.REESE LIBRARY 
 
PRELIMINARY SURVEY 
 
PRINTED BY 
 
 SPOTTISVYOODE AND CO., NEW-STREET SQUARE 
 LONDON 
 
PRELIMINARY SURVEY 
 
 AND 
 
 ESTIMATES 
 
 BY 
 
 THEODORE GRAHAM GRIBBLE 
 
 CIVIL ENGINEER 
 
 t. L 
 
 JTY 
 
 LONDON 
 LONGMANS, GREEN, AND CO, 
 
 AND NEW YORK : 15 EAST i6' h STREET 
 1891 
 
 A // rights reserved 
 
s 
 
INTRODUCTION 
 
 THE Anglo-Saxon race, in sundry climates and conditions, 
 and under divers forms of government, is unquestionably 
 pre-eminent to-day in the civilisation of the world. 
 
 It is not alone beca'tis^e they are the greatest traders, but 
 because they are at the same time the greatest navigators 
 and engineers of the world that the English-speaking nations 
 hold the proud primacy of race. Whether it be at the 
 first appearance of railways upon the Eastern hemisphere, or 
 the first cable-knot between the old world and the new, or 
 the development of virgin continents, and the carrying of a 
 luxurious civilisation into the heart of nature's wilderness, 
 the Anglo-Saxon is always at the front. 
 
 The strengthening of the Anglo-Saxon bond from year 
 to year is more attributable to improved means of com- 
 munication than to sentiment. The constantly conflicting 
 interests of commerce, the intense rivalries of handicrafts, 
 the minor jealousies of social life, fostered by selfish iso- 
 lation, produce barriers which would increase and gulfs 
 which would widen ; but the iron horse, the ocean grey- 
 hound, and the subtler electric fluid are for ever making 
 the men who speak the same tongue shake hands again. 
 To the pioneer surveyor, however, the field available for 
 
TI Preliminary Survey 
 
 new enterprise is rapidly becoming less, whilst the number 
 of surveyors increases. It becomes, therefore, more and 
 more important for those who leave our shores to possess 
 the handiest and most efficient instruments, to know the best 
 and most rapid methods of using them, and to understand 
 the diverse conditions of the countries to which they go. 
 
 The members' list of the British Institution of Civil 
 Engineers has now reached the colossal total of about six 
 thousand. These figures alone would serve to show the 
 extent of the demand for foreign employment, for certainly 
 there is not enough home work to go round so many ; but, 
 especially in the department of surveying, the Institution list 
 gives but little idea of the number of young men who are 
 issuing year by year from pupilage or college with their eye 
 on our distant dependencies. 
 
 It is furthermore a noteworthy fact that, especially for 
 surveyors, although the field of engineering enterprise is be- 
 coming greater and greater, the colonial door is closed to 
 young Englishmen, just as soon as men can be trained abroad. 
 Both in Canada and Australia, a diploma is needed to 
 qualify a man to practise as a land-surveyor ; the studies 
 for which cannot be easily pursued in England. 
 
 In the case of India the Government have met the 
 difficulty by giving their men the special training needed 
 for that country at Cooper's Hill College, but the door is 
 closed to others. 
 
 In the Colonies the reason of this is because in the first 
 place Australians are independent in their ideas, but also 
 very much because the young English surveyor is too often 
 an importer of instruments of which he knows little into a 
 country of which he knows less, so they prefer to educate 
 their engineers on the spot. 
 
Introduction vii 
 
 Things have changed for the better, no doubt, but 
 about twenty years ago it seemed as if the English engineer 
 were educated as much as possible in things he could not 
 use, and as little as possible in things which would be 
 needed by him in a new country. The writer enjoyed the 
 last two years of the lectures of one of the most celebrated 
 professors of engineering of his day, and purchased the whole 
 of that scientist's textbooks ; but it is a significant fact that 
 one of the former students earned his living by explaining after 
 the lecture what the professor had meant to convey to his 
 hearers. Finding a year or two afterwards that he could 
 get the kind of information he wanted in a smaller compass 
 and simpler language the writer parted with his little library 
 of textbooks. 
 
 The same drawbacks attended the pupil in the engineer's 
 office as the student in the university. 
 
 Men were not then made to keep their levels in adjust- 
 ment, but allowed to run to the nearest instrument-maker. 
 They were never taught the American method of levelling 
 or curve-ranging, and the road and railway making which 
 they learned was that which was suitable to a country 
 like England, but of little use for the Colonies. The con- 
 sequence has been that when they arrived there they were 
 thrown upon their own ingenuity, and produced a conglome- 
 rate of different types of construction upon different gauges, 
 which has been the reverse of profitable to the investors and 
 without reflecting much credit upon themselves. 
 
 On the Canadian Pacific Railway the writer rarely met 
 a young engineer fresh from England who could quickly 
 adjust his level or theodolite or who knew anything of the 
 American system of curve-ranging or had the least notion 
 of telemetry. 
 
viii Preliminary Survey 
 
 It has been the fashion to criticise America for her 
 cheap railways, her numerous gauges, her erroneous 
 curve-ranging, and in fact everything that was not the way 
 we do it in England. This has been the language of those 
 who have either not been there or who have not understood 
 the methods adopted there when casually observing them 
 with a biassed judgment on a passing visit. 
 
 The American is just an Anglo-Saxon like ourselves, 
 only with a little more liberty and a great deal more scope. 
 He is not at all ashamed to come and learn from the old 
 country what age and experience have qualified her to teach 
 him, but in the handling of a virgin colony, with great 
 undeveloped resources, we may do well to learn of him. 
 
 In simplicity of survey practice, uniformity of gauge, 
 types of bridges and of rolling stock, the American engi- 
 neer may be profitably (though not slavishly) imitated in 
 the work of opening out a new sphere of enterprise such 
 as our recently acquired colonies, and it is to be hoped 
 that, profiting by past experience, English engineers will fuse 
 their ideas into something like uniformity and produce a 
 harmonious construction. 
 
 The methods of surveying considered in the following 
 pages are by no means exclusively American. In the 
 class of work formerly called telemetry, but now tacheo- 
 metry, we have to go to Italians, French, and Germans for 
 most of the original conceptions and the best modern 
 developments. Comparatively few English engineers 
 really practise these methods unless they have learned them 
 abroad, although some are thoroughly proficient in them. 
 
 The title of this book, ' Preliminary Survey,' is American, 
 and answers somewhat to our ' Parliamentary Work ; ' but it 
 covers a wider range, in fact the whole science of surveying 
 
Introduction ix 
 
 in condensed form with the exception of those minute 
 details where very great accuracy is needed. 
 
 The object in view has been to present to the young 
 engineer going abroad a handy vade-mecum which with the 
 necessary tables will enable him to carry out a survey in a new 
 country rapidly, correctly, and according to the ideas and 
 requirements of the people. It has also been sought to furnish 
 in the first and third chapters an aide-memoire to the expe- 
 rienced surveyor for his assistance in roughly estimating the 
 cost of the proposed works, and so to guide his decision in 
 the case of alternative routes and situations. 
 
 Considerable use has been made of standard authorities 
 on both sides of the Atlantic, but the subject matter is in 
 the main the result of actual experience. The necessary 
 compactness of such a work has made it eclectic. Some 
 methods have been passed over with slender comment, 
 although occupying much space in other textbooks. On 
 the other hand such subjects as tacheometry, computation 
 by diagram and slide-rule, signalling, &c., which are as yet 
 hardly known to the general public except in pamphlet form, 
 are here treated of at considerable length. An attempt has 
 been made to explain the elements of astronomy, as far as 
 they are needed in the simple problems used by the surveyor, 
 in such a manner as will be understood by those having no 
 previous knowledge of the subject, and a great many of the 
 definitions which take up much space in ordinary textbooks 
 have been placed in a glossary. No tables are given which 
 are to be found in the Nautical Almanac or in ordinary 
 mathematical tables, as these have to form part of the 
 surveyor's impedimenta. 
 
 The following extract from the statute book of the 
 Dominion of Canada will give a fair idea of what the 
 
x Preliminary Sutvey 
 
 pioneer surveyor in any of the colonies should know, both 
 in theory and practice. Both in Australia and India survey 
 practice is carried on very much in the American manner. 
 The subjects enumerated in the Canadian statute are not 
 treated so much in detail in this work, in order to leave 
 space for other subjects, such as tacheometry and curve- 
 ranging, which are equally useful to the railway man. 
 
 The author desires to express his acknowledgments for 
 a great deal of useful material to the following gentlemen 
 who have kindly given their courteous permission to use 
 tables, maps, diagrams, and formulae in works of which they 
 are either the authors or custodians : 
 
 James Forrest, Esq., Secretary Inst. C.E. and editor of 
 ' Minutes of Proceedings.' 
 
 Captain Wharton, Hydrographer to the Navy, and author 
 of ' Hydrography.' 
 
 A. M. Wellington, Esq., C.E., editor of 'Engineering 
 News,' New York, and author of standard works referred t'o 
 in the text. 
 
 John C. Trautwine, Esq. (jun.), editor of Trautwine's 
 < Pocket Book.' 
 
 Other authorities on different subjects have been also 
 referred to, and acknowledged in different parts of the book. 
 
 The calculations in chapters three and eight have been 
 very kindly checked by an old friend, Mr. William T. 
 Olive, Resident Engineer on the Manchester Main Drainage ; 
 most of the other figures have been checked in one way or 
 another, but it is possible in a first edition that errors may 
 still remain undetected, and any information as to mistakes 
 in the text, figures, or diagrams will be gladly welcomed by 
 the author. 
 
Introduction xi 
 
 QUALIFICATIONS OF THE DOMINION LAND AND 
 TOPOGRAPHICAL SURVEYOR 
 
 Excerpt. 49 Victoria, Chapter 17. Royal assent, May 25, 
 1883. (Dominion of Canada.} 
 
 99. No person shall receive a commission from the 
 Board of Examiners authorising him to practise as a Dominion 
 land surveyor until he has attained the full age of twenty- 
 one years, and has passed a satisfactory examination before 
 the said Board on the following subjects ; that is to say : 
 Euclid first four books, and propositions first to twenty-first 
 of the sixth book ; plane trigonometry, so far as it includes 
 solution of triangles ; the use of logarithms, mensuration of 
 superficies, including the calculation of the area of right-lined 
 figures by latitude and departure, and the dividing or laying 
 off of land ; a knowledge of the rules for the solution of 
 spherical triangles, and of their use in the application to 
 surveying of the following elementary problems of practical 
 astronomy. 
 
 1. To ascertain the latitude of a place from an observa- 
 tion of a meridian altitude of the sun or of a star. 
 
 2. To obtain the local time and the azimuth from an 
 observed altitude of the sun or a star. 
 
 From an observed azimuth of a circumpolar star, when 
 at its greatest elongation from the meridian, to ascertain the 
 direction of the latter. 
 
 He must be practically familiar with surveying opera- 
 tions, and capable of intelligently reporting thereon, and be 
 conversant with the keeping of field notes, their plotting 
 and representation on plans of survey, the describing of 
 land by metes and bounds for title, and with the adjust- 
 
xii Preliminary Survey 
 
 ments and methods of use of ordinary surveying instruments, 
 and must also be perfectly conversant with the system of 
 survey as embodied in this Act, and with the manual of 
 standing instructions and regulations published by the 
 authority of the Minister of the Interior from time to time 
 for the guidance of Dominion land surveyors. 
 
 102. Any person entitled to receive, or already possess- 
 ing a commission as Dominion land surveyor, and having 
 previously given the notice prescribed in clause 98 of this 
 Act, may be examined as to the knowledge he may possess 
 of the following subjects relating to the higher surveying, 
 qualifying him, in addition to the performance of the duties 
 declared by this Act to be within the competence of 
 Dominion land surveyors, for the prosecution of extensive 
 geodetic or topographic surveys, or those of geographic 
 exploration, that is to say : 
 
 1. Algebra, including 'quadratic equations, series, and 
 calculation of logarithms. 
 
 2. The analytic deduction of formulas of plane and 
 spherical trigonometry. 
 
 3. The plane co-ordinate geometry of the point, straight 
 line, the circle, and ellipse, transformation of co-ordinates, 
 and the determination, either geometrically or analytically, 
 of the radius of curvature at any point in an ellipse. 
 
 4. Projections : the theory of those usually employed in 
 the delineation of spheric surfaces. 
 
 5. Method of trigonometric surveying : of observing 
 the angles and calculating the sides of large triangles on 
 the earth's surface, and of obtaining the differences of 
 latitude and longitude of points in a series of such triangles, 
 having regard to the effect of the figure of the earth. 
 
 6. The portion of the theory of practical astronomy 
 
Introduction xiii 
 
 relating to the determination of the geographic position of 
 points on the earth's surface, and the direction of lines on 
 the same, that is to say : 
 
 Methods of determining latitude. 
 
 a. By circum-meridian altitudes. 
 
 b. By differences of meridional zenith distance (Talcott's 
 method). 
 
 c. By transits across prime vertical. 
 Determination of azimuth. 
 
 a. By extra-meridional observations. 
 
 b. By meridian transits. 
 Determination of time. 
 
 a. By equal altitudes. 
 
 b. By meridian transits. 
 Determination of differences of longitude. 
 
 a. By electric telegraph. 
 
 b. By moon-culminating stars. 
 
 7. The theory of the instruments used in connection 
 with the foregoing, that is to say, the sextant or reflecting 
 circle, altitude and azimuth instrument, astronomic transit, 
 zenith telescope, and the management of chronometers ; also 
 of the ordinary meteorological instruments, barometer 
 (mercury and aneroid), thermometers, ordinary and self- 
 registering, anemometer, and rain gauges, and on his know- 
 ledge of the use of the same. 
 
 15 GREAT GEORGE STREET, LONDON, S.W., 1890. 
 
CONTENTS 
 
 INTRODUCTION 
 
 PAGB 
 
 Extent of the demand for surveyors. Necessity for adaptation to 
 the requirements of a new country. Objects aimed at in the 
 present work. Statutory qualifications of the land and topo- 
 graphical surveyor for the Dominion of Canada v 
 
 CHAPTER I 
 GENERAL CONSIDERATIONS 
 
 Qualifications of the pioneer surveyor. Subject matter of a pre- 
 liminary report. Extent to which general considerations affect 
 the location. Tables of resistance to traction from gradients 
 and curvature. Abt system. Cape railways. Rule for finding 
 amount of traffic necessary to pay a given dividend. Maximum 
 amount of business with a single line. Arrangement of curves 
 and gradients. Tables of ditto on American and Australian 
 railways. Rough estimates for railways, highways, and tram- 
 ways. Cost of plant ........ i 
 
 CHAPTER II 
 
 ROUTE-SURVEYING OR RECONNAISSANCE 
 
 Pioneering for railway location in America. Methods suit- 
 able to different circumstances. Sketching. Photography. 
 The plane-table and prismatic compass. The meridian by the 
 plane-table. Traversing with plane-table and stadia. Traverse 
 with passometer, aneroid, and pocket altazimuth- Profile and 
 
xvi Preliminary Survey 
 
 PAGE 
 
 contours by ditto. Surveying with the sextant. Range-finding 
 with a two-foot rule. Distance-measurement by time, by 
 horses' and camels' gait, by patent log, by revolutions of the 
 propeller. Mapping by plane-construction, by conical projec- 
 tion, by stereographic projection. Mercator's projection. . 30 
 
 CHAPTER III 
 
 HYDROGRAPHY AND HYDRAULICS 
 
 Coast-lining. Boat-survey. Running survey from the ship. 
 Distance-measurement by gun-fire. Gnomonic projection. 
 Symbols in charting. Sun -signalling. Flag-signalling. The 
 Morse code. Tides and currents. Hydraulics. Kutter's 
 formula and diagram. Discharge from tanks, cisterns, and 
 weirs. Diagram for ditto. Trautwine's approximate rule for 
 discharge of pipes under pressure. Ditto for discharge over 
 weirs. Horse-power of falling water. Efficiency of water- 
 wheels. Horse-power of a running stream. Hydrostatic pres- 
 sure and diagram. Notes on dredging, dredging plant, boring, 
 concrete and dock work . . . . . . .81 
 
 CHAPTER IV 
 GEODETIC ASTRONOMY 
 
 Compared with nautical astronomy. General principles. ' Gain- 
 ing or losing a day.' Classification of methods. Observations 
 for determining the meridian. By equal altitudes. By cir- 
 cumpolar stars in same vertical. By time interval of culmina- 
 tion of circumpolar stars. By pole star at elongation. By 
 solar azimuth. Observations for local mean time and longi- 
 tude. Table for reducing arc to time, and vice versd. Time 
 by solar transit. By solar hour-angle. Sidereal hour-angle. 
 Absolute methods ' of determining the longitude. Jupiter's 
 satellites. Lunar occultation. Lunar distance. Terrestrial 
 difference of longitude. Moon- culminating stars. Observa- 
 tions for latitude. Rules for different cases in both hemispheres. 
 By circumpolar stars. By meridional altitude of a fixed star. 
 By meridional altitude of the sun. By an altitude of the sun 
 cut of the meridian. By two altitudes of the snn or a star. 
 Graphic latitude . . . . . . . . .116 
 
Contents xvii 
 
 CHAPTER V 
 
 TACHEOMETRY 
 
 PAGE 
 
 Derivation. Definitions. History and first principles of stadia 
 measurement. Method of putting in stadia-hairs to any 
 required reading. Analogy of tacheometry with celestial 
 parallax. Limits of error. Different types of telescope. 
 Different methods of holding the stadia- staff. Comparison of 
 results. Surveying with the tacheometer. Auxiliary work to 
 ditto. Description of the author's methods in the Sandwich 
 Islands. Reduction of the traverse to difference of latitude 
 and departure. Ditto to longitude and latitude. Levelling 
 with the tacheometer. Contouring. Plotting the profile. 
 Sketch of a Hawaiian ravine with part of a horse-shoe curve 
 and contours. Tacheometric curve-ranging. Mr. Lyman's 
 conclusions on stadia-telescopes. The plane-table and stadia. 
 Time occupied in tacheometric survey . . . . .15^ 
 
 CHAPTER VI 
 
 CHAIN-SURVEYING 
 
 Application to preliminary survey. Chaining. Sources of error. 
 Setting out a square. Deflection distance. Surveying with 
 chain only. With chain and cross-staff. Fieldbook. Areas. 
 Traverse with transit and chain. Different methods of fieldbook. 
 Curve-ranging with the chain only. Krohnke's tangential 
 system. Jackson's six-point equidistant system. Kennedy and 
 Hackwood's method. Method by offsets without a table . 194 
 
 CHAPTER VII 
 
 CURVE-RANGING WITH TRANSIT AND CHAIN 
 
 Properties of the circle and general nomenclature. Confusion 
 arising from diversity of terms. Advantages of decimal gradua- 
 tion. Fundamental problems. Different methods of keeping 
 the fieldbook and specimens. Curve-ranging by tangential 
 angles. Dalrymple-Hay's curve-ranger. Parallel tangents. 
 
 a 
 
xviii Preliminary Survey 
 
 PAGK 
 
 Reverse curves. Turn-outs. Transition curves. The ' tramway ' 
 spiral. The ' horse-shoe ' spiral. The 'mountain ' spiral. The 
 ' trunk-line ' spiral ........ 209 
 
 CHAPTER VIII 
 
 GRAPHIC CALCULATION FOR PRELIMINARY ESTIMATES 
 
 Objects of preliminary measurement for estimates. Use of 
 diagrams. The slide-rule for ordinary arithmetic. Proportion. 
 Multiplication. Division. Involution and evolution. The 
 slide-rule as a universal decimal scale. Calculation of slopes 
 and gradients with tables and slide-rule. Table of squares 
 and square- roots. Table of railway sleepers for all gauges. 
 Measurement of earthwork by diagram and slide-rule. Measure- 
 ment of iron bridges by diagram and slide-rule. Stone and 
 brick bridges by diagram and slide-rule. Service diagram for 
 railways and tramways. Centrifugal force. Reduction of 
 thermometer scales. The slide-rule as a universal measurer. 
 Circles. Areas. Volumes. Weights and measures, with 
 tables. Tree-timber. Permanent way. Angle and tee iron. 
 Round and square iron. The slide-rule as a ready reckoner. 
 Wages table. British and foreign money . . . .241 
 
 CHAPTER IX 
 
 INSTRUMENTS 
 
 Levels and their adjustment. Levelling. Level-staves. Theo- 
 dolites. The ' Ideal ' tacheometer. The sketch-board plane- 
 table. Pocket altazimuths. Passometers and pedometers. 
 Sextants. The solar compass. The heliostat and heliograph. 
 Principles of telemeters. Eckhold's omnimeter. The Wagner- 
 Fennel tacheometer. The Dredge-Steward omni-telemeter. 
 The Weldon range-finder. The simplex range-finder. The 
 Bate range-finder. The aneroid barometer. The boiling-point 
 thermometer. Hypsometry and diagram. Drawing instru- 
 ments. The slide-rule. The protracting tee-square. Im- 
 provised protractors. The eidograph and pantagraph. The 
 planimeter. Stanley's computing scales. The station-pointer. 
 Books. Mathematical tables. Field and MS. books. 
 Stationery . ' ". ;..-., . . . . . 286 
 
Contents xix 
 
 APPENDIX 
 
 PAGJt 
 
 Functions of right-angled plane triangles. Functions of oblique 
 plane triangles. Functions of right-angled spherical triangles. 
 Functions of oblique spherical triangles. Tables for azimuth 
 by circumpolar stars. Table for transition curves Tables of 
 specific gravity 373 
 
 GLOSSARY .......... 391 
 
 INDEX . 416 
 
LIST OF PLATES 
 
 PLATE 
 
 I. FIG. 26, KUTTER'S FORMULA . . . To face page 100 
 
 II. FIGS. 28, 29, THEORETICAL VELOCITY IN 
 FEET PER SECOND, MILES PER HOUR 
 
 DUE TO GIVEN HEAD . . ,, ,, IO6 
 
 III. FIG. 30, COEFFICIENT in IN TRAUTVVINE'S 
 
 FORMULA FOR FLOW IN PIPES . ,, ,, 108 
 
 FIG. 31, COEFFICIENT r IN TRAUTVVINE FOR 
 
 FLOW OVER WEIRS . . . . ,, ,, 108 
 
 IV. FIG. 32, HYDROSTATIC PRESSURE . ,, ,, no 
 V. FIGS. 75, 76, EARTHWORK . . . . ,, ,, 252 
 
 VI. FIGS. 80, 81, IRON BRIDGES . . . ,, ,, 256 
 
 VII. STONE ARCHES ,, ,, 258 
 
 VIII. TIMBER TRESTLES 259 
 
 FIG. 82, SERVICE DIAGRAM FOR TRAMWAYS On ,, 261 
 
 FIG. 83, ,, ,, RAILWAYS ,, ,, 262 
 
 IX. FIGS, no, iii, BAROMETRICAL PRESSURE . To face ., 350 
 
PRELIMINARY SURVEY 
 
 AND 
 
 ESTIMATES 
 
 
 
 CHAPTER I 
 
 GENERAL CONSIDERATIONS 
 
 THE following remarks will be more applicable to railway 
 reconnaissance, though much of the principle contained in 
 them is also that which guides the surveyor for trunk roads 
 for military or commercial transportation. 
 
 QUALIFICATIONS OF THE SURVEYOR 
 
 The man who is first in the field should be a man of 
 wide range of experience rather than a minute technologist. 
 He is usually given much discretionary power as to his 
 location. He has also advisory powers, or rather duties, 
 which are great responsibilities. He is called upon to report 
 upon the scheme from a bare possibility down to a desirable 
 investment. Before engaging his services, the promoters 
 have generally made up their minds that there must be 
 * money in it,' and they want, like most other people, to 
 obtain a maximum of good showing for a minimum amount 
 of outlay. 
 
 The surveyor is generally disposed to favour a new 
 
2 Preliminary Survey 
 
 undertaking, because, however much or little money there 
 may be in it for him, there is likely .to be ' work in it,' and he 
 has often to resist the natural tendency to make too good 
 a showing. 
 
 There are several considerations which likewise influence 
 him in this direction. Shareholders always expect a sanguine 
 report, and take discount off it in any case ; so that a mode- 
 rate report is to them a bad one. There is a moral certainty 
 that, however carefully a walk-over survey may be made, a 
 revised location will show a material improvement in the 
 line of economy or efficiency, or both, and therefore the 
 surveyor is tempted to make allowance for this in his trial 
 profile. He is, perhaps, well aware that the nearer his profile 
 resembles the surface of a billiard table the better he will 
 please his employers. 
 
 When the country is so rough that chaining is out of the 
 question unless he is able to adopt one of the rapid methods 
 to which these pages are meant to draw attention a large 
 element of conjecture enters into his calculations, and he 
 is naturally disposed to conjecture favourably rather than 
 critically. 
 
 SUBJECT MATTER OF A RAILWAY REPORT 
 
 The surveyor is generally called upon to advise his 
 promoters 
 
 1. Whether any kind of line is feasible. 
 
 2. Whether it is likely to be profitable. 
 
 3. What type of railway would be most suitable, and 
 what style of rolling-stock. 
 
 4. He is to furnish a plan, profile, and estimate of one 
 or more routes which he considers eligible. 
 
 All these points are closely connected. One kind of 
 line is feasible where another is wholly impracticable ; a light, 
 cheap railway will often yield a handsome dividend where a 
 heavy line would never emerge from the hands of a receiver. 
 
General Considerations 3 
 
 On the other hand, a light railway built to carry heavy traffic 
 will probably be wedged out of existence by a higher-class 
 competing line. The style of rolling-stock procurable to 
 handle the business often regulates the location as much as 
 the location rules the rolling-stock. The route is dependent 
 on the topography to a great extent, but the situation of towns 
 with which communication is necessary often overrides the 
 consideration of topography. 
 
 It is only experience which can enable an engineer to 
 form rapidly and correctly the general idea of the class of 
 line suited to the circumstances. If, as is often the case, 
 gauge and rolling-stock are fixed factors in the problem, 
 there remain the questions of grades and curves, which must 
 be to a large extent dependent upon the topography, and it 
 is there that the judgment of the surveyor is most needed, 
 both in the limiting and the arrangement of these vital 
 elements of a railway. 
 
 With regard to the first point of feasibility. This has 
 almost dropped out of the reckoning of to-day. It may be 
 taken for granted that a railway can be constructed nearly 
 anywhere. The only insurmountable difficulties to railway 
 projects are, first, lack of funds ; second, opposition of 
 vested interests. It is another thing when we come to the 
 question of 
 
 Whether it will pay. Here the engineer has to study, 
 i. What the existing traffic of the district is, and how it 
 would be likely to be affected by the introduction of a rail- 
 way. 2. What is the probability of the traffic being handled 
 by some other means of transit in competition. 3. What rates 
 can be commanded, and whether it will be in the main a 
 through or a local traffic. 4. What is the outlook for 
 development of the business, with any possibly counteracting 
 causes. 5. Probabilities of another competing railway in 
 the future. 
 
 All these subjects dovetail themselves into the actual 
 reconnaissance of the route ; engineering difficulties give 
 
 B 2 
 
4 Preliminary Survey 
 
 way to commercial exigencies and vice versa, until the sur- 
 veyor has evolved his ideas of a line with maximum efficiency 
 at minimum cost which will command the maximum amount 
 of business. 
 
 The climate in which the undertaking is to be carried on 
 is a very important consideration. In tropical climates the 
 use of timber is to be avoided where possible, on account 
 of predatory insects and the rapid decay produced by alter- 
 nations of hot sun and heavy rains. The rainy season 
 regularly changes the trickling rivulet into the mighty river, 
 and these actions of the weather greatly affect the construction 
 of culverts and bridges, and therefore, indirectly, the location. 
 In some places streams are turned into tunnels to save build- 
 ing culverts, and an overflow channel provided at the junction 
 of an embankment and side-hill for abnormal freshets. 
 
 In cold climates snowsheds are a very costly item, and 
 the study of the principles of drifting snow will often modify 
 the location. 
 
 The general topography also radically affects the location. 
 It rules both gradients and curvature and the type both of 
 gauge and equipment. 
 
 If the land falls toward the seaboard, with a heavier export 
 than import trade such as a mineral railway which only 
 takes back lumber, agricultural produce, and so forth the 
 gradients can be steeper than would be otherwise permis- 
 sible ; the rule being then to adopt that which can be sur- 
 mounted as a contrary grade by the light traffic. 
 
 The method of ' bunching,' or concentrating the severe 
 gradients in order to handle them specially, is a very im- 
 portant one. The best policy for a new country is to carry 
 long trains as far as possible with one engine, and then to 
 divide them on a turn-out and take them over the climb in 
 sections, or else to provide an assistant engine for the 
 district. 
 
 The following table (No. XXIV. of Mr. Wellington's 
 standard work on American ' Railway Location ') shows the 
 
General Considerations 
 
 engine ton-mileage required to move i ton of net load (ex. 
 engine) .TOO miles on a level, except for a rise of 2,400 feet 
 on different grades, worked with assistant engines : according 
 to the average daily experience of American railways. 
 
