say 5 ft. 8 in., and a great part of all the men would be of nearly this height, one quarter being not more than 11⁄2 inches shorter and one quarter not more than l1⁄2 inches taller. The line of the heads would be nearly horizontal, but would gradually slope more and more, until at one end we should have the comparatively few dwarfs and at the other the few giants. These relations can be illustrated by the bell shaped curve, whose properties are well known.
Five feet eight inches is the average height of men, and the number of men of that height (within say 1/10 in.) is proportional to the line OY. The number of men say 11⁄2 inches (within 1/10 in.) larger than the average by an amount equal to the probable error or 11⁄2 in. is proportional to the line PQ, and the number of men within these limits, one quarter of all the men, is proportional to the area OYQP which is one quarter the area of the curve. The number of men 6 ft. 2 in. in height—who depart from the mean by 6 in., or four times the probable error—would be OU, only 1/50 as many as are 5 ft. 91⁄2 in. in height, and but three in a thousand of all men would be taller than 6 ft. 2 in.
Now applying this to the collective traits giving efficiency, we have one half of all men coming within the limit OP which may be taken as a unit of measure. The total number of men surpassing the average by four times the amount of the average departure would be about 300,000. Most of us may hope to fall within this group. The thousand preeminent men filling the extreme area of the curve would begin at a point six times the average departure, and the relative excellence of the greater men on the list can also be expressed numerically.
Turning now to the distribution of these eminent men in time and race we may review statistics not wholly devoid of interest. The number of great men born in each half century since the beginning of history is shown in the accompanying curve. In still more remote ages there were leaders of men, gods, prophets and heroes, whose names are forgotten or obscured, and at the beginning we have four names, representing rather work than persons Zoroaster, Homer, Hesiod,