LIBRARY UNIVERSITY OF CALIFORNIA. A PRACTICAL TREATISE ON MINE SURVEYING BY THE SAME AUTHOR Third Edition, Revised and Enlarged MINING An Elementary Treatise on the Getting of Minerals With 596 Diagrams and Illustrations Crown %vo, gs. net LONGMANS, GREEN, AND CO. LONDON, NEW YORK, AND BOMBAY VENTURE COLLIERY. BLACK VEIN, REFERENCE. Fences on, Surface, - Line of fault, irv coal Footpaths ^-. BoJLor faulty cooJU Cart-tretcTcs si-^-.-_-.-_-r Coal/wor'JceGisTtizdecL Underground, Rocucis in, Coa.1/ ==. ScaJLe, 10 Chains to 1 Iruch, (As this is only a portion of the IERY PLAN. the boundary of the estate is not shown.) Frontispiece. A PRACTICAL TREATISE ON MINE SURVEYING BY ARNOLD LUPTON \i MINING ENGINEER, CERTIFICATED COLLIERY MANAGER, SURVEYOR, MEMBER OF THE INSTITUTION OF CIVIL ENGINEERS, MEMBER OF THE INSTITUTION OF MECHANICAL ENGINEERS, MEMBER OF THE INSTITUTE OF MINING ENGINEERS, MEMBER OF THE INSTITUTE OF ELECTRICAL ENGINEERS, FELLOW OF THE GEOLOGICAL SOCIETY, FELLOW OF THE SOCIETY OF ARTS, ETC-, LATELY PROFESSOR OF COAL MINING AT THE VICTORIA UNIVERSITY (YORKSHIRE COLLEGE, LEEDS), AND SOMETIME EXAMINER IN MINE SURVEYING TO THE CITY AND GUILDS OF LONDON INSTITUTE WITH ILLUSTRATIONS OF THE f UNIVERSITY J OF LONGMANS, GREEN, AND CO. 39 PATERNOSTER ROW, LONDON NEW YORK AND BOMBAY 1902 AH rights reserved * PREFACE THIS book has been prepared with the intention of assisting students in learning the art of Surveying. The author, during the twenty-one years of his Professorship in the Mining Depart- ment of the Yorkshire College, had to teach a great many students the elements of this art, and for that purpose put together various notes. As a former Examiner in Mine Survey- ing to the City and Guilds of London Institute also, the author gained a considerable insight into the needs of students. He has added to his own experience as a practical surveyor by reading a number of books on surveying and papers published in the transactions of various scientific societies both in this and other countries. Where it has been thought advisable to reproduce extracts or drawings from these, acknowledgment will be found in the text. Whilst primarily the object the author has had in view has been the preparation of an elementary text-book, he has endeavoured to make the book of value as a reference book to the more advanced parts of the subject, and the chapters dealing with Trigonometrical Plotting, Hypsometry, Method of finding the True North, Metalliferous Mine Surveying, Photographic Surveying, Prospecting with the Magnetic Needle, etc., have been included with this purpose in view. The reader should endeavour, as far as possible, to get practical experience of the instruments and in the method of using them, and the author would recommend such of his readers as have not done so to view the collection of surveying instruments at the South Kensington Museum, London. 182313 vi PREFACE. The author would like to acknowledge the uniform courtesy shown to him by those members of the Government Depart- ments (Koyal Observatories, Greenwich and Kew, the Ordnance Survey Office, the Meteorological Office, etc.) who have supplied him with various information, and also his thanks to the various makers of surveying instruments herein described. The tables of Logarithms, Antilogarithms, Squares, Sines, Cosines, Tangents, etc., which form a portion of the appendix, are taken from a work on Elementary Physics by Mr. John Henderson, D.Sc. (Edin.), A.I.E.E., F.K.S.E., to whom the author is indebted for permission to reproduce them. In conclusion, the author wishes to state that professional engagements might have entirely prevented him from com- pleting this work had it not been that among his assistants he numbered some experienced surveyors, and he thinks it fair to acknowledge the valuable assistance he has had from them, especially from Mr. Herbert Perkin. He would also like to thank those of his friends who have undertaken the revision of various parts of the work. Any corrections or additions which suggest themselves to the reader will be gratefully acknowledged. AENOLD LUPTON. G, DE GREY ROAD, LEEDS, July, 1901. CONTENTS CHAPTKK I. NEED AND ADVANTAGES OF ACCURATE PLANS, ETC. II. THE MEASUREMENT OF DISTANCES . . . ... ,. . 5 III. METHOD OF SURVEYING ON THE SURFACE BY MEANS OF CHAIN AND POLES . /. ' ' i . ,-s 18 IV. INSTRUMENTS FOR MEASURING ANGLES 43 V. INSTRUMENTS FOR PLOTTING LENGTHS AND ANGLES .... 85 VI. GEOMETRY, TRIGONOMETRY, LOGARITHMS 96 VII. SURFACE SURVEYING WITH THE THEODOLITE ., 112 VIII. UNDERGROUND SURVEYING 129 IX. METHODS OF PLOTTING AN UNDERGROUND SURVEY .... 149 X. METALLIFEROUS MINE SURVEYING . . . 181 XI. METHODS OF CONNECTING SURFACE AND UNDERGROUND SURVEY 188 XII. LEVELLING . . . ..." 201 XIII. CONSTRUCTION OF PLANS 255 XIV. MEASUREMENT OF MINERAL TONNAGES CALCULATION OF CON- TENTS OF PIT-HILLS CALCULATION OF EARTHWORK, ETC. . 279 XV. SURVEYING BORE-HOLES 288 XVI. MISCELLANEOUS 307 XVII. PROSPECTING FOR MINERALS BY MEANS OF THE MAGNETIC NEEDLE 349 XVIII. METHODS OF FINDING TRUE NORTH, OR GEOGRAPHICAL MERIDIAN 356 viii CONTENTS. PAGE APPENDIX EXAMINATION QUESTIONS VARIOUS 371 CITY AND GUILDS or LONDON INSTITUTE 375 SURVEYORS' INSTITUTION EXAMINATION PAPERS . . 386 THE LAW AND MINE SURVEYING ... 391 ATTRACTION or THE MAGNETIC NEEDLE BY IRON . . . . . . 393 MATHEMATICAL TABLES . . . . . . 396 INDEX . 409 MINE SURVEYING CHAPTER I. NEED AND ADVANTAGES OF ACCURATE PLANS, ETC. MINE surveying is necessary for two reasons : In the first place, a map or plan, and section, are necessary to guide the miner in his daily work, so that when the workings have extended over a considerable area, it may be seen at a glance which parts of the mineral have been got and which remain to get ; in what direc- tion the roads go, how far apart they are one from another ; how machinery can be best arranged for underground haulage ; how the ventilation of the mine may be most economically conducted ; and how the drainage may be effected. The plan should also show the direction of faults, and where the mineral has been found good, or where inferior or unworkable. The section will show the inclination of the bed or vein, and the height above or depth below any given datum- line. Contour-lines on the plan give the same information for the whole mine. In the second place, the plan of the mine (see Frontispiece) is required to show the position of the underground workings with regard to objects and boundary-lines on the surface. To take mineral from under- neath the land without the previous sanction of the landowner may be treated as felony, and, if it is done through accident or inadvertence, may be punished with a heavy fine. It is, there- fore, of the highest importance that the owner or tenant of the mine should not only have an accurate plan of the boundary of the estate under which he has a licence to work, but an equally accurate plan of the underground workings, drawn upon the same paper (or other drawing material) that is used for the plan showing the boundaries, fences, buildings, roads, streams, and other notable objects above ground. A mining plan is MINE SURVEYING. therefore, generally speaking, incomplete unless it is also a plan of the land above ; and a mining surveyor is therefore not competent for the entire production of a mining plan unless he understands land surveying as well as mine surveying. It frequently happens, however, that the plan of the surface is made by a land surveyor, and the plan of the mine by a mine surveyor; and this combination often produces very accurate results. In some respects it is better that the whole of the plan should be made by one surveyor, who is responsible for the accuracy of the combination of underground and surface work, and in this case that person should be the mine surveyor, as he is the man who possesses the additional knowledge of the mine which is necessary for a proper survey. Meridian Line, Even in case the mine surveyor is relieved from the work of land surveying by having an accurate map of the estate put into his hands, he cannot delineate upon it the workings of the mine unless he has some knowledge of land surveying, because he will require to mark upon the plan a meridian line to which his underground survey must be referred. This meridian line may be drawn north and south in the geographical meridian, or line of longitude ; or it may be drawn in the direction of the magnetic pole, or it may be some other line which is marked out both on the surface and in the mine below, in the same vertical plane. None of these lines can be correctly marked upon the surface plan without some knowledge of land surveying. It is, therefore, necessary that the mine surveyor should be instructed in the art of land surveying. Every art in which it is possible to achieve perfection has a fascination for the human mind, and surveying is one of these arts. Degree of Accuracy attained. The accuracy with which the survey may be made is only limited by the skill and care of the sur- veyor, provided he has the opportunity of using the most suitable instruments which are made; and, as a general rule, the surveyor obtains the accuracy necessary for his purpose. It is, however, perhaps also true that, as a general rule, he is not much more accurate than is necessary. Thus, in a mine of large extent, the workings of which are neither near a boundary nor near to some important building which must not be disturbed, an error of half a chain in the position of any part of the workings is by no means uncommon. NEED AND ADVANTAGES OF ACCURATE PLANS. 3 Reasons for Great Accuracy. On the other hand, when approaching some important building, or when approaching a boundary which must not be passed under a heavy penalty, and which must yet be reached because the owner of the mine does not wish to sacrifice any portion of the mineral which is his, then minute accuracy is often attained. In some metalliferous mines great value attaches, perhaps sometimes reaching 1000, to a single square yard of ground, and in such a case it is necessary that the plan should be so accurate that no rival skill can detect an error. If the owner of a mine inadvertently crosses the boundary, and gets mineral to which he has no right, he may be obliged to pay in damages nearly the whole market price of the mineral, possibly ten times the royalty ordinarily payable, so that in the case of a seam of coal, he might be fined to the extent of two or three shillings per square yard for every yard in thickness. In order to avoid crossing the boundary, there are only two courses one is to leave a considerable margin of the mineral inside the boundary, and the other is to have a plan of extreme accuracy, and to mark out the limits of workings underground upon this plan from day to day. To leave a wide margin of coal or other mineral ungot, unless it is required for the purposes of a permanent barrier, involves a corresponding loss and waste of mineral. An accurate plan is also necessary for engineering reasons. It may be necessary to drive an underground road or tunnel from one pit to some other pit, and a serious loss may result if the mark aimed at is not hit in the centre. For reasons of safety an accurate plan is much to be desired. Abandoned workings may be full of water, and if the plan of these abandoned workings does not show them all and in a correct position, the workings from some new mine may inadvertently break in upon accumulations of water, and thus lead to fatal, and financially disastrous results. It is, therefore, in the highest degree desirable that mine surveyors should habituate themselves to the making of accurate plans, because a habit of carelessness, once acquired, is difficult to throw off when minute accuracy is necessary. It is, however, not the surveyor who requires to be impressed with the importance of an accurate plan, it is rather those who have to pay for his services, and they do not always see where 4 MINE SURVEYING. they get any return for an expenditure on carefully made maps and plans. It thus happens at some collieries that hundreds of pounds are annually wasted which would be saved by the employment of a careful surveyor, not merely to make a plan of the roads after they are driven, but to set out the roads in the right direction. The cause of this waste is easily explained : without an accurate plan, showing the existing workings, faults, and inclination of the seam, it is impossible to lay out the roads so that the shortest length of road may suffice ; hence an unnecessary number of roads, and these roads crooked, are often made. Also, even if the roads are correctly schemed, they will not be made in the direction intended unless the workmen are guided by marks carefully fixed by the surveyor. Each yard of road in the mine costs so much to make, varying accord- ing to circumstances in coal-mines N from 2s. to 20s., and in metalliferous mines and cross-measure drifts from 10s. to 10 ; it also costs so much to maintain, and then there is the cost of transit. Thus in a mine raising 300,000 tons of coal a year, the cost of making and maintaining roadways of all kinds, and of haulage, may, combined, easily amount to 20,000 a year. If the length of the roadways is 5 per cent, longer than necessary, the cost will be increased in a corresponding degree, or to the extent of 1000 a year. In many cases the costs are on a higher scale, and, of course, the loss from unnecessary lengths of road is correspondingly increased. It is absolutely certain that the money spent on the produc- tion of accurate plans and contours, and sections giving every engineering and geological detail, is repaid many times over (tenfold to a hundredfold) every year in the ordinary course of working. CHAPTER II. .,, THE MEASUREMENT OF DISTANCES. CHAINS, TAPES, POLES, MEASURING-WHEELS. THE instruments generally used by the mine surveyor are as follows : Measuring-chains. Gunter's chain is that usually employed for land surveying and in coal-mines. This chain (see Figs. 1 and 2) is 66 feet long, or the eightieth part of a mile. It is divided into 100 parts, called "links." 100,000 square "links," or 10 square chains, equal 1 acre. The chain is con- structed either of iron, steel, or brass wire. If made of steel wire, it is about T V inch in diameter. A chain-length is composed of a hundred pieces of wire, which have a loop at each end, and are 6 inches in length. These pieces are united by three short links, about f inch, internal measurement, made of flat wire. Fia 1 '~ Gu c n h t ^ n 8 measurin s- Swivel to 3O 4O FIG. 2. Gunter's measuring-chain (enlarged view). These three short pieces and the long pieces make up a length of nearly 8 inches, or exactly 7 '92 inches. At each end of the chain the 6-inch piece is shortened to about 4 inches ; 6 MINE SURVEYING. then comes a small link, and then a brass handle, making up the total length of 7*92 inches. Measuring from the outside of the handle for a length of 10 links, the end of the tenth link is in the centre one of the three small loops connecting two 6-inch pieces. Attached to the centre loop is a small brass tag, with one prong, which indicates a length of 10 links from the end of the chain. Measuring 10 links further, another brass tag is similarly attached to the chain ; but this second tag has two prongs. At the end of the next 10 links is another brass tag, which has three prongs ; at the end of the next 10 links is a similar brass tag, with four prongs ; the end of the next 10 links is the centre of the chain, and has a simple round-ended brass tag. Each end of the chain is constructed in the same way, measuring from the outside of the handle to the centre, so that the same tag may count 40 or 60, according as it is before or after the centre, 30 or 70, 20 or 80, 10 or 90. At 25 links from each end of the chain, instead of the three simple loops connecting two 6-inch pieces, there is one loop and two swivel-jointed loops, so that if the chain has got twisted it may be untwisted. The swivel-joint also marks the length of 25 or 75 links. At the centre of the chain is another swivel link ; this is marked by the round-ended tag above mentioned. Sometimes 10 links at each end of the chain are made of brass, so that the end of the chain may be held near a magnetic compass without attracting the needle. If the chain is made of brass or iron wire instead of steel wire, it is about inch thick. For ordinary mine surveying it is desirable to have a good strong chain. Engineers often use a chain 100 feet long, divided into links of 1 foot in length. Where a section is being levelled, it is convenient to have the lengths in feet, because the altitudes are measured in feet. The use of 100-foot chains is making headway, and has much to recommend it. Whenever the term "link" is used, however, Gunter's link of 7'92 inches is the one referred to. In the Cornish mines a chain 10 fathoms, or 60 feet, in length is used, the chain being divided into 120 parts, each 6 inches in length, and marked with a tag every 6 feet (i.e. every fathom). Tapes. A 66-foot painted tape, divided on one side into feet and inches, and on the other into links, is very convenient for measuring offsets, and the width and height of roads. The best THE MEASUREMENT OF DISTANCES. 7 kind of tape is the " metallic " tape, made with fine brass wires interwoven with vegetable fibre. Steel Tapes. Where great accuracy is desired, steel tapes may be used. The steel tape, being one continuous ribbon of metal, is less liable to stretch than the chain. One side is marked with feet and inches, and the other with links. Steel tapes have to be carefully used, in order to avoid breaking, and must be cleaned after use, or the marking will become obliterated by rust. Sometimes a tape much longer than 100 links is used. Mr. Eckley B. Cox, of Drifton, Pennsylvania, showed the writer a steel tape 500 feet in length. This tape was very light, about ^ inch broad and T V inch thick. Every tenth foot was marked with a piece of brass wire soldered on with white solder, the number of each mark being shown by figures on the solder. The tape is carried on a reel, from which the required length may be unwound. One end of the tape is held at one station, and the distance to the other is read off upon the tape to the nearest 10-foot mark ; from this mark to the station the length is measured by a 6-foot staff marked in feet and decimals of a foot. By the use of this long tape, the entire length of a line can be measured at one operation to the hundredth part of a foot, and the errors due to marking off chain-lengths on rough and uneven ground are thus avoided. When measuring large tracts of outlying country, where portability and lightness are of great importance, what is known as a compound steel band chain is often used. It consists of two or more separate steel bands, each one chain long. These can be joined together by swivels and hooks, and used in lengths of one, two, or more chains. The first chain of each set is divided into links in the usual manner ; but the other chains are not subdivided. The bands are wound up on a steel cross. Measuring-poles. For measuring short lengths poles are often used, divided into links by painting alternate lengths of one link black and white. The divisions of the pole are some- times in feet for architectural purposes; and for measurements of extreme accuracy, the divisions are subdivided into tenths. As a general rule, poles are only used for measuring offsets to the line measured by the chain. For this purpose a 10-link (or, in the alternative, a 10-foot) pole is most convenient. In some 8 MINE SURVEYING. cases the base-line for a trigonometrical survey has been measured along a line, carefully levelled for the purpose, by means of poles laid end to end, so as to avoid the errors due to the inaccuracy of chains or tapes. Measuring-rods have been so constructed that the length is uniform for all temperatures. These are made by using a rod compounded of two side by side, one brass and the other iron, which have an unequal expansion. At each end is a cross-piece, projecting on one side, with a centre-mark so placed that the centre-marks maintain an equal distance during variations of temperature. .