THE COLLECTION. BOOKS III, IV 
369 
this way. If d be the diameter of the sphere, set out two 
straight lines x, y such that d, x, y are in the ratio of the sides 
of the regular pentagon, hexagon and decagon respectively 
described in one and the same circle. The smaller pair of 
circles have r as radius where r 2 = ^y 2 , and the larger pair 
have r' as radius where r' 2 = -|ic 2 . 
(e) In the case of the dodecahedron the same four parallel 
circular sections are drawn as in the case of the icosahedron. 
Inscribed pentagons set the opposite way are inscribed in the 
two smaller circles; these pentagons form opposite faces. 
Regular pentagons inscribed in the larger circles with vertices 
at the proper points (and again set the opposite way) determine 
ten more vertices of the inscribed dodecahedron. 
The constructions are quite different from those in Euclid 
XIII. 13, 15, 14, 16, 17 respectively, where the problem is first 
to construct the particular regular solid and then to ‘com 
prehend it in a sphere ’, i. e. to determine the circumscribing 
sphere in each case. I have set out Pappus’s propositions in 
detail elsewhere. 1 
Book IV. 
At the beginning of Book IV the title and preface are 
missing, and the first section of the Book begins immediately 
with an enunciation. The first section (pp. 176-208) contains 
Propositions 1-12 which, with the exception of Props. 8-10, 
seem to be isolated propositions given for their own sakes and 
not connected by any general plan. 
Section (1). Extension of the theorem of Pythagoras. 
The first proposition is of great interest, being the generaliza 
tion of Eucl. I. 47, as Pappus himself calls it, which is by this 
time pretty widely known to mathematicians. The enunciation 
is as follows. 
‘If ABC be a triangle and on AB, AG any parallelograms 
whatever be described, as ABDE, AG EG, and if DE, EG 
produced meet in H and HA be joined, then the parallelo 
grams ABDE, ACFG are together equal to the parallelogram 
1 Vide notes to Euclid’s propositions in The Thirteen Books of Euclid's 
Elements, pp. 478, 480, 477, 489-91, 501-8. 
b b 
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