CHAPTER XVII.

THE LAW OF ERROR.

To bring error itself under law might seem beyond human power. He who errs surely diverges from law, and it might be deemed hopeless out of error to draw truth. One of the most remarkable achievements of the human intellect is the establishment of a general theory which not only enables us among discrepant results to approximate to the truth, but to assign the degree of probability which fairly attaches to this conclusion. It would be a mistake indeed to suppose that this law is necessarily the best guide under all circumstances. Every measuring instrument and every form of experiment may have its own special law of error; there may in one instrument be a tendency in one direction and in another in the opposite direction. Every process has its peculiar liabilities to disturbance, and we are never relieved from the necessity of providing against special difficulties. The general Law of Error is the best guide only when we have exhausted all other means of approximation, and still find discrepancies, which are due to unknown causes. We must treat such residual differences in some way or other, since they will occur in all accurate experiments, and as their origin is assumed to be unknown, there is no reason why we should treat them differently in different cases. Accordingly the ultimate Law of Error must be a uniform and general one.

It is perfectly recognised by mathematicians that in each case a special Law of Error may exist, and should be discovered if possible. “Nothing can be more unlikely than that the errors committed in all classes of observations should follow the same law,”[276] and the special Laws of Error which will apply to certain instruments, as for instance the repeating circle, have been investigated by Bravais.[277] He concludes that every distinct cause of error gives rise to a curve of possibility of errors, which may have any form,--a curve which we may either be able or unable to discover, and which in the first case may be determined by *à priori* considerations on the peculiar nature of this cause, or which may be determined *à posteriori* by observation. Whenever it is practicable and worth the labour, we ought to investigate these special conditions of error; nevertheless, when there are a great number of different sources of minute error, the general resultant will always tend to obey that general law which we are about to consider.

[276] *Philosophical Magazine*, 3rd Series, vol. xxxvii. p. 324.

[277] *Letters on the Theory of Probabilities*, by Quetelet, translated by O. G. Downes, Notes to Letter XXVI. pp. 286–295.

*Establishment of the Law of Error.*

Mathematicians agree far better as to the form of the Law of Error than they do as to the manner in which it can be deduced and proved. They agree that among a number of discrepant results of observation, that mean quantity is probably the best approximation to the truth which makes the sum of the squares of the errors as small as possible. But there are three principal ways in which this law has been arrived at respectively by Gauss, by Laplace and Quetelet, and by Sir John Herschel. Gauss proceeds much upon assumption; Herschel rests upon geometrical considerations; while Laplace and Quetelet regard the Law of Error as a development of the doctrine of combinations. A number of other mathematicians, such as Adrain of New Brunswick, Bessel, Ivory, Donkin, Leslie Ellis, Tait, and Crofton have either attempted independent proofs or have modified or commented on those here to be described. For full accounts of the literature of the subject the reader should refer either to Mr. Todhunter’s *History of the Theory of Probability* or to the able memoir of Mr. J. W. L. Glaisher.[278]

[278] *On the Law of Facility of Errors of Observations, and on the Method of Least Squares*, Memoirs of the Royal Astronomical Society, vol. xxxix. p. 75.

According to Gauss the Law of Error expresses the comparative probability of errors of various magnitude, and partly from experience, partly from *à priori* considerations, we may readily lay down certain conditions to which the law will certainly conform. It may fairly be assumed as a first principle to guide us in the selection of the law, that large errors will be far less frequent and probable than small ones. We know that very large errors are almost impossible, so that the probability must rapidly decrease as the amount of the error increases. A second principle is that positive and negative errors shall be equally probable, which may certainly be assumed, because we are supposed to be devoid of any knowledge as to the causes of the residual errors. It follows that the probability of the error must be a function of an even power of the magnitude, that is of the square, or the fourth power, or the sixth power, otherwise the probability of the same amount of error would vary according as the error was positive or negative. The even powers *x*^{2}, *x*^{4}, *x*^{6}, &c., are always intrinsically positive, whether *x* be positive or negative. There is no *à priori* reason why one rather than another of these even powers should be selected. Gauss himself allows that the fourth or sixth power would fulfil the conditions as well as the second;[279] but in the absence of any theoretical reasons we should prefer the second power, because it leads to formulæ of great comparative simplicity. Did the Law of Error necessitate the use of the higher powers of the error, the complexity of the necessary calculations would much reduce the utility of the theory.

