186 There are no fewer than eight different triangles which have for their angular points poles of the sides of a given triangle ABC; but there is only one triangle in which these poles A', B', C', lie towards the same parts with the opposite angles A, B, C, and this is the triangle A'B'C', which is known under the name of the polar triangle. 12. A spherical polygon is any portion of the sphere's surface included B by more than three arcs of different great circles, as ABCDE. one another; because the right-angled triangles OKA, OKB have their hypotenuses O A, O B each a radius of the sphere, and the side OK common to both (I.13.). Therefore, in this case the section is a circle having the centre K. Therefore, &c. Cor. 1. The radius of a great circle is the same with the radius of the sphere; and the radius-square of a small circle is less than the radius-square of the Esphere by the square of the perpendicular, which is drawn to its plane from the centre of the sphere (I. 36. Cor. 1.). Cor. 2. Every diameter of a great circle is likewise a diameter of the sphere. 13. Opposite points on the surface of the sphere are those which are opposite extremities of a diameter of the sphere. It is evident that the arcs which join two such points with any third point on the sphere's surface, are parts of the same great circle, and are together equal to a semicircumference (see the second figure of def. 9.) PROP. 1. Every plane section of a sphere is a circle; the centre of which is either the centre of the sphere, or the foot of the perpendicular which is drawn to the plane from the centre of the sphere. The substance of this proposition has been already given in the corollary to Book IV. Prop. 8; and the following demonstration is only a statement at greater length of the reasoning from which it was there inferred. If the plane pass through the centre O of the sphere, as PA P', the distance OA of any point A in the circumference of the section, from the point O, will be the same with a radius of the sphere, and therefore the section will be a circle having the centre O. And if the plane do not pass through the centre, the distances AK, BK, of any two points A, B, in the circumference of the section, from K the foot of the perpendicular O K, will be equal to PROP. 2. Either pole of a circle of the sphere is equally distant from all points in the circumference of that circle; whether the direct or the spherical distance be understood. Let A B C (see the figure of prop. 1.) be any circle of a sphere which has the centre O, and let OK be drawn perpendicular to the plane ABC, and produced to meet the surface of the sphere in P; then, if A, B be any two points in the circumference of the circle A B C, and if the straight lines PA, PB, as also the spherical arcs PA, PB be drawn, the line P A shall be equal to the line PB, and the arc PA to the arc PB. Join KA, KB. Then, because K is the centre of the circle A B C (1.), the right-angled triangles P K A and PK B have the two sides P K, KA of the one equal to the two sides PK, KB of the other, each to each; therefore, (I. 4.) the hypotenuse PA is equal to the hypotenuse PB. And because, in equal circles, the arcs which are subtended by equal chords are equal to one another (III. 12. Cor. 1.), the arc PA is likewise equal to the arc PB. And in like manner it may be shown that the other pole P' is also equidistant from A and B. In this demonstration it is supposed that the point K does not coincide with the point O, or that the circle in question is not a great circle. If, however, ABC is a great circle, the angles POA, POB are right angles, and therefore equal to one another (I. 1.), from which the equality of the chords PA, PB and of the arcs PA, PB will follow as before. Therefore, &c. Cor. 1. Hence any circle of a sphere may be conceived to be described from either of its poles as a centre with the spherical distance of that pole as a radius. For, if this distance be carried round the pole, its extremity will lie in the circumference of the circle. Cor. 2. The distances of any circle from its two poles are together equal to a semicircumference. Cor. 3. A great circle is equally distant from its two poles; but this is not the case with a small circle. For if A B C be supposed to be a great circle, the angles POA, P' OA will be right angles, and therefore equal to one another, so that the polar distances PA, P'A will be likewise equal (III. 12.); but if ABC be a small circle, the angles POA, POA will be, one of them less, and the other greater than a right angle, and therefore