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That the accelerating force is here falsely assumed is manifest from taking the case of the angle of the plane being equal to zero; in which case, s = 0, and consequently viz. the accelerating force is the same as if there were no second body x, which is obviously erroneous.

Many other instances of this kind occur in various problems, which render the investigations of little value: but the experiments may be made useful, and therefore deserve the attention of engineers. Art. 23. An Introduction to the Method of Increments, expressed

by a new Form of Notation ; shewing more intimately its Relation to the Fluxional Analysis. By Peter Nicholson. 8vo. pp. 130. Davis and Dickson. 1817.

The method of increments, as well as the method of Auxions, is a native of our own soil: but, like the latter, it has been neglected by Englishmen. Both have been transplanted into foreign countries, and cultivated with peculiar care and perseverance; which have again been amply rewarded by a rich harvest of important scientific discoveries. Montucla observes that, had the human mind always pursued the path which appears the most natural, the theory of increments would have preceded that of flux. ions; because it seems much more direct to be carried from the consideration of finite differences to that of differences indefinitely small, than to pass from the latter to the former. Such, however, was the fact; the first distinct notion of the doctrine of increments not appearing till the year 1715, in a work intitled “ Methodus Incrementorum," by Dr. Brooke Taylor; and the only other English writer on the subject being Emerson, who published his “Method of Increments," in 1763. More than a century has therefore passed, since the first discovery of this most important branch of analysis, without any attempt (except the solitary one above mentioned) being made by an English mathematician to promote and cultivate a discovery that does honour to our country. The subject, however, has not been neglected: but, on the contrary, as we have above observed, it has been pursued with the utmost assiduity by our continental neighbours, and Taylor's Theorem is “ miliar in their mouths as household words;" while nine-tenths of our English students are scarcely acquainted even with the signification of the expression, much less with any method of investigating its principles, or with its extensive application.

Mr. Nicholson's Introduction may be of some utility, by bringing the subject before a certain class of readers, who might otherwise remain quite ignorant of this doctrine: but we doubt whether it be well calculated to inspire them with any desire for persever, ing in their studies, by exhibiting the beauties and simplicity of its principles, and pointing out the road to more profound and interesting researches.

To have written, with any prospect of success, on the theory of increments, would have required a more extensive acquaintance with

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the works of modern foreign authors than Mr. N. appears to possess; for this is not one of those subjects that can be studied, comprehended, and illustrated by the same person and at the same time. Yet such appears to have been the attempt in the present instance, as will be seen by the following abstract from the author's preface; from which our readers will be able to appreciate both his style of writing (which is far from being of the most correct kind) and his qualifications for treating on one of the most abstruse branches of analysis.

Having by ardent desire, in early youth, acquired a knowledge of most of the Mathematical Sciences, by application to books, principally from the works of that celebrated mathematician, Mr. Emerson, but, from the want of similar society at that time, and a desire of applying the scientific knowledge I then possessed to useful purposes, I directed my attention to Architecture; and what I have done in this art will sufficiently appear in the works that have been published under my name.

After an interval of twenty-five years, about two years ago, my attachment was again renewed to the scientific parts of mathematical knowledge; and being desirous of understanding the method

of increments, (fluxions having been an early acquirement,) I directed my attention thereto, and the following Work is a part of the result.

• When I first went upon the study, it was not with a view of publication, but with a desire of obtaining a knowledge of the summation of series by this method; but, having, by the doctrine of combinations, discovered the decomposition of powers, or their resolution into progressions, and progressions into powers, and also the same for their reciprocals, I became anxious of writing on the subject, and more particularly so, knowing that it would create a greater interest in the study.

• But, in proceeding with the Work, improvements so often occurred, that I was frequently obliged to re-write many of the articles, such as were not printed off, till satisfied; and this will account for the want of uniformity that is to be found in the subsequent pages.

As an introduction was only at first intended, the Work will, perhaps, agree with the title, the higher branches of the method of Increments not being treated upon; though the principles delivered are capable of a much more extensive application.'

