Tft 549 .Wi4 Copy 1 % PAPERS PEEPAEED iOE TKE USE OP THB TIIOMASON CIVIL ENGINEERING COLLEGE R O O R K E E. No. VII. SURVEYING ! i ROORKEE FEINTED AT THE THOJIASON CIVIL ENGINEERING COLLEGE PRESS. MUCCCLXIIL. 360 Coptei. Prtce 1-4-0 THOiviAS< Class jrLK_£M. ,ROORKEE, S( Booku-t SMITHSONIAN. DEPOSIT BAD. Papet Roorkee. ABBOTT 01] _ ^^ TV jiii^L. FOUNDATIONS Rs. 0-8-0. ACCOUNT of the GANGES CANAL, Es. 0-8-0. BOILEAU'S TRAVERSE TABLES, Rs. 1-4-0. „ TABLES of WAGES and RENT, &c., Rs. 1-0-0. SCANTLINGS for BEAMS for FLAT ROOFS, Rs. 0-4-0. LOG. SIN^ i P, (for COMPUTING TIME,) Rs. 0-8-0. CAPE'S GEOMETRY, Rs. 1-0-0. MECHANICS, Rs. 2-0-0. LOGARITHMS, with TABLES of SQUARES, CUBES, &c., Cash Rs. 1-0-0 ; Credit, 1-5-4. CATALOGUE, GEOLOGICAL MUSEUM, Parts I and II, Rs. 0-12-0. DESCRIPTION of HALL'S PATENT BRICK-MAKING MACHINE, Rs. 0-8-0. ELEMENTS of HINDUSTANI GRAMMAR, Rs. 0-2-0. EXAMINATION QUESTIONS, Rs. 0-8-0. GLOSSARY of TERMS RELATING to BUILDING STONES, &c., Rs. 0-1-0. 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This small work has Keen prepared primarily for the use of the Students of the Thomason Civil Engineering College, but it is hoped that it may be found useful to Overseers en Public Works generally in the Department Public Works, and others who are unable to purchase the more expensive works on the subject. With the above object, the compiler has endeavoured to make the work as complete as possible for all the more ordi- nary kinds of Surveying; and it is trusted that the insertion of certain minute details, (which have perhaps been omitted as unnecessary in larger works,) and the order in which the subjects have been placed — the chapter on each branch being complete in itself — may render the work particularly useful to begin- ners. It professes to be nothing but a compilation, the greater part having, by kind permission, been extracted from that valuable work, " Surveying for India," by Lieut.-Cols. Smyth and Thuillier. The Chapters on the Traverse Table, which it is believed are not to be found complete in any other book on Surveying, have been transferred in original. Erome's " Outlines of a Trigonome- trical Survey," Jackson's "Military Surveying," and Baker's " Land and Engineering Surveying," have also been made use of. I. P. W. ROOEKEE, March 2nd, 1865. INDEX. CHi-PTEE PAGE I. SUSVETII^^G- WITH THE ChAIIS- ONLY, 1 Measuring Chains — Directions for using them — The Cross Staff and Offset Eod— The Perambulator— The Field- book — Example of a Chain Survey — Plotting the Survey. II. The Prismatic Compass, 13 Description and Method of using the Prismatic Compass The Surveying Compass — Method of Surveying with the Prismatic or Surveying Compass — Plotting the Sur- vey — The Plane Table — Method of using it — Problem for finding your position from three known points. III. The Theodolite, 29 The Vernier — Description of the Y Theodolite and its Adjustments — Everest's Double Arc Theodolite and its Adjustments — Method of observing with the Theodolite — Parallax of the Wires — Kepeating an Angle — Hints on the use of a Theodolite — Traversing with the Theo- dolite — Plotting the Traverse — Circular Protractor with Vernier. IV. The Teaveese System, 50 Explanation of the mode of Surveying by Traverse — De- monstration of the proof of a Survey by Traverse — Amount of Error allowed, and method of distributing the corrections — Method of Plotting by Traverse — De- monstration of the Universal Theorem — The Traverse Table — Computation of the different Columns. Yl CHAPTER PAGE V, The Pocket Sextant, 80 Description of the Pocket Sextant, and Method of adjust- ing it — The Artificial Horizon — Demonstration of the principal of the Construction of the Sextant — Angles taken by the Sextant not horizontal — Examples of find- ing inaccessible heights and distances by the Sextant — The Optical Square. VI. Levelling, . , , 90 Definition of. Levelling — Curvature of the Earth — Re- fraction — Compound and Simple Levelling — Levelling Staves — The Y Level and its Adjustments — Gravatt's ' or the Dumpy Level and its Adjustments — Troughton's Level and its Adjustments — Method to be pursued in li Levelling a tract of Country — Levelling with the Theo- I dolite — Contouring — Table of Corrections for Curvature j and Eefraction. i \ VII. Kailwat Curves, Useful Problems in Suevet- iNG, AND Scales, 119 Simple Curves — Compound Curves — Serpentine Curves — I Curves of Deviation — Useful Problems in Surveying — Simple Scales — Diagonal Scales — Methods of reducing or enlarging Drawings — The Pentagvaph. i VIII. Teigonometeical Surveying, 150 Selection and Measurement of a Base Line — Selection of Station — Observations of Angles — Reducing to the I Centre — Laying down the Triangulation on paper — Forms used in Field-book — Forms used in Calculation Book. CHAPTER I. THE MEASUEING CHAIN. Theee are two kinds of chains in general use for survey purposes ; one divided into feet, and the other into links. The former is generally a 50 or 100 feet chain, of which the first would only be used on account of its portabilifcy. The latter is 63 feet long, and is divided into 100 parts, each part being equal to a link, or 7*92 inches ; it is called a G-aiiter's chain, and is specially adapted for measuring areas which are required to be computed in acres, roods, &c., for which it offers great facility, as its length is equal to four poles or perches, so that one square chain is equal to 16 square per- ches, or one-tenth of an acre. The length of a straight line must be found mechanically by the chain, and it is the most difficult operation in surveying. The sur- veyor, therefore, cannot be too careful in guardiug against, rec- tifying, or making allowances for every possible error, for, on the exactness of this measurement, the correctness of his work depends. The chain, however useful and necessary, is liable to many errors — first, in itself; secondly, in the method of using it ; and, thirdly, in the uncertainty of pitching the arrows. Every possible precau- tion must therefore be used. If the chain be stretched too tight, the rings will give, the arrows incline, and the measured line will be shorter than it reaUy is ; • on B 2 the other hand, if it be not drawn sufficiently tight, the measure obtained, will be too long. If the chain is a new one, it should, invariably, be measured daily until it lias stretcbed to its utmost ; if an old one, and which a surveyor will find by experience, to be always preferable, once in every three or four days is sufficient. A careful and correct sur- veyor, will, however, compare it daily, the mean of two compari- sons should then be taken as the length, for the work done in the interval. ^ Chains have been known to stretch as much as three inches in a day's work ; this, though trifling, in one chain, would be found of material consequence, after measuring 400 or 500 chains during the day, amounting, as such an error would, to nearly one chain and a half in the whole distance measured. The true length of a chain line measured with an incorrect chain, is easily found from the following proportion : — As the length of a correct chain is to the length of the chain used, so is the measured distance to the true one. For example, suppose the length of the chain to be J.00-32 feet or links, and the measured line 1050 feet or links, then, 100 : 10032 : : 1050 : 1053-36, the true length of the line. To measure a chain, the ordinary offset rod, will, for general pur- poses, be sufficiently accurate ; two should be used in the following manner : — Stretch the chain pretty tightly on a level piece of ground, fixing two stout wooden pins at each end, in the handles of the chain ; then, lay down the two rods from one end, keep the second stationary, and, taking up the first, place it beyond the second, then keeping that stationary, take up the second and place it beyond the third, until the end of the chain be reached, when the decimals de- ficient, or in excess of the nearest foot, mav be measured with a Marquois, or other small, scale. It is a common practice to allow chainmen too much latitude in measuring lines, i. e., the surveyor is satisfied to come up at the end of the line measured, count the numher of links up to the station, depending entirely on the rear chainman for a correct account of the number of chains measured. This, even were the account of chains correct, (which is always doubtful,) can never be a satisfac- torily measured line. Unless the surveyor follows in rear of his chainmen, and keeps a continued watch on them, the probabilities are, that his work will have to be measured over again. A surveyor should accustom himself to follow his rear chainman, and satisfy himself as he is progressing, that he is measuring straight. To ensure the chainmen proceeding in as straight a line as possible, it is always well for the leading chainman to check the direction of the rear chainman, 'hj keeping the latter and the back station, (on which there is invariably a flag), in a straight line with himself. The 7*ear chainman does this, as he directs the leading one with the forward station, and thus, by a mutual check, great accuracy is obtained. Care must also be taken to see that the chainmen place the pins in the ground properly. The rear man should bring the handle of the chain up to the pin left in the ground, and the fore man, after getting in line, and stretching the chain, should put the pin in the ground, inside, and up against the end of the handle ; otherwise, if both pins be inside, or both outside of the handles, the thickness of the handle is gained, or that of the pin lost in each chain, which, when the distance measured is great, would amount to something considerable. If the ground is too hard to admit of the pin being driven in, a cross should be scratched on the ground, arid the piu laid down pointing to the intersection. Eleven arrows should be used instead of ten, as is generally the custom, for in the latter case, when the chain arrives at the end of the tenth arrow, thus denoting ten chains as measured, the chain is stopped!, and liable to be shifted ; whereas, with eleven arrows, one arrow always remains a fixture in the ground, and is never brought into the account, thus preventing the possibility of the chain being shifted whilst the other ten arrows are being taken to the leading chainman. Directions for using the Chain. — Flags are first to be set up at the places whose distances are to be obtained ; the place where the measurement is commenced may be called the first station, and that measured to, the second station. Two men hold the chain, one at each end. On the chain being stretched in the direction of the second station, the leader who is provided with eleven arrows, drives one firmlj into the ground, the rear chainman holding the other end at the first station ; he then proceeds in the direction of the seco7id station, until the rear chainman has arrived at the first arrow, when the latter directs the former in a liae with the first station, and a second arrow is firmly driven in, the rear chainman then takes up the fi.rst arrow, counts one chain as measured, and proceeds on until the eleven arrows are expended, one of which remaiuing in the ground, the other ten are sent on to the leading chainman. The exchange of the arrows is always notified by the rear chainman call- ing out with a loud voice, so many tens. The surveyor here marks in his Field-book that one change has been made, or 10 chains, or 1000 links measured. The chaiumen then proceed onwards, until another change has been made and entered, and so on, mark- ing every change until the second station be arrived at, when the number of arrows in the hand of the rear chainman will denote the number of chains, which, together with the odd links, and the number of changes that may have been made between the two stations, will make up the entire length of the line. THE CBOSS STAFF, AND OFFSET EOD. When the boundary of a survey has turns and bends in it, as is generally the case, it is not necessary to measure round every such turn and bend. The best and most usual way is, to proceed in a straight line from one principal corner to another, and when oppo- site to any bend in the boundarj, to measure the rectangular dis- tance, termed the ojfset, from the chain line to the bend, noting the same, together with the distance on the chain line from whence such offset was made. These offsets are geuerallv" measured with an offset staff or rod of tea feet. G-reat care is reijuired on the part of the surv^ejor in measuring offsets, for, unless the offset is taken at right angles with the chain line, the perpendicular measured for determining its area will be too long, and a correct result will not be obtained. A very convenient inslrument, called the " cross staff, " which can be made up bj any Bazaar Carpenter, is used for the purpose oi taking offsets. It consists of a piece of wocd, about six inches square and an inch and a half in thickness, fixed on the end of a staff about five feet in length, with an iron spike at the end, for the convenience ofplantingitinthe ground. The square piece on the top has two grooves ab and cd in it, about half an inch deep, at right angles with each other, made with a common saw. This instrument being placed any where on the chain line, if one groove be directed to the forward or back station, the other will of course give the per- pejidicular to the chain line. A well practised sur- veyor can, however, generally tell a right angle for an offset, with- out the assistance of this instrument. The best method of measuring offsets is, for the offset man to walk along the boundary, and to give a signal to the chain party, whenever he comes to a bend or corner ; the surveyor then places himself on the chain line in a rectangular position with the offset man, when the latter, measuring down towards him, gives in the length of the offset in rods, and returns immediately to the boun- dary to take up the next bend. A good offset man should never be taken off his work, for, by constant practice, he knows exactly when and where an offset is required. THE PEEAMBULATOB. This instrument is very useful for measuring roads, level plains, and everything where expedition is required. It does not give, however, a very correct measure in going over uneven surfaces, which is one of its principal objections ; and it is, therefore, only applica- ble to road and route surveys, where great accuracy is not essential. The following figure represents the English Pattern Perambulator, which consists of a wheel of wood A, shod or lined with iron to prevent the wear ; a short axis is fixed to this wheel, which com- municates motion by a long pinion fixed in one of the sides of the carriage B, to the wheel-work C, included in the box part of the instrument. For portability, the wheel A, is separable. In this iastrument the circumference of the wheel A, is eight feet three inches, or half a pole ; one revolution of this wlieel turns a single threaded worm once round ; the worm takes into a wheel of 80 teeth, and turns it once round in 80 revolutions ; on the socket of the wheel is fixed an index, which makes one revolution in 40 poles, or one furlong ; on the axis of this worm is fixed another worm with a single thread, that takes into a wheel of 40 teeth ; on the axis of this wheel is another worm with a single thread, turning about a wheel of 160 teeth, whose socket carries an index that makes one revolution in 80 furlongs or 10 miles. On the dial plate there are three graduated circles, the outermost is divided into 220 parts, or the yards in a farlong: the next into 40 parts, the number of poles in a furlong ; the third into 80 parts, the number of furlongs in ten miles, every mile being distin- guished by its proper Eoman figure. STJEVEFIiS-G BY THE CHAIN" ONLY. In making a survey with the chain only, we are confined to one, and the simplest geometrical figure, viz., the triangle, for of all plane geometrical figures, it is the only one of which the form can- not be altered, if the sides remain constant. That the triangle pos- sesses this property, is evident from the Theorem, (Euclid, vii. 1.,) which proves that " Upon the same base, and on the same side of it, there cannot be two triangles that have their sides, which are terminated at one extremity of the base, equal to one another, and likewise those which are terminated in the other extremity, equal to one another." The surface to be measured is therefore to be divided into a series i\ 8 of imaginary triangles ; and in this division it must be borne in mind that the triangles are to be as large, with reference to the whole surface to be measured, as is consistent with the nature of the ground ; for, by such an arrangement, we are acting on the important prin- ciple in all surveying operations, that it is well always to work from lolwle to part, and rarely from part to whole. The sides of these triangles are first measured, and as a neces- sary check on this part of the work, a straight line is in addition measured from one of the vertices to a poiut in or near the middle of the opposite side. This fourth line is called a tie-line, and is an efficient means of detecting errors, if any have been committed in the measurement of the sides of the triangle. This fourth measure- ment is made in accordance Avith a maxim which ought invariably to be acted upon in all surveying operations, viz., that where ac- curacy is aimed at, the dimension of the main lines, and the positions of the most important objects, should be ascertained or tested by at least two processes independent one of the other. AVithin the larger triangles, as many tie-lines and smaller triangles are to be measured as may be necessary to determine the position of all the objects embraced in the survey. The directions of the lines forming the sides of these secondary triangles are so selected or disposed that they shall connect and pass close by, as many objects as possible, so that the offsets to be measured from them may be as short and as few in number, as practicable. If the sides of these secondary triangles be in any case so distant from the objects whose positions are to be determined, as to require a length of offset greater then one or two chains, it then becomes advisable to construct, either on the whole or part of the side of the triangle as a base, a small offset triangle with the sides so dis- posed that they shall either embrace, or pass very near to the ob- jects to be measured by their intervention. The disposition and general combination of these triangles de- manding care and judgment, it is customary, previous to commene- 9 ing anj measurement, to walk over the ground for the purpose of obtaining a general knowledge of the surface, ajid of the relative positions of the most conspicuous objects. The acquisition of this knowledge depending on the coup d^oeil,is much assisted by an eye- sketch drawn with rapidity, and showing some of the principal roads, streams, temples, &c. This hand-sketch is not drawn to any scale, and its object is attained if it simply bear a general resemblance to a plan of the ground, as it will thereby assist the memory in the distribution of the surface into triangles. The sides of the larger ti'iangles are to pass as close as possible to the external boundaries to be surveyed ; the triangles should, moreover, be made to approach, as nearly as practicable, to the form of equilateral, avoiding with care very acute or very -obtuse angles, because the further the form of the triangle is removed from the equilateral, the greater will be the alteration in the form of the figure and its area, should any error have been committed in the measurement of any one of the sides. The triangles having thus been disposed to the greatest advantage, marks or pegs are placed in the grou.nd at each vertex of the tri- angles; the general form or position is then noted on the hand-sketch previously made, and distinctive letters are wiitten on the diagram at each point of intersection ; this arrangement admits of easy re- ference in the Field-book, or on the ground, to any triangle or part of a triangle. The points of intersection of all straight lines, as well as the vertices of the triangles, are always points measured to or fro^n ; they are called station points, and the lines connecting them, sta- tion lines, thereby distinguishing them from the simple offset lines. Stations are generally expressed by letters, main stations by capital or roman letters. A, B, C, &c., and secondary stations by small let- ters, a, h, c, &c. The hand-sketch, or rough diagram, is usually made in a Field- c 10 book, i. e., a book in which every minute step of the operations gone through, is to be entered with precision at the time. This Fie)d-book should be of a convenient size for the pocket, having the page ruled with a central column ; this central column is intended for all actual lines measured, and by commencing from the bottom of the page, the page becomes a smaller representation of the reality, with the line measured from you, and the offsets at their respective distances on that line, taken at so many links to the right or to the left, as they actually are on the ground and noted to the right or left of the central column. In keeping the Field-book, it first should ever be remembered that the central column is virtually but one line representing the Chain, the space within the column being merely required for the J I several distances on the Chain, whence the offsets are taken, and j|[ secondly, that all offsets read either way outward from the centre column, in the same way as they are measured outward from the Chain ; if the station line, therefore, should be crossed on the ground by a road or any boundary meeting it obliquely, its representation or type in the Field-book must not be made to pass obliquely across the middle column, but must arrive at one side of the column and leave it at the other, at points precisely opposite, as it would do were the middle column merely of the thickness of a line ; inatten- tion in this particular, causes much confusion in the relative position of offsets. To preserve uniformity, as it is more natural to measure from left to right, the place measured from is put on the left of the central column at the bottom of the line, and the station measured to is put at the top to the riglit ; the points of commencement and ter- mination of the line can thus be immediately seen. The book should be interleaved with blotting paper and the en- tries made in ink or inked in the same day on return from the field ; this should be rigidly adhered to, as any doubtful point is more likely to be remembered, or any error corrected, if the Field-book is ^0 © OTzcci^.^ :/06' Izn^ A7Vt-» t e AhrL'jtpore -v^> Ci^lttrra^^doTV ^ O i'Z i wdo 1^1 ^ / ChyOfJu^ Svvr'i^-^if Clos^-'d J" 213 JIC ib i4 Wcz^^ \ W\^A- A ?2/ ri 1 C ^1-4 400 / 68 zgo 260 15 / ^^# 11 inked in at once and not left for two or three days ; the pages should also be numbered for facility of reference and each day's work dated. In taking offsets to corners of boundary marks or other objects, mark the relative position of the corner or object as to the chain line, and generally be careful to make the Field-book as much as possible ?ifac simile of the ground itself, with each boundary mark, &c., placed on the book, as to the central column, considered always as one line, in the same position as they stand to the chain on the ground ; no time is gained to the surveyor by hurrying over the notes in the Field-book, a little care in the Field saving mucli trou- ble in office. It cannot be too strongly impressed on the surveyor that the work which he is called upon to perform depends for its accuracy in a very great measure on the order, system, and neatness bestowed on all the steps whether of delineation or measurement : proper atten- tion in keeping the Field-book saves much time in plotting, and guards againsts the errors unavoidably arising from reference to a confused Field-book ; moreover, care bestowed in the first essays, will amply reward the surveyor, by giving accuracy of eye, freedom and steadiness of hand, qualities indispensible to his success. A specimen is here given of a Field-book, and the plan made from the Field-book, as an example of a chain survey. The line AB was first measured and offsets taken to the principal objects ; a and h were marked as points from which to run cross or tie-lines, but were afterwards found to be unnecessary. Next, the lines B C and C A, were measured, the points c in the former, and d, e in the latter, were noted, and then the cross lines, c d and h e were chained to enable ofi'sets to be taken to the objects in the middle of the triangle, and also to serve as check lines when plotting the survey ; these cross lines may be left nntil the sides of all the principle triangles are measured, and then each triangle may be filled in afterwards, but the points in the main line from which it is intended to run them must be marked at the time of measuring the sides of c 2 l-Z 10 I.UIllll.llii 10 20j 'I" "■■'"'"■■' the |. oipal triangles, or a great deal of unnecessary measure- ment will be entailed. Thus are all the sides of the triangles measured in succession, and their dimensions with the additional assistance of the offsets, give the means of ascertaining all boun- daries, external and internal, positions of houses, &c., and of finding the area of the whole and of every part, by direct computation from the Field-book. The method of plotting a chain survey is so self evident that a few words of explanation will suffice. In the above example, lay down on paper a line equal to A B, taken from any scale of equal parts, from the same scale take the length A C with a pair of com- passes, and with this as a radius and A as a centre, describe an arc ; now, taking E C as a radius and B as a centre, describe an arc, cutting the first, their intersection will give station C ; in the same way lay down the triangles C D B and C D E. Now, commence with A B, and mark off the distances given in the centre column of the Field-book, at the same time setting off the ofi"sets ; for this purpose a small cardboard or paper scale, of the shape shewn in diagram, will be found very useful ; by means of the middle scale either of the short arms can be placed at the required distance from the station and the offsets marked ofi-from them. In the same way proceed with the sides A C and B C, and then fill in the triangle by means of the cross lines c d and B^. Proceed with the other triangl similar mannar until the whole is completed [es m a y I lyuACE OF Kumc e\r g,oRE EXAMPLE A SURVEY EET TO AN INCH DO :oo eoo 700 boo spo loot CHAPTER II THE PEISMATIC COMPASS. The use of this little instrument is to measure horizontal angles only, or rather to take the bearings of objects, when the angle can be deduced from the two bearings ; and from its portability it is par- ticularly adapted for filling in the detail of a map, where all the principal points have been correctly fixed by means of the Theodo- lite. In the figure A re- presents the compass box, and B the card, which being attached to the magnetic needle, moves as it moves, round the agate centre, a, on v.hich it is suspended. The circumference of the card is usually divided to 15' of a degree, but it is doubtful whether an an- gle can be measured by it even to that degree of accuracy : c is a prism, which the observer looks through in observing with the instrument. The perpendicular thread of the sight- vane, E, and the divisions on the card appear together on looking through the prism, and the division with which 14 the thread coincides, when the needle is at rest, is the magnetic azimuth, or bearing, of whatever object the thread may bisect. The prism is mounted with a hinge-joint, D, by which it can be turned over to the side of the compass box, that being its position when put into the case. The sight-vane has a fine thread or horse hair stretch- ed along its opening, in the direction of its length, which is brought to bisect any object, by turning the box round horizontally ; the vane also turns upon a hinge-joint, and can be laid flat upon the box, for the convenience of carriage. F is a mirror, made to slide on or oil* the sight-vane E ; and it may be reversed at pleasure, that is, turned face downwards ; it can also be inclined at any angle, by means of its joint, d; and it will remain stationary on any part of the vane, by the friction of its slides. Its use is to reflect the image of an object to the eye of the observer when the object is much above or below the horizontal plane. AVhen the instrument is employed in observing the azimuth of the sun, a dark glass must be interposed, and the colored glasses represented at G- are intended for that pur- ■pose ; the joint upon which they act allowing them to be turned down over the sloping side of the prism-box. At e is shown a spring which being pressed by the finger at the time ot observation, and then released, checks the vibrations of the card, and brings it more speedily to rest. A stop is likewise fixed at the other side of the box, by which the needle may be thrown off its centre ; "which should always be done when the instrument is not in use, as the constant playing of the needle would wear the point upon w^hich it is balanced, and upon the fineness of the point much of the accurac}^ of the instrument depends, A cover is adap- ted to the box, and the whole is packed in a leather case, which may be carried in the pocket without inconvenience. Prismatic compasses are now made with a silver graduated rim to the card, which is a vast improvement, and they should not be less than four inches in diameter. The method of using the instrument is very