MEAN, a middle state between two extremes ; thus we have an arithmetical mean, geometrical mean, mean distance, mean motion, &c. An arithmetical mean is half the sum of the extremes : thus, if 2 and 12 be the extremes, then 2 + 12 2 7 is the arithmetical mean : likewise be twcen a and b it is a+ Geometrical mean, usually called a mean proportional, is the square root of the product of the two extremes c therefore, to find a mean proportional between two given ex tremes, multiply these together, and ex tract the square root of the product. Thus, a mean proportional between 6 and 24 is 12 ; for %,/ 6 x 24= 144 = : and between x and y it is x y. The arithmetical mean is greater than the geo metrical mean between the same two ex tremes : thus, between 6 and 24 the geo metrical mean is 12; but the arithmetical mean is 6 + 15. Or, generally, let 2 a be the greater and b the less ; then a + b is greater than b, or multiply 2 ing both by 2 ; a + b is greater 2 b : for squaring both we have a' + 2 a b b3 greater than 4 a b; for take away 4 a b and a' — 2 a b b' greater than 0: or 7—n- greater than 0 by the supposition.
To find a mean proportional, geometri cally, between two given right lines, a and b, (Plate Miscel. X. fig. 6.) join the two given lines together at x in one con tinued line, a b; upon the diameter a b deScribe a semicircle ax b, and erect the perpendicular a x, which will be the required mean proportional ; for, by a well-known theorem in geometry, axxxbis equat to x:', or x ; x b, To find two mean proportionals. be tween two given extremes : t' 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 greater ; then extract the cube root out of each product, and the two roots will be the two mean propiirtionals sought. Thus the two mean proportionals between a and b are if a' b and Va : or between 2 and 16 the mean pro portionals are and = 4 and 8.