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On the Theory of Probabilities*

Published online by Cambridge University Press:  18 August 2016

John F. W. Herschel
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Extract

The theory of Probabilities has been characterized by Laplace, one of those who have contributed most largely to its advance,—as “good sense reduced to a system of calculation;” and such, no doubt, it is. But it must be especially noticed that there is hardly any subject to which thought can be applied, which calls for so continuous an application of that excellent quality, or in which it i s easier to make mistakes from simple want of circumspection. And, moreover, that its reduction to calculation is attended with difficulties of a very peculiar nature, such as occur in no other application of mathematical analysis to practical subjects, arising out of the great magnitudes of the numbers concerned, which defeat the ordinary processes of arithmetical and logarithmic calculation, by exhausting the patience of the computer, and require special methods of approximate evaluation to bring them within the compass of human industry. These methods form a conspicuous feature of the general subject, and have furnished scope for very extraordinary displays of mathematical talent and invention. That very large numbers will inevitably be concerned in questions where numerous and independent contingencies may take place, and in any order or mode of combination, will be apparent to any one who considers the astonishing fecundity of such combinations numerically estimated, when the combining elements are many. For example, the number of possible “hands” at whist (regard being had to the trump) is 1,270,027,119,200.

Type
Reprint
Information
Journal of the Institute of Actuaries , Volume 15 , Issue 3 , October 1869 , pp. 179 - 218
Copyright
Copyright © Institute and Faculty of Actuaries 1870

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References

page 183 note * Recherches sur la Probabilité des Jugemens en Matiére Criminelle et en Matiére Civile; précédées des RéglesGénéralesdu calcul des Probabilités. Paris, 1837.

page 186 note * This phrase appears to us rather too strong. While the same Office will grant insurances for £100 and for £10,000 at its own risk, it can scarcely be said that the subdivision of risks is carried to an extreme point.—ED. J. I. A.

page 187 note * 1850.

page 191 note * That is, the increase or diminution in one of which may take place without increasing or diminishing the other. On this, the whole force of the proof turns. (H. 1857.)

page 191 note † Sir Joshua Reynolds, in his celebrated Lectures to the Royal Academy, has laid it down as the fundamental principle of the pictorial art, that beauty of form and feature consists in their close approximation to the mean or average conformation of the human model. Were this the case, ugliness ought to be extremely rare, and the highest degrees of beauty those of the most ordinary occurrence, a conclusion contrary to all experience. (H. 1857.) Another consequence follows, viz., that in designing the original prototype of the human form and face, the designer had not un view especially the production of what men call beauty, but some one or more objects of greater importance to the wellbeing of the total organism. The animus of making the beautiful thing, in that sense, was absent. The capability of beauty having been secured in the plan of the oiganiza-tion, it seems as if it were intended that the perfection of personal beauty like the highest genius or the most exalted goodness should occur but rarely in our species.

page 193 note * Sur I'Homme et sur la Développement d e ses Facultés; on Essai de Physique Sociale. Paris, Bachelier, 1835.

page 193 note †

page 194 note * Adopting this distinction, which appears to us not only scientifically correct, but practically convenient, it follows that we must speak of the “average duration of life at a given age,” instead of the “mean duration”—the phrase introduced by Dr, Fair.—ED. J. I. A.

page 201 note * So again, an examination of the elements of all known cometary orbits has disclosed a tendency to direct or eastward motion, increasing in the degree of its prominence with the approach to coincidence of the orbit with the plane of the ecliptic, —and especially marked in the cases where calculation has assigned elliptic elements to "the orbit. Here we have a tendency pointing to a cause, still unknown, but with whose effects we are so far familiar that we can trace its action throughout the planetary system, with only two known exceptions among its most remote and insignificant constituents, an d those of a very undecided character.

page 202 note * The chances against throwing an ace only nine times in succession, are ten millions t o one.

page 201 note † Essai de Phys. Sociale, i, 57. Citing Hofacker and Sadler in corroboration.

page 203 note * Encyclop. Metropol, Article Sound, § 323, et seq.

page 204 note * Catalogus Novus Stellarum duplicium,&c. Dorpati, 1827.

page 204 note † Ibid., p. xxxii., Introduction. Each of M. Struve's classes is doubled; since each constituent of a double star counts as a separate case.

page 205 note * 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 15o south declination, which number, for the reason in the foregoing note, has to be doubled.

page 205 note † London, Ed. and Dub. Philosoph. Magazine, &c. Aug., 1849.

page 209 note * The vast multitude of illegitimate births in France would seem to be traceable in great measure to the difficulties thrown in the way of marriage by requiring the expressed consent of a great number of relatives of both parties to its celebration.