2 All the All the S, C: S, A:: AB: BC. And Angles SidesAB, S, C: S, B::AB: AC:Whence A, B, C. AC, BC. if the Angles are given, the Proportions of the Sides may be found, but not the Sides them* felves, unless one of them be first known. The two The Angles AB:BC::S, C: S, A; which therefore may be found. When AB, BC, A and B. AB the Side oppofite to C, the and C, the Angle oppofite 3 to one of them. given Angle is longer than BC the Side oppofite to the fought Angle, the fought Angle is lefs than a right one. But when it is fhorter, because the Sine of an Angle, and that of its Complement to two Right Angles, is the fame, the Species of the Angle A must be first known, or the Solution will be ambiguous. Vid.Fig. to Prop.13.BC+AB: BC-AB:: TA+CTA-C T, Whence is known theDifference of the Angles A and C, whose Sum is given; and fo (by the Prob. following Prop. 14.) the Angles themfelves will be given. Via. Fig. B. Let the Perpendicular be drawn from the Vertex to the Bafe, and find the Segments of the Bafe by Prop. 14. viz. Make as BC: AC+AB:: AC-AB: DC-DB. And fo BD, DC, are given from this Analogy; and thence the Angles ABD, ADC, will be given by the Refolution of Right-angled Triangles. THE ELEMENTS OF Spherical Trigonometry. I DEFINITION S. T HE Poles of a Sphere are two Points in the Superficies of the Sphere that are the Extremes of the Axis. II. The Pole of a Circle in a Sphere, is a Point in the Superficies of the Sphere, from which all Right Lines that are drawn to the Circumference of the Circle, are equal to one another. III. A great Circle in a Sphere, is that whofe Plane paffes thro' the Center of the Sphere, and whofe Center is the fame of that of the Sphere. IV. A fpherical Triangle is a Figure comprehend ed under the Arcs of three great Circles in a Sphere. V. A fpherical Angle is that which, in the Superficies of the Sphere, is contained under two Ares of great Circles; and this Angle is equal to the Inclination of the Planes of the faid Circles. PRO PROPOSITION I. Great Circles ACB, AFB, mutually bifeet each F other. OR fince the Circles have the fame Center, their common Section fhall be a Diameter of each Circle, and fo will cut them into two equal Parts. Coroll. Hence the Arcs of two great Circles in the Superficies of the Sphere, being lefs than Semicir cles, do not comprehend a Space; for they cannot, unless they meet each other in two oppofite Points in a Semicircle. PROPOSITION II. If from the Pole C of any Circle AFB, be drawn a Right Line CD to the Center thereof, the faid Line will be perpendicular to the Plane of that Circle. Vid. Fig. to Prop. 1. L ET there be drawn any Diameters EF, GH, in the Circle AFB; then because the Triangles CDF, CDE, the Sides CD, DF, are equal to the Sides CD, DE, and the Base CF equal to the Bafe CE; (by Def. 2.) then (by 4. El. 1.) fhall the Angle CDF Angle CDE; and fo each of them will be a Right Angle. After the fame Manner we demonftrate that the Angles CDG, CDH, are Right Angles; and fo (by 4. El. 11.) CD shall be perpendicular to the Plane of the Circle AFE. W.W.D. Coroll. 1. A great Circle is diftant from its Pole by the Interval of a Quadrant; for fince the Angles CDG, CDF, are Right Angles, the Measures of them, viz. the Arcs CG, CF, will be Quadrants. 2. Great Circles that pass thro' the Pole of fome other Circle, make Right Angles with it; and contrariwife, if great Circles make Right Angles with fome other Circle, they fhall pass thro' the Poles of that other Circle, for they muft neceffarily pafs thro the Right Line DC, PRO PROPOSITION III. If a great Circle ECF be defcribed about the the HE Arcs AC, AF, (by Cor. 1. Prop. z.) are Quadrants, and confequently the Angles ADC, ADF, are Right Angles. Wherefore (by Def. 6. El. 11.) the Angle CDF (whofe Measure is the Arc CF) is equal to the Inclination of the Planes ACB, AFB, and alfo equal to the fpherical Angle CAF, or CBF. W. W.D. Coroll. 1. If the Arcs AC, AF, are Quadrants, then fhall A be the Pole of the Circle paffing thro' the Points C and F; for AD is at Right Angles to the Plane FDC (by 14. El. 11.) 2. The vertical Angles are equal, for each of them is equal to the Inclination of the Circles; alfo the adjoining Angles are equal to two Right Angles. PROPOSITION IV. Triangles fhall be equal and congruous, if they have two Sides equal to two Sides, and the Angles comprehending the two Sides alfo equal. PROPOSITION V. Alfo Triangles fhall be equal and congruous, if one Side, together with the adjacent Angles in one Triangle, be equal to one Side, and the adjacent Angles of the other Triangle. PROPOSITION VI. Triangles mutually Equilateral, are alfo mutually Equiangular. PROPOSITION VII. In IfofcelesTriangles, the Angles at the Bafe are equal. PRO PROPOSITION VIII. And if the Angles at the Bafe be equal, then the Triangle fhall be Ifofceles. These four laft Propofitions are demonftrated in the fame Manner, as in plain Triangles. PROPOSITION IX. Any two Sides of a Triangle are greater than the third. FOR OR the Arc of a great Circle is the shortest Way, between any two Points in the Superficies of the Sphere. PROPOSITION X. A Side of a fpherical Triangle is less than a Semicircle. LET AC, AB, the Sides of the Triangle ABC be produced till they meet in D; then shall the Arc ACD, which is greater than the Arc AC, be a Semicircle. PROPOSITION XI. The three Sides of a spherical Triangle are lefs than a whole Circle. OR BD+DC is greater than BC, (by Prop. 9.) and adding on each Side BA+AC, DBA+ DCA; that is, a whole Circle will be greater than AB+BC+AC, which are the three Sides of the fpherical Triangle ABC. PROPOSITION XII. In any Spherical Triangle ABC, the greater Angle A is fubtended by the greater Side. AKE the Angle BAD Angle B; then shall MAKE AD BD (by 8 of this) and fo BDC=DA +DC, and thefe Arcs are greater than AC. Wherefore the Side BC, that fubtends the Angle BAC, is greater than the Side AC, that fubtends the Angle B. U 3 PRO |