simple. First raise 15 the prism in its socket h, until you obtain distinct vision of the di- visions on the card, and standing at the place where the angles are to be taken, hold the instrument to the eye, and looking through the slit c, turn round till the thread in the sight-vane bisects one of the objects whose azimuth, or angular distance from any other ob- ject is required ; then by touching the spring e, bring the needle to rest, and the division on the card which coincides with the thread on the vane, will be the azimuth or bearing of the object from the North or South points of the magnetic meridian. Then turn to any other object and repeat the operation ; the difference between the bearing of this object and that of the former, will be the angular distance of the objects in question. Suppose the former bearing to be 40^" 30' and the latter 10° 15', both East or both West, from the North or South, the angle will be 30^ 15'. Prismatic compasses may be used with or without a stand. The divisions in some instruments are numbered from 0° to 180*^ South counting Eastward, and thence to 180° North counting "West- ward, others are numbered 6\ 10*^, 15°, &c., round the circle to 330°, 90^ representing East, 180° South, 270^ West, and 360° North. These are by far the best, as the least liable to error in recording the results in a Eield-book, and are more generally understood by Natives. This instrument must be held or set up as nearly horizontal as possible, in order that the card may play freely ; also it must not be used near iron, or by a surveyor wearing steel spectacles. The variation of the needle must always be attended to, for if the fixed points above alluded to have been surveyed on the true meridian of the earth, the variation of the needle must be added to, or deduct- ed from, the observed bearing to obtain the true meridional bearing of the line. 16 THE SURTEYIKG COM T ASS. The Siirveyiug Compass consists of a compass-Lox, magnetic needle and two plain sights, perpendicular to the meridian line in the box, one of which has a longitudinal slit through which the surveyor lines the horse hair on the object of which the bearing is required; it is used for the same purpose as the Prismatic Com- pass, for filiiug in the interior detail of a survey by means of bear- ings. The sights are attached in various ways for portability, but are now generally made to turn down on a hinge, in order to lessen the bulk of the in- strument and render it more convenient for carriage, the diameter of the box varies from three and a half to four and five inches. "With- in the box is a graduated circle, the upper surface of which is divided into de- grees only, and numbered 10°, 20^ 30 ^ &o., up to 360'". The bottom of the box is divided into four parts or quadrants, each subdivided into 90^ num- bered from the jSTorth and South points each way, to the East and West points. The dis- advantage of a surveying compared with a prismatic compass is, that when looking through the sights, the angle is not presented to the eye, but it is necessary after fixing the sights on an object to read off the actual number of degrees pointed out by the 17 needle. It is evident from this that the surveying compass cannot be used without a stand, which is another disadvantage. This in- strument is however well adapted for first instruction of a Native, and is the same in principle as all the Vernacular Compasses made expressly for Native use. METHOD OF STJBVETINQ WITH THE PEISMATIC OR SURVEYING COMPASS. Let the annexed plan represent a survey of roads to be per- formed by the Prismatic or Surveying Compass. Having fixed on a starting point A, set the compass up there, and send a man with D 18 a, flagstaff to B, as far down the road as can be seen, and let it be placed in such a position that a long forward shot may be obtained the next time. Now, take the bearing of B, and proceed to chain from A to B, taking offsets to the sides of the road and any re- markable objects, precisely as in a chain survey. Having arrived at B the compass must be again set up, and a flagstaff having been sent to C, its bearing must be obtained ; then the line B C is to be measured, and so on. Angles must also be taken to any conspicuous objects that are out of reach of an offset, such as the corner of the house in the figure ; bearings from two points are sufficient to fix it, but a third should be taken as a check. The Pield-book is kept as in the chain survey, the only addition being the bearings. The angles to the next station are called the "forward bearings," the first of these should be written in the centre column immediately above station No. 1, and afterwards to the right or left of the centre column, according as the angle is to the right or left of the last direction ; this wdll pre- vent any mistake in plotting it. It is only necessary to take angles from every other station — for instance, we might go to B, and from there observe the beariugs of both B A and B C, then to D, and observe D C and D E ; but, it is advisable to take angles from every station, for it prevents confusion in the Field-book, and also the trouble of adding or subtracting ISO'* from every other angle, which in the other case would be necessary before the work could be plotted. For plotting a compass survey we require a protractor, which is an instrument for laying off angles. Protractors are either rect- angular, circular, or semicircular ; the former are most commonly used. They are made of either ivory, boxwood, or brass, about six inches long by two wide, and should be numbered in two rows, the outside one from 0° to 180°, and the inside from 180° to 360°. To plot the above survey, having fixed on a convenient spot on 19 the paper for the stai'ting point A, (i. e.^ so that the survey may be contained in the paper, and as nearly in the middle as possible,) draw a line through it to represent the magnetic meridian, place the protractor to the right of this line, the edge coinciding with it, and the centre at the point A ; now, with a pencil, mark the required angle, and draw a line through this point and A, this will represent the first bearing ; on this line, produced if necessary, set off from any scale of equal parts, the length of AB, and through B draw the line NBS parallel to NAS, to represent another meridian, and plac- D 2 20 ing the protractor as before, lay off the angle NBC — set off BC the required length; and proceed with each angle in the same way until the end of the line EA should coincide with the starting point A. If we wish to measure an angle greater than 180°, the protractor must be placed to the left of the meridian, as at D, and the second row of figures used. Having completed the circuit and found it to be correct, w^e now proceed to lay off the offsets. The prismatic compass is very useful for what is called filling in a survey. The plotting of this kind of work is usually done in the field, each angle being laid down as soon as taken, and each distance and offset as soon as measured, so that no Field-book is required. A piece of paper with the work already done, such as the above circuit of road plotted on it, is placed in a sketching case, faint parallel lines having been ruled over it in the direction of the meridian, a quarter or Kalf an inch apart ; then, when an angle is taken, the protractor is placed parallel to these lines, (which can be done near enough for this kind of work by the eye,) and the angle is measured off. The following method of finding one's place in a survey with the prismatic compass will be found use- ful — referring to Mg. on page 17 — Suppose we wish to start the fil- ling in from a point Gr, and not from any of our former stations, the first thing is to find the position of the point G- on our paper ; to do this, we take the bearings of any two convenient stations, in this case D and E ; now, to find the bearing of our position y^'ow^ D and E, we have only to add 180° to, or substract it from, the bearings taken to D and E, and to protract the angles thus obtained from D andE — their intersection will be the point required. For instance, we find the bearings of D and E from Gr to be 100° and 205° respectively ; the former being less than 180°, we add that number to it, and the latter being greater, we substract it from it, and obtain 280° and 15° respectively, placing a protractor at D and E, plot these angles, and their intersection gives the point G. It will be as \vell to take a bearincf to a third station as a check. Observe that 21 the nearer the two bearings meet at a right angle, the more accurate will be the station determined. If a large area had to be surveyed by the prismatic compass, it would be advisable to fix some points in it by triangles, starting from a measured base line ; these points should not be more than half a mile or a mile apart. Although they cannot be fixed very accurately when the angles are measured with an instrument which only reads to half degrees, yet with care they can be sufficiently so to serve as a check in the filling in, and oifer good points from which to start the circuits of traversing. THE PLANE TABLE. The Plane table used in India is of a very simple construction, it consists of an ordinary drawing board, varying from fifteen to twenty-four inches square, mounted on a tripod stand, and is move- able about an axis which goes through the head of the stand, and is fastened on the other side by a nut. Oak, teak, and toon, are the best woods for plane tables, deal and other soft woods imbibe moisture quickly and expand across the grain. A magnetic needle, in a compass box, is some- times attached to the table, it serves to point out the direction, and acts as a check upon the sights ; but, it mast be remembered, that it is never to be used inde- pendently. It is a dan- gerous thing in the hands 2?. of natives, who are apt to set their table up by it alone, without ever verifying its position by sights. There is also an index or ruler, made of brass, iron, or wood, about the length of the diagonal of the table, at each end of this is a sight similar to those in the surveying compass ; one of the edges of the ruler is chamfered, and is called the fiducial edge ; the ver- tical hair and this edge are usually in the same plane, but this is not necessary. "When required for use, a sheet of paper is stretched on the board by first wetting it, and then glueing down the edges. To use the tahle. — Fix it at a convenient part of the ground, and make a point on the paper to represent that part of the ground. E;un a fine steel pin or needle throught this point into the table, against which you must apply tbe fiducial edge of the index, moving it round till you perceive some remarkable object, or mark set up for that purpose. Then draw a line from the station point, along this edge of the index. Now set the sights to another mark or object, and draw that station line, and so proceed till you have obtained as many angular lines as are necessary from this station. The next requisite, is the measure or distance from the station to as many objects as may be necessary by the chain, taking at the same time the offsets to the required corners or crooked parts of the edges, setting off all the measures upon their respective lines upon the table. Now remove the table to some other station, whose distance from the foregoing was previously measured ; then lay down the objects which appear from thence and continue these operations till your work is finished, measuring suck lines as are necessary, and determining as many as you can by intersecting lines of direc- tion, drawn from different stations. The use of* the instrument will be better understood from the following example, as given by Simms. 23 In the annexed diagram, let the points marked A, B, C, &c., be a few of an extensive series of stations, either fixed or temporary, the relative situations of which are required to be laid down npon the plan. Select two stations, as I and K, (considerably distant from each other,) as the extremities of a base line, from which the great- est number of objects are visible ; then, if the scale to which the plan is to be drawn is fixed, the distance, IK, must be accurately measured, and laid ofi" upon the board to the required scale ; other- wise a line may be assumed to represent that distance, and at some subsequent part of the work, the value of the scale thus assumed must be determined, by measuring a line for that purpose, and comparing the measurement with its length, as represented on the plan. Set up the instrument at one extremity of the base, suppose at I, and fix a needle in the table at the point on the paper represent- ing that station, and press the fiducial edge of the index gently PA agftiust the needle. Turn the table about until the meridian line of the compass-card coincides with the direction of the magnetic jieedle, and ia that position clamp the table firm. Then always keeping the fiducial edge of the index against the needle, direct the sights to the other station K, and by the side of the index draw a line upon the paper, to represent, the base IK ; when, if the scale is fixed, the exact length must be laid off, otherwise the point K may be assumed at pleasure on the line so drawn. But it is sometimes necessary to draw the base line first, when required, on some particular part of the board, so as to admit of the insertion of a greater portion of the survey ; when this is the case, the index must be laid along the line thus drawn, and the table moved till the further end of the base line is seen through both the sights ; then fix the table in that position, and observe what reading on the compass-card (or bearing) the needle points to, for the pur- pose of checking the future operations, and also for setting the table parallel to its first position, wherever it may afterwards be set up. It should be observed, that in placing it over any station, that spot on the table representing such station, and not the centre of the table, should be over the station on the ground : it may be so placed by dropping a plumb-line from the corresponding point on the underside of the table. Having fixed the instrument and drawn the base line, move the the index round the point I, as a centre, direct the sights to the station A, and keeping it there, draw the line IA along the fidu- cial edge of the index. Then direct in the same manner to B, and draw the line IB ; and so proceed with whatever objects are visi- ble from the station, drawing lines successively in the direction of C, D, E, &c., taking care that the table remains steady during the operation. This done, move the instrument to the station K, and placing the edge of the index along the line IK, turn the table about till the sights are directed to the station I, which if correctly done, the 25 compass-needle will point to the same bearing as it did at the for- mer station (in our example it was set to the meridian). Now, move the needle from I, and fix it in the point K ; lay the edge of the index against the needle, and direct the sights in succession to the points A, B, C, &c., drawing lines from the point K, in their several directions, and the intersection of these lines, with those drawn from the point I, will be their respective situations on the plan. To check the accuracy of the work, as well as for extending the survey beyond the limits of vision at I and K, the table may be set up at any one or more of the stations thus determined, as at E ; the needle being now fixed in the point E, on the board, and the edge of the index placed over E and I, (or K), the table may be moved round till the station, I, is seen through both the sights, and then clamped firm; the compass will now again (if all be correct), point outjts former bearing, and any lines drawn from E, in the di- rection of A, B, C, &c., in succession, will pass through the intersec- tion of the former lines, denoting the relative places of those objects on the board ; but, should this not be the case with all, or any of the lines, it is evident that some error must exist, which can be detected only by setting the instrument up, and performing similar operations at other stations. Having a number of objects laid down upon the plan, the situa- tion of any particular spot, as the bend of a road, &c., may at once be determined, by setting the instrument up at the place, and turning the table about till the compass has the same bearing as at any one of the stations. Clamp the table firm, and it will now be parallel to its former position, if no local attraction prevents the magnetic needle from assuming its natural position at the difi'erent stations. Eix a needle in the point representing one of the sta- tions, and resting the edge of the index against it, move the index till the station itself is seen through both the sights, and then draw a line on that part of the paper where the point is likely to fall. Ee- E 26 move the needle to another point or station on the board, and resting the index against it, direct the sights to the corresponding station on the ground, and draw a line along the edge of the index ; the point where this line intersects the last, will be the situation on the paper of the place of the observer. But, as a check upon the accuracy of the work, a third, or even a fourth line should be drawn in a similar manner in the direction of other fixed points, and they ought, also, to intersect in the same point. In this manner, the plane-table may be employed for filling in the details of a map ; setting it up at the most remarkable spots, and sketching by the eye what is not necessary should be more particularly determined, the paper will gradually become a repre- sentation of the country to be surveyed. The following problem, extracted from AVaugh's " Instructions for Topographical Surveying," may be found useful: — To fix a plane table in position at an unknown point x, by means of three points A,B,C, whose positions are laid down on the plane table, and represented by a,i,c, respectively. Pix a pin in the point h on the plane table, and placing the ruler against it and the point a, with the object and sight towards a {vide Fig. 1,) ; turn the table about, until the point A is intersected, then clamping the table in this position, turn the ruler and inter- sect the point Q, with the edge of the ruler still against the pin at I, and draw the line hn — now remove the pin to the point a, and unclamp the table — place the ruler agaiust the pin at a, and the point h, and turn about the table until the point B is intersect- ed, {vide Fig. 2,) ; clamp the table again and having intersected the point C as before, draw the line an through the intersection p of the line an and hm ; draAv the line cpq^ passing through the point c, and placing the edge of the ruler against this line, unclamp the table once more, and turn it about until the point C is intersected {vide Fig. 3,) ; now clamp the table, and it will be in position, and the unknown point co will be situated on the line cpq^ ; to find 28 tins point it is merely necessary to intersect either of the points as A, and draw the line Aax, and the accuracy of the operation is tested by intersecting the other point B and drawing the line JBlx which should intersect the line Aax^ on the line c^oq^ thus giving the position of x on this line. The demonstration of this problem is evident to those acquaint- ed with the same problem in Plane Trigonometry, it only remains to be remarked, that when the point C with regard to the point x is situated on the other side of the line AB, or below it, the lines an and hn will intersect on the opposite side of the line db, to that on which c is ; and if the point x be situated within the tri- angle ABC, these lines (an and hn) will diverge instead of con- verge, in which case they must be prolonged on the opposite direc- tion, until they intersect for the pointy. N.B. — The accuracy of the result depends on the length of the line cp. 29 CHAPTER III. THE YEENIEE. The Vernier is a contrivance for measuring oiF parts of the space between the equidistant divisions on the limb of a divided circle, arc, or any graduated scale ; it obtains this object by measuring the differences between the divisions of two approximating scales, one of which is fixed, and called the primary scale, the other moveable, and called, the vernier. If a number of parts equal to ^ — 1,* be taken from the primary scale, and a space equal to this be transferred to the moveable scale, and divided into n parts, these parts will each be smaller than the first by the n^^ part of a division on the primary scale. For let a = length of a division on the primary scale, h = length of a division on the moveable scale. Then by hypothesis {n — V) a =z nh ov 71 a — a =. nh and a — - n a 7 0. That is, h a division on the moveable, is smaller than a a division on the fixed scale, by the ;^^'^ part of a. This is the principle of the vernier. * Islote. — All this applies equally if w + 1 parts be taken, but it is more usual to take » — 1. 30 Suppose you have a theodolite, the horizontal circle of which is divided into 360°, and each degree into half degrees, or 30 minutes, and you want to construct a vernier to read to one minute. Y 30 25 20 15 10 5 ^ TTT-J i'4. I's 1*2 A I'b -9'- «'• v'- el- si- *!• a'- ei- Take the length of 29 half-degrees on the horizontal circle, and divide that distance into 30 equal parts ; this forms the vernier, which is marked for convenience, 5, 10, 15, 20, 25, 30, from one end, or the zero. Set the vernier, which is so constructed as to slide evenly along the graduated limb of the instrument, to the horizon- tal circle, so that zero of the former may be in contact with zero of the circle ; then the last division, marked 30, of the vernier will, of course, agree with 14iJ®, or 29 half-degrees of the circle, and the proportion of each division of the vernier, will be to a division of the circle, as 29 to 30. If the zero of the vernier be moved from the zero of the circle, then the first coincidence that takes place between a division of the vernier with one on the circle, indicates the number of minutes passed over. To read off an angle on the horizontal circle, use a magnifying glass, and notice how many de- grees have been passed over by the zero of the vernier. Eor example : Let us suppose that the arrow at zero of the vernier has passed the 21st degree of the circle : then, for the number of minutes in addi- tion, look along the vernier, until one of its divisions is found to agree exactly with a division on the circle below it : we will sup- pose, that the 14ith division of the vernier does so : then the angle is21M4'. I will give an example to show how to find the proper number of divisions for the vernier scale. Suppose the primary scale is divided to 10 minutes, and I wish to construct a vernier to read 10 seconds. 31 Then using the same letters as before, or 60a — 60d = a or 59 « = 60 5 That is I must take 59 divisions from the primary scale, and divide them into 60 parts for the vernier. THE THEODOLITE. As an angular' instrument the Theodolite has from time to time received such improvements, that it may now be considered the most important one employed in surveying. They are of various modes of construction, but it will only be necessary here to de- scribe the two patterns in general use on the Indian surveys. Description of the Y Theodolite. This instrument (as represented in the next pagej , consists of two circular plates, A and B, called the horizontal limb, the upper, or vernier plate. A, turning freely upon the lower, both having a horizontal motion by means of the vertical axis, C. This axis consists of two parts, external and in- ternal, the former secured to the graduated limb, B, and the latter to the vernier plate, A. Their form is conical, nicely fitted and ground into each other, having an easy and a very steady motion ; the external centre also fits into a ball at D, and the parts are held together by a screw at the lower end of the internal axis. The diameter of the lower plate is greater than that of the upper one, and its edge is chamfered off and covered with silver, to receive the graduations : on opposite parts of the edge of the upper plate, or 180° apart, a short space, «, is also chamfered, forming with the edge of the lower plate a continued inclined plane : these spaces are, likewise, covered with silver, and form the verniers. The lower limb is usually graduated to 30 minutes of a degree, and it is sub- divided by the vernier to single minutes, which being read off by 32 the maguifjing glass, E, half or even quarter minutes can easily be estimated. The parallel plates, F tiud Gr, are held together by a ball and socket at D, and are set firm and parallel to each other by four milled-headed screws,* three of which, b, 5, i, are shown in the figure ; these turn in sockets fixed to the lower plate, while their * Nearly all Thaodolites are now made with three foot-screws, as hereafter described for the Everest's Theodolite. Thej^ possess great advantages over the old parallel plate screws, which can only be appreciated by a person who has used both. :33 heads press against the under side of the upper plate, and being set in pairs, opposite each other, they act in contrary directions ; the instrument by this means is set up level for observation. Beneath the parallel plates is a female screw adapted to the staff head, which is connected by brass joints to three mahogany legs, so constracted, that when shut up, they form one round staff secured in that form for carriage, by rings put on them ; and when opened out they make a very firm stand, be the ground ever so uneven. The lower horizontal limb can be fixed in any position, by tight- ening the clamping screw, H, which causes the collar, c, to embrace the axis, C, and prevent its moving ; but, it being requisite that it should be fixed in some precise position more exactly than can be done by the hand alone, the whole instrument, when thus clamped, can be moved any small quantity by means of the tangent screw, I, which is attached to the upper parallel plate. In like manner, the upper or vernier plate can be fixed to the lower, in any position, by a clamp, which is also furnished with a tangent-screw. The motion of this limb, and of the vertical arc, hereafter to be described, is sometimes effected by a rack and pinion ; but this is greatly inferior where delicacy is required, to the slow motion produced by the clamp and tangent-screw. Upon the plane of the vernier plate, two spirit levels, d, d^ are placed at right angles to each other, with their proper adjusting screws : their use is to determine when the horizontal limb is set level : a compass also is placed at J. The frames, K and L, support the pivots of the horizontal axis of the vertical arc (or semicircle), ]M, on which the telescope is placed. The arm which bears the microscope, N, for reading the altitudes or depressions, measured by the semicircle, and denoted by the vernier, e, has a motion of several degrees between the bars of the frame, K, and can be moved before the face of the vernier for read- ing it off. Another arm clamps the opposite end of the horizontal axis by turning tlie screw, 0, and has a tangent-screw at P, by 1-' 34 whicli the vertical arc and telescope are moved very small quantities up or down, to perfect the contact when an observation is made. One side of the vertical arc is inlaid with silver, and divided to single minutes by the help of its vernier ; and the other side shews the difference between the hjpothenuse and base of a right-angled triangle, or the number of links to be deducted from each chain's length, in measuring up or down an inclined plane, to reduce it to the horizontal measure. The level, which is shown under and parallel to the telescope, is attached to it at one end by a joint, and, at the other, by a capstan-headed screw, f, which, being raised or lowered, will set the level parallel to the optical axis of the teles- cope, or line of collimation ; the screw, g, at the opposite end, is to adjust it laterally, for trus parallelism in this respect. The teles- cope has two collars, or rings, of bell metal, groimd truly cylindrical, on which it rests in its supports, 7^, Ti, called Y's, from their resem- blance to that letter ; and it is confined in its place by the clips, ^, i, which may be opened by removing the pins, j, y, for the purpose of reversing the telescope, or allowing it a circular motion round its axis, during the adjustment. In the focus of the eye-glass are placed three lines, formed of spider's web, one horizontal, and two crossing it, so as to include a small angle between them ; a method of fixing the wires which is better than having one perpendicular wire, because an object at a dis- tance can be made to bisect the said small angle with more certainty that it can be bisected by a vertical wire. The screws adjusting the cross wires are shown at m : there are four of these screws, two of which are placed opposite each other, and at right angles to the other two, so that by easing one and tightening the opposite one of each pair, the intersection of the cross wires may be placed in ad- justment. The object-glass is thrust outwards by turning the milled-head, Q, on the side of the telescope, that being the means of adjusting it to shew an object distinctly. 