 TABLE I. Traction on Grades. 
 
 
 
 Engine ton-mileage per ton of 
 
 Rate of 
 
 Length 
 
 Length of 
 
 net load moved 100 miles 
 
 incline 
 
 incline 
 
 track 
 
 While on 
 
 While on 
 
 
 
 
 incline 
 
 level track \ 
 
 feet per mile 
 
 miles 
 
 miles 
 
 
 
 24 
 
 IOO 
 
 
 
 1-056 
 
 056 
 
 30 
 
 60 
 
 40 
 
 0-862 
 
 O'2IO '072 
 
 80 
 
 30 
 
 70 
 
 0760 
 
 0-369 -129 
 
 100 
 
 2 4 
 
 7 6 
 
 0755 
 
 0-400 T55 
 
 120 
 
 20 
 
 80 
 
 0-766 
 
 0-42I -187 
 
 150 
 
 16 
 
 84 
 
 0-803 
 
 0-442 1-245 
 
 2OO 12 
 
 88 
 
 0-900 
 
 0-463 1-363 
 
 ' It would be seen that the rate of incline had an incon- 
 siderable influence on the motive power required, for the 
 reason, largely, that the length of the run on which large 
 power was required decreased paripassu with the increase 
 of rate, which was not the case with through grades. 
 
 ' In this table moreover it was assumed that the total 
 length of the road remained uniform at 100 miles, whatever 
 the rate of grade adopted for the high-grade section. This 
 is ordinarily quite out of the question, the lower grade being 
 usually attainable only by adding so much further develop- 
 ment within an approximately uniform air-line distance. 
 
 4 Assuming, for example, that in the above table eighty 
 miles of level track was essential in any case, and that in the 
 remaining air- line distance of twenty miles, any one of the 
 above rates of pusher-grades from twenty-four to 200 feet 
 per mile was obtainable, but only by development a rather 
 extreme assumption, but sufficient for illustration the 
 table would thus read : ' 
 
Preliminary Survey 
 
 TABLE II. Traction on Grades. 
 
 Rate of 
 
 Length 
 
 Engine ton-mileage per ton of net 
 load moved between the incline 
 
 grade. 
 
 Incline Level 
 
 TO- rasr 
 
 While on 
 level track 
 
 Total 
 
 ft. per mile 
 
 miles 
 
 miles 
 
 miles 
 
 
 
 24 
 
 IOO 
 
 80 
 
 1 80 I-056 
 
 0-421 
 
 '477 
 
 30 
 
 60 
 
 80 
 
 140 
 
 0-862 
 
 0-421 
 
 283 
 
 80 
 
 30 
 
 80 
 
 110 
 
 0760 
 
 O'42I 
 
 181 
 
 IOO 
 
 24 
 
 80 
 
 IO4 
 
 0755 
 
 0-421 
 
 176 
 
 1 2O 
 
 2O 
 
 80 
 
 IOO 
 
 0766 
 
 0-42I 
 
 187 
 
 150 
 
 16 84 
 
 IOO 
 
 0-803 
 
 0-442 
 
 245 
 
 2OO 
 
 12 88 
 
 IOO 
 
 O-QOO 
 
 0-463 
 
 363 
 
 TABLE III. Adjustment of Gradients for Assistant Engines, according 
 to the Average Daily Performance on American Railways. (H. M. 
 Wellington. ) 
 
 \ 
 
 
 Grade at which the same train can be drawn by the aid of 
 
 Ruling 
 grade 
 
 One assistant engine 
 
 Two 
 
 assistant engines 
 
 worked by 
 
 
 
 one engine 
 in feet 
 
 Heavier by 
 Of equal ' Ofeaual 
 
 Heavier by 
 
 per mile 
 
 weight on 
 drivers 20 P er 40 per , 
 
 weight on 
 drivers 
 
 20 per 40 per | 
 
 
 
 cent. cent. 
 
 
 cent. 
 
 cent. 
 
 
 
 
 
 1 
 
 level 
 
 24 
 
 29 
 
 33 
 
 46 
 
 54 
 
 62 
 
 IO 
 
 42 48 53 70 
 
 80 
 
 90 
 
 20 59 66 72 92 
 
 104 
 
 116 
 
 30 76 84 91 
 
 "3 
 
 126 
 
 138 
 
 40 
 
 92 
 
 101 
 
 109 
 
 133 
 
 147 
 
 1 60 
 
 fo 
 
 107 
 
 122 
 
 117 126 
 133 142 
 
 152 
 169 
 
 167 
 
 185 
 
 180 
 199 
 
 70 
 
 136 
 
 148 
 
 158 
 
 185 
 
 201 2l6 
 
 80 
 
 15 
 
 162 
 
 173 201 
 
 217 232 
 
 90 
 
 164 176 
 
 187 216 
 
 232 
 
 247 
 
 IOO 
 
 177 189 201 
 
 230 
 
 247 26l 
 
 no 
 
 190 202 214 
 
 
 
 
 
 120 
 
 203 
 
 215 
 
 227 
 
 
 
 
 
 I 3 
 
 215 
 
 227 
 
 239 
 
 
 
 , 
 
 
 
 I4O 
 
 227 
 
 239 
 
 251 
 
 
 
 
 
 
 
 'SO 
 
 238 250 
 
 262 
 
 
 
 
 
 - 
 
General Considerations 7 
 
 Caution. In calculating the increase of motive power due 
 to severe gradients, the wear and tear on locomotives, such 
 as the ' thrashing ' of an engine up a steep incline by an 
 inexperienced driver, is an item which, though difficult to 
 calculate, should be allowed for by a large margin. 
 
 The assumptions in the above table are that the rolling 
 friction on the level is 10 Ibs. per ton ; for lower frictions 
 the gradients are proportionally lessened. The gradients 
 are compensated for curvature. 
 
 A good method of overcoming steep gradients is by 
 the Abt rack system. The special feature about it is that 
 sections of mountain line can be worked thus without 
 changing gauge or altering the rolling stock. The loco- 
 motives do not depend on adhesion, therefore they can be 
 much lighter just where the construction of bridges is the 
 most serious item. 
 
 The Abt system is also specially adapted to short branch 
 mountain lines. 
 
 The notes and memoranda that the surveyor wants are 
 to give him a general but accurate idea of the alternative 
 advantages of the different schemes that arise, and it is 
 with that object that the chapter on Graphic Calculation 
 has been added ; condensed to its utmost limit. 
 
 In selecting a route and deciding upon the class of line 
 for a railway scheme it should be considered 
 
 First: If a competing line, how to obtain a pronounced 
 superiority to the existing one, either on the score of 
 efficiency or economy. 
 
 Second : If the first in the district, how without wasteful 
 expenditure to secure primacy. That is to obtain a line which 
 will so handle the existing and prospective business as to 
 hold its own, and that the best, of the business. 
 
 In order to ensure that his line will be suitable to the 
 future mode of working it, the surveyor should be acquainted 
 both with the ordinary and the special types of rolling-stock 
 that are to be used. In new countries it is essential to 
 
8 Preliminary Survey 
 
 economy that the engines and cars should be flexible, not 
 only as regards side-play, but also, if one may coin the term, 
 ' up and down ' play. 
 
 It is furthermore necessary that level crossings should 
 be permitted over all country highways, and, when on the 
 level prairie, over existing railways also. 
 
 Station buildings should be very primitive, and the 
 booking performed on the train. 
 
 It was stated by Mr. J. C. Mackay at the Institution of 
 Civil Engineers in 1886, that 'the present railways of the 
 Cape Colony had been constructed on a lavish scale with 
 rails weighing 60 Ibs. per yard, and expensive stations, some 
 of them costing over 2o,ooo/., while the railways alone had 
 cost 8,ooo/. per mile. This great expenditure had been 
 incurred for the sake of conveying one tram per day, in some 
 cases only one train every other day, and the consequence 
 was that the revenue did not pay one half the interest on 
 the loan, after deducting working expenses, and the working 
 of these lines was obliged to be carried on in such a manner 
 that the bullock waggon competed successfully with the rail- 
 ways. 
 
 'At Kimberley, with its 1,500 passengers and 350 tons 
 of goods per mile of railway per annum, a line with rails of 
 60 Ibs. per yard, and expensive rolling-stock and stations, 
 had been adopted.' 
 
 The writer is not in a position to verify at the moment 
 the accuracy of these figures, nor to state to what extent 
 they may be modified by subsequent development of 
 the districts ; but as they stand, reflecting most adversely 
 upon the judgment of the promoters and the engineers, it 
 should be added as a qualification that they only serve to 
 show one side of the question, but that which needs to 
 be most emphasised for a new country the danger of put- 
 ting old-country ideas upon young-country shoulders. The 
 counter-evil of putting down poor lines where there is busi- 
 ness for good ones, probably to pad the promoters' pockets, 
 
General Considerations g 
 
 has plenty of illustrations both at home and abroad, and the 
 engineer is too often compelled against his judgment to 
 make both his location and construction suit the ' spirit of 
 the times.' 
 
 The former evil of wasteful or even premature expendi- 
 ture is one which greatly checks the inflow of fresh capital 
 into a country. A receiver is a perfect scarecrow to fresh 
 enterprise. Purely speculative railway-making is as great a 
 hindrance to bona fide undertakings as jerry-building in its 
 smaller sphere ; whether it come in the form of too good or 
 too bad a line of railway. 
 
 A single line with properly arranged passing places, 
 rails 30 to 40 Ibs. per yard, engines of 15 to 20 tons, in easy 
 country, can be built for 4,ooo/. per mile, including the 
 equipment, in almost all parts of the globe ; provided that 
 the line starts from the seaboard, or from a place in rail- 
 connection with the seaboard. 
 
 This line, properly located, is capable of handling 1,000 
 tons of freight per day, and is, therefore, even with low rates, 
 in a position to yield a handsome profit to the investors. 
 Putting net receipts at \d, per ton-mile, it would return 9 \ per 
 cent, on the cost of construction with that volume of business. 
 
 APPROXIMATE RULE FOR FINDING THE AMOUNT OF 
 TRAFFIC REQUIRED TO PAY FIVE PER CENT. ON THE 
 COST OF CONSTRUCTION OF A RAILWAY. 
 
 Assumptions. Tariff, 75^. per ton-mile for passengers 
 and freight. Passengers reckoned at two tons each. 
 Expenses -50^. per ton-mile. Net receipts, -25^. per ton- 
 mile. 365 days to the working year. 
 
 Traffic in tons per diem = cost in per mile x '1315. 
 Example : On a road costing 4,ooo/. per mile, the traffic 
 needed in tons per diem=4,ooo x '1315 = 526. The amount 
 of traffic required varies inversely as the net value of the 
 receipts per ton-mile. 
 
io Preliminary Survey 
 
 Therefore for any other net value such as -53^. per ton- 
 mile, the amount of traffic is found by the slide-rule in the 
 following manner : 
 
 Place the given value -53^. on the upper scale of the 
 slide under the '25 on the rule. Find the required multiplier 
 0-62 on the rule opposite to the 1315 on the slide. Leave 
 the brass marker at 62 on the rule, and make a i of the 
 slide coincide with it. Then the result, 248 tons, will be 
 found on the rule opposite to 4,000!. cost, or 186 tons 
 opposite to 3,ooo/. cost and so on. 
 
 CONVERSE RULE 
 
 To find the percentage on cost of construction when 
 the argument is : 
 
 1. The tonnage of freight per day (average of 365 days). 
 
 2. The net profit per ton-mile in decimals of a penny. 
 
 3. The cost of construction of one mile in sterling. 
 Multiply the tonnage by the profit. Divide the product 
 
 by the cost of construction. Multiply the quotient by 
 152-1. Result is the percentage. 
 
 Example. 
 
 Data are : Daily tonnage . . . .542 
 Net receipts per ton-mile . -47^. 
 Cost of construction per mile . 46507. 
 
 BY SLIDE-RULE, USING LOWER SCALES OF SLIDE AND 
 RULE 
 
 Place a i of the slide over the 542 (tonnage) on the 
 rule. Place the brass marker at 47 (receipts) on the slide. 
 Place 465 (cost) on the slide at the marker. Place the 
 marker at the right-hand i of the slide. Place the left-hand 
 i of the slide at the brass marker. Read the percentage 
 
General Considerations II 
 
 8-34 on the rule opposite to the constant 152-1 on the 
 slide. 1 (The operation can be done in about 35 seconds.) 
 
 Mr. Robert Gordon in his paper on Economic Construc- 
 tion of Railways, ' Min. Proc. Inst C.E.,' gives a note on 
 the problem of the maximum capacity of a single track, 
 quoting from Mr. Thompson of the New York, Pennsylvania, 
 and Ohio Railroad ('Railway Gazette,' 1884, 1-43). He 
 says that for trains running an average of twenty miles per 
 hour, the most economical speed for freight, the maximum 
 is reached with stations 37 miles apart, when with an 
 allowance of six minutes' detention for each train crossed, and 
 eight minutes extra for each passenger train passed, it is 
 found that the limit is reached when the time of detention 
 equals that of running in the 24 hours, which gives sixty 
 trains per day both ways. For fifty trains and over, the 
 track should be doubled between the termini and the 
 next stations. In practice it is found that grades limit the 
 number of cars run in a train, so that, if forty loaded cars be 
 the ordinary number on the level, only twenty are taken 
 over an undulating country by a single engine. Actually, 
 on Mr. Thompson's division, 98 miles in length, the 
 Standard engine takes nineteen cars, the Mogul twenty- 
 three cars, and the Consolidation thirty-three cars each per 
 train. 2 
 
 The type of railway is affected first of all by grade. There 
 
 1 For the explanation of the principle of the slide-rule see pp> 
 242, &c., also 361, &c. 
 
 2 The total annual expenses on railroads in the United States usu- 
 ally range between 65 and 130 cents (2s. %\d. and $s. $d.} per train- 
 mile, that is, per mile actually run by trains. Also between I and 2 
 cents (| and id.} per ton of freight and per passenger carried one mile. 
 When a road does a very large business, and of such a character that 
 the trains may be heavy and the cars full (as in coal-carrying roads), the 
 expense per train-mile becomes large, but that per ton or passenger 
 small ; and vice versa, although on coal-roads half the train-miles are 
 with empty cars. Trautwine's ' Engineer Pocket Book. ' , 
 
 See also, at p. 21, Table of ' Earnings of American Railways. ' 
 
1.2 Preliminary Survey 
 
 can be no question that there are many light narrow-gauge 
 railways which are earning a good return on the capital 
 which could not have kept their heads above water as 
 standard- gauge railways. It is true, on the other hand, that 
 a very large mileage of narrow gauge is every year being 
 converted into standard gauge, in order to avoid breaking 
 bulk. Surveyors are not gifted with prophecy to know what 
 the changes of the next fifty years will be, but a fair amount 
 of experience will enable them to tell whether the line will 
 always remain a feeder, or whether it is likely to form part of 
 a trunk line. 
 
 The narrow gauge enables the engineer to adopt curves 
 of very small radius, but then he must keep to a flexible 
 type of locomotive and car which involves lighter loads and 
 less speed ; consequently, less business done for the same 
 labour and superintendence. 
 
 A theoretically perfect railway is an air-line on a dead 
 level between two points, and yet, apart from its cost, in 
 nine cases out of ten it would not be the best line. 
 
 A great deal of the sinuosity of a well laid-out railway in 
 a new country is productive. It carries the trains to where 
 the business is or where it is going to be. 
 
 Sometimes the local traffic is the major part of the busi- 
 ness. It generally commands a higher rate than through 
 traffic, and it would be a serious mistake to straighten a line 
 to catch through traffic of small bulk carried at cut-rates and 
 by so doing to lose a steady monopoly of a lucrative local 
 business. 
 
 Bad gradients are worse than sharp curves ; the latter 
 can be to a great extent mitigated in their discomfort by 
 well-made carriages ; in their resistance by flexible locomo- 
 tive frames ; and in their danger by careful signalling. But 
 gravity no skill can dispose of, and bad gradients have killed 
 many a promising line. 
 
 Considerable opportunity exists in every line ot railway 
 for arrangement of the curves and gradients so as to make 
 
General Considerations 
 
 them as little objectionable as possible. The first question 
 for the surveyor is whether or not the construction is to 
 be homogeneous. If, for instance, he has to traverse level 
 prairie for fifty miles, then cross a range of hills 3,000 feet 
 high within an air-distance of another fifty miles, then another 
 stretch of prairie of fifty miles, he has to consider whether 
 it would be advisable to develop his mountain section for 
 better gradients or whether he should arrange the line for 
 a different class of locomotives in the three sections. "All 
 this affects the location. 
 
 The two main points to be borne in mind with regard to 
 curvature are the speed at which trains will have to run and 
 the kind of rolling-stock which will have to be carried. The 
 following Tables IV. to IX. from Mr. Wellington's book 
 already referred to will be found useful in fixing the limit 
 of curvature and gradient to be adopted on a new line. The 
 American curve-nomenclature is explained in Chapter VII. 
 
 TABLE IV. Resistance on Curves. 
 
 
 
 Length of wheelbase 
 
 Resistance on 4 curve 
 at 10 miles per hour 
 
 Type of engine 
 
 Weight 
 
 
 
 
 
 
 
 
 
 
 Rigid 
 
 Total 
 
 Total 
 
 Ibs. per 
 ton 
 
 Ibs. per 
 degree 
 
 American 
 
 101,000 
 
 7' 6" 
 
 21' I0f" 
 
 1963 
 
 39-0 
 
 975 
 
 Ten-wheel . 
 
 123,000 
 
 12' 5 " 
 
 23' 8" 
 
 175 
 
 28-4 
 
 7-1 
 
 Consolidation 
 
 136,000 
 
 H' 6| 
 
 21' I" 
 
 1850 
 
 2O 'O 
 
 5'o 
 
 Note. Consolidation engines are made to run round a Wye (see 
 p. 240) with curves of 136 feet radius without any trouble. 
 
 Curvature per mile on some of the Railways in the Rocky 
 Mountains. 
 
 Average 
 degree 
 per mile 
 
 34 
 
 Name of Railway 
 
 Colorado Central . 
 Virginia, Truckee . 
 Union Pacific 
 Texas Central 
 Southern Pacific , 
 
 Length of 
 
 section. 
 
 miles 
 
 22 
 65 
 
 H3 
 142 
 
 420 
 278 
 
 59 
 
 247 
 
 63-6 
 
1.4 
 
 Preliminary Survey 
 
 TABLE V. Curvature and Grades on Sections of 
 Eastern Trunk-roads. 
 
 Name of road 
 
 Miles 
 
 Curvature 
 per mile 
 
 Per cent, 
 curved 
 
 Degrees 
 per mile 
 
 Per cent, 
 level 
 
 o. w 
 |1 
 
 Rise and fall 
 per mile 
 
 Ruling 
 Grades 
 
 New York Central 
 
 296-6 
 
 0-78 
 
 ig'8 
 
 iS'i 
 
 ,4-8 
 
 1-8 
 
 3 '8 
 
 21-23 
 
 Boston and Albany 
 
 202 'o 
 
 1*55 
 
 56'o 
 
 72-5 
 
 o'o 
 
 9'3 
 
 T 3'7 
 
 So 
 
 Concord and Ports 
 mouth .... 
 
 4o'o 
 
 1-41 
 
 37 '2 
 
 
 3'8 
 
 19*0 
 
 i5'3 
 
 80 
 
 Ulster and Delaware 
 Montrose Penna . 
 Summary of N. East 
 ern States . . 
 
 73 '2 
 27-2 
 
 5,372 
 
 3'6o 
 6 '35 
 i'88 
 
 41-0 
 49-0 
 
 35'5 
 
 80-5 
 24*1 
 
 55*9 
 
 13-6 
 3 '9 
 22-8 
 
 24'! 
 
 4'7 
 
 ji'6 
 9-6 
 
 13*0 
 
 160-142 
 95-74 
 
 The column ' Rise per mile ' gives the average excess 
 of rise over the fall in one mile. The next column gives the 
 feet of rise and fall. Thus, if a road rose 500 feet and fell 
 200 in 100 miles, it would be given above : Rise per mile 
 3*0, Rise and fall 2-0. The first quantity is an unavoidable 
 necessity, due to difference of level at the termini. 
 
 From the same work, on the authority of Mr. M. N. 
 Forney, locomotive expert, it is stated that : (> , 
 
 TABLE VI. Curve Limits far faced Wheel Bases. 
 
 Feet rad. 
 
 Axles 3 feet apart will roll in a curve of 67 
 ,, 4 >- . 91$ 
 
 ,, 5 M 'S3 
 
 ,,6 174.1 
 
 ,, 7' ., M 251 
 
 8 ,, ,, ,, 337 
 
 ,, 9 479 
 
 10 ,, 6 43 | 
 
 CENTRIFUGAL FORCE 
 
 On any three curves having radii as i, 2, 3, the centri- 
 fugal force at any given velocity is as 3, 2, i ; but the 
 coefficient of safety against overturning or disagreeable 
 effect is as v/3, ^2, V 1 = 173, 1-41, roo. 
 
General Considerations 
 
 TABLE VII. Giving for various curves the inferior and superior limits 
 of speed within which the centrifugal force is more or less objection- 
 able or dangerous. 
 
 Curve 
 
 Maximum and minimum limits of speed 
 in miles per hour 
 
 Degree 
 
 Minimum. Having 
 Radius in feet no disagreeable 
 effect 
 
 Maximum. On the 
 point of overturning 
 the vehicle 
 
 2 
 
 2,865 
 
 41-39 
 
 130-89 
 
 4 
 
 i433 
 
 29-27 
 
 92-55 
 
 6 
 
 955 
 
 23-90 
 
 75*57 
 
 8 
 
 717 2070 
 
 65-44 
 
 10 
 
 574 18-51 
 
 5 8 -54 
 
 12 
 
 478 16-90 
 
 53-43 
 
 14 
 
 410 15-64 
 
 49'47 
 
 16 
 
 359 14-63 
 
 46-28 
 
 18 
 
 3i9 1378 
 
 43^ 
 
 20 
 
 288 13*09 
 
 41'39 
 
 22 
 
 262 12-48 
 
 39-46 
 
 24 
 
 240 
 
 "95 
 
 3778 
 
 26 
 
 222 
 
 11-61 
 
 36-72 
 
 28 
 
 2O7 
 
 1 1 -06 
 
 34-98 
 
 3 
 
 193 
 
 10-69 
 
 33-80 
 
 40 
 
 146 
 
 9-25 
 
 29-27 
 
 50 
 
 "5 
 
 8-28 
 
 26-lS 
 
 60 
 
 96 
 
 7-56 
 
 23-90 
 
 Rule. For the centrifugal force in Ibs. per ton of 2,000 Ibs. 
 C= -02 3348 V 2 D, where C= centrifugal force in Ibs. per ton, 
 V= velocity in miles per hour, and D= degree of curvature 
 from which the following table is made. 
 
 TABLE VIII. Curve Limits at Different Speeds. 
 
 Speed in 
 miles per 
 
 Degree of curvature 
 
 hour 
 
 
 
 
 
 
 
 l0 
 
 5 
 
 10 
 
 15 
 
 20 
 
 10 
 
 2-33 
 
 11-67 
 
 23-35 
 
 35*02 
 
 46-70 
 
 2O 
 
 9-34 
 
 4670 
 
 93*39 
 
 140-09 
 
 18678 
 
 30 
 
 2I'OI 
 
 I05-07 
 
 210-13 
 
 315-20 
 
 420-26 
 
 40 
 
 g 
 
 60 
 
 37-36 
 
 58-37 
 84*05 
 
 18678 
 291 "85 
 42O'26 
 
 373-57 
 583-70 
 840-53 
 
 560-35 
 
 875-55 
 1,260-8 
 
 747-14 
 I,l67-4 
 
 1,681-1 
 
 70 
 
 II4-4 
 
 572-03 
 
 1,144-05 1,706-08 
 
 2,288-1 
 
1 6 Preliminary Survey 
 
 The centrifugal force varies directly as the degree of 
 curvature. 
 
 The heavy division lines mark the assumed maximum 
 limit of speed for safety, when the centrifugal force is=W. 
 
 On the 4 per cent, grades of the Mexican Railway, re- 
 versed curves of 150 feet radius were worked for a year 
 with ordinary locomotives. 
 
 Narrow-gauge railways have rarely been constructed with 
 curves sharper than 24 in the United States, but there are 
 a few as sharp as 30 in Colorado. 
 
 In the writer's location of 3 ft. gauge railways in the 
 Sandwich Islands ; curves of 40 (146 '2 ft. radii) were the 
 minimum, but many of them were more than a semicircle ; 
 previous railways had been constructed with 75 ft. radii in 
 that district, but in most cases without necessity. The trains 
 run at ten miles an hour with perfect safety, though it can 
 hardly be called comfortable. There is practically no 
 danger of the trains overturning because the loss of speed 
 due to curvature keeps them well within the limits of safety. 
 For rules on the same subject applicable either to feet or 
 Gunter's chains, see p. 264. 
 
 There is chronic trouble to railway managers from curves 
 of unnecessary sharpness, put in either to save the trouble 
 of a second revision, or from lack of experience on the part 
 of the surveyor. 
 
 It is true that rolling-stock can be made to go round 
 almost anything, but not without suffering from it. 
 
 It is a curious fact that single-line railways which have 
 
 DOWN Vj, >- 
 
 UP 
 FIG. i. 
 
 their crossing stations arranged in the usual way, as Fig. i, 
 wear the locomotive wheels more on one side than the 
 
General Considerations 17 
 
 other. This arises from the trains always entering the 
 stations faster than they leave them. Probably, if the 
 crossing stations were made as shown in Fig. 2 this would 
 not take place. It is a preferable arrangement, except 
 
 DOWN vm => 
 
 FIG. 2. 
 
 where the through express trains run past the station with- 
 out stopping, in which case the usual arrangement is to 
 be preferred unless very flat curves are used in the switch 
 on Fig. 2. 
 
 EFFECT OF INCREASED DISTANCE FROM DEVELOPMENT, 
 AND OF CURVATURE UPON WORKING EXPENSES 
 
 The following notes and Table IX. are from Mr. Welling- 
 ton's book, being deduced from extensive American statistics. 
 
 Fractional changes of distance increase or decrease 
 expenses by only 25 to 40 per cent, of the average cost of 
 operating an equal distance. 
 
 600 degrees of curvature will waste about 50 per cent, as 
 much fuel as the average burned per mile run. 
 
 The lowest probable limit of curve-resistance at ordinary 
 speeds in ordinary curves is about Ib. per ton per degree 
 of curvature. With worn rails and rough track it may be as 
 high as f Ib. per ton. 
 
 Curve-resistance per degree of curve is very much 
 greater on easy than on sharp curves, so that when, for ex- 
 ample, the resistance is i Ib. per ton on a i curve, it may 
 be 6 Ibs. to 8 Ibs. per ton on a 10 curve, and not more than 
 15 to 18 Ibs. on a 40 or 50 curve. 
 
 The almost uniform increase in cost in the first three 
 main divisions of the line is principally due to grades and 
 curves, which get worse as the line stretches inland. 
 
 c 
 
1 8 Preliminary Survey 
 
 TABLE IX. Running Expenses on Pennsylvania Railroad affected 
 by Cuivattire. 
 
 Average cost per train 
 mile in cents 
 
 Repairs 
 
 Fuel 
 
 Stores 
 
 Total 
 
 Eastern division . 6-42 
 
 6-93 i -ic 
 
 14-50 
 
 Middle division . j 8-87 7-00 1-07 
 
 16-94 
 
 Western division . 9*25 7-59 
 
 I'39 
 
 18-23 
 
 Mountain and 
 
 
 
 
 
 Tyrone ... 6-60 
 
 7-04 
 
 0-83 
 
 J 4'47 
 
 COMPENSATION FOR CURVES 
 
 Some of the various rules used in compensating steep 
 gradients for curvature are given by Mr. Robert Gordon in 
 ' Min. Proc. Inst. C.E.,' vol. Ixxxv. : 
 
 'The best American practice invariably allows com- 
 pensation when the curve falls on a gradient by lessening 
 the inclination as the sharpness of the curve increases. 
 Some difference of opinion exists amongst the authorities as 
 to the amount of reduction required, but the average given 
 is 0*05 per i oo ft. per degree of curvature. 
 
 'This practice varies, however, and Mr. A. A. Robinson, 
 , who has had great experience on steep gradients, gives as 
 follows : 
 
 TABLE X. Compensation for Curves. 
 