[279] *Méthode des Moindres Carrés. Mémoires sur la Combinaison des Observations, par Ch. Fr. Gauss. Traduit en Français par J. Bertrand*, Paris, 1855, pp. 6, 133, &c.

By mathematical reasoning which it would be undesirable to attempt to follow in this book, it is shown that under these conditions, the facility of occurrence, or in other, words, the probability of error is expressed by a function of the general form ε^{–*h*^{2} *x*^{2}}, in which *x* represents the variable amount of errors. From this law, to be more fully described in the following sections, it at once follows that the most probable result of any observations is that which makes the sum of the squares of the consequent errors the least possible. Let *a*, *b*, *c*, &c., be the results of observation, and *x* the quantity selected as the most probable, that is the most free from unknown errors: then we must determine *x* so that (*a* - *x*)^{2} + (*b* - *x*)^{2} + (*c* - *x*)^{2} + ... shall be the least possible quantity. Thus we arrive at the celebrated *Method of Least Squares*, as it is usually called, which appears to have been first distinctly put in practice by Gauss in 1795, while Legendre first published in 1806 an account of the process in his work, entitled, *Nouvelles Méthodes pour la Détermination des Orbites des Comètes*. It is worthy of notice, however, that Roger Cotes had long previously recommended a method of equivalent nature in his tract, “Estimatio Erroris in Mixta Mathesi.”[280]

[280] De Morgan, *Penny Cyclopædia*, art. *Least Squares*.

*Herschel’s Geometrical Proof.*

A second way of arriving at the Law of Error was proposed by Herschel, and although only applicable to geometrical cases, it is remarkable as showing that from whatever point of view we regard the subject, the same principle will be detected. After assuming that some general law must exist, and that it is subject to the principles of probability, he supposes that a ball is dropped from a high point with the intention that it shall strike a given mark on a horizontal plane. In the absence of any known causes of deviation it will either strike that mark, or, as is infinitely more probable, diverge from it by an amount which we must regard as error of unknown origin. Now, to quote the words of Herschel,[281] “the probability of that error is the unknown function of its square, *i.e.* of the sum of the squares of its deviations in any two rectangular directions. Now, the probability of any deviation depending solely on its magnitude, and not on its direction, it follows that the probability of each of these rectangular deviations must be the same function of *its* square. And since the observed oblique deviation is equivalent to the two rectangular ones, supposed concurrent, and which are essentially independent of one another, and is, therefore, a compound event of which they are the simple independent constituents, therefore its probability will be the product of their separate probabilities. Thus the form of our unknown function comes to be determined from this condition, viz., that the product of such functions of two independent elements is equal to the same function of their sum. But it is shown in every work on algebra that this property is the peculiar characteristic of, and belongs only to, the exponential or antilogarithmic function. This, then, is the function of the square of the error, which expresses the probability of committing that error. That probability decreases, therefore, in geometrical progression, as the square of the error increases in arithmetical.”

[281] *Edinburgh Review*, July 1850, vol. xcii. p. 17. Reprinted *Essays*, p. 399. This method of demonstration is discussed by Boole, *Transactions of Royal Society of Edinburgh*, vol. xxi. pp. 627–630.

*Laplace’s and Quetelet’s Proof of the Law.*

However much presumption the modes of determining the Law of Error, already described, may give in favour of the law usually adopted, it is difficult to feel that the arguments are satisfactory. The law adopted is chosen rather on the grounds of convenience and plausibility, than because it can be seen to be the necessary law. We can however approach the subject from an entirely different point of view, and yet get to the same result.

Let us assume that a particular observation is subject to four chances of error, each of which will increase the result one inch if it occurs. Each of these errors is to be regarded as an event independent of the rest and we can therefore assign, by the theory of probability, the comparative probability and frequency of each conjunction of errors. From the Arithmetical Triangle (pp. 182–188) we learn that no error at all can happen only in one way; an error of one inch can happen in 4 ways; and the ways of happening of errors of 2, 3 and 4 inches respectively, will be 6, 4 and 1 in number.

