DEMONSTRATION. Fig. XVI. In the Right-angled Spherical Triangle ABC, Right-angled Fig. XVII. at C, and Acute-angled at A and B. The Measure of S BAC { the Acute Angle, ABC or BDE } is the Arch { EF HI (by the fourth hereof) But the Arches EF and ED together, are equal to a Quadrant; therefore the Arches FE and HI, added together, are more than a Quadrant; and confequently, the Angles answering to those Arches, namely, the Angles B AC and ABC together, are more than a Quadrant. But the Angle ACB is a Right Angle, by the Propofition: Therefore, in the Spherical Triangle ABC, confifting of Two Acute Angles, the Three Angles together are greater than Two Right Angles. Again, In the Acute-angled Triangle K L M ) NO Lis the Arch V X But these Three Arches NO, VX, and RQ together, are more than Two Quadrants: For PQ, and PV, (being the Complements of the Two Arches, QR and V X) added together, are less than the Arch NO, by the Proposition: Therefore the Arch NO, being the Measure of the third Angle, is more than the Complements of the other Two Angles added together; and consequently, the third Angle is greater than the Complements of the other Two Angles. And therefore, In Acute-angled Spherical Triangles, the Three Angles are greater than Two Right Angles. Which was to be demonstrated. CHAP. II. Such Affections of Great Circles of the Sphere, as relate to the Great Circle of the Sphere, Is such a Circle, as divideth the whole Body of the Globe or Sphere, into Two Equal 2. A Spherical Triangle is that part of the Superficies of the Fig. Globe, as lyes between the Arches of Three Great Circles of the XVII. Parts. 2. A Sphere interfe&ting one another. 3. A Spheric Angle is the fame with the mutual Aperture or Inclination of the Plains of such Two Great Circles which conftitute the Angle. 4. When one Circle falls upon another Circle, or when the Arches of Two Great Circles interfect each other, the Sum of the Angles made thereby is equal to Two Right Angles: And the Vertical Angles made thereby are mutually Equal. 5. In all Spherical Triangles, the Greater Angle is always oppos'd to the Greater Side. 6. An Isofceles Triangle, hath its Two Angles at the Base mutually Equal; and, on the contrary, if a Triangle hath Two Angles Equal, it hath Two Sides also Equal. 7. Two Triangles mutually Equilateral, are also Equiangular one to the other. 8. If there be Two Triangles, and in each one Angle, and the Two Sides including it, respectively equal: Or, if One Side, and the Two Angles adjacent, be severally equal, then are those Two Triangles equal. 9. An Arch of a Great Circle, is the shortest Distance between Two Points on the Surface of a Globe: And so, any Two Sides of a Spherical Triangle taken together, are Greater than the Third. 10. All Great Circles cut each other into Two equal Parts; for their common Section is a Diameter of the Sphere, and confequently the Two Sections of the Peripheries of Two Great Circles are at a Semicircle's Distance. Hence it follows, That 11. Every Side of a Spherical Triangle, is less than a Semicir. Fig. cle. So DB is less than the Semicircle DBC or DAC. 12. The oppofite Angles at the Sections of Two Circles, are Equal; as the Angle at D, is equal to that at C; for the fame Plains constitute both Angles. 13. In any Spherical Triangle, if the Sum of the Legs containing an Angle be Greater, Equal to, or Leffer than a Semicircle, the internal Angle at the Bafe, is (accordingly) Greater, Equal to, or Leffer than the outward opposite; and confequently, the Sum of the Two internal Angles at the Base, are Greater, Equal to, or Leffer than Two Right Angles. XVIII. Fig. XVIII. Fig.XIX. DEMONSTRAΤΙΟΝ. If DB and BA together be Greater, Equal or Leffer than DC, then BA is Greater, Equalto, or Leffer than BC; and therefore the Angles at C and Dare Greater, Equal or Leffer than the Angle BAC; and the Angles B D A and DA B, Greater, Equal to, or Leffer than the Angles BAC and DAB, equal to Two Right Angles. COROL'LARY. In an fofcheles Triangle, if one of the Equal Legs be Greater, Equal to, or Leffer than a Quadrant, the Angle at the Base is Greater, Equal to, or Leffer than a Right Angle. 14. The Sum of the Three Sides of a Triangle is less than a Whole Circle, or 360 Deg. For BA is Less than BC and AC. Therefore D B, D A and B A together, are Leffer than DBC and DAC. 15. If from the Point of an Angle, as a Pole, you describe a Great Circle; or, if you describe a Circle at 90 Deg. on the angular Point, the Ark of that Circle so described, which is intercepted between the Legs of the Angle, is the Measure of that Angle. 