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 XII.

INSTRUMENTS TO MEASURE SUBTENSE OR TANGENTIAL ANGLES TO ASCERTAIN DISTANCES—HISTORICAL NOTES OF THE METHOD—PRINCIPLES INVOLVED—STADIUM MEASUREMENT, DIRECT AND BY THE ORDINARY TELESCOPE—CORRECTIONS FOR REFRACTION OF THE OBJECT-GLASS—STANLEY'S SUBTENSE DIAPHRAGM—ANALLATIC TELESCOPE OF PORRO—TACHEOMETERS—STADIUM—FIELD-BOOK—OMNIMETER AND ITS FIELD-BOOK—BAKEWELL's SUBTENSE ARRANGEMENT.

553.—Direct Subtense Measurement of Distances, by an Instrument, depends upon our powers of measuring the image of a distant staff or stadium, or the divisions marked thereon as they appear at the focus of the telescope. If the stadium is placed at right angles to the direction of one of two sight lines which subtend a given angle, the number of units divided upon the stadium cut by these lines will be proportional to units of length of base or cotangent for a constant focus of the telescope; so that if we can measure at a fixed angle the number of equal units of measurement of a stadium correctly, we can obtain its exact distance; and whether this method is more or less exact than chain measurement will depend entirely upon the perfection with which either of these operations may be practically performed.

554.—The Origin of the Invention of Subtense Surveying was thought to be due to Wm. Green, an optician of Great Moulton Street, London, who was awarded a premium for its invention by the Society of Arts in 1778. He published a pamphlet giving a description of his method in 1778.[28] This subject he pressed upon the notice of professional men at the time, and his method has continued in use in this country ever since. His refracting telescope, which alone has remained in use, formed part of the theodolite. A micrometer was placed in the focus of the eye-piece of the telescope, which revolved a quarter turn in its axis to read angles vertically or horizontally. He constructed his micrometer with lines fixed at a given distance apart, and by a second method with the lines adjustable. For this adjustment a fine line was ruled upon one side of two pieces of glass. The ruled sides were placed face to face, so as to be at the same focus. One of the lines was adjustable by a micrometer screw. His staff was 20 links in length by 4 inches in width, divided decimally into 1000. His description of the manner of using his instrument will give a general idea of working the others which have been derived from it—tacheometers, omnimeters, etc.—and this is worthy of note, as the invention, though generally attributed to him, was not his:—

555.—"To find the contents of a field with either of the instruments described, let the telescope be placed so that the observer may see all its angles from his station. If near the centre of the field the better. The person who carries the scale (staff) is to go all round the field, stopping at every angle, and to place the scale at right angles to the axis of the telescope (passing) from corner to corner (from right to left if required) with the help of a signal by the observer. After the distances all round the field are taken (by measurement of the image of the micrometer) and all the angles included betwixt them, with the theodolite, plot it out in the usual manner, e.g., with a nonius protractor. Describe a circle, and on this circle set off all the angles from the centre through each point upon the circumference. Set off the length of every line by a scale of equal parts. These points will give the limits of the field, which may be laid out in trapeziums, triangles, etc., and measured from the same scale of equal parts. The surveyor will comprehend how easily the contents of the field are found by trigonometrical calculation, since by this method there are two sides and one included angle given.

"The common method of measuring with the chain, besides the inaccuracies to which it is liable, does only give the length of the surface of the ground between two objects, and therefore not its proper distance, unless the surface be straight and no object to hinder its being measured from one end to the other. How often this is practicable I leave to the consideration of those who are most accustomed to measure lines, and doubt not that upon the whole they will find the telescope method has besides ease, accuracy, and universality, necessity itself to recommend it."

He points out the utility of the system for levelling, as "both distance and inclination may be taken at the same time." He finds by experiment that the accuracy of the method exceeds that which he could reasonably expect by calculations deduced from theory, by several circumstances in its favour being inseparable from it.

"The observer's station is the centre of circle whose radius is the distance required, which is obtained by measuring the length, that is, the tangent or subtense, of the small arcs whose limits are defined by viewing their image in the focus of a telescope between two points there placed, and moving them up and down until they appear to touch the very extremities of said limits exactly. The manner of seeing is natural and by practice will become habitual, and therefore continually approach nearer to perfection.

"Thus may any surveyor in less than two hours take all the dimensions of an irregular polygon necessary for obtaining its area, if it be as much as 80 or 100 acres and limited by twenty or thirty unequal sides."

Green points out that if the subtense angle is taken horizontally, atmospheric refraction error is eliminated. He proposes to use both reflecting and refracting telescopes. With the reflector he possibly obtained accurate results, but with the refracting telescope he does not appear to have recognised a constant correction which is necessary and important.

It has since been found that in 1778 the Danish Academy of Sciences awarded a prize to G. F. Brander for a similar device, which he had applied to his plane-table, six years before. Its real discoverer was James Watt, who used it in 1771 for measuring distances in the surveys for the Tarbert and Crinan Canals. In James Patrick Muirhead's Life of James Watt, he gives a statement by Watt himself that he constructed his instrument in 1770 and showed it to Smeaton in 1772.

