D G E duct in question cannot be greater than with the segment, and the other part the sector; for then would it be greater (17. Cor. 4.) a sphere of which AG is than some circumscribed solid, and the diameter. therefore (12.) the base of the sector Therefore, &c. greater than the convex surface of such PROP. 23. a solid, which (19.) is impossible: neither can it be less, for then would it be Every double-based spherical segless also than some inscribed solid; which ment is equal to the half of a cylinder is impossible, because, not only is the having the same altitude with the segbase of the sector greater (19.) than the ment and a base equal to the sum of convex surface of such a solid, but the its two bases, together with a sphere of radius CH is likewise greater than the which that altitude is the diameter. perpendicular CE. Therefore it must Let FH KG be any portion of the sebe equal to the sector. micircle ADB, by the revolution of Therefore, &c. which about the diameter A B, a doublePROP. 22. based segment, having the altitude GK, is generated: the segment shall be Every spherical segment, upon a equal to the half of a cylinder having single base, is equal to the half of a cy- the same altitude and a base equal to linder having the same base and the the sum of its two bases, together with sume altitude, together with a sphere, of a sphere of which G K is the diameter. which that altitude is the diameter. For, in the first place, because the Let ADFG be any circular half- solid, generated by the revolution of the segment, by the revolú. circular segment FH, A DA tion of which about the is equal to the excess diameter AB, a spheri of the difference of the cal segment having the sectors generated by H K altitude AG is gene CAH and CAF rated : the spherical seg C above the solid gement shall be equal to nerated by the trithe half of a cylinder angle CFH, it may having the same base and be shown by the B 'B altitude, together with a same steps as in the sphere, of which A G is the diameter. last proposition, that the solid generated Join AF; and from the centre C by the revolution of FH is equal to draw CE perpendicular to AF. Then, ** GK ~ FH2, i. e. if F L be drawn because the solid, generated by the seg- parallel to GK, to XGK (FL? + ment ADF, is equal to the difference L H2). But (I. 22.) F L is equal to G K, of the solids generated by the sector and LH is equal to the difference of FG CADF and the triangle CAF, the and H K (I. 22.): therefore (I. 33.), former of which (21.) is equal to one- LHis equal to FG?+H K? – 2 FGX third of the product of CA by its base HK, and į axGK (FL’+LH”), or the AG * 2 * X CA (20.), and the latter solid generated by the segment FH, is (12.) to one-third of the product of CE equal to į XGK (FG2+H K2-2 FG by the convex surface generated by YHK) + 5*XG K3. To this add the A F viz. (12.) A G * 2 * CE; the truncated cone generated by the trasolid, generated by the segment ADF, pezoid FG KH; which (11.) is equal is equal to jr-X AG (ČA– CE), to jrXGK (FG2 + H K2 + F Ĝ X i. e. I + XAG X A Es, or to * XAG HK): therefore, the double-based segX AF, because (I. 36. Cor. 1.) C Al ment in question is equal to į r X CE2 is equal to A E2, and A E GK (3 FG? + 3 H K2) + à r + GK3; (III. 3.) to a fourth of A F2. To this or to X GK (F G2 + H K2) + ir add the cone generated by the triangle XG K3; the first part of which is the A FG, which (9.) is equal to ir X AG half of a cylinder (4. Cor. 1.) having X FG?: therefore, the spherical seg- the altitude GK, and a base š x F G2 ment in question is equal to £ T XAG + = x H K?, equal to the sum of the (A F3 + 2 FG%), i. e. to į X AG (3 FG% bases of the segment, and the other + AG2) because AF is equal to AG2 + part a sphere of which G K is the diaFGo (III.36.). And it X AG(3 FG% meter (17. Cor. 4.). + AG) is equal to įr X AG * F G2 + Therefore, &c. ** X AG; the first part of which is Cor. It appears from the demonstra(4. Cor. 1.) the half of a cylinder having tions of this and the preceding proposithe altitude AG and the same base tion, that the solid generated by the re volution of any circular segment about two bases and a mean proportional a diameter of the circle, is equal to between them (IV.33.). Therefore, the ** X GK FH2; G K being that por- spherical orb is equal to the sum of tion of the diameter, which is intercepted three pyramids, having their common between two perpendiculars drawn to it altitude equal to the thickness of the from the extremities of the segment, orb, and for their bases its exterior and and FH the chord which is the base of interior surfaces, and a mean proporthe segment. tional between them. PROP. 24. Therefore, &c. Every spherical orb is equal to the PROP. 25. sum of three pyramids having their common altitude equal to the thickness Every spherical ungula is to the of the orb, and for their bases its ex whole sphere, as the angle between its terior and