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[German version] (νεῦσις/ neûsis, ‘inclination’, in the mathematical sense: ‘verging’) is a geometric operation that cannot be performed with a compass and ruler alone. It allows problems  that lead to cubic and other higher equations (for example, cube duplication, angle trisection, squaring the circle) to be solved geometrically. A neûsis construction is necessary when a straight line through a given point is supposed to intersect two given lines so that the distance between the points of intersection is equal to a certain distance. Nicomede…

(Διόδωρος, Διόδορος; Diódōros, Diódoros). Well-known representatives of the name: the philosopher D. [4] Kronos, the mathematician D. [8] of Alexandria, the universal historian D. [18] Siculus, the early Christian theologian D. [20] of Tarsus. [German version] [1] Athenian fleet commander in the Peloponnesian War Athenian, fleet commander with Mantitheus at the end of 408-407 BC at the Hellespont with a sufficient number of ships, so that Alcibiades [3] was able to sail to Samos and Thrasyllus and Theramenes to Athens (Diod. Sic. 13,68,2). (Traill, PAA 329550; Develin 171). Kinzl, …

(Μενέλαος/Menélaos, Attic Μενέλεως/Menéleos; Latin Menelaus). [German version] [1] Ruler of Sparta, married to Helena A significant character in the cycle of myths about the Trojan War ( Troy: Cycle of myths). A younger brother of Agamemnon, who ruled the most significant power centre in Greek myth, Mycene, by marriage to Zeus's daughter Helen ( Helene [1]; their only child was a daughter, Hermione) M. became king of a region in the Eurotas valley with its capital Sparta and Amyclae [1], which was significant…

(Αρισταιο̃ς; Aristaîos). [German version] [1] Greek rural deity Rural deity linked with sheep, the discovery of olive oil and honey, hunting, healing, prophecy and the end- ing of a period of drought on Ceos (cf. Apoll. Rhod. 2,500 ff.). In literature he is famous for the death of his bees, which occurred because he was responsible for the death of  Euridices, and he successfully searched for ways to restore the bee populations (Verg. G. 4,315-558). A. is a complex figure who can be found in Central Greece, in Arcadia, on Ceos and in Cyrene. He was the husband of Auto…

(ὁ τοῦ κύκλου τετραγωνισμός/ 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…

[German version] (Θεύδιος; 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…

[German version] (Eυτόκιος; 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]…

[German version] [1] see Groma see  Groma (surveying) Folkerts, Menso (Munich) [German version] [2] see Clocks see  Clocks (time measurement) Folkerts, Menso (Munich) [German version] [3] Arithmetic technical term Arithmetic technical term from Greek numerical theory. The term was adopted from geometry, where the 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…

[German version] I Greek (Φίλων/ 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. …

[German version] The ‘Method (Ἔφοδος; 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…

[German version] In the same way as postulates, axioms were presumably introduced in conflict with Eleatic philosophy in order to enable the acceptance of the existence of the manifold [1. 322-325; 2; 3]. According to Aristotle, axioms are central to every branch of knowledge, particularly the law of contradiction and the principle of the excluded middle (Metaph. Γ 3,1005a19-b27); for the organization of the fundamental principles of a branch of knowledge wherein proof is sought see Aristot. An. p…

Folkerts, Menso (Munich) [German version] A. Introduction (CT) Taking their point of departure from the mathematical accomplishments of the Egyptians and Babylonians, the Greeks remodelled mathematics into a deductive system based on a theory of proof. For the Greeks, unlike for their predecessors, mathematics was a science practiced for its own sake, which also investigated its own foundations; thus, practical considerations and directly numerical problems faded into the background. The main accomplish…

[German version] [1] Prisoner of war from Pedasa Prisoner of war from Pedasa, who, according to Hdt. 8,104f., as a eunuch, had become one of the closest confidants of  Xerxes I and is said to have taken his revenge on the slave trader Panionius (Ath. 6,266e attests that the story was well known).  Eunuchs Wiesehöfer, Josef (Kiel) Bibliography 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 …

(Λεωδάμας; 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…

(Διοκλῆς; 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…

Epithet of many gentilicia [1]. Famous persons: the jurist Salvius I. [1]; the doctor I. [2]; the emperor I. [11], called ‘Apostata’; the bishops I. [16] of Aeclanum and I. [21] of Toledo. [German version] [1] L. Octavius Cornelius P. Salvius I. Aemilianus Roman jurist, 2nd cent. AD Jurist, born about AD 100 in North Africa, died about AD 170; he was a student of  Iavolenus [2] Priscus (Dig. 40,2,5) and the last head of the Sabinian law school (Dig. 1,2,2,53). I., whose succession of offices is preserved in the inscription from Pupput, provi…

(Εὔδοξος; 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…

(Θεαίτητος; 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…

(Νικόμαχος/ 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 …

(Ἑλικών; 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…

[German version] [1] Of Alexandria, mathematician and engineer, 1st cent. AD (Ἥρων; 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…

(Λέων; 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…

[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…

(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…

(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…

(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 …

(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…

[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…

(Ἱπποκράτης; 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…

(Νίκων; 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…

(Θέων; 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…

I. Mesopotamia [German version] A. Numerical systems Before systems for representing numbers in writing were (further) developed, counting stones, known as 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…

(Ἀπολλώνιος; 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…

(Μέναιχμος/ 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…

(Νικομήδης; 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) [German version] [2] N. I of Bithynia King from 280 BC S…

[German version] (μεσολάβιον; 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 ). Hippocrates [5] of Chios Folkerts, Menso (Munich) Bibliography mes T. L. Heath, A History of Greek Mathematics, Vol. 2, 1921, 258-260.

(Γέμινος; Géminos) [I]. [German version] [1] Astronomer and mathematician Astronomer and mathematician from the school of Posidonius. Almost nothing is known about his life. The height of his creativity was around 70 BC. It is generally accepted that he lived in Rhodes. The only fully extant treatise by G. is the ‘Introduction to Astronomy’ (Εἰσαγωγὴ εἰς τὰ φαινόμενα). It is in the tradition of  Eudoxus and  Aratus [4]. Similarly to the later writing by  Cleomedes, it is an elementary textbook on astrono…