος/ Theodósios). [German version] [I 1] Greek mathematician and astronomer, 2nd/1st cent. BC Greek mathematician and astronomer. Folkerts, Menso (Munich) [German version] I. Life and works According to Str. 12,4,9, T. was one of the most important men in Bithynia; the birthplace Tripoli given in the Suda (s. v. Θ.) may relate to another T. As Strabo also names T.’ sons as important mathematicians, T. must belong in the 2nd half of the 2nd cent. BC, or, at the latest, the 1st half of the 1st. …


(385 words)

Author(s): Folkerts, Menso (Munich)
[German version] (Δεινόστρατος; Deinóstratos) D. is mentioned in Eudemus' list of mathematicians as the brother of Menachmus, who was a pupil of Eudoxus (Procl. in primum Euclidis elementorum librum comm., p. 67,11 Friedlein). He therefore lived in the middle of the 4th cent. BC  Pappus of Alexandria reports (4,30, p. 250,33-252,3 Hultsch) that to square the circle D. used a curve that was accordingly called the quadratrix (τετραγωνίζουσα). This curve, said to have already been used by Hippias of Elis for the trisection…


(734 words)

Author(s): Graf, Fritz (Columbus, OH) | Folkerts, Menso (Munich)
(Αὐτόλυκος; Autólykos). [German version] [1] Son of Hermes and Chione Son of Hermes and Chione (or Philonis, who also bore the singer  Philammon to Apollo, Hes. fr. 64,14). He was included in various mythical family circles, as the father of  Odysseus' mother Anticlea (Hom. Od. 11,85), of  Jason's mother Polymede (Apollod. 1,107) and of Aesimus, the father of  Sinon. He gives the newborn Odysseus his name, and it is whilst hunting with his sons on Mount Parnassus that Odysseus receives the wound in his th…


(162 words)

Author(s): Folkerts, Menso (Munich)
[German version] (Θυμαρίδας; Thymarídas). Mathematician from Paros; according to Iamblichus (v. P. 104), T. was an early Pythagorean (Pythagorean School). He defined 'unity' (μονάς/ monás; i.e. the One that generates all the natural numbers) as περαίνουσα ποσότης ( peraínousa posótēs, 'limiting quantity'; Iambl. in Nicomachi arithmeticam introductionem 11,2-5) and called prime numbers εὐθυγραμμικός ( euthygrammikós, 'rectilinear'; ibid.  27,4), because they can only be set out in one dimension. The name 'Flower of T.' (Θυμαρίδειον ἐπάνθημα, Thymarídeion epánthēma) is gi…


(196 words)

Author(s): Frey, Alexandra (Basle) | Folkerts, Menso (Munich)
(Κάρπος; Kárpos). [German version] [1] Son of Zephyrus and a certain Hore Handsome youth, son of Zephyrus and of a certain Hore ( Horae). He organizes a swimming race with  Calamus, his best friend, but drowns in the event. In mourning, his friend kills himself and is turned into reeds. C. is turned into a crop of the field (Nonnus, Dion. 11,385-481). Frey, Alexandra (Basle) [German version] [2] C. of Antioch Mathematician A mathematician, who lived presumably in the 1st or 2nd cent. AD. Information on him is given in four fragments by Pappus (8,3), Proclus (in Euc…


(7,286 words)

Author(s): Knell, Heiner (Darmstadt) | Folkerts, Menso (Munich) | Baumhauer, Otto A. (Bremen) | Schmitz, Winfried (Bielefeld) | Blume, Horst-Dieter (Münster) | Et al.
[German version] I Greek (Θεόδωρος; Theódōros). [German version] [I 1] Of Samos, Greek architect, bronze sculptor and inventor, Archaic period Multitalented Greek inventor, architect, bronze sculptor and metal worker ( toreutḗs; Toreutics) of the Archaic period from Samos (for the occupational image cf. architect). His father was Telecles (Hdt. 3,41; Paus. 8,14,8; 10,38,6) or according to other sources (Diog. Laert. 2,103; Diod. Sic. 1,98) Rhoecus [3]; his name is so frequently mentioned in conjunction with the latter that …


(19,876 words)