interior surfaces, and a mean planes to four right angles; and its proportional between them. lune, or convex surface, is to the surface For, a spherical orb is the differ- of the whole sphere in the same ratio. ence between two concentric spheres. Let ADBE be an ungula of a sphere Now, if a pyramid be described having having the centre C and the diameter its base equal to the exterior surface, А. or surface of the larger sphere, and B its altitude equal to the radius of that surface, this pyramid will be equal AB; and from Clet CD, CE be to the whole sphere (IV. 32. Cor. 1. drawn in the planes A DB, A E B, perand 17.). And if, from this, there be pendicular to A B (I. 44.): the ungula cut off, by a plane parallel to the ADB E shall be to the whole sphere, as base, a pyramid, having its altitude the angle D CE to four right angles. equal to the radius of the interior sur- For, since the plane DCE is perface, the two pyramids will be to one pendicular to A B (IV.3.), the angle, another as the cubes of any two homolo- which measures the inclination of any gous edges (IV. 34. Cor.); or, since it two planes passing through A B, may may be shown that their altitudes are be drawn in that plane at the point to one another in the same ratio with C (IV. 17. Schol.); and, if any two the homologous edges, as the cubes of of these angles at C be equal to one their altitudes, (IV. 27. Cor. 3.), that is, another, the dihedral angles which they as the cubes of the radii of the spheres, or measure will be equal (IV.17.), and there. (18.) as the spheres. Therefore, because fore the ungulas, which have those dihethe larger pyramid is equal to the larger dral angles, may be made to coincide, and sphere, the smaller pyramid is equal to are equal to one another. Now, let the the smaller sphere (II. 18.); and the dif- angle D C E be divided into any number ference of the two pyramids is equal to of equal angles D C F, FCG, &c.; and the difference of the two spheres, that is, therefore the dihedral angle DAB E into the frustum is equal to the spherical the same number of dihedral angles by the orb. And, because the larger base of planes A CF, ACG, &c. (IV.17.); and the frustum is equal to the surface of the the ungula A DB E into as many equal larger sphere, it may be shown that its ungulas A DBF, AD BG, &c. by the smaller base is equal to the surface of same planes. Then, if the angle DCF the smaller sphere, exactly in the same be contained in the four right angles manner as it has been already shown, about C any number of times exactly, that the content of the smaller pyramid or with a remainder, the ungula ADBF is equal to the content of the smaller will be contained in the whole sphere the sphere; also the altitude of the frustum same number of times exactly, or with is equal to the thickness of the orb. a remainder. Therefore the ungula But the frustum is equal to the sum ADBE is to the whole sphere as the of three pyramids, having the same angle DC E to four right angles (II. altitude with it, and for their bases its def. 7.). And a similar proof may be applied the two extremities of a diameter, any to show that the lúne ADB E is to the number of great circles may be made to surface of the whole sphere in the same pass, for they are in the same straight ratio, viz. that of the angle D C E to four line with the centre of the sphere (IV. right angles. 1. Cor. 4.). Therefore, &c. 2. If a sphere is cut by a plane which Cor. 1. Every spherical ungula is does not pass through the centre, the equal to one-third of the product of the section is called a small circle of the radius by its lune or convex surface. sphere; the radius of such a section Cor. 2. In the same or in equal being less than that of the sphere. spheres, any two ungulas are to one A circle, it is plain, may be made to another as the angles between their pass through any three points in the planes. And the same may be said of sphere's surface; and it will be a great any two lunes. or a small circle, according as its plane Scholium. passes through the centre of the sphere, or otherwise. We might here add the proportions of 3. The axis of any circle of the sphere similar segments, sectors, orbs, ungulas, is that diameter of the sphere which is and of their convex surfaces. The perpendicular to the plane of the circle; reader will, however, easily perceive, and the extremities of the axis are called from the demonstration of prop. 15, that the poles of the circle.. if similar spherical segments and sectors be defined to be such as are generated such as have their planes parallel. 