A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

SERENUS 519
520 COMMENTATORS AND BYZANTINES
survive in Greek. Tannery, as we have seen, conjectured
proof, that the said oblique sections cutting all the generators
that, in like manner, the first six of the thirteen Books of
are equally ellipses whether they are sections of a cylinder or
Diophantus's Arithmetica survive because Hypatia wrote
of a cone. He begins with a more general definition ' ' of a
commentaries on these Books only and did not reach the
cylinder to include any oblique circular cylinder. ' If in two
others.
equal and parallel circles which remain fixed the diameters,
while remaining parallel to one another throughout, are moved
The first writer who calls for notice in this chapter is one
round in the planes of the circles about the centres, which
who was rather more than a commentator in so far as he
remain fixed, and if they carry round with them the straight line
wrote a couple of treatises to supplement the Conies of
joining their extremities on the same side until they bring it
Apollonius, I mean Serenus. Serenus came from Antinoeia
back again to the same place, let the surface described by the
or Antinoupolis, a city in Egypt founded by Hadrian (a. d.
straight line so carried round be called a cylindrical surface!
117-38). His date is uncertain, but he most probably belonged
to the fourth century A.D., and came between Pappus
The cylinder is the figure contained by the parallel circles and
the cylindrical surface intercepted by them ; the parallel
and Theon of Alexandria. He tells us himself that he wrote
circles are the bases, the axis
a commentary on the Conies of Apollonius. 1 is the straight line drawn
This has
through their centres; the generating straight line in any
perished and, apart from a certain proposition of Serenus
' position is a side. Thirty-three propositions follow. Of these
the philosopher, from the Lemmas ' preserved in certain manuscripts
of Theon of Smyrna (to the effect that, if a number of
Prop. 6 proves the existence in an oblique cylinder of the
parallel circular sections subcontrary to the series of which
rectilineal angles be subtended at a point on a diameter of a
the bases are two, Prop. 9 that the section by any plane not
circle which is not the centre, by equal arcs of that circle, the
parallel to that of the bases or of one of the subcontrary
angle nearer to the centre is always less than the angle more
sections but cutting all the generators is not a circle
remote), we have only the two small treatises by him
;
the
entitled
next propositions lead up to the main results, namely those in
On the Section of a Cylinder and On the Section of a Cone.
Props. 14 and 16, where the said section is proved to have the
These works came to be connected, from the seventh century
property of the ellipse which we write in the form
onwards, with the Conies of Apollonius, on account of the
affinity of the subjects, and this no doubt accounts for their
QV 2 :PV.P'V = CD 2 :CP 2 ,
survival. They were translated into Latin by Commandinus
in 1566 ;
the first Greek text was brought out by
and in Prop. 17, where the property is put in the Apollonian
Halley along
with his Apollonius (Oxford 1710), and we
form involving the latus rectum,
now
QV = PV
have 2 . VR (see figure
the
definitive text edited by
on p. 137 above), which is expressed
Heiberg (Teubner
by saying that the square
1896).
on the semi-ordinate is equal to the rectangle applied to the
(a) On the Section of a Cylinder.
latus rectum PL, having the abscissa PV as breadth and falling
short by a rectangle similar to the rectangle contained by the
The occasion and the object of the tract On the Section of
diameter PP f and the latus rectum PL (which is determined
a Cylinder are stated in the preface. Serenus observes that
by the condition PL . PP'= DD'
many persons who 2 and is drawn at right angles
were students of geometry were under the
to PV). Prop. 18 proves the corresponding property with
erroneous impression that the oblique section of a cylinder
reference to the conjugate diameter DD' and the corresponding
latus rectum t
was different from the oblique section of a cone known as an
and Prop. 19 gives the main property in the
ellipse, whereas it is of course the same curve. Hence he
form QV 2 :PV.P'V = Q'V' 2 :PV. P'V. Then comes the
thinks it necessary to establish, by a regular geometrical
proposition that '
it is possible to exhibit a cone and a cylinder
1
Serenus, Opuscula, ed. Heiberg, p. 52. 25-6.
which are alike cut in one and the same ellipse ' (Prop. 20).