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 II.

27

we get
a.AP2 +. BP2 + ac.CP2 + d. DP2, &c.,
= a.AO2 + b.BO2 + c.C02 + d. DO2 + &c.,
+ (a + b + o + d, &c.)OP2.
This Proposition evidently includes the last.
Cor. 1.-The locus of a point, the sum of the squares
of whose distances from any number of given points,
multiplied respectively by any system of constants
a, b, c, d, is a circle, whose centre is the centre of
mean position of the given points for the system of
multiples a, b, c, d.
Cor. 2.-The sum of the squares for any system of
multiples will be a minimum when the lines are drawn
to the centre of mean position.
Prop. 12.-From the Propositions vi. and ix. it follows
that, if a line is divided into any two parts, the rectangle
of the parts is a maximum, and the sum of their squares
is a minimum, when the parts are equal.
Cor.-If a line be divided into any number of parts,
the continued product of all the parts is a maximum,
and the sum of their squares is a minimum when they
are all equal. For if we make any two of the parts
unequal, we diminish the continued product, and we
increase the sum of the squares.
SECTION II.
EXERCISES,
1. The second and third Propositions of the Second Book are
special cases of the First.
2. Prove the fourth Proposition by the second and third.
3. Prove the sixth by the fifth, and the tenth by the ninth.
4. If the L C of a A ACB be I of two right L s, prove
AB2 = AC2 + CB2 - AC. CB.
5. If C be I of two right L s, prove
AB2 = AC2 + CB2 + AC. CB.