# 1911 Encyclopædia Britannica/Serenus of Antissa

**SERENUS** “of Antissa,” Greek geometer, probably not of
Antissa but of Antinoeia or Antinoupolis, a city in Egypt founded
by Hadrian, lived, as may be safely inferred from the character
and contents of his writings, long after the golden age of Greek
geometry, most probably in the 4th century, between Pappus
and Theon of Alexandria. Two treatises of his have survived
viz. *On the Section of the Cylinder* and *On the Section of the Cone*,
the Greek text of which was first edited by Edmund Halley
along with his Apollonius (Oxford, 1710), and has now appeared
in a definitive critical edition by J. L. Heiberg (*Sereni Antissensis*
*opuscula*, Leipzig, 1896). A Latin translation by Commandinus
appeared at Bologna in 1566, and a German translation
by E. Nizze in 1860–1861 (Stralsund). Besides these works
Serenus wrote commentaries on Apollonius, and in certain MSS.
of Theon of Smyrna there appears a proposition “of Serenus
the philosopher, from the Lemmas” to the effect that, if a
number of rectilinear angles be subtended, at a point on a
diameter of a circle which is not the centre, by equal arcs of that
circle, the angle nearer to the centre is always less than the
angle more remote (Heiberg, preface, p. xviii.).

The book *On the Section of the Cylinder* had for its primary object
the correction of an error on the part of many geometers of the time
who supposed that the transverse sections of a cylinder were different
from the elliptic sections of a cone. When this has been done,
Serenus, in a series of theorems ending with Prop. 19 (ed. Heiberg),
shows in Prop. 20 that “it is possible to exhibit a cone and a cylinder
cutting one another in one and the same ellipse.” He then solves
problems such as—“given a cone (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 formed shall be similar ellipses” (Props. 23, 24).
In Props. 27, 28 he deals with sub contrary and other similar sections
of a scalene cylinder or cone. He then gives the theorems: “All the
straight lines drawn from the same point to touch a cylindrical (or
conical) surface, on both sides, have their points of contact on the
sides of a single parallelogram (or triangle)” (Props. 29, 32). Prop.
31 states indirectly the property of a harmonic pencil.

The treatise *On the Section of the Cone*, though Serenus claims originality for it, is unimportant. It deals with the areas of triangular sections of right or scalene cones by planes through the vertex, finding. *e.g.* the maximum triangular section of a right cone and the maximum triangle through the axis of a scalene cone, and solving, in some easy cases, the problem of finding triangular sections of given area. (T. L. H.)