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. 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. 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), 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,1 "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 ; and with four methods similarly. 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

1 Gauss, translated by Bertrand, p. 25.

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.

Even when we have no perception of the points at which error is likely to enter, we may with advantage. vary the construction of our apparatus in the hope that we shall accidentally detect some latent cause of error. Baily's purpose in repeating the experiments of Michell and Cavendish on the density of the earth was not merely to follow the same course and verify the previous numbers, but to try whether variations in the size and substance of the attracting balls, the mode of suspension, the temperature of the surrounding air, &c., would yield different results. He performed no less than 62 distinct series, comprising 2153 experiments, and he carefully classified and discussed the results so as to disclose the utmost differences. Again, in experimenting upon the resistance of the air to the motion of a pendulum, Baily employed no less than 80 pendulums of various forms and materials, in order to ascertain exactly upon what conditions the resistance. depends. Regnault, in his exact researches upon the dilatation of gases, made arbitrary changes in the magnitude of parts of his apparatus. He thinks that if, in spite of such modification, the results are unchanged, the errors are probably of inconsiderable amount; but in reality it is always possible, and usually likely, that we overlook sources of error which a future generation will detect. Thus the pendulum experiments of Baily and Sabine were directed to ascertain the nature and amount of a correction for air resistance, which had been entirely misunderstood in the experiments by means of the seconds pendulum, upon which was founded the definition of the standard yard, in the Act of 5th George IV. c. 74. It has already

been mentioned that a considerable error was discovered in the determination of the standard metre as the tenmillionth part of the distance from the pole to the equator (p. 314).

We shall return in Chapter XXV. to the further consideration of the methods by which we may as far as possible secure ourselves against permanent and undetected sources of error. In the meantime, having completed the consideration of the special methods requisite for treating quantitative phenomena, we must pursue our principal subject, and endeavour to trace out the course by which the physicist, from observation and experiment, collects the materials of knowledge, and then proceeds by hypothesis and inverse calculation to induce from them the laws of nature.

BOOK III.

INDUCTIVE INVESTIGATION.

CHAPTER XVIII.

OBSERVATION.

ALL knowledge proceeds originally from experience. Using the name in a wide sense, we may say that experience comprehends all that we feel, externally or internally— the aggregate of the impressions which we receive through the various apertures of perception-the aggregate consequently of what is in the mind, except so far as some portions of knowledge may be the reasoned equivalents of other portions. As the word experience expresses, we go through much in life, and the impressions gathered intentionally or unintentionally afford the materials from which the active powers of the mind evolve science.

No small part of the experience actually employed in science is acquired without any distinct purpose. We cannot use the eyes without gathering some facts which may prove useful. A great science has in many cases

risen from an accidental observation. Erasmus Bartholinus thus first discovered double refraction in Iceland spar; Galvani noticed the twitching of a frog's leg; Oken was struck by the form of a vertebra; Malus accidentally examined light reflected from distant windows with a