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.

A SEQUEL TO EUCLID.

Join PH, and produce it to meet the circumference of
the larger 0 in E. Draw QF |I to PE. Join EF,
which will be the common tangent required.
Dem.-The lines HE and QF are, from the construction, equal; and since they are II, the fig. HEFQ is a c;.-. the Z PEF = PHQ = right angle;.'. EF is a tangent
at E; and since Z EFQ = EHQ = right angle, EF is a
tangent at F. The tangent EF is called a direct common tangent.
If with P as centre, and a radius equal to the sum
of the radii of the two given Os, we shall describe a
0, we shall have a common tangent which will pass
between the Os, and one which is called a transverse
common tangent.
Prop. 8.-If a line passing through the centres of two
circles cut them in the points A, B, C, D, respectively;
then the square of their direct common tangent is equal
to the rectangle AC. BD.
Dem.-We have (see last fig.) AI = CQ; to each add
IC, and we get AC = IQ. In like manner, BD = GQ.
Hence AC. BD = IQ. QG = E2.
Cor. l. —If the two Os touch, the square of their
common tangent is equal to the rectangle contained by
their diameters.
Cor. 2.-The square of the transverse common tangent of the two Os = AD. BC.
Cor. 3.-If ABC be c
a semicircle, PE a I
to AB from any point R
P, CQD a ( touch-
ing PE, the semicircle
ACB, and the semicircle on PB; then, if A P B
QR be the diameter
of CQD, AB. QR = EP2.
Dem. PB. QR =PQ2 (Cor. 1)
AP. R = EP2-_ PQ2; (6)
therefore, by addition, AB. QR = EP2.
Cor. 4.-If two Os be described to touch an ordi