556.—Subtense Instruments, as that originally made by Green, are of some form of theodolite, the telescopes of which are constructed to measure either the angle subtended by the chord of a small arc or the tangent of the same. For convenience the tangent is more generally taken upon a graduated stadium or staff, which is erected for measurement perpendicularly to the horizon, the principle of which is shown in the following scheme:—

Fig. 239.—Diagram of tangential angle measurement.

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Let AC, Fig. 239, be a horizontal line; BC a stadium set up vertically. Then if the angle BAC and the height BC are known, the distance of AC can be easily calculated. For any intermediate distance between A and C a vertical will be in length proportional to this distance. Let de be at one-third the distance from A; then the line de will be one-third the length of BC. If we divide BC into three parts and place the stadium at fg two-thirds the distance from A, the angle dAe given by an instrument subtending a fixed angle will cut the staff at the second division, equal to two thirds the staff, which demonstrates the principle of all tacheometers, Cleps, etc.

If the tangent be made a constant equal to the length of the stadium BC, and this stadium be placed at another position, say de or fg; then the angle subtended by its entire length will vary in a manner that can only be estimated by trigonometrical calculation.

In case of reading two distant marks on the stadium only for the subtense, the single central web of the telescope being directed first to one and then to the other of these webs, the distance is calculated as follows:—

Given the tangent BC and the angle BAC, required the distance AC. Let the angle BAC be represented by D; then—

CA	= cotan D, or CA = CB × cotan D.
CB

Reducing by logarithms, we have—

log CA = log CB + L cotan D - 10.

For example, make CB 14 feet, and the angle D 2° 45' 50, thus:—

log CB	= 1·146128
L cotan D - 10	= 1·316265
log CA	= 2·462393,	or CA = 290 feet.

The above gives the principles followed with instruments of the theodolite class simply; but arrangements are made in omnimeters and similar instruments to read the tangent directly and determine the height CB in equal parts, so that observation of the heights BC gives rectangular co-ordinates and thus saves reduction from degrees of arc.

In practice the staff or stadium is made of the greatest length convenient for portability. With a telescopic staff, 14 or 16 feet is commonly used. If a unit tangent be not employed, the foot is divided into 100 parts, each of which parts, with the tacheometer, represents 1 foot of the base, and the whole staff 1400 or 1600 feet. The ordinary Sopwith staff, art. 263, answers the purpose, but art. 268 better.

557.—Measuring Distances by the Ordinary Telescope by Measurement of its Focal Image.—When we apply a refracting telescope to measure a subtense angle by webs fixed in the diaphragm, vision is not direct as in the scheme Fig. 239, but subject to bending caused by the refractive quality of the lens, art. 58, the telescopic focus varying with the distance from the staff. Thus with a 12-inch telescope there will be a difference of about ·25 inch in the focus, whether the staff is held at 50 or 500 links from the telescope; and this difference of focus is equal to a difference of base or cotangent between the points A and C in the last figure, so that these distances do not remain proportional to the fixed unit of the tangent or stadium. It is important to go carefully into this subject of the use of subtense webs in the ordinary telescope, as the necessary correction does not appear to have been recognised by English writers on instruments, and no doubt this is the principal reason that subtense measurement has not been more practised in this country.

558.—At the commencement of the last century, Riechenbach, a Bavarian engineer, pointed out a method still in use on the Continent. The author is indebted to the kindness of Lord Rayleigh for the following demonstration of Riechenbach's formula:—

Fig. 240.—Subtense diagram.

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Let Fig. 240 AB = s, ab = i, OA = d, Ob = r. O is the optical centre of the object-glass; ab a pair of webs at variable distance r from O according to telescopic focus; f focus for parallel rays. Then by similar triangles s/d = i/r or d = rs/i, r is found by optical laws to vary in the proportion of 1/r + 1/d = 1/f. We may therefore eliminate the variable r by substituting its value r = fd/(d - f), by which we find d = sf/i + f, which gives the true correction; and the distance from the axis of the instrument will be d = sf/i + f + c where c is the constant distance of the object-glass from the axis of the instrument. It is usual to place the vertical axis of a theodolite central between the object-glass and the diaphragm at solar focus, so that the constant c becomes f/2.

It is seen that sf/i represents the direct subtense, whereas the refraction, which is a constant, gives f and the position of the object-glass f/2. Riechenbach's formula being true for parallel rays is evidently also true for any subtense with refraction for the staff at any distance. We may therefore adopt a plus constant of 1½f, which added to the apparent subtense is found to produce no error. Thus with a telescope of 1 foot solar focus, and using the decimal system of notation, as before mentioned, if a stadium or distinct scale be placed at 301½ feet distance from the centre of the instrument, and the webs or points of the diaphragm be adjusted to read 3 feet = 300 divisions, every subtense may afterwards be taken as number of divisions read + 1½ feet for distance in feet. If the subtense is to be taken in links or metres the dividing of the stadium will be to these measures, but the constant remains the same 1½ feet always.

559.—When the line of sight is inclined from the horizon and the stadium is held erect—a convenient method commonly followed upon the Continent—the reading becomes in excess of the true reading, in the ratio of the cosine of the angle of the stadium, represented by a line tangent to the sight-line subtended to the foot of the stadium, as shown in the following diagram.

Fig. 241.—Diagram of vertical stadium on an incline.

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Thus, Fig. 241, let the portion cut by the lines AB, S' be the reading of the stadium; then

S'(cos a) = S.