35 A brass plummet and line are packed in the box with the The- odolite, to suspend from a hook under its centre, by which it can be placed exactly over the station from whence the observations are to be taken : likewise, if required, two extra eye-pieces for the teles- cope, to be used for astronomical observations ; the one inverts the object, and has a greater magnifying power, but having fewer glasses possesses more light ; the other is a diagonal eye-piece, which will be found extremely convenient when observing an object that has a considerable altitude, the observer avoiding the unpleasant and painful position he must assume, in order to look through the teles- cope when either of the other eye-pieces is applied. A small cap containing a dark-coloured glass is made to apply to the eye end of the telescope, to screen the eye of the observer from the intensity of the sun's rays, when that is the object under observation. A magnifying glass mounted in a horn frame, a screw driver, and a pin to turn the capstan-screws, for the adjustments, are also furnished with the instrument. The adjustments. 1st. TJw line of collimation, — The line of collimation in a Telescope is an imaginary line joining the centres of the object and eye-glasses ; it is evident, therefore, that the situation of this line holds a fixed relation to the tube and its appendages, so long as the object and eye-glasses maintain their fixity ; it is obvious then that, technically speaking, to speak of correcting the line of collimation is wrong ; what we ivisli to do is to bring the intersection of the cross wires into this line of colli- mation ; what we really do efi'ect, by^the following adjustment, is to bring the intersection of the cross wires into the line joining the centres of the Y's, which, if the instrument be properly made, should coincide with the line or axis of collimation. To do this, fix the intersection of the cross wires on some well defined distant object, turn the telescope half round in the Y's, and if the in- tersection of the cross wires still remains on the object, the instru- ment does not require adjustment in this respect ; but, if the Y 2 •36 horizontal wire be above or below the object, bring it back again to the point, half by means of the diaphragm screws, above and below the telescope, by first loosening one and then screwing up the other, and half by the tangent screw of the vertical arc, turn the telescope again half round in the Y's, and, if again off the object, repeat the operation till correct ; in the same way if on turning the telescope half round the vertical wire be to the right or left of the object, proceed in the same manner, moving half by the diaphragm screws at the side, and half by the tangent screw of the horizontal limb, and repeat till perfect. The intersection of the wires will now remain on the object during a complete revolution of the telescope in the Y's. Ind. The upper level. — The object of this adjustment is to set the bubble tube under the telescope parallel to the rectified line of collimation, (or what has been shewn to be the same thing, the line joining the centres of the Y's,) open the clips of the Y's, and bring the bubble to the centre of its run by the tangent screw of the vertical arc, now reverse the telescope in its Y's, turning it end for end, and if the bubble still remain in the centre, it is in ad- justment*; but, if not, bring it back to the centre, half by the adjusting screw at one end of the bubble, and half by the tangent screw of the vertical arc. E-everse the telescope again, and if still not correct, the operation must be repeated until it is so. ^rd. Verticality of the axis. — The third adjustment is to set the bubbles attached to the horizontal plate, so that when in the centre of their run, they shall indicate the verticality of the axis, and, therefore, the horizontality of this plate. To do this having placed * The proof of this, viz : That on inverting the telescope, if the bubble re- main in the centre, it is parallel ; and that if it goes to one end or the other, it is not parallel to the line joining the centres of the Y's, is as follows: — suppose them to be parallel, then, since the bubble is in the centre of its ran the tube is horizon- tal ; and, therefore, the aforesaid line is horizontal. Now, on reversing, if the bubble is out of the centre, the tube is not horizontal, but the line joining the axis of the Y's has not been moved and is still horizontal ; therefore, our first supposi- tion that these two were parallel is not true ; but, on reversing, did the bubble again come to the centre, our supposition must be true. 37 the telescope parallel to two of the foot screws, bring the 'upper bubble to the centre of its ran, turn the instrument half round on its axis, and if the bubble be not still in the centre, bring it back half by the tangent screw of the vertical arc and half by the foot screws, repeat this until it remains in the centre both "ways, then turn the telescope a quarter round, bringing it over the third foot screw*, and by it bring the* bubble to the centre, the axis will now be vertical: to complete the adjustment, bring the bubbles attached to the horizontal plate to the centre of their runs by means of the capstan-headed screws at their ends. There only remains to deter- mine the zero of altitude ; when the axis has been made vertical and the telescope is horizontal, the vernier of the vertical arc should read zero, if it does not, the vernier plate may be unscrewed and placed so as to do so, but the better way is to note as an index error whatever the vernier may read and to apply it to all vertical angles. The other description of Theodolite used in India was designed by Sir G-eorge Everest, late of the Bengal Artillery and Surveyor Greneral of India ; it is known as " Everest's Pattern." There are two patterns, the double arc and the single are, both of which with their adjustments, will be here described. The accompanying diagram represents an instrument of the first kind. The horizontal circle, or limb, A, of this instrument consists of one plate only, which, as usual, is graduated at its circumference. The index is formed with four radiating bars, a, h, c, d, having ver- niers at the extremities of three of them, marked. A, B, and C, for reading the horizontal angles, and the fourth carries a clamp, e, to fasten the index to the edge of the horizontal limb, and a tangent- screw, f, for slow motion. These are connected with the upper works which carry the telescope, and turning upon the same centre * Or if an instrument with four foot screws, over the two remaining ones. 38 shdWany angle through which the telescope has been moved. The instrument has also the power of repeating the measurement of an angle ; for the horizontal limb being firmly fixed to a centre, move- able within the tripod support, E,, and governed by a clamp and tangent screw, s, can be moved with the same delicacy, and secured with as much firmness, as the index above it. The tripod support, which forms the stand of the instrument, has a foot- screw at each extremity of the arms which form the tripod ; the heads of the foot-screws are turned downwards, and have a flange (or shoulder) upon them, so that when they rest upon the triangular plate fixed upon the stafi'-head, another plate locks over the flange, and being acted upon by a spring, retains the whole in- strument firmly upon the top of the staff". The advantage of the tripod stand is, that it can easily be disengaged from the top of the staff", and placed upon a parapet or other support, in situations where the staff cannot be used. The telescope is mounted in the following manner.— The horizon- tal axis, L, and the telescope, M, form one piece, the axis crossing the telescope about its middle, and terminating at each extremity 89 in a cylindrical pivot. The pivots rest upon low supports, (only one of them, D, being visible in the figure,) carried out from the centre, on each side, by a flat horizontal bar, P, to which a spirit- level, Gr, is attached for adjusting the axis to the horizontal plane. The vertical angles are read off on two arcs of circles, H, H, which have the horizontal axis at their centre, and being attached to the telescope, move with it in a vertical plane. An index, upon the same centre, carries two verniers, I, I, and has a spirit-level, K, attached to it, by which it can be set in a horizontal position, so that whatever position the telescope, and consequently the graduated arcs, may have, when an observation is made, the mean of the two readings will denote the elevation or depression of the object observed, from the horizontal plane. On the upper part of the telescope is fixed a narrow box, containing a magnetic needle, for observing the bearings of objects. Adjustments. 1st. The loiver level. — Turn the telescope so that the lower level may be parallel to two foot screws, and by their motion bring the bubble to the centre of its run. If it remain so, on turning the telescope half round, the level is correct : but if not, the tube carrying the spirit glass is not perpendicular to the axis of motion, or, in other words, is not parallel to the horizontal plate. To remove this error, the bubble must be brought to the middle, half by the foot screws, and half by the adjusting screws at the ends of the level. Having perfected this, the levelling of the instrument is completed by turning the telescope a quarter round, so that one end of the level may be over the third foot screw, by which the bubble is to be brought to the centre of its run. 2nd. The line of Collimation in Azimuth. — Having levelled the instrument as above, and having intersected some distant and well defined object with the cross hairs of the telescope, clamp the horizontal plates. Invert the telescope, and if the object be still intersected, the collimation is perfect. If not, correct half the error* in azimuth by the screws that give horizontal motion to the 40 diaphragm carrying the cross hairs, and the other half by the tangent screw, giving motion in azimuth to the instrument. Bepeafc till perfect. Two of the four screivs, ty icliicli the diapliragm is generally secured in the telescope, are only necessary therefore for this adjustment. Srd. The Zero of Altitude. — The above operations having been performed, bring the bubble of the upper level to the centre of its run, by the screws which retain the index, intersect an object and take the mean of the readings of the vertical arcs. Having inverted the telescope, and brought the bubble of the upper level to the middle of its run, obtain , in a similar manner, a second mean. Set the verniers of the vertical arcs so that the mean of the readiiigs shall equal the mean of the two means just found, and cause the cross hairs to intersect the object by means of the screws that retain the index, the bubble of the upper level, being thus thrown out, must be brought back to tlie centre of its run, by the adjusting screws at its extremities. E-epeat till perfect. The second kind differs from the Double Arc Theodolite, as its name implies, chiefly from having only a single arc ; on this account, vertical readings can only be taken on one face of the in- strument, for on inverting the telescope (by turning the axis end for end) and bringing the screws whicli retain the index under- neath, the index and the arc will be found to be on opposite sides of the axis. In other respe(;ts the instrument differs little from the double arc ; the single arc usually has only two, instead of three, verniers to read the horizontal plate ; and the level attached to the vertical index is fixed by the maker, and does not therefore admit of adjustment. The first two adjustments iu this instrument are similar to those for the double arc. ^rd. The Zero of Altitude. — As before explained vertical read- ings cannot be taken with inverted positions of the telescope, it 41 therefore becomes necessary to use other means than those adopted with the Double Arc Everest, to fix the zero of altitude. Set up the instrument, and having levelled it, send a levelling staff with the vane set to the height of the axis of the telescope to any convenient distance, and having taken the elevation or depres- sion of the vane, change the places of the instrument and staff, level the former and set the vane of the latter to the present height of the telescope — again, take the reading of the vane. Now, the mean of this and the former reading will give the true inclination of one place to the other. Set the vernier to this mean, and by means of the screws which hold the index, bring the telescope to bear on the vane. This should complete the adjustment, but to see that the operation has been gone through correctly, it is ad- visable to repeat the process. THE METHOD OE 0BSEEYI:N"G WITH THE THEODOLITE. To level the Instrument. — The instrument being placed exactly over the station from whence the angles are to be taken, by means of the plumb-line suspended from its centre, it must fir^t be set approximately level by the legs of the stand, and then the levelling must be completed by the foot screws, Q, Q, Q, thus ; place the level, Gr, in a direction parallel to two of the foot screws, and by their motion, turning them both inwards, or both outwards, ac- cording as you want the bubble to go to the right or left, bring it to the centre of the tube : then turn the instrument a quarter round, so as to bring the bubble over the third foot-screw, and turn it to the right or left, until the bubble also becomes station- ary in the middle in this position. In performing this last opera- tion, the level, will be perhaps, thrown out on moving the instru- ment back to the first position, in Avhich case it must be again l-evelled. "When the bubble remains stationary in the middle G 42 during a whole revolution, the instrument is ready for obser\'a- tion. When you wish to intersect any ohjecfc, first move out the eye piece u.ntil you can see the cross wires distinctly, then by means of the focussing screw move the object-glass out until on moving the eye about, the image of the object ceases to move off the inter- section, of the wires and to have a fluttering, and undefined appear- ance. A picture of the field of view is formed in the telescope, and until it coincides with the plane of the wires this motion will take place ; just as when looking out of a window, the position of the bars with respect to the landscape change according as we change our position ; but if the landscape was painted on the glass (i. e., was in the same plane as the bars) this would not be the case. By moving the object-glass in and out, or as it is called focussing, we move the imaginary picture backwards and forwards in the telescope until it coincides with the plane of the wires, when there can be no mo- tion ; and the eye glass, which is merely a magnifier, having been set to see the wires distinctly, will also shew the picture so. The above motion is called par allajc, which may be defined as the apparent an- gular motion of an object arising from change of our point of view. To observe an Angle. — By means of the clamp, ^, and tangent- screw, y, (see last fig.) set the vernier marked. A, to 360° ; then turn the limb round, and with the lower clamp and tangent-screw, s, fix the cross-wires in the telescope on any object. Then loosen the upper clamp, e, and turn the upper limb round, fixing the cross- wires by the same clamp and tangent-screw on any other object ; the angle subtended can be then read 05" on the instrument. Another Iletliod. — Clamp the lower horizontal limb firmly in any position, and direct the telescope to one of the objects to be observed, moving it till the cross-wires and object coincide ; then clamp the upper limb, and by its tangent-screw make the intersection of the wires nicely bisect the object ; now read ofi" the two verniers, the degrees, minutes, and seconds of (either) one, which call A, and the 43 minutes and seconds only of the other, which call B, and take the mean of the readings, thus : A=142^ 36' 30* B= „ 37 Mean=142 36 45 'Next release the upper plate, and move it round until the telescope is directed to the second object, (whose angular distance from the first is required,) and clamping it, make the cross-wires bisect this object, as was done by the first ; again read off the two verniers, and the difference between their mean, and the mean of the first reading, will be the angle required. To repeat an Angle. — Leave the upper plate clamped to the lower, and release the clamp of the latter ; now move the whole instrument (bodily) round towards the first object, till the cross-wires are in contact with it ; then clamp the lower plate firm, and make the bi- section with the lower tangent-screw. Leaving it thus, release the upper plate, and turn the telescope towards the second object, and again bisect it by the clamp and slow motion of the upper plate. This will complete one repetition, and if read off, the difference between this, and the first reading will be double the real angle. It is, however, best to repeat an angle four or five times ; then the difference between the first and last readings (which are all that it is necessary to note) divided by the number of repetitions will be the angle required. The magnetic bearing of an object is taken, by simply reading the angle pointed out by the compass-needle, when the object is bi- sected : but it may be obtained a little more accurately by moving the upper plate (the lower one being clamped) till the needle reads zero, at the same time reading off the horizontal limb ; then turning the upper plate about, bisect the object and read again ; the difter- ence between this reading and the former will be the bearing re- quired. G 2 44 In taking angles of elevation or depression, it is scarcely necessary to add, that the object must be bisected bj the horizontal ^Yire, or rather by the intersection of the wires, and that, after observing the angle with the telescope in its natural position, it should be repeated with the telescope turned half round in its Y's, that is, with the level uppermost : the mean of the two measures will neutralize the effect of any error that may exist in the line of colli- mation. The altitude and azimuth of a celestial object may likewise be observed with the theodolite, the former being merely the elevation of the object taken upon the vertical arc, and the latter, its hori- zontal angular distance from the meridian. We here suggest a few hints on the use of these delicate instru- ments. 1st. They must not be handled roughly. In taking them in and out of the box, it should be done with the greatest care, not knock- ing them against the sides of the box or forcing them into their positions within it ; the boxes are so constructed, that the instru- ment fits exactly into its own place, and unless it settles down of itself, forcing it will throw the instrument out of adjustment. 2nd. J^ever permit a Native Surveyor to apply oil to any part of the instrument, under the idea that it will work easier ; a new in- strument will perhaps work stiff at first ^ but a very few days' use will rectify it, the application of oil is nothing but a restmg place for dust that is always flying about in the field : this dust works up into the various screws, wears them, and at the end of six months the instrument requires repair, or is next to useless ; if oil be neces- sary, it should be applied by the assistant, and then wiped off as dry as possible. 3rd. Always throw the needle off its centre by the stop fixed on one side of the box, when the instrument is not in use, as the constant playing of the needle wears the pivot upon which it is balanced; and on the fineness of this point depends the accuracy of 45 the bearing. This is equally applicable to the Prismatic Compass and Circumferentor. 4th. Always wipe the dust off the instrument on commencing and finishing a day's work, with a camel hair brush, as this will tend to prevent any accumulation of dirt about it : a Surveyor should partly be judged of by the state of his instruments, 5th. "When once the variation of the needle is ascertained, never remove the box from off the telescope, for unless it be screwed on again, in the exact position it originally was, the variation of the needle will apparently alter. 6th. On the care a Surveyor takes of his Theodolite, depends much of the accuracy of his work ; if he neglect and be careless about the former, he will one day have to lament over the accumu- lated errors of the latter. A survey of roads, &c., can be made with the Theodolite in the same way as that already described with the prismatic compass, and much more accurately, as the smallest Theodolites read to one minute of a degree, also another source of error is avoided, viz., that likely to occur from taking every angle with the magnetic meri- dian, the variation of which from the true meridian is not the same at different places, and in the same place at different times. la making such a survey wdth the Theodolite, called traversing, we should proceed precisely in the same manner as that previously laid down for the prismatic compass, the only difference being in the method of taking the angles which we will endeavour to explain. The Theodolite must be set up at the first station and levelled by means of the foot screws, the upper and lower horizontal plates must then be clamped at zero, and the whole instrument turned about until the magnetic needle steadily points to the NS line of the compass box, and then fixed in that position by tightening the clamping screw of the lower plate. iS'ow release the upper plate and direct the telescope to any objects that it may be advantageous to fix by intersection without 46 direct measurement to them, and having intersected them with the cross wires note the readings of the vernier in your Field-book, taking care to read from the same vernier as was before set to zero ; lastly, take the angle to your forward station where a staff* must be held for the purpose ; now leaving the horizontal plate clamped at that angle move your instrument to your second station, the dis- tance between the two must be measured and offsets taken. Arrived at the 2nd station, by means of the plumb-line place the theodolite over the mark where the staff was, and having levelled it, unclamp the bottom plate and with the vernier still at the last for- ward reading, turn the instrument bodily round and intersect the staff placed at the first station which is now the back station ; again tighten the clamp-screw and the instrument is now fixed in the same relative position as it was at the first station, and the upper plate may now be released to take angles to any conspicuous objects and to read the forward angle to station 3. To see that the above has been gone through correctly, after releasing the upper plate set the index to zero and the compass should, as in the first instance, coin- cide with the NS line of the compass box ; if not, an error has been committed in taking the last forward angle, or else the plate must have been moved from its position before the back station had been bisected : when this is the case it is necessary to return and exa- mine the work at the last station. If this be done every time the instrument is set up, a constant check is kept upon the progress of the work ; and this indeed is the most important use of the compass. Having thus proved the accuracy of the last forward angle, the an- gles from the 2nd station may be taken. At the 3rd, and every succeeding station, a similar operation must be performed, bisecting the back station with the instrument reading the last forward an- gle ; then the bearings taken to any conspicuous objects, and lastly the forward angle must be measured. After having fixed the teles- • A staff should be intersected as near the ground as possible, this prevents an error occurring if the staff be not placed perpendicular in the ground. 47 cope on the back station by clamping the lower plate, great care must be taken to prevent it from having the least motion whilst taking the other angle. Objects already fixed by intersections should continue to be observed to, so long as they are in view, as they serve as checks on the accuracy of the work. Prom this method of traversing, it will be easily seen that the angles taken at every other station, that is to say the 2nd, 4th, 6th, &c., are 180^ out, and require that number to be added to or subtracted from them (according as they are less or greater than 180°) before they can be plotted. The reason of this is obvious, at the 2nd station when we intersect the back station with the vernier at the last forward reading, the NS line of the compass box becomes parallel to the position that line occupied at the 1st station, but the N" now points where S did, and therefore the position of the theo- dolite is 180^ out. At the 3rd station it will be 180® out, from what it was at the second, and therefore in the same position as at the first. If the relative situations of some conspicuous points were pre- viously fixed by triangulation, there would be no necessity to have recourse to the magnetic meridian at all, as a line connecting the starting point with any visible fixed object, may be assumed as a working meridian ; the reading of the compass needle, should be noted at the first station when any such fixed object is bisected, the vernier of the horizontal plate reading zero, then at every succeed- ing station, upon the theodolite being set to zero, the compass needle should read the same as at the first. Indeed, even if no such points be fixed, it is better to use a line through your first station, and any conspicuous object as a working meridian, in preference to using the magnetic and if no object be available it would be better to lay down a referring mark. Eor, as before explained, the position of the North is continually varying, so that suppose you want to go over your work again or to start from the same place in a difierent direction, with the same meridian, having used the mag- 48 netic, you cannot be certain of doing so with any degree of cor- rectness. The method of plotting described for the compass survey is liable to some inaccuracies of practice, on account of having a new meri- dian for every particular angle to be laid down, and on account of laying off every new line from the point of termination of the pre- ceding one, whereby any little inaccuracy that may happen in laying down one line is communicated to the rest. As the angles taken by the theodolite are so much more accurate than those taken by the compass, it is as well to employ a more accurate means of laying them down, and this is gained to some extent ^by using the circular card protractor. This consists of a circle divided to quar- ter degrees, marked out on a square card, the centre part of which is cut out for the purpose of carrying on the protraction within it. The circle is numbered in two rows, like numbers being opposite to one another, i. e., the zero of the second row begins at the 180" of the first. The method of using this protractor is as follows : — A line having been taken to represent the meridian, place the protrac- tor on the paper with the zero and 180° points coinciding with it, and fix it in this position with weights or drawing pins to prevent its shifting, now apply a parallel ruler to the degree corresponding to the angle which it is wished to plot, one end coinciding with the number on the first row, and the other with the like number in the second row, on the opposite side of the circle, then slide the rule up to the point from which the angle is to be plotted and draw a line, which will represent the direction of the required angle. As soon as you have plotted to the extent permitted by the space cut out in the protractor, you must remove the protractor, draw a new meridian parallel to the first and apply the protractor to it as before. This method saves the trouble of shifting the protractor at every bearing, and also ensures greater accuracy in plotting, as a great number of bearings being laid down from one meridian, a trifling 49 error in the direction of one line does not affect tlie next ; the -accn- racy of the plot, howeyer, depends much upon using a ruler that moves truly parallel, which it is well to look to before using this method of plotting. A still more accurate method of plotting is by the traverse system, which will be explained in the next chapter. H 50 CHAPTER IV. THE TRAYEESE SYSTEM. A TEAYERSE may be defined as a circuitous route performed ou leaving any place on the earth's surface, by stages, iu different directions, and of various lengths, with a yIcav of arriving at any other place situated in any direction with reference to the former, and at any distance therefrom which cannot be reached in the direction of the shortest line connecting them. The angles which the stages or station lines form with the meridian are called '' bear- ings" the quantity of Northing or Southing made in each distance , is called the difference of latitude, and the amount of Easting or Westing is termed the dej^arture, . When the bearing corresponds with the meridian, or with the perpendicular to it, there will in the former case be nothing but difference of latitude, and in the latter nothing but difference of departure, and the distance measured will itself, express the amount of Northing or Southing, or of Easting or Westing due to the change of position. The perpendicular to the meridian coincides, at first, with the small circle of latitude. When the distances are great, the deviation of these two becomes sensible, being the differ- ence between the base and hypothenuse of a right-angled spheric triangle. In ordinary survey work the difference is scarcely sensible. When, hoAvever, the bearing does not correspond with the meri- dian or with the perpendicular to it, there will be for every dis- tance measured a certain corresponding change both in latitude 51 and longitude (or departure) ; and as these will, with reference to their particular distance, answer the condition of our definition, they may, with propriety, be termed the traverses of the distances : "We will therefore define : 1st. Meridians as North and South lines, which are supposed to pass through every station of a survey, running parallel to each other.