Fig. 140 shows at C a portion of the theodolite circle as seen through the micrometer microscope. P is the movable pointer, M the micrometer head, and I the index or reading line.

To use the micrometer the first steps are to carefully focus the pointer P by means of the eye-piece until it appears clear and perfectly sharp, and set the reflector at the bottom of the microscope so that it reflects sufficient light to illuminate the divisions on the circle. Then, by turning the micrometer head M, set the pointer P to the centre of its travel, so that it covers the V cut in the bottom of the slide, and leave the 0 of the micrometer head exactly opposite the index line I. Now proceed in the same manner with the other microscope. After setting the microscopes as described above, lightly clamp the lower clamp screw of the instrument and release the upper one. Now revolve the upper part of the theodolite until 360 degrees on the circle appears exactly under the pointer of one of the microscopes. The other will then be pointing to 180 degrees, and the instrument is set ready for measuring the first angle.

We will presume now that a bearing has been taken by the telescope and it is required to read the angle, and that on inspection of the micrometer it is seen to be in the position illustrated at Fig. 141, viz., between 227 and 228 degrees. Now as the degree is subdivided into 6 parts, each of these subdivisions must represent 10 minutes of arc, therefore the pointer is situate between 227° 30′ and 227° 40′. It is now necessary to measure exactly the distance of the pointer from the division 227° 30′, which is done in the following manner, by means of the micrometer head M.

This micrometer head is so constructed that one complete revolution of it causes the pointer to exactly travel over the space of one division on the circle.

The head itself is divided into 10 primary parts, which indicate single minutes, and these are subdivided into 6 parts of 10 seconds each, therefore in order to measure the exact position of the pointer in Fig. 141 it is only necessary to turn the head M until the pointer is exactly over the previous division of the circle (as shown in Fig. 142) and read the distance on the micrometer head M. In this case the head has been turned through six main divisions of 1 minute = 6 minutes and two subdivisions of 10 seconds = 20 seconds, giving a total reading of 6′ 20″, this, added to the circle reading of 227° 30′, gives 227° 36′ 20″, which is the correct reading of the angle.

It will be seen that this method is very much simpler and a great deal more accurate than any form of vernier reading, and also that its greater accuracy permits the use of smaller instruments. Thus a 5-inch micrometer reading theodolite is more accurate than a 6-inch one with verniers.

Six-inch micrometer theodolites are usually divided to read to 5 seconds of arc. The method of reading is the same as described above, but in this case the circle is divided to spaces of 5 minutes each and the micrometer head to 5 main divisions of 1 minute, each of these having 12 subdivisions of 5 seconds, which it is possible to again subdivide by estimation and so measure angles to 2½ seconds.

Another feature in favour of micrometer reading instruments is the ease with which they can be adjusted. With verniers, should they get out of adjustment through damage, the instrument must be returned to a maker; with micrometers, if through rough usage or accident, it is found that after bringing the pointers to the centre of their V's and setting the micrometer heads to 0 they are not exactly opposite one another (180 degrees apart), then their setting has become disturbed and must be readjusted in the following manner:—First bring the V of one micrometer to the 360° on the circle, then see if the V of the opposite micrometer is exactly at 180°, if not this can be easily set to it by means of the small adjusting screw which will be found at the left end of the micrometer box, that is, the opposite end to the divided head. Having examined the V's and adjusted them if necessary, the next step is to set the pointers P exactly to 360° and 180° respectively, in which position the divided heads should both read 0; if they do not do so reset them as follows: Take a screw-driver and slacken the small screw which is in the centre of the divided head; this will free the divided rim so that it can be turned without shifting the position of the pointer. Turn the divided rims until they read exactly 0 at the index line and retighten the screws. This completes the adjustment.

346.—Clamp and Tangent Adjustment.—The vernier reading to the circle, when this was adjusted by the hand, was scarcely practicable at nearly its full value until the discovery of the clamp and tangent screw motion was made. This useful invention is due to Helvetius, the celebrated astronomer of Danzig (about 1650). By this mechanical arrangement the circle or arc is left quite free to move about its axis until the clamp is screwed down, which then fixes it firmly. The fixing arrangement of the clamp is attached to a solid part of the instrument, but is so constructed that when it is clamped it may yet be moved without unclamping, in relation to the fixed part of the instrument, by the tangent screw which, as its name indicates, is placed in a direction tangential to the circle or arc. This arrangement may take many forms in detail, two of which, the most general and especially adapted to surveying instruments, will be described.

