S P H 



6. When any two fides, a, b, and their contained angle C, 



are given, then we have 



cot. i a . cot. ^ b + cof. C 

 cot. 1 E = -^^ 



7. When the three fides are given, we may alfo, inftead 

 of the formula in art. ;, make ufe of the following elegant 

 theorem, djfcovered by Simon L'HuUier, viz. 



tan. i E = ^/ (tan. 



J + 4 + f 



tan. 



+ *■ 



tan. 



~a^b + c- 

 tan. 



)• 



One or other of thefe rules will apply to all cafes in 

 which the fpherical excefs will be required. 



We (hall conclude this fhort article with one or two ex- 

 amples of its application, as given by general Roy, Phil. 

 Tranf. 1790, p. 172. 



Names of Stations. 



Hanger-hill Tower (a) 

 Hampton Poor-houfe (i) 

 King's Arbour \c) 



Obferved Angles. 



42° 2' 32" 



67 55 39 

 70 I 48 



179 59 59 



Diflance in Feet. 



(a) from {b) = 38461.12 

 {a) from (c) = 27404.7 



Hence, making the diftance from (a) to (f ) the bafe of 

 the triangle, the perpendicular on that bafe will be equal 

 to 38461.12 X fin. 42° 2' 32"; and, therefore, the area of 

 the triangle = 



bafe X perp. ^ r 011/ 

 J— ^ = 24704.7 X 19230.56 X fin. 42' 2' 32". 



Computation. 



Log. fm. 42° 2' 32" = 



Log. 24704.7 = 



Log. 19230.56 = 



9.8258661 

 4.3927761 

 4.2839906 



18.5026328 



Taking away 10 for radius, we have 8.5026328 for the 

 log. of the area in feet. 



Whence from 

 Take conft. log. 



Correfp. numb. .14992 



8.5026328 

 9-3267737 



1.7158591 



Whence the fpherical excefs in feeonds is .14992", or 

 «>''. I 5 nearly. 



Again, as a fecond example. 



Names of Stations. 



Hundred Acres {^) 



Hanger-hill Tower (e) 

 St. Anne's Hill (/) 



Vbferved Angles, 



5f5^'3S"-7S 

 68 24 44 



57 36 39-5 



179 59 59.25 



S P H 



Diftance in Feet. 



id) from {e) = 7 '934-2 

 (d) from (/) =: 79211.22 



Log. fin. 53° 58' 35".75 

 Log. 35967.1 

 Log. 79211.22 



Sum minus radius 

 Subtraft conft. log. 



= 9-9078237 



= 45559054 



= 4.8987866 



= 9.3625207 



== 9-3267737 



Correfp. numb. 1.0858 = 0.0357470 



Whence the fpherical excefs is i".o858. 



In this manner the computed fpherical excefs will enable 

 the obferver to examine the accuracy of his obfervations, 

 and in fome degree to correft them ; after which he may 

 proceed to calculate the fides of the triangles by the rules 

 of fpherical trigonometry, or by Legendre's theorem, viz. 

 " a fpherical triangle being prupofed, of which the fides 

 are very fmall, relatively to the radius of the fphere ; if, 

 from each of the angles, one-third of the excefs of the fum 

 above two right angles be fubtrafled, the angles fo dimi- 

 nifhed may be taken for the angles of a reftilinear triangle, 

 the fides of which are equal in length to thofe of the pro- 

 pofed fpherical triangle." 



Spherical Geometry, the doftrine of the fphere ; parti- 

 cularly of the circles defcribed on the furface thereof, with 

 the method of projefting the fame on a plane ; and meafuring 

 their arcs and angles when projefted. 



Spherical Numbers. See Circular Numbers. 



Spherical Trigonometry. See Spherical Trigonome- 

 try. 



SPHERICITY, the quality of a fphere; or that by 

 which a thing becomes fpherical or round. 



The fphericity of pebbles, fruits, berries, &c. as alfo of 

 drops of water, quickfilver, &c. and of bubbles of air under 

 water, &c. Dr. Hook takes to arife from the incongruity 

 of their particles with thofe of the ambient fluid, which 

 prevents their coalefcing ; and by prefling on them, and 

 encompafling them all round equally, turns them into a 

 round form. 



This, he thinks, appears evidently from the manner of 

 making fmall round fhot of feveral fizes, without cafting 

 the lead into any moulds ; from drops of rain being formed, 

 in their fall, into round hail-itones ; and from drops of 

 water faUing on fmall duft, fand, &c. which foon produce 

 an artificial round mafs ; and from the fmall, round, red- 

 hot balls, formed by the collifion or fufion of flint and fteel, 

 in ftriking fire. 



But all thefe cafes of fphericity fecm better accounted for 

 from the great principle of attraftion ; by which the parts 

 of the fame fluid drop, &c. are all naturally ranged as near 

 the centre as poffible, which necefiarily induces a fpherical 

 figure ; and, perhaps, a repelling force between the par- 

 ticles of the drop, and of the medium, contributes alfo not 

 a little thereto. See Attraction and Cohesion. 



SPHERICS, the doSrine of the fphere, particularly of 

 the feveral circles defcribed on its furface ; with the method 

 of projefting the fame in piano. 



The principal matters (hewn here are as follow : 



I . If a fphere be cut in any manner, the plane of the 

 feftion will be a circle, whofe centre is in the diameter of 

 the fphere. 



Hence, i. The diameter HI {Plate 11. Trigononutry, 

 Jig' 9.) of a circle, pafling through the centre C> is equal 



to 



