the common axis of the instrument; its motion is therefore perpendicular to the horizontal plates. It is graduated like the plate B, and by means of a venier, fixed above the compass, can be read off to minutes. It has its clamp-screw O, and its adjusting-screw P; and is used for taking vertical angles.

The telescope is placed above the vertical circle in two receptacles, called Y's from their shape, and secured above by two clips i, i. It has a long spirit-level fixed beneath, and parallel to it. The objectglass is adjusted to the sight by the screw Q, and the eye-glass by moving it backwards and forwards with the hand.

Between what are called the two parallel plates F and G are fixed two pairs of conjugate screws, b, b, &c. by means of which the upper one F can always be made horizontal. These two plates are connected by the ball and socket D; and to the lower one G are fixed the three legs by which the whole instrument is supported.

To take a horizontal Angle with the Theodolite.

The bubbles in the two levels d, d, by a proper opening of the three legs of the Theodolite, should be made nearly central, and the plumbet, suspended by a hook under the body of the instrument, should be also made to hang above the station, at which the angle is to be taken : then unclamp the whole instrument by means of the screw H, the other motions being kept clamped. Set one of the two levels on the horizontal plate A, over one of the pairs of opposite screws b, b, and the other level will be over the other pair, because the pairs of screws as well as levels are conjugate, i. e. at right angles to each other. If both the bubbles in the levels be not in the centre, loosen one of the conjugate screws and tighten the corresponding one till the bubble be accurately adjusted. Loosen and tighten the other pair, if required, till the same result is obtained with respect to the other level. If the last operation throw the former adjustment out, repeat the adjustments on each pair of screws, till both shall be level.

Now clamp the whole instrument, and unclamp the venier plate A; set the broad arrow of the venier to 360°, or zero, on the plate B, and reclamp the plate A. This must be carefully done by the microscope E, and the adjusting-screw I.

Again unclamp the whole instrument, and turn it to the left of the two stations, between which the angle is to be taken, till the intersection of the cross wires in the telescope cut the flag or other object in the station as accurately as can be done with the hand; then clamp the screw H, and by slowly turning the screw I the greatest accuracy may be obtained.

Now unclamp the upper plate A, and turn it round till the cross wires in the telescope cut the object in the second station; then clamp and adjust with the two screws attached to the plate A, till having obtained perfect accuracy, read off the angle by means of the venier with the glass E: in the same way, read off the angle with the other venier, and the mean of the two will be the correct angle.

To take a vertical Angle.

The horizontal plates being set level, as already explained, bring the bubble of the telescope-level to the centre of its tube, observing, at the same time, whether the zero point of the vertical circle coincides with that of the venier by the microscope N. These points being found to coincide, raise or depress the telescope till the cross wires cut the required object: then clamp with the screw O, and adjust with the screw P, and having obtained perfect accuracy, read off the angle, which will be an angle of depression, if the broad arrow be between the zero of the vertical circle and the object-glass of the telescope; and an angle of elevation, if beyond them.

ADJUSTMENTS.

I. To adjust the Line of Collimation.

Direct the telescope to some well-defined object at a great distance; adjust it till the intersection of the wires cut it accurately; then turn the telescope on its axis, and observe whether the centre of the wires still continue to cut it during a whole revolution. If it does it is in adjustment; if it does not, the line of collimation, or optical axis of the telescope, is not in the line joining the centres of the eye and object-glasses.

To correct this error, turn the telescope on its axis, and by means of the four conjugate screws, m, &c., correct for half the error, alternately loosing one screw and tightening its opposite one.

II. To adjust the Axis of the Level with that of the Telescope. Make the telescope perfectly level by the tangent screw P; then reverse the telescope in the Y's: if the level remains the same, it is in adjustment; if not, correct for half the error by the screw ƒ at the end of the level, and for the other half by the screw P. Replace the telescope in its former position, and correct again, if necessary.

For a description of the venier see page 319.

III. To make the Axes of the Levels on the Venier Plate

parallel to it.

Set one of the levels over one pair of the conjugate screws, b, b, &c., then the other level will be over the other pair; make both the bubbles come to their respective centres, turn the instrument half round, and if the bubbles deviate from their centres, correct half the error by the small screws on the levels, and the other half by the screws b, b, &c. Repeat this operation till the bubbles are central in every position throughout a whole revolution of the instrument.

NOTE. There are other adjustments of the theodolite, such as that of the vertical circle, and its venier; but a good theodolite rarely requires these adjust

ments.

