very liable to animadversion. It is a sort of mechanical demonstration, depending on a property of the Parabola, namely, that the deflection from the tangent is in that curve equal to half the second fluxion of the ordinate. This is generalized by a very concise process, which we think will only be acceptable to those who are in a hurry to pass from general principles to the particular application of them. The problems which follow on resisting media, may be considered as furnishing a very good analytical commentary on a great part of the second book of the Principia. The next problem of importance respects those cases of maxima and minima, where certain conditions of a curve remain invariable, while certain others are the greatest or the least possible. This differs from a problem in which maxima and minima were formerly treated of, because in those the function was given of which the greatest or the least value was to be found. Here the function itself is the thing required, under the condition that when certain analytical operations are performed on it, the result shall be a given quantity; and that when certain other operations are performed on it, the result shall be a maximum or a minimum. This most curious and difficult problem, about which Euler, an author no less remarkable for conciseness than perspicuity, has written an entire volume, and for the solution of which the new Calculus of Variation was found necessary by Lagrange-this problem is despatched by our author in less than eight pages; for he seems to have made it a rule that the conciseness of his discussion should be in the direct proportion of the difficulty of the subject. The last chapter extends to nearly 100 pages, and is occupied by a collection of miscellaneous problems, not of great difficulty, but very interesting in themselves, well calculated to il lustrate the rules that have been laid down, and resolved for the most part with a great deal of elegance and perspicuity. The execution of this part of the work will readily be allowed to deserve considerable praise. On the whole, the defects of this book as an elementary treatise, do not consist in teaching any thing that is false, but in not teaching all that is true and important to be known. These defects at the same time are carefully concealed: And the book, in one respect, is very skilfully composed it lays down just as much of general principles and general methods as is sufficient for the solution of the particular problems that follow; and the student who reads the former, and proceeds afterwards to the latter, naturally imagines that the system is complete, and that the rules he has made himself master of, are sufficient for the solution of all the import ant problems to which the Calculus can be applied. In this, how VOL. XXVII. NO. 53. G ever, if he proceed further in his mathematical studies, he will find himself sadly disappointed; and we are certainly guilty of no exaggeration when we affirm, that after being a perfect master of all that is contained in this treatise, he will not find himself prepared for reading the first six pages of the Mechanique. Celeste. None of the new discoveries are so much as mentioned in it; we doubt if it contains any thing that was not known a hundred years ago; and we are sure that it does not contain many important things that were well known at that period. It is not only the discoveries of the foreign mathematicians which are omitted to be mentioned, but many of those of our own country, and even of Newton himself, are very superficially treated. Thus, the doctrine of Series, and the Integration of Fluxionary Quantitics by Approximation, subjects which have been very fully and successfully treated by the English mathematicians, are very slightly mentioned, and certainly are not explained in such a manner, that in the use of them, the pupil can proceed a single step beyond the point to which he is conducted by the hand of his master. The truth of all these remarks will strike any one very forcibly, who shall compare this volume with any elementary treatise of the same size, and on the same subject, that has appeared within the last fifty years in France, Germany or Italy. The success which the work has nevertheless had, the rapidity with which it has arrived at a second edition, and the support. it seems to have met with in the University where the mathematical science of this island is supposed to be most concentrated, are abundant proofs, that, in the higher mathematics, we have not of late made the same progress as the neighbouring nations. It is certainly a curious problem with respect to national genius, whence it arises, that the country in Europe most generally acknowledged to abound in men of strong intellect and sound judgment, should, for the last 70 or 80 years, have been inferior to so many of its neighbours in the cultivation of the science which requires the greatest and most steady exertions of the understanding; and that this relaxation should immedi ately follow the period when the greatest of all mathematical discoveries had been made in that same country. This is a paradox not altogether impossible to be explained, and to the consideration of which, having no room at present for such a dis cussion, we shall be glad hereafter to return. ART. V. Voyage de Humboldt et Bonpland. Quatrième Partic Astronomie. 2 vol. 4to. Paris, 1810. T HOUGH the greater part of the different works to which the travels of BARON DE HUMBOLDT have given rise have been already taken notice of in this Journal, yet one has been omitted which must be considered as the basis of a great deal that is contained in the rest, namely, the two volumes of Geographical and Astronomical Observations above announced. Without including this, the analysis which we have offered of the other parts, may justly be regarded as incomplete, because of a reference to many things, of which the evidence and grounds of information are not clearly pointed out. Being, however, at present unable to enter into the detail which these two volumes so well deserve, we must satisfy ourselves with such a brief statement as may make some amends for an omission which we cannot pretend to justify. It will thence be evident, that to the many other claims which this illustrious traveller has to the gratitude of the scientific world, there must be added that of having contributed more than any single observer to the improvement of geographical knowledge; and that, too, in parts of the earth among the most inaccessible and the least known. M. HUMBOLDT, in setting out on his travels, took care to furnish himself with a set of excellent