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
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:—
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;22 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,