We may infer that the error of two inches is the most likely to occur, and will occur in the long run in six cases out of sixteen. Errors of one and three inches will be equally likely, but will occur less frequently; while no error at all, or one of four inches will be a comparatively rare occurrence. If we now suppose the errors to act as often in one direction as the other, the effect will be to alter the average error by the amount of two inches, and we shall have the following results:--

Negative error of 2 inches 1 way. Negative error of 1 inch 4 ways. No error at all 6 ways. Positive error of 1 inch 4 ways. Positive error of 2 inches 1 way.

We may now imagine the number of causes of error increased and the amount of each error decreased, and the arithmetical triangle will give us the frequency of the resulting errors. Thus if there be five positive causes of error and five negative causes, the following table shows the numbers of errors of various amount which will be the result:--

+----------------------+-------------------+---+-------------------+ | Direction of Error. | Positive Error. | | Negative Error. | +----------------------+-------------------+---+-------------------+ | Amount of Error. |5, 4, 3, 2, 1| 0 | 1, 2, 3, 4, 5| +----------------------+-------------------+---+-------------------+ |Number of such Errors.|1, 10, 45, 120, 210|252|210, 120, 45, 10, 1| +----------------------+-------------------+---+-------------------+

It is plain that from such numbers I can ascertain the probability of any particular amount of error under the conditions supposed. The probability of a positive error of exactly one inch is 210/1024, in which fraction the numerator is the number of combinations giving one inch positive error, and the denominator the whole number of possible errors of all magnitudes. I can also, by adding together the appropriate numbers get the probability of an error not exceeding a certain amount. Thus the probability of an error of three inches or less, positive or negative, is a fraction whose numerator is the sum of 45 + 120 + 210 + 252 + 210 + 120 + 45, and the denominator, as before, giving the result 1002/1024. We may see at once that, according to these principles, the probability of small errors is far greater than of large ones: the odds are 1002 to 22, or more than 45 to 1, that the error will not exceed three inches; and the odds are 1022 to 2 against the occurrence of the greatest possible error of five inches.

If any case should arise in which the observer knows the number and magnitude of the chief errors which may occur, he ought certainly to calculate from the Arithmetical Triangle the special Law of Error which would apply. But the general law, of which we are in search, is to be used in the dark, when we have no knowledge whatever of the sources of error. To assume any special number of causes of error is then an arbitrary proceeding, and mathematicians have chosen the least arbitrary course of imagining the existence of an infinite number of infinitely small errors, just as, in the inverse method of probabilities, an infinite number of infinitely improbable hypotheses were submitted to calculation (p. 255).

The reasons in favour of this choice are of several different kinds.

1. It cannot be denied that there may exist infinitely numerous causes of error in any act of observation.

2. The law resulting from the hypothesis of a moderate number of causes of error, does not appreciably differ from that given by the hypothesis of an infinite number of causes of error.

3. We gain by the hypothesis of infinity a general law capable of ready calculation, and applicable by uniform rules to all problems.

4. This law, when tested by comparison with extensive series of observations, is strikingly verified, as will be shown in a later section.

When we imagine the existence of any large number of causes of error, for instance one hundred, the numbers of combinations become impracticably large, as may be seen to be the case from a glance at the Arithmetical Triangle, which proceeds only up to the seventeenth line. Quetelet, by suitable abbreviating processes, calculated out a table of probability of errors on the hypothesis of one thousand distinct causes;[282] but mathematicians have generally proceeded on the hypothesis of infinity, and then, by the devices of analysis, have substituted a general law of easy treatment. In mathematical works upon the subject, it is shown that the standard Law of Error is expressed in the formula

*y* = *Y*ε^{-*cx*^{2}},

[282] *Letters on the Theory of Probabilities*, Letter XV. and Appendix, note pp. 256–266.

in which *x* is the amount of the error, *Y* the maximum ordinate of the curve of error, and *c* a number constant for each series of observations, and expressing the amount of the tendency to error, varying between one series of observations and another. The letter ε is the mathematical constant, the sum of ratios between the numbers of permutations and combinations, previously referred to (p. 330).

