Glossary of Greek & Latin Arithmetical Terms

Polytonic Greek, Boethian Latin & Arithmetical Lexicon of Introductio Arithmetica

Systematic lexicon of 27 technical concepts governing discrete multitude (poson), continuous magnitude (pēlikon), the four mathemata (quadrivium), ratio species, figurate gnomons, Same and Other, and the Ten Means.

philosophia (φιλοσοφία)

  • Greek: φιλοσοφία | Boethian Latin: philosophia / amor sapientiae

  • Rendering: love of wisdom | Hoche Loci: I.1 (pp. 1–2)

Term coined according to Nicomachus (I.1.1) by Pythagoras, restricting ‘wisdom’ (σοφία) to exact scientific knowledge of eternal, immaterial realities.

poson / plethos (ποσόν / πλῆθος)

  • Greek: τὸ ποσόν / τὸ πλῆθος | Boethian Latin: multitudo / quantitas discreta

  • Rendering: multitude / discrete quantity | Hoche Loci: I.2.4–5, I.3.1 (pp. 4–6)

Discontinuous assemblage rooted in a definite minimum (the monad) and advancing indefinitely by addition without upper bound; object of arithmetic (per se) and music (ad aliud).

pelikon / megethos (πηλίκον / μέγεθος)

  • Greek: τὸ πηλίκον / τὸ μέγεθος | Boethian Latin: magnitudo / quantitas continua

  • Rendering: magnitude / continuous quantity | Hoche Loci: I.2.4–5, I.3.1–2 (pp. 4–6)

Coherent spatial extension bounded as a finite whole yet infinitely divisible without reaching a minimum part; object of geometry (stationary) and sphaerics/astronomy (in motion).

tessares methodoi / mathemata (τέσσαρες μέθοδοι / μαθήματα)

  • Greek: τέσσαρες μέθοδοι / μαθήματα | Boethian Latin: quadrivium (Boethius, De Inst. Arith. I.1)

  • Rendering: the four mathematical sciences / Quadrivium | Hoche Loci: I.3–5 (pp. 5–12)

Arithmetic, Music, Geometry, and Astronomy (Sphaerics): the ‘ladders and bridges’ (κλίμακες καὶ γέφυραι, I.3.6) conducting the eye of the soul from sensible flux to intelligible being.

synanairei men, ou synanaireitai de (συναναιρεῖ μὲν, οὐ συναναιρεῖται δέ)

  • Greek: συναναιρεῖ μὲν, οὐ συναναιρεῖται δέ | Boethian Latin: simul perimit sed non simul perimitur

  • Rendering: ontological priority by non-reciprocal elimination | Hoche Loci: I.4–5 (pp. 9–12)

Criterion establishing Arithmetic as ‘mother’ (μητρόπολις) of the Quadrivium: abolishing absolute number abolishes relative ratio (Music) and numbered figures (Geometry), whereas abolishing triangle or octave leaves pure number intact.

arithmos (ἀριθμός)

  • Greek: ἀριθμός | Boethian Latin: numerus

  • Rendering: number | Hoche Loci: I.7.1 (p. 13)

Defined threefold by Nicomachus (I.7.1): limited multitude (πλῆθος ὡρισμένον), a combination of monads (μονάδων σύστημα), or a flow of quantity composed of monads (ποσότητος χύμα ἐκ μονάδων συγκείμενον).

monas (μονάς)

  • Greek: μονάς | Boethian Latin: unitas

  • Rendering: monad / unit | Hoche Loci: I.7–8, II.6–7 (pp. 13–15, 82–85)

The indivisible principle and seed (σπέρμα) of all number, containing potentially (δυνάμει) every even, odd, polygonal, and solid form prior to actualization.

koskinon Eratosthenous (κόσκινον Ἐρατοσθένους)

  • Greek: κόσκινον Ἐρατοσθένους | Boethian Latin: cribrum Eratosthenis

  • Rendering: Sieve of Eratosthenes | Hoche Loci: I.13.2–9 (pp. 29–35)

Algorithmic winnowing of the odd progression 3, 5, 7, 9, 11, … by striding p places from each odd prime p to mark off secondary/composite numbers and isolate primes.

protos kai asynthetos (πρῶτος καὶ ἀσύνθετος)

  • Greek: πρῶτος καὶ ἀσύνθετος | Boethian Latin: primus et incompositus

  • Rendering: prime and incomposite | Hoche Loci: I.11 (pp. 25–27)

