On the Principles of Arithmetic and On the Principles of Music

VII. Also, every number is the midpoint original: medietas; here referring to the arithmetic mean of those placed around it and joined to it in a natural arrangement; and those which are beyond those two that are joined at the middle—if they are added together—the aforementioned number is also the middle portion of them;
5 and again of those which are joined in the second place beyond them, since they themselves are also added, this former number sits in the place of the midpoint; and this will continue until the unity In Neo-Pythagorean arithmetic, "unity" (the number 1) is often considered the "seed" of number rather than a number itself. is encountered and sets a limit. For instance, if one were to set down the number
10 five original: quinarium, on either side around it are 4 above In this context, "above" and "below" refer to the numbers preceding and following in the natural sequence: 1, 2, 3, 4, 5, 6, 7, 8... and 6 below. Therefore, if these are joined, they make 10, of which the number 5 is the midpoint. Furthermore, those which are around them—that is, around 6 and 4—namely 3 and 7, if these same are joined, the number 5 is their midpoint;
15 again, of those placed on either side, if they are joined, even these are double the number 5; for above 3 is 2, and above 7 is 8; therefore, if these are joined, they make 10, of which 5 is again the midpoint. This same thing happens in all numbers, until one can arrive
20 at the limit of unity; for unity alone does not have two terms around itself, and for that reason it is the "midpoint" only of that which is near itself. For next to 1, only the number two original: binarius is naturally established, of which unity is the middle part. Wherefore it is established that unity is the first of all those numbers which are in the natural
25 arrangement, and that it is rightly recognized as the mother of the whole plurality, however extended it may be.

1 Title omitted in d. 3 and those who are above those two in f. 4 joined, with u written over a in r. || are composed in f. 7 this [hic] in a, [is] with h deleted by erasure in l. 9 made a limit in c, d, f, l; made a limit and [or] sets [ponat] written above the line in b. || In following the sequences 1, 2, 3 up to 10 in the margin of f, l, r; moreover 1, 2, 3 up to 11 in the context of f. 10 original 4 in d. 11 are in a, b, d, l, r; were in c, s. || they will make in a, b, c, l, s. || five [quinarius] in c, f, r, s. 13 three in a, f, l, s. || joined if in c, d, r, s. || are in a. 14 it will be in d. 15 they are joined in d, it is joined in r. 16 double in c. || three in d. || two in f. || 6 are 7 in c. 17 are in a, b, c, d, l, r. 18 the number five in f. 20 however in b, d, f, l, r. || around in f, l. 22 one in c, f, s. 25 extensively in c, d, f, l.