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therefore, we feel authorized in recommending this treatise to-the public, as exhibiting by far the most complete view of the doctrine of life annuities, assurances, reversions, &c. that ha"s yet appeared in any language.
From a work where so much is really accomplished, it is natural to wish that every blemish may be removed. Before, therefore, we proceed to select some useful quotations from this treatise, we shall offer a few critical observations, of which we shall be glad to find Mr. Baily availing himself in a new edition.
1st. With respect to the notation. It is, on the whole, extremely well contrived; being at once fitted to prevent circumlocution, and ambiguity,—at least, after due attention has been paid to the observations on the subject of this notation, at pp. xxxiv—xxxix of the preface. But to render the advantages accruing from this notation complete, the author should either have exhibited all its peculiarities in one place, or in one place Irave referred to the various pages of the work, in which those peculiarities are introduced. This is attempted on the pages just referred to; but the illustration is not complete,^—as the author will soon find by comparing his preface with his Work. (Seepages 121, 125, 182, &c.) We would suggest the propriety of having a concise view of the notation, with references to the pages where its individual peculiarities are nlore fully explained, printed upon a single leaf; which may be cut out, and moved on in the book, so as at once to serve as a mark of the reader's progress, and to assist his recollection. We have frequently recommended a similar plan to young readers of French mathematical works, where almost every author introduces some peculiarity of notation, (often, it must be confessed, for no good purpose); and have uniformly been told that the expedient was found beneficial. ■ .
2ndly. Our author has sometimes deviated from the established rules of Mathematicians, never to affirm bat where they demonstrate, and never to assert in one place what is to be demonstrated afterwards. Thus, at p. 117, he asserts the equality of two quantities, 'for reasons,' he says, 'that will hereafter appear.' These 'reasons' may appear 'hereafter;' and we believe we should not find much difficulty in assigning them. But we are not aware that they appear in Mr. Baily's treatise; at least there is no reference; nor have we happened to detect them in the course of our perusal. Thus again, at p. 139, Mr. B. says, 'now, I have found, from a number of repeated trials that the value of' a particular quantity the subject of investigation, may 'be safely expressed' by oae or other of two formulae which are specified. Now we have found from a number of repeated trials, that mathematicians are • an unbelieving brood,' who think nothing can 'be safely expressed' that cannot be satisfactorily demonstrated. Mathematical science is not the science of ipse dixils. It takes nothing on authority. Mr. Baily should have selected a fair sample of his 'repeated trials' to enable his readers to trace the limits of error, before they are called upon to adopt his theorems. In consequence of our author having failed to establish conviction on this point, his table of the comparative accuracy of bis own results, and Mr. Morgan's, stands for nothing. It is sufficiently plain, that Mr. Morgan, by taking it as an equal chance in all ages that B will die after C may err egregiously; but we have no proof that Mr. Baily does not err in some degree: so that, as we are furnished with no measure of that degree, there can be no fair comparison instituted of the accuracy of these calculators with regard to the problem under discussion. Mr. Baily's noie, at p- 109, is open to similar objections. What arc the 'other quantities which very muck diminish the error?' And how may the 'proper allowances be made' in the cases adverted to?
3dly. We think our author much too brief in his explication of the laws of chance, and the application of these laws to the probabilities of human existence. That a thing so infinitely variable as the duration of human life should be subjected to mathematical research and computation, is » circumstance in itself so curious, that the developcment of the principles employed to educe certainty ' from so much apparent uncertainty, is certainly deserving of more than usual care. In this respect Baron Maseres's treatise is extremely perspicuous and satisfactory; and it would have been gratifying to us, had his circumspection and fulness of illustration, as to this particular, been imitated by Mr. Baily.
4thly. Our author, we apprehend, has sufficient opportunities of displaying his talents as an analyst, without expanding censure into an infinite series; and throwing a term upon almost every other page, as he has done through nearly a third of the present work. When a writer like Mr. Morgan, who has devoted the greater part of his life to the cultivation of one branch of analysis, and who is not celebrated for his knowledge of any other department of literature or science, betrays great negligence, inelegance, and inaccuracy; and when at ihe same time his works are appealed 10 by the public as decisive authority, it is highly proper that his principal errors should be pointed out. Yet we should like to see this done, without any indications of asperity or triumph. Some of the instances adduced by Mr. Baily of Mr. Morgan's awkwardness and inaccuracy of investigation are certainly very extraordinary; and several of the comments that accompany these instances are both curious and important; as at pp. 186, 219, 274, 290. We have not space for many quotations: but our readers shall not be deprived of the following.
'Is it not singular that, after the unlimited censure which Mr. Morgan has cast upon the "wretched" hypothesis of De Moivre, he should (as editor of Dr. Price's Observations on Reversionary Payments) suffer the Fortieth Table in that work to remain without a comment? particularly.as he must well know, because he hns taken much pains to prove, that the values in that table are extremely erroneous, and in many cases are more than one third of their true value too much. And is it not more singular that, to this very hour, it should serve to determine the value of such assurances at the Equitable Society, the business of which Mr. Morgan has so long and so ably conducted! Il is true, that, the concerns of that society being, for the most part, established upon such fair and truly equitable principles, it little signifies &>w muck is paid for an assurance provided every one pays in proportion: but it must be evident that, in the present case, the rest of the society are benefited at the expense of those who assure on the contingency mentioned in this problem. And it is amusing to observe how blindly the other established offices, as well as the new ephemeral Companies, have followed this error of their great prototype.' Note, pp. 187—188.
