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Any direct solution to this curious problem, except by means of the analytical formulæ employed above, would be exceedingly tedious and operose.
Er. 7. Demonstrate that sin 18° (− 1 + √5), and sin 54° = cos 36° is =
cos 72° is = R
Er. 8. 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°3′; and thus that the tables may be continued by addition only.
Ex. 9. 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. 10. Demonstrate that the sum of the squares of the sine and versed sine of an arc, is equal to the square of double the sine of half the arc.
Ex. 11. Demonstrate that the sine of an arc is a mean proportional between half the radius and the versed sine of double the arc.
Ex. 12. Show that the secant of an arc is equal to the sum of its tangent and the tangent of half its complement.
Er. 13. 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. 14. How must three trees, A, B, C, be planted, so that the angle at A may be double the angle at B, the angle at в double that at c; and so that a line of 400 yards may just go round them?
Ex. 15. 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?
Er. 16. 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. 17. The sides of a triangle are to each other as the fractions,, what are the angles?
Er. 18. 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°.
Er. 19. Demonstrate that 4 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. 20. Convert the equations marked XXXIV into their equivalent logarithmic expressions; and by means of them and equa. IV, find the angles of a triangle whose sides are 5, 6, and 7.
General Properties of Spherical Triangles. .
ART. 1. Def. 1. Any portion of a spherical surface bounded by three arcs of great circles, is called a Spherical Triangle.
Def. 2. Spherical Trigonometry is the art of computing the measures of the sides and angles of spherical triangles.
Def. 3. A right-angled spherical triangle has one right angle the sides about the right angle are called legs; the side opposite to the right angle is called the hypothenuse.
Def. 4. A quadrantal spherical triangle has one side equal to 90° or a quarter of a great circle.
Def. 5. Two arcs or angles, when compared together, are said to be alike, or of the same affection, when both are less than 90°, or both are greater than 90°. But when one is greater and the other less than 90°, they are said to be unlike, or of different affections.
ART. 2. The small circles of the sphere do not fall under consideration in Spherical Trigonometry; but such only as have the same centre with the sphere itself. And hence it is
that spherical trigonometry is of so much use in Practical Astronomy, the apparent heavens assuming the shape of a concave sphere, whose centre is the same as the centre of the earth.
3. Every spherical triangle has three sides and three angles and if any three of these six parts, be given, the remaining three may be found, by some of the rules which will be investigated in this chapter.
4. In plane trigonometry, the knowledge of the three angles is not sufficient for ascertaining the sides: for in that case the relations only of the three sides can be obtained, and not their absolute values: whereas, in spherical trigonometry, where the sides are circular arcs, whose values depend on their proportion to the whole circle, that is, on the number of degrees they contain, the sides may always be determined when the three angles are known. Other remarkable differences between plane and spherical triangles are, 1st. That in the former, two angles always determine the third; while in the latter they never do. 2dly. The surface of a plane triangle cannot be determined from a knowledge of the angles alone; while that of a spherical triangle always can.
5. The sides of a spherical triangle are all arcs of great circles, which, by their intersection on the surface of the sphere, constitute that triangle.
6. The angle which is contained between the arcs of two great circles, intersecting each other on the surface of the sphere, is called a spherical angle; and its measure is the same as the measure of the plane angle which is formed by two lines issuing from the same point of, and perpendicular to, the common section of the planes which determine the containing sides: that is to say, it is the same as the angle made by those planes. Or, it is equal to the plane angle formed by the tangents to those arcs at their point of intersection. 7. Hence it follows, that the surface of a spherical triangle BAC, and the three planes which determine it, form a kind of triangular pyramid, BCGA, of which the vertex G is at the centre of the sphere, the base ABC a portion of the spherical surface, and the faces AGC, AGB, BGC, sectors of the great circles whose intersections determine the sides of the triangle.
Def. 6. A line perpendicular to the plane of a great circle, passing through the centre of the sphere, and terminated by
two points, diametrically opposite, at its surface, is called the aris of such circle; and the extremities of the axis, or the points where it meets the surface, are called the poles of that circle. Thus, PGP' is the axis, and P, P', are the poles, of the great circle CND.
If we conceive any number of less circles, each parallel to the said great circle, this axis will be perpendicular to them likewise; and the points P, P′, will be their poles also.
8. Hence, each pole of a great circle is 90° distant from every point in its circumference; and all the arcs drawn from either pole of a little circle to its circumference, are equal to each other.
9. It likewise follows, that all the arcs of great circles drawn through the poles of another great circle, are perpendicular to it: for, since they are great circles by the supposition, they all pass through the centre of the sphere, and consequently through the axis of the said circle. The same thing may be affirmed with regard to small circles.
10. Hence, in order to find the poles of any circle, it is merely necessary to describe, upon the surface of the sphere, two great circles perpendicular to the plane of the former; the points where these circles intersect each other will be the poles required.
11. It may be inferred also, from the preceding, that if it were proposed to draw, from any point assumed on the surface of the sphere, an arc of a circle which may measure the shortest distance from that point, to the circumference of any given circle; this arc must be so described, that its prolongation may pass through the poles of the given circle. And conversely, if an arc pass through the poles of a given circle, it will measure the shortest distance from any assumed, point to the circumference of that circle.
12. Hence again, if upon the sides, AC and BC, (produced if necessary) of a spherical triangle BCA, we take the arcs CN, CM, each equal 90°, and through the radii GN, GM (figure to art. 7) draw the plane NGM, it is manifest that the point c will be the pole of the circle coinciding with the plane NGM: so that, as the limes GM, GN, are both perpendicular to the common section GC, of the planes AGC, BGC, they measure, by their inclination, the angle of these planes; or the arc NM measures that angle, and consequently the spherical angle
13. It is also evident that every arc of a little circle, described from the pole c as centre, and containing the same number of degrees as the arc MN, is equally proper for mea
suring the angle BCA; though it is customary to use only arcs of great circles for this purpose.
14. Lastly, we infer, that if a spherical angle be a right angle, the arcs of the great circles which form it, will pass mutually through the poies of each other: and that, if the planes of two great circles contain each the axis of the other, or pass through the poles of each other, the angle which they include is a right angle.
These obvious truths being premised and comprehended, the student may pass to the consideration of the following theorems.
Any Two Sides of a Spherical Triangle are together Greater than the Third,
This proposition is a necessary consequence of the truth, that the shortest distance between any two points, measured on the surface of the sphere, is the a great circle passing through these points.
The Sum of the Three Sides of any Spherical Triangle îs Less than 360 degrees.
For, let the sides AC, BC, (fig. to art. 7) containing any angle A, be produced till they meet again in D: then will the arcs DAC, DBC, be each 180°, because all great circles cut each other into two equal parts: consequently DAC + DBC = € 360°. But (theorem 1) DA and DB are together greater than the third side AB of the triangle DAB; and therefore, since CA + CB +DA+ DB = 360°, the sum CA + CB + AB is less than 360°. Q. E. D.
The Sum of the Three Angles of any Spherical Triangle is always Greater than Two Right Angles, but Less than Six.
For, let ABC be a spherical triangle, G
the centre of the sphere, and let the chords of the arcs AB, BC, AC, be drawn: these chords constitute a rectilinear triangle, the sum of whose three angles is equal to two right angles. But the angle
at B made by the chords AB, BC, is less than the angle aвc, formed by the two tangents Ba, BC, or less than the angle of