(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.

Contains also (I), under the head of the de determinate sectione of Apollonius, **lemmas** which, closely examined, are seen to be cases of the involution of six points; (2) important **lemmas** on the Porisms of Euclid (see PoRIsM); (3) a lemma upon the Surface Loci of Euclid which states that the locus of a point such that its distance from a given point bears a constant ratio to its distance from a given straight line is a conic, and is followed by proofs that the conic is a parabola, ellipse, or hyperbola according as the constant ratio is equal to, less than or greater than i (the first recorded proofs of the properties, which do not appear in Apollonius).

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.