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If we calculate μ from an equation corresponding to (1) of Art. 109, we have
1. If the sides of a spherical triangle AB, AC be produced to B', C', so that BB', CC' are the semi-supplements of AB, AC respectively, shew that the arc B'C' will subtend an angle at the centre of the sphere equal to the angle between the chords of AB and AC.
2. Deduce Legendre's Theorem from the formula
A tan2 2
sin (a + b − c) sin † (c + a − b)
3. Four points A, B, C, D on the
surface of a sphere are
joined by arcs of great circles, and E, F are the middle points of the arcs AC, BD: shew that
cos AB + cos BC + cos CD + cos DA = 4 cos AE cos BF cos FE.
4. If a quadrilateral ABCD be inscribed in a small circle on a sphere so that two opposite angles A and C may be at opposite extremities of a diameter, the sum of the cosines of the sides is constant.
5. In a spherical triangle if A=B= 2C, shew that
6. ABC is a spherical triangle each of whose sides is a quadrant; P is any point within the triangle: shew that
cos PA cos PB cos PC + cot BPC cot CPA cot APB = 0,
tan ABP tan BCP'tan CAP = 1.
7. If O be the middle point of an equilateral triangle ABC', and P any point on the surface of the sphere, then
(tan PO tan OA) (cos PA + cos PB + cos PC)2 =
cos PA+cos2PB+cos PC-cosPAcos PB-cosPB cosPC-cosPC cosPA.
8. If ABC be a triangle having each side a quadrant, O the pole of the inscribed circle, P any point on the sphere, then
(cos PA + cos PB + cos PC)2= 3 cos2 PO.
9. From each of three points on the surface of a sphere arcs are drawn on the surface to three other points situated on a great circle of the sphere, and their cosines are a, b, c; a', b', c'; a", b′′, c". Shew that ab"c' + a'bc" + a′′b'c = ab'c" + a'b'c + a′′bc'.
10. From Arts. 110 and 111, shew that approximately
log ẞ = log a + log sin B-log sin A +
(cot A - cot B).
11. By continuing the approximation in Art. 106 so as to include the terms involving *, shew that approximately
cos A = cos A' - By sin'A' By (a" - 36 - 37") sin3A'
12. From the preceding result shew that if A = A' ÷ @ then approximately
X. GEODETICAL OPERATIONS.
114. One of the most important applications of Trigonometry, both Plane and Spherical, is to the determination of the figure and dimensions of the Earth itself, and of any portion of its surface. We shall give a brief outline of the subject, and for further information refer to Woodhouse's Trigonometry, to the article Geodesy in the English Cyclopædia, and to Airy's treatise on the Figure of the Earth in the Encyclopædia Metropolitana. For practical knowledge of the details of the operations it will be necessary to study some of the published accounts of the great surveys which have been effected in different parts of the world, as for example, the Account of the measurement of two sections of the Meridional arc of India, by Lieut.-Colonel Everest, 1847; or the Account of the Observations and Calculations of the Principal Triangulation in the Ordnance Survey of Great Britain and Ireland, 1858.
115. An important part of any survey consists in the measurement of a horizontal line, which is called a base. A level plain of a few miles in length is selected and a line is measured on it with every precaution to ensure accuracy. Rods of deal, and of metal, hollow tubes of glass, and steel chains, have been used in different surveys; the temperature is carefully observed during the operations, and allowance is made for the varying lengths of the rods or chains, which arise from variations in the temperature.
116. At various points of the country suitable stations are selected and signals erected; then by supposing lines to be drawn connecting the signals, the country is divided into a series of triangles. The angles of these triangles are observed, that is, the angles which any two signals subtend at a third. For example, suppose A and B to denote the extremities of the base, and C a
signal at a third point visible from A and B; then in the triangle ABC the angles ABC and BAC are observed, and then AC and BC can be calculated. Again, let D be a signal at a fourth point, such that it is visible from C and A; then the angles ACD and CAD are observed, and as AC is known, CD and AD can be calculated.
117. Besides the original base other lines are measured in convenient parts of the country surveyed, and their measured lengths are compared with their lengths obtained by calculation through a series of triangles from the original base. The degree of closeness with which the measured length agrees with the calculated length is a test of the accuracy of the survey. During the progress of the Ordnance Survey of Great Britain and Ireland, several lines have been measured; the last two are, one near Lough Foyle in Ireland, which was measured in 1827 and 1828, and one on Salisbury Plain, which was measured in 1849. The line near Lough Foyle is nearly 8 miles long, and the line on Salisbury Plain is nearly 7 miles long; and the difference between the length of the line on Salisbury Plain as measured and as calculated from the Lough Foyle base is less than 5 inches (An Account of the Observations...page 419).
118. There are different methods of effecting the calculations for determining the lengths of the sides of all the triangles in the survey. One method is to use the exact formulæ of Spherical Trigonometry. The radius of the Earth may be considered known very approximately; let this radius be denoted by r, then if a be the length of any arc the circular measure of the angle which the arc subtends at the centre of the earth is The formulæ of
Spherical Trigonometry give expressions for the trigonometrical
functions of so that may be found and then a.
practice is always very small, it becomes necessary to pay
attention to the methods of securing accuracy in calculations which involve the logarithmic trigonometrical functions of small angles (Plane Trigonometry, Art. 205).
Instead of the exact calculation of the triangles by Spherical Trigonometry, various methods of approximation have been proposed; only two of these methods however have been much used. One method of approximation consists in deducing from the angles of the spherical triangles the angles of the chordal triangles, and then computing the latter triangles by Plane Trigonometry (see Art. 105). The other method of approximation consists in the use of Legendre's Theorem (see Art. 106).
119. The three methods which we have indicated were all used by Delambre in calculating the triangles in the French survey (Base du Système Métrique, Tome III. page 7). In the earlier operations of the Trigonometrical survey of Great Britain and Ireland, the triangles were calculated by the chord method; but this has been for many years discontinued, and in place of it Legendre's Theorem has been universally adopted (An Account of the Observations ... page 244). The triangles in the Indian Survey are stated by Lieut.-Colonel Everest to be computed on Legendre's Theorem. (An Account of the Measurement... page CLVIII.)
120. If the three angles of a plane triangle be observed, the fact that their sum ought to be equal to two right angles affords a test of the accuracy with which the observations are made. We shall proceed to shew how a test of the accuracy of observations of the angles of a spherical triangle formed on the Earth's surface may be obtained by means of the spherical excess.
121. The area of a spherical triangle formed on the Earth's surface being known in square feet, it is required to establish a rule for computing the spherical excess in seconds.
Let n be the number of seconds in the spherical excess, 8 the number of square feet in the area of the triangle, r the number of