-+ Pacing. Distances are sometimes measured by pacing. With a little practice a surveyor may learn to step a yard, and in this way to measure distances with an error not greatly exceed- ing 5 per cent. The ordinary pace is much shorter, being, say, 30 to 33 inches. There is a great difficulty, however, in count- ing the paces, as it is difficult to maintain concentrated atten- tion. Paces may be counted by means of a pedometer, an instrument which registers the movements of the body made in walking, thus counting the paces. A man may educate himself to take a pace of even length uphill and downhill, the natural tendency being to take a long pace downhill and a short pace uphill. To maintain, however, uniformity of pace, a man of average height should adopt a pace not exceeding 2 feet 9 inches ; and then, with practice, he may maintain this for the whole day both uphill and downhill. Measuring-wheel. A measuring- wheel may also be used, with a counter to record the number of revolutions. The wheels of any carriage, whether propelled by steam, horse, or hand- power, or an ordinary bicycle or tricycle fitted with a counter, will do. The circumference of the wheel being known, say 10 feet, the distance traversed will be the number of revolutions multiplied by 10 feet. Of course, this will only give the dis- tance with approximate accuracy, but for many purposes, such as a preliminary geological survey, this accuracy might be quite sufficient. For still less accurate measurements, there are other means, such as the speed of a steamer on a river or lake. Accuracy of Steel Tape. For any purposes required by the mining engineer, a steel tape is sufficiently accurate. The expansion of steel between the temperature of freezing and boiling water is rather more than 1 in 1000, say 1*2 ; and the THE MEASUREMENT OF DISTANCES. 9 expansion in length for 1 is about 6'4 parts in a million, and for 50 is about 3'2 parts in 10,000, or, say, one part in 3125. In temperate regions a variation of 50 is as much as is to be expected ; in England this is an extreme variation. Suppose the steel tape to be tested and found correct at a temperature of 50 , 1 then for a variation of 10 either higher or lower, the variation would be about 6'4 parts in 100,000, or, more correctly, 1 in 15,625. Where extreme accuracy is required, this correc- tion should be made. To enable it to be done more readily, Mr. W. F. Stanley of London makes a patent band chain handle adjustment, in which, by means of a screw, the chain or band can be lengthened or shortened as desired. A scale on the handle also enables adjustment to be made for variation in temperature during the performance of the work. A steel tape *37 inch wide and "01 inch thick, 66 feet long, when laid out on a pavement, requires a pull of about 4 Ibs. to draw it straight over the slight inequalities of the pave- ment. A total pull of 8 Ibs. will stretch it T V beyond the mark made at the 4-lb. pull. A total pull of 12 Ibs. gives a total stretch of a bare eighth ; a total pull of 16 Ibs. gives a stretch of a good eighth ; and a total pull of 20 Ibs. stretches the chain fV beyond the mark made with the 4-lb. pull. The steel tape is not suitable for rough usage, and is therefore only used for the main lines of an important survey, and for those details which it is necessary to mark on the plan with extreme accuracy, or for measuring the base-line of a trigono- metrical survey on the surface. For the ordinary work of a mining survey a strong chain is the best measuring instrument. Testing a Chain. Before beginning a survey, and frequently during the survey, it is necessary to test the chain, to see that it is the right length. The importance of this will be understood when the reader considers that if the links of a chain are joined by three rings, then there are eight wearing surfaces for each link, or 800 in a chain-length. If each should wear the y^ part of an inch, this means that the chain is lengthened by 8 inches. For the purpose of testing, a flat piece of pavement or piece of level ground beside a straight wall should be carefully measured with a pole or foot rule, and a chisel-mark put on 1 Messrs. Chesterman claim that their steel tapes are practically accurate at 62 Fahr., and say the expansion is about -008 inch in 100 feet for each degree. io MINE SURVEYING. every tenth link ; the chain is then drawn tight over or against these marks. If any section of the chain is too long, it is shortened by taking out a loop ; if any section is too short, it is lengthened by putting in a loop ; the two ends of the piece of wire forming the loop are not welded together, so that the link can be easily opened with a chisel and closed with a hammer. A few hours' work with the chain over rough ground, where the chain has to be pulled tight to draw it into a straight line, or to set it free from some obstruction against which it has caught, may be sufficient to stretch the chain an inch or more. A care- fully tested steel tape is a very convenient instrument for testing the accuracy of a chain in the absence of any more certain fixed measure. Method of Chaining. When measuring on the surface with a chain, the method is as follows : The line to be measured having been marked out with poles, the chain is managed by two men the leader and the follower. The leader takes one end of the chain, and draws it in the direction of the pole towards which he is steering; the follower holds the other end of the chain at the peg or mark where the line begins. The leader carries ten arrows ; these arrows (see Fig. 3) are pins made of iron wire about y\ inch m diameter, pointed !$O *' /s '~ 1 a ^ one en ^ an( ^ f rme( l m ^o a ring 2 inches in diameter at the other end, and may be FIG. 3. Arrow. any convenient length from 13 inches to 20 inches ; to render them more conspicuous, a piece of coloured ribbon is tied at the top of each. The follower directs the leader to the right or left until the chain is drawn tight in an absolutely straight line for the next pole ; the leader then places an arrow at the end of the chain, and lets the chain lie upon the ground until directed to drag it forward. In case there are two marks in the requisite line behind the leader, he can put himself in direction by turning round so as to face the follower, and then moving the chain till he has placed it in a line with the guiding poles or pegs. Whilst the chain is lying on the ground, offsets can be taken to any building or other object to the right or left, or the distance of any fence, ditch, pathway, etc., that is crossed by the chain may be exactly noted. On receiving a signal, the leader drags the chain forward another length, putting a second arrow in the ground. When signalled forward again, the follower takes up THE MEASUREMENT OF DISTANCES. n the first arrow and advances to the second arrow, and so on ; thus the number of chains measured is always the same as the number of arrows in the hands of the follower. When the tenth arrow has been placed in the ground, the leader drags the chain forward and lets it lie upon the ground in its proper position until the follower has picked up the tenth arrow and handed the whole ten to the leader, who must never receive from the follower less than ten arrows. Any breach of this rule will probably lead to confusion. In order to mark the end of the chain when the leader has no arrow in his hand, he must make a mark with a wooden peg. After receiving from the follower the ten arrows, he puts one down beside this peg, thus marking the end of the eleventh chain. Measuring Rough Ground. In measuring over hillocky ground and through fences, copses, etc., it is necessary to draw the chain straight between the arrows, otherwise the length will measure greater than it really is. In order to make it straight (that is, nearly straight), it is necessary to pull tight, though violent pulling is unnecessary and injurious. In measuring up or down a bank, the length of the slope being greater than the horizontal distance, the measured length will be too great for a plan. In order to measure the correct horizontal length for a plan, it is usual, when measuring downhill, for the follower to hold his end of the chain on the ground, and for the leader to fix a pole vertically in the ground at some convenient length, and then to hold the chain on a level with the starting-point against this staff, and read the length ; the horizontal distance is thus measured in steps (see Fig. 4). This method is only adopted for very short slopes, or in case the surveyor has no instrument for measuring the inclination. In the case of a long uniform slope, the length of the slope is measured by drawing the chain straight down it, the angle of the 'slope is taken with a suitable instrument, and the length of the slope as measured is reduced by calculation to the true horizontal distance before putting the length on the plan. It is sometimes a good practice to put a wooden peg into the ground at the end of every tenth chain, from which measure- ments can be taken at some later period of the survey. Taking a Line through Obstructions, The measurement of a line is often hindered by some obstruction, such as a stone 12 ML\E SURVEYING. wall. In this case it is necessary to measure up to the stone wall, which is say 48 links distant from the last peg, the thick- FIG. 4. Measuring in steps. ness of the wall is found by measuring on to the top to be say 3 links, making the distance through the wall 51 links. The follower then, taking hold of the fifty-first link, holds it against the foot of the wall, and directs the leader as before where to fix the end of the chain. In a country containing many trees, it is often difficult to set out a line which may not lead into the trunk of a tree. In such a case there are three courses to be adopted : the first is to cut down the tree ; the second is to end the line at the tree ; and the third (see Fig. 5) is as follows : Measure at right angles C A Q S FIG. 5. Obstruction to survey-line. to the line an offset (A to B) longer than the width of the obstruction, and at 2 chains back measure another offset (C to D) of equal length, and at 4 chains back a similar offset (E to F) ; three poles set up at the end of these offsets will be in THE MEASUREMENT OF DISTANCES. 13 a straight line parallel with the main line. This line is then con- tinued for a distance of 6 chains past the obstruction, and three offsets, PQ, RS, TU, set out from this parallel line in the opposite direction to the three original offsets ; three poles set up at the end of these three offsets at Q, S, U, will be in a straight line, and a continuation of the original line. The same course may be adopted if the original line runs into a building. Chaining Underground, When chaining in the mine, arrows are not usually employed, because the ground is too hard for them to pierce ; the end of the chain is usually marked on stone or rail with a piece of chalk, and the number of the chain written by the side of the mark ; the leader chalks on a piece of stone the figures 1, 2, 3, 4, 5, etc., up to 10, or marks (/, //, ///, ////, /////, //////, ///////, etc.), and then begins again. It is, however, seldom that the lines in an underground survey reach a length of 10 chains. This system of marking the length leads to many errors ; the attention of the leader and follower and surveyor may be called off, and the number of chain-lengths forgotten; and it would save many errors in measurement if the system of arrows adopted by surface surveyors was copied in the mine. Instead of an arrow, a simple ring of metal would suffice ; the end of the chain would be marked by the chalk as usual, and the ring of metal laid down beside it would form an automatic counter of the number of chain-lengths, the leader starting with 10 rings in his possession, and the follower, taking the rings up, will know the number of chain-lengths by the number of rings he holds. At the end of each line the follower would give up his rings to the leader, who would always start with ten rings. So many of the lines measured in mining surveys are, however, less than 1 chain in length, and the length so seldom exceeds 5 chains, that mining surveyors as a rule have not thought it worth while to adopt such a system ; but the writer's experi- ence leads him to think that it would lead to a considerable saving of time. It rarely happens that the end of any line to be measured corresponds exactly with the end of the chain ; therefore, except in these rare cases, the chain should be drawn forward past the dial or mark indicating the end of the line, and then the exact distance to the mark read off upon the chain. If this rule is always observed, it will be conducive to accuracy 14 MINE SURVEYING. of measurement, as the chain will always be read from the follower's end. Surveying-poles. These are used for marking out the line to be measured, and generally vary in length from 10 to 15 links ; a 12-link pole is a very convenient length. It is generally made of pine (see Fig. 6), about 1-J- inch diameter at the base, gradu- Blcuck Red, White FIG. 6. Surveying-pole. Black ally tapering to f inch at the top ; the base is shod with iron , about 9 inches in length, ending in a point ; with this iron point a hole can be made, even in hard ground, in which the pole can be fixed. It is necessary that the pole should be perfectly vertical, as it frequently happens that only the top of it can be seen over hedges or other obstructions ; therefore, if the top is not over the point, the line will not be set out straight. Fig. 7 shows a surveyor in the act of fixing a pole in line with two other poles. The surveyor, desiring to mark out a line, fixes two poles in the desired direction, at a convenient distance apart, say 20 to 50 yards; he then fixes a third pole in the same line at a further distance of say 20 to 50 yards ; if these poles are in a straight line, when standing behind one pole at a distance of say 10 yards, and closing one eye, the other two poles should be invisible. A fourth pole is now fixed in the same line. The first pole can now be taken out and placed in advance, forming the fifth mark ; then the second can be taken up and placed in front, forming the sixth mark, and so on ; by means of these four poles a straight line of any length can be marked out across the country. If three poles are always in the ground, it will be at once evident if one of them has got moved. In practice a good deal of care is required to keep the line quite straight, as it is not always easy to fix the poles perfectly plumb, or they may be blown on one side by the wind, or may FIG. 7. Fixing a pole. THE MEASUREMENT OF DISTANCES. be inaccurately fixed to the extent of half an inch. If the third pole is 60 yards in advance of the first pole, and half an inch out of its correct position, that is a deflection of 1 in 60 X 36 X 2, or 1 in 4320. 'This deflection in a small survey might not be very serious, but the deflected line may be deflected still further in the same direction, and the error of 1 in 4000 may soon be increased to 1 in 1000. For setting out long and important lines, the eye of the surveyor is often assisted by the telescope mounted on a theodolite stand. With a good instrument and great care almost perfect accuracy may be maintained in poling out a line. Sometimes small flags about a foot square are fastened to the top of the poles to make them more conspicuous. The poles are all painted black, red, and white in alternate lengths of 1 link (or 1 foot), so as to make them more easily visible, and this also fits them for use as measuring-poles. For special purposes, as, for instance, for use in a large trigonometrical survey, poles of extra length and strength are used; these are maintained in a vertical position by means of guy ropes (see Fig. 8) fastened to pegs in the ground, or to weights. Sometimes a pole is fixed on a wooden frame. It happens very frequently that it is necessary to range a straight line between two fixed points, neither of which is visible from the other, or, if visible one from the other, it is in- convenient to go to either of them so as to range out the line from the beginning; but whilst one of these points is invisible from the other, they are both visible from an elevated piece of ground between them. The surveyor and his assistant proceed to this intermediate position, and, each , , , . holding a pole and standing about . 8. Pole fixed by guy ropes. 50 yards apart, face each other, placing themselves as nearly as they can guess in a line between the two fixed points, A and B (Fig. 9). The surveyor at D', looking towards A, motions 16 MINE SURVEYING. the assistant at C' into the line AC'D' ; his assistant at C' looking towards B, motions the surveyor into the line BD"C'. As the surveyor is moved towards the line BDC, the assistant D , has to be moved at the same C '_..- -' & ' H ' time towards the line ACD, ^--- --.'.'-- ~.