or, by the binomial theorem, cos.3 a = 1 — — a2 (rejecting as

=cos.3 a very nearly: and if a were expressed

or, in logarithms,

cos.a; or sin. aa sin. 1′′ cos.3 a;

log. sin. a = log. a + log. sin. 1′′ + log.cos. a (A).

From this formula, while a (in seconds) is less than 4 degrees we may obtain the value of log. sin. a with great precision as far as seven places of decimals. Therefore, in any terrestrial triangle whose sides (in seconds) are represented by a, b, and c, and whose angles, opposite to those sides, are A, B, and C, if a were a given side, and all the angles were given by observation or otherwise, we should have (in order to find one of the other sides, as b, from the theorem sin. A : sin. a sin. B sin. b (art. 61.)), by substituting the above value of log. sin. a,

log. a + log. sin. 1"+

log. cos. a+ log. sin. Blog. sin. A = log. sin. b;

in which, since a is very small, its cosines vary by very small differences, and log. cos. a may be taken by inspection from the common tables. From the value of log. sin. b so found, that of b (in seconds) may be obtained by an equation corresponding to (A) above: thus

log.blog. sin. b— log. sin. 1′′-log. cos. b;

in which, since b is very small, log. cos. b may be found in the common tables, that number being taken in the column of cosines, which is opposite to the nearest value of log. sin. b in the column of sines.

If a were expressed in feet it might be converted into the corresponding arc in terms of radius (= 1) on dividing it by R, the earth's mean radius in feet: and into seconds by the formula

400. A formula for the reduction of any small arc of the terrestrial sphere to its chord may be investigated in the following manner: Let a be a small arc expressed in terms of radius (1); then (Pl. Trigon., art. 46.) we have

a3

sin. a = a

6

(rejecting powers of a above the third);

hence

1

2 sin. a, or chord. a, a a3, and a - chord. a=4a3:

= —

1 24

24

an equation which holds good whether a and chord. a be expressed in terms of radius (= 1) or in feet.

1

If a were given in seconds we should have (a sin. 1") equal to the difference between the arc and its chord in terms of radius (=1); or in logarithms,

3 (log. 1′′) a+log. sin. 1") - log. 24=log. (a — chord. a) in terms of rad. (=1);

3

or if a were given in feet we should have() for the value of the same difference; or in logarithms,

3 (log. a— log. R) — log. 24 = log. (a— chord. a) in terms of rad. (=1).

401. The reduction of an angle of a spherical triangle to the corresponding angle between the chords of the sides which contain it, may be thus effected:

G

D

A

E

C

Let the curves AB, AC be two terrestrial arcs constituting sides of the triangle ABC, and let their chords be the right lines AB, AC: let o be the centre of the sphere, and draw OG, OH parallel to AB, AC: draw also OD and OE to bisect the arcs in D and E, cutting the B chords AB and AC in d and e. Then, since the angles AOB, AOC are bisected by od and Oe, the angles Ado and Aeo are right angles; therefore the alternate angles DOG, EOH will also be right angles, and DG, EH will each be a quadrant; also the arc GH or the angle GOH will be equal to the plane angle BAC between the chord lines. The sides AG, AH and the included spherical angle at A being known, the arc GH which measures the reduced or plane angle BAC may be computed by the rules of spherical trigonometry (as in art. 64.); and in like manner the angles between the chords at B and c might be computed. The excess of each spherical angle above the corresponding angle of the plane triangle formed by the chords of the terrestrial arcs is thus separately found; and it is evident that the sum of the three reduced angles will be equal to two right angles if the spherical angles have been correctly observed.

402. The third method of computing the sides of terrestrial triangles is the application of a formula which was inves

tigated by Legendre, who, in seeking what must be the angles of a plane rectilineal triangle having its sides equal to those of a triangle on the terrestrial sphere, arrived at the conclusion that, neglecting powers of the sides higher than the third, each of the angles of the former triangle should be equal to the corresponding angle of the spherical triangle diminished by one third of the spherical excess found as above shown (art. 396.). This proposition may be demonstrated in the following manner. (See Woodhouse, " Trigonometry.") Let A, B, and C, as in the above figure, be the angles of a spherical triangle, and a, b, c expressed in terms of radius (1) be the sides opposite to those angles; also let p represent half the sum of those sides. Then, (art. 66. (11)), sin. p sin. (p—a).

