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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.
The distance fr and the angles DEN, NFf, being thus known, we shall have (th. 6 cor. 3 Geom.)
fFP = 180°
So that, in the right-angled triangle ƒFP, two angles and one side are known; and therefore the appropriate spherical excess may be computed, and thence the angle FPf and the sides fP, FP. Resolving next the right-angled triangle eEP, we shall in like manner obtain the position of the point E, with respect to the meridian Ax, and to its perpendicular AY; that is to say, the distances Ee, and Ae=AP-ep. And thus may the computist proceed through the whole of the series. It is requisite however, 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 facilitate 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 calculations. 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 operations, 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 plane 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 contemplated 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, 1806, pa. 130, that the difference 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 governments. As by M. Maupertius, Clairaut, &c, in Lapland, 1736; by M. Bouguer and Condamine, at the equator, 1736— 1743; by Cassini, in lat. 45o, 1739-40; by Boscovich and Lemaire, lat. 43°, 1752; by Beccaria, lat. 44° 44′, 1768; by Mason and Dixon in America, 1764-8; by Major 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, and Colonel Mudge, in England, from 1784 to the present time. The three last mentioned of these surveys are doubtless the most accurate and important.
The trigonometric 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, M. M. Cassini, Mechain, and Legendre, met General Roy and Dr. (now 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 expence of the Honourable Board of Ordnance, in order to perfect the geography of England, and to determine 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 a
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 timber, 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 Colonel 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 23 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.
13. For the determination of angles, the French and Swedish 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. General Roy and Colonel 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 supe riorities 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 formulæ of computation, the actual employment 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 has been observed by a competent judge,' "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. In proof of the great correctness of the English survey, we shall state a very few particulars, besides what is already mentioned in art. 12.
General Roy, who first measured the base on HounslowHeath, measured another on the flat ground of RomneyMarsh in Kent, near the southern extremity of the first series of triangles, and at the distance of more than 60 miles from the first base. The length of this base of verification, as actually measured, compared with that resulting from the computation through the whole series of triangles, differed only by 28 inches.
Colonel Mudge measured another base of verification on Salisbury plain. Its length was 36574-4 feet, or more than 7 miles; the measurement did not differ more than one inch from the computation carried through the series of triangles from Hounslow Heath to Salisbury Plain. A most remarkable proof of the accuracy with which all the angles, as well as the two bases, were measured!
The distance between Beachy-Head in Sussex, and Dunnose in the Isle of Wight, as deduced from a mean of four series of triangles, is 339397 feet, or more than 644 miles. The extremes of the four determinations do not differ more than 7 feet, which is less than 13 inches in a mile. Instances of this kind frequently occur in the English survey*. But we have not room to specify more. We must now proceed to discuss the most important problems connected with this subject; and refer those who are desirous to consider it more minutely, to Colonel Mudge's "Account of the Trigonometrical Survey;" Mechain and Delambre, "Base du Systéme Métrique Décimal;" Swanberg, "Exposition des Opérations faites en Lapponie;" and Puissant's works entitled "Geodesie" and "Traite de Topographie, d'Arpentage, &c."
Problems connected with the detail of Operations in Extensive Trigonometrical Surveys.
It is required to determine the Most Advantageous
1. In any rectilinear triangle ABC, it is, from the proportionality of sides to the sines of their opposite angles, AB BC: sin C: sin A, and consequently AB. sin ABC. sin c. Let AB be the base, which is supposed to be measured without perceptible error, and which therefore is assumed as constant; then finding the extremely A
* Puissant, in his "Geodésie," after quoting some of them, says, "Neanmoins, jusqu'à présent, rienn'egale en exactitude les opérations géodesiques qui ont servi de fondement à notre système métrique." He, however, gives no instances. We have no wish to depreciate the labours of the French measures; but we cannot yield them the preference on mere assertion.