On the Institution of Arithmetic

the terms constituted above them, as is the case in this order: 2, 6, 10, 14. For 2 and 14 joined together grow into 16, which the number six coupled with ten will produce. And this occurs in more numerous terms, having taken the beginning from the middle, until it comes to the extremes.

On the evenly odd number and its properties. Chap. II.

A woodcut depicts a decorative, ornate initial 'I' formed with twisting floral and foliate scrollwork within a square frame.

The evenly odd number is fashioned from both, and it is concluded in the place of a mean by a doubled extremity, such that it differs from both, yet is joined to either by a kinship. This number is such that it divides into equal parts, and each part can be divided into other equal parts, and sometimes the parts of the parts are divided; but that uniform division does not progress all the way to the unit, as are 24, 7, 28 original: "24. 7. 28." - Note: The text likely refers to 24 and 28, with the 7 being a potential transcription error for another number, though it remains in the translation as per the OCR.. For these can be divided into halves, and their parts in turn can be divided into other halves without any doubt. There are also certain other numbers whose parts receive other divisions, but the division itself does not reach all the way to the unit. Therefore, in the fact that it admits more than one section, it has a similarity to the evenly even number; but it is separated from the evenly odd original: "pariter impari". However, in that it does not carry that section all the way to one, it does not refute the evenly odd, but it is disjoined from the evenly even. It happens, moreover, to this number both to have things that the former ones do not have, and to obtain both things that they receive. And it has indeed what both do not have: that while in one only the greater term divides, and in another only the lesser term does not divide, in this one neither the greater term alone receives division, nor does the lesser term alone separate from division. For they both resolve into parts and that section does not come all the way to the unit; but before the unit, a term is found which you cannot cut. It obtains also what those others receive: that certain of its parts correspond to the denomination which, according to its genus, relates to its proper quantity, namely in the likeness of the evenly even number. Other parts, however, take a contrary denomination of their proper quantity, namely to the form of the evenly odd number. For in the number 24, the quantity of the part denominated by an even number is even. For the fourth part is 6, the second is 12, the sixth is 4, the twelfth is 2, which names of parts do not differ from the parity of the quantity. Conversely, they are denominated when the third part is 8, the eighth is 3, and the twenty-fourth is 1, which denominations, since they are even, find odd quantities. And when the sums are even, the denominations are odd. Such numbers are born in such a way that they designate their substance and nature even in their very own generation, being procreated from evenly even and evenly odd numbers. For all evenly odd numbers were born from the previously posited odd numbers in order, while evenly even numbers arise from a double progression. Therefore, let all natural odd numbers be set in order...

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