The Arithmetic of Boethius

departs from the highest simplicity of unity, is composed, and necessarily consists of itself. It should therefore not seem surprising if number, concerning which the human mind philosophizes as the first thing initiated, is asserted by the Pythagoreans to consist of itself. Nor did this escape Plato, who established the infinite and the finite as the principles of numbers. He called the multitude deserted by unity the infinite, but the union by which every number has it that it is called one, the finite. Nor did he intimate anything other than those Pythagoreans who thought it consisted of the one and the other, of the even and the odd. From which it is further explored that number consists of units and their union. Furthermore, it should be noted that this composition of number has the greatest moment for Divine theoriai contemplations/theories, in which part the artifice of our mind is known to correspond to Divine artifice. To these it seems should be added that Nicomachus, in one or another place of the second book, composes every number of the even and the odd, where he takes number aggregately, for the entire body and series of numbers. And there is no doubt that the series of numbers has alternating even and odd numbers, and thus number and the series of numbers consist of the even and the odd. But the discussion of this same matter must be deferred to that place.

ON THE DEFINITION AND DIVISION OF NUMBER: AND various definitions of the even and the odd. CHAP. III.

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FIRST, it must be defined what number is. Number is a collection of units: or, a heap of quantity produced from units. 7 Pointing hand icon The first division of this, therefore, is into odd and even. 8 Pointing hand icon And the even is that which can be divided into two equal parts, with no middle unit intervening. The odd, however, is that which no one divides into equal parts without the aforementioned one intervening in the middle. 9 And this definition of such a kind is common and well-known. That one, however, according to the Pythagorean discipline, is such: An even number is that which, under the same division, can be divided into the greatest and the smallest—the greatest in space, the smallest in quantity—according to the contrary passions of these two kinds. An odd number, however, is that to which this cannot happen, but to which a natural section into two unequal sums belongs. This is the exemplar: as if any given even number is divided, a greater one (as far as the spaces of division are concerned) will not be found than a discrete half. But in quantity, no partition is smaller than that made into a double. As if the even number 8 is divided into 4 and 4, there will be no other division that makes larger parts. Furthermore, there will be no other division that divides the whole number in smaller quantity. For in a division into two parts, there is nothing smaller. For when one has parted the whole by a triple division, the sum of space is decreased, but the number of the division is increased. But what was said, "according to the contrary passions of two kinds," is of this sort. For we have taught that quantity increases into infinite pluralities, but spaces, that is, magnitudes, are diminished into infinite smallnesses. And therefore, the opposite happens here. For this division of the even is greatest in space, and smallest in quantity. According to an older manner, there is another definition of an even number. An even number is that which [can be divided] into two equal [parts], and...