PROPOSITION VI. 68. To investigate formulæ convenient for logarithmic computation, for determining two sides of any spherical triangle when there are given the other side and the two adjacent angles. In a spherical triangle ABC, the angles at B and C and the side BC being given; it is required to find the sides A B and AC. From the formulæ (a') and (b') in Prop. III., omitting the accents, after multiplying both members of the latter by cos. A, and substituting as in the last Proposition, there will be obtained the equation cos. AB sin. A=cos. C sin. B+ cos. BC cos. B sin. C .......(m') Also from the formulæ (c') and (b) in Prop. III., or, which is the same, writing C for B and B for c in the last equation, there is obtained cos. AC sin. A=cos. B sin. C+ cos. BC cos. C sin. B. (n'). Adding together (n') and (m'), and afterwards subtracting (n ́) from (m') we get, on transforming, as in Art. 67., sin. A cos. (AB + AC) cos. (AB — AC) = sin. (B+ c) cos. BC .... (p'), and sin. A sin. (AB + AC) sin. (AB — AC) = sin (B-C) sin.2 BC . . . . (9). Then, dividing (q′) by (p') there is obtained sin. (AB+ AC) sin. 1 (AB—AC) _sin. (B-C) cos. = 1 (B–C) cos. (AB+ AC) cos. (AB-AC) sin. (B+c) cos. (B+C) tan.2 BC.... 21 This last equation being first multiplied by (s), and afterwards divided by (s), in the last Proposition, there will be obtained after the necessary reductions Thus the values of AB and AC may be separately found. 66 Napier's 69. The investigation of formulæ expressing the relations between an angle and its orthographical projection on a plane inclined to that of the angle may be made as follows. If the vertices of the given and projected angles are to be coincident, the sides containing the given angle and its projection may be conceived to be in the planes of two great circles of a sphere whose centre is the angular point, and which intersect each other in a line passing through that point. Thus, let acb be the given angle, then, zc being any line passing through c, if the planes of two circles pass through zc and the C/ b B/ B lines ca, cb, and meet a plane passing through C perpendicularly to zc, the intersections of the circles with the latter plane will be in the lines CA, CB, and the angle ACB will be the orthographical projection of acb. Now the arc ab, of a great circle, measures the angle acb; and the arc AB measures the projected angle ACB; therefore the arcs Aa, вb, or the angles aCA, bCB being given, their complements za, zb are known; and in the spherical triangle a zb, with the three sides za, zb, ab, the angle azb may be computed by one of the formulæ (a), (b), or (c), Prop. I., or by one of the formulæ (1), (11), or (111) Prop. IV., and consequently its equal ACB is found. If one of the sides, as cb, were coincident with a side, as CB, of the reduced angle, since zb would then be a quadrant, its sine would be equal to radius, and its cosine to zero: therefore, one of the formulæ (a), (b), or (c), Prop. I. would give or cos. azb= cos. ab sin. Za' or = Cos. ab (radius being unity), cos. Aca: cos. acb :: rad. (= 1): cos. ACB. A particular formula for the reduction of the angle ach, when Aa and Bb are small arcs, will be given in the chapter on Geodesy (art. 397.). If the arcs Aa, Bb, or the angles ACa, BCb be equal to one another, each of them may be considered as the inclination of the plane acb to ACB, and the reduction may be made thus: — imagine a plane, as ac'b, to pass through ab parallel to AC B, and the straight lines, or chords, ab, AB to be drawn; then the angle ac'b will be equal to ACB, and the triangles a c'b, ACB will be similar to one another: therefore c'b CB chord ab chord AB, or as 2 sin. arc. ab : 2 sin.arc. A B. But therefore or c'b: CB:: sin. zb: rad., or as cos. Bb: radius ; 70. If it were required to find the length of an arc, as AB of a great circle from the given length of the corresponding arc ab, of a small circle, having the same poles (of which let z be one), and consequently (Sph. Geom., 1 Cor. 1 Def.) parallel to it, the sectors ACB, ac'b being similar to one another, we have c'a: CA ab: AB. But za is the distance of the small circle from its pole z; and, the radius of the sphere being unity, c'a is the sine of that distance; therefore the above proportion becomes Conversely ab = AB sin. Zα = AB cos. Aα. If the chord of the arc ab were transferred