The inclined distance is then equal to

(f/i)S'(cos a) + f + c

and the horizontal projection of that distance or

a = ((f/i)S'(cos a) + f + c)cos a;

or as f + c is small and the angle generally small also, f + c may be taken equal to (f + c) cos a. Then

a=(f/i)S' cos² a + f + c.

Fig. 242.—Stanley's patent subtense diaphragm.

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560.—The Subtense-diaphragm of the author, Fig. 242, forms the eye-piece of a theodolite. It has movable indices which are separated according to a scale formed by calculation upon the data of the above formulÆ. By this, distances may be taken in the horizontal plane for land of any inclination without after calculation. This result is obtained by observing the angle of inclination upwards or downwards on the theodolite and setting the micrometer to this angle before reading the subtense distance. The reading is taken by points which are arranged to measure the subtense 1 to 100, so that the ordinary Sopwith staff may be used. The diaphragm at zero appears as an ordinary subtense-diaphragm. It may be observed that this diaphragm may be used as a good check, as distances may be taken over any irregularities of intervening incline and give the true base for the entire distance.

561.—If the mean contour distance is required from station to station, this may be taken directly by subtense from the staff-reading held at right angles to the axis of the telescope. The means of doing this, devised by the author, is to place a sight director of a special form upon the side of the staff, Fig. 243. This small piece of apparatus is shown attached to the staff. It consists of a small telescope three inches long attached at right angles to the staff by means of a dovetail slide fitting when in use as shown. The staff-holder sights the tacheometer through the short telescope, which can only be seen to appear therein by moving the staff until it is approximately at right angles to the direction of the tacheometer.

Fig. 243.—Sight director for stadium.

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562.—The Anallatic Telescope.—In this telescope the focus is constant, and consequently the tangential measurements indicated by the numerical qualities subtended by a constant angle are directly proportional to the base, so that there is no constant to be added. The invention of this instrument and its modern application to subtense measurement was due to Professor J. Porro, of Milan, who put it to practical test in 1823,[29] in an instrument termed a tacheometer. The telescope will be best understood by the following details:—

The object-glass O, Fig. 244, is made of a focus that falls well in front of the axis of the instrument CC', so that the rays cross before falling upon the anallatic lens A, the optical arrangement being such that if the rays fell direct without any refraction they would reach the axis and subtend angles therefrom inversely proportional to the distance of the stadium. The object-glass and anallatic lens are of the same focus, so that the rays after crossing from equal refraction may emerge parallel in the space A to M. The stop at S and at the axis CC' cuts off eccentric rays that would otherwise give internal reflections from the telescope tube. The eye-piece, represented by MF, may be made to pick up the image of the stadium in front of it upon an ordinary webbed diaphragm or upon ruled glass. The diaphragm webs are fixed, or the glass surface engraved with three or five horizontal lines and one vertical. The outer horizontal lines are used generally as the subtense lines, and the central line for levelling and taking altitudes. The vertical line is used for triangulating on the surface of the ground.

Fig. 244.—Diagram of anallatic telescope.

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563.—There is an adjustment made by sliding tubes to bring the object-glass and anallatic lens within mutual focus to ensure the parallelism of the emergent rays and to adjust magnification. This is commonly effected by means of a rack and pinion, moved by a separate key kept in the instrument case, but which should not be touched after the instrument is once adjusted by the maker, except in the case of accident. It is much better made without this rack adjustment and permanently fixed by the maker, as if it has the adjustment it is likely to be tampered with and thus defeat its object. The eye-piece adjusts to distance from the object-glass in the ordinary manner of the surveying telescope—by rack and pinion.

564.—The eye-piece of the anallatic telescope is generally made of much higher power than those ordinarily employed for levels and theodolites—25 to 30 diameters is usual. Where a diaphragm is used the subtense lines are commonly placed on a slip of glass in two or three sets, so that greater magnitude of image may be taken for objects at distances of from 2 to 7 chains with the 14-feet staff, or that the staff may be read at greater distances than 14 chains. This series of lines is distinguished as 50, 100, and 200, Figs. 245, 246 and 247; so that with this as great a distance as 28 chains with a 14-feet staff may be estimated, but this is beyond the safe power of the instrument. The intermediate line, as shown Fig. 220, is valuable in all cases for levelling. The advantage of the increased power of the eye-piece is more than neutralized by the loss of light.

Figs. 245, 246, 247.—Subtense lines ruled on glass.

Fig. 248.—Adjustable point diaphragm with stadia points.

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565.—While many civil engineers are satisfied with a single percentage pair of subtense lines the author much prefers using the point system, arts. 237 to 239. In this case the diaphragm, as made by the author, possesses two systems of adjustment; that shown Fig. 248 at a for the single point for altitudes, and the pair of points separated by the spring ss for subtense angles. These points adjust by separate screws top and bottom with a milled-headed key f. The two verticals are fixed permanently. These points are all made of platino-iridium, which possesses the hardness and elasticity of spring steel, and is at the same time, as far as is known, perfectly non-corrosive. In case of any light dust or moisture resting upon the points, it is perfectly safe to brush them lightly with a soft camel-hair brush to clean them. Where the 200 factor is required, a mean may be taken of two observations above and below the central point. Where 50 is required, the vertical points may be adjusted to this.

