eastern part of the mountains of New Granada; finally, a geological picture of the singular volcano of Jorullo, which sprung out of the earth in the month of September 1759, surrounded with several thousands of volcanic cones of 100 toises in height.

Tables of the Divisors for all the Numbers of the 2d Million. By M. Burckhardt.—We have already spoken of this work, which appeared on the first of January, 1814. We take the opportunity of announcing that the third and fourth millions are in the press.

Treatise on the Differential and Integral Calculus, by M. Lacroix; 2d Edition, revised and augmented, volume second. About 850 pages. Paris, Madame Veuve Courcier.-The author, in a short advertisement, gives the motives which led him to deviate in some points from the plan pointed out in the preliminary discourse to the first volume. In his opinion the general theory of the conditions of integration ought to follow immediately the methods relative to the integration of functions, with a single variable quantity; because it is to this integration that is reduced that of any differential functions whatever, when they satisfy the conditions of integrability. In what concerns partial differential equations, he has made some considerable transpositions, in order to prevent repetition, and to show more clearly the properties and the connexion of the different processes proposed by the great mathematicians of our time, to treat this kind of equations. By this new arrangement, by his remarks on the integrals to which this new edition gives new developements, he has endeavoured to throw light on a subject which had not yet been sufficiently elucidated. He is at pains to point out the difficulty.

The calculus of variations is treated with all the details which the importance and singularity of the method required. To make its nature sufficiently understood, it became necessary to point out the principal attempts that have been made to unite it with the principles of the differential calculus. This the author has executed in his last chapter, where the method of variations is presented in all the generality and simplicity which the first symbols employed by M. Lagrange, and the consideration of infinitely small quantities, give it.

Philosophical Essay on Probabilities, by M. le Compte Laplace. A Volume in 4to. of 96 pages, Paris, Madame Veuve Courcier.M. Laplace, after having in a first work treated this interesting and difficult subject like a consummate mathematician, and having singularly enriched it by new methods, more general and fruitful than those of the great mathematicians who had made it the subject of their meditations, examines it here in a point of view purely philoso❤ phical. Without the assistance of analysis, without supposing the reader acquainted with any thing more than arithmetic and the first elements of algebra, he explains the principles and the general results of that theory. He makes it originate from suppositions the most simple, combined so as only to require the degree of attention of which every man is capable, who has the habit of

reflecting. He then applies the principles to the most important questions of life, which, in general, are nothing else than problems of probability. He treats successively of hope, of games, of the unknown inequalities which may exist between chances supposed equal; of the laws of probability which result from the indefinite multiplication of events; of the calculation of probabilities applied to the inquiry into events and their causes; of the means which we must choose among the results of a great number of observations; of the tables of mortality and the mean length of life; of marriages and any kind of associations; of benefices depending on the probability of events; of the choice and decisions of assemblies; of the illusions in the estimate of probabilities; of the different means to approach to certainty.

The work is terminated by an historical notice on the calculation of probabilities, in which the attempts and discoveries of the mathematicians who have applied themselves to this subject are stated and appreciated with great impartiality. Nobody was better entitled than Laplace to draw up this notice, nor more interested that it should be well done.

