Mean, an homonymous word, the chief uses of which may be divided thus. (1) A verb with two principal applications, to intend, purpose or design, and to signify. This word is in O.E. maenan, and cognate forms appear in other Teutonic languages, cf. Du. meenen, Ger. meinen. The ultimate origin is usually taken to be the root men-, to think, the root of "mind." (2) An adjective and substantive meaning "that which is in the middle." This is derived through the O. Fr. men, meien or moien, modern ntoyen, from the late Lat. adjective medianus, from medius, middle. The law French form mesne is still preserved in certain legal phrases (see Mesne). The adjective "mean" is chiefly used in the sense of "average," as in mean temperature, mean birth or death rate, &c.
"Mean" as a substantive has the following principal applications; it is used of that quality, course of action, condition, state, &c., which is equally distant from two extremes, as in such phrases as the "golden (or happy) mean." For the philosophic application see Aristotle and Ethics.
In mathematics, the term "mean," in its most general sense, is given to some function of two or more quantities which (1) 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.
From the sense of that which stands between two things, "mean," or the plural "means," often with a singular construction, takes the further significance of agency, instrument, &c., of which that produces some result, hence resources capable of producing a result, particularly the pecuniary or other resources by which a person is enabled to live, and so used either of employment or of property, wealth, &c. There are many adverbial phrases, such as "by all means," "by no means," &c., which are extensions of "means" in the sense of agency.
The word "mean" (like the French moyen) had also the sense of middling, moderate, and this considerably influenced the uses of "mean" (3). This, which is now chiefly used in the sense of inferior, low, ignoble, or of avaricious, penurious, "stingy," meant originally that which is common to more persons or things than one. The word in O. E. is gemaene, and is represented in the modern Ger. gemein, common. It is cognate with Lat. communis, from which "common" is derived. The descent in meaning from that which is shared alike by several to that which is inferior, vulgar or low, is paralleled by the uses of "common." In astronomy the "mean sun" is a fictitious sun which moves uniformly 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 termed mean-solar or clock time; it is regular 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.
The arithmetical mean of n quantities is the sum of the quantities divided by their number n. The geometrical mean of n quantities is the nth root of their product. The harmonic mean of n quantities is 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 = 2 (a+c); the geometrical mean b places a, b, c in geometrical progression, i.e. in the proportion a: b ::b :c or b' = ac; and the harmonic mean places the quantities in harmonic proportion, i.e. a :c :: a - b :b - c, or b = 2acl (a+c). The contraharmonical mean is the quantity b given by the proportion a: c :: b - c :a - b, i.e. b = (a 2 - -c2)/(a+c). 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.