347.—The above illustrations, Figs. 143, 144, represent a clamp and tangent motion in two sections at right angles to each other. This form is common to vertical circles and arcs generally, of a theodolite, arc of sextant, circles upon some mining-dials, protractors, and many other instruments. Fig. 145 is partly a front elevation of the same, but with part of the clamp screw A cut off. The stem of the tangent screw is shown in section at E. In all the figures L is the limb of the circle or arc. This has a groove at its under side at G, into which a fillet of the clamping piece C is inserted to make the clamp slide freely about the periphery of the circle when the clamping screw A is loose. A spring is sometimes inserted to open the clamp between the sliding piece K and the clamp C. FF, Figs. 143, 144 is the tangent nut to E. This nut is sawn down and has a cross screw to keep sufficient tightness to prevent loss of time, and yet to allow the tangent screw to work pleasantly at the same time that it holds the circle and vernier quite dead to the position to which it is adjusted by the screw. The tangent nut F has to move to the direction horizontal to the plane of the tangent screw; therefore it has an axis vertical to the plane of the clamp. This is shown at K. The axis is held down firmly by a nut and a washer fitted with a square hole, to prevent the nut unscrewing. The tangent screw has a collar fitting or shank at the tangent boss B, which is turned down from the full-sized metal of the screw. The fellow collar on the outer side of the boss is formed by the shank of the milled head of the tangent screw D. The hole through the milled head is made square, so that it can be adjusted up to the boss without risk of after unscrewing by friction by the screw E. This is tightened up by means of a screw-driver applied at E. The boss B has a vertical axis N, similar to the tangent nut, and is attached to a solid part of the instrument by the washer and nut shown at 0.

348.—The above construction is solid and good, and will bear considerable wear; but there is a little delicacy of touch required to adjust the collars to the boss and to give pleasant tightness to the screw; a better plan is to dispense with the split in the tangent nut and the inner collar turned on the tangent screw, and place a spiral spring over the tangent screw which follows the adjustment, or in some cases a long bow spring may be conveniently used in place of the spiral. These plans answer very well: one of them will be presently described for axis clamping. In place of the groove at G the clamp is sometimes constructed to move on an arm direct from the axis of the circle. This is on the average a pleasanter motion, but in complex instruments it would often interfere with the motion of other necessary parts.

349.—Axis Clamp and Tangent.—This is generally used to bring the horizontal axis of an instrument to bearing, and is made independent of the circle and vernier. The ordinary form, which is very effective when properly constructed, is shown Fig. 146. This form is used for clamping the vertical axis of a theodolite, mining-dial, Y-level, and some other instruments. The clamp C surrounds the axis as a collar, from which two lugs in the same casting are projected at a. These are brought tight upon the outer axis socket B by means of the screw W, which has a wing-nut head to give good purchase. In the construction of this form of clamp the collar should be fitted and ground to its bearings with the lug in the solid, and the cut at a be sawn through afterwards.

350.—The tangent screw adjustment is shown at T, moved by the milled head M, the boss E being fixed to the instrument. This part of the arrangement is just the same as that described above for a vernier tangent. Objection has sometimes been made to this form of clamp, that it tends to become weak after a time from the constant clamping and releasing, which causes loss of elasticity in the metal. When this occurs it is no doubt due to the metal of the clamp not being good gun-metal; or, if brass, not thoroughly pressed or hammered before the piece is made up. A plan, in not uncommon use in Germany, of avoiding this supposed source of weakness is to bring up a tumbling piece direct on the axis by a screw. This is shown in Fig 147, screw W; tumbling piece A. This produces a direct clamp upon the axis socket B′. The clamp ring CC′ is made loose on its socket.

351.—In practice it is found impossible to clamp the axis of a theodolite without disturbing the centre more or less. In some experiments the author made he found the direct or tumbling piece clamp Fig. 147, although it holds firmly, disturbs the centre much more than the clasping clamp Fig. 146. Therefore when the former is used the clamp should be upon a strong flange. This increases weight, and it can scarcely be so well for a portable instrument. In all cases, in the construction of the instrument, clamps should be fitted and screwed down before the centre is ground and finished. This ensures the centre being made correct in its clamped position, in which it will afterwards be used.

The arrangement Fig. 147 shows also a spring S falling upon a stud at E, fixed upon a part of the instrument upon which it acts as a fulcrum. The spring should be of hard rolled German silver. In this case the tangent screw needs no split or other adjustment to make it tight, as all loss of time is taken up by the spring.[14] The plan is found practically to answer fairly; but unless this is very carefully made there is a want of solidity in the movement which a well-fitted, direct-acting tangent screw possesses.

