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.

13

Cor.-If from any number of points JLs be let fall on
any line passing through their mean centre, the sum of
the Is is zero. Hence some of the I-s must be negative, and we have the sum of the Is on the positive
side equal to the sum of those on the negative side.
Prop. 18.- We may extend the foregoing Definition as
follows:-Let there be any system of points A, B, C, D, 8c.,
and a corresponding system of multiples a, b, c, d, pc.,
connected with them; then divide the line joining the points
AB into (a + b) equal parts, and let AG contain b of these
parts, and GB contain a parts. Again, join G to a third
point C, and divide GC into (a + b + c) equalparts, and let
GI contain c of these parts, and HC the remaining parts,
and so on; then the point last found will be the mean
centre for the system of multiples a, b, c, d, ic.
From this Definition we may prove exactly the same
as in Prop. 17, that if AL, BL, CL, &c., be the Ls
from the points A, B, C, &c., on any line L, then
a. AL + b. BL + c. CL + d. DL + &c.
= (a + b + c + d + &c.) times the J from the centre of
mean position on the line L.
DEF.-If a geometrical magnitude varies its position
continuously according to any law, and if it retains the
same value throughout, it is said to be a constant; but if
it goes on increasing for some time, and then begins to
decrease, it is said to be a maximum at the end of the
increase: again, if it decreases for some time, and then
begins to increase, it is a minimum when it commences to
increase.
From these Definitions it will be seen that a maximum value is greater than the ones which immediately
precede and follow; and that a minimum is less than
the value of that which immediately precedes, and less
than that which immediately follows. We give here a
few simple but important Propositions bearing on this
part of Geometry.