[The present paper is a translation of the second part of the third book of Pappus's Mathematical Collection. Pappus's date is uncertain, but 300 a.d. may be taken as an approximation to it.

Throughout the translation I have used the word “progression” as a rendering of the Greek μεσότης, which has no English equivalent. The only other alternative was to employ the term mediety, from the Latin medietas.

The account of the various progressions given by Nicomachus, in his Arithmetical Introduction, differs somewhat from that of Pappus. I hope to have something to say about Nicomachus in a future paper.]