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Classical Arithmetic, Lesson 02. Book I, Chapter 2

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Lesson

On the Definition of Number

Thales (624-548 BC) defined number conformably to the doctrine of the Egyptians by whom he was instructed, to be a collection of monads. But Pythagoras(570-495 BC) defined it to be the extension and energy of the spermatic reasons contained in the monad. Or otherwise, to be that which prior to all things subsists in a divine intellect, by which and from which all things are co-ordinated, and remain connumerated in an indissoluble order. Others, however, of his followers defined it to be a progression from unity through the magnitude of it.  But Eudoxus (408-355 BC) the Pythagorean says, that number is definite multitude, in which definition he distinguishes the species from the genus. The followers of Hippasus (530-450 BC) who were called Acousmatici said that number is the first paradigm of the fabrication of the world; and again, that it is the judicial instrument of the god who is the demiurgus of the universe. But Philolaus (470-385) says that number is the most excellent and self-begotten bond of the eternal duration of mundane natures.

On the Monad

But the monad is in discrete quantity that which is the least, or it is the first and common part of discrete quantity, or the principle of it. According to Thymaridas (400-350 BC) indeed it is bounding quantity; since the beginning and end of everything is called a bound. In certain things however, as in the circle and the sphere, the middle is called the bound. But those who are more modern, define the monad to be that according to which everything that exists is called “one”. To this definition however, the words, “however collected it may be,” are wanting. The followers of Chrysippus (279-206 BC) assert confusedly, that the monad is one multitude: for the monad alone is opposed to multitude. But certain of the Pythagoreans said that the monad is the confine of number and parts; for from it as from a seed and an eternal root, ratios are contrarily increased and diminished; Some through a division to infinity being always diminished by a greater number; but others being increased to infinity, are again augmented. Some likewise have defined the monad to be the form of forms, as comprehending in capacity or power, [i. e. causally] all the reasons which are in number. But it is considered after this manner, in consequence of wholly remaining in the reason of itself; as is likewise the case with such other things as subsist through the monad.

The monad, therefore, is the principle and element of numbers, which while multitude is diminished by subtraction, is itself deprived of every number, and remains stable and firm; since it is not possible for division to proceed beyond the monad. For if we divide the one which is in sensibles into parts, again the one becomes multitude and many; and by a subtraction of each of the parts we end in one. And if we again divide this one into parts, the parts will become multitude; and by an ablation of each of the parts, we shall at length arrive at unity. So that the one, so far as one, is divisible and indivisible. For another number indeed when divided is diminished and is divided into parts less than itself. Thus, for instance, 6 may be divided into 3 and 3, or into 4 and 2, or into 5 and 1. But the one in sensible things, if it is divided indeed, as a body it is diminished, and by section is divided into parts less than itself, but as a number it is increased; for instead of the one (whole) it becomes many (parts). So that according to this the one is divisible.  For nothing in sensible things which is divided, is divided into parts greater than itself; but that which is divided into parts greater than the whole, and into parts equal to the whole, is divided as number. Thus, if the one which is in sensible things, be divided into six equal parts, as number indeed, it will be divided into parts equal to the whole, viz. into 1, 1, 1, 1, 1, and 1; and also into parts greater than the whole, viz. into 4 and 2, for 4 and 2 as numbers are more than one. Hence the monad as number is indivisible. But it is called the monad, either from remaining immutable, and not departing from its own nature; for as often as the monad is multiplied into itself, it remains the monad; since once one is one.; and if we multiply the monad to infinity, it still continues to be the monad. Or it is called the monad, because it is separated, and remains by itself alone apart from the remaining multitude of numbers.

Since therefore number is the connective bond of all things, it is necessary that it should abide in its proper essence, with a perpetually invariable sameness of subsistence; and that it should be compounded, but not of things of a different nature. For what could conjoin the essence of number, since its paradigm joined all things. But it seems to be a composite from itself. Moreover, nothing appears to be composed from similars; nor yet from things which are conjoined by no analogy, and which are essentially separated from each other. Hence it is evident, since number is conjoined, that it is neither conjoined from similars, nor from things which mutually adhere by no analogy. Hence the primary natures of which number consists, are the even and the odd, which by a certain divine power, though they are dissimilar and contrary, yet flow from one source, and unite in one composition and modulation.

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