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.

11

Con. —Let ABC be the A: let fall the 1 CD; cutoff
BE = AD; join EC; bisect the Z EDC by the line DF,
meeting EC in F; through F draw a 11 to AB, cutting
the sides BC, AC in the points G, IH; from G, H let fall
the IJs GI, HJ: the figure GIJSH is a square.
Dem.-Since the EDC is bisected by DF, and the
Z s K and L right angles, and DF common, FK = FL
(xxvi.); but FL = GH (Prop. 13, Cor. 1), and FK = GI
(xxxiv.);.-. GI = GI, and the figure IGHJ is a square,
and it is inscribed in the triangle.
Cor.-If we bisect the Z ADC by the line DF',
meeting EC produced in F', and through F' draw a
line 1! to AB meeting BC, and AC produced in G', H',
and from G', H' let fall is G'I', H'J' on AB, we shall
have an escribed square.
Prop. 15.-To divide a given line AB into any number
of equal parts.
Con.-Draw through A any line AF, making an Z
with AB; in AF take any point C, and cut off CD, DE,
EF, &c., each = AC, F
until we have as many
equal parts as the \
number into which we
want to divide AB-
say, for instance, four c
equal parts. Join BF;
and draw CG, DHI, El,
each II to BF; then A G H
is divided into four equal parts.
Dem.-Since ADH is a A, and AD is bisected in C,
and CG is I1 to DH; then (2) AH is bisected in G;
A~. AG = GIH. Again, through C draw a line II to AB,
cutting DIT and EI in K and L; then, since CD = DE,
we have (2) CK = KL; but CK = GH, and KL = HI;.. GIH = HI. In like manner, HI = IB. Hence the
parts into which AB is divided are all equal.
This Proposition may be enunciated as a theorem as follows:If one side of a A be divided into any number of equal parts, and
through the points of division lines be drawn 1I to the base, these U1 s
will divide the second side into the same number of equal parts,