An odd number measured solely by the common monad (1) and receiving no proper fractional part besides the paronymous fraction named from itself (e.g. 3 has only 1/3; 5 has only 1/5).

teleios arithmos (τέλειος ἀριθμός)

  • Greek: τέλειος ἀριθμός | Boethian Latin: numerus perfectus

  • Rendering: perfect number | Hoche Loci: I.14–16 (pp. 36–44)

A number exactly equal to the sum of its proper aliquot divisors, standing in virtuous equality between abundant (ὑπερτελής) excess and deficient (ἐλλιπής) defect: 6, 28, 496, 8128, … generated by 2^(p−1)(2^p − 1) when 2^p − 1 is prime.

isotes (ἰσότης)

  • Greek: ἰσότης | Boethian Latin: aequalitas

  • Rendering: equality | Hoche Loci: I.17, I.23, II.1–2 (pp. 44, 64–75)

Unvarying 1 : 1 relation admitting no species or subdivision; proved in I.23 and II.1–2 to be the primordial ‘mother and root’ (μήτηρ καὶ ῥίζα) from which all five species of inequality unfold and into which all resolve.

pollaplasios (πολλαπλάσιος)

  • Greek: πολλαπλάσιος | Boethian Latin: multiplex

  • Rendering: multiple ratio | Hoche Loci: I.18 (pp. 46–49)

First species of greater inequality (k : 1 for k >= 2: duple 2:1, triple 3:1, quadruple 4:1, …), where the greater contains the lesser an exact integer number of times.

epimorios (ἐπιμόριος)

  • Greek: ἐπιμόριος | Boethian Latin: superparticularis

  • Rendering: superparticular / epimoric ratio | Hoche Loci: I.19 (pp. 49–55)

Second species of greater inequality ((n + 1) : n), containing the whole lesser term plus one aliquot part 1/n: hemiolios 3:2 (fifth), epitritos 4:3 (fourth), epogdoos 9:8 (whole tone).

hemiolios (ἡμιόλιος)

  • Greek: ἡμιόλιος | Boethian Latin: sesquialter

  • Rendering: sesquialter (one-and-a-half, 3 : 2) | Hoche Loci: I.19, II.29

First species of superparticular ratio (1 + 1/2 = 3 : 2), governing the Pythagorean musical consonance of the Fifth (διὰ πέντε, e.g. 9 : 6 and 12 : 8).

epitritos (ἐπίτριτος)

  • Greek: ἐπίτριτος | Boethian Latin: sesquitertius

  • Rendering: sesquitertian (one-and-a-third, 4 : 3) | Hoche Loci: I.19, II.29

Second species of superparticular ratio (1 + 1/3 = 4 : 3), governing the Pythagorean musical consonance of the Fourth (διὰ τεσσάρων, e.g. 8 : 6 and 12 : 9).

epogdoos (ἐπόγδοος)

  • Greek: ἐπόγδοος | Boethian Latin: sesquioctavus

  • Rendering: sesquioctave (9 : 8, the whole tone) | Hoche Loci: I.19, II.29 (pp. 145–147)

Superparticular ratio 9 : 8 measuring the musical whole tone (τόνος), the exact quotient separating the Fifth (3 : 2) from the Fourth (4 : 3) between 9 and 8 in 6 : 8 :: 9 : 12.

epimeres (ἐπιμερής)

  • Greek: ἐπιμερής | Boethian Latin: superpartiens

  • Rendering: superpartient ratio | Hoche Loci: I.20–21 (pp. 55–59)

Third species of greater inequality ((n + r) : n with 2 <= r < n and gcd(n, r) = 1), containing the whole lesser plus several aliquot parts, e.g. epidimoiros 5 : 3 (1 + 2/3), epitrimoiros 7 : 4 (1 + 3/4).

pollaplasiepimorios & pollaplasiepimeres

  • Greek: πολλαπλασιεπιμόριος / πολλαπλασιεπιμερής | Boethian Latin: multiplex superparticularis / multiplex superpartiens

  • Rendering: multiple superparticular (kn + 1 : n) & multiple superpartient (kn + r : n) | Hoche Loci: I.22–23 (pp. 59–71)

Fourth and fifth species of greater inequality (k >= 2), completing Nicomachus’s exhaustive fivefold taxonomy of every rational inequality m : n > 1.

anastrophe (ἀναστροφή)

