probability of an error not exceeding a certain amount. Thus the probability of an error of three inches or less, positive or negative, is a fraction whose numerator is the sum of 45 + 120 + 210 + 252 + 210 + 120 + 45, and the denominator, as before, giving the result 1002/1024. We may see at once that, according to these principles, the probability of small errors is far greater than of large ones: the odds are 1002 to 22, or more than 45 to 1, that the error will not exceed three inches; and the odds are 1022 to 2 against the occurrence of the greatest possible error of five inches.
If any case should arise in which the observer knows the number and magnitude of the chief errors which may occur, he ought certainly to calculate from the Arithmetical Triangle the special Law of Error which would apply. But the general law, of which we are in search, is to be used in the dark, when we have no knowledge whatever of the sources of error. To assume any special number of causes of error is then an arbitrary proceeding, and mathematicians have chosen the least arbitrary course of imagining the existence of an infinite number of infinitely small errors, just as, in the inverse method of probabilities, an infinite number of infinitely improbable hypotheses were submitted to calculation (p. 255).
The reasons in favour of this choice are of several different kinds.
- It cannot be denied that there may exist infinitely numerous causes of error in any act of observation.
- 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.
- We gain by the hypothesis of infinity a general law capable of ready calculation, and applicable by uniform rules to all problems.
- 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;7 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
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).