-'-to--"- : ~- : '~ :: *"&--- -Q and this movement is con- * D B tinned until the two lines ex- FIG. 9. Setting out a straight line between ,.i * ;! -i/u A^PID two points nSt visible from each other. actlv coincide, then AC D B form one straight line. With a little practice this operation can be performed in two or three minutes. Where great accuracy is required, a theodolite may be used to check the positions C and D, first erecting it at D to ascertain if C is in the straight line ACD, and then erecting it at C to ascertain if D is in the straight line BDC ; a central position E may be marked out with a peg, and a centre line accurately fixed; a transit theodolite may then be fixed over this and directed towards A ; when the telescope is reversed the cross-hairs should be upon the station B. Although four poles are sufficient with which to mark out a line, it is usual to have more, perhaps seven or eight, in one line. With six or seven poles standing in a line there is less chance of a divergence from the original direction, because although three poles are sufficient if there are no accidents, still if two of these should be accidentally knocked a little on one side, the direction would be lost. As each pole is pulled up, a peg is put into the hole to mark the place, so that the line may be easily found another day. The kind of peg that is used varies according to the circumstances of the case ; sometimes a small twig cut from a hedge is the best kind of mark, as it is not likely to attract attention ; on other occasions a piece of wood about 18 inches long and 1J- inch square, pointed at one end and flat at the top, may be driven in. For a permanent station it is, however, necessary to have a stake which cannot be easily withdrawn, say 3 feet long and 4 inches square, driven down till the top is but little above the ground, with a cross-mark nicked in the top to show the line of survey. Pegs of this kind, however, should not be put dpwn in a place where they will interfere with agricultural work, such as mowing-machines, but should be placed by the side of a hedge or ditch, where they will be no impediment and attract no notice. THE MEASUREMENT OF DISTANCES. TABLE SHOWING THE EQUIVALENT VALUES OF VARIOUS MEASUREMENTS. LINEAL MEASURE (LENGTH). Mile. Chains. Yards. Feet. Links. Inches. 1 80 1760 5280 8000 63,360 0125 i 22 66 100 792 000568 04545 1 3 4-545 36 000189 i -01515 333 1 1-515 12 000125 01 22 66 1 7-92 0000158 00126 0278 0833 126 1 SQUARE MEASURE (AREA). Acres. Roods. Perches. Sq. yards. Sq. feet. Sq. inches. 1 4 160 4840 43,560 6,272,640 25 1 40 1210 10,890 1,568,160 00625 025 1 30i 272J 39,204 0002( 66 000826 0331 1 9 1,296 000023 0000918 00367 111 1 144 000000159 00000064 -0000255 000772 00694 1 / 27,878,400 sq. feet. 1 square mile = ] 3,097,600 sq. yards. 640 acres. Acres X '0015625 = sq. miles. Sq. yards x '000000323 = sq. miles. 10 sq. chains = 100000 sq. links 1 acre. 46,656 cub. inches = 27 cub. feet = 1 cub. yard. NOTE. The above tables will be found to comprise many of the data needed by the surveyor. To use the tables : Suppose it is required to convert yards into links. On referring to the table we find 1 yard is equal to 4'545 links, so by multi- plying yards by 4-545 we get the equivalent distance in links. Other units of measurement may be converted in a similar manner. CHAPTEE III. METHOD OF SURVEYING ON THE SURFACE BY MEANS OF CHAIN AND POLES. FOR the purpose of making a survey on the surface of an estate of moderate size, say 1000 acres, it is not necessary to have expensive instruments. A score of straight poles, a good G-unter's chain, ten arrows, an off-set staff or tape, and some pegs to mark the stations, are all the instruments required ; a compass and theodolite may be very useful and advan- tageous, but they are not absolutely necessary. The method of making a survey with chain and poles may be explained in the fol- lowing manner : Let Fig. 10 be the plan of an estate on level ground, of triangular form ABC. From the point A, B is visible; fix a pole at A and another at B, and measure the distance A to B. From the point B, C is visible ; fix a pole at C, and From the point C, A is visible ; The measurements are entered A to B, length 600 links B to C, 900 C to A, 800 . Line FIG. 10. Simple surface survey. measure the distance B to C. measure the distance C to A. in a book, thus Line No. 1 No. 2 ... No. 3 The survey is now complete, and, if the lines have been measured METHOD OF SURVEYING ON THE SURFACE. 19 accurately, these three measurements are sufficient for the production of an accurate plan. It will be seen that A, B, and C are angles, and that the figure measured is a triangle (from Lat. trcs, tria, three, and angulus, an angle, meaning a figure with three angles). The length of each side has been measured, but not the angles; so that if only two of the sides had been measured the lines could not be drawn on paper in their correct position as regards each other. Having got the lengths of the three sides, however, they can be plotted with the aid of an elementary knowledge of geometry. Plotting a Triangular Survey. The method is explained by reference to Fig. 10. The line No. 2, being the longest, is drawn on the paper, and the length marked by means of the s 921 312- 46 282^ 252, ^ 200 921 Line from to 395 435 55 62 300 (10) 61 250 55 @ 3510 48 100 50 35 448 Line (21) from ^ to ^ 7) Line (20) continued FIG. 16 (6). conti nued. Line@ from gg to 511 511 (40) (300) 120 56 56 2320 _ 963 - line from 1028 1550 Line from to Line 480 continued FIG. 16 (7). continued. 618 = 545 (28) 570 467 1032 30 NO. 2 PIT ---57 1799 " J54_ WINDING ENGINE HOUSE ^ NO.l PIT. '61 928 61 852 . 756 780 550 M 56 WINDING ENGINE HOUSE *& - 618 N0.2 PIT. 1169 65] Line (2?) continued 732 772 )] fro.i PIT 480 371 N0.2 PIT Line (32) from -7^- to ^~ 375 W from 535 420 IS 319 789 FIG. 16 (8). continued. . 673 709 to 300 -@ 322 94 333 . 300 _ 420 "35 Line (39\Tie from JL to _*20_ 250 N0.2 PIT to Line(37)Tiefrom^to-^ ' 83 = 1311 Line to @ METHOD OF SURVEYING ON THE SURFACE. 39 FIG. 16 (9). continued. to ML 34) (12) NOTE. In order to save space, and also for convenience and rapidity of booking, the starting and finishing points and junction of one line with another are ex- 770 419 ^ pressed as fractions ; e.g. " Line (16) from /^\ to ^ '' means that line (16) starts from a station left at 770 links along line (?), and ends at a station 419 links in line (is) ; the number of the line being enclosed in a circle and appearing as the denominator of the fraction. To indicate a station, its length as read off from the chain line is enclosed in a circle; eg. the length (609) in line (15) is a station, line (g) crossing at this point (1350) links from the starting-point (i.e. of line (14)). the station at 1010. In measuring from this to the intersection of line 34, two hedges are crossed, and the survey-line No. 14, so that there is little possibility of a mistake in identifying the station from which the measurement was taken. In the same way the position of the station 1550 on the base-line, where No. 26 ends, is found by remeasuring a portion of the base-line. The same care has to be taken in crossing other lines, as, for instance, the tie-line No. 34 crosses lines 14, 1, 13, 29, 33, 32, and 27 ; and the position of the intersection of all these lines must be noted with the same care as was taken in noting the intersection on the base-line. By this careful noting of intersections, the detection of any error in the survey is a certainty ; and not only that, but the place of the error is quickly discovered, and the length which has been inaccurately measured or incorrectly entered in the note-book can be measured over again, otherwise the surveyor might have to waste a great deal of time in remeasuring lines that had been accurately measured the first time. Complete Note-book. Figs. 15 and 16 give the whole of the survey-book from which the plan Fig. 14 has been plotted, and 40 MINE SURVEYING. the student is recommended to plot this survey without referring to Figs. 14 and 17 until he has finished. Fig. 17 shows the order in which the triangles are plotted. Railway Surveying. The mining engineer has frequently to set out railways for mineral traffic, and every surveyor ought to understand how to survey the country where it is proposed to make a railway. Fig. 18 shows a piece of country through which it is proposed to make the line which is shown by the FIG. 17. Showing the order in which the survey-lines given in Figs. 15 and 16 are plotted. thick black curve, and it is necessary to make a plan of the fields, etc., through which it passes. The main lines of the sur- vey are marked 1, 2, 3, 4, 5, 6, 7, 8. It will be seen that the proposed line of railway starts in a direction north-west, then turns to north-east, and again to south-easfc; and the piece of ground surveyed is a strip about 12 chains wide, follow- ing the curve of the railway. By the careful measurement of the triangles, line 4 is accurately placed in relation to METHOD OF SURVEYING ON THE SURFACE. 41 line 1, and line 6 is accurately placed in relation to line 4, and the survey may thus be continued for a good many miles with great accuracy. It is essential that all parts of the survey should be connected by two or more lines, so that all the FIG, 18. Preliminary railway survey. measurements are checked. The lines Nos. 1 to 8 are the main lines ; numerous other lines run alongside the fences and com- plete a network of triangulation that eliminates all chance of undetected errors. 42 MINE SURVEYING. When the student has once mastered the principles on which the plans Figs. 14 and 18 are made, he understands the whole theory of an ordinary surface survey of an estate; and practice, combined with the requisite physical and mental faculties, only is required to make him a competent land surveyor. CHAPTEE IV. INSTRUMENTS FOR MEASURING ANGLES. IT is characteristic of the man of science to use every means at his command for testing the accuracy of his observations. Keferring to Fig. 18, plan of a railway survey, it is obvious that if the bearings of some of the main lines were taken, that is to say of lines 1, 4, and 6, they would be a check upon the accuracy of the triangulation, especially if it were continued for a long distance, say 10, 20, or 100 miles. For this reason surveyors commonly use a magnetic compass to take the bearings of their main lines. With this information they can quickly correct any very serious blunder that might have been made either in the measuring or in the plotting of the survey. Magnetic Compass. The construction of the magnetic compass is based on the well-known fact that a very light steel bar, like a knitting-needle, which has been magnetized, will, if balanced at its centre on a fine point, turn so that one end points to the north and the other end to the south ; whichever way the needle is placed originally, it is always the same end which seeks the north, that end is therefore called the north end (meaning the north- seeking end) of the needle, and the other end the south end of the needle. The direction in which the needle points is not, however, towards the north pole (i.e. towards the pole star), but it is towards the magnetic pole. To an observer in England this magnetic pole is west of the true or geographical north pole. A person standing at Greenwich and looking due north would have the magnetic pole a little to the left of his line of sight. The difference between magnetic and true north, or the angle between the magnetic meridian and the geo- graphical meridian, is called magnetic declination. Declination of the Needle. On the 1st of January, 1901, the magnetic needle at Greenwich pointed in a line about 16 26' west of the true or geographical north. The magnetic pole is constantly moving its position. Three hundred years ago the magnetic north in England was east of true north ; it moved gradually westward until the year 1818, when the needle near 44 MINE SURVEYING. London pointed about 24 38' west of the north pole. Since then it has been gradually returning eastwards. The move- ment in England is now approximately at the rate of 6' to 8' a year, or roughly 1 in 8J years. Apparently the present rate of movement is rather slower than the average of the last 36 years, which has been fairly regular, averaging during that period about 8 minutes a year in the neighbourhood of London. 1 Variation of the Declination. The declination of the needle from the true north is not the same for all places ; thus whilst the declination may be 16 26' west at Greenwich, and about the same at Worthing in Sussex, and Newmarket in Cambridge, it would be about 17 34' at Torquay, or Kidderminster, or Leeds, or MiddlesborOugh, and 18 35' at Pembroke, or Conway, or Barrow, and 19 30' at Glasgow. The lines of equal declina- tion (or isogonic lines) for the British Isles are shown on a map published annually as a supplement to the Colliery Guardian. 2 A somewhat similar map on a reduced scale is shown in Fig. 19. This map has been prepared by reference to the elaborate paper by Professors Riicker and Thorpe, published in the Philosophical Transaction*, 1890. The isogonic lines drawn on this map represent average declinations ; there are a great many local variations due to various causes, such as the mag- netic character of the rocks, of which no account is taken in the diagram. The direction of these lines is north-easterly, and a person travelling along one of these lines, say from Torquay in Devonshire to Leeds in Yorkshire, and using the magnetic compass, would find the same decimation from the true north along the whole line, but in journeying from London to Liverpool there would be a change in the declination every mile. By way of illustrating the use of this map, a surveyor in the Warwick- shire Coalfield will find that the isogonal marked 19 in 1886 passes through that district, and that the declination in January, 1901, was 17 15'. A surveyor in the Liverpool district is on another isogonal, marked 20 in 1886, and 18 15' in 1901. Half- 1 Everyday there is a slight movement, known as the diurnal variation. Accord- ing to Professor H. Stroud, M.A., D.Sc., the needle reaches its westerly maximum deviation at 1 p.m., and its maximum easterly at 10 p.m. (in the southern hemi- sphere east and west must be interchanged). This variation is about 10 minutes, or a sixth of a degree. For ordinary bearings this slight variation may be neglected, but when fixing the north point on a plan, or in other cases where extreme accuracy is desired, account must be taken of this variation. The reader is referred to the paper by Professor Stroud, in the Proceedings List Mining Engineers, vol. vii. p. 268. 2 May be obtained from the Colliery Guardian office, 49, Essex Street, Strand, W.C . INSTRUMENTS FOR MEASURING ANGLES. 45 II 10 9 6 7 6 S 4- 3 2 101 2 FIG. 19. Magnetic chart for the British Isles, showing the lines of equal magnetic declination, as laid down by Professors Riicker and Thorpe in 1886. The dotted lines were obtained by joining up the points where equal declinations were found ; the full lines show the average or mean lines. The figures printed at the ends of the curves are the declinations as appended by Professors Rttcker and Thorpe in 1886. The present declination can be obtained approxi- mately at any time by deducting 7 minutes per year for every year since that date. 46 MINE SURVEYING. way between these two isogonals, that is, in the neighbourhood of Crewe, the decimation in January, 1901, was a mean between 17 15' and 18 15', that is to say, 17 45'. If a person were travelling along a line of latitude from east to west in England, the declination would increase as he went westward at the rate of about 1 in 100 miles on the south coast, and in the latitude of Berwick at the rate of 1 in about 70 miles. If instead of travelling from east to west, he were to travel (in England) from north to south, that is to say, along a line of longitude, the variation would be about 1 in 300 miles in the longitude of London, and 1 in 200 miles in the longitude of Falmouth and Milford Haven. If he were to travel north-west, as from London through Oxford and Cheltenham to Aberyst- with, the variation would be on the average at the rate of 1 in rather more than 80 miles, and travelling from Whitby to Carlisle the declination would change at the rate of about 1 in 70 miles. If the traveller should get on board a ship and sail round the world, he would find that in some places the declination is west, in others east, and that in some places there is no declination, that is to say, the needle points in the direction of the geographical north. For the guidance of mariners and others, maps are prepared which show the declination of the needle in all parts of the world that have been explored by civilized man. A reduced map prepared from the Admiralty chart corrected up to 1900, is shown in Fig. 20. 1 In whatever locality a surveyor may find himself, the Admiralty chart will show him the decimation of the needle, and the local rate of variation. He can, however, always ascertain the declination of the magnetic needle from the geographical meridian by observation, provided he knows how to mark out a line due north and south. For particulars of the various methods of finding the true north, the reader is referred to the chapter dealing with that subject. In the magnetic compass the surveyor has an instrument with which he can observe how far any line which he may have marked out varies in direction from a line drawn towards the magnetic north, or, as it is usually called, the magnetic meridian ; if he is able to observe the angle that each line makes with the magnetic meridian, he can easily calculate the angle that each line makes with the other lines, thus he can calculate the angles 1 The Admiralty charts may be had from the agent, 31, Poultry, London, E.C. INSTRUMENTS FOR MEASURING ANGLES. 47 o O5 00 i >-. - II c bo 2.2 "S Q 2 o ^ a j g O s la 48 MINE SURVEYING. at the intersection of lines 1 and 4 in Fig. 18, and of lines 4 and 6, and can thus check the accuracy with which these lines have been laid down on the plan. Mariner's Compass. Before proceeding to further detail as to the method of using the magnetic needle, it will be well to describe some of the numerous forms of magnetic compass. The mariner's compass is the form most generally known, and of greatest use, because by means of it all the fleets of the world are steered across the ocean. The novice, looking at a mariner's compass (see Fig. 21), might fail to learn that it had anything to do with the magnetic needle, because no needle is visible. The magnetized steel bar (or needle) is covered with a card, and, being supported at its centre on a sharp-pointed pivot, is free to revolve, and the card, being attached, moves with it; the instrument is enclosed in a brass case with a glass window, so that it is sheltered from the wind; the compass-holder is suspended in brass hoops (gim- bals), so that the horizontal posi- tion of the card may not be disturbed by the motion of the ship. Inside the box or case are two marks above and in a line FIG. 21.-Mariner's compass. with the ce ~ ntre of * n e card, and on opposite sides of the card : these two marks are placed in a line parallel with a line drawn through the centre of the vessel. If, then, the ship is pointing towards the magnetic pole, these two marks will coincide with the direction of the magnetic needle ; if the ship is turned to the right of the magnetic pole, these two marks will be pointing in a line north-east of the magnetic meridian ; and if the ship is turned the other way, it will point north-west of the magnetic meridian. In order that the direction in which the ship is pointing may be ascertained without delay, the card is divided by marks called "points;" there are thirty-two points in the circum- ference, eight in each quadrant, so that each point is an arc of 11 15'. Thus : north, north by west, north north-west, north- west by north, north-west, north-west by west, west north-west, west by north, and west begins or ends another quadrant. The other quadrants are similarly divided, and the outer rim of the INSTRUMENTS FOR MEASURING ANGLES. 