  • Greek: ἀναστροφή | Boethian Latin: conversio ordinis

  • Rendering: reversal of terms | Hoche Loci: I.23.8–15 (pp. 66–71)

Reversing a geometric triple (a, b, c) -> (c, b, a) before applying Nicomachus’s three-term command N1, yielding N2(a, b, c) = (c, b+c, a+2b+c) and branching multiples into superparticulars and superpartients.

gnomon (γνώμων)

  • Greek: γνώμων | Boethian Latin: gnomon

  • Rendering: gnomon | Hoche Loci: II.7–12 (pp. 85–99)

Each successive term g(s, k) = 1 + (s − 2)(k − 1) of an arithmetic progression whose border-addition preserves self-similar s-gonal figure from one rank to the next.

polygonos arithmos (πολύγωνος ἀριθμός)

  • Greek: πολύγωνος ἀριθμός | Boethian Latin: numerus polygonius

  • Rendering: polygonal number | Hoche Loci: II.8–12 (pp. 86–99)

Plane number P(s, n) = n + (s − 2)n(n − 1)/2 generated by summing n terms of the arithmetic progression with step d = s − 2; decomposes into T_n + (s − 3)T_{n−1} (II.12).

heteromekes (ἑτερομήκης)

  • Greek: ἑτερομήκης | Boethian Latin: parte altera longior

  • Rendering: heteromecic / oblong number | Hoche Loci: II.17–19 (pp. 108–117)

Plane product n(n + 1) whose sides differ by exactly unity (2, 6, 12, 20, 30, …), generated by summing consecutive even numbers 2 + 4 + … + 2n = 2T_n; strictly distinguished from general promecic rectangles n(n + k) with k >= 2.

tauton kai thateron (ταὐτὸν καὶ θάτερον)

  • Greek: ταὐτὸν καὶ θάτερον | Boethian Latin: idem et alterum

  • Rendering: the Same and the Other | Hoche Loci: II.17–20 (pp. 108–119)

Pythagorean–Platonic principles manifested arithmetically: odd numbers and squares n^2 embody Sameness (constant 1:1 side ratio), while even numbers and heteromecics n(n+1) embody Otherness (changing side ratio n:(n+1)).

analogia / mesotes (ἀναλογία / μεσότης)

  • Greek: ἀναλογία / μεσότης | Boethian Latin: proportio / medietas

  • Rendering: proportion / mean | Hoche Loci: II.21–28 (pp. 119–144)

Strictly, the identity or binding together of two or more ratios (λόγων σύλληψις), unfolding across three classical archetypes (arithmetic, geometric, harmonic) and seven subcontrary/Neo-Pythagorean means to reach the Pythagorean Decad of Ten Means.

harmonike mesotes (ἁρμονικὴ μεσότης)

  • Greek: ἁρμονικὴ μεσότης | Boethian Latin: medietas harmonica

  • Rendering: harmonic mean | Hoche Loci: II.25–26 (pp. 131–136)

Third classical mean (z − m):(m − x) = z:x, or m = 2xz/(x + z); called ‘subcontrary’ by earlier Pythagoreans and renamed ‘harmonic’ by Archytas and Hippasus because it generates musical consonances and governs the Cube (6, 8, 12).

hypenantia (ὑπεναντία)

  • Greek: ὑπεναντία | Boethian Latin: subcontraria

  • Rendering: subcontrary mean | Hoche Loci: II.25.1, II.28.3–5 (pp. 131, 139–141)

Originally the archaic name for the harmonic mean; in II.28 applied specifically to the Fourth Mean (m − x):(z − m) = z:x, e.g. (3, 5, 6), where m = (x^2 + z^2)/(x + z), and analogously to the 5th and 6th means.

teleiotate analogia (τελειοτάτη ἀναλογία)

  • Greek: τελειοτάτη καὶ μουσικωτάτη ἀναλογία | Boethian Latin: perfectissima proportio / maxima et perfecta symphonia

  • Rendering: the Most Perfect Proportion | Hoche Loci: II.29 (pp. 144–147)

Four-term solid proportion 6 : 8 :: 9 : 12 (ϛ, η, ϑ, ιβ) crowning Book II, Ch. 29: three-dimensional, uniting Arithmetic (9), Harmonic (8), and Geometric (6·12 = 8·9 = 72) symmetry, the Cube (6 faces, 8 vertices, 12 edges), and all Pythagorean consonances (2:1, 3:2, 4:3, 9:8).