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 I.

7

lines, be given in magnitude and position, and if m, n,
p, q, &c., be any system of multiples, all positive, or
some positive and some negative, and if the area of
m times A ABP + n times CDP +p times GHP + &c.,
be given, the locus of P is a right line.
Cor. 4.-If ABCD be a quadrilateral, and if P be a
point, so that the sum of the areas of the As ABP,
CDP is half the area of the quadrilateral, the locus
of P is a right line passing through the middle points
of the three diagonals of the quadrilateral.
Prop. 9.-To divide a given line AB into two parts,
the difference of whose spuares shall be equal to the square
of a given line CD. E D
Con.-Draw BE at
right angles to AB,
and make it = CD;
join AE, and make
the Z AEF = EAB;
then F is the point
required. B
Dem.-Because the Z AEF = EAF, the side AF
= FE;.-. AF2 = FE2 = FB2 + BE2;.-. AF - FB2 = BE2;
but BE2 = CD2;.. AF2 - FB2 = CD2.
If CD be greater than AB, BE will be greater than AB, and
the L EAB will be greater than the L AEB; hence the line EF,
which makes with AE the L AEF = L EAB, will fall at the
other side of EB, and the point F will be in the line AB produced.
The point F is in this case a point of external division.
Prop. 10.-Given the base of a triangle in magnitude
and position, and given also the difference of the squares of
its sides, to find the locus of its vertex.
Let ABC be the A whose base AB is given; let fall
the ~L CP on AB; then
AC2 = AP2 + CP2; (xlvii.)
BC = 3BP2 + CP2;
therefore AC2 - BC2 = AP2- BP2;
but AC2 - BC2 is given;.-. AP2 - BP2 is given. Hence
AB is divided in P into two parts, the difference of
whose squares is given;.'. P is a given point (9), and
the line CP is given in position; and since the point C