maining angle, AEC will be equal to DEB. In the same manner it may be proved that the angle AED is equal to CEB.

Cor.-Hence if any number of arcs of circles intersect each other, all the angles formed about the point of intersection are together equal to four right angles.

PROPOSITION IV.

25. The arc of a great circle, between the pole and the circumference of another great circle, is a quadrant.

Let ABC be a great circle, and P its pole; if PC, an arc of a great circle, pass through P and meet ABC in C, the arc PC is a quadrant.

Let the circle, of which PC is an arc, meet ABC again in A, and let AC be the common section of the planes of these great circles, which will pass through E, the centre of the sphere: Join PA, PC.

Because AP=PC, (def.), and equal straight lines in the same circle, cut off equal arcs, the arc AP the arc PC; but APC is a semicircle, therefore the arcs AP, PC, are each of them quadrants.

E

B

Cor. 1. If PE be drawn, the angle AEP is a right angle; and PE, being at right angles to every line it meets with in the plane of the circle ABC, is at right angles to that plane. Therefore the straight line drawn from the pole of any great circle to the centre of the sphere is at right angles to the plane of that circle; and, conversely, a straight line drawn from the centre of the sphere perpendicular to the plane of any great circle meets the surface of the sphere in the pole of that circle.

Cor. 2. The circle APC has two poles, as has been shown in art. 21., one on each side of its plane, which are the extremities of a diameter of the sphere perpendicular to the plane APC; and no other points but these can be poles of the circle APC.

PROPOSITION V.

26. If the pole of a great circle be the same with the intersection of other two circles, the arc of the first circle intercepted between the other two, is the measure of the spherical angle which the same two circles make with one another.

P

Let the great circles AP, BP, on the surface of the sphere of which the centre is O, intersect each other in P, and let AB be an arc of another great circle of which the pole is P, AB is the measure of the spherical angle APB.

B

Join PO, AO, BO; since P is the pole of AB, PA, PB are quadrants, and the angles POA, POB are right; therefore the angle AOB is the inclination of the planes of the circles PA, PB, and is equal to the spherical angle APB; but the arc AB measures the angle AOB, therefore it also measures the spherical angle APB. Cor. If two arcs of great circles, PA, PC, which intersect each other in P, be each of them quadrants, P will be the pole of the great circle which passes through A and B, the extremities of those

P

arcs. For since the arcs PA and PB are quadrants, the angles POA, POB are right angles, and PO is therefore perpendicular to the plane AOB, that is, to the plane of the great circle which passes through A and B. The point P, therefore, is the pole of the great circle which passes through A and B.

PROPOSITION VI.

27. An angle made by any two great circles of the sphere is equal to the angle of inclination of the planes of these circles.

n m

B

Let BAE be a spherical angle made by two great circles CBA, CEA; then will this angle be equal to the angle of inclination of the planes of those circles. For, take the arcs AB, AE, each equal to 90°, or a quadrant, and through the points B, E draw the arc of the great circle BE, and from D, the centre of the sphere, draw DB, DE.

Then, because AB, AE are quadrants, A and C are the poles of the circle of which BE is a part, and the lines DB, DE are each perpendicular to the common section AC; consequently BDE is the angle of inclination of the planes CBA, CEA. But since DB, DE are equal, being radii of the same sphere, the angle BDE, which is measured by the arc BE, is equal to the angle BAE, which is measured by the arc BE, is equal to the angle BAE, which is measured by the same arc.

And if FH be drawn in the plane CBA, and FG in the plane CEA, each perpendicular to the common section AC, the angle HFG, which is equal to the angle BDE, will also be equal to the angle BAE.

Cor. The angle BAE made by two great circles of the sphere BA, EA, is equal to the angle n A m, formed by two tangents drawn from the angular point A, one in each plane, these tangents being each perpendicular to the diameter AC.

PROPOSITION VII.

28. The distance of the poles of any two great circles of the sphere is equal to the angle of inclination of the planes of those circles.

Let AEB, CED be two great circles, and P, P' their poles; then will the arc PP' be equal to the angle of their inclination AOC or BOD.

For, since P is the pole of the circle AEB, C and P' of CED, the arc PA will be equal to PC, being each quadrants, or 90°; and if PC, A which is common to each, be taken away, the remaining arc, PP', which is the distance of two poles, is equal to CA, the measure of the angle of inclination AOC.

PROPOSITION VIII.

P

29. The circumference of a secondary is at right angles to the cir cumference of its great circle at the point of intersection.

B

The direction of the circumference of a great circle at any point being the same as the direction of its tangent at that point, the angle OBT, (figure prop. V.), is a right angle, BT being a tangent to BP

9

at the point B. POB is also a right angle, and the arc PB is in the plane POB, therefore the direction of the circumference PB at B must be parallel to PO. But PO is perpendicular to the circle ABC; therefore the circle PBP' is at B perpendicular to the circle ABC; hence the arc PB at B is at right angles to AB at B. For the same reason PAB is also a right angle.

Cor. 1.—If a great circle, PBP', be perpendicular to ABC, and BP, BP' be taken each equal to a quadrant, or 90°, P, P' are the poles of the circle ABC.

Cor. 2.-If any two great circles, PAP', PBP', be perpendicular to the circle ABC, they meet at the poles P, P' of that circle.

PROPOSITION IX.

