(12) Enough upon that side of the question. But how does the perfection of numbers, lifeless things, depend upon their particular unity? Just as all other inanimates find their perfection in their unity.

If it should be objected that numbers are simply non-existent, we should point out that our discussion is concerned with beings considered from the aspect of their unity.

We may again be asked how the point- supposing its independent existence granted- participates in perfection. If the point is chosen as an inanimate object, the question applies to all such objects: but perfection does exist in such things, for example in a circle: the perfection of the circle will be perfection for the point; it will aspire to this perfection and strive to attain it, as far as it can, through the circle.

But how are the five genera to be regarded? Do they form particulars by being broken up into parts? No; the genus exists as a whole in each of the things whose genus it is.

But how, at that, can it remain a unity? The unity of a genus must be considered as a whole-in-many.

Does it exist then only in the things participating in it? No; it has an independent existence of its own as well. But this will, no doubt, become clearer as we proceed.

(3) Now the number 300 is, 3 by 100. Ten is allowed to be the perfect number. And 8 is the first cube, which is equality in all the dimensions - length, breadth; depth. "The days of men shall be," it is said, "120 (rk) years." And the sum is made up of the numbers from r to 15 added together. And the moon at 15 days is full.

(6) This child once undertook and completely succeeded in raising the number 8 progressively up to the sixteenth power—in naming the result, 281,474,976,710,656 he was absolutely correct in every figure. He could raise any given number progressively up to the 10th power, with so much speed that the person putting down the figures on paper would frequently request him to manifest less speed. He gave instantly the square root of 106,929, and the cube root of 268,336,125. He could give the prime factors of very large numbers, and could detect large prime numbers instantly. Once asked how many minutes there were in forty-eight years, and before the question could be written down he answered "25,228,800", adding "and the number of seconds in such period is 1,513,728,000." The child, when questioned concerning his ability to give such answers, and to solve such difficult problems, was unable to give such information. He could say that he did not know how the answer came into his mind, but it was evident from watching him that some actual process was under way in his mind, and that there was no question of mere trick of memory in his feats. Moreover, it is important to note that he was totally ignorant of even the common rules of arithmetic, and could not "figure" on slate or paper even the simplest sum in addition or multiplication. It is interesting to note the sequel to this case, i.e., the fact that when a few years later the child was sent to the common schools and was there instructed in the art of written arithmetic, his power began to vanish, and eventually it left him altogether, and he became no more than any other child of his age. It seemed as if some door of his soul had been closed, while before it had stood ajar.

(5) Another marked case is that of Zerah Colburn, the mathematical prodigy, whose feats attracted the attention of the scientific world during the last century. In this case, the child under eight years of age, without any previous knowledge of even the common rules of arithmetic, or even of the use and powers of the Arabic numerals, solved a great variety of arithmetical problems by a simple operation of the mind, and without the use of any visible symbols or contrivances. He could answer readily a question involving the statement of the exact number of minutes or seconds in any given period of time. He could also state with equal facility the exact product of the multiplication of any number containing two, three, or four figures by another number consisting of a like number of figures. He could state almost instantly all the factors composing a number of six or seven places of figures. He could likewise determine instantly questions concerning the extraction of the square and cube roots of any number proposed, and likewise whether it was a prime number incapable of division by any other number, for which there is no known general rule among mathematicians. Asked such questions in the midst of his ordinary childish play, he would answer them almost instantly and then proceed with his play.

(20) For the motion of the sun from solstice to solstice is completed in six months - in the course of which, at one time the leaves fall, and at another plants bud and seeds come to maturity. And they say that the embryo is perfected exactly in the sixth month, that is, in one hundred and eighty days in addition to the two and a half, as Polybus the physician relates in his book On the Eighth Month, and Aristotle the philosopher in his book On Nature. Hence the Pythagoreans, as I think, reckon six the perfect number, from the creation of the world, according to the prophet, and call it Meseuthys and Marriage, from its being the middle of the even numbers, that is, of ten and two. For it is manifestly at an equal distance from both.