THE VENIER.

The venier is a contrivance for measuring the fractional parts of the half-degrees on the lower plate of the theodolite. The space occupied by 29 or 31 of these half-degrees is used to form the scale of the venier, and divided into 30 parts; whence it is obvious that the difference of one division on the venier and one on the lower plate will beth in excess or defect of the 1° on the lower plate, or a single minute; and the difference between 2, 3, 4, &c. divisions on the two plates, will be 2, 3, 4, &c. minutes. Hence the following methods of finding the minutes corresponding to a fractional part of a degree in any given angle, &c.

1. To find the minutes in the fractional part of a degree of a given angle by the venier.

If the broad arrow of the venier is between a full degree and a half-degree, say between 40° and 4010; then the angle is 40° and as many minutes as are shown by the number of the first division-line of the venier that coincides with a division on the lower plate, reading forward as the degrees are numbered. If the broad arrow be between 4010 and 41°, then the angle is 40° 30′ plus the number of minutes shown by the first coincidence of the divisions of the venier and those on the lower plate. If the coincidence take place at the 13th division on the venier, then the angle in the former case is 40° 30', and in the latter 40° 30′ + 13′ : = 40° 43'.

2. To set the theodolite to an angle of any number of degrees and minutes.

If the required angle be 48° 28', bring the venier till the broad arrow shall be in such a position within the half-degree above 48°, that the division numbered 28 on the venier shall coincide with the same numbered line on the lower plate; the theodolite is then set to the required angle.

PROBLEM I.

To measure a base Line across a wide River.

Let DB A be the direction of the line, which has been measured to the edge b of the river; it is required to find the width ba of the river.

This may be done by various methods according to circumstances.

sin. A = cos. C = 53° 8'............................. 9.77812

ba = 768 (36 + 43) = 689 links, the breadth of the river.

The distance B A may be found by construction. See Solution of right-angled plane Triangles, Case I. ex. 2.

=

NOTE. If the angle A CB had been 45°, the angle A would also have been 45°, and therefore, B A B C. In this case no calculation, or construction, would be required. A cross-staff, having also sights placed at this angle, is sometimes used for thus expeditiously solving this Problem, when the nature of the ground admits of the line B C being measured of a sufficient length for the purpose.

2. The distance B A may be found independently of Trigonometry, thus. Having measured the perpendicular B C = 576 links, as in the last example, range and measure a line CD perpendicular to CA, and meeting the base line DB A in D; measure DB = 432 links: then (Euc. VI. 8.)

* The edges b and a of the river are not always found preferable for the places

of the flags, on account of obstructions.

NOTE. This calculation may be readily performed on the ground, so that the measurement of the base line D B A may be continued.

3. Should obstructions prevent the measurement of the perpendicular CD, let the line C D = 576 be measured as before; at any convenient distance BH = (suppose) 200 links on the base line, range and measure HI perpendicular to DB A, till the point I shall be in a direct line with the flags at A and C, and let HI= 726 links; then, (see last fig.)

As HI- BC: BH:: BC: BA*,

that is, 726 - 576 200 :: 576: 768 links =
В НХ В С 200 × 576
HI · ВС

or BA=

=

150

BA,

=768 links.

4. Moreover, should impediments prevent the lines B C, HI being measured perpendicular to HB A, measure B C to make any angle with HA, and at a convenient distance BH measure HI parallel to B C, which may be done by erecting two equal perpendiculars to B C, the point I being in a right line with A and C as before.

5. Lastly, if the nature of the ground be such that none of the four preceding methods can be adopted, neither to the right nor to the left of the base line, measure BC in the most convenient direction, and take the angles A B C, A C B with the theodolite; then by taking the sum of these angles from 180°, the angle A will become known, whence, by Case I. of oblique-angled triangles :

As sin. A BC:: sin. ¿C: BA.

This problem may also be solved by construction.

=

Examples for Practice.

1. BC 308 links (fig. to first method) is measured perpendicular to B A, and C D is ranged perpendicular to CA, meeting DA in D, the distance BD being 400 links, and the flags at B and A being at the water's edge; required the distance B A.

Ans. By the second method B A is found = 237 links nearly.

* This rule is derived from similar triangles; for HI: BC:: HA: BA, and BC HABA :: BC: BA, i. e. HI BC: HB:: BC: B A.

HI

Q. E. D.

Y