What Number Is; and Whether Unity Is a Number

The Necessity of Number

Since it has been said above that Music is a science which considers Numbers and Proportions, it seems to me that now is the time to begin to reason of such things — especially since from the first origin of the world (as is manifestly seen, and as the Philosophers affirm) all things created by God were ordered by Him with Number; indeed Number was the principal exemplar in the mind of the Creator. Whence it is necessary that all things, whether they exist separately or together, are comprehended by number and subject to it: so necessary is it that if it were taken away, first all things would be destroyed, and afterward there would be taken from man (as Plato holds) prudence and knowledge — since he would be unable to render account of anything he had in the intellect or in the memory; the arts would be lost, and there would be no longer any need to speak or write anything concerning Music, for the entire foundation of it would be annulled, having no greater firmness than that of numbers. Number sharpens the intellect, strengthens the memory, directs the intellect toward speculation, and conserves all things in their proper being. Moreover, the blessed God gave it to man as an instrument necessary to all his reason and discourse.

In the Sacred Letters, an infinite number of most marvelous and divine secrets come to be discovered through the means of numbers — of the knowledge and understanding of which (as Augustine holds) without the aid of numbers we would certainly be deprived. Our Savior, as is seen in the Gospel in many places, observed them; and all the ceremonies of the Written Law are comprehended through number. So that (as Augustine also says) Numbers and Music are found honorably placed in Scripture in several passages. Whence one should not marvel if the Pythagoreans esteemed that in numbers there was a certain something divine. So that for what we have said, and for what could further be said in discoursing with the intellect, Number is supremely necessary.

The Definition of Number

And although many have defined it, nonetheless Euclid of Megara seems to me to have described it most excellently, saying: Number is a multitude composed of several Unities. Which Unity, although it is not itself a number, is nonetheless the principle of number — and from it every thing, whether simple or composite, corporeal or spiritual, comes to be called One: for just as one cannot call any thing white except by reason of whiteness, so one cannot call any thing one except by reason of Unity. Which Unity is so contained within the thing that is, that that thing conserves itself in its proper being precisely insofar as it contains Unity within itself — and on the contrary, when it ceases to be one, it then lacks its being. And in this, Unity is no different from the Point, which is a minimal indivisible in a line: for just as when the Point is moved (as some hold) it makes the line, and is not on that account called a Quantity but rather the principle of Quantity; so Unity is not a number, even though it is the principle of number.

Unity, Principle, and the Ternary

And just as the end is not, and cannot be called, except in relation to the principle, so the principle cannot be except it has relation to the end. And therefore it is to be noted that a thing is not called a principle except by reason of the end; nor an end except in relation to the principle — so that from the principle to the end one cannot arrive except through the middle. It will therefore be necessary that every thing which is whole and complete should contain within itself beginning, middle, and end — all three of which are contained in the Ternary number, called by the Philosopher Perfect for this reason. Whence Unity, lacking the middle and the end, cannot be called a number, but only the principle of those numbers which are disposed in natural order — for the natural disposition of numbers is this: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, an order which can be continued to infinity by adding Unity.

Which Unity, because from it every quantity has its beginning — whether continuous or discrete — is called the Genetrix, that is, the principle, origin, and common measure of every number: since each number contains Unity within itself more than once. So for example the Binary, which follows immediately after Unity, is formed only through the joining of two Unities, from which the Binary itself results — the first even number. And to this Binary, then adding Unity, the Ternary is formed — the first odd number; from which with Unity appended the Quaternary is made, called the evenly even Number; and from this and Unity is produced the Quinary, called the Incomposite Number — and so the others of diverse species, proceeding to infinity.