 Per 100 feet 
 per degree 
 
 * Rate of maximum grade o to i in 166-0 compensation 0-06 
 
 ,, ,, ,, i in 166 to i in 62-5 ,, 0-05 
 
 ,, ,, iin62'5toiin 33-5 ,, 0-04 
 
 * Mr. Blinkensdorfer gives 0*03 to 0-07 in the same limits ; 
 while Mr. Wellington allows 0*06 on all maximum curves. 
 The practice also of widening the gauge on curves varies 
 much. Some engineers allow only the same play of \ inch 
 that is given on straight lines ; while others increase it \ 
 inch and more on curves. But opinion is unanimous in 
 requiring a tangent between reverse curves, and sharp 
 
General Considerations 
 
 curves are eased off at both ends. In some cases gradients 
 also are eased at the approaches.' 
 
 The following tables are taken from Mr. Trautwine's 
 ' Pocket- Book : ' 
 
 TABLE XI. 
 
 ; 
 
 spine United 
 
 
 States Railroads. 
 
 \ .'-. 
 
 Name of company 
 
 Per mile 
 of road 
 
 Per train 
 mile 
 
 Per cent, 
 of receipts 
 
 
 $ 
 
 cents 
 
 
 Lehigh Valley, 1872 . 
 
 
 
 6 5| 
 
 Pennsylvania Central, 1869 
 
 32,000 
 
 
 
 
 Philadelphia and Reading, 1859 . 
 
 
 
 
 54i 
 
 1868. 
 
 17,200 
 
 144 
 
 
 Connecticut, average of all the R. R., t 
 1861 \ 
 
 3,7Sl 
 
 95 
 
 57 
 
 Massachusetts, average of all the \ 
 R. R., 1861 J 
 
 3,785 
 
 85 
 
 60 
 
 * New York State, average of all the ) 
 R. R., 1867 1 
 
 13,856 
 
 
 
 76 
 
 New York Central, average of all the ) 
 R. R., 1867 ) 
 
 15,620 
 
 170 
 
 77^ 
 
 English R. R., averages for 1856-7-8 
 
 
 
 66 
 
 50 
 
 Scotch ,, ,, ,, 
 
 
 
 56 
 
 44 
 
 Irish ,, ,, ,, 
 
 
 
 5 2 
 
 40 
 
 TABLE XII. Statistics of several United States Narroiv-gauge 
 Railroads for 1884. (Poor's Manual. ) 
 
 ] 
 
 
 
 Rolling-stock 
 
 "B 
 
 
 
 
 _ 
 
 
 
 | 
 
 ' 
 
 .> 
 
 
 oad and eq 
 t per mile 
 
 mual earni 
 lile of road 
 
 1 
 
 penses -f- 
 s earnings 
 
 o 
 
 o 
 
 
 
 o 
 
 
 1 
 
 8! 
 
 ra 
 
 
 *- S3 
 
 3 e 
 
 C t, 
 
 MP 
 
 
 
 ** 
 
 
 
 PH 
 
 
 
 fei 
 
 o 6 
 
 V: 
 
 a 
 
 %& 
 
 M 
 
 
 
 
 
 
 
 
 3 
 
 O 
 
 
 
 Bridgton & Saco | 
 Riv., Me. . . [ 
 
 2 
 
 16 
 
 2 
 
 2 
 
 i 
 
 16 
 
 $ 
 12,167 
 
 $ 
 
 I,Tt2 
 
 $ 
 834 
 
 '75 
 
 Profile and Fran- ) 
 conia, N. H. . f 
 
 3 
 
 H 
 
 3 
 
 7 
 
 
 
 6 
 
 15,43 
 
 1,346 
 
 640 
 
 48 
 
 Catnden and Mt. | 
 Ephraim, N. J. f 
 Bradford and ) 
 Kinzua, Pa. . j 
 
 3 
 3 
 
 6 
 39 
 
 5 
 
 5 
 
 2 
 
 6q 
 
 13,645 
 14,922 
 
 2,868 
 i,793 
 
 2,642 
 r,7i7 
 
 92 
 '96 
 
 Denver and Rio ) 
 Grande . . . J 
 
 3 
 
 1,685 
 
 239 
 
 "5 
 
 7- 
 
 5,676 
 
 35,ooo 
 
 3,519 
 
 2,573 
 
 '73 
 
 C 2 
 
.20 
 
 Preliminary Survey 
 
 TABLE XIII. Items of Total Annual Expenses for Maintenance and 
 Operation of all the Railroads of the United States in 1880. (Poor's 
 Manual. ) 
 
 
 
 $ Per cent, 
 per mile total 
 
 Percent, of 
 earnings 
 
 Repairs of road, bed, and track 
 
 451 1 1 -23 6-82 
 
 Renewals of rails .... 
 
 197 
 
 4-89 
 
 2-97 
 
 Renewals of ties .... 
 
 122 
 
 3 '04 
 
 i-85 
 
 Repairs of bridges .... 
 
 102 i 2-55 
 
 i-55 
 
 Repairs of buildings 
 
 87 2-17 1-32 
 
 Repairs of fences, crossings, &c. 
 Telegraph expenses 
 
 17 -42 
 
 41 i "oi 
 
 25 
 
 62 
 
 Taxes ...... 
 
 152 3-77 2-29 
 
 Maintenance of road and real i 
 estate ' 
 
 1,169 
 
 29-08 
 
 17-67 
 
 Repairs, c. of locomotives 
 
 249 
 
 6-19 
 
 376 
 
 Repairs, &c. of passenger, baggage, j_ 
 and mail cars . . . . J 
 
 120 
 
 2'99 
 
 1-82 
 
 Repairs, c. of freight cars 
 
 257 
 
 6-40 
 
 3-89 
 
 Repairs, c. of rolling-stock \ 
 
 
 
 
 (including renewals and addi- 1 
 
 627 
 
 I5-58 
 
 9 '47 
 
 tions) . . . . . J 
 
 
 
 
 Passenger train expenses . 
 
 137 
 
 
 2-07 
 
 Freight train expenses 
 
 330 
 
 8-21 
 
 4'99 
 
 Fuel for locomotives 
 
 374 
 
 9-31 
 
 5-66 
 
 Water-supply, oil, and waste . 
 
 70 
 
 1-74 
 
 i -06 
 
 Wages of locomotive runners and ) 
 firemen . . . . . i" 
 
 310 
 
 772 
 
 4-69 
 
 Agents, and station service and ) 
 supplies . . . . . ) 
 
 45i 
 
 11-23 
 
 6-82 
 
 Salaries of officers and clerks . 
 
 139 
 
 3-46 
 
 2-10 
 
 Advertising, insurance, legal ex- ) 
 
 
 6 
 
 1-8*7 
 
 penses, stationery and printing . J 
 
 123 
 
 3 
 
 1 07 
 
 Damages to persons and property 
 
 40 
 
 98 
 
 60 
 
 Sundries ..... 
 
 250 
 
 6-22 
 
 378 
 
 Running and general expenses 
 
 2,224 
 
 55'34 
 
 33-64 
 
 Aggregate annual expenses 
 
 4,019 
 
 lOO'OO 
 
 6078 
 
General Considerations 
 
 21 
 
 TABLE XIV. Gross Annual Earnings per Mile, per Passenger Mile, 
 and per Ton Mile, of some of the Principal United States Railways 
 in 1880. 
 
 _ 
 
 Length 
 in miles 
 
 From 
 passen- 
 gers per 
 mile of 
 
 From 
 passen- 
 gers per 
 passen- 
 
 From 
 freight 
 per 
 mile 
 _r 
 
 From 
 freight 
 per 
 ton 
 
 
 
 road 
 
 ger 
 
 mile 
 
 ot 
 road 
 
 mile 
 
 Pennsylvania R. R. 
 
 1,806 
 
 4,700 
 
 $ 
 0242 
 
 $ 
 15.615 
 
 0089 
 
 New York Central 
 
 994 
 
 6,651 
 
 O2OO 
 
 21,794 
 
 0086 
 
 Central Pacific . 
 
 2,447 
 
 2,237 
 
 0303 
 
 4,577 
 
 0249 
 
 Chicago, Burl., and Quincey. 
 
 1,805 i i,532 
 
 O24O 
 
 7,202 -on i 
 
 Philadelphia and Reading . 
 
 780 
 
 3,429 
 
 O2OI 
 
 17,200! 'Oi6i 
 
 Union Pacific 
 
 1,215 
 
 2,624 
 
 0320 ! 7,154 
 
 0199 
 
 Atchison, Topeka, and \ 
 Santa Fe . . .1" 
 
 i,398 
 
 1,144 
 
 0606 3,974 
 
 0209 
 
 Average of United States 
 
 87,801 
 
 1,641 
 
 0251 
 
 4,740 -0129 
 
 TABLE XV. Annual Earnings and Expenses of the 
 above roads in 1880. 
 
 
 
 Length in 
 miles 
 
 Gross earn- 
 ings per 
 mile 
 
 Expenses 
 per mile 
 
 Expenses 
 -r- gross 
 earnings 
 
 Pennsylvania R.R. . . . 
 
 1, 806 
 
 $ 
 20,315 
 
 $ 
 12,267 
 
 585 
 
 New York Central . . . 
 
 994 
 
 28,445 
 
 17,969 
 
 609 
 
 Central Pacific .... 
 
 2,447 
 
 6,814 
 
 3,340 
 
 470 
 
 Chicago, Burlington, & Qu. 
 Philadelphia and Reading . 
 
 1,805 
 780 
 
 8,734 
 20,629 
 
 4,454 
 n,754 
 
 497 
 568 
 
 Union Pacific 
 
 I 21 S 
 
 q,778 
 
 4, CQ7 
 
 426 
 
 Atchison, Topeka, & Santa ) 
 Fe 1" 
 
 1,398 
 
 5,n8 
 
 2,408 
 
 458 
 
 Total United States . . . 
 
 87,801 
 
 6,611 
 
 4,019 
 
 608 
 
 ESTIMATES 
 
 A few estimates of roads, railways, and tramways, will 
 now be given, which will to some extent fix the ideas on 
 what must necessarily be subject to very great diversity, 
 according to the circumstances of each case. 
 
 The surveyor would do well to make rough shots at his 
 estimate on his first walk-over, so as to guide him in his 
 
22 Preliminary Survey 
 
 choice of alternative routes, and even the most approximate 
 ' aide-memoires ' are often of great service. 
 
 The tendency of even experienced men is to over-esti- 
 mate rough country and under-estimate easy country. A 
 big gulch or canon is apt to scare most men, and swaggering 
 viaducts float across their mind's eye, which often by patient 
 reconnaissance melt down to one or two reverse curves and 
 a ' bit of a trestle.' 
 
 An ordinary American railway of about 100 miles in 
 length in moderately easy country will require about 15,000 
 cubic yards of earthwork per mile. 
 
 Mr. Trautwine's estimate, made quite a number of years 
 ago, is near enough for a rough shot to-day. 
 
 It is as follows for a single line in United States : 
 
 Gauge 4' 81". Labour $\ 75 = js. per day. 
 
 Grubbing and clearing (average of entire road) 3 acres $ 
 
 at $50 150 
 
 Grading 20,000 cubic yards of earth at 35 cents . 7,000 
 
 Ditto 2,000 cubic yards of rock at $i . . 2,000 
 Masonry of culverts, drains, abutments of small 
 
 bridges, retaining walls, 400 cubic yards at $8 . 3,200' 
 
 Ballast 3,000 cubic yards broken stone at $i . . 3,000 
 
 Cross-ties 2,640 at 60 cents delivered . . . 1,584 
 
 Rails (60 Ibs. to a yard) 96 tons at $30 delivered . 2,880 
 
 Spikes 150 
 
 Rail-joints ........ 300 
 
 Subdelivery of material along the line . . . 300 
 
 Laying track 600 
 
 Fencing (average of entire road) supposing only one 
 
 half of its length to be fenced . . . . 450 
 Small wooden bridges, trestles, sidings, road -crossings, 
 
 cattle-guards, &c. ...... 1,000 
 
 Land damages 1,000 
 
 'Engineering, superintendence, officers of Co., sta- 
 tionery, instruments, rents, printing, law expenses, 
 
 and other incidentals ..... 2,386 ' 
 
 26,000 
 
 This amount is only extended to units to bring the total to a lump 
 
General Considerations 23 
 
 Add for depots, shops, engine-houses, passenger and 
 freight stations, platforms, wood sheds, water stations 
 with their tanks and pumps, telegraph, engine-cars, 
 weigh scales, tools, &c. ; also for large bridges, tunnels, 
 turnouts, &c. (Trautwine.) 
 
 Mr. R. C. Rapier, of the well-known firm of Ransomes 
 and Rapier, in his 4 Remunerative Railways ' gives an 
 estimate for the equipment of a metre-gauge single- line 
 railway, 40 miles long, including 40 Ib. rails, wooden sleepers, 
 seven engines weighing 15 tons each on six wheels, turn 
 tables, tanks, water-cranes, weigh-bridges, sheerlegs, signals, 
 35 passenger carriages and brake vans, 150 freight waggons 
 of different kinds, workshop-fittings, and stores. The total 
 for 40 miles 86,7087., or for i mile 2,i68/. 
 
 The same author gives another estimate for the equip- 
 ment of a forty-mile 3 ft. 6 inch gauge-railway with 45 Ib. 
 rails, eight engines weighing 18 tons on six wheels, and a 
 similar list of materials to the metre-gauge, somewhat in- 
 creased. The total for forty miles 98,8407., or for one mile 
 2,47 1/. 
 
 Where it is necessary, as in England, to put up gate- 
 keepers' houses, and sometimes signal-interlocking gates at 
 level crossings, it is often as cheap, having regard to future 
 expenses of operating the line, to go over or under the 
 
 road. 
 
 Gatekeepers house. 
 
 s. d. 
 
 Wages 15^. per week = 5 per cent, on capital ot 780 o o 
 House ........ 220 o o 
 
 ;i,ooo o o 
 
 Even where there is no help from the ground the earth- 
 work can generally be got under 20,000 cubic yards. 
 
 A Iter native bridge. 
 
 s. d. 
 
 20,000 cubic yards earthwork at ga. . . 750 o o 
 
 One bridge ..*.... 250 o o 
 
 ;i,ooo o o 
 
* - 
 
 B| 
 
 o\ 
 
 k 
 
 ca 
 
 ^ 
 
 1 
 
 ^ 
 
 
 vo" 
 
 tC 
 
 * 
 
 
 
 
 ^ 
 
 
 ^ 
 
 VO 
 
 g 
 
 tx 
 
 % 
 
 CO 
 
 *cj 
 
 f^. 
 
 ^ 
 
 
 CO 
 
 t 1 *-. 
 
 O 
 
 o 
 
 H 
 
 
 
 H 
 
 1 
 
 q 
 
 Os 
 
 g 
 
 CO 
 
 & 
 
 ll 
 
 Ii 
 
 21 
 
 a| 
 
 S2 f* 
 
 1 
 tt 
 
 H-i ^o " 
 
 S 
 
 C tr! 
 
 3 
 -0 
 
 
 o 
 ri 
 
 oji.S 
 
 c-G 3 
 
General Considerations 25 
 
 A railway 87 miles long was completed for the Nizam of 
 Hyderabad about the year 1885, for under 6,ooo/. per mile. 
 It was promoted by the Indian Government, and built 
 under the supervision of the Government engineer. The 
 gauge was 5 ft. 6 inches, the rails were 66^ Ibs. per yard, of 
 steel, and the sleepers steel also. The price included 
 equipment and fencing. 
 
 ROADS 
 
 The Grand Trunk carriage road of the Bengal Presi- 
 dency cost approximately 5007. per mile. The work was 
 done mainly by starving poor during famine time, under the 
 administration of Lord William Bentinck about 1835. The 
 width was 40 feet, metalled in the centre 16 feet wide with 
 either broken stone or a natural concretion of carbonate of 
 lime, called ' Kunkur,' rammed by hand (there being no 
 steam rollers in those days). The cost of laying the metal 
 at an average lead of one mile was i62/. 6s. od. per mile, 
 and cost of repairs and maintenance of ditto 337. i2s. od. 
 per annum. Total cost of maintenance was 5<D/. per annum. 
 (Gen. Tremenheere, ' Min. Proc. Inst. C. E.' vol. xvii.) 
 
 The main roads of South Australia are described by 
 Mr. Charles T. Hargrave in ' Min. Proc. Inst. C.E.,' vol. 1. 
 and he gives the average cost with a metalled way 18 feet 
 wide and a 60 feet right of way as follows : 
 
 Earthwork in cuttings &c., 800 yards at I s. . 40 o o 
 Culverts, nine at 15 cubic yards each; 135 
 
 cubic yds. at 12s. . . . . . 8100 
 
 Wheelguards and posts ; 18 sets of 2 posts and 
 
 one guard at 40^. per set . . . . 36 o o 
 Forming the metal bed ; 80 chains, 8s. . . 32 o o 
 Bottom metal or soling, 4" thick, 15 cubic yards 
 
 per chain for 80 chains ; 1,200 cubic yards 
 
 at 4.5-. . . . . . . 240 o o 
 
 Metal 2^" thick, 22 cubic yards per ch. for 80 
 
 ch. ; 1,760 cubic yards at "js. . . . 616 o o 
 Blinding (i.e. thin top dressing of gravel) 7 cubic 
 
 yards per ch : for 80 ch. : 560 cubic yards 
 
 at is 28 o o 
 
 Rolling eight days at 255. . . . . 10 o o 
 
 Carried forward . . .1,083 
 
26 Preliminary Survey 
 
 Brought forward . . 1,083 o o 
 To this must be added fencing (not always done) 
 
 80 ch. at 8 rods = 640 rods at 5 s. . . 160 o o 
 And if land has to be purchased ; 8 acres per 
 
 mile at 5/. . . . . . . 40 o o 
 
 ^1,283 o o 
 
 For the first six miles out of Adelaide during ten years, 
 300 to 400 cubic yards of metal were needed per mile per 
 annum. At fifty miles from the city only slight repairs were 
 needed. 
 
 Annual cost of clearing culverts and weeding 3/. per mile 
 for each side of road. On 326 miles constructed the cost 
 of maintaining the metalled portion was i22/. per annum. 
 
 The price of labour was $s. 6d. to 6s. per day ; masons 
 and carpenters Bs. to gs. 
 
 A temporary road for conveyance of railway materials 
 into the bush costs from 507. to zoo/, per mile. 
 
 STONE-CRUSHERS 
 
 ' Blake's or Blake-Marsden Stone-Crushers ' vary in size 
 and cost, from a 10" x 8" machine breaking stone of that size 
 to the extent of 3^ cubic yards per hour, nominal horse-power 
 3, total weight including screening apparatus 5 tons 6 cwts., 
 price i57/., up to a 30" x 13", breaking stone of that size to 
 the extent of 14 cubic yards per hour, 16 horse-power, weight 
 1 6 tons 2 cwt, price 4407. 
 
 ROAD-ROLLERS 
 
 A i5-ton ' Aveling and Porter Roller ' costs about 6507. 
 It will roll on an average 1,100 square yards per day. The 
 cost of rolling, including all charges, is somewhere between 
 \d. and id. in England ; and the cost of binding material 
 about $d. per square yard. 
 
 The two last estimates are gathered from Mr. Boulnois' 
 Municipal Surveyor's Handbook.' 
 
General Considerations 27 
 
 TRAMWAY ESTIMATES 
 
 The following estimates of tramway construction in 
 the United States are from the report of the committee 
 of the American Street Railway Association at the Min- 
 neapolis Convention in October 1889, upon 'The Con- 
 ditions necessary to the Financial Success of Electricity as a 
 Motive Power.' 
 
 They are for a single line ten miles long equipped com- 
 plete with fifteen cars. 
 
 CABLE-ROAD 
 
 
 Road-bed, rails, conc'vit cable .... 140,000 
 
 Power station . ...... 25,000 
 
 Cars 3,ooo 
 
 Total for ten miles ..... ; 168,000 
 
 ELECTRICAL OVERHEAD-WIRE CONSTRUCTION 
 
 
 Road-bed and rails ...... 14,000 
 
 Wiring ........ 6,000 
 
 Cars ......... 12,000 
 
 Power station ....... 6,000 
 
 Total for ten miles ..... ,38,000 
 
 STORAGE BATTERIES 
 
 (also termed Secondary Batteries or Accumulators} 
 
 
 Road-bed and rails ...... 14,000 
 
 Cars 15,000 
 
 Power station 6,000 
 
 Total for ten miles ..... ^"35,000 
 
 * In the above cases of electrical construction the motor 
 car would be capable of pulling one or two tow-cars if 
 necessary. These figures your committee has no doubt will 
 
28 
 
 Preliminary Survey 
 
 be found to be cal- 
 culated within a 
 reasonable limit of 
 cost.' 
 
 Figs. 3 and 
 4, excerpt ' Min. 
 Proc. Inst. C. E.' 
 vol. Ixxxv., will 
 give some idea of 
 the way difficulties 
 of ascent are over- 
 come by curve- 
 development, and 
 the ground plan of 
 Burlington shops 
 will show the way 
 the lines are laid 
 out for the engine 
 depots in America. 
 
 The actual 
 measurement for 
 the preliminary 
 estimate is much 
 facilitated by dia- 
 grams, some of 
 which will be found 
 in the chapter on 
 ' Graphic Calcula- 
 tion.' These ope- 
 rations follow upon 
 the actual survey, 
 whether it be 
 merely a route- 
 survey or a tele- 
 metric survey, and 
 are therefore de- 
 scribed later on. 
 
General Considerations 
 
 29 
 
 $ ! 
 " 
 
 11 
 
 
 ; 
 
 ri 
 
 31 
 
30 Preliminary Survey 
 
 CHAPTER II 
 
 ROUTE-SURVEYING OR RECONNAISSANCE 
 
 BEFORE going into the subject of ' Route-survey ' in detail, 
 the first steps of American pioneer railway-making will be 
 briefly described. 
 
 The Nipissing division of the Canadian Pacific Railway 
 will furnish a fair illustration of American location. It is on 
 the north shore of Lake Superior, and it was there that the 
 writer gained his first experience of that kind of work. A 
 network of lakes and rivers leads the Indian, on his fishing 
 and hunting expeditions, clear across the watershed of 
 Northern Ontario down to Hudson's Bay. The first pioneer- 
 ing was done by Mr. W. R. Ramsey, with an Indian guide 
 and one or two white men. Taking only the aneroid and 
 prismatic compass, they followed up the course of several 
 rivers with a canoe, often having to carry it over long ' por- 
 tages,' and undergoing considerable hardships, until they 
 emerged at the junction point with another survey, proceed- 
 ing from Port Arthur on the other side of the great lake. A 
 sketch-map was made for the approval of the general 
 manager, showing the topography to a small scale, and the 
 length of line. 
 
 The next party, headed by Mr. Ramsey, consisted of the 
 usual preliminary location party : 
 
 1 Transit-man 2 Axe-men 
 
 2 Levellers 2 Chain-men 
 2 Rod-men i Slope-man 
 i Picket-man i Cook 
 
Route- Survey ing 3 1 
 
 The modus operandi is then as follows : 
 
 The leader keeps ahead of the party, revising his previous 
 survey with aneroid and compass, and keeping the whole of 
 the operations under control. Every morning he indicates 
 the direction to the transit-man, who has a line cut through 
 the bush and chains it, driving stakes at every hundred feet. 
 The leveller follows on the transit-man, making the profile 
 (or section as we call it in England), whilst another takes the 
 side-slopes of the hills with a clinometer. The field-work 
 is plotted in tent at night. The profile is greatly facilitated 
 by printed profile paper, which dispenses with all scaling. 
 
 Under favourable circumstances a mile or a mile and a 
 half a day can be surveyed in this manner, but on the division 
 here referred to only an average of ten miles a month was 
 possible with one party. At times there were four parties in 
 the field to distribute the work. Nearly all of it was gone 
 over three or even four times, always saving money on the 
 construction. A hundred pounds extra in survey will often 
 save a thousand pounds in construction. 
 
 The present chapter is devoted to methods which do 
 not require chaining, but more on the American methods of 
 location will be found in Chapters VI. and VII. 
 
 When the country through which a road or railway is to 
 be made is not well known specially when the inhabitants 
 are unfriendly, but in almost every case of projected work in a 
 new country it is necessary to obtain, cheaply and speedily, 
 one or more surveys of alternative practicable routes or of 
 the general topographical features of the area. 
 
 A chained survey would be out of the question, and even 
 a telemetric traverse would generally be too expensive. A 
 fairly accurate map is required, which will be made by one 
 man at the rate of ten to twenty miles per day, and which 
 will serve the projectors with sufficient information to guide 
 them in approving the route favoured, in capitalising their 
 enterprise, and in preparing their prospectus. 
 
 The class of work most suitable for this object has been 
 
32 Preliminary Survey 
 
 more thoroughly studied and practised, though with a dif- 
 ferent object, by military than by civil engineers, and the 
 best instruments and books extant are from those sources. 
 Failing time or money for chaining or triangulation, it is 
 necessary to have recourse for linear measurement to records 
 of pace of men, gait of horses, or speed of river steamers 
 taken t with as much care as possible. Military movements 
 are made with trained regularity, and no one who has not 
 studied the subject would believe what great precision is 
 attained in making maps of marches the distances of which 
 are laid down simply and solely from either time-records or 
 pace-measurers. 
 
 These maps, which are called reconnaissances, route- 
 surveys, or flying surveys, vary in accuracy, from a rapid 
 field-sketch relying upon a trained eye for estimation of dis- 
 tance and a knowledge of perspective for the filling in of 
 detail, to a survey correct within a margin of from one to five 
 per cent. ; based upon true trigonometrical and traverse prin- 
 ciples and preserved from cumulative error by astronomical 
 observations similar to those which determine the position 
 of a ship at sea. 
 
 .The term * a mere sketch ' is often applied rather con- 
 temptuously to what may be a most valuable and perhaps the 
 only record of the topography of an important position. 
 
 Military engineers know best the value of a good sketch. 
 They have to do their work under fire or in danger of 
 surprise, and a man who can dash off a sketch of the enemy's 
 position, the points of vantage, and the best line of attack, 
 will by so doing provide in a few moments information which 
 may decide the final issue of the struggle. 
 
 Surveyors have more time at their command, and their 
 disposition is rather to depreciate methods which overstep 
 the bounds of rigid mechanical exactness. The standing 
 orders of our English Parliament for deposited plans of 
 public works call for a high standard of accuracy, and justly 
 so in a country like this, possessing already excellent 
 
Route- Surveying 3 3 
 
 maps published by the Ordnance Department. It was 
 due to the existing vested interests that they should be 
 protected from the attempts of speculative promoters 
 to interfere with their property without fully and clearly 
 representing in all its bearings what the effect of such 
 interference would be. The standing orders, however, per- 
 mit of the framing of memorials of opposition containing 
 frivolous allegations which as such are frequently overridden 
 by the Examiner. 
 
 Surveyors who get their training in England do 
 not consequently have much opportunity of practising 
 either route-survey or field-sketching, and so, specially 
 in regard to the latter, do not know if they have a 
 talent for it or not. Artists are born, not made, but there 
 are few engineers who are so wanting in 'eye' that they 
 cannot greatly increase their efficiency as pioneers by a 
 course of study and practice in sketching. Even when the 
 route survey is carried out with all possible care so as to 
 approach to the purely mechanical system, a knowledge 
 of sketching will be most useful in filling in the adjacent 
 country. Lake, wood, bluff, canon, or river will, by a few 
 dashes of the pencil, give the promoters at home something 
 more for their minds to feed upon than a system of straight 
 lines with the bearings written up. It should of course be 
 evident from the map, when part is sketched and part scale- 
 able, which is the reliable portion. 
 
 The art of free-hand drawing can hardly be dwelt upon 
 in a work of this kind. A very few lessons in perspective 
 combined with a thorough course of practice in the field is 
 the only way to obtain the needed skill. Even in the course 
 of telemetric survey, the sketching of detail is a great help 
 both in the plotting of contours and as an accessory to the 
 finished map. 
 
 Photography is also becoming a useful aid to the sur- 
 veyor. The little 'Detective' camera .recently introduced 
 is only a little larger than a carriage clock ; it is carried in 
 
34 Preliminary Survey 
 
 the hand without needing a tripod or other paraphernalia. It 
 is directed and focussed by observing a reflected image of the 
 view on the top of the camera. It has also the advantage 
 over previous inventions of the kind that the views may 
 be removed one by one instead of in batches. 1 
 
 The scales adopted in route-surveying vary from six 
 inches to the mile, to thirty or more miles to the inch. If 
 to a small scale, it is usual to plot in miles of latitude and 
 longitude, so as to correspond with the daily astronomical 
 observations. 
 
 The route survey is essentially a traverse, checked where 
 possible by triangulation with compass angles and range- 
 finder, or simply by rays drawn upon the sketch-board or 
 plane-table. 
 
 SURVEYING WITH THE SKETCH-BOARD OR PLANE-TABLE 
 
 Those who have become proficient in the use of the 
 plane-table are generally enthusiastic in favour of it, and 
 there can be no doubt that for a topographical survey of 
 a large area, based upon the known stations of a primary 
 triangulation, it is both rapid and accurate. During the 
 season of 1886-7 the U.S. Geological Survey mapped in 
 this way an area of 56,000 square miles with a staff of 160 
 men, at an average cost of i2s. per square mile. 
 
