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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

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

**Source:**Brill’s New Pauly

## 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

**1**T. L. Heath, The Thirteen Books of Euclid's Elements, vol. 1, 21925, 189

**2**A. Hug, s.v. Ῥόμβος (

*rhombus*), RE 1 A, 1069. [German version] [2] See Top see Top [German version] [3] See Rho…

**Source:**Brill’s New Pauly

## Neusis

(124 words)

*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…

**Source:**Brill’s New Pauly

## 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…

**Source:**Brill’s New Pauly

## 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…

**Source:**Brill’s New Pauly

## 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]…

**Source:**Brill’s New Pauly

## 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…

**Source:**Brill’s New Pauly

## 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…

**Source:**Brill’s New Pauly

## Axiom

(143 words)

**Source:**Brill’s New Pauly

## Mathematics

(3,425 words)

**Source:**Brill’s New Pauly

## 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…

**Source:**Brill’s New Pauly

## 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…

**Source:**Brill’s New Pauly

## 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…

**Source:**Brill’s New Pauly

## 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…

**Source:**Brill’s New Pauly

## 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…

**Source:**Brill’s New Pauly

## 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…

**Source:**Brill’s New Pauly

## 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…

**Source:**Brill’s New Pauly

## Geminus

(723 words)

*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…

**Source:**Brill’s New Pauly

## 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

*pi*(thus [1. 258,22]), provided his own solution to the problem of the duplication of the cube [1. 76-78; 4. 266-268] and rejected the Quadratrix of Hippias [5] of …

**Source:**Brill’s New Pauly