711 



SPHERICAL TRIOOS'OMETRY. 



710 



not exist ; if it be equal to unity, the triangle U right angled at B, ami 

 may be treaU-d as a right-angled triangle. But if sin o sin A be leoa 

 than unity, laid .- from ain i = sin 6 ain A, and x and B from 



in '. 



Un x tan b cot A, ain B - ain A -. . 



ain a 



There are two value* of B, supplement* of each other, both of which 

 are possible answer*. Lot B be the one which is leu than a right 

 angle, and B' that which is greater. 



First, when A is acute and a lees than li. Calculate y from cos y = 

 coa a -7- coa :, and b't and o'r from 



tan* 

 cot b't = co 6 tan A, COB o'j = ^ n ~^- 



There are two triangles which satisfy the data ; in the first, 

 e = x y, B' is the angle opposite to b, and c 6** a"z ; in the 

 second, e = x + y, B is the angle opposite to 6 and o b"z + a'z. 



Secondly, when A is acute and a equal to or greater than 6. Cal- 

 culate exactly as in the last case, but there is only one triangle which 

 satisfies the data, namely, the second of the preceding. 



Thirdly, when A is obtuse, in which cage there is no triangle, unless 

 a be greater than b. Calculate (using A as more convenient) 



sin 6 

 sin i = sin 6 sin A', tail x = tan b cos A, ain B = sin A' JJTJ-^ 



Use the value of B which is less than a right angle ; calculate J'c and 

 a": from 



cot 6': < 



tan z 



cos b tan A', cos a'z = ; . 



tan a 



Then e = y - x, and c = 0*2 I':. 



In the case in which one or both of the sides are greater than a right 

 'angle, which rarely, if ever, occurs, it is best to have recourse to one 

 of the adjacent triangles described at the beginning of this article, and 

 to use it in the same manner as the supplementary triangle has been 

 used. It is not however necessary to dwell on this point. 



"u^filfneni. Given two angles (A and B) and a side opposite to one 

 of them (a) ; required the remaining parta. Let A' and B be the sides 

 of a triangle, and a' the angle opposite to A'. Find c' the remaining 

 side, and b' and c' the remaining angles ; then c is the remaining 

 angle of the original triangle, and b and c the remaining sides. 



All the cases would need some subdivision to adapt them to cal- 

 culation, if it were really often required to solve triangles with very 

 large sides and angles. But in application it generally happens that 

 the reasoning of the previous part of the process is so conducted as to 

 throw the calculation entirely upon triangles which have at least two 

 sides and two angles severally less than a right angle. Divide a great 

 circle into three parts, and we have the extreme limit of a spherical 

 triangle : the sum of its sides being 360, and the sum of its angles 

 six right angles. But a triangle which should be very near to this 

 limit would be best used in reasoning, and solved in practice, by means 

 of one of the other seven triangles into which its circles divide the 

 sphere. And if one of the sides should be greater than two right 

 angles, the remainder of the hemisphere would be the triangle on 

 which calculation is employed. And it is to be understood that all 

 the formulae are demonstrated only for the case in which every side is 

 less than two right angles. At the same time we should recommend 

 the beginner to procure a small sphere, and to habituate himself to the 

 appearance of all species of triangles. 



The area of a spherical triangle is singularly connected with the sum 

 of its angles, on which alone it depends, the sphere being given. Let 

 any two triangles, however differently formed, have the sum of their 

 angles the same, and they must have the same area. If the angles be 

 measured in theoretical unite [ANGLE], the formula is as follows : r 

 being the radius, 



Area = r 5 (A + B + c T), 



which gives the number of square units in the area, the radius being 

 expressed in corresponding linear units. But if the angles be measured 

 in degrees and fractions of a degree, the formula ia 



Area = '01745329252 r 3 (A + B + c 180). 



