# MEAN

, a middle state between two extremes: as a mean motion, mean distance, arithmetical mean, geometrical mean, &c.

*Arithmetical* Mean, is half the sum of the extremes.
So, 4 is an arithmetical mean between 2 and 6, or between
3 and 5, or between 1 and 7; also an arithmetical
mean between *a* and *b* is or .

*Geometrical* Mean, commonly called a mean proportional,
is the square root of the product of the two
extremes; so that, to find a mean proportional between
two given extremes, multiply these together, and extract
the square root of the product. Thus, a mean proportional
between 1 and 9, is ; a mean
between 2 and 4 1/2 is also; the
mean between 4 and 6 is ; and the
mean between *a* and *b* is √*ab.*

The geometrical mean is always less than the arithmetical
mean, between the same two extremes. So
the arithmetical mean between 2 and 4 1/2 is 3 1/4, but the
geometrical mean is only 3. To prove this generally;
let *a* and *b* be any two terms, *a* the greater, and *b* the
less; then, universally, the arithmetical mean
shall be greater than the geometrical mean √*ab,* or
greater than 2√*ab.* For, by
squaring both, they are ;
subtr. 4*ab* from each, then ,
that is - - - .

*To find a Mean Proportional
Geometrically,* between two given
lines M and N. Join the two
given lines together at C in one
continued line AB; upon the
diameter AB describe a semicircle,
and erect the perpendicular CD;
which will be the mean proportional
between AC and CB, or
M and N.

*To find two Mean Proportionals* between two given
extremes. Multiply each extreme by the square of
the other, viz, the greater extreme by the square of
the less, and the less extreme by the square of the
greate<*>; then extract the cube root out of each product,
and the two roots will be the two mean proportionals
sought. That is, √^{3}*a*^{2}*b* and √^{3}*ab*^{2} are the two
means between *a* and *b.* So, between 2 and 16, the
two mean proportionals are 4 and 8; for , and .

In a similar manner we proceed for three means, or
four means, or five means, &c. From all which it
appears that the series of the several numbers of mean
proportionals between *a* and *b* will be as follows: viz,
one mean, √*ab;*
two means, √^{3}*a*^{2}*b,* √^{3}*ab*^{2};
three means, √^{4}*a*^{3}*b,* √^{4}*a*^{2}*b*^{2}, √^{4}*ab*^{3};
four means, √^{5}*a*^{4}*b,* √^{5}*a*^{3}*b*^{2}, √^{5}*a*^{2}*b*^{3}; √^{5}*ab*^{4};
five means, √^{6}*a*^{5}*b,* √^{6}*a*^{4}*b*^{2}, √^{6}*a*^{3}*b*^{3}, √^{6}*a*^{2}*b*^{4},
√^{6}*ab*^{5};
&c, &c.

*Harmonical* Mean, is double a fourth proportional
to the sum of the extremes, and the two extremes
themselves *a* and *b:* thus, as the harmonical mean between *a* and *b.* Or it is
the reciprocal of the arithmetical mean between the
reciprocals of the given extremes; that is, take the
reciprocals of the extremes *a* and *b,* which will be
1/*a* and 1/*b;* then take the arithmetical mean between
these reciprocals, or half their sum, which will be
or ; lastly, the reciprocal of this is
the harmonical mean: for, arithmeticals
and harmonicals are mutually reciprocals of each
other;
so that if *a, m, b,* &c be arithmeticals,
then shall 1/*a,* 1/*m,* 1/*b,* &c be harmonicals;
or if the former be harmonicals, the latter will be
arithmeticals.

For example, to find a harmonical mean between
2 and 6; here *a* = 2, and *b* = 6; therefore
the harmonical
mean sought between 2 and 6.

In the 3d book of Pappus's Mathematical Collections
we have a very good tract on all the three
sorts of mean proportionals, beginning at the 5th proposition.
He observes, that the Ancients could not
resolve, in a geometrical way, the problem of finding
two mean proportionals; and because it is not easy
to describe the conic sections in plano, for that
purpose, they contrived easy and convenient instruments,
by which they obtained good mechanical constructions
of that problem; as appears by their writings;
as in the Mesolabe of Eratosthenes, of Philo,
with the Mechanics and Catapultics of Hero. For
these, rightly deeming the problem a solid one, effected
the construction only by instruments, and Apollonius
Pergæus by means of the conic sections; which others
again performed by the *loci solidi* of Aristæus; also
Nicomedes solved it by the conchoid, by means of|
which likewise he trisected an angle: and Pappus himself
gave another solution of the same problem.

Pappus adds definitions of the three foregoing different
sorts of means, with many problems and properties
concerning them, and, among others, this
curious similarity of them, viz, *a, m, b,* being three
continued terms, either arithmeticals, geometricals, or
harmonicals; then in the
Arithmeticals, *a* : *a* :: *a* - *m* : *m* - *b;*
Geometricals, *a* : *m* :: *a* - *m* : *m* - *b;*
Harmonicals, *a* : *b* :: *a* - *m* : *m* - *b.*

Mean-*and-Extreme Proportion,* or *Extreme-and-Mean
Proportion,* is when a line, or any quantity is so divided,
that the less part is to the greater, as the greater is
to the whole.

Mean *Anomaly,* of a planet, is an angle which is
always proportional to the time of the planet's motion
from thé aphelion, or perihelion, or proportional to
the area described by the radius vector; that is, as the
whole periodic time in one revolution of the planet,
is to the time past the aphelion or perihelion, so is
360° to the Mean anomaly. See Anomaly.

Mean *Axis,* in Optics. See Axis.

Mean *Conjunction* or *Opposition,* is when the mean
place of the <*>un is in conjunction, or opposition, with
the mean place of the moon in the ecliptic.

Mean *Diameter,* in Gauging, is a Mean between
the diameters at the head and bung of a cask.

Mean *Distance,* of a Planet from the Sun, is an
arithmetical mean between the planet's greatest and
least distances; and this is equal to the semitransverse
axis of the elliptic orbit in which it moves, or to the
right line drawn from the sun or focus to the extremity
of the conjugate axis of the same.

Mean *Motion,* is that by which a planet is supposed to
move equably in its orbit; and it is always proportional
to the time.

Mean *Time,* or Equal time, is that which is measured
by an equable motion, as a clock; as distinguished
from apparent time, arising from the unequal motion
of the earth or sun.