352.—The French generally in all their superior instruments clamp upon a flange carried out from the lower rim of the socket, with the screw placed longitudinally to the axis. When this plan is very carefully carried out, so that the clamping has neither tendency to raise or lower the socket-piece, it is no doubt very good. In large instruments, where weight is no object and the flange may be made large, it is certainly the best plan. In such cases the clamp may be released as a free fitting to prevent the possibility of strain. Fig. 148 shows the French plan attached to a tribrach: S socket, F flange, C clamping screw, T tangent screw. The tangent in this arrangement acts against a spiral spring contained in a tube A, which gives a very steady motion to the instrument.

353.—Some particulars of the care required in the manufacture of the tangent screw were given, art. 22. The test for the equality of this screw, which is important when it moves a vernier, is to loosen its clamp and to see whether it works equally, firmly, and smoothly at all parts when it is turned down from end to end. The test for its straightness is to screw down the clamp, then to notice any little mark on the milled head of the tangent screw, or make a slight mark upon it, and to place this mark uppermost, and then to take a reading with the vernier, then to turn the milled head a quarter turn and take another reading, and again another quarter, and so on consecutively. By comparing the rates of reading of the vernier at the quarter turns, if we find these equal the screw is straight. A little allowance is necessary for imperfect work. If the work is very bad at some quarter turns there will be an advance at the opposite quarter of nearly double the proper mean quantity.

354.—For Testing and Adjusting the Fitting of the Tangent Screw.—The clamp should be tightened down and the ball B, Fig. 144, held tightly between the thumb and forefinger; then, by using a gentle reciprocating motion in the direction of the tangent just sufficient to move the circle, if there is any looseness in the screw or the ball fitting B it will be felt as a jar, or technically, a slight loss of time. If this be in the ball B it can be taken up by the screw E at its end. If it be in the screw it can be taken up by the cross clamp screw. If it be in neither of these, it may be in one or both of the axes N and K. In this last case it will need refitting. It appears a somewhat simpler test with a theodolite to lightly press the telescope on one side of the eye-piece and take a reading of the vernier, and then to press the other side and again take a reading. This, possibly, indicates loss of time in the clamp and tangent if there is found any difference in these readings; but this would not be with any certainty, as the fault might be in some other part of the instrument. It, nevertheless, is a simple plan to test the whole instrument, including the clamp and tangent, although this does not localise any defect there may be in any special part of it.

355.—Use and Wear of the Clamp.—The common fault of a novice when he commences to use an instrument is that he applies too much violence to all clamping parts. Thus we find the lower parallel plate of an instrument soon becomes deeply indented, and the clamp of the tangent screw often strained, or its screw worn loose by extreme clamping. The best rule to avoid this with a clamp is to make a personal test of how little force is required to produce sufficient hold for the action of the tangent screw, and when this is found out to try to clamp only slightly in excess of this. A novice scarcely recognises the power of a screw. It is, perhaps, a fault of some makers of giving much too large heads to clamp screws which to a certain extent permits this overstraining from clamping. In discussing this matter with a scientific civil engineer upon an instrument which had been very much strained, to which small clamping screw heads were suggested, this gentleman replied that he looked to the optician to "supply instruments, not brains," and made the user responsible; but, really, a young surveyor is generally so intent on the object of his work that he cannot consider the mechanical details of his instrument, to which his attention possibly has never been properly directed; so that there is a policy in cutting off possibility of injury to the instrument where this can be conveniently done.

