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## Division of angles and circles

(923 words)

*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…

**Source:**Brill’s New Pauly

## Diophantus

(1,146 words)

*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…

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## Land surveying

(895 words)

*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…

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## Polyidus

(381 words)

*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…

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## Attalus

(2,358 words)

*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 …

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## Aristarchus

(2,018 words)

*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…

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## Serenus

(635 words)

*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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## Hippocrates

(5,685 words)

*Hippokrátēs*). [German version] [1] Father of Peisistratus, from Brauron Father of Peisistratus. H. is presumed to have come from Brauron, the later deme of Philaidai, and traced his ancestry back to Neleus (Hdt. 1,59; 5,65; Plut. Solon 10; 30). Patzek, Barbara (Wiesbaden) Bibliography Traill, PAA 538385. [German version] [2] Son of Megacles from Athens, approx. 6th cent. BC Son of the Alcmaeonid ( Alcmaeonids) Megacles from Athens, born around 560 BC, H. was the brother of Cleisthenes, the father of Megacles and Agariste [2] and thus th…

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## Nicon

(255 words)

*Níkōn*). [German version] [1] Theban military leader, 413 BC Theban leader of 300 Boeotian hoplites who, together with some Spartan units, crossed over to Sicily in 413 BC in order to defend Syracuse (Thuc. 7,19,3). Beck, Hans (Cologne) [German version] [2] Comedy writer, 4th/3rd cent. BC Comedy writer of the 4th or 3rd cent. BC; there is a preserved fragment of a play

*Kitharōdós*, in which apparently the direct speech of a non-Greek slave is quoted (fr. 1). Bäbler, Balbina (Göttingen) Bibliography

**1**PCG 7, 1989, 38. [German version] [3] Co-founder of the anti-Roman alliance of…

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## Theon

(2,323 words)

*Théōn*). [German version] [1] Greek painter from Samos, 300 BC and after T. of Samos was a Greek painter of the Hellenistic Period, who was active around and after 300 BC. His skill as a creator of images and the successful way in which his paintings were composed were praised in handbooks of rhetoric (e.g. Quint. Inst. 12,10,6) as examples to be followed. The viewer's creative imagination and intuitive understanding were meant to be stimulated at the same time by means of the artistic

*phantasía*(Lat.

*ingenium*, 'image creation'; Phantasia), so that the viewer might imagine e…

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## Number

(5,221 words)

*calculi*or tokens, were used in arithmetic. As first-order representations they enabled operations such as increasing, decreasing, combining, separating, and distributing. Their relationship to the numerical notations recorded in the oldest ‘texts’ (

*c*. 3300 BC; Uruk) is still discussed [2]. The numerical signs in these texts do not represent absolute numbers but context-dependent units of count…

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## Quadrature of the circle

(1,369 words)

*ho toû kýklou tetragōnismós*, Latin

*quadratura circuli*). [German version] I. The nature of the problem The quadrature of the circle is one of the three 'classic problems' (the other two being the trisection of an angle, cf. division of angles and circles, and the duplication of the cube) of ancient Greek mathematics. The problem is to find the side

*x*of a square such that its area is equal to the area of a circle with radius

*r*using a geometric procedure; that is, to determine the value of the variable

*x*in the equation

*x*2 = π

*r*2. Accordingly, the solution to the q…

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## Theudius

(210 words)

*Theúdios*). Mathematician and philosopher from Magnesia, probably 4th century BC. The only information about him comes from the catalogue of mathematicians in Proclus's [2] commentary on Euclid [1. 67, Z. 12-20]. T. is mentioned there after Eudoxus [1] and before Philippus of Medma, who was a pupil of Plato [1]; Therefore, T. was probably a contemporary of Aristotle [6]. According to Proclus, T., Menaechmus [3] and Deinostratus conducted research together at the Academy (

*Akadḗmeia*), improved the arrangement of the

*'Elements'*, and put many limited pr…

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## Eutocius

(168 words)

