ity bent upwards, and expanding into an oval cistern, open at bottom like the ordinary barometer, and filled with almond-oil coloured red. Close beside this, there is placed, on the right side, a fixed scale, to which a slide is adapted for the purpose of reading the height of the barometer, and also a graduated scale of fathoms for ascertaining heights. On the opposite side is a delicate thermometer to observe the temperature, and the instrument is used in the following manner :— Observe the temperature by the thermometer, and set the index upon the sliding scale opposite to the degree of temperature upon the fixed scale, then the height of the oil indicated on the sliding scale will be the pressure of the atmosphere in inches of mercury. By the fathom scale also, by subtracting the number of fathoms indicated at the under station from that indicated at the upper, the height of one place above another will be found. This must be corrected for the difference between the mean temperature of the two stations and 32°, or the factor by which the approximate height must be multiplied according to the sum of the two temperatures, may be taken from the following table: 72 PART II. SPHERICAL TRIGONOMETRY. SECTION I. Definitions, Principles, and General Properties. 1. Spherical Trigonometry is that branch of mathematics by which we are enabled, in all cases, where three of the six parts of a triangle formed by arcs of great circles in the surface of a sphere are given, to compute or determine the other three. 2. 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 value; whereas, in spherical trigonometry, when the sides are circular arcs, whose value depends 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. Among other remarkable differences between plane and spherical triangles are, (1.) That in the former, two known angles always determine the third; while in the latter they never do. (2.) The surface of a plane triangle cannot be determined from a knowledge of the angles alone; while that of a spherical triangle always can. 3. A sphere or globe is a round body formed by the revolution of a semicircle about its diameter, which remains fixed. 4. The centre of the sphere is the same with that of the revolving semicircle. 5. The axis of the sphere is the straight line about which the semicircle relolves. PROPOSITION I. 6. If a sphere be cut by a plane, the section will be a circle. E Let the sphere AEBF be cut by the plane ADB; then will the section ADB be a circle. Draw the chord, or diameter of the section AB, perpendicular to the section ADB, and through the centre C draw the axis of the sphere ECGF, which will (Euc. III. 3.) bisect the chord AB in the point G. Also join CA, CB; and draw CD, GD, to any point D in the perimeter of the section ADB. Then, because CG is perpendicular to the plane ADB, it must be perpendicular both to GA and D GD. Hence CGA and CGD are two right-angled triangles, having F the perpendicular CG common, and the hypotenuse CA equal to the hypotenuse CD, being both radii of the same sphere; therefore their sides GA, GD, are also equal. In like manner, it may be shown, that any other line drawn from G to the circumference of the section ADB, is equal to GA, or GB, and consequently that section is a circle. Cor.-If a sphere be cut by a plane through the centre, the section is a circle, having the same centre with the sphere, and equal to the circle by the revolution of the half of which the sphere was described. For all the straight lines drawn from the centre to the surface of the sphere are equal to the radius of the generating semicircle. Therefore the common section of the spherical surface, and of a plane passing through its centre, is a line lying in one plane, having all its points equally distant from the centre of the sphere, and is consequently the circumference of a circle, having for its centre the centre of the sphere, and for its radius the radius of the sphere, that is, of the semicircle by which the sphere is described. It is therefore equal to the circle of which that semicircle is a part. 7. Any circle formed from the section of a sphere, by a plane through its centre, is called a great circle of the sphere. Cor.-All great circles of the sphere are equal, and any two of them bisect each other. They are all equal, because they have all the same radii, as has just been shown, and any two of them bisect one another; for, as they have the same centre, their common section is a diameter of both, and therefore bisects both.* 8. The pole of a great circle of the sphere is a point in the surface of the sphere equidistant from every part of the circumference of that circle. 9. A spherical angle is an angle on the surface of a sphere contained by the arcs of the two great circles which intersect each other, and is the same as the inclination of the planes of, or tangents at the point of intersection to, these great circles. 10. A spherical triangle is a figure on the surface of a sphere formed by the intersection of three arcs of great circles, each of which is less than a semicircle. 11. A right-angled spherical triangle has one right angle; the sides about the right angle are called legs, and that opposite the right angle is called the hypotenuse. 12. A quadrantal spherical triangle has one side equal to a quadrant, or 90°. 13. An oblique-angled spherical triangle has none of its angles right. 14. Spherical triangles are also called equilateral, isosceles, or scalene, according as they have three sides equal, two sides equal, or all the three sides unequal. 15. Two arcs, or angles, when compared together, are said to be alike, or of the same affection, when both are less or both are greater than 90°. But when one is less and the other greater than 90°, they are said to be unlike, or of different affections or characters. 16. Every spherical triangle has three sides and three angles; Hence the intersections of the circumference of two great circles are two points diametrically opposite to each other. K and if found. any three of these six parts be given, the other three may be 17. A lune is a part of the surface of a sphere contained by the semicircumferences of the two great circles. 18. A small circle of the sphere is that whose plane does not pass through the centre of the sphere. 19. The small circles of the sphere do not fall under the consideration of 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. 20. The sides of a spherical triangle are all arcs of great circles, which, by their intersection on the surface of a sphere, constitute that triangle. 21. If ABDG be a great circle of the sphere whose centre is C and PCP' a diameter of the sphere perpendicular to its plane, the points P, P' are the poles of that circle. And if the small circle a, b c d be per- A pendicular to PP', we call P, P' the poles of that small circle also. 22. The great circles PAP', PGP' passing through the poles P, P' of the great circle ABDG, are called secondaries to that circle. PROPOSITION II. P b B P 23. If two arcs of circles meet each other, they make two angles, which are together equal to two right angles. Let the arc AB meet the arc CD in the point B; then will the two angles ABC, ABD be equal to two right angles. For, suppose the arc BE to be perpendicular to CD, then the angles EBC, EBD are right angles. And since the angle EBD is equal to the angles C EBA, ABD, the three angles, EBC, EBA, ABD, are equal to the two right angles. E B But the two angles EBC, EBA, are equal to the angle ABC; whence the two angles, ABC, ABD, are also equal to two right angles. PROPOSITION III. 24. If two arcs of a circle intersect each other, the vertical or opposite angles will be equal. Let the two arcs, AB, CD, intersect each other A in E, then will the angle AEC be equal to DEB, and AED to CEB. For since the arc AE meets the arc CD, the angles AEC, AED are together equal to two right angles, (Prop. II.) And because the arc DE meets the arc AB, the angles DEB, DEA are also equal to two right-angles. Taking away from each the common angle AED, and the re |