On the Institution of Arithmetic

...as has been said, have obtained a nature and power of immutable substance, they are truly and properly said "to be." Wisdom, therefore, professes the science of these things—that is, those that are proper and are called essences by their own name. There are two parts of 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 disconnected from itself, determined by parts, and gathered, as it were, into one assembly, such as a flock, a people, a chorus, a heap, and whatever things have parts terminated by their own extremities and are distinct from the limit of another. The proper name for these is "multitude." Again, some of the multitude are "by themselves," such as three or four, or a square, or any number which, as it is, needs nothing else. Others do not stand by themselves, but refer to something else, such as a double, a half, a sesquialter a ratio of 3:2, or a sesquitertian a ratio of 4:3, and whatever is such that it cannot exist unless it is related to another. Regarding magnitude, some are staying and lack motion, while others are those which turn in constant mobile rotation and do not rest at any time. Arithmetic speculates upon that multitude which is by itself. Music understands the temperaments of modulation that refer to something else. Geometry promises the knowledge of immobile magnitude. The skill of the astronomical discipline claims the science of the mobile. If an inquirer lacks these four parts, he cannot find the truth; and without this contemplation of truth, one should not correctly...

...be wise. For wisdom is the knowledge and entire comprehension of those things which are true. If anyone scorns this—that is, these paths of wisdom—I declare to him that he is not philosophizing correctly. For philosophy is the love of wisdom, which he would have previously despised by scorning these. I also think it must be added that the whole power of multitude, having progressed from one limit, grows into infinite increments of progression, whereas magnitude, beginning from a finite quantity, does not receive a limit in division. For it undergoes infinite sections of its body. Philosophy, therefore, voluntarily rejects this infinity and indeterminate power of nature. For nothing that is infinite can be gathered by science or comprehended by the mind. But reason itself has taken for itself the things in which it could exercise its skill as a seeker of truth. For it chose from the plurality of infinite multitude the limit of finite quantity, and having rejected the section of interminable magnitude, it demanded for itself definite spaces for cognition. It is certain, therefore, that whoever has passed these over has lost the entire doctrine of philosophy. This, therefore, is that quadrivium by which those must travel for whom a more excellent soul, with the senses we were born with, is led to the more certain things of intelligence. For there are certain degrees and certain dimensions of progressions by which one can ascend and progress, so that the eye of the soul, which Plato Greek philosopher (427–347 BC) says is worth more than many bodily eyes to be saved and constituted, might be able to be traced or inspected by that light alone. These disciplines, I say, illuminate this eye again, having been submerged and blinded by bodily senses. Which of these, therefore, is to be learned first, if not that which holds a principle and, as it were, a maternal portion toward the others? This is arithmetic. For it is prior to all, not only because God, the creator of this worldly mass, had this as the first exemplar of his reasoning, and established all things that the reasoning of a maker found concord through numbers of an assigned order, but arithmetic also declares this to be prior: that whatever things are prior by nature, when these are removed, the posterior ones are destroyed at the same time. But if the posterior ones perish, nothing of the state of the prior substance is changed. For example, an animal is prior to a human. For if you remove "animal," the nature of "human" is immediately destroyed. If you remove "human," "animal" will not perish. And conversely, those things are always posterior which bring in something else with them; those are prior which, when they are spoken, pull nothing of the posterior with them. Just as in the same human: for if you say "human," you also name "animal" at the same time. For "human" is the same as "animal." If you say "animal," you have not simultaneously introduced the species of "human." For "animal" is not the same as "human." This same thing seems to occur in geometry or in arithmetic. For if you remove numbers, from where would come the triangle or square or whatever is turned in geometry, all of which are denominative of numbers? But if you remove the square and the triangle, and all geometry is consumed, the names of three and four and other numbers will not perish. Again, when I have said any geometric form, the name of numbers is implicitly contained in it. When I have said numbers, I have not yet named any geometric form. But how much prior the power of numbers is to music can be proven most of all from the fact that not only are those things prior by nature which stand by themselves rather than those which refer to something else, but even that musical modulation itself is noted by the names of numbers. And the same can happen in this as was predicted in geometry. For the diatesseron musical fourth, the diapente musical fifth, and the diapason musical octave are named from the names of the preceding number. The proportion of the sounds themselves against one another is found by no other numbers. For the sound which is a diapason consonance is gathered by the double proportion of number. The modulation which is diatesseron is composed by epitrite 4:3 ratio collation. The consonance they call diapente is joined by an hemiolia 3:2 ratio mean. What is epogdous 9:8 ratio or whole tone in numbers is a tone in music. And so that I do not pursue individual things with the labor of this work, the following chapters will show without any doubt how much prior arithmetic is. It precedes spherical science and astronomy as much as the two remaining disciplines precede this third by nature. For in astronomy, there is the circle, the sphere, the center, the parallel circles, and the middle axis, all of which are the concern of the geometric discipline. Wherefore, it is possible from this to show the power of geometry is older, because all motion is after rest, and by nature, stillness is always prior. The motion of mobiles...