356.—Use and Wear of the Tangent Screw.—Seeing that the axis of an instrument is quite free to the extent of the loss of time on the tangent screw which holds it, and that this freedom, by any slight touch of the telescope, may cause a difference of reading—in some cases of several minutes of arc—it becomes important to observe that the tangent screw is in good order. This matter considered at its full value, we may wonder, perhaps, what kind of work may have been done with the tangent screw loose and worn down in its central part, as we find it in many old instruments sent for repair. A great amount of the common defects we find in worn tangent screws might have been prevented by using certain precautions; and even the much-worn tangent screws would sometimes go on fairly by a different method of use from that to which they have evidently been submitted. The wear of a tangent screw is due principally to the fact that this screw is necessarily oiled to make it work freely, and that the oiled part being exposed to dust, this dust attaches itself and works into the thread with the oil so as to cut both the screw and the nut. Precaution is necessary that this should be obviated as far as possible. One precaution may be taken, that when the screw is oiled, say once in three months, the parts outside the nut should be cleaned off quite dry with a few strands of thread. The oil left in the nut, if the screw has been turned through it, will be quite sufficient to lubricate the screw. Another better precaution is to use only one part of the screw for a period, say one month. The screw may be divided mentally into three parts—near part, middle part, and end part. If one part only be used for a period, and the vernier be set in using the instrument so that not more than about 1° of motion is required of the screw, no grit can be carried far into the centre of the nut; and if the precaution of cleaning the screw with thread be taken every time the instrument is returned to its case after a day's work, the screw being left at about the same place on the screw and nut, it will keep true with little wear. When another part of the screw is taken into use, this part should be first cleaned with thread and then oiled with watch oil, after which the former position of the nut should be cleaned quite dry with thread. Treated in this manner a tangent screw will last, in constant wear, for ten years or so, keeping in fairly good order. Where a spring is used to take up loss of time there is less risk, and the only precaution necessary is to be sure the spring continues to act properly. There is generally, however, a little more wear with a spring than with a free thread.

357.—If the instrument be not touched after the tangent is set, and there is no wind to cause vibration, the instrument will read correctly although the tangent may be out of order. But after the adjustment by the tangent screw, which may cause a disturbance, it is always necessary to set the microscope to the vernier. This is one important reason why the microscope should move as softly as possible, and that it is advisable to centre it upon the axis. Where any doubt of the quality of the tangent exists, the telescope should be reobserved for verification of its position after reading, which is also undoubtedly the safest in all cases.

358.—Some contrivances have been applied to tangent screws to prevent wear from dust, and also to take up the nut after wear. A very good plan, common in American instruments, is to insert the end part of the screw beyond the nut in a closed tube. This entirely prevents dust from resting on this part; and if the precaution be taken to clean the exposed part of the screw after use it is very effective for preservation. This plan the author has combined with a spring arrangement, which appears to render it very safe from loss of time and much wear. This arrangement is, however, a little expensive to make, therefore can only be applied to high-class instruments. Fig. 149, C nut, through which tangent screw passes; B tangent boss, A milled-head, H covering tube to the point of the screw, GG′ EE′ pair of telescopic tubes which cover the screw. A German silver or platinum spring works inside these tubes, keeping a constant separating pressure between C and B to take up any loss of time in the screw.

359—Free Tangent Screw.—There is always a risk of a tangent screw of any fixed kind producing a certain amount of strain upon the instrument, therefore, where practicable, it should be made free. The illustration, Fig. 150, shows the form of free tangent the author now applies to many instruments. The centre stud is clamped to the lower part of the instrument by the screw shown in dotted lines. To the left hand a piston containing a spiral spring carries a pressing-rod against which the screw to the right hand works.

360.—Loss of Time by Wear of the nut is variously taken up when no spring is used. One plan was shown of splitting it up. A plan common in Germany is to make the nut in two pieces, which are brought up by two screws. This is a very effective plan. The author has found a tumbling piece arrangement also effective. Fig. 151, S section of tangent screw, T tumbling piece moved by the adjusting screw, shown above, for wear of the tangent screw. This adjusting screw A should be tapped tight without oil, and put together dry to prevent its receding by pressure.

361.—Hypotenuse and Base.—Other trigonometrical values besides the division of the circle into equal parts are occasionally placed on instruments for special purposes. The most common of these is the scale of difference of hypotenuse and base, which is generally placed upon the back of the vertical arc of a theodolite and upon some dials and clinometers. The division for this purpose is generally done by hand. The scale gives a percentage difference for certain angles. Thus when used with chain measurement, it gives the number of links of the chain to be deducted per chain of 100 links for the inclination of land that the theodolite or other instrument indicates in following the surface contour.

362.—A Horizontal Scale of Tangents was placed upon the surveying theodolites by Ramsden. This was divided upon a scale carried by the vernier plate, which read to the zero line (0°) of the limb. It is found in practice more accurate to take the tangent to any curve from a scale of tangents, as, for instance, that in Molesworth's pocket-book, and set this off upon the limb by means of the vernier.

363.—Gradient Scale.—Civil engineers engaged on railway work occasionally have a scale of gradients upon the back of the vertical arc 1 to 100, 150, 200, etc. These are better read from the circle with vernier from a table of gradient arcs.