Surveying and Levelling Instruments, Theoretically and Practically Described. / For construction, qualities, selection, preservation, adjustments, and uses; with other apparatus and appliances used by civil engineers and surveyors in the field.

CHAPTER VI.

DIVISION OF THE CIRCLE AND METHODS EMPLOYED IN TAKING ANGLES—DIVIDING ENGINE—SURFACES FOR GRADUATION—VERNIER—VARIOUS SECTIONS—READING MICROSCOPES—SHADES—MICROMETERS—CLAMP AND TANGENT MOTIONS—OF LIMBS—OF AXES—USE AND WEAR—DIFFERENCE OF HYPOTENUSE AND BASE.

304.—Division of the Circle.—Sexagesimal Division.—All true surveying instruments depend, as their special function, upon taking the direction, or angular position, of surrounding objects or definite parts of the surface of the earth from positions which are at first accurately measured or ascertained. The instruments required for such work must possess an accurately divided circle or arc, with means of subdividing the visible divisions of this to greater closeness than any possible method of drawing lines simply would permit. The lines upon the circle in general practice in Great Britain are divided into degrees, which are subdivided to 30, 20, 10, or 5 minutes, according to the size of the instrument, and arranged for further subdivisions by means of a vernier into minutes or 30, 20, or 10 seconds of arc. Upon large circles, say of 10 and 12 inches diameter, and with modern 5, 6, and 8 inch diameters, angular displacements in the direction of the telescope are ultimately read off with a microscope by means of a screw with divided head, termed a micrometer, placed tangentially to the divided circle; or by a series of lines placed at equal distances apart in front of an eye-piece or within a microscope; but in the ordinary portable instruments, or those that a surveyor can personally carry about the country, the ultimate subdivisions of the circle are still generally made by a vernier scale only, which will presently be described, although the smaller modern micrometer reading instruments are slowly but surely coming into favour for all high class work.

305.—Centesimal Division.—Ten to fifteen years ago on the Continent generally, and in America occasionally, the division of the circle into 400-grades and ½-grades, and the subdivision of these decimally to centigrades, appeared to be coming more and more into use, particularly with the more extended use of the tacheometer. Under this system it will be seen that the right angle subtends 100 grades. This division, with its centesimal parts, was found to blend conveniently with logarithmetical calculation and to permit the free use of the slide rule with great saving of time over ordinary calculation, but it is now very little used.

The decimal division of the ordinary degree of 90 to the quadrant greatly facilitates the calculation compared with what is necessary with the sexagesimal division into minutes and seconds, and the reading of the verniers is much simpler and less liable to errors; moreover, the mental conversion of the sexagesimal division into decimals of the same degrees is much simpler than the conversion into the centesimal degrees of 100 to the quadrant.

306.—Dividing Engine.—This important tool is used for cutting the graduations on all surveying instruments. If possible a position should be secured for it on a ground floor at a mile or more distance from any railway, and at a good distance from roads upon which there is heavy traffic, as small vibrations are sufficient to cause unpleasant working and some error in the division of large instruments. For very accurate work some makers divide at night for the sake of stillness. The principles of construction of this machine, as at present in general use, were invented by Jesse Ramsden, of which an account was printed by the Board of Longitude in 1777. Refinements of detail have been added to the invention, and the steady action of steam or electric power has been applied in place of the foot, but otherwise the machine remains practically the same. Therefore a brief description of this machine as originally invented will be sufficient for the purposes of this work, which is not intended to fully describe the tools used in the manufacture of instruments.

307.—Ramsden's Engine consists of a circular brass surface plate, made generally of 36 inches diameter. This plate is supported from below upon a hollow vertical axis, which moves in an adjustable collar placed at its upper end and in a conical point or pivot at its base. The pivot rests in a cup of oil and supports the weight of the plate and axis, so that this part rotates with little friction. The outer edge of the surface plate is cut with 2160 teeth or threads, into which an endless or tangent screw works, so that the plate can be revolved any desired quantity by means of the screw. Six turns of the tangent screw moves the plate 1°. The head of the tangent screw is divided as a micrometer into 60 parts; therefore the movement of one of the divisions of this head revolves the plate 10 of an arc. A ratchet wheel of 60 teeth is attached to the tangent screw, and so arranged that by reciprocating motion applied to a rack which works into it the circle can be advanced any multiple of 10. Motion is given to the tangent screw by a catgut over a pulley worked by the foot. The work is centred and clamped down upon the surface plate. While the divisions are being cut this surface plate remains for the time quite stationary.

