A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

of
THE COLLECTION. BOOK V 395
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 lj 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 if to a semicircle such
that CH = HE, and if CD, EF be drawn perpendicular to the
diameter AB, then (CD + EF)CE = 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 semicircle
;
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,
Section (5).
sin # + sin?/ .
cos y — cos x
, .
= cot 4(03 — y).
Of regular solids with surfaces equal, that is
greater which 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
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 XIV 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 octahedron
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 socalled
Little Astronomy, that is, the smaller astronomical
treatises which were studied as an introduction to the great
Syntaxia 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 TpLnXtvpov, three-side.