On the Institution of Arithmetic

that is, thirty-two and four. For thirty-two is the fourth part of the total sum, and four is the thirty-second. Again, above these parts, sixty-four is the second part, and two is the sixty-fourth part. This continues until the extremities make the limit, which it is not doubted enjoy the same correspondence. For every sum is once 128, and one is the 128th. But if we posit odd terms, that is, sums—for I name terms as sums—according to the nature of the odd, one mean can be found, and one will be its own counterpart. For if this order is posited, 1, 2, 4, 8, 16, 32, 64, there will be only one mean, that is, eight. This eight is the eighth part of the total sum and converts to itself in denomination and quantity. And in the same way, as mentioned above, the terms around them give each other mutual names according to their proper quantities and exchange their titles. For four is the sixteenth part of the total sum, and sixteen is the fourth. And again, above these terms, thirty-two is the second part of the total sum, and two is the thirty-second part; and once the whole sum is 64, the sixty-fourth is found to be the unit. This, therefore, is what was said: that all its parts are found to be evenly even in both name and quantity. This also, by much consideration and much constancy, is perfected by divinity: that the smaller sums arranged in order and piled up on top of each other in this number are always equal to the following one minus one. For if you add one to these two that follow, they make three, which falls short of the four by one. And if you add four to the superiors, they are seven, which are surpassed by the following eight by only a unit. But if you add those same eight to the aforementioned, fifteen will be made, which would exist as the quantity of the number sixteen, were it not hindered by the lesser unit. But the first generation of number also preserves and guards this. For the unit, which is the first, is more contracted than the two subsequent ones by only a unit; hence it is no wonder that the total growth of the sum agrees with its own principle. But this consideration will be most useful to us in recognizing those numbers which we will demonstrate to be superfluous, deficient, or perfect. For there, the accumulated quantity of the parts is compared to the limit of the whole number. That, too, we can by no means forget to mention: that in this number, with the parts multiplied and corresponding to each other, the greater extremity and the sum of the same number are created. And first, if the arrangements are even, the means are multiplied, and then those that are above them, up to the aforementioned extremities. For if the arrangements are even, according to the nature of the even, they will contain two terms in the middle, as in that arrangement of numbers in which the extreme term ends at 128. For in this number, the means are eight, that is, and sixteen, which, multiplied among themselves, will create the sum of the greater by increasing the plurality. For if you multiply eight times sixteen or sixteen times eight, the sum of 128 increases. And these numbers that are above them, if they are multiplied, do the same