5tlily. There is one class of problems which Mr. Baily has inadequately, and, indeed, incorrectly solved: we mean that class, in which it is required to determine the present value of a given sum payable at the time when any number of lives become extinct. Mr. Baily, like most of his predecessors, finds the present value of the sum payable at the end of the year, instead of at the time when the lives become extinct. This, it is true, simplifies the inquiry, but it leaves the real problem unsolved. The investigator thus takes that to be an annual chance which is really momentary, and allows as much discount of money on the value of the chance in the first moment of the year as in the last. It cannot require much of Mr. Baily's acuteness to know that this must be incorrect: and indeed it has long ago been shewn, that in estimating the assurance on a single life, the error is mora than a sixty-seventh part of the result.* In more lives the error becomes increased or decreased, according to circumstances which need not here be traced. Mr. Baily in some cases seems aware of the inadequacy of his own solution: but he
* See Eel. Rev. Vol. V. p. 955, and Leybourn's Math. Rep, N. S. No. VII. there referred to.
rather evades the difficulty, than fairly-meets it in a manner worthy of his ingenuity. Thus, at p. 429, where he inquires, 'what is the present value of a legacy of oflOO to be received on the extinction of any one of three' given lives, he says, * I consider a legacy as not due till the end of the year in which the testator dies: for it is seldom paid immediately.' And hence he solves a very different question to the one he has proposed. It does r.ot follow that any one of I he three lives mentioned in the example, must of consequence be the life of the testator; and even if the individual life which actually becomes first extinct should be that of the testator, the value of the legacy found by Mr. Baily's rule is not the correct value, because it is not the value of the sum payable at the death of the person, but at the end of the year in which he dies. Had Mr. Baily found the value of the sum payable ayear after the extinction of the life, be would have produced a result corresponding more accurately with the usual custom on these occasions; yet even this would not have been the answer to the question he proposed to resolve. (To he concluded in the next Number.)
Art. III. Lectures on Painting, delivered at the Royal Academy of Arts. By the late John Opie, Esq. Professor in Painting to the Royal Academy. To which are prefixed, A Memoir by Mrs. Opic, and other Accounts of Mr. Opie's Talents and Character. 4-to. pp. 90, 18§. Price 11. Is. Longman and Co. 1809.
HTHE volume before us affords a striking contrast to the Works of Barry, reviewed in our last number. -However highly we might rate the professional merits of that unfortunate artist, we could not but feel disgusted with the extravagance of bis self complacency, and the presuming ignorance of his editor. But in the present instance we have had to listen to the modest and feeling tribute paid by surviving worth to departed talent, and to contemplate the literary relics of a man, whose unobstrusive merit raised him gradually from obscurity to fame; who betrayed no symptoms •of envy at a rival's praises, who passed through life without an enemy, though not without misfortune, and who died honoured and lamented.
Mrs. Opie's memoir of her husband, although extremely interesting, is not quite satisfactory. It is less a memoir than an elogc, although not so much exaggerated as effusions of this description usually are, and giving upon the whole, perhaps, a juster notion of the man, of his excellences-arid nis defects, than we could have expected from conjugal partiality. It is the production of a delicate, feeling, and modest mind; and is alike creditable to the soundness of the writer's judgement and the correctness of her taste. A slight sketch of Opie's professional life, by Mr. P. Hoare, is subjoined, from which we select the following particulars. ,
< John Opie was born in May, 1761, in the parish of St. Agnes, about seven miles from the town of Truro. His father and grandfather were respectable master carpenters in that neighbourhood.—He was early remarkable for the strength of his understanding, and for the rapidity with which he acquired all the learning that a village school could afford him. When ten years old, he was not only able to solve many difficult problems of Euclid, but was thought capable of instructing others: and such was his increasing confidence in his own superior powers, that he had scarcely reached his twelfth year, when he set up an evening school in St. Agnes, and taught arithmetic and writing.
'About the tenth year of his age seeing one of his companions, whose name was Mark Oates, engaged in drawing a butterfly, he looked eagerly, in silence, at the performance: on being asked what he was thinking of, he replied, "he was thinking that he could draw a butterfly, rf he was to try, as well as Mark Oates." He accordingly made the experiment, and triumphed.'—« It happened soon afterwards, that his father be;ng employed in the repairs of a gentleman's house in Truro, young Opie attended him: in the parlour hung a picture of a farm-yard, probably of humble execution, but of sufficient merit to attract his notice, and he took every opportunity of stealing from his father's side to contemplate the beauties of this performance, which in his eye, were of the highest class. His father, catching him in one of these secret visits, corrected him 5 but this had little effect; he was soon again at the door of the parlour, where being seen by the mistress of the house, he was, by her interference, permitted to view the picture without interruption. On his return home in the evening, his first care was to procure canvass and colours, and he immediately began to paint a resemblance of the farm-yard. The next day he returned to the house, and again in the evening resumed his task at home. In this manner, in the course of a few days, by the force of memory only, he transmitted to his own canvas a very tolerable copy of the picture.
♦ He now commenced professed portrait painter, and went to many of the neighbouring towns, with letters of introduction to all the principal families resident in them. One of these expeditions was to Padstow, whither he set forward, dressed, as usual, in a boy's plain short jacket, and carrying with him all proper apparatus for portrait-painting. Here, * amongst others, he painted the whole household of the ancient and respectable family of Prideaux, even to the dogs and cats of the family. He remained so long absent from home, that some uneasiness began to arise on his account, but it was dissipated by his returning dressed in a hand• some coat, with very long skirts, laced ruffles, and silk stockings. Oft seeing his mother, he ran to her, and taking out of his pocket twenty guineas, which he had earned by his pencil, he desired her to keep them; adding, that, in future, he should maintain himself
On the subject of Opie's settlement in London, little more