in this,—that the grouping of a large number of separate values that have arisen from causes of the same kind and with the modifications repeatedly mentioned, may be correctly represented by a mathematical formula, the so-called Law of Errors. This is especially characterised by the fact that it contains but one unknown quantity. This unknown quantity measures the relative compactness of the distribution of the separate values around their central tendency. It therefore changes according to the kind of observation and is determined by calculation from the separate values.
Note. For further information concerning this formula, which is not here our concern, I must refer to the text-books on the calculation of probabilities and on the theory of errors. For readers unfamiliar with the latter a graphic explanation will be more comprehensible than a statement and discussion of the formula. Imagine a certain observation to be repeated 1,000 times. Each observation as such is represented by a space of one square millimeter, and its numerical value, or rather its deviation from the central value of the whole 1,000 observations, by its position on the horizontal line p q of the adjoining Figure 1.
For every observation which exactly corresponds with the central value one square millimeter is laid off on the vertical line m n. For each observed value which deviates by one unit from the central value upward one square millimeter is laid off on a vertical line to right of m n and distant one millimeter from it, etc. For every observed value which deviates by x units above (or below) the central value, one sq. mm. is placed on a vertical line distant from m n by x mms., to the right (or left, for values below the central value). When all the observations are arranged in this way the outer contour of the figure may be so compacted that the projecting corners of the separate squares are transformed into a symmetrical curve. If now the separate measures are of such a sort that their central value may be considered as a constant as conceived by physical science, the form of the resulting curve is of the kind marked a and b in Fig. 1. If the middle value is a statistical constant, the curve may have any sort of a form. (The curves a and b with the lines p q include in each case an area of 1,000 sq. mms. This is strictly the case only with indefinite prolongation of the curves and the lines p q, but these lines and curves finally approach each other so closely that where the drawing breaks off only two or three sq. mms. at each end of the curve are missing from the full number.) Whether, for a certain group of observations, the curve has a
