Prop. X. (64.) Problem. 'An arch of a circle, in the surface of a given sphere, being given, to find the poles of the circle. > Find (Art. 63.) two straight lines equal to the distances between any point in the given arch and its two poles : then, from the extremities of the given arch, as poles, at distances equal to the two straight lines, so found, describe (art. 59.) two circles in the sphere : the intersections of the circumferences of the circles, so described, are the poles of the circle, to which the given arch belongs. For, the poles (Art. 31.) are in each of the circumferences, and therefore they are in the intersections of the circumferences. (65.) Cor. A given arch of any circle, in a sphere, may be produced, in the sphere's surface (Art. 64. and 59.) Prop. XI. (66.) Problem. Two points in a sphere's surface, being given, to describe a circle, in the sphere, which shall pass through them, and shall have the direct distance between its pole and its circumference,' equal to a given straight line * In the following articles, when two points on a sphere's surface are directed to be joined, it is intended that they should be joined by arches of great circles, unless the contrary be specified. From each of the two given points, as a pole, and at a distance equal to the given straight line, describe (Art. 59.) a circle in the sphere: then, from either of the two points, in which the circumferences of the circles, so described, cut one another, as a pole, describe (Art. 59.) a circle passing through either of the two given points ; and it will, manifestly, pass through the other given point, and have the distance between its pole and its circumference equal to the given straight line. (67.) Cor. 1. If the given straight line be either greater than a diameter of the given sphere, or less than the half of the straight line which joins the two given points in the sphere's surface, the construction fails; and, in that case, the problem is evidently impossible. (68.) Cor. 2. An indefinite number of arches of great circles may, by means of Art. 63, be drawn through any given point, in a sphere's surface. Prop. XII. (69.) Problem. A spherical angle being given, in the surface of a given sphere, to find an arch of a great circle, in the sphere, which shall measure it. Let EPF be a spherical angle, in the surface of the sphere PAPD: It is required to find an arch of a great circle, which shall measure the angle EPF. Find (Art. 63.), the direct distance between a great circle of the sphere and either of its poles ; which is done, independently of any great circle having been actually described: then, from P as a pole, at the distance so found, describe (Art. 59.) the great circle ABD; and (Art. 65.) produce the two arches PE and PF, until they meet the circumference ABD, in the points A and B. The arch AB (Art. 54.) measures the spherical angle EPF. PROP. XIII. (70.) Problem. In the surface of a given sphere, to draw an arch of a great circle, which shall pass through a given point, in that surface, and be at right angles to the circumference of a given circle, in the sphere. Find (Art. 64.) either pole of the given circle describe (Art. 66.) a great circle, of the sphere, passing through the pole thus found, and through the given point: and (Art. 50.) its circumference shall be at right angles to the circumference of the given circle. PROP. XIV. (71.) Problem. A spherical angle, in the surface of a given sphere, being given, to make a plane rectilineal angle, which shall be equal to it. a From the angular point of the given spherical angle, as a pole, describe (Art. 59.) any circle, in the sphere; describe, also, (Art. 61.) a circle, in a plane, that shall be equal to the circle first described, in the sphere : and in this latter circle, place (E. 1. 4.) a straight line equal to the direct distance between the two points, in which the circle in the sphere, first described, cuts the arches containing the given angle; which distance (Art. 59.) may be considered as given : then shall the plane rectilineal angle subtended by this straight line, at the center of the circle in which it is placed, be equal (E. 28. and 27. 3. and Art. 54.) to the given spherical angle. PART I. THE ELEMENTS OF Spherical Geometry. SECTION III. ON THE GENERAL RELATIONS OF THE SIDES AND ANGLES OF SPHERICAL FIGURES. DEFINITION. (72.) A Spherical Triangle is a figure, on the surface of a sphere, contained by three arches of great circles, in the sphere, each of which arches is less than the semi-circumference of a great circle *. * A portion of the sphere's surface may be bounded by three arches of great circles, of which arches, one may be greater than the half of the circumference ;' and the angle opposite to it may be greater than two right angles. There might, indeed, if the restriction laid down in the above definition were removed, be no fewer than eight spherical triangles formed, by joining three given points on a sphere's surface. But, as the main object of Spherical Geometry is to elucidate Spherical Trigonometry, and as the determination of the unknown parts, of such a trilateral figure, is always reducible to the solution of a spherical triangle, such as we have defined it to be, the properties of the former kind of figure are not investigated in this Treatise. |