30. In an isosceles spherical triangle the angles at the base are equal.

Let ABE (figure prop. VI.) be a spherical triangle, having the side AB equal to the side AE, the spherical angles ABE, ABE are equal.*

Cor. 1.-Hence, if two of the angles of a triangle be equal, the sides opposite to them are likewise equal.

Cor. 2.-A perpendicular drawn from the vertex of an isosceles spherical triangle to the base, bisects both the base and the vertical angle, except when the two sides are quadrants; in which case there are an indefinite number of perpendiculars.

PROPOSITION X.

31. If the three sides of one spherical triangle be equal to the three sides of another, each to each, the angles which are opposite the equal sides are equal.

PROPOSITION XI.

32. If two sides and the included angle of one spherical triangle be equal to two sides and the included angle in another, these two triangles are equal.

PROPOSITION XII.

33. If from the angles of a spherical triangle, as poles, there be described on the surface of the sphere three arcs of great circles, which, by their intersections, form another spherical triangle, each side of this new triangle will be the supplement of the measure of the angle which is at its pole, and the measure of each of its angles the supplement to that side of the primitive triangle to which it is opposite.

PROPOSITION XIII.

34. If the three angles of one spherical triangle be equal to the three angles of another, each to each, the sides which are opposite to the equal angles are equal.

PROPOSITION XIV.

35. If a side and two adjacent angles of one spherical triangle be

* The demonstrations, which may be seen in Playfair's or Legendre's Geometry, are omitted, as they would swell this work too much, but may perhaps appear in a more complete treatise on trigonometry that has been long meditated.

equal to a side and two adjacent angles of another, each to each, their remaining sides and angles will be equal.

PROPOSITION XV.

36. The sum of any two sides of a spherical triangle is greater than the third side, and the difference of any two sides is less than the third side.

Cor. The shortest distance between any two points on the surface of a sphere is the arc which passes through these points.

PROPOSITION XVI.

37. The greater side of any spherical triangle is opposite to the greater angle, and the less side to the less angle.

And, in a similar manner, it may be shown that the less side is opposite to the less angle, and the less angle to the less side.

PROPOSITION XVII.

38. The sum of the three sides of any spherical triangle is less than the circumference of a circle, or 360°; and the difference of any two sides is less than 180°.

PROPOSITION XVIII.

39. The sum of the three angles of every spherical triangle is greater than two right angles, or 180°, and less than six, or 540°. Cor. The sum of any two angles of a spherical triangle is greater than the supplement of the third angle.

For the angles A+B+C, being greater than two right angles, or than ACB+ACG, if ACB or C be taken away, the sum of the remaining angles A+B, will be greater than ACG.

PROPOSITION XIX.

40. If the sum of any two sides of a spherical triangle be equal to, greater, or less than a semicircle, the sum of their opposite angles will, accordingly, be equal to, greater, or less than two right angles, and conversely.

And, in a similar manner, it may be shown, that if the sum of the two angles B and C be equal to, greater, or less than 180°, the sum of the opposite sides, AB and AC, will also be equal to, greater, or less than 180°.

Cor. 1.-If each side of a spherical triangle be equal to, greater, or less than 180°, each of the angles will, accordingly, be right, obtuse, or acute, and conversely.

Cor. 2. Half the sum of any two sides of a spherical triangle is of the same kind as half the sum of their opposite angles.

PROPOSITION XX.

41. In any right-angled or quadrantal spherical triangle, the legs or sides are of the same kind or affection as their opposite angles, and conversely.

The same will also hold if the triangle be quadrantal; for its sides and angles being the supplements of the angles and legs of the polar triangle, which in this case is right-angled, the similarity will be the same as before.

PROPOSITION XXI.

42. In any right-angled spherical triangle the hypotenuse is less or greater than 90°, according as the two legs, or the two angles, or a leg and its adjacent angle, are alike or unlike.

SECTION II.

Solution of Spherical Triangles.

HAVING given a view of the general principles and properties of spherical triangles, the solution of the various problems in spherical trigonometry ought necessarily to follow. These problems may be resolved either by geometrical construction or by arithmetical calculation. There are various methods of construction, but the most simple and generally employed is the stereographic, in which all the circles of the sphere are represented by straight lines or circles.

Of the Stereographic Projection of the Sphere.

DEFINITIONS.

I. To project an object, as it is commonly called, is to represent every point of that object upon the same plane as it appears to the eye in a certain position.

II. That plane upon which the object is projected is called the plane of projection, and the point where the eye is situated, the projecting point.

III. The stereographic projection of the sphere is that in which a great circle is assumed as the plane of projection, and one of its poles as the projecting point.

IV. The great circle, upon the plane of which the projection is made, is called the primitive.

V. By the semitangent of any arc is meant the tangent of half that arc.

VI. The line of measures of any circle of the sphere is that diameter of the primitive, produced indefinitely, which is perpendicular to the line of common section of the circle and the primitive.

VII. The projection, or representation of any point in the sphere, is the point in which the straight line drawn from it to the projecting point intersects the plane of projection.

THEOREM I.

Every great circle of the sphere, which passes through the projecting point, is projected in a straight line, passing through the centre of the primitive; and every arc of it, reckoned from the other pole of the primitive, is projected into its semitangent.*

Cor. 1.-Every small circle, which passes through the projecting point, is projected into that straight line which is its common section with the primitive.

* For the investigation of the properties of this method of projection, see Gregory's or Keith's Treatises of Trigonometry, and West's Mathematics, published under the care of Professor Leslie.