1

cos. 2A, or (1 + cos. A) =

=

sin. b sin. c

developing (Pl. Trigon., art. 46.) the second member as far as the third powers of p, a, b, c, we get

(1 + cos. a) =

or =

=

1

p (p − a) { 1 — ¿ (p2 + ( p − a)2 } '1 — † (b2 + c2) '

bc

!p (p − a ) { 1 — ↓ ( p2 + ( p − a)2) } (1 + ¿

or again,

bc

=

PP-1-— ↓ (p2 + (p − a)2 — b2 — e2)}.
p (p a ) { ¿
— —

bc

But p = (a+b+c) and p − a = 1⁄2 (b + c − a) :

substituting these values in the second co-efficient of P (p—a),

bc

the numerator of that co-efficient will be found to be equal to twice the product of (a + c—b) and 1⁄2 (a + b — c), that is to 2(p-b) (p-c): therefore

1⁄2 (1 + cos. ▲) =P (p − a) _ p (p—a) (p—b) (p − c) - _ c)....... A)

Now, in a plane triangle whose sides are expressed by a, b, c, and of which the angles opposite to those sides are represented by A', B', C', we have (Pl. Trigon., art. 57.)

cos.2 ↓ A', or ↓ (1 + cos. A') = P(p − a);

also, by the rules of mensuration, the square of its area is

equal to 62 c2 sin.2 A', and to p (pa) (p—b) (p −c) ; therefore

1 b2 c2 sin.2 a′ = p(p − a) (p —b) (p − c), or 11⁄2 bc sin.2 A'= p (p − a) (p—b) (p −c)

3 bc

Substituting the equivalents in equation (A) we have

(1 + cos. A) =

or

(1 + cos. A') - be sin.2 A', cos. A = cos. A' - b c sin.2 A'.

sin. A' sin. h,

Again, assuming a = A' + h, we have (Pl. Trigon., art. 32.) cos. A cos. A' cos. h

or, since h is very small, cos. A = cos. A′

hence h sin. A' be sin.2 A', or h =

=

h sin. A':

bc sin. A'.

But, as above, be sin. A' represents the area of a plane triangle of which the curve lines a, b, c, considered as straight are the sides: consequently

h = (area), and A = A + (area) or A'A - (area). In like manner B'B (area), and c'c- (area). · }

Now the area of such triangle considered as on the surface of the sphere, and the radius of the latter being unity, has been shown to be equal to the excess of the arcs which measure the spherical angles, above half the circumference of a circle (the arcs being expressed in terms of radius = 1): hence, in employing the third method above mentioned, each angle of the spherical triangle on the surface of the earth must be diminished by one third of the spherical excess, in order to obtain the corresponding angle of the plane triangle, in which the lengths of the straight sides are equal to the terrestrial arcs whether expressed in seconds or in feet.

403. When a base line has been measured, or when any side of a triangle has been computed, it becomes necessary to reduce it to an arc of the meridian passing through one extremity of the base or side; and therefore the angle which the base or side makes with such meridian must be observed. For this purpose, as well as with the view of obtaining the latitude of a station by the meridian altitude, or zenith distance, of a celestial body, or of making any other of the observations which depend on the meridian of the station, the position of that meridian should be determined with as much accuracy as possible. Since the Nautical Almanacs now give at once the apparent polar distances and right ascensions of the principal stars, it is easy (art. 312.) to compute the moment at which any one of these will culminate; and a first

approximation to the position of a meridian line on the earth's surface may be made in the following manner.

A well-adjusted theodolite having a horizontal and a vertical wire in the focus of the object glass may have its telescope directed to the star a little before the time so computed; then, causing the vertical wire to bisect the star, keep the latter so bisected by a slow motion of the instrument in azimuth till the moment of culmination. At that instant the telescope is in the plane of the meridian; and the direction of the latter on the ground may be immediately indicated by two pickets planted vertically in the direction of the telescope when the latter is brought to a horizontal position.

But, employing a transit telescope or the great theodolite with which the terrestrial angles for geodetical surveys are taken, a more correct method of obtaining the position of the meridian is that of causing the central vertical wire of the telescope to bisect the star a Polaris at the time of its greatest eastern or western elongation from the pole; that is, about six hours before or after the computed time of culminating; and then, having calculated the azimuthal deviation of the star from the meridian, that deviation will be the angle between the plane of the meridian and the vertical plane in which the telescope moves. Consequently, by the azimuthal circle of the theodolite, the telescope can be moved into the plane of the meridian; whose position may then, if necessary, be correctly fixed by permanent marks.

N

404. To obtain the star's azimuth at the time of its greatest elongation from the pole, let NPZ be the direction of the meridian in the heavens; zs that of the vertical circle in which the telescope moves at the time of such elongation: let P be the pole, s the star, and z the zenith of the station. Then, in the spherical triangle ZPS, right angled at s, we have (art. 60. (e)), Rad. sin. PS sin. PZ sin. PZS,

and PZS is the required azimuth. The above method possesses the advantage of being free from any inaccuracy which, in the former method, might arise on account of the error of the clock; since the star at s appears for a moment stationary in the telescope, and consequently it can be bisected with precision at that moment by the meridional wire. The method of bringing a transit telescope, or that of a great theodolite, accurately to the meridian, at any time, or of determining its deviation from thence, has been explained in arts. 94, 95, 96.

405. When a meridian line had not been previously de