The fourfold division of the sciences is stated here, in the first chapter of the Arithmetica, and it is the reason this volume is one book rather than two bound together. Boethius distributes the whole of mathematics across two axes: multitude and magnitude, each taken at rest and in relation. Arithmetic studies multitude per se; music, multitude ad aliquid — "quae ad aliquid, musici modulaminis temperamenta pernoscunt"; geometry, magnitude unmoving; astronomy, magnitude in motion. Music is therefore not an art placed beside arithmetic but arithmetic applied to relation — number considered as ratio rather than as quantity. The claim that follows is unusually strong for a textbook: "Quibus quattuor partibus si careat inquisitor, verum invenire non possit." Lacking these four, the inquirer cannot find the truth. This sentence, transmitted through the cathedral schools, is what makes music a subject of the medieval university at all, and it is why a treatise on intervals opens with a treatise on integers.
Boethius, De institutione arithmetica libri duo · De institutione musica libri quinque, ed. Gottfried Friedlein (Leipzig: Teubner, 1867) — read from page images, with the critical apparatus separated from the text
The original is shown on its own: no English is held for this issue yet.
| INST. ARITHM. I, 1. 9 scunt. Horum ergo illam multitudinem, quae per se est, arithmetica speculatur integritas, illam vero, quae ad aliquid, musici modulaminis temperamenta pernoscunt, inmobilis vero magnitudinis geometria notitiam pollice- tur, mobilis vero scientiam astronomicae disciplinae pe- 5 ritia vendicat. Quibus quattuor partibus si careat inqui- sitor, verum invenire non possit, ac sine hac quidem spe- culatione veritatis nulli recte sapiendum est. Est enim sapientia earum rerum, quae vere sunt, cognitio et in- tegra comprehensio. Quod haec qui spernit id est has 10 semitas sapientiae ei denuntio non recte esse philoso- . phandum, siquidem philosophia est amor sapientiae, quam in his spernendis ante contempserit. Illud quoque ad- dendum arbitror, quod cuncta vis multitudinis ab uno progressa termino ad infinita progressionis augmenta con- 15 crescit. Magnitudo vero a finita inchoans quantitate modum in divisione non recipit; infinitissimas enim sui corporis suscipit sectiones. Hanc igitur naturae infini- tatem indeterminatamque potentiam philosophia sponte repudiat. Nihil enim, quod infinitum est, vel scientia ?o potest colligi vel mente comprehendi, sed hinc sumpsit sibi ipsa ratio, in quibus possit indagatricem veritatis exercere sollertiam. — Delegit enim de infinitae multitu- dinis pluralitate finitae terminum quantitatis et intermina- bilis magnitudinis sectione reiecta definita sibi ad cogni- ?5 tionem spatia depoposcit. Coustat igitur, quisquis haec praetermiserit, omnem philosophiae perdidisse doctrinam. Hoc igitur illud quadruvium est, quo his viandum sit, qui- bus excellentior animus a nobiscum procreatis sensibus
This issue carries a scholarly apparatus: 422 notes on the editor’s variant readings. Scholar · $29 a month, by application — see what it adds →