[Illustration]

To show the close correspondence of this general law with the special law which might be derived from the supposition of a moderate number of causes of error, I have in the accompanying figure drawn a curved line representing accurately the variation of *y* when *x* in the above formula is taken equal 0, 1/2, 1, 3/2, 2, &c., positive or negative, the arbitrary quantities *Y* and *c* being each assumed equal to unity, in order to simplify the calculations. In the same figure are inserted eleven dots, whose heights above the base line are proportional to the numbers in the eleventh line of the Arithmetical Triangle, thus representing the comparative probabilities of errors of various amounts arising from ten equal causes of error. The correspondence of the general and the special Law of Error is almost as close as can be exhibited in the figure, and the assumption of a greater number of equal causes of error would render the correspondence far more close.

It may be explained that the ordinates NM, *nm*, *n′m′*, represent values of *y* in the equation expressing the Law of Error. The occurrence of any one definite amount of error is infinitely improbable, because an infinite number of such ordinates might be drawn. But the probability of an error occurring between certain limits is finite, and is represented by a portion of the *area* of the curve. Thus the probability that an error, positive or negative, not exceeding unity will occur, is represented by the area M*mnn′m′*, in short, by the area standing upon the line *nn′*. Since every observation must either have some definite error or none at all, it follows that the whole area of the curve should be considered as the unit expressing certainty, and the probability of an error falling between particular limits will then be expressed by the ratio which the area of the curve between those limits bears to the whole area of the curve.

The mere fact that the Law of Error allows of the possible existence of errors of every assignable amount shows that it is only approximately true. We may fairly say that in measuring a mile it would be impossible to commit an error of a hundred miles, and the length of life would never allow of our committing an error of one million miles. Nevertheless the general Law of Error would assign a probability for an error of that amount or more, but so small a probability as to be utterly inconsiderable and almost inconceivable. All that can, or in fact need, be said in defence of the law is, that it may be made to represent the errors in any special case to a very close approximation, and that the probability of large and practically impossible errors, as given by the law, will be so small as to be entirely inconsiderable. And as we are dealing with error itself, and our results pretend to nothing more than approximation and probability, an indefinitely small error in our process of approximation is of no importance whatever.

*Logical Origin of the Law of Error.*

It is worthy of notice that this Law of Error, abstruse though the subject may seem, is really founded upon the simplest principles. It arises entirely out of the difference between permutations and combinations, a subject upon which I may seem to have dwelt with unnecessary prolixity in previous pages (pp. 170, 189). The order in which we add quantities together does not affect the amount of the sum, so that if there be three positive and five negative causes of error in operation, it does not matter in which order they are considered as acting. They may be intermixed in any arrangement, and yet the result will be the same. The reader should not fail to notice how laws or principles which appeared to be absurdly simple and evident when first noticed, reappear in the most complicated and mysterious processes of scientific method. The fundamental Laws of Identity and Difference gave rise to the Logical Alphabet which, after abstracting the character of the differences, led to the Arithmetical Triangle. The Law of Error is defined by an infinitely high line of that triangle, and the law proves that the mean is the most probable result, and that divergencies from the mean become much less probable as they increase in amount. Now the comparative greatness of the numbers towards the middle of each line of the Arithmetical Triangle is entirely due to the indifference of order in space or time, which was first prominently pointed out as a condition of logical relations, and the symbols indicating them (pp. 32–35), and which was afterwards shown to attach equally to numerical symbols, the derivatives of logical terms (p. 160).

*Verification of the Law of Error.*

The theory of error which we have been considering rests entirely upon an assumption, namely that when known sources of disturbances are allowed for, there yet remain an indefinite, possibly an infinite number of other minute sources of error, which will as often produce excess as deficiency. Granting this assumption, the Law of Error must be as it is usually taken to be, and there is no more need to verify it empirically than to test the truth of one of Euclid’s propositions mechanically. Nevertheless, it is an interesting occupation to verify even the propositions of geometry, and it is still more instructive to try whether a large number of observations will justify our assumption of the Law of Error.

