SOLID ANGLE. 



hence, all thofe geometers, from the time of Euclid, who 

 have confined their attention principally to the magnitude of 

 the plane angles, inftead of their relative pofitions, have 

 never been able to develope the properties of this clafs of 

 geometrical quantities ; but have affirmed, that no fohd angle 

 can be faid to be half or double of another ; and have fpoken 

 of the bifedtion, trifedtion, &c. of fqlid angles, even in the 

 fimpleft cafes, as impoffible problems. But all this fuppofed 

 difficulty vaniflies, and the doftrine of folid angles becomes 

 fimple and univerfal in its application, by zilumingjp/ierical 

 furjfaces for their meafure, jull as circular arcs are alfumed 

 for the meafure of plane angles. This idea feems to 

 have firft occurred to Albert Girard, and publifhed by 

 him in his " Inventions Nouvelles en I'Algebra," an account 

 of which is given at p. 7, vol. iii. Montucla's Hiftory of 

 Mathematics. The principle of this method, however, 

 feems to have efcaped the attention of geometers, till it was 

 recently brought forward again in vol. iii. of Hutton's Courfe 

 of Mathematics, the writer of which, it fliould be obferved, 

 was not aware of the fubjedl having been previoufly treated 

 on the fame principles by Girard. We fliall, in what fol- 

 lows, avail ourfelves of the latter article. 



Imagine, then, that from the fummit of a fohd angle, 

 (formed by the meeting of three planes,) as a centre, any 

 fphere be defcribed, and that thofe planes are produced till 

 they cut the furface of the fphere ; then will the furface of 

 the fpherical triangle, included between thofe planes, be a 

 proper meafure of the folid angle, made by the planes at 

 their common point of meeting, for no change can be con- 

 ceived in the relative pofition of thofe planes, without a 

 correfponding and proportional mutation in the furface of 

 the fpherical triangle. If, in like manner, the three or 

 more furfaces, which by their meeting conftitute another 

 folid angle, be produced till they cut the furface of the 

 fame, or an equal fphere, whofe centre coincides with the 

 fummit of the angle ; the furface of the fpheric triangle, or 

 polygon, included between the planes which determme the 

 angle, will be a correft meafure of that angle. And the 

 ratio wtiich fubfitls between the areas of the fpheric trian- 

 gles, polygons, or other furfaces thus formed, will be ac- 

 curately the ratio that fubfifts between the fohd angles, 

 conftituted by the meeting of the feveral planes or fur- 

 faces at the centre of the fphere. 



Hence the comparifon of folid angles becomes a matter of 

 great eafe and fimplicity ; for, fince the areas of fpherical 

 triangles are meafured by the excefs of the fums of their 

 angles, each above two right angles, and the areas of fphe- 

 rical polygons, of n fides, by the excefs of the fum of their 

 angles above (2 n — 4) right angles, it follows, that the 

 magnitude of a trilateral folid angle will be meafured by 

 the excefs of the fum of the three angles, made refpeftively 

 by its bounding planes above two right angles ; and the 

 magnitude of fohd angles, formed by « bounding planes, 

 by the excefs of the fum of the angles of inclination of the 

 feveral planes above (2 « — 4) right angles. 



Ai to folid angles, limited by curve furfaces, fucli as the 

 angles at the vertex of cones ; they will manifeitly be mea- 

 fured by the fpheric furface, in the fame manner as angfes 

 determined by plane furfaces, are meafured by the triangles or 

 polygons they mark out upon the fame, or an equal fphere. 

 In all cafes, the maximum limit of folid angles will be the 

 plane towards which various planes determining fuch angles 

 approach, as they diverge fartiier from each other about the 

 fame fummit ; the fame as a right line is the maximum limit 

 of plane angles, being formed by the two bounding.lines, 

 vfhen they make an angle of 180°. The maximum limit of 

 fohd angles is meafured by the furface of the hemifphcrc, in 



Vol. XXXIII. 



like manner as the maximum limit of platie angles is meafured 

 by the arc of a femicircle. The folid right angle (the angle, 

 for example, of a cube) is 4: = (5)' of the maximum fohd 

 angle ; while the plane right angle is half the maximum 

 plane angle. 



The analogy between plane and folid angles being thu» 

 traced, we may proceed to exempHfy this theory by a few 

 inilances ; affuming 1000 as the numeral meafure of the 

 maximum folid angle = 4 times 90^ folid = 360° folid. 



I. The folid angles of right prifms are compared with 

 great facility. For of the three angles made by the 

 three planes, which by their meeting conftitute every fuch 

 fohd angle, two are right angles, and the third is the fame 

 as the correfponding plane angle of the polygonal bafe ; on 

 which, therefore, the meafure of the folid angle depends. 

 Thus, with refpeft to the riglit prifm with an equilateral 

 triangular bafe, each fohd angle is formed by planes, which 

 relpeftively make angles of 90°, 90', and 60^^. Confe- 

 quently, 90"^ + 90° + 60° — 180° = 60°, is the meafure 

 of fuch angle, compared with 360°, the maximum angle, 

 and is therefore one-fixth of the maximum angle. 



A right prifm, with a fquare bafe, has, in like manner, 

 each folid angle = 90° + 90° + 90° — 180° = 90°, which 

 is one-fourth of the maximum angle ; and thus it may be 

 found that each folid angle of a right prifm, with an equi- 

 lateral 



igle = 



2 m 



Hence it may be (hewn, that each folid angle of a re. 

 gular prifm with triangular bafe, is ha/f each folid angle 

 of a prifm of an hexagonal bafe. Each with regular 



fquare bafe = I of each with regular ofkagonal bafe. 

 pentagonal =4 ... decagonal, 



hexagonal = -J ... duodecagonal. 



4 m-gonal = ... nt-gonal. 



in — 2 



Hence, again, we may infer, that the fum of all the 

 folid angles of any prifm of triangular bafe, whether that 

 bafe be regular or irregular, is half the lum of the folid 

 angles of a prifm of quadrangular bafe, regular or irregular ; 

 and the fum of the fohd angles of any prifm of 



tetragonal bafe = I fum of angles of pentagonal bafe. 

 pentagonal bafe = 4 - - . hexagonal, 

 hexagonal bafe =: ^ • - - heptagonal. 

 m — 2 



»(-gonal bafe ;= 



m — I 



(m + i)-gonal. 



2. Let us now compare the folid angles of the five regular 

 bodies. In thcfe bodies, if m be the number of fide;; of 

 each face ; n the number of planes wliich meet at eacli 



R r fohd 



