GEODESIC OPERATIONS, latitude on an ellipsoid of revolution, these curves are plane and circular. 8. The situation of a place is determined, when we know either the individual perpendicular to the meridian, or the individual parallel to the equator, on which it is found, and its position on such perpendicular, or on such parallel. Therefore, when all the triangles, which constitute such a series as we have spoken of, have been computed, according to the principles just sketched, the respective positions of their angular points, either by means of their longitudes and altitudes, or of their distances from the first meridian, and from the perpendicular to it. The following is the method of computing these distances. Suppose that the triangles ABC, BCD, &c. (see the fig. to art. 6) make part of a chain of triangles, of which the sides are arcs of great circles of a sphere, whose radius is the dis. tance from the level or surface of the sea to the centre of the earth; and that we know by observation the angle cax, which measures the azimuth of the side Ac, or its inclination to the meridian AX. Then, having found the excess E, of the three angles of the triangle Acc (cc being perpendicular to the meridian) above two right angles, by reason of a theorem which will be demonstrated in prob. 8 of this chapter, subtract a third of this excess from each angle of the triangle, and thus by means of the following proportions find Ac, and cc. sin (90): COS (CAC sin (90° E) sin (CAC E) AC AC; The azimuth of AB is known immediately, because BAX = AMB 180°- M'AB ABM+ E. To determine the sides AM', BM', a third of E must be de. ducted from each of the angles of the triangle ABM'; and then these proportions will obtain : viz. sin (180°--M'AB ABME): sin (ABM' -E) :: AB: AM', E) AB: BM. M'AB ABME) sin (M'AB sin (1080 In each of the right-angled triangles Abв, 'dD, are known two angles and the hypothenuse, which is all that is necessary to determine the sides ab, bв, and м'd, do. Therefore the distances of the points B, D, from the meridian and from the perpendicular, are known. 9. Proceeding in the same manner with the triangle ACN, or M'DN, to obtain AN and DN, the prolongation of CD; and then with the triangle DNF to find the side NF and the angles DNF, DFN, it will be easy to calculate the rectangular coordinates of the point F. VOL. II, 10 The distance fr and the angles DEN, NFf, being thus known, we shall have (th. 6 cor. 3. Geom.) frr = 180° EFDDFN - NFf. So that, in the right-angled triangle frr, two angles and one side are known; and therefore the appropriate spherical excess may be computed, and thence the angle Frf and the sides fP, FP. Resolving next the right-angled triangle CEP, we shall in like manner obtain the position of the point E, with respect to the meridian Ax, and to its perpendicularAx; that is to say, the distances Ee, and Ae = AP —er. And thus may the computist proceed through the whole of the series. It is requisite how. ever, previous to these calculations, to draw, by any suitable scale, the chain of triangles observed, in order to see whether any of the subsidiary triangles ACN, NFP, &c, formed to faci litate the computation of the distances from the meridian, and from the perpendicular to it, are too obtuse or too acute. Such, in few words, is the method to be followed, when we have principally in view the finding the length of the portion of the meridian comprised between any two points, as a and X. It is obvious that, in the course of the computations, the azimuths of a great number of the sides of triangles in the series is determined; it will be easy therefore to check and verify the work in its process, by comparing the azimuths found by observation, with those resulting from the calcu lations. The amplitude of the whole arc of the meridian measured, is found by ascertaining the latitude at each of its extremities; that is, commonly by finding the differences of the zenith distances of some known fixed star, at both those extremities. 10. Some mathematicians, employed in this kind of opera. tions, have adopted different means from the above. They draw through the summits of all the triangles, parallels to the meridian and to its perpendicular; by these means, the sides of the triangles become the hypothenuses of right-angled triangles, which they compute in order, proceeding from some known azimuth, and without regarding the spherical excess, considering all the triangles of the chain as described on a plain surface. This method, however, is manifestly defective in point of accuracy. Others have computed the sides and angles of all the triangles, by the rules of spherical trigonometry. Others, again, reduce the observed angles to angles of the chords of the respective arches; and calculate by plane trigonometry, from such reduced angles and their chords. Either of these two methods is equally correct as that by means of the spherical excess so that the principal reason for preferring one of these to the other must be derived from its relative facility. As to the methods in which the several triangles are contem. plated as spheroidal, they are abstruse and difficult, and may, happily, be safely disregarded: for M. Legendre has demonstrated in Mémoires de la Classe des Sciences Physiques et Mathématiques de l'Institut, 1506, p. 130, that the dif ference between spherical and spheroidal, angles is less than one sixtieth of a second, in the greatest of the triangles which occurred in the late measurement of an arc of a meridian between the parallels of Dunkirk and Barcelona. 11. Trigonometrical surveys for the purpose of measuring a degree of a meridian in different latitudes, and thence inferring the figure of the earth, have been undertaken by different philosophers, under the patronage of different go. vernments. As by M. Maupertuis, Clairaut, &c. in Lapland, 1736; by M. Bouguer and Condamine, at the equator, 1736– 1743; by Cassini, in lat. 45, 1739-40; by Boscovich and Lemaire, lat. 43, 1752; by Beccaria, lat. 44 44, 1768; by Mason and Dixon in America, 1764-8; by Colonel Lambton, in the East Indies, 1803; by Mechain, Delambre, &c. France, &c., 1790-1805; by Swanberg, Ofverbom, &c. in Lapland, 1802; and by General Roy, Colonel Williams, Mr. Dalby, General Mudge, and Colonel Colby, in England, from 1784 to the present time. The three last mentioned of these surveys are doubtless the most accurate and important. The trigonometrical survey in England was first commenced, in conjunction with similar operations in France, in order to determine the difference of longitude between the meridians of the Greenwich and Paris observatories; for this purpose, three of the French Academicians, MM. Cassini, Mechain, and Legendre, met General Roy and Sir Charles Blagden, at Dover, to adjust their plans of operation. In the course of the survey, however, the English philosophers, selected from the Royal Artillery officers, expanded their views, and pursued their operations, under the patronage, and at the expense of the Honourable Board of Ordnance, in order to perfect the geography of England, and to deter mine the lengths of as many degrees on the meridian as fell within the compass of their labours. 