^{53} “Pseudo-Ṭūsī,” *Taḥrīr*, pp. 125–7:Google Scholar

“Any two groups of magnitudes equal in number, whatever the number may be, each pair from a group according to the ratio of a pair from the other group, with the ratio ordered, are in an *ex aequali* relation when, if the first from the first group is greater than the last from it, the first from the second group is greater than the last from it; and if equal, it is equal; and if less, it is less.

Let *A, B, G, D, E, Z* be two groups of the same number, and the ratio of *A* to *B* be as the ratio of *D* to *E* and the ratio of *B* to *G* as the ratio of *E* to *Z*. I say that if *A* is greater than *G, D* is greater than *Z*; and if *A* is equal to *G, D* is equal to *Z*; and if *A* is less than *G, D* is less than *Z*.

The proof is to let *A* be greater than *G*. Then, because the ratio of *D* to *E* is as the ratio of *A* to *B*, and the ratio of *A* to *B* is greater than the ratio of *G* to *B* according to proposition eight, then by proposition twelve, the ratio of *D* to *E* is greater than the ratio of *G* to *B*. But the ratio of *G* to *B* is as the ratio of *Z* to *E*. Thus the ratio of *D* to *E* is greater than the ratio of *Z* to *E* by using proposition twelve. Therefore, by proposition ten, *D* is greater than *Z*.

And if *A* is equal to *G*, then, because the ratio of *D* to *E* is as the ratio of *A* to *B*, and *A* is equal to *G*, the ratio of *G* to *B* is as the ratio of *A* to *B*, according to proposition nine. Thus, by proposition eleven, the ratio of *D* to *E* is as the ratio of *G* to *B*. But the ratio of *Z* to *E* is as the ratio of *G* to *B* by inversion. Thus, the ratio of *D* to *E* is as the ratio of *Z* to *E*, according to proposition eleven. Therefore, *D* is equal to *Z* by proposition nine.

And if *A* is less than *G*, then, by inversion, the ratio of *E* to *D* is as the ratio of *B* to *A*. But the ratio of *B* to *A* is greater than the ratio of *B* to *G*, according to proposition eight. Thus, by proposition twelve, the ratio of *E* to *D* is greater than the ratio of *B* to *G*. But the ratio of *B* to *G* is as the ratio of *E* to *Z*. Thus the ratio of *E* to *D* is greater than the ratio of *E* to *Z*, by using proposition twelve. Therefore, by proposition ten, *D* is less than *Z*. The proposition is established, and that is what we wanted to show.”

“<Any two groups of magnitudes equal in number, whatever the number may be,> each pair from a group according to the ratio of a pair from the other group, and the ratio being perturbed, are in an *ex aequali* relation when, if the first from the first group is greater than the last from it, the first from the other group is greater than the last from it; and if it is equal, it is equal; and if it is less, it is less.

Let *A, B, G, D, E, Z* be two groups of magnitudes of the same number, and the ratio of *A* to *B* be as the ratio of *E* to *Z* and the ratio of *B* to *G* as the ratio of *D* to *E*. I say that if *A* is greater than *G, D* is greater than *Z*; and if it is equal, then it is equal; and if it is less, it is less.

The proof is that, because the ratio of *E* to *Z* is as the ratio of *A* to *B*, and *A* is greater than *G*, the ratio of *A* to *B* is greater than the ratio of *G* to *B* by proposition eight. Thus, the ratio of *E* to *Z* is greater than the ratio of *G* to *B* according to proposition twelve. But, by inversion, the ratio of *G* to *B* is as the ratio of *E* to *D*. Thus, the ratio of *E* to *Z* is greater than the ratio of *E* to *D* by using proposition twelve. Therefore, *D* is greater than *Z* by proposition ten.

But if *A* is equal to *G*, then, because the ratio of *E* to *Z* is as the ratio of *A* to *B* and *A* is equal to *G*, the ratio of *G* to *B* is as the ratio of *A* to *B* by proposition seven. By proposition eleven, the ratio of *E* to *Z* is as the ratio of *G* to *B*. But by inversion, the ratio of *E* to *D* is as the ratio of *G* to *B*. Thus, by proposition eleven, the ratio of *E* to *Z* is as the ratio of *E* to *D*. Therefore, *D* <and> *Z* are equal by proposition nine.

And if *A* is smaller than *G*, then, by inversion, the ratio of *E* to *D* is as the ratio of *G* to *B*. But the ratio of *G* to *B* is greater than the ratio of *A* to *B* by proposition eight. Thus, by proposition twelve, the ratio of *E* to *D* is greater than the ratio of *A* to *B*. But the ratio of *A* to *B* is as the ratio of *E* to *Z*. Thus the ratio of *E* to *D* is greater than the ratio of *E* to *Z* by using proposition twelve. Therefore, *D* is less than *Z* by proposition ten. And that is what we wanted to show.”