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feet (six metres) in length, the whole being easily transported by four



Let a and b be the given sides, and c the included angle, which is nearly 180°. Put c = 180° — 6.* The formula is


c = a + blog.

(2.6264222 + log. a + log. b + ar. co. log. (a + b) +2 log. 4.)

The angles of the triangle of the great or primary chain are observed in England and this country with an instrument called


That employed on the Coast Survey of the United States was made by Troughton and Simms, of London, under the direction of the late Mr. Hasler. It consists of a horizontal limb thirty inches in diameter, made narrow and light, with a vertical rim below, to strengthen it, supported by conical arms from beneath a central drum, upon which the telescope is mounted on pillars, as in the transit instrument. The telescope is of four feet focal length. The limb is graduated to 5 spaces, and numbered to 360°. Three reading microscopes, reading to single seconds at intervals of 120°, are sustained above the limb, by horizontal arms projecting from the central drum. Three of the six horizontal arms which support the limb extend beyond the limb, and are provided with foot screws, upon which the instrument rests, on short pillars or legs, which fit each into one of three holes 120° apart, in a wooden frame strongly trussed.

The method of observing is as follows. The instrument is directed to the signal at one of the distant vertices of the triangle whose angles are to be measured, upon which the middle vertical wire is adjusted with great accuracy, and the reading taken by the three microscopes, the degrees for only one of them. The telescope is then inverted by turning it on its horizontal or supporting axis, and the instrument turned on its vertical axis 180°, till the middle vertical wire again coincides with the signal, when the reading of the three microscopes is again taken. The legs

* being very small, the formula preceding (2) Art. 21, App. I., putting for sin its value in (1), becomes, putting sin or for its equal cos c, neglecting 04, &c.,

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Whence the formula in the text, in which 0 is expressed in minutes

are then shifted in the holes of the wooden frame, which changes the position of the limb 120°, and the same operation gone through as before, and so on.* As there are three holes for the legs, the whole number of readings on a single signal will thus amount to 18, the indiscriminate mean of which will involve a compensation for all instrumental errors.†


Eminences are usually chosen, and, if possible, such as to permit the signals, which serve to mark the stations, being seen against the sky. A very small object is thus made visible at a very great distance.

Permanent monuments of masonry or pottery are sunk below the surface of the ground far enough to be undisturbed by ploughing, with an orifice in the top, in which to insert a signal staff. The top of this staff supports some object to which the telescope is directed. Spheres made of barrel hoops, covered with white muslin, have been found to answer well in mountainous regions, being visible fifty miles, but not so well in low grounds, and near the sea. Tin cones of an angle adapted to reflect advantageously the light of the morning and evening sun, were used by Mr. Hasler, who gives (Trans. Am. Phil. Soc., 1825), the mode of reducing in a simple manner the observation to the axis of the cone, a reduction depending on the relative position of the cone to the sun and the observer. Steeples, circular or polygonal towers, windmills, &c., have been employed, and the methods of correcting the observations upon these are given by Puissant.§ At night, signals have been used formed of

*This is for the purpose of measuring the same angle upon different parts of the limb.

+ The French employ a large repeating circle, with which the angles are observed, in oblique planes, passing through the objects and the eye. These require to be reduced to the horizon by methods given by Puissant (Traité de Geodesie). The theodolites employed on the Coast Survey, besides the 30 inch, are one of 24 inches, and one of 18 inches, by Troughton and Simms, and a number of 14, 12, 10, and 6 inch repeating circles by Troughton and Simms, and Gambey, of Paris.

A formula easily deducible from Hasler's is

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§ Traité de Geodesie, a standard work on this subject, to which we shall have frequent occasion to refer.

lamps, placed at the focus of a parabolic reflector, or behind a lens at the focus of parallel rays. The most perfect is the Drummond light, which may be seen at the distance of seventy miles.

