the more convenient to employ them as they are, or ought to be always present to the mind of the calculator. But since they involve operations which are frequently laborious, and which require a minute attention to accuracy at every step, mathematicians have found it necessary to investigate formulæ in which the differences only between the observed and reduced angles may enter; and thus the reductions are effected by the employment of logarithms extending to a few places of decimals, while the errors of the calculation have small influence on the result. The practice of using such formulæ, when possible, in preference to the direct processes, prevails, indeed, throughout the whole of the calculations connected with geodetical, as well as astronomical operations. 395. The following is an investigation of a formula for reducing any observed angle to the centre of a station.

Let A, B, C be the centres of three stations, and let E (near B) be the place of observation. Here AB and the angle BAC are supposed to be accurately known by previous operations; EB is supposed to be measured, and the angles CEB, CEA to be observed; and it is required to deterthe angle CBA.

H/

B

Produce EB indefinitely towards H; then

E

B

H

C

F

the angle CBH=CEB+ECB, and angle ABH=AEB+EAB:

whence by subtraction, we have

(

=AB

sin. ACB/

sin. CAB).

sin. ACB sin. CAB:: AB: CB

Substituting this value of CB, we obtain

sin. ECB =

BE sin. CEB sin. ACB

A B sin. CAB

and hence, sin. ECB sin. EAB =

BE sin. CEB sin. ACB-sin. A E B sin. CAB

For sin. ACB putting sin. (CAB+CBA), the sine of an angle being equal to the sine of its supplement; and for CBA substituting CEA, to which on account of the smallness of the difference it may be considered as equal: also for AEB putting its equal CEB-CEA, the last equation becomes

sin. CEB sin. (CAB+CEA)—sin. CAB sin. (CEB—CEA) Developing (Pl. Trigo., art. 32.) sin. (CAB+CEA) and sin. (CEB-CEA), and then simplifying the whole, we have

BE sin. CEA

sin. ECB-- sin. EAB=

sin. (CAB+CEB).

AB sin. CAB

But the angles ECB and EAB being very small, their values in terms of the radius (1) may be put for their sines; and hence

ECB-EAB (in seconds)

BE sin. CEA
AB sin. CAB sin. 1"

sin. (CAB+CEB).

The second member of this equation, and consequently the first, will be negative when the sum of the angles CAB, CEB exceeds two right angles.

If B were on the other side of E, as at B'; by producing EB' towards H', the equivalent of the angle AEB' would be CEB'+CEA, and we should have

ECB'-EAB' (in seconds)

=

B'E sin. CEA

AB' sin. CAB' sin. 1′′

sin. (CAB'-CEB′).

In like manner if F were the place of observation, instead of C, we should have

And these are the corrections to be applied with their proper signs to the observed angle CEA (in equation (A) above), or AFB in order to have the required angle ABC or ACB.

396. When the three angles of a triangle on the surface of the earth are taken with a theodolite, and such an instrument has always been used in the English surveys, each angle being in a plane parallel to the horizon of its vertex; that is, being the inclination of two vertical planes passing through

the angular point and the two distant stations, it must be considered as appertaining to a spheroidal triangle: consequently the sum of the three must exceed two right angles by a certain quantity which, when the sides are several miles in length, becomes sensible; and should the observed excess be equal to that which is determined by computation, the correctness of the observations would thereby be proved. In the tract on Spherical Geometry (prop. 20.) it is shown that the area of a spherical triangle is equal to the rectangle contained by half the diameter of the sphere and a line equal to the difference between half the circumference and the sum of the arcs which measure the angles of the triangle, and this may be expressed by the formula AR (a+b+c-T); where A is the area of the triangle, B the radius of the earth supposed to be a sphere, π=3.14159; and a, b, c are the several arcs (expressed in terms of a radius 1) which measure the angles of the triangle: hence

=

In employing the theorem for the purpose of verifying the observed angles of any triangle on the surface of the earth, the radius of the latter may be considered as equal to 20886992 feet. And the area of the triangle being generally expressed in square feet, the value of the spherical excess found by the above formula would be expressed in feet (linear measure): therefore in order to obtain the excess in seconds, the term should be divided by R sin. 1" (equal to an arc, in feet, subtending an angle of one second at the centre of the earth), and thus the excess in seconds will be equal to

A

R2 sin. 1"

R

It

may be observed that the area A is supposed to be computed in square feet from the ascertained lengths of the sides, by the rules of mensuration, as if the surface of the triangle were a plane.

