4. In the triangle C D E, we have the side C D, and the adjacent angles; hence we obtain C E.

The accuracy of the work is proved by the length of CE, obtained by two distinct operations, being brought out without any variation.

If for any cause it has been found advisable to commence the triangulation before the base has been measured, the sides of the triangles may be calculated from an assumed base, and afterwards corrected for the difference between this imaginary quantity and the real length of the base line. Or, as was found to be the case with one of the Indian bases*, if the length of the base has originally been, from the want of access to correct standards, incorrectly reduced, the triangulation may be easily rectified; the property of similar angles readily points out the method to be employed.

*The base measured near Gooty, for the Indian survey*, was found, after a careful comparison of the chains with the standard brass scale, to require a small correction. A standard chain referred to in that operation had been carefully tested some years before, and laid aside by Colonel Lambton, who relied on the belief that its length would remain invariable; however, a considerable period after the calculation of the triangles depending on the Gooty base had been completed, he was led again to test the length of the standard chain, and found that it must have been slightly too long at the time of the measurement of the base. This had led to erroneous results, most minute it is true, but still such as Colonel Lambton thought fit to notice and rectify. In the length of a degree, due to latitude 11° 37′ 49′′, the required correction amounted to 1.25 fathom, the length of the degree having been originally calculated at 60480-3 fathoms, and the corrected length being 60481-55.

* Asiatic Researches, vol. xiii., and Philosophical Transactions, 1823.

EXERCISES FOR CALCULATION *.

Given A H = 27404-2 feet, and the angles of the several triangles as below, required their sides.

Reduction of Angles to the Centre of the Station.

In extending a series of triangles in populous neighbourhoods, wooded districts, or occasionally under other circumstances, instead of planting moveable signals at each point of observation, it will be found more convenient and more economical to select permanent well-defined objects, such as steeples, towers, windmills, &c., for the principal

* Trigonometrical Survey., vol. i., pages 82, 83.

stations in the triangulation. But when a choice is made of such objects, the theodolite or circular instrument can seldom be placed in the centre or axis of the station. The observer, in such cases, approaches as near to the centre as he can with advantage, and calculates the quantity of error which the minute displacement may occasion. But, as a general rule, it will be found expedient to take pains to select objects, such as towers, &c., that will admit of the theodolite or instrument being placed in the centre of the station, the corrections for eccentricity being so troublesome and so complex, as to consume much time in additional observations, and present additional chances of error in the aggregate. I subjoin in a note* examples of some of the precautions and corrections used in great surveys.

*

Suppose it be required to determine the angle A CB which the remote objects A and B subtend at C, the centre of the permanent station; the instrument is placed in the immediate vicinity at the point D, and the distance DC with the angle ADC noted, while the principal angle A D B is observed; the central angle A C B may then be computed from the rules of trigonometry *.

When the instrument is placed at the base of a tower or high permanent object, the centre of which is the true vertex of the triangle, it sometimes happens that the centre of the station cannot itself be seen, but the direction of that centre from the axis of the instrument is required for the purpose of measuring the angle A D C. The direction of the centre is found as follows:

* See Notes to LESLIE's Trigonometry, page 469, and PUISSANT's Géodesie, pages 182, et seq.

H

Sketch of Permanent Objects used as Stations.

When observations are made to churches, towers, or other permanent objects, it is desirable to make a slight

First, supposing the base of the station or signal to be rectangular; from the extremities of the diameter H F, draw the lines DF, DH, and measure their lengths; then on D F, take any point ƒ and from D H, cut off D h, so that D F:DH:: Df: D h, then f h is parallel to FH, one of the diameters. Bisect fh in o, join Do; Do will be in the direction D C required.

B

Supposing the base of the tower or signal to be circular: draw the tangents D T, D t, by sweeping the telescope of the theodolite round, until the visual ray describes a line touching the circumference of the tower. From D T, D t, cut off equal lines D F, Df, join Ff, and bisect it in o, the line D o joining D and o will be in the direction of the centre C. The points F and ƒ should be chosen as near as possible to the tower, in order that Ff may be as long as possible.

If the instrument be placed within the circumference of the circle; through D, the axis of the instrument, draw the chord E F bisected in D. On D erect the perpendicular DC; DC will be the direction required.

C

F

sketch of their general form and appearance from the point whence they are observed; and if the signal be irregular in its outline, a mark should be made showing the part of the object intersected, in order to avoid errors when the same object has to be viewed from another distant station, or if it should be necessary to re-observe the angles at a future time.

If the position of the point D be such that both extremities of a diameter of the base cannot be seen from it, or if the base of the station be a regular polygon of any number of sides, the direction DC is obtained as follows:

Then in the triangles D I C, D E o, we have

DI: IC: DE: E o, or

point of intersection of D C with F E, and consequently the direction D C, is obtained.

FORM OF OBJECTS UNDER OBSERVATION.

Attention is also to be paid to the form of objects or signals under observation. Those which do not terminate in a point, whether presenting the form of a truncated pyramid, or a rectangular top, may lead to errors, when they are illuminated obliquely by the sun, causing thereby the observer to direct his telescope not to the centre of the signal but