49 card is divided into 360. The direction in which the ship travels is seen by reading on the card the position of the fixed marks in the box ; one mark represents the bow of the vessel, and the other the stern ; the bow mark is red. The card of the mariner's compass is shown in Fig. 22. FIG. 22. Card of mariner's compass. Prismatic Compass. A somewhat similar compass, called the prismatic compass (see Fig. 23), is used by land surveyors, but a light divided circle (generally made of aluminium) is often substituted for the card over the needle ; it is also fitted with sights : one sight, A, has a slit ; the opposite sight, B, has an opening, down the middle of which is stretched a hair or other fine thread. This slit and hair are placed in the direction of the line of survey, and the bearing is read by a pointer in the box, which is in the same line as the sights. The instrument is made so that the bearing may be read whilst it is held in the hand. In such a case it is necessary to read the bearing at the same instant that the sights come into the line. That this may be MINE SURVEYING. done, a reflecting prism is placed just below the top of the slit A. By means of this prism the marks on the graduated circle are reflected into the eye, and the mark which coincides with the line of sight is the bearing. This method, of course, only suffices for rough approximations to the bearing. Where accuracy is required, the compass must be placed on a stand, and in some cases this stand is made of a single stick, the pointed end of which is placed in the ground, and on the upper end is a ball-and-socket joint, by means of which the compass can be levelled. In some cases a tripod stand is used, and this is suitable for underground work. In order that the correct bearing may be read, it is necessary that the circle should be marked as if the north end of the needle were the south end. Suppose the observer is look- ing towards a staff, light, or other mark north of him, the north end of the needle will, of course, be at the opposite side of the compass- box to the observer ; therefore the observer can only read the south end. If this end is marked "south," the observer would be apt to book that reading, and afterwards imagine that he had proceeded in a southerly direction. To avoid such an error, FIG 23. Prismatic compass. the reading he observes should give him the direction in which he is moving, and therefore the letter N should be placed at the centre of the southern semicircle, and the letter S at the centre of the northern semicircle, and the east and west marks should be put in their correct positions relatively to the north and south marks, that is to say, the letter E will be at the side which is really the west, and the letter W at the side which is really the east. Graduation of Circle. The division of the circle into points as used by the mariner is not required by the surveyor. The circumference is divided into degrees only, each degree being the three hundred and sixtieth part of the circumference. Counting from the N. end of the card, which is 0, and pro- ceeding, say, towards the E. mark, the first quadrant, up to 90, INSTRUMENTS FOR MEASURING ANGLES. is called north-east ; the second quadrant, from 90 to 180, is south-east ; the third quadrant, from 180 to 270, is south-west ; the fourth quadrant, from 270 to 360, north-west. In order, however, to facilitate the plotting, it is a common plan to count both ways, from both the south and the north ends ; thus from north to west the degrees may be figured (on an inner ring of figures) from to 90, and from north to east also from to 90 ; from south to west in the same way the figures go from up to 90, and the same from south to east ; so that the bearings are always read so many degrees from the meridian line, say 40 north-west or 40 north-east, as the case may be ; or, if the observer is proceeding in a southerly direction, he might be going 30 south-west or 30 south-east, meaning that the bearing is the direction of a line proceeding from the centre pivot of the compass through a mark on the circumference 30 from the meridian line. The compass is made in various sizes from 1J inch diameter up to 6 inches ; the common size is about 2J inches. The weight of the card or metallic circle on the needle is, however, some objection to the use of this form of compass. FIG. 24. Hedley dial with outside vernier. Dial Joint. h iiiiiliiii i li Vernier Clamp HlUed head to turn, Vernier FIG. 25. Details of Hedley dial with outside vernier. INSTRUMENTS FOR MEASURING ANGLES. 53 Miner's Dial. The dial is the instrument generally used by mining surveyors for taking bearings and angles. It differs from the two preceding forms of compass in this important respect that the card or graduated circle is stationary, and the needle swings clear of it. One of general utility is shown in Figs. 24 and 25. Dials are made in various sizes, from a small pocket one, up to one carrying a needle 18 inches long (these large ones being for special work only). For general work the most usual size has a needle about 4^ inches long ; occasionally a 6-inch needle is used. In France the needle is usually a thin flat bar, wide at the centre, and the sides gradually converging to a point at the extremities (a, Fig. 26). In 'England it is common to use a needle rectangular in cross- section, and nearly the same thickness throughout. Just at the middle it is a little wider, and near the ends it is drawn down to a fine edge (6, Fig. 26). Sometimes, instead of drawing the end / [ - Q \ down to an edge, a line is marked on the top to repre- sent the middle of the needle (c, Fig. 26). A piece of agate (stone) is securely fixed in a brass cap Screwed into FlG 2 6.-VarietiTs of compass needle, a hole drilled through the middle of the needle from top to bottom, and in this agate a conical hole is drilled from the under side nearly through ; this agate rests on the sharp point of hard steel of the pivot that carries the needle. The agate is hard enough to resist the cutting effect of the steel point. The needle is free to revolve round the point in a horizontal plane. It is essential that the friction on the point should be reduced to a minimum, as the magnetic force is very small, and is insufficient to overcome any but the smallest frictional resistance. The needle has to be so weighted that, when magnetized, it is evenly balanced on the steel point or pivot; a small piece of brass clipping the needle firmly, but capable of sliding along it, enables the balancing to be done accurately. 54 MINE SURVEYING. In course of years a needle is apt to lose its magnetism, and requires to be remagnetized. This may be done -by taking- out the needle and unscrewing the cap. The north pole of a strong permanent bar magnet is then stroked down the needle from the centre to the south end. The needle is then turned round, and the south pole of the magnet is stroked from the centre to the north end of the needle. The needle is then turned over, and the process repeated on its under side. It is important that the agate cap of the needle, and the steel pivot on which it works, should be kept free from dust. The pivot should also be kept sharp, so as not to interfere with the free movement of the needle. The top of the needle is level with the upper surface of a graduated circle which is fastened on to the dial-plate, and this upper surface is about J- inch above the bottom of the dial. The graduations are carried down the vertical side of the circle. This circle is divided into degrees, and if the end of the needle is not opposite one of the divisions, the surveyor has to estimate as nearly as he can the fraction of the degree beyond the last mark, thus : -, J, f, ^, --, f , and -J ; the bearing being, say, south- east 21| or , as the case may be. Many surveyors do not profess to read to eighths on a dial of this size (4i-inch needle), and would only book quarters, as 21|. It is, however, possible, with a well-marked dial and a well-made and properly magnetized needle, to read to even one- eighth of a degree, which means that, supposing the bearing is booked by the surveyor as 21J, it is possible that he may be deceived, and that the real bearing is 21^ or 21f ; but the error need not be more than % either way. E. and W. reversed. In the ordinary dial (Figs. 24 and 25) the letter E is put on the west side, and the letter W on the east side of the dial-plate or graduated circle. The graduations are read by the help of two systems of figuring. The outer set of figures are marked 10, 20, etc., up to 360. These figures go from north to east, and continue round the way the sun travels ; thus W. is at 90, S. at 180, and E. at 270. The other system of figuring is on the inside ring, and refers to the quadrants, as 10, 20, 30, up to 90. Thus, counting from the north to the right hand are 10, 20, 30, etc., N.W. ; starting from the north to the left hand are 10, 20, 30, etc., N.E.; starting from the south to the right hand are 10, 20, 30, etc., INSTRUMENTS FOR MEASURING ANGLES. 55 S.E. ; and starting from the south towards the left-hand are 10, 20, 30, etc., S.W. Mode of using the Dial. There are two sights on the dial in a line with the north and south marks on the dial-plate. These are shown in Fig. 25, and consist of folding arms hinged at the point i, so as to fold down when not in use. Each sight has in it a broad opening and a slit, and down the centre of the broad opening is stretched a hair. The observer takes a sight by placing his eye at the slit, and moving the sights until the hair in the opening opposite is exactly in the centre of the object to which he is sighting. When taking inclinations, the circular holes shown are sighted in a similar manner. In using the dial the north (or N.) sight is always turned in the direction the surveyor is going. If he happens to be sighting a station behind him, then the south (or S.) end of the dial is turned towards this station ; if he happens to be going in a direction magnetic north, the north end of the needle will point exactly to 360 or of the graduated circle over the letter N ; if he happens to be going north-east, the line of sight will be to the right hand of the north end of the needle. To read the bearing, the surveyor looks at the north end of the needle, and reads the bearing against which it points, say 21 N.E. ; but, whichever way the surveyor goes, he must bear in mind to turn that end of the dial which is marked N. (for north) in the direction in which he is going, and to read the bearing from the north end of the needle. The north end of the needle is indicated by a mark upon it ; it sometimes consists of a notch, and sometimes of a brass cross-bar. Hedley Dial. The kind of dial most commonly used, and perhaps the most convenient form that is made, is known as the Hedley dial, and it is this form of dial which is illustrated in Figs. 24 to 28. The distinctive feature of this dial is that the sights are not fixed on the dial-plate, but to a separate ring outside, carried on bearings on each side of the centre dial- plate ; the circle carrying the sights can thus be moved up and down through an arc of about 60 either way, so that a sight can be taken up or down a very steep place. Attached to the instrument is a semicircle for measuring vertical angles, the arm J (Fig. 24) is fixed to a projecting end of the axis which carries the movable ring to which the sights are attached. The semicircle is fastened by two studs to this ring, and is therefore 56 MINE SURVEYING. inclined to the same degree as the line of sight, when it is taken FIG. 27. Face of dial, showing Halden's method of measuring inclinations. through the small round eye-hole and the cross-hair of the opposite sight. The arm J always remains in a ver- tical position as long as the surface of the dial is kept level, and a pointer at the end of the arm en- ables the angle of elevation or depression to be read. The semicircle is gra- duated in quadrants, zero being at the centre, and the graduations extending to 90 each way. There is a clamping-screw at the lower end of the arm J, by which the sights can be clamped at any desired angle of inclination. Messrs. Halden make a dial with an improved arc for measuring vertical angles. Instead of an external attachment, FiG.'.27A. Caeartelli's dial, showing semicircle for measuring inclinations. INSTRUMENTS FOR MEASURING ANGLES. 57 which may get broken, the graduated arc is on the base of the compass-box, the traversing finger working on a centre near the E. point of the dial (see Fig. 27) . Another form of inclinometer is shown in Fig. 2?A. This is made by Messrs. Casartelli, and con- sists of a graduated brass semicircle, in the same line as the sights, which folds down on one side of the dial-box when not in use. Dial with Inside Vernier. The dial is generally so made that it could be used for measuring angles if the needle was taken away, or if, owing to the presence of iron or other magnetic metal or rock, it cannot be used. One form of this is shown in Fig. 28. On the inside of the dial-box is fastened an index or FIG. 28. Hedley dial with inside vernier. vernier, the on the vernier being in the same line as the dial- sights. The dial-box is so made that it can be moved round independently of the dial-plate. If the dial-plate is firmly clamped and the sights moved to the east or west, the mark on the inside rim of the box moves with them, and the angle of movement can be read on the graduated circle. The use of the vernier is that fractions of degrees may be accurately read. The ordinary dial vernier reads to 3', or ^ part of a degree. The dial-plate is graduated and figured as already described. When using this dial for taking bearings with the needle, the mark in the centre of the vernier must be over the north or zero point of the graduated dial-plate (as 58 MINE SURVEYING. shown in Fig. 28), and it can be kept in this position by means of a brass pin, which is put up through the bottom of the dial-box and the dial-plate. When it is desired to use the vernier for taking angles, this brass pin is pulled out and the clamping-screw slackened. The sights are moved by means of a milled head on a pinion, the teeth of which fit into a rack on the inner side of the dial-box. By means of this pinion the sights can be easily moved to the required extent. Outside Vernier. In another, and in some respects superior, form of this dial (Figs. 24 and 25) there are two graduated circles : one inside the dial-box, e (Fig. 25), to be used for taking bearings with the needle; and the other outside the dial-box, / (Fig. 25), to be used when taking angles without the needle. This outside graduated circle is immovably fixed to the vertical axis, while all the other parts of the dial above it can revolve (by the action of the rack and pinion). The outer graduated circle is covered by a brass rim outside the compass- box, which conceals it from view, except at one place where this rim is partly cut away so as to expose the graduations for a length of say 30. On the movable dial-box is fixed the vernier h (Fig. 25), on which is a centre-mark. For the sake of convenience in reading, this vernier is not exactly under either of the sights, but is a little on one side, and at the beginning of a survey the centre-mark of the vernier is placed opposite the zero on the graduated external circle. If the dial-sights are then looking north and south, any movement to the east or west will be measured in degrees and minutes by the movement of the mark on the vernier from the zero point. The advantages of this form of dial are first, that the outer graduated circle and vernier can be easily read ; second, that the sights are always in a line with the north-and-south line on the dial-plate, and therefore the needle can always be swung and a true bearing observed (in case there is no attraction), whereas with the dial with inside vernier a loose-needle bearing could not be read until the ring carrying the sights had been put back into its original position, with the centre-mark of the vernier opposite the north-and-south line of the dial. It is essential that the dial should be placed level, and for that reason two spirit-levels, at right angles to each other, are generally placed on the body of the dial (as shown in Figs. 24 and 25). INSTRUMENTS FOR MEASURING ANGLES, 59 The spirit-levels may also be placed on the limb to which the sights are attached (as shown in Fig. 28), and, although more liable to get broken in this position, they do not interfere with the swinging of the needle. Dials are generally made of brass, but aluminium dials are now being made, and are preferred by some on account of their great lightness. Dial-joint. The dial is generally carried on a tripod stand, to which it is attached by a coupling, having a ball-and-socket joint (see Fig. 25). Above the ball is a strong brass pillar, a, which fits into a socket, b, which may be screwed on and off from the under side of the dial. The dial and socket are free to revolve round this pin or vertical axis, but can be fixed in one position by means of a clamping-screw, c. Below the ball is a clamp, d, by means of which the vertical axis can be tightened in the required position, and by which it can be slackened to admit of adjustment. This ball-and-socket joint, and the upper swivel movement, generally give satisfaction if they are kept in good order, but it is necessary that they should be cleaned from time to time and used with care. Some sur- veyors of great experience condemn this joint because of the insecure attachment of the dial by a screw, which may lead to an inaccurate survey, and also because of the absence of any convenient mode of levelling the head of the tripod holding the lamp-cup. There are, however, other modes of attachment. The ordinary parallel plates, such as are used with the theodolite (Fig. 37), may be substituted for the ball-and-socket joint. There is also the Hoffman joint, made by Davis of Derby (see Fig. 29). By turning the milled- head screws a, a from left to right, the two concentric balls B and D are liberated, and the dial can then be approximately levelled up ; on turning the Screws in the opposite direction, FlG " 29- Hoffman levelling-joint. the joint is clamped, and the final adjustment may be made by turning opposite screws in reverse directions just as required. 