*Eutókios*) The mathematician E. of Ascalon was presumably born around AD 480; the widespread assumption that he was a pupil of the architect Isidorus of Miletus is hardly plausible [1. 488]. He wrote commentaries on three works of Archimedes [1] (

*Perì sphaíras kaì kylíndrou*, Περὶ σφαίρας καὶ κυλίνδρου,

*kýklou métrēsis*, κύκλου μέτρησις,

*Perì epipédōn isorrhopiôn*, Περὶ ἐπιπέδων ἰσορροπιῶν, text editions [3. 1-319]) as well as on the first four books of Apollonius'

*Kōniká*(Κωνικά) [13] (dedicated to Anthemius [3], text edition [4. 168-361]…

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## Gnomon

(272 words)

*gnomon*describes the shape of an angle bar that remains when a smaller square is removed from a larger square. The Pythagoreans represented arithmetic series with geometrically arranged dots (pebbles) in the form of figures, so t…

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## Philo

(5,673 words)

*Phíl*

*ōn*). [German version] [I 1] Athenian politician Athenian from Acharnae who was exiled by the Oligarchic regime in 404 BC (Triakonta). During the civil war, he lived as a

*metoikos*(resident without Attic citizenship) in Oropos awaiting the outcome of events. Following his return, when he applied to join the

*boulḗ*he was accused of cowardice and other misdemeanours at a dokimasia investigation (Dokimasia) (Lys. 31; possibly 398 BC). Walter, Uwe (Cologne) Bibliography Blass, vol.1, 480f. Th.Lenschau, A. Raubitschek, s.v. P. (2), RE 19, 2526f. …

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## Mechanical method

(255 words)

*Éphodos*) of Archimedes [1] is our source for his mechanical method from which he derived geometric formulas. To compare the surfaces of two figures, he disassembled each into an infinite number of parallel lines and balanced them on a scale. On one side of the scale, one surface is hung up at one point, i.e., as a whole. On the other side, the surface is hung up along the entire arm, i.e., each layer remains where it is and acts with a different leverage. When ea…

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## Axiom

(143 words)

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## Mathematics

(3,425 words)

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## Hermotimus

(132 words)

**1**P. Briant, Histoire de l'empire perse de Cyrus à Alexandre, 1996, 283-288

**2**P. Guyot, Eunuchen als Sklaven und Freigelassene, 1980, Register s.v. [German version] [2] Mathematician from Colophon of Colophon, mathematician. He continued …

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## Leodamas

(261 words)

*Leōdámas*). [German version] [1] Athenian orator, c. 400 BC The Athenian L. of Acharnae, a skilful orator (Aristot. Rh. 2,23,25 1400a 31-35), was rejected at his

*dokimasia*of 382 BC to assess his candidacy for the eponymous archonship ( Archontes) because of his political role prior to 403 (Lys. 26,13f.). Engels, Johannes (Cologne) Bibliography PA 9076 Davies 13921, p. 523 LGPN 2, s.v. Leodamas (2) Traill, PAA 605085. [German version] [2] Athenian rhetor, 4th cent. BC Son of Erasistratus of Acharnae, outstanding Athenian rhetor, student of Isocrates (Plut. Mor…

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## Diocles

(2,746 words)

*Dioklês*). [German version] [1] Hero in Megara Hero in Megara. He supposedly died in battle, bravely covering a youth with his shield. At his grave boys competed for who could give the sweetest kiss. This agon, which took place every spring, was called Dioclea (Schol. Pind. Ol. 7,157; 13,156a; Theoc. 12,27-33 with Schol.: Aition). Perhaps the kisses represented farewell kisses repeated in the cult of the hero ([1]; to the contrary [2]). According to Schol. Aristoph. Ach.774 the agon was founded…

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## Iulianus

(4,648 words)

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## Eudoxus

(1,483 words)

