A history of Greek mathematics - Wilbourhall.org

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SERENUS 523
Observing that the sum of x 2 and r 2 — x 2 + h 2 is constant, we
and therefore A, is a maximum when
see that A 2 ,
x 2 = r 2 - x 2 + Ir, or x 2 = J (r 2 + li 2 ) ;
and, since x is not greater than r, it follows that, for a real
value of x (other than v), h is less than r, or the cone is obtuseangled.
When h is not less than r, the maximum triangle is
the triangle through the axis and vice versa (Props. 5,8);
when k = r, the maximum triangle is also right-angled
(Prop. 13).
If the triangle with base 2 c is equal to the triangle through
the axis, h 2 r 2 = c 2 (r 2 — c 2 + h 2 ) }
or (r 2 — c
2
) (c2 — h
2
)
= 0, and,
since ch 2 r 2 ,
and the triangle with base 2x is greater than either of the
equal triangles with bases 2r, 2c, or 2 h (Prop. 11).
In the case of the scalene cone Serenus compares individual
triangular sections belonging to one of three classes with other
sections of the same class as regards their area. The classes
are
(1) axial triangles, including all sections through the axis;
isosceles sections, i.e. the sections the bases of which are
(2)
perpendicular to the projection of the axis of the cone on the
plane of the base ;
(3) a set of triangular sections the bases of which are (a) the
diameter of the circular base which passes through the foot of
the perpendicular from the vertex to the plane of the base, and
(6) the chords of the circular base parallel to that diameter.
After two preliminary propositions (15, 16) and some
lemmas, Serenus compares the areas of the first class of
triangles through the axis. If, as we said, p is the perpendicular
from the vertex to the plane of the base, d the distance
of the foot of
this perpendicular from the centre of the base,
and 6 the angle which the base of any axial triangle with area
A makes with the base of the axial triangle passing through
p the perpendicular,
A =?V(£> 2 + d 2 sin 2