308.—The dividing knife is attached to a swinging frame having a reciprocating motion. The forward extent of its swing is regulated by a detent wheel with teeth of varied heights, which, as they are brought by the mechanism consecutively forward, stop the knife at a definite position; so that the cuts upon the circle—technically the limb—are regulated for lengths to represent 10 degrees, 5 degrees, degrees and parts. In the use of this dividing machine the divider who worked it had alternately to press his foot upon a treadle and then pull a cord attached to the dividing knife frame. These motions are now performed by self-acting mechanism. For full particulars and details of the dividing engine see Troughton's Memoir, Phil. Trans., 1809: Memoirs of the Royal Astronomical Soc., vol. v. p. 325; vol. viii. p. 141; vol. ix. pp. 17 and 35. For various plans that have been tried see Holtzapffel's Turning and Mechanical Manipulation, pp. 651–955.

309.—The Material upon which the limb or circle of an instrument is divided is almost uniformly of silver, except for mining survey instruments, which need a very strong cut. Silver being dense and of extremely fine crystallisation, or grain, as it is technically termed, bears a uniform smooth cut with sharp outline. Occasionally circles or arcs are divided on platinum, certainly the best metal, as it keeps constantly clean; but it is expensive. The verniers are then made either of this metal or of gold. The silver of the circle, when this metal is employed, is rolled down from a surfaced cast plate of about ·25 inch in thickness to about ·045 inch, by means of which it becomes uniformly dense and fine grained. In all cases possible, that is, upon all flat internal surfaces, the silver is placed in an undercut groove and planished down to fill the groove without any other fixing being necessary. This plan of insertion is employed for all vertical circles—the horizontal circle of Everest's theodolite, limbs of sextants, box sextants, etc. In Fig. 117 the silver is shown at A, in the section to which it is drawn by a plate after it is cut in slips. It is shown placed in its groove B ready for planishing down. By this method certainty of dense surface is obtained for the future division.

310.—Upon bevelled edges and outer surfaces the rolled silver is planished to form, and then soldered to the metal of the part of the instrument to be divided. The surface, after being made as dense as possible by planishing or otherwise is turned to form and stoned to surface ready for the dividing knife.

Fig. 117.—Insertion of silver in circle.

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311.—Graduating.—The object aimed at by the skilful divider is to obtain as deep a sharp-edged cut as possible, which shall be at the same time as fine as it can be read clearly by the microscope with which it is to be used. This matter is most important to the possessor of the instrument afterwards for use, as in the atmosphere the silver soon forms an oxide and a sulphuret upon its surface which has to be cleaned off; and at every cleaning a portion of the silver is necessarily removed, so that in old or badly divided instruments the divisions become dull or lost from this reason.

Fig. 118.—Piece of charcoal.

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312.—After the instrument is divided it is engraved with figures and stoned off with fine blue-stone, and finally finished with willow or pearwood charcoal, which has just sufficient cut in it to leave a hard edge to the division lines.

313.—It may be useful to the surveyor, far from aid of the optician, to know that divisions on silver which are much oxidised may be brought up to sharp lines by the use of a piece of fine-grained charcoal, sharpened by a clean file to a chisel point. This should be frequently dipped in water, and rubbed lightly with the flat of its end surface, Fig. 118, keeping the motion of the hand in the direction of the circumference of the circle. The piece of charcoal before being used should be first tried upon a piece of plain, smooth metal—an old coin which is worn smooth will do—to see that it is not scratchy. No kind of polishing powder should in any case be used for cleaning limbs or verniers, as this is sure to rub down the edges of the cuts and thereby ruin the divisions of the instrument.

314.—It must be understood that the above directions are not intended for the ordinary cleaning of the circle for an instrument in general use, as such would be injurious to it. In the ordinary daily use of the circle, if it is not in any case touched by the hand, and is kept carefully brushed with a large, soft camel-hair brush when taken from the case, and the same when returned to it, it will keep a long time in an excellent state. If the circle is slightly tarnished, this tarnish may be removed by a piece of quite clean wash leather; but the brush is always the safest if sufficient. If the vernier gets grubby against the circle, a piece of clean thin writing-paper may be passed between these parts, which will clear out any dirt or grit there may be between sufficiently.

