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• Serene (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 in1860-1861(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 rectilineal 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. IN hen 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 fora ed stall be sin ilar ellipses" (Props. 23, 24). In Props. 27, 28 he deals with subcontrary 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 cores 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.)