It will be evident, from this extract, that the author was too much in haste in committing his labours to the press; and the volume itself shews this fact in a much clearer point of view, by the changes of principles and changes of notation which occur in many of its pages. Indeed, we have little doubt that Mr. N. himself would be glad of an opportunity of retracing his own steps, and of giving to the whole a more uniform dress than that in which it at present appears.

The notation adopted at page 54. is ingenious, but it ought to have been employed throughout the work: the introduction, also, on the decomposition of powers, displays an originality of thinking: b:it, like the other parts, we suspect that it was committed to the press before it had been completely investigated and developed. We know not whether we may expect to see a second edition : but, if this should be the case, we have little doubt of finding the book much improved.

NATURAL HISTORY. Art. 24 Elements of Conchology, according to the Linnæan Sys

tem; illustrated by Twenty-eight Plates, drawn from Nature, by the Rev. E. J. Burrow, A.M. F.L.S. Mem. Geal. Soc. 8vo.

pp. 263. 16s. Boards. R. and A. Taylor. 1815. Art. 25. The Elements of Conchology; or Natural History of

Shells : according to the Linnæan System. With Observations on modern Arrangements. By Thomas Brown, Esq. Captain Forfar Regt., Fellow of the Linnæan Society, Member of the Wernerian Natural History Society, &c. Illustrated with Nine coloured Engravings. 8vo. pp. 174. Sewed. Lackington and Co. 1816.

Of these two publications, the first bespeaks much familiar conversancy with the subject, united with scholar-like precision and elegance of style, critical ingenuity, and appropriation of remark. The introduction, which slightly touches on some of the general topics of testaceology, is followed by a series of neat explanations of the technical terms; and the generic and specific descriptions are then successively unfolded, in a very masterly manner,

and with due references to the plates. A list of English trivial names is also subjoined, for the accommodation of purchasers and collectors of shells. We are next presented with a well arranged catalogue of writers on conchology, principally extracted from a paper in the viith vol. of the Linnaan Transactions, by Dr. Maton and the Rev. Mr. Rackett; and, lastly, with a short table of colours, in which the author endeavours to express the Latin terms by corresponding English epithets.

Of the plates, it is observed that they are all, without any exception, drawn from nature, and as accurately as an uninstructed hand could, with much care, accomplish. They are intended to illustrate the forms of every natural subdivision of each genus, and will admit of being coloured by the student in conchology, from other specimens, either as an embellishment, or as a very beneficial exercise towards acquiring a more intimate knowledge of this peculiar science. A few drawings have been added, with a view to exemplify the different stages of growth in shells; to point out a few among many species of doubtful genus; and to offer to the curious conchologist such rare or non-descript shells as have recently come under the observation of collectors."

Mr. Burrow has taken pains not only to trace most of the generic names to their origin, but to sprinkle the systematic portions of his composition with various observations; which agreeably relieve the monotony of abridged definitions and descriptions, and impart an interest and an air of originality to the whole performance.

Captain

Captain Brown has executed a similar task, with more regard to neatness than to correctness of typography. To his copious list of errata, we might easily add at least an equal number, besides such ungrammatical expressions as many genera constitutes,' — "bivalve shell are united,'-'the hinges of all the Arca,'' that fine saw-like edge which unite,'~' the colours also are of service, &c. though it cannot always be depended on,'—the exterior surface are covered,'- which it can oblique,' &c. &C.-We are told that

