beginning of the series, are not < 1, provided they ultimately become so. For the value of the latter part of the continued fraction is incommensurable, as has been proved; and if we perform the operations indicated, S will also be incom mensurable. Now it has been proved (Prob. 99) that, m and n being any whole numbers, (2r-1) n is <1. If therefore be commensurable, its tangent will be incommensurable; but it = 1, and therefore π, and consequently π, is incommensurable. A TREATISE ON SPHERICAL TRIGONOMETRY. GENERAL PROPERTIES OF CIRCLES OF THE SPHERE. 1. THE boundary of every plane section of a sphere is a circle. If the cutting plane pass through the center, this is evident; and in this case the section is called a great circle, and is determined when any two points on the surface of the sphere through which it passes are given. All great circles are equal to one another, since they have the same radius, namely that of the sphere; and they all bisect one another, since their planes intersect in diameters of the sphere. Hence the distance of the points of intersection of two great circles measured on the sphere is a semi-circumference. If the cutting plane does not pass through the center of the sphere, from 0 (fig. 1) the center of the sphere drop upon it the perpendicular OC, and join C with any point A in the boundary of the section; then AC NAO-OC, which is invariable; therefore the boundary of the section is a circle whose center is C; and it is called a small circle. Arcs of small circles are very rarely used; and when, hereafter, an arc of a circle is mentioned, an arc of a great circle, unless the contrary be specified, is invariably intended; and in most cases it is employed to denote the angle which it subtends at the center of the sphere, no regard being had to the radius of the sphere. 2. If OC be produced both ways to meet the surface of the sphere in Pand P', then the line PA=√PC+AC, which is invariable. 173 Also if PAM be an arc of a great circle passing through P and A, since in equal circles equal straight lines cut off equal arcs, the length of the arc PA is invariable. Therefore the distance of P is the same from every point in the perimeter of the circle AB, whether measured along the straight line, or the arc of a great circle, drawn from it to the point. The point P, and the point P' which has evidently the same property, are called the nearer and more remote poles of the circle AB; being the extremities of that diameter of the sphere which is perpendicular to the plane of the circle. They are also the poles of all circles of the sphere whose planes are parallel to ACB. If MN be the great circle of which P is the pole, since OP is perpendicular to the plane MON, and the angle POM is consequently a right angle, the distance of P from every point in the boundary of MN, measured on a great circle, is a quadrant. 3. The angle at which two arcs of great circles intersect on the surface of the sphere (in the same way as for any other curves) is the angle between the straight lines touching them at the point of intersection, and consequently is the same as the angle between the planes in which the arcs lie; for, as the tangents or touching lines are situated in the same planes, respectively, with the arcs, and are perpendicular to the radius of the sphere which is the intersection of those planes, the angle between the tangents is the same as the angle contained between the planes. Thus, let two arcs of great circles PA, PB, (fig. 1.) intersect in P, and let two tangents be drawn to them, viz. PD which will be in the plane POA, and PE in the plane POB; then since PD, PE, are both perpendicular to PO, ▲ APB = ▲ DPE = inclination of the planes in which the arcs are situated. 4, Let the arcs PA, PB, be produced to meet the great circle of which P is the pole in M, N, and any small circle of which P is the pole in A, B; and join OM, ON, CA, CB, then since PD, PE are respectively parallel to OM, ON, LDPE=MON; and therefore LAPB = = 4 MON=arc MN, employing the arc, according to a preceding remark, to express the angle which it subtends at the center of the sphere. This shews that if two arcs containing any angle be produced till each is a quadrant, the arc joining their extremities (which will be a portion of the great circle of which their point of intersection is the pole) will measure the angle they include. Again, AC the radius of the small circle AB =0A sin AOP= OA sin PA; and length of arc AB AC x circular measure of ACB = = OA sin PA x circular measure of MON = length of arc MNx sin PA; which shews that the length of the arc of a small circle intercepted between two great circles passing through its pole, is proportional to the sine of the polar distance of the small circle. 5. The planes of all great circles passing through P will contain OP, and therefore be perpendicular to the plane MON; therefore all great circles passing through P will cut the great circle MN, of which Pis the pole, at right angles. Great circles. which pass through the pole of another great circle are called secondaries to the latter, or meridians. Hence, a great circle MN being given, its pole P is determined either (1) by measuring an arc MP equal to a quadrant on any great circle perpendicular to MN; or (2) by drawing any two great circles not in the same plane MA, NB, at right angles to MN, and producing them till they intersect in P. And, conversely, if a point on the sphere be such that an arc of a great circle drawn from it perpendicular to a proposed circle is a quadrant, or such that quadrants of two different great circles are intercepted between it and the proposed circle, then that point is the pole of the proposed circle. 6. The arc joining the poles of two great circles subtends an angle at the center of the sphere equal to their angle of inclination; and the point of intersection of the great circles (i. e. the extremity of the diameter in which their planes intersect) is the pole of the great circle in which their poles lie. Let P and Q (fig. 2) be the poles of AC, BC, two great, circles whose planes intersect in the diameter OC, then each of the angles POC, QOC, is a right angle, therefore CO is perpendicular to the plane POQ, and consequently C is the pole of the great circle PQAB passing through P and Q; and since each of the angles POA, QOB is a right angle, 7. The portion of the surface of a sphere contained by three arcs of great circles which cut one another two and two, is called a spherical triangle; the three planes in which the arcs lie forming a solid angle at the center of the sphere. The objects of investigation in Spherical Trigonometry are the relations subsisting between the angles at which the three plane faces containing a solid angle are inclined to one another, and the angles which the lines of intersection of those plane faces, or the three edges of the solid angle, form with one another. Let 0 (fig. 3) be the vertex of a solid-angle contained by the three plane faces AOB, BOC, COA, and let the arcs of great circles AB, BC, CA be the intersections of these planes with the surface of a sphere described from center O with any radius; then the inclinations of the planes AOB, BOC, COA; to one another, are identical with the angles A, B, C, of the spherical triangle; and the three angles at O contained between the edges of the solid angle OA, OB, OC are proportional to the sides AB, BC, CA. On this account the spherical triangle ABC, which is called the base of the solid angle at O, may be employed with great advantage in conducting the investigation of the relations of the angles of inclination of the faces and edges of the solid angle to one another; which relations, as has been said, are the proper objects of our research. The sense in which the spherical triangle is employed being once understood, we may transfer our attention from the solid angle to the triangle in which its faces cut the sphere, and the solid angle need not be represented in our diagrams; but we must still keep in mind that, not the arcs forming the sides of |