On the Institution of Arithmetic

and under these, beginning from four, all the doubles, and let them be in this way.

These things, therefore, being thus posited, if the first grows from the multiplication of the first, that is, if the quaternion multiplies the ternary, or if the same first multiplies the second, that is, the octonary multiplies the ternary, or if the same first multiplies the third, that is, 16 multiplies the ternary, and so on until the last; or if the second multiplies the first and the second, or if the second multiplies the third and so on until the extremity; or if the third, beginning from the first, passes through to the extremity. And in this way, the fourth and all those higher in order, if they multiply those that are beneath them in the arrangement, will procreate all impariter pares evenly odd numbers. Let us take such an example of this matter: if you multiply three by four, 12 will result; or if 5 multiplies four, the number 20 will grow; or if again 7 multiplies 4, 28 will grow; and this up to the end. Again, if 8 multiplies 3, 24 will be born. If 8 multiplies 5, 40 will be produced; if 8 multiplies 7, 56 will be collected. And in this manner, if all the lower doubles are multiplied by the higher ones, or if the higher ones multiply those same lower ones, you will find that all that are born are evenly odd numbers. And this is the admirable form of this number: that when this arrangement and description of the numbers is examined, the property of the numbers is found according to the width of the evenly odd numbers and the length of the evenly even numbers. For two extremities are equal to two means, or two doubled extremities are equal to one mean. In length, however, it designates the property of the evenly even number. For that which is contained under two means is equal to that which is completed under the extremes, or that which is born from one mean is equal to that which is contained under both extremities. The description, however, which is posited beneath, was made in this way. However many times the ternary number has multiplied in the order of evenly even numbers, whoever were procreated from it are arranged in the first row. Again, those who were born by the quinary multiplying them were constituted in the second place. After those, however, whom the septenary procreated by multiplying the others, we wrote them down in the third place, and we completed the remaining part of the description in the same way.

In this following formula, the similarity of the evenly even and the evenly odd to the evenly odd is shown.