the distances PA, P'A will be unequal. PROP. 3. Equal circles of the sphere have equal polar distances; and conversely. Let A B C and A'B'C' (see the figure of prop. 1.) be any two equal circles of the sphere; K, K' their centres, and P, P' their poles; then, if the radius KA is equal to the radius K'A', the polar distance PA shall be equal to the polar distance P'A'; and conversely. For, if O be the centre of the sphere, and OK, KP be joined, OK will be perpendicular to the plane ABC (1.), and therefore (def. 3.) O K, K P will lie in the same straight line; and in like manner OK will be perpendicular to the plane A'B'C', and OK', K' P' will lie in the same straight line. Join OA, PA and OA', P'A'. Then, because the right-angled triangles O KA, OK'A' have the hypotenuse OA equal to the hypotenuse O A', and the side KA equal to the side K' A', the angle KO A or POA is equal to the angle K'O A' or P' O A' (I. 13.); and therefore, also, the arc PA (III. 12.) is equal to the arc P'A'. And, conversely, if the arc PA be equal to the arc P'A', the angle POA will be equal to the angle POA (III. 12.); and, therefore, because in the right-angled triangles OK A, O K'A', the hypotenuse OA is equal to the hypotenuse O A', and the angle KOA to the angle K'O A', the radius KA is equal to the radius K' A' (I. 13.). In the foregoing demonstration it is supposed that the points K and K' do not coincide with the point O, that is, that the circles in question are not great circles of the sphere. If, however, the circles are great circles, the angles POA, POA are right angles, and therefore the arcs PA, P'A' quadrants and it is evident that, conversely, circles whose polar distances are quadrants pass through the centre of the sphere, that is, are great circles of the sphere, and are equal to one another. Therefore, &c. Cor. Circles whose polar distances are together equal to a semicircumference are equal to one another (2. Cor. 2.) PROP. 4. bisect one another. Any two great circles of the sphere For, since the plane of each passes through the centre of the sphere, which is also the centre of each of the great circles, their common section is a diameter of each; and circles are bisected by their diameters. Therefore, &c. Cor. 1. Any two spherical arcs may points, which are opposite extremities of be produced to meet one another in two a diameter of the sphere. which pass through the same point may Cor. 2. Any number of spherical arcs be produced to pass likewise through the opposite point. PROP. 5. The spherical arc which is drawn from the pole of a great circle to any point in its circumference is a quadrant of a great circle, and is at right angles to the circumference. Let the point P be the pole of a great circle ABC: let any point A be taken in the circumference ABC, and let A PA be joined by the spherical arc PDA: the arc PDA is a quadrant, and at right angles to the circumference ABC. Take O the centre of the sphere, and join O P, O A. Then, because (def. 3.) OP is at right angles to the plane AB C, the angle POA is a right angle (IV. def. 1.); and, therefore, the arc PDA is a quadrant. Again, because OP is at right angles to the plane ABC, 188 the plane OPDA is at right angles to the plane ABC (IV. 18.); and, therefore, the arc PDA is at right angles to the circumference ABC (def. 7. and def. 8.). Therefore, &c. Cor. 1. If two great circles cut one another at right angles, the circumference of each shall pass through the poles of the other. Cor. 2. If the spherical distances of a point P in the surface of the sphere from two other points A and C in the same surface which are not opposite extremities of a diameter be each of them equal to a quadrant, P shall be the pole of the great circle which passes through the points A and C. For, if O be the centre of the sphere, the angles POA and POC will be right angles, because the ares PA and PC are quadrants; and, therefore, P O is at right angles to the plane OA C (IV. 3.); for which reason PO must be the axis, and P the pole of the great circle which passes through A and C (def. 3.). Every spherical angle is measured by the spherical arc which is decribed from the angular point as a pole, and intercepted between the sides of the angle. Let BAC be any spherical angle, and from the point A, as a pole, let a great circle be described cutting the sides A B, AC in the points M, N respectively: the spherical angle BAC shall be measured by the arc M N. A M N Take O the centre of the sphere, and join OA, OM, ON. Then, because A is the pole of the spherical arc M N, the plane MON