tance, so as to subtend an angle of half a minute, which experience has shown to be the least allowable

are not.

*

To make the tops of the signal-masts conspicuous, flags may be attached to them; white and red, if to be seen against the ground, and red and green if to be seen against the sky. The motion of flags renders them visible, when much larger motionless objects. But they are useless in calm weather. A disc of sheetiron, with a hole in it, is very conspicuous. It should be arranged so as to be turned to face each station. A barrel, formed of muslin sewn together four or five feet long, with two hoops in it two feet apart, and its loose ends sewn to the signal-staff, which passes through it, is a cheap and good arrangement. A tuft of pine boughs fastened to the top of the staff, will be well seen against the sky.

In sunshine, a number of pieces of tin nailed to the staff at different angles, will be very conspicuous. A truncated cone of burnished tin will reflect the sun's rays to the eye in almost every situation. But a "heliotrope," which is a piece of looking-glass, so adjusted as to reflect the sun directly to any desired point, is the most perfect arrangement.

For night signals, an Argand lamp is used; or, best of all, Drummond's light, produced by a stream of oxygen gas directed through a flame of alcohol upon a ball of lime. Its distinctness is exceedingly increased by a parabolic reflector behind it, or a lens in front of it. Such a light was brilliantly visible at 66 miles distance.

(385) Observations of the Angles. These should be repeated as often as possible. In extended surveys, three sets, of ten each, are recommended. They should be taken on different parts of the circle. In ordinary surveys, it is well to employ the method of "Traversing," Art. (373). In long sights, the state of the atmos

phere has a very remarkable effect on both the visibility of the signals, and on the correctness of the observations.

When many angles are taken from one station, it is important to record them by some uniform system. The form given below is convenient. It will be noticed that only the minutes and seconds of the second vernier are employed, the degrees being all taken from the first.

When the angles are "repeated," Art. (370), the multiple arcs will be registered under each other, and the mean of the seconds shewn by all the verniers at the first and last readings be adopted.

(386) Reduction to the centre. It is often impossible to set the instrument precisely at or under the signal which has been observed. In such cases pro

Fig. 265.

AR

会

as near as possible to C, and measure the angle RDL. It may be less than RCL, or greater than it, or equal to it, according as D lies without the circle passing through C, L and R, or within it, or in its circumference. The instrument should be set as nearly as possible in this last position. To find the proper correction for the observed angle, observe also the angle LDC, (called the angle of direction), counting it from 0° to 360°, going from the left-hand object toward the left; and measure the distance DC. Calculate the distances CR and CL with the angle RDL instead of RCL, since they are sufficiently nearly equal. Then

CD. sin. (RDL + LDC)

CD. sin. LDC*

RCL = RDL +

CR. sin. 1"

CL. sin.1"

The last two terms will be the number of seconds to be added or subtracted. The Trigonometrical signs of the sines must be attended to. The log. sin. 1′′=4.6855749. Instead of dividing by sin. 1", the correction without it, which will be a very small fraction, may be reduced to seconds by multiplying it by 206265. Example. Let RDL 32° 20′18′′.06; LDC=101° 15' 32" .4; CD = 0.9; CR = 35845.12; CL = 29783.1.

The first term of the correction will be +3.750, and the second term-6".113. Therefore, the observed angle RDL must be diminished by 2′′.363, to reduce it to the desired angle RCL.

Much calculation may be saved by taking the station D so that all the signals to be observed can be seen from it. Then only a single distance and angle of direction need be measured.

'

T

F

Fig. 260

T

It may also happen that the centre, C, of the signal cannot be seen from D. Thus, if the signal be a solid circular tower, set the Theodolite at D, and turn its telescope so that its line of sight becomes tangent to the tower at T, T'; measure on these tangents equal distances DE, DF, and direct the telescope to the middle, G, of the line EF. It will then point to the centre, C; and the distance DC will equal the distance from D to the tower plus the radius obtained by measuring the circumference.

