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

SERENUS 521
Serenus then solves such problems as these : Given a cone
(or cylinder) and an ellipse on it, to find the cylinder (cone)
which is cut in the same ellipse as the cone (cylinder)
(Props. 21, 22); given a cone (cylinder), to find a cylinder
(cone) and to cut both by one and the same plane so that the
sections thus made shall be similar ellipses (Props. 23, 24).
Props. 27, 28 deal with similar elliptic sections of a scalene
cylinder and cone ;
there are two pairs of infinite sets of these
similar to any one given section, the first pair being those
which are parallel and subcontrary respectively to the given
section, the other pair subcontrary to one another but not to
either of the other sets and having the conjugate diameter
occupying the corresponding place to the transverse in the
other sets, and vice versa.
In the propositions (29-33) from this point to the end of
the book Serenus deals with what is really an optical problem.
It is introduced by a remark about a certain geometer,
Peithon by name, who wrote a tract on the subject of
parallels. Peithon, not being satisfied with Euclid's treatment
of parallels, thought to define parallels by means of an
illustration, observing that parallels are such lines as are
shown on a wall or a roof by the shadow of a pillar with
a light behind it. This definition, it appears, was generally
ridiculed ; and Serenus seeks to rehabilitate Peithon, who
was his friend, by showing that his statement is after all
mathematically sound. He therefore proves, with regard to
the cylinder, that, if any number of rays from a point outside
the cylinder are drawn touching it on both sides, all
the rays
pass through the sides of a parallelogram (a section of the
cylinder parallel to the axis)—Prop. 29— and if they are
produced farther to meet any other plane parallel to that
of the parallelogram the points in which they meet the plane
will lie on two parallel lines (Prop. 30) he adds that the lines
;
will not seem parallel (vide Euclid's Optics, Prop. 6). The
problem about the rays touching the surface of a cylinder
suggests the similar one about any number of rays from an
external point touching the surface of a cone ;
these meet the
surface in points on a triangular section of the cone (Prop. 32)
and, if produced to meet a plane parallel to that of the
triangle, meet that plane in points forming a similar triangle
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