To measure the things required. The Angle D and the Side DC are measured as in the laft, and the Side BD is measured by laying a Ruler to the Pole of the oblique Circle a, and upon the Point D, which will cut the primitive Circle. in c; then measure Be upon the Line of Chords, and you will find it 42° 09'; and fo much is the Side BD. CASE II. The three Sides given, to find the Angles. Let the Side BD 42° 09', BC 30° 00', and CD 24° 04' be given, to make the Triangle. On the primitive Circle make BD 42° 09′ the greater Side, then at the distance of BC 30° 00' draw a Parallel, and at the distance of CD 24°04' draw another Parallel, and thro the Point of Interfection draw the two oblique Circles; and it is done. B D If the Great Circle a b be drawn thro the two Poles of the oblique Circles, and Lines be drawn from the faid Poles to the Center or Pole of the primitive Circle, these three will conftitute the Triangle a b c whofe Sides are equal to the Angles of the other Triangle BCD, and contrariwife the Angles of a b c are equal to the Sides of BCD, only the greater Angle in the one, is equal to the Complement of the greater Side in the other to 180 degrees; that is, the Side be is equal to the Angle CBD, and the Side ac to the Angle CDB, and the Side ab is equal to the Complement of the Angle BCD. CASE 12. The three Angles being given, to find the Sides. This Cafe is the fame as the laft in effect; for it is but changing the Angles into Sides, as is above fhewed, (and as it is demonstrated Chap. 6. Def. 6.) and making a Triangle of thofe Sides, as in the last Cafe; and it is done. Thus have I fhewn how to refolve all Cafes of fpherical Triangles, both right and oblique-angled, by Projection. ΑΝ APPENDIX СНАР. I. The Refolution of all the Cafes of Plain Triangles, both right and obliqueangled, by Natural Arithmetick. I. And first, of right-angled Triangles, A THE RULE. LWAYS divide 172 (with a competent Number of Cyphers annexed) by the Degrees and Decimal parts of a Degree contained in the leffer acute Angle, 2. From the Square of this Quotient always fubtract 3, and out of the Remainder extract the fquare Root. 3. Subtract this fquare Root from the double of the Quotient, of the Remainder fhall be the Hypothenufe. 4. Sub. 4. Subtract the double of the Hypothenufe from the faid Quotient, the Remainder fhall be the greater of the Sides; the leffer Side being always aflumed 1.000, &c. From these affumed Sides we may easily find the true Sides, by the Rule of Three. CASE I. In the Triangle ABC, right-angled at A, there are given the Hypothenufe BC, and the acute Angles, B and C; to find the Bafe AC, and Perpendi cular AB. And the Side AB is fuppofed 1. By these three aflumed Sides, the true Sides may be found (by Lib. 6. Pr. 4. of Euclid) by the Rule of Three; thus: |