566.—In adjusting the lines, webs, or points to a given subtense, the anallatic lens may be moved to give more or less angular displacement or magnification of the image. Greater accuracy is obtainable when the staff is held normal to the line of sight instead of vertical. If the staff be held incorrectly in the inclined position at great angles of elevation or depression, the resulting error is very much smaller than in the case of an equal variation of the staff from the true vertical position. When adjustment is made upon a distant stadium at small angles of elevation or depression, the subtense of the small arc will vary so little from a tangent to one of its radii that the one or other may be taken without sensible error. The plan originally proposed by Green of placing a sight tube through the stadium at right angles to its face, as a means of keeping it in the chord of the arc, is as good as any other, but is more cumbersome than that described art. 561. If the vertical stadium be preferred, this may be set up by the small level, Fig. 109, p. 163.

567.—It is well to note that with the anallatic telescope the stadium must not be so near that the rays from the object-glass do not cross in front of the anallatic lens or the subtense will appear much increased, so that there is a fixed nearness at which this form of telescope can be used, say 50 feet. For this reason engineers generally prefer an ordinary telescope, making use of the addition of a constant. The author also prefers the plain telescope, as being more correct according to his experiments where the constant is correctly allowed. There are many advantages in the use of a plain open telescope instead of the anallatic telescope for tacheometers, among them the following may be mentioned. More light reaches the eye because there are fewer lenses; there is no intermediate lens requiring adjustment and which becomes dirty and bedewed and is inaccessible for cleaning, and for the same dimensions of the telescope greater power can be obtained. A larger telescope and of higher power is of great advantage in subtense measurement, but the full advantage is not obtained in the anallatic telescope. The idea which appears to be still common that an ordinary open telescope will not give accurate results at all distances by means of stadia readings, plus the distance of the anterior principal focus of the object-glass from the axis of the instrument, is entirely erroneous. When a staff is held at any distance in front of the object-glass of an open telescope, an inverted image of the staff is formed at the conjugate focus which subtends an angle at the corresponding nodal point of the lens, equal to that subtended by the staff at the other nodal point. If a diaphragm with two stadia points or webs be placed at this conjugate focus the ratio i/f' = the ratio l/D; where i is the space between the stadia points, l the height on the staff which these points appear to intercept when viewed through the eye-piece accurately focussed on them, D the distance of the staff from the object-glass, and f' the distance of the diaphragm from the object-glass. Now in this equation i is a fixed space, l is the observed height on the staff, and both f' and D are variables, of which it is desired to find the value of D. From the laws of optics it is also known that 1/f' + 1/D = 1/F where F is the principal focal length of the lens. Therefore f' = FD/(D - F) for all values of D. Substituting this value of f' in equation (1) we get i × (D - F)/FD = l/D; and multiplying both sides by D, i × (D - F)/F = l. ? D - F = (F/i)l and D = (F/i)l + F which is true for all distances. But this distance is measured from the object-glass, and the distance S required by the surveyor is that from the axis of the instrument, and it is therefore necessary to add that of the object-glass from the axis d. ? S = D + d = (F/i)l + F + d, and F + d is the constant of the instrument = c. ? S = (F/i)l + c.

When the range is greater than that at which the divisions of an ordinary levelling staff can be clearly read with the stadia points, target stadia rods or targets fixed to a levelling staff are used. It is usual to use plain targets fixed with their centre lines at exactly 10 or 20 feet apart or other convenient distance, and the angle subtended by these is measured by a micrometer diaphragm. The reviser, in conjunction with Mr. C. W. Scott, B.A.I., A.M.I.C.E., has designed a micrometer diaphragm which has been proved to give very accurate results. It is made to revolve, so that either horizontal or vertical stadia rods may be measured, and it is fitted with fine fixed platino-iridium points, which are much more satisfactory than webs or lines engraved on glass. These are fixed on one side of the diaphragm, two each 1/200 part of the principal focal length of the object-glass above and below the axial point. On the other side of the diaphragm is a movable point which can be traversed over the fixed points by a micrometer screw, every complete turn of which moves the point over a distance equal to 1/1000th of the principal focal distance and the head of the micrometer being divided into 100 parts, it reads to the one hundred-thousandth part of the same; while a small star-wheel records the number of complete revolutions, five of which cover the space between any two of the fixed points. In using this micrometer with say a 10-foot target, let the lower target cross-bar be clamped to the level staff at 2 feet, and the upper target cross-bar at 12 feet. Direct the axial point to the centre between the targets at 7 feet and read the angle, then bring the nearest fixed point to the top or bottom mark by means of the tangent screw, and bring the micrometer point to the other mark by the micrometer screw. The micrometer reading is the reading on the divided head plus the hundredths indicated on the star-wheel plus 500 for each included complete space between the fixed points. See whether the micrometer reads up or down, and set the fixed point to the lower or upper mark on target accordingly. To obtain the distance from the axis of instrument, divide 100,000, multiplied by the length of the target by the micrometer reading, and add the constant of the instrument S = 100,000l/x + c where x is the micrometer reading, and l the length of the target. The tacheometer which the author has lately made has a plain open telescope, but this is of the same size as that used upon the Porro system, and consequently it gives much more light and better definition.

