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observations, any preponderance, however small, among the efficient causes in action — that it becomes applicable to those complicated cases in which we find it resorted to. As an instance of this nature, we shall take a phenomenon which has engaged the attention of all who have written on probabilities, from Laplace downwards; one which has been much insisted on by M. Quetelet, and on whose acknowledged obscurity his inquiries have at length thrown a ray of light; viz., the excess of the number of births of male over that of female infants. As a matter of observation, the phenomenon is indisputable; but it requires the assemblage of a great number of instances to bring it out into evidence. In individual experience, or in the birth registers of a parish or small town, the tendency to excess on the male side is quite overlaid and concealed by accidental irregularities. It is otherwise when those of great cities or whole nations are consulted. The irregularities then disappear by mutual destruction, and the result exhibits the tendency in question in its full prominence. If we extract from the population returns of England and Wales the total numbers of registered births in the seven years, from 1839 to 1845 inclusive, we find 1,863,892 males and 1,772,491 females, the excess being 91,401 on the male side, or 105.157 males to 100 females. Suppose it were urged that this may, after all, be a purely accidental excess. It might be said, not without apparent plausibility, that as it would be the height of improbability to expect in so vast a number an exact equality, so, on the other hand, an excess of 91.401, which, though a large number in itself, is yet but 2, per cent. on the total number of cases, does not seem so very improbable. To this theory replies that, where such high numbers are concerned, it is so : — that the case assumed in the objection is identical with that of drawing 3,636,383 balls out of an urn containing black and white balls in equal proportion and infinite in number, and that the expectation of drawing such an excess of one colour in such a number, so far from a mere moderate unlikelihood, is, in fact, equivalent, supposing the chances equal, to the expectation of throwing an ace 613 times successively, with a single fair die.* Even on a total of 20,000 births we might bet many thousand millions to one that the same relative preponderance would not be found, were the chances even.

It is abundantly evident, therefore, that we have here arrived at proof of a tendency which must be taken as a law of human

* The chances against throwing an ace only nine times in succession, are ten millions to one.

VOL. XCII. NO. CLXXXV.

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nature under the circumstances in which it exists, at least in this country; and the constancy with which the proportion is maintained in successive years, and even in different nations, is not less striking than the fact itself, and shows it to be a result of deep-seated causes, acting with almost absolute uniformity on great masses of mankind. Thus in the seven years from which the above ratio has been concluded, taking them seriatim, we find 104.8, 104•7, 105.3, 105.2, 105.4, 105-4, 105•2, on totals averaging about half a million each; while in France a similar comparison gives 105.9, 105.7, 106.1, 106.2, 105.8, 105.9, 105.9, on nearly double the total numbers. As to the causes of this most striking phenomenon, much speculation has, of course, prevailed; but the inquiries of M. Quetelet into the statistics of marriage have rendered it extremely probable* that the relative ages of the parents very materially influence the sex of the offspring, and that the effect is therefore a resultant one, due to this physiological cause, acting through the medium of all those prudential and moral considerations which in civilised states determine the relative ages of parties contracting marriage. This view of the subject is strongly corroborated by a separate examination of the registers of illegitimate birth, which indicate an excess of only 3 instead of 5 per cent.

The causes, or tendencies indicative of causes, which may be disclosed by the assemblage and comparison of numerous recorded instances, are classed by M. Quetelet under three heads : constant, variable, and accidental. The latter class may be considered as entirely eliminated by their mutual destruction when vast numbers are concerned, and the whole series of collected cases is so investigated as to afford a single result. The same process also will in great measure destroy the effect of variable causes, if their variation be periodical in its law, and the observations be made indifferently in all the phases of their period. It is the peculiar property, however, of causes of this latter description, through whatever train of circumstances their action is propagated, ultimately to emerge to view in manifestations equally periodical with the causes themselves. In cases of dynamical action this peculiarity is susceptible of demonstration, and has been so demonstrated under the name of the

principle of forced vibrations;' and experience abundantly proves its general applicability to every case of indirect action, whether physical or moral. To those, therefore, who personally watch the developement of phenomena, and register effects as

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* Essai de Phys. Sociale, i. 57. Citing Hofacker and Sadler in corroboration.

1850.

Detection of periodical Causes.

35

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they arise with sufficient exactness, such causes will be detected, and their periods at the same time disclosed by the periodical fluctuations they occasion; or they may be searched for, if suspected to exist overlaid by accidental errors, by dividing the series of observed results into groups, differing in phase (i. e., dividing the extent of the period suspected into several equal portions, and grouping the results observed in each together).

The influence of the periodical cause suspected will then become apparent under the form of differences in the mean results of the several groups. Of this process every part of science teems with examples. In astronomy we owe to it the grand discoveries of aberration of light, the nutation of the earth's axis, the separation of the effects of the sun and moon on the tides, and an infinity of others; in meteorology, that of the diurnal and annual fluctuations of the barometer; in magnetism, the daily and annual changes in the direction and intensity of the magnetic forces; and in statistics, the annual oscillations observable in all the great elements of population, which the researches of M. Quetelet have placed in a distinct light.

