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be the impression of all who have examined the evidence for this theory.
It is not our intention to conclude the subject in the present number; the length of this article (for such articles should not be very long) warns us to conclude for the present by finishing our account of the difficulties which have been placed in the way of the student, previously to entering upon the consideration of the subject matter of the treatise.
The Théorie des Probabilités consists of three great divisions. 1. An introductory essay, explanatory of general principles and results, without any appearance of mathematical symbols. 2. A purely mathematical introduction, developing the analytical methods which are finally to be employed. 3. The application of the second part to the details of the solution of questions connected with probabilities. The first of these has been also published in a separate form, under the title of Essai Philosophique, &c., and is comparatively well known. Our business here is mostly with the second and third. The arrangement will seem simple and natural, but there is a secret which does not appear immediately, and refers to a point which distinguishes this and several other works from most of the same magnitude. The work is not an independent treatment of the subject, but a collection of memoirs taken 'verbatim from those which the author had previously inserted in the Transactions of the Academy of Sciences. Thus in the volume for 1782, appears a paper on the valuation of functions of very high numbers, with an historical and explanatory introduction. Now this introduction being omitted, the rest of the memoir is, substantially, and for the most part word for word, inserted in the work we are now describing. And the same may be said of other memoirs published at a later period: so that the Théorie des Probabilités, first published in 1812, may be considered as a collection of the various papers which had appeared in the Transactions cited from 1778 up to 1812.
This materially alters the view which must be taken of the treatise, considered as intended for the mathematical student. It also makes a change in the idea which must be formed of the real difficulty of the subject, as distinguished from that which is actually found in reading Laplace. The course taken has both its advantages and disadvantages; on which it may be worth while to say a few words.
Of the highest and most vigorous class of mathematical students, it may be easily guessed that they are most benefitted by the works which are least intended for them. Complete digestion and arrangement, so far from being essential to aid them in the formation of power, are rather injurious. The best writer is he who shows most clearly by his process where the difficulty lies, and who meets it in the most direct manner. All the artifice by which the road is smoothed and levelled, all the contrivance by which difficulty is actually overcome without perception of its existence, though a desirable study for the proficient, and most useful with reference to the application of the science, is a loss of advantageous prospect to the student who wishes to become an original investigator. An officer who has never seen any but well-drilled soldiers, may command an army of them: but he who would raise an army must have been used to the machine he wishes to create in every stage of its process of creation, from a disorderly assembly of clowns up to a completely organized force. It is on such a ground as this that we take our stand, when we say that Euler, from the almost infantine simplicity with which he presents the most difficult subjects, and Lagrange, from the unattainable combination of power and generality which he uses for (more than through) the student, are not the best guides for one who would practice investigation. It is Laplace whose writings we should recommend for this purpose, for those very reasons which induce us to point him out as one of the most rough and clumsy of mathematical writers. A student is more likely, pro ingenio suo, to be able to imitate
Laplace by reading Laplace, than Lagrange by Lagrange, or Euler by Euler.
In the next place, of all the works which any one has produced, the most effective for the formation of original power are those which lie nearest to his own source of invention. All the difference between analysis and synthesis will exist, for the most part, between the memoir in which the discoverer opened his views for the first time, and the ultimate method which he considered as most favourable for their deduction from his first principles. Hence we should recommend to the student to leave the elementary works and the arranged treatises as soon as possible, and betake himself to the original memoirs. He will find them not only absolutely more clear than compilations from them, but what is of much more importance, they state with distinctness what has been done on each particular point, and what is attempted to be done. If there should arise confusion from the student not perceiving that he is employed upon an isolated part of a whole which is not yet complete, there are safeguards in the Memoir which do not exist in the Treatise. Take any work on the differential calculus, from the time of Leibnitz downwards, and the formality of chapters, distinction of subjects, and treatment of nothing but what is complete, or appears so, will leave the impression that the whole is exhausted, and that all apparent difficulty arises from the student not being able to see all that is presented to him.
Now the fact is that in many cases the obstacle is of another kind, namely, that the reader is not made aware that there is more to be looked for than is presented. The assertion, je n'en sais rien, by which Lagrange frequently astonished those who imagined that a grand mathematician knew every thing, is frequently embodied in the spirit, or enspirited into the body, of a memoir, but seldom into that of a formal treatise. It happened to us not long ago to be very much puzzled with the account of a process given in the great work of Lacroix, one of the best of methodical writers. Chance threw in our way the original memoir of Legendre, from which the process was taken, and we found that, word for word nearly, the former writer agreed with the latter, so far as he went. But a few sentences of omission in which the original writer had limited himself, were, it should seem, inconsistent with the vastness of the general design indicated in the heading of the excellent compiler's chapter. The difficulty vanished at once, since it merely arose from venturing to hint to ourselves, in the way of doubt, precisely what the original writer had proposed as a limitation.