Encke has given an excellent instance of the correspondence of theory with experience, in the case of observations of the differences of Right Ascension of the sun and two stars, namely α Aquilæ and α Canis minoris. The observations were 470 in number, and were made by Bradley and reduced by Bessel, who found the probable error of the final result to be only about one-fourth part of a second (0·2637). He then compared the numbers of errors of each magnitude from 0·1 second upwards, as actually given by the observations, with what should occur according to the Law of Error.

The results were as follow:--[283]

+-------------------------+--------------------------+ | | Number of errors of each | | Magnitude of the errors | magnitude according to | | in parts of a second. +-------------+------------+ | | Observation.| Theory. | +-------------------------+-------------+------------+ | 0·0 to 0·1 | 94 | 95 | | ·1 " ·2 | 88 | 89 | | ·2 " ·3 | 78 | 78 | | ·3 " ·4 | 58 | 64 | | ·4 " ·5 | 51 | 50 | | ·5 " ·6 | 36 | 36 | | ·6 " ·7 | 26 | 24 | | ·7 " ·8 | 14 | 15 | | ·8 " ·9 | 10 | 9 | | ·9 " 1·0 | 7 | 5 | | above 1·0 | 8 | 5 | +-------------------------+-------------+------------+

[283] Encke, *On the Method of Least Squares*, Taylor’s *Scientific Memoirs*, vol. ii. pp. 338, 339.

The reader will remark that the correspondence is very close, except as regards larger errors, which are excessive in practice. It is one objection, indeed, to the theory of error, that, being expressed in a continuous mathematical function, it contemplates the existence of errors of every magnitude, such as could not practically occur; yet in this case the theory seems to under-estimate the number of large errors.

Another comparison of the law with observation was made by Quetelet, who investigated the errors of 487 determinations in time of the Right Ascension of the Pole-Star made at Greenwich during the four years 1836–39. These observations, although carefully corrected for all known causes of error, as well as for nutation, precession, &c., are yet of course found to differ, and being classified as regards intervals of one-half second of time, and then proportionately increased in number, so that their sum may be one thousand, give the following results as compared with what Quetelet’s theory would lead us to expect:--[284]

+------------+--------------------+------------+--------------------+ |Magnitude of| Number of Errors |Magnitude of| Number of Errors | | error +------------+-------+ error +------------+-------+ | in tenths | by | by | in tenths | by | by | |of a second.|Observation.|Theory.|of a second.|Observation.|Theory.| +------------+------------+-------+------------+------------+-------+ | 0·0 | 168 | 163 | -- | -- | -- | | +0·5 | 148 | 147 | -0·5 | 150 | 152 | | +1·0 | 129 | 112 | -1·0 | 126 | 121 | | +1·5 | 78 | 72 | -1·5 | 74 | 82 | | +2·0 | 33 | 40 | -2·0 | 43 | 46 | | +2·5 | 10 | 19 | -2·5 | 25 | 22 | | +3·0 | 2 | 10 | -3·0 | 12 | 10 | | -- | -- | -- | -3·5 | 2 | 4 | +------------+------------+-------+------------+------------+-------+

[284] Quetelet, *Letters on the Theory of Probabilities*, translated by Downes, Letter XIX. p. 88. See also Galton’s *Hereditary Genius*, p. 379.

In this instance also the correspondence is satisfactory, but the divergence between theory and fact is in the opposite direction to that discovered in the former comparison, the larger errors being less frequent than theory would indicate. It will be noticed that Quetelet’s theoretical results are not symmetrical.

