Unit I — Philosophy, the Quadrivium, the Odd Sieve & Perfect Numbers — Critical Commentary

Greek Philology, Exact Number-Theoretic Analysis & Boethian Quadrivium Reception · Book I, Chs. 1–7, 13, 16

Nicomachus I.13 & I.16 — The Odd Koskinon of Eratosthenes & Perfect Numbers Top: Sieving odd numbers by strides of 3 and 5 · Bottom: Dyadic Euclid–Nicomachus generation of 6, 28, 496, 8128 I.13 · Odd Progression (3 … 35): Stride-3 arcs (rose) & Stride-5 arcs (blue) 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 Gold circles = Prime & incomposite (πρῶτοι καὶ ἀσύνθετοι) · Crossed = Secondary & composite (δεύτεροι καὶ σύνθετοι; 9 = 3², 15 = 3×5) I.16 · Perfect Numbers (τέλειοι): Equal to the Sum of Their Proper Aliquot Parts 6 = 2¹(2² − 1) = 1 + 2 + 3 1 2 3 Whole = 6 (first perfect number, in units) 28 = 2²(2³ − 1) = 1 + 2 + 4 + 7 + 14 1+2 4 7 14 Whole = 28 (second perfect number, in tens) Dyadic Rule: P = 2ᵖ⁻¹ · (2ᵖ − 1) p = 2 : 1+2 = 3 (prime) → 2·3 = 6 p = 3 : 1+2+4 = 7 (prime) → 4·7 = 28 p = 5 : 2⁵ − 1 = 31 (prime) → 16·31 = 496 p = 7 : 2⁷ − 1 = 127 (prime)→ 64·127 = 8128 p = 11: 2¹¹ − 1 = 2047 = 23 × 89 Composite! Breaks naive 10⁴ alternation
Exact odd-sequence Sieve of Eratosthenes (I.13) and Mersenne prime generation of Perfect Numbers 6, 28, 496, 8128 (I.16).

I. From Nicomachus’s Tessares Methodoi to Boethius’s Quadrivium (I.1–6)

  1. Poson vs. Pēlikon and Aristotle’s Categories: In aristotle:cat:4b20 (Categories 6, 4b20), Aristotle divided quantity (to poson) into discrete (number and speech) and continuous (line, surface, body, time, place). Nicomachus refines the terminology by reserving poson (multitudo) strictly for bounded discrete multitude and pelikon (magnitudo) for bounded continuous magnitude. Five centuries later, Proclus reproduces Nicomachus’s fourfold schema verbatim in his Prologue to Euclid (proclus:in-euclid:35-37, pp. 35.17–36.25 Friedlein), while Boethius renders Nicomachus’s “four methods” (tessares methodoi), “bridges” (gephyrai), and “ladders” (klimakes) by the newly minted Latin compound quadruvium (“fourfold crossroads”) at De Institutione Arithmetica I.1 (boethius:friedlein:1867, p. 7.25).

  2. Asymmetric Co-Removal (*Synanairein*, I.4–5): Nicomachus’s proof that Arithmetic precedes Geometry, Music, and Astronomy adapts the Academic and Peripatetic test of ontological priority: \(A\) is prior by nature to \(B\) when abolishing \(A\) abolishes \(B\) (synanairei), whereas abolishing \(B\) leaves \(A\) intact (ou synanaireitai). Every polygon, polyhedron, and musical ratio presupposes pure cardinal integerhood (triangle presupposes \(3\); hemiolios presupposes \(3\) and \(2\)), whereas integers require neither spatial extension nor audible strings. In the thirteenth century, Thomas Aquinas analyzes this exact Boethian-Nicomachean hierarchy in Super Boethium De Trinitate q. 5 (aquinas:in-boethium-trinitate).

  3. Number in the Mind of the Creator (I.6.1): By relocating Plato’s Forms into the dianoia of the World-Creating God (en tēi tou kosmopoiou theou dianoiāi) as a cosmic preliminary sketch (prokentēma) and archetype (archetypon paradeigma), Nicomachus forged the exemplarism linking Middle Platonism to Christian, Islamic, and Renaissance mathematical cosmology down to Luca Pacioli (pacioli:divina-proportione:1509) and Johannes Kepler.

II. Why Nicomachus’s Sieve Operates Strictly on Odd Numbers (I.13)

In geometor.nicomachus.sieve_of_eratosthenes, we preserve a crucial historical feature of Greek arithmetic often obscured in modern textbooks:

  • For Nicomachus, neither the monad (\(1\), the “seed” rather than an actual multitude) nor the dyad (2, the principle of the even genus, D’Ooge p. 204 n. 1) belongs to the genus of prime and incomposite numbers, which Nicomachus defines strictly as a subspecies of the odd!

  • Consequently, Eratosthenes's Sieve sets out only the odd numbers \(a_m = 2m + 3\) (\(m = 0, 1, 2, \dots\)). When sieving with an odd prime \(p\) (3, 5, 7), one passes over \(p - 1\) odd entries—advancing by stride \(p\) in the odd array, which equals \(2p\) in natural integers (3 -> 9 -> 15 -> 21, 5 -> 15 -> 25 -> 35)—automatically skipping all even multiples \(2p, 4p, 6p\).

  • Furthermore, Nicomachus misstates (or leaves ambiguous in I.11–13) one subtlety caught by Euclid: two odd numbers can both be secondary/composite in themselves and yet not be coprime (e.g., 15 and \(21\) share common measure \(3\)), which is why Nicomachus illustrates his third class specifically with \(9 = 3^2\) and \(25 = 5^2\) and confirms coprimality via Euclidean anthyphairesis (reciprocal subtraction, I.13.11–13; cf. Elements VII.1–2).

III. Perfect Numbers & Nicomachus’s Two Famous Conjectures (I.16)

In geometor.nicomachus.euclid_nicomachus_perfect_numbers, Euclid's theorem (Elements IX.36) generates even perfect numbers whenever \(M_p = 2^p - 1\) is prime:

\[P_p = 2^{p-1}(2^p - 1) \quad \text{for } p \in \{2, 3, 5, 7\} \implies P_p \in \{6,\; 28,\; 496,\; 8128\}.\]

In I.16.3, however, Nicomachus extrapolates from these first four values two celebrated empirical generalizations (transmitted unchallenged through Boethius and medieval Europe for over a millennium; see heath:hgm:1921, Vol. I, p. 100 and dooge:1926, p. 210 n. 2):

  1. One perfect number per decimal order of magnitude (units, tens, hundreds, thousands, …): False from the fifth perfect number onward! Since \(M_{11} = 2^{11} - 1 = 2047 = 23 \times 89\) is composite, there is no five-digit, six-digit, or seven-digit perfect number; the fifth perfect number (\(p = 13\)) leaps to eight digits: \(2^{12}(2^{13} - 1) = 33{,}550{,}336\).

  2. Strict alternation of terminal digits 6, 8, 6, 8, ...: Partially true, partially false! Every even perfect number does end in either 6 or 8 (provable mod \(10\) from \(p \equiv 1\) vs. \(3 \pmod{4}\) for odd primes \(p\)), but they do not alternate strictly: the fifth (\(p = 13\), \(33{,}550{,}336\)) and sixth (\(p = 17\), \(8{,}589{,}869{,}056\)) both end in 6!