4. Parallel circles of a sphere are by similar circular segments and sectors, their surfaces will be as the squares of the same axis and poles; for a straight It is evident that parallel circles have the radii, and their contents as the cubes line which is perpendicular to one of two of the radii. And the same may be said parallel planes is perpendicular to the of similar spherical orbs, defined to be other likewise (IV. 11.). It may also be such that the radii of their exterior and observed that two parallel circles caninterior surfaces are to one another in not both of them pass through the centre the same ratio ; and of similar ungulas defined to be such as have their dihedral be great circles of the sphere. of the sphere, that is, they cannot both angles equal to one another. These four definitions may be illustrated by referring to the figure of prop. 1. in which P A P is a great circle, BOOK VI. ABC a small circle, PO P the axis, § 1. Of great and small circles of the and A'B'C', A B C are parallel circles and P, P the poles of the circle A B C, Sphere.-2. Of Spherical Prian of the sphere. gles.-13. Of equal Portions of Spherical Surface, and the Measure of 5. Any portion of the circumference solid Angles.-94. Problems. of a great circle is called a spherical § 1.-Of great and small Circles of the Two points are said to be joined on Sphere. the surface of the sphere when the spheDef. 1. If a sphere is cut by a plane rical arc between them is described; and which passes through the centre, the this arc is called the spherical distance section is called a great circle of the of the two points, in order to distinguish sphere; the radius of such a section it from their direct distance, which is the being the greatest possible, the same, straight line which joins them. The namely, with the radius of the sphere. spherical distance of opposite extremities From this definition it is evident that of a diameter of the sphere is evidently a great circle may be made to pass half the circumference of a great circle: through any two points in the surface of but the spherical distance of any other a sphere; and that, if the two points be two points is less than a semicircumnot opposite extremities of a diameter, ference, being always the lesser of the only one great circle can be made to two arcs into which they divide the pass through them, for its plane must great circle which passes through them. pass through the centre of the sphere, 6. The polar distances of any circle and only one plane can be made to pass of the sphere are the spherical arcs through three points which are not in the which join any point in the circumsame straight line (IV. 1.) But through ference with the two poles of the circle. arc. B arcs. a By the polar distance (singly) the lesser neal acute angle, and a spherical obtuse of these two arcs, or distance from angle by a rectilineal obtuse angle. the nearer pole, is generally to be under- 9. A spherical triangle is a portion of stood. the sphere's surface included by three 7. If the arcs AB, arcs of different great circles, as A B C. AC of two great cirB Every spherical tri A cles meet one another angle, ABC, has three in a point A, they sides, viz. the containing are said to form at arcs AB, AC, and BC, that point a spherical and three angles A, B, angle BAC and C. C A spherical angle In the spherical triangles here conis greater or less, sidered, it is supposed that each of according to the opening between its the sides is less than a semicircumcontaining arcs: thus the angle B A C is ference. For, the greatest spherical greater than the angle D AC by the an- distance at which two points can be gle BAD. placed is a semicircumference; and if Every spherical angle is measured by any arc, as PAP', P the plane angle which measures the in- be taken equal to clination of the planes of the containing a semicircumference, For it is easy to perceive, that if its extremities P, P this inclination is the same in any two will be extremities of А) spherical angles, they may be made to a diameter POP of coincide, and therefore are equal to one the sphere, and thereanother. If, therefore, the dihedral an fore the same great gle made by the planes of one spherical circle will pass angle contain any sub-multiple of the through both of them, dihedral angle made by the planes of and any third point Q on the sphere's another a certain number of times ex- surface, so that the arcs Q P and QP actly, or with a remainder, the first will be arcs, not of different circles, but spherical angle will contain a like sub- of the same circle. multiple of the other the same number Any three points on the sphere's surof times exactly or with a remainder; face may be assumed for the angles of and, therefore, the spherical angles are a spherical triangle (see def. 5.), proto one another (II. def. 7.) as the dihe- vided they are not in the same great dral angles made by their planes, and circle, nor any two of them opposite to have the same measures with them.