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.
Among French works the Traité Élémentaire du Calcul des Probabilités, by S. E. 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), commence 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, or which always lies in one direction. We sometimes require to be roused from a false feeling of security, and to be urged to take suitable precautions against such occult errors. “It is to the observer,” says Gauss,23 “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 the same 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 great 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 1/4; and with four methods similarly 1/8. If each method be affected, as is always the case, by several independent sources of error, the probability becomes much greater 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 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 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. If we measure over and over again the same angular magnitude by the same divided circle, maintained in exactly the same position, it is evident that the same mark in the circle will be the criterion in each case, and any error in the position of that mark will equally affect all our results. But if in each measurement we use a different part of the circle, a new mark will come into use, and as the error of each mark cannot be in the same direction, the average result will be nearly free from errors of division. It will be better still to use more than one divided circle.