THE COLLECTION. BOOK V
395
of the sphere, Pappus quotes Archimedes, On the Sphere and
Cylinder, but thinks proper to add a series of propositions
(chaps. 20-43, pp. 362-410) on much the same lines as those of
Archimedes and leading to the same results as Archimedes
obtains for the surface of a segment of a sphere and of the whole
sphere (Prop. 28), and for the volume of a sphere (Prop. 35).
Prop. 36 (chap. 42) shows how to divide a sphere into two
segments such that their surfaces are in a given ratio and
Prop. 37 (chap. 43) proves that the volume as well as the
surface of the cylinder circumscribing a sphere is 1-| times
that of the sphere itself.
Among the lemmatic propositions in this section of the
Book Props. 21, 22 may be mentioned. Prop. 21 proves that,
if C, E be two points on the tangent at H to a semicircle such
that GH = HE, and if CD, EF be drawn perpendicular to the
diameter AB, then (CD + EF) GE = AB. DF; Prop. 22 proves
a like result where C, E are points on the semicircle, CD, EF
are as before perpendicular to AB, and EH is the chord of
the circle subtending the arc which with CE makes up a semi
circle ; in this case (CD + EF) CE = EH . DF. __ Both results
are easily seen to be the equivalent of the trigonometrical
formula
sin {x + y) + sin {x — y) = 2 sin x cos y,
or, if certain different angles be taken as x, y,
sin x + sin y , , . .
= cot 4 [x — y).
cosy— cos x
Section (5). Of regular solids with surfaces equal, that is
greater ivhich has more faces.
Returning to the main problem of the Book, Pappus shows
that, of the five regular solid figures assumed to have their
surfaces equal, that is greater which has the more faces, so
that the pyramid, the cube, the octahedron, the dodecahedron
and the icosahedron of equal surface are, as regards solid
content, in ascending order of magnitude (Props. 38-56).
Pappus indicates (p. 410. 27) that ‘some of the ancients’ had
worked out the proofs of these propositions by the analytical
method; for himself, he will give a method of his own by
