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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 See also:

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

diameter of a circle which is not the centre, by equal arcs of that circle, the See also:

ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

angle nearer to the centre is always less than the angle more remote (Heiberg, See also:

PREFACE (Med. Lat. prefatia, for classical praefatio, praefari, to speak beforehand)

preface, p. xviii.). The See also:

BOOK

book On the Section of the Cylinder had for its See also:

PRIMARY

primary See also:

OBJECT

object the correction of an See also:

ERROR (Lat. error, from errare, to wander, to err)

error on the See also:

PART

part of many geometers of the See also:

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 See also:

SERIES (a Latin word from serere, to join)

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 See also:

ELLIPSE (adapted from Gr. EXkei 'tc, a deficiency, iXXeirely, to fall behind)

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 See also:

PLANE

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 See also:

DRAWN

drawn from the same point to See also:

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

touch a cylindrical (or conical) See also:

SURFACE

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 See also:

PROPERTY

property of a See also:

HARMONIC

harmonic See also:

PENCIL (Lat. penicillus, brush, literally little tail)

pencil. The See also:

TREATISE

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 See also:

AXIS (Lat. for " axle ")

axis of a scalene cone, and solving, in some easy cases, the problem of finding triangular sections of given See also:

AREA

area. (T.

End of Article: SERENUS

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