SERENUS

This article appears in Volume V24, Page 663 of the Encyclopedia Britannica.

Encyclopedia Britannica - Main :: SCY-SHA

SERENUS " of Antissa," Greek geometer, probably not' of Antissa but of Antinoeia or Antinoupolis, a city in

Egypt

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

THEON
THEON, AELIUS

of Alexandria. Two treatises of his have survived, viz. On the Section of the Cylinder and On the Section of the

Cone

CONE (Gr. Kwvos)

, 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

CRITICAL (in alphabetical order of authors)

edition by J. L. Heiberg (Sereni Antissensis opuscula,

Leipzig

LEIPZIG

, 1896). A Latin translation by Cornmandinus appeared at Bologna in 1566, and a German translation by E. Nizze in 186o-1861 (Stralsund). Besides these works Serenus wrote commentaries on Apollonius, and in certain MSS. of

Theon

THEON
THEON, AELIUS

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

DIAMETER (from the Cr. &a, through, µErpov, measure)

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

CONE (Gr. Kwvos)

. When this has been done, Serenus, in a

series

SERIES (a Latin word from serere, to join)

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 subcontrary and other similar sections of a scalene cylinder or cone. He then gives the theorems: " All the straight lines

drawn

DRAWN

from the same point to

touch

TOUCH (derived through Fr. toucher from a common Teutonic and Indo-Germanic root, cf. " tug," " tuck," O. H. Ger. zucchen, to twitch or draw)

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

AXIS (Lat. for " axle ")

of a scalene cone, and solving, in some easy cases, the problem of finding triangular sections of given area. (T. L. H.)

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