repeat it with some detail. Given a horizontal plane; above this plane, upon the same vertical and at known distances, are placed two points, whose shadows upon the horizontal plane can be followed. Around the common projection of these points, as a centre, two arcs of circles are described, whose radii are such that they can intersect the traces of the shadows of the points themselves before and after the meridian passage of the sun. The observation consists in measuring the chords of the ares obtained by joining the intersections of the traces of the shadows with the circles; knowing the length of the chords, the radii of the arcs, and the heights of the points whose shadows are observed, we can, by a simple formula, calculate the latitude of a place. Either the two points whose shadows are projected may be two spheres, fixed upon the same vertical thread which traverses their centres, or we may use small circular openings pierced in metallic plates, and placed so that their centres are found on the same vertical. The shadows of these objects are small ellipses, whose centres may easily be found. The variation in the declination of the sun, between the afternoon and morning observations, occasions only a very small error, its effect being, in great part, eliminated from the final formula. The errors that may be made in measuring the various data above enumerated can occasion very small errors in latitude. The principal source of error is a possible want of cxact horizontality in the plane on which the shadow is cast. In fact, an error of about two degrees in the inclination of this plane may produce an error of one degree in latitude. If, however, the plane be horizontal in a north and south direction, but incline in an east and west direction, the effect of the latter inclination may be neglected. The elimination of this latter source of error is due in part to the adoption of two points and shadows. The same advantages do not inhere in the employment of one point and one shadow.-Bulletin Hebdomadaire, XV., 578. DETERMINATION OF LATITUDE AND TIME. In a communication on the method of determining the time by means of two observed altitudes of any celestial body, Vice-admiral Von Wullerstorff-Urbair states that the method of determining the latitude by means of observations of stars, at equal zenith distances, was proposed and applied by him in 1848, at the Naval Observatory in Venice, but has in later times been widely adopted, and is known in America as Talcott's method. Admiral Von Wullerstorff-Urbair shows that the same system may be applied with accuracy to the determination of time; and quotes a note from Palissa, at the Naval Observatory at Pola, who says that this method was applied by him in December, 1873, and gives results whose value is equal to those deduced from the transit instruments. The method is specially to be recommended to travelers, since by means of the same theodolite both time and latitude may be accurately determined. The formulæ given by Von Wullerstorff-Urbair seem scarcely so convenient in practice as those taught for many years past by Dollen and the Russian geographers, and which were published in full some years ago by Smysloft.-—"Mitth.” Austrian Ilydrogr. 0f, II., 129. THE COMPUTATION OF THE AREAS OF IRREGULAR FIGURES. There often occurs a necessity for determining from a drawing the superficial contents of planes bounded by curved lines. This is the case, for instance, in the determination of the superficial contents of the water-lines of vessels. In such computations, ordinarily, we employ somewhat rude approximations, as in Simpson's or Stirling's methods. The latter author has given two methods: the first depending upon the principle that the portion of a curved line, between any two ordinates, may be considered as a portion of a parabola of the second degree. In the second niethod, given by the same author, the curve is considered as a portion of a parabola of the third degree. These three methods may be supplemented by other methods depending upon formulæ developed by Gauss, Cotes, and others. But in general all these methods are somewhat more difficult of application than that known as Simpson's, which is far more frequently employed than any other. A very decidedly better way has been proposed by the Russian mathematician, Tchebitcheff, whose method is simpler than either of those just mentioned, and, although less accurate than that of Gauss, is more accurate than those of Cotes, Stirling, and others. Indeed, a greater simplicity of application than this method offers is scarcely to be demanded, and its accuracy surpasses the ordinary B 2 necessities of the arts. In order to represent as closely as possible a curved figure by a series of polygons, Tchebitcheff takes six terms of the integral formula corresponding to six ordinates selected in the following manner: Let the surface be inclosed by the curved line A B C D, and the straight line A D; subdivide A D at E. The one half of A E, multiplied by the measured values of its ordinate, is then to be set off on either side of E, thus marking the places where new ordinates are to be measured, which are themselves to be multiplied by one half of A E; the products again set off on either side of E, and then a third pair of ordinates measured. In this way three pairs, or six ordinates, are obtained, whose values have a certain relation to each other and to the given curved line. The desired area is found by multiplying one sixth of the sum of these ordinates by the length of the line A E D. Other methods generally give results somewhat less than the truth. The method of Tchebitcheff generally gives larger results than the others. — “Mittheilungen" Austrian Hydrogr. Office, 1874, p. 