Our remarks on the two remaining Numbers must be extremely brief. The Mensuration contains little which is not generally found in elementary works, except the Section on Isoperimetry; which appears to be chiefly taken from Legendre, and is reduced to a neat and scientific form. Those plane and solid figures, the mensuration of which depends on the principles of the common Geometry, are treated of in the text; while those which depend on the principles of the fluxional calculus, have rules given for them, without demonstrations, in an appendix. We think the introduction of these rules, in any shape, is a violation of the method which the author prescribed to himself, at the commencement of his work; but it is a violation which the exigency of the case seems to render indispensable. Although ostensibly introduced for the use of those who may be pursuing a more limited course of private study, we believe that, even by those for whom the work is primarily designed, the subjects of this appendix will be taken up with more advantage here, than in that part of the Course, to which the demonstrations more properly belong. The mind is never less fitted for remembering and applying practical rules, than when it is primarily engaged in long and dificult analytical investigations. For the practical mathematician, such a summary is absolutely necessary. Could we even suppose him, in ordinary cases, so well versed in the principles of Fluxions, as to be able to comprehend the reasons of these rules, he would still want a synopsis of them for purposes of reference. To have the various rules for the areas of the conic sections,-for the superficies and solids generated by their revolution,--for guaging casks, &c. dispersed through a volume of Fluxions, just where the investigation of these surfaces and solids might happen to fall, would, in practice, render them entirely useless.

The Number just published, containing Navigation and Surveying, to which is prefixed the Mensuration of heights and distances, although it does not contain all the details which would be requisite for a practical navigator or surveyor,' is sufficiently copious for the use of those whose object is rather to learn principles, than the minute rules which are called for in professional practice.' The elementary principles of Navigation are delivered with a clearness and method, which will not disappoint the expectations of those who have seen our author's former publications. What most deserves our notice at present, is the view given of the principles of plane sailing and the plane chart,—a subject on

which the works of former writers on Navigation are full of confusion and inconsistency. To give an instance of this confusion:-Nicholson, of whose treatise Webber has given an abstract, in describing the construction of the plane chart, directs, that the meridians should be laid down at the same distance from each other with the parallels of latitude. Under plane sailing, he defines departure to be the distance, on the plane chart, between two meridians. If so, departure and difference of longitude are identical. But when he comes to middle latitude sailing, he represents departure as equal to the meridional distance, reckoned, not on the equator, but on a parallel of latitude, between that which is left, and the one arrived at. Most of the writers whom we have consulted, agree in defining plane sailing to be the art of navigating a ship upon principles derived from the supposition of the earth's being an extended plane.' Hence they correctly infer, that, in consequence of the sphericity of the earth, the calculations derived from the principles of plane sailing must be inaccurate. But as Mr. Day has very justly remarked, if there were any incorrectness in plane sailing, it would extend to Mercator's sailing also,'—since the difference of longitude is determined by the departure, calculated by plane sailing;-whereas the calculations in Mercator's sailing, for a single course, are universally admitted to be correct. All this disorder has been remedied by Mr. Day, by denying that plane sailing is founded on the principles of the plane chart, and assigning a new definition of departure. He considers it as the sum of the indefinitely small deviations from successive meridians, estimated each on the parallel of latitude where the deviation takes place. Bowditch has indeed given, in substance, these views of departure under middle latitude sailing; but, in defining the terms of plane sailing, he has expressed himself in a manner liable to the same objections with that of his predecessors. Departure,' according to him, is the east or west distance a ship has made from the meridian of the place she departed from, and in the plane chart is the same with the difference of longitude.' According to the obvious sense of this definition, departure cannot be accurately found by the principles of plane trigonometry; but, according to that of Mr. Day, the principles of plane sailing are mathematically accurate. This accounts. for the correctness of Mercator's sailing; for it will be seen, by consulting the common demonstrations of Mercator's first theorem, that it is of departure, in Mr. Day's sense of the term, that this theorem is proved.

Mr. Day proposes a method of constructing the plane chart, different from the one generally laid down by writers on this subject. Instead of making the meridional distances equal to those of the parallels at all latitudes, he would diminish them in the ratio of the cosine of the middle latitude of the chart to radius. Nicholson, Bowditch, et ceteri, in laying down the plane chart, consider the earth as an Archimedes cylinder; Mr. Day considers it as composed of an indefinite number of narrow cylinders, having a common axis, and the curve surface of each, at its middle, coinciding with the surface of the earth. The latter method has the advantage of representing the objects on the earth's surface much more nearly in their natural shape, than the former. But some single method is evidently wanted, which shall be equally applicable to small and to large portions of the earth's surface; and this can be only that which preserves the distance of the meridians the same, from the equator to the poles.

