(9) A Collection of **Lemmas**, consisting of fifteen propositions in plane geometry.

Halley added in his edition (1710) a restoration of Book viii., in which he was guided by the fact that Pappus gives **lemmas** "to the seventh and eighth books" under that one heading, as well as by the statement of Apollonius himself that the use of the seventh book was illustrated by the problems solved in the eighth.

Pappus then enumerates works of Euclid, Apollonius, Aristaeus and Eratosthenes, thirty-three books in all, the substance of which he intends to give, with the **lemmas** necessary for their elucidation.

In 1749 was published Apollonii Pergaei locorum planorum libri II., a restoration of Apollonius's lost treatise, founded on the **lemmas** given in the seventh book of Pappus's Mathematical Collection.

Of the eight books which made up his original treatise, only seven are certainly known, the first four in the original Greek, the next three are found in Arabic translations, and the eighth was restored by Edmund Halley in 1710 from certain introductory **lemmas** of Pappus.