An elementary treatise on cubic and quartic curves, by A. B. Basset.

86

SPECIAL CUBICS.

142. Newton has given the following geometrical construction
for drawing a cissoid.
The side CB of a right angle is of constant length 2c. The
side CA passes through a fixed point A, whilst the extremity B
moves along a fixed straight line whose distance from A is equal
to CB. Then the locus of the middle point P of CB is a cissoid.
Let 0 be the middle point of AD, then AO = OD =c; also
let CAD = 0, x= OM, y= PM. Then
~/ B c - x = MD = c sin 0,
y cos 0 + (c + x) sin 0 = CP =c,
p
^A / AM D whence eliminating 0, we get
x (x2 + y2) = 2cy2,
~C ~ and therefore the locus of P is a cissoid.
143. The cissoid was invented by the
Greek geometer Diodes for the purpose of obtaining a geometrical
construction for solving the problem of finding two mean proportionals between two straight lines; or, as it is sometimes
called, the duplication of the cube. This construction, combined
with Newton's method of drawing the curve, enables the problem
to be solved by the aid of mechanical appliances.
Let a and b be two straight lines, then it is required to
determine x and y such that
a/x = xly = y/b,
which requires that a2b = x3.
Let OA = a, OD = b; join AD meeting the cissoid
x (x2 + y2) = ay
in P. Join OP and produce it to meet the
Q asymptote in Q. Then A Q is the required
line.
From the equation of the cissoid
0 M A
PM2 OM
\OM- AM'
Bt PM AQ
But 0i11 a
OM-ii a'