530. ASTRONOMICAL WORK AT CORDOBA. In his annual report, as Director of the National Observatory of the Argentine Republic, for the year 1874, Dr. Gould states that the three principal undertakings of that observatory, viz., the uranometry, the zones, and the smaller catalogue of stars, have satisfactorily advanced toward their completion. An inevitable delay having occurred in the publication of the first mentioned of these works, the opportunity was seized to revise some portions of it--a revision which indicates that the accuracy attained is quite commensurable with Gould's original hopes and expectations. Having secured the necessary funds, it is now expected that in the course of the present year the publication of the charts will be completed. These will be thirteen in number, comprising the whole of the southern heavens. The total number of stars whose positions and magnitudes will be given will be not far from 8500. With reference to the zones of stars, he reports that some 12,500 additional observations have been made, bringing the total number up to 82,537. It is not improbable that the number of observations yet to be made will swell the total to more than 100,000; which work he then (March, 1875) hoped would be completed by the end of July, 1875. The greatest hinderance to the prosecution of this undertaking consists in the difficulty of securing the services of an adequate number of trained astronomical computers. Of the large number of stars observed in these zones, a small portion have been selected as fit to form a special catalogue of brighter stars. This catalogue includes nearly 5000 stars, and some 12,400 observations upon these were made during the year. Dr. Gould adds that not one hour of unclouded sky between sunset and midnight was lost by his assistants during the whole time of his recent visit to the United States, notwithstanding that other observations were also going on by night, and continual computations by day. The equatorial telescope has been as busily employed as the meridian circle. Coggia's comet was observed from the 27th of July to the 18th of October. Standard Cordoba time has been given regularly from the observatory without a single case of failure; and latterly the exact Buenos Ayres time has been telegraphically transmitted to that city for the convenience of the shipping. Meteorological observations have been conducted and reported regularly to the Meteorological Office. Dr. Gould's corps of assistants has consisted of four persons, with occasional aid from others competent to act as copyists and computers. The assiduity of the labors of all concerned is abundantly testified to by the record of their results. Annual Report, March, 1875. PROPERTIES OF TIIE TETRÆDRON. In an exhaustive memoir by Dostor on the application of determinants to certain problems in solid geometry, we find the following theorems relating to tetrædrons: The sine of a triedral angle is equal to the product of the sines of two of its faces, multiplied by the sine of the inclosed diedral angle. Again, in every tetrædron each face is equal to the sam of the products which we obtain by multiplying each of the three other faces by the cosine of its inclination to the first face. And, again, in every tetrædron the faces are to each other in the same proportion as the sines of the supplements of the opposite triedral angles. The volume of the tetrædron is equal to one sixth of the produet of three contiguous edges multiplied by the sine of the triedral angles formed by these edges. Its volume may also be expressed as equivalent to multiplying one half of the product of two opposed edges by the sine of the angle comprised between them, and by one third of the shortest distance between these faces. In the regular tetrædron, the radius of the circumscribed sphere is triple the radius of the inscribed sphere.-Grunert's Archiv, LVII., 113. ORBIT OF THE DOCBLE STAR 42, COMÆ BERENICES. The star 42, Comce Berenices, was discovered to be double in 1826 by the elder Struve, but it appeared single in 1833, since which time it has been observed regularly either by the discoverer or by his son, Otto Strnye, as well as by other astronomers. Since 1826 it has four times presented the appearance of a single star, one of the bodies being actually occulted by the other. The very accurate observations of Otto Struve made since 1840, after having been corrected for the personal errors peculiar to his observations, and which have been most carefully investigated by himself, have sufficed to enable him to determine with very considerable accuracy the position and apparent dimensions of the relative orbits of these stars. The plane of their orbits coincides so nearly with the line joining them to the sun, that we can not certainly state that there is any appreciable inclination between the two. We have therefore to adopt 0° as the inclination between the line of sight and the orbit of the stars, and there results 11°, or the mean of all observed directions, as the angle between the ascending node and the declination circle. The remaining elements of the orbit of the stars, viz., the mean annual motion, the eccentricity, the major axis, the time of passage through the periaster and the angle in the orbit between the periaster and the ascending node, must all be deduced from micrometric measures of the relative distances of two stars. Observations of this nature are proverbially so difficult that up to this time astronomers have avoided employing them when position angles could be used instead. The great accuracy of Struve's micrometric observations, however, is fully illustrated by the remarkable agreement between the observed distance and those computed in accordance with the numerical values |