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' O A 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 A B C (1.), and therefore (def. 3.) OK, 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, O K' A' have the hypotenuse OA equal to the hypotenuse O A', and the side K A 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 P' O A' (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. Any two great circles of the sphere bisect one another. 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 be produced to meet one another in two points, which are opposite extremities of a diameter of the sphere. Cor. 2. Any number of spherical arcs which pass through the same point may be produced to pass likewise through the opposite point. PA be joined by the spherical arc PDA: the arc PDA is a quadrant, and at right angles to the circumference A B C. 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 A B 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, the plane OPDA is at right angles to the plane A B C (IV. 18.); and, therefore, the arc PDA is at right angles to the circumference A B C (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 arcs PA and PC are quadrants; and, therefore, PO is at right angles to the plane O AC (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 C M 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. of another, the latter shall likewise be If one triangle be the polar triangle the sides of either triangle shall be the polar triangle of the first; and the supplements* of the arcs which measure the opposite angles of the other. and let A', B', C' be those poles of the Let A B C be any spherical triangle, sides B C, AC, A B, which lie towards the same parts of the arcs B C, AC, A B, 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. A 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 E D B C N 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, AA 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'. 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 MO, NO are each of them perpendicular to O A. Therefore the angle M ON 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 arc N M be produced to R, so that RN may be a quadrant, and to Q, so that 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' C' (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 times called supplementary triangles, *From this property polar triangles are some 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 B C. And hence, taking AC from each, A B alone is greater than the difference of A C and BC. 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 AB 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. be shown that all the sides of any Cor. 2. In the same manner it may 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 A B C 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, O C. Then, because the solid angle at O is contained by three plane angles AOB, AOC, and B O C, the two A O B and A O C are together greater than the third B OC (IV. 19.). 0 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 arc between them. See Book I. prop. 10. Scholium. 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 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, ОВ, О С. From the 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 A O C (III. 12.); and, because the triangles S O 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 B C, 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 S BT is equal to the angle S CT(I.7.). Therefore, also, the spherical angle ABC is equal to the spherical angle ACB. Next, let the angle ABC be equal to the angle A CB: the side A B shall be equal to the side A C. 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. PROP. 12. be greater than another, the opposite If one angle of a spherical triangle side shall likewise be greater than the side opposite to that other; and conversely. See the demonstration of Book I. Prop. 11. Cor. If one side B C of a spherical triangle A B C be produced to D, the R exterior angle A CD shall be equal to, or less than, or greater than, the interior and opposite angle ABC, according as the sum of the two sides AB, 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 CD is equal to, or less than, or greater than the angle at D, according as AC 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. PROP. 13. If two spherical triangles have two sides of the one equal to two sides of the other, each to each, and likewise the included angles equal; their other angles shall be equal, each to each, viz. those to which the equal sides are opposite, and the base, or third side, of the one shall be equal to the base, or third side, of the other. There are here two cases for consi deration; first, that in which the equal sides A B, A C and DE, DF lie in the same direction; and, secondly, that in which they lie in opposite directions. The first case may be demonstrated by superposition, after the same manner as Book I. Prop. 4, to which, for brevity's sake, the reader is referred: the second case as follows: triangles, which have the two sides AB, AC equal to the two sides DE, DF, each to each, viz. AB to D E and AC to DF, but DE lying in a direction from DF, which is the reverse of that in which AB lies from AC, and let them likewise have the angle B A C equal to the angle EDF: their other angles shall be equal, each to each, viz, A B C to DEF; and ACB to DFE, and the base B C shall be equal to the base E F. Take O the centre of the sphere: from A draw A P perpendicular to the plane OB C, and produce it to meet the surface of the sphere in A'. Join PO, PB, PC, and OB; from P draw PQ perpendicular to OB; join A Q, A B, AC, A' Q,*. ,* A' B, A' C ; and draw the spherical arcs A' B, A/C, A'A. Then, because in the right-angled triangles APO, A'PO, the hypotenuse A O is equal to the hypotenuse A'O, and the side PO common to both, the remaining sides AP and A'P are equal to one another (I. 13.): and because in the right-angled triangles A PQ, A' PQ, the side AP is equal to the side A'P, and the side P Q common to both, the hypotenuse A Q is equal to the hypotenuse A' Q, and the *A'Q is omitted in the figure. angle A Q P to the angle A' Q P (I. 4.); and in the same manner it may be shown that A B is equal to A'B, and AC to A'C. Now, because A P is perpendicular to the plane O B C, and that PQ is perpendicular to the line OB in that plane, AQ is likewise perpendicular to OB (IV. 4.); and for the like reason A'Q is perpendicular to the same O B. Therefore, the angles AQ P, A'QP measure the dihedral angles formed by the planes OA B, OB C, and OA' B, OBC (IV. 17.), or, which is the same thing, (def. 7.) the spherical angles A B C and A'BC; and because, as has been already demonstrated, the angle AQP is equal to the angle A'QP, the spherical angle ABC is equal to the spherical angle A'B C. In the same manner, it may be shown that the spherical angles ACB and A'CB are equal to one another. And because the straight line AB is equal to the straight line A' B, the arc AB is equal to the arc A'B (III. 12. Cor. 1.); and, for the like reason, the arc AC is equal to the arc A'C. Therefore, in the isosceles spherical triangle BA A', the angle B A'A is equal to the angle B A A', and in the isosceles spherical triangle C A A' the angle CA'A is equal to the angle CA A (11.); and, consequently, the whole (or, if the points B, C, are on the same side of the arc AA', the remaining) angle BAC is equal to the whole or remaining angle BA C. Therefore, the triangles A' BC, ABC have their several sides and angles equal to one another, but lying in a reverse order. Now, because A' B and D E are each of them equal to A B, they are equal to one another; and, for the like reason, A'C is equal to D F, and the angle BAC to the angle EDF. Also, the equal parts lie in the same direction from one another in these two triangles, case, the base B C is equal to the base A'BC, DEF. Therefore, by the first EF, and the angles A'B C, A'CB to the angles DEF and DFE respectively. And, because the angles ABC, A CB are equal to the angles A'B C, A'CB, each to each, the former angles are likewise equal to D E F and DFE respectively. Therefore, &c. PROP. 14. If two spherical triangles have two angles of the one equal to two angles of the other, each to each, and likewise the interjacent sides equal; their other sides |