397. In the French surveys, instead of a theodolite, the repeating circle was used for taking the angles between the stations; and when all these were not equally elevated above the general surface of the earth, considered as a sphere or spheroid, it became necessary to reduce the observed, to horizontal angles. For this purpose it was merely required to find the angles, at the zenith of each station, in a spherical triangle of which the three sides are given: viz. the angular distance between the other stations and the angular distances

of these last from the zenith; and a rule for determining such angle at the zenith has been given in art. 69. But the following investigation leads to a formula by which the difference between the observed angular distance of two stations and the corresponding horizontal angle may with ease and accuracy be obtained.

B

Let A and B be the two stations between which the observed angle is taken, and c the place of the observer: then z being the zenith of c in the celestial sphere, let zCA", ZCB" be parts of two vertical circles passing through C and the two stations. Let CA"B" be the plane of the observer's horizon, and imagine the lines CA, CB to be produced to A' and B' in the heavens: then A'CB' is the observed angle and A"CB", or the spherical angle at z, is the required horizontal angle.

C

A

Α'

Now let the observed angle A'CB' be represented by C, and the observed altitudes A'A", B'B" by a and b respectively; then, by either of the formulæ (a), (b), or (c), art. 60.,

or since the arcs a and b are small, substituting a and b for

art. 46.), we have, rejecting powers of a and b higher than the

second, cos. z =

cos. C-ab

62, or again, by division,

1

2 2

cos. z=cos. c —ab + 1⁄2 (a2 + b2) cos. C.

Next, let z be represented by c+h; then (Pl. Trigo., art. 32.) cos. z=cos. C cos. h-sin. c sin. h: but because his small, the arc may be substituted for its sine and its cosine may be considered as equal to unity; therefore

cos. Z cos. C-h sin. c:

equating these values of cos. z, we get

In this equation, a, b, and h are supposed to be expressed in arcs; therefore, in order to have the value of h in seconds

when a and b are expressed in seconds, each of the two terms in the second member of the equation must be multiplied by sin. 1". Hence the value of z may be found.

If one of the stations as B were in the horizontal line CB",

we should have b=0, and in this case h =

a2 2

cotan. C.

398. In computing the sides of triangles formed by the principal stations on the earth's surface, when all the angles have been observed and one side measured or previously determined, three methods have been adopted. The first consists in treating the triangles as if they were on the surface of the sphere and employing the rules of spherical trigonometry. In the second method the three points of each triangle are imagined to be joined by lines so as to form a plane triangle; then, the given side being reduced to its chord and the spherical angles to those which would be contained by such chords, the other chord lines are computed by plane trigonometry, and subsequently converted into the corresponding arcs of the terrestrial sphere or spheroid. third method consists in subtracting from each spherical angle one third of the spherical excess, and thus reducing the sum of the three angles of each triangle to two right angles; then, with the given terrestrial arc as one side of a plane triangle, computing the remaining sides by plane trigonometry, and considering the sides thus computed as the lengths of the terrestrial arcs between the stations. The first and third methods present the greatest facilities in practice, and all may be considered as possessing equal accuracy.

The

399. The computations of the sides of the triangles by the first of the above methods might be performed by making the sines of the sides proportional to the sines of the opposite angles: but in so doing a difficulty is felt on account of the imperfection of the logarithmic tables; for the sides of the triangles being small, the increments of the logarithmic sines are so great as to render it necessary that second differences should be used in forming the correct logarithms. This labour may be avoided by using a particular formula, which is thus investigated :

Let a be any terrestrial arc expressed in terms of radius (=1); then (Pl. Trigo., art. 46.)

sin. aa- a3 (rejecting higher powers of a)

Now cos. a = (1 − sin.2 a); hence cos.3 a = (1 − sin.2 a)3: