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Character of the Author.
tment of mathment of ideas particulars, since lose sight of
caution all who desire to enter upon the more general and intricate parts of the theory, never for an instant to lose sight of special examples and numerical particulars, since nothing can exceed the bewilderment of ideas experienced by the tyro in this department of mathematics, who trusts himself with both feet off the ground to the whirl of symbols and notations in which those who are accustomed to ride these storms know how to guide their course, and even seem to feel a wild and fierce delight in the turmoil.
There is, however, a very large portion of those who desire to know something of the results at which thinking men have arrived in this as in all other departments of knowledge, to whom a book full of mere algebraic formulæ and calculations must remain for ever sealed. These are not necessarily or generally persons of despicable acquirements or intellect; nor is this their curiosity to be slighted as devoid of a reasonable object or motive. They desire to understand with a view to apply. Mathematicians, in common with men of high science in all departments, have long since begun to perceive that they have to address a mixed audience of a highly important and respectable character — an audience by no means disposed to treat them with derision or distrust, but on the contrary, to regard them as their fitting instructors in matters within the scope of their legitimate pretensions, if only they will condescend to make themselves intelligible. Learned jargon such an audience will not endure. Charlatanerie of every description it can detect and chastise. Common-sense statement driven home by pointed illustration, and an earnest endeavour to inform, are what it eagerly desires, and in such a spirit is assuredly entitled to receive at the hands of those able to afford it.
The work now before us is conceived on these principles, and on this view of the duty devolving on those who have advanced beyond the ordinary limits of knowledge, to pause occasionally in their onward career, and inform the world, in plain terms and without exaggeration, whither they have got, and what they see beyond, which may make it worth while either for themselves to continue in the track, or for others to follow in it; as well as to render easy and intelligible to all whom it may concern the practical application of the information acquired. Its author is a teacher well worth listening to, and may claim attention on the excellent ground that he has himself approached his subject in a practical manner, through a long and severe apprenticeship to the actual collection of data in a great variety of departments, and to the deduction
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from them of definite results of unmistakeable value and import, by the rules and principles he professes to teach.
M. Quetelet has been long and advantageously known as an ardent and successful cultivator of science. No one has exerted himself to better effect in the collection and scientific combination of physical data in those departments which depend for their progress on the accumulation of such data in vast and voluminous masees, spreading over many succeeding years, and gathered from extensive geographical districts, --- such as Terrestrial Magnetism, Meteorology, the influence of climates on the periodical phenomena of animal and vegetable life, and statistics in all the branches of that multifarious science, political, moral, and social. Peculiar facilities (or rather opportunities which he has improved into facilities) have been afforded him for such researches by his position in his own country, where he has filled the leading and responsible office (there at least, as in France, so considered) of perpetual Secretary to the Belgian Academy of Sciences, as well as that of Director of the Royal Observatory of Brussels, an institution which owes its establishment mainly to his solicitations, and its remarkable efficiency as a physical observatory' entirely to his activity and perseverance. Placed as a member of the Central Statistical Commission in direct communication with his government, whose confidence he deservedly possesses, he has been enabled to suggest and carry out a variety of useful and important improvements, both in the forms and objects of statistical registry. The center of an immense correspondence, he has moreover succeeded in inspiring numerous and able coadjutors, not only in Belgium but in other countries, with a similar zeal, and impress them with his views and secure their aid in carrying out a system of definite and simultaneous observation. No one threw himself with more entire devotion into the system of combined magnetic and meteorological observation set on foot by the British and other governments, and which has been productive of, and continues to produce, such useful and valuable results to science. And it will not cease to be remembered that while in one special branch of combined inquiry (that directed to tracing the progress of atmospheric waves across Europe and the Atlantic), France stood aloof and furnished not one solitary instance of co-operative observation (thus interposing herself as a desert might have done between England and the rest of the Continent), Belgium, influenced by M. Quetelet's example and appeals, supplied corresponding and very valuable observations from five
General form and Division of the Work.
stations.* By many who may be little able to estimate such claims on our attention, our author will yet be regarded with interest as the preceptor of a prince whose conduct and virtues have endeared him to every Englishman.
In considering the manner in which he has executed his task, we beg to protest in limine against the form of letters to an illustrious personage in which it appears, and of which (though as he informs us, so originally written) a very moderate amount of subsequent alteration would have divested it. Although nothing which can be considered adulatory occurs in any of them, yet every reader must feel that a certain portion of each letter is out of place as regards his own information, and, so far, an interruption to the consecutiveness of his thoughts. It is a certain quantity of non-luminous matter, interposed between the author and his meaning, which serves no good purpose, since it neither pleases nor relieves. The objection is general against all such artifices of cominunication as letters dialogues — catechisms, &c., if the subject be a scientific one and the object of the work didactic. They are like pebbles in the bed of a stream, which may make it sparkle and please the eye and ear when the thought is but loosely engaged. But the welling waters of scientific lore should be clear, glassy, and unrippled, offering their inmost depths to a quiet and contemplative gaze, and neither distracting by murmurs nor dazzling by irregular reflections.