*Eúdoxos*). [German version] [1] of Cnidus Ancient mathematician and astronomer One of the most important ancient mathematicians and astronomers, he was presumably born in 391/390 BC (on the problem of dating see [7. 137-139]). He studied mathematics with Archytas [1] and medicine with Philistion. At the age of 23 he went to Athens and it is said that, among other things, he attended lectures there with Plato. At the expense of his Cnidian friends, presumably in 365/4 [11], E. went with a letter…

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## Theaetetus

(1,081 words)

*Theaítētos*). [German version] [1] T. of Athens, mathematician, c. 400 BC Mathematician, a native of Athens, pupil of Theodorus [2] of Cyrene and later a member of Plato's Academy (

*Akadḗmeia*). In Plato's [1] dialogue named after him, T. appears (together with the aged Theodorus [2]) as about fifteen years old in 399 BC; he was therefore born

*c.*414. Plato describes him as gentle, courageous and quick to apprehend. After he had been wounded in the battle of Corinth, T. contracted dysentery and died in 369. T. contributed substantially to the theory of irrational quantiti…

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## Nicomachus

(1,669 words)

*Nikómachos*). [German version] [1] Healing hero See Gorgasus and Nicomachus Schmitz, Winfried (Bielefeld) [German version] [2] Athenian official, 410-404 BC Allegedly the son of a slave and only later accepted as an Athenian citizen. In 410-404 BC, N. led the commission for recording the laws (

*anagrapheîs tôn nómōn*). Exiled under the Thirty (

*triákonta*), he returned in 403 and again became

*anagrapheús*. In 399/8 BC, N. was accused of manipulating the laws, thus e.g. contributing to the sentencing of the demagogue Cleophon [1] in 404, evading his …

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## Helicon

(372 words)

*Helikṓn*). [German version] [1] Mountain range in central Greece Mountain range in central Greece, dividing the Copais Basin and the upper Cephissos Valley from the Gulf of Corinth (cf. Str. 9,2,25; Paus. 9,28,1-31,7). The western part of the H. belonged to Phocis and the eastern part to Boeotia. The highest elevation is the peak of the Palaiovouno (1,748 m). Few passes lead over the H., which is rich in springs and forests and was famed for its herbs. The H. has large areas that were used in anti…

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## Hero

(1,389 words)

*Hḗrōn*). [German version] A. Life H. of Alexandria, mathematician and engineer. No details of his life are known. He lived after Archimedes [1], whom he quotes, and before Pappus, who quotes him. In the

*Dioptra*, ch. 35, H. describes a method for determining the time difference between Rome and Alexandria by observing the same eclipse of the moon at both locations. It is quite likely that this eclipse occurred in AD 62 and that H. probably observed it himself in Alexandria [10. 21-24]. Folkerts, Menso (Mu…

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## Leon

(1,337 words)

*Léōn*). Cf. also Leo. Byzantine emperor Leo [4-9]. Sicilian place name L. [13]. [German version] [1] Spartan king, 6th cent. BC Spartan king, Agiad ( Agiads), grandfather of Cleomenes [3] I (Hdt. 5,39); is said to have been successful in war together with his fellow king Agasicles in the early 6th cent. BC, but to have been defeated by Tegea (Hdt. 1,65). Sparta is said to have already achieved

*eunomía*(‘good order’) before his time [1. 45ff.]. Welwei, Karl-Wilhelm (Bochum) Bibliography

**1**M. Meier, Aristokraten und Damoden, 1998. [German version] [2] Tyrant of Phlius, 6th cent. BC Tyran…

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## Theodosius

(3,100 words)

*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. …

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## Deinostratus

(385 words)

*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…

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## Autolycus

(734 words)

*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…

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## Thymaridas

(162 words)

*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…

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## Carpus

(196 words)

*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…

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## Theodorus

(7,286 words)

*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 …

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## Ptolemaeus

(19,876 words)