So far then, as the great work we have before us preserves the actual contents of the original memoirs, it must be looked upon as very wholesome exercise for the student. But there are still some defects, arising from not completing the plan. The short historical notice and general explanation is omitted, in consequence, we suppose, of the humiliation which the writer of a treatise would feel, were he compelled to name another man. The extravagance of an original memoir lights the candle at both ends; not only is an author permitted to say clearly where he ends, but also where he began. Did Stirling give a result which might have afforded a hint as to the direction in which more was to be looked for? Laplace may and does confess it in the Transactions of the Academy. But the economy of a finished work will not permit such freedoms; and while on the one hand the student has no direct reason for supposing that there ever will be any body but Laplace, he has, on the other, no means of knowing that there ever was any body but Laplace.
In the next place, the difficulty of the subject is materially increased by the practice of placing general descriptions at the beginning, instead of the end. Our present work begins with a tremendous account of the theory of generating functions, which we doubt not has deterred many a reader, who has imagined that it was necessary to master this first part of the work before proceeding to the rest. And why is this obstacle placed in the way? Because there was an old memoir ready to reprint from. And where in the subsequent part of the work is it used ? In some isolated problems connected with gambling, which in the first place might be omitted without rendering the material part of the work more difficult; and in the second place are applications of the theory of generating functions of so simple a character, that the preliminaries connected with it might be discussed in two pages. And in what future part of the work do the very tedious (though skilful) methods of development become useful which are formally treated in the introductory chapter ? Nowhere.
Hence the reader may begin to suspect that the difficulty of this work does not lie entirely in the subject, but is to be attributed in great part to the author's method. That such difficulty is in part wholesome, may be very true; but it is also discouraging, unless the student be distinctly informed upon its cause and character. Believing as we do that in spite of all we have said, the Théorie des Probabilités is one of the points to which the attention of the future analyst should be directed, as soon as the subject is in any way within his power, we shall here finish what we have to say on the character of the work, and proceed in a future article with that of its results.
ART. V.-1. De la Démocratie en Amérique. Par Alexis de
Tocqueville, Avocat à la Cour Royale de Paris, l'un des Auteurs du Livre intitulé “ Du Système Pénitentière aux Etats
Unis." Bruxelles, 1835. 2. The Americans in their Moral, Social, and Political Relations.
By Francis J. Grund. London, 1837.
HE United States of North America have been, and are, TH
every day, becoming more and more the subject of European attention. The experiment of a new form of Government there in progress—the magnificent scale on which that experiment has been essayed—the contrasts and comparisons which occur between the state of things there and in the countries of the old continent-all these points furnish matter for infinite and incessant speculation, and diversity of opinion. Theorists and practical men,” political parties of every shade and hue, all, with common consent, refer to America for examples to bear out their
respective doctrines. The speculatist enlarges upon the leading principles of the American constitution, and their development present and future. He compares them with the institutions of his own hemisphere, and strains the comparison in favour of either, according to his own predilections. The man, who eschews theory, pursues those principles through all the details of their working, and predicts boldly on the fate of the grand experiment from the operation, for good or evil, of its minor parts. The Conservative pounces
eagerness upon any defect that he can discover,--and every casual failure, or imperfect success, of the provisions for good government, is loudly proclaimed to the world; while all that gives hope of obtaining that great end, is passed over by him in most significant silence. The Reformer, according to the degree of his liberalism, appeals with moderate, or triumphant confidence to the example of America, to prove, that the freer the institutions—the happier and the more prosperous must a nation be.
It is in England that this attention has been peculiarly excited, as might naturally be expected, from the affinity that exists between her and her quondam colonies. Almost equally naturally this attention has been a good deal sharpened by jealousy. The parent state did not easily brook to be outstripped by her offspring in the development of the long (and wilfully) misunderstood art of good government. England had been too long and too much in advance of other nations, to be content with now seeing herself left behind. Neither was it at first a pleasant sight to behold American fleets stretching into seas once furrowed by British keels alone, and bearing the fruits of American industry and enterprize, to vie with, and perhaps excel, the productions of the mother country in remote markets that had hitherto been supplied solely by the latter. The incipient ill-feeling was assiduously fanned by the enemies of freedom, whose foul interest it was, that division and disunion should exist between the promoters of universal liberty. Although, happily, now on the wane, this jealousy yet subsists, and so strong as not to be surmounted without an effort by many men even of liberal minds. But the majority of this description have long conquered the unworthy feeling, and rejoice sincerely in the prosperity of America. With them, other and better motives give the impulse to an examination of her condition and prospects. They know that the great experiment in progress in that country, is one fraught with the deepest and most intimate interest to the wellbeing not only of the existing generation, but of millions yet unborn. The past history of mankind details many sorrows and much suffering, the consequence and effects of evils inherent in