A comparatively small portion of the work, the first and least extensive only of four divisions into which it is broken, and an appendix in the form of notes containing tables and formulæ, are devoted to the theory of Probabilities in the abstract, and to the illustration of its fundamental axioms and propositions; all which have been so repeatedly and so well laid down and elucidated in the various treatises we already possess, that it is hardly possible to place them in any very new and more than usually striking light. The distinction between mathematical and moral expectation belongs to this part of the subject, and can hardly be put more pointedly than it was originally done by Buffon, who first called attention to it.
If two men were to determine to play for their whole property' [supposed equal, and with equal risks), what would be the effect of the agreement ? The one would only double his fortune, and the
* Subsequently increased to 70 stations over all parts of Europe, held in correspondence with Brussels. (Rapport addressé au Ministre de l'Intérieur sur l'état et les travaux de l'Obs. R. de Bruxelles, 1845.)
other would reduce his to nought. What proportion is there between the loss and the gain? The same that there is between all and nothing: the gain of the one is but a moderate sum ; the loss of the other is numerically infinite, and morally so great that the labour of his whole life may not suffice to restore his property.'.
It was on such considerations that Daniel Bernouilli was led to propose, as a rule for estimating the value of a very small pecuniary or other material advantage, its relative value as compared with the total fortune of the party benefited, and for the moral as distinguished from the mathematical expectation of such advantage, that relative value multiplied by the probability of its accruing. On this or some equivalent mode of estimation is founded the principle of the subdivision of risks, which, rightly understood, so as to preserve their absolute independence while multiplying their number, is the best guarantee of commercial security. It is by such subdivision carried to an extreme point, that insurance and annuity offices thrive, and that benefit societies might do so, were it not for the single great risk which the dishonesty of entrusted agents throws in. their way as a fearful stumbling-block.
In the case of savings' banks, this is, in fact, the only risk ; and, as experience has too recently and abundantly shown, a most imminent and fatal one. To annihilate this risk by a perpetual and searching superintendence, carried even to the utmost stretch of suspicious vigilance, obnoxious as it may appear, is the paramount duty of all who connect themselves with them as managers or trustees. Of the general benefit of such institutions, which, by guaranteeing the security of the produce of successful exertions, tend to cherish habits of industry, prudence, and frugality, no one can entertain. a doubt. It is in this point of view that a certain considerable amount of national indebtedness, so far from meriting denunciation as an evil, ought to be regarded as an indispensable element and engine of civilisation. In its practical working it resolves itself into the establishment of a savings' bank on a vast scale, administered with what may be considered a perfect exemption from the consequences of dishonesty in its officials, and subject only to the inconvenience (no doubt a considerable one) of its deposits being withdrawable only at a market value, — but that market the fairest, readiest, and openest which can anywhere exist. Yet it is too commonly forgotten by those who deprecate taxation, while insisting on the objects for which taxation is instituted, and which alone it can secure, that the interest on savings' bank deposits is derivable only from that source, and
Means and Limits.
that every depositora tax-holder
asur actions certain low, and
that every depositor is as truly (and in some respects even more emphatically) a tax-holder — as the proprietor of consols.
To render the consequences of our actions certain and calculable as far as the conditions of humanity will allow, and narrow the domain of chance, as well in practice as in knowledge, is so thoroughly involved in the very conception of law and order as to make it a primary object in every attempt at the improvement of social arrangements. Extensive and unexpected fluctuation of every description, as it is opposed to the principle of divided and independent risks, so it also, by consequence, stands opposed to the most immediate objects of social institutions, and forms the element in which the violent and rapacious find their opportunities. Nothing, therefore, can be more contrary to sound legislative principle than to throw direct obstacles in the way of provident proceedings on the part of individuals (as, for instance, by the exorbitant taxation of insurances), or to encourage a spirit of general and reckless speculation, by riding unreservedly over established laws of property, for the avowed purpose of affording a clear area for the developement of such a spirit on a scale of vast and siinultaneous action. The sobering influence of an upper legislative assembly, refusing its sanction to the measures demanded, or spreading them over time, can alone repress or moderate these epidemic outbreaks of human cupidity: and its mission is abandoned, and its functions pro tanto abdicated, if it retreat from the performance of this duty.
The first and most important application of the calculus of Probabilities (since it applies to all departments of science, and affords a measure of the degree of precision attained in all numerical determinations) is that which relates to means and limits, and which forms the second division of M. Quetelet's work. A general idea of the sort of questions contemplated in this department of the theory, and the kind of relations they involve, may be conveyed by the following simple case. Suppose a man to throw stones at random, and without any aim : from the marks left by any given number of them, however great, on a wall, we could obtain no impression, or a fallacious one, of his intention. All that we could conclude from their evidence would be, that, if he aimed at anything, it was not a point in the surface of the wall, and that only stray missiles had struck it. But, suppose he had been practising with a rifle at a wafer on the wall; which, being subsequently removed, we were required to indicate at once the situation it had occupied, and his skill as a marksman. It is obvious enough that, from the evidence of a great number of shot-marks, both might be deter
VOL. XCII. NO. CLXXXV.