368
PAPPUS OF ALEXANDRIA
the other, he. it is shown that broken lines, consisting of
several straight lines, can be drawn with two points on the
base of a triangle or parallelogram as extremities, and of
greater total length than the remaining two sides of the
triangle or three sides of the parallelogram.
Props. 40-2 show that triangles or parallelograms can be
constructed with sides respectively greater than those of a given
triangle or parallelogram but having a less area.
Section (4). The inscribing of the five regular solids
in a sphere.
The fourth section of Book III (pp. 132-62) solves the
problems of inscribing each of the five regular solids in a
given sphere. After some preliminary lemmas (Props. 43-53),
Pappus attacks the substantive problems (Props. 54-8), using
the method of analysis followed by synthesis in the case of
each solid.
(a) In order to inscribe a regular pyramid or tetrahedron in
the sphere, he finds two circular sections equal and parallel
to one another, each of which contains one of two opposite
edges as its diameter. If d be the diameter of the sphere, the
parallel circular sections have cl' as diameter, where d 2 = | d' 2 .
(b) In the case of the cube Pappus again finds two parallel
circular sections with diameter d' such that d 2 =-|d' 2 ; a square
inscribed in one of these circles is one face of the cube and
the square with sides parallel to those of the first square
inscribed in the second circle is the opposite face.
{(■) In the case of the octahedron the same two parallel circular
sections with diameter d' such that d 2 = f d' 2 are used; an
equilateral triangle inscribed in one circle is one face, and the
opposite face is an equilateral triangle inscribed in tire other
circle but placed in exactly the opposite way.
(d) In the case of the icosahedron Pappus finds four parallel
circular sections each passing through three of the vertices of
the icosahedron; two of these are small circles circumscribing
two opposite triangular faces respectively, and the other two
circles are between these two circles, parallel to them, and
equal to one another. The pairs of circles are determined in
