S P H 



S P H 



ac, and B C = i ^ ; then will A = «, B = J, and C = c : 

 the demonitrations of which coincide with thofe of the like 



of the feverai circles, lines, Sec. of the fphere, infcribed on 

 them. In the middle, upon the axis of the fphere, is a ball, 

 T, reprefenting the earth, on whofe furface are the circles, properties in plain triangles ; the theorems of the congruencjr 

 &c. of the earth. The fphere is made to revolve about the of reftihnear triangles extending to all other curvilinear, 

 faid axis, which remains at reft ; by which means the fun's 

 diurnal and annual courfe about the earth are reprefented ac- 

 cording to the Ptolemaic hypothefis ; and even by means 

 of it, all problems relating to the phenomena of the fun and 

 earth are folved, as upon the celeftial globe, and after the 

 fame manner, which fee defcribed under Globe. 



Sphere, The Copernlcan, (reprefented Plate XX. /IJlro- 

 nomy. Jig. 7- ) is very different from the Ptolemaic, both in 

 its conititution and ufe ; and is more intricate in both. In- 



Its 



deed, the inftrument is in the hands of fo few people, and 

 its ufe fo inconfiderable, except what we have in the other 

 more common inftruments, particularly the globe and Ptole- 

 maic fphere, that we (hall be eafily excufed the not taking 

 up room with any defcription of it. 



Spheres, Harmony of the. See Harmony. 



Spheres, Obliquity of the. See Obliquity. 



Sphere, Rea'ifying of the. See Rectifying. 



Sphere, Dialling. See Dialling. 



Spheres, polifhed fpherical maifes of a mixed metalline 

 compofition, ufed in optics. The manner of making them 

 is as follows. 



Take of pure tin, three pounds ; copper, one pound ; 

 melt thefe two metals together, and when in fufion call upon 

 the mafs fix ounces of burnt tartar, and an ounce and half 

 of faltpetre ; and laftly, a quarter of an ounce of alum, and 

 two ounces of arfeuic : let all thefe matters evaporate and 

 burn away, and then cail the pure metal into the figure of a 

 fphere, and it will be capable of a high and elegant polifh. 

 Neri's Art of Glafs, p. i66. See STEEL-G/a^/. 



SPHERICAL Angle, is the mutual inclination of two 

 planes, by which a fphere is cut. 



Thus, the inclination of the two planes C A F and C E F, 

 {Plate II. Trigonometry, Jig. i.) forms the fpherical angle 



The meafure of a fpherical angle, ACE, is an arc of 

 a great circle A E, defcribed from the vertex C, as from a 

 pole, and intercepted between the legs C A and C E. 



Hence, i. Since the inclination of the plane C E F to 

 the plane C A F is every where the fame, the angles in the 

 oppofite interfeitions, C and F, are equal. 



2. Hence the meafure of a fpherical angle A C E is de- 

 fcribed by the interval of a quadrant A C or E C from the 

 vertex C, between the legs C A, C E. 



If a circle of the fphere A E B F {Jg. 2.) cut another, 

 C E D F, the adjacent angles, A E C and A E D, are equal 

 to two right ones ; and the vertical angles, A E C and DEB, 

 are equal to one another. The former likewife holds of fe- 

 verai angles formed on the fame arc C E D, at the fame 

 point E. Hence, any number of fpherical angles, as A E C, 

 A E D, D E B, B E C, &c. made on the fame point E, are 

 equal to four right angles. See Spherical Triangle. 



Spherical Triangle, a triangle comprehended between 

 three arcs of great circles of a fphere, interfering each other 

 in its furface. 



Spherical Triangles, Properties of. I. If in two fpheri- 

 cal triangles {Plate II. Trigonometry, Jig. 3.) A B C and 

 aic, A —a, BA — in, and C A = c a; then will B, i, 

 and the fides including the angles, be refpeftively equal ; 

 the whole triangles are equal : that is, BC = ic, B = i, 

 and C = f . 