6o MINE SURVEYING. Another variety of levelling-joint (shown in Fig. 2?A) has a ball held between two plates which can be tightened or slackened by turning a thumb-screw. On the top of the ball is a strong brass pillar, fitting into a socket fixed on the under side of the dial ; in the socket is a clamping-screw. Each tripod has FIG. 30. Bullock's levelling-joint. fixed to it the levelling-joint. Two lamp-cups, fitted with cross- levels, are used to hold the object-lamps, and by means of these levels the brass pillar is set, so that when the dial (as in fast- needle work) is moved and placed upon it, its face will be level. INSTRUMENTS FOR MEASURING ANGLES. 61 Some surveyors who have used most kinds of dials strongly recommend the joint above described. Bullock's ball-and-socket joint is shown in Figs. 30 and 30A. FIG. 30A. Bullock's le veiling-joint. On reference to the figure, it will be seen that three adjusting screws, a, converge on to a cone, b, to which is attached the ball ; then, by tightening or slackening these screws, the top 62 MINE SURVEYING. may be thrown to any reasonable angle, and so enable the operator to obtain an accurate adjustment. One advantage of this joint is that when all the screws are touching the cone, the top cannot be thrown out of adjustment. Dial-legs. The tripod head is carried on three legs, usually about 4 feet 6 inches long; when these legs are spread out, the dial is at a convenient height to read ; for low roads shorter legs are used. The long legs are often jointed in the middle, so that by unscrewing the lower half the legs remain about 2 feet 3 inches in length. These legs may be jointed again, for thin seams or very low places in the roads, for which places legs 12 or 15 inches in length are used. Telescopic legs are sometimes made, and are convenient in low, narrow, and rough places. It is important that the legs should be attached to the tripod head in such a manner as to preclude the possibility of slackness, whilst they must not be too stiff for convenient use. The three kinds of head commonly employed are shown in Figs. 31, 32, FIG. 31. Ordinary dial tripod. FIG. 32. Improved form of dial tripod. FIG. 33. Theodolite tripod. 33. In Fig. 31 the legs get slack if they are kept long in a dry place, but they can be tightened by soaking the joints in water, which causes the wood to swell. In Fig. 32 the split end of the legs can be tightened over the brass projection by means of the thumb-screw. In Fig. 33, which is the method adopted in the tripod stand for theodolites, the joints can be tightened or slackened as much as desired by turning a nut upon a screw. This seems to be the strongest and best method. Lamp -cups. In fast-needle dialling two lamp-cups are usually employed. These are shown in Fig. 25, and consist of shallow cups of suitable diameter to receive the lamp. One of the cups is provided with levels, and this is always used in fixing the front legs ready to receive the dial. INSTRUMENTS FOR MEASURING ANGLES. 63 Various Dials. Many modifications of the dial are made. In one of these a telescope is substituted for the simple slit and hair-'sight, the sights being made detachable, so that the tele- scope may be taken off and the ordinary slit and hair-sights substituted, as shown at A and B (see Fig. 34). The telescope FIG. 34. Hedley dial with telescope. is advantageous for work where extreme accuracy is required, because the lamp, candle, or other mark can be clearly seen, and the meridian line of the dial turned precisely on the centre of the light, whereas with the slit and hair an error amounting to the thickness of the hair or the width of the slit may be easily made. The possible error, however, from this source, if the hair is properly fixed, is not more than 1 in 1200, or less than ^V of a degree, so it is only for special cases, either where very long sights are taken or where special accuracy is required, that the telescope is useful. It is obvious that a single telescope is, in some respects, not so convenient as the ordinary sights, which are made double for looking either backward or forward, and where the telescope is supported in the manner shown in Fig. 34, it is necessary to take it out of the holders and reverse it for the back sight. Dial with Eccentric Telescope. In surveying without the needle or " fast needle," as it is called the angle can be read 64 MINE SURVEYING. with great accuracy by means of the outside vernier ; but if the needle is used there may be some difficulty in reading it, in case the line of sight should correspond with the magnetic north, as the needle will then lie immediately below the tele- scope. This difficulty is got over by placing the telescope on one side of the dial instead of over the centre, as shown in Fig. 35. There is, however, a drawback attending this form, because the line of sight through the telescope is not directly parallel with the direction of a line from the centre of the dial to the lamp. This, however, may be got over by placing the lamp to be looked at at an equal distance away from the mark, and on the same side of the mark. The plan of having the telescope on one side has not only the advantage of leaving the top of the dial quite clear and taking up less headroom, but has the further advantage of permitting the telescope to be moved through a complete FIG. 35.-Dialith eccentric telescope gQ ag fo j k (Kindly lent by Messrs. W. F. Stanley & Co., Ltd.) backwards or forwards, up or down, as required, or at any intermediate inclination. When the telescope is fixed on one side of the centre, it is called eccentric. This eccentricity of the telescope need not be taken into account when reading the degrees on the vertical circle ; it is only when measuring a horizontal angle or transferring a horizontal line of sight from a plane on another level either above or below, that the eccentricity has to be considered. In dialling "loose needle " (that is, using the needle to read the bearings) for ordinary purposes, the eccentricity need not be considered, because the error in reading, whatever it may be in the back sight, is corrected in the fore sight. In fast- needle work, however, the eccentricity has to be considered. The surveyor must so arrange the lamp or other object viewed through the telescope that it is exactly as far from the centre of its tripod stand as is the telescope from the centre of the dial, and the object viewed must be on the same side of the centre of the stand as the telescope. Any failure to attend to INSTRUMENTS FOR MEASURING ANGLES. this may lead to very serious errors. This liability to error has discouraged the use of this form of instrument. By the use of an eccentric lamp-holder, the line of sight from the tele- scope to the lamp is exactly parallel to the line from the centre of the dial to the centre of the tripod stand which is in the line of survey, and therefore the eccentricity of the tele- scope leads to no error. Combined Mining Dial, Level, and Theodolite. This instrument, which has only recently been brought out, is shown in Fig. 35A. The chief feature is the method of supporting the telescope in cranked gim- bals, thus enabling a sight to be taken vertically up- wards or downwards. For fast-needle work, two outside verniers are used, reading to single minutes. The vertical circle has a clamp and tangent, and is also divided to read to minutes. Hanging Compass. An old- fashioned kind of compass, which is still used in some places, is shown in Fig. 36, In this case the compass-box, instead of resting on a tripod stand, is suspended by a cord in such a manner that the box is always level, and the needle free to revolve. The cord is FIG. 36. Hanging compass. Stretched from end tO end Of (Kindly lent by Messrs. W.F.Stanley^Co., LM.^ the line of which the angle has to be taken, and the reading of the compass-needle shows the bearing of this cord. FIG. 35A. Combined raining dial, level, and theodolite. (Kindly lent by Messrs. W. F. Stanley & Co., Ltd.} 66 MINE SURVEYING. The following recommendations, in addition to those already made, may be of use to purchasers of dials : (1) There should he two verniers where very accurate work is required. (2) The plate which carries the vernier should he clamped with a proper grip, and not merely by the point of a screw. (3) The dial should not be attached to the stand by a screw which may unscrew unknown to the surveyor. (4) Spirit-levels should have a white backing. Vernier. Called after the inventor Pierre Vernier. This is a small movable scale running parallel to the fixed scale on the dial, theodolite, protractor, barometer, etc. The use of the vernier is to facilitate the reading of the exact, position of some mark which elides upon the scale or close to it. For instance, in the case of a dial the centre mark is indicated by an arrow- head which moves round the circumferentor when the sights of the dial are moved, as in fast-needle dialling. There is a similar mark indicated by an arrow-head on the plate that revolves above the graduated circle of the theodolite. In the case of a barometer, the moving mark is the top of the column of mercury. This mark may be placed exactly opposite one of the marks of the graduated circle in a dial or theodolite, or on a straight barometric scale, in which case no vernier is required ; but if the mark comes to some position between the graduations, then the vernier is useful in reading the exact position between the two divisions of the scale. In the case of a dial, the sliding scale or vernier is fixed to that part of the dial which revolves round the graduated circle, and the arrow-headed mark is generally in the centre of the vernier, say 20 divisions on the vernier corresponding to 19 divisions on the graduated circle. If the divisions of the graduated circle are equal to one degree, then the divisions of the vernier are each equal to -?/g of a degree, so that when the centre mark on the vernier is set opposite one degree of the circumferentor, the nearest division of the vernier to the right or left of the centre mark will be -$ of a degree short of reaching to the corresponding mark on the graduated circle. If, therefore, the arrow-mark is moved 5 \ f of a degree to the right, the next division on the vernier to the arrow-mark will coincide to the corresponding mark on the graduated circle. If the arrow-mark should be moved -./$ of a degree, the second division of the vernier will be in line with the corresponding INSTRUMENTS FOR MEASURING ANGLES. 67 division upon the graduated circle, and so on; therefore, in order to read the exact distances that the arrow-mark is from the degree from which it has moved, it is necessary to look for the line on the vernier that happens to coincide with one of the Reading 67 51'. Reading 281 12 FIG. 36A. Vernier readings. divisions of the graduated circle. If that division is 6 from the arrow-head, then the arrow-head is ~$$ of a degree past the degree on the graduated circle from which it has been moved. Since the degree contains 60 minutes, the twentieth part of a degree 68 MINE SURVEYING. is three minutes, and therefore if the sixth division on the vernier scale corresponds with a line on the graduated circle, the arrow-head is 18 minutes past the degree. It will he seen that the principle of the vernier is that the space between any two divisions on the vernier scale is a small fraction less than the space between any two divisions on the fixed scale, and therefore if one division line on the vernier scale is exactly opposite a line on the fixed scale, to bring the next line on the vernier scale opposite the next line on the fixed scale it 'must be moved through the small fraction above named. It should be noted that the divisions on the vernier scale may be spaced farther apart than those on the fixed scale. An illustration of three readings of the vernier is given in Fig. 36A. Theodolites. The principle of theodolite construction is similar to that of the improved Hedley dial, with outside graduated circle shown in Fig. 34 ; but the details of construc- tion are very different, as may be gathered from Fig. 37. In the theodolite a telescope is always used, and mining theodo- lites are generally constructed as transit instruments; that is to say, the telescope can be turned all round in its bearings, so as to look either forward or backward. The telescope is generally carried on a vertical framework, a (Fig. 37), attached to and standing above the horizontal compass-box I at a sufficient height to allow the telescope to be reversed. The graduated circle c, for measuring vertical angles, is fixed on one of the telescope trunnions, while a pointer, d, carrying verniers is fixed to the framework. This circle can be clamped by means of the screw x. On the plate to which the telescope framework is attached are two verniers, e, c, at opposite sides ; below this plate is another carrying the horizontal graduated circle /, which can be clamped to the vertical axis of the instrument by the screw h. The upper plate can also be clamped to the lower plate. Spirit-levels are placed on the telescope and on the upper horizontal plate. A 5-inch theodolite will read both vertical and horizontal angles to 1', and an 8-inch theodolite to -' ; a 12-inch theodolite will read to 1". Mining theodolites are seldom bigger than 6 inches : the 5-inch is big enough for convenience (a 5 -inch transit theodolite weighs from 12 Ibs. to 14 Ibs. without the legs). By a 5-inch theodolite is meant one in which the horizontal graduated circle is 5 inches in diameter. With this instrument, the compass-needle, being INSTRUMENTS FOR MEASURING ANGLES. 69 underneath the framework carrying the telescope, is not easily observed ; it is therefore only occasionally used for taking the "bearing of a base-line, or for noting the approximate direction of lines ; the chief use of the instrument being for measuring FIG. 37. Transit theodolite. the angles, both vertical and horizontal, made by one line with the next. Another variety of theodolite construction is shown in Fig. 37A. The standards carrying the telescope, which are usually made in separate parts screwed together, are here all in one solid casting. The axis and standards are also in one casting, so that displacement of the axis is impossible. 70 MINE SURVEYING. Instead of the ordinary compass-needle, a trough compass is sometimes substituted, shown in Fig. 38 (and shown in position in Fig. 3?A). In this narrow box or trough the compass- needle is only free to revolve a few degrees on either side of the FIG. 37A. Stanley's theodolite. meridian, and it is merely used for fixing the theodolite in the magnetic meridian, this line serving as a base from which the bearings of the other lines can be calculated. Considerable FIG. 38. Trough compass. accuracy may be obtained in fixing the instrument in the magnetic meridian, because it is possible to see a very slight divergence of the needle from the N. and S. marks on the compass-box. INSTRUMENTS FOR MEASURING ANGLES. Another kind of compass (Fig. 39) was made for the author, useful only for the purpose of setting the telescope in the meridian ; it is fixed below the bottom plate of the theodolite. In this case the needle is very short only 2 inches and is not suspended at the centre, but near to one end, the short end being thick and balancing the longer end, the thin end of which conies opposite a nick in the tube when the instrument is turned in the magnetic meridian, and the position of the needle is accurately observed by means of a microscopic eyepiece. FIG. 39. Improved form of trough compass. Theodolites are generally made with parallel plates (see #, g, Fig. 37), by which the instrument can be levelled. A Hoffman head or other form of ball-and-socket joint, however, is some- times used, which also has four adjusting screws. The ball- and-socket joint enables the instrument to be levelled whilst the tripod stand is on very irregular ground. With the parallel plates alone there might be some difficulty in adjusting the instrument. Use of Theodolite Underground. For the purpose of illumi- nating the cross-hairs of the telescope, which, owing to the darkness of the mine, would be otherwise in- visible, one of the trunnions is made hollow, a lens being screwed into the outer end. Oppo- site this glass is fixed the bull's-eye of a small oil-lamp, the light from which passes down the hollow trunnion till it meets a reflector, consisting of a polished steel face about T V inch in diameter, placed within the telescope, by which the light is reflected on to the cross- FlG- 40. Lamp for hairs. For use in mines containing fire-damp, the small lamp for illuminating the cross-hairs must be enclosed in gauze, similar to that used for safety- lamps, and also shielded against the effects of strong currents, so as to comply with the conditions of the Mines Regulation Act (see Fig. 40). illuminating the cross-hairs of theo- dolite. MINE SURVEYING. In the absence of the hollow trunnions, the cross-hairs may be seen by the light of a lamp held near the object-glass. Sextant. This is an instrument for taking angles either in a vertical or a horizontal plane. It is used in surveying new countries, and for nautical and military surveying (Fig. 41). To measure the angle at the intersection of two lines, the tele- scope is directed upon an object in line No. 1. By means of a movable reflector fitted on the instru- ment and connected to the vernier, another object, in line No. 2, is at the FIG. 41. Sextant. FIG. 42. Box sextant. (Kindly lent by Messrs. W. F. Stanley tfc Co., Ltd.') same time brought into the same line of vision ; the angle through which the reflector is moved is measured by the vernier, and the angle between the two objects is read on the graduated arc. Small sextants, called box sextants (Fig. 42), are often made about 3 inches in diameter, so arranged that they can be conveniently packed in a pocket- case. The instrument is carried in the hand, but, owing to the fact that two objects are brought simultaneously into the line of vision, the angle formed by the two lines of sight may be read with some approach to accuracy. 1 Henderson's Rapid Traverser. Mr. James Henderson has recently patented a very simple instrument (see Fig. 43) for measuring and recording the angles of a survey. It consists of a circular metal table, on the top of which is fixed, by means of several small brass screw-nuts and bolts, a thin disc of cel- luloid or other suitable material, about 10 inches in diameter. 1 For description of the sextant and mode of using, the reader is referred to Hints to Travellers, published by the Koyal Geographical Society, also Surveying Instruments, by W. F. Stanley. INSTRUMENTS FOR MEASURING ANGLES. 