Author(s): Ameling, Walter (Jena) | Mehl, Andreas (Halle/Saale) | Zahrnt, Michael (Kiel) | Günther, Linda-Marie (Munich) | Schottky, Martin (Pretzfeld) | Et al.
(Πτολεμαῖος/ Ptolemaîos). Personal name meaning 'warlike' (not 'hostile'), first recorded in Hom. Il. 4,228; the name occurred in Macedonia in the 5th and 4th cents. BC, from where it spread to Thessaly, still in the 4th cent. (IG IX 2, 598). It became prominent with the Lagid dynasty, and became common, not only in Egypt, where it may at first have indicated solidarity with the dynasty, but also elsewhere. It underwent many deformations and transmutations. Ptolemies Famous persons: P. [1] I Soter, P. [6] III Euergetes; P. [22], the son of Caesar; the scientist Claudius P. [65]. Ameling, Wa…


(603 words)

Author(s): Folkerts, Menso (Munich)
[German version] (Ὑψικλῆς; Hypsiklês). Hellenistic mathematician and astronomer. From the introduction to book 14 of Euclid's ‘Elements’ written by him, it follows that H. lived in Alexandria around 175 BC. It is attested by MSS that he composed what later was added as book 14 to the ‘Elements’ of  Euclides [3] (ed. [1]). Like bk. 13 it deals with the inscribing of regular bodies into a sphere and was thought of as an explanation to a lost work of  Apollonius [13] about dodecahedra and icosahedra. H. shows that the planes th…


(2,119 words)

Author(s): Folkerts, Menso (Munich) | Degani, Enzo (Bologna)
[1] of Syracuse C. 287-232 BC [German version] A. Life A. was born in 287 BC in Syracuse, son of the astronomer Phidias. He was friends with King Hieron II, and later with his son Gelon. A. probably spent some time in Alexandria; he later sent on his writings to the mathematicians (Conon, Dositheus, Eratosthenes) who were working there. In Syracuse, A. studied problems of mathematical and physical theory, but also their practical applications; the machines and physical apparatus which he built (e.g. the s…


(2,633 words)

Author(s): Rhodes, Peter J. (Durham) | Döring, Klaus (Bamberg) | Folkerts, Menso (Munich) | Zaminer, Frieder (Berlin) | Di Marco, Massimo (Fondi Latina) | Et al.
(Εὐκλείδης; Eukleídēs). [German version] [1] Athenian archon in 403/2 BC Athenian archon in 403/2 BC. During his year in office Athens made a new start following the Oligarchy of the Thirty (e.g., see And. 1,87-94) and, among others, officially adopted the Ionian alphabet (Theopomp. FGrH 115 F 155). Rhodes, Peter J. (Durham) Bibliography Develin 199 LGPN 2, Εὐκλείδης (9). [German version] [2] of Megara Student of Socrates Student of Socrates, founder of the  Megarian School; born between 450 and 435, probably died early in the 360s. In Plato's Phaedon (59c) E. is named among those …

Pappus of Alexandria

(727 words)

Author(s): Folkerts, Menso (Munich)
(Πάππος Ἀλεξανδρεύς; Páppos Alexandreús). [German version] I. Life Eminent Greek mathematician. Based on his calculation of a partial solar eclipse for the year AD 320, it is assumed that P. lived in the first half of the 4th cent. (on this and on erroneous dating in the Suda see [2. 2-4]). Folkerts, Menso (Munich) [German version] II. Works The most important surviving work is the Συναγωγή/ Synagōgḗ, customarily cited as the Collectio (ed. [1], French translation [3], edition and English translation of book 7 [2]). Of the 8 books, the first is wholly lost, the se…

Duplication of the Cube

(1,109 words)

Author(s): Folkerts, Menso (Munich)
(κύβου διπλασιασμός/ kýbou diplasiasmós according to Eratosthenes, in [1. 88,16]). [German version] I. General The duplication of the cube ─ besides the  division of angles and circles and the  squaring of the circle ─ belongs to the three classic problems in Greek  mathematics. The challenge is such: to find ─ through the use of geometry ─ for a given cube with a side-length of a (and thus the volume of a 3) the side x of another cube whose volume is twice as big as that of the given cube. The problem is therefore to find the value of x, to which applies: x 3 = 2 a 3 (that is: x = a 32). The problem thus…


(744 words)