315.—The Vernier Reading Index.—This is one of the most important inventions ever applied to instruments of precision for measuring upon the circumference of the circle. It was invented or brought into practical use by Pierre Vernier, a native of Ornans, near BesanÇon, in Burgundy. The first publication of the invention appears in a pamphlet published in Brussels in 1631, Construction, Usage, et Proprietes du Quadrant Nouveau de Mathematique. This invention was possibly foreshadowed, as it is mentioned by Cristopher Clavius in his Opera Mathematica, 1612, vol. ii. p. 5, and vol. iii. p. 10; but he did not propose to attach it permanently to read into an arc, that is, to place it in its practical form.

316.—The value of the vernier as a means of reading small quantities depends upon the fact that the eye cannot separate lines, drawn at equal distance apart, of above a certain degree of closeness, there being a point for all vision where such lines appear to mix with the ground upon which they are drawn and form a tint; therefore, an index reading into such close lines would be, unless under extreme magnification, most indefinite; whereas the eye can see a single separate line clearly and detect any break in it. The vernier for reading subdivisions depends upon the functions of the eye having power to detect any break in an otherwise straight line, so that a line that appears without a break may be taken as the index of reading from among others that appear broken or separated. It is found in practice that a line as fine as it can be clearly seen will appear broken in its continuity with another equally fine line, if at the meeting the rectilinear displacement is as much as ·25 to ·2 part of the width of the line. It therefore follows that we may read closer by displacement of parts of a single line than by any possible series of lines that can be drawn in spaces apart upon a surface; so that if we can arrange lines in such a manner that they open out or separate into distinct lines to admit of this principle, we obtain the full value of the unbroken single line reading, and this is the principal aim of the vernier.

317.—On the same principle that we can find the straight or most direct line of a series of lines to take as our index, we can also estimate the amount of the displacement of our selected line, if this does not read perfectly straight from the vernier division to the circle division. This small difference is detected in practice by many experienced surveyors, so that a vernier reading nominally to minutes only is recorded n' + 15, 30 or 45, that is to 15. There is no doubt that this will be approximate, but it may be much nearer than the even minutes, say to the 30 on a 5-inch, or the 15 on a 6-inch sharply divided circle.

Fig. 119.—Origin of vernier scale.

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318.—The Vernier Scale, as employed by Vernier, was divided to read minutes upon a circle or limb divided to half degrees, by taking thirty-one divisions of the scale and dividing these in thirty equal parts for a separate scale to read against it. This plan is now termed an inverse reading, the reading being the reverse to the direction of that of the arc. In modern practice the vernier to read minutes is divided to the length of 29 half degrees, and this length is subdivided into thirty equal parts: consequently, where the vernier and scale are placed edge to edge or reading to reading, every division of the vernier advances consecutively on the scale one-thirtieth of the half degree, that is = 1' of arc on the scale divided to half degrees. In the above diagram, Fig. 119 represents the scale and vernier at the position from which the description is taken, wherein the vernier is shown to cover 29 half degrees or 14° 30', and this length is divided into thirty parts. The consecutive advance of the vernier on the scale is shown + 1' for each half degree. In this position of the vernier, or at a similar position in relation to any other half degree of the circle the arrow placed at the zero of the vernier reads direct into the degree or half degree, so that this reading must be n° or n° 30' at any equivalent position in relation to any line on the limb.

Fig. 120.—Vernier scale, reading 23° 12'.

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319.—In Fig. 120 the arrow upon the vernier scale is shown reading at a position beyond 23°, which we then know must be 23° n'. Now, if we look along the vernier, the lines of this and the scale appear coincident at the twelfth division of the vernier; consequently, the n' is 12', and the reading is altogether 23° 12'.

