A second way of arriving at the Law of Error was proposed by Herschel, and although only applicable to geometrical cases, it is remarkable as showing that from whatever point of view we regard the subject, the same principle will be detected. After assuming that some general law must exist, and that it is subject to the principles of probability, he supposes that a ball is dropped from a high point with the intention that it shall strike a given mark on a horizontal plane. In the absence of any known causes of deviation it will either strike that mark, or, as is infinitely more probable, diverge from it by an amount which we must regard as error of unknown origin. Now, to quote the words of Herschel,‍[281] “the probability of that error is the unknown function of its square, i.e. of the sum of the squares of its deviations in any two rectangular directions. Now, the probability of any deviation depending solely on its magnitude, and not on its direction, it follows that the probability of each of these rectangular deviations must be the same function of its square. And since the observed oblique deviation is equivalent to the two rectangular ones, supposed concurrent, and which are essentially independent of one another, and is, therefore, a compound event of which they are the simple independent constituents, therefore its probability will be the product of their separate probabilities. Thus the form of our unknown function comes to be determined from this condition, viz., that the product of such functions of two independent elements is equal to the same function of their sum. But it is shown in every work on algebra that this property is the peculiar characteristic of, and belongs only to, the exponential or antilogarithmic function. This, then, is the function of the square of the error, which expresses the probability of committing that error. That probability decreases, therefore, in geometrical progression, as the square of the error increases in arithmetical.”

Laplace’s and Quetelet’s Proof of the Law.

However much presumption the modes of determining the Law of Error, already described, may give in favour of the law usually adopted, it is difficult to feel that the arguments are satisfactory. The law adopted is chosen rather on the grounds of convenience and plausibility, than because it can be seen to be the necessary law. We can however approach the subject from an entirely different point of view, and yet get to the same result.

Let us assume that a particular observation is subject to four chances of error, each of which will increase the result one inch if it occurs. Each of these errors is to be regarded as an event independent of the rest and we can therefore assign, by the theory of probability, the comparative probability and frequency of each conjunction of errors. From the Arithmetical Triangle (pp. [182]–188) we learn that no error at all can happen only in one way; an error of one inch can happen in 4 ways; and the ways of happening of errors of 2, 3 and 4 inches respectively, will be 6, 4 and 1 in number.

We may infer that the error of two inches is the most likely to occur, and will occur in the long run in six cases out of sixteen. Errors of one and three inches will be equally likely, but will occur less frequently; while no error at all, or one of four inches will be a comparatively rare occurrence. If we now suppose the errors to act as often in one direction as the other, the effect will be to alter the average error by the amount of two inches, and we shall have the following results:‍—

Negative error of 2 inches1 way.
Negative error of 1 inch4 ways.
No error at all6 ways.
Positive error of 1 inch4 ways.
Positive error of 2 inches1 way.

We may now imagine the number of causes of error increased and the amount of each error decreased, and the arithmetical triangle will give us the frequency of the resulting errors. Thus if there be five positive causes of error and five negative causes, the following table shows the numbers of errors of various amount which will be the result:‍—

Direction of Error.Positive Error.Negative Error.
Amount of Error.5, 4, 3, 2, 101, 2, 3, 4, 5
Number of such Errors.1, 10, 45, 120, 210252210 120, 45, 10, 1

It is plain that from such numbers I can ascertain the probability of any particular amount of error under the conditions supposed. The probability of a positive error of exactly one inch is 210/1024, in which fraction the numerator is the number of combinations giving one inch positive error, and the denominator the whole number of possible errors of all magnitudes. I can also, by adding together the appropriate numbers get the 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]).