1. Method by the Canon of Logarithmetic Sines, &c. 2. Method by the Canon of Natural Sines, Tangents, &c. 3. Method by Lord Napier's five Circular Parts. 4. Method by the Sector. 5. Method by the Sliding Rule. 6. Method by Gunter's Scale and Compaffes. 8. Method by Spherical Projection. 9. Method by Orthographic Planisphere, or Analemma. 10. Method by the Stereographic Planifphere. Of thefe ten Methods, five are common to both Plain and Spherical Triangles; fuch are the 1, 2, 4, 5, 6; the other five are peculiar to Spherical Triangles only. And having already applied the five Common Methods to the Solution of Plain Triangles, and there fhewed the Nature and Conftruction of the Numbers in the Canon, both of Logarithmetic and Natural Sines, &c. and alfo given a large Defcription and Manner of making the Lines on the Inftruments, I fhall refer the Reader to that Volume, to be informed of those Matters; and fhall here only apply them to the Solution of Spherical Right-angled Triangles, with the other five Methods, in the Order I have above exhibited them. And as there are but fix Cafes of Right-angled Spherical Triangles, whofe Analogies are really diverfe from each other, it will be fufficient to exemplify them in every Method here mentioned; and thereby the young Trigonometer will be provided with an infallible Clue, to conduct him thro' all the abftrufe and meanderous Windings of the Mazy Labyrinth of Spherical Geometry, with fafety, eafe, and pleasure. The The Six Cafes I here mean are as follows; 1. Given the Hypothenuse, and one Oblique Angle; As in Synop. Cafe 1, and 2. 2. Given the Hypothenufe, and one of the Legs; As Cafe 3, and 4. 3. Given one Leg, and an adjacent Angle; As Cafe 5, and 7. 4. Given one Leg, and an oppofite Angle; As Cafe 6, and 8. 5. Given both the Legs; As Cafe 9. 6. Given both the Oblique Angles; As Cafe 10. These Cafes, with all the Varieties, I fhall fhew how to refolve by all the Methods above, as in the following Chapters. CHA P. IX. The First Method of Solving Right-Angled Spherical Triangles, by the Canon of Logarithmetic Sines, and Tangents. TH HE Reader is fuppofed to have his Eye on the Scheme, and the Proportions and Anafo gies in the foregoing Synopfis, as he proceeds in the Subfequent Solutions; and thus will he be fure to understand at once both the Theory and Practice of his Art. VOL. II, Cafe The Analogy R: SC :: sBC: sBA. That is, As Radius 8 10. Is to the Sine of the Angle C 56 57 = 9.9233405 So the Sine of the Hypoth. BC=44 52 = 9,8484720 To the Sine of the Log. BA36 15 ≈ 9.7718125 2. To find the other Leg AC. The Analogy RtBC :: csG: AG. That is, As Radius 10. To the Tangent of the Hyp. BC=44 52=9.9979787 So is the Co-fine of the Angle C-56 57-9.7366918 To the Tangent of the Leg AC 28 30=9.7346705 3. To find the Angle B. The Analogy R: csBC: tC ctB. That is, As Radius 10. To the Co-fine of the Hyp. BC 44 52 9.8604730 So is the Tang, of the Angle C-56 57-10.1865775 To the Co-Tan. of the Ang. B=42 34 10.0370505 Cafe Cafe 2. The Hypothenuse 8 1 BC= 44 52 Given and one of the Legs, as BA = 36 15 1. To find the other Leg AC. The Analogy esBA: ResBC: csAC. That is, ୪ As the Co-fine of the Leg BA=36 15=9.9065745 Is to the Radius. 10. So is the Co-fiine of the Hyp. BC 44 52=9.8504730 To the Co-fine of the Leg AC28 30=9.9438985 2. To find the Angle B. The Analogy BC R BA: csB. That is, 10. As the Tang. of the Hypoth. BC=44 52=9.9979787 3. To find the Angle C. The Analogy s BC: R:: sBA: SC. That is, 10. As the Sine of the Hypoth. RC=44-52=9.8484720 is to the Radius fo is the Sine of the Leg BA÷36 13=9.7718150 To the Sine of the Angle C56 57-9.9233430 VOL. II, P 2 Cafe Cafe 3. Given One Leg, as - BA= BA= 36 15 B42 34 1. To find the Hypothenufe B C. The Analogy R: csB:: ctAB: ctBC. That is, As Radius IO. Is to the Co-fine of the Angle B=42 34 9.8672673 fo is the Co-tang. of the Leg BA=36 15=10.1347596 To the Co-tang. of the Hy. BC=44 52=10.0020369 2. To find the other Leg A C. The Analogy R: SBA :: tBtAC. That is, As Radius " 10. Is to the Sine of the Leg BA=36 15 9.7718150 fo is the Tangent of the Angle B 42 34 9.9629494 To the Tangent of the Leg AC=28 30=9.7347644 3. To find the other Angle C. The Analogy Rcs BA :: sB: cs C. That is, As Radius ୪ 10. A Is to the Co-fine of the Leg BA=36 15=9.9065745 fo is the Sine of the Angle B=42, 349.8302342 To the Co-fine of the Angle C=56 57=9.7368087 Cafe |