110 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[April, 



and Ah c he respectively turned about tlie lines A D and A 6. Then 

 it is manifest, that \\hn\ the points r and I are made to coincide, the 

 radii C c and C t coincide also, and form one of the edges of the; trian- 

 gular pyramid, as may be seen by elevating the corres|)onding planes 

 in the preceding diagram ; and Ijy this means the figare is rccomposed 

 in so far as respects its constituent planes. Another step of the com- 

 position is eliected by bringing into coincidence the straight lines A a, 

 A/,nndV/,'Dg; and when Ac falls upon irf the structure is com- 

 jdete, both as respects the bounding planes and the angles which mea- 

 sure their inclinations. 



It now remains to calculate the several parts of the pyramid, on the 

 supposition tliat the angles at the vertices of the planes j-t^sand sC D 

 are given; and in order to this, 



Let c ^ rCs, the angle at the vertex of the plane iCs, which cor- 

 responds with a portion of the face of the arch, 

 6 =:s C D, the angle at the vertex of the plane s C D, which 

 corresponds with a portion of the plan or base, and is per- 

 pendicular to r C s, 

 « =^ D C /, the angle at the vertex of the plane DC/, which is 

 a portion of the bed or coursing joint, and subtends the in- 

 clination of the planes rCs and s C D, 

 B = A / D, tlie angle that measures the inclination of the 

 planes r C is and DC/, 

 and C = A 6 c, the angle that measures the inclination of the planes 



DC/ and 8 CD. 

 This notation being agreed on, let C A be made the radius ; then by 



the definitions of trigonometry, A a and C a are respi'ctively the sine 

 and cosine of the angle A Co, while A D is the tangent of the angle 

 A C D. But by the construction. A/ is equal to A a, and conseijuently 

 A/is equal to the sine of the angle >Cs; therefore, by the |)rin- 

 ciples of plane trigonometry, we have 



A/ : AD : : rad. : tan. Ay D ; that is, sin. e : tan. 6 : : rad. : 

 tan. B r= tan b, cosec c. 



Here we have determined the angle of inclination between the 

 planes rCs and DC/, and a similar process will discover the angle 

 A 6 c, or the inclination between the planes 8 C D ami D C /. Thus, 

 since C A is radius, A 6 is the sine of the angle s C D to that radius, 

 and by construction, Ac is equal to the tangent of the angle A Co, for 

 Ac is equal to A/i, and Ah is evidently the tangent of the angle A Ca 

 to radius C A ; therefore, by plane trigonometry, we get 

 A6 : Ac : : rad. ; tan. A 6c; that is, sin. 6 1 tan. c : : rad. ; tan. 

 C = tan. e, cosec. 6. 



M'e have next to determine the angle hCd in the plane DC/, and 

 fortius purpose it is only necessary to recollect, that Cg' is equal to 

 the cosine of A C a, and C D equal to the secant of A C D ; hence we 

 have 



CD 



; rad. : COS. D C g ; that is, sec. 6 



cos. c 



rad. 



COS. 



a ^= COS. 6, COS. c. 



And exactly in the same manner, if any other two of the ))arts be 

 given the rest may be found, and the several results when calculated 

 and reduced to their simplest form, are respectively as exhibited in 

 the following table : 



Table offormulcefor calculating the several parti of a right angled triangular pyramid standing on a spherical base. 



Values qfffie angle a, at the vertex 

 of the hypotheiuisal plane B VC. 



sin. a = sin. b cosec. B. 

 sin. a = sin. c cosec. C. 

 tan. o = tan. b sec. C. 

 tan. o = tan. c sec. B. 

 cos. a = cos. b cos. c. 

 COS. o = cot. B cot. C. 



Values of the angle B, mbtended by 

 the base or plane A V C. 



sin. B — sin. b cosec. a. 

 sia. B = sec. c cos. C. 

 tan. B = sec. a cot. C. 

 tan. B = tan. b cosec. c. 

 cos. B = cot. a tan. c, 

 COS. B = cos. b sin. C. 



Values of the angle c, at the vertex 

 of the perpendicular plane A V B. 



Values of the angle C, subtended 

 by the perpendicular plane AV B. 



sin. C = sin. c cosec. a. 

 sin. C = sec. b cos. B. 

 tan, C = sec. a cot. B. 

 tan. C = tan. c cosec. b. 

 cos. C = cot. tan. b. 

 cos. C = cos. c sin. B. 



Values of the angle i, at the vertex 

 of the plane or base A V C. 



sin. i — sin. a sin. B. 

 sin. i = tan. c cot. C. 

 tan. i = tan. a cos. C. 

 tau. i = sin. c tan. B. 

 COS. i = cos. a sec. c. 

 COS. 4 = cos. B cosec. C. 



The above fable contains the simplest forms of the equations neces- 

 sary for resolving the dilVercnt cases and varieties of right angled 

 spherical triangles, as they depenil upon the triangular pyramiil 

 VBAC. It is designed to preclude the necessity of either learning 

 by rote or investigating the various rules and proportions connected 

 W'ith this branch of the subject; for by simply referring to that com- 

 partment of the table which cimtains the values of the quantity sought, 

 an expression will be found denoting the precise operation to be per- 

 formed for the value of the required term. Thus lor exanqile. Suj)- 

 pose that in the right angled spherical triangle B AC, the base AC^= 

 h, and the perpendicular U A ^ c are given, and it is required to find; 



1. The hypothennse BC' = o. 



2. The angle AB C = B contained between the hypothenuse B C 

 and perpendicular B A, or that which is subtended by the base A C. 



To find the hypothennse BC := u, refer to that compartment of the 

 table that contains the values of the hypothennse, ami select that ex- 



pression which exhibits a combination of the given quantities 6 and c 

 This is readilv perceived to be No. 5, the only case in which the two 

 given terms form au equation with the one required ; hence we get 

 COS. a ;= COS. 6 cos. c. 



And the numerical operation denoted by this expression, may, w hen 

 converted into words, be read in the following manner : — 



Multiply the natural cosine of tlie given base, by the natural cosine of 

 the gmu jJirjiindicular, and the product mill give the natural cosine of 

 the hypolhtu tine. 



The multiplication of trigonometrical quantities is however a very 

 laborious process, miless the contracted method of decimal multiplica- 

 tion is resorted to; and since very few of our practical mechanics have 

 taken the trouble to familiarize themselves with the application of 

 that method, the necessity of employing it may be entirely superseded 

 by the use of logarithms. The rule will then be as follows: — 