* 2nd. The difference of latitude or the Northing or Southing of any line, as the distance that one end of a line is North or South from the other end. 3rd. The departure of any line, as the perpendicular distance from one end of the line to a meridian passing through the other end. It is proved in Euclid I. 32. Cor. I. " that all the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides," or, in other words, that — In any rectilineal Jigure, the sum of all the interior angles, is equal to twice as many right angles as tlie figure has sides, less four right angles. The Traverse System is a method of computation by rectangular co-ordinates, and is applicable to any mode of surveying whatever such as Honte Surveys, Kail way Lines, Navigation Courses and the like, where every station is fixed by the distances on the meridian and perpendicular, and this is essential to Gale's System, which may be termed the periphery measuring or perimetrical method. By throwing a series of angles over the face of a country, and forming a network of large circuits, the liability to error is reduced witliin the narrowest limits, which the means at disposal permit. This angular Circuit System, in extensiye operations in a country like * Tliese meridians are not really parallel, but converge towards the poles of tlie earth, but so insensibly as not to be worthy the notice of a Surveyor's opt-rations within a very linnited space. In extended operations, however, as in India, w'herc whole provinces come within the Traverse System of the Revenue Survey, the paral- lelism of the meridians must be preserved, to carry out in practice the accuracy of the above Thcoiem. fi 2 52 India, with instruments of the best construction and moderate power and size, can alone enahle a surveyor to carry out in prac- tice the theoretical accuracy of the Traverse, and permit, by the aid of logarithmic calculation, an approximation to the proof req[uired, viz : — 1st. "That the sums of all the interior angles shall be equal to twice as many right angles as the figure has sides, less four right angles," and — 2nd. As regards the linear measurements, " That the sums of the Northings be equal to the sums of the Southings, and the sums of the Eastings be equal to the sums of the Westings ; which latter wiU be presently explained. However correctly distances may be measured, unless the angular work is also correct, the result will be unsatisfactory, but with both these data accurately determined, the proof will be certain, 53 and it will be observed, how admirably each step in the work proves the other, and what confidence the system gives to a sur- veyor who has no need whatever to put any of his work on paper, but with his Traverse correct, may produce his map at any future period with undoubted certainty. "We will now proceed to explain the mode of Surveying by Traverse. Draw any figure such as ABCDEFGrHIJA, representing the sides of an irregular Polygon. If the Theodolite is first set up at the station A, and the interior angle JAB is observed, and then at B, observing the interior angle ABC, at C, the interior angle BCD, and so on all round the poly- gon, then will the sum of all the interior angles, JAB, ABC, BCD, &c., be equal to ten times two right angles (the figure having ten sides) less four right angles or 180^ X 10 — 360^ — 1440°. In practice it will be found that this result caunot be exactly attained, and that the sum of the angles will generally amount tO' two or three minutes more or less ; to meet this, a correction of one minute in every four or five angles, additive or substractive as the case may need, is generally necessary to obtain the result re- quired. Having thus proved the angular work correct, the next operation is to obtain the bearings of the several sides of the polygon or angles subtended by these sides with the meridian. This is either done by the magnetic needle on the theodolite or by astronomical observation, but as all should progress on the true meridian of the earth, we shall therefore treat only of true meridional bearings or angles formed by each line with the true meridian. If the Theodo- lite were adjusted in the plane af the meridian on every station of a survey, w^e should find no difficulty in obtaining the true bearing of each line, but as this would be very troublesome and next to impos- sible, it is only necessary in practice to obtain the correct bearing 5J. of the first line of a survey, from which by the assistance of the an- gular work the bearings of the other lines can be deduced. This true bearing being once established has only to be checked and corrected by similar means after every 50 or 100 square miles of country traversed, and it will be found seldom to exceed from 5 to 10 minutes of a degree from the true meridian, Avhereas, if the magnetic needle was used, an error of 15 or 30 minutes is scarcely traceable in a single ol:)servation, and where so many instruments are in use, all giving a different magnetic variation, it is plain that without this method of deducing the azimuths from the angular observations, the utmost confusion would arise. Bide. — To the bearing of the line preceding that of which the bearing is sought, add the inward angle formed by these two lines, and the sum increased or diminished by ISO"", according as it may be less than, or in excess of 180^, will be the bearing of the next line sought. Before proceeding to prove the aboA^e rule, we will first premise that, in all modern theodolites the divisions are numbered round the circle, from 0'' to 360°, so that the bearing of any object between 0° and 90° is reckoned North-East, between 90' and 180^ South-East, between 180° and 270^ South- West, and between 270° and 360° North- West, 0°, 90°, 1S0°, 270^, being respectively due North, South, East, and West. This method of reckoning the bearings of objects is by far the most convenient for practice, as without the necessity of making iise of the letters to denote the direction, the bearing is known at once by the number of degrees contained in the arc. Let the bearing of the line AB in the following figure be given, as found by astronomical observation. Ih find the learing of tlic line BC. — Produce AB to a, and CB to c. The two meridians NS, and N^S' being parallel, the angle WBa is e(iual to the angle NAB, if to the anolc ^'B« or arc NV^ wc add 5^ the interior angle of the polygon ABC, or its equivalent in arc aS^c, we obtain the angle formed by the line CB with the meridian N'S^ or arc N^«S'c, if then the angle cBC or 180° be deducted from this, thus reversing the direction of the line, we have left the angle !N^BC or bearing of the line BC with the meridian ]N'^S\ To find the hearing of the line CD. — Produce BC to 6, and DC to d. The two meridians N^S' and N^S^ being parallel, the angles IS'^BC and N^C6 are equal. If to the angle Is'^Cb or arc Wh, we add the interior angle of the polygon BCD, or its equivalent in arc ftS'J, we obtain the angle formed by the line DC with the meridian IS'^S^ or arc Wh^-d, if tlien the angle dCJ) or 180° be deducted from 5G this, thus reversing the direction of the line, we have left the angle WCD or bearing of the line CD with the meridian WB\ To find the hearinj of the line BE. — Produce CD to e, and ED to/. The two meridians N'S' and N^S^ being parallel, the angles N^CD and WJ)e are equal. If to the angle WDe or arc N^d we add the interior angle of the xjolygon CDE, or its equivalent in arc ffi we obtain the angle formed by the line ED with the meridian N'S=5 or arc ^^efi if then the angle /DE or 180° be added to this, thus reversing the direction of the line, we obtain the angle N^DE or bearing of the line DE with the meridian N^S^ To find the leaving of the line EF. — Produce DE to y, and FE to h. The two meridians N^S^ and N^S* being parallel, the angles, N^DE and WE^g are equal. If to the angle WEg or arc NViS*^" we add the interior angle of the polygon DEE or its equivalent in arc gl^^h, we obtain the angle formed by the line EE, with the meridian JST^S^ or arc N*AS*^T^Vi from which if we deduct the angle 7iEF or 180^, thus reversing the direction of the line we have left, the ungle N*EE or bearing of the line EE with the meridian N*S*. And so on, this proof may be carried through every line of the polygon, until we come to the last line JA, when its bearing added to the interior angle JAB -J- or — 180*, as the case may require, will give the original starting bearing of the line AB. "We have been thus prolix in explaining how the bearings of the above four lines of the polygon are obtained, as they contain cases in each quadrant of the circle, BO, being a South-East bear- ing, CD, North-East, DE, North- AVest, and EF, South- West ; the same rule is however applicable to the remaining sides.* * If tlie sum of the preceding bearing and forward angle after deducting 180° amounts to more than 360", deduct 360° from the total, the remainder will be the bearing of the next line. 57 THE PROOF OF THE TEA.VERSE, AMOITI^T OF EREOR ALLOWED, A:XD METHOD OF CORRECTIOI^. We now offer for consideration the following Theorem, viz. : That in every Survey, correctly taken y the sum of the distance gone North from a certain ^oint, will he equal to the sum of the distances returned South to the same point, and that the sum of the distances gone East, will he equal to the sum of the distances returned West. The truth of the above is self-evident, for the meridians within the limits of an ordinary Survey having no sensible difference from parallelism, it muBt necessarily follow, that if a person travel any way soever within such limits, and at length come round to the place where he set out, he must have travelled as far to the ISTorth as to the South, and to the East as to the "West, though the prac- tical surveyor will always find it difficult to make his work close with this perfect degree of exactness. We will, however, explain this more fully with the assistance of a diagram. Let the line NS run due North and South, and EAV due East and West. If we fix on the point A, as a starting point, and a person walks from A to h, on the line NS, say 400 yards, and wishes to return to A, he must walk back 400 yards ; in going therefore from A to h, and back from h to A, he has walked 400 yards North and returned 400 yards South. In the same manner if he fixes on the point 5 as a starting point, and walks to 13 on the line EW, say 300 yards to get back to h, he must return 300 yards ; in walking therefore to B, he has gone 300 yards East, and returned 300 yards West. Supposing now, he walks from A to B, say 500 yards in the direc- tion of the line AB, he will then have gone North from A, 400 yards, and East from A, 300 yards. In a continuation of the figure, having walked or measured from A to B, he proceeds on and measures from B to C, in doing so, he I 58 goes a certain distance Soutb and a certain distance East of B to arrive at C, thence he measures to D, going a ceriain distance north and east of C to arrive at D, from D he measures to E, from E to E, and so on, going north, south, east, or west, from the pre- ceding station as the direction of the line may he, until he arrives back at his original starting point A. In making this tour, there- fore, he has gone the same distance north as he has returned south, and the same distance east as he has returned west. Let the vertical and horizontal lines drawn through the several stations A, B, C, &c., represent, the former, a series of meridians or north and south lines or lines of longitude; and the latter, a series of east and west lines or lines of latitude : as these lines of latitude and longtitude are all respectively perpendicular and jmral- 59 iel to each other, it follows that the angle formed by the intersec- tion of the meridian lino of one station, and the latitudinal line of the next station as at 6, k^ Z, m^ &c., must be a. right angle or 90^. JSTow supposing all the lines AB, BC, &c., to have been carefully measured with a chain, and that haying obtained the bearing of the line AB, by astronomical observation, we have deduced the bearings of all the other lines by the rule (page 53) ; we then have the data in each line, of a side and two angles to find the other two sides. Eor instance, in the triangle ASB, we have the side AB, and the two angles 5AB, A6B, (the latter being invariably 90° or a right angle,) to find the other two sides Ah and 6B, the former being the difierence of latitude, and the latter the departure of the station B from A. In like manner, in the triangle B/tC, we have the side BC, and the two angles CB/c (obtained by deducting NBC from 180°) and 'EkQ (a right angle), to find the other two sides BA; and Cky the former being again the difference of latitude, and the latter the departure of the station C from B, and so on for every line round the figure. The object of calculating all the sides of these several right- angled triangles on each line, is to obtain the difference of latitude and departure of each station from the preceding one, which difference being found, the sums of all differences of latitude of lines going North, must equal the sums of all differencea of latitude of lines going South, or A6 + CZ + Dwi + G^ + J5 = Bh + E;^ + Fy + H^^ + Ir and the sums of all differences of departure of lines going East must equal the sums of all differences of departure of lines going "West, or IB 4- kC + ZD + rZ 4- sA = wE + ??F + yG +^H + rjl and if this is not the result of the above calculations, the Survev has net been truly taken. GO We have before stated, that in the measurement of angles, a certain correction is allowed in practice, to obtain the result of the Theorem which forms the basis of the work, so also in the measurement of Chain lines, a correction is necessary to meet the errors, that notwithstanding the greatest care, will occur. In actual practice, the columns of latitude and departure will not balance exactly, for inaccuracies must arise from observations and chaining in the field, which no care could obviate. To adjust these differences, previous to defining the meridian distances, the rule is, that should the discrepancy amount to one-fifth of a Pole or five Links for every station, it will be clear an error has been made in the field measurements, which must be discovered by a re-survey. When difierences, however, are within these limits, the amount of error allowed is one link in ten chains, additive or subtractive from the sums of the Northings and Southings to correct the lati- tude, and from the sums of the Eastings and Westings to correct the departure. This error must be apportioned among each of the distances of the Survey by the following proportions, viz. : As the sum of all the distances is to the whole error, so is each distance to its correction. This must be done for the latitudes and also for the departures, and is entered in a column appropriated to each, called the ]N"orth and South correction, and the East and West correction ; the cor- rection, thus determined, must be placed, collaterally, with the distance to which it refers, without distinguishing as to JSTorth, South, East, or West. Having found the several corrections for each of the latitudes and departures, add them together severally, and see whether their total agrees with the whole error, and if so, proceed to allot the corrections. If the error be an excess of IN'orthings, substract each correction from its collateral Northing or add it to the colla- teral Southing : if an excess of Easting, add to the Westing and 61 substract from the Easting ; the corrected sums of the corrected latitudes and departures will then be found exactly to agree. We here subjoin an example: — .2 m Bearings. Dis- tances. North. c o 9 o South, ;-. East. o o ;-! o West. ^ 1 o o E + + A 63° 45' 17-68 7-83 •040 • . • , , 15-85 •080 • • • . B 67° 45' 6-37 2-61 •014 • • • 5-6S •028 . C 47° 30' 3-S6 2-61 •008 . • • 2-S4 •016 • • • , ^ D 284° Ou' 1-4-63 3-54 •033 . . . . . . . , , 14-19 -066 E 212° 00' 19-73 . . . 16-73 .045 . . . • • 10-46 •099 + — + SuQiS, . . . 62-27 16-59 •095 16-73 16-59 •045 24-37 •124 24-65 24-37 •165 Difference, . . . • • • •14 •28 In the above, the error is -{- 'IJi in the South and + -28 in the West, we will now divide this error proportionately among the several distances, by the rule previously given, viz. : — As the sum of the distances : the whole error : : each distance : its parti- cular correction, or 62-27 : -M : : 17-68 : "040 : : 637 : -014 : : 3-86 : "008 : : 14-63 : '033 : : 1973 : -045, for the ]S'orthings and Southings. And 62-27 : -28 : : 17-68 :' '080 : : 637 : '028 : : 3*86 : '016 : : 14-63 : -066 : : lS-73 : -090 for the Eastings and Westings. It will be observed that the sum of the several corrections as above apportioned, amounts to '14 in the Xorthings and Southings or -095 + -045 = -14, and to -28 in the Eastings and Westings or -124 H- -1-56 = -28. This is the method of subdividing an error in theory, but in practice, an approximation is sufficient, the proportion of error to each line being made without reference to calculation, the error when it is below the maximum allowed, of one link in ten chains being equally divided between the two 62 columns of Northings and Southings and of Eastings and AVest- ings, and generally thrown into the longest lines. The example given, however, must not be understood as a specimen of the real extent of correction on such small distances, we have taken ample figures merely to serve the purpose of illustration. AV"e have omitted mentioning here the several methods given in other works on " Surveying by the Traverse System " of finding unknown distances, by addiug up the Northings and Southings, and the Eastings and Westings of a polygon, and applying the difference of the two severally, as the latitude and departure of the unknown line and thence finding the chain distance. Poly- gonometry as given in Hutton's Mathematics, Vol. 3, and other books, treat of these methods, and to them we refer the reader for further information. THE METHOD OP PLOTTING BY TRAVERSE. These difierences of latitude and departure, or distances on the meridian and perpendicular of each station from the preceding one are not only applicable to the proof of fieldwork, but are sub- servient also to the plotting and computation of the area of the Survey, which will now be explained. All the distances on the meridian of each station from the pre- ceding one, North or South, and all the departures of each station, from the preceding one, East or "West, can be referred to the meridian of the first station or starting point of the survey, viz., station A. Eor instance, on tlio meridian of A, ( Fig. Page 58, ) for the line AB, the distance North is A6 and the departure East is 6B ; on the meridian of B, for the line BC, the distance South is B/f, and the departure East is kC ; deduct the distance that B is North of A, from the distance that C is South of B, and we obtain the distance that C is South of A, or B^' — Ah ~ Ac; in like 6S manner add the distance that is East of B, to the distance that B is East of A, and we obtain the distance that C is East of the meridian of A, or ^-^13 = cC Again, on the meridian of C for the line CD, the distance ISTorth is CI, and the departure East is ID, deduct the distance that C is South of A, from the distance that D is North of C, and we obtain the distance that D is JSTorth of A, or C^ — Ac = Ac?; in like manner add the distance that D is East of C, to the distance that C is East of A, and we obtain the distance that T> is East of the meridian of A, or ZD + cC =dJ). On the meridian of D, for the line DE, the distance North is Din and the departui'e TVest is wE, add the distance that E is North of D to the distance that D is North of A, and we obtain the distance that E is North of A, or Dm + Ac? = Ae : also, deduct the distance that E is "West of D from the distance that D is East of A, and we obtain the distance that E is East of A or dD — m'E =: eE, and so on all round the figure until arrived back at A, when the distance that A is North of J, the preceding station, deducted from the distance that J is South of A, or Aj — Js, and the distance that A is East of J, deducted from the distance that J is West of A, or J J — 5 A will leave no remainder, proving that the calculation has been correctly made. The line EG- it will be perceived crosses the meridian of A, in this case, it is only necessary to deduct the distance that E is East of A from the distance that Gr is "West of E, to obtain the distance thet a is "West of A or y^ - E/ = ^a. To plot, therefore, all these station points, draw a meridian line, and another perpendicular to it, representing the East and West direction. Eix on any point on this meridian line for the station A, lay off with a pair of common compasses and a scale of equal parts the distance Ab North of A, draw a lino parallell to the East and West line through the point h, lay off the distance 6B, East, 64 and join the points A and B, we thus obtain the bearing and distance of the line AB. Next lay off the distance Ac South of A, draw a line parallel to the East and West line, through the point c, lay off the distance cC East, and join the points B and C, thus obtaining the bearing and distance of the line BC. Then lay off the distance A.cl North of A, and with a parallel to the East and West line through the point d, lay off the distance dD East, join C and D, and we obtain the bearing and distance of the line CD, and so on all round the figure, observing that when the distances on the perpendicular are West of the meridian of A or starting point, they are laid oft" West on the plot. The reduction of the distances on the meridian and perpendicular of each station to the first station or starting point is therefore easily effected by a simple addition or substraction, and may be comprised in the fol- lowing rule. Rule. — When the distances run North of the first station, add them one to another, until they change to South, then deduct them one by one until the Southing exceeds the Northing, when deduct the latter from the former, changing the denomination to South ; all distances then going South, are added and those going North deducted, and so on, until arrived back at the original starting point. Likewise in the distances on the perpendicular, when the dis- tances run East of the meridian of the first station, add them one by one until they change to West, then deduct them until the Westing exceeds the Easting, when deduct the latter from the former changing the denomination to West ; all distances then going West are added, and those going East deducted, and so on, until arrived back at the original starting point. This method of plotting is by far the most perfect, and the least liable to error of any that has been contrived ; it may appear to require more labour, than the common method by angular protrac- tion or protraction by Bearings, on account of the computations 65 required, but these are made with so imicli ease and expedition by the help of Traverse Tables,* that this objection would vanish, even if they were of no other use than for plotting, but as we have already said, they are subservient also to finding the area, and which cannot be ascertained with equal accuracy in any other way ; when this is considered, it will be found to be attended with less labour on the whole than the common method. One great advantage in the above method of plotting is, that if a station be incorrectly plotted, it does not affect in the least, the correctness of the other stations, which is not the case when plot- ting with a common protractor by bearings or angles, where an error made in plotting one line is carried on through the series. THE TJK^IYEESAL THEOKEM. We must now look on the distances on the perpendicular above computed of each station, from the first in the series or starting point, as the sides of certain figures which multiplied into the dis- tances on the meridian between each station, will give certain products, from which the area of the figure is derived by an easy computation from the following : rXITEESAL THEOREM. If the simi of the distances of an East and West line of the two ends of each line of a Siorvey^from any meridian lying entirely out of, or running through the Survey, he multijolied hy the JSToethinq- or SozTHTSOr made on each respective line; the difference heticeen the * A. set of Traverse Tables has beai published by Major J. T. Boileau, Bengal Engineers, to every minute and degree of the quadrant, and these Tables are now in general use in the Revenue Surveys; we, therefore, refer the Surveyor to this work, in which he will find the method of using them fully explained and much valuable information regarding the application of the system to general purposes. A new edition of these Tables, carefully revised, is much wanted. 66 sum of the North Peoducts and the sum of the South Peoduots, ivill he double the area of the Survey. To explain this Theorem, it is necessary to take the three dif- ferent cases that present themselves separately, which are — 1st. "When the meridian is to the West of the polygon and lying entirely out of the Survey, 2nd. When the meridian is to the East of the polygon and lying entirely out of the Survey. 3rd. When the meridian runs through the polygon a portion of the Survey lying East and West of it. Case 1st. When the meridian is to the West of the Polygon and lying entirely out of the Survey. Let ABCDEFA be any polygon, NS an indefinite straight, or meridian, line. Draw perpendiculars 0A, feB, cC, ^D, eE, /F, from the extremities of each side of the polygon, meeting ^ the line JSTS, at a, b, c, d, e, f then will the distances ah, he, cd, de, ef fa, be the meridional distances corres- ponding to the sides AB, EC, CD, DE, EF, FA. If, therefore, we multiply the sura of the perpendi- culars at the extremities of each Northing or ascending side of the polygon by the i 67 meridional distance corresponding to that side, and place the products in one column, calling them North Products, and if we multiply the sums of the perpendiculars at the extremities of each Southing or descending side of the polygon, by the meridional dis- tance corresponding to that side, and place the products in another column, calling them South Products, then will the sum of the South Products deducted from the sum of the North Products be double the area of the polygon, that is : {aA + &B X rt& + cC + ^D X cc? + p -Tf.-^ ^ -^ t- oo'—i'^'ffleoco"*— luo '-^ CO CM --< <-! "M (M - •sai-tas 3i[} ui uon -■Bjs }s.Tg sqj mo.ij uopins qoBa je sjni.iBcIap .10 acina -ipuacljad aqi uo S30Lre:)siG[ C-lOOOCOOOCOiOO ^v vs -^ i^- XI cp '^ tx: t^ to itM OC CiO t^ •<* CI '.'5 kO : JO UBTpUaiU 91^ uo S30UT3^SI(J 5 CO-^OCit— OOCOOKJO CO 7* t^ "*■ -p <^0 00 C3 -< « m •uoipgjio^ I * I I • • « • • 1 p 03 CO Cft ■"# » * * ' cj 0 1~-. * ■•<*i * ^t^) 9 on; ut pgjnsBaoi se saouB^STQ to H u iO<=>COO(M>iOCOO-im j^ 1— ' »o CO ri< <^^ a: oc (N »-> CM CM — " — ' 1 -SUOIlE^g 1 <220QK!faOEi-(^<: 10 w o as X bo O C>3 + E< The following simplification of the . method of calculating columns 4, 6, 7, 8, 9, 10, 11, 12, 13, and 14 will, it is hoped, place the matter beyond doubt ; and by comparing each given quantity with the diagrams, the mode of obtaining its corresponding result will be easily understood. To obtcnn Column 4 of tlie Table from Column 2. G-iven the Astronomical Then, Bearg. AB 50° „ BC 140 „ CD 21 ,, DE 315 „ EF 191 ,, FG 248 „ GH 333 „ HI 202 IJ 144 „ JA 47 Bearing of the line AB, 50® 12' X. E. as found by observation, or otherwise. (Diagram, page 55.) 12' + 23 + 33 -f 14 + 56 + 57 + 13 + 5S 4- 01 + 19 + ZB 270°11'=32C^ 23' — 1S0°=140' ,,G 61 ;0=201 „ D 113 41=135 5, E 56 42 =371 „ F 237 01=423 ,,G264 21=513 „ H 49 40-382 „ I J21 03=324 „ J 83 1S2 = 27 „ A182 53=230 33 -^180 =251 14 4-180 =315 56 —ISO =191 57 —ISO =248 18 —180 =333 58 —ISO =202 01 —180 =144 19 —180 = 47 12 —180 = 50 23'S.E. or Bearg . BC 33 N.E. or jj CD 14 N.W . or j> DE 5 6 S.W. or )> EF 57 S.W. or 5> FG 18 N.W or )> GH 58 S.W. or ••> HI 01 S.E. or >> \i 19 N\E. or J> J A 12 N.E. or J> AB To oltain Columns 6, 7, 8. and 9. Biile. — As Eadius : Distance : : Cosine of Bearing : Latitude and : : Sine of Bearing : Longitude or Departure. Latitude =^ Distance X Cos. Bearing ; or Departure = Distance x Sin. Bearing. See Diagram, page 58. Line AB Lat. A 6 = AB Cos. A , BC )> B k - BC »> B , CD j> C I = CD )) C , DE j> D m = DE M D , EF »' E n = EF J> E , FG 9} F y = FG Jt F , GH •y G V = GH )> G , HI J5 H Q = HI V H , IJ »> I r = IJ '> I , JA J» J s = JA }> J Long h E = AB £ ine A M h C = BC M B >J I D= CD >J C )i m E = DE J> D J> n F = EF >> E J) y G - FG !» F 5 ) V H r: Gil >> G 'J a I = HI :» H J5 r J = IJ j> I J» s A =JA !> J 76 JExamples of Columns 6, 7, 8 and 9. On the Line AB given, Bearing N.E. 50^ 12' Distance 8.35. Cosine. Sine. Bearing 50°12' 9-806254 9-885521 Distance 8-35 0-921686 0'9216S6 •727940 5-34 Lat. '807207 = 6-4 1 D3p. The above example bj Major Boileau's Tables (See Note, page 65). Latitude. Departure. Bearing 50°] 2' 5-J2 .... 800 ... . 6*15 •18 ... . 30 ... . -23 Distance 8-35 -04 ... . 