of the reflecting circle; formulæ for the reduction to the centre of the station, and to the horizon; formula for computing the latitudes, longitudes, and azimuths, of the summits of the triangles; formula for the resolution of spheroidal triangles; expressions of the various radii of curvature relative to the ellipsoid of revolution; computation of the difference of le vel, both by the trigonometric and the barometric methods; determination of the figure of the earth by observations on pendulums, with the theory of pendulums vibrating in resist ing media; geodesic operations in the Isle of Elbe; approximative calculus of distances, with allowances for the various reductions necessary to arrive at their exact determination. 1. Book II. contains an analysis of the projections of the sphere, and the construction of charts and maps. It is divided into three chapters, comprizing researches into the properties of the stereographic and orthographic projections of the circles of the sphere; application of the theory to the construction of maps of the world, and to the projection of particular maps and charts, according to the conic developement, the cylindric developement, the projections of Flamsteed, Cassini, and Lorgna. Book III. describes, in three chapters, the detail of geodesic operations, and of questions relative to land-surveying. Here we have descriptions of the several instruments employed, with applications of the principal methods; solutions of the problems relative to the evaluation of surfaces, the principles of polygonometry, and the division of lands. The fourth book exhibits the theory and practice of levelling, in four chapters: here the principles of levelling are first deduced from the equation of equilibrium of a fluid mass; then the difference between the true and apparent level is explained, and the effects of terrestrial refraction are computed; the different kinds of levels are next described, and their practical use shewn; the book being terminated by rules for the computation of terraces, and the cubature of solids.. - In book V. the author treats of the reduction of charts, maps, and plans, by pantographs, micrographs, &c. and directions for the descriptive memoirs of the topographer, and statistic observer. The work is terminated by two supplements, on-the centre of oscillation,-some new formulæ for spherical and spheroidal triangles, and the determination of the area of a spheroidal zone; with five tables, relative to the spherical excess, and to the degrees of latitude and longitude on a spheroid whose compression is represented by 1 From the preceding analysis, our readers will perceive that M. Puissant's treatise includes the discussion of many interesting topics. We can assure them, too, that many of the ins vestigations and disquisitions are conducted with no small ability. But there is great inequality in our author's manner. He possesses mathematical knowledge enough to furnish matter for an excellent treatise; but not logical skill sufficient to mould his materials into the most commodious and useful shape. It is not a little singular, that, with only one or two exceptions, the most elementary discussions, such for example, as those relating to the mensuration of surfaces and solids, and to the common operations in land-surveying, are very clumsily conducted; while the profounder investigations, and the nicer and more difficult operations, are usually managed with much dexterity and judgement. There are, however, some of the ele mentary disquisitions, and especially, those which regard the division of lands, and the different operations in projection of the sphere, which we have not seen so well handled in any other work. We were also much pleased with the specimen our author gives of the collection of questions that should be asked in order to acquire the most interesting information respecting the climate, topography, and statistics, of any idepartment or country: its length, alone, prevents our translating it for insertion here. Many of the analytical formulæ exhibited in different parts of this volume are highly curious and valuable: but the limits to which we are obliged to confine ourselves will only allow us to extract a few of them. Such of our readers, as are conversant in the theories by which the more extensive operations of the measurement of degrees on the meridian are regulated, know that much depends upon what is termed the spherical excess, that is, the excess of the sum of the three angles of any triangle measured, above two right angles; this excess, in fact, being a measure of the surface of the triangle. Let A, B, C, be the angles of a spherical triangle, a, b, c, the sides opposite the angles A, B, C, respectively, two right angles, r the radius of a great circle of the sphere, and R" the number of seconds comprised in radius; then the spherical excess may be ascertained by the following elegant theorem first given by Simon Lhuillier of Geneva: See Legendre's Geometry, p. 320; Bonnycastle's Trigonometry, p. 394. But