320.—Learning the reading of the vernier is very similar to that of the clock, wherein a child at first gets confused by the difference of value of the minute hand and the hour hand. In the case of the vernier we have only to get clearly in our minds that the degree reading and the vernier reading are quite distinct processes, in which the vernier reads minutes only, and this by coincidence of lines only, and that it has nothing to do with degrees, which are indicated by the arrow only. The arrow may be assumed to be placed on the vernier scale to save an unnecessary line of division; but this practically might just as well be placed quite outside of it, as it has nothing whatever to do with the vernier reading.

Fig. 121.—Vernier scale, reading, 23° 47'.

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321.—It is important to make this matter of reading the vernier clear; therefore in Fig. 121 the index arrow and vernier are shown reading past a half degree. At this position the arrow reads 23·30 on the limb + the vernier, or 23° 30' + n' of the vernier reading. We find the coincident line of the vernier with the limb is at 17, therefore the reading is 23° 30' + 17' or 23° 47'.

322.—The principle of the vernier, upon which it takes its reading from the coincidence of lines, as just stated, points out that the figuring of values of points of coincidence may be varied at discretion, and the zero index may be in any convenient position. The above described is the common reading to the theodolite and many other instruments. In mining dials and some other instruments the zero is placed in the centre. We may, for example, take a central reading with a vernier reading to 3', wherein the circle being divided into degrees; the vernier is then, necessarily, in the direct method, divided into twenty divisions (20 × 3 = 60) which correspond with nineteen degree marks of the circle. With a central reading the vernier in this case is figured 30, 45, 0, 15, 30. This is rather a simple reading, as the zero to which an arrow is attached gives the true bearing, and it is readily seen to which degree it refers.

Fig. 122.—Vernier reading centrally to 3'.

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Fig. 123.

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In Fig. 122 the 45 of the vernier is coincident with a line of the limb, this must, therefore be 45'; and as the index arrow is past 44°, it is 44° 45'. If the vernier had read the division next past the 45, the division being to 3', this reading would have been 44° + 45' + 3' = 44° 48'. The same principles may be applied to any subdivision. Circles are commonly divided by the vernier in various ways to give readings from 5' to 5.

Theodolites reading to 30 seconds are usually divided degrees and thirds of degrees on the circle and minutes and halves on the vernier, as illustrated (Fig. 123), the reading in this case being 153 degrees 40 minutes on the circle and 8 minutes 30 seconds on the vernier, giving a total reading of 153° 48' 30.

A 20 second reading usually has divisions of 20 minutes on the circles and these are subdivided into minutes and thirds by means of the vernier.

Fig. 124.

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28 degrees 40 minutes on the circle and 12 minutes 20 seconds on the vernier, giving a total of 28° 52' 20.

A 10 second reading is designed in the same manner as the above, but each division of the circle is 10 minutes instead of 20 minutes, with minutes and sixths on the vernier. Fig. 125 is an illustration of this, showing a reading of 7° 16' 30.

Fig. 125.

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323.—For Centesimal Division the vernier to read minutes is generally divided 50 into 49 for the half grades, for small circles 4 inches to 5 inches. For larger circles, 6 inches to 8 inches, verniers are cut 25 to 24. The circle is then divided to ·25. Where there is space for five divisions to the grade, ·20, the third decimal place, may be estimated or read exactly to ·005 by a vernier 40 to 39, or more closely if desired by a micrometer, to be described presently.

Figs. 126, 127.—Sections of scales and vernier for circular readings.

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Figs. 128, 129.—Sections of scales and vernier for circular readings.

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324.—Surfaces of Limb and Vernier.—To get a perfect reading of a vernier the scale and vernier should be brought into contact upon a plane. This, for many reasons, is impossible in a great number of cases upon an instrument, from the conditions of its construction, convenience of vision, and in some cases for want of means of ensuring durability of the edges which work together. Therefore verniers and scales are more commonly constructed upon the methods shown in section Figs. 126, 127, where VV are verniers, LL limbs. The plan shown in section Fig. 128 gives a nice reading on a new instrument; but the part of the edge not covered by the vernier is open to accident, or if nearly covered by a part of the instrument, open to the introduction of gritty dust, which wears the meeting line open, and thereby causes loss of edge to edge reading. Fig. 129 shows a section we find on some French instruments. This plan was introduced by the late Colonel A. Strange for the section of the limb reading of theodolites for India, but it was found in practice awkward to use upon this instrument, as it required unpleasant stooping to read it. It is, nevertheless, one of the best permanent vernier readings, as the division remains constant under the amount of wear occasioned by the sliding of the vernier upon its circle.