5 .... -03 5-3< 6-41 To ohtain Column V^from Columns 6 and 7. Line AB Dist. on Merd. of A to B or A6 = N 5-33 „ BC „ „ B „ C „ m = S 7-47 S 2-14 Diflf. /Distance on Meridian of „ CD „ „ C „ D „ C« = N 14-84 \ A to C or Ac N 12-70 „ r „ „ DE „ „ D „ E „ Dm = N 979 \ of A to D or Ad N 22-49 Sum r \ of. „ EF „ „ E „ F „ En = S 4-42 \ of A to E or Ae N 18-07 Difif.r „ „ FQ „ „ F „ G „ Fj/ = S 4-G9 \ of A to F or A/ N 13-38 „ / „ „ GH „ „ G„H„G;) = N 945 \ of A to G or A^ N 22-83 Sum/ „ „ HI „ „ H „ I „ Hg = S 14-83 „ \ of A to H or AA N 800Diff./ „ ,, IJ „ „ I « J „ Ir = S 13-28 „ \ of A to lor A» S 5-28 „ / „ M _ ,t ., JA „ „ J M A „ J« = N 5-28 ,, \ of A to J or hj 0-00 77 To obtain Column 11 from Columns 8 and 9. Line AB Departure of A to B or 6D = E 6-42 It BC o „ CD „ » FG >i GH „ >» HI „ B ,, C ., iC = E 6.11 E 12-60 Sum = Dep. of A to C or eO C „ D „ /D = E 5-86 E 18-46 „ „ „ A „ D „ dD D „ E „ ^E = W 9-70 E 8-76 Diff. „ „ A „ E „ Ee E „ F „ nF = W 0-93 E 7 83 „ „ „ A „ F ,,/F F „ G „ A or N 6-42 X E N. E. & S. W. N. W. & S. E. ine 5-33 ~ 3-42 )) BC &B -}- cC X ho „ S 19-02 X E 7-47 = „ 14-20 CD cC + rfD X erf „ N 31-06 X E 14-84 - 46-08 j> DE d\) + eE X rfe „ N 27-22 X E 9-79 = 26-63 )) EF eE + /F X ef „ s 16-59 X E 4-42 = „ 7-33 FG* y'F - ^G X /5- V S 3-47 X E 4-69 = „ 1-63 GH .-G + /iH X gh „ N 13-46 X W 9-45 = „ 12-71 HI /iH + il X /« „ s 24-48 X VV 14-83 = 36-29 j» IJ il + iJ X ^/ ,, s 21-13 X w 13-28 - 28-06 >) JA iJ X iA M N 5-75 X w Sl 5-28 = „ 3-03 ms, . , 140-48 38-90 38-90 DifF. . . 101-58 Acres = double the area of ABCDEFGHIJ A. * The Co-ordinates here changing from East to West, or crossing tire Meridian of Station A, the dijference between the pairs instead of the sum is taken, by which means the area of the Parallelogram F.r G^, is properlj' balanced, the portion lying to the West of the Meridian of A, being cancelled by a portion on the East side, leaving a difference in favour of the latter, and which accordingly remaining East, and multiplied by a Southing, forms a South Product. 79 Consequently, (bB X b^ -{- cC ■]- dD X. cd -\- dD ■\- eE X de + h\i + il X hi + il + J J Xij) — (bE+TU X be + eE +/F X ef + fF-gG X fg + gG + hti X gh+jJ xJA) = Double the area of the figure ABCDKFGHIJA, Page 69. We have now endeavoured to explain the Traverse Table, and though all the operations which have been given at length for the sake of explanation appear laborious, they are performed with the greatest facility, the whole, with the exception of Colamns 6, 7, 8 and 9, being obtained by a simple addition or Subtraction, and Columns 13 and 14 by multiplication. Such is the traverse system of surveying, and to use Mr. Adams'* words, "the superior accuracy and ease with which every part of the process is performed, cannot, it is imagined, fail to recommend it to every practitioner." Geometrical Essays, by George Adams, 1313. 80 CHAPTER V. THE POCKET SEXTANT The Pocket Sextant combines numerous valuable properties : it measures an angle to one minute of a degree, requires no sup- port but tbe hand, may be used on horseback, maintains its adjust- ment long, and is easily re-adjusted when put out of order. It will determine the latitude by a meridian altitude to one minute ; and an approximation may even be made with it to the longitude, by means of lunar observations. Parther, it is very portable, forming when shut up, a circular box under three inches in diameter, and only li inches deep. o The above figure represents the instrument screwed to its box, for convenience of holding in the hand, and with the telescope 81 drawn out. A, is the index arm, haying a vernier adjusted to the graduated arc B, which latter is numbered to 140°, but the sextant will not measure an angle greater than about 125*^. The index is moved by the milled-head C, acting upon a rack and pinion in the interior. Two mirrors are placed inside ; the large one, or index mirror, is fixed to, and moves with the index : the other, called the horizon glass, is only half- silvered. The proper adjustment of the instrument depends on these glasses being parallel when the index is at zero— while they are, at the same time, perpendicular to what is termed the plane of the instrument, represented by its upper surface or face. To observe whether the instrument is in perfect adjustment, remove the telescope by pulling it out, and supply its place with a slide far the purpose, in which is a small hole to look through : then place the index accurately at zero, and direct the instrument, holding it horizontally, towards the sharp angle of a building, not less than half a mile distant, applying the eye so as to see both through the hole in the slide, and also through the un- silvered part of the horizon glass : the same object ought then to be so reflected from the index mirror fcothe silvered part of the horizon glass, as to seem but one with the object seen direct : if such be not the case, a correction becomes necessary, which is thus per- formed : D is a key, removeable at pleasure, that fits two keyholes, the one at «, the other at h. Apply this key at a, and gently turn, until the reflected object, and the one seen direct, seem but as one. The glasses are then parallel. The next point is to examine whether the horizon glass is per- pendicular to the plane of the instrument. For this purpose, hold the sextant horizontally, and look at the distant horizon ; then, if any adjustment be wanted, two horizons will appear, or the reflec- ted one will be higher or lower than the one seen direct ; should this be the case, apply the key at &, so as to bring the two horizons together. It must be observed that the large, or index mirror, be- ing correct by construction, it can want no alteration. M 82 By looking at the sun, we can always satisfy ourselves with respect to the adjustments; the telescope has a dark glass at the eye end, and with this on, we have only to place the index at zero, and using the telescope, to look at the sun — when, provided the instrument is in exact adj ustment, one perfect orb only will be seen. If the reflected image project beyond the other, then correc- tion is necessary. The full moon will answer as well as the sun for this purpose ;'but the dark glass at the eye end of the telescope must then be removed. The instrument is provided with two other dark glasses, which sink out of the way by raising two little levers at/*. It has been mentioned above, that for trying the adjustments of the sextant, an object must be half a mile off; this is on account of what is called the parallax of the instrument, occasioned by the ne- cessity of placing the eye of the observer on one side of the index mirror. Could we look from the middle of it, there would be no parallax ; which is the angle subtended by the point of vision, and centre of the index glass, when observing any near object ; conse- quently, as the distance of an object is increased, this angle dimi- nishes, and at length, becomes as nothing when compared with it. Half a mile is considered sufficient for all error to vanish, but at half that distance, it is scarcely perceptible. To take an angle, the observer looks either through the telescope, or hole in the slide (having previously raised the levers of the dark glasses at /), at the left hand object, holding the sextant horizon- tally in his left hand ; with his right, he turns the milled head C, until the other object, reflected from the index glass, appears upon the silvered part of the horizon glass, exactly covering or agreeuig with the left hand object, seen direct through the unsilvered por- tion of the horizon glass ; the angle is then obtained by the ver- nier to one minute. If the required angle be a vertical one, the sextant is held in a vertical position, by the right hand, while the left turns the milled head C, until the object is brought down to the horizon. 83 When the altitude of a celestial body is taken at sea, it is brought down, as the term is, to the natural horizon, and the measure of the angle, or height of the object, is read off upon the graduated are ; but on land, the natural horizon can seldom be used, on account of its irregularity ; recourse is then had to, what is called, an artificial horizon, such as a vessel containing water, mercury or other fluid. The observer then places himself in a situation, to see the reflected image of the sun, or other body, in the fluid : he has only then to bring down the image, as reflected from the index glass, until it reaches its reflection in the fluid : tlie altitude will then be ^aZfthe number of degrees, indicated by the graduated arc, subject to certain corrections, not necessary to be explained here. The reason of only taking half the number of degrees^ [will^be seen from the following explanation :— ' Let AB represent the surface of the quicksil- ver contained in a wood- en trough, whose plane is continued to C; DEF, the roof, in which are fixed two plates of glass, DE and EF, whose sur- faces are plane and paral- lel to each other, and Q the sun at S, whose alti- tude is required. Now the ray SH, proceeding from the sun's lower limb to the surface of the quicksilver, will be reflected thence to the eye in the direction of HGr, and the upper limb of the sun's image, reflected from the quicksilver, will appear in the line GH, continued to E-; and it is a well known principle in catoptrics, that the angle of incidence, SHA or SHC, is equal to M 2 84 *s the angle of reflectioUj GHB ; and as the angle AHE, or CHE, is the opposite angle of GHB, it is, therefore, equal to it, and to the angle SHC, the altitude of the sun's lower limb above the horizon- tal plane : so that, if we suppose the angle SHE, measured by a sextant, to be SO"", the altitude of the sun's lower limb will be 40°, subject to the corrections, as above. The principle of the construction of the sextant may Be ga- thered from the following demonstration. Let ABC represent a Sextant, having an index, AG, (to which is attached a mirror at A,) moveable about A as a centre, and denoting the angle it has moved through, on the arc, BC : also let the half-silvered (or horizon) glass, a h, be fixed parallel to AC ; now a ray of light, SA, from a celestial object, S, im- pinging against the mirror, A, is re- flected off at an equal angle, and striking the half-silvered glass at D, is again reflected to E, where the eye likewise receives, through the transparent part of that glass, a direct ray from the horizon. Then the altitude, SAH, is equal to double the angle, CAG, Jiieasured upon the limb, BC, of the instrument. Por the reflected angle, BAG, (or DAP) = the incident angle, SAT, and the reflected angle iDE == the incident «DA — DAE ~ DEA, because al i^ parallel to AC. jN'oav, HAI = DFA r= (FAE + EEA), and DAE, being equal to DEA, it follows that HAI = (DAE + EAE). From HAI and (DAE -f FAE) take the equal angles, SAI and DAF, and there reiuains SAH = 2 FAE, or 2 GAC ; or, in other words, the angle of elevation, SAH, is equal to double the angle of inclination of the two mirrors, DGA, being equal to GAC. Hence the arc on the limb, BC, although only the sixth part of 85 a circle, is divided as if it were 120°, on accouut of its double being required as the measure of CxlB, and it is generally extend- ed to 140° The chief and indeed onl}^ objection to the sextant as a survey- ing- instrument, arises from the ang-les taken with it not being: always like those measured by the theodolite and compass, liori- zontal ones. If the theodolite be set truly level, we can take angles all round its circle, no matter whether one object be high and another low, and these angles will be what are termed hori- zontal or azimuthal angles ; so that Avere we to take angles from object to object and complete the circle, the sum of all these angles ought to be 360^, or the measure of a circle. But if the angles were to be measured by a reflecting instrument, they would not amount precisely to 360°, unless taken upon a perfectly level plane ; owing to this circumstance, that to take an angle by the sextant, the two objects have to be brought into contact, viz,. — the reflected one and that seen direct, it is necessary for the observer to hold his instrument, not strictly horizontal, but in the plane of the two objects, or in such a position as will enable him to form the contact ; and, therefore, if one point is elevated very much above the other, the sextant must be held at a corresponding in- clination Avith the horizon. Angles so taken require a reduction to horizontal ones. But as the sextant is never used to lay down points for a iTigonometrcal Survey of importance, it rarely occurs that the reduction is required ; indeed, to eff'ect it with accuracy is attended with considerable difiiculty, as the angles of elevation or depression must be known, a matter always difiicult of attain- ment with the sextant. It is better to avoid, if possible, the necessity of this reduction by selecting stations neither much elevated or depressed. The height and distance of objects, as walls or buildings, whether accessible or otherwise, may be obtained in a very simple and ex- 86 peditious manner with the sextant, by means of the little table below : — Multiplier. Angle. Angle. Divisor 1 . . . . . . 45° 00' . . . . . . . 45° 00' . . . ... 1 2 . . . ... 63 26 ... . ... 26 34 . , . ... 2 3 . . . ... 71 34 . . . ... 18 26 . . . ... 3 4 . . . ... 75 58 . . . ... 14 02 . . . ... 4 5 . . . . . . 78 41 . . . ... 11 19 . . . ... 5 6 . . . . . . 80 32 . . . ... 9 28 . , . ... 6 8 . . . . . . 82 52 ... . ... 7 08 . . . ... 8 10 . . . . . . 8i 17 ... . ... 5 43 . . . ... 10 ■ Make a mark upon the object, if accessible, equal to the height of your eye from the ground. Set the index to one of the angles in the table, and retire on level ground, until the top is brought by the glasses to coincide with the mark ; then, if the angle be greater than 45°, multiply the distance by the corresponding figure to the angle in the table ; if it be less, divide — and the product, or quoti- ent, will be the height of the object above the mark. Thus, let EB be a wall, whose height we want to know ; and 26° 34' the angle selected. Make a mark at D equal to the height of the eye ; then step back from the wall, until the top at E is brought down by the glasses to coincide with the mark : measure the distance AB, namely, from your station to the wall, and divide that distance by 2, the figure corresponding to 26"* 84', this will give the height DE, to which BD must be added. The par alia jc of the instrument exerts an influence on measure- ments of this kind, from the object being near. To correct it, we have only to ascertain its amount, by placing the index at zero, and looking through the instrument at the top of the wall ; when, if influenced by parallax, it will appear as a broken line ; but by moving the index a little way on tbe arc of excess or to the left of c,-"' 87 zero, the broken line will reunite, and the adjustment be effected. When any quantity is taken thus on the arc of excess, the amount must be deducted, when setting the instrument to any of the tabular angles. "When the object is inaccessible — set the index to the greatest of the divisor angles in the table, that the least distance from the object will admit of, and advance or recede, till the top of it be brought down by the sextant to a level with the eye ; at this place, set up a staff, equal to the height of the eye. Then set the index to one of the lesser angles, and retire in a line from the object, till the top be brought to coincide with the staff, set up to indicate the height of the eye ; place a mark here, and measure the distance between the two marks ; this, divided by the difference of the figure opposite the angles used, wdll give the height of the object above the height of the eye or mark. For the distance multiply the height of the object by the numbers against either of the angles made use of, and the product will be the distance of the object from the place where such angles was used. The above will be understood better by means of a diagram. Let AB be a wall, not to be approached nearer than C ; and that we find, upon trial, that this distance admits of our using the angle 45° ; assume a point E on the wall, as the height of the eye ; then the index being set to 45°, fix yourself so that the glasses shall bring the top A to coincide with E. At this point, place a staff, 88 CGr, equal to the height of the e3'e. Now select any one of the lesser angles from the tables — 18° 2G', for instance, and retire until the point A agrees with the top of the staff CGr, which occurs at P. Place a mark at F, and measure the distance from P to Gr ; which, divided by 2, the difference of the numbers opposite to the angles used, will give AE— to which add BE = CGr, the height of the eye, and the total height AB is obtained. Then," for the dis- tance — the height AE, multiplied by 3, its corresponding figure, will give the length EE ; and AE multiplied by 1, will, in like manner, give GrB = AE in this instance. Horizontal distances, as well as heiglits, may be found by means of this table where the ground is level ; but as we may not always be able to measure in a direction at right angles to the distance we wish to ascertain, (as in the acompanying diagram), I prefer using a more independent method. Suppose AB to be the breadth of a rirer which it is desired to find ; send a flag in any direction a- long the bank of the river to C, and with the sextant measure the angle ABC, (A being any object on the opposite bank of the river), set the index to half the supplement of the angle, ABC, and proceed in the direction BC, until the glasses of the instrument shew a flag staff at B, and the object A, in contact. Suppose D to be this point, then will BD be equal to AB, the breadth sought. The reasoning of the above is very simple, the supplement of ABC is EBA, which is equal to BDA + BAD, and since we make 89 BDA half EBA, tlierefore, BAD is also equal half EBA, and, therefore, BDA = BAD, and therefore BD = BA. These methods of determining heights and distances are valuable, as the operations are speedily performed, and with tolerable accu- racy ; while it enables us to dispense with logarithmic tables and trigonometry. The pocket sextant is very useful, when taking offsets ; set the index to 90°, and walk along the station line ; then, when you wish to ascertain at what point any mark or object becomes perpendicu- lar to the station line, you have only to look through the sextant at the left hand object, and move forward or backward until the two objects, namely, the offset mark, and that on your station line, are brought to coincide. Or, if you wish to lay off a line at right- angles to another, send your assistant with a staff in the required direction, and having set the index at 90°, cause him to move right or left until his staff and your other mark are made to agree. An instrument which has been made specially for the above purpose and which, where attainable, has quite superseded the use of the cross staff, is the " Optical Square." It is made of brass, and contains the two principal glasses of the Sextant, viz., the index and horizon glasses, fixed at an angle of 45° ; hence while viewing an object by direct vision, any other, forming a right angle with it at the place of the observer, will be referred by reflection, so as to coincide with the object viewed. This instrument has the advan- tage of great portability, not being larger than an ordinary sized hunting watch. N 90 CHAPTER VI. LEYELLII^G. Leyelling- is the art of tracing a Hue at the surface of the earth, which shall cut the directions of gravity everywhere at right angles. If the earth w^ere an extended plane, all lines representing the direction of gravity at eveiy point on its surface would be parallel to each other ; but, in consequence of its figure beiug that of a sphere or globe, they everywhere converge to a point within the sphere which is equi- distant from all parts of its surface ; or, in other words, the direction of gravity invariably tends towards the centre of the earth, and may be considered as represented by a plumb-line when hanging freely, and suspended beyond the sphere of attraction of the surrounding objects. Eor the better elucidation of the subject, I consider it best to give the reader some idea of the method of levelling before enter- ing into minute detail of the instruments employed, and of their adjustments, which latter will be better understood, when some no- tion is formed of the ends which they are intended to accomplish. Therefore, in the following explanations in speaking of a spirit level, the reader, w^ho is unacquainted with this description of instru- ment, must imagine a telescope similar to that of the theodolite, and mounted in somewhat the same way, the horizontal wire of which, when the instrument is levelled, will trace out a right line parallel to the horizon. 91 In tlie above- diagram let the straigJit line AB represent the sur- face of the earth, upon the supposition of its being an extended plane, the direction of gravity at the points, A, I, and B, would be represented] bj the lines, AC, ID, and BE, all parallel to each other, and at right angles to the horizontal line AB. JS'ow, if the surface was undulatorj, as shown by the curved line AB, and it was required to make a section representing it, an instrument capable of tracing out a line parallel to the horizontal line AB, (as a spirit-level,) might be set up any where on the surface, as at I, and staves being placed or held along the line, as at a, h, c, d, Sfc.^ the different heights above the ground where such staves wxre in- tersected by the lines so traced out, would at once show the relative level of all these points, Avith regard to the horizontal line, as a datum or standard of comparison. But as the earth is a globe, its circumference must be circular, as IKL, in the annexed figure : the straight line AB, will, there- fore, not represent the surface of the earth, but the sensible horizon of an observer stationed at the point I, to which point it is a tangent, being at right angles to the radius of the circle (or semi-diameter of the earth) IC. A line which is parallel to the sensible horizon of the observer, is the line traced out bv our spirit-levels, and is a tangent to the earth's surface at that point only where the instrument is set up : thus, A B is 2 92 N B 13 a tangent at I, and DE a tangent at P; such being the fact, the difference of level between any two points cannot be de- termined by simple re- ference to a horizon- tal line, since every point on the surface of the globe (however near to each other) has a distinct horizon of its own. If the earth were everywhere surrounded by a fluid at rest, or that its surface was smooth, regular and uniform, every point thereon would be equally distant from the centre ; but, in conse- quence of the undulating form of the surface, places and objects are differently situated, some further from, and others nearer to the centre of the earth, and consequently, at different levels. The operation of levelling may, therefore, be defined as the art of finding how much higher or lower any one point is than another, or, more properly, the difference of their distances from the centre of the earth. Eeferring to our last figure, we have seen that the line AB is a true horizontal or level line at the point I, but being produced in the direction A or B it rises above the earth's surface; and, al- though it may appear to be level, as seen from I, yet, it is above the true level, (which is represented by the circumference of the circle,) at every other point, and continues to diverge from it, the further it is produced ; at Gr, the apparent line of level, as the horizontal line AB is called, is above the true level, by the distance Gil, and at M by the distance MN, the difference leing equal to tlie excess of the secant of the arc of distance alove the radius of the earth. 93 The difference, GrH or MN, between the true and apparent level may be thus found. Put t in the following diagram for the tangent IH, o' for the ra- dius, cl of the earth, and ^ for GH, the excess of the se- cant of the arc of distance above the radius ; IH being considered as equal to IG- ; then ■f2 (r + ccy = r- + t or 5^' + 2rx + x"" = r' + f and2r.r + x' = f or (2r + ^^) X = i' But, the diameter of the earth 2r is so great with respect to the quantity (x) sought at all distances to which a common levelling operation usually extends, that 2r may be taken for 2r-\-x without sensible error ; we then have 2rx ^ f and 2/' Or in tuords : — The difference (x) between the true and apparent level is eqiialtothe sguare of the distance (f) divided hy the diameter* of the earth {2r) ; and, consequently, is always iirajportional to the square of the distance. The mean diameter of the earth is 791G miles and the excess of the apparent above the true level for one mile .-. = >,. ^^^ of a mile, or 8"004i inches, at two miles it is four times that quantity or 32-016 inches, and so on increasing in proportion to the square of the dis- tance. If we reject the decimal •OOJi and assume the difference between the true and apparent level for one mile, to be exactly 8 inches, or two-thirds of a foot, there arises the following convenient form for 9-i computing the correction of level due to the curvature of the earth, for distances given in miles, viz, — — , D being the distance in miles. A very easily remembered formula, derived from the above for the correction for curvature in feet is two-thirds of the square of the distance in miles ; and another, for the same in inches is the square of the distance in chains divided hy 800. Prom the above it will be seen that the effect of taking the apparent instead of the true level, is to make the level of the point observed fo, lower with reference to the point observed from, than it really is, so that the correction for curvature must be added. Eor instance, suppose I find the apparent level of a point G50 feet distance to be + 6*52 feet ; that is to sa}^, 6'52 feet higher than the point from which I observed, the correction for curvature for that distance is '01 ; and, therefore, the true level of the point is + 6'53 ; but supposing the apparent level be — 652 ; then the true level will be — 651. But this effect, due to the earth's curvature, is modified by another cause arising from optical deception. This second correc- tion, viz. — terrestial refraction, has the contrary Q^ect oi elevating the apparent place of any oljject above its real place. The rays of light bent from their rectilnear direction in passing from a rare into a denser medium, or the reverse, are said to be refracted, and this causes an object to be seen in the direction of the tangent to the last curve. Every difference of level, accompanied as it must be with a difference of density in the strata of the atmosphere, will have, corresponding to it, a certain amount of refraction, and as the curve described by each ray of light is concave next the earth ; the tangent to the curve will lie above it, and consequently, the object will appear more elevated than it really is. A simple rule to correct the error occasioned by refraction, is to diminish the effects of the earth's curvature by one-seventh of itself, this is not quite correct, but will be found sufiTiciently so in most cases. 95 A table of curvature and refraction and both combined, in which form they are generally given, will be found at the end of the chapter. It must be borne in mind that if these corrections are applied directly to the reading of the staff, they must be used in an opposite way to what they are when applied to the difference of level, viz., the curvature must be subtracted, and the refraction added ; for the lower the point observed to, the greater is the read- ing on the staff; if, therefore, we were to add the correction for curvature to the reading on the staff', we should be making that point lower still, instead of counteracting the effect which has already made it too low. But in ordinary levelling, i. e. with a spirit level and two staves, these corrections are very seldom applied in practice, for under a distance of 700 feet, they are inappreciable, and the injurious effects that they might have in a succession of stations during a long day's levelling, are counteracted by placing the instrument mid- way between the staves, the effect of curvature is thus altogether removed as well as that of refraction, as the latter will affect both observatious alike, unless under peculiar circumstances of weather, over which tlie observer has no control. The fact of placing the instrument equally distant from each staff, has also other advan- tages which will be pointed out hereafter. The method of proceeding in levelling is shewn in the annexed figure . , — ~ZlA ^ hr~ A C ^ ^ & Suppose it were required to find the difference of level between the points A and Gr ; a Staff is erected at A, the instrument is set up at B, another staff at C, at the same distance from B, that B, is from A, and the readings of the two staves are noted. The instru- ment iy then conveyed toD, and the staff which stood at A, is now 90 removed to E, the stafi C, retaining its former position, and from being the forward staff at the last observation, is now the back staff : the readings of the two staves are again noted, and the in- strument removed to F, and the staff C, to the point Gr, the staff at E, retaining the same position now becomes in its turn the back staff, and so on to the end of the work, which may thus be extended many miles : the difference of any two of the readings will show the dift'erence of level between the places of the back and forward staff; and the difference between the sum of the back-sights, and the sum of the forward-sights will give the difference of level be- tween the extreme points, thus : E Jack-Sights. Fore-Sights Ft. Dec. Ft. Dec. A, and C, 10-46 11-20 Cj 5? E, 11-33 8-00 E, „ Gr, Sums 7-42 7-91 2921 2711 27-11 Difference of level, 2*10 Bhowing that the point Gr, is 210 feet, higher than the point A. The foregoing process is called compound levelling. The following is an example of simple levelling, being preformed at one operation and therefore subject to the correction for curvature and refraction to obtain a correct result. 