 1 A camera should be preferred which does not require the removal 
 and changing of the slide every time. Abrahams Co. make the 
 * Ideal ' camera, which is quite suitable for the purpose. If the surveyor 
 is on one side of a wide canon and a train is passing along the other 
 side, he can get a good representation of the whole hillside by taking 
 views of the train all along its passage, the steam from the engine will 
 make a good landmark and especially just as the train goes into tunnel. 
 The views can afterwards be pieced together ; the length of the train 
 will serve as a measure of relative distances, and considerable informa- 
 tion be placed on record by a few seconds' field-work. The slides 
 can be preserved and developed at leisure or sent home for develop- 
 ment, but the developers are provided in soluble pellets for foreign 
 
Route- Surveying 3 $ 
 
 The author's practice in surveying for railways has led 
 him to the persuasion that the plane-table is more useful 
 as an adjunct than as a universal instrument. He uses the 
 modification of Captain Verner's military sketch-board plane- 
 table either buckled on the wrist or mounted on its light 
 tripod, and in this form finds it one of the most valuable 
 portions of the surveying outfit. 
 
 When the country is very magnetic, the needle becomes 
 unreliable, and the plane-table may have to be used for the 
 whole of the field-work. 
 
 Under favourable conditions, it is possible to do pretty 
 much the same class and quantity of work with the 
 plane-table alone as with the prismatic compass and field 
 notes. 
 
 With the plane-table, the work is all done in the field, 
 which has the advantage of exhibiting the errors liable to 
 arise with either class of work, and so to enable them to be 
 eliminated on the spot. It has the disadvantage of occupy- 
 ing time in mapping during daylight, which might be wholly 
 given to fieldwork, and the mapping done either at night 
 or when the sun is too powerful, or when too wet to go out. 
 It has the further disadvantage of disclosing to unfriendly 
 natives the object of the traveller's journey, being much more 
 conspicuous than a prismatic compass. It has the advantage 
 of affording to the skilful surveyor a complete method of 
 triangulation, producing results of an accuracy (even over ah 
 extended area) very little short of that obtained with good 
 theodolites ; its horizontal angles being determined by rays 
 aided if desired by the telescopic sight-rule, and indepen- 
 dent of magnetic variation. It has the disadvantage of being 
 with difficulty checked when carrying forward a continuous 
 traverse, especially so when the bases have to be short, as 
 in a crooked road with high hedges. There are generally 
 some salient points on which rays can be taken as in- 
 dependent checks, but without them every mistake in angle 
 is perpetuated. 
 
 D 2 
 
36 Preliminary Survey 
 
 For an extended triangulation a measured or calculated 
 base is required as in geodetic survey, whether working 
 with the plane-table or compass ; and where the needle 
 can be relied upon, the angles can be taken with the same 
 accuracy with either, but in the case of a continuous traverse, 
 the compass has the advantage of giving the magnetic 
 bearing every time independently of the previous work. 
 
 When the plane-table is used as a universal instrument, 
 a stadia telescope is attached to the sight-rule, which has 
 to be transported in a separate case. The class of work 
 performed by it under these circumstances answers to the 
 telemetric survey with the transit, but when passing rapidly 
 through a country, it is much more cumbersome to transport 
 a heavy plane-table with sight-rule and telescope than to 
 shoulder a theodolite. 
 
 A few remarks will now be made upon the method of 
 triangulation with a large-sized plane-table over an extended 
 area, after which the subject of the plane-table will be 
 only considered in the form suitable to preliminary railway 
 or road survey with the smaller instrument described in 
 Chap. IX. 
 
 TRIANGULATION WITH THE PLANE-TABLE 
 
 Where a base is known by the exact geographical posi- 
 tions of two towns or villages a few miles from one another, 
 and commanding a good view of surrounding country, an 
 almost unlimited tract may be correctly surveyed from it. 
 When this is not the case, the two extremities of the base 
 have each to be located astronomically, as described on 
 Chap. IV., and the distance between them calculated by 
 problem on p. 150. The accuracy of the succeeding 
 triangulation will depend fundamentally upon this astrono- 
 mical work. The latitude should be easily obtainable within 
 a sixth of a mile, and by careful repetitions to about 100 
 yards of the earth's surface, so that if the watch be not 
 reliable to give Greenwich time to a second, the more the 
 
Route-Surveying 37 
 
 base approximates to a true meridional line the better for 
 the results. 
 
 To obtain practice in England the plane-tabler should 
 locate two suitable points for his base from the six-inch Ord- 
 nance map of the district, which gives parallels and meridians 
 to every single second or about thirty yards. They should 
 be selected as far apart as possible, say from two to seven 
 miles. 
 
 A common military plane-table 30 inches x 24, of one- 
 inch panelled deal, will map the whole county of Yorkshire 
 on a scale of four miles to the inch. It can afterwards be 
 compared with the shilling edition of W. H. Smith & Son's 
 reduced Ordnance maps on that scale. 
 
 The paper should be pasted on the board by its 
 margins, well damped so as to stretch as tight as a drum 
 when dry. The parallels and meridians should then be 
 drawn as described on p. 75, and the base line AB laid 
 on from the data taken from the six-inch Ordnance map. 
 The table is then carried to station A, and, with the sight- 
 rule touching the base line, the board is turned on its axis 
 until the opposite base-station B is brought into the exact 
 line of sight. The compass, which should be of the portable 
 trough-needle type, is then placed in one corner of the 
 map, and carefully turned to and fro until the needle is at 
 rest at the centre of its run. A fine pencil line is then drawn 
 round the compass box with the letter N. (mag.) opposite 
 the north point of the needle. 
 
 With the plane-table firmly clamped, rays are then taken 
 from A to all prominent points in view ; they should not 
 be drawn right across the paper, but merely fixed by a fine 
 pencil line in the margin of the map about half an inch long, 
 with marks denoting the point of observation and the point 
 observed, thus A/A t to correspond with descriptive entries in 
 the Field Book, shown overleaf. 
 
 When all possible rays have been taken from A, the 
 instrument is shifted to B, and 'set in meridian ' by a back 
 
Preliminary Survey 
 
 FlELDBOOK 
 
 Date 
 
 | 
 
 "-, HE 
 
 Ob- 
 ject 
 
 Latitude 
 
 Longitude 
 
 Ane- 
 roid 
 
 Varia- 
 tion of 
 compass 
 
 Remarks 
 
 o 
 
 ' 
 
 * 
 
 o 
 
 / 
 
 " 
 
 1890 
 
 
 
 
 
 
 
 
 
 <* 
 
 
 
 Jan. i 
 
 3 P.M. A 
 
 B 
 A> 
 
 
 
 
 
 
 
 640 
 
 i 9 i W. 
 
 Spire of Hockley 
 Parish Church 
 
 
 1 
 
 A, 
 
 
 
 
 
 
 
 
 
 Clump of Beeches 
 
 
 
 
 
 
 
 
 
 
 
 
 near Hockley 
 
 ray on A. The compass-box is placed on its delineated 
 position on the paper, and if the sight-rule has been cor- 
 rectly adjusted, and there be no local magnetic deviation or 
 sufficient diurnal variation to account for it, the needle will 
 still be at rest at the centre of its run. If not, its position 
 should be measured by the graduation on the compass-box 
 as an addition to or deduction from the initial variation, and 
 an entry made of the variation at B. 
 
 In selecting the next base C by intersection of two rays 
 from A and B, a point should be chosen which, in addition 
 to advantages of commanding position, should form as 
 nearly as possible with AB an equilateral triangle ; the 
 reason being that in any triangulation the more acute the 
 angles, the less reliable must of necessity their intersections 
 be. and an equilateral triangle is the only one in which 
 no included angle is less than 60. 
 
 When primary points have thus been located all over 
 the map, the filling in of roads and other detail is done 
 either with the prismatic compass and passometer, as shown 
 in the example on p. 54, or, if more accuracy is required, 
 with compass and tape. This subsidiary survey is plotted 
 on tracing paper, and pricked through on to the map. This 
 method is preferable to traversing in the detail with the 
 plane-table itself as described on p. 41 because it saves 
 the map from getting soiled, and is an independent check' 
 (when the tape is used) upon the other work. 
 
Route- Survey ing 39 
 
 The main practical points to be observed in using the 
 table are to set it up firmly upon its tripod ; to level it per- 
 fectly true with a circular bubble ; to be very exact with 
 its setting in meridian ; to keep the pencil finely pointed ; 
 to draw the rays with the utmost nicety, and above all 
 things not to get hurried. 
 
 ROAD AND RAILWAY WORK 
 
 The work required for preliminary road and railway 
 survey resolves itself into two classes : the first a reconnais- 
 sance of such rapidity that even stadia measurements take 
 too long, and, except at intervals, there has to be as little 
 time as possible devoted to erecting, levelling, and adjusting 
 of instruments of any kind ; the second, a telemetric location 
 survey, which will give the levels sufficiently close to plot a 
 profile from them, and this is quicker and better done by 
 the transit-telemeter than by the plane-table. 
 
 The 'Verner' sketch-board is described on p. 318, to 
 which the reader's careful attention is directed; its use will 
 be first explained when accompanied by a prismatic compass 
 on reconnaissance, and then some few further illustrations 
 will be given of its more extended application when en- 
 larged and developed into the regular surveying plane-table 
 for the sake of those who may wish to make the most of it. 
 
 VARIATION OF THE COMPASS BY THE PLANE-TABLE 
 
 Before starting, the variation of the compass should be 
 ascertained either by an observation of the solar azimuth, as 
 explained on p. 132, or, failing suitable instruments for that 
 purpose, by equilinear shadows on the sketch-board, as shown 
 by Fig. 5. Erect the instrument on its wooden tripod, having 
 the roll of mapping paper tightened up into its place. Fix the 
 brass stile in the hole provided for it near the compass-box, 
 and adjust the board level and stile vertical by the clino- 
 
Preliminary Survey 
 
 S Q QS 
 
 meter or otherwise. Draw the centre line of the table across 
 the paper from headpiece to tailpiece, and ' set ' the table so 
 that the needle shall be at rest in line with this centre line. 
 
 Check once more the adjust- 
 ments by clinometer, and 
 the instrument is ready. 
 
 About an hour or two 
 before noon place a mark 
 with a pencil at the extremity 
 of the shadow cast by the 
 stile, and from the base of 
 the stile as a centre and with 
 a radius equal to the length 
 of the shadow describe a 
 circle right round the board. 
 When the sun begins to 
 drop again in the afternoon, 
 watch the shadow until 
 it once more touches the 
 circle. Mark it, and bisect 
 the chord drawn between 
 the two shadow-points in 
 the circle, and draw a line 
 to the base of the stile from 
 the centre of the chord. 
 This will be very nearly the 
 astronomical meridian, and 
 the angle between it and the centre-line will be the variation 
 of the compass. The error, as explained on p. 135, arises from 
 the altered declination of the sun during the lapsed time, 
 which varies from o to i angular minute per hour. The 
 error may at all times be neglected for this class of work. 
 When the weather is uncertain it is best to take two or three 
 points, say at 2 hours, ij hours, and i hour before the 
 meridional passage, in order to ensure getting the sun in the 
 afternoon again. 
 
 FIG. 5. 
 
Route- Surveying 1 4 1 
 
 SKETCHING AND PLANE-TABLING 
 
 The sketch-board is used upon two distinct principles. 
 When buckled on to the wrist, the correctness of the align- 
 ment depends on the little compass in the head-piece, checked 
 as much as possible by bearings taken with the prismatic com- 
 pass. No back and forward rays can, of course, be taken. The 
 board has to be ' set ' at every point of observation by the 
 'working meridian,' which is a line drawn across the compass- 
 box with an index at its end, by which the compass-box can 
 be turned in any direction. The circle round the compass is 
 graduated into divisions of ten degrees each, and the index is 
 placed in such a position that when the needle is under it the 
 board will be directed in the general line of route. When once 
 fixed the index is never touched unless the survey begins to run 
 off the paper. It is a fiducial line by which to adjust or ' set ' 
 the board whenever a sight is taken ; and considerable 
 practice is required to hold the board steady on the wrist 
 with the needle truly under the index whilst the sight is 
 being taken. It is also quite an awkward business to keep 
 the edge of an ordinary sight-rule or ruler at the station- 
 point whilst it is being rotated to take a sight. It was to 
 meet this difficulty that the needle-point sight-rule, described 
 in Chap. IX., was contrived, which has proved a great 
 convenience, and much better than merely sticking a needle 
 in the station. Most surveyors use elastic bands for keeping 
 the ruler in position, but a thin string with a spring under- 
 neath is neater. 
 
 The ' working meridian ' is fixed upon before starting 
 by means of any existing map, or, failing all such data, by 
 inspection of the ground. For instance, if a route-survey be 
 required from London to Birmingham we draw a pencil line 
 between the two cities upon an ordinary atlas, and, running 
 up the line to the intersection of a parallel, we find the 
 astronomical bearing to be roughly 50 N.W. Supposing 
 also that the variation of the compass has been just deter- 
 
42 Preliminary Survey 
 
 mined, as already described, to be 20 W., the magnetic 
 bearing of the air-line between the two cities will be 30 N. W. 
 This is called the c Line of Direction,' and is marked as such 
 on the headpiece, corresponding with the direction in which 
 the paper has to be fed forward upon the rollers. The words 
 * Line of correspond with a due northerly direction ; that is 
 to say, when the index is placed in line with the ' Line of 
 
 FIG. 6. 
 
 Direction,' and the needle is brought under it with its north 
 pole towards the words ' Line of,' the board will be held in 
 position for running due north. 
 
 We want to run a course of 30 N.W., and therefore fix 
 the index 30, that is, three divisions to the right of the words 
 * Line of,' so when the needle is brought to rest under the 
 index, the ' Line of Direction ' will point to Birmingham. 
 
 The width of the paper being ten inches, it will take in 
 
Route- Survey ing 43 
 
 20 miles on either side of the air-line to a scale of four miles 
 to the inch, and this would take in the whole map if we were 
 following one of the main highways. 
 
 If the line should run off the paper a ' cut-line ' must be 
 drawn and a fresh start made in the middle of the paper with 
 the same line of direction. This method of using the sketch- 
 board is illustrated by Fig. 6. The magnetic bearing of the 
 line of direction should be written up before commencing, 
 so that the index, if accidentally shifted, may be replaced. 
 
 The prismatic compass is of great use in checking the 
 angles taken in this manner with the sketching-board. The 
 bearings of at least the main lines of the traverse can be 
 taken, and whenever important intersections are wanted upon 
 salient points they should be likewise checked with the pris- 
 matic compass. It is quite possible to have too many instru- 
 ments, but a compass clinometer, as described in Chap. IX., 
 is no inconvenience and most valuable. However skilfully 
 the board is held in line, the slightest jog to the arm may, 
 imperceptibly to the sketcher, twist the table several degrees. 
 The value of the sketching-board is not lessened, but on the 
 contrary much enhanced, by assisting it with the prismatic 
 compass. The same amount of detail can be filled in, but 
 with increased accuracy. 
 
 SKETCHING WITH THE AID OF MAPS 
 
 Where existing maps of any kind are available, they 
 should be made all possible use of. An enlargement from 
 an atlas, however imperfect, should be laid down to scale, and 
 unless the map is thoroughly reliable, like an Ordnance map, 
 it is best to draw the enlargement in pencil, and to plot in 
 ink, or vice versa, so that the divergences may be at once 
 apparent. To copy detail from the existing map would only 
 confuse, but a check of alignment and distances is of great 
 assistance. 
 
 It frequently happens that during a rapid traverse a case 
 
44 Preliminary Survey 
 
 arises demanding more careful treatment than can be given 
 with the board on the wrist. A metallic telescoping tripod 
 is buckled to the bag of the sketch-board, or, when riding, is 
 attached to the saddle. It is only sixteen inches long when 
 shut up, and gives no trouble. By it the sketch-board can be 
 used as an ordinary plane-table, but being light and vibratory 
 it is only used in emergencies ; for continuous plane-tabling 
 the wooden tripod should always be employed. 
 
 In this form, the work can be done in fairly easy country 
 at the rate of twenty miles a day. It is something more 
 than a sketch and something less than a survey, but seeing 
 that it takes hardly any more time to a practised man than 
 if he simply took notes, it has the great advantage of graphic 
 representation over mere literary description. 
 
 ACCURATE PLANE-TABLING 
 
 The second method of using the sketch-board mounted 
 on its wooden tripod differs in no way from the ordinary 
 plane-table except that the instrument is smaller. 
 
 The principle of this kind of surveying is the geometrical 
 law of similar figures, by which when a single side is known 
 all the rest are determined by their positions in the figure. 
 It is a graphic triangulation resting on the same mathe- 
 matical theory as telemetry. The method of setting up the 
 table has been already explained. 
 
 The U.S. Geological and Coast Survey have covered 
 immense tracts of country with plane-table work and use 
 instruments of large size 24 inches by 30 is the maximum. 
 They are fitted with elaborate joints for levelling them true, 
 and furnished with sight-rules carrying stadia telescopes 
 with vertical arcs for measuring angles of inclination. The 
 figure on p. 42 of a traverse by the first method will also 
 serve for the second. The preliminaries are all the same, 
 the only difference being that the board is ' set ' at each new 
 station by a backsight on the previous station with the sight- 
 
Route- Surveying 
 
 45 
 
 rule. The precision attainable in this way is very great. 
 Check angles should still be taken with the prismatic compass, 
 especially where the bases are short, but the rays are gene- 
 rally more accurate than the compass-bearings, particularly 
 so if there is local attraction to the needle. The compass- 
 bearings, both with the needle in the headpiece (which is 
 set to the * working meridian ' precisely as before) and those 
 by the prismatic compass, are simply checks and nothing 
 more. 
 
 In taking a sight, the ray should be projected at the edge 
 of the paper about half an inch long ; it should be marked 
 with the signs of the back and fore station thus : A/B if 
 taken from one main station to another, or B/B t &c. if taken 
 from a main station to some outside point. The signs should 
 
 FIG. 7. 
 
 also be entered in a book of description (see p. 38) specify- 
 ing what the points represent, and accompanied by little 
 sketches, especially when the points are not very sharply 
 defined, such as distant villages, hill-peaks, or river-bends, 
 so that when arriving at the next station and taking the 
 backsights of intersection, the memory may be assisted as 
 to the precise spot viewed on from the last station. These 
 little sketches are sometimes made upon the map, but it is 
 not such a good plan, as they are liable to come in the way 
 of succeeding rays. 
 
 The flag on Fig. 6, p. 42, is shown as an out-station, but 
 the board can be moved to its vicinity and backsights taken 
 to all the previous stations on flag poles. The coincidence of 
 
46 Preliminary Survey 
 
 these rays with the foresight will prove the accuracy of the 
 work and close the traverse. 
 
 The whole principle of plane-tabling is here embodied. 
 The stadia work, levelling and contouring, which are added 
 by means of the accessory instruments to the larger tables, 
 do not form an essential part of the instrument itself, and 
 will be treated of under telemetry. 
 
 AUXILIARY PLANE-TABLING 
 
 The method of using the plane-table as an adjunct to 
 the tacheometer for filling in detail will be next described. 
 In Fig. 7 a tacheometer traverse is being carried along a 
 main road, and a building with enclosure is proposed to 
 be filled in with the plane-table, being situated upon a 
 cross road, and it being desired to avoid a deviation with 
 the tacheometer. A point E l is fixed in the vicinity of the 
 building by a sight with the tacheometer, if possible, within 
 100 feet of the further corners. The plane-table, worked by 
 an assistant, is then set up at B t , and aligned by a ray on B. 
 The line BB L is then laid down on the paper without any 
 reference to the compass. 
 
 The distance BBj is not absolutely necessary, but, being 
 known, it is better to plot it, and so locate B on the paper 
 to enable a checksight to be taken upon it if the table has 
 to be moved to another sub- station. If all the corners are 
 within 100 feet, they can be taped without moving the 
 table and laid off to scale on rays taken with the sight-rule. 
 If the table has to be moved to another sub-station in order 
 to command the whole of the detail, it can be triangulated 
 as already described, but it is generally quicker to locate 
 several points with the tacheometer as plane-table stations 
 than to sub-triangulate with the plane-table. 
 
 LOCATION OF THE INSTRUMENT 
 
 The subject must not be dismissed without touching 
 upon the well-known three-point problem for locating the 
 
Route- Survey ing 47 
 
 instrument, although it should not be used when a more 
 direct method is possible. 
 
 It frequently happens that, when surveying with the aid 
 of a map, it becomes necessary to determine the position 
 of the instrument on the ground from one or more points 
 indicated on the map and transferred from it to the plot. 
 If only one or two points are known, the problem cannot be 
 solved without the compass. 
 
 Case i. When one point is known. Maps are generally 
 constructed having the astronomical north at the top of the 
 sheet. Ordnance maps and ordinary atlases are made thus, 
 but not so the parish maps. When this is not the case, the 
 
 .- 
 
 position of the astronomical or magnetic meridian or both 
 must first be determined on the ground from the known 
 point in the manner already described. 
 
 When the geographical alignment of the map is known, 
 and the variation of the compass has been ascertained, the 
 magnetic meridian should be marked on the map at the 
 known point, a bearing taken to it with the prismatic 
 compass, and plotted backwards from it, i.e. 1 80 bearing. 
 A base line is then run at right angles if possible, but if the 
 ground will not admit of it, as nearly square as possible, and 
 the bearing taken. The base line is measured, and from 
 its extremity another bearing is taken to the known point. 
 
48 Preliminary Survey 
 
 Then laying down the two bearings from the point towards 
 the observer's position on the plot, take the measured 
 distance by scale and place it with the dividers, in its proper 
 bearing, where the extremities will coincide with the two rays 
 from the known point. If the base cannot be run square, the 
 bearing of the base has to be run out with a parallel ruler. 
 It is also convenient to lay off the base in an even number 
 of feet or yards as 100 or 1,000, for then the distance can 
 be read off from a table of tangents. 
 
 Case 2. When two points are known, the prismatic 
 compass is used in the same way, but a base is dispensed 
 with. The two known points are plotted from the map in 
 their proper position upon the plane-table, their bearings 
 
 A 
 
 'a 
 
 FIG. 9. 
 
 taken and plotted backwards as before from magnetic 
 meridians drawn through the known points. Their inter- 
 section is the locus of the instrument. 
 
 Case 3. When three points are known the plane-table 
 can be located by the sight-rule alone. 
 
 Let A, B, C be the three known points and #, 6, c their 
 position on the plot. It is required to locate upon the 
 plot the point of observation and to set the table so that 
 the plot shall have its lines parallel to those in nature, or, as 
 it is termed, the table be ' in meridian.' It is obvious that 
 rays through A, B^, or Cc will intersect somewhere, and 
 by Euc. vi. 2 we know that in any triangle, #AB, #BC, 
 or rAC, when the sides AB, ab ; AC, ac ; BC, be, are 
 proportional they are parallel. Therefore, if by adjustment 
 
Route- Surveying 
 
 49 
 
 of the table we find the position in which the three rays Aa, 
 B<, Gr, intersect in one point, the plot will be parallel and 
 the table in meridian, and the point of observation correctly 
 located on the plot. 
 
 When the instrument is not c in meridian,' a ' triangle of 
 error ' is formed as shown on the figure, the elimination of 
 which adjusts the table. There is one exception, when the 
 
 of error 
 
 FIG. 10. 
 
 point of observation is situated in the circumference of a 
 circle described about the three known points, the triangle 
 of error disappears for any position in that arc. This is at 
 once seen by being able to rotate the table without causing 
 any error, and some other point must be chosen from which 
 when located the required point can be determined. See 
 more on the three-point problem by station-pointer in 
 Chapter IX. 
 
 THE PLANE-TABLE AS A RANGE-FINDER 
 
 Fig. 8, p. 47, illustrates the principle of range-finding 
 described on p. 338. The plane-table will, carefully handled, 
 determine this range or distance of an object with as great 
 accuracy as with some optical range-finders. Military engi- 
 neers do not use it for this purpose because the enemy is 
 fond of making a target of it. 
 
 E 
 
5o Preliminary Survey 
 
 MEASUREMENTS OF DISTANCE 
 
 Setting aside the chain, the methods available for deter- 
 mining the distances upon route survey are of three kinds. 
 Firs^ mere judging by the eye and hand, for which some data 
 will be presently given of value to the sketcher. Secondly ', the 
 ancient method of pace-counting, to which may be added 
 the trocheometer for wheeled vehicles, the time-measure- 
 ment of horses' gait, and the patent-log records combined 
 with time measurement when steaming on a river. 
 Thirdly ', the telemetric or optical measurement of distance, 
 which, including the subject of military range-finding, is 
 treated of in the chapter on Tacheometry, also in the chapter 
 on Instruments. It belongs in its practice more to the 
 route survey, though in principle it is to be classed with 
 tacheometry. 
 
 First. Judging by the eye is a faculty which is an im- 
 portant part of the capabilities of the sketcher. A good 
 shot will sometimes come pretty near the distance by look- 
 ing at it as a range for a fowling-piece ; a cricketer by the 
 distance he can throw a ball. By holding the hand across 
 the field of view at arm's length, so as to cover the height of 
 a man, a horse, or a man on horseback either with the palm 
 or with one or more fingers, we have a measurement some- 
 what on the stadia principle and of some assistance in 
 guessing. 
 
 The second method, of pace-counting &c., may be 
 brought to a very fair degree of accuracy for the purpose 
 of reconnaissance, and we will first touch upon the 
 passometer or pace-measurer. It is in appearance like a 
 watch, the mechanism consisting of an escapement carried 
 by a loaded lever, which is shaken by the shock of the 
 step and returns by means of a light spring ; the escape- 
 ment actuates a train of wheelwork, and moves an index on 
 the dial to record either the number of paces or the actual 
 
Route- Surveying 51 
 
 mileage. If the latter, an adjustment is provided to make 
 the index read correctly to any given length of pace and 
 instrument is called a pedometer. For long bases, the 
 mileage indicator is as useful as the other; but for short 
 distances, and varying inclinations, the counting index is 
 preferable as no instrumental adjustment is needed, but 
 scales of paces are drawn upon the board to their ascer- 
 tained value under different conditions of travel, with their 
 respective designations. The needle-point sight-rule is 
 provided with a broad chamfered edge upon which, when 
 using only one or two scales of paces, they can be 
 gummed with stamp-margins so as to have the working- 
 scale at all times on the paper and avoid dividers. This is 
 dispensed with by using the Mannheim slide-rule as sight- 
 rule and scale : see p. 245. 
 
 The passometer does nothing more than count the 
 paces ; the accuracy of the measurement depends upon the 
 regularity with which the walker can pace. A course of 
 training in this is indispensable, but can be easily gained 
 during times of recreation. The chief points are to walk 
 naturally without trying to step yards or any other specified dis- 
 tance, to hold the body erect, and to maintain the same speed. 
 
 The best way to arrive at reliable data of pacing is to 
 walk over a piece of road where the mile-stones are cor- 
 rectly indicated. A piece of railway will do very well for 
 flat walking if the sleepers and ballast are avoided, keeping 
 to the side of the bank or cutting. A turnpike road is 
 better. For hill walking a six-inch Ordnance map will give 
 the contour lines crossing the public roads, from which the 
 gradients can be calculated. 
 
 The time of one's walk has a great deal to do with the 
 length of step. By educating oneself into the same rate of 
 time uphill or downhill, fresh or tired the length of 
 pace will be much more uniform than it is ordinarily. 
 Gradients as steep as i in 40 do not then make any differ- 
 ence on the average length. 
 
 E 2 
 
Preliminary Survey 
 
 The following trials were made up and down a road 
 varying from i in 20 to level, but all uphill one way, and 
 down the other. 
 
 
 
 Dis- 
 tance, 
 feet 
 
 Number of paces 
 
 Value in feet of 
 100 paces 
 
 Uphill 
 A to B . 
 B to C . 
 C to D . 
 
 2,900 
 2,080 
 820 
 
 ist time 
 1,091 
 
 783 
 284 
 
 and time 
 
 1,161 
 
 705 
 285 
 
 3rd time 
 ^IS 2 
 
 
 
 ist time 
 266 
 266 
 289 
 
 2nd time 
 268 
 
 295 
 
 287 
 
 3rd time 
 2 5 6 
 285 
 291 
 
 Total 
 
 5,800 
 
 2,158 
 
 2,151 
 
 2,144 
 
 284 
 
 741 
 1,163 
 
 269 
 
 269-5 
 
 271 
 
 Downhill 
 
 D to C . 
 C to B . 
 
 B to A . 
 
