Scholium. If the circumference which passes through the points D, B should not cut the edge of the tower or perpendicular AE, but only touch it, it would admit of only one solution, and that point which would answer the conditions would be the point of contact; but if the circle should not reach the per pendicular, the question would be impossible. EXAMPLES FOR PRACTICE. Ex. 1. Given the angles of elevation of any distant object, taken at three places on a level plane, no two of which are in the same vertical plane with the object; to find the height of the object, and its distance from either station. Let A, B, C, be the three stations, K the object, and KH perpendicular to the plane of the triangle ABC. Put BC-a, AC=b, AB=c, HAK=α, K B H HBK=8, HCK=7, and HK=x; then the angles AHK, BHK, CHK being right angles, we have AH=x cot. a, BH=x cot. B, CH=x cot. y; whereby from the given data the required may be found. Ex. 2. Given a=30° 40', B=40° 33', y=50° 23'; find x, when the three stations are in the same straight line, AB being 50° and BC=60 yards. Ans. 77.7175 yards. Ex. 3. Demonstrate that sin. 18° cos. 72° is = R(-1+ ✓5), and sin. 54° cos. 36° is=1 R (1+√5). Ex. 4. Demonstrate that the sum of the sines of two arcs which together make 60°, is equal to the sine of an arc which is greater than 60°, by either of the two arcs: Ex. gr. sin. 3'+sin. 59° 57'sin. 60° 30'; and thus that the tables may be continued by addition only. Ex. 5. Show the truth of the following proportion: As the sine of half the difference of two arcs, which together make 60°, or 90°, respectively, is to the difference of their sines; so is 1 to 2, or 3, respectively. Ex. 6. Demonstrate that the sum of the square of the sine and versed sine of an arc, is equal to the square of double the sine of half the arc. Ex. 7. Demonstrate that the sine of an arc is a mean proportional between half the radius and the versed sine of double the arc. Ex. 8. Show that the secant of an arc is equal to the sum of the tangent and the tangent of half its complement. Ex. 9. Prove that, in any plane triangle, the base is to the difference of the other two sides, as the sine of half the sum of the angles at the base, to the sine of half their difference: also, that the base is to the sum of the other two sides as the cosine of half the sum of the angles at the base, to the cosine of half their difference. Ex. 10. How must three trees, A, B, C, be planted, so that the angle at A may double the angle at B, the angle at B double that at C; and so that a line of 400 yards may just go round them? Ex. 11. In a certain triangle, the sines of the three angles are as the numbers 17, 15, and 8, and the perimeter is 160. What are the sides and angles? Ex. 12. The logarithms of two sides of a triangle are 2.2407293 and 2.5378191, and the included angle, is 37° 20′. It is required to determine the other angles, without first finding any of the sides? Ex. 13. The sides of a triangle are to each other as the fractions,,: what are the angles? Ex. 14. Show that the secant of 60°, is double the tangent of 45°, and that the secant of 45° is a mean proportional between the tangent of 45° and the secant of 60°. Ex. 15. Demonstrate that four times the rectangle of the sines of two arcs, is equal to the difference of the squares of the chords of the sum and difference of those arcs. Ex. 16. Convert formulæ, Chap. III, into their equivalent logarithmic expressions; and by means of them and formulæ B, Chap. III, find the angles of a triangle whose sides are 5, 6, and 7. Ex. 17. Being on a horizontal plane, and wanting to ascertain the height of a tower, standing on the top of an inaccessible hill, there were measured, the angle of elevation of the top of the hill 40°, and the top of the tower 51°: then measur ing in a direct line 180 feet farther from the hill, the angle of elevation of the top of the tower was 33° 45': required the height of the tower. Ans. 83.9983 feet. Ex. 18. From a station P there can be seen three objects, A, B, and C, whose distance from each other are known, viz. AB=800, AC=600, and BC=400 yards. There are also measured the horizontal angles APC=33° 45', BPC=22° 30'. It is required, from these data, to determine the three distances PA, PC, and PB. Ans. PA=710.193, PC=1042.522, PB=934.191 yards. SPHERICAL TRIGONOMETRY. Having demonstrated in the treatise on Spherical Geometry, several important properties of the circle of the sphere, and of spherical triangles, we shall now proceed to deduce various relations which exist between the several parts of a spherical triangle. These constitute what is called Spherical Trigonometry; and enables us, when a certain number of the parts are given, to determine the rest. The first formula which we shall establish, serves as a key to the rest, and is to spherical trigonometry what the expression for the sine of the sum of two angles is to plane trigonometry. CHAPTER I. 1. To express the cosine of an angle of a spherical triangle in terms of the sines and cosines of the sides. Let ABC be a spherical triangle, O the centre of the sphere. Let the angles of the triangles be denoted by the large letters A, B, C, and the sides opposite to them by the corresponding small letters, a, b, c. At the point A, draw AT a tangent to the arc AB, and At a tangent to the arc AC. Then the spherical angle A is equal to the angle TAt be tween the tangents, (Spher. Geom. Prop. VII.) Join OB, and produce it to meet AT in T. sec.c+sec.2 b. 2 sec. c sec. b cos. a, - (1) Again, in triangle TAt TAT+Ať - 2AT. At cos. TAt .. =sec.' c+sec.❜b - 2 sec. c sec. b cos. a - 2 tan. c tan. b cos. A=2-2 sec. c sec. b cos. a 2. To express the cosine of a side of a spherical triangle, in terms of the sines and cosines of the angles. Let A, B, C, a, b, c, be the angles and sides of a spherical triangle; A', B', C', a', b', c', the corresponding qualities in the Polar triangle, Then by (a), |