On the Principles of Arithmetic and On the Principles of Music

PRINCIPLES OF ARITHMETIC I, 1.

...brought their names? Continuing from the previous page, Boethius is arguing that things which derive their names from another are naturally posterior to the source of that name. But indeed, if you were to remove the square and the triangle and all of geometry were consumed, the names "three" and "four" and of other numbers will not perish. Conversely, when I mention a geometric form, the name of numbers is simultaneously implied within it; but when I mention 5 numbers, I have not yet named any geometric form.

Regarding Music, the degree to which the power of numbers is prior can be proven most clearly from this: not only are those things prior by nature which exist through themselves, compared to those which are relative to something else, but even musical modulation itself 10 is marked by the names of numbers, and the same thing can happen in this science as was previously said of geometry. For the "Fourth," the "Fifth," and the "Octave" original: Diatessaron, diapente, diapason. These are the fundamental musical intervals. are named from the names of the preceding numbers. The proportion of the sounds themselves relative to one another is likewise found through no other 15 means than by numbers. For the sound which exists in the "Octave" harmony is gathered from the proportion of a double number The ratio 2:1; the modulation which is the "Fourth" is composed of a four-to-three ratio epitrita: a ratio where the larger number contains the smaller plus one-third of it, such as 4:3; the harmony they call the "Fifth" is joined by a three-to-two ratio hemiolia: a ratio where the larger number contains the smaller plus one-half of it, such as 3:2; what in numbers is 20 an eight-to-nine ratio epogdous: a ratio of 9:8, where the larger contains the smaller plus one-eighth; in music, this defines a whole tone is the same as a "tone" in music. And so that I do not labor to pursue every single point, the following parts of this work will demonstrate without any doubt how much prior arithmetic is.

Indeed, it precedes "Spherics" The study of the celestial sphere and astronomy by as much as the two remaining disciplines [geometry and music] naturally precede this third one. For in astronomy 25 there are circles, the sphere, the center, parallel circles, and the central axis, all of which are the concern of the discipline of geometry. Therefore, it is also possible to show from this the senior power of geometry: that all motion comes after rest, and by nature, stability is always prior, whereas moving things [are posterior]...