*The Probable Mean Result.*

One immediate result of the Law of Error, as thus stated, is that the mean result is the most probable one; and when there is only a single variable this mean is found by the familiar arithmetical process. An unfortunate error has crept into several works which allude to this subject. Mill, in treating of the “Elimination of Chance,” remarks in a note[285] that “the mean is spoken of as if it were exactly the same thing as the average. But the mean, for purposes of inductive inquiry, is not the average, or arithmetical mean, though in a familiar illustration of the theory the difference may be disregarded.” He goes on to say that, according to mathematical principles, the most probable result is that for which the sums of the squares of the deviations is the least possible. It seems probable that Mill and other writers were misled by Whewell, who says[286] that “The method of least squares is in fact a method of means, but with some peculiar characters.... The method proceeds upon this supposition: that all errors are not equally probable, but that small errors are more probable than large ones.” He adds that this method “removes much that is arbitrary in the method of means.” It is strange to find a mathematician like Whewell making such remarks, when there is no doubt whatever that the Method of Means is only an application of the Method of Least Squares. They are, in fact, the same method, except that the latter method may be applied to cases where two or more quantities have to be determined at the same time. Lubbock and Drinkwater say,[287] “If only one quantity has to be determined, this method evidently resolves itself into taking the mean of all the values given by observation.” Encke says,[288] that the expression for the probability of an error “not only contains in itself the principle of the arithmetical mean, but depends so immediately upon it, that for all those magnitudes for which the arithmetical mean holds good in the simple cases in which it is principally applied, no other law of probability can be assumed than that which is expressed by this formula.”

[285] *System of Logic*, bk. iii. chap. 17, § 3. 5th ed. vol. ii. p. 56.

[286] *Philosophy of the Inductive Sciences*, 2nd ed. vol. ii. pp. 408, 409.

[287] *Essay on Probability*, Useful Knowledge Society, 1833, p. 41.

[288] Taylor’s *Scientific Memoirs*, vol. ii. p. 333.

*The Probable Error of Results.*

When we draw a conclusion from the numerical results of observations we ought not to consider it sufficient, in cases of importance, to content ourselves with finding the simple mean and treating it as true. We ought also to ascertain what is the degree of confidence we may place in this mean, and our confidence should be measured by the degree of concurrence of the observations from which it is derived. In some cases the mean may be approximately certain and accurate. In other cases it may really be worth little or nothing. The Law of Error enables us to give exact expression to the degree of confidence proper in any case; for it shows how to calculate the probability of a divergence of any amount from the mean, and we can thence ascertain the probability that the mean in question is within a certain distance from the true number. The *probable error* is taken by mathematicians to mean the limits within which it is as likely as not that the truth will fall. Thus if 5·45 be the mean of all the determinations of the density of the earth, and ·20 be approximately the probable error, the meaning is that the probability of the real density of the earth falling between 5·25 and 5·65 is 1/2. Any other limits might have been selected at will. We might calculate the limits within which it was one hundred or one thousand to one that the truth would fall; but there is a convention to take the even odds one to one, as the quantity of probability of which the limits are to be estimated.

Many books on probability give rules for making the calculations, but as, in the progress of science, persons ought to become more familiar with these processes, I propose to repeat the rules here and illustrate their use. The calculations, when made in accordance with the directions, involve none but arithmetic or logarithmic operations.

The following are the rules for treating a mean result, so as thoroughly to ascertain its trustworthiness.

1. Draw the mean of all the observed results.

2. Find the excess or defect, that is, the error of each result from the mean.

3. Square each of these reputed errors.

4. Add together all these squares of the errors, which are of course all positive.

5. Divide by one less than the number of observations. This gives the *square of the mean error*.

6. Take the square root of the last result; it is the *mean error of a single observation*.

7. Divide now by the square root of the number of observations, and we get the *mean error of the mean result*.

8. Lastly, multiply by the natural constant 0·6745 (or approximately by 0·674, or even by 2/3), and we arrive at the *probable error of the mean result*.

Suppose, for instance, that five measurements of the height of a hill, by the barometer or otherwise, have given the numbers of feet as 293, 301, 306, 307, 313; we want to know the probable error of the mean, namely 304. Now the differences between this mean and the above numbers, *paying no regard to direction*, are 11, 3, 2, 3, 9; their squares are 121, 9, 4, 9, 81, and the sum of the squares of the errors consequently 224. The number of observations being 5, we divide by 1 less, or 4, getting 56. This is the square of the mean error, and taking its square root we have 7·48 (say 7-1/2), the mean error of a single observation. Dividing by 2·236, the square root of 5, the number of observations, we find the mean error of the *mean* result to be 3·35, or say 3-1/3, and lastly, multiplying by ·6745, we arrive at the *probable error of the mean result*, which is found to be 2·259, or say 2-1/4. The meaning of this is that the probability is one half, or the odds are even that the true height of the mountain lies between 301-3/4 and 306-1/4 feet. We have thus an exact measure of the degree of credibility of our mean result, which mean indicates the most likely point for the truth to fall upon.

