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.

110

110 ~A SEQUEL TO.EUCLID. r

In like manner, OR=,
but we have proved ON2 + 01R' r2;
therefore + =r
or * +(it + 8)2 (R - 8)2 r
This Proposition is an important one in the Theory of
Elliptic Functions (see iDur'ge, Theorie der.Ellitihe
Funetionen, p. 18 5). Our proof is as simple and elementary as could be desired. For another proof, by R1. F.
Davis, i\LA., see Educational Timi~es (reprint), vol. xxxii.
Prop. 15.-If ABC lbe a jplane triangle, AD, BE, CF its3
per~pendieulars, 0 their _point of intersection, then the four
circles whose centres are A, B, 0,0
and the squares of whose radii are
respectively equal to the rectangles
AO. AD, BO. BE, CO. CF, E
OA. OD, are mutually orthogonal.D
Dem.-AO. AD +BO. BE0
-AF. AB + BF. BA. = AB 2.
Hence the sum of the squaresAF
of the radii of the ODs whose AF 3
centres are the points A, B = AB2;.. these ODs cut
orthogonally. Similarly the ODs whose centres are C
and A cut orthogonally.
Again, let us consider the fourth 0D, whose centre is
the point 0, and the square of whose radius is = to the
rectangle OA. OD. Now, since OA and OD are measured in opposite directions, they have contrary signs;. the rectangle OA. OD is negative, and the 0D has a
radius whose square is negative; hence it is imaginary;
but, notwithstanding this, it fulfis the condition of intersecting the other ODs orthogonally. For AO. AD +
OA. OD = AO. AD - AO. OD = A 02; that is, the