 820 
 2,080 
 2,900 
 
 286 
 726 
 1,156 
 
 296 
 725 
 
 1,121 
 
 287 
 287 
 251 
 
 277 
 287 
 26l 
 
 288 
 28l 
 249 
 
 Total 
 
 5,800 
 
 2,168 
 
 2,142 
 
 2,188 
 
 267 
 
 270-5 
 
 26 S 
 
 Average of the three times . 
 Range of one time from average . 
 Maximum range of shortest distance) 
 from average .... I 
 
 2-687 fe et per pace 
 i -4 per cent. 
 
 10-9 per cent. 
 
 These trials were under unfavourable conditions as 
 regards gradient, and are given to show results attainable by 
 an unpractised walker. See also closing error on p. 55. 
 They demonstrate very clearly the tendency of pacing 
 towards uniformity over long distances even when there may 
 be great variations over short lengths. 
 
 The distance was only a little over a mile, but this fact 
 becomes more apparent on daily journeys of ten to twenty 
 miles, in which the total error can easily be kept within 
 from i to 2 per cent. 
 
 When particular exactness is needed, and an assistant 
 is present, it is advisable every two or three miles to check 
 the rate by taping a stretch of 300 paces or so, in order to 
 detect changes due to fatigue, rough or slippery roads, c. 
 
Route- Surveying 5 3 
 
 A series of scales may be constructed for use on various 
 gradients, but it is less confusing to work to one scale and 
 make a marginal note in the fieldbook to guide in making 
 the corrections when plotting, or calculating the latitude 
 and departure. 
 
 The plotting may be done by the protractor, but the 
 principle of working to latitude and departure is more exact, 
 and the calculations are done as quickly with the slide-rule 
 as if the angles were laid off with the protractor. There is 
 a harmony, moreover, between this process and the daily 
 astronomical observations, both being a reference to rect- 
 angular co-ordinates. 
 
 The analogy of traversing with navigation should be 
 thoroughly studied ; even down to curve-ranging as will be 
 shown later on. In the traverse for route-survey the work 
 is nothing more than the dead-reckoning ; only instead of 
 suffering from liability to error in under-currents, slip of 
 screw, and what not, it is troubled with magnetic aberra- 
 tion, irregularities of pace, and 'personal error.' The 
 astronomical observation comes in to help out the land 
 surveyor with the addition of the frequent sighting of 
 landmarks whose geographical position is known, and the 
 number of which becomes every year greater and greater. 
 The astronomical work is sometimes performed with the 
 Hadley's sextant, but the surveyor finds a greater range of 
 usefulness in the transit theodolite. 
 
 SURVEYING WITH THE SEXTANT 
 
 The Hadley's sextant is a favourite instrument with 
 travellers, who learn its use from one of the ship's officers 
 when getting to their destination, and then employ it for 
 traversing on land. It will not take angles any more 
 correctly than they can be plotted direct upon a plane- 
 table ; it is necessary to correct the angles when they are 
 taken between points at considerable difference of level, 
 whereas the plane-table gives the horizontal projection at 
 
54 
 
 Preliminary Survey 
 
 once. It is much slower in sighting, and needs careful 
 sketches and entries in the fieldbook to avoid mistakes. 
 Its great advantage is its portability, and an immense deal 
 of good work may be done with it, but only the principles 
 of adjusting it are given in the chapter on instruments, its 
 use being very simple. 
 
 CLOSED PASSOMETER TRAVERSE 
 
 In order to exhibit the degree of accuracy attainable 
 with the passometer and prismatic compass alone, the closed 
 traverse illustrated by Figs. IT, 12, was made in the course of 
 
 Scale of Feet 
 
 1000 
 
 3000 R- 
 
 FIG. ii. 
 
 daily walks and visits to friends, and without making correc- 
 tions for sloping ground. It has to be remembered that 
 not only does the pace vary in length according to the slope, 
 
Route- Surveying 
 
 55 
 
 but a deduction has to be made for the horizontal projec- 
 tion of the distance. As going uphill tends to shorten the 
 step, and increase the number of them to the mile, the 
 error is aggravated by the projection. Going downhill it is 
 diminished. The table for deductions due to projection is 
 
 L - ~^\\Ciosing error 25 /& JUICES . 
 
 FIG. 12. 
 
 given in the chapter on chaining, but the pedestrian must 
 make his own table of pace-variation from actual experi- 
 ment. 
 
 The roads of Sevenoaks are both hilly and tortuous, and 
 therefore represent an unfavourable case for a route-survey. 
 
56 Preliminary Survey 
 
 Hardly any check angles could be taken on account of 
 the obstructions to view. It will serve to show a degree 
 of accuracy which can be at least equalled in more favour- 
 able situations. The 'closing error' of 25 J- paces has been 
 purposely left in the plot, and the process by which it should 
 be distributed over the bases is placed upon a separate 
 figure. The closing error does not represent the maximum 
 error j it is an average one, arising in large measure from 
 the sloping ground ; but there is also a slight twist in the 
 plot, so that in one place where both pace and angle error 
 assist one another, there is a divergence of fifty paces from 
 the truth in the neighbourhood of one of the cross roads. 
 The total closing error amounts to J per cent, of the 
 periphery, the maximum error to about \\ per cent. On 
 a day's march of twenty miles, the error at the next solar 
 observation would be barely detected ; but as each observa- 
 tion is independent, cumulative error of pace-measurement 
 is removed within the instrumental limits of about \ mile of 
 latitude, or, supposing an exact chronometer, of longitude 
 either, but failing the chronometer, the error in longitude 
 may be much more, as already explained in Chapter I. 
 
 DISTRIBUTION OF CLOSING ERROR 
 
 This must not be confounded with fudging, which means 
 correcting by guesswork. Distribution of error is to a great 
 extent dissipation of its amount all over the plot. The 
 process is almost self-explanatory on Fig. 12. The total 
 periphery is laid out on a straight line marking each station, 
 and an ordinate is laid off at the extremity equal to the 
 closing error. The extremity of this ordinate being con- 
 nected with the other end of the line, a triangle is formed the 
 ordinates to which at every station represent (on the assump- 
 tion of the error being gradually cumulative) the correction 
 to be applied at each point in a direction parallel to that of 
 the line between the two divergent ends of the traverse. 
 
Route- Survey ing 57 
 
 Where the error is small in proportion to the total periphery 
 this process reduces it to an unscaleable quantity on any 
 single base line. 
 
 SCALE OF PACES 
 
 The construction of a scale of paces is as follows. Let 
 us suppose that we wish to produce a map upon a scale of 
 six inches to the mile. A chained base of 1,000 feet is paced 
 and repaced until the average has been found to be for 
 instance 357 paces. We then lay down our scale of miles, 
 say three inches for half a mile, in furlongs and chains, and 
 calculate by slide-rule the value of 1,000 paces in furlongs 
 and chains. 1 
 
 Taking this amount from the mile scale we lay it down 
 as our scale of 1,000 paces and subdivide it as follows : 
 (Fig. 1 3, p. 59). From one end of it, we erect a perpendicular, 
 and selecting some convenient decimal boxwood scale such 
 as a i oft. to the inch, we adjust it so that one end is at the 
 extremity of the pace-scale, and some multiple of ten (in this 
 case the 40) on the perpendicular ; we then draw a line form- 
 ing the hypotenuse of a triangle and tick off every four 
 divisions of the boxwood scale so that we have subdivided 
 our hypotenuse into ten equal parts. All we have to do 
 
 1 The two proportions are as follows : 
 
 Let x be the number of paces in a mile, and y the number of chains in 
 i ,000 paces. 
 
 1,000 : 5,280 :: 357 : x; or* - 357 x 5,280 
 
 x: 80:: 1,000 : y- Q iy = .to_2LL. 
 
 Using the lower scales of rule and slide. Place the right-hand i of 
 the slide over the 528 of the" rule. Place the brass marker at the 357 
 on the slide. Without displacing the marker, bring the 80 of the slide 
 to the marker and read off the result, 42 ch. 44 links, on the slide oppo- 
 site the left-hand i of the rule. 
 
a 
 * 
 
 C/3 <? 
 
 Is 
 
 OJOL 
 
 SDUBJSIp IBJOJ, 
 J3J3UIOUJI3 I 
 
 ssBdtuo'5 
 
 
 jug piojauy 
 
 . , 
 
 8J 
 
 UOJJBJS jo 
 
Route- Surveying 
 
 Scale of MileS;One-Stx inches 
 
 iMile 
 
 CH 
 
 > 
 
 
 
 
 1 
 
 
 
 
 *H 
 
 Scale of Paces . 1000-4? 44 Chains 
 
 
 
 00 * 
 
 500 
 
 '* 
 
 oa fttj.9 
 
 < ' 
 
 _^ 
 
 
 0' 
 
 I t 
 
 ""--., 
 
 
 
 \ f 
 
 ''--. 
 
 
 
 \ t 
 
 " ~ -i. __ 
 
 
 
 J / 
 
 ' 4 
 
 
 
 ) 
 
 ^ ~ - M 
 
 
 
 S; 
 
 "*--.., 
 
 
 
 ! ; 
 
 
 
 
 SCALES TO FIG. 13. 
 
 then is to rule down perpendiculars to the pace-scale and it 
 will be divided into spaces of 100 paces each. See also 
 direct scaling by slide-rule, p. 245. 
 
 FIELDBOOK 
 
 The fieldbook, on opposite page, requires but a very brief 
 explanation. The backsights are taken to equalise minute 
 errors of observation, to locate and remove them when 
 important, and to detect magnetic deviation. 
 
 To reduce the meridional bearings to azimuths from 
 north or south point, the following table may be used. 
 
 TABLE XVII. Reduction of Azimuths 
 From o to 90, azim. = bearing unaltered : N. E. 
 From 90 to 180, azim. = 180 bearing: S. E. 
 From 1 80 to 270, azim. = bearing- 1 80 : S.W. 
 From 270 to 360, azim. = bearing 360 : N. W. 
 [The author's pocket altazimuth has both graduations. See Chapter IX. ] 
 
 The error at closing is seen to be 2 2 paces to the north 
 and 13 to the west. We will represent it thus '\T7\ Then 
 
 (i) to find the direction and magnitude of the line itself we 
 have tangent angle d = ^. By slide-rule as before = '591. 
 
6o Preliminary Survey 
 
 Keeping the brass marker at the '591 we shift the tangent 
 scale to its initial position and find under the marker the 
 angle 30 35' N. W. 
 
 (2) To find the length of the closing error L= v/22 2 +i3' 2 
 For 22 2 bring the brass marker to 22 on the lower scale of 
 this rule, the upper index will then correspond with 484 on 
 the upper scale. Similarly with the lower index at 1 3, the 
 square 169 is found on the upper scale. Adding the two 
 together =65 3 ; direct the upper index to 653 on the upper 
 scale and the square root 25-5 is read off from the lower. 
 It will be seen that involution and evolution are performed 
 by simple inspection without using the slide, and this forms 
 one of the most important uses of the slide-rule. l 
 
 (3) To get latitude and departure. Place the right hand 
 extremity of the sine-scale of the slide under the distance, 
 and read off the latitude from the rule above the com- 
 plement of the angle, and the departure opposite the 
 angle itself. Thus in the first entry the distance=7o. Place 
 the extremity of the sine-scale opposite a 7 on the upper 
 scale of the rule. The reduced bearing is 19!, of which 
 the complement is 70^ ; opposite these two angles on the 
 slide we shall find the departure and latitude respectively. 
 
 Check-sights are very useful in such work as this to 
 correct twists The house shown on the plot was filled in 
 entirely by angles. When engaged in filling in new roads 
 to an old but accurately triangulated map such as an old 
 Ordnance Survey the errors are localised by first plotting the 
 work in the usual way and then superimposing a tracing of 
 it upon the Ordnance Map ; the errors of the new work 
 are thus narrowed within the limits of the nearest reliable 
 points and the whole made very nearly as correct as the rest 
 of the Ordnance Map. 
 
 1 This operation can be also performed by placing the sine-scale 
 with the angle 30 35' under the 13 of the rule, and the answer will 
 be found opposite the right-hand I of the slide. 
 
Route- Surveying 6 1 
 
 PROFILE 
 
 The profile or section is produced from readings of the 
 aneroid barometer at every station. The distances are laid 
 off and the heights ruled up as in ordinary levelling. In the 
 case of a closed traverse, there will generally be a ' closing 
 error' of levels which has to be equalised or distributed 
 similarly to the closing error of the traverse. Some of the 
 sources of error are explained in the chapter on instruments, 
 but they are usually cumulative and approximately uniform. 
 They can only be treated as such, and the total periphery of 
 the base being laid off upon a horizontal line representing 
 the true datum, the amount of the closing error is laid off 
 on a perpendicular at its extremity either above or below 
 according as the last reading is less or greater than it ought 
 to be. A ' false datum ' is then drawn from the starting 
 point at the end of the perpendicular, and the levels are scaled 
 from the false datum, which is afterwards erased. 
 
 CONTOURS 
 
 The contours are laid on by the pocket-altazimuth, check- 
 angles being taken along the bases and in other directions. 
 The pocket altazimuth is fully explained in Chapter IX. 
 When we know the elevation of the point of observation 
 and the slope of the ground in any direction we can plot the 
 contours from the cotangent of the angle (that is the tangent 
 of the complement) which represents the horizontal distance, 
 corresponding to one foot of difference of level. Thus let 
 the slope be 10 of depression. To find cot 10. Place 
 the tangent-scale in its initial position and brass marker at 
 10. Reverse the slide and place the right-hand i of the 
 slide at the index. Read the answer 5-67 on the slide 
 opposite the left-hand i of the rule. If we want contours 
 at every 10 feet the horizontal equivalent will be ten times 
 this, i.e. 56*7 feet. Inasmuch as the aneroid readings are in 
 
62 
 
 Preliminary Survey 
 
 feet it is best to have a feet scale upon the plot as well as 
 the two already mentioned for setting out the contours. 
 
 This is by no means the only way of contouring. The 
 contours of the Ordnance Survey are taken with the level, 
 and dots are placed on the map where the staff was held. 
 They are either plotted from cross sections or from field 
 
 400 
 
 30Q 
 
 200 
 
 IQOO 675 320300 oFeet DatuMWFttabotvOniJJat.- 
 
 _StatK>n4 32 \ For Plan see Fig. II 
 
 froftle of &John*s Road 
 Jfor : Scale 6 Inches =On& MIL& 
 Vert, Scale 1O0F*-OneMite 
 
 FIG. 14 
 
 tracings on which the location of the level points are estab- 
 lished by tape measurements from hedges, buildings, &c. 
 The cross section is the most laborious and hardly the more 
 exact method of the two. If the contours are needed before 
 the plan is plotted, it is unavoidable, but when the principal 
 contours are at a hundred or even fifty feet interval it be- 
 
Route- Surveying 6 3 
 
 comes a very tedious operation in hilly country. By the 
 second process, the level is kept at nearly the same collima- 
 tion going round the hill until it comes round to the same 
 point again, or leaves the region of the survey. Telemetric 
 contouring with the level (for which see p. 197) is suffi- 
 ciently accurate for all ordinary purposes and is indepen- 
 dent of any plan. Hill-sketching is often done in the 
 form of contours by the eye, instead of hachures, which 
 convey but little idea of the topography ; such contours by 
 being close or far apart show at once the relative steepness 
 or flatness of the ground. 
 
 The examples of profile and contouring given, Figs. 14, 
 15, and 16, with the fieldbook are, one from the traverse, 
 Fig. n, the other from a walk through Knole Park. The 
 instruments used were the aneroid barometer, pocket 
 altazimuth, and passometer. 
 
 The aneroid was first examined between Ordnance bench- 
 marks, with results as follows : 
 
 Ordnance Aneroid 
 
 Ordnance Benchmark at ' The Vine ' to 
 
 Ordnance Benchmark Railway Bridge 
 
 London, Chatham, and Dover Railway 187-8 185 
 
 Ordnance Benchmark at ' The Vine ' to 
 
 Ordnance Benchmark Railway Tavern 2OO'8 205 
 
 Ordnance Benchmark at Railway Bridge 
 
 to Ordnance Benchmark at Railway 
 
 Tavern 13-8 20 
 
 On the profile, the discrepancies between the aneroid 
 readings and the elevations calculated by the altazimuth 
 were so small that they were not distributed. In the con- 
 touring, on the other hand, the aneroid had to be used with 
 less time allowance for settling, and needed considerable 
 correction from the altazimuth. The profile was taken 
 whilst walking up the hill with a friend without detaining 
 him beyond two or three minutes. The only entries made 
 in the fieldbook at the time were the station column, 
 vertical angle, pedometer, aneroid, and remarks. 
 
6 4 
 
 Preliminary Survey 
 
 If a cumulative error of the aneroid is discovered from 
 benchmarks, or from returning to the starting point on a 
 closed traverse, it must be eliminated as already described 
 before entering its readings in the column provided for the 
 purpose. When in the field its readings can be entered in 
 
 Feet Horizontal 
 
 iqoo ' . w 
 
 
 Feet Vertical 
 
 FIG. 16. 
 
 the column for remarks. When this is done, the aneroid 
 levels rule the profile ; the altazimuth angles are only relied 
 upon for the portion of profile between the aneroid 
 readings. 
 
 The advantage of using a clinometer or altazimuth in 
 conjunction with thejaneroid, especially in the form recom- 
 

 
 
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66 Preliminary Survey 
 
 mended in Chapter IX. is, first, rapidity, and secondly, accu- 
 racy of detail. The aneroid could be used throughout, but 
 is not reliable for small differences of level ; for instance, on 
 a stretch of almost dead level, the aneroid might show jumps 
 of a few feet here or there which did not exist at all, but the 
 altazimuth is graduated to show a difference in vertical 
 angle of TO', which equals i foot in 333. On the other 
 hand, the clinometer angles on a long stretch will soon 
 run into a very serious error if unchecked by the aneroid. 
 For more on the aneroid see Chapter IX. 
 
 The use of the slide-rule for the calculations cannot be 
 over-estimated, it is simply invaluable. To begin with the 
 scales. Tt is tedious enough to transfer with dividers from 
 a scale of paces to a scale of feet, but when it comes to 
 three or more scales of paces for uphill, level, and downhill 
 at varying inclinations, the work would be interminable, 
 and recourse has to be had to percentages of addition or 
 deduction as already described. But with the slide-rule it 
 simply means a gentle tap upon the slide, and the new scale 
 is there ; no calculation whatever is needed. Take, for 
 instance, the scale for the slope of St. John's Road, on which 
 533 paces measured 1,605 ^ eet - We use tne upper scales of 
 the slide and rule, and bring the 533 of the slide to the 
 1,605 of the rule ; the distances in feet corresponding to any 
 number of paces can then be read off directly from the 
 slide. If then the rate alters to 543 paces to 1,611 feet, we 
 have nothing more to do than to slide forward to that ratio. 
 For rise and fall the operation is almost instantaneous, being 
 analogous to that for latitude and departure (see Chapter V. 
 P- I73-) 
 
 CONTOURING WITH POCKET ALTAZIMUTH AND ANEROID 
 
 The accompanying diagram of contouring (on p. 64) will 
 illustrate first of all the adjustment of aneroid error already 
 alluded to. The traverse was made with the passometer 
 precisely as on Fig. 1 2, but the correction of the closing error 
 
Route- Surveying 67 
 
 was done by the slide-rule, and as this is by far the most 
 satisfactory method, as well as much quicker, it will now be 
 explained. 
 
 The northings and southings being added up as before, 
 the difference of latitude amounted to 1,812 feet south and 
 1,764 feet north. Placing the difference of the two, 48, on 
 the upper scale of the rule opposite to 1,812 on the slide, 
 the correction for the first of the northings, 141, on the base 
 9-10 will be given on the scale opposite 141 on the slide, 
 and equals 37. 
 
 .The departures being corrected in the same way, the 
 plotting of the traverse is proceeded with as before, by 
 rectangular co-ordinates from a N. and S. line, and an E. and 
 W. line, the total latitude or departure being scaled for each 
 station from the starting point afresh, so as to avoid the 
 errors certain to arise in scaling from point to point. 
 
 When the traverse is complete, the total periphery is laid 
 out on a horizontal base as before described. It was found 
 in this case that on returning to station 3, where the traverse 
 closed, there was a discrepancy of fifty feet in the aneroid 
 readings, arising from change in the weather (see Fig. 16). 
 
 As the instrument read too low at closing, the error was 
 laid off above the true datum, and the aneroid levels scaled 
 up from the false datum. The profile from the aneroid is 
 shown by a dotted line. The difference of level by clino- 
 meter angles was then worked out by slide-rule as before, 
 and came to a total difference of 14 feet more than the 
 aneroid. The rises and falls were then reduced in the same 
 way as the errors of latitude and departure by the slide-rule. 
 The maximum difference of level existed at the two ex- 
 tremities of the work, and amounted by the aneroid to 80 
 feet. This was checked by a second visit With the aneroid 
 to the two extreme points, and found to be correct. The 
 rises and falls were then squared with this total by placing 
 the total error of 14 feet in ratio on the slide-rul^ with 80, 
 and the error subdivided over the whole traverse propor- 
 
68 Preliminary Survey 
 
 tionally to each rise or fall. The full line on the profile 
 shows the final correction, and the erratic nature of aneroid 
 readings, which is never got rid of even in the most delicate 
 instruments. The line 10-11 for instance is shown by the 
 aneroid as an ascent, whereas it was really a descent, and if 
 it had not been for the altazimuth would not have been de- 
 tected. To be perfectly sure of the main difference of level 
 by the aneroid, the surveyor should repeat his visits to the 
 two principal positions several times. On extended survey he 
 should have several aneroids, and despatch an assistant on 
 horseback with an aneroid to go to and fro between two 
 objective points until a fair average is obtained. When 
 long base lines are measured by the passometer, say over 
 400 paces, the clinometer angles are of little value, and the 
 aneroid alone has to be followed. 
 
 The contours are plotted in two different ways : First, 
 on lines the extremities of which are known in elevation ; 
 secondly, on lines of which the slope alone is known by the 
 clinometer. 
 
 Let us take the base-line 3-4, the ends of which were 
 found to be 393 feet and 404 feet high respectively. The 
 contours being required at every 5 feet, we want to know 
 where the 395 and 400 contours cross the line. The distance 
 was 369 feet, and the difference of level 1 1 feet. Placing the 
 1 1 on the rule opposite the 369 on the slide, we look for 2 
 and 7 on the rule, and find opposite to them on the slide their 
 horizontal equivalents, namely 67 and 235, which we tick off 
 upon line 3-4, measured from 3, and so on. This plan is 
 adopted for all the base lines of the traverse, and also for any 
 points which have been triangulated. In the present case a 
 series of points a, b, c, d, e, &c., were determined along the 
 rising ground in the middle of the plot by intersecting com- 
 pass-rays, and the slope taken with the clinometer. The 
 distances were then scaled from the plan, and the difference 
 of level calculated as before with the slide -rule. The points 
 were all trees of sufficient prominence to be identified from 
 
Route- Surveying 69 
 
 successive stations, but these are among the least eligible of 
 points, being badly defined. Low cottages, whose height 
 above the ground varies but little, are amongst the best 
 points to choose. The contours were then filled in on the 
 rays from the primary stations to these points, and also on 
 lines connecting the points themselves. 
 
 It is convenient to tick off the distance on a slip of 
 paper, marking each tick with its elevation ; then apply the 
 distance to the scale, and mark off from it the distances of 
 the contours without using dividers. 
 
 The second method of plotting contours, on lines of 
 which the direction and slope alone are determined by the 
 altazimuth, is for the outlying ground in order to show the 
 trend of the contours after they leave the closed traverse. 
 The process is as simple with the slide-rule as the former 
 one. We take a bearing in the direction in which we desire 
 the slope (sometimes we require several from the one station), 
 using either a tree or similar mark, which we can describe 
 for identification in the fieldbook, or else send out a flag. 
 
 We then take a vertical angle to the same point and 
 book it likewise. 
 
 When we have plotted the rest of the traverse and con- 
 tours between fixed points we lay off radial lines from the 
 respective stations in the directions of the independent com- 
 pass-shots. 
 
 Then with the slide-rule we place the scale of tangents 
 under the upper scale of the rule, and note the percentage 
 of the slope (i.e. 100 times the tangent). Thus if the slope 
 be 3 25' we find the percentage to be 5 "97, which is the 
 same thing as the tabular tangent of '0597 to a radius i. 
 It is also the same thing as saying that on that slope every 
 horizontal stretch of 100 feet has a rise of 5*97 feet. We 
 then reverse the slide to show the scale of numbers, and, 
 placing the 5-97 of the rule opposite the 100 on the slide, 
 we can read off the horizontal equivalents for each difference 
 of level. Thus, supposing the elevation of the point from 
 
70 Preliminary Survey 
 
 which the line was taken was 473, and the slope +, we 
 should want for 5ft. contours horizontal equivalents for 
 2ft., 7ft., 1 2ft., and so on, which we find at 33, 117, and 
 201. They should be plotted on with the paper-slip from 
 the scale as before. 
 
 When a slide-rule is not available, the foregoing expla- 
 nation will serve equally well for the use of tables, which is 
 of course much more lengthy. A set of scales of horizontal 
 equivalents should be made for every degree of slope. To 
 save pricking off odd distances, the scales are constructed 
 from the tabular cotangents of the slopes which are the 
 horizontal equivalents to a difference of level of i foot. 
 They can also be laid off graphically by erecting a perpen- 
 dicular scale at one end of the profile in the manner shown 
 on Fig. 1 6, p. 64, and drawing horizontals cutting the base- 
 lines at the required elevations. 
 
 DISTANCE-MEASUREMENT OTHER THAN ON FOOT. 
 
 Hitherto only pedestrian operations have been dwelt 
 upon, and for several reasons the surveyor should keep to 
 his feet where he can. He is more independent and can 
 better give his attention to his work when he has no animal 
 to look after ; he can use the sketch-board as a plane-table, 
 and take steadier sights with the prismatic compass. But 
 it often happens that he has to ride and to depend on his 
 animal for the measurement of his distances. It is better to 
 have a horse who walks well than a fast one. A well-trained 
 cavalry horse will, according to Captain Verner, R.A., walk 
 with much greater regularity than a man. Horses will fall 
 into an even pace much better when two are together. The 
 walking gait is measured over a chained base. The trot is 
 counted similarly ; each rise in the saddle being a ' trot.' 
 As a rough approximation three yards may be counted to 
 each ' trot.' A canter requires exceptional horsemanship to 
 be used to any extent for distance-measurement. 
 
Route- Survey ing 7 1 
 
 When steaming on a river, the speed is measured by 
 revolutions of the screw or paddle, by the patent log, or by 
 time only. In any case the speed of the current has first to 
 be measured before starting, and periodically afterwards. 
 A good method is by patent log combined with timing for 
 short distances. 
 
 Mr. George Kilgour, M.I.C.E., made a survey for 200 
 miles in length with a steam launch between the first and 
 second cataracts of the Nile in five days, for the Soudan 
 Railway. He adopted time-measurement of speed, for which 
 he had three scales, full speed, half speed, and dead slow ; 
 the value of the scales was determined by an accurately 
 measured base on the centre line of the vessel, at the two 
 ends of which, near the bow and stern, two lines of sight 
 were placed square with the centre line, by which the time 
 was registered in which the vessel passed some well-defined 
 point on the bank, such as a cocoanut tree, at the three 
 rates of speed, The survey was made from the deck with 
 a plane-table kept constantly in the meridian by the com- 
 pass, and astronomical observations were taken at night. 
 
 DISTANCE-MEASUREMENT BY THE RANGE-FINDER. 
 
 The most recent improvements in this class of instru- 
 ment are described in Chapter IX. but a few further 
 remarks will here be made upon their use. 
 
 There are cases when they can be used for measuring all 
 the bases of a traverse. All that they require is some well- 
 defined point to view on and an accurately measured sub- 
 base at station. When the bases themselves are not measur- 
 able in this way there are sometimes well-defined points close 
 to those bases, from which the distances to them can be 
 accurately determined and so serve equally well. All these 
 instruments require considerable practice in order to obtain 
 reliable results every time. Those which have no lens 
 power require a good and quick eye and a steady hand. 
 
72 Preliminary Survey 
 
 The most important feature about the range-finder is that 
 it is a rapid means of measuring inaccessible distances ; 
 and across rough country where pacing is either impossible 
 or very inaccurate, it is as reliable as on level ground. A 
 range-finder is always valuable as a check on salient points 
 oif the centre line. 
 
 Unless exceptionally proficient in its use it is not advisable 
 to take long shots except as rough checks. It is better to 
 keep the bases within 1,000 feet, if possible. An error of 
 one foot in sub-base means fifty feet in distance. Two Weldon 
 range-finders are better than an optical square and a range- 
 finder because then they can be used either in the manner 
 explained in Chapter IX. or with the two acute angles or 
 with one right angle and one acute. 
 
