A history of Greek mathematics - Wilbourhall.org

;
(r
+
x
ON THE SPHERE AND CYLINDER, II 49
How this ingenious analysis was suggested it is not possible
to say. It is the equivalent of reducing the four unknowns
h, hf, k, k' to two, by putting h = r + x, h' = r—x and h' = y,
and then reducing the given relations to two equations in x, y,
which are coordinates of a point in relation to Ox, Oy as axes,
where is the middle point of AA\ and Ox lies along 0A\
while Oy is perpendicular to it.
Our original relations (p. 47) give
h + k
-., ah' r — x ah r + x , m
7
y = fc = —r- = a j k = 77 = a j and — =
•
t-.—^
h r + x h r — x n h +k
We have at once, from the first two equations,
Icy = a y = a ,
whence
and
{r + x)y = a (r — x),
(x + r) (y + a) = 2 ra,
which is the rectangular hyperbola (4) above.
. m
6
' n
,
h + k '\ r — x/
""
W + kf
+ x)(l + -^—)
(r-x)(l+^-)
r + x,
whence we obtain a cubic equation in x,
which gives
(r + x) 2 {r + a — x) = — (r — &)
2 (r + c& + x),
m / w/^+a + arv
— (r— o;W
2
x) 2 (~ -I =(r + a) 2 —
w v V r
2
.
=
>
,
t, ,
But
1/
~^— =
a .
whence
y + r — x r + a + x
r — & r + # r — a; r + #;
and the equation becomes
— (y + r — x)
2
= (r + a) 2 — x
2
,
which is the ellipse (3) above.
1523.2 E