568.—Tacheometers consist essentially of any form of theodolite that is provided with means for reading distances by its telescope. Stadia work is simply another name for tacheometry, which is derived from the Greek tacheos (quickly), and metreo (I measure), and signifies the art of measuring rapidly. The graduation of the arcs and circles of these instruments is sometimes made upon the centesimal system, the circle reading 400 grades, which are subdivided to half grades to read with the vernier or micrometer to centigrade minutes of ·01 grade. The centesimal system facilitates calculation, and permits a free use of a logarithmic slide rule of a special kind. In France, where working with this system at one time became more general, we have very complete centesimal trigonometrical tables adapted to the tacheometer published in stereotype,[30] but it has not gained favour, and very few instruments are now so divided. A compromise which has found a certain amount of favour is the decimal division of the ordinary degree of 90 to the quadrant; this 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 degree is much simpler than the conversion into the centesimal degree of 100 to the quadrant. Any instrument divided sexagesimally can be converted by simply changing the vernier if the divisions on the limb are degrees or half degrees. The theodolite, Fig. 169, the author made specially for a tacheometer. Any theodolite may be converted into a tacheometer by fitting it with a subtense diaphragm. A modern tacheometer should be a high-class theodolite in which every possible refinement is included.

569.—The tacheometer, although manufactured for many years for export, has been very little used in this country. The instrument to be described, shown Fig. 249, is the author's latest pattern. It is made with sexagesimal division or ingrades, to read by the verniers to 20 or to centigrade minutes. The telescope is of much larger and of higher power than that of the ordinary theodolite. For a 6-inch instrument the telescope is of 11 inches focus, with an object-glass of 1¾ inches aperture. The eye-pieces are of the Ramsden form of powers 18 and 25. The points in the diaphragm are set to cut 100 divisions of the stadium at 100 units + constant of the measurement intended to be taken, links, feet, or metres. This precludes distant measures, say of over 15 chains, where a 16-feet stadium is used, but they are made adjustable so that they may be set, if desired, for any other subtense, although this is not recommended. It is doubtful whether the subtense method can be considered as reliable at a distance of over 1500 links; or at any rate we must assume that much greater accuracy can be obtained by dividing distances greater than this into two by an intermediate station for observation, independently of the additional convenience of having the staff-holder within easy distance of communication.

Fig. 249.—Stanley's 6-inch tacheometer.

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570.—Where points are not used in the diaphragm or where lines are preferred, these may be divided upon glass in fine lines as Fig. 246; or spider webs may be used, but these are more difficult to set exact for stadia.

571.—Stadium.—Any accurate levelling staff will answer for the stadium, but the ordinary Sopwith, Fig. 99, is slightly confusing. A more open reading is generally recommended—that shown Fig. 102, p. 155, which the author designed for the purpose, answers perfectly. It is better to read the stadium low, as there is less vibration; but it is not often possible or at any time advisable to read it from the bottom—1 foot up is generally most convenient. Readings are taken and recorded of each subtense web, or point, separately, and the difference of reading subtracted for the subtense of tangent. With a point diaphragm for taking the subtense angle a fair certainty of accuracy of measurement of distance within ·002 may be assured, which is much nearer than can be attained by average chaining, taking six times the labour.

572.—The General System of Working the Tacheometer, with sufficient detail for practice, would take too much of our limited space to be given here. We now have several good works published in Great Britain, in addition to the able paper by Mr. Brough before mentioned, such as The Tacheometer: Its Theory and Practice, by Mr. Neil Kennedy; Surveying, by Whitelaw; Aid to Survey Practice, by L. D'A. Jackson, &c. There is a small work published in New York giving some details.[31] There are complete works in French, Italian, German, and Spanish. In French, Leves de Plans a la Stadia, by M. J. Moinot, engineer to the Paris, Lyons, and Mediterranean Railway, gives very complete instructions for all conditions of country, upon surveys which he has personally carried into practice with this instrument.

There are several tacheometers made upon the Continent, of more complicated forms than those herein described, but they do not produce better work.

573.—Field-books for the tacheometer are ruled in various ways in columns, which vary in number in different books from twelve to twenty. The French generally have fourteen columns, giving the number of the station, time, heights of line of collimation above point levelled, numbers of points selected, horizontal and vertical angles observed, reading of subtense webs and their differences, height of staff by reading central web, and columns for calculations and remarks; most English forms are more simple.

Fig. 249a.

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574.—A convenient protractor in which the equivalent of surface reading is taken from a scale upon its lower part directed from the centre of the protractor is here shown.

Fig. 250.—Perspective view of 5-inch omnimeter.

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575.—The Omnimeter is one of the class of instruments in which the tangent to a radius proceeding direct from the axis of the telescope is represented by the stadium made of constant length, the subtense angle varying with the distance. The omnimeter is the invention of Chas. A. C. Eckhold, a German engineer, described in the provisional British patent[32] as "a person living in Alexandria." The instrument as originally devised consisted of a kind of theodolite to which the subtense tangential system was added as an entirely separate part. The important part of the provisional specification shows that the principle of the invention consists in the use of two sights to the instrument, one a telescope to sight the object, and the other a powerful compound microscope to read divisions upon a tangential scale. The telescope and microscope are firmly united together in parallel position with their axes exactly crossing the transverse axis of the theodolite, so as to move together through the same angle by the motion of the telescope in traversing the azimuth. A delicate level is placed upon the telescope, and when the bubble is in the centre of its run the scale is truly at right angles to the axis of the microscope. The scale in the early instruments stood vertically at the extreme edge of the instrument in a position lateral to the object-glass of the telescope. It was finely divided to millimetres, and read the intervals of the divisions by means of a micrometer screw with a vernier.

