But the sine of the sum of A and B is the same as that of their supplement C, or of the angle contained by the sides a and 3, and ". - A . consequently 9 is the third part of sinc, the area of the tri angle, or the third part of the excess of the angles of the spherical, above those of the plane triangle. Wherefore the sines of the sides, which, in the spherical triangle, are as the sines of their opposite angles, are likewise proportioned, in the plane triangle, to the sines of those angles, inoreasing each by the common excess. It is hence evident, that the angles of the plane triangle are obtained from those of the spherical, by deducting from each the third part of the excess above two right angles, as indicated by the area of the triangle. The whole surface of the globe being proportioned to 720°, -- - 720° that of a square mile will correspond to TZ1956×7513 or 1 the #555 part a second. Hence each angle of the small spherical triangle requires to be diminished by as sinc 18. Another problem of great use in the practice of delicate surveying, is to reduce angles to the centre of the station. Instead of planting moveable signals at each point of observation, it will often be found more convenient to select the more notable spires, towers, or other prominent objects which occur interspersed over the face of the country. In such cases, it is evidently impossible for the theodolite or circular instrument, although brought within the cover of the building, to be placed immediately under the vane. The observer approaches the centre of the station as near, therefore, as he can with advantage, and calculates the quantity of error which the minute displacement may occasion. Thus, suppose it were required to determine the angle AOB which the remote object A and B subtend at O, the centre of a permanent station : The instrument is placed in the immediate vicinity at the point C, and the distance CO, with the angle of deviation OCA, are noted, while the principal angle ADCB is observed. The central angle AOB may hence be computed from the rules of trigonometry; but the calculation is effected by simpler and more expeditious methods. Since (I. 30. El.) the exterior angle ADB is equal both to AOB with OAC, and to ACB with OBC; it is evident that AOB = ACB + OBC—OAC. But the angles OBC and OAC, being extremely small, may be considered as equal CO to their sines, and (art. 5. Note 14) sin OBC = GB sinBCO, CO and sinoAC = OR sin ACO; wherefore the angle AOB at - sinBCO sin ACO the centre, is nearly equal to ACB + co(*.* #) - ACO in ACO =AcB+co (*#o–o) callihedi. tances AC and BC of the point of observation, a and b, the distances AO and BO of the centre a' and bo; the displacement CO, and the angle ACO of deviation m and 4, while the subtended angles ACB and AOB are denoted by C and C, and the opposite angles ABO and OAB by A and B; then C' sin(C sin - C -}. m. ( 272 ;to - o 3438. If the centre O lies on AC, the correction of the observed angle, expressed in mi m nutes, will be merely (#sinc) 3438', But the problem admits of a simpler approximation. Let a circle circumscribe the points A, O, and B, and cut AC in E. The angle AOB = (III. 16. El.) AEB = ACB + CBE; CE but sincLE = EB sinACB, and sino EC = sinaEO or ABO CO is equal to GE sincoF or AEO-ACO, and hence by com CO sinACB sin(ABO-ACO bination sin CBE = onaco-aco • Since, therefore, EB is nearly equal to OB, and the small angle CBE may be regarded as equal to its sine, the correction to be add ed to the observed angle is denoted in minutes by # sinCsin(A–p) sin A vanish when p becomes equal to A, or the angle ABO equals ACO; in which case, the point of observation C coincides with E, or lies in the circumference of a circle that passes through the two remote points A and B and centre of the station. To place the instrument at E, therefore, would only require to move it along CA, till the angle AEO be equal to ABO. Both these methods for the reduction of an angle to the centre are given by Delambre; but, in his calculations, he generally preferred the last one, as being simpler and sufficiently accurate for practice. The investigation, however, will be found to be now considerably shortened. 3438. This quantity, it is evident, will entirely 19. The accuracy of trigonometrical operations must depend on the proper selection of the connecting triangles. It is very important, therefore, in practice, to estimate the variations which are produced among the several parts of a triangle, by any change of their mutual relations. Suppose two of the three determining parts of a triangle to remain constant, while the rest undergo some partial change; and let, as before, the small letters a, b and c denote the sides of the triangle, and the capitals A, B and C their opposite angles. Case I.--When two sides a and b are constant. Since the angles A and B, after passing into A + AA and B + AB, must have their sines still proportional to the oppo- - - - ... • - sin A sin B site sides, it is evident that sin(A + AA) = sin(B+ ABj' and sin(A + AA)—sin A sin(B+AB)—sinB . °onsequently sinia AA) + in A- sin(B+ABj+sinb’ wherefore, by alternation and art. 7. Note 12., tan AA tan(A + 4AA) ‘ian AB = Ian (BIAB). Next, in the incremental triangle formed by the sides c, c-HAc, and the contained angle AA, (art. 1. Note 12.) f J. #: - —o, and hence reciprocally, 2. – “.. =––oto‘ian AAT cos(B+TAB) In like manner, from the incremental triangle contained by the sides c, c-H Ac and the angle AB, it follows that #Ac – c-H Ac Ian AB TT cos(A+TAA) Again, the base of the incremental isosceles triangle contained by the equal sides b, b, and the vertical angle AC, is (art. 15. Note 12.) 2b sin; AC; wherefore, in the incremental triangle formed with the same base and the sides c and c-HAc, After the same manner, it will be found that, sinkAA — a cos(B+}ABs -. sin. ACT c-ETAcT’ Multiply the expressions of art. 4. into those of art 3. and #Ac b sin(A+ AA) “sin. ACT Teos, AB Multiply likewise the expressions of art. 2, and 5, and #Ac asin(B+ AB) in ACT cos, AAT' If, in all the preceding formulae, the increments annexed to the varying quantities be omitted, there will arise much simpler expressions for the differentials. Case II.--When one side a, and its opposite angle A, are constant. Since (art. 5. Note 12.) *—= -*. , it is evident that - sin A T sin B a sinB-5 sinA, and taking the differences by art. 1. of Note 10. A5 sinA = 2a sin; ABcos(B+ 4AB), whence sin; AB sin A Tcos(B+HAB)” and consequently, by art. 5, of Note 12. |