16. The Poles of the Sides of any Triangle G HD, constitute another Triangle nx m, which we may call Supplemental to the Triangle GHD, for the Supplements of the Angles and Sides of the Triangle n x m are equal to the Sides and Angles of the Triangle G H D. DEMONSTRATION. From the Points G, H, D, as Poles, describe Three Great Circles, xAY, RTmn, xBnZ; then is Y m equal to a Quadrant, and equal to A x, because m is the Pole of HGY, and x or E, the Pole of GA; therefore m x, equal to A Y, equal to the Supplement of CA, equal to the Angle HGD; and the Quadrant Zn, equal to BX; therefore nx, equal to B Z, equal to the Supplement of the Angle HDG, and n T, equal to a Quadrant, equal to m R; therefore n m, equal to TR, equal to the Supplement of the Angle DHG. Now that the Triangle n Em, constituted between the Three next Poles, hath its Three Sides and Angles, equal to the Angles and Sides of the Triangle GHD, fave that the Greatest Side nm is the Supplement of the Greatest Angle H, and the Angle E, of the Side GD. 17. Any 17. Any Angle of a Triangle, with the Difference of the other Fig.XIX. Two, is Leffer than Two Right Angles: For xn-is Leffer than x m and nm, that is, Two Right Angles, wanting D, is Leffer than Two Right Angles, wanting G, and Two Right Angles, wanting H. Therefore, G and H wanting D, is less than Two Right Angles. 18. If Two Triangles are mutually Equiangular, they are also mutually Equilateral, for, because they are Equiangular, their Supplemental Triangles are Equilateral (by the 16th) and therefore Equiangular (by the 7th). And therefore the proposed Triangles are Equilateral (by the 16tb.) 19. The Three Angles of every Spherical Triangle, are Greater than Two Right Angles, and Leffer than Six Right Angles. For, nx and x m and nim together, are Leffer than Four Right Angles, (by the 14th.) that is, Six Right Angles, wanting D, and G, and H, leffer than Four Right Angles: That is, Two Right Angles are leffer than D, and G, and H. Also, the Sum of the Internal-angles is less than the Sum of the Internal and External Angles taken together, for both of them make but Six Right Angles. 20. Of several Arches of Great Circles falling from the fame Point of the Spheres Surface, on another Circle, the Greateft is that which paffeth through the Pole of the Circle; and the next to this, is Greater than that which is farther off For suppose P Fig. XX the Pole of the Circle CD, and w the Pole of DPC; then is AD Greater than AB, AB Greater than AE, A E Greater than AC: And the Ark BwC Greater than the Ark BP, and BP Greater than B D. 21. A Great Circle paffing through the Poles of another Great Circle, cuts it at Right Angles; And on the contrary, If it cut it at Right Angles, it paffeth through its Poles: The Angle PBO is equal to a Right Angle, equal to PGD, equal to PD B, equal to w A C. 22. In an Oblique-angled Triangle, if the Angles at the Base are like, or of the Same Kinds; that is, both Acute, or bothObtufe, the Perpendicular falls Within the Triangle, and the Quadrantal Arch without. But if they be unlike, the Perpendicular falls Without; and the Quadrantul Arch Within the Triangle. For the Triangle AEF hath the Angles at E and F Acute, and the Perpendicular AC falls Within, and the Quadrantal Arch Aw Without. Alfo, the Triangle BAG hath the Angles at B and G, Obtufe, and the Perpendicular AD Within, and the Quadrantal 12 A Fig. XX. Arch A w Without. But the Triangle BAE hath the Angles at Band E of different Kinds; and the Perpendicular A C falls Without, and the Quadrental Arch Aw Within, the Triangle. Fig.XXI. I. Diagram A. CHAP. III. Of the Menfuration, or Solution, of Right-angled N a Right-angled Spherical Triangle, there are (besides the CIRCULAR. Side A B, Viz. The Complement of the Angle at B, Complement of the Angle at C, Complement of the Side B C. But the Right Angle at A is set aside from being any of the II. In the Solution of a Right-angled Spherical Triangles, there III. These Three Terms (namely, the Two that are Given, and the Third which is Required) must be first looked upon according to their Circular Parts. IV. Of which, One is named the Middle (or Mean) Part; the other Two are called the Extream Parts, borrowing their Appellation from the Scituation of the Terms themselves: For, of Three Terms, One must (of neceffity) be in the Middle, and the other Two in the Extreams: Therefore the Circular Part |