A history of Greek mathematics - Wilbourhall.org

THE CONICS, BOOKS IV-V 159
most remarkable of the extant Books.
It deals with normals
to conies regarded as maximum and minimum straight lines
drawn from particular points to the curve. Included in it are
a series of propositions which, though worked out by the
purest geometrical methods, actually lead immediately to the
determination of the evolute of each of the three conies ;
that
is to say, the Cartesian equations to the evolutes can be easily
deduced from the results obtained by Apollonius.
be no doubt that the Book is
a veritable geometrical tour de force.
There can
almost wholly original, and it is
Apollonius in this Book considers various points and classes
of points with reference to the maximum or minimum straight
lines which it is possible to draw from them to the conies,
i. e. as the feet of normals to the curve. He begins naturally
with points on the axis, and he takes first the point E where
AE measured along the axis from the vertex A is \p, p being
the principal parameter. The first three propositions prove
generally and for certain particular cases that, if
in an ellipse
or a hyperbola AM be drawn at right angles to AA' and equal
to J p, and if CM meet the ordinate PN of any point P of the
curve in H, then PN 2 = 2 (quadrilateral MANH)
;
this is a
lemma used in the proofs of later propositions, V. 5, 6, &c.
Next, in V. 4, 5, 6, he proves that, if AE = \p, then AE is the
minimum straight line from E to the curve, and if P be any
other point on it, PE increases as P moves farther away from
A on either side ;
from P,
he proves in fact that, if PN be the ordinate
(1) in the case of the parabola PE 9 - = AE 2 + AN 2 ,
(2) in the case of the hyperbola or ellipse
PE 2
= A& + AN* •
AA
AA
A r,
where of course p = BB' 2 /AA\ and therefore (AA / ±p) / A A'
is equivalent to what we call e 2 ,
P
,
the square of the eccentricity.
It is also proved that EA' is the maximum, straight line from
E to the curve. It is next proved that, if be any point on
the axis between A and E, OA is the minimum straight line
from to the curve and, if P is any other point on the curve,
OP increases as P moves farther from A (V. 7).