325.—With the reading planes shown in section Fig. 126 we require great care to bring the eye, whether open or through the microscope, directly radial with the centre of the circle at the line into which the vernier cuts. If we read the line in the slightest degree one-sided it is quite possible to make a difference of a minute on a 5-inch or 6-inch circle. This is the section of the general reading plane of theodolites, where, from the necessary height of the telescope, the limb has to be placed much lower than the eye. With this section the circle comes fairly square to a comfortable position for reading. It will be noticed that there is a slight lap shown to the vernier over the limb at a, Fig. 126, which is always found in new instruments of this section. It gives an allowance for wear between the vernier and the limb caused by the fretting of the metals together, as also by the intrusion of grit, which is always present in instruments used in the open air. The lap should not be great, and it should be nearly equal along the edge of the vernier, although it is a difficult matter for the maker to get it perfectly so.

Fig. 127 is a section of the reading planes common to sextants and parts of many instruments. This plan requires the same care to obtain a truly perpendicular reading to the division as that described above for Fig. 126.

326.—In the very best of work there is at all times a certain amount of error, both between the divisions themselves, and in the place of the axis in relation to the centre of the divided circle, and of the position of the vernier in relation to both these. It therefore becomes necessary, where exactness is required, to place at least two verniers to read opposite sides of the circle. These bisect every reading through the axis of the instrument, and detect very small errors in the work, as well as personal errors of the observer, of which the mean reading of the minutes or seconds only may be taken and used for correction to mean position. Where very great precision is aimed at, three or even five verniers are sometimes placed round the circle, and the mean reading is taken of the small differences in minutes or seconds, after calculation for correction, to find the direct position of the axis of the telescope required for the record of the observation.

327.—Reading Microscope.—The microscope usual for reading the vernier is either a simple plano-convex lens of short focus or a Ramsden eye-piece of the kind described for observing lines on the diaphragm of a telescope, art. 82. Frequently the microscope, technically called the reader, is made of a compound form, sometimes with a diagonal prism or mirror. It is uniformly mounted in such a manner that it may move concentrically to the divided circle into which it reads. In English instruments it is placed normal to the surface of the vernier, so that following its curvature it may read opposite any line upon it. In French instruments the reader is frequently placed obliquely, so as to look along the line of the limb into that of the vernier, which is said to be advantageous in certain lights.

328.—In theodolites for reading the horizontal circle, the reader is sometimes mounted to slide in an undercut groove near the circumference of the limb to follow its curvature. This motion is not pleasant; it is better in this and all cases of vernier reading, if possible, to mount the reader on frame-work proceeding directly from and moving upon the axis. Where it is practicable, it is much better to have two readers where there are two verniers, and in all cases to have one to each vernier, than to shift one reader about after the instrument is placed in position, which is liable to disturb it. With opposite readers mounted on a pair of arms formed of one piece of metal, where these bisect the circle working through its axis, by the setting of one reader truly normal to the coincident division of the vernier the opposite reader will be set also; so that this does not only save time, but the instrument need not be touched for reading the second vernier. The same principle should be applied to any greater number than two verniers as nearly as it may be practical.

329.—Instruments that have to be packed in cases for conveyance should always have readers removable from the instrument, with proper fittings in the case provided for them, or they should be hinged to turn up to a secure position, the latter being a more expensive but a much better way. It is better also, if possible, to remove the light frame with the reader if this does not turn up, so that it cannot be injured in replacing the instrument in its case.

Fig. 130.—Reader fixed normal to surface.

Fig. 131.—Jointed reader to set to any angle.

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330.—Fig. 130 shows a good rigid form of reader for an oblique plane of division:—V vernier, L limb. This reader is placed on an arm radial from the centre of the instrument, more generally in pair with an opposite reader. The connection with the arm is commonly made for portability with a dovetail slide fitting to the reader, sprung by a saw-cut down it to ensure constant contact after wear, as shown in section Fig. 132; N arm of reader, O fitting to arm. The better form is shown in Fig. 131. In this the arm is jointed, so that the reader out of use is turned up into the central part of the instrument. This plan admits of adjustment of the reader for reflection of light from the division, or for reading down the lines if preferred. The magnifying power of either of these microscopes is generally two to three diameters. The adjustment of the glasses should be such as will produce a flat field (Ramsden's principle, p. 41), so that several divisions of the vernier and limb may be read sharply when it is in focus, although the central division only is taken for the reading.

