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Neusis

(124 words)

Author(s): Folkerts, Menso (Munich)
[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…

Diodorus

(3,891 words)

Author(s): Kinzl, Konrad (Peterborough) | Riedweg, Christoph (Zürich) | Döring, Klaus (Bamberg) | Badian, Ernst (Cambridge, MA) | Bartels, Jens (Bonn) | Et al.
(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, …

Menelaus

(2,514 words)

Author(s): Visser, Edzard (Basle) | Zahrnt, Michael (Kiel) | Ameling, Walter (Jena) | Wandrey, Irina (Berlin) | Folkerts, Menso (Munich) | Et al.
(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…

Aristaeus

(716 words)

Author(s): Schachter, Albert (Montreal) | Folkerts, Menso (Munich)
(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…

Quadrature of the circle

(1,369 words)

Author(s): Folkerts, Menso (Munich)
(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…

Theudius

(210 words)

Author(s): Folkerts, Menso (Munich)
[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…

Eutocius

(168 words)

Author(s): Folkerts, Menso (Munich)
[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]…

Gnomon

(272 words)

Author(s): Folkerts, Menso (Munich)
[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…

Philo

(5,673 words)

Author(s): Walter, Uwe (Cologne) | Döring, Klaus (Bamberg) | Ameling, Walter (Jena) | Knell, Heiner (Darmstadt) | Folkerts, Menso (Munich) | Et al.
[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. …

Theaetetus

(1,081 words)

Author(s): Folkerts, Menso (Munich) | Albiani, Maria Grazia (Bologna)
(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…

Nicomachus

(1,669 words)

Author(s): Schmitz, Winfried (Bielefeld) | Zimmermann, Bernhard (Freiburg) | Hoesch, Nicola (Munich) | Nesselrath, Heinz-Günther (Göttingen) | Albiani, Maria Grazia (Bologna) | Et al.
(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 …

Helicon

(372 words)

Author(s): Freitag, Klaus (Münster) | Folkerts, Menso (Munich) | Hurschmann, Rolf (Hamburg)
(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…

Hero

(1,389 words)

Author(s): Folkerts, Menso (Munich) | Waldner, Katharina (Berlin)
[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…

Leon

(1,337 words)

Author(s): Welwei, Karl-Wilhelm (Bochum) | Cobet, Justus (Essen) | Schmitz, Winfried (Bielefeld) | Engels, Johannes (Cologne) | Folkerts, Menso (Munich) | Et al.
(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…

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…

Diophantus

(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…

Theodorus

(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 …

Ptolemaeus

(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…

Hypsicles

(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…

Archimedes

(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…
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