from A to B' on the arc AB, we should have chord ab=c'a. 2 sin. 1⁄2 a c'b, and chord A B'= CA. 2 sin. & ACB': but chord ab chord AB'; therefore When the arc ab is small, we have, nearly, the angle ACB′ = a c'b cos. Aa; or, conversely, a cb = ACB' cos. A a 71. When two great circles, as PA, PB, make with each other a small angle at P, their point of intersection, and when from any point as M, in one of these, an arc Mp of a great circle is let fall perpendicularly on the other; also from the same point M, an arc Mq, of a small circle having P for its pole, is described; it may be required to find approximatively the value of pq, and the difference between the arcs Mq and Mp. p P Let c be the centre of the sphere, and draw the radii CM, cp; also imagine a plane MC'q to pass through Mq perpendicularly to PC, and let it cut the plane MCp in the line MN. Then, since the plane MC'q is perpendicular to PC, it is perpendicular to the planes PCB, PCA (Geom., 18 Planes); the plane MCP is also perpendicular to PCB; therefore the line of section MN is perpendicular to PCB and to the lines B A M c'q, cp, which it meets in that plane; and the angle MC'q is equal to BPA or BCA, the inclination of the circles PA, PB to one another. Now, if the radius of the sphere be considered as unity, we shall have MC'sin. PM, and cc' C'N (= MC' cos. MC'N) C'N (CC' tan. PCP) cos. PM; then sin. PM cos. P, also therefore sin. PM cos. P cos. PM tan. Pp, cos. PM tan. Pp; But 1-cos. P= 2 sin. 2P (Pl. Trigon., art. 36.), and since P is supposed to be small, the arc which measures the angle may be put for its sine; therefore 1-cos. P becomes (P being expressed in seconds so that P sin. 1" may represent sin. P) equal to p2 sin.2 1". Also, for sin. (PM-PP) may be put (PM-Pp) sin. 1", or pq sin. 1", and in the denominator, Pp may be considered as equal to PM; therefore, for the denominator there may be put sin. PM cos. PM, or its equivalent sin. 2 PM. Thus we obtain, approximatively, P2 sin. 21" = PM-PP sin. 2 PM sin. 1", or p2 sin. 2 PM sin. 1"=pg (in seconds). Again, since MN = sin. Mp, on developing Mp in terms of its sine (Pl. Trigon., art. 47.), neglecting powers higher than the third, we have Subtracting from this the above equation for мp, we obtain = ain.2 PM MN3 cotan.2 PM; or expressing Mp in seconds, and putting Mp sin. 1" for MN, we have (in arc) Mp3 sin.31" cotan.2PM, and, in seconds, = Mp3 sin.2 1" cotan.2 PM. 72. THE longitudes and latitudes of celestial bodies are not, now, directly obtained from observations, on account of the difficulties which would attend the adjustments of the instruments requisite for such a purpose: but on land, particularly in a regular observatory, the positions of the sun, moon, planets, and fixed stars are generally determined by the method which was first practised by Römer or La Hire. This consists in observing the right ascensions by means of a transit telescope and a sidereal clock, and the declinations by means of a circular instrument whose plane coincides with that of the meridian: the longitudes and latitudes, when required, are then computed by the rules of trigonometry. It will be proper therefore, in this place, to explain the nature of the instruments just mentioned, their adjustments and verifications, and the manner of employing them. It is not intended, however, to describe at length the great instruments which are set up in a national observatory, but merely to indicate them, and to explain the natures and uses of such as, being similar to them and of more simple construction, may without great risk of injury be transported to foreign stations, where regular observatories do not exist, in order to be employed by persons charged with the duty of making celestial observations, either for the advancement of astronomy itself or in connection with objects of geodetical or physical inquiry. 73. The Sidereal Clock is one which is regulated so that the extremity of its hour-hand may revolve round the circumference of the dial-plate in the interval of time between the instants when, by the diurnal rotation of the earth, that intersection of the traces of the equator and ecliptic which is designated the vernal equinox, or the first point of Aries, appears successively in the plane of the geographical meridian, on the same side of the pole. This interval is called a sidereal day, |