S P H 



a quadrant. Hence, if the two fides be lefs than a qua- 

 drant, the two angles are acute. 



14. If in a fpherical triangle the feveral fides be each 

 greater than a qua Irant ; or only two of them greater, 

 and the third be equal to a quadrant, the feveral angles are 

 obtufe. 



I J. If in an oblique-anoular fpherical triangle two fides 

 be lefs than a quadrant, and the third greater ; the angle 

 oppofite to the greateft will be obtufe, and the reft 

 acute. 



Spherical Triangles, Refolution of. See Triangle. 



Spherical Afironomy, th.U part of aftronomy which 

 confiders the univerfe fuch as it appears to the eye. See 

 Astronomy. 



Under fpherical aftronomy, then, are comprehended all 

 the phenomena and appearances of the heavens and hea- 

 venly bodies, fuch as we perceive them, without any enquiry 

 into the reafon, the theory, or the truth of them. By which 

 it is diftinguilhed from tbcor'ical aftronomy, which con- 

 fiders the real ftrufture of the univerfe, and the caufes of 

 thofe phenomena. 



In the fpherical aftronomy, the world is conceived to be 

 a concave fpherical furface, in whofe centre is the earth, or 

 rather the eye, about which the vifible frame revolves, with 

 ilars and planets fixed in its circumference. And on this 

 fuppolilion all the other phenomena are determined. 



The theorical aftronomy teaches us, fr^.m the laws of op- 

 tic?, &c. to correft this fcheme, and reduce the whole to a 

 jufter fyllem. 



Spherical Compajes. See Compasses. 



Spherical Excefs, in Trigonometry, is the excefs of the 

 )uni of the three angles of any fpherical triangle above three 

 right angles ; which excefs in fcconds, multiplied by the 

 radius of the fphere, is equal to the area of the triangle, as 

 was firft ftiewn by Albert Girard. 



It is extremely defirable, in geodetic operations, to be 

 able to afccrtain the fpherical excefs from other principles 

 than thofe of the obferved angles ; thefe being very fubjeft 

 to fmall inaccuracies, from the effeft of refraftion near the 

 horizon : and nothing feems bettter calculated for this pur- 

 pofe, than the foregoing property of the area of the triangle ; 

 for this being, as we have ftated above, equal to the excefs 

 multiplied by the radius of the fphere, therefore, converfely, 

 the area being known or computed on other principles, the 

 excefs may thence be determined, and the accuracy of the 

 obferved angles fubmitted to the tell thence obtained. 

 Now as fuch triangles as occur, even in the molt extenfive 

 lurveys, differ but little from reftilinear ones, their areas 

 may be computed as if they really were fuch, with fcarcely 

 any perceptible error ; and hence this important objeft is 

 at once obtained. 



The application of Albert Girard's theorem was firft 

 made by general Roy, or rather by Mr. Dalby, his afiiftant, 

 and publifticd in the Ptiilofophical Tranfadlions for 1790, 

 p. 171, where we have the following rule : " From the area 

 of the logarithms of the triangle, computed as a plane one, 

 in feet, fubtract the coiiftant logarithm 9-3267737, and the 

 remainder is the logarithm of the excels above 180° in fe- 

 conds nearly." This rule is very general, and, being the 

 firft application of this principle to geodetic computations, 

 is highly creditable to its inventor ; but it is not always the 

 moft concifc ; and other rules have fince been given by dif- 

 ferent authors, which are applicable to every variety of data. 

 We can, however, in this place only enumerate fome of the 

 principal ones, and muft refer the reader for their invefti. 

 gations to the works from which they have been felefted. 



Vol. XXXIII. 



S P H 



I. The fpherical excefs may be found in feconds by the 



R"S 

 expreflion E = ; where E is the excefs ; S^is the fur- 



iar.fin. B = ia- "" ~^ "^T ; r is the radius of the 



face of the triangle = i i c . fin. A = | a i . fin. C = 

 fin. B . fin. C 

 ^''" fin. (B + C)' 

 earth, in the fame meafure as a, b, c, the fides of the triangle ; 

 and R" = 2o6264".8, the feconds in an arc equal in length 

 to the radius. 



If this rule be applied logarithmically, then log. R" =: 



2. Hence is readily deduced general Roy's rule, as above 

 given, TOz. 



log. E" = log. (area in feet) — 9-3267737. 



3. Since S = iif . fin. A, we fiiall manifellly Hstc 



R" 



E = b c . fin. A. Hence, if from the vertical aHgle 



2 r' * 



B, of- a fpherical triangle ABC, we demit the perpendi- 

 cular B D, upon the bafe AC, dividing it into two feg- 

 ments a and /3, we (hall have b ^^ 01, ■\- ^\ and hence, 



R" 



2r' 

 R" 

 2r' 



f (a -f- |S) fin. A 



«c fin. A + /3 c fin- A. 



But the two right-angled triangles being in all praftical 

 cafes nearly redlilinear, gives <x = /i cof. C, and ^■=.c cof. A ; 

 whence we have 



R" R" 



E = -^— a c fin. A . cof. C + ■ <:' fin. A cof. A. 



2 r' 2r' 



In like manner, the triangle ABC, which is itfelf fo 

 fmall as to differ but a little from a plane triangle, gives 

 c fin. A = a fin. C, alfo fin. A . cof. A = i fin. 2 A, and 

 fin, C . cof. C = I fin. 2 C ; therefore, finally, 



R" R" 



E = -^— a' . fin. 2 C -F -— - c'- fin. 2 A. 

 4r- 4r'^ 



From which theorem a table may be formed, whence the 

 fpherical excefs may be found ; entering the table with each 

 of the fides above the bafe and its adjacent angle as argu- 

 ments. 



4. If the bafe b, and height h, of the triangle be given, 



R" 

 then we have evidently E = i** — —. Hence refuitt 



the following fimple logaritiimic rule : Add the logarithm 

 of the bafe of the triangle, taken in feet, to the logarithm 

 of the perpendicular, taken in the fame meafure, and dcduft 

 from the fum the conftaiit log. 9.6278037; and the re- 

 mainder will be the common logarithm of the fpherical 

 excefs in feconds and decimals. 



5. Again, when the three fides of the triangle are given, 

 then 



log. E = i ( log. t -»- log. (j - <7) 4- log. (/ - *) -f- log. 



(/-<:)) -9 32'57737; 

 where x — 5 fum of the fides, or .f = (a -f i -| f) -r- I. 



The two latter formula- are in faA only pirticular cafes 

 of Roy's general theorem. 



' 3T 6. When 



