Arithmetic, Geometry, and Music

A Preface in which the division of mathematics is discussed. Chapter 1.

Preface in which the division of mathematics is discussed.

What wisdom is.

Among all the men of ancient authority who flourished, led by the pure reason of the mind under the guidance of Pythagoras, it is clearly established that no one can reach the peak of perfection in the disciplines of philosophy unless such nobility of prudence is tracked by a sort of fourfold path original: "quadriuiio" — the quadrivium (arithmetic, geometry, music, and astronomy).. This will not escape the mind of one who rightly considers it. For wisdom is the apprehension of the truth of those things which are and which possess an immutable substance. We call those things "to be" which do not grow by intention, nor are diminished by retraction, nor are altered by variations, but which guard themselves in their own constant force by the supports of their own nature. These things are qualities, quantities, forms, sizes, smallnesses, equalities, relations, actions, dispositions, places, times, and whatever is found somehow joined to bodies. These things are indeed of an incorporeal nature and flourish with the force of an immutable substance; yet, by participation in a body, they are changed, and by the touch of variable things, they pass into mutable inconstancy. Therefore, since these things, as has been said, possess an immutable substance and force by nature, they are said to be truly and properly. Wisdom, therefore, professes the knowledge of these things which are properly what they are, and which are named by their own essential name.

Essence is twofold.

There are two parts to essence: one is continuous and joined to its own parts, not distributed by any boundaries, such as a tree, a stone, and all the bodies of this world, which are properly called magnitudes. The other is disjointed and determined by parts, gathered together as if into a heap or council, such as a flock, a people, a chorus, a pile, and whatever has parts terminated by its own extremities and distinct from the boundary of another. The proper name for this is multitude. Furthermore, some parts of multitude are through themselves, such as three or four, or a square, or any number which needs nothing to exist. Others do not consist through themselves but are referred to something else, such as double, half, sesquialter a ratio of 3:2, or sesquitertian a ratio of 4:3, and whatever is of such a kind that, unless it is related to another, it cannot exist. As for magnitude, some are remaining and lacking motion, while others are those which are turned by constant mobile rotation and do not rest at any time.

Arithmetic as primary.

Arithmetic, therefore, contemplates that multitude which is through itself. Music provides the temperaments for that which is related to something. Geometry promises knowledge of immobile magnitude. The skill of the astronomical discipline claims the science of mobile [magnitude]. If inquiry lacks these four parts, it cannot find the truth, and without this speculation of truth, no one can be truly wise. For wisdom is the knowledge and entire comprehension of those things which truly are.

Philosophy.

Whoever scorns these, I declare that he does not philosophize rightly along these paths of wisdom. For if philosophy is the love of wisdom, he has scorned it beforehand in scorning these. I also judge it must be added that the whole force of multitude, having progressed from one term, grows to infinite increases of progression, whereas magnitude, starting from a finite quantity, does not receive a limit in division. For it receives infinite sections of its own body. Therefore, philosophy willingly rejects this infinity of nature and indeterminate power. For nothing that is infinite can be gathered by science or comprehended by the mind. But from this, reason itself has taken where it could exercise its skill in the investigation of truth. For it has chosen the limit of finite quantity from the plurality of infinite multitude, and having rejected the sectioning of interminable magnitude, it demands for itself definite spaces for cognition. It is clear, therefore, that whoever skips these has lost the entire doctrine of philosophy. Therefore, that fourfold path is where those must walk for whom the superior mind is led away from the senses created with us toward the more certain things of the intellect. For there are certain

degrees and dimensions of progression by which one can ascend and progress, so that the eye of the mind—which (as Plato says) is more worthy to be saved and established by many bodily eyes, because truth can be tracked or seen by that light alone—may be illuminated again by these disciplines, having been submerged and blinded by bodily senses. Which of these, therefore, must be called first, except that which holds the position of a beginning and a womb, as it were, to the others? This is arithmetic. For this is prior to all, not only because God, the creator of this worldly mass, had this as the first exemplar of his reasoning and constituted everything by it—which, through the reason of a craftsman, found harmony through numbers of assigned order—but arithmetic is also declared prior because whatever things are prior in nature are taken away at the same time as those which are posterior, if the former are removed. If the posterior things perish, nothing is changed in the status of the prior substance, just as an animal is prior to a man. For if you remove an animal, the nature of man is immediately deleted. If you remove a man, an animal will not perish. Conversely, those things are always posterior which bring along something else with them, and those are prior which, when spoken of, bring nothing of the posterior with them, just as in the case of man. For if you say "man," you also name "animal" at the same time. For man is the same thing as animal. If you say "animal," you have not simultaneously brought in the species of man. For an animal is not the same thing as man. This same thing seems to occur in geometry or arithmetic. For if you take away numbers, where is the triangle or square or whatever is involved in geometry, all of which are denominative of numbers? But if you take away the square and the triangle, and all of geometry is consumed, the words "three" and "four" and other numbers will not perish. Again, when I have said a geometric form, the name of numbers is implicit in it. When I have said numbers, I have not yet named any geometric form. But that music is prior in the force of numbers can be proven most strongly from this: not only because those things are prior in nature which stand through themselves rather than those which are referred to something else, but also because music's very modulation is annotated by the names of numbers. And the same can happen in this, as was predicted for geometry. For diatessaron the perfect fourth, diapente the perfect fifth, and diapason the octave are named from the names of the preceding number. The proportion of the sounds themselves against each other is found only through numbers and nothing else. For which sound is the diapason harmony, the same is collected by the proportion of a double number. Which diatessaron modulation is composed by the epitrite ratio of 4:3 collation. Which they call diapente symphony is joined by the hemiolia ratio of 3:2 mean. That which in numbers is the epogdous ratio of 9:8 is the same as the tone in music. And that I may not labor in pursuing individual points, the things following in this work will show without any doubt how much prior arithmetic is. Spherical and astronomical [knowledge] precedes it only as much as the other two disciplines precede this third one by nature. For in astronomy, there are circles, a sphere, a center, parallel circles, and a central axis, all of which are the concern of the geometric discipline. Wherefore, it is also possible to show from this that the force of geometry is older, because all motion is after rest, and a state is always prior by nature. Astronomy is the doctrine of mobile things, while geometry is [the doctrine] of immobile things, or because the motion of the stars themselves is celebrated by harmonic modulations. Wherefore it is also constant that the force of music precedes the course of the stars in antiquity, than which the former appears to be more ancient. Yet, the entire nature of numbers is the basis upon which every course of the stars and every astronomical reason is constituted. For thus we collect risings and settings; thus we guard the slowness and velocity of the wandering stars the planets; thus we recognize eclipses and the multiple variations of the moon. Wherefore, since the force of arithmetic is prior, as has been made clear, let us take the beginning of our disputation from here.

Concerning the substance of number. Chapter 2.