and therefore the Three Sides, A G, A H and H G, fhall be Fig. XV. XVI. If the third Angle of a Spherical Triangle, having Two Right As in the Spherical Triangle G H I, Acute-angled at G, the third Side HI is less than a Quadrant. But in the Spherical Triangle AGI, Obtufe-angled at G, the third Side AI is more than a Quadrant. XVII. An Oblique Spherical Triangle, confifteth fimply of Acute or XVIII. A Spherical Triangle, with Two Obtufe Angles, and One As if the Angles at A and D, be fuppofed Acute; then the Triangle BC D, with Two Obtufe Angles at B and C, and One Acute Angle at D, is oppofite to the fimple Acute-angled Triangle ABC. XIX. A Spherical Triangle, with Two Acute Angles, and One Obtufe, is oppofite to a Spherical Triangle, fimply Obtufe angled, & contra. As if the Angles at A and D, be fuppofed Obtufe; then the Triangle ABC, with Two Acute Angles at B and C, and One Obtufe Angle at A, is oppofite to the fimply Obtufe-angled Triangle BDC. XX. The Three Angles of every Spherical Triangle, are greater. than Two Right Angles. This is evident in Spherical Triangles, having more Right, or Obtufe Angles, than One: But in Acute-angled Triangles, it may be thus Demonftrated. DE DEMONSTRATION. Fig. XVI. In the Right-angled Spherical Triangle A B C, Right-angled at C, and Acute-angled at A and B. Fig. XVII. ABC or BDE is the Arch { EF HI The Measure of S BAC Again, In the Acute-angled Triangle K L M The Measure of the Acute Angle M K L is the ArchV X LMKS R Q But these Three Arches N O, V X, and R Q together, are more than Two Quadrants: For PQ, and P V, (being the Complements of the Two Arches, Q R and V X) added together, are less than the Arch N O, by the Propofition: Therefore the Arch N O, being the Measure of the third Angle, is more than the Complements of the other Two Angles added together; and confequently, 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 demonftrated. CHA P. II. Such Affections of Great Circles of the Sphere, as relate to the Great Circle of the Sphere, Is fuch 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. Sphere interfecting one another. 3. A Spheric Angle is the fame with the mutual Aperture or Inclination of the Plains of fuch 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 Ifofceles Triangle, hath its Two Angles at the Bafe mutually Equal; and, on the contrary, if a Triangle hath Two Angles Equal, it hath Two Sides alfo Equal. 7. Two Triangles mutually Equilateral, are alfo Equiangular one to the other. 8. If there be Two Triangles, and in each one Angle, and the Two Sides including it, refpectively equal: Or, if One Side, and the Two Angles adjacent, be feverally equal, then are thofe Two Triangles equal. 9. An Arch of a Great Circle, is the shortest Distance between Two Points on the Surface of a Globe: And fo, 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 lefs than a Semicircle. So DB is lefs than the Semicircle D BC 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 conftitute 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 Baft, is (accordingly) Greater, Equal to, or Leffer than the outward oppofite; and confequently, the Sum of the Two internal Angles at the Bafe, are Greater, Equal to, or Leffer than Two Right Angles. Fig. XVIII. Fig. XVIII. Fig.XIX. DEMONSTRATION. If DB and B A 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 B A Č; and the Angles B D A and DA B, Greater, Equal to, or Leffer than the Angles B A C and D A B, equal to Two Right Angles. COROL'LARY In an ofcheles 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 Lefs than B C and A C. Therefore DB, DA and B A together, are Leffer than D BC and DA C. 15. If from the Point of an Angle, as a Pole, you defcribe a Great Circle; or, if you defcribe a Circle at 90 Deg. on the angular Point, the Ark of that Circle fo defcribed, 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 H D, conftitute 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 HD. DEMONSTRATION. From the Points G, H, D, as Poles, defcribe Three Great Circles, x AY, RTmn, xBn Z, then is Y m equal to a Quadrant, and equal to Ax, becaufe m is the Pole of H GY, and x or E, the Pole of G A; therefore m x, equal to A Y, equal to the Supplement of CA, equal to the Angle H G D; and the Quadrant Zn, equal to B X; therefore n x, 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 D H G. Now that the Triangle n Em, conftituted between the Three next Poles, hath its Three Sides and Angles, equal to the Angles and Sides of the Triangle GHD, fave that the Greateft Side nm is the Supplement of the Greatest Angle H, and the Angle E, of the Side G D. 17. Any 17. Any Angle of a Triangle, with the Difference of the other Fig.XIX. Two, is Leffer than Two Right Angles: For x n is Lesser than xm and nm, that is, Two Right Angles, wanting D, is Leffer than Two Right Angles, wanting G, and Two Right Angles, wan ting H. Therefore, G and H wanting D, is lefs than Two Right Angles. 18. If Two Triangles are mutually Equiangular, they are alío 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 n m 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. Alfo, the Sum of the Internal-angles is lefs than the Sum of the Internal and External Angles taken together, for both of them make but Six Right Angles. 20. Of feveral 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 fuppofe P Fig. XX. the Pole of the Circle C w D, and w the Pole of D PC; then is AD Greater than A B, A B Greater than A E, A E Greater than AC: And the Ark B & C Greater than the Ark B P, 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 P G D, 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 fame Kinds; that is, both Acute, or both Obtufe, the Perpendicular falls Within the Triangle, and the Quadrantal Arch without. But if they be unlike, the Perpendicular falls Without; and the Quadrantal Arch Within the Triangle. For the Triangle A EF hath the Angles at E and F Acute, and the Perpendicular AC falls Within, and the Quadrantal Arch Aw Without. Alfo, the Triangle B AG hath the Angles at B and G, Obtufe, and the Perpendicular A D Within, and the Quadranta! I 2 |