Hence the difference of level on a sloping height of 2100 links of Gunter's surveying chain, or 2100 x 0.661386 feet, is 19 feet. When a spirit-level exactly adapted to this purpose is not at hand, if there is a theodolite to be had, it will perform the operation, though it is not quite so convenient,

In case of levelling for canals, the process is not different, only the canal is carried on an exact level, by judiciously choosing the situa tion winding round rising grounds, conveying it across ravines by aqueduct bridges, and allowing it to descend at particular points by means of locks. Roads ought to be carried along a level line as nearly as possible, and only having gentle acclivities and declivities. This may be readily obtained by following routes somewhat circuitous in uneven parts of the country, taking the advantage of ravines, water-courses, and the sides of lakes; for a greater distance on a road nearly level is productive of less expense of animal strength, than by passing over considerable elevations. All very quick turns in the road, particularly when entering upon a bridge, ought to be avoided, as the danger from centrifugal force, which may be easily estimated by the formula, Part III., Sec. IV., is considerable. The justice of these remarks may be readily appreciated by considering many parts in most of our public roads which have hitherto been constructed upon the very worst principles, having been intrusted to what are called practical men, who are frequently the mere slaves of custom.

SECTION IV.

Rules and Formula.

When two angles of a plane triangle are known, the third may be found, consequently, for general purposes, it is unnecessary to measure the third angle. But when great accuracy is required, or when the sides on the surface of the earth are large, they become spherical arcs, and then the third angle should always be measured as a check upon the results. If the sum of the three angles of any triangle actually measured amount to 180° very nearly, it is probable that they have been taken with great accuracy, and the excess or defect must be distributed among the three angles according to the judgment of the observer, by applying such a part of the error to each as appears most likely upon a due consideration of all the circumstances of the case to be the most probable quantity. If all the angles be taken with equal care, one-third of the error must be applied to each angle with its proper sign, so as to reduce their sum to 180°+e exactly, e being the spherical excess to be afterwards explained.

Let the spherical excess, or e=0".49, and

When the ordinary instruments are employed where the error of reading may be 30" or so, then it will be sufficient to apply onethird of the excess above, or defect from 180° to each of the angles, so as to reduce their sum to 180° exactly. In conducting geodetical operations, the triangle should be so chosen, if possible, as to produce the most accurate conclusions. To diminish the probability of error, the following rules should be observed:

I. When one side only of a triangle is to be determined, the measured base should be nearly equal to the required side.

II. When two sides of a triangle are to be determined, the triangle should, if possible, be equilateral.

III. When the base cannot be equal to one or to both the required sides, it should be as long as possible, and the two angles at the base equal, and not less than twenty or thirty degrees.*

IV. When the centre of the instrument cannot be placed in the vertical line occupied by the axis of a signal, the observed angles must be reduced to it by an appropriate formula. Let C be the centre of the station, such as a tower, P the place of the centre of the instrument, by which the angle subtended by A B at

P is to be measured. Let the angle A P B be observ- B
ed, and the distance C P be measured, it is required
to find C, the measure of the angle A CB? Suppose
APB P, BPC= p, CP=d, A C D and B C
=D'.

=

=

Since the exterior angle of the triangle API is equal to the sum of the two interior and opposite

=

P

angles, AIB P+IA P, and of the triangle BI C, the exterior angle A I B C + CB P. Making these two values of A I B equal, by transposition, we have C-PIAP-C B P. But the triangles CAP, CBP give sin C AP sin I A P =

СР

AC

- sin A PC

d sin (P+P); sin C B P= СР

D

BC

sin BP C =

d sinp. And since the angles C A P, C B P, are, by hypothesis,

D

always very small, their sines may be substituted for their arcs,

hence, C-P=

D
sin

d sin (P+p)

d

sin 1"

{

sin (P+p)

p

};

; or R" being the length of an arc in

seconds equal to the radius, or 206264′′.8, then

*For a demonstration of these properties, see vol. III. of Hutton's Course of Mathematics.

C-PR" d x

{

sin (P+p)

Ꭰ

_ simp}.

If this be developed,

we have C-P=R" d sin P, sin (A—p).

Cor. 1.-If c fall upon the line P B, then C-P=

R" d sin P.

Ꭰ

Cor. 2.-In operations where the angles are measured by a common theodolite or pocket-sextant, R' may be used instead of R". Cor. 3.-When the theodolite cannot conveniently be placed at the same height as the top of the signal observed,

d4=

R"dh sin A
Ꭰ

In which A is the observed zenith distance, R" the length of an arc equal to the radius in seconds, h the difference between the height of the centre of the circle and the top of the signal, and D the distance.

The use of this formula cannot be embarrassing, provided the signs of sin p, and sin (P+p) be properly attended to, as is illustrated by the following example:-Let the observed angle P be 43° 52′ 49′′.44, p = 264° 41′ 24′′, d = 10.706 feet, D = 57508 feet and D' = 66750 feet, required the reduction?

(2.)

(1.)

Log R

5.314425

log d

1.032860

Scholium.-The error in height or distance arising from an error of one minute in measuring the horizontal or vertical angle when the altitude is small, or the height small, in comparison of the dis

tance.