this theorem, however elegant it may be, considered as an analytical formula, is very confined in its application to practical purposes. It was, therefore, an important problem, to investigate a theorem which should furnish a mean of deducing the third angle of a triangle when circumstances were not favourable for the measurement of that angle. M. Puissant, by a very simple process, discovers the following theorem for the excess. R" E R" Չ When the value of is given in this form, it is easy to compute a table which will give the spherical excess in two parts. Such a table in fact has been given by Delambre (Base du Système métrique décimal). M. Puissant exhibits a table in the present work, by means of which the spherical excess may be determined from knowing b the base and h the height of the triangle the corresponding theorem is the ancient measures are used, we have R" ɛ = R" = ɛ i (2) bh. When log. =1.98.527, and 0".00000, 00096, 66 b h. r 2 or, according to the new metrical system, :: log. 1.89509, and 0". 00000, 00078, 54 bh. ε The chapter on determining the figure of the earth by the vibrations of pendulums in different latitudes, contains a sum. mary view of the truly ingenious method devised by the illustrious author of the Mécanique Céleste for this purpose: the general theorem he lays down for the length of the seconds pendulum in metres, in any latitude is..... 0.739502 + 0.004208 sin. It also contains a curious disquisition, by M. Poisson, on the motion of pendulous bodies in resisting media, where a theorem, first published by Bouguer (Figure de la Terre, p. 341), has met with a satisfactory demonstration; M. Poisson having now shewn that the time of the ascending semi-oscillation is diminished by the effect of the resistance of the medium, by the same quantity as that of the descending semi-oscillation is augmented: whence it results that the time of the entire oscillation is the same as if the motion had been in чисис. 6 t The chapter on Polygonometry contains a few curious results, deduced principally from the valuable work of Lhuillier on that subject. As Lhuillier's treatise is scarcely known in this country, we shall extract one or two of the simplest theorems; such as may be understood without diagrams. 1. The square of any side whatever of a plane polygon is equal to the sum of the squares of all the other sides, minus twice the product of all the other sides multiplied two by two, and by the cosines of the angles which they include, 2. Let a, b, c, d, e, represent the sides of a pentagon, of which the angle A is comprehended between the sides e and a, B between a and b, C between b and c, and so on; then the surface S of the polygon may be found by the formula below. S ab sin B + D) } ac sin (BC) + ad sin (B+C+D) +bc sin Cbd sin (C + D) + cd sin D. Analogous formulæ are given for other polygons. The chapter on the division of lands contains many useful problems, as we have before remarked. But we were surprised to find one solved by a tedious process of two pages, which has been done by Simpson in his Geometry in less than 10 lines: it is to divide a triangle into two parts which shall have a given ratio, by a minimum right line: in this case the right line must cut off an isosceles triangle; as is well known to almost every school-boy in England. But the problem seems quite new to the French mathematicians, being 'un de ceux qui sont énoncés dans le 8e numéro de la Correspondance sur l'Ecole Polytechnique!" Our last citation must relate to a more important subject, and one on which we owe almost all we know to the continental mathematicians. The first supplement contains the investigations by which M. Henry deduced, from the spheroidal theory, the chief results that are applicable to the grand geodesic operations. To shew the application of his formula to a practical case of considerable moment, this mathematician proposes the following problem. Knowing The semi-axis major, a, of the meridian ellipse, or the radius of the equator. The semi-axis minor, b, or the semi-axe of revolution. The line, s, of shortest distance between two points on the surface of the earth. The latitude, a, of one of the extremities of that line. The angle, A, which that line makes with the meridian passing through its extremity. The number of seconds, r, contained in the radius of the circle whose value is such that r = To find 1 very nearly. The latitude,, of the other extremity of the line of shortest distance. The angle, a, which that line makes with the neridian passing through that extremity. The difference of longitude, w, of the meridians terminating such line. The arc of the parallel, μ, comprised beween those meridians under the latitude sought. The arc of the meridian, 6, comprised between the parallels to the equator passing through the extremities of the line. The following are two systems of analogous formule which completely solve this problem. : II. } { tán x b sin x sinn sing t b -b2 sín2x [Ur sin U cos (2v+U)] sin (u+v) sin sin sin x sin v sin v |