73 Fixed on to the upper surface of the table and above the celluloid disc, by means of a centre-pin passing through, is a cross-bar, called an alidade, one side of which is bevelled. At each end of this cross-bar is a sight similar to the ordinary dial sight. By means of the usual clamping-screws, the table FIG. 43. Henderson's rapid traverser. carrying the celluloid disc can be clamped to the stand, and the alidade, with the sights attached, can also be clamped to the table, when required. The disc is divided into five concen- tric rings, slightly scratched or grooved on the celluloid ; and the bevelled edge of the alidade is notched out so as to afford to 74 MINE SURVEYING. each ring on the disc a certain length of bevelled edge, each length being distinguished by a number. The object of these concentric rings is not only to permit separate surveys to be accomplished on one disc, but to avoid overcrowding of direction-lines in any particular spot on the FIG. 44. Henderson's rapid traverser, showing quadrant. disc. The semicircle for reading angles in a vertical plane with ordinary sights or telescope can be attached when required (see Fig. 44). The instrument is based on what is known as the plane-table system of surveying ; unlike the plane table, INSTRUMENTS FOR MEASURING ANGLES. 75 however, it is not intended that the rapid traverser should be used for plotting the survey in the field, but this is done afterwards, in the office, with the aid of a parallel ruler and scale. The table is levelled by means of two spirit-levels, one of which is fixed on the alidade, and the other a small portable one which is carried in the pocket. The magnetic meridian is taken, at any convenient point in the course of the survey, by means of a trough-compass placed temporarily against the back edge of the alidade. The actual direction of the lines of sight is indicated by making a pencil- mark on the disc, and at the conclusion of the survey the disc is taken off and the directions of the lines ruled off it on to the plan. For future reference the disc itself may be kept, or else the magnetic bearings of the lines can be read off by means of a protractor and entered in the field-book, when the celluloid disc can be cleaned with soap and water or indiarubber, and so made ready for a future survey. The discs are now being made of enamelled zinc instead of celluloid. Tacheometer (see Fig. 45). This is an instrument used for measuring distances without a chain or tape. The ordinary tacheometer is similar to a theodolite, the only radical difference being in the telescope, in the diaphragm of which are fixed marks which can be directed to a graduated staff, such as a levelling-staff. The further the staff is from the instrument, the greater number of feet or inches will be seen between the two marks in the telescope. These marks may be made either of cobweb, like the ordinary hairs in the diaphragm of the theodolite, or of fine metallic points (in the later forms of instru- ment, lines engraved on a glass diaphragm are substituted for these hairs or wires) ; and they are placed at such a distance apart that the vertical height of an object between those two lines or points is some fraction, say 1 per cent., of the horizontal distance from the observer to the object. Thus if the vertical height on the graduated staff between the two points is 1 foot, the staff is 100 feet distant ; if the vertical height is 10 feet, the staff is 1000 feet distant. According to the kind of work which it is intended to do, these points can be placed nearer together or further apart. The accuracy with which measurements can be made in this way depends upon the power FIG. 45. Tacheometer (Troughton and Simms). INSTRUMENTS FOR MEASURING ANGLES. 77 of the telescope and of the microscopic eye-piece, and also upon the fineness of the points or cobweb used. Where it is possible to chain, the surveyor will, of course, employ this method in preference to the tacheometer, if great accuracy is required; but where, owing to the roughness or impassability of the ground, the measurement cannot be taken in this way, the tacheometer is of great use, and also for approximate measure- ments it is convenient. With a telescope of moderate power (magnifying, say, fifteen diameters), and for distances not exceeding 500 feet, tacheo- meter-measurements, on a bright day, should be correct to 1 per cent. ; for shorter distances, say under 300 feet, the error should not exceed J per cent. ; with a more powerful telescope the error may be much less. Some engineers have claimed that the error has never exceeded 1 in 2000 ; but for such a degree of accuracy a very fine instrument and great care in using are necessaiy. It is stated by surveyors of experience that a telescope mag- nifying fortyfold will read a staff to ^tro foot at a distance of 660 feet ; and, supposing the arrangement of hairs in the dia- phragm is such that 1 foot on the staff represents 100 feet hori- zontal distance, this means a possible error of J foot in 660, or an error of 1 in 1320. There is no doubt that with a good tele- scope great accuracy may be obtained with the tacheometer. Measurement of Distances with Ordinary Theodolite. It is possible to measure distances with the theodolite without the aid of two cross-hairs or other marks, by simply measuring the vertical arc subtended by a staff of given length. To measure lengths in this manner, direct the horizontal hair to the bottom of the staff or to some fixed mark above the bottom, and then, by means of the tangential screw, direct the hori- zontal hair to the top of the staff or some fixed mark, say 10 feet above the lower mark. Having read the angle, the distance can be calculated. Assuming that the staff is held vertically, and that the ground is level, the 10 feet will represent the chord of the arc. If the angle measured, for instance, was 1, the natural chord is 0'017453 ; then the distance may be found by the following sum : 0-017453 : 1 : : 10 : 572*96. This method is not so handy as that with two hairs, because the calculation is longer, and it involves two readings with the telescope, and there is, perhaps, an additional chance of error ; still, it is one 78 MINE SURVEYING. that may be easily used in the absence of a tacheometrical attachment to the theodolite. It follows, then, that when using a 10-foot staff, an error of one minute in the reading at a distance of 573 feet would mean an error of V f * na ^ distance, or nearly 10 feet. The ordinary 5-inch theodolite is only graduated to read to minutes ; but there is no reason why an error of one minute should be made in the reading. The error in the reading should not exceed half that ; and it is not necessary that there should be any material error. The longer the staff, the less will be the error for a given length ; but it is evident that for the accurate measurement of long lengths it is necessary to have a theodo- lite graduated to read to 10". With such an instrument and a 10-foot staff, the error, instead of being 1 per cent., will be reduced to \ per cent., or 1 in 600. In comparing the accuracy of tacheometer-measurements with that of ordinary chaining, it should be borne in mind that over rough ground, whether on the surface or in the mine, an error of half a link to the chain is very easily made, unless the surveyor gives the most careful personal attention to the laying out of the chain. Some tacheometers are constructed on a slightly different principle. Instead of fixed points or cross-hairs at the dia- phragm, between which is seen a length of a graduated staff, varying in exact proportion with the distance the staff is from the object-glass, a staff of fixed length is used, and at the dia- phragm is a slide carrying a cross-hair, which can be raised or lowered by means of a screw until the whole length of the staff, or of two very clear marks on the staff, are included between two cross-hairs. The movement of this slide depends on the distance the staff is away; the further the staff is from the object-glass, the less the movement of the slide. This move- ment is measured by the turns of a screw, on the head of which is a scale ; the distance corresponding with any given move- ment of the screw is marked upon the scale, so that no calcula- tion has to be made. In taking the observation, the two cross-hairs are so placed that one entirely obscures the other, and are directed towards one of the marks on the staff; the telescope is then clamped, and the requisite movement of the micrometer screw is made. Many tacheometers are so made that the distance as read INSTRUMENTS FOR MEASURING ANGLES. 79 on the scale requires no correction; in others a correction is necessary, owing to the fact that the distance measured by a tacheometer of the simplest kind is from the principal focus of the object-glass, whilst the distance required is from the centre of the instrument at which the angles are measured ; therefore the distance, as read off the staff, has to be corrected by the addition of a constant quantity equal to the sum of the focal distance of the object-glass, and the length from the object-glass to the centre of the theodolite. Thus, in using the theodolite with the fixed points, it is observed that the length of the graduated staff between them is, say, 2 feet ; if the points have been adjusted so that the factor for length is 100, then the distance is 2 x 100 = 200 + the length between the object- glass and the centre of the telescope (say 6 inches) -f the focal length (say 12 inches), or the required length is 201*5 feet. If the lengths are required in links, the staff should be graduated in links and decimals. Tacheometer Measurements in Hilly Country. When the tacheometer is used for measuring lengths on a level country, the staff will, of course, be held in a vertical line. If, however, FIG. 46. Tacheometrical measurements in hilly country. the ground is steeply inclined, then some consideration is necessary. In the first place, the telescope may be fixed quite level, and the staff held vertical, in which case the distance measured will be the horizontal length between the telescope and the staff (Fig. 46) ; of course, in this case, the length measured is limited by the height of the staff for the back sight, and the height of the telescope above the ground for the fore sight. In the second place (see Fig. 47), the telescope may be directed in a line parallel with the inclination of the ground, and the staff held at right angles to the inclination of 8o MINE SURVEYING. the ground ; then the distance measured will be the length of the slope and not the horizontal distance, which would have to be calculated by means of an observation of the angle made by the telescope with a horizontal line. In the third place (Fig. 48), the staff may be held vertical, and the telescope inclined at FIG. 47. Tacheometrical measurements in billy country. the same angle as the average slope of the ground, in which case the length measured will be greater than the length of the slope, and a correction will have to be made, owing to the greater length of the staff visible between the cross-hairs. Perhaps the best practice on steep gradients is to hold the staff FIG, 48. Tacheometrical measurements in hilly country. at right angles to the incline ; for moderate inclines the errors due to not holding the staff exactly in the correct position are very slight when this method is employed. For further information on the subject of tacheometry, the reader is referred to Mr. T. G. Gribble's excellent book on Preliminary Survey (Longmans, Green and Co.). Prismatic Stadia-telescope. 1 An ingenious modification of the ordinary stadia-telescope (tacheometer) is to use a glass 1 Robert H. Richards, Boston, America, Inst. M.E. Montreal Meeting, February 1893 ; also Glen Summit Meeting, October, 1891. INSTRUMENTS FOR MEASURING ANGLES. 81 prism or wedge. A ray of light passing through a prism is deflected, the amount of deflection depending on the angle enclosed by the two sides of the prism at their apex if prolonged. If, therefore, half the object-glass of a telescope is covered with a prism, and a graduated staff is observed, the figures on one side will be seen in their correct position ; on the other side they will be seen out of place, owing to the deflection caused by the prism. Thus, if with the uncovered half of the object-glass the cross-hairs of the telescope appear to cut the figure 3, with the covered half the cross-hair may appear to cut the figure 5, showing that the deflection of the rays of light caused by the prism is measured by 2 feet on the staff if the staff is distant 100 feet. This deflection is equal to an angle of about 1 9'. If, therefore, the staff were moved to a distance of 200 feet from the telescope, the deflection, being at the same angle, would cover 4 feet of the staff; and if the staff were moved to a distance of 300 feet, the deflection would cover 6 feet of the staff, and so on. This angle of deflection being ascertained, it follows that the distance at which the staff is held from the telescope can be calculated from the amount of deflection as read on the staff. Thus If the figure read with one half of the telescope is 3, and with the other half 4, the distance is 50 3 5 100 >, ' ,, 3 ,, ,, 6 ,, 150 8 7 200 3 8 250 ,, ,, 3 ,, ,, 9 ,, 300 and so on, every foot of deflection on the staff representing 50 feet of distance from the telescope, every T V foot representing 5 feet ; T ^ foot, 0'5 foot ; and T ^VTF foot, 0'05 foot. Mr. Kobert H. Kichards has tried various telescopes in which the deflection of the prism varies from 1 foot of staff in 100 feet in length, to 3 feet of staff in 100 feet in length. The greater the deflection, the greater the accuracy with which the amount of it can be read ; on the other hand, the greater the deflection, the longer the staff required for any given distance. Mr. Pdchards considers a telescope magnifying thirty dia- meters suitable for reading the staff at a distance of 1000 feet, 82 MINE SURVEYING. and for distances up to 2500 feet, with a specially constructed sliding target staff. For a sight of 1000 feet and a prism de- flecting 1 per cent., a staff about 12 feet long is required. Mr. Richards also recommends the use of what he calls the optical vernier. This may be understood by reference to Figs. 49 and 50. This is a staff about 6 inches wide, and a height necessary for the distance it is intended to read ; it is painted half white and half black. On the upper left-hand side is a vernier painted in white ; the rest of the left-hand side of the staff is black, and the main scale is painted in black opposite this.' It is divided into lengths repre- senting 50 feet of horizontal distance, which are num- bered 1, 2, 3, 4, 5, 6, etc. ; this means six fifties, or 300 feet. Each fifty is divided by five equidistant diamond points, represent- ing 10 feet. The vernier is also divided so that five points shall cover a space equal to four points on the main scale. The main scale is seen through the un- covered half of the telescope, SELF-READING TARGETS AS SEEN BY THE EYE SELF READING TARGET AS SEEN THROUGH THE PRISM in connection with a prismatic stadia telescope. 4-V.p vprm'pv ^11^ Vtililltil FIG. 49. staff used FIG. 50. Method of prism. The prism deflects reading ditto. the vernier, and it is thrown , ., down opposite some figure on the main scale. In Fig. 50 the zero of the vernier is apparently past the third point below 4 ; 4 means 4 times 50, or 200 ; the three points on the main scale are each 10 feet, therefore the distance is 230 + a fraction of 10. The second point of the vernier from zero is exactly opposite one of the points on the main scale ; each point of the vernier counts 2, therefore the fraction is T V x 10, or 4 feet. So that the total distance is 234. The INSTRUMENTS FOR MEASURING ANGLES. heights on the main scale, representing 50 feet of distance, have been found by experiments with the prism. Tape Target. At distances greater than 1000 feet, the figures on the staff cannot be read, and Mr. Eichards recommends a tape target, the distance being read by the assistant carrying the tape. This target is shown in Fig. 51. The telescope' is directed towards the centre of three diamond points on one target ; the other target is moved along the tape, in accordance with signals given by the surveyor, until its deflected image becomes opposite to the image seen through the uncovered portion of the object-glass ; the two centre diamonds of each set of three correspond when the targets are set at the correct distance apart ; the two outer diamonds do not correspond, and the distance of their points apart should be equal for each pair, as shown in TAPE-TARGETS AS SEEN BY EYE FIG. 51. Tape with movable targets. TAPE TARGETS BEING READ BY THE PRISM. FIG. 52. Method of reading movable targets. Fig. 52. The assistant reads the distance on the tape, and books the figure, and perhaps signals the reading to the surveyor. All the systems of tacheometry above described necessitate the use of a staff on which the graduations can be read through a telescope, or on which are movable marks which can be read by an assistant who adjusts the marks in accordance with signals received by flags or otherwise from the surveyor. It is, however, very convenient to use a range-finder, with 84 MINE SURVEYING. which the surveyor is independent of any markings upon a staff. The ordinary method of triangulation with the theodolite from a measured base is a kind of range-finding, and for exact work it is the best method known. For approximate calculations, such as are used sometimes by military engineers, a tape or cord of given length may be carried by two observers, each carrying a box sextant, and reading simultaneously the angle formed by the base-line and the object whose distance they wish to ascertain. Range-finder. An ingenious adaptation of the prism has been devised by Professors Barr and Stroud. In this instru- ment the measured base is a short tube, 3 feet long, held by the surveyor in his hand, or fixed on a tripod. The tube is held at right angles to the line of sight (see Fig. 53). It contains the v' g./ fl" 5e -^ FIG. 53. Barr and Stroud's range-finder. equivalent of two telescopes, one at each end of the tube or base, with the requisite optical appliances for seeing the two fields of view in juxtaposition one over the other. Eays of light from the object viewed enter through openings, V.