Fig. 132.—Section of movable arm fitting to reader.

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331.—Surface Reflection to Reader.—In reading with the microscope the silver surface, from its brightness in certain lights, gives unpleasant reflections which render the reading difficult. In practice the hand or a piece of white paper is used to shade the open vernier in such cases. In large instruments a piece of ground glass is fixed in a frame over the vernier, which throws a soft light, producing the effect of a dead surface upon the silver, or the light is reflected from a cardboard or ivory surface. Fig. 133 shows a common form of microscope for reading a vertical circle, by which the light is reflected from a white surface surrounding the field-glass end of the reader.

Fig. 133.—Reflecting surface reader.

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332.—Shades for Vernier.—It is very general on the Continent to place the divided reading of the circle and its vernier on a plane perpendicular to the axis, Fig. 128, and to place the reader at a fixed angle for down-the-line reading, the object-glass of the reader being constructed to focus parallel rays. In this way the division of the circle is followed into its vernier or vice versa. In this case the silver may be shaded by ground glass, which gives a soft, pleasant reading in most lights. The general arrangement is shown, Fig. 134; L limb, V vernier, S shade of ground glass, M reader. Objection is made to glass shades by civil engineers as being too delicate and liable to fracture, with risk of the particles of glass getting into the working parts of the instrument. To obviate this the author has made the shade of a piece of thin horn or transparent ivory, which appears to answer very well and to save this risk.

Fig. 134.—Oblique reading microscope with shade, French plan.

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333.—For ordinary instruments with no provision for shading, a piece of transparent horn about 2¼ inches by 1¼ inches may be carried in the waistcoat pocket, and will be found a great comfort if held over the vernier when the lines appear glary, or the horn may be placed in a pocket frame with the case containing reflector for bubble reading, Fig. 52. In large theodolites, used for geodetic surveys, the object-glass of the micrometer microscope is sometimes surrounded by a thin belt of turned ivory. This throws a very soft light upon the divisions.

334.—Micrometer Microscope, for Reading Subdivisions.—Where more exact reading is required than is possible with the vernier, as in the case of the reading of circles 10 inches or more to seconds, a micrometrical microscope is employed, which gives means of measuring the distance from line to line of the division upon the limb by the displacement of a web, point, or line moved by a fine screw with a divided head.

The great demand of late years for reducing the size and increasing the accuracy of theodolites has induced the highest class makers to introduce micrometer reading instruments of six, five, and even four-inch circles, and their accuracy is far greater than is possible with any instrument of the same size that reads by verniers. Of course the workmanship in these instruments has to be of a higher order, and the reviser estimates the accuracy of the micrometer through magnification and the necessary refined workmanship to be at least four times as great as the vernier reading, with the advantage that the micrometer is much more certain and easier to read.

335.—The construction of the reading micrometer as originally designed by Troughton has not been materially modified in those in general use. Certain refinements have been introduced for astronomical work: these are sometimes expensive and often cumbersome, so that they need not be considered in relation to surveying instruments.

336.—In all cases where micrometers are used, the structure of the framework of the instrument which carries them should be made extremely rigid, as very minute deflections or vibrations render the reading to seconds of arc impossible. The number of micrometers applied to a circle is generally 2, 3, or 5.

337.—If a circle is to be read by micrometers, the vernier is generally dispensed with. The circle is usually divided to read in 5'. The first approximate reading used to be taken by a single index line with the aid of the ordinary reader, Fig. 130. From the index line the degrees or minutes were taken to the last 5' line indicated. Since the introduction of high-class engraving machinery the figuring is made at each degree and is clearly read in the microscope, so that the index reader is unnecessary. This engraving is quite a nice piece of work, as to figure from 0 to 360 means nearly a thousand figures, and on a 5-inch circle these have to be less than 1/100th of an inch high. Only the highest class makers are able to do this work. When a microscope is adjusted to one line it should be observed that all the other microscopes upon the same circle should also read exactly to a line that should be true from microscope to microscope to the arc they subtend between each other.

