Even the most patient and exhaustive investigations will sometimes fail to disclose any reason why some results diverge from others. The question again recurs—Are we arbitrarily to exclude them? The answer should be in the negative as a general rule. The mere fact of divergence ought not to be taken as conclusive against a result, and the exertion of arbitrary choice would open the way to the fatal influence of bias, and what is commonly known as the “cooking” of figures. It would amount to judging fact by theory instead of theory by fact. The apparently divergent number may prove in time to be the true one. It may be an exception of that valuable kind which upsets our false theories, a real exception, exploding apparent coincidences, and opening a way to a new view of the subject. To establish this position for the divergent fact will require additional research; but in the meantime we should give it some weight in our mean conclusions, and should bear in mind the discrepancy as one demanding attention. To neglect a divergent result is to neglect the possible clue to a great discovery.
Method of Least Squares.
When two or more unknown quantities are so involved that they cannot be separately determined by the Simple Method of Means, we can yet obtain their most probable values by the Method of Least Squares, without more difficulty than arises from the length of the arithmetical computations. If the result of each observation gives an equation between two unknown quantities of the form
ax + by = c
then, if the observations were free from error, we should need only two observations giving two equations; but for the attainment of greater accuracy, we may take many observations, and reduce the equations so as to give only a pair with mean coefficients. This reduction is effected by (1.), multiplying the coefficients of each equation by the first coefficient, and adding together all the similar coefficients thus resulting for the coefficients of a new equation; and (2.), by repeating this process, and multiplying the coefficients of each equation by the coefficient of the second term. Meaning by (sum of a2) the sum of all quantities of the same kind, and having the same place in the equations as a2, we may briefly describe the two resulting mean equations as follows:—
(sum of a2) . x + (sum of ab) . y = (sum of ac),
(sum of ab) . x + (sum of b2) . y = (sum of bc).
When there are three or more unknown quantities the process is exactly the same in nature, and we get additional mean equations by multiplying by the third, fourth, &c., coefficients. As the numbers are in any case approximate, it is usually unnecessary to make the computations with accuracy, and places of decimals may be freely cut off to save arithmetical work. The mean equations having been computed, their solution by the ordinary methods of algebra gives the most probable values of the unknown quantities.
Works upon the Theory of Probability.
Regarding the Theory of Probability and the Law of Error as most important subjects of study for any one who desires to obtain a complete comprehension of scientific method as actually applied in physical investigations, I will briefly indicate the works in one or other of which the reader will best pursue the study.
The best popular, and at the same time profound English work on the subject is De Morgan’s “Essay on Probabilities and on their Application to Life Contingencies and Insurance Offices,” published in the Cabinet Cyclopædia, and to be obtained (in print) from Messrs. Longman. Mr. Venn’s work on The Logic of Chance can now be procured in a greatly enlarged second edition;[297] it contains a most interesting and able discussion of the metaphysical basis of probability and of related questions concerning causation, belief, design, testimony, &c.; but I cannot always agree with Mr. Venn’s opinions. No mathematical knowledge beyond that of common arithmetic is required in reading these works. Quetelet’s Letters form a good introduction to the subject, and the mathematical notes are of value. Sir George Airy’s brief treatise On the Algebraical and Numerical Theory of Errors of Observations and the Combination of Observations, contains a complete explanation of the Law of Error and its practical applications. De Morgan’s treatise “On the Theory of Probabilities” in the Encyclopædia Metropolitana, presents an abstract of the more abstruse investigations of Laplace, together with a multitude of profound and original remarks concerning the theory generally. In Lubbock and Drinkwater’s work on Probability, in the Library of Useful Knowledge, we have a concise but good statement of a number of important problems. The Rev. W. A. Whitworth has given, in a work entitled Choice and Chance, a number of good illustrations of calculations both in combinations and probabilities. In Mr. Todhunter’s admirable History we have an exhaustive critical account of almost all writings upon the subject of probability down to the culmination of the theory in Laplace’s works. The Memoir of Mr. J. W. L. Glaisher has already been mentioned (p. [375]). In spite of the existence of these and some other good English works, there seems to be a want of an easy and yet pretty complete mathematical introduction to the study of the theory.