Introductio Arithmetica (Ἀριθμητικὴ Εἰσαγωγή) — Quadrivium, Ratios, Gnomons & Cube Harmony

Synchronized 4-Track Book Codex (English · Greek Hoche 1866 · Summary · Critical Notes)

Introductio Arithmetica (Ἀριθμητικὴ Εἰσαγωγή, Books I–II) of Nicomachus of Gerasa (c. 60–120 CE) is the foundational Neo-Pythagorean treatise that organized the Four Mathematical Sciences (mathemata, rendered quadruvium by Boethius), unfolded all rational ratios from primordial Equality (\(1 : 1 : 1\)), constructed polygonal and solid numbers by arithmetic gnomons, proved the Sum-of-Cubes Theorem (\(\sum_{k=1}^n k^3 = T_n^2\)), and crowned \(6 : 8 :: 9 : 12\) as the Most Perfect Three-Dimensional Proportion (teleiotate analogia) uniting the Cube and the Pythagorean musical scale.

Every unit is presented across four synchronized tracks—English (Martin Luther D'Ooge's public-domain translation, 1926; dooge:1926), Greek (Hoche 1866) (Richard Hoche's Teubner critical edition, 1866; hoche:1866), Summary, and Critical Notes (pairing interactive .euclid-graphic SVG diagrams and exact SymPy / Rational Trigonometry proofs with Boethius's De Institutione Arithmetica [boethius:friedlein:1867] and the GEOMETOR canon).

Units I–IV — Books I & II of the Introduction to Arithmetic

Introductio Arithmetica

• Unit I — Philosophy, the Quadrivium, the Odd Sieve & Perfect Numbers
• Unit II — Five Species of Inequality & the Equality Unfolding Tree
• Unit III — Inverse Reduction, Polygonal Gnomons, Same/Other & Sum of Cubes
• Unit IV — The Ten Neo-Pythagorean Means & the Cube’s Most Perfect Proportion

Critical Apparatus & Canon Concordance

Critical Apparatus

• Master Chapter, Boethius & GEOMETOR Canon Concordance
• Bibliography & Critical Editions
• Mathematical & Philosophical Authorities
• Glossary of Greek & Latin Arithmetical Terms