Fig. 251.—Details of omnimeter, showing section of microscope and scale.

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576.—With the instrument as originally constructed, it was found that the delicate scale, protruding vertically to the extreme edge of the instrument, was very liable to injury unless supported by heavy metal work, which rendered the instrument cumbersome. A great improvement was made in this instrument, which brought it to its modern form, by placing the tangent scale in a horizontal position, where it could be firmly fixed upon the vernier plate as shown Fig. 251 S, and reading the scale by means of a reflecting prism P in the eye-piece of the microscope. In this improved instrument, as the microscope and telescope are still united on one axis so that they move at equal angles to each other, it is clearly indifferent whether the scale be placed vertically or horizontally, provided it is placed truly at right angles to the microscope when the axis of the telescope is horizontal. The scale, which is 4 inches long, is placed in a sliding fitting to adjust longitudinally to its position by means of a micrometer screw. In the English instrument the scale is divided into 100 parts for calculation. The divisions are subdivided by shorter lines, making the actual division 200. The micrometer screw has 50 threads to the inch, and moves over one of the divisions of the scale only. The micrometer head is divided into 100, numbered at the tens; a vernier placed against the head subdivides each of these divisions into 5, making the total micrometer 500 for one complete revolution. The total division of the 4-inch scale therefore becomes: 200 (divisions of scale) × 500 (micrometer) = 100,000 in 4 inches. The scale is placed centrally to the instrument, so that when the telescope is level the microscope is vertical, and reads 50,000 when in perfect adjustment.

Fig. 252.—Details of prismatic eye-piece.

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577.—The general appearance of the instrument resembles the transit theodolite, already described art. 368, in every way except for the addition of the microscope and scale, shown in perspective in Fig. 250. The details of construction of the microscopic apparatus may be followed in Fig. 251. T telescope with sensitive level B mounted upon it; R body of microscope connected solidly upon the same axis as the telescope, shown in half section. The eye-piece is placed at right angles to the microscope and telescope, and reads through the reflection of a prism P to the face of the instrument. The details of the eye-piece are shown in section Fig. 252. The tangential scale is shown in section Fig. 251 S with the micrometer with edge reading vernier at M. The compass of the instrument C is of the trough form, and placed on the opposite side to the level to be used after transitting the telescope from the position in which it is shown in the figure. The axis of the connected telescope and microscope is exactly 6 inches above the surface of the tangential scale S.

Fig. 253.—Omnimeter webs; a telescope, b microscope.

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578.—The telescope diaphragm is generally webbed with one horizontal and two vertical webs, Fig. 253 a, the altitude reading being taken from the top of the horizontal web, and the horizontal angular position from the centre of the interval between the vertical webs. The microscope diaphragm b has two horizontal webs, and reads from the centre of the interval, which is judged by the eye. Observed in this manner, there is no error due to covering angle subtended by the webs themselves. The most exact reading is obtained with a fine point.

579.—Reading of the Tangent Scale.—As the micrometer divides half a principal division into 500, the complete figured divisions are therefore divided into 1000. This is done for the sake of decimal notation. In reading it is only necessary to observe that the shorter or half division is 500, which must be added to the micrometer reading when it is past this division; as for instance 65½ reading is 65,500, and say the micrometer reads 234 past this, the reading is then clearly 65,500 + 234 = 65,734, just as before described for reading half degrees with the vernier.

580.—Value of the Scale taken in Rectangular Coordinates.—The radius from the transverse axis of the telescope to the tangent surface of the scale is exactly 6 inches. The scale is 4 inches divided into 100,000 parts, as it is read with the aid of the micrometer and vernier. The radius therefore in terms of the scale would be at 6 to 4, that is 150,000. By this we see that the divisions of the scale by the angle subtended give tangents, the value of each division of which is the reciprocal of this on 150,000 of the radius or base to any unit we may select. If we make the unit 1 foot, then one division represented by a unit of change of position of the vernier reading, and consequently of equal angular change in the direction of the axis of the telescope, would give a tangent of 1 foot upon a stadium placed at 150,000 feet distance. If the stadium were made 10 feet, as is usual, the same angular magnitude would be traversed in ten times this distance, or over 280 miles, making the value of the units of the vernier 1,500,000. This will give a general idea of the delicacy of the instrument so far as constructive principles are concerned, and not its performance.

581.—The Stadium is marked off in a number of feet, links, or metres, according to the unit taken for measurement of the surface of the land. The English stadium is generally formed of a 14-feet levelling staff, with the surface painted with a ground of plain white. At 10 feet apart two black bands about 2 inches wide are painted in, leaving in the centre of each band a clear white line of about one-tenth of an inch in width. These white lines are carefully set to 10-feet standard centre to centre. But a better plan is to have two equilateral triangles painted, with their apices meeting to the centre. An intermediate 5-feet line is drawn in black, which is found convenient for near measurements, to avoid too great angular displacement of the telescope. When the measurement is in chains, 15 links or 20 links may be taken for the distance of the lines apart to give the tangent. For metre measurement 3 metres are commonly taken for the stadium division. These are in each case subdivided. The lowest stadium reading should be 1 foot at least from the ground to avoid grass and other obstructions.

