Fundamentals of Arithmetic; Fundamentals of Music

De principalitate unitatis On the primacy of unity

VII. Every number is also the mean of those numbers placed around it and joined in a natural arrangement. If those that are above the two joined in the middle are added together, the aforementioned number is also the middle portion of them; and again, of those that are joined in the second place above, since they themselves are also composed, the preceding number is in the place of the mean, and this will continue until the unity encountered puts an end to it. For instance, if one posits the number five, on either side around it are 4 below and 6 above. If these are joined, they make 10, of which the number 5 is the mean. Those, however, that are around them, that is, around 6 and 4, namely 3 and 7, if they are joined, the number five is their mean. Again, of those that are placed on either side, if they are joined, these are also the doubles of the number five; for above 3 is 2, and above 7 is 8. If these are joined, they make 10, of which 5 is again the mean. This same thing occurs in all numbers, until one can arrive at the limit of unity; for unity alone does not have two terms around it, and therefore it is the mean only of that which is near it. For next to 1, only the number two is naturally established, of which unity is the middle part. Therefore, it is certain that unity is the first of all numbers that are in natural arrangement, and it is rightly recognized as the progenitor of the entire, however extensive, plurality. 5, 10, 15, 20, 25