, 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. 4ab 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, √3a2b and √3ab2 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, √3a2b,3ab2; three means, √4a3b,4a2b2, √4ab3; four means, √5a4b,5a3b2, √5a2b3; √5ab4; five means, √6a5b,6a4b2, √6a3b3, √6a2b4, √6ab5; &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.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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