^ V 2 , at each end of the tube, and are reflected at right angles along the axis of the telescope by means of the reflectors H^ H 2 . The observer places his right eye at the eye-piece K 2 , and, by means of the arrangement of prisms at J, sees two images of the object, one above the other, but not in line with each other. By the movement of an achromatic glass prism, M, of small angle along the axis of one of the telescopes, the two images of the object whose range is required are brought into exact alignment, when the position of the prism furnishes a measure of the range, which is read off by the left eye on a scale, B, attached to the prism, and moving with it. The surveyor has, therefore, no calculations to make, but simply sets his instrument upon the object, such as a staff, church, house, tree, fence corner, candle, lamp, etc., and then, after adjusting the two images of the object in exact alignment, reads the distance as written on the instrument. A similar instrument has been adopted in H.M Navy, and is now installed on most of the battle-ships and cruisers. CHAPTEE V. INSTRUMENTS FOR PLOTTING LENGTHS AND ANGLES. IN considering the use of instruments for plotting angles, it will be well to refer to the plan of an estate shown in Fig. 14. On this plan the bearing of No. 1 line is marked " North 10 East," which means that the direction of the line from the starting-point is going towards the north-east, and the exact bearing is 10 east of north ; the bearing of No. 2 line is also given as N. 30 W., and line No. 12 is S. 74 W. If these lines are laid down according to the bearings so marked, and for the lengths measured, they will take up their correct position as regards each other, and it will not be necessary to use the compasses for the purpose of plotting them. If, however, the lines have been already plotted from the measurements only, the bearings can be used as a check on the accuracy of the survey and of the plottings, because the relative positions of the lines, as shown by the bearings, will be the same as that shown by the triangular measurements. One use, therefore, of an instrument for taking these bearings is to check the accuracy of the survey ; the second use is, perhaps, more important, and that is to ascertain the direction of the survey-lines with regard to the magnetic meridian, and for most mineral plans it is necessary to have the magnetic meridian, or "north point," as it is commonly called, marked with extreme care. In the production of a plan, two distinct classes of instru- ments are necessary. These are, first, the instruments pre- viously described for measuring lengths and angles on the ground, and second, the instruments for drawing or plotting upon paper the above-mentioned lengths and angles. Scales. The instrument for drawing the lengths is called the scale : it consists of a straight piece of hard material, either 86 MINE SURVEYING. ivory, wood, metal, or cardboard; it is generally a little more than 12 inches long, and is divided into equal parts to suit the purpose required. For an ordinary English mining plan it is usual to have a scale of chains, the measurements being taken with the Gunter's chain. Thus it may be desired that 1 chain in length shall be represented by a length of 1 inch on the plan ; then the scale will be divided into inches. If, however, this would produce too big a plan, -J inch may be used to represent 1 chain, and the scale will therefore be divided into half-inches, or it may be divided into thirds, fourths, fifths, sixths, eighths, or tenths of an inch, each division intended to represent 1 chain. The most common scale for mining plans is that in which % inch represents 1 chain, commonly called a 2-chain scale, which means that 1 inch on the plan is equal to 2 chains measured in the field or mine. In the Coal-Mines Eegulation Act of 1887 it is mentioned that the scale of a colliery plan must not be less than 25'344 inches to the mile (which is equivalent to 3157 chains to 1 inch). 1 This seems to give sufficient latitude as to the size of scale to be adopted ; in many mines a scale of 3 chains to 1 inch is used; in others, a scale of 1 chain, and sometimes half a chain to the inch. Having divided the scale into chain-lengths, each chain- length is then subdivided into tenths, each tenth representing 10 links. The surveyor, in plotting a length more than 10 and less than 20 links, must divide the space by his eye, as smaller graduations are not generally used. The edge of the scale is bevelled, so that the dividing marks on the edge of the scale touch the paper. It is found convenient to have on the opposite edge of the scale to that on which the chain-scale is divided, a feet-scale. This is a scale in which sixty- six divisions on the feet-edge measure the same distance as 100 divisions on the opposite or chain-edge. This enables the scale to be used for taking off measurements in feet from a plan which has been plotted in links. The use of feet and links on the same scale, however, often leads to confusion and error. Another scale is also used, called an offset scale. It is generally 2 inches in length, graduated in the same manner as 1 The exact wording of the Act of 1887 is as follows: "Every such plan must be on a scale of not less than that of the Ordnance Survey of twenty-five inches to the mile, or on the same scale as the plan for the time being in use at the mine." INSTRUMENTS FOR PLOTTING LENGTHS AND ANGLES. 87 the long scale, but the divisions begin and end exactly at the ends of the scale. It is used in the manner shown in Fig. 54, to mark off lengths at right angles to the lines drawn on the plan. The scale is laid down on the paper along the line FIG. 54. Scale and offset. (Kindly lent by Messrs. W. F. Stanley and Co., Ltd.) representing the survey-line ; the offset scale is then placed so as to measure lines at right angles, and is moved along the scale to the division representing the required distance on the survey-line ; the length of the offset can then be marked off by means of the shorter scale. In constructing a plan, the scale is usually drawn upon it, and thus, if any serious shrinkage of the paper takes place, measurements may be made by means of this scale. Ivory is much liked for scales, because of the clearness of the lines, but boxwood is cheaper and less easily broken ; metal is not much used, partly, perhaps, because of its greater tendency to expand or contract with variations of the temperature. The expansion of brass between freezing point and boiling point is -o- of its original length, which is equal to Tr w part for each degree of temperature, or to the expansion of o-^Tr part for a rise of 40 in temperature, that is to say, the scale would expand 1 inch in a length of 2250 inches, or ^V part of an inch in a length of 100 inches. This amount of expansion is not very serious, especially as the temperature of a drawing office in England does not usually vary as much as 40 ; it is seldom that drawing is done in an office of a less temperature than 50 or a higher temperature than 65, hence the expansion would be only that due to 15, or ^V inch in a total length of 100 inches ; therefore the expansion of brass does not seem to 88 MINE SURVEYING. be a sufficient reason why it should not be used. A more practical objection is that metal scales soil the drawings. Ordnance Maps. The survey of the United Kingdom was commenced by order of the Government about the year 1784. The survey has been published in maps of various scales, viz. 1 inch to the mile, or -yihnr; 6 inches to the mile, or TTrhnr; and 25'344 inches to the mile, or TT/OTT- Town plans, on scales of 10| feet to a mile and 5 feet to a mile, are also published of the principal towns. On these maps are shown the various boundaries of the counties, unions, parishes, etc. The first two series show the contour-lines, and are particularly useful for the purpose of deciding as to the best route to adopt for lines of railway, and the positions of shafts, buildings, etc. They also show the lines of latitude and longitude. The oVoij scale maps can be obtained either plain or with the buildings and rivers coloured. The fields are all numbered, and the area of each field in acres is either printed on the map or can be obtained for each parish, published in book form. A plan made by mounting the various sheets of the -r^Vo- map covering the royalty is sometimes used on which to mark underground workings. By application to the Director-General of the Ordnance Survey Office, Southampton, however, tracings from the original plotted plans can be obtained, and these are much more accurate for this purpose, as the printed maps often shrink a good deal. Owing to this latter fact, measurements from the Ordnance plans should be made with the printed scale given on each sheet. Geological maps are also published on the 1-inch and 6-inch scales, and give a great deal of valuable information as to the faults, dip of the measures, and other geological features of the country. Compasses. Compasses are generally used to set off the distances from the base-line as previously explained ; these are shown in Fig. 55. They are made in various sizes, ranging from 2J inches to 9 inches long. There are points at the end of each limb, needle-points are the best ; one limb is jointed, so that the needle-point can be taken out, and a pencil, a, or pen, b, substituted ; one or more lengthening pieces, c, can be added to this limb, so as to increase the length that can be set out. When this length is insufficient, beam compasses are used. These are formed with a beam, or piece of wood, and are shown INSTRUMENTS FOR PLOTTING LENGTHS AND ANGLES. 89 in Fig. 56. At one end of this beam is fastened a screw-clip, a, carrying a point at right angles to the beam, and about 2 inches long. A similar clip, b, carrying a pencil is slipped on to the beam, and is moved along till the required distance from the point fixed at the other end is obtained. It is then clamped, FIG. 55. Compasses. (Kindly lent by Messrs. W. F. Stanley and Co., Ltd.) and an exact adjustment for length is made with an adjusting screw on the point-holder; then, with the fixed point as the centre, a circle may be described with the pencil-point. Several beams of say 2, 4, and 6 feet in length are kept for use with these compasses. Straight-edge. For the purpose of ruling a straight line FIG. 56. Beam compasses. (Kindly lent by Messrs. W. F. Stanley and Co., Ltd.) from one point to another, a straight-edge is used ; a metal straight-edge is the best, not being liable to warp. Steel straight-edges require to be kept bright, and are sometimes nickel-plated. Though not absolutely necessary, it is a good thing to have bevelled edges to the ruler. Parallel Ruler. A parallel ruler (Fig. 57) is much used by mining surveyors ; it is generally made of metal, as a considerable 90 MINE SURVEYING. weight is advantageous ; it consists of a bar from 2 inches to 3 inches wide, and from -/^ inch to y'V inch in thickness, with bevelled edges, and varying from 6 inches up to 2 feet in length. In this bar are cut two holes within a short distance FIG. 57. Rolling parallel ruler. (Kindly lent by Messrs. W. F. Stanley and Co., Ltd.) of each end ; on the upper side of the bar are fixed two rollers, fixed on a long spindle, the ends of which are carried in brackets ; the lower sides of the rollers project a little way through the bar, so that the bar may roll along. Each of these rollers is the same diameter, and is roughened with longitudinal cuts to prevent it from slipping. These rollers being the same diameter, if there is no slipping, the two ends of the bar will move at the same rate and the same distance when rolled along over the paper. Thus, if the ruler is held in a given position, and a line drawn, and it is then carefully rolled across the paper, and another line drawn, the two lines will be parallel one to the other. Fig. 58 shows another form of parallel ruler, known as the sliding-bar parallel ruler, but for the purposes of a mine surveyor the rolling parallel ruler will be found FIG. 58.-Sliding parallel ruler. to be the . m st efficient. (Kindly lent by Messrs. W. F. Stanley and <>., d.) Drawing-pencil. 111 plot- ting a survey the lines are drawn with a hard-lead pencil cut to a fine point. Pencils are made in varying degrees of hardness, the most useful being that marked H.H. The Koh-i-noor pencil is highly recommended. Pricker, Distances and stations are generally marked off the scale with a needle-pointed pricker, the point of the needle making a much finer and more permanent mark than the point of the pencil. In this way a length may be marked on the 2-chain scale with an error not exceeding 1 link ; thus if the actual distance measured was 8 chains 55 links, the prick- INS TR UMENTS FOR PL TTING LENG THS AND A NGLES. 9 1 mark made with the needle might possibly be 8 chains 54 links or 8 chains 56 links, but, in either case, it would be within a link of the correct distance. Set-squares. A large set-square is useful 1 for marking out X FIG. 59. Set squares. (Kindly lent by Messrs. W. F. Stanley and Co., Ltd.) lines at right angles to one another ; such lines are required for plotting lengths ascertained by trigonometrical computation ; the larger this set-square, the greater the degree of accuracy with which the cross-lines can be drawn. The draughtsman is recommended to use one not less than 12 inches long on each of the square sides. The two most usual forms of set-square are shown in Fig. 59. Protractor. For plotting angles a graduated circle marked in a similar way to the dial, called a protractor, is used (see Fig. 60). 1 These may be made of brass, and vary from 8 to 12 inches in dia- meter ; the 8-inch protractor is graduated to half-degrees, FIG. 60. Brass protractor. {Kindly lent by Messrs. W. F. Stanley and Co., Ltd.) and the 12-inch protractor to quarter-degrees, smaller frac- tions of a degree having to be estimated ; the protractor being so much larger than the dial, the fractions of a degree can be estimated with greater 1 For plotting circle readings of the needle, the numbers on the protractor should count the reverse wav of those on the dial. 9 2 MINE SURVEYING. accuracy, and therefore there should be no serious errors in plotting from this cause. It is, however, difficult with an 8-inch protractor to divide a degree without some error, which may very likely amount to ^; the thickness of a needle-prick is about J on an 8-inch protractor, so that for very accurate work a simple 8-inch pro- tractor is not sufficient. By using a 12-inch protractor the accuracy is increased in the proportion of 2 to 3 ; but for very accurate work a protractor fitted with a vernier with, folding arms, clamp, and tangent-screw is sometimes used (Fig. 61). FIG. 61. Brass protractor with folding arms. (Kindly lent by Messrs. W. F. Stanley and Co., Ltd.) By means of the vernier the arms may be adjusted to 1', that is to say, to the sixtieth part of a degree. At the end of each arm is a sharp pricker, which can be pressed down to mark the paper. If this instrument is well constructed and properly used, the angles can be marked out with great accuracy. It is, however, a common practice to use a cardboard pro- tractor (Fig. 62). The graduated circle is printed on to a stout card, and is generally 12 or 15 inches in diameter. The divisions are made to read inward from the circumference, instead of outwards as with other protractors, the centre space of the card being entirely cut away. In using cardboard protractors it is not necessary to prick off the angle, as the parallel ruler can be placed upon the protractor at the right angle, and then rolled to the required INSTRUMENTS FOR PLOTTING LENGTHS AND ANGLES. 93 place, provided, of course, that the work is within the circum- ference of the protractor. For plotting underground surveys, where the lines are usually short and close together, these protractors are very convenient. A modified form of cardboard protractor has been designed by Mr. R. F. Percy, 1 and is shown in Fig. 63. It is made of FIG. 62. Cardboard protractor. thin pasteboard. Parallel north-and-south lines are, with the greatest care and accuracy, drawn at intervals of 2 or 3 inches, and at the ends of all these parallels, on the left at the north edge, and on the right at the south edge, divergences of 1 and fractions of 1 are indicated (Fig. 63). The part within the 1 Transactions Fed. Institute of Mining Engineers, vol. xiii. p. 585. 94 MINE SURVEYING. divided circle is, as usual, cut away, and the plotting is executed within that space. The parallel meridians allow the protractor to be placed exactly where it is needed, very few meridian-lines being required \_J FIG. 63. Percy's form of cardboard protractor. on the plan. The divergences marked at the ends of the parallel lines will allow the protractor to be twisted for declination, so as to bring the meridian to the date of the survey. Drawing-pens. Fig. 64 shows a drawing-pen; it has two STANLEY LONDON x> FIG. 64. Drawing-pen. (Kindly lent by Messrs. W. F. Stanley and Co., Ltd.) pointed blades, kept apart by a spring ; the distance apart can be adjusted by turning a milled-head screw. It is supplied with ink by means of a brush or pen, and when used should be held nearly upright between the thumb and forefinger. After being used some time, the nibs become blunt, and will require sharpening on an oil-stone ; this is an operation requiring some skill and practice. Curves. For drawing curved lines, such as railway curves, it is found useful to have ruling-edges made of pear wood or cardboard. These are cut to arcs of circles with radii varying from 1 to 250 inches, and are sold in sets. Weights and Pins. To hold the plan while working at it, drawing-pins may be used, but these injure the plan. Lead or iron weights are more commonly used by mine surveyors ; they are of oblong form, and covered with cloth or leather so as not to soil the paper. INSTRUMENTS FOR PLOTTING LENGTHS AND ANGLES. 95 Colours and Brushes. For inking-in the finished plan, Indian ink is used. This is generally sold in hexagonal or octagonal sticks, and is ground into liquid ink by rubbing with water upon some kind of palette. The rubbing is continued until a line drawn with the ink dries quite black. Lines drawn with the best ink, however, are liable to run when colour is washed over them, so the lines should be as fine as possible. Liquid Indian ink may be obtained which overcomes this defect, but it is hardly so good to draw with as the stick ink. Water-colours are used for colouring drawings ; they are supplied in cakes, and are ground in the same way as Indian ink. The best kind of brushes for colouring are those made of sable hair. Drawing-paper. It is important that the best drawing-paper should be used for mining plans. That known as Whatman's is very good. The sizes in which sheets of drawing-paper can be obtained are Demy 20 inches by 15| inches Medium 22f ,,17 Boyal 24 19^ Imperial ... 30 ,, ,,22 Double elephant ... 