The reader should observe that as the object in these calculations is only to gain a notion of the degree of confidence with which we view the mean, there is no real use in carrying the calculations to any great degree of precision; and whenever the neglect of decimal fractions, or even the slight alteration of a number, will much abbreviate the computations, it may be fearlessly done, except in cases of high importance and precision. Brodie has shown how the law of error may be usefully applied in chemical investigations, and some illustrations of its employment may be found in his paper.[289]

[289] *Philosophical Transactions*, 1873, p. 83.

The experiments of Benzenberg to detect the revolution of the earth, by the deviation of a ball from the perpendicular line in falling down a deep pit, have been cited by Encke[290] as an interesting illustration of the Law of Error. The mean deviation was 5·086 lines, and its probable error was calculated by Encke to be not more than ·950 line, that is, the odds were even that the true result lay between 4·136 and 6·036. As the deviation, according to astronomical theory, should be 4·6 lines, which lies well within the limits, we may consider that the experiments are consistent with the Copernican system of the universe.

[290] Taylor’s *Scientific Memoirs*, vol. ii. pp. 330, 347, &c.

It will of course be understood that the probable error has regard only to those causes of errors which in the long run act as much in one direction as another; it takes no account of constant errors. The true result accordingly will often fall far beyond the limits of probable error, owing to some considerable constant error or errors, of the existence of which we are unaware.

*Rejection of the Mean Result.*

We ought always to bear in mind that the mean of any series of observations is the best, that is, the most probable approximation to the truth, only in the absence of knowledge to the contrary. The selection of the mean rests entirely upon the probability that unknown causes of error will in the long run fall as often in one direction as the opposite, so that in drawing the mean they will balance each other. If we have any reason to suppose that there exists a tendency to error in one direction rather than the other, then to choose the mean would be to ignore that tendency. We may certainly approximate to the length of the circumference of a circle, by taking the mean of the perimeters of inscribed and circumscribed polygons of an equal and large number of sides. The length of the circular line undoubtedly lies between the lengths of the two perimeters, but it does not follow that the mean is the best approximation. It may in fact be shown that the circumference of the circle is *very nearly* equal to the perimeter of the inscribed polygon, together with one-third part of the difference between the inscribed and circumscribed polygons of the same number of sides. Having 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.[291]

[291] Quetelet, *Letters*, &c. p. 116.

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 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 source of error is far to be preferred to others which trust to probabilities for the elimination of the error. As Flamsteed says,[292] “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.

[292] Baily, *Account of Flamsteed*, p. 56.

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:[293]′“--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.[294]

[293] Gould’s *Astronomical Journal*, Cambridge, Mass., vol. ii. p. 161.

[294] Philadelphia (London, Trübner) 1863. Appendix, vol. ii. p. 558.

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 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. Professor Roscoe found that the presence of phosphorus caused serious and almost unavoidable errors in the determination of the atomic weight of vanadium.[295] 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.[296]

[295] Bakerian Lecture, *Philosophical Transactions* (1868), vol. clviii. p. 6.

[296] *Results of Observations at the Cape of Good Hope*, p. 283.

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 symmetrical and characteristic form of the curve of error. If so, 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 *a*^{2}) the sum of all quantities of the same kind, and having the same place in the equations as *a*^{2}, we may briefly describe the two resulting mean equations as follows:--

(sum of *a*^{2}) . *x* + (sum of *ab*) . *y* = (sum of *ac*), (sum of *ab*) . *x* + (sum of *b*^{2}) . *y* = (sum of *bc*).

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;[297] 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, 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.

[297] *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.

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,[298] “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.

[298] Gauss, translated by Bertrand, p. 25.

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.

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;[299] 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 ten-millionth part of the distance from the pole to the equator (p. 314).

[299] Jamin, *Cours de Physique*, vol. ii. p. 60.

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 IV.

INDUCTIVE INVESTIGATION.