Fig. 254.—Ruling of omnimeter field-book.

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582.—Field-book.—The field-book as shown above, Fig. 254, was recommended by the inventor.

583.—Mode of Operating with the Omnimeter.—Carefully set the instrument up at its station in perfect adjustment as a theodolite, noting the departure point upon the scale reading through the microscope. Place the stadium in a vertical position at the point to which measurements are required. Direct the telescope so that the horizontal web cuts the upper line of the stadium, and lightly clamp it. Now read the microscope and record the reading as observed in the field-book. Unclamp the telescope and take the reading of the lower point of the staff and record this. Record the bearing of the instrument on the horizontal circle as with a theodolite.

584.—To Determine the Horizontal Distance in Feet.—Divide the constant radius of 1,500,000 given before by the difference of the two readings of the stadium mark, which are 10 feet apart. For example:—

First reading	of scale	67,500,	micrometer	235 =	67,735
Second	"	64,000,	"	450 =	64,450
				Difference	3,285;

then

1,500,000 3285 = 456·6 feet distance.

The process is somewhat simplified by logarithms, as we have only the log. of the difference to subtract from the constant, the 1,500,000 mantissa of which is 1,760,913. Thus—

log. 1,500,000	6·1760913
log. 3,285,000	3·5165354
	2·6595559	= 456·6 feet.

585.—To Determine Horizontal Distance in Chains the stadium should be marked as just described for feet, but at 20 links distance from line to line. Then the radius 150,000 × 20 gives 3,000,000. Taking for example, readings as before with difference of 3285 we have—

3,000,000 3285 = 913·2, or 9 chains 13·2 links distance.

To Determine Horizontal Distance in Metres, the stadium is divided to 4 metres. Then radius 150,000 × 4 = 600,000. Taking, for example, difference of reading as before 3285 then

600,000 3285 = 182·64 metres.

586.—Levelling—Taking Altitudes.—To take the elevation of the staff above the level of the instrument, subtract the reading of the scale, when the axis of the telescope is level, from the lower reading of the staff on the scale, and divide by the distance difference, as found by the method discussed before, then multiply this by 10 feet. Thus taking the lower reading as before 64,450 and the constant for the level position of the instrument, say 50,010, we then have—

Lower	reading	64,450
Level	"	50,010
	Difference	14,440

then

14,440 3285 × 10 = 43·96 feet nearly.

The heights, in relation to the position of the instrument, are positive or negative according as the scale readings are greater or less than the constant level reading or departure point.

587.—Work of the Omnimeter.—The perfection of the principles of the omnimeter would lead anyone to infer that work might be done with it of the highest degree of accuracy. The testimony of the greatest authorities show by comparison that it is unable to compete in this respect with the best made tacheometers. A large number of these instruments are employed in India. Colonel Laughton reports upon it—"It has been found to give very accurate heights of buildings, etc., also to be wonderfully accurate when used as a levelling instrument; but it is not so accurate for measuring distances over 600 feet, and even at this distance the error sometimes amounts to as much as 1 foot. It is recommended as admirably adapted for city surveys and traversing, also in hilly and jungly countries, and for railway and similar purposes."[33]

588.—Wherein the instrument fails to give exact results is no doubt in the difficulty of its manipulation. For taking two readings, which are necessary for every operation in distance, the instrument has necessarily to be set twice, the hand being placed upon the micrometer for the second observation while the attention is upon the sighting of the telescope; and even when the readings are taken by the telescope, the microscope has to be separately adjusted to read the micrometer scale. In the repetition of these processes it is almost impossible to avoid some slight disturbance of centre by pressure. In distant readings atmospheric changes giving difference of refraction occur quickly, so that there is more risk of error from two separate observations than if the observations of the subtense webs are taken simultaneously, as with the tacheometer. Further, any defect in workmanship or wear tells seriously against the readings of the instrument. Its advantages are theoretically that a wide angle is subtended by the stadium with the omnimeter in short distances which must be in every way an advantage. Further, since the early general use of the omnimeter, the tacheometer has been greatly improved, particularly in providing it with a larger and better object-glass so as to obtain greater field of view, that fairly near stations may be taken with it that were formerly only possible of reading with the omnimeter. The manufacture of omnimeters is now very limited; the subject is only retained in this edition because there are still some hundreds of these instruments in use.

589.—Improvement in the Omnimeter.—One improvement in this instrument by Mr. W. N. Bakewell, M.Inst.C.E., consists in turning the body of the microscope to a right angle at the position of the transverse axis of the omnimeter, and placing a reflecting prism at the angle. By this means the eye-pieces of the telescope and the microscope are brought side by side, greatly facilitating the joint readings. A second improvement is in making the scale 1,000,000 instead of 150,000, which much facilitates calculation, but it is doubtful if these improvements will stay the declining popularity of the omnimeter.

Fig. 255.—Bakewell's tangential arrangement to a theodolite.