40 ,, ,, 27 Antiquarian 53 ,, ,,31 ,, Mounted plan paper can also be obtained in continuous rolls in widths varying from 27 inches to 60 inches, or paper can be mounted to order to make a plan of any size. For a large permanent plan the best paper mounted on strong brown holland will cost as much as 5d. to Sd. a square foot. The thickness of the paper and holland together varies from about T V inch up to about ^ inch ; ^V inch makes a very good plan. Tracings. Copies of drawings are usually made on tracing- paper or tracing-cloth, which are transparent. These may be obtained in continuous rolls the same as the drawing-paper. CHAPTER VI. GEOMETRY, TRIGONOMETRY, LOGARITHMS. BEFORE proceeding to consider the method of surveying on the surface by means of angles, or of underground surveying which is always done by means of instruments for measuring angles, it will be necessary to consider the relations of the sides and angles of a triangle to each other, which are ascertained by the science of Trigonometry. A slight knowledge of Geometry is also necessary. The definitions given below are taken from Euclid's Elements. Fig. 65 shows a circle ; the point A, from which it has been described, is called the centre of the circle. The diameter of a circle is a straight line drawn through the centre, terminated both ways by the circumference (BC, Fig. 65). The radius of a circle is a straight line drawn from the centre to the circumference (AB, Fig. 65). The circumference of a circle is the line described by the pencil of the compass when it is re- volved round a point. A chord is any straight line drawn across the circle from cir- cumference to circumference, not passing through the centre (DE, Fig. 65). An arc is that part of the circumference of a circle which lies between the two ends of a chord (DFE, Fig. 65). An angle is formed when two straight lines, not in the same FIG. 65. Circle : diameter, radius, chord, arc. GEOMETRY, TRIGONOMETRY, LOGARITHMS. 97 straight line, meet together. The unit adopted in measuring angles is the degree. The circle is divided into 360 degrees (written ) ; each degree is subdivided into sixty equal parts, called minutes (written ') ; and each minute is subdivided into sixty equal parts, called seconds (written "). The circle is also divided into four equal parts, called quadrants, each containing 90 degrees (CAP, BAF, BAG, CAG, Fig. 65). The measure of any angle (CAM, Fig. 65) is the number of degrees covered by the arc CH. A right angle encloses 90 degrees ; a straight line at right angles to another straight line is said to be a perpendicular (Pig. 66, (1)). An obtuse angle contains more than 90 (Fig. 66, (2) ). (2) (3) (4) (6) FIG. 66. Angles and triangles. An acute angle contains less than 90 (Fig. 66, (3) ). A triangle is a figure contained by three straight lines. An equilateral triangle has three equal sides, and three equal angles (Fig. 66, (4) ) ; an isosceles triangle has two sides equal (Fig. 66, (6) ) ; a right-angled triangle is that which has one of its angles a right angle (Fig. 66, (5) ). A square has four equal sides, and all its angles are right angles. A rectangle has all its angles right angles, but not all its sides equal. A trapezium is a plane figure contained by four straight lines, of which no two are parallel. Parallel straight lines are those which, if produced both ways, would never meet. H 98 MINE SURVEYING. The following theorems are also taken from Euclid, and should be thoroughly mastered : (1) When a straight line meets another straight line, the angles formed are together equal to two right angles. Eef erring to Fig. 67, the two angles ABC and ABD together equal two right angles, or 180, so that if _^_^ we know the number of degrees D B C in one angle, we can find the FIG. 67. Elementary geometry. magnitude of the other by sub- traction. (2) If two straight lines cut one another, the vertical or opposite angles are equal. Thus in Fig. 68 the angle AEC FIG. 68. Elementary geometry. equals the angle DEB, and the angle AED is equal to the angle CEB ; and the four angles are together equal to 360 ; therefore, if one angle is known, the other three can be calculated. (3) If a straight line, EF, fall on two parallel straight lines AB and CD, the angles AGH and GHD B are equal, the angles EGB and GHD are equal, and the two angles BGH and D GHD are together equal to two right angles (see Fig. 69). (4) The angles at the base of an isosceles triangle FIG. 69. Elementary geometry. (see Fig. 66, (6) ) are equal to one another. (5) The three angles of a triangle are together equal to two right angles, or 180 ; therefore, knowing the two angles, we can get the third by subtraction. \ GEOMETRY, TRIGONOMETRY, LOGARITHMS. 99 (6) Any two sides of a triangle must be together greater than the third. (7) In any right-angled triangle, the square which is described on the side opposite the right angle is equal to the sum of the squares described on the sides containing the right angle. Fig. 70 shows a right-angled triangle ; then AB 2 = AC 2 -f BC 2 . Suppose AC is 80, BC is 100; then to find AB AB 2 = (80) 2 + (100) 2 /. AB = 128-1 In the same way, we can find AC or BC, if we have the other two sides of the triangle given. 80 B wo <- D FIG. 70. A right-angled triangle. B C FIG. 71. Elementary geometry. (8) If one side of a triangle be produced, the external angle is equal to the sum of the two opposite internal angles. The angle ABC is equal to the sum of the angles DAB and ADB (Fig. 71). (9) In every triangle equal sides subtend (or are opposite to) equal angles, the greatest side subtends the greatest angle, and the least side -the least angle. Practical Geometry. (1) To bisect a line AB (Fig. 72) ; that is, to divide it into two equal parts. From A and B, with any radius greater than the half of AB, describe arcs cutting each other in c and d. From c draw a straight line to d, and it will bisect the line AB. (2) To draw a line perpendicular to a given line AB at a point C in the line (Fig. 73). From C, with any radius, cut the line AB in c, c ; from c, c, with any radius greater than half cc, describe arcs cutting in d ; draw the line Cd, and it will be perpendicular to AB. 100 MINE SURVEYING. (3) To draw a line perpendicular to a given line AB, from a point C above or below the line (Fig. 74). B A B FIG. 72. Method of bisecting a line. FIG. 73. To draw a perpendicular line. The description and letters of the last problem apply to this figure also. (4) To draw a line perpendicular to a given line AB, at its extremity (Fig. 75). A B FIG. 74. To draw a perpendicular line. FIG. 75. To draw a perpendicular line. From B, with any radius, describe an arc having its extremity c in the line AB. From c, with the same radius, cut the arc in d ; and from d, with the same radius, cut the arc in e. From d and e, with the same radius, describe arcs cutting in/. Draw the line /B, and it will be perpendicular to the line AB at its extremity. GEOMETRY, TRIGONOMETRY, LOGARITHMS. 101 (5) Through a given point C to draw a straight line parallel to a given straight line AB (Fig. 76). A' B FIG. 76. To draw a parallel line. From any point B in the line AB describe an arc CA, and from the centre C, with the same radius, describe the arc BD, and make the arc BD ^ equal tq^the arc AC. Then the line joining CD is parallel to the line AB. (6) To construct a triangle, its three sides being given (Fig. 77). Let the sides be 50, 75, and 60. Draw a line AB, and mark off the length AC equal to 50 on the scale; then, with centre A and radius 75, draw an arc, and from the centjre C, with the radius 60, draw another A 50 c B FIG. 77. To construct a triangle, three sides being given. Then join AD and CD, and arc, cutting the first arc in D. ACD is the required triangle. (7) To construct a triangle when two of its sides and the angle between them are known (Pig. 78). Let the two sides be 30 and 40, and the angle included 45. Then draw a straight line AB, and mark off a length AC equal to 40, and, by means of a pro- tractor, make the angle CAD equal to 45, and make AD equal to 30. Then, by joining DC, the triangle is completed. A 40 C B FIG.- 78. To construct a triangle, two sides and the included angle being given. 102 MINE SURVEYING. Trigonometry deals with the relative measures of the sides and angles of triangles. Let ABC be any angle (Fig. 79), then in one of the lines containing the angle take any point D, and from D draw DE perpendicular to AB. Then we have formed a right-angled triangle BDE, and the side DE is called the perpendicular; the side BD, which is opposite the right B E A an gi e> i s called the hypotenuse, FIG. 79. Relations between sides -i ji * DC , P <,]]*,] th p and angles of a triangle. ancl tne S1Cle Dt 1S callea tne base. From these three sides we can form six ratios or fractions as follows : (1) BE = Perpendicular ^ EBQ BD hypotenuse /r\ BE base (2) = r- ,, ., cosine ,, ,, BD hypotenuse /0 \ ED perpendicular By inverting the above three ratios, we obtain three more, as follows : (4) _r- > = '- is called the cosecant of the angle ABC ' D E perpendicular / e \ BD hypotenuse (5) ~ = -^^ secant /\ BE_ These trigonometrical ratios are always the same for the same angle, but are different for different angles. In some cases these ratios i.e. sine, cosine, etc. may be represented in magnitude by single lines. For instance, referring to Fig. 80, suppose the circle to have been drawn with a radius of 1 Then the sine of the angle ABC is = F P = FD and the cosine ,, ^ = = FB oLJ GEOMETRY, TRIGONOMETRY, LOGARITHMS. 103 and the tangent of the angle ABC is ^ ^ = cotangent 1 secant cosecant 1 = AC HE_HE_ HB~ F BC = BC = BA " i BE = BE = HB i It will be seen that by referring all these ratios to a radius of 1, we are able to measure their values for any angle. Thus in Fig. 80 the angle ABC is drawn 60, and if the line FD be measured with the same scale that AB was drawn with, it will be found to be 0'866 ; therefore the sine of 60 (to radius 1) is 0'866. In the same way the other ratios can be arrived at. \30 90?" B- FIG. 80. Trigonometrical functions. -- 6 OO Links - -> FIG. 81. Use of trigonometrical ratios. Tables of these ratios may be got in which the values of the natural sines, cosines, etc., have been worked out for all angles. The word "natural sine" is used to distinguish it from the logarithmic sine. The natural sines are the actual values of the ratios, while the logarithmic sine is the logarithm of that ratio. EXAMPLES. (1) Let Fig. 81 represent a triangular field. The base EB is known to be 6 chains, also the angle EBD 30; then to find the side ED. We know that ^^ = tangent of EBD- On referring to our book of tables, EB we find the natural tangent of 30 is 0-5773503. Then = 0-5773503; but EB = 6 chains = 600 links. Then ED = 600 x 0-5773503 = 346-41. An*. Since the angle HEB = the angle ABC. IO4 MINE SURVEYING. In a similar manner, by working out the equation ~~ cosine 30, we can find the other side BD- (2) At a point 100 yards from the foot of a building, I measure the angle of elevation of the top, and find that it is 23 15' : what is the height of the building ? Let Fig. 82 represent the pro- blem ; E D is the unknown height. The length BE is known to be 100 yards, and the angle EBD to be 23 15'. FIG. 82. Use of trigonometrical ratios. Then ^g = tan 23 15'. ED From the table of tangents we find that tan 23 15' = 0-4296339. .*. ED = 100 x 0-4296339 = 43 yards (nearly) which is the required height. Of course, both these problems could have been solved by plotting ; but unless the scale had been very large, the results would not have been nearly so accurate. Logarithms. Logarithms are used to facilitate calculations. The logarithm of a number is the power to which an invari- able (or constant) number, called the base, has to be raised to equal the given number. In common logarithms the base is 10, and the power to which 10 has to be raised to produce any number is the logarithm of that number. Thus 10 X 1, or 10 1 = 10 /. 1 = log. 10 10 x 10, or 10 2 = 100 /. 2 = log. 100 10 X 10 x 10, or 10 3 = 1000 /. 3 = log. 1000 10 X 10 x 10 X 10, or 10 4 = 10000 .'. 4 = log. 10000 and so on. It is proved by algebra that 10 = 1 and O'l or T \, = 10' 1 and 0-01 or T J = 10 ~ 2 0-001 or T(5 V (T = 10- and so on. = log 1 - 1 = log. 0*1 - 2 = log. O'Ol - 8 = log. O'OOl Thus we see that the logarithm of a number greater than 1 and less than 10 is a positive decimal ; and the log. of a number GEOMETRY, TRIGONOMETRY, LOGARITHMS. 105 between 10 and 100 is greater than 1 and less than 2 ; that is to say, will be 1 + a decimal, and so on. We see also that the logarithm of any number between 1 and O'l is negative, and would lie between and 1, and can be written - 1 + a decimal ; and the log. of a number between O'l and O'Ol can be written - 2 + a decimal; and so on. A logarithm consists of two parts the integral, or whole- number part, which is called its characteristic , and the decimal part, which is called the mantissa. The mantissa of the logarithm may be found in a table of logarithms, but the characteristic is found as follows : (a) If the number whose logarithm is sought is greater than unity, the characteristic is always one less than the number of figures it contains ; thus (c) 1 (m) The logarithm of 43758 = 4*6410575 4375-8 = 3-6410575 43-758 = 1-6410575 4-3758 = 0-6410575 etc. (b) If the number is less than unity, the characteristic is minus or negative, and is found by adding one- to the number of cyphers between the decimal point and the first significant figure; thus GO O) Log. 0-43758 = 1-6410575 0-043758 = 2-6410575 0-00043758 = 4'6410575 Many good tables of logarithms can be obtained ; the author often uses Chambers's, 2 which, in addition to giving the loga- rithms of all the numbers from 1 to 108000, contain an excellent explanation of their use, from which some of these illustrations are taken. 3 I. To perform multiplication by logarithms. Add the logarithms of the factors, and the sum will be the logarithm of the product. 1 c = characteristic ; m = mantissa. 2 Chambers's Mathematical Tables, published by W. & R. Chambers. 3 Babbage and Callet's Tables give logarithmic sines, cosines, etc., worked out to 10 seconds. OF THE- ( UNIVERSITY OF 106 MINE SURVEYING. EXAMPLES. (1) Multiply 9999 by 999. Log. 9999 = 3-9999566 999 = 2-9995655 Sum = 6-9995221, which is the log. of 9989001. Am. (2) Multiply 0-03902, 59-716, and 0-00314728. Log. 0-03902 = 2-5912873 59-716 = J.-7760907 0-00314728 = 3-4979353 Sum = 3-8653133, which is the log. of 0-007333533. Ans. II. To perform division by logarithms. From the logarithm of the dividend subtract that of the divisor, and the remainder will be the logarithm of the quotient. EXAMPLES. (1) Divide 371-49 by 52-376. Log. 371-49 = 2-5699471 52-376 = 1-7191323 Difference = 0-8508148, which is the log. of 7-092752. Ans , (2) Divide 241-63 by 4-567. Log. 241-63 = 2-3831509 4-567 = 0-6596310 Difference^ 1-7235199, which is the log. of 52-90782. Ans. III. To raise a number to any power by logarithms. Multiply the logarithm of the given number by the index of the power to which it is to be raised, and the product will be the logarithm of the required power. (1) Find the cube of 30-7146, written thus : (30'7 146) 3 . Log. 30-7146 = 1-4873449 3 4-4620347, which is the log. of 28975-75. Ans. (2) What is the value of 9-163 4 ? Log. 9-163 = 0-9620377 4 3-8481508, which is the log. of 7049-38. Ans. IV. To extract any root by logarithms. Divide the logarithm of the given number by the index of the root to be extracted, and the quotient will be the logarithm of the required root. (1) Find the cube root of 12345, written thus : V 12345. Log. 12345 = 4-0914911. 3)4-0914911 1-3638304, which is the log. of 23-11162. Ans. GEOMETRY, TRIGONOMETRY, LOGARITHMS. 107 (2) Find the fourth root of 0-0076542. Log. 0076542 = 3 8838998 = T-4709749 To divide a negative characteristic, add such a quantity to the characteristic as will make it divisible without a remainder, and prefix an equal number to the decimal part of the logarithm. Thus, in the example, add 1, and you get 4 + 1-8838998 -r- 4 = T-4709749, which is the log. of 0-295784. Ans. In calculations in which sines, cosines, etc., occur, and logarithms are to be used, then the logarithmic sine, cosine, etc., must be used. They can be obtained from Chambers's Tables. The logarithmic sine is obtained by finding the logarithm of the number representing the natural sine, and adding 10 to its characteristic. For example, if the reader refers to his book of tables, he will find that the natural sine of 30 is 0-5000000. The logarithm of 0'5 is 1 '6989700, but to avoid the inconvenience of the negative characteristic, 10 is added, and so we arrive at log. sine 30, which is equal to 9'6989700. In using log. sines, cosines, etc., the 10 which has thus been added is always deducted again, as in the following example : To find ED, page 104, Example 2. ED = 100 x tangent 23 15' v ] og. ED = log. 100 + log. tan 23 15' - 10 = 2 + 9-6330985 - 10 = 1-6330985, which is the log. of 42-964 " ED = 42-964. Ans. The Solution of Triangles. In every triangle there are six parts, viz. three sides and three angles. If any three of these parts are given, one of which must be a side, the remain- ing parts can be found, the process being known as the " solution " of the triangle. It will be at once seen that this information is of great service to the surveyor, who is able, by observing the angles of his triangles, to calculate AT. i 4.1- j.u -j j j.i FIG. 82A. Solution of the lengths of the sides, and thus triangles. check the measured distance. In cases also where it is not practicable or necessary to measure one of io8 MINE SURVEYING. the sides, its length can be calculated from the other known parts of the triangle. In order to shorten the formulae, the three angles of the triangle will be referred to as A, B, and C, and the three sides opposite them a, b, and c, respectively (see Fig. 82A). Case 1. Given the three sides a, b, and c, to find the angles. Let s = half the sum of the three sides. Then ton - a) These formulae will give us the angles A and B. The angle C = 180 - A - B. EXAMPLE. The three sides of a triangle are : a = 750 links ; I = 835 links ; and c = 679 links. Find the angles A, B, and C. Here S = 75 + 8 -! 5 + 679 = 1132 A /( 6) (s - c) then tan w = * / i ~-^ ^ ' 2 'v s(s a) 1132 835) (1132 -679) 1132(1132-750) 297 x 453 lT32~~x~382 = \/0-31 11321. = 0-5577921, which is the natural tangent of the angle 29 9' 9" ~ = 29 9' 9" and the angle A = 58 18' 18" T> / "2 = \/ tan 8(8 - /(U32-750)(1132-i;7I> 1132(1 132 -835) / 382 x 453 V H32 x 291 453 ~Wl - V6-5I47053 = 0-7174290, which is the natural tangent of the angle 35 39' 24-5" 2 = 35 39' 24-5" and the angle B = 71 18' 49" and C = 180 - 58 18' 18"-7118' 49"= 50 22' 53" GEOMETRY, TRIGONOMETRY, LOGARITHMS. 109 Case 2. To solve a triangle, having given two angles and a side. In any triangle the sides are proportional to the sines of the opposite angles. a b c S " ~ shTA " sin B ~ sin C Let A and C be the given angles and b the given side. Then the angle B = 180 - A - C. To find the sides a b sin A ~~ sm B _ b sin A ~slrTF from which we get the side a. _ sin C ~ sin B 6 sin c .-. c = s D sm B from which we get the side c. EXAMPLE. In a triangle ABC, the angle A = 50, the angle C = 66, and the side a is 1000 yards. Find the remaining sides and angle. The angle B = 180 - 50 - 66 = 64 To find the sides sin A sin B 1000 & ' sin 50 " sin 64 6 1000 x sin 64 ' l ~ sin 50 _ 1000 x 8987940 0-7660444 898-7940 _ 0=7660444 -i c and -f-fi sin C sin B & sin c c = -^ ?s- sin B 1173-29 x sin 66 c = sin 64 _ 1173-29 x 09135455 0-8987940 = 1192*5 i io MINE SURVEYING. Case 3. Given any two sides b and c, and the angle A between them, to find the remaining side and angles. The angles (B + C) = 180 - A, from which we get (B + C), and tan - = j - cot ^, from which we get (B - C) ; and