(4) On another principle, 120 is a triangular number, and consists of the equality of the number 64, [which consists of eight of the odd numbers beginning with unity], the addition of which (1, 3, 5, 7, 9, 11, 13, 15) in succession generate squares; and of the inequality of the number 56, consisting of seven of the even numbers beginning with 2 (2, 4, 6, 8, 10, 12, 14), which produce the numbers that are not squares Again, according to another way of indicating. the number 120 consists of four numbers - of one triangle, 15; of another, a square, 25; of a third, a pentagon, 35; and of a fourth, a hexagon, 45. The 5 is taken according to the same ratio in each mode. For in triangular numbers, from the unity 5 comes 15; and in squares, 25; and of those in succession, proportionally. Now 25, which is the number 5 from unity, is said to be the symbol of the Levitical tribe. And the number 35 depends also on the arithmetic, geometric, and harmonic scale of doubles - 6, 8, 9, 12; the addition of which makes 35. In these days, the Jews say that seven months' children are formed. And the number 45 depends on the scale of triples - 6, 9, 12, 18 - the addition of which makes 45; and similarly, in these days they say that nine months' children are formed.

(6) And the numbers introduced are sixfold, as three hundred is six times fifty; and tenfold, as three hundred is ten times thirty; and containing one and two-thirds (epidimoiroi), for fifty is one and two-thirds of thirty.

(546) the laws which regulate them will not be discovered by an intelligence which is alloyed with sense, but will escape them, and they will bring children into the world when they ought not. Now that which is of divine birth has a period which is contained in a perfect number, 1 but the period of human birth is comprehended in a number in which first increments by involution and evolution [ or squared and cubed] obtaining three intervals and four terms of like and unlike, waxing and waning numbers, make all the terms commensurable and agreeable to one another. 2 The base of these (3) with a third added (4) when combined with five (20) and raised to the third power furnishes two harmonies; the first a square which is a hundred times as great (400 = 4 × 100), 3 and the other a figure having one side equal to the former, but oblong, 4 consisting of a hundred numbers squared upon rational diameters of a square (i.e. omitting fractions), the side of which is five (7 × 7 = 49 × 100 = 4900), each of them being less by one (than the perfect square which includes the fractions, sc. 50) or less by 5 two perfect squares of irrational diameters (of a square the side of which is five = 50 + 50 = 100); and a hundred cubes of three (27 × 100 = 2700 + 4900 + 400 = 8000). Now this number represents a geometrical figure which has control over the good and evil of births. For when your guardians are ignorant of the law of births, and unite bride and bridegroom out of season, the children will not be goodly or fortunate. And though only the best of them will be appointed by their predecessors, still they will be unworthy to hold their fathers’ places, and when they come into power as guardians, they will soon be found to fail in taking care of us, the Muses, first by under-valuing music; which neglect will soon extend to gymnastic; and hence the young men of your State will be less cultivated. In the succeeding generation rulers will be appointed who have lost the guardian power of testing the metal of your

(1) As then in astronomy we have Abraham as an instance, so also in arithmetic we have the same Abraham. "For, hearing that Lot was taken captive, and having numbered his own servants, born in his house, 318 (tih)," he defeats a very great number of the enemy.

(2) What, then, of the "Number of the Infinite"?

To begin with, how is Number consistent with infinity?

Objects of sense are not unlimited and therefore the Number applying to them cannot be so. Nor is an enumerator able to number to infinity; though we double, multiply over and over again, we still end with a finite number; though we range over past and future, and consider them, even, as a totality, we still end with the finite.

Are we then to dismiss absolute limitlessness and think merely that there is always something beyond?

No; that more is not in the reckoner's power to produce; the total stands already defined.

In the Intellectual the Beings are determined and with them Number, the number corresponding to their total; in this sphere of our own- as we make a man a multiple by counting up his various characteristics, his beauty and the rest- we take each image of Being and form a corresponding image of number; we multiply a non-existent in and so produce multiple numbers; if we number years we draw on the numbers in our own minds and apply them to the years; these numbers are still our possession.