Theoretical and practical Astronomy, by M. Delambre; three vols. 4to. Paris, Madame Veuve Courcier. 1814.-We have announced the abridgement of this work, which appeared in 1813, in one volume 8vo. We have said that the plan of the two treatises is the same. It remains to point out what was suppressed in the abridgement. What distinguishes the complete treatise is, in general, a great variety of questions, of solutions, and formulas, more details on the construction, use, and verification of instruments. Thus, in the article Spherical Trigonometry, will be found a number of formulas, expressing the relation between five and six parts of the same triangle, the analysis of the different methods in use for the resolution of triangles, a more complete and methodical collection of differential expressions, and, finally, a proof triangle calculated with the greatest detail, and which may serve to verify all the formulas imaginable. In the article Gnomonics, besides the new formulas for the description of the horary lines and the arcs of the signs, the whole gnomonic plan is reduced to a single formula, which has only one linear variable quantity, which only affects one of the terms of which it is composed. In the article Refractions, there is a synthetical solution of the problem of the shortest twilight, much more complete, and much easier, than all those drawn from the differential calculus. The chapter on The Diurnal Motion offers solutions, mostly new, of the most useful problems; that of corresponding heights is terminated by tables of correction of a new form, followed by useful remarks on the use of logarithmic tables, which we wish to substitute for tables calculated in natural numbers. When treating of The Elliptical Motion of the Earth, will be found the comparison of the different hypotheses contrived to account for the inequalities of the sun, new solutions of the problem of Kepler, a great number of elliptic formulas, and

all those of M. Gauss, otherwise demonstrated and brought to a notation, which it has been attempted to render more clear. All these formulas are illustrated by numerical examples. The same attention has been paid to the parallaxes, the eclipses, and to every thing that concerns the planets and the comets. In the article Tables of the Sun will be seen the means employed to construct the last tables, the reductions applied to observations made with the repeating circle, the method of observing an equinox and a solstice, with tables to facilitate all these operations. The Theory of Eclipses is explained in different manners, partly new, which have the advantage of being easier and more general. The application of them has been made to the eclipses of 1764, of which it has served to determine the curves of entrance and exit.

In the chapter on The Planets is shown how by means the most simple we obtain the first approximation of an unknown orbit, and how this first sketch may be afterwards perfected. The Transits of Mercury and Venus are treated in a way altogether new, which leads by a more direct and certain way to a knowledge of the parallaxes. In the article Rotation, seven different solutions of that problem are collected, and, for an example, are taken eleven observations of a spot of the sun. The Theory of Saturn's Ring, which terminates this chapter, offers the most exact and simple means, either to predict the phenomena, or to determine the elements from observations. The chapter on The Comets is long. The known formulas are presented so as to serve equally for the parabola and the ellipse. The methods are explained properly to facilitate the construction and ensure the accuracy of the general tables. Examples are given, calculated according to the most accredited methods; and one is explained, which recommends itself by the following advantages: it employs only the calculations most familiar to the astronomer; the longitudes and latitudes observed are, as in all the other methods, the primary data; but they enter only into the first calculations; the succeeding ones employ only heliocentric places; and the elements found may be perfected by the totality of the observations, by equations nearly of the same conditions as those of the planets: the calculations are made as the observations succeed, and the astronomer who has discovered a comet, may on the day itself of the third observation, ascertain its elements. This chapter is terminated by a catalogue of the orbits of all the comets hitherto observed, and by the general tables of the parabolic movements of different form, according as we take for data the anomaly or the days since the perihelion. In the chapter on The Measure of the Earth will be found new calculations on the irregularities of the arc of the meridian, measured in France and in Spain. In the chapter on Nautical Astronomy there are numerous solutions of the most important problems; and the work terminates by a chapter on The Calendar, in which we must point out a fault in the drawing up, which may in some rare circumstances occasion an error of seven days in the result of the calculus. We

allude to the formulas given by M. Gauss to find Easter for any year of the Italian and Gregorian calendar. We satisfy ourselves with noticing here that the formulas are not complete, we shall republish them elsewhere with more exactness, and we shall join to them new formulas still more expeditious, and which, besides, give us the dominical letter, the golden number, and the epact of the year. See the Connoissance Des Temps of 1807, p. 307.

Exercises in the Integral Calculus. Fourth Part, by M. le Chevalier Legendre. This new part is divided into two sections. In the first M. Legendre has completed the theory which he explained in the second part of his exercises. He has particularly attached himself to explain with all the requisite minuteness the properties of a function, which is the mutual connecting link of a multitude of transcendental quantities, and the source from which flow all the formulas which concern the comparison of these transcendentals, their reduction, and their evaluation. The author has already satisfied himself, that he was not mistaken when he hoped that this theory, considered under a new point of view, and augmented by a great number of new formulas, might fix the attention of mathematicians, and that they would see a new branch of analysis brought almost to perfection.

