1 Fig.XXI. B. CHAP. IV. The Analogies, or Proportions, for the Solution of the feveral Cafes of Right-angled Spherical Triangles, by the Universal Proposition. Fa OR the Performance hereof, I shall make use of this Rightangled Spherical Triangle A B C, Right-angled at A, the Quantities of whose Sides and Angles are adfixed to their respective Circular Parts in the Diagram noted with B, in Fig. XXI. And also in this Table, both in Sexagenary Degrees and Minutes; and in Decimal Parts also. Fig.XXI. Cafe I. And in every Cafe I shall diftinguish the Two Given Terms, (be- The XVI Cases of Right-angled Spherical Triangle, Resolved, As Radius 90 Deg. To Sine C, 56 Deg. 52 Min. So Sine BC, 66 Deg. 30 Min. To Sine B A, 50 Deg. 10 Min. 10 0000000 9.9229334 9.9623978 19.8853312 CASE CASE II: The Side Adjacent A C, Extream Conjunt. Fig. XXI. As Co-tangent B C, 23 Deg. 30 Min. To Radius, 90 Deg. So Co-fine C, 33 Deg. 8 Min. To Tangent C A, 51 Deg. 30 Min. : 96383019 10. 19.7376611 Cafe II. 10.0993592 Cafe III. CASE III. The other Angle B, Extream Conjunct. 9.814 9.8147277 As Co-Tangent C, 33 Deg. 8 Min. To Radius, 90 Deg. So Co-fine B C, 23 Deg. 30 Min. To Co-Tangent B, 31 Deg. 25 Min. The Hypotenuse BC, and Side A-C, given, to find CASE IV. The Oppofite Angle at B, Ex. Disj. 4 T So Sine CA, 51 Deg. 30 Min. 19.8935444 CASE V. The Adjacent Angle C, Middle Part. CASE VI. The other Side A B, Extream Disjunct. 9.9311466 Cafe V. 10. 10.0993948 9.6383019 X9.7376967 Cafe VI. 9.7941496 10. 19.6006997 $9.8065501 Fig. XXI. The Side A C, and the Angle opposite there to B, being Given; To find CafeVII. CASE VIII. The other Angle at C, Extream Disjunct. Cafe IX. CASE IX. The Hypotenuse B C, Extream Disjunct. Cafe X. The Side CA, and the Angle C, adjacent thereto, given; To find Cafe XI. CASE XI. The other Angle B: Middle Part. To Co-fine B, 31 Deg. 25 Min. 19.7170830 CASE CASE XII. The Hypotenuse BC, Extream Conjunct. Fig.XXI. The Two Sides, AC and AB, given; To find CASE XIII. Either Angle, as C: Extream Conjunt. 10.0993948 Cafe XII. 10. 19.7376611 9.6382663 Cafe XIII. 10.0787534 10. 19.8935444 9.8147910 10. 9.7941496 9.8065575 29.6007071 CASE XIV. The Hypotenuse CB: Middle Part. The Two Angles B and C, given; To find CASE XV. Either of the Sides, as AC: Extream Disjunct. Cafe XV. As the Radius, 90 Deg. 9.9229334 10. 19.7170526 9.7941192 Cafe CASE XVI. The Hypotenuse BC: Middle Part. To Co-tangent C, 33 Deg. 8 Min. So Co-tangent B, 31 Deg. 25 Min. To Co-fine BC, 23 Deg. 30 Min. 10. 9.8147277 9.7859004 19.6006281 Fig.XXI. Note, These are the Proportions answerable to the Univerfal Propofition, yet may (many of them) be varied; so that the Radius may be brought into the first Place, and that by the latter part of the foregoing Corollary: Which fays Radius is a Mean Proportional between the Tangent of an Arch, and the Tangent Complement of the Same Arch. -So that (in the XIIIth CASE) where it is faid, As the Tangent BA, Is to Radius: .... It is all one, as if you should fay, ! : To the Co-tangent of C, 33 Deg. 3 Min. 19.8157910 The like Course may be taken in the Second, Third, Tenth and Twelfth CASES. And thus you have the whole Doctrine of the Dimension of Right-angled Spherical Triangles, performed by Help of this one Catholick Propofition. I CHAP. IV. Some Pranotions concerning Oblique-angled, Spherical N Oblique-angled Spherical Triangles there are XII Cafes, Ten of which may be refolved by the Universal Proposition; but then the Oblique Triangle must be reduced into Two Right-angled Triangles by help of a Pérpendicular let fall, fometimes within, sometimes without, the Triangle: And to know whether it fall within or without, the subsequent Rules are to be observed. RULE I. If the Angles at the Base of the Triangle be both of the Same Affection, that is, both Acute or Obtuse; the Perpendicular let fall from the Vertical Angle shall fall within: But if of different Affections, without. As |