97 If it were required to drain a Jheel A, by making a cut to a stream at B, a distance of 20 chains : let a level be set up at C, and directed to a staff held upright at the edge of the water at B. The horizontal line CD represents the line of sight which would cut the staff at D, the reading being 17'4;4 feet ; the height of the instru- ment above the ground was- 4i feet, and the depth of the Jheel 10 feet ; therefore the difference of level between the bottom of the Jheel and the surface of the stream was as follows : Ft. Dec. Reading of the staff 1 7*4i Height of instrument i 4'00 Depth of Jheel lO'OO Curvature and Refraction for 20 Chains. 1 see Table, at end of chapter, ,,.,,./ * ' • 14-03 Difference of level, ». 3-4] LETELLI^■& STAVES. Levelling staves are of various patterns, but thev may be divided into two classes. Levelling staves with, vanes ; and those on which the scale of feet is painted in a bold conspicuous manner, which ena- bles the observer to note the reading himself. In the former, the vane, which is a piece of wood painted half black and half white, and which slides on the staff, is moved up or down by an assistant according to the signals of the observer, until the horizoDtal wire of the telescope bisects it, when the height should be read off by the staff-man. A second vane is placed at a height of six feet from the first ; when the readings are greater than six feet, the upper vane is used and six feet is added to the reading on the staff. In levelling with the theodolite and for contouring, this description of staff may be used with advantage, but for ordinary levelling, it is greatly inferior to the staves which can be read by the observer, especially in this country where the staff-man cannot be trusted to give the reading ; the staff has to be brought for the leveller to read, thus much time is lost, and the leveller is always in the hands of the staff-man, who by accident or on purpose may shift the vane ; o 9S also it takes some time to adjust the vane to tlie proper height, especially if the staff-man is not accustomed to the work. The only advantage of these staves is, that the vane can be bisected at a greater distance than the figures of the others can be read. The second description of levelling staves are made, either in one piece, or two or more pieces, which have joints like those of a fi.sh- ing rod, and fit one into the other (a very objectionable pattern), or telescopic. I prefer the first for use, about twelve feet long. The telescopic staves are more portable, and equally good if they do not get out of order, but they are very liable to do so in this country from the swelling of the wood ; they are, moveover, three times the expense ofthe simple staff in one length. Several different ways of marking levelling staves have been tried, to enable them to be clearly read. The best is that pattern designed by Mr. Conybeare, the Engineer of the Vehar Water-works, which can be read at a greater distance than any other staff I know of. Having got, so far, I will now proceed to treat of levelling instru- ments and their adjustments. theYievel. The accompanying figure represents the Y level. A, is an achromatic telescope, resting upon two supporters, which in shape resemble the letter Y, and are consequently called the ys. The lower ends of these supporters are let perpendicularly into a strong bar, which carries a compass box, C. This compass box is convenient for taking bearings, and has a contrivance for throwing the needle off its centre, when not in use. One of the Y suppor- ters is fitted into a socket, and can be raised or lowered by the screw B. Beneath the compass box, which is generally in one piece with the bar, is a conical axis passing through the upper of two parallel l^lates, and terminating in a ball supported in a socket. Immedi- ately above this upper parallel plate is a collar, which can be made 99 to embrace the conical axis tightly, by turning the clamping screw E, and a slow horizontal motion may then be given to the instru- ment by means of the tangent screw D. The two parallel plates, are connected together by the ball and socket already mentioned and are set firm by four mill-headed screws, which turn in sockets fixed to the lower plate, while their heads press against the under side of the upper plate, and thus serve the purpose of setting th© instrument up truly level. Beneath the lower parallel plate is a female screw, adapted to the stafi'-head, which is connected by brass joints with three ma- hogany legs, so constructed, as to shut together, and form one round staff, a very convenient form for portability, and, when opened out, to make a firm stand, be the ground ever so uneven. The spirit level, I I, is fixed to the telescope by a joint at one end, and a capstan-headed screw at the other, to raise or depress it for adjustment. The telescopes of levelling instruments are provided with cross wires, fitted to a diaphragm, as before described for the theodolite and moveable by the same means. The cross wires in the diaphragm of the level are usually arranged as shewn in the diagram ; the horizontal wire marks the intersection of the horizontal visual ray o 2 100 with the staff; the two vertical wires serve to direct the telescope so that the staff shall be seen between them, and thus be in the axis of the lenses; and by their means the observer can judge whether the staff is being held vertical. The Y level has. three adjust- ments, viz.: — 1st. — T7ie line of Collimation ; i. €. to bring the horizontal wire into the line of collimation of the telescope. First, let us see what would be the effect on levelling operations if this were not done. Let A B, represent a section of the plane of the wires, and C the proper position of the horizontal wire ; the telescope is set truly horizontal, and on its being turned in the direction of a staff, a portion of the latter comes into the field of view, forming a picture on a smaller scale than the original, at A B ; then the division of the staff cor- responding to the point C is on a level with the axis of the telescope ; but suppose the horizontal wire is at D, it will cut two staves, placed at equal distances from the instrument, at points differing from the true level points, by as many divisions of the staff as are included between C and D, and since the staves are at equal distances, the scale of the picture formed in the teles- cope, and, therefore, the number of divisions between C and D, will be the same in both cases ; and the reading on each staff will differ from the true level point by the same quantity, they will therefore give the true differences of level between the two places. But if the staves are at unequal distances, the reverse of this takes place, for the scale of the picture, and, therefore, the number of divisions included between C and D alter as the distance of the staff is 101 altered ; if, therefore, the horizontal wire be not at the point C, the readings obtained where the staves are at unequal distances from the instrument, will not give the true difference of level between the two places. To effect this adjustment, make the horizontal wire coincide with some well defined distant object, then invert the telescope by turning it half round in its Ys, and if the wire has moved off the object, bring it back, half by the adjusting screws above and below the telescope, and half by the foot screws ; repeat this until the wire coincides with the object in both positions of the telescope, when the adjustment will be complete. 2nd. — To adjust the lubhle tube. Open the clips of the Ys and placing the telescope parallel to two of the foot screws, by their means bring the bubble to the centre of its run ; now reverse the telescope in the Ys by turning it end for end, and if the bubble still remain in the centre, it is in adjustment ; but if not, bring it back to the centre, half by the capstan-headed screw at one end of the bubble tube, and half by the foot screws. Eeverse the teles- cope again, and if still not correct repeat the operation until it is so. The bubble tube will now be parallel to the line of coUimation. 8rd. — To set the telescope at right angles to the horizontal axis ; so that when this axis is vertical, the line of coUimation may de- scribe a horizontal plane. To do this, bring the telescope over two of the foot screws and by their means bring the bubble attached to the telescope to the centre of its run, turn the telescope half round on its axis, and if the bubble should then leave the centre, bring it back, half by the adjusting screw B which raises or depresses the Y, and half by the foot screws ; repeat this till perfect, and then turn the telescope a quarter round, and by means of the third foot screw bring the bub- ble again to the centre of its run. The bubble should now remain in the centre during a complete revolution, and the instrument will be in adjustment. 102 The Y level is more easily adjusted, but from their superior compactness, increased optical power, and greater stability of the adjustments, Grravatt's and Troughton's levels are more generally used. geavatt's oe the dtjmpt leyel. This instrument is furnished with an object glass of large aper- ture and short focal length, and sufficient light being thus obtained to admit of a higher magnifying power in the eye-piece, the advan- tages of a much larger telescope are obtained, without the incon- venience of its length. The diaphragm is carried by the internal tube a a, which is nearly equal in length to the external tube. The external tube T T, is sprung at its aperture, and gives a steady and even motion to the internal tube, which is thrust out, and drawn in. to adjust the focus for objects at different distances by means of the mill-headed screw A. The spirit level 1 1, is placed above the telescope, and attached to it by capstan-headed screws, one at either end, for the purpose of adjusting it. 103 A cross level ^, is placed upon the telescope at riglit angles to the principal level I I, by v^hich we are enabled to set the instru- ment up at once, with the axis nearly vertical. It is only to be used for thus levelling the instrument, approximately. The mirror M mounted upon a hinge joint, can be placed at the end of the level, 1 1, so that the observer, while reading the staff, can at the same time see that the instrument retains its proper position, a precaution by no means unnecessary in windy weather, or on bad springy ground. The telescope is attached to the horizontal bar by capstan-headed screws, B B, space being left between the bar and the telescope for the compass box C. D is the clamping screw, and E the tangent or slow motion screw; but sometimes these levels are made without either, as shewn in the Troughton level. Almost all levels, of every description, are now made with three foot-screws instead of the old parallel plate screws. As the telescope of this is fixed and not moveable in Ys, it is not possible to collimate it the same way as the Y level. I have al- ready explained that if the horizontal wire be not in the line of collimation, then even though the axis of the telescope be horizon- tal, the wire will not trace out a liorizontal line ; but more than this the wire will not even trace out a right line ; for example, (see fig. next page,) with the level at A, if the wire be fixed on d of the staff B, then it will not cut the point E, on the staff C, directly in a line with d, but will cut some point above or below as /"or g. The rea- son of this is as follows, let A repre- a | — — . — : — \c sent the proper position (i. e. in the line of collimation) of the horizontal wire, and CA the direction of the axis of a pencil of light passing though the object glass and coming to its focus at A. Then the axis of the tube of the telescope being set truly horizontal, the line AC, is also truly horizontal, and every point bisected by the horizontal wire will be situated on the prolongation of the line AC. lOi Suppose now the position of tlie diaphragm carrying the wires to have become deranged, so that the horizontal wire is moved to B, then every point intersected by this wire will be on the prolonga- tion of the line BC. Buit in order to see clearly things at different, moderate distances through a telescope, the object glass must be di'awn in or thrust out, the point C will thus alter in position, and therefore the prolongation of BC will not be the same line for short as for long distances. But if the horizontal wire be at A, the point C moving along AC, or the axis of the telescope^ the prolonga- tion of AC will always remain the same. On the above has been founded G-ravatt's method of collimating. To discover if the wire traces out a right line or no. Pirst find the horizontal line, a' h" c", then if the point e be in a straight line with the points d and a', the triangles a' l" d and a' c" e, will be similar, and e c" will be to d V as a c" to a' h" . To find this horizontal line, having chosen a tolerably level piece of ground, place the staves A, B, C, at equal distances, about 150 or 200 feet apart, on pegs driven in the ground ; now place the level at D, half way between A and B, and having levelled it, read the staff A, then turn it round, and if necessary, having re-levelled it, read staff B, then since the level is equally distant from A and B, the points a and h tbus obtained will be on the same level, what- ever may be the errors of adjustment in the instrument* : proceed similarly at E, half way between B and C, and we shall obtain the • The reason of this is that the enof in adjustment will aflfect the readings on each stafF equally , to find the difFerence of level we subtract these readings, one from the other, the error will therefore be eliminated. 105 two points y and c on the same level. The difference of the two readings on staff B is 5 })\ add this (with its proper sign) to the readings on staff 0, and this gives the point g\ "We now have the three known points, a h c', equidistant from the earth's centre, and a line passing through them will be horizontal. Now take the level to A, and placing it so that the height of the axis of the telescope may "be measured by the staff A when per- pendicular, read the staffs B and C. From A.a\ the height of the axis of the telescope, subtract A« the height of the point a, this will give aa, apply this with its proper sign to B6 and Co', and we obtain the two points h" , c" on a level wdth a\ !N'ow subtract W)" and Qc" from the last readings on B and C, respectively, and we get h"d and c"e, the former of which should be half the latter, since ah" = J a'c" ; but, if not, the wire is out of adjustment and re- quires correcting. To do this, loosen the upper and tighten the lower screw holding the diaphragm plate, or vice versa^, and take a another reading to B and C, (and if the instrument has been moved at all, A.a' must be re-measured ; this will affect B5" and Co" which must be corrected accordingly,) again subtract from these read- ings B5" and Cc", and if Vd is not now half c" e, the operation must be repeated until it is so. The second adjustment is to put the level tube parallel to the line of collimation. After perfecting the above adjustment, if the instru- ment be not moved, all we have to do to effect this, is to direct the horizontal wire to the point c'^ (by means of the foot screws) on the staff C, and since this point being on the same level as a', the line of collimation is horizontal, bring the bubble to the centre of its run (by the adjusting screws at one end of the tube) and it will be horizontal, and, therefore, parallel to the line of collimation. * Care must be taken to make a note of wliat was done in each trial, such as " gave top screw one quarter turn to the right or left," for guidance in perfecting the ad- justment, after having seen whether the previous operation has made it better or worse. 106 The third adjustment, viz., setting the line of colliraation per- pendicular to the axis, is performed in the same way as in the Y level. Instead of there being a screw at one end though for the purpose of making this adjustment, there are at each end three screws, secured bj a covering plate which must first be removed ; the two outside ones pusb up, and tbe middle one pulls down, that end of the telescope. T E OTTGHTOITS LEVEL, In tbis level, tbe telescope T, rests close down upon the horizontal bar 55, the spirit level II, is permanently fixed to tbe top of the telescope, and does not, therefore, admit of adjustment, and the compass box C is supported over the level by four small pillars at- tached to the horizontal bar. This construction makes the in- strument very firm and compact. This instrument may be adjusted in the same manner as that described for the Dumpy level, witb the exception that it does not allow of the second adjustment, the level tube as before described, being permanently fixed by the maker. When the axis of collimation has been made horizontal, then if the bubble is far from the centre, the instrument must be rejected as a bad one ; but if it be only a little out, the error can be calculated 107 by means of the scale on the bubble tube when the staves are placed at unequal distances, in which case only, the readings would be aiFected by it. Having discovered by Grravatt's method that the instrument is a good one, i. e. that the level tube has been correctly fixed, should the instrument again get out of adjustment, we can rectify it by the following method, which when practicable, is both simple and accurate. Drive t^fo stakes into a pond or other con- venient piece of water, at about a distance of one hundred yards one from the other, with their heads level with the surface of the water, the stakes should be so placed that you can put your instru- ment in a line with them and close to one of them. Now, having levelled your instrument, and staves being held upon the stakes read the nearest one, then if the further one does not give the same reading, the horizontal wire must be moved up and down, until it does so, by means of the diaphragm screws. This method does not apply to the Dumpy level, as we have no means of ascertaining if the level is correct ; and, therefore, of levelling it. Having now given a description of the ordinary levelling instru- ments, I will return to the practice of levelling. The following will explain the method to be pursued in levelling a tract of country. In the first instance the stafi'-holder must place his stafi* on the Bench mark* from whence the levels are to commence. The Sur- veyor must then set up his spirit-level in the most suitable spot which presents itself, from whence he can have an uninterrupted view, not only of the stafi" at; the back station, but also for a consid- erable distance in the direction he wishes to carry his levels. The * In the practice of levelling, it is usual to leave at convenient intervals, what are called Bench marks; these mostly consist of permanent objects, such as stumps of trees, rrilestones, &c., on which it is usual to cut a distinguishing mark, that it may be known hereafter. Their use is chiefly for future reterence, in the event of its being necessary, either to check the levels by repetition, to change the direction of the line of levels from any point, or to take up and continue the levels at the commencement of a day's work, a Bench mark having been left at the close of the day preceding. 108 station selected should not in any cas9 exceed 4 or 5 chains, for when long distances are taken, unless both the back and forward stations are equally distant from the instrument, errors w411 gradu- ally creep in, which in a long series of levels, are liable, by their accumulation, to be of serious consequence. The proper station being determined on, and the level adjusted for observation*, it must be directed to the back staff and the foot and decimal fraction of a foot, with which the central part of the horizontal wire appears to be coincident, noted w^ith all possible exactness, and entered in the proper column of the Pield or Obser- vation Book ; as soon as it is registered, look to see that the spirit bubble has not returned from its central position, and then repeat the observation, to ensure that no mistake has been made in noting it. The back observation being made, turn the Telescope round in the forward direction ; then look at the spirit bubble, and if it has at all changed its position, by receding towards either end of the tube, bring it back to the centre by the parallel plate screws, then observe what division on the staff is intersected by the cross-wire, and enter the reading in the proper column of the Pield-book. Having entered it, verify it by a second observation, which will complete the first levels. The first levels being completed, the Surveyor, passing the man who holds the forward staff, proceeds to some convenient spot to set the instrument a second time, which, as before remarked, should not be more than 4 or 5 chains distant ; * The Level must be adjusted for observation in the following order; First draw out the eyepiece of tlie Telescope till the cross-wires are perfectly defined ; then, di- recting it to the staff, turn the focussing screw on the side of the Telescope, till the smallest graduations on the statF are likewise clearly distinguishable; that these two adjustments be very carefully and completely performed, is of more con- sequence than is generally supposed, for upon them depends the existence or non- existence of parallax, to remove which has already been explained at page 42. The ebove operations having been gone through, bring the spirit bubble into the centre of its glass tube, which position it must retain unmoved in every direction of the instrument ; this is accomplished in the same manner as in the Theodolite, by bringing the Telescope successively over each pair of the parallel plate screws, and giving them motion, screwing up one, while uniicrevving the other to an equal extent. 109 the man who held the staff at the back station lilvewiso proceeds still further onwards to take up a new station, and as "nearly as possible at the same distance from the instrument, as the instru- m.ent is from the staif, which has now become the back station. The instrument is tben again adjusted, and the same process follow- ed as above described, until arrived at the end of the series. The foregoing description of the method of taking levels is general, and applies equally to every kind of levelling operation. The following is the form of Field-book used for entering the observations, &c. .2 CO ;-< Back. Forward. ■ (5 0) > s p5 Reniarks. 6 o a 13 1 ■CD c P5 .s C P o / I o / 1 200 271-30 5-85 3-50 87-00 200 2-35 100-00 2 200 269-30 4-75 4-50 91-00 200 •25 102-35 4-3 S-l o 200 267-15 2-88 8-75 89-50 262 5-87 102-60 o 4 100 231-30 0-18 9-63 48-00 100 9-45 96-73 5 150 221-30 0-06 6-80 16-30 200 6-74 87-28 G 200 181-30 4-71 2-18 301-30 200 2-53 80-54 7 200 96-30 11-50 0-18 262-30 200 11-32 83-07 d 8 300 74-00 11-00 1-48 248-45 300 9-52 94-39 5 9 4U 71-30 7-32 1123 184-00 413 3-91 103-91 Q Sometimes a column is inserted between the back and forward readings, in which is noted the height of the instrument at each station, thus furnishing the level of the ground at that point also. It is usual to refer all levels to some fixed datum line, which is easily recognisable, suck as the mean level of the sea ; in England, Trinity Higli AVater Mark ; in Calcutta, the sill of the stone on the Tide Guage at Kyd's Dockyard, &c. The reduced levels shew the height above or below such datum line. Where this is not done, 110 ifc is usual for convenience sake to take an imaginary datum line, some even number of feet above or below the first, or any other station, and reduce all the levels to this, the object is to save the necessity of using signs (+ or — ) in the reduced levels. "Where levels are made for the formation of a section it is neces- sary that the distance between the levelling staves be measured, as well as the bearing observed of each staff to enable the Surveyor to plot and draw the section, but in running or check levels, there is no necessity for the chain or compass, the object of check levels being only to obtain the difference of level between certain inter- mediate and the extreme points of the section previously made, to check its accuracy. It is also immaterial by what route we pro- ceed from one point to another, so that such spots may be selected for the stations as are most convenient for the purpose, and may afford opportunity of checking any intermediate points on the sec- tion line. The Pield-book required therefore for check levels, is merely a simple entry of back and fore-sights, the difterence of the sums of which will be the difference of level between the extreme points of the Section line. In plotting sections of levels a larger scale is generally used for the vertical than for the horizontal distances. Ey this means space is economised in the length of the section, and the slopes of the ground, especially in roads, railroads, canals, &c., the depth of cuttings, height of embankments, &c., are shewn with much greater clearness than if the two scales were equal. Ten to one is a con- venient proportion between the scales. LEVELLIlS^a "WITH THE THEODOLITE. In taking sections across broken irregular ground intersected by ravines, this system of operation is recommended as being much more easy and rapid than tracing a series of short horizontal lines with the spirit level. Where, however, this latter instrument can be used with tolerable facility, it should always be preferred. Level- Ill ling for sections by angles of elevation and depression -with the theodolite is thus performed. The instrument is set up at one extremity of the line previously marked out by banderoles or long pickets at every change of the general inclination of the ground ; and a levelling-staif, with the vane set to the exact height of the optical axis of the Telescope, being sent to the first of these marks, its angle of depression or elevation is taken, and by way of insuring accuracy, the instrument and staff are then made to change places, and the vertical arc being clamped to the onean of tlie two readings^ the crosswires are again made to bisect the vane. The distances may either be chained before the angles are observed, marks being left at every irregu-larity on the surface where the levelling staff is required to be placed ; or both operations m.ay be performed at the same time, the vane on the staff being raised or lowered till it is bisected by the wires of the Telescope, and the height on the staff noted at each place. The accompanying sketch explains this method : — A and B are the places of the instrument, and of the first station on the line, where a mark equal to the height of the instrument is set up ; between these points the intermediate positions », 6, c, f?, for putting up the levelling staff are determined by the irregularities of the ground. The angle of depression to B is observed, and if great accuracy is required, the mean of this and the reciprocal angle of elevation from B to A is taken, and the vertical arc being clamped to this angle, the Telescope is again made to bisect the vane at B. On arriving at B, after reading the height of the vane at a^ J), c, Sfc, 112 and measuring the distances A a, ^c, the instrument must be brought forward, and the angle of elevation taken to C, the same process being repeated to obtain the outline of the ground between B and C. In laying the section down upon paper, a horizontal line being drawn, the angle of elevation and depression can be protracted, and the distances laid down on these lines ; the respec- tive height of the vane on each staff being then laid off from these points in a vertical direction, will give the points a, h, c, Sfc, marking the outline of the ground. A more correct way of course is to calculate the difference of level between the stations, w^hich is the sme of the angle of depression or elevation to the hypothenusal distance AB considered as radius, allowing in long distances for curvature and refraction. Instead of only taking the single angle of depression to the distant Station B, and noting the heights of the vane at the inter- mediate Stations, a, h, c, Sfc, angles may be taken to mark the same height as the instrument set up at eacli of these intermediate points, which will equally afford data for laying down the Section ; but the former method is certainly preferable. The details may be kept in the form of a Field-book, but for this species of levelling the measured distances and vertical heights can be written without confasion on a diagram, leaving the corrections for refraction and curvature (when necessary) to be applied when the section is plotted. "Where a number of cross sections are required, the Theodolite is particularly useful, as so many can be taken without moving the instrument. It is also well adapted for trial sections, where minute accuracy is not looked for, but where economj^, both of time and money, is an object. The theodolite is likewise used in running check levels, to test the accuracy of those taken in detail with a S2nrit-level. Eecipro- cal angles of elevation and depression, taken between bench marks, whose distances from each other are known, afford a proof of the 113 general accuracy of the work ; and if these points of reference are proved to be correct, it may safely be inferred that the intermedi- ate work is so likewise. CO^'TOURIKG. The last description of levelling by the spirit level to be noticed, is the method of tracing in'strumentally, horizontal sections termed "contours," either round a group of isolated features of ground for the formation of plans for drainage, sanitary, railway, or other