To extend further the applications of this theory, he has inserted at the end of the first section a more extended table than that which terminated the second part. The new logarithms are exact to the 12th decimal, in order that the cipher of the last order may never be in error more than one unity, or at most two. This has given occasion to rectify, and almost to double in extent, a table which Euler had given in his differential calculus, for the sums of the reciprocal powers of the natural numbers.

In the second section will be found different researches, which form a sequel to the third part. A great number of formulas are demonstrated, either entirely new or recently discovered. Among the last are several definite integrals given by M. Bidonc, in the Memoirs of Turin. The author has also presented some new views on the summation of different series, and on the formulas which serve to give the sum of a series of which the general term is given.

It is impossible for us to dwell longer on a work of pure analysis, and almost entirely composed of formulas. See what we have said formerly on the first part of this work. We shall take this opportunity to rectify a passage in our notice of 1810.

In giving an account of the second memoir on Elliptical Transcendentals, we have denoted by the word Loxodromic, a species of spiral which M. Legendre there considers, and one of the properties of which is to be the shortest road between two points situated under two different meridians or parallels. This acceptation of the word Loxodromic is not that of navigators and mathematicians; but it appeared to us more conformable to the etymology of the word, which is an oblique route. There is no straight

or orthogonal route, but that which takes place in the direction of a meridian or parallel. In the first case we cut all the parallels at right angles, in the second all the meridians. Every other route would traverse both under angles almost always oblique. The spiral considered by M. Legendre had been already analyzed by M. Dusejour, who had given the formula of the variable angle, and remarked further, that this curve would cross the equator in different points at each demi-revolution which it made round the spheroid. It was from this singular property, as well as from the continual variation of the angle, that it appeared to us to merit the name of Loxodromic. The Loxodrome of sailors, on the contrary, makes a constant angle with all the meridians which it crosses, and it is not the shortest road between two given points. Here, then, are two striking differences between the two curves. Hence they ought not to be confounded. But the danger is not great, as the one is only employed in navigation and the other in geodesy; so that the difference of the problems points out sufficiently the choice of the formulas. For the Loxodromic of sailors, see the 36th chapter of our Treatise on Astronomy.

Præcipuarum Stellarum inerrantium Positiones media ineunte Sæculo 19, ex Observationibus habitis in Specula Panormitana al Anno 1792 ad Annum 1814, ex regia Tygographia Militari.-This new work of M. Piazzi is dedicated to the Institute of France, of which the author is one of the oldest correspondents. In this new edition, in which the number of stars is 7646, without counting those whose positions have not yet received the greatest degree of precision, M. Piazzi has not chosen to adopt any thing which he had not himself verified. He determined the right ascension of the fundamental stars by a direct comparison with the sun. The others have been deduced, as usual, by the difference of their passage over the meridian, observed a great number of times, and the mean result has been taken. Unfortunately, the size of the volume, and unfavourable circumstances, have obliged him to suppress all these comparisons, and even the observations on which they are founded. The declinations suppose the mean refractions of the fifth book of the work on the observatory of Palermo, the latitude 38° 6′ 41′′, and a total precession of 58.388", which leaves 50-2066" for the precession in longitude.

The annual motions comprehend the proper motions, whenever it was possible to find in the ancient catalogues positions sufficiently certain.

The notes which accompany this catalogue offer many curious remarks on the stars, whose motions had not yet been observed, or of which the brilliancy appears to increase and diminish periodically.

This extract, which we have been obliged to abridge, will be sufficient to show how precious this new catalogue must be to astronomers, who already made constant use of the first edition.

The following memoirs have been approved by the Class :