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.

88

A SEQUEL TO EUCLID.

Cor. 1.-Any ( passing through the points C and
D will be cut orthogonally by the ( described on AB
as diameter.
Cor. 2.-The points C and D are inverse points with
respect to the 0 described on AB as diameter.
DEF.-If C and D be harmonic conjugates to A. and B,
AB is called a harmonic mean between AC and AD.
Observation.-This coincides with the the algebraic Definition of harmonic mean.
For AC, AB, AD being three magnitudes, we have
AC: CB:: AD: BD;
therefore AC: AD:: CB: BD;
that is, the 1st is to the 3rd as the difference between
the 1st and 2nd is to the difference between the 2nd
and 3rd, which is the algebraic Definition.
Cor.-In the same way it can be seen that DC is a
harmonic mean between DA and DB.
Prop. 3.-The Arithmetic mean is to the Geometric
mean as the Geometric mean is to the Harmonic mean.
Dem. —Upon AB as diameter describe a 0; erect
EF at right angles to AB through C; draw tangents to
the ( at E, F, meeting in D; wF
then, since the A OED is rightangled at E, and EC is L to/ \0
OD, we have OC. OD = OE2 A B D
= OB2. Hence, by Prop. 1, C
and D are harmonic conjugates
to A and B. Again, from the
same A, we have OD: DE:: DE: DC; but OD =
< (DA + DB) = arithmetic mean between DA and DB;
and DE is the geometric mean and DC the harmonic
mean between DA and DB.
Cor.-The reciprocals of the three magnitudes DA,
DO, DB are respectively DB, DC, DA, with respect to
DE2; but DA, DO, DB are in arithmetical progression.