Fig. 135.—Side elevation of Troughton's micrometer.

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Fig. 136.—Section of micrometer.

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Fig. 137.—Micrometer slide.

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338.—The Micrometer, as it is now technically termed to include the whole piece of apparatus, is a compound microscope consisting of three lenses, with measuring apparatus at the mutual foci of the field-glass and of the two lenses which form the eye-piece. The field-glass, which is placed nearest the divided arc, is generally an achromatic microscopic lens of an inch or more in focus. The eye-piece is of the Ramsden form, Fig. 16. By the construction of the compound microscopic arrangement the eye of the observer may be placed at any convenient distance from the limb, and any desired magnification may be obtained to assure micrometric nicety of measurement. The engravings represent the micrometer, Fig. 135 in side elevation, Fig. 136 longitudinal section, and Fig. 137 the micrometrical slide, which is shown partly in section for demonstration in all the figures; a the micrometer, q microscope body tube. This has a male screw outside at b', upon which there are two collars dd' with capstan heads. These collars hold the microscope upon the reading frame b at any required distance from the limb to secure proper focal adjustment. g objective tube. This screws into the body tube and permits adjustment of the objective to the division of the limb and the micrometer index web by the milled head s. This tube has a locking nut i to secure it from after movement when it is once properly adjusted. h an achromatic object glass of half an inch or over in focus. e the casing that receives the eye-piece which screws into the outer plate of the micrometer. f the eye-piece, generally made about one inch long. This slides by friction in its cell to produce distinct vision of the spider lines in the micrometer.

339.—The micrometer frame, Fig. 137, a has a fixed scale or comb, with five or more points or teeth formed upon it, and a movable sliding frame, upon which a spider web or webs are inserted and cemented in finely engraved lines to form an index, brought as nearly as possible to the mutual focal plane of the object-glass and the eye-piece. The index web frame has a fine screw of about a hundred threads to the inch tapped into it. The micrometer screw, divided drum, and milled head are now generally constructed as shown in Fig. 137. Two springs press upon the index frame and the outer frame, and thus keep the drum up to its collar. The drum r is divided upon its edge into sixty equal parts, to read seconds of arc generally to a single line index. The screw is moved by the milled head beyond the drum, so that the divided surface of the drum need not be touched.

340.—The portion of the arc measured being generally 5', the distance of it, as it appears at the magnified image of the arc at the position of the index of the micrometer, is made to correspond with five turns of the micrometer screw, the head of which divides each turn into 60. By this means the 5' is divided into 300, that is, to single seconds, and by approximation of the interspaces on the micrometer head, as far as the reading is concerned, to fractions of a second. The fixed scale, or comb, as it is termed, is commonly placed in the focus of the eye-piece with five webs upon it, fixed to agree with five turns of the screw or a rack with points at the bottom. These webs or rack divide the 5' of arc in minutes, and indicate the number of revolutions of the screw, as shown by the displacement of its index line. A pair of lines or webs are commonly placed in modern instruments at 1' part, to ensure certainty of reading by the mean of two observations.

341.—The magnitude of 5' of arc depends necessarily upon the radius of the divided circle; therefore the microscope of the micrometer has to be made to suit the division it is required to subdivide—that is, using the same micrometer, the smaller the circle the higher the magnifying power is required to be to take register by the same screw. Within a wide range the micrometer is perfectly adjustable, to ensure exactness upon this point, by varying the distance of the object-glass from the limb, for which purpose the microscope is made adjustable by the pair of screws dd' which clamp it to its standard as already mentioned. The principle of this adjustment is easily seen, for if we place the object lens at a distance equal to its solar focus from the limb, the image will emerge in parallel lines; but as we cause it to recede from the limb, the image may be brought to any position within the tube greater than the solar focus of the objective of the microscope. The image is therefore brought to a position where it may be picked up conveniently by the eye-piece. In this manner we have only to make the adjustment of the object-glass from the limb such as the space of any pair of divisions of the limb may be magnified up equal to the displacement of five turns of the screw for seconds measurement.

