A history of Greek mathematics - Wilbourhall.org

522 COMMENTATORS AND BYZANTINES
(Prop. 33). Prop. 31 preceding these propositions is a particular
case of the constancy of the anharmonic ratio of a
pencil of four rays. If two sides AB, AC of a triangle meet
a transversal through D, an external point, in E, F and another
ray AG between AB and AG cuts DEF in a point G such
that ED : DF = EG : GF, then any other transversal through
D meeting AB, AG, AG in K, L, M is also divided harmonically,
i.e. KB DM : = KL : LM. To prove the succeeding propositions,
32 and 33, Serenus uses this proposition and a
reciprocal of it combined with the harmonic property of the
pole and polar with reference to an ellipse.
(f3) On the Section of a Gone.
The treatise On the Section of a Cone is even less important,
although Serenus claims originality for it. It deals mainly
with the areas of triangular sections of right or scalene cones
made by planes passing through the vertex and either through
the axis or not through the axis, showing when the area of
a certain triangle of a particular class is a maximum, under
what conditions two triangles of a class may be equal in area,
and so on, and solving in some easy cases the problem of
finding triangular sections of given area.
This sort of investigation
occupies Props. 1-57 of the work, these propositions
including various lemmas required for the proofs of the
substantive theorems. Props. 58-69 constitute a separate
section of the book dealing with the volumes of right cones
in relation to their heights, their bases and the areas of the
triangular sections through the axis.
The essence of the first portion of the book up to Prop. 57
is best shown by means of modern notation. We will call h
the height of a right cone, r the radius of the base ; in the
case of an oblique cone, let p be the perpendicular from the
vertex to the plane of the base, d the distance of
the foot of
this perpendicular from the centre of the base, r the radius
of the base.
Consider first
the right cone, and let 2 x be the base of any
triangular section through the vertex, while of course 2r is
the base of the triangular section through the axis.
A be the area of the triangular section with base 2x,
A = x V (r 2 — x 2 + h 2 ).
Then, if