* 
SERENUS 
531 
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 pro 
blem. 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 treat 
ment 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