THE CONICS, BOOK I 
135 
ndicular to 
it and pro- 
nove, while 
>ass succes- 
.raight line 
lique or, as 
ling to the 
definitions 
fe stated as 
diameter to 
sects all the 
llel to any 
anight line 
c the curve 
an ordinate 
ou) to the 
die primary 
giving the 
¡ting all the 
en straight 
• meets the 
r ippOia) to 
ines drawn 
dlel to any 
; again the 
Ha/meters in 
di of which 
3 particular 
lords which 
e same way 
our modern 
rchimedes’s 
from an 
Apollonius 
le vertex to 
the centre of the circular base. After proving that all 
sections parallel to the base are also circles, and that there 
is another set of circular sections subcontrary to these, he 
proceeds to consider sections of the cone drawn in any 
manner. Taking any triangle through the axis (the base of 
the triangle being consequently a diameter of the circle which 
is the base of the cone), he is careful to make his section cut 
the base in a straight line perpendicular to the particular 
diameter which is the base of the axial triangle. (There is 
no loss of generality in this, for, if any section is taken, 
without reference to any axial triangle, we have only to 
select the particular axial triangle the base of which is that 
diameter of the circular base which is * 
at right angles to the straight line in 
which the section of the cone cuts the 
base.) Let ABC be any axial triangle, 
and let any section whatever cut the 
base in a straight line DE at right 
angles to BG\ if then PM be the in 
tersection of the cutting plane and the 
axial triangle, and if QQ' be any chord 
in the section parallel to DE, Apollonius. 
proves that QQ' is bisected by PM. In 
other words, PM is a diameter of the section, 
careful to explain that, 
Apollonius is 
‘ if the cone is a right cone, the straight line in the base {DE) 
will be at right angles to the common section {PM) of the 
cutting plane and the triangle through the axis, but, if the 
cone is scalene, it will not in general be at right angles to PM, 
but will be at right angles to it only when the plane through 
the axis (i.e. the axial triangle) is at right angles to the base 
of the cone ’ (I. 7). 
That is to say, Apollonius works out the properties of the 
conics in the most general way with reference to a diameter 
which is not one of the principal diameters or axes, but in 
general has its ordinates obliquely inclined to it. The axes do 
not appear in his exposition till much later, after it has been 
shown that each conic has the same property with reference 
to any diameter as it has with reference to the original 
diameter arising out of the construction; the axes then appear