DISCOVERY OF THE CONIC SECTIONS 111 
r PERGA 
ONIUS 
naechmus. 
»roblem of the 
luplication of 
he is credited 
e epigram of 
nus whereas 
e rectangular 
lutions. The 
ink of obtain- 
3 no informa- 
)f a section of 
ich of course 
7 he the right 
the attention 
>ed, would be 
a, cylinder by 
ccurred, as it 
solid was cut 
ould presum- 
be made to 
•f the elonga- 
¡ctions of the 
3 most easily 
er; it would 
ler the curve 
perty as that 
ve seen, the 
observation that an ellipse can be obtained from a cylinder 
as well as a cone is actually made by Euclid in his Phaeno- 
mena: ‘ if says Euclid, ‘ a cone or a cylinder be cut by 
a plane not parallel to the base, the resulting section is a 
section of an acute-angled cone which is similar to a dope 69 
(shield),’ After this would doubtless follow the question 
what sort of curves they are which are produced if we 
cut a cone by a plane which does not cut through the 
cone completely, but is either parallel or not parallel to 
a generator of the cone, whether these curves have the 
same property with the ellipse and with one another, and, 
if not, what exactly are their fundamental properties respec 
tively. 
As it is, however, we are only told how the first writers on 
conics obtained them in actual practice. We learn on the 
authority of Geminus 1 that the ancients defined a cone as the 
surface described by the revolution of a right-angled triangle 
about one of the sides containing the right angle, and that 
they knew no cones other than right cones. Of these they 
distinguished three kinds; according as the vertical angle of 
the cone was less than, equal to, or greater than a right angle, 
they called the cone acute-angled, right-angled, or obtuse- 
angled, and from each of these kinds of cone they produced 
one and only one of the three sections, the section being 
always made perpendicular to one of the generating lines of 
the cone; the curves were, on this basis, called ' section of an 
acute-angled cone ’ (= an ellipse), ‘ section of a right-angled 
cone’ (= a parabola), and ‘section of an obtuse-angled cone ’ 
(= a hyperbola) respectively. These names were still used 
by Euclid and Archimedes. 
Menaechmuss probable procedure. 
Menaechmus’s constructions for his curves would presum 
ably be the simplest and the most direct that would show the 
desired properties, and for the parabola nothing could be 
simpler than a section of a right-angled cone by a plane at right 
angles to one of its generators. Let OBC (Fig. 1) represent 
1 Eutocius, Comm, on Conics of Apollonius.