But among accumulated masses of results, without any attempt at subdivision into periodic groups, the influence of periodical causes may start into evidence on a general inspection of the differences from a mean result, after a totally different manner. We have seen that these differences present inter se a definite and perfectly cognisable law of arrangement, so long as their causes are purely casual. Any deviation from this law among the differences of the observed values from the mean, then, becomes at once an indication of a determining tendency, and will very often, by the character of the deviation, lead to a wellgrounded surmise of the nature of its cause. For instance, if a sudden falling off in the number of observed differences, beyond certain limits either way from the mean, accompanied with some degree of improbable accumulation at or about those limits, should be noticed, it may be taken as a certain indication of a periodical disturbing influence, having those limits for the maximum and minimum of its effect.

Again, if at any particular point in the scale of results arranged in order of magnitude we should notice a sudden and marked irregularity confined to a small extent, we may be sure that it arises from the action of some single, powerful, and exceptional influence. Thus, from the undue accumulation of conscript measurements below the standard height of 5 feet 2 inches, accompanied with a deficiency to the extent of 2275 cases in the two inches just above that standard, M. Quetelet is led to conclude that an influence foreign to the subject — in fact, a fraudulent practice, favouring the escape of the shorter men, has prevailed to that extent in the formation of the official returns he has employed as the basis of his calculations. (P. 98. Transl.)

Astronomy affords us a very remarkable example of this nature, which we adduce, by reason of a singular misconception of the true incidence of the argument from probability which has prevailed in a quarter where we should least have expected to meet it. The scattering of the stars over the heavens, does it offer any indication of law ? In particular, in the apparent proximity of the stars called “double,' do we recognise the influence of any tendency to proximity, pointing to a cause exceptional to the abstract law of probability resulting from equality of chances as respects the area occupied by each star? To place this question in a clear light, let us suppose that, neglecting stars below the seventh magnitude, we have measured the distance of each from its nearest neighbour, and calculated the squares of the sines of half these distances, which therefore stand to each other in the relative proportion of the areas occupied exclusively by each star. Suppose we fix upon a circular space of 4" in radius as the unit of superficial area, and that we arrange all the results so obtained in groups, progressively increasing from 0 by the constant difference of one such unit. Now the fact, to which M. Struve originally called attention *, and on which we believe all astronomers are agreed, is, that the first of these groups is out of all proportion richer than any of the others; and that the numbers degrade in the groups adjacent with excessive rapidity; so that, for example, calculating on the numbers given by Struve t, we find the first group to contain 180 cases ; the next three 68, or on an average 22 each ; the next twelve 70, or 6 each on an average; and the next forty-eight only 94 in all, averaging 2 to each ; while a general average † would assign only one star to 540,000 such units of area. The case, then, is parallel to that of a target of vast size, marked out into 6700 millions of equidistant rings, riddled with shot marks in the bull's eye, and with a tolerable sprinkling in the first 50 or 60 rings, beyond which the whole area offers nothing for remark indi

* Catalogus Novus Stellarum duplicium, &c. Dorpati, 1827.

f Ibid., p. xxxii., Introduction. Each of M. Struve's classes is doubled, since each constituent of a double star counts as a separate case.

$ Taking 12,400 as the number of stars of the magnitudes and within the region of the heavens contemplated, viz. from the North Pole to 15° south declination, which number, for the above reason, has to be doubled.

.1850.

A priori Argument respecting double Stars.

37

cative of any particular local tendency, though clotted all over with marks, in the sparing manner above described. Any one who should view such a target, bearing in mind what is said above, must feel convinced that a totally different system of aiming had been followed in planting the interior and exterior balls.

Such we conceive to be the nature of the argument for a physical connexion between the individuals of a double star prior to the direct observation of their orbital motion round each other. To us it appears conclusive; and if objected to on the ground that every attempt to assign a numerical value to the antecedent probability of any given arrangement or grouping of fortuitously scattered bodies must be doubtful*, we reply, that if this be admitted as an argument, there remains no possibility of applying the theory of probabilities to any registered facts whatever. We set out with a certain hypothesis as to the chances : granting which, we calculate the probability, not of one certain definite arrangement, which is of no importance whatever, but of certain ratios being found to subsist between the cases in certain predicaments, on an average of great numbers. Interrogating Nature, we find these ratios contradicted by appeal to her facts; and we pronounce accordingly on the hypothesis. It may, perhaps, be urged that the scattering of the stars is un fait accompli, and that their actual distribution being just as possible as any other, can have no à priori improbability. In reply to this, we point to our target, and ask whether the same reasoning does not apply equally to that case? When we reason on the result of a trial which, in the nature of things, cannot be repeated, we must agree to place ourselves, in idea, at an epoch antecedent to it. On the inspection of a given state of numbers, we are called on to hold up our hands on the affirmative or negative side of the question, Bias or No bias? In this case who can hesitate ?

Accidentally variable causes overlay altogether the evidence of regular action, so that the elimination of their influence is in all cases synonymous with the extension of knowledge. It is not, however, to this or to any other calculus that we can look for special rules of conduct in this part of inductive inquiry beyond the simple precept of collecting facts in great numbers, and employing mean results in lieu and to the exclusion of single observations wherever numerical magnitude is concerned. This precept is, however, of infinite use in all cases where we test the efficacy of a presumed cause by the numerical corre

* London, Ed. and Dub. Philosoph. Magazine, &c. Aug. 184

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