 DISTANCE-MEASUREMENT BY A TWO-FOOT RULE 
 
 The following method needs no instruments beyond a 
 two-foot rule and a decimal scale. One of the old-fashioned 
 carpenter's rules, with a brass slide to it, graduated on the 
 one side with the logarithmic scale and on the other with 
 inches and tenths, will answer the purpose. Or if a Mann- 
 heim rule is at hand with its bevelled millimetre scale it will 
 give a closer reading, or, best of all, a ten-inch slide-rule of 
 the author's pattern described in Chapter IX. with a fifty scale 
 on its bevelled edge. This will give an even dividend-number 
 and as close a reading as it is possible to have. 
 
 Dividends are given for all three scales below. 
 
 The two-foot rule is held with the eye at the centre of 
 the joint, and the legs are spread so as exactly to intercept a 
 known sub-base. It is convenient to measure distances up to 
 300 feet with a sub-base of five feet or ten feet in the form of 
 a pole, with a pair of discs or crossheads painted black and 
 white, held by an assistant either on foot or on horseback 
 at the point whose distance is to be measured. 
 
 For longer distances, a base of fifty feet should be run 
 
Ron te- Su rveying 7 3 
 
 out with a tape by two assistants, who will hold a flag 
 or a disc at each end of the base. The line of sight to the 
 centre of the base should be at right angles to it. This is 
 easily done with a single assistant by a sighting-piece at 
 the middle of the pole, but when a base is to be taped it 
 is easier to set it at right angles to a line of sight to one 
 or other of its extremities. In the latter case, if the angle is 
 small the error will be hardly scaleable ; but to be exact 
 the aperture of the legs of the rule should be measured on 
 the square from one extremity to the line of the other leg 
 produced. 
 
 All that is required is to divide the dividend in the table 
 according to base and scale by the measurement of the 
 aperture of the legs of the rule. The only advantage in this 
 case of a slide-rule is to perform the division, but that is so 
 simple that it can be done mentally when using an English 
 scale. 
 
 DIVIDEND-NUMBERS 
 
 
 
 Base 5 feet 
 
 j Base 10 feet 
 
 Base 50 feet 
 
 Scale of inch and tenths 
 Scale of millimetres 
 Scale of inch and fiftieths 
 
 600 
 
 1,524 
 
 3,000 
 
 1,200 
 
 3,048 
 
 6,000 
 
 6,OOO 
 15,240 
 3O,OOO 
 
 Example. Viewing on a ten -foot base, measured the 
 aperture of a two-foot rule twenty-seven fiftieths of an 
 inch; required the distance. 
 
 6000 r t 
 
 Ans. - - = 2 22 '2 feet. 
 27 
 
 Good guesses at distance can be made by similar means 
 without anything more than a base which is itself guessed at. 
 Perhaps some reader will pooh-pooh such guesswork as 
 this, but no one who knows what it is to be without any 
 assistant for measurement and to need some help to the 
 mere guess at distance in perhaps a very deceptive piece of 
 country will under-value such a method as this when they 
 
74 Preliminary Survey 
 
 have tried it. In even slightly civilised countries, people 
 still build with some system, and- mud cabins, log huts, or 
 snake fences go pretty much in sizes. 
 
 In countries where they build in brick, one-storey and 
 two-storey houses bear considerable uniformity of height 
 per storey. Then a man on horseback, or a cow or any 
 other animal, may be taken for a guessed base. Even trees 
 in old wooded countries, though varying individually up to 
 any size, form as a forest a line of approximately equal 
 height which can be ascertained for different species and 
 used as a base. The suggestive fact is that any height of 
 that kind is much easier guessed at than a distance, because 
 in the one case there is something to go by, in the other 
 nothing. 
 
 MAPPING 
 
 We come now to the various contrivances for producing 
 a correct graphic representation of the fieldwork upon 
 paper. 
 
 The only absolutely true map is a terrestrial globe, but 
 as we cannot carry globes about with us we have recourse 
 to the principles of projection, which are quite numerous in 
 variety, but are all of them artificial representations of a 
 spherical, or more properly spheroidal, surface upon a plane. 
 
 If the survey extends over a large area, it becomes neces- 
 sary to adopt some method of projection by which, in the 
 first place, the distances are reduced to sea-level, and in the 
 second place, the meridians converged or distorted so as to 
 allow for curvature. 
 
 When the survey is a continuous traverse of a railway 
 route this is not necessary. It is not the object of the rail- 
 way surveyor to know the sea-level dimensions ; he needs 
 the actual length of his road wherever it may be. The dif- 
 ference in length between a degree of latitude at sea-level 
 and at 528ft. (3^ mile) elevation is only about nine feet. 
 At an elevation of 5,280 feet (one mile) it would be about 
 
Route- Siirvey ing 75 
 
 92 feet, and the distance-measurement practicable on route- 
 survey does not come nearer than that. 
 
 Neither does the surveyor want a distorted map, but one 
 to which he can apply a scale throughout ; he therefore does 
 not need to take account of the earth's curvature, but plots 
 his traverse on a horizontal plane. When the area over which 
 a triangulation extends is not large, the surveyor is still able 
 to adopt one of two methods of plane construction. 
 
 In the first the meridians and parallels of latitude are all 
 parallel straight lines at right angles to one another. 
 
 In the second the parallels of latitude are straight lines, 
 but the meridians are converging straight lines, or, if great 
 accuracy is needed, curved lines. 
 
 Limits may be given merely to fix the ideas within which 
 to use the first or second method, of say 1,000 square miles 
 for the first and 100,000 for the second. 
 
 When the survey is in high latitude the spheroidal form 
 of the earth much more affects the map than near the equator, 
 in which region a belt could be projected all round the globe 
 by the first method without sensible error. 
 
 Taking as an illustration of the extreme limit given for 
 the use of the first method, 1,000 square miles ; this area 
 would be contained in a square of which the side was not 
 quite half a degree ; let us lay down two plots of squares of 
 which the side is 2, which will embrace an area of about 
 16,000 square miles, or sixteen times the limit given, and 
 examine from it what the error would amount to in using 
 the first method, at a mean latitude of 32. 
 
 First. By mean longitude. The length of a degree of 
 latitude at 32 is 68-90 statute miles (see table, p. 175), and 
 the length of a degree of longitude at that latitude is == 5870 
 miles (see table). In the centre of the paper draw a hori- 
 zontal line to represent the middle parallel of latitude, and 
 through its centre erect a perpendicular to represent the cen- 
 tral meridian 5 of longitude, and lay off upon it 68-90 miles 
 above and below, and through the ends draw parallels to 
 
7 6 
 
 Preliminary Survey 
 
 represent 31 and 33 of latitude. Lay oft on the middle 
 parallel 5870 miles on each side of the centre, and for the 
 first method rule up verticals to represent the meridian of 
 4 and 6 longitude. The whole figure will then represent 
 four equal rectangular quadrilaterals. 
 
 For the second method lay off on the 33 parallel two 
 lengths of 58-09 miles, and on the 31 parallel two lengths 
 of 59*32 miles, being the length of a degree of longitude at 
 the respective latitudes (see Table XXV. p. 175), and com- 
 plete the figure. 
 
 The error of contraction at 31 will be seen to be = 1-2 
 
 FIG. 17. 
 
 FIG. 18. 
 
 miles, and the error of expansion at 33= 1-22 miles by the 
 first method. No error exists on lines running due north 
 and south. A diagonal through one of the quadrilaterals is 
 subject to an error of about half a mile, but on a diagonal 
 clear through the figure there will be hardly any error, 
 because the contraction in the upper almost exactly balances 
 the expansion in the lower. 
 
 At the same latitude a similar figure bounded by a ^ degree 
 of latitude and covering an area of 1,000 square miles would 
 
Route- Surveying 
 
 77 
 
 have an error of expansion at the top of the sheet of about fifty 
 yards and a similar error of contraction at the bottom. 
 
 This matter of possible error has been gone into numeri- 
 cally to fix the ideas before commencing to plot a survey 
 without wasting time in deciding upon whether to use pro- 
 jection or not. 
 
 The second method, of straight converging meridians, 
 may be used with quite sufficient accuracy up to a latitude 
 of 65 for stretches of 100,000 square miles, and there is not 
 much work done above this latitude anywhere. 
 
 It may be said, therefore, that the method of plane con- 
 struction meets all the ordinary requirements of the surveyor, 
 but in case he may be called upon to reduce extensive surveys 
 to atlas scale it may be as well to explain the principles of 
 
 CONICAL PROJECTION 
 
 A globe may be conceived to be wholly contained 
 inside a cylinder or partly contained inside a hollow cone. 
 
 For purposes of projection the cylinder must have a 
 diameter equal to that of the globe, but the cone must be 
 
 FIG. 19. 
 
 of such dimensions that its sides will be tangential to the 
 radius of the sphere at the point of contact. Supposing 
 the earth to be thus contained ; a belt of a few degrees on 
 either side of the equator might be conceived to be un- 
 
78 Preliminary Survey 
 
 wrapped or developed on the cylinder without sensible error. 
 This answers to plane parallel construction. 
 
 Similarly a belt of the cone may be developed as shown 
 on Fig. 19^, with converging meridians and with curved 
 parallels. Maps of continents are drawn in the atlas upon 
 this principle, and being of large extent the apex of the cone 
 is determined, and the radial parallels of latitude drawn 
 direct from it, with trammels. 
 
 Example. It is required to project by conical projection 
 a belt of 10 longitude, say from 20 to 30 east, whose 
 middle parallel of latitude is 50. The width to be 10, i.e. 
 from 45 to 55 ; the scale 100 miles to the inch. Draw 
 the scale of miles at the foot of the paper. Draw a hori- 
 zontal base-line in the middle of the paper, fix its centre, 
 and draw a perpendicular through it from top to bottom. 
 The base will represent the chord of the middle parallel, 
 and the perpendicular the central meridian 25 longitude E. 
 Calculate the length of the chord by following formulae. 
 
 Radius of Cone = ra.d. of earth x cotan lat. = 3,950 x 
 cot 5o=3,3i4=R. 
 
 Central ang/e = tota\ longitude x sin latitude = 10 x 
 sin 5o=7'66 / =A. 
 
 =R sin -=3,314 xsin 3-8 3 ' = 22i miles. 
 
 2 2 
 
 A 
 
 Versed f=R R cosin =7-4 miles. 
 
 Lay off - - on each side of the centre, and 221* 
 
 2 
 
 less 7*4 on the perpendicular, and 7*4 below the centre. 
 Then from the apex draw the middle parallel through the 
 extremities of the chord and the versed sine, and join the 
 apex to the two ends of the chord for the two extreme 
 meridians 20 and 30 east, upon which, on either side of 
 the middle parallel, lay off distance, = the latitude for each 
 degree (see table on p. 175), and describe the arcs of the 
 remaining parallels. 
 
Route- Survey ing 79 
 
 i ' 
 
 Then for the meridians draw the bottom chord to the 
 parallel 55, subdivide it into ten portions, and draw radial 
 lines to the apex. 
 
 If the radius of the cone be inconveniently long for 
 plotting, lay off the extreme meridians by protracting an 
 angle at each end of the chord to middle parallel equal to 
 
 l In this case the angle would be 86-17. Draw 
 
 2 
 
 top and bottom chords to parallels 45 and 55, and sub- 
 divide each into ten equal portions, through which divisions 
 draw the converging meridians. 
 
 STEREOGRAPHIC PROJECTION 
 
 is that in which the great circle of a sphere is assumed as 
 the plane of projection, and one of its poles as the projecting 
 point. In terrestrial maps it is used for representing a 
 hemisphere. The plane of projection is termed the primi- 
 tive. Projections of great circles drawn through the pole of 
 projection are straight lines, and all others are circles. The 
 centres of all great circles passing through any point in the 
 plane of the primitive are situated in a straight line called 
 the locus. 
 
 In the stereographic projection of the eastern hemisphere, 
 the primitive is usually the great circle of longitude passing 
 through the 2oth western and i6oth eastern meridians, and 
 the pole of projection is on the equator at 70 east longi- 
 tude. 
 
 The principal use of this projection to the surveyor is 
 for astronomical problems, such as that of 'graphic lati- 
 tude,' p. 158, or the chart of circumpolar stars on p. 393. 
 The explanation of this projection is given in detail in 
 Chambers's 'Practical Mathematics,' and in Heather's 
 ' Instruments ' in a more general manner. 
 
So Preliminary Survey 
 
 MERCATOR'S PROJECTION 
 
 is a development of the earth's surface by elongation of the 
 meridian, so that a ship's course will always appear as a 
 straight line, and is the projection used in the published 
 Admiralty chart. They are, however, constructed on the 
 principle of 
 
 GNOMONIC PROJECTION 
 for which see p. 89. 
 
8i 
 
 CHAPTER III 
 
 HYDROGRAPHY AND HYDRAULICS 
 
 WHEN a company is formed to develop the resources of 
 some new country like Africa, it often falls to its lot 
 either to make new harbours or improve natural ones, to 
 canalise rivers or to deepen them ; and the pioneer surveyor 
 has to be prepared at least to make a hydrographic chart 
 of a considerable degree of accuracy, to measure the dis- 
 charge of streams and rivers, or possibly to undertake trial 
 borings, soundings, or even dredging on a small scale for 
 the purpose of estimating the cost of a proposed work. 
 
 This chapter will be occupied with short descriptions of 
 methods adopted in carrying out such operations, keeping 
 strictly to preliminary work. 
 
 HYDROGRAPHY ON LAND 
 
 Some of the naval surveyor's methods differ but very 
 little from that of the land surveyor. He chains base-lines, 
 plants permanent trigonometrical stations, triangulates with 
 the theodolite, and reduces his work to sea-level. It does 
 not come within our province to follow up such lengthy 
 methods as these. We will content ourselves with the 
 subject technically called coast-lining, which signifies the 
 mapping of the shore-line, either from the ship, aided or 
 unaided by points on shore, by boats, by traverses on foot, 
 or by combinations of all the above methods ; together with 
 a short paragraph on Boat-survey of rivers. 
 
 G 
 
82 Preliminary Survey 
 
 COAST-LINING ON FOOT 
 
 This, where practicable, is the best way of rigorously 
 determining all the small indentations, creek-mouths, 
 directions of small streams, &c. Boating tends to over- 
 sight in these particulars, whilst from the ship itself only 
 the general outline can be obtained. The first thing is to 
 obtain fixed points on the shore visible from one another 
 from which to fill in the intervening detail. These may be 
 salient points in nature, such as prominent rocks, trees, or 
 houses, or else they may be beacons, cairns, or flagpoles, 
 whitewash marks, or any other artificial stations. 
 
 Their position is determined by astronomical observa- 
 tion, and the base is measured by difference of latitude on a 
 meridional line or by a meridian distance. These problems 
 are described in Chapter IV. 
 
 The plane-table is not much used by naval men, but it 
 might be made a very useful instrument, both on board 
 ship and on shore. They do, however, use an ordinary 
 field-board for plotting in details of coast-line from primary 
 points, or ' fixes ' as they call them, plotted previously on 
 the board. The intervening work between the primary 
 points is put in by telemetry ; they use a ten-foot pole 
 with a pair of discs as a base, the angle of which is measured 
 by a sextant or micrometer. 
 
 The pole is maintained square with the observer by a 
 directrix at its middle, or else it is swayed slowly in a hori- 
 zontal position until the observer has measured the maxi- 
 mum angle. 
 
 Another plan, where the ground is sufficiently open to 
 admit it, is to use a 500 feet lead-line run out square from 
 one end of the line to be measured. Fig. 20 is taken from 
 Captain Wharton's ' Hydrographical Surveying, 7 and repre- 
 sents a method largely used by Lieutenant W. U. Moore in 
 the survey of the Fiji Islands, and performed by two officers 
 only. Starting at A officer No. i moves on to B, leaving 
 
Hydrography and Hydraulics 
 
 FIG. 
 
 officer No. 2 at A with the lead-line, who fixes a flag at A 
 and runs out his line either with an optical square or a ' 3, 
 4, and 5 ' line, and plants another flag at its extremity. 
 Officer No. i then measures the angle between the flags 
 = 500 x cot 0. Each officer then takes sextant angles to 
 salient points pre- 
 viously determined 
 along the shore, so 
 as to make intersect- 
 ing rays for subsidiary 
 'fixes.' They then 
 travel to meet one 
 another, filling in de- 
 tail by bearings with 
 the prismatic com- 
 pass, and distances 
 with the micrometer 
 and ten-foot pole or sextant and ditto. The writer never 
 having done this kind of work cannot speak positively, but 
 he thinks it probable that the plane-table would be found 
 preferable for the filling in of detail between the extremities 
 of the base. The subsidiary fixes form independent checks 
 to the detail, and if a plane-table were used errors could be 
 more easily eliminated at the time. 
 
 SURVEYING WITH BOATS 
 
 Rock-bound coasts, or rivers with banks covered with 
 dense jungle, are best surveyed with boats. The use of 
 boats in coast-lining does not much alter the methods 
 already mentioned ; we will therefore confine ourselves to 
 boat-survey of rivers. A steam pinnace is used whenever 
 one is obtainable, and provided with a compass and patent 
 log. The latter is attached to the gunwale, and the fan 
 towed astern. The prismatic compass stands on a tripod 
 in the stern. The velocity of the current is measured 
 from time to time by anchoring in midstream. A current - 
 
 G 2 
 
8 4 
 
 Preliminary Survey 
 
 meter should be added to the equipment as being more 
 adapted for that purpose than the patent log, especially 
 with slow currents. 
 
 When a large-scale plan of a wide river has to be made, 
 several boat-parties (Fig. 21) work in concert, four if pos- 
 sible ; two on either side of the river, triangulating their 
 way up from point to point. Two starting-points are esta- 
 blished at the mouth of the river, whose relative position 
 and distance has been determined with the utmost possible 
 accuracy. Two of the boats remain at these points, and 
 the other two move up the river to convenient stations, and 
 each of the four boats takes sextant angles to the other 
 
 FIG. 21. 
 
 three. Here the point C is fixed from line AB by the 
 angles BAG and ABC, and the point D by the angles 
 ABD and BAD ; the angles ACB, ACD taken from 
 C and the angles BDC, BDA taken from D, form 
 independent checks. The shore-line may be sketched 
 by the boats A and B when moving up stream to take the 
 places of C and D. If the sounding has to be done 
 thoroughly, the boats should return by a diagonal course 
 AD, BC, or else the boats C, D, when first moving up the 
 river, can fill in the shore-line with the patent log and 
 compass angles, and also take soundings along the lines 
 AC,BD, leaving the boats A, B, only the diagonal sound- 
 ings AD, BC to take. 
 
Hydrography and Hydraulics 85 
 
 SURVEYING FROM THE SHIP 
 
 A great deal of very good charting is done without any 
 connection with the shore, except the landing of a boat to 
 make sound signals. This class of work is termed ' Running 
 Survey.' It is the most rapid though the least accurate of 
 any of the methods which are considered worthy of being 
 called surveying. The mere sketch independent of any- 
 thing more than guesses at distance is not now resorted to, 
 although many islands exist in the Pacific, as well as por- 
 tions of the mainland of Australasia, and other continents, 
 where the charts are still, to a large extent, sketchy, and it 
 is the important duty of the Hydrographical Department of 
 the British Navy to diminish year by year the ' terra incog- 
 nita,' which presents its ofttimes hidden dangers to our 
 world-covering mercantile marine. 
 
 Running survey is often assisted by known points on 
 shore. If there are, for instance, three sharply defined 
 peaks whose geographical position has been accurately 
 determined, the work is both facilitated and improved. 
 For by the station pointer, see p. 367, the ship's place is 
 located at any time as long as they are kept in view. As 
 the ship moves forward, her path being thus clearly defined 
 upon the chart, subsidiary cuts are made by angles with 
 the sextant to salient points upon the coast and the detail 
 sketched in. From fifty to a hundred miles of coast-line 
 can be thus put in in a day. 
 
 When no assistance of this kind can be obtained from 
 the shore, the base is obtained by sound-signals. The 
 vessel is maintained as nearly as possible in the same 
 position, whilst a party on shore and a party on board 
 alternately fire guns within sight of one another, so that 
 the time between flash and report may be taken by the 
 chronometer on either side. During this operation angles 
 are being taken by the sextant to all the important points, 
 which can be seen both from the shore station and from 
 
86 Preliminary Survey 
 
 the ship. When the length of the base is determined by 
 the sound signals, the landing party returns on board, the 
 various peaks are located on the chart by intersections, and 
 henceforth serve as known points from which to identify the 
 ship's place at any time as she travels forward. 
 
 Sound travels at the rate of about 1,090 feet per second 
 at a temperature of 32 F. and increases its velocity at the 
 rate of 1*15 feet according to some authorities, and 1*25 
 according to others, for each degree Fahrenheit of rise in 
 temperature, or vice versa for fall of temperature. 
 
 The object of signalling from both ends of the base is 
 to counteract the effect of the wind, which greatly affects 
 the accuracy of the result. It is not, however, always re- 
 sorted to. 
 
 The firing is done by prearranged signal, such as the 
 dipping of a flag, in order that no time may be wasted in 
 looking out. 
 
 Chronometer watches beat five ticks to two seconds. 
 English lever watches nine ticks, and Geneva watches ten ticks. 
 
 When watching for the warning signal with a telescope, 
 the watch should be tied to the ear with a handkerchief, 
 and the counting commenced thus : nought, nought, 
 nought, until the flash is seen, then one, two, three, to ten. 
 At each ten one of the fingers is put down, so that the ten 
 fingers will represent one hundred beats or forty seconds 
 with a chronometer=about eight and a quarter miles, 
 which is longer than bases are usually measured in this way. 
 
 The following example is from Captain Wharton's 
 ' Hydrography,' p. 64. 
 
 * In meaning the result the arithmetical mean is not strictly 
 correct, as the acceleration caused by travelling with the 
 wind is not as great as the retardation caused in the opposite 
 direction, as in the latter case the disturbing cause has clearly 
 acted for a longer period. The formula used is 
 
 7+7 
 
Hydrography and Hydraulics 87 
 
 where T is the mean interval required, / the interval 
 observed one way, /' the interval observed the other way. 
 The mean interval thus found multiplied into the velocity 
 of sound for the temperature at the time will give the required 
 distance. 
 
 ' As an example let us suppose A and B the two ends of 
 the base to be measured. 
 
 * At A have been observed 
 
 46 beats with watch beating 5 beats to 2 seconds. 
 
 47 35 33 33 " 
 
 46 33 53 33 33 
 
 Mean 46*33 beats= 18*532 seconds. 
 
 81 beats with watch beating 9 beats to 2 seconds. 
 
 82 
 
 33 
 
 Mean 82 beats =18*2 2 2 seconds. 
 Mean at A= 18*376 seconds. 
 ' At B have been observed 
 85 beats with watch beating 9 beats to 2 seconds. 
 
 "7 3? 33 33 33 
 
 88 
 
 Mean 86*66 beats= 19*258 seconds. 
 47 beats with watch beating 5 beats to 2 seconds. 
 
 47 
 
 4" 33 33 33 35 
 
 Mean 47*33 beats= 18*932 seconds. 
 Mean at 6 = 19*095 seconds. 
 
 Then working, T = f ; = 18*728 seconds. 
 
 ' Temperature 80 F., at which velocity of sound= 
 i, 1 45* 2 feet per second x 18*728 seconds =2 1,448 feet' 
 
 The temperature must be taken in the open air with the 
 thermometer shaded from the direct rays of the sun, but not 
 in too cool a spot or it will not give the true temperature of 
 the free air. 
 
88 
 
 Preliminary Survey 
 
Hydrography and Hydraulics 89 
 
 The small island shown on Fig. 22 is a copy from a 
 recently made chart kindly lent by Lieutenant Vernon 
 Brooke Webb, one of the hydrographers of H.M.S. ' Dart ' 
 on her expedition to New Guinea. It will serve to illustrate 
 the rapidity of running survey ; the time actually occupied 
 was nineteen hours, no special hurry having been made 
 over the work. The distance covered is about sixty 
 miles. The method was running survey, aided by several 
 fixed stations on a neighbouring island, from which the 
 ship's place could be located whilst on one side of the island 
 by the station-pointer. The vessel steamed at about four 
 miles an hour, towing the patent log all the time, and taking 
 soundings where shown. These being pretty deep occupied 
 about ten minutes each. On the further side of the island 
 all the distance-measuring was done by patent log. The 
 hills were put in by sextant angles, both for position and 
 elevation. The rivers were shot in with compass angles. 
 The inhabitants being cannibals, the island was not explored 
 inland. The graduation is a little displaced in order to bring 
 the margin within the sheet. 
 
 The following are the notes appended to this chart : 
 
 Current; *-, ebb; <-' , Hood; kn, knots; H.W.F. & C ix h 
 o m ., springs rise 5 ft., neaps, 3^ f t ; B., bay ; C., cape ; Cr., creek ; 
 D., doubtful; H' 1 ., head; H r ., harbour; I., island; L., lake; P., 
 port ; P'., point ; R., river ; Rk., rock. 
 
 bk., black; cl. , clay; crl. , coral ; f. , fine ; g. , gravel ; h. , hard ; 
 m., mud ; r., rock ; s., sand ; sh., shells ; St., stones. 
 
 Figures on the land show the heights in feet. 
 
 Bearings to the marks and views are magnetic. 
 
 Soundings in fathoms. 
 
 Magnetic variation in 1890, nearly stationary. 
 
 GNOMONIC PROJECTION 
 
 The Admiralty charts are now all constructed upon this 
 projection and published on the Mercator's projection. 
 
9O Preliminary Survey . 
 
 Gnomonic projection is very similar to conical projection, 
 but the plane of projection is not the development of a 
 conical zone ; it is a true plane touching the sphere only at 
 one point, viz., the middle point of the central parallel. The 
 convergency of the meridians is measured by the difference 
 of true bearing of one point from the other at the ex- 
 tremities of the map. 
 
 It is the only projection in which all great circles are 
 represented by straight lines, so that spherical distances from 
 point to point are scaleable all over the chart when it has 
 been properly graduated. 
 
 This does not mean that one scale may be used through- 
 out, but, unless in very high latitudes, considerable portions 
 of the chart are practically to one scale and the graduations at 
 the side give the scale at any point. 
 
 The actual length of any projected line measured on 
 a meridian from the middle parallel is proportional to the 
 tangent of the latitude. 
 
 It will be seen from the following illustration, that at 
 a scale of 10 miles to the inch in 45 latitude, a stretch of 
 250 miles square, or 62,500 square miles, would not differ 
 in measurement from a plane construction with converging 
 meridians ; because the lengths of the tangents at that scale 
 are sensibly equal within 2 on either side of the middle 
 parallel to the developments of the spherical distance. The 
 difference at 2 between tangent and arc is -0000222, and if 
 we consider -01 inch as the limit of scaleable quantity, we 
 get 37 - 5 feet as the limit of radius at which error is appreci- 
 able, which is equal to about 8| miles to the inch. 
 
 The natural scale, or the proportion which the chart 
 lineally bears to the actual size in nature, is obtained by 
 dividing the number of inches in the nautical mile at the 
 latitude by the number of inches corresponding to one mile 
 on the chart. The result will be the denominator of a fraction 
 whose numerator is one. 
 
 Thus, supposing the scale to be i '8 inches to a mile in 
 
Hydrography and Hydraulics 9 1 
 
 latitude 3, we divide 72,552 (the number of inches in a 
 mile, see Table XXVI., p. 175) by 1-8 ; this gives ^QW as 
 the natural scale, which should be noted on all sheets that 
 are not graduated. 
 
 ILLUSTRATION OF LIMIT UP TO WHICH PLANE CON- 
 STRUCTION DOES NOT SENSIBLY DIFFER FROM GNO- 
 
 MONIC PROJECTION. 
 
 Scale 10 miles per inch =- 3 -j 1 6oir At a mean terrestrial 
 radius of 20,888,628 feet the length of radius to scale= 
 32-968 feet. The length of i from central parallel on 
 central meridian='57546 feet. The length of 2 1*15127 
 feet. 
 
 But the mean length of a degree of latitude =69 -05 
 miles, and at 10 miles per inch=6'9o5 inches or '5754 feet. 
 Therefore the projection is sensibly equal to the develop- 
 ment of the spherical distance within at least 3 of latitude. 
 As it is not likely that the surveyor would be called upon, 
 unless with proper notice for obtaining his special map 
 equipment, to carry out so extended a survey as to need 
 the exactitude of the above method, it will not be further 
 dwelt upon. 
 
 He will find a thorough description of the gnomonic 
 projection in Capt. Wharton's ' Hydrography,' Murray's, 
 Albemarle Street. 
 
 SYMBOLS IN CHARTING 
 
 The annexed specimen list of symbols, taken by Capt. 
 Wharton from the Admiralty Manual, shows the authorised 
 delineation, and is reproduced here almost verbatim. 
 
 The Days of the Week. 
 