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590.—Bakewell's Tangential Arrangement to a Theodolite for Measuring Distances.—This arrangement, which gives the distances by direct reading without calculation, was devised by Mr. W. N. Bakewell to extend the power of an ordinary 6-inch transit theodolite fitted with subtense webs. The observations are made on marks at 3 feet and 13 feet on an ordinary Sopwith staff—a 10-feet base, as is usual with the omnimeter. Any other base may be used if the distances registered are proportionally altered, or the scale may be divided to suit. It was first applied by the author to a theodolite that had been in good service, without the necessity of making any structural alterations in the instrument.[34] The measuring apparatus consists of a tangent screw impinging upon a radial plane, with micrometer and vernier. The details will be readily comprehended from the engraving, Fig. 255, and the following full description.

591.—The transverse axis of a theodolite, upon the opposite side of the telescope to that upon which the vertical arc is fixed, is turned down to a cylindrical surface true with the pivots. A collar A, which fits the cylindrical surface, is slit up on one side to enable it to be clamped firmly to any position of the axis by a clamping screw B. The collar is connected in the same gun-metal casting with the radial arm C that terminates at T in a plane, which is made truly radial with the transverse axis of the telescope. This radial arm C has a long German-silver spring S at the opposite side to the radial plane, which keeps it up firmly in contact with the point of the micrometer screw. A screw is cut on the drum of the micrometer D; on the spiral the scale of distances is engraved; and readings are taken from a line on the index I which slides on the bar E. The scale being one of reciprocals the divisions are at unequal distances, so a vernier cannot be used; consequently at long ranges where the divisions are close, the subdivisions must be estimated. Where this is too rough a method, resort must be had to calculation. The outer end of the drum D is divided into 200, and reads by vernier V carried by the arm E in 5 or thousandths of a revolution. The micrometer screw has twenty-five threads to an inch, and the radius of the arm C is 4 inches. One complete revolution of the screw is one-hundredth of the radius, and using a base of 10 the radius factor is 1000 × 100 × 10 or 1,000,000; consequently Barlow's or any other table of reciprocals can be used, and the distances obtained, by inspection with comparatively little labour. This additional part has not range sufficient for altitudes, being available for about 2 degrees only. The distance may be taken as a subtense or small tangential angle at a radius which, with the azimuthal angle taken by the vertical arc of the theodolite, will give altitude by its sine and horizontal distance by its cosine in the usual manner. The principle is that of the omnimeter, and it possesses the same objection for perfect performance, that the theodolite has to be handled twice for the two observations necessary.

592.—The Gradienter Screw.—This is no doubt a simplified copy of Bakewell's tangential arrangement and is shown at Fig. 256.

Fig. 256.—Gradienter screw.

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It is a micrometer screw fitted to a tangent arm, which can be clamped to the trunnion of telescope when the latter is in any position.

The screw is cut of a value that causes the web of the telescope to move 50/100 of a foot at 100 feet distance for each revolution, and the head of the screw is divided into 50 parts, consequently each division upon the head represents a movement of the cross web of the telescope of 1/100 of a foot upon a scale placed at 100 feet distance. The scale on the arm over the gradienter screw indicates the number of complete revolutions of the head, therefore, if the screw be revolved two whole revolutions the two divisions covered on this scale indicate 50/100 × 2 = 1 foot to the 100 feet.

To establish any grade with this screw.—Set the gradienter head to zero, then level the telescope and clamp the gradienter arm. Suppose grade required be 1·75. Turn the gradienter head through three whole revolutions, which will equal 150, then go on turning through 25 of the divisions on the head and the total movement will be 1·75, the required grade.

For Measuring Distance.—First with a staff for moderate distances. Any space on the staff covered by two complete revolutions of head is 1/100th part of distance, thus, if the difference between the two readings be 3·475 feet the staff is distant 347·5 feet.

Second Method.—For long distances with any rod of known length, such as a 20-foot stadia rod. Send out a man with the rod which he holds vertical at place to be measured. Then measure its length with the gradienter screw; say it takes 2 revolutions and 45 divisions over, thus 2 revolutions = 100 and 45 extra divisions = 145. Then—

20·00 feet 1·45 × 100 = 1379·3 feet.

Another instance.—Suppose the man at a distance has no stadia rod. He simply holds up any stick, say a walking stick. Measure this in telescope. Say it subtends 1 revolution and 28 divisions. This = 78. When your man comes in with the stick, measure its length. Say it was 3·25 feet. Then—

3·25 feet 0·78 × 100 = 416·6 feet.

The above illustrations are for readings taken approximately level. If there be much elevation or depression the angle must be read and the difference of hypo and base calculated and the stadia rod or staff must be inclined so that its face is at right angles to the line of sight from telescope. This can be done by the rod man inclining the staff or rod until the shortest reading is given if a staff be used, or the longest measurement is recorded by the gradienter screw head if a stadia rod be used. It is better in this case to have the staff fitted with a director (see art. 561), so that the person holding the staff may sight into the telescope of the instrument, thus ensuring the staff being exactly at right angles to the line of sight.

No constant should be added with either this, Bakewell's, or omnimeter measurements, as the angles are taken from the centre of the instrument. This gradienter screw has the same fault as mentioned for the two foregoing, viz., that all readings are taken by two movements of the instrument.

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