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Let fall a perpen. on the side adjacent to the angle sought.
sum X tan diff. of the sides tan base
Tan 1 sum or 1 diff. of the seg. of the base. S
Cos angle sought tan adj. seg. X cot adja. side.
Will be obtained by finding its correspondent angle, in a triangle
Questions for Exercise in Spherical Trigonometry.
Ex. 1. In the right-angled spherical triangle BAC, right-
Here, by table 1 case 1, sin c
COS B =
Or, log sin c = log sin c-log sin a + 10.
log sin a =
log sin c =
log sin 83°56′ = 9.9985556
Here c is acute, because the given leg is less than 90°.
log tan α =
log tan 78°20′
log cos B =
log cos 27°45'
в is here acute, because a and c are of like affection. Lastly, 10+ log cos a == 10+ log cos 78°20′ ⇒ 19∙3058189 log cos c 9.3564426 log cos 76°52′ :
log cos b =
log cos 27°8′
where b is less than 90°, because a and c both are so.
Ans. a 113°55′, c = 28°51′, B = 104° 8'.
Ex. 6. Given в ≈ 48o, c = 64935'; required the other
cos b =
Ex. 7. In the quadrantal triangle ABC, given the quadrantal side a = 90o, an adjacent angle c=42° 12′ and the opposite angle A64° 40'; required the other parts of the triangle?
Ex. 8. In an oblique angled spherical triangle are given the three sides viz. a 56° 40′, b 83° 13,c= 114° 30′ :
to find the angles.
Here, by the fifth case of the table 2, we have sin (1s-b). sin (s—c)
sin b. sin c
Or, log. sin ¦ ▲ — log sin (s—b)+log sin (1s—c) + ar. comp.
9.3418385 0.0030508 0-0409771 19-2274413
Then, by the common analogy,
As, sin a
sin 24° 15′9:6137206
sin 56°40′...log9-9219401 . sin 48°31'
log sin 114°30 ..log
And so is, sin c
To, sin c
sin 125°19′... log 9.9116507. So that the remaining angles are, B=62°56′, and c
To, sin A..
2dly. By way of comparison of methods, let us find the
log tan 1 (b+a) = log tan 69°56′
Subtract log tan clog tan 57°15'
Their sum = 19·8103420
tan 79° 49′Xcot b:
Therefore, since cos a
The other two angles may be found as before. The preference is, in this case, manifestly due to the former method. triangle, are given
Ex. 9. In an oblique-angled sphericalectively, and the
two sides equal to 114° 40' and 56° 30′
Ans. Side opposite first angle, 56° 40'; other side, 83° 12'; third angle 62° 54'.
The sum rejecting 10 from the index
Ex. 11. Given two sides, equal 114° 30′, and 56° 40′ and their included angle 62° 54′: to find the rest.
Ex. 12. Given two angles, 125° 20′ and 48° 30′, and the side comprehended between them 83°12′ to find the other parts.
Ex. 13. In a spherical triangle, the angles are 48° 31', 62° 56′, and 125o 20': required the sides?
opposite the former, 62° 42′ and 58° 8; and a side
Ex. 14. Given two angles, 50° 12′,
to find the other parts.
Ans. The third angle is either 130°56′ or 156°14'.
Side betw. giv. angles,
either 119°4′ or 152°14'. either 79°12′ or 100°48′.
Ex. 15. The excess of the three angles of a triangle, measured on the earth's surface, above two right angles, is 1 second; what is its area, taking the earth's diameter at 79573 miles?
Ans. 76-75299, or nearly 763 square miles. Ex. 16. Determine the solid angles of a regular pyramid, with hexagonal base, the altitude of the pyramid being to each side of the base as 2 to 1.
Ans. Plane angle between each two lateral faces 126°52′11′′1 between the base and each face 66°35′12′′i.
The max. angle being 1000.
Solid angle at the vertex 114-49768 )
ON GEODESIC OPERATIONS, AND THE FIGURE OF
General Account of this kind of Surveying.
ART. 1. In the treatise on Land Surveying in the first volume of this Course of Mathematics, the directions were restricted to the necessary operations for surveying fields, farms, lordships, or at most, counties; these being the only operations in which the generality of persons, who practise this kind of measurement, are likely to be engaged: but there are especial occasions when it is requisite to apply the principles of plane and spherical geometry, and the practices of surveying, to much more extensive portions of the earth's surface; and when of course much care and judgment are called into exercise, both with regard to the direction of the practical operations, and the management of the computations. The extensive processes which we are now about to consider, and which are characterised by the terms Geodesic Operations and Trigonometrical Surveying, are usually undertaken for the accomplishment of one of these three objects. 1. The finding the difference of longitude, between two moderately distant and noted meridians; as the meridians of the observatories at Greenwich and Oxford, or of those at Greenwich and Paris. 2. The accurate determination of the geographical positions of the principal places, whether on the coast or inland, in an island or kingdom; with a view to give greater accuracy to maps, and to accommodate the navigator with the actual position, as to latitude and longitude, of the principal promontories, havens, and ports. These have, till lately, been desiderata, even in this country: the position of some important points, as the Lizard, not being known within seven minutes of a degree; and, until the publication of the board of Ordnance maps, the best country maps being so erroneous, as in some cases to exhibit blunders of three miles in distances of less than twenty.