en- gineering purposes ; or over a whole tract of country with a view of giving a mathematical representation of the surface of the ground in connection with a national, or other extensive and accurate sur- vey. Contour lines give a most prefect delineation of the ground, and they are the only part of a survey which will remain unaltered in the lapse of ages, hills and valleys being much more permanent things than houses, roads and boundaries, which cease to give accu- rate information in a few years and require revision at a great cost. It would be useless expense to increase the number of contour lines on mountain ground where no probable demand either for roads or drains exists ; and on the other hand in districts which are nearly level, contours only at great diiference of altitude would be of little practical utility. In waste lands, contours tend to a knowledge of the best mode of improvement, as the levels are connected with each other throughout the country, and referred to the sea as a datum line. As a general system, however, contouring can scarcely be said to be applicable to India, where the mountains are inaccessible and for the most part untrodden, and the wastes impenetrable and im- pervious, from the denseness of the jungle and rankness of the vegetation. The undulations and round smooth downs of Englaud are here wanting, and the vast extent of the country leaving but few points fixed by the great triangulation, the operation, so simple 114 ou the Ordnance Survey of England, would be one of much diffi- culty in this country, where there is so little to mark the inequali- ties of the surface until the stupendous hills I'ise suddenly and pre- cipitously above the general level. A few remarks on the system, however, which has become so common in England, w^ill not be misplaced. " The method of tracing these contours in the field is thus per- formed. Banderoles or long pickets are first driven, one at the top and another at the bottom of such slopes as best define the ground, particularly the ridge lines and watercourses ; should no such sensible lines exist, they must be placed at about equal intervals apart, regulated by the degree of minutise required, and the variety in the undulations of the surface of the ground. A short picket being driven on the level of the intended upper (or lower) line of contours, and in line between two of the ban- deroles, the level is placed so as to command the best general view of this first line and adjusted, care being taken that its axis is not so low as to cut the ground below the picket (or so high as to be above the top of the levelling-staff", if the lower contour is the first traced) ; the staff is then placed at this picket, and the vane raised or lowered till it is intersected by the cross-wires of the Telescope, the staff is then shifted to another point on about the same level, and in the line between the next two pickets, and the staff itself moved up or down the slope till the vane again coin- cides with the cross-wires, at which spot another picket is driven. This operation is continued, till it is necessary to move the level to continue the same upper contour lines, when (the staff" beiog placed at one of the pickets just driven) the vane is again raised or lowered to suit the next position of the axis of the instrument and kept at this height, as before, for the continuation of the line. To trace the next lower contour line, it is merely necessary to raise the vane on the staff, five, ten, or whatever number of feet may be the vertical distance determined upon, and proceed as 115 before. When the level itself has to be moved to lower ground, it must be so placed that its axis will cut the ground above one of the pickets of the line just marked out, and the same quantity of li?e or ten feet added to the reading of the staff at this picket, will give the height of the vane for the next lower horizontal line. " The use of driving all these pickets, marking out the contours nearly in the same line down the slopes, becomes evident when they are to be laid down on the plan, the places of the original banderoles or long pickets being fixed with reference to each other, it is only necessary to measure between them, entering the distances on these lines, with the offsets to the right or left to the different short pickets marking the horizontal lines." The instrument, best adapted for contouring where a rapid delineation of country is au object frequently of greater importance than accuracy, is the water-level its best recommendation being the facility with which it can be made and requiring no adjustment when using it. The following description is taken from " Erome on Surveying." The Erench water-level is much used, on the continent, in taking sections for military purposes. It possesses the great advantage of never requiring any adjustment, and does not cost the one-twentieth part of the price of a spirit-level. From having no telescope, it is impossible to take long sights with this instrument ; and it is not of course susceptible of very minute accuracy : but on the other hand, no gross errors can creep into the section, as may be the case with a badly adjusted spirit-level, or a theodolite used as such, the horizontal line being adjusted by nature without the interven- tion of any mechanical contrivance. As this species of level is not generally known in England, the following description is given ; which, with the assistance of the sketch, will enable any person to construct one for himself without further aid than that of common workmen to be found in every village. a 5 is a hollow tube of brass about half an inch in diameter, 116 and about three feet long, c and d are short pieces of brass tube of larger diameter, into which the long tube is soldered, and are for tlie purpose of receiving the two small bottles eand/, the ends of which, after the bottoms have been cut off, by tyiiig a piece of string round them when heated, are fixed in their positions with putty or white lead ; the projecting short axis g works (in the instru- meut from v\'hich the sketch was taken) in a hollow brass cylinder h, which forms the top of a stand used for observ- ing with a repeating cir- cle ; but it may be made in a variety of ways so as to revolve on any light portable stand. The tube, when required for use, is filled with water (coloured with lake or indigo), till it nearly reaches to the necks of the bottles, which are then corked for the convenience of carriage. On setting the stand tolerably level by the eye, these corks are both with- drawn (which must be done carefully and when the tube is nearly level, or the water will be ejected with violence) and the surface of the water in the bottles being necessarily on the same level, gives a horizontal line in whatever direction the tube is turned, by which the vane of the levelling-staff is adjusted. A slide could easily be attached to the outside of c and d, by which the intersection of two cross wires could be made to coincide with the surface of the water in each of the bottles ; or floats, with cross hairs made to rest on the surface of the fluid in each bottle, the accuracy of their intersection being proved by changing the floats from one bottle to the other ; either of these contrivances would render the instrument more accurate as to the determination of the horizontal 117 line of sight ; though cue of its great merits, quickuess of execu- tion, would be impaired by the first, and its simplicity affected by either of them. For detailed sections on rough ground, where the staff is set up at short distances ajpart, it is well qualified to super- sede the spirit-level ; and is particularly adapted to tracing contour lines. 118 COKEECTIONS FOR CURVATURE AND EEFRACTION. Showing the difference of the Apparent and True Level in Feet, and Decimal parts of Feet, for Distances in Feet, Chains, and Miles. Correction in Feet. Correction in Feet. '3 r^ V a a; C r; M S3 rt ? c3 • r-t tM U V- -^ Pi 3 ^ C3 4; Pi 13 aj UC3 IH Ut >- -C3 to S-i u t--d "ti o o o o o - ^ (^ Ph Ti Q t^ fi^ fi- i p; Correction in Feet. 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 1950 2000 •00024 •00054 •00096 •00149 •00215 •00293 ■00383 "00484 •00598 ■00724 '00861 •01010 •01172 '01345 '01531 '01728 "01938 '02159 '02392 '02638 '02895 "03164 '03445 "03738 "04043 "04361 "04689 "05030 "05383 "05748 '06125 '06514 "06914 "07327 "07752 "08188 "08637 "09098 '09570 00004 00008 00013 00021 00031 ■00042 00055 ■00069 •00085 ■00103 ■00123 •00144 ■00167 •00192 ■00219 •00247 •00277 •00308 ■00333 •00377 •00414 ■00452 ■00492 '00534 •00578 •00623 •00670 •00719 •00769 •00821 •00875 ■00931 •00988 •01047- •01107 ■01170 01234 01300 •01367 000-20 00046 •00083 •00128 ■00184 •00251 •00328 •00415 •005 13 •00621 •00738 •0OS66 •01005 •01153 •01312 •01481 01661 01851 02059 •02261 •02481 •02712 •02953 •03204 •03465 •03738 •04019 •04311 •04614 •04927 ■05250 •05583 •05926 •0628(1 •06645 •07018 •07403 •07798 •08203 1-0 1-5 2 2^5 3^0 3-5 4-0 4-5 5-0 5-5 6-0 6-5 7^0 7-5 8^0 8^5 9-0 9-5 lO^O 10^5 iro 11-5 12-0 12-5 13-0 13^5 14^0 14-5 15^0 15^5 16-0 16-5 17^0 17^5 18^0 18^5 19^0 19^5 20-0 00010 00024 Ci0042 •00065 •00094 •00128 ■00167 •00211 ■00261 ■00315 •00375 •00440 00511 •00586 •006G7 •00753 •00844 •00940 •01042 '01149 •01261 •01378 •01501 ■01628 ■01761 •01899 •02043 •02191 •02345 •02504 •02668 •02837 •03012 •03192 •03377 •03567 •03762 •03963 •04169 00001 00003 00006 00009 00013 00018 00024 00030 00037 00045 ■00054 ■00063 •00073 •00084 •00095 •00108 •00121 •00134 •00149 •00164 •00180 •00197 00214 00233 ■00252 •00271 •00292 •00313 •00345 •00358 •00381 •00405 •00430 •00456 •00482 •00509 •00537 •00566 •00596 00009 ■00021 00036 00056 00081 001 10 ■00143 ■00181 •00224 ■00270 ■0032! •00377 ■00438 •00502 •00572 •00645 •00723 •00806 00893 •00985 •01081 01181 •01287 •01395 •01509 •01628 ■01751 •01878 •02010 •02146 •02287 •02432 •02582 •02736 •02895 ■03058 •03225 •03397 •03573 1 1| 2 2h 3 H 4 4| 5 5^ 6 H 7 71 ' 2 8 H 9 9h 10 11 12 13 14 15 16 17 18 19 20 •0417 •1668 •3752 •6670 1 5008 2^6680 4-1688 6^0030 8-1708 10-6720 13^5468 16-6750 20^1769 24^0120 28^1809 32^6830 37^5190 42-68S0 48^1910 54^0270 60^1971 66^7000 80-7070 96-0480 112^7230 130^7320 1500750 170-7520 192 7630 216^10S6 246-7870 266^8000 •0060 •0238 •0536 •0953 •2144 •3811 •5955 •8561 M673 1-5246 1-9295 2-3821 2^8824 3-4303 4-0258 4'6690 5-3599 6-0997 6-8844 7-7181 8-5996 9-5286 11-5296 13 7211 16-1033 18-6760 21-4393 24-3931 27-5376 30-8727 34-3981 38^1143 •0357 •1430 •3216 •5717 1^2S64 2-2869 5733 1469 0035 1474 11-5773 14-2929 17^2945 20^5817 24^1551 28-0143 321591 36-5883 41-3066 46^3089 51-5975 57-1714 69-1774 82-3269 96-6197 1 12-0560 128-6357 146-3589 165-2254 1 85^2359 206-3889 228-6857 119 CHAPTEE YII. E A I L W A T C r E Y E S . Curves aro not necessary for plain roads, but it is always a much neater method to join two straight portions of any road by a re- gular curve than bj a mere bent line ; and it is also to be borne in mind that it may b@ desirable to convert a common road iuto a railway, in which case considerable trouble and expense wiU be saved, if all the alterations of directions have been made of regular curves. Curves may be laid out in several ways, a few only of the most useful methods for ordinary practice are given here. The length of radius to be given to a curve is manifestly inde- terminate, for if AB, EC be two y'>^ portions of a line of road meeting in E, and it be required to unite them by a circular arc DE, we have by plane trigonometry, DB = DO tan. DOE, and since DOB is constant, DB varies aa DO, we may therefore either fix the point D and its corresponding point E in the other lines, and find the radius DO by the above equation ; or we may assume the length of the radius DO, and thus determine the length of the tangent DB, that is, the distance of D from the intersection E of the two lines. 120 lu practice the first method M'iil generally he necessary, for if any obstruction P, as a aycII or house, be situated near the line, the commencement of the arc must be taken so that P may be en- tirely within or without the curve. Again if there be no features of the ground to determine the question, as often happens in India, frcm the open nature of the country, a good method will be to fix the point D at some convenient distance from A the commence- ment of the road, with reference to the division of the road into miles, furlongs, &c., remembering however that the radius thus de- termined should never be less than i mile, and that ceteris parihtcs, the greater the radius the better the road. The number of cords must depend on the amount of curvature ; the fewer chords there are, the less trouble there will be in laying out the curve, while on the other hand they must not be so long as to have any sensible inclination to one another. The versed-sine of half the angle subtended by the chord at the centre, gives the greatest deviation of the chord from the arc ; and by finding this versed-sine for difi'erent numbers of chords, the Engineer will, generally, after two or three trials, be enabled to fix their number. METHODS OF LATIlN^a OUT THE CURVE. \st Method, — Centre invisible, no angular instrument required. Find the length of the sine of 1° to the ^. Be given radius, and lay ofi" this distance in continuation on the straight line as .s^ from B to C, and from the point thus \ obtained, lay ofi" at right angles the \ versed-sine to the same augle and ra- dius. The first point D in the curve is thus fixed and by producing the chord BD (" 2 sin 30' X radius,) to a distance DE (= BD cos 1°,) and from the point E setting off at right angles EF = BD sin 1°, a 121 second point is tlius obtained. By continuing to produce tlie last chord in a similar manner and setting oil the offset, the remaining points are established. 2ncl. Iletliod. — Same conditions as 1st. vSometimes the ground without the curve, only, is adapt- ed for chain measurem.ents. In which case, when the curve is not a long one i. e., does not ex- ceed say one-quarter of its ra- dins, the following method may be found useful. Lay off equal distances Ap^, p^ p^, p^ p^ &c., from A along AC^ and perpen- dicular offsets p>i ^'-1 !>■> ''^^2' ^^-f the points m^ m.^, &c., fixing the curve. To find these offsets. p)^ m^ = Ap^ tan. CAw?j 2?^ m^ = 2^ X p^ m^ p^ mn — - n^ X p^ m^ Por draw m^ n^, m^ n^ &c., perpendicular to AO Then v.i^ n^ == Ap^ = An^ (2r—An^ = An^ X 2r nearly, since A??^ is verj^ small compared with r =^ p^ «?i X 2r w^ n."^ — Ap.^ cz: 2- Aj)^ = An^ X "Ir nearly = p^ m^ X 2r or p^ m^ =^ ^2? p^ m^ similarl}^ p^ m^ =z S^ 2\ '^i &C. =. &G. Case. II. Jhj ilie same method to lay out ilie curve when it is a long one. In a long curve, the tangents, if prolonged to their point 122 of meeting, would necessarily fall at a great distance from the curve, thus giving an incon- venient length to the offsets which in prac- tice should never ex- ceed two chains. To remedy this incon- venience, the curve must be divided into two or more parts, by introducing one or more additional tangents, and thus the offsets may be confined within their proper limits. In the annexed figure the curve AC is divided into two unequal parts at B ; at which point the tangent DBE is introduc- ed to meet the tangents AD, CE in D and E. The tangent AD must first be measured to an extent not exceeding one-eighth of the radius AO. Then in the right-angled triangle ADO, AO, DO are given from which the angle ADO can be formed, this angle, being doubled, gives the angle ADB which determines the direction of the tangent DBE. If no angular instrument be at hand to lay ofi:' this angle, the length of CE can be calculated and by measuring off the distance CE, the point E will be obtained, and by joining D and E, the tangent DBE. This done, the ofi:sets to the curve may be laid ofi" as in the last case the order of offsets being inver- ted in DB and again in BE. drd. Method. — Sa)ne conditions as first. Sometimes it may be most convenient to lay out the curve by oft'scts from its chord or chords, where obstructions, on its convex side, prevent the use of the proceeding method. Let ACB, be a portion, or the whole, of a railway curve ; HA, a tangent at its commencement ; TC, a tan- gent to its middle point C, Take if possible the chord AB, an even number of chains ; find the successive ofisets to the radius 123 AO and tlie tangent TC (r- AD = one-half AB,) the last offset TA will be = CD; from CD subtract the suc- cessive offsets, and the remainders will be the offsets p^ <2'i <^c. which must be set off in an inverted order from A to D, and their order must be again inverted in setting them off from D to B. If the curve be not yet completed, the operation may be continued hj taking other chords, as BE.* ^tJi. Metliod. — Centre visible and accessible, -no angular instrument required. This method may sometimes be used with advantage especially for curves of large radii, find the centre of the curve by the intersections of the perpendiculars to the tangents at the points of contact, place a signal staff at that intersection, and also at the intersection of the straight lines produced. Divide these lines into any convenient number of equal parts, and set off at each in the direction of the centre a distance — \Jr^ + d'^—r, r being the radius of the curve, and d the distance from the point of con- tact to the point on the straight line. 6tJi. Metliod. — Centre visible and accessible, angular insttn^nenf necessary. Where the curve is quick and the ground over which it passes is hilly, but at the same time is commanded from the points cf contact and centre, this method is attended with great advantages, in as much as the points are established independently of each other, and are free from all error which may arise (and which in the previous method must be allowed for) from the slop- • jVo/e.— It will be seen that the tangent TC, is not used in the operation further than to explain the nature of the method of obtaining the offsets. R % 121. ing of the ground. This and the succeeding two are the simplest methods of all, as they require no calculation or measurement of offsets. They depend on the well known properties of the circle, that the angle contained by a tangent and a chord is equal to the angle in the alternate segment, and the angle at the centre is double of that at the circumference. The appropriate radius having been selected, the angle which a chord of about the length which it is required to have the points apart, would subtend at the circumference, (^^ e. the angle in the alternate segment) is computed . , „ , . f, ,.. ■ -, d X rad. of tables , irom the lormula, sine oi the angle :== , where. « 2 r = length of chord, and r = radius of curve : then supposing r = 20 chains, the angle which a chord of 100 feet would subtend would be 2° 10' 15'' ; with an ordinary theodolite this angle could only be laid off approximately ; an angle of 2^ would therefore be adopted the points due to it being 92-13 feet apart. This being chosen then as the angle, two theodolites would be set up one at B the other at 0, that at B ha- ving its telescope directed ^^ .^ on D, and that at O on B and having clamped their lower plates, the points in the curve wnll be obtained by the intersection of the arcs formed by moving them though 2^ and 4° re- spectively ; for the angle DB p == angle in alter- nate segment of the circle =: one-half the angle at the centre BOj?. An assistant must of course move by signal a flagstaff J until it is intersected by both theodolites when he will put down a picket to finally mark the spot. 6th. Metliod. — Centre invisible, angular instrument required. The pcints in the curve in this method are fixed by intersections 125 from the points of contact of the tangents ; first find the angle ACB, suppose it to be 2a, then AOC = 90" — a, and if the re- quired curve is to be composed of n chords, and Ap^ is the first of these, Pi ^2 ^^® second, &c., Then Z. CA;^, =-. ^^""""^ n and /. CBp, = (90" -a) ^-^ of therefore a flagstafl" be moved until it comes into the intersec- tion ^^^ of Ajt^ and By^, ^^ will be a point in the curve. Similarly Z. Cj\. j^o =: 2 Z CB«, = (90-a)'^^ and so on. 7tJi. Metliod.—Same conditions as sixtli,~lt will often happen that the services of a second person capable of using the theodo- lite are not available, in which case the above method cannot be applied, the curve must tben be laid out by theodolite and chain, calculate the lengths of the chord Ap^, Ap.^, Ap.^, AF, (see last fig.) lay off banderoles along the lines Ax^, Ax^ &c. (the angles CA:rp QAx^ &c. being successively /3, 2 /3, 3 /3 &c. and /3 being found as already explained) and measure the chords along them. Proceed similarly from the point B. The point E the centre of the curve 126 slioulcl of course coincide with tlie same point as measured from A; if any small difference exist, the mean of the two points should be taken. To find the lengths of chords ^ • A0», A^i = 2 sin — ^e^ A^2 = 2 sin AO/>j &C. r= &C. This method has the disadvantage of requiring a seperate cal- culation for each chord, but this will not be of consequence when the number is small. It posessess also the farther disadvantage that an error made in measuring the length of any chord will not be detected ; on the other hand, any error in one chord will not affect the accuracy of the rest, as would be the case in many of the methods ordinarily employed. Tlie Compound Curve, consists of two, three or more portions of arcs of different radii, and is adopted where the line is required to pass through given points to avoid obstructions, or where a princi- pal station or terminus is required. Case I. Tojiiid the radius of the compound curve, the starting point and one radius being given. From the given point B in the tangent AB, draw the given radius BO perpendicu- lar to AB ; and draw the curve to some point C, w^iere it is found convenient to change the radius ; draw the radius OC, and perpendicular there- to draw CT', meeting the tan- gent DT in T'; make TC = TC, and from C draw C\y at right angles to TC, meeting CO, prolonged if necessary, in 0' ; then 0' is the centre of the arc CC of the curve, conformable to the nature of tangents. 127 Case II. One of tlie two radii of the_ compound curve, and its sta7'ting and closing points heing given, to find tJie other radius. Let AB, CD be the tangents, B and Q' the starting and closing points of the curve. Draw the perpendiculars BO = C'H = given radius, to the tangents ; joiu OH, and bisect it in E; draw EO' per- pendicular to OH, meeting C'H prolonged in O' ; join 00' and pro- long it till OC ==: C'H: then the points 0,0', are the centres of the arcs BC, CC, which constitute the curve, O'C = O'C being the radius required. The Serpentine Curve, is used in railways when obstructions or some other cause' render its adoption preferable ; it consists of two circular arcs of different or the same radii, having their convex sides turned in opposite directions, like the letter S, whence it is some- times called the S curv^e ; the two portions of the curve have a com- mon normal at their point of junction, and therefore a common tan- gent at the same point. This curve affords the most easy means of joining two parallel or nearly parallel, portions of a line of railway. Case I. One radius and its tangential point heing given, to find the other radius and tangential point of the Serpentine Curve. Erom the given tangential point C draw the radius CO perpen- dicular to the tangent CD, and draw the curve CGr to some point Gr Avhere it is found convenient that it should have its point of .contrary flexure; throuj;h OGr draw the normal OGO'; from G draw GT at right angles to OGO' to meet the tangent AT; make 128 TB = TGr; aiid draw BO' perpendicular to AT, meeting OGO' in O' ; then O' is the centre, and O'B = OG- is the radius of the curve BGr, as is evident from the nature of tangents. Case II, When the tangential points and o?ie of the radii are given to find the other radius. From the given tangential points C and B draw CO, BH respec- tively perpendicular to the tangents CD and BA, and equal to the given radius ; join OH and bisect it in P'; draAV FO' at right angles to OH, jmeeting HB prolonged in O', and join 00' ; meeting O'Gr =: O'B, then is the centre, and O'B =: O'Gr is the radius of the portion BGr of the curve, as required. Case III. When the two portions have the same radius, to deter- mine that radius, the tangential points and their distance heing given. Let AB, CD be the tangents. B and C the given tangential points, and BC the given distance, draw Bo = Co' respectively perpendicu- lar to AB, CD and of any convenient length ; through o, parallel to BC draw o q indefinitely ; with the compasses apply o'o" = 2 Co' -"^0 — 2Bo ; through C,o" draw Co" O, meeting BO prolonged in O ; and though O, parallel to o"o' , draw 00', meeting Co' prolonged in O'; then 00' are the centre, and OB and O'C are the equal radii of the serpentine curve BGC, the common normal of the por- tions Ba, GC of the curve, beiug OGO' rzr 2B0 = 2C0'. Curve of Deviation. In some cases it may be necessary to make 129 a given deviation from a straight line of railway, so that the works may avoid a building or other obstruction situated on or near it, this is done by means of three curves as follows. Let ABCD be a straight portion of the rail\^'ay, Ji a building or other obstruction on the line. Take HQ of a sufiicient length for a deviation, that the line may avoid the object at h ; and tiirough Q draw a curve GQGr' of radius QO' eq^ual to, or greater than one mile. Draw also two curves BGr, Gr'C, of like radius, to the first curve at Gr- and Gr', and the line at B and C ; then the lines 00' and O'O"' joining the centres of the curves, will pass through their contrary points of flexure at Gr and Gr'. Put r = common radius OB = O' Q = O'T, and d = required deviation = HQ ; then BH = H C ~ V*^ C^ ^ — d) and the four equal chorda BG-, CG-, &c., are each equal to s/d r. Having given these various methods of determining the radii and common normals, indicating the positions of the tangent pointa of the parts of the Compound, Serpentine and Deviation curves, the manner of laying out the curves themselves by the previous me- thods, according to circumstances, will be readily seen, recollect- ing, that when junction point of curves of different radii occur ; to commence the operation afresh, by using the radii and tangent of the respective portions of the curve. USEFUL PROBLEMS IN SURVEYING. Problem I. To draw upon the ground a straiglit line through two given points. Plant a picket, or staff, at each of the given points, then fix an- other between them, in such a manner that when the eye is plac-ed s 130 at tlie edge of one staff, the edges of the other two may coincide with it. The line may then be prolonged by fixing np other staves. The accuracy of this operation depends grciitly on fixing the staves upright, and not letting the eye be too near the staff from ^vhence the observation is made. Peob. II. To walJc in a straight line from a proposed ^oint to a given ohject. Fix upon some point, as a bush, or a stone, or any mark that you mtiy find to be in a line with your given object, and walk forward, keeping the two objects strictly in line, selecting a fresh, mark when you come within 20 or 30 paces of the one upon which YOU have been inovino^. Observe — that to walk in a direct line, it is always necessary to have two objects constantly in view. Peob. III. To trace a line in tlie direction of two distant points. Let two persons separate to about 50 or 60 paces ; then, by alternately motioning each other to move right or left, they soon get exactly into line with the distant objects : or, for greater accu- racy, they may hold up staves. In sketching ground, it is constantly necessary to get in lino between two objects : if these are not very distant, a well-drilled soldier can alway do so within a few paces (near enough for sketching purposes) by fronting one object exactly, and then fiicing to the right about ; when, if he finds himself accurately fronting the other object, he will be tolerably well in line with them. . A right angle may also be formed very nearly by fronting an object, and then facing to the rigid or hft. Peob. IV. IIoio to lag off a perpendicular loith the chain. Suppose A the point at which it is required to erect a right angle 131 fix an arrow iuto the ground at A, througli the ring of the. chain, marking twenty links ; measure foi^fi/ links on tlie line AB, and pin down the end of the chain firm- ly at that spot, then draw out the remaining eighty links as far as the chain will stretch, holding by the centre fifty -link brass ring as at C — the sides of the triangle are then in the proportion of three, four, and five, and consequently CAB must be a right-angle. An angle equal to any other angle can also be marked on the ground, with the chain only, by measuring equal distances on the sides containing it, and then taking the length of the chord : the same distances, or aliquot parts thereof, will of course measure the same angle. Peob. lo avoid an olstacle, sucli as a house, in your chain line. The usual way of avoiding an obstacle of only a chain or two in len^^th such as a house, is by turning off to the right or left at right angles till it is passed, and then returning in the same manner to the origi- nal line. A more convenient method is to measure on a line making an angle of 60^ with the ori- c gmal du'ection a distance suf- ficient to clear the obstacle, and to return to the line at the same angle, making CD = BC the distance BD is then equal to either of these measured lines. 