astronomical instruments, well adapted to the purposes of a traveller, and constructed by the best artists of Paris and London. These he applied in the course of his journeys in America to a variety of objects, the solution of which does no less credit to his judgment than the observations themselves do to his industry and perseverance. The position of any point on the earth's surface relatively to the whole globe, is determined by three things, its latitude, its longitude, and its height above the level of the sea. When any one of these is unknown, the position is evidently not determined; and there must always be the whole of a certain line, or a certain surface, in any part of which the point referred to may exist. The fixing, therefore, of all these three conditions, is necessary to constitute an accurate geographical determination, or such a one as may suit the purposes both of Physical and Geometrical science. The number of positions thus determined in the work before us, amounts to two hundred and thirty-five. See Tableau des Positions Geographiques, tom. I. de page 5ème a page 33ème.. In the same volume, page 295, is found a table containing the heights of points in the Cordillera of the Andes, amounting in all to four hundred and fifty-three. This table contains not only the heights of many of the most elevated points in the greatest chain which is yet known to traverse the surface of the earth, but also the levels of many of the great plains encompassed by them; of the remarkable passes by which the ridges are crossed; of the sources of the rivers, and many of the most remarkable points in their courses; of the limit also of everlasting snow; of the growth of certain plants; of the levels at which certain diseases begin or cease--and to all these are added, observations on the mineralogy, climate, soil, &c. All this is reduced into a table occupying not more than 40 pages. We doubt if an equal quantity of geographical and physical knowledge was ever before brought within so small a compass; and are confident, at any rate, that no such mass of informa tion was ever accumulated by the personal observation of a single individual. The first volume also contains a curious memoir on the astronomical refractions of the torrid zone, corresponding to elevations above the horizon of less than ten degrees Vol. I. p. 110. It is at these small elevations, as is well known, that the principal irregularities in refraction occur, and where, from the observations of Bouguer, it was supposed that the refractions of the temperate and torrid zones did not follow the same law. M. HUMBOLDT has shown, that his observations, and those which Dr MASKELYNE made at Barbadoes, which last had not before been calculated, agree perfectly with one another; and also serve to prove, that the supposed difference of the two zones does not exist, but that the refracting power of the atmosphere is the same in both, if the difference of temperature be taken into account. This is a simplification of great value. For the determination of longitudes, the same author has used a method that had been almost entirely neglected, founded on observing the declination of the moon. This method he found very useful; and has shown it to be a resourse which, on many occasions, may be resorted to with great advantage. The measurement of heights by the barometer, when combined with angles of elevation, depression, and azimuth, affords the means of making a trigonometrical survey of a country, and of determining both the bearings and the horizontal distances of the different points on its surface. The lines, which serve in this case as bases of the different triangles, are vertical instead of horizontal; and, though smaller than the hori zontal bases might be made, yet if there are a great number of them, their combination may lead to results which possess very considerable accuracy. This is called, by M. HUMBOLDT, the Hypsometric method, and was used for determining the po sition of Vera Cruz, relatively to the city of Mexico, as well as of several other places. The principal geographical determinations, however, result from the reunion or combination of all these different methods. Having ascertained a certain number of longitudes of places where he had resided a considerable time, (Cumana, Santa Fé de Bogota, Quito, Lima), by the satellites of Jupiter, Lunar distances, eclipses of the Sun, and the passage of Mercury over the disc of that luminary, which last he had the good fortune to observe from the beginning to the end of it, at Lima, in November 1802, he afterwards used the chronometer, to connect a vast number of other points with those which were thus determined, and which served as a basis to the whole survey, Vol. II. p. 421. The observations thus made, were all calculated upon the spot; and the first results were published, partly in the Connoissance des Tems, and partly in the Astronomical Journal of the BARON DE ZACH; but, in order that the public might have the full advantage of them, and that they might be the more easily made subservient to the construction of his maps, he engaged MR OLTMANS, a young geometer of Berlin, to revise the whole of his journals, and to make all the calculations anew, employing the Lunar Tables of BURG, and correcting them at the same time by the passages of the moon over the meridian, as observed at Greenwich. MR OLTMANS calculated, in this manner, seven hundred positions, which the Institute of France has recognized as the greatest mass of materials for astronomical geography which at present exists, awarding, at the same time, to MR OLTMANS, in consequence of his work, the prize for Astronomy in 1809. In order that men of science may be enabled to judge of the accuracy of the whole, the observations themselves are all published, even to the smallest angle that was measured. To be assured of his own accuracy, M. HUMBOLDT, on his return to Europe, determined the latitude of Paris with the same instruments which he had used in his travels; and found that, in a series of ten observations, the error only twice amounted to three seconds, Vol. I. Introduction, p. xii. He has also compared his observations with those made by the Spanish astronomers about the same time, and not published till six years afterwards. One may see the singular agreement which prevails among them, by looking into the Introduction, Vol. I. p. xxxv. He has altered the longitude of Quito, by nearly one degree; and, what is very remarkable, and the best verification of his results, is, that the observations of the Academicians who measured the arch of the meridian in Peru, when they are cal |