Again, if in two fpherical triangles A = o, C = <:, and 

 A C = ar; then will B z= 6, AB = ab, and B C = 6c. 



Laltly, if ia two fpherical triangles AB — ab, AC = 



circular, paraboHcal, Sec. provided their fides be fimilar. 

 See Triangle. 



2. In an equicrural triangle ABC (Jig. 4.) the angles at 

 the bafe, B and C, are equal ; and if in any triangle, the 

 angles B and C, at the bafe B Q, are equal, the triangle is 

 equicrural. 



3. In every fpherical triangle, each fide is lefs than a femi. 

 circle : any two fides taken together are greater than the 

 third ; and all the three fides together are lefs than the peri- 

 phery of a great circle ; and a greater fide is always oppofed 

 to a greater angle, and a lefs fide to a lefs angle. 



4. If in a fpherical triangle B A C (^^■5-)^two legs, 

 A B and B C, taken together, be equal to a femicircle ; 

 the bafe A C being continued to D ; the external angle 

 BCD will be equal to the internal oppofite one B A C. If 

 the two legs together be lefs than a femicircle, the external 

 angle BCD will be greater than the internal oppofite one 

 A ; and if the legs be greater than a femicircle, the external 

 angle BCD will be lefs than the internal oppofite one A ; 

 and tjie reverfe of all thefe holds, viz. if the angle BCD 

 be equal to, greater, or lefs, than A, the fides A B and 

 B C are equal to, greater, or lefs, than a femicircle. 



5. If in a fpherical triangle ABC, two fides, A Band BC, 

 be equal to a femicircle, the angles at the bafe, A and C, 

 are equal to two right ones : if the fides be greater than a 

 femicircle, the angles are greater than two right ones ; and 

 if lefs,'lefs; and converfely. 



6. In every fpherical triangle, each angle is lefs than two 

 right ones ; and the three together are lefs than fix right 

 angles, and greater than two. 



7. If in a fpherical triangle BAC [Jig. 5.) the fides 

 A B and A C be quadrants, the angles at the bafe, B and 

 C, will be right angles. And if the intercepted angle A 

 be a right angle, B C will be a quadrant : if A be obtufe, 

 B C will be greater than a quadrant ; and if acute, lefs ; 

 and converfely. 



8. If in a fpherical reftangular triangle, the fide B C 

 (Jig. 6.) adjacent to the right angle B, be a quadrant, the 

 angle A will be a right angle ; if B E be greater than a 

 quadrant, the angle A will be obtufe ; and if B D be lefs 

 than a quadrant, the angle A will be acute ; and converfely. 



9. If in a fpherical reftangular triangle, each leg be either 

 greater or lels than a quadrant, the hypothenufe will be lefs 

 than a quadrant ; and converfely. 



10. If in a fpherical triangle ABC {Jig. 7.) reftangular 

 only at B, one fide, C B, be greater than a quadrant, and the 

 other fide, A B, lefs, the hypothenufe, A C, will be greater 

 than a quadrant ; and converfely. 



11. If in a fpherical oblique-angular triangle A C B 

 (Jig. 8.) both angles at the baie, A and B, be either obtufe 

 or acute ; the perpendicular C D, let fall from the third 

 angle C, to the oppofite fide A B, falls within the triangle : 

 if one of them. A, be obtufe, and the other, B, acute, 

 the perpendicular falls without the triangle. 



12. If in a fpherical triangle AC, all the angles. A, B, 

 and C, be acute, the fides are each lefs than a quadrant. 

 Hence, if in an oblique-angular fpherical triangle, one fide 

 be greater than a quadrant, one angle is obtufe, -viz. that 

 oppofite to this fide. 



13. If in a fpherical triangle A C B, two angles, A and 

 B, be obtufe, and the third, C, acute ; the fides, A C and 

 C B, oppofite to the obtufe angles are greater than a qua- 

 drant ; and that oppofite to the acute angle, A B, is lefs than 



a qua- 



I 