132 Pros. YI. To find the length of the line AB accessible only at loth ends. Having fixed on some convenient point O, measure BO and AO ; and prolong those lines till OG — OB, and OD == OA ; then the distance between the points D and C ^Yill be equal to AB, for the sides of the triangle COD, BOA, about the equal angles at are respectively equal, therefore the third sides CD, BA, will also be equal. Peob. Yll. To find the distance of an inaccessible object by means of a rhombus. With a line or measuring tape, Avhose length is equal to the side of the intended rhombus, lay down one side BA in the direction BO, and let BC another side be in any conveni- ent direction : fasten two ends of two of those lines at G and A; then the other ends (at D) being kept together, and the lines stretched on the ground, those lines AD, CD, will form the other two sides of the rhombus. Set up a mark at E, where OG, AD, intersect; and measure ED ; then the sides of the triangles BDC, CBO, being respective- ly parallel, the triangles will be similar : hence, ED BO. Suppose the side of the rhombus is 100 feet, and ED = 11 ft, 7 in.— then, ll/y : 100 : : 100 : 863 feet nearly = BO. DC :: CB 133 If be ground be nearly level, a rhombus, whose side is 100 feet, will determine distances to the extent of 300 yards within a very few of the truth. Peoe. VIII. To find flie length of the line AD, inaccessible at the point D» The measurement of the line AD, supposed to be run for the de- termination of a boundary, is stopped at B by a river or other obstacle. The point E is taken up in the line at about the estimated breadth of the obstacle from B ; and a mark set up at E at right angles to AD from the point B, and about the same distance as BE. The theodolite being adjusted at E, the angle EEC is made equal to BEE, and a mark put up at C in the line AD ; EC is then evident- ly equal to the measured distance EB. If the required termination of the line should be at any point C, its distance from B can be determined by merely reversing the order of the operation, and making the angle BEE' equal to BEC, the distance BE' being subsequently measured. There is no occa- sion in either case to read the angles. The instrument being level- led and clamped at zero, or any other marked division of the limb, is set on B : the z^^j^cr plate is then undamped, and the telescope, pointed at E, Avhen being again clamped, it is a second time made to bisect B ; releasing the plate, the telescope is moved towards D till the vernier indicates zero, or whatever number of degrees it was first adjusted to and the mark at C has then only to bo placed 134 in the line AD, and bisected hy tlie intersection of the cross wires of the telescope. If it is impossible to measure a right angle at B, from some local obstruction, lay off arj convenient angle ABE and set up the theodolite at E. Make the angle EEC equal to one half of ABE, and a mark being set up at C in the prolongation of AB, BC is evidently equal to BE, which must be mea- sured, and which may at the same time be made subservient to the purpose of dQlineating the boun- dary of the river. PnoB. IX. To find ilie distance to any inaccessihJe looinf, on tlie oilier side of a river, ivithout the use of any instrument to measure ancdes. Prolong AB to any point D; making BC equal to CD; lay off the same distances in any direction Dc — c5 : — mark the intersection E of the lines joining Be and Ql : mark also E the intersection of DE pro- duced, and of A5— produce Db, and BE, till they meet in a and 135 Peob. X. To find the ^oint of intersection of two Ihies meeting in a lal<:e or river, and tlie distance DIB to the point of meeting. Erom any point E on tlie line AX draw FD, and from any other point E draw ED, produce both these lines to H and G, making the prolongations either equal to the lines themselves, or any aliquot part of either length suppose one-half; join GIT, and produce it to O, where it meets the line CB, then OH is one-half of EB, and OD equal to half of DB ; which results give the point of intersection B, and the distance to it from D. PnoB. XL To fi'iid the hsiglit of a poi:it on an inaccessible hill ivithout the me of instruments. Drive a picket three or four feet long at H, and another at L, where the top of a long rod ED is in a line with the object S from the point A (the heads of these pickets be- ing on the same level) ; mark also the point C, where the head of the rod is in the same line with S, from the top of any other picket B, and measure AE and BC ; lay 136 off tlie distance BC from E to I, and the two triangles ADJ and ASB, are evidently similar, whence PS AB ~Ab HI 1 AP __ AB HI HO ^^^ AF —' A6~HO PS therefore DE. ^^ and AP AF.-^. HO Pbob. XII. To find your place in a Survey Let A and B be two sta- tions, whose places are fixed, and we want to determine the point C. Take the bear- i^ ing of A, 123° N.W. : having done which, we know, tliat C bears from A, 128° S.E. Adjust the protractor at A, by means of the east and west parallel lines, and lay off 128^ S.E. the bearing of C ; Avhich point C must, we know, lie somewhere in the line thus obtained. Next, take the beariTig of B 63° N.E., and having adjusted the protractor at B, lay of 63' S.AV., and where a line drawn from B (to represent this bearing) cuts the line or bearing drawn from A, is the required station C. The above may be put into short rule : thus — To find your sfa- Hon hy ohservations talcen to two j^oints already known. Protract Note. That the nearer your tno bearings meet at a right angle, tlie more correct will the station be determined: and also, that when a third fixed point can be seen, a bearing to it will serve to corrohoraie your other observations; and a point so obtained, namely, by the exact meeting of three bearings, becomes as good as any other point. Tlie above is a very useful problem iudeed, indispensable when sketching ground and filling in a survey. 137 from those points the opposite bearings to what you observe, and their intersection fixes the place sought. For example, if the bear- ing to a point be 20° N.E., protract from that point 20° SAY., &e. Peoe. XIII. To reduce the offset-piece ABODE to a riglit angled triangle AEc, hy an equalizing line Ec, loith the parallel ruler. Draw the indefinite line Ac nerpendicular to AE. Lav the parallel ruler from A to C ; hold the near side of the ruler firmly, and move the further side to B, which will cut Ac? at a^ where a mark must be made. Lay the ruler from a to D, and the further side thereof being now held fast, bring the near side to C, marking A i^> ^^-j of a secon- dary, and consequently -j^, y^-^, &c., of a primary division. If it be necessary, with the same representative fraction -^th, to construct a scale of feet and inches, and to read diagonally to xoth of an inch, we divide AC into tivelve equal parts and proceed as before. Example. — Construct a scale of one mile to 4 inches in yards, and reading diagonally to 10 yards, to measure 2,000 yards. Here the representative fraction = — = -- ^ 1,760x36 440x36 1 X 2,000 50 . . , X = .f!^^ = .f^L. =z 4-54 inches. 440X36 2,000x36 440 11 A length of 4*54 inches is divided into two equal parts, each repre- senting 1,000 yards, the left hand one is subdivided into 10 equal 144 parts, each representing 100 yards. Ten parallel lines and tlie diagonals, &c., as before, complete the scale to fulfil the conditions. Comparative Scales.-— Jn examining a drawing or map when the scale is constructed according to the linear measure of any Foreign country, it may often be necessary to determine the varions dis- tances in English, or other measure, in which case another scale is prepared, having an}?- convenient unit of measure. This second scale is termed "a comparative scale" to the first, and is very easily obtained. Example. — A Map of the Crimea is given, on which the scale is of E-ussian versts ; it is required to construct a comparative scale shewing English miles. Beferring to the scale of versts we observe that 50 versts meas- ure 6"35 inches. Hence as the verst = 1,166"6 yards, is the representative fraction of the scale, and 50x1,166-6x36 ^ assuming that the required scale be sufficiently long to measure 30 miles, proceeding as before wdth simple or plain scales — 1 6"35 x we nave = 50 X 1,166*6 X 36 30 X 1,760 x 36 X = 6;35x^30 x_l^ ^ ^.,^^ .^^^^^ 50x1,166-6 The length of 5*748 inches is subdivided into three equal parts, each representing 10 miles, and the left hand primary division is subdivided into 10 equal parts, each of which is one mile. The scale is completed as before. Each of these scales having the same re- presentative fraction a-soVsT) ^^^ correctness of the comparative scale cannot be doubted. ON COPYING PLANS, MAPS, &C. Theee are several methods of doing this when the copy is to be of the same size as the original, such as placing the plan to be copied 145 with a sheet of paper over it on a tracing glass, placed in such a position that a strong light may fall on it from behind, and then tracing it off, or by placing a sheet of thin paper, having its under side blacked (by rubbing finely powdered black lead, or soft lead pencil over it,) on the sheet of paper that is to receive the copy, the original^ being placed over both, and the whole made steady by pla- cing weights thereon. All the lines of the copy must how be careful- ly passed over with a fine tracing point, and with a pressure propor- tionate to the thickness of the paper. The paper beneath will re- ceive corresponding marks, forming an exact copy, which may afterwards be inked in. "When the drawing is to be reduced or enlarged, the pentagraph or the method of copying by squares must be resorted to. The Pentagraph consists of four rulers, AB, AC, DE, and 'EF, made of stout brass. The two longer rulers, AB, and AC, are connected together at A, and have a motion round it as a centre. The two shorter rulers are connected in like manner with each other at F, and with the longer rulers atD and E, and, being equal in length to the portions AD and AE of the longer rulers, form with them an accurate parallelogram, ADEE, in every position of the instrument. Several ivory castors support the instrument, par- allel to the paper, and allow it to move freely over it in all direc- tions. The arms, AB and DP, are graduated and marked J J, &c., and have each a sliding index, which can be fixed at any of the divi- sions by a milled-headed clamping screw, seen in the engraving. The sliding indices have each of them a tube, adapted either to slide on a pin rising from a heavy circular weight called the fulc- rum, or to receive a sliding holder with a pencil or pen, or blunt tracing point, as may be required. When the instrument is correctly set ; the tracing point, pencil, and fulcrum will be in one straight line, as shown by the dotted line in the figure. The motions of the tracing point and pencil are U 14G then, each compounded of two circular motiona, one about the fulcrum, and the other about the joints at the ends of the rulers upon which they are respectively placed. Ihe radii of these motions form sides about equal angles of two similar triangles, of which the straight Hue BC, passing through the tracing point, pencil, and fulcrum, forms the third side. The distances passed over by the tracing point and pencil, in consequence of either of these motions, have then the same ratio, and, therefore, the dis- tances passed over, in consequence of the combination of the two motions, have also the same ratio, which is that indicated by the setting of the instrument. vi:®* Our diagram represents the pentagraph in the act of reducing a plan to a scale of half the original. For this purpose the sliding indices are first clamped at the divisions upon the marks marked |; the tracing point is then fixed in a socket at C, over the original drawing ; the pencil is next placed in the tube of the sliding index upon the ruler DF, over the paper to receive the copy ; and the fulcrum is fixed to that at B, upon the ruler AB. The instrument being now ready for use, if the tracing point at C be passed deb- U7 cately and steadily over every Hue of the plan ; a true copy, but of one-half the scale of the original, will be marked by the pencil on the paper beneath it. The fine thread represented as passing from the pencil quite round the instrument to the tracing point at C, enables the draughtsman at the tracing point to raise the pencil from the paper, whilst he passes the tracer from one part of the original to another, and thus to prevent false lines from being made on the copy. The pencil holder is surmounted by a cup, into - which sand or shot may be put, to press the pencil more heavily on the paper, when found necessary. If the object were to enlarge the drawing to double its scale, then the tracer must be placed upon the arm DF, and the pencil at C ; and if a copy were required, of the same scale as the original, then, the sliding indices still remaining at the same divisions upon D¥, and AB, the fulcrum must take the middle station, and the pencil and tracing point those on the exterior arms, AT5 and AC, of the instrument. Let figure 1, in the annexed engraving represent a plan of an estate, which it is required to copy upon a reduced scale of one- half. The copy will therefore be half the length and half the breadth, and consequently will occupy but one-fourth the space of the original. JF{^. 3. Fls. 2. m, u 2 us Draw the lines PI, FGr at right angles to each other ; from the point r towards I and Gr, set off any number of equal parts, as Ya, a I, 1) c, &c., on the line E I, and F i, i Jc, k I, &c., on the line F Gr : froui the points in the line F I, draw lines parallel to the other line FGr, as a a, h h, c c, &q., and from the points on FGr, draw lines parallel to FI, as i i, k 7c, I I, &c., which being sufficiently extended towards I and Gr, the whole of the original drawing will be covered witli a net- work of small but equal squares. Next draw upon the paper intended for the copy, a similar set of squares, but having each side only one-half the length of the former, as is repre- sented in figure 2. It will now be evident that if the lines AB, BO, CD, &c., figure 1, be drawn in the corresponding squares in figure 2, a correct copy of the original will be produced, and of half the original scale. Commencing then at A, observe where in the original the angle A falls, which is towards the bottom of the square, marked d e. In the corresponding square, therefore, cf the copy, and in the same proportion towards the left hand side of it, place the same point in the copy : from thence tracing where the curved line AF crosses the bottom line of that square, which crossing is about two-fifths of the width of the square from the left hand corner towards the right, and cross it similarly in the copy. Again, as it crosses the right hand bottom corner in the second square below d e, describe it so in the copy : find the position of the points similarly where it crosses the lines ff and ^' (/, above the line I Z, by comparing the distances of such crossings from the nearest corner of a square in the original, and similarly marking the required crossings on the corresponding lines on the copy. Lastly, determine the place of the point B, in the third square below (J h on the top line ; and a line drawn from A in the copy, through these several points to B, will be a correct reduced copy of the original line. Proceed in like manner with every other line oi^ 149 the plan, and its various details, and you will have the plot or drawing, laid down to a small scale, j^et bearing all the proportions in itself exactly as the original. It may appear almost superfluous to remark, that the process of enlarging drawings, by means of squares, is a similar operation to the above, excepting that the points are to be determined on the smaller squares of the original, and transferred to the larger squares of the copy. The process of enlarging, under any circumstances^ does not, however, admit of the same accuracy as reducing. 150 CHAPTEE VIII. TEiaONOMETBICAL SUEYEYING. The basis of an accurate survey must necessarily be an extended system of Triangulatlon, the preliminary step in wbicb is the careful- measurement of a Base Line on some level plain : at eacb extremity of this base, the angles are observed between several surrounding objects previously fixed upon as Trigonometrical Stations ; and also those subtended at each of these points by the base itself. The distances of these stations from the end of the base line and from each other are then calculated, and laid down upon paper, forming so many fresh bases from whence other Trigonometrical points are determined, until the entire tract of country to be surveyed is covered over with a net-work of triangles, of as large a size as is proportional to the contemplated extent of the survey, and the quality and power of the instruments employed. The interior detail between these points is filled up either by measurement with the chain and theodolite, or by partial measurement, (principally of the roads,) and by sketching the remainder with the assistance of some portable instrument. For the description of the regular Trigonometrical Survey of a country, the reader must refer to larger works on the subject. What will be described here, is such a survey as might be made 151 with a 5-iiicli theodolite, if the surveyor had some few square miles of country to survey accurately. In fixing upon an appropriate site for the measurement of a Base Line, a level piece of ground should obviously be selected, v^here both ends of the base v^^ould be visible from the nearest trigonome- trical points. It should also be as near the centre of the survey as possible, but this is not absolutely necessary. For a survey of the extent abovementioned, it should be about 2,000 feet long, and the sides of the triangles may be increased to a mile or more. The pro- cess of measuring the base line is as follows: — The theodolite being set up at one end, the inclination of the ground, as far as it con- tinues the same, is measured by sending a staft' with the vane set to the height of the instrument, to the point where the change of inclination takes place, the distance between these points is then measured carefully with the chain, both forwards and backwards. The chain must be compared with a standard bofore measuring the base line, and afterwards, and the mean of the measurements taken for its true length. The theodolite is then removed to the place where the staff was held, which is called the second station, and the angle of elevation of the portion of the base line already measured re-taken, as well as that of the next portion, up to the 3rd station. The distance between the 2nd and 3rd stations is then measured twice as before, and in this way the whole length of the base line is measured, and also the inclinations of the ground with a view to re- ducing it to one horizontal line. Form A, at the end of the Chapter, shews the method of entering these observations in the Field-book. The trigonometrical stations must be chosen with a view to the formation of loell-conditloned triangles, i. e. triangles none of whose angles are less than 30° ; the nearer the triangle approaches to the equilateral the better. The sides of the triangles should increase as rapidly as possible from the measured base. The accompanying sketch shews how this is to be managed without admitting any ill- conditioned triangles. 152 AB is supposed to be the measured base, and C and D the nearest Trigonometrical points. All the angles being observed and the length of AB having heen measured, the other sides of the triangles DAB, CAB may be calculated. We can then calculate DC from the two tri- angles DAC, DBC, (having the two sides and included angle of each given,) one calculation act- ing as a check upon the accu- racy of the other. This line DC is again made the base from which the distances of the tri- gonometrical stations E and E are computed from D and C, and these lines ED, EC, DP, CE, can be used as fresh bases for ex- tending the triangulation, or if these be not sufficiently large, the length of EP can be calculated and used as a base. This is the usual method of starting from a base line, unless the nature of the ground to be surveyed, interferes. The remainder of the trigonometrical stations must be arranged over the whole survey, as the nature of the country will best allow, and care must be taken that no point in the survey is too far from some one of these stations. The best form of signal for a station is a basket, covered with canvas and white-washed over, fixed on the top of a lamhoo. The observations being made to the bamboo immediately under the basket, or if the stations be very far from one another, to the centre of the basket. These should be fixed in trees, or if there are no trees near the spot where it is wished to have a station, the bamboo 153 should be lashed to a hdlee, and the end of the latter sunk a foot or two in the ground, and the whole made firm and steady with ropes pegged into the ground, after the manner of a flag staff. Having selected all the stations and placed the signals, all the angles of the triangles must be observed with the theodolite, and for obtaining the relative altitudes of the ground at the different stations, the vertical angles also. The angles are observed from each station in succession, as follows : — The theodolite is centered over the point on the ground, marking what is called the station dot, which is vertically under the signal.* It is then levelled, and the A vernier being set to 360°, or zero, the intersection of the wires is made to bisect one of the stations, by turning the whole instrument round, the bottom plate is then clamped ; the vernier plate is now loosed and each signal that is visible is intersected in turn, and the horizontal and vertical angles read off and entered in the field-book (see form Bj. In the hori- zontal angles, the columns A,"B,C, are the minutes read off from the three verniers, the degrees being all read from the same vernier. In the vertical angles, columns A and B are the minutes read ofl: on the two arcs of the Everest theodolite ; if a T theodolite be used, there will only be one column. Having completed the circuit, and on re-intersecting the first station, the vernier should of course read 360° ; if it does not, the under plate could not have been properly clamped, or the whole instrument must have been moved, and as it is not possible to find where the error occurred, all the angles must be re-observed. To prevent the omission of this check, the first station is re-entered after all the others. The 2nd set of angles are * This is found in the following manner. — Set up the theodolite at a short dis- tance from the signal, and having levelled it, fix the intersection of the wires on the signal, clamp the lower plates and bring ilie telescope down till it intersects the ground a foot or so beyond the signal. Make a mark at this point, and stretch the chain from the theodolite to it. Now remove the theodolite to another point, so that the direction between it and the signal, is about a right angle with the last direction and repeat the operation ; the point where these two lines intersect will be vertically under the signal. 154 observed witli tlie face of the instrument reversed, by which means any error in collimation is eliminated, and the angles commence from 180° instead of 360°, tbis tends to eliminate the errors that must necessarily exist in the division of the arc, by taking the angles at different portions of it, and taking their means. If the instrument has only two verniers, three sets of angles should be taken.* The height of the theodolite, and of the signal, from the ground must be measured, as also the height of the station if the angles should be observed from any artificial elevation, such as the roof of a house. These data are necessary for calculating the altitudes. Sometimes church steeples, prominent buildings, or other mark- ed objects, are peculiarly adapted for trigonometrical stations, but they have this disadvantage that the theodolite can seldom be set up immediately over the point observed. "When such is the case, a station called a satellite is chosen, as near as possible to the prin- cipal station, and the angles are taken from this point. It is clear that these angles before they can be employed in the computation of the triangulation, will be required to be transferred to the princi- pal station, that is they must be reduced to what they would have been had they been observed from that point — this is technically called tJie reduction to tlie centre. The angles are entered in the field-book in a similar manner to that of the ordinary stations, the necessary extra data being the distance of the principal from the satellite station and its direction. The observation of the angles completes the field work portion of the trigonometrical part of the survey. The various calculations must now be made and entered in the calculation book. The Base Line is first reduced to one horizontal line (see Form C) and its true length obtained. The horizontal and vertical angles are then meaned out from the field book, and entered in the calculation book (see Form D). The angles taken from satellite stations must next • If the Instrument be a Y theodolite, the 2nd set must be taken with tlie telescope inverted. ■■P IIIIIIIIIIIIIIIIIIIIICIIIIIII • I > I .53 be reduced to the centre, and then we have all the data ready for computing the sides of the triangles and the relative heights of the stations. The principle on which the angles of satellite stations are re- duced to the centre is as follows :— Suppose P to be the principal, S the satellite station, ABC three other trigonometrical stations. The theodolite having been set up at S, and suppose, for conveni- ence of illustration, that when set at 360° it was directed on P, the readings of A, B and C are taken. Let PS he produced both ways to m and n. J^ow the direction of A from S is represented by the angle mSA, and the same referred to P, by the angle mPA, but the angle mVA = «^SA -[- SAP. If, therefore, we find the angle SAP, and add it to the angle mSA, we shall obtain the angle mPA or the direction of A from P, which was required. To find the angle PSA, the necessary correction, we have the angle PSA, the side SP, and we can obtain the side PA, which is only required approximately since it is so much larger than SP., either by construction or by calculation, by the latter most cor- rectly. In the case of the next station B, the angle mS'B (taken in the direction of the dotted circle) is the angle we have observed, and the one we require is the angle mPB ; and the latter is less than the former by the small angle at B.* In this case, therefore, we must find the angle B and subtract it from the reading of B, to obtain what it would have been, if observed from P instead of S. * For the /. hPB = ^ «SB — Z. B, and adding 180° to each /i rwPB = Z. wiSB — Z. B. X 2 i5G We have, therefore, this rule that standing at the satellite and fa- cing the principal station, the corrections for all the angles in the Qnglit semicircle are additive, for those in the left semicircle the cor- rections are stiitractive. Form E, shews the way of reducing the angles taken from a satellite station. In the first part, the column headed ' observed angles' gives the angles as taken from the field- book ; the next two columns are for finding the angle PSA, PSB, &c., (by deducting the reading of the principal station from that of each trigonometrical station successively, or vice versa, as the case may require,) which are then used below for the computation of the cor- rections, viz. the small angles A,B,C, &c. These being found are entered above, and applied to the " observed angles," which gives the " corrected angles " shewn in the last column ; in the example given, all the angles lying in the left semicircles, the corrections are all subtractive. These corrected angles can now be used for the determination of the angles of the triangles, as the angles in Form D. The horizontal distances, i. e. the sides of the triangles are now calculated (see Forms F and G). Each side should if possible, be calculated from two difierent triangles, and the mean of the values obtained taken as the true one ; and each station should be fixed by at least three lines. When all the sides have been computed they should be entered in a table (see Eorm H). The last calculation is that of the relative heights of the difi'erent stations (see Eorm K.) Erom the vertical angle and the horizon- tal distance we obtain the difierence in altitude between the axis of the telescope of the theodolite and the signal of the station ob- served. To this when cleared of curvature and refraction (see re- marks on page 94, also table page 118) must be added the height of the theodolite at the time of observation, and when the height of the signal observed is deducted, we have the difference of level in the ground at these points. The triangulation must now be laid down on paper, very carefully, 157 by the aid of beam compasses, taking the mean length of the sides from the table of distances. When this skeleton triangulation is completed, the interior details must be filled in, first by traversing the roads or other conspicuous lines with the theodolite, and after- wards by sketching with the plane table or compass, as before des- cribed in a former part of this work. 158 < t4 o o CD CO O O O a c >» > cc o o t>0 CQ oi P CO 11 0:3 (14 i n rt C C <; 1-1 «t-i 0) »« »o O) CO CO CI o> a> o o + I o o + I I + o o + I o o I + o o CO •-**— ^ o n ]59 es <« 00 o "^ ■n ^ c ta to j= (U o ^ o u s o o ^ "S cS Vi Q us o o CO (£4 s o e p o I ^" = S « .2 » I^ U3 ^ o «^ o 01 • JA OU ^ a CO s o ^ C4 ^ ^ o 0) J3 d 6 6 £ O .^ ^ O 0i 0) is c O •o o a. o H o CC c O t: 73 •o o lO OS n . 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