De institutione musica III.11: Demonstratio Archytae superparticularem in aequa dividi non posse eiusque reprehensio.
A superparticular proportion cannot be cut into equal parts by a proportional number interposed between. This will be demonstrated firmly later. For the demonstration which Archytas gives is far too weak. It is of this kind. Let there be, he says, a superparticular proportion AB; I take in the same proportion the least numbers C, DE. Since, then, C and DE are the least in the same proportion and are superparticular, the number DE exceeds the number C by one part of itself. Let this be D. I say that D will not be a number, but a unit. For if D is a number and is a part of it, the number D measures DE; therefore it will also measure E, whence it comes about that it measures C also. The number D will therefore measure both the numbers C and DE, which is impossible. For those which are least in the same proportion to any other numbers are prime to each other, and keep only unity as their difference. D is therefore a unit. The number DE therefore exceeds the number C by a unit. Therefore no middle number falls in which would cut that proportion equally. Hence it comes about that between those which hold the same proportion as these, no middle number can be placed which would cut the same proportion equally.
And according to the reasoning of Archytas, no middle term falls in a superparticular proportion that would divide the proportion equally, because the least numbers in the same proportion differ only by a unit; as if the least numbers in a multiple proportion also did not obtain the same difference of a unit, although we see that there are many multiples, besides those placed at the roots, between which a middle term cutting the same proportion equally can be fitted. But he who has carefully examined arithmetical numbers understands this more easily. It must be added that it happens as Archytas thinks only in the superparticular proportion; it must not, however, be said universally. Now let us turn to what follows.