SERENUS of Antissa, an ancient Greek geometer, the author of two treatises— De Sectione Cylindri et Coni, libri duo— which Halley has published in Greek and Latin along with his edition of the Conics of Apollonius of Perga. Great difference of opinion has existed as to his date: Halley says in his preface to the Conics, “We know nothing of Serenus except that he was born at Antissa, a town in the island of Lesbos; and that, besides his book On the Section of the Cylinder, and another On the Section of the Cone, he wrote commentaries on Apollonius; and that he lived before Marinus—the pupil of Proclus—as appears from the preface of Marinus to the Data of Euclid.” Montucla says vaguely that Serenus lived within the first four centuries of the Christian era. Chasles places him about the same time as Pappus. Bretschneider pointed out that Antissa was completely destroyed by the Romans in 167 B.C., and inferred thence that Serenus lived c. 220- 180 b.c. To this inference it has been fairly objected by Cantor, after F. Blass, that the name Serenus is Latin and that Antissa had been rebuilt at the time of Strabo. The statement of Halley that “he lived before Marinus” has been since repeated by many writers; but Heiberg has pointed out (Rev. Crit. d'Hist. et de Litt., 1881, p. 381) that the passage referred to in support of this statement is faulty, and that the name of Serenus is certainly not to be found in it. Th. H. Martin, in his edition of the Astronomy of Theon of Smyrna (Paris, 1849), has published a fragment which in the MS. follows the text of Theon and is headed From the Lemmas of the Philosopher Serenus. This is unquestionably the same as Serenus of Antissa, to whom this appellation “philosopher” is given in the titles of the two treatises edited ...by Halley. No conclusion, however, can be drawn from this as to the date of Serenus, for the extract is not given by Theon but by an anonymous scholiast. M. Paul Tannery in an elaborate paper (Bull. des Sc. Math. et Astron., 2d series, vii., 1883) has shown from the character of Serenus’s writings that he lived long after the brilliant period of Greek mathematics, and that he must be placed chronologically between Pappus and Hypatia, consequently in the 4th century. This determination of the date of Serenus is accepted by Cantor (Zeitschrift für Math. und Phys., August 1885, p. 124).
In the treatise On the Section of the Cone, which is the less important of the two books, Serenus, as he tells us in the preface, was the first to take up the particular branch of that subject with which he deals. In it he treats of the area of a triangle formed by cutting a cone, right or scalene, on a circular base by a plane through the vertex. He shows how “to cut a right cone whose axis is not less than the semi-diameter of the base by a plane through the vertex so that the triangle thus formed shall be equal to a given triangle” (Prop. 8), or “a maximum” (Prop. 13). He then considers the case of the scalene cone, solves the problem “to cut a given scalene cone by a plane through the vertex so as to form an isosceles triangle” (Prop. 21), and shows that, “of the triangles which are formed by cutting a scalene cone through the axis, the greatest is the isosceles, the least that which is at right angles to the base of the cone; of the rest, however, that which is nearer the greatest is greater than one more remote” (Prop. 22). The general questions for a scalene cone, corresponding to the problems for the right cone (Props. 8 and 13), and which depend on solid loci for their solution, are not attempted. These have been solved by Halley in his edition of Serenus, p. 68 sq.
In his preface to the treatise On the Section of the Cylinder, Serenus tells us that many geometers of his time supposed that the transverse sections of a cylinder were different from the elliptic sections of a cone, that he thought it right to refute this error and to prove that these sections were of the same kind. Having established this in a series of theorems ending with Prop. 18, he shows in Prop. 19 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. 20, 21); “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. 22, 23); “given a cylinder cut in an ellipse, to construct a cone having the same base and altitude as the cylinder, so that the section of it by the same plane is an ellipse similar to the ellipse of the cylinder” (Prop. 25). In Props. 26-29 he shows how to cut a scalene cylinder or cone in an infinite number of ways by two planes—which are not parallel—so as to form similar ellipses (subcontrary sections). He then gives some theorems: “all the straight lines drawn from the same point to touch a cylindrical surface, on both sides, have their points of contact on the sides of a single parallelogram” (Prop. 31); “all the straight lines drawn from the same point to touch a conical surface, on both sides, have their points of contact on the sides of a single triangle” (Prop. 34). This last is proved by means of Prop. 33, where we find, indirectly stated, the property of an harmonic pencil.
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