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the two objects, and n sin (H+ h). tan. A-sin2 (H-h). cot A. then x (the correction) = n. sec. H. sec. h. The value of n is given in tables computed for the purpose of facilitating this calculation for every minute of H and h, and for every ten minutes of A. When the altitudes differ more than 2° or 3° from zero, the following formula is to be used in preference :
the reduced angle
✓ (sin S―d). sin (1⁄2 S—♪).
S being the sum of the angle observed, and the two zenith distances; and and the respective zenith distances of the objects *.
All observed horizontal angles are, however, essentially spherical angles; and in every triangle measured on the surface of the earth, the sum of the three angles must, if taken correctly, be more than 180°. The lines containing the observed angles are in fact tangents to the sphere (supposing the earth to be one), whereas to obtain the three points considered as vertices of a plane triangle, the angles must be reduced to the value of those contained between the chords of the arcs constituting the sides of the spherical triangle. The correction for this spherical excess, though too minute to be applied to angles observed with moderate sized instruments, being completely lost in the unavoidably greater errors of observation, should be however calculated in the principal triangles, which is easily done on the supposition that the area of a spherical triangle, whose sides are immeasurably small compared with the whole sphere, may be considered identical with that of a plane triangle, whose sides are of the same length as those of the spherical, and whose angles are each diminished by one-third of the spherical excess; from which theorem, demonstrated by Legendre, and known by his name, is deduced the
* For the investigation and application of these formulæ, see vol. i. "Puissant, Traité de Géodesie," page 174; "Géodesie, par Francœur," pages 128 and 435; and Dr. Pearson's "Practical Astronomy," vol. ii. page 505. Hutton's formula is the same, except that it is expressed in terms of the altitude instead of the zenith distances. See also Woodhouse's "Trigonometry," page 220, and the corrections to the observed angles in the first volume of the "Base Métrique."
S form or for the excess in seconds, R": where S denotes the R area, and R the radius of the earth *.
The earth being considered a perfect sphere whose radius is 21,008,000 feet; one second of space 101·43 feet, and (101·43) ' the square feet in a square second.-R the radius 206264,8 area in feet seconds, and the expression becomes (101·43) × (206264,8) × 206264,8; or in logarithms, Log area—4,0123486-5,3144251 = Log area-9,3267737 for the spherical excess in seconds †.
On the Trigonometrical Survey of England, the spherical excess was constantly calculated, not solely for the purpose of diminishing the observed angles by the amount, but to correct the observations. Thus, in one of the large triangles in Dorsetshire, the sum of the three angles was 0-5 less than 180°; the calculated spherical excess amounted to 1"-29, showing an error of 1"-79 in the observation, and in many of the triangles this error was more considerable. One-third of the error thus found, added to each of the angles, corrects them as angles of a spherical triangle, and onethird of the spherical excess deducted from each of these corrected spherical angles converts them into the angles of a plane triangle ready for calculation, and the sum of whose angles is 180°, as is seen in the example below.
One-third of the spherical excess has here been deducted from each angle, but it might have been calculated for each separately,
*R" may be considered identical with
See "Puissant," vol. i. page 100.
+ Woodhouse arrives at the same result at the termination of a long investigation of this correction." Trigonometry," page 229.
by reducing the angles of the spherical triangles to the angles formed by the chords. (Woodhouse, page 239; Base du Système Métrique, &c.) Thus there are three modes of solving the large triangles of a survey, first, by calculating them as spherical triangles with the corrected spherical angles; secondly, by computing them as rectilinear triangles with the angles of the chords; and thirdly, by Legendre's method of reducing each angle by onethird of the spherical excess; this latter method is by far the most expeditious. In the "Base du Système Métrique," the sides of the triangles were computed by all three methods. On the Ordnance Survey they were formerly mostly calculated by the second, and checked by the third, but at present the last of these modes, that by Legendre's formula, is the only one used.
This subject is treated at length in Puissant, vol. i. pages 100, 117, and 223, and also in the account of the Trigonometrical Survey, in Professor Young's, and Woodhouse's Spherical Trigonometry; and in various other works.
When the theodolite cannot be placed exactly over the station *, a correction for this eccentricity, termed the "Reduction to the Centre," becomes necessary.
