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sect. 5.]

Arrangement and Formation of the Series.

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§ 5. In support of an affirmative answer to the former of these two questions, several different kinds of proof are, or might be, offered.

(I.) For one plan we may make a direct appeal to experience, by collecting sets of statistics and observing what is their law of distribution. As remarked above, this has been done in a great variety of cases, and in some instances to a very considerable extent, by Quetelet and others. His researches have made it abundantly convincing that many classes of things and processes, differing widely in their nature and origin, do nevertheless appear to conform with a considerable degree of accuracy to one and the same^[1] law. At least this is made plain for the more

↑ Commonly called the exponential law; its equation being of the form $y=Ae^{-hx^{2}}$ The curve corresponding to it cuts the axis of $y$ at right angles (expressing the fact that near the mean there are a large number of values approximately equal; after a time it begins to slope away rapidly towards the axis of $x$ ; (expressing the fact that the results soon begin to grow less common as we recede from the mean); and the axis of $x$ is an asymptote in both directions (expressing the fact that no magnitude, however remote from the mean, is strictly impossible; that is, every deviation, however excessive, will have to be encountered at length within the range of a sufficiently long experience). The curve is obviously symmetrical, expressing the fact that equal deviations from the mean, in excess and in defect, tend to occur equally often in the long run.
Illustration from Logic of Chance

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 vortex 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.

[1] Commonly called the exponential law; its equation being of the form $y=Ae^{-hx^{2}}$ The curve corresponding to it cuts the axis of $y$ at right angles (expressing the fact that near the mean there are a large number of values approximately equal; after a time it begins to slope away rapidly towards the axis of $x$ ; (expressing the fact that the results soon begin to grow less common as we recede from the mean); and the axis of $x$ is an asymptote in both directions (expressing the fact that no magnitude, however remote from the mean, is strictly impossible; that is, every deviation, however excessive, will have to be encountered at length within the range of a sufficiently long experience). The curve is obviously symmetrical, expressing the fact that equal deviations from the mean, in excess and in defect, tend to occur equally often in the long run.
Illustration from Logic of Chance

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 vortex 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.

[1]