the author was induced to undertake this simple work, because there was no elementary book on shells in the English language, with the exception of Da Casta's, (Costa's,] whose arrangement is now generally exploded. From this unqualified assertion, we may infer that the author was ignorant of the treatise to which we have just adverted. — The adoption of method and nomenclature has doubtless accomplished much : but, when Captain Brown broadly affirms that every object can be traced with certainty and facility to its proper place in the System of Nature,' we can, with certainty and facility, trace the sentiment, and even the language, to Stewart's Elements of Natural History. Such an unlimited position, however, obviously involves an inference to which no sober-minded naturalist will feel disposed to subscribe ; namely, that artificial arrangments of natural objects have attained to practical perfection. It is, likewise, completely at variance with the humiliating reflection which occurs in a subsequent passage of the work before us. • Most shells,' says the writer, are very different in their young and adult state, both in form and colour ; the student ought, therefore, to make himself well acquainted with the generic characters, to enable him to discriminate the difference. But, indeed, this is often a dificult matter, even to those who have studied the science for many years. The same remark, we apprehend, forcibly applies to all the other departments of animated na. ture. — The Captain's catalogue of testaceological writers is somewhat meagre; and his explanations of the parts of shells, and of the scientific terms employed in conchology, might have been more conveniently incorporated in one division, than separated by the interjected definitions of the genera and species.

With the above exceptions, however, these Elements of Conchology will be found to exhibit a sufficiently accurate and perspicuous outline of the Linnéan system of a limited, but interesting, department of zoology. The author's explanations, especially when exemplified by figures, are particularly distinct and satisfactory; and his practical hints, which chiefly relate to the use of the instrument called the spoon, in gathering shells, and to the most approved methods of cleaning them, though compressed within the compass of a very few pages, will not be overlooked by collectors.

Captain Brown's directions for detecting the tricks of dealers, repairing deformities in shells, &c. will also be found of considerable service; and the rather because these particulars are seldom introduced into elementary manuals. The plates are handsome delineations of the explanatory details of the text.

MISCELLANEOUS. Art. 26. Time's Telescope for 1817; or a complete Guide to the

Almanack; containing an Explanation of Saints' Days and Holy Days; with Illustrations of British History, &c.; Antiquities; Notices of obsolete Rites and Customs; and Sketches of come parative Chronology, astronomical Occurrences, &c. To which is prefixed an Introduction, containing the Principles of Zoology. 8vo. gs. Boards. Sherwood and Co.

We have already (see Vol. Ixxxi. M. R.) noticed the preceding volume of this amusing and instructive performance; and we have now little to add to, or to deduct from, the encomiums which we deemed it our duty to pass on the contents of that part: the plan being still the same, and the execution and arrangement as nearly as possible on the same model. One of the most marked distinctions between the two volumes is the substitution of an introduction to the Principles of Zoology for the former introduction to Botany; while the theory of Astronomy also here takes the place of the preceding Historical Sketches, and is still divided under the heads of the several months in the same manner as before, to which arrangement we offered some objections in our former notice.

We shall not consider it as requisite for us to continue our report of this annual publication. Art. 27. Metrology; or an Exposition of Weights and Measures,

chiefly those of Great Britain and France; comprizing Tables of Comparison, and Views of various Standards; with an Account of Laws and Local Customs, parliamentary Reports, and other important Documents. By P. Kelly, LL.D. Master of the Finsbury-square Academy, London. 8vo. pp. 154. Lackington and Co. 1816.

Our readers are already acquainted (M. R. Vol. Ixvi.) with a work of some magnitude by Dr. Kelly on the subject of Commercial Exchanges, and we now introduce to their notice a small performance on a topic connected with the same species of inquiry. Dr. K. begins with observations on the standards afforded us by nature, such as the length of a pendulum in different latitudes, the length of different degrees of the meridian, the uniformity of the diurnal rotation of the earth, &c.: whence he proceeds to an account of the measurement of a meridian line in France, and to an explanation of the new system of weights and measures established in that country in 1795, which he subsequently compares with the weights and measures of England. We have next a statement of the Report of a Committee of the House of Commons on our weights and measures in 1758, and an abstract of our statutes on that subject for several centuries back. After an account of the weights and measures of Scotland and Ireland, the book concludes with a copy of the Report of the Committee of the House of Commons made so lately as 1814, and with the outline of the bill which passed the Commons in the last year, but was afterward lost in the Upper House. - The work is evidently intended for nothing beyond an elementary treatise, and seems sufficiently likely to

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