The best signal when the sun shines is one employed at present on the Coast Survey of the United States, called a heliotrope. It consists of a common telescope, mounted upon a three-legged stand, horizontally. It is accompanied by a man called a heliotroper, who directs the telescope to the tent in which the great theodolite is placed. Upon the eye end of the telescope is supported a small plane mirror, which has motion round both a vertical and horizontal axis, so as to be capable of being placed in a position to reflect the sun in any direction, at pleasure. The heliotroper attends and turns the mirror continually, so as to reflect the sun in a direction parallel to the axis of the telescope, which he accomplishes by causing the rays to pass through two perforated discs, supported like the mirror, on the top of the telescope tube, one being near the object end, and the other between it and the mirror.* * The signal pole is supported by a wooden tripod at least its height, and the heliotrope is placed at a short distance from it, in a line with the theodolite station. All the signals visible from the station at which the great theodolite is placed, are observed every day for several weeks. The instrument is then moved to a new station, and by this means all the angles of every triangle are repeatedly observed.


It sometimes happens from the nature of the signal employed, that the theodolite cannot be placed at the axis of the signal called the centre of the station. The process of determining what the observed angle would have been with the instrument so placed, from observation with the instrument placed at a short measured distance from the proper point, is called reducing to the centre of the station.

Let c in the diagram be the centre of the station, o the place of the instrument. From the observed angle AOB required the angle ACB.. Make AOB w, BC = 9, AC = d, ACB =x,OC=r, COB = y.


AIB = +IAO, and AIB = x + CBO

The heliotroper is on duty till 10 A.M. and after 3 P.M., the atmosphere in the middle of the day being too unsteady near the earth's surface for good observations. If the centre of the station be inaccessible, this distance must be calculated from measurements which can be made by methods which the student will easily devise for any given case.

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which is the formula for the correction to be applied to the observed angle w, to obtain the required angle x.

The distance being small, d and g are computed with the angle w.


The excess of the sum of the angles of a spherical triangle over two right angles is technically called the spherical excess. One third of the spherical excess being subtracted from each of the angles of a spherical triangle occupying but a small portion of the surface of the sphere, a plane triangle may be formed with the resulting angles, and with rectilinear sides, equal in length to the curvilinear sides of the spherical triangle, and A, B, C being the angles, s the area of the triangle, and r the rad. of the earth (or better the rad. of curvature, see App. VI. p. 366),

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*This is Legendre's theorem, the demonstration of which, by Lagrange, is as follows:-The sides of a spherical triangle being a, b, and c, and the radius of the

sphere r,
which put equal to a, B, y, and by Art. 82,

the similar triangle on the sphere, whose radius is 1, will have for sides

a b c

COS a cos ẞ cos y

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and similar expressions obtain for cos ẞ, cos y, sin y .. (1) becomes


which is the formula of verification, the last term being expressed in seconds.

If the sum of the three observed angles exceed 180° by

observations are correct.

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7" sin 1"


As the spherical excess is expressed in terms of the area of the triangle, that must be computed, which may be done with sufficient accuracy by considering the spherical triangle as plane, and applying to it one of the formulas of Art. 19, App. I.

The angles of the plane triangle, whose sides are equal in length to those of the given spherical triangle, being found by means of the spherical excess, as above explained, and the length of one of the sides of the

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Transferring (1-8-y?) to the numerator, by giving it the exponent developing as far as to terms of the fourth degree inclusive, (2) becomes a1 + B1+ y1 — 2a2 ß2 — Qa2 y3 — 2ßa ya


B2 + y2 — a3



Replacing the values of a, ß, y, (3) may be expressed as follows:

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If the same lengths, a, b, c, be sides of a plane triangle, and a' be the angle opposite a, by Art. 69,

COS A' =

b2 + c2 — a2

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Raising both members of (5) to the square, and substituting 1 - sin A' for cos2 A'

— 4 b2 c2 sin2 a′ = a1 + b1 + c1 — Qa2 b2 — Qa2 c2 —— 262 c2=N

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Let A=A+x, x being the difference between the spherical and plane angle,

sin A' sin x

COS ACOS A' cos x- sin a' sin x = cos A' √1 — sin? x Putting z, which is very small, for sin x, and rejecting the second power of x,

COS A COS ▲'- -x sin A'

Combining (6) and (7), which have the same first members

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