396 
PAPPUS OF ALEXANDRIA 
synthetical deduction, for which he claims that it is clearer 
and shorter. We have first propositions (with auxiliary 
lemmas) about the perpendiculars from the centre of the 
circumscribing sphere to a face of (a) the octahedron, (b) the 
icosahedron (Props. 39, 43), then the proposition that, if a 
dodecahedron and an icosahedron be inscribed in the same 
sphere, the same small circle in the sphere circumscribes both 
the pentagon of the dodecahedron and the triangle of the 
icosahedron (Prop. 48) ; this last is the proposition proved by 
Hypsicles in the so-called ‘ Book XIY of Euclid ’, Prop. 2, and 
Pappus gives two methods of proof, the second of which (chap. 
56) corresponds to that of Hypsicles. Prop. 49 proves that 
twelve of the regular pentagons inscribed in a circle are together 
greater than twenty of the equilateral triangles inscribed in 
the same circle. The final propositions proving that the cube 
is greater than the pyramid with the same surface, the octa 
hedron greater than the cube, and so on, are Props. 52-6 
(chaps. 60-4). Of Pappus’s auxiliary propositions, Prop. 41 
is practically contained in Hypsicles’s Prop. 1, and Prop. 44 
in Hypsicles’s last lemma; but otherwise the exposition is 
different. 
Book VI. 
On the contents of Book VI we can be brief. It is mainly 
astronomical, dealing with the treatises included in the so- 
called Little Astronomy, that is, the smaller astronomical 
treatises which were studied as an introduction to the great 
Syntaxis of Ptolemy. The preface says that many of those 
who taught the Treasury of Astronomy, through a careless 
understanding of the propositions, added some things as being 
necessary and omitted others as unnecessary. Pappus mentions 
at this point an incorrect addition to Theodosius, Sphaerica, 
III. 6, an omission from Euclid’s Phaenomena, Prop. 2, an 
inaccurate representation of Theodosius, On Days and Nights, 
Prop. 4, and the omission later of certain other things as 
being unnecessary. His object is to put these mistakes 
right. Allusions are also found in the Book to Menelaus’s 
Sphaerica, e.g. the statement (p. 476. 16) that Menelaus in 
his Sphaerica called a spherical triangle TpinXtvpov, three-side.