 Sunday . . . Sun's day . . Sun 
 
 Monday . . . Moon's day . . Moon ]> 
 Tuesday . . . Teut's day . . Mars $ 
 Wednesday . . . Woden's day . . Mercury $ 
 Thursday . . . Thor's day . . Jupiter 2/. 
 Friday . . . Friga's day . . Venus $ 
 Saturday . . . Saturn's day . . Saturn h 
 
92 Preliminary Survey 
 
 The following Signs are used in the Fieldbook*. 
 
 Cutting ' vX'!' ^^P^ Embankment 
 
 ^ 
 
 Objects in line, called transit . . . . . 
 Station where angles are taken ... 
 
 Zero, from which angles are measured . ' 
 
 Single altitude sun's lower limb .... 
 
 ,, ,, ,, upper limb . . . . 
 
 Double altitude sun's lower limb in artificial horizon 
 
 ? > upper ,, ,, ,, " 
 
 Sun's right limb . . . . . , | 
 
 Sun's left limb ....... | 
 
 Sun's centre . . . . . . . . CD 8 
 
 . ^ 
 
 Right extreme, or tangent, as of an island 
 
 Left 
 
 Zero correct . . . . . . . . Z K 
 
 Windmill ........ & , 
 
 Water level . . . . . . . w. 1. 
 
 Whitewash ........ W. W. 
 
 Bridge ....... ' )= ^ 
 
 In Colouring. 
 
 Sand ..... Gamboge, dots black. 
 Low water, sand edge . . Gamboge, dots carmine. 
 Mud, dry low water . . . Neutral tint, edge of fine black dots. 
 
 r- i j i i ) I Burnt sienna and carmine mixed 
 
 Coral, dry low water or any rocky I 
 
 , . j lor wash ; same darker tor edg- 
 
 ground covering and uncovering 
 
 in S- 
 
 Cliff . . . . . Dark neutral tint. 
 
 Roads ..... Burnt sienna. 
 
 /Either a faint wash of cobalt all 
 
 over the area included within 
 Fathom lines up to five fathoms . -<' the fathom line, or a narrow 
 
 edging of the same colour inside 
 
 the dots of the fathom line. 
 
Hydrography and Hydraulics 93 
 
 SIGNALLING WITH HELIOSTAT OR HELIOGRAPH 
 
 These instruments are described in Chapter IX. Their 
 use is also sufficiently explained to dispense with anything 
 further here. 
 
 The following description of the methods used is nearly 
 verbatim from the ' Military Handbook of Signals.' 
 
 The Morse code is that which both military and naval 
 officers use. It has nothing but a combination of dots and 
 dashes for its basis. 
 
 The duration of the flash is what constitutes it a dot 
 or a dash, and practice is required both to give uniform 
 durations and to read the signals. An unpractised signaller 
 at the receiving station can take down the flashes by dot 
 and dash on paper and afterwards read them off. In giving 
 the signals, the transmitter should count, say, ten for a dash 
 and three for a dot, or less when in good practice. Counting 
 the duration of the dot as a unit, the pause between each 
 letter should be three, and between each word six units. 
 
 TABLE Will. Alphabet. 
 
 A .- N . 
 
 B . O- 
 
 C . . P . 
 
 D . Q 
 
 E R 
 
 F S 
 
 G -- . T- 
 
 (! U 
 
 I . . V . . . 
 
 J . - W . - 
 
 K . - X . . 
 
 L . . Y . 
 
 M- Z -- 
 Numbers. 
 
 1 
 
 2 
 
 3 ... -- 
 
 4 .... 
 
 5 ..... 
 
94 Preliminary Survey 
 
 Punctuation. 
 
 Full stop 
 
 Preparative and erasure, a continuous succession of dots. 
 
 The preparative sign is to call attention. To answer it 
 the received either gives the 
 
 General Answer 
 
 a single dash, or else he gives the code letter of his name 
 or station. 
 
 When intended as an erasure to signify that a mistake 
 has been made by the transmitter, it should be answered by 
 the erasure signal. 
 
 The Break signal I I is used between the 
 address and text of a message, and after the text if the name 
 and address of the sender are to be signalled. 
 
 The Completion signal V E but sent as a 
 group, not two letters, denotes the completion of a message. 
 
 ' Repeat ' I M I given continuously 
 
 The Figure signal F I means that figures 
 
 are intended. 
 
 The Figure Completion signal F F - - -- 
 means that figures are done. 
 
 Indicator - - is sent at commencement and 
 conclusion of message. It is answered by the same signal. 
 
 All right R. T. 
 
 Go on ....... G. 
 
 Move to your right . . . . . R. 
 
 Move to your left . . . . L. 
 
 Move higher up or further off . . . H. 
 
 Move lower down or closer . . O. 
 
 Stay where you are . S. R. 
 
 Separate your flags . . . . S. F. 
 
 Use blue flag B. 
 
 Use white flag W. 
 
 Use large flag . . , . , L. F. 
 
Hydrography and Hydraulics 
 
 95 
 
 Use small flag S. 
 
 Your light is bad . . . . L. B. 
 
 Turn off extra light T. O. L. 
 
 Wait M. Q. 
 
 Say when you are ready . . . . K. Q. 
 
 I shall signal without expecting answers . K. K. K. K. 
 
 In America the signal for all right is O. K., supposed to 
 have been invented by a Mr. Joshua Billings. 
 
 Some of the above signals refer to flag and light signals. 
 The lime-light apparatus is not described in this work, but 
 the flag system will be here explained. 
 
 FLAG-SIGNALLING 
 
 FIG. 23. 
 
 The Morse alphabet is used, but the dots and dashes 
 are made by movements of the flag and not simply by 
 exposing it. 
 
 Either large or small flags are used. They are both 
 made of the same material, a sort of muslin, and of two 
 colours, white with a blue horizontal strip for use with a dark 
 background and dark blue for use with a light background. 
 
 The large flags are three feet square mounted on a pole, 
 five feet six inches long, one inch diameter at the butt and 
 tapering to half an inch at the top. 
 
96 Preliminary Survey 
 
 The small flags are two feet square mounted on a pole 
 three feet six inches long, three-quarter inch diameter at the 
 butt and tapering to half an inch at the top. 
 
 ARMY FLAG DRILU 
 
 Large flags. The signaller may work from left to right 
 or from right to left, or may turn his back to the station to 
 which he is signalling, according to the direction of the 
 wind, so that the flag may be waved from the normal posi- 
 tion against the wind. 
 
 To make a dot. Wave the flag from the normal position 
 a, Fig. 23 to a corresponding position, b, Fig. 24 on the 
 opposite side of the body, and without any pause back to #, 
 Fig. 23, keeping the left elbow close to the side. 
 
 To make a dash wave the flag from a, Fig. 23, to c, Fig. 
 25, so that the point of the pole nearly touches the ground, 
 still keeping the left elbow close to the side and straightening 
 the right arm ; make a short but distinct pause in this posi- 
 tion and then return to a, Fig. 23. 
 
 When signalling a letter, say R, the flashes repre- 
 senting it should be made in one continuous wave of the 
 flag, taking particular care that no pause is made when at 
 the normal position. Thus to make R wave the flag from #, 
 Fig. 23, to b, Fig. 24, back to a, Fig. 23, and without any 
 pause down to <r, Fig. 25 ; slight pause at c (see instructions 
 for making a dash), back to a (Fig. 23) ; then without pause 
 to b, Fig. 24, and back to the normal position a, Fig. 23. 
 A pause equal to the length of a dash should be made at 
 the normal position a, Fig. 23, between each letter of a word 
 or a group of letters. When the word or group is finished 
 the flag pole is lowered and the flag gathered in with the 
 left hand. 
 
 A slight pause should be made at the normal position 
 before commencing a word or group. In receiving a 
 
Hydrography and Hydraitlics 97 
 
 message the flag should be lowered and gathered in until 
 required for answering. 
 
 In order to keep the flag always exposed while moving 
 it across the body to form the flashes the point of the pole 
 should be made to describe an elongated figure of 8 in the 
 air. 
 
 With clear sending and under favourable conditions these 
 flags can be seen and read with the ordinary service telescope 
 at distances up to twelve miles. 
 
 Uniform unbroken backgrounds are always better than 
 broken ones. Bare earth rocks and trees form the darkest 
 backgrounds, and these appear darker or lighter in propor- 
 tion to their distance in rear of the object projected on them. 
 If far off the background will be lighter, if immediately be- 
 hind the object it will be darker. Sky forms the lightest 
 background and then water and distant land ; green or 
 stubble fields form an intermediate class of backgrounds 
 against which light or dark objects appear almost equally 
 visible. As dark backgrounds appear darker in proportion 
 as they are nearer to the object, so white backgrounds, such 
 as chalk cliffs, whitewashed walls, &c., appear lighter under 
 the same conditions. The position of the sun should always 
 be taken into consideration in applying the foregoing obser- 
 vations. All backgrounds become lighter when the sun is 
 opposite to them and darker when it is behind them. An 
 exception to this rule is, however, found in the light mists 
 which rise from valleys towards evening or the smoke of 
 habitations, which both form a lighter background than the 
 surrounding country, whatever be the position of the sun. 
 The most favourable conditions for flag-signalling are a 
 clear atmosphere and a clouded sun. 
 
 STANDARD OF EFFICIENCY 
 
 For the standard of efficiency the minimum rates of 
 reading correctly from and sending a test message with the 
 different instruments are as follows. 
 
 H 
 
98 Preliminary Sutvey 
 
 Large flag at the rate of . .9 words per minute 
 
 Small flag ,, ,, ... 12 ,, ,, 
 
 Lamp ,, ,, . .10,, ,, 
 
 Heliograph ,, ,, . .10,, ,, 
 
 The degree of accuracy is tested by the percentage of 
 letters read correctly. A man is reckoned as 
 
 Very accurate who can read 97 per cent, correctly 
 Accurate . . . -95 ,, ,, 
 
 Fairly accurate . . -93 >> ,, 
 
 Inaccurate . . . .90 ,, ,, 
 
 Very inaccurate . below 90 ,, ,, 
 
 TIDES AND CURRENTS 
 
 The soundings on charts are given in fathoms of depth 
 at low water of ordinary spring tides because those are the 
 times of highest high water and lowest low water. For the 
 theory of tides the reader must study works of a different 
 character from the present one. Some definitions of terms are 
 given in the glossary. No satisfactory data can be obtained 
 from a casual investigation ; observations should be "made 
 at different times of the day, month, and year, and the results 
 registered in a systematic manner. The pioneer surveyor is 
 not supposed to have as much time as this at his command, 
 and we shall therefore confine ourselves to practical details 
 for a preliminary examination of tidal phenomena. 
 
 DATUM 
 
 A fiducial point should be chosen on shore as near as 
 possible to the position where the register is to be kept, but 
 safe from any encroachment of the sea. A hard rock point 
 or a benchmark on a house or tree will do. If there is only 
 a sandy beach sometimes a piece of old iron pipe 6" dia- 
 meter is obtainable, which filled with Portland cement 
 concrete makes a very durable datum. 
 
Hydrography and Hydraulics 99 
 
 ' r vV : ; I/?/ %N 
 
 . - A iV'J. K\.r 
 
 TIDE GAUGE 
 
 If a pier is available, a planed board is lashed to it 
 marked in feet and tenths, the former being painted red, 
 white, and blue alternately so as to be read at a distance. 
 If there is no pier, a pile must be driven and stayed. To 
 the side of the pile the planed board is attached, or else a 
 simple maximum and minimum device is constructed of a 
 casing made with four boards and containing a float main- 
 tained in position by guides working in slits and having 
 indicators one above a guide, the other below the opposite 
 guide. These indicators have to be set twice in the twenty- 
 four hours and are then self-registering. Or else if left for 
 any length of time they will leave on record the maximum 
 and minimum during that period. 
 
 The registration is made from simple inspection of the 
 tide-board, the zero of which has been previously levelled 
 with a levelling instrument from the datum point on 
 shore. 
 
 There are many elaborate and expensive self-registering 
 tide-gauges for use on permanent works of construction or 
 important hydrographical work. 
 
 A very beautiful repertory of tidal instruments was ex- 
 hibited by Sir William Thomson at the Institution of Civil 
 Engineers in 1881 in connection with his lecture on that 
 subject. These comprehended a tidal harmonic analyser, 
 a tide predicter, and a model of the No. 3 Clyde Tide-gauge. 
 Wave disturbance is almost entirely annulled in the floater ; 
 clockwork mechanism records by either an ink or a pencil 
 marker the movement of the tide upon a revolving diagram 
 like the steam-engine indicator. The abstruse calculations ot 
 the harmonic analysis are replaced by those of an automatic 
 integrator, and the time, not merely of high or low water, but 
 the position of the water-level at any particular port, is pre- 
 dicted for any time of day of any future year. 
 
 H 2 
 
loo Preliminary Survey 
 
 The velocity of currents at sea are generally taken by a 
 ship at anchor with a current-meter. They are also ascer- 
 tinued by observing a drifting boat from two points upon 
 the shore. When far out at sea they are calculated by 
 comparing the day's run, ascertained astronomically with the 
 figures of the patent log, making allowance for instrumental 
 errors. 
 
 An elementary explanation of tidal phenomena, together 
 with high and low water at the principal ports of the United 
 Kingdom and tidal constants for minor ports, is given in 
 Whitaker's Almanack, which should form part of the travelling 
 library of every surveyor. 
 
 The velocity of river currents is best taken by current- 
 meter, unless sufficient time is available for obtaining the 
 data required by Kutter's formula. 
 
 Floats are either surface- floats, of wood or wax or vertical 
 tin tubes loaded at bottom. In large rivers the average of 
 the vertical floats, which gives the approximate mean velocity 
 at that section, has been found to be about -9 of the surface 
 velocity. 
 
 HYDRAULICS 
 
 The numerous problems of more or less complexity 
 occurring in hydrostatic and hydraulic science are peculiarly 
 suitable to treatment by the graphic method, and by the use 
 of large-scale diagrams may be solved with a greater degree 
 of mathematical precision than is actually necessary with 
 any formula, by reason of the element of uncertainty which 
 in hydraulics must always attach to the resisting power of 
 the conduit. For the approximate estimates which the pre- 
 liminary surveyor has to prepare, the small-scale diagrams 
 on Plate I. Fig. 26, and Figs. 28, 29 will be found amply 
 sufficient ; any slight inaccuracies due to scale will not pro- 
 duce variations in the final result as great as those between 
 two authorities like Beardmore and Kutter as exhibited 
 
PLATE L 
 
 FIG. 26. 
 
 Kutter's formula for flow in open channels. Diagram 
 for finding coef. C in formula V=C x JRx S. 
 
 Note. The values of N are from 'or to '04. 
 
 ,, S ., '00002510 "01. 
 
 ,, R ,, o to 60 feet 
 
 Intermediate values may be interpolated by eye. 
 
 Note i. The coefficient is to be taken from the thickened, right hand side of 
 the vertical scale. 
 
 Note 2. If this diagram is in frequent use with dividers, a piece of dull-back 
 tracing cloth, gummed over it by the four corners, will protect it. 
 
Hydrography aud Hydraulics 101 
 
 later on. The formula of the latter engineer has been 
 adopted as being the most comprehensive and in accordance 
 with recent experiments. Messrs Ganguillet and Kutter, 
 Swiss hydraulic engineers, have of late years developed the 
 theory of flowing water in open channels, and have furnished 
 a formula of greater complexity than most of the preceding 
 ones, but held by modern authorities to be superior in 
 accuracy. However perfect the formula may be, an element 
 of uncertainty must always exist as far as rivers are concerned, 
 because the coefficient of roughness in the conduit, or n as it 
 is called by Kutter, has to be obtained, to some extent by ex- 
 periment but largely by judgment based upon actual study 
 of the ground or channel. Plate I. Fig. 26 for finding the 
 coefficient C is constructed for English measurement and 
 will embrace all ordinary cases, but where much work is 
 required it will be found advisable to construct a diagram 
 to a large scale, for which full directions are given in 
 ' Trautwine's Pocket Book.' 
 
 It is evident that the varying velocities in any conduit of 
 water must arise from the force of gravity modified by the 
 resistances of the surface ; from natural roughness, vege- 
 table growths, bends, &c. ; the consistency of the fluid, the 
 wind, and it may be other causes. 
 
 Kutter's formula, like those of other good authorities old 
 and new, is based upon the theory that the resistances to 
 flow are directly proportional to the area of the surface 
 exposed to the flow (which would be, in the case of pipes 
 running full, the entire internal surface) and to the square 
 of the velocity. The formula stands thus for English 
 measurement : V=C >/ RS^. In which V=velocity in feet 
 per second ; C a coefficient derived from three independent 
 variable quantities the resistance of surface, hydraulic mean 
 depth, and slope ; R is the hydraulic mean depth ; S is the 
 slope or sine of the angle of slope. 
 
 The velocity V is the mean velocity of the whole stream, 
 
IO2 Preliminary Survey 
 
 and is the only reliable means of ascertaining the discharge. 
 Experimental determinations of surface velocity by floats 
 are very useful, but are generally too local to give more than 
 an approximate result. The mean velocity varies in large 
 rivers from '85 to "95 of the surface velocity found by 
 floats. 
 
 The coefficient C is obtained directly from Fig. 26 by 
 drawing lines from the horizontal scale of R, that is the 
 hydraulic mean depth, to the intersection of the hyperbolic 
 slope curve S with the radial ' roughness ' line n. This 
 line will cut the right-hand side of the vertical scale OC at 
 the value of C. The dotted line shown on the diagram is 
 drawn from a hydraulic mean depth of 50 feet to the inter- 
 section of n '02 with slope of i in 2,500 ; the coefficient C= 
 123 is halfway above the first subdivisions from 120, each of 
 them being 2. 
 
 Any intermediate curves or radial lines may be interpo- 
 lated by the eye with quite sufficient accuracy. 
 
 The value of n is obtained from the following table. 
 
 TABLE XIX.-- Artificial Channels of Uniform Cross-Section'. 
 
 Coefficient 
 of roughness, 
 
 n 
 
 Sides and bottom lined with well-planed timber . -009 
 Sides and bottom rendered in cement, glazed ware, 
 
 and very smooth iron pipes . . . -oio 
 
 I to 3 cement mortar, or smooth iron pipes . -oil 
 Unplaned timber or ordinary iron pipes . . -012 
 Ashlar or brickwork . . . . . -013 
 
 Rubble -017 
 
 Channels subject to Irregularity of Cross-Section. 
 
 Canals in very firm gravel ..... 'O2O 
 Canals and rivers of tolerably uniform cross-section, 
 slope, and direction in moderately good order 
 and regimen, and free from stones and weeds -025 
 Having stones and weeds occasionally . . '030 
 
 In bad order and regimen, overgrown with vege- 
 tation, and strewn with stones and detritus -035 
 
Hydrog rapJiy and Hydraulics 103 
 
 Messrs. Trautwine and Hering use -015 as a value of n 
 for ordinary brick sewers instead of -013 given by Kutter, in 
 consideration of the ' usual rough character of sewer brick- 
 work.' This remark would not apply to the London and 
 Paris sewers or to those of many other large cities, where 
 the brickwork is exceedingly good ; but unless properly main- 
 tained the coefficient would soon increase. 
 
 The hydraulic mean depth R is equal to the 
 
 area of wet cross-section 
 length of wet perimeter dbco. 
 
 In the case of a circular culvert or pipe running full, R= 
 
 2 rJ 
 
 ~~ =- or - where r=radius and ^/=diameter. When 
 2-rrr 2 4 
 
 half-full R is also = - and the velocity will be the same. 
 
 4 
 At -75 of the diameter, i.e. f full, R=D x -3. When j- 
 
 full R=Dx'i5. To obtain the hydraulic mean depth of 
 irregular conduits such as rivers, the cross-section must 
 be taken by soundings and the area measured. 
 
 The slope is usually termed the sine of the angle made 
 by the average bed of the channel with the horizon. At the 
 small angle of most rivers and sewers the sine is sensibly the 
 same as the tangent, but why it is called the sine is not clear 
 to the writer, because slopes are usually from convenience 
 described in the ratio of base to perpendicular, and not from 
 hypotenuse to perpendicular. 
 
 The length of the conduit is always supposed to be 
 measured on the level, and it would seem to be proper to 
 take the ratio from that rather than from the sloping length. 
 The table of tangents corresponding to some leading slopes 
 of feet per mile given on p. 248 will be found sufficient by 
 
IO4 Preliminary Survey \ 
 
 the aid of the slide-rule to obtain any required tangent, and 
 the method of interpolating is there explained. Both the 
 hydraulic mean depth and the tangent of slope are often 
 very small decimal quantities of which the square root has 
 to be extracted. Another short table is given, on p. 249, of 
 leading values of squares of decimals by aid of which the 
 operation is greatly facilitated. The diagram only gives 
 slopes up to '01, that is, i in 100. Above that the coefficient 
 C remains the same. The formula upon which the diagram 
 for finding the coefficient C is constructed is for English 
 measure 
 
 0028 1 \ 
 
 slope 
 
 C= 
 
 / - '002Xr\ 
 
 xn 
 
 slope / 
 
 v/mean radius in feet 
 
 When the velocity in feet per secondjhas been obtained 
 from the diagram and formula V= C s/RS, the discharge, Q, 
 is obtained in cubic feet per minute by the simple equation 
 Q=V x 47-1 2 D 2 for circular sewers where D=diam. in feet. 
 
 In irregular channels and generally Q=area in square 
 feet x vel. in feet per second x 60. 
 
 Example i. What is the velocity in feet per second and 
 discharge in cubic feet per minute in a circular brick sewer 
 running full ; 2 feet diameter ; slope S = io feet per mile= 
 
 RS= -000945 V^S=' 37 ; c y diagram 4 is found 
 to be (using a coefficient of -015 for )=85, and =85 x 
 0307 = 2-6 feet per second. 
 
 Q=2-6 X47'i2 x 4=490 cubic feet per minute. 
 
 Example 2. What will be the velocity in feet per 
 second and discharge in cubic feet per minute of a V- flume 
 of unplaned timber, sides sloping 60 and bottom board 10" 
 wide in the clear, the slope being i in 50 and the depth of 
 water 9 inches ? 
 
HydrograpJty and Hydraulics 
 
 105 
 
 The whole calculation can be done in a few minutes by 
 the slide-rule and diagram. The angle of slope of sides gives 
 the horizontal spread from the line of tangents=5'22 inches, 
 from which we find area='7o8 square feet and wet perimeter 
 
 2-5 feet; R=-'l-?=- 2 83 feet ; n from Table XIX. = -oi2 ; 
 
 
 
 S='02. C is found from Fig. 26=98; RS = '283X'O2 
 = 00566; \/RS = -o75 ; and =98 x '075 = 7-35 feet per 
 second; Q = 7o8 + 7-35 x 60=313 cubic feet per minute. 
 
 The following comparison of the flow of circular sewers 
 running full, calculated from Beardmore's and Kutter's 
 formulae, has been made with assistance as regards the former 
 from the well-executed diagrams of Mr. W. T. Olive 
 Resident Engineer on the Manchester Main Drainage, 
 published by the Inst. C. E. in their minutes of 
 Proceedings, vol. xciii. 
 
 It will be seen that the older formula produces results 
 nearly midway between those by Kutter's rule with the 
 coefficient '015 and -oi for roughness, but that in the larger 
 culverts the discharges by Kutter's formula gradually gain 
 upon those of Beardmore, until they are considerably ahead 
 even with the coefficient '015 for roughness. 
 
 TABLE XX. Comparison of Beardmore's and Kutter's formultc. 
 
 \ 
 
 
 
 Discharge in 
 
 Diameter 
 in inches 
 
 Slope in 
 feet per mile 
 
 Discharge in c. ft. 
 per min. 
 (Beardmore) 
 
 Discharge in c. ft. 
 per min. for 
 brickwork; ='oi5 
 (Kutter) 
 
 c. ft. per min. 
 for glazed- 
 ware or iron ; 
 
 ='OI 
 
 
 
 
 
 (Kutter) 
 
 6 
 
 
 12\ 
 
 9'3 
 
 I5-5 
 
 6 
 
 50 
 
 38 
 
 29-4 
 
 49 
 
 6 
 
 150 
 
 66 
 
 43'4 
 
 727 
 
 24 
 
 10 
 
 540 
 
 490 
 
 
 
 24 
 
 50 
 
 I,2IO 
 
 1,112 
 
 
 
 96 
 
 IO 
 
 17,100 | 20,850 
 
 
 
 96 
 
 15 
 
 21,000 
 
 25 ? 633 
 
 
 
io6 Preliminary Survey 
 
 DISCHARGE FROM TANKS, PIPES, CISTERNS, AND WEIRS. 
 
 The diagrams, Plate II., of theoretical velocity due to 
 different heads will need no explanation. 
 
 They are furnished for use when no slide-rule is at hand ; 
 otherwise it is quicker to work out the velocity by the slide- 
 rule than to scale it on the diagram, and the velocities for 
 fractional values are given with the same rapidity as those 
 of integral values, more accurately than can be scaled from 
 a diagram, and without the interpolation needed with a 
 table. 
 
 Example. What is the theoretical velocity due to a head 
 of 15*65 feet? Place the right hand i of the slide under 
 the upper 15 feet of rule and read the velocity 31-8 feet per 
 second on the lower scale of the rule opposite to 8*03 on 
 the lower scale of the slide. 
 
 Example 2. What is the theoretical head due to a 
 velocity of 135 feet per second? Place the 8-03 on the 
 Jower scale of the slide over the 135 of the rule and read the 
 head 282*5 feet on the upper scale of the rule opposite to 
 the right-hand i of the slide. l 
 
 Rule i. For tanks and cisterns flowing into the open air 
 Let V = the theoretical velocity in feet per second due to 
 head, given on Fig. 28. 
 
 A==area of aperture in square feet. 
 
 D= diameter of circular aperture in square feet. 
 
 C=coefficient of friction due to nozzle or opening 
 given by table. 
 
 VV=actual velocity of discharge in feet per second. 
 
 Q= discharge in cubic feet per minute. 
 
 Then VV=VxC 
 
 and Q=VVx6oAor 
 
 1 If the velocity is wanted in miles per hour instead of in feet per 
 second, use the number 5'475 instead of 8-03. For heads from i to 10 
 feet use the left-hand half of the rule, and from 10 to 100 the right- 
 hand half. 
 
PLATE II. 
 
 II 1 
 
 Ocn 
 
 Formula. -V =8-03 v/H 1 '. 
 
 M- 5-475 VH U ." 
 Where H = head in feet. 
 
 ,, V=vel.' in feet per second. 
 
 ,, M = vel. in miles per hour. 
 
 Note. The vel. in feet per second is the only one which can be obtained from 
 insoection of the diaprnm t-hp vel. in miles ner hour must he taken with dividers 
 
Hydrography and Hydraulics 107 
 
 Values of C for 
 different orifices 
 
 Circular or rectilinear openings in thin iron plate in 
 
 bottom or sides of either vertical or inclined tanks= . '62 
 
 Short tube projecting outwards, bore \ the length . . '80 
 Ditto bore -^ of the length . . . . . .70 
 
 Short tube projecting inwards ..... -68 
 
 Converging tube, outside of tank ; angle 13^ ; velocity 
 
 at narrow end ....... '94 
 
 Diverging ditto, outside of tank ; angle 5 ; velocity 
 
 at narrow end . . . . . . . '92 
 
 Example. What are the velocity and discharge in feet 
 per second and cubic feet per minute respectively in the 
 following case ? 
 
 A tank is maintained by a ball cock with a depth of 
 water of 7^ feet. A short cylindrical tube projects from the 
 bottom of 2\ inches inside diameter and 2 feet i inch long. 
 The velocity due to 7*5 feet head is found by Fig. 28 to 
 be 22 feet per second. C is an intermediate between -80 
 
 and 70, ' of orifice being equal to T V. Interpolating 
 length 
 
 with the slide-rule we find C=7i and VV=22 x 71 = 15-6 
 feet per second. The diameter 2^ inches ='2 08 feet and 
 discharge Q=VV x 47*12 D 2 or=i5'6 x 47*1 2 x '2o8 2 . 
 Ans. VV=i5'6 feet per second. 
 Q=3r8 cubic feet per minute. 
 
 Rule 2. For iron pipes under pressure approx. mean vel. 
 in feet per second = 
 
 coeff. m (Fig. 30) x _ 
 
 V diam. in ft. x total head in ft. 
 
 This approximate rule of the late Mr. Trautwine takes into 
 account velocity head, entry head, and friction head. It 
 was not used by him for very long pipes with low heads. 
 The limit given by him was 1,000 diameters, beyond which 
 he neglects entry head and treats the flow by the formula 
 for an open channel. It is not applicable to very high 
 
io8 Preliminary Survey 
 
 pressures either, but is suitable to ordinary reservoirs having 
 straight iron pipe conduits. 
 
 The diagram Fig. 30 gives values of coefficient ;;/ for various 
 diameters of pipes in feet. The horizontal scale gives values 
 
 of the A / _ _ for which the ordinates are values of m. 
 