Supposing the error of the measured angle to be l'.

Error arising from the eccentricity of the telescope of a theodolite to the vertical circle additive to elevations, but subtractive from depressions.

The ordinary 5-inch theodolites have an eccentricity of about 1.86 inch, and the error at 1000 feet is about 32".0. These may enable observers to allow for such small errors when thought necessary.

When signals are circular or polygonal towers, various methods may be employed to find the true angle, from a due consideration of the nature of the case, which, to any one possessing a knowledge of the elements of geometry, will readily occur.

V. The angles measured in an inclined plane should be reduced to the horizontal plane,

In this case the altitudes must be also observed, and then there is formed a spherical triangle, of which the three sides are given to compute the angle at the zenith, which may be performed by the rules of spherical trigonometry.

VI. A spherical triangle being proposed, of which the three sides are very small compared with the radius of the sphere; if from each of its angles, one-third of the excess of the sum of its three angles, above two right angles be subtracted, the angles so diminished may be taken for the angles of a rectilineal triangle, whose sides are equal in length to those of the proposed triangle.

To find the spherical excess when the three sides are given in feet. 1. Rule. To the constant logarithm 1.349380, add the logarithm of half the sum of the three sides, the logarithms of the three differences between these sides and that half sum, half the sum of these five logarithms will be the logarithm of the spherical excess in seconds.

2. To the logarithm of the area of the triangle taken as a plane one in feet, add the constant logarithm 0.674690; the sum is the logarithm of the excess above 180° in seconds.

3. If the base and perpendicular of a triangle be given. To the logarithm of the base in feet, add the logarithm of the perpendicular, and the constant logarithm 0.373660; the sum will be the logarithm of the spherical excess in seconds.

The spherical excess amounts to one second for an area of 76 English square miles, whence, if the area in square miles be known, the spherical excess may be readily obtained by dividing it by 76.

VII. To reduce a base on an elevated level to that at the surface

of the sea.

Let r represent the radius of the earth, corresponding to the base b at the level of the sea, and r+a the radius referred to the level of the measured base B; then it is obvious that r+a: r :: B : b=Bx Hence, B — b = B — B- -=Bx -=Bx

a

r+a

r

r+a

α

r+a

(+, &c.). But the radius of the earth being very great in

comparison of the difference of level a, we have the correction & sufficiently accurate, by retaining the first term. Hence, d=Bx.

a

r

Rule. By logarithms. To the logarithm of the measured base in feet, add the logarithm of its height above the sea, and the constant logarithm 2.680110; the sum will be the logarithm of a number of feet which, taken from the measured base, will be that at the level of the sea required.

VIII. To determine the horizontal refraction from observation.

Rule. From the measure of the intercepted terrestrial arc subtract the sum of the two depressions at its extremities; half the remainder is the refraction. If by reason of the smallness of the contained arc, one of the objects has an elevation instead of a depression, then the depression must be taken from the sum of the contained arc and elevation; half the remainder is the refraction.

If-d' becomes an elevation, it changes its sign, and becomes c+e-d +e, and in that case R = 2

(2.)

The exact quantity of terrestrial refraction is very variable. It is estimated by Dr Maskelyne at one-tenth of the intercepted arc, by Delambre at one-eleventh, by General Mudge at one-twelfth, and by Legendre at one-fourteenth at a mean state of the atmosphere. In peculiar circumstances it varies very considerably from this, as from one-sixth to one-eighteenth of the contained arc. If one-twelfth of the contained arc be allowed, then, since the length of a second is about 100 feet in moderate distances, 1200 of the distance in feet will be the refraction in seconds.

IX. To find the angle made by a given line with the meridian. With a good instrument measure the greatest and least angular distance of the pole star from the vertical plane in which the given line is situated; half the sum of these two measures will be the angle required, or if the exact place of the pole star be computed, the true azimuth may be found at any time in the same manner as the latitude. See page 112.

This may also be done, though less accurately, by computing the azimuth of the sun, or a star, when on the line, from an altitude taken for that purpose.

X. In addition to what has already been said relative to finding the latitude of the place, we may add here, that the same thing may be very accurately obtained, by observing the greatest and least altitude or zenith distance of a circumpolar star, and correcting them for the effects of refraction; half the sum of the altitudes, thus corrected, will be the latitude, or half the sum of the zenith distances will be the colatitude.

XI. To determine the ratio of the earth's axes, and their actual magnitude from the measure of a degree of the meridian in two given distant latitudes, supposing the earth a spheroid generated by the rotation of an ellipse about its minor axis.

Let d and d' be the measures of two degrees, d being the least, or that nearest the equator, l and l' the latitudes of their middle points, t the semitransverse axis of the meridian or radius of the equator, c the semiconjugate or semipolar axis, e the excess of the equatorial radius above the polar semiaxis, and 7° 57°.2957795, the number of degrees in an arc are equal to the radius.

Then, é =

e

r° (d'—d)
3 sin (l') sin (l'—l)
d'-d

=

And ==3d sin (+1) × sin ('—')

(1.)

(2.)