12. It is not our province to enter into the history of these surveys: but it may be interesting and instructive to speak a little of the instruments employed, and of the extreme accuracy of some of the results obtained by them. These instruments are, besides the signals, those for measuring distances, and those for measuring angles. The French philosophers used for the former purpose, in their measurement to determine the length of the metre, rulers of platina and of copper, forming metallic thermometers. The Swedish mathematicians, Swanberg and Ofverbom, employed iron bars, covered towards each extremity with plates of silver. General Roy commenced his measurement of the base at Hounslow-Heath with deal rods, each of 20 feet in length. Though they, however, were made of the best seasoned tim. ber, were perfectly straight, and were secured from bending in the most effectual manner; yet the changes in their lengths, occasioned by the variable moisture and dryness of the air, were so great, as to take away all confidence in the results deduced from them. Afterwards, in consequence of having found by experiments, that a solid bar of glass is more dilatable than a tube of the same matter, glass tubes were substituted for the deal rods. They were each 20 feet long, inclosed in wooden frames, so as to allow only of expansion or contraction in length, from heat or cold, according to a law ascertained by experiments. The base measured with these was found to be 27404-08, feet, or about 5-19-miles. Several years afterwards the same base was remeasured by General Mudge, with a steel-chain of 100 feet long, constructed by Ramsden, and jointed somewhat like a watch-chain. This chain was always stretched to the same tension, supported on troughs laid horizontally, and allowances were made for changes in its length by reason of variations of temperature, at the rate of 0075 of an inch for each degree of heat from 62° of Fahrenheit: the result of the measurement by this chain was found not to differ more than 2 inches from General Roy's determination by means of the glass tubes : a minute difference in a distance of more than 5 miles; which, considering that the measurements were effected by different persons, and with different instruments, is a remarkable confirmation of the accuracy of both operations. And further, as steel chains can be used with more facility and convenience than glass rods, this remeasurement determines the question of the comparative fitness of these two kinds of instruments. Still greater improvements, however, in the construction of apparatus for the measurement of a base, are now ready for introduction into the survey, by its scientific and indefati gable conductor Colonel Colby. 13. For the determination of angles, the French and Swe. dish philosophers employed repeating circles of Borda's construction: instruments which are extremely portable, and with which, though they are not above 14 inches in diameter, the observers can take angles to within 1" or 2" of the truth. But this kind of instrument, however great its ingenuity in theory, has the accuracy of its observations necessarily limited by the imperfections of the small telescope which must be attached to it. Generals Roy and Mudge made use of a very excellent theodolite constructed by Ramsden, which, having both an altitude and an azimuth circle, combines the powers of a theodolite, a quadrant, and a transit instrument, and is capable of measuring horizontal angles to fractions of a second. This instrument, besides, has a telescope of a much higher magnifying power than had ever before been applied to observations purely terrestrial; and this is one of the superiorities in its construction, to which is to be ascribed the extreme accuracy in the results of this trigonometrical survey. Another circumstance which has augmented the accuracy of the English measures, arises from the mode of fixing and using this theodolite. In the method pursued by the Continental mathematicians, a reduction is necessary to the plane of the horizon, and another to bring the observed angles to the true angles at the centres of the signals: these reductions, of course, require formule of computation, the actual em.. ployment of which may lead to error. But, in the trigonometrical survey of England, great care has always been taken to place the centre of the theodolite exactly in the vertical line, previously or subsequently occupied by the centre of the signal the theodolite is also placed in a perfectly horizontal position. Indeed, as was observed by professor Playfair, "In no other survey has the work in the field been conducted so much with a view to save that in the closet, and at the same time to avoid all those causes of error, however minute, that are not essentially involved in the nature of the problem. The French mathematicians trust to the correction of those errors; the English endeavour to cut them off entirely; and it can hardly be doubted that the latter, though perhaps the slower and more expensive, is by far the safest proceeding." 14. With a view to facilitate the observation of distant stations, many contrivances have been adopted; among which those recently (1826) invented by Lientenant Drum. mond, R. E. deserve peculiar notice of these, one is appli cable by day, the other by night. The first, which consists in employing the reflection of the sun from a plane mirror as a point of observation, was first suggested by Professor Gauss; and the result of the first trials made in the survey of Hanover proved very successful. Recourse was had to this method on some occasions that occurred in the Trigonometrical Survey of England, where, from peculiar local circumstances, mnch difficulty was experienced in discerning the usual signals. Even as a temporary expedient, and under a rude form, viz. that of placing tin plates at the station to be observed in such a manner that the sun's reflection should be thrown |