In the triangle ABC, suppose C the station where the instrument cannot be set up. If at any convenient point D, the angles ADB and ADC are taken, and the distance CD measured, the angle ACB can be thus determined.
Where mills, churches, and other marked objects are selected as trigonometrical points, which are otherwise peculiarly well adapted, but on which the theodolite cannot be set up, this reduction becomes necessary if angles are required to be taken from them. Temporary trigonometrical stations are easily formed of three or four pieces of scantling 10 or 12 feet long, framed together as in the sketch, with a short pole projecting vertically upward from the apex of the pyramid. A plummet suspended from this gives the exact spot on
which to set up the theodolite. Long poles, which can be removed when it is required to adjust the theodolite over the station, answer the same purpose. Two circular disks of iron or other metal on the top of a pole, placed at right angles to each other, form very good marks for observation.
The necessity for the above correction is not of common occurrence, as in the principal triangles stations
are generally selected from whence observations can be made; and in those of the secondary order, the measurement of the third angle is not considered imperative.
In observing the angles for triangulation, too much care cannot be bestowed upon the adjustments of the instrument. These are briefly as follows for the 5 or 7-inch theodolites used in fixing points in the interior, and for traversing. The large theodolite, 3 feet in diameter, known by the name of its maker, Ramsden †, is fully described in the "Trigonometrical Survey;' and the peculiarities in the construction and management of the
* Instead of deducing the angle at the station on which the instrument cannot be set up from that observed at any spot convenient to it, it is often found more expeditious, particularly if there are many observations made, to correct the other angles of the triangles; this latter method is generally now practised on the Ordnance Survey.
An instrument of the same size has since been made by Messrs. Troughton and Simms for the survey of India, as also another for the Ordnance Survey. A theodolite of 18 inches diameter upon a repeating stand was constructed by General Mudge, with an idea of its superseding the larger theodolite, the weight and size of which rendered its carriage an affair of difficulty; but the advantage of repetition (so desirable in single observations) possessed by moderate sized instruments does not appear to compensate for the diminished size of the circumference of the horizontal circle. Theodolites of 24, 18, 12, 10, 9, and 8 inches diameter are also used on the Ordnance Survey, as well as those of smaller dimensions, of 7 and 5 inches.
other large instruments with which the angles of the principal and secondary triangles are observed, are soon understood by any officer conversant with the adjustment of the smaller class, which he most generally has to work with, and which is therefore the one selected for description.
The first adjustment is for the line of collimation, and consists in making the cross wires* in the diaphragm of the telescope coincide with the axis of the supports in which the telescope rests; the proof of which is their intersection remaining constantly fixed upon some minute, well-defined, distant point, during an entire revolution of the telescope upon its own axis in the Ys, which are left open for the purpose. When this intersection on the contrary forms a circle round the object, the wires require adjusting. They are generally placed crossing each other, at an angle inclined to the horizon of about 45°, and the operation is facilitated by first turning the telescope partly round, till they appear horizontal and vertical; half the divergence of each of these lines from the point is then corrected by the screws near the eye-piece, working in the diaphragm, loosening one screw as that opposite to it is tightened. One or two trials will perhaps be required, the diaphragm being moved in the contrary direction to that which in the inverting eye-piece it appears to require.
The second adjustment is for the purpose of setting the level
* Platinum wire is the best adapted for the purpose, though cobwebs are generally used by surveyors; and as they are liable to break from the slightest touch, it is necessary that every person using a theodolite should be able to replace them himself. They must be stretched tight across the diaphragm, and confined in their places (indicated by faint notches on the metal) by gum, or varnish, the latter of which is to be preferred on account of its not being affected by the humidity of the atmosphere. The following simple and ingenious mode of fixing these cobwebs, which to a novice is often a difficult and tedious operation, was mentioned to me by Mr. Simms, who constructs all the mathematical and astronomical instruments for the Ordnance Survey. A piece of wire is bent into a shape something like a fork, the opening a b being rather larger than the diameter of the diaphragm. A cobweb being selected, at the extremity of which a spider is suspended, it is wound round the fork in the manner represented in the sketch, the weight of the insect keeping it constantly tight. The web is thus kept stretched ready for use; and when it is required to fix on a new hair, it is merely necessary to put a little gum or varnish over the notches on the diaphragm, and adjust one of the threads to its proper position.