A rough graphic representation of the curve is given above. For the benefit of those unfamiliar with mathematics one or two brief remarks may be here appended concerning some of its properties. (1) It must not be supposed that all specimens of the curve are similar to one another. The dotted lines are equally specimens of it. In fact, by varying the essentially arbitrary units in which x and y are respectively estimated, we may make the portion towards the vertex of the curve as obtuse or as acute as we please. This consideration is of importance; for it reminds us that, by varying one of these arbitrary units, we could get an ‘exponential curve’ which should tolerably closely resemble any symmetrical curve of error, provided that this latter recognized and was founded upon the assumption that extreme divergences were excessively rare. Hence it would be difficult, by mere observation, to prove that the law of error in any given case was not exponential; unless the statistics were very extensive, or the actual results departed considerably from the exponential form. (2) It is quite impossible by any graphic representation to give an adequate idea of the excessive rapidity with which the curve after a time approaches the axis of x. At the point R, on our scale, the curve would approach within the fifteen-thousandth part of an inch from the axis of x, a distance which only a very good microscope could detect. Whereas in the hyperbola, e.g.
the rate of approach of the curve to its asymptote is continually decreasing, it is here just the reverse; this rate is continually increasing. Hence the two, viz.
the curve and the axis of x, appear to the eye, after a very short time, to merge into one another.
[6] As by Quetelet: noted, amongst others, by Herschel, Essays, page 409.
[7] Proc.
R. Soc.
Oct. 21, 1879.
[8] We are here considering, remember, the case of a finite amount of statistics; so that there are actual limits at each end.
[9] It must be admitted that experience has not yet (I believe) shown this asymmetry in respect of heights.