TRIGONOMETRY.

767

The Solution of the Cases of Oblique Spherical Triangles, (fig. 19 and 20.)

Case

Given

Sought

Solution

7

Two angles A, ACB, andthe side AC between them

Either of the othersides, suppose BC

As rad. ; co-sine AC : : tang. A : co-tang. ACD (by theorem 4,) whenceBCD is also known; then, as co-sine BCD : co-sine ACD ; ; tang. AC :tang. BC (by theorem 1.)

8

Two angles A, B, and a sideAC opposite to one ofthem

The side BC oppositethe other

As sine B : sine AC :: sine A : sine BC (by theorem 1.)'

9

Two angles A, B, and aside AC opposite to oneof them

The side AB betweenthem

As rad.: co-sine A : : tang. AC : tang. AD (by theorem 1,) and as tang.

B : tang. A; ; sine AD : sine BD (by theorem 4,) whence AB is also known.

10

Two angles A, B, and a sideAC opposite to one ofthem

The other angle ACB

As rad. : co-sine AC : ; tang. A ; co-tang. ACD (by theorem 4,) and asco-sine A : co-sine B : : sine ACD : sine BCD (by theorem 3,) whenceACB is also known.

11

All the three sides AB,AC, and BC

An angle suppose A

As tang. £AB : tang.^i2£; : tang,- C ~ B £ ; tang. DE, the distance

of the perpendicular from the middle of the base (by theorem 6,) whenceAD is known; then, as tang. AC ; tang. AD ; ; rad. : co-sine A (bytheorem 1.)

12

All the three angles A, B,and ACB

A side, suppose AC

ABC + A , ABC—A ACB

As co-tang.---: tang. ---; ; tang.—-—: tang, of the angle

included by the perpendicular and a line bisecting the vertical angles;whence ACD is also known; then (by theorem 5,) tang. A : co-tang.ACD : : rad. : co-sine AC.

.The following propositions and remarks concerning sphericaltriangles (communicated to Dr. Hutton by Dr. Maskelyne) will

also render the calculation ol them perspicuous • 1

1. A spherical triangle is equilateral, isoscelar, or scalene ac-

cording as it has its three angles all equal, or two of them equalor all three unequal; and vice versa. ^ ’

2. The greatest side is always opposite the greatest angle, and thesmallest side opposite the smallest angle.

3. Any two sides taken together are greater than the third.

4. It the three angles are all acute, or all right, or all obtuse;the three sides will be, accordingly, all less than 90°, or equal to90’, or greater than 90°; and vice versa.

•5. If ( r om the three angles A, B, C, of a triangle ABC,(fig. 21,) as poles, there be described upon the surface of thesphere, three arches of a great circle DE, DF, FE, forming by‘he intersections a new spherical triangle DEF ; each side of thet ' ew ’ triangle will be the supplement of the angle at its pole; andeac h angle of the same triangle will be the supplement of the side°P|)osite to it in the triangle ABC.

6, In any’triangle ABC, or AbC, (fig. 22,) right-angled in A,

. s ‘> The angles at the hypothenuse are always of the same kindp* their opposite sides; 2dly, The hypothenuse is less or greater. 311 a quadrant, according as the sides including the right angleo le of the same or different kinds; that is to say, according aslg se same sides are either both acute or both obtuse,'or as one isp u 'e and the other obtuse. And vice versa, 1st, The sides in-^ uding the right angle are always of the same kind as their op-jip s 'te angles; 2dly, The sides including the right angle will be ofe same or different kinds, according as the hypothenuse is lessmore than 90°; but one at least of them will be of 90°, if the■>P° l henuse he so.

pit 7 s ball here present the reader with a concise view of the ap-in Catl0n °f spherical trigonometry, in the solution of problemsst ru a8 !. ron °my, geography, navigation, dialing, &c. The con-hiin l 0n of a ‘Etngiain for presenting at one view the greatesttnicM rS of s P lie rical triangles, in order to the solution of astrono-st e . problems, is that of fig. 1, Plate CLXXXI. which is aridi a °®I a P* lica l projection of the sphere on the plane of that me-l) ^NQS, which is called the solstitial colurc. For the defini-

tions of the several circles ot the split, c. occ .astronomy, ailuGeography. In this scheme, AiQ is the equinoctial, and itstwo poles, N, S. EL is the ecliptic, and its poles P, T. HO isthe horizon, its poles Z, the zenith, and D the Nadir. ZD, theprime vertical, and its poles FJ, O. Ess is the tropic of cancer*y? L, the tropic of capricorn. 0 is the sun’s place in the eclip-tic ; N©S a hour-circle, and ZQD a verticle circle passingthrough the sun 0. PRT a circle of longitude, and NRS acircle : of declination passing through a star at K. Now these vari-ous circles of the sphere by their intersections, form many spheri-cal triangles, both rectangular and oblique- We shall first specifynose tig it-angled tnangles (hat are of most use in astronomy,

h, wl ich e BC is .1,7' 7 st ’. '7 BC0, right-angled at 11;

. Inch BC is the sun s right ascension, or its complement to thepoint G. CO IS the sun s longitude, or complement to C. Bqis the suns decimation; and the angle BC© the obliquity ofthe ecliptic. .Secondly, the triangle A©C, right-angled at A *in which the side C0 is the sun’s longitude in the ecliptic from’C ; the side A ©, its altitude above the horizon. AC is the azimuth from the point C. And the angle AC© is the sum of theco-latitude and obliquity of the ecliptic, viz. /EH -j-^EE. Thirdl •Let the sun be in the tropic of cancer, and in the point G in theprime vertical, when due East or West. Then in the triangleVCG, right-angled at V, the side VC is the hour from six. 'flu-side VG = /EE, the sun’s declination. The side CG the sun’saltitude when East or West. And the angle VCG = A£Z f]latitude of the place. Fourthly, On the same day the sun is’at 1°at the hour of six precisely;'then in the triangle ICM r j 0 i lt ’angled at M; the hypothenuse Cl is the sun’s declination - fllfthe sun’s altitude at six; and CM the azimuth from C u

angle ICM = NO = NIZ, the latitude of the place. Fifthly 1the same day the sun is in the horizon at K, and ib Pr e -J<■„ °‘l

the triangle WCK right-angled at W; in which WK is the W,declination. CW is the ascensional difference, or tl 1P t„„„. Vsix, of its rising and selling; CK, the amplitude of risiiw and set’

ting from the East to West. And the an<de WCK_(fo ;= 7

co-latitude of the place. Sixthly, In the triangle NKO,’right!angled at O, he side NO is the latitude; the side NK the co de’clination; and the side KO the co*anipHtude or azimuth from O,

And*