A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

82

A SEQUEL TO EUCLID.

is an inscribed triangle given in species. Hence, if M
be the point of intersection of circles described about
the As PAQ, QDR, the A MAD is given in species. —
See Demonstration of (III., 17).
In like manner, if N be the point of intersection of
the Os about the As QAP, PBS, the A ABN is given
in species. Hence the ratios AM: AD and AN: AB
are given; but the ratio of AB to AD is given, because
the figure ABCD is given in species. Hence the ratio
of AM: AN is given; and M, N are given points;
therefore the locus of A is a circle (7); and where this
circle intersects the circle PAQ is a given point. Hence
A is given.
Cor.-A suitable modification of the foregoing, and
making use of (III., 16), will enable us to solve the
cognate Problem-To describe a quadrilateral of given
species whose four vertices shall be on four given
lines.
(6). Given the base of a triangle, the difference of the
base angles, and the rectangle of the sides, construct it.
(7). Given the base of a triangle, the vertical angle, and
the ratio of the sum of the sides to the altitude: construct
it.
SECTION II.
CENTRES OF SIMILITUDE.
DEF.-If the line joining the centres of two circles be
divided internally and externally in the ratio of the radii
of the circles, the points of division are called, respectively,
the internal and the external centre of similitzde of the
two circles.
From the Definitions it follows that the point of
contact of two circles which touch externally is an internal centre of similitude of the two circles; and the
point of contact of two circles, one of which touches
another internally, is an external centre of similitude.
Also, since a right line may be regarded as an infinitely