*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…

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## Hypsicles

(603 words)

*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…

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## Archimedes

(2,119 words)

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## Euclides

(2,633 words)

*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 …

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## Pappus of Alexandria

(727 words)

*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…

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## Duplication of the Cube

(1,109 words)

*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*32). The problem thus…

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## Zenodorus

(744 words)

*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 …

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## Apollonius

(7,446 words)

*Apollṓnnios*). [German version] [1] Dioiketes of Ptolemy II (259-245 BC) Of Caria, possibly Ptolemaic o

*ikonomos*there in 267 BC. Was

*dioiketes*of Ptolemy II from April /May 259 until the end of 245; in 252 escorted Berenice to her wedding to Antiochus II. At a critical transition period A. found himself responsible for the economy of the kingdom of the Ptolemies, adapting the fiscal system to the monetary economy of the Lagids, for which purpose he was entrusted with the management of finances and the co…

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## Menaechmus

(496 words)

*Ménaichmos*). [German version] [1] M. of Sicyon Greek historian and antiquary, 4th cent. BC Greek historian and antiquary of the 4th cent. BC. Author of a Pythian history (

*Pythikós*), which was superseded by a list of victors of the Pythian Games at Delphi composed by Aristotle (T 3) and therefore must have existed in the early 330s (cf. Syll.3 275). A history of Alexander, (

*Historía hē katá ton Makedóna Aléxandron*) is entirely lost (T 1), while only fragments remain of a local history of Sicyon (

*Sikyōniká*). Fragments of a treatise ‘On Artists (

*Perí technítōn*, F 3-6; 9) deal pr…

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## Nicomedes

(1,542 words)

*Nikomḗdēs*). [German version] [1] Spartan commander, 458 or 457 BC Member of the Spartan royal family of the Agiadae, son of Cleombrotus [2], brother of Pausanias, the victor of Plataeae. In 458 or 457 BC, N. led a Spartan army as the guardian of his underage nephew Pleistoanax to support the inhabitants of the Doris region against the Phocians and on the return march defeated the Athenians near Tanagra (Thuc. 1,107,2-108,1; Diod. Sic. 11,79,4-80,6; Plut. Cimon 17,4-9; Plut. Pericles 10,1-4). Welwei, Karl-Wilhelm (Bochum) …

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## Mesolabium

(99 words)

*mesolábion*). A mechanical device invented by Eratosthenes [2] to establish graphically the two geometric means

*x*and

*y*between two given lines

*a*and

*b*(as in the relationship

*a*:

*x*=

*x*:

*y*=

*y*:

*b*). The mesolabium enabled the mechanical solution of the problem of the duplication of the cube (‘Delian problem’): if

*b*= 2

*a*, then

*x*is the desired solution of the equation for the duplication of the cube (

*x3*=

*2a3*).…

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## Geminus

(723 words)

*Géminos*) [I]. [German version] [1] Astronomer and mathematician Astronomer and mathematician from the school of Posidonius. …

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## Sporus

(279 words)

*Spóros*) or Porus (Πόρος;

*Póros*). It is unclear whether the two individuals of this name living around AD 200 are in fact the same person (v. [5]). S. or Porus wrote a (lost) compilation, Κηρία (

*Keria*), with extracts on the quadrature of the circle and the duplication of the cube [4. 226]. He criticized Archimedes' [1] approximation of the number

*…*

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## Rhombus

(103 words)

*rhómbos*). [German version] [1] Geometric shape In the plane, a rectangle with four sides of equal length but with unequal angles (

*i.e.*, with two acute and two obtuse angles; Euc. 1, Def. 22; Censorinus, DN 83,14 Jahn). In three dimensions, a rhombus is the solid of revolution consisting of two cones with the same base (Archim. De sphaera et cylindro 1, def. 6). Folkerts, Menso (Munich) Bibliography

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**Source:**Brill’s New Pauly