stances, the direction of the meridian with reference to the triangulation is to be laid down. It is necessary, therefore, to observe the direction of the meridian with respect to the original base, or some one of the sides of the principal triangles, from which the azimuthal distance of each part is given. When treating on the subject of longitude, we shall give an account of the methods that may be adopted to ascertain with precision the direction of the meridian at any one station; we shall here describe some ready methods by means of which its direction can be obtained with approximate accuracy.

MERIDIAN LINE.

Fix the theodolite at one of the stations used in the triangulation, and some hours before mid-day direct the telescope so that the cross wires shall touch the upper or lower limb of the sun in the east; note the horizontal and vertical readings of the arcs;-repeat the operation at short intervals, taking care to direct the intersection of the cross wires to the same limb of the sun that was before observed, and note all the readings in their regular

succession.

Again, in the afternoon, when the sun descends westwards, clamp the vertical arc to the last reading, and note the horizontal angle at the time of the sun's limb touching the intersection of the cross wires. The vertical arc being clamped in succession in the descending series of the vertical angles, all the horizontal readings at the time of each successive intersection are entered. The point on the horizontal limb half way between all the readings will give the angle to which the vernier is to be placed, in order that the telescope may point to the position occupied by the sun at noon. A picket driven into the ground in that direction serves to mark the meridian line, and the angle, formed between it and any side of the triangles having the

selected point for a vertex, being taken, the azimuthal direction of each and all the sides of the triangles is obtained *.

The same method, however, is applicable without correction to the observation of a fixed star; and the pole star, from the facility with which it is identified, is frequently

*This method would be quite correct if the sun moved constantly in the same parallel, but the change in his declination between the time occupied by the observations renders necessary a minute correction to be explained hereafter, but which in ordinary surveying operations may be altogether neglected. This mode of determining the meridian line by double altitude of the sun, is based on the assumption that the sun crosses the meridian when it has attained its greatest elevation. Supposing the sun's declination to remain constant in the interval between the first and last observations, it may be demonstrated as follows, that when it crosses the meridian its elevation is greatest.

Let EPRQ be a great circle passing through the pole and the zenith of the observer, and consequently his meridian line,

Let HR be the true horizon,

EQ the equator,

S S' the sun's daily path in half the interval between the observations, assumed parallel to E Q.

In the triangle PZS, PZ + ZS > PS, but PSPS' because SS' parallel to EQ; therefore

PZ+ZS > PS', take away PZ which is common to both terms, there remains ZS > ZS'.

But ZHZO, therefore

ZH-ZS' > SO-ZS. or

S'H> SO, that is, the altitude of the sun when in the meridian greater than his altitude in another part of his daily course.

selected for the purpose, being observed at the time of its greatest apparent eastern and western elongation. But if the telescope of the theodolite be not powerful enough to observe the star under these conditions,

(as one of the observations must generally be made by daylight,) a very close approximation may be had by remembering that the pole star very nearly reaches the true meridian, when it is in the same vertical plane with the e or Alioth, or the star in the tail of the Great Bear, which is nearest to the quadrilateral. The vertical position can be ascertained by means of a plummet. To see the cross wires in the field of the telescope at the

Pole Star.

κα

same time with the star, a faint light should be placed near the object glass. When the pole star has been brought correctly into the central part of the angle formed by the intersection of the cross wires, the horizontal limb is firmly clamped, and the telescope brought down to the horizon; and a light, seen through a small aperture in a board, and held at some distance by an assistant, is moved according to signals, until it is bisected by the wires. A picket driven into the ground underneath the light serves to mark the meridian line for reference

by day, when the angle formed between it and the side of the triangle may be measured.

The true situation of the North Pole may also be nearly ascertained by the following indications of the stars near it. In the first place, as shown. above, a straight line drawn from the pole star to the star Alioth, or e, in the

Pole

Great Bear, passes through the pole, and a perpendicular

to this line at the pole passes through the small star nearest the pole. Finally the stars called the "Pointers," in the above-named constellation, point almost directly towards the pole. The pole star is distant from the pole about 23 degrees.

When no angular instrument is at hand, an approximate meridian line may be set out as follows:

Drive a thin staff or picket vertically on a level piece of ground, a gravel walk for instance. Several hours before noon measure the length of shadow thrown by the picket, and from the base of the staff as a centre, with the

North

length of the shadow as

Picket.

direction of the meridian.

radius, describe an arc of a circle from west to east. About the same interval of time after mid-day, observe the point where the extremity of the shadow again coincides with the arc; a line drawn from the centre of the staff to the middle point of the arc thus intersected, will be nearly in the It will be better to describe

three or four such arcs at different elevations of the sun, and to make use of the mean of their central points to trace the meridian line.

OF PROTRACTING THE TRIANGULATION.

In protracting an extended triangulation, which has been conducted with the minute precautions indicated in this chapter as necessary to ensure great accuracy, it will be better to lay down the triangles from the lengths of their sides, than by measuring the angles; because measures of length can be taken from a scale, and transferred to the plan with more exactness than angles can be

pricked off from a protractor. If, however, the triangulation is to be laid down on paper, which is subject to changes from difference in the state of the atmosphere, having a greater effect on the accuracy of the plotted work than the difference between the degrees of exactness of triangles protracted by the length of their sides, or by measurement of the angles, it is not essential to adopt the course here indicated. Hence, for ordinary surveys, the triangulation is more frequently plotted by means of the angles, using, for the sake of expedition, a circular protractor. There are various kinds of Protractors, among which I select the following, which is found expeditious and accurate.

The Circular Protractor.

"It consists of an entire circle, A A, connected with its centre by four radial bars, a a, &c. The centre of the metal is removed, and a circular disk of glass fixed in its place, on which are drawn two lines crossing each other at right angles, and dividing the small circle into four quadrants, the intersection of the lines denoting the centre of the protractor. When the instrument is used for laying down an angle, the protractor must be so placed on the paper that its centre exactly coincides with or covers the angular point, which may easily be done, as the paper can be seen through the glass centre-piece.

"Round the centre, and concentric with the circle, is fitted a collar, b, carrying two arms, cc, one of which has a