Against Those Who Have Written on Disease Patterns

​In this way also, if a period of eleven hours is presented, you will say that there are twenty-four eleven-day period types, which is “duodeciman” (twelfth-day fevers). And in this way also, if the period is of thirteen hours, twenty-four “quatuordeciman” (fourteenth-day fevers). Likewise, even if the presented period is greater than twenty-four hours, if indeed it has no common measure with that of twenty-four—just as it has happened in the case of twenty-five—you will say that there are twenty-four “vicesimosextan” (twenty-sixth-day) types (just as indeed also twenty). But if there is some common measure of both, just as in the case of twenty-seven the third is [the common measure], you will say that there are eight “deciman” (tenth-day fevers), since the third of twenty-four is eight, and of twenty-seven, nine. For it is clear that the period of the deciman type is of nine completed days, which produce a number of two hundred and sixteen hours, of which the presented number of twenty-seven hours is an eighth part. Therefore, this method is subordinate to the common one mentioned before according to the diagram, differing from that one by this much: by the ability, even without the diagram, to find some first number of hours measured by daily periods, of which the presented [number] is a part. And this theorem has been demonstrated also by Euclid in the *Elements*, and it is its seventh book. And one must, taking the greater common [measure] of both numbers, examine how many times this common measure measures each of them; for example, in the case of fifteen and twenty-four, the first common measure is the number three; and this measures fifteen by five, and twenty-four by eight. If then you multiply five by twenty-four, or eight by fifteen, you will find the same number produced, as of one hundred and twenty hours, and clearly of five days. And because of this, you will call the type “sextan” (sixth-day fever). According to the same argument, if the presented number is of thirteen hours, both this and that of twenty-four will measure in common the multiple arising from both of each other, which is that of three hundred and twelve. (Since thirteen is one of the numbers called “prime”. For they address in this way those of which there is no common measure except for the unit.) The number, therefore, which both measure will be three hundred and twelve. For this has the twenty-four hours thirteen times. And the type will be of thirteen complete days, that is, “quatuordeciman” (fourteenth-day fever); and since this [number, thirteen] measures three hundred and twelve twenty-four times, the multitude of the types will be twenty-four. At any rate, you now have also this proof of the method, although I did not wish to teach you useless things; for it would have been more beautiful to look down upon them entirely, remembering the Pythian command that it is proper to spare time.