Mean

MEAN. The adjective "mean" is chiefly used in the sense of "average," as in mean temperature, mean birth or death rate, etc.

In astronomy (q.v.) the "mean sun" is a fictitious sun which moves at a uniform rate in the celestial equator and has its right ascension always equal to the sun's mean longitude. The time recorded by the mean sun is mean-solar or clock time; it is reg ular as distinct from the non-uniform solar or sun-dial time. The "mean moon" is a fictitious moon which moves around the earth with a uniform velocity and in the same time as the real moon. The "mean longitude" of a planet is the longitude of the "mean" planet, i.e., a fictitious planet performing uniform revolutions in the same time as the real planet.

Mean in Mathematics.

The term "mean," in its most general sense, is given to some function of two or more quantities which (I) becomes equal to each of the quantities when they themselves are made equal, and (2) is unaffected in value when the quantities suffer any transpositions. The three commonest means are the arithmetical, geometrical and harmonic ; of less importance are the contraharmonical, arithmetico-geometrical and quadratic.

The arithmetical mean of n quantities is the sum of the quan tities divided by their number n. The geometrical mean of n

quantities is the nth root of their product. The harmonic mean of 71 quantities is the reciprocal of the arithmetical mean of their reciprocals. The significance of the word "mean," i.e., middle, is seen by considering 3 instead of n quantities ; these will be denoted by a, b, c. The arithmetic mean b, is seen to be such that the terms a, b, c are in arithmetical progression, i.e., b=1(a+c), the geo metrical mean b places a, b, c in geometrical progression, i.e., in the proportion a:b::b:c or and the harmonic mean places the quantities in harmonic proportion, i.e., a : c a—b : b—c, or b=2ac/(a+c). The contraharmonical mean is the quan tity b given by the proportion a : c : : b—c : a—b, i.e., The arithmetico-geometrical mean of two quantities is obtained by first forming the geometrical and arithmetical means, then forming the means of these means, and repeating the process until the numbers become equal. They were invented by Gauss to facilitate the computation of elliptic integrals. The quadratic mean of n quantities is the square root of the arithmetical mean of their squares.