inches in diameter; and to eliminate as much as possible, every source of error, great numbers of operations are made on each station, the readings being made on different points of the arc. Usually from 40 to 60 observations are made for each angle-one measurement, with the telescope direct, and one with it reverted, constituting a complete observa tion. With these precautions, it has been found that the error in a primary triangle (where the sum of its three an gles has been compared with 180°), has fallen much within 3 seconds. The error of 3 seconds has been adopted as the highest admissible limit of error.

17. Observations are also made at the principal stations upon the pole-star, and other stars near the pole, for the purpose of determining the angle, made by the sides of the triangle with the meridian. In minor surveys, and in a secondary triangulation, the operations are much less elaborate; still, every precaution is to be taken to insure the greatest attainable accuracy. As a general rule, all the angles of every triangle, should be measured, if possible.

18. To illustrate the manner of carrying on a minor triangulation, let us refer to the plan of the harbor [plate 6], in which AB is the measured base, C, D, E, &c., triangulation points, at which signals have been erected. Commence the triangulation at A, the west end of the base; and for convenience in plotting, it would be well to make the line, passing through the 0 point, and 180° parallel, in each position of the instrument, to the base AB. Having brought the 0 of the vernier to the 0 of the limb, clamp the vernier plate, and direct the upper telescope to the signal at B, and clamp the limb. observation as in the following table:

OBSERVATION AT STATION A.

Enter the

Having recorded the reading of the first vernier, and the minutes and seconds of the second vernier, unclamp the vernier plate, and direct the telescope to the station at E, and record both verniers, as before. Again unclamp the vernier plate, and direct the telescope on the signal at G; and then read and record, as before.

Having determined the angles subtended by all the signals visible from A, let the theodolite be removed to B. Bring the 0 of the vernier I to 180° on the limb, and direct the telescope on the signal at A-the line (0°, 180°) will then be parallel to its first position, and the limb may be clamped. Read now the angles to the signals at A, E, C, &c., and record as before.

If the theodolite is now removed to the station E, the line (0°, 180°), may be made parallel to its first position, by adding 180° to the reading of the first vernier, from A to E, and then directing the telescope on the signal at A. The line (0°, 180°), will thus be made parallel to AB, and the reading may be made and recorded as before; and so on until all the stations have been visited, and the angles measured. From the field records, the angles BAE, EAG, ABE, EBG, &c., may be easily deduced, the whole may be plotted on paper, or the several sides may be com puted trigonometrically. It may be observed that the line (0°, 180°), has been made parallel to the base line at each station; where great accuracy is required, this cannot be done, since a single reading is insufficient to give the angle. The angle is then determined, as directed in the previous article, or by means of the principle of repetition.

19. To illustrate this principle of repetition, suppose the 0 of the vernier to coincide with the 0 of the limb, and the telescope to be directed, from the station A, upon one of the objects, as the signal at B. Clamp the limb, and unclamp ing the vernier plate, direct the telescope on the second ob ject, as the signal at E. If we now clamp the vernier plate, and unclamping the limb, direct the telescope on the signal at B, the iine (0, 180°), cf the limb, will make

and unclamping the vernier plate, direct the telescope on the signal at E. The reading will evidently be equal to twice the angle BAE, and if we repeat the operation, the reading will be three times the angle, and so on. After ten repetitions, if we add 360° each time the 0 of the vernier passes the 0 of the limb, the final reading will be ten times, the angle BAE, affected with the joint errors of the ten observations, and one-tenth of this will be the read. ing required, to a greater degree of accuracy than could probably be attained by a single observation.

20. The method of reading angles, by this mode, is as follows:

Angles at station A, between signals B (left), and E

(right.)

21. After the triangulation is completed, the interior may be filled up by the aid of the Compass, or the planetable.

USE OF THE COMPASS.

22. When the secondary and tertiary triangles have been considerably multiplied, the compass is taken in hand. The field-notes may be kept in the following manner. Divide a page of the note-book into two equal parts, by two parallel lines near to each other, and consider each part as

spaces, and the middle space is generally smaller than either of the others, which are equal.

The notes begin at the bottom of the first page, and run up the page to the top. They then commence again at the bottom of the next page, and run up to the top; thence to the bottom of the third page, and thus, for as many pages as the work may require.

When the compass is used in the way we are about to explain, the distances to objects which lie on the right or left of the courses, are determined by means of offsets.

The beginning of every course is designated in the middle column by 0, and the bearing is entered directly above. The other figures of the middle column, express the distances from the beginning of each course to the offsets, and those in the side columns indicate the lengths of the offsets, or the distances to objects on the right or left of the compass lines.

To explain more definitely the manner of using the compass on the field, let us suppose that we have determined the prominent points and longer lines with the theodolite. Place the compass at A (Plate 6), and take the bearing of the line AE, which is S 12° W.

AE any distance, as Aa equal to 130 yards, and make an offset to the lake, which we measure and find to be 50 yards. Enter the 130 in the middle column, and as the lake lies on the right (in going from A to E), we insert the 50 in the right hand column.

We then measure along the line AE to b, 350 yards from A. Here we make a second offset to the lake, and find it to be equal to 100 yards. Having entered the distances in the notes, we measure to q, the point where the line AE crosses the creek, and we enter the distance from A, 415 yards.

At d, we lay off an offset on the left, to the pond, 70 yards at e, an offset to the mouth of the creek, 150 yards: and at E, where the course terminates, an offset to the lake, of 160 yards. The entire distance from A to E is 800 yards.

At E, we take the bearing to H, which is N 50° E. Having measured along this line to f, 315 yards, we make an offset to the pond, on the left, of 50 yards, and to the shore, on the right, of 90 yards. Having entered these distances, we recommence the notes at 315 below, which we suppose to be at the bottom of the second page. Having reached H, the extremity of the course, we enter the entire distance from E, 680 yards. We next take the bearing to I, S 52° E. We then measure the distances to m, n, p, and I, and enter them, together with the offsets, as in the notes.

23. It is also well to make, in the columns on the right and left, such sketches of the ground, fields, houses, creeks and rivers, as will afford the means of making an accurate delineation on paper.

THE PLANE-TABLE-ITS USES.

24. Pl. 3, Fig. 1. The plane-table consists of two parts; a rectangular board CDBA, end a tripod EHG, to which it is firmly secure 1.

Directly under the rectangular board are four milled