342.—The two points where the divisions and their images are situated are termed the conjugate foci of the lens, and the magnifying power is proportional to these distances; thus, if we call the distance of the object, that is the limb, from the object lens f, and the distance of the focal plane of its image within the tube F, the image will exceed that of the object in the ratio of Ff, or F/f will represent the magnified image. By this method it will be seen that the expression F/f will have an increased value, if we either increase F or diminish f, which we have to consider in the construction of the microscope to bring it to the conditions under which it will adjust to bring the micrometer screw exactly to its required reading.

Fig. 138.—Grubb's plan of securing micrometer screw.

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343.—It is very general in instruments at the present time to tap the micrometer screw directly into the micrometer frame, and to make the drum and milled head a part of the screw. In this case a very soft motion may be given to the screw by dividing its nut longitudinally and bringing the parts together with a certain amount of spring. Sir Howard Grubb, of Dublin, has placed a spring ball fitting, as shown Fig. 138 at EE', over the screw upon his astronomical instruments, which gives a very soft motion to the screw. These refinements are very important, as it is not desirable that any undue pressure should be put upon a delicate instrument which under all conditions must be made rigid enough to resist it, and the greater the pressure required to bring the instrument to bearing the stronger it must be made.

Fig. 139.—Stanley's micrometer slide.

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344.—Stanley's Micrometer.—The author has made an arrangement in which the screw has a long, double tubular sliding stem, Fig. 139. The inner stem which carries the milled head has a groove cut down it, into which a stud from the inside of its covering tube slides. This arrangement permits the milled head to be pressed inwards or outwards in turning it without any pressure coming upon the micrometer greater than the friction upon the sliding tube, and that of a weak spring which keeps the stem nearly extended in its tube. A simple Hook's joint H is formed at the head of the screw, so that no part of the weight of the hand comes upon the screw. A tubular guard-piece T prevents the milled head hanging down too far when out of use. When the screw is used it is lifted to about the centre of the guard tube. With this arrangement, as no practical weight or pressure comes upon the micrometer from handling it, the supporting frame-work may be made much lighter than is necessary with any other form of micrometer.

345.—The author prefers to form the micrometer scale and the index of fine lines engraved upon parallel worked glass for surveying instruments. This avoids the risk of breaking webs, and, what is much more important, he finds that with engraved lines on glass he is able to bring the scale and index exactly and permanently into the plane of mutual foci of the object-glass and eye-piece by placing the lines upon the same faces of glass, thus avoiding the great difficulty of focussing to guess-work of an intermediate position between two sets of webs at different distances.

The strip of glass A is fixed by a clamp and two screws to the side of the micrometer box. The slip B is ground and polished to fit A. B is carried by the micrometer frame F, which holds it in a clamp by two screws. A spring, not shown, presses B against A, so that any displacement of the micrometer lines may be made by the milled head. The lines upon A are adjusted to the position of the circle they are intended to read at exactly 5' or other quantity.

For the smaller instruments which will be much more frequently used by the surveyor a simpler form of reading is used, and as the reviser is convinced that in future this form of reading will gradually replace the vernier for all high-class work, a full description of this very simple reading is here given. The reviser is confident, after many years of practice for the most accurate form of index, that a point certainly stands first, a pair of webs or lines on glass, between which the division is seen, second; and a single web or line on glass placed over the division, third. The comb mentioned in art. 339 is done away with, and one revolution of the micrometer screw made to carry the index over one division of the limb. For clearness the engravings show only a 10 reading; for a 5 reading the divisions on the limb are to 5' instead of 10', and the micrometer head is divided and figured accordingly.

Fig. 140.—Stanley's micrometer reading.

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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.

Fig. 141.

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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.

Fig. 142.

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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.

Fig. 143, 144.—Sections of clamp and tangent in two directions.

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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.

Fig. 145.—Elevation and part section of clamp and tangent.

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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.

Fig. 146.—Clamp and tangent to a vertical axis.

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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.

Fig. 147.—Clamp and tangent to vertical axis, German plan—HunÄus.

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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.

Fig. 148.—French axis clamp and tangent.

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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.

Fig. 149.—Protected tangent screw with helical spring.

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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.

Fig. 150.—Free tangent adjustment.

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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.

Fig. 151.—Tumbling piece adjustment for wear of tangent screw.

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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.

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