In book v., after an interesting preface concerning regular poly gons, and containing remarks upon the hexagonal form of the cells of honeycombs, Pappus addresses himself to the comparison of the areas of different plane figures which all have the same perimeter (following Zenodorus's treatise on this subject), and of the volumes of different solid figures which all have the same superficial area; and, lastly, a comparison of the five regular solids. Incidentally Pappus describes the 13 other polyhedra bounded by equilateral and equiangular but not similar polygons, discovered by Archimedes, and finds, by a method recalling that of Archimedes, the surface and volume of a sphere.
According to the preface, book vi. is intended to resolve diffi culties occurring in the so-called /..uxpin IcarpovoiwEvos. It ac cordingly comments on the Sphaerica of Theodosius, the Moving Sphere of Autolycus, Theodosius's book On Days and Nights, the treatise of Aristarchus, On the Sizes and Distances of the Sun and Moon, and Euclid's Optics and Phaenomena.
The preface of book vii. explains the terms analysis and syn thesis, and the distinction between theorem and problem. Pappus then enumerates works of Euclid, Apollonius, Aristaeus and Eratosthenes, 33 books in all, the scope of which he intends to describe, with the lemmas necessary for their elucidation. With the mention of the Porisms of Euclid we have an account of the relation of porism to theorem and problem. In the same preface are included (a) the famous problem known as "Pappus's Prob lem," which formed, so to say, the "text" of Descartes's Geome tric; (b) the theorems which were rediscovered by and named after Paul Guldin, but appear to have been discovered by Pappus him self. Book vii. contains also (I), under the head of the de deter
minata 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 gives a complete proof of the focus-directrix property of the three conic sections.
Lastly, book viii. treats principally of mechanics, the properties of the centre of gravity, and some mechanical powers. Inter spersed are some questions of pure geometry. Proposition shows how to draw an ellipse through five given points.