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 SECOND.
SECTION I.
ADDITIONAL PROPOSITIONS.
Prop. I.-IJ ABC be an isosceles triangle, whose equal
sides are AC, 130; and if ClD be a line drawn from 0 to any,
point D in the base AB3;
then AD.DB=BC2_CD.C
Dem.-Let fall the I
CE; then AB is bisected
in E and divided un-E
ecqually inD;AE D
therefore AD. DB + ED2 =EB2;
adding to each side EC2;
therefore AD. DB ~ CD 2 =BC2; (I. xlvii"
therefore AD. DB =1302 - CD2.
Cor.-If the point be in the base produced, we shall
have AD. BD = CD2 _ 0132. If we consider that DB
changes its sign when P passes through 13, we see
that this case is included in the last..Prop. 2.-If ABC be any triangle, P the middle point
of AB, then AC2 + 130 2 = 2AD2 + 2DC2.