But it is more than time that these remarks were brought to a close. We cannot take our leave of this work, however, without expressing our earnest wish, that it may receive a general admission into our public seminaries. It will be found sufficiently extended for the use of those in which mathematical science is pushed the farthest; and, if more extended than is consistent with the plan adopted by others, several of the less elementary Sections may be omitted, without essential inconvenience. We trust, however, that those directors of the education of youth who may adopt this system, will rather elevate their own standard, than mutilate and depress that of our author.* Should it be thus instrumental in giving pure science a higher rank than it has hitherto held, in our systems of public education, it would be none of the least of its beneficial effects.

While we recommend the system of Professor Day to the public institutions of our country, we consider it as no less adapted to the use of the private student. The full and luminous manner, in which it is written, will render it an invaluable acquisition to those whose curiosity, or whose professional business, leads them to take a general survey of the elements of science, but whose circumstances require them

* In saying this, we would not be understood to depreciate the importance of classical, rhetorical, or ethical studies. So far from depressing them to make room for a more extended mathematical course, we think it desirable that the tone of each of these pursuits alike should be elevated; and that in most of our public institutions, such an elevation is practicable.

to be wholly, or in part, their own instructors. We know of no work whatever, which we should so soon put into the hands of one who was desirous of being introduced to the different branches of mathematics without a living guide. The illustrations given by our author, indeed, are almost precisely the same with those which the learner would gain from an able instructor. He has scarcely left any room for the subject to be rendered plainer by verbal explanation.

To the professed mathematician, this work cannot, from its design, be expected to supersede the more extensive systems already published. Yet even to him, we conceive that it would not be altogether useless. If it furnishes the experienced instructor with no new principles, it might at least aid him in reducing his acquisitions to the best method,array known truths in a clearer light,--and suggest those illustrations by which they may be most successfully communicated to others.

ART. II.-Travels in France, during the years 1814-15. Comprising a Residence at Paris, during the Stay of the allied Armies, (in 1814,) and at Aix, at the Period of the Landing of Bonaparte. 2 vols. 12mo. pp. 571. 16s. Boards. Longman and Co. 1815.-From the Monthly Review.

A

CIRCUMSTANCE not very common in authorship distinguishes this production: it consists of two independent parts, the tours being performed by separate travellers, and the narrative composed by different writers. A volume is allotted to each, the first being occupied with a journey in the north of France, and a residence of some months at Paris; while the second relates a similar journey in the south, and a stay at Aix, in Provence, from December 1814 to the following March. The authors of both volumes are anonymous, but appear evidently, from the frequency of their allusions to Scottish, and sometimes to Highland customs, to come from the north of the Tweed. The plan of combining the labour of two individuals in one publication possesses the advantage of bringing before the reader, at once, the materials which would otherwise have been sought in separate books; and it affords an opportunity of treating the more interesting particulars, such as national manners, the state of political feeling, the tone of society, &c. at considerable length. In the present case, also, the ground travelled over in the two volumes is sufficiently distinct for the purpose of novelty, and yet sufficiently connected for that of conjunction in the mode of

publication. The details appear thus considerably better in a combined than they would in a separate form, although much might have been done to improve them in the important points of condensation and order.

Vol. I. opens with the journey to Paris in May 1814, a few weeks after the north-east part of France had been occupied by the Russian, Prussian, and Austrian troops. The travellers, on conversing with the peasants, found that the character of Bonaparte, though certainly not in favour, was by no means such an object of hatred as they anticipated. Public affairs occupy the middling and lower orders much less on the opposite side of the channel than with us; while to analyze the duplicity and detect the manœuvres of the late ruler would require a stretch of reflection greatly beyond the intellectual patience of a Frenchman. The writer and his friends were more gratified on reaching Paris, and on enjoying an opportunity of seeing the most remarkable sovereigns and generals in Europe collected within the walls of that metropolis.

It is fortunately superfluous for us to enlarge on the appearance, or on the character of the emperor Alexander. We were struck with the simplicity of the style in which he lived. He inhabited only one or two apartments in a wing of the splendid Elysée Bourbon-slept on a leather mattress, which he had used in the campaign-rose at four in the morning, to transact buisiness -wore the uniform of a Russian general, with only the medal of 1812, (the same as is worn by every soldier who served in that campaign, with the inscription, in Russ, Non nobis sed tibi, Domine); had a French guard at his door went out in a chaise and pair, with a single servant and no guards, and was very regular in his attendance at a small chapel, where the service of the Greek church was performed. We had access to very good information concerning him, and the account which we received of his character even exceeded our anticipation. His humanity was described to us as almost unparalleled. He repeatedly left behind him, in marching with the army, some of the medical men of his own staff, to dress the wounds of French soldiers whom he passed on the way; and it was a standing order of his, to his hospital staff, to treat wounded Russians and French exactly alike '

The King of Prussia was often to be seen at the Parisian theatres, dressed in plain clothes, and accompanied only by his son and nephew He is known to be exceedingly averse to public exhibitions, even in his own country. He had gone through all the hardships and privations of the campaigns, had exposed himself with a gallantry bordering on rashness in every engagement; his son and nephew always by his side; his coolness in action was the subject of universal admiration.-We had the good fortune of