should then make it a point of prime importance to discover the exact nature and amount of such an error, and either prevent its occurrence for the future or else introduce a corresponding correction. In many researches the whole difficulty will consist in this detection and avoidance. of sources of error. Thus Professor Roscoe found that the presence of phosphorus caused serious and almost unavoidable errors in the determination of the atomic weight of vanadium1. Sir John Herschel, in reducing his observations of double stars at the Cape of Good Hope, was perplexed by an unaccountable difference of the angles of position as measured by the Seven-feet Equatorial and the Twenty-feet Reflector Telescopes, and after a careful investigation was obliged to be contented with introducing a correction experimentally determined.

Even the most patient and exhaustive investigations will sometimes fail to disclose any reason why some results diverge in an unusual and unexpected manner 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 most fatal influence of bias, and what is commonly known as the 'cooking' of figures. It would amount in most cases to judging fact by theory instead of theory by fact. The apparently divergent number may even prove in time to be the true one. It may be an exception of that peculiarly valuable kind which upsets our false theories, a real exception, exploding apparent coincidences, and opening the way to a wholly new view of the subject. To establish this position for the divergent fact will of course require additional research; but in the meantime we should give it a fair

r Bakerian Lecture, 'Philosophical Transactions' (1868), vol. clviii. p. 6. s 'Results of Observations at the Cape of Good Hope,' p. 283.

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 single Method of Means, we can yet obtain their most probable amounts 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 only need two observations giving two equations; but, for the attainment of greater accuracy, we may take a series of observations, and then reduce the equations so as to give only a pair with average coefficients. This reduction is effected by, firstly, 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 secondly, by repeating this process, and multiplying the coefficients of each equation by the coefficient of the second term. Thus 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 a3). 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 only need additional mean equations to be obtained by multiplying by the third, fourth, &c., coefficients. As the numbers

are in any case only approximate, it is usually quite unnecessary to make the computations with any great degree of accuracy, and places of decimals may therefore 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 and the Law of Error.

Regarding the Theory of Probability and the Law of Error as constituting, perhaps, the most important subjects of study for any one who desires to obtain a complete comprehension of logical and 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 from Messrs. Longman. No mathematical knowledge beyond that of common arithmetic is required in reading this work. Quetelet's 'Letters,' already often referred to, also form a most interesting and excellent popular introduction to the subject, and the mathematical notes are also of value. Sir George Airy's brief treatise On the Algebraical and Numerical Theory of Errors of Observation 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 pro

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found and original remarks concerning the theory generally. In Lubbock and Drinkwater's work on Probability,' in the Library of Useful Knowledge, we have a very concise but good statement of a number of important problems. The Rev. W. A. Whitworth has given, in an interesting little work entitled 'Choice and Chance,' a number of good illustrations of the calculations both in the theories of 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. 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 introduction to the study of the theory of probabilities.

Among French works the 'Traité Élémentaire du Calcul des Probabilités,' by S. F. Lacroix, of which several editions have been published, and which is not difficult to obtain, forms probably the best elementary treatise. Poisson's Recherches sur la Probabilité des Jugements,'. (Paris, 1837), commences with an admirable investigation of the grounds and methods of the theory. While Laplace's great Théorie Analytique des Probabilités' is of course the Principia' of the subject, his Essai Philosophique sur les Probabilités' is a popular discourse, and is one of the most profound and interesting essays ever published. It should be familiar to every student of logical method, and has lost little or none of its importance by lapse of time.

Detection of Constant Errors.

The Method of Means is absolutely incapable of eliminating any error which is always the same, and which always lies in one direction. We sometimes require to be aroused from a false feeling of security, and to be urged

to take suitable precautions against such occult errors. 'It is to the observer,' says Gausst, that belongs the task of carefully removing the causes of constant errors,' and this is quite true when the error is absolutely constant. When we have made a number of determinations with a certain apparatus or method of measurement, there is a great advantage in altering the arrangement, or even devising some entirely different method of getting estimates of the same quantity. The reason obviously consists in the improbability that exactly the same constant error will affect two or more different methods of experiment. If a discrepancy is found to exist, we shall at least be aware of the existence of error, and can take measures for finding in which way it lies. If we can try a considerable number of methods, the probability becomes considerable that errors constant in one method will be balanced or nearly so by errors of an opposite effect in the others. Suppose that there be three different methods each affected by an error of equal amount. The probability that this error will in all fall in the same direction is only; and with four methods similarly. If each method be affected, as is always the case by several independent sources of error, the probability becomes very great that in the mean result of all the methods some of the errors will partially compensate the others. In this case, as in all others, when human foresight and vigilance has exhausted itself, we must trust the theory of probability.

In the determination of a zero point, of the magnitude of the fundamental standards of time and space, in the personal equation of an astronomical observer, we have instances of such fixed errors; but as a general rule a change of procedure is likely to reverse the character of the error, and many instances may be given of the value of this precaution.

t Gauss, translated by Bertrand, p. 25.