is perpendicular to OA (def. 3.), and M O, NO are each of them perpendicular to O A. Therefore the angle MON measures the dihedral angle MOAN (IV. 17.), or which is the same thing, (def. 7.) the spherical angle MAN or BAC. Therefore, the arc MN which measures the angle M O N, measures also the spherical angle B A C. Therefore, &c. Cor. The angle contained by two spherical arcs is measured by the distance of their poles, which lie towards the same parts of the arcs. For, if the are N M be produced to R, so that RN may be a quadrant, and to Q, so that QM may be a quadrant, QR will be equal to MN (I. ax. 3.). And the points Q, R are the poles of A M, AN respectively, because Q M, QA, as also RN, RA, are quadrants (5. Cor. 2.). PROP. 7. If one triangle be the polar triangle of another, the latter shall likewise be the polar triangle of the first; and the sides of either triangle shall be the supplements* of the arcs which measure the opposite angles of the other. Let A B C be any spherical triangle, and let A', B', C' be those poles of the sides B C, AC, A B, which lie towards the same parts of the arcs B C, A C, AB, with the opposite angles A, B, C, respectively, so that A' and A lie towards the same parts of B C, B' and B towards the same parts of A C, and C and C towards the same parts of AB: that is, (def. 11.) let A'B'C' be the polar triangle of ABC: the triangle A B C shall, likewise, be the polar triangle of A' B'C', and the sides of either triangle shall be the supplements of the arcs which measure the opposite angles of the other. For, in the first place, B' being the pole of A C, A B' is a quadrant (5.); and C' being the pole of AB, A C is likewise a quadrant: therefore (5. Cor. 2.) A is the pole of B'C'. Also, B it is upon the same B E D с side of B'C' that A' is: for, because A' and A are upon the same side of BC, and that A' is the pole of B C, A' A is less than a quadrant; and because A is the pole of B' C', and that A A' is less than a quadrant, A and A' are upon the same side of B'C'. And, in the same manner, it may be shown that B is the pole of A'C', and B, B' upon the same side of A' C'; and that C is the pole of A' B', and C, C' upon the same side of A' B'. Therefore, the triangle A B C is the polar triangle of A' B' Č' (def. 11.). Next, let the arc B'C' be produced both ways, if necessary, to meet the arcs AB, AC (produced likewise if necessary) in the points D, E, (4 Cor. 1.). Then, because A is the pole of the arc B' C', the spherical angle BAC is measured by DE (6.). Again, because B' is the pole of A C, B' E is a quadrant; and for the From this property polar triangles are some. times called supplementary triangles, SECTION 2.-Of Spherical Triangles.' PROP. 8. The angles which one spherical arc makes with another upon one side of it But these angles are respectively measured by the arcs A B, A C, and B C. Therefore AB and AC are together greater than BC. And hence, taking AC from each, AB alone is greater than the difference of A C and B ̊C. Therefore, &c. Cor. 1. The three sides of a spherical triangle are together less than the circumference of a great circle. For, if A B and A C be produced to meet in D, the arcs A B D, ACD will be semicircumferences; but B C is less than B D and DC together; therefore, A B, A C, and BC are together less than ABD and ACD, that is, less than the circumference of a great circle. Cor. 2. In the same manner it may be shown that all the sides of any spherical polygon are together less than are either two right angles, or are to the circumference of a great circle. gether equal to two right angles. See the Demonstration of Book I. Prop. 2. Cor. 1. If two spherical arcs cut one another, the vertical or opposite angles will be equal to one another. See the Demonstration of Book I. Prop. 3. Cor. 2. If any number of spherical arcs meet in the same point, the sum of all the angles about that point will be equal to four right angles. PROP. 9. Any two sides of a spherical triangle are together greater than the third side; and any side of a spherical triangle is greater than the difference of the other two. A Let ABC be a spherical triangle; the sides BA and AC shall be together greater than B C; and A B alone shall be greater than the difference of AC and B C. Take O the centre of the sphere, and join O A, O B, OC. Then, because the solid angle at O is contained by three plane angles 9 AO B, AOC, and B O C, the two AO B and A O C are together greater than the third BOC (IV. 19.). D B This is likewise evident from IV. 20. Scholium. By help of this proposition, it