Arithmetica

...despised before he has found them. ¶ I also think this should be added: that every power of multitude discrete quantity, such as numbers, proceeding from the starting point of unity, grows toward the infinite increases of progression. Yet magnitude continuous quantity, such as a line or solid, beginning from a finite quantity, does not receive a limit in division. For it admits the most infinite sections of its own body. Therefore, philosophy of its own accord rejects this nature of infinity and indeterminate power.
Nothing in that which is infinite can either be seen to be gathered or comprehended by the mind

For nothing that is infinite can be gathered by science or comprehended by the mind. But from here, reason itself took those things in which it could exercise its skill as a seeker of truth. For it chose a limit of finite quantity from the plurality of infinite multitude; and having rejected the sectioning of terminable magnitude, it demanded for itself defined spaces for knowledge. It is certain, therefore: whoever has overlooked these things has lost the entire doctrine of philosophy.
Praise of Mathematics

This, therefore, is that quadrivium four-fold path by which those must travel whose more excellent mind is led from the senses born with us to the more certain things of intelligence. ¶ For there are certain steps and certain dimensions of progressions by which one can ascend and proceed, so that that eye of the mind, which (as Plato says) is more worthy to be saved and established than many bodily eyes,
The ascent to the wisdom of Plato

because truth can be investigated or inspected by that light alone. This eye, I say, submerged and blinded by the bodily senses, these disciplines illuminate again. Which of these, therefore, must be learned first, if not that which holds the principle and, in a way, the position of a mother to the rest? 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 own reasoning, and established all things according to this, whatever things, through the fashioning of reason, found the harmony of assigned order through numbers; but Arithmetic is also declared prior by this, that whatever things are prior by nature, if these are removed, the later things are simultaneously taken away. But if the later things perish, nothing of the state of the prior substance is changed, just as "animal" is prior to "man." For if you take away "animal," the nature of man is also immediately destroyed. If you take away "man," the animal will not perish. And on the contrary: those things are always later which bring another thing along with themselves.
Those things are always later which bring anything else along

Those things are prior which, when they are spoken of, draw nothing of the later things with them. As in that same example of man. For if you say "man," you will simultaneously name "animal." For man is the same thing as animal. If you say "animal," you have not simultaneously brought in the species of man. For animal is not the same thing as man. This same thing is seen to occur in geometry or in arithmetic. For if you take away numbers, from where comes the triangle or square or whatever is treated in geometry? All of these are derivatives of numbers. But truly, if the square and triangle are removed and all geometry is consumed, the names of "three" and "four" and other numbers will not perish. Again, when I have spoken of some geometric form, the name of numbers is simultaneously implied in it. When I have spoken of numbers, I have not yet named any geometric form. Indeed, how much the power of numbers is prior to Music can be proved most greatly from this: that not only are those things prior by nature which exist through themselves rather than those which are referred to something else, but even the musical modulation itself is marked by the names of numbers.

And the same thing can happen in this as was said in geometry. For the diatessaron the musical interval of a fourth, the diapente the fifth, and the diapason the octave are named from the names of the preceding numbers. The proportion of the sounds themselves toward each other is also found only in numbers and no others. For the sound which is in the diapason octave symphony is gathered by the same double proportion of number a 2 to 1 ratio.
24 / 12 / 6 / 8 / Octave / Fourth / 24 / 12 / 3 /

The modulation which is the diatessaron fourth is composed by the epitrita 4 to 3 comparison. That which they call the diapente fifth symphony is joined by the hemiolia 3 to 2 mean. That which is the epogdous 9 to 8 ratio in numbers is the same as the tone in music. And lest I labor to follow every single point, the following parts of this work will show without any doubt how much prior arithmetic is. Indeed, it precedes spherical science and astronomy as much as the two remaining disciplines precede this third one by nature. For in astronomy, there are circles, the sphere, the center, and parallel circles, and the middle axis. All of these are the concern of the geometric discipline. Wherefore it is also possible to show from this the more ancient power of geometry: because all motion is after rest, and by nature stillness is always prior. But astronomy is of mobile things; geometry is the doctrine of immobile things. Or also, that motion of the stars itself is celebrated by harmonic modulations. Wherefore it is certain that the power of music precedes the course of the stars in antiquity.
Music or Astronomy surpasses the others in antiquity

There is no doubt that nature surpasses Arithmetic, since it seems more ancient than those things which it is. Properly, however, the very nature of numbers has established every course of the stars and every astronomical reasoning. For thus we gather the rising and setting of the stars; thus we guard the slowness and velocity of the wandering stars; thus we recognize the eclipses and multiple variations of the moon. Wherefore, since the power of Arithmetic is prior, as has become clear, let us take from here the beginning of our discussion.

THE COMMENTARY OF GIRARDUS RUFFUS ON THE TWO BOOKS OF BOETHIUS’S ARITHMETIC.

COMMENTARY ON THE FIRST CHAPTER OF THE FIRST BOOK.

flow

A decorative initial "P" is illustrated with floral motifs and a geometric border, signifying the start of the commentary.
The first thing in which the author of this work spends his effort in showing is this: namely, that mathematics original: "mathesim" does not obtain the last rank of dignity, even if you look at the subject matter and the material which he discusses in his theorems. Though I might meanwhile pass over in silence the manner of discourse in which it excels the rest, those who yield to the dignity of science say it is ascribed to Aristotle. He hints at this in these words while discussing the soul referring to Aristotle's "De Anima": "supposing knowledge to be a fine and honorable thing, but one more than another either according to its precision, or by its being of better and more wonderful things." That is, those who judge science to be good and
Mathematics excels from the dignity of the subject and the reason of the argument and cuts off the principles of wisdom

honorable certainly prefer one to another. Such as that which is either more exact, or of those things which are both better and more excellent. For of beings, some are separated from matter: therefore subject to no change. In this genus are those most blessed minds which tower over the world. Others, on the contrary, are given over to matter and change. The physicist embraces such things. There are also those which walk
A two-fold division of science here

in a middle path between both, as they are not entirely immune to matter, yet they do not stick to it everywhere. Rather, they are by their own nature incorporeal and immutable; yet by the touch of matter, and as if by a soldering, they are open to change. Of this genus are forms and accidents: which mathematics encloses in its circuit.