The reasons in favour of this choice are of several different kinds.
1. It cannot be denied that there may exist infinitely numerous causes of error in any act of observation.
2. The law resulting from the hypothesis of a moderate number of causes of error, does not appreciably differ from that given by the hypothesis of an infinite number of causes of error.
3. We gain by the hypothesis of infinity a general law capable of ready calculation, and applicable by uniform rules to all problems.
4. This law, when tested by comparison with extensive series of observations, is strikingly verified, as will be shown in a later section.
When we imagine the existence of any large number of causes of error, for instance one hundred, the numbers of combinations become impracticably large, as may be seen to be the case from a glance at the Arithmetical Triangle, which proceeds only up to the seventeenth line. Quetelet, by suitable abbreviating processes, calculated out a table of probability of errors on the hypothesis of one thousand distinct causes;[282] but mathematicians have generally proceeded on the hypothesis of infinity, and then, by the devices of analysis, have substituted a general law of easy treatment. In mathematical works upon the subject, it is shown that the standard Law of Error is expressed in the formula
y = Y ε –cx2,
in which x is the amount of the error, Y the maximum ordinate of the curve of error, and c a number constant for each series of observations, and expressing the amount of the tendency to error, varying between one series of observations and another. The letter ε is the mathematical constant, the sum of ratios between the numbers of permutations and combinations, previously referred to (p. [330]).
To show the close correspondence of this general law with the special law which might be derived from the supposition of a moderate number of causes of error, I have in the accompanying figure drawn a curved line representing accurately the variation of y when x in the above formula is taken equal 0, 1/2, 1, 3/2, 2, &c., positive or negative, the arbitrary quantities Y and c being each assumed equal to unity, in order to simplify the calculations. In the same figure are inserted eleven dots, whose heights above the base line are proportional to the numbers in the eleventh line of the Arithmetical Triangle, thus representing the comparative probabilities of errors of various amounts arising from ten equal causes of error. The correspondence of the general and the special Law of Error is almost as close as can be exhibited in the figure, and the assumption of a greater number of equal causes of error would render the correspondence far more close.