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.

BOOK Vt.

M

sum of the squares of the radii of the circles whose
centres are at the points A, 0 = AG'. Hence these
circles cut orthogonally.
Observation.-In this Demonstration we have made the A
acute-angled, and the imaginary 0 is the one whose centre is at
the intersection of the Is, and the three others are real; hut if
the A had an ohtuse angle, the imaginary 0 would he the one
whose centre is at the ohtuse angle.
Prop. 16. ---If four circles be mutually orthogonal, and
if any figure be inverted with respect to eacis of the four
circles in successwon, the fourth, inversion will coincide, with
the original figure. C
Dem.-It will plainly be P
sufficient to prove this iPro
position for a single point,
for the general Proposition
will then follow. Let the
centres of the four ODs be
the angular points A, B,
C of a A, and 0 the intersection of its -Ls: the A F
squares of the radii -will
he AB.AF, 13A.BF, -
CO. OF, CF. CO. Now
let P be the point we operate on, and let I' he its inverse with respect to the 0) A, and IP" the inverse of
I? with respect to the 0D B. Join P"O and CP meeting
in P"'. Now, since F' is the inverse of P with respect
to the 0 A, the square of whose radius is AB. AF,
we have AB.AF =AP.AP';.-.theALAFP is equiangular to the A,- AP'B;.-. Z AFP = AP'B: in like
manner the L BFP" = APTB.,. the A s AFP, BP"/F
are equiangular,.-. rectangle AF. FB PPF. FPP".
Again, because 0 is the intersection of the -Ls of the
AL ABC, AF. FB = CF. OF. Hence CF.0OF =
PF. PiP", and the Ls OFP and OPT" are equal, since
the L s AFP and BFP" are equal;.. the A\ sP"FTO and
UPP are equiangular, and the Ls OP"P and PCF are
equal; hence the four points C,?"', F, P"` are concyclic;