this knowledge, we ought of course to act upon it, instead of trusting to probability.

We may often perceive that a series of measurements tends towards an extreme limit rather than towards a mean. In endeavouring to obtain a correct estimate of the apparent diameter of the brightest fixed stars, we find a continuous diminution in estimates as the powers of observation increased. Kepler assigned to Sirius an apparent diameter of 240 seconds; Tycho Brahe made it 126; Gassendi 10 seconds; Galileo, Hevelius, and J. Cassini, 5 or 6 seconds. Halley, Michell, and subsequently Sir W. Herschel came to the conclusion that the brightest stars in the heavens could not have real discs of a second, and were probably much less in diameter. It would of course be absurd to take the mean of quantities which differ more than 240 times; and as the tendency has always been to smaller estimates, there is a considerable presumption in favour of the smallest.1

In many experiments and measurements we know that there is a preponderating tendency to error in one direction. The readings of a thermometer tend to rise as the age of the instrument increases, and no drawing of means will correct this result. Barometers, on the other hand, are likely to read too low instead of too high, owing to the imperfection of the vacuum and the action of capillary attraction. If the mercury be perfectly pure and no appreciable error be due to the measuring apparatus, the best barometer will be that which gives the highest result. In determining the specific gravity of a solid body the chief danger of error arises from bubbles of air adhering to the body, which would tend to make the specific gravity too small. Much attention must always be given to one-sided errors of this kind, since the multiplication of experiments does not remove the error. In such cases one very careful experiment is better than any number of careless ones.

When we have reasonable grounds for supposing that certain experimental results are liable to grave errors, we should exclude them in drawing a mean. If we want to find the most probable approximation to the velocity of

1 Quetelet, Letters, &c. p. 116.

sound in air, it would be absurd to go back to the old experiments which made the velocity from 1200 to 1474 feet per second; for we know that the old observers did not guard against errors arising from wind and other causes. Old chemical experiments are valueless as regards quantitative results. The old chemists found the atmosphere in different places to differ in composition nearly ten per cent., whereas modern accurate experimenters find very slight variations. Any method of measurement which we know to avoid a scurce of error is far to be preferred to others which trust to probabilities for the elimination of the error. As Flamsteed says,1 " One good instrument is of as much worth as a hundred indifferent ones." But an instrument is good or bad only in a comparative sense, and no instrument gives invariable. and truthful results. Hence we must always ultimately fall back upon probabilities for the selection of the final mean, when other precautions are exhausted.

Legendre, the discoverer of the method of Least Squares, recommended that observations differing very much from the results of his method should be rejected. The subject has been carefully investigated by Professor Pierce, who has proposed a criterion for the rejection of doubtful observations based on the following principle: 2-observations should be rejected when the probability of the system of errors obtained by retaining them is less than that of the system of errors obtained by their rejection multiplied by the probability of making so many and no more abnormal observations." Professor Pierce's investigation is given nearly in his own words in Professor W. Chauvenet's "Manual of Spherical and Practical Astronomy," which contains a full and excellent discussion of the methods of treating numerical observations.3

Very difficult questions sometimes arise when one or more results of a method of experiment diverge widely from the mean of the rest. Are we or are we not to exclude them in adopting the supposed true mean result of the method? The drawing of a mean result rests, as I

1 Baily, Account of Flamsteed, p. 56.

2 Gould's Astronomical Journal, Cambridge, Mass., vol. ii. p. 161. 3 Philadelphia (London, Trübner) 1863. Appendix, vol. ii. p. 558.

have frequently explained, upon the assumption that every error acting in one direction will probably be balanced by other errors acting in an opposite direction. If then we know or can possibly discover any causes of error not agreeing with this assumption, we shall be justified in excluding results which seem to be affected by this cause.

In reducing large series of astronomical observations, it is not uncommon to meet with numbers differing from others by a whole degree or half a degree, or some considerable integral quantity. These are errors which could hardly arise in the act of observation or in instrumental irregularity; but they might readily be accounted for by misreading of figures or mistaking of division marks. It would be absurd to trust to chance that such mistakes would balance each other in the long run, and it is therefore better to correct arbitrarily the supposed mistake, or better still, if new observations can be made, to strike out the divergent numbers altogether. When results come sometimes too great or too small in a regular manner, we should suspect that some part of the instrument slips through a definite space, or that a definite cause of error enters at times, and not at others. We 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. Roscoe found that the presence of phosphorus caused serious and almost unavoidable errors in the determination of the atomic weight of vanadium.1 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.2

Professor

When observations are sufficiently numerous it seems desirable to project the apparent errors into a curve, and then to observe whether this curve exhibits the symmet1 Bakerian Lecture, Philosophical Transactions (1868), vol. clviii. p. 6.

2 Results of Observations at the Cape of Good Hope, p. 283.

If so,

rical and characteristic form of the curve of error. it may be inferred that the errors arise from many minute independent sources, and probably compensate each other in the mean result. Any considerable irregularity will indicate the existence of one-sided or large causes of error, which should be made the subject of investigation.

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 ac), (sum of be).

(sum of a2). x + (sum of ab) . y (sum of ab). x + (sum of b2) . y 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; it contains a most interesting and able discussion of the metaphysical

The Logic of Chance, an Essay on the Foundations and Province of the Theory of Probability, with especial reference to its Logical Bearings and its Application to Moral and Social Science. (Macmillan), 1876.