The angle A + B + c - 180 is called the tpherical exctu, and it may 

 be found at once from the sides by the formula 



i a i i c 

 **" ~sT **" ~T~ ; 



so that the area of a triangle is easily found from its side*. If a 

 spherical triangle were flattened into a plane one, without any altera- 

 tion ( the length* of its sides, it is obvious that the sum of the angles 

 would undergo a diminution, being reduced to 180. The angles 

 would not fliininuh equally; but, if the sphericity of the o 

 triangle were small, or if it occupied only a small part of the sphere, 

 tlin diminutions which the several angles would undergo in tli. 

 "f I- mi; flattened would be so nearly equal, that it would !. useless, 

 for any practical purpose, to consider them a unequal For a triangle 



f small sphericity, then, it may be assumed that in being flattened, 

 each of iU angles loses one-third ..f thn spherical excess. This pro- 

 a is one of considerable use in the measurement of a degree of 

 the meriili in. 



BPHKROQRAPH, an instiiiineiil invented in 1 856, for facilitating 

 the practical use of spherics in navigation, Ac., being a contrivai; 

 constriietinR, without dividers or scales, any p< ric triangle, 



and reading oil' the measures of the parts required; thus in most 

 oases saving much time and labour. The degree of accuracy of the 

 instrument is limited only by ,t it has been found by navi- 



gators that circles of 5-inch radius will work any question which arises 

 at sea, siil!':cient.|y near for the practical purposes of the navigator. 



The description of the instrument will be better understood by 

 giving some general preparatory hints as to the names of the ordinary 

 lines of the sphere. 



In th.' following figure, No. 1, the observer is supposed to be at the 

 centre c of a hollow transparent sphere, on which' tin- usual lines are 

 drawn as upon a terrestrial globe. He would see the sun as at o upon 

 the globe's surface amongst these lines. Rejecting all superfluous 



Fig. 1. 



circles, &c., and confining our description to such parts as affect the 

 sun's position at the time, we have in fig. 1 what is called a pro- 

 jection of the sphere on the plane of the meridian, the primitive circle 

 representing the meridian of the place of the observer, H being the 

 south part of the horizon H n, and n being the north, z will be the 

 zenith, N the nadir, c will be the east or west point of the horizon, i' 

 the north pole, s the south pole of the world ; a I parallel of altitude 

 in which the sun is at the moment of observation, d c the sun's decli- 

 nation, t iv a parallel of 18 distance below the horizon, limiting twilight 

 to the period at which the sun is traversing from s where he sets, to c 

 where he will be at midnight, d being his place at noon ; H d being 

 the meridian altitude, p K being the latitude (represented by the 

 height of the pole above the horizon), then p z will be the co-latitude, 

 o n the sun's altitude as measured upon the azimuth circle z ; o z 

 will be the zenith distance ; of being the sun's declination as measured 

 upon the hour circle r/s, o P will be the polar distance, E Q the equator, 

 z i the amplitude of the sun at setting the angle R r t being the time 

 of sun-set, the angle z p o the time of observation. The small circle 

 d i is the semi-diurnal arc, or half the length of the day, and s c the 

 semi-nocturnal arc, or half the length of the night (c falling within R w 

 there will be no real night, only twilight), P s is the six o'clock hour- 

 circle, z N the prime vertical, &c. In the sphorograph only five terms 

 are principally used, namely, latitude, declination, time, azimuth, and 

 altitude, and what precedes will have fully prepared the mind of the 

 reader for the application of the instrument to practical purposes. 



The instrument is composed of two pieces of stout card-board, nicely 

 attached, and revolving concentrically upon a pin carefully turned to 

 work without lateral motion in an ivory collar. The upper card has 

 the ruled part, on which the lines are described, formed of stout trans- 

 parent tracing paper, and it is ruled like fy. 2 ; having only azimuths 



and parallels of altitude (it has an oval space cut so as to enable the 

 observer to put pencil marks on the under sphere), while the undo 