If the signal be rectangular, measure DE, DF. Take any point G on DE, and on DF set off DH

DF
DE

= DG Then is GH parallel to EF, (since

DG: DH :: DE: DF) and the telescope directed to its middle, K, will point to the middle of the

diagonal EF. We shall also have DC = DK

DE
DG

Any such case may be solved by similar methods.

* For the investigation, see Appendix B.

D

Fig. 267.

The "Phase" of objects is the effect produced by the sun shining on only one side of them, so that the telescope will be directed from a distant station to the middle of that bright side instead of to the true centre. It is a source of error to be guarded against. Its effect may however be calculated.

(387) Correction of the angles. When all the angles of any triangle can be observed, their sum should equal 180.* If not they must be corrected. If all the observations are considered equally accurate, one-third of the difference of their sum from 180°, is to be added to, or subtracted from, each of them. But if the angles are the means of unequal numbers of observations, their errors may be considered to be inversely as those numbers, and they may be corrected by this proportion; As the sum of the reciprocals of each of the three numbers of observations Is to the whole error, So is the reciprocal of the number of observations of one of the angles To its correction. Thus if one angle was the mean of three observations, another of four, and the third of ten, and the sum of all the angles was 180° 3′, the first named angle must be diminished by the fourth term of this proportion; + 1 + † : 3′ : : : 1′ 27′′.8. The second angle must in like manner be diminished by 1′ 5′′.9; and the third by 26".3. Their corrected sum will then be 180°.

66

1

3

1

10

3'

3

It is still more accurate but laborious, to apportion the total error, or difference from 180°, among the angles inversely as the "Weights," explained in Art. (369). On the U. S. Coast Survey, in six triangles measured in 1844 by Prof. Bache, the greatest error was six-tenths of a second.

(388) Calculation and platting. The lengths of the sides of the triangles should be calculated with extreme accuracy, in two ways if possible, and by at least two persons. Plane Trigonometry may be used for even large surveys; for, though these sides are really arcs and not straight lines, the difference will be only one

* If the triangles were very large, they would have to be regarded as spherical, and the sum of their angles would be more than 180°; but this "spherical excess" would be only 1" for a triangle containing 76 square miles, 1' for 4500 square miles, &c.; and may therefore be neglected in all ordinary surveying operations.

twentieth of a foot in a distance of 113 miles; half a foot in 23 miles; a foot in 34 miles, &c.

The platting is most correctly done by constructing the triangles, as in Art. (90), by means of the calculated lengths of their sides. If the measured angles are platted, the best method is that of chords, Art. (275). If many triangles are successively based on one another, they will be platted most accurately, by referring all their sides to some one meridian line by means of "Rectangular Coordinates,” the Method of Art. (6), and platting as in Art. (277.) In the survey of a country, this Meridian would be the true North and South line passing through some well determined point.

(389) Base of Verification. As mentioned in Art. (380), a side of the last triangle is so located that it can be measured, as was the first base. If the measured and calculated lengths agree, this proves the accuracy of all the previous work of measurement and calculation, since the whole is a chain of which this is the last link, and any error in any previous part would affect the very last line, except by some improbable compensation. How near the agreement should be, will depend on the nicety desired and attained in the previous operations. Two bases 60 miles distant differed on one great English survey 28 inches; on another one inch; and on a French triangulation extending over 500 miles, the difference was less than two feet. Results of equal or greater accuracy are obtained on the U. S. Coast Survey.

(390) Interior filling up. The stations whose positions have been determined by the triangulation are so many fixed points, from which more minute surveys may start and interpolate any other points. The Trigonometrical points are like the observed Latitudes and Longitudes which the mariner obtains at every opportunity, so as to take a new departure from them and determine his course in the intervals by the less precise methods of his compass and log. The chief interior points may be obtained by "Secondary Triangulation," and the minor details be then filled in by any of the methods of surveying, with Chain, Compass, or Transit, already explained, or by the Plane Table, described in Part VIII.