Author(s): Folkerts, Menso (Munich) | Matthaios, Stephanos (Cologne) | Bringmann, Klaus (Frankfurt/Main) | Neudecker, Richard (Rome)
(Ζηνόδωρος/ Zēnódōros). [German version] [1] Greek mathematician, probably at the beginning of the 2nd cent. BC Greek mathematician, probably at the beginning of the 2nd cent. BC [5; 6. 604 f.]. He wrote a work 'Isoperimetric figures' (Περὶ ἰσοπεριμέτρων σχημάτων, Perì isoperimétrōn schēmátōn) in which he proved that of all figures of the same circumference the circle has the greatest area, and formulated the proposition that of all bodies of the same surface area the sphere has the greatest volume [3; 4; 7]. Substantial parts of the …

Land surveying

(895 words)

Author(s): Folkerts, Menso (Munich)
[English version] The writings of the Roman surveyors ( agrimensores) deal with their various areas of activity: measurement of areas; limitation, i.e. division by orthogonal boundaries; creation of land registers and general parceling maps; functioning as a judges or experts in land law, particularly in boundary disputes; collaboration in religious ceremonies; units of length and area, weights and determining area and volume. Mathematical questions are dealt with most notably by Balbus' work Expositio et ratio omnium formarum (ca . AD 100), the anonymous Liber podismi and a wo…


(381 words)

Author(s): Käppel, Lutz (Kiel) | Börm, Henning (Kiel) | Folkerts, Menso (Munich) | Zimmermann, Bernhard (Freiburg)
(Πολύιδος/ Polýidos, Latin Polyidus). [German version] [1] Mythical seer and miracle-worker from Corinth ('of wide learning'). Mythical seer and miracle-worker from Corinth (cf. Cic. Leg. 2,33), descendant of Melampus [1] (Pherecydes FGrH 3 F 115a; Paus. 1,43,5), spouse of Eurydameia, father of Euchenor (Hom. Il. 13,663-668; cf. Cic. Div. 1,89), Cleitus [2], Astycrateia and Manto (not identical with the seeress Manto). His powers are testified to by numerous accounts of his assitence: in Corinth, for example…


(2,358 words)

Author(s): Badian, Ernst (Cambridge, MA) | Mehl, Andreas (Halle/Saale) | Folkerts, Menso (Munich) | Dingel, Joachim (Hamburg) | Neudecker, Richard (Rome) | Et al.
(Ἄτταλος; Áttalos). [German version] [1] Friend of  Philippus, rival of Alexander the Great at the court of his father Friend of  Philippus who did not punish him for an insult inflicted on Pausanias. At the wedding of his niece Cleopatra (II) to Philippus (337 BC) he called  Alexander [4] the Great a nothos (illegitimate son) and was attacked by him, whereupon Alexander and Olympias were banned (Plut. Alex. 9 among others). With his father-in-law (Curt. 6,9,18) Parmenion, he commanded the invading army in Asia. After Philippus' death, Alexander …


(2,018 words)

Author(s): Meier, Mischa (Bielefeld) | Pressler, Frank (Heidelberg) | Folkerts, Menso (Munich) | Montanari, Franco (Pisa)
(Ἀρίσταρχος; Arístarchos). [German version] [1] Athenian politician (end of the 5th cent. BC) Athenian politician, in 411 BC the most embittered opponent of the demos among the 400 Oligarchs in Athens (Thuc. 8,90,1). A. participated in the fortification of  Eetioneia when he was strategos (Xen. Hell. 2,3,46). After the regime was toppled, he betrayed the border fortification Oenoe to the Boeotians (Thuc. 8,98), for which he was executed in 406 (?) (Lycurg. Leocr. 115; PA, 1663; Traill PAA, 164155). Meier, Mischa (Bielefeld) [German version] [2] of Tegea Tragedian, 5th cent. BC Tragedi…


(635 words)

Author(s): Touwaide, Alain (Madrid) | Schmidt, Peter Lebrecht | Folkerts, Menso (Munich)
[German version] [1] Quinctius S. Sammonicus Author of a collection of recipes (also Quintus Serenius). Author of the Liber Medicinalis, a collection of therapeutic recipes which can be neither dated nor identified; Q. has at times been identified with S. [2] Sammonicus or with his son (Septimius [II 6] S. Sammonicus; both died at the beginning of the 3rd cent. AD). The collection (dating between the 2nd and 4th cents. AD) cannot be chronologically ordered with any accuracy. It is written in hexameters and contain…
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