* one another, that is, opposite extremities 8. When one spheri of a diameter of the sphere. cal arc standing upon 10. Two spherical triangles are said to another makes the ad be symmetrical, when the sides of the jacent spherical angles equal to one another, each of them is called a spherical right angle, and the arc which stands upon the other is said to be perpendiculur, or at one are equal to the sides of the other, right angles to it. each to each, but in a reverse order, as The terms acute and obtuse are likewise ABC and DEF. applied to spherical angles, in the same 11. If ABC is any sense as in Book I. def. 11. spherical triangle, It is evident that a spherical right an- and the points A', B', A gle is measured by a rectilineal right an- C' are those poles of gle, a spherical acute angle by a rectili- the arcs B C, AC, AB, respectively, which lie upon the * Hence a spherical angle has been defined by B some writers to be identical with the dihedral angle same sides of them B of its planes ; while others have extended to it the with the opposite angeneral definition of the angle in which two curves cut one another, considering it the same with the gles A, B, C, and the triangle A'B'C' plane rectilineal angle of the tangents at the point is completed: this triangle A' B' C' A ; for the latter angle, being contained by perpen is said to be the polar triungle of the diculars to the common section (A, measures the dihedral angle of the planes. triangle ABC. a D B с E A A. с D a There are no fewer than eight differ- one another; because the right-angled ent triangles which have for their angu- triangles OKA, OK B have their hypolar points poles of the sides of a given tenuses O A, O B each a radius of the triangle ABC; but there is only one sphere, and the side OK common to triangle in which these poles A', B', C', both (I.13.). Therefore, in this case the lie towards the same parts with the op- section is a circle having the centre K. posite angles A, B, C, and this is the Therefore, &c. triangle A'B'C', which is known under Cor. 1. The radius of a great circle is the name of the polar triangle. the same with the radius of the sphere; 12. A spherical polygon and the radius-square of a small circle is? any portion of the is less than the radius-square of the sphere's surface included B Esphere by the square of the perpendicuby more than three arcs of lar, which is drawn to its plane from different great circles, as the centre of the sphere (I. 36. Cor. 1.). ABCDE. Cor. 2. Every diameter of a great 13. Opposite points on the surface of circle is likewise a diameter of the the sphere are those which are opposite sphere. extremities of a diameter of the sphere. It is evident that the arcs which join two PROP. 2. such points with any third point on the Either pole of a circle of the sphere is sphere's surface, are parts of the same equally distant from all points in the great circle, and are together equal to a circumference of that circle; whether semicircumference (see the second figure the direct or the spherical distance be of def. 9.) understood. Let ABC (see the figure of prop. 1.) PROP. 1. be any circle of a sphere which has the centre o, and let OK be drawn perpenEvery plane section of a sphere is a dicular to the plane ABC, and procircle; the centre of which is either the duced to meet the surface of the sphere centre of the sphere, or the foot of the in P; then, if A, B be any two points perpendicular which is drawn to the in the circumference of the circle ABC, plane from the centre of the sphere. and if the straight lines PA, PB, as The substance of this proposition has also the spherical arcs PA, PB be been already "given in the corollary to drawn, the line P A shall be equal to the Book IV. Prop. 8; and the following line PB, and the arc P A to the arc PB. demonstration is only a statement at Join KA, KB. Then, because K is greater length of the reasoning from the centre of the circle A B C (1.), the which it was there inferred. right-angled triangles PK A and PKB have the two sides P K, K A of the one If the plane pass through the cen. equal to the two sides PK, K B of the tre 0 of the sphere, as PAP', the other, each to each; therefore, (I. 4.) P 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 P A is likewise equal to the arc PB. And in like man ner it may be shown that the other pole K B 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 quesdistance O A of any point A in the tion is not a great circle. If, however, circunference of the section, from the ABC is a great circle, the angles POA, point 0, will be the same with a radius PO B are right angles, and therefore of the sphere, and therefore the section equal to one another (I. 1.), from which will be a circle having the centre O. the equality of the chords PA, PB and And if the plane do not pass through of the arcs PA, PB will follow as the centre, the distances AK, BK, of before. any two points A, B, in the circumfer- Therefore, &c. ence of the section, from K the foot of Cor. 1. Hence any circle of a sphere the perpendicular 0 K, will be equal to may be conceived to be described from B A/ A |