may be shown that the shortest distance of two points on the surface of a sphere, measured over that surface, is the spherical 10. Scholium. are between them. See Book I. prop. PROP. 10. The three angles of a spherical triangle are together greater than two right angles, and less than six right angles. For the arcs which measure the three angles together with the three sides of the polar triangle are equal to three semicircumferences (7.), or six quadrants: therefore, the former alone are less than six quadrants, and consequently the angles which they measure are less than six right angles. Again, the sides of the polar triangle are less than a whole circumference, or four quadrants (9 Cor. 1.): therefore, the arcs before mentioned are greater than two quadrants, and consequently the angles which they measure greater than two right angles. Therefore, &c. Cor. 1. A spherical triangle may have two or even three right angles, or two or even three obtuse angles. For, it is evident from the demonstration of the proposition, that the sum of the angles depends upon the magnitude of the sides of the polar triangle, and since the sum of these last may be any whatever less than four quadrants, the sum of the angles of the original triangle may be any whatever greater than two, and less than six right angles. Cor. 2. If one side of a spherical triangle be produced, the exterior angle will be less than the sum of the two interior and opposite angles. For the exterior angle, together with its adjacent interior angle, is only equal to two right angles (8.); but the two interior and opposite angles, together with the same angle, are greater than two right angles. PROP. 11. B B C D T S If two sides of a spherical triangle be equal to one another, the opposite angles shall be likewise equal; and conversely. Let A B C be a spherical triangle, having the side AB equal to the side AC; the angle ACB shall likewise be equal to the angle A B C. Take O the centre of the sphere, and join O A, From the OB, O C. point C, in the plane A O C, draw CS at right angles to CO (and, therefore (III. 2.), touching the arc CA in C) to meet O A produced in S: at the points B and C draw BT and CT, touching the arc BC, and meeting one another in T, and join BS, ST. Then, because the arc AB is equal to AC, the angle AOB is equal to the angle AOC (III. 12.); and, because the triangles SO B, SOC have two sides of the one equal to two sides of the other, each to each, and the angles SO B, SOC which are included by those sides equal to one another (I. 4.), the base S B is equal to the base S C, and the angle S B O'to the angle S CO, that is, to a right angle. Therefore, BS touches the arc A B in B (III. 2.). And, because the spherical angles A B C, ACB are measured by the plane angles of the tangents at B and C (see def. 7. note) they are measured by the angles S BT, SCT respectively. But, because TB and TC are tangents drawn from the same point B to the arc BC, TB is equal to TC (III. 2. Cor. 3.). Therefore, the triangles SBT and SCT have the three sides of the one equal to the three sides of the other, each to each, and consequently the angle SBT is equal to the angle SCT (I. 7.). Therefore, also, the spherical angle ABC is equal to the spherical angle A C B. Next, let the angle ABC be equal to the angle AC B: the side A B shall be equal to the side AC. For, if the polar triangle A' B'C' be described, its sides A B and A' C' which are supplements to the measures of the equal angles (7.) will be equal; and, therefore, by the former part of the proposition, the spherical angle at C' is equal to the spherical angle at B'. But the sides AB and AC are supplements to the measures of these angles (7.). Therefore, also, AB is equal to A C. Therefore, &c. exterior angle A CD shall be equal to, or less than, or greater than, the interior and opposite angle A B C, according as the sum of the two sides A B, AC is equal to, or greater than, or less than, the semicircumference of a great circle. For, if B A and B C be produced to meet one another in D, the angles at B and D will be equal to one another, having for their common measure the measure of the same dihedral angle (def. 7.); and BAD will be a semicircumference. But, by the proposition, the angle A C D is equal to, or less than, or greater than the angle at D, according as A C is equal to, or greater than, or less than AD. Therefore, the angle ACD is equal to, or greater than, or less than the angle at B, according as A B and AC are together equal to, or greater than, or less than a semicircumference. |