 V L + 54 D 
 
 D being diameter in feet, H= total head in feet, L=length 
 in feet. The value for -200 serves for all ratios above it. 
 Intermediate values are to be taken by interpolation. The 
 results are always in excess of those where there is no entry 
 head and all the fall is in the pipe itself. 
 
 Rule 3. Discharge over weirs (Eytelwein). 
 
 Q=discharge in cubic feet per minute. 
 
 L= length of overfall in feet. 
 
 H=head in feet. 
 
 Q=204 L x/jp 
 
 Remark. The head is measured by ascertaining the 
 difference of level between the crest of the weir and the 
 surface of the water before it commences its chute- curve. 
 
 The formula is for discharge over a thin plate or a weir 
 with a sharp edge. A slight current towards the weir makes 
 hardly any difference in the result. 
 
 Example. Required the discharge over a weir in thin 
 plate length 200 feet. H=i'5 feet. 
 
 Q=204 x 200 N/!^^ 74,954 cubic feet per minute. 
 
 Rule 4. (Approximate John C. Trautwine) see Dia- 
 gram, Fig. 31, for C. 
 
 Q=CxLxVxH. 
 
 Where Q= discharge in cubic feet per minute. 
 C= coefficient per diagram. 
 L=length of weir in feet. 
 V= theoretical velocity due to H in feet per 
 
 second from Fig. 28. 
 H=head in feet measured as in preceding rule. 
 
 This formula gives results somewhat less than the pre- 
 ceding one for sharp-edged weirs, and is therefore on the 
 
PLATE III. 
 
 200 
 
 4FT PIPE 
 
 10 -05 
 
 Flow of water ; n pipes. 
 Values of 
 
 /DxTTJ 
 V L + 54l> 
 
 FIG. 30. 
 
 01 
 
 HEAD IN INCHES 36 30 25 20 15 10 5 
 
 Discharge ovei weirs. 
 
 FIG. 31. 
 
 A = crest 2 inches thick. 
 A. A. ^crest sharp edge. 
 A.A.A. = crest 3 feet thick, smooth ; sloping outward and downward from i in 12 t< 
 
 i in 18. 
 A.A.A.A.=crest 3 feet thick, smooth and level. 
 
Hydrography and Hydraulics i 09 
 
 safe side ; it also meets all the ordinary cases by means of 
 the coefficient C with quite a sufficient degree of accuracy for 
 preliminary work. 
 
 Example. What would be the discharge over the same 
 weir described in the preceding example only with a crest 
 3 feet thick, smooth and level ? 
 
 Q=i8 - 4 x 200 x 9 -8 x i -5=54,096 cubic feet per minute. 
 
 Over a sharp-edged weir it would be 67,000 instead of 
 the 74,954 of Eytelwein's formula, but with a crest 2'' thick 
 it would be 80,000 by Trautwine. 
 
 In the case of reservoirs, it is generally easier to deter- 
 mine the discharge over the weir by that of the supply con- 
 duit by Kutter's formula. 
 
 HORSE-POWER OF FALLING WATER 
 
 QH. 
 
 33000 
 
 Where Q= discharge in cubic feet per minute, and H= 
 height of fall or head over a turbine or other motor. 
 
 Example. Over a fall 16 feet in vertical height, 800 cubic 
 feet of water are discharged per minute. 
 Hp= 8ooxi6x62- 5 Hp 
 
 33000 
 
 EFFICIENCY OF WATER-WHEELS 
 
 For undershot wheels x result found as above by '25 to '33 
 breast wheels ,, ,, by -5. 
 
 ., overshot ,, ,, by -5 to 75. 
 
 ., turbines ,, ,, by 75 to '85. 
 
 HORSE-POWER OF A RUNNING STREAM 
 
 HP== QxHx62 :5== . Qoi 
 
 33000 
 
 Where Q= discharge in cubic feet per minute actually 
 impinging upon the float or bucket ; H theoretic head, due 
 to velocity of stream found by Fig. 28. 
 
no Preliminary Survey 
 
 Thus, if the floats of an undershot wheel, driven by 
 current alone, be 5 feet x i foot, and the velocity of stream 
 = 210 feet per minute, or 3^ feet per second, of which the 
 theoretical head is -19 feet. Q=5 square feet x 210=1,050 
 cubic feet per minute ; H='i9 ; 
 
 and HP = I0 5 ox ' 19x62 '5 = - 37 8. 
 
 33000 
 
 The wheels only realise about -4 of this power, from 
 friction and slip. 
 
 The last three rules are from Trautwine's ' Pocket Book.' 
 
 HYDROSTATICS 
 
 Water pressure is always normal to the surface pressed, 
 i.e. in flat surfaces perpendicular to them, and in curved 
 surfaces in line with the radius vector, or, which is the same 
 thing, perpendicular to the tangent at that point. 
 
 Whatever the inclination of the surface or the extent of 
 its immersion or distance of its submersion, the following 
 rule holds good. 
 
 Rule i. Pressure in pounds=62'5 AD. 
 Where 62'5=the assumed weight of a cubic foot of water. 
 A = area of surface pressed in square feet. 
 D= depth of centre of gravity of surface pressed 
 
 below the surface of the water. 
 
 Rule 2 For pressure in pounds per square inch, multiply 
 the depth in feet by -434. 
 
 Rule 3. For tons per square foot, multiply the depth in 
 feet by -0279. 
 
 Rule 4. Total pressure from surface in tons on a section 
 i foot wide=D 2 x '0139. 
 
 For the depth in feet, at which any given pressure exists : 
 Divide pounds per square inch by '434. 
 pounds per square foot by 62-5. 
 tons per square foot by -0279. 
 Plate IV., Fig. 32, gives the normal pressure per square 
 
PLATE IV. 
 
 PRESSURE PER SQUARE FOOT 
 
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 FIG. 32. 
 
 Note i. The left- and right-hand scales have no connection with one another. 
 The Ibs. on the left-hand scale refer solely to pressures per square foot. The tons on 
 the right hand refer solely to total pressure from surface. 
 
 Note 2. If this diagram is in frequent use with dividers, a piece of dull-back 
 tracing-cloth, gummed over it by the four corners, will protect it. 
 
Hydrography and Hydraulics 
 
 in 
 
 foot of any immersed surface due to any given depth 
 of its centre of gravity below the surface of the water 
 thus : 
 
 FIG. 33. 
 
 It also gives the total pressure upon a plane standing 
 upright in the water, such as a sluice-gate, the plane being 
 one foot wide, thus : 
 
 DEPTH 
 
 FIG. 34. 
 
 In order to combine two diagrams in one, the left-hand 
 vertical scale gives normal pressure per square foot in 
 divisions of 100 pounds each, and measures the ordinates to 
 the straight line. The right-hand vertical scale gives tons 
 total pressure with each subdivision =-i ton, and measures 
 the ordinates to the curved line. 
 
 Example i. What is the pressure per square inch on a 
 water-main where the head is 300 feet? Ans. 300 x "434= 
 130 pounds per square inch. 
 
 Example 2. What is the total pressure on a lock-gate 
 20 feet wide, with a head of 8 feet ? Ans. 20 x 8 x 8 x '01395 
 = 17-85 tons, or by Plate IV. = 20 x -9=18 tons. 
 
 To divide a vertical surface under hydrostatic pressure, 
 
1 1 2 Preliminary Survey 
 
 such as a sluice gate, into sections of equal pressure by 
 horizontal lines. 
 
 Let N= number of sections, then there will be N i 
 lines, the distance of which from the top will be D,, D 2 , D 3 , 
 DJJ.!. Let H= height of water producing the pressure. 
 Then 
 
 D.-lj|VN (N-l). 
 
 Example. Let H=2o feet and N=4. 
 *>i=5>/4; D 2 =5N/8; D 3 =5N/i2. 
 = 10 ; =14-1 ; =17-3. 
 
 COST OF DREDGING 
 
 Price of labour 4s. per day, no allowance for interest or 
 depreciation of plant. 
 
 In from 5 to 10 feet of water, $d. to 6d. per cubic yard. 
 
 In from 10 to 20 feet of water, 6d. to yd. per cubic yard. 
 
 These prices include the removal of the material -j to i- 
 a mile. 
 
 DREDGING PLANT 
 
 The dredging plant used at Leith Docks which remove 
 about 110,000 tons of silt every year cost i3,ooo/. 
 
 The dredging plant used at St. Nazaire, Loire, France, to 
 remove 400,000 cubic yards of silt per annum cost 29,8007. 
 
 The dredging of soft material in small quantity can be 
 done with a bag-spoon (Fig. 35). This is simply a bag, B, 
 of canvas or leather with an iron ring at its mouth. It has 
 a fixed handle, h, by which it is thrust down into the mud ; 
 another man draws it along by therope g. and a third hauls 
 it up when full by the rope e (Trautwine). 
 
Plydrography and Hydraulics 1 1 3 
 
 FIG. 35. 
 
 BORINGS 
 
 in common soils or clay may be made 100 feet deep in a 
 day or two by a common wood -auger \\ inches in diameter 
 turned by two to four men with 3-feet levers. This will 
 bring up samples (Trautwine). Weale's series, ' Well Sink- 
 ing and Boring,' contains a detailed description of boring 
 tackle. 
 
 CONCRETE 
 
 At Aberdeen harbour 16,000 cubic yards of concrete 
 were deposited in jute bags at a cost, including plant, super- 
 intendence, and maintenance, of i/. 9^. \od. per cubic yard. 
 
 13,000 cubic yards of concrete were deposited in mass at 
 a cost of i/. 9-f. 6d. per cubic yard. The concrete consisted 
 of i part of Portland cement, 3 of sand, and 4 of shingle. 
 
 At Port Said breakwater, concrete in blocks weighing 
 about 20 tons was deposited, forming two breakwaters, one 
 9,800 feet long, the other 6,233 f eet l n g> at a cost of i^ S s - n 
 per cubic yard. The concrete was composed of 3^ of sand 
 and gravel to i of shell lime, made in boxes and left two 
 months to set before being used. 
 
 DOCK WALLS 
 
 The cost of the Hull dock-wall, 43 feet deep to bottom 
 of foundations, cost 1 9/. ys. per lineal foot ; the South West 
 India docks, 41 feet deep, i2/. \os. ; the Penarth Extension,' 
 
 i 
 
114 Preliminary Survey 
 
 52 feet deep without excavation, 261. is. ; the cost of the 
 Dublin quay- wall, 47 feet deep, 4o7. per lineal foot. 
 
 The cost of the West India Dock wall per cubic yard 
 was i2s. 6d., of the Penarth Extension wall 17^., of the 
 Portsmouth wall i7. 2s., of the Chatham wall 75. io^7. 
 
 COST OF DOCKS PER ACRE 
 
 The Antwerp docks, of about 50 acres, having about 
 3,700 lineal yards of quay and a basin lock with entrances 
 59 feet wide, depth of sills at entrance 23 feet, cost about 
 4,8oo/. per acre of water area. 
 
 The Joliette basin at Marseilles, of 54 acres contained by 
 2,300 lineal yards of quay, cost 640,0007. or nearly i2,ooo/. 
 per acre of water area. 
 
 At Leith, the old docks cost 28,5007. per acre, the 
 Victoria Dock 27,0007., the Albert dock 20,4007., and the 
 Edinburgh dock 24,0007. per acre. 
 
 The Chicago breakwaters of cribwork filled with rubble, 
 30 feet wide and 30 feet deep to bottom of foundations, cost 
 227. i8.r. per lineal foot. 
 
 The New York quay walls of pilework hearting filled 
 with rubble, cut stone face-wall protected at toe by riprap 
 and riprap backing, 43 feet deep to bottom of excavation, 
 cost 497. i6s. per lineal foot. 
 
 The New York jetties or landing stages for ocean-going 
 steamers of timber un-creosoted cost about $s. 6d. per square 
 foot of area. 
 
 GRAVING DOCKS 
 
 Twenty-five of the Liverpool Graving Docks, having an 
 aggregate length of 12,490 feet, cost 940,0007. =about 757. 
 per lineal foot of floor length. 
 
 Graving docks have been built for vessels of 2,000 tons 
 for under 2o,ooo7. 
 
Hydrography and Hydraulics 1 1 5 
 
 The Somerset Dock at Malta, 468 feet long, cost 
 i5o,ooo/.,=about 3207. per foot of floor length. 
 
 At per cubic yard of capacity they vary from i/. to 4/., 
 the most recent docks at Portsmouth averaging about 2/. 
 
 The sluicing basin at Honfleur, having an area of 143 
 acres, cost 2oo,ooo/. 
 
 Warehouses of brick with iron columns and fireproof 
 doors cost from 4^. to %d. per cubic foot of total capacity. 
 
 I 2 
 
1 1 6 Preliminary Survey 
 
 CHAPTER IV 
 GEODETIC ASTRONOMY 
 
 THE title of this chapter might convey the idea of a much 
 more comprehensive treatment of the subject than would 
 be possible or desirable in a book of the kind. The rough 
 astronomical calculations which lie within the average 
 surveyor's attainable knowledge or within the power of his 
 instruments can hardly be justly termed astronomy ; never- 
 theless in its most elementary form the subject matter of this 
 chapter will be what it is stated, and an attempt has been 
 made in the glossary and also in this chapter to describe the 
 principles in a manner which will be intelligible to anyone 
 possessing but very little previous knowledge of the subject. 
 
 Surveyors who desire to pursue the study of astronomy 
 further will find all they seek in Chauvenet's ' Spherical and 
 Practical Astronomy,' clearly and simply explained. 
 
 If a work of a more elementary, and less expensive cha- 
 racter is desired, the volume on ' Practical Mathematics ' in 
 Chambers's Educational Course, will be found both useful 
 and handy. 
 
 Some of the most commonly used formulae of plane and 
 spherical trigonometry are given in the appendix to the pre- 
 sent volume, but the demonstration of them is necessarily 
 omitted; they will nearly all be found, however, in Chambers's 
 ' Mathematics.' 
 
 Geodetic astronomy differs but little from nautical astro- 
 nomy. It is the science of determining by observation and 
 calculation of celestial phenomena the position and course 
 on land. It differs from nautical astronomy only in affording 
 
Geodetic Astronomy 117 
 
 the facilities of ' terra firma ' for some operations which are 
 not practicable at sea, and in being generally debarred from 
 the use of those which depend upon the sight of the natural 
 horizon. 
 
 In a treatise on Preliminary Survey the science of geodesy 
 cannot with propriety be treated fully, neither can the 
 analysis of the problems of plane and spherical trigonometry 
 involved therein be closely pursued ; but the object of this 
 chapter will be to explain general principles sufficiently to 
 make the formulae intelligible, and to furnish examples of 
 every problem likely to be useful to the pioneer. A great 
 deal of explanation has been placed in the glossary, which 
 has been arranged alphabetically to be easily referred to, and 
 should be studied before commencing this chapter. 
 
 The position of any point on the earth's surface is deter- 
 mined astronomically by finding its relation to the pole and 
 to some arbitrarily chosen meridian, and these two relations 
 are termed latitude and longitude. 
 
 LATITUDE 
 
 Let us first take the latitude. Everyone has learnt how 
 to find the pole star by the ' pointers ' ; they perhaps also 
 know that the altitude of the pole star represents roughly 
 the latitude of the place. We can find our place on a map 
 by measuring the distances of latitude and longitude on the 
 circles described upon it for that purpose, but the only fidu- 
 cial point given us by nature as a starting-point from which 
 to map the world is the north pole. The map of the earth 
 is made from the map of the heavens, but the map of the 
 heavens is first made from the axis of the earth's rotation. 
 
 It is the discovery of a stationary point in the heavens 
 called the celestial pole which determines the position of the 
 terrestrial pole and terrestrial equator ; and conversely the 
 position of the observer relatively to the celestial pole deter- 
 mines the position relatively to the terrestrial equator. 
 
1 1 8 Preliminary Survey 
 
 A moment's thought will serve to grasp this. If I were 
 standing at the north pole, the celestial pole would be over- 
 head, because I should be standing in the axis of the earth's 
 rotation. If I were at the equator, it would appear on the 
 horizon ; consequently if I measured the altitude of the 
 pole star it would be in the first case 90, in the second case 
 o. (see Altitude, Glossary). And these are the respective 
 latitudes, measured from the terrestrial equator. It is evi- 
 dent, then, that an altitude of a celestial body taken when 
 passing the meridian will, if we know its distance from the 
 pole, enable us to determine the latitude by a simple sub- 
 traction sum. 
 
 And now, leaving for the present the subject of latitude, 
 let us touch upon the principles on which the calculations 
 of terrestrial longitude are based. 
 
 LONGITUDE * 
 
 Unlike latitude, which as we have seen can be deter- 
 mined independently of almanacs or chronometers, longi- 
 tude depends upon an arbitrary fiducial point. Both in the 
 heavens and earth there is nothing fixed by which to deter- 
 mine the easterly or westerly positions without either a time- 
 table or a watch or both. 
 
 The earth performs a rotation in twenty-four hours of 
 sidereal time (see ' Sidereal Time,' Glossary). A star 
 which crossed the meridian of Greenwich to-day at mid- 
 night will cross the iSoth meridian, that is over the Fiji 
 Islands, at twelve sidereal hours later, and will recross 
 Greenwich meridian twelve hours later still. Any celestial 
 phenomenon happening, independently of our meridian, 
 to the star, such as an occultation of it by the moon, would, 
 although in itself an instantaneous phenomenon, be seen at 
 different local times in different places, and the difference of 
 local time corresponds exactly with that arc of the earth's 
 rotation which would bring successively the two points of 
 1 See Glossary. 
 
Geodetic Astronomy 119 
 
 observation under the same celestial meridian ; in other 
 words, is equal to the difference of terrestrial longitude. 
 
 Phenomena such as those just described are very useful 
 in their place, but the chief basis of calculation is the sun, 
 whose transit across the meridian can be observed with the 
 utmost exactness, and having by chronometer the time when 
 he crossed the meridian at some known place, we can at 
 once, without any mathematics, determine the difference of 
 longitude by observing his transit and reducing the differ- 
 ence of time to difference of arc. We use the meridian of 
 Greenwich as our fiducial point, and perhaps the time is 
 not far distant when all the nations will agree upon one 
 common meridian. 
 
 Reserving further explanation of the methods available 
 to the surveyor for ascertaining his longitude, we will add to 
 these prefatory remarks an attempt to explain the curious 
 fact which travellers puzzle over when they first cross the 
 Pacific Ocean ; viz., that of 'losing' a day in going one way 
 and gaining it when going the other way. 
 
 As the sun appears to go round from east to west, a 
 place east of Greenwich, say Paris, will have sunrise a little 
 before Greenwich, or west, as Dublin, a little afterwards. In 
 other words Paris time is ahead of, and Dublin time behind, 
 Greenwich. If we could travel round the world across 
 America as quick as lightning, and left Greenwich at mid- 
 day of January i, 1890, when passing New York we should 
 find the early risers at breakfast on New Year's morning. 
 At San Francisco and Honolulu they would still be all in 
 bed, and at Fiji if we kept to the same calendar we should 
 find them at midnight of December 31. Fiji is just half 
 way round, being on the iSoth meridian. Instead of its 
 being midnight of New Year's eve, however, it would be 
 styled in that place midnight of January i, and the reason 
 for this we will explain by supposing that instead of going 
 via America, we had gone via Calcutta in our lightning 
 flight. Leaving Greenwich at the same time, viz. midday 
 
I 20 Preliminary Survey 
 
 of January i, passing Calcutta we should find them 
 already at their New Year's evening dinner. At Melbourne 
 they would be just going to bed, and at Fiji it would be 
 midnight of January i. It will be seen therefore that the 
 Fijians might keep their time twelve hours back or twelve 
 hours ahead, according as they chose to consider them- 
 selves in the Western or Eastern Hemisphere. Conse- 
 quently they have the option of one day's date. The writer 
 has been informed by a missionary that the keeping of two 
 Sundays running has been of frequent occurrence in the 
 palmy days of evangelisation upon that group, but they have 
 finally elected to consider themselves in the Eastern Hemi- 
 sphere in order to be in the same calendar as Australasia. 
 
 Now to take the passage of a vessel circumnavigating 
 from Greenwich eastwards ; she would keep putting on her 
 time every day at noon until her time was twelve hours 
 ahead of Greenwich at Fiji, and correct with local time and 
 date there. A vessel circumnavigating westwards from 
 Greenwich would have been putting back her time all the 
 while, until at Fiji she was twelve hours behind Greenwich, 
 and consequently correct with local time there, but one day 
 behind in date. 
 
 If they both happened to arrive at Fiji at midnight of 
 January i, 1890, the eastward navigator would be correct 
 to time and date, but the westward navigator's date would 
 be midnight of December 31, consequently when he passes 
 over into the Eastern Hemisphere he will have to skip a day 
 and call it January i. The eastward navigator passing 
 into the Western Hemisphere will have to put back a day 
 and call it December 31. 
 
 CLASSIFICATION OF METHODS 
 
 There are three different circumstances in which a surveyor 
 is ordinarily placed as regards his astronomical work, modi- 
 fying the methods which he can use and the degrees of 
 accuracy which he can obtain. 
 
Geodetic A stronomy 1 2 1 
 
 1. When starting from the coast and proceeding with a 
 carefully measured telemetric survey inwards. 
 
 2 . When starting from the coast and making a route- 
 survey inwards. 
 
 3. When carrying on an inland survey and having to 
 locate his starting-point. 
 
 The most disadvantageous circumstances, short of having 
 no instruments to all, in which a surveyor can be placed are 
 to be 
 
 1. Far inland and out of telegraphic communication. 
 
 2. Possessed of a poor watch. 
 
 3. Unable to revisit his points of observation. 
 
 4. Compelled to move forward rapidly in an easterly or 
 westerly direction. 
 
 As will be explained presently, the surveyor is very 
 dependent upon Greenwich time for his longitude. This 
 is preserved upon important geodetic work with the utmost 
 care. A number of chronometers in padded cases, swing- 
 ing on patent joints, wound every day at the same time 
 by the same individual with the same number of turns, 
 and constantly compared with one another, enable the 
 observer to obtain his longitude with the same accuracy as 
 his latitude. Very different is the case with the preliminary 
 surveyor. He is rarely able to transport more than one 
 chronometer, he often does not possess even a semi-chrono- 
 meter watch, and so when out of range of the telegraph has 
 to resort to what are termed * absolute methods' of deter- 
 mining his longitude. Apart from the great inconvenience of 
 transporting high-class chronometers whilst on rapid tran- 
 sit, their very delicacy renders them less useful under 
 rough usage than semi-chronometer watches, two or more 
 of which in a surveying party are the best instruments 
 available. ' Absolute methods ' of finding the longitude are 
 either performed with the sextant by 'lunar distance' or 
 by ' lunar altitudes ; ' with the transit as in the method 
 termed 'moon-culminating stars'; or with the telescope 
 
122 Preliminary Survey 
 
 alone as by eclipses of Jupiter's satellites or the occultation 
 of a fixed star by the moon. Sextant observations will 
 only be explained briefly, in principle ; the surveyor rarely 
 possesses anything of that description larger than a pocket 
 sextant, which is of no use for such purposes. 
 
 In the first case, when starting from the coast on a 
 telemetric survey, the daily traverse is reduced to latitude 
 and departure by slide-rule or by a table, such as that in 
 Chambers's mathematical tables. A very close check is thus 
 kept on the astronomical work, and the longitude and 
 latitude by account is used like the dead reckoning at sea 
 for determining the argument of the daily observation. The 
 daily observations should be if possible four. 
 
 One for time to correct the watch's rate. 
 
 One for longitude by watch. 
 
 One for latitude by fixed star. 
 
 One for azimuth by sun or star. 
 
 If any celestial phenomena are available for ' absolute 
 longitude' they should not be missed. 
 
 If only one observation can be taken it should be that 
 for azimuth, and it can be made en route from the sun 
 without interrupting the work for more than five or ten 
 minutes. No continuous telemetric traverse should be 
 carried forward into an unknown country without a daily 
 check upon the direction. A slip of one degree which may 
 be set down to change of magnetic deviation may produce 
 an error of a thousand feet in a single day's work. 
 
 In the second case, when starting on a route survey 
 from a known point, the observations should be the same 
 but more strictly kept up every day, because the survey 
 covers so much more ground, and the check by latitude 
 and departure is much rougher ; otherwise the principle is 
 the same. 
 
 In the third case, when the location of the starting-point 
 has to be determined independently, the surveyor must 
 take time to get reliable data to begin with ; he should stay 
 
Geodetic Astronomy 123 
 
 two or three days in the same place to take and reduce a 
 series of observations. The first thing is to determine the 
 meridian, which is best done by equal altitudes of a fixed 
 star. From this, local mean time is obtained by observing 
 the transit of a known star. Latitude is then had from the 
 pole star or a circumpolar star such as one in the Great Bear 
 or Cassiopea. Longitude is determined by the successive 
 transits of the moon and a fixed star close together, or else 
 by an eclipse of Jupiter's satellites or the occultation of a 
 star by the moon. This is supposing he has not got Green- 
 wich time by chronometer to start with. 
 
 Three days of favourable weather and careful work will 
 give the surveyor who has nothing more than a six-inch 
 transit to work with, a very fair approximation to his true 
 geographical position. 
 
 The starting-point, once fixed, should remain as the 
 basis of the whole survey. It is probable that errors will be 
 found later on by bringing greater precision to bear upon it, 
 but in order to preserve symmetry in the work the starting- 
 point should be assumed as correct for graduating the sheets, 
 so that any error found afterwards will be a constant through- 
 out the work. The local mean time and Greenwich date 
 at starting-point once determined, can be recovered at any 
 time by returning to the spot and observing a star's transit. 
 
 The constant check upon the watch by means of 
 sidereal time cannot be too strongly recommended. No 
 mathematics are needed, and no time wasted in waiting. 
 By it the poorest time-keeper can be made valuable, and the 
 best chronometers are none the worse for being checked by it. 
 
 OBSERVATIONS FOR DETERMINING THE TRUE MERIDIAN. 
 
 First and foremost amongst the uses of the stars to the 
 surveyor is the determination of the true north and south 
 line or meridian of the place. He needs it for running his 
 survey line true, and he needs it above all things for his 
 other astronomical calculations. 
 
124 Preliminary Survey 
 
 There are still old-fashioned people who think the 
 magnetic compass is quite near enough for a railway, and 
 star-gazing a superfluous luxury ; they are generally of the 
 same sort who think the chain is the only practical way of 
 surveying, and a transit instrument only a means of saving 
 lazy engineers their due amount of manual labour. We will 
 not enter into controversy with them for fear of being hit 
 too hard, but proceed to touch upon the various ways of 
 determining the meridian. 
 
 A rough and ready method has been explained in the 
 chapter on route-surveying at p. 40. 
 
 The solar compass as described on p. 328 gives the true 
 bearing of any points observed, and the true north and 
 south line. 
 
 Some surveyors have found very good results attainable 
 with simple sun-shadows reduced to true north and south 
 line by Davis's tables from the time of day recorded opposite 
 to each shadow. 
 
 All such operations, though useful in their place, are 
 very approximate. 
 
 A watch keeping apparent time, and held so that the 
 hour hand points towards the sun, will roughly give the 
 meridian, midway between the sun and the xn of the 
 watch. 
 
 MERIDIAN BY EQUAL ALTITUDES OF A STAR 
 
 The simplest of the really accurate methods is by obser- 
 vation of a fixed star at the same altitude, east and west of 
 the meridian. The movement being perfectly uniform, 
 the successive bearings are taken from any clearly defined 
 terrestrial point, and the mean bearing corresponds exactly 
 with the meridian. 
 
 A good plan is to adjust the transit for collimation and 
 bubble in daylight upon a good solid position, then to ad- 
 just the horizontal limb so that the vernier and compass 
 needle both stand at zero. Then, clamping the external 
 
Geodetic A stronomy 125 
 
 axis and releasing the parallel plates, take the exact bearing 
 on the same vernier of some well-defined terrestrial point at 
 a considerable distance, which will be its magnetic bearing. 
 On no account use the compass of the instrument for this 
 bearing. 
 
 Cover over the instrument until the stars begin to appear. 
 Choose a bright star as early as possible and about 30 to 
 45 from the meridian either north or south and about 20 
 to 40 above the horizon. 
 
 If it is a circumpolar star taken to the north, be careful 
 to note from its position relatively to the pole whether it is 
 on its way to upper or lower culmination. 
 
 If on its upper path, the first altitude must be observed 
 to the east like the southern stars, but if on its lower path 
 it must first be observed to the 
 westward as shown on Fig. 36. 
 
 This method requires no 
 mathematics whatever. Book 
 the altitude of the star from the 
 vertical limb in its first position, 
 and opposite to it the magnetic 
 bearing from the horizontal limb ; 
 not from the compass. Unclamp 
 the vertical arc and bring the 
 telescope bubble to the centre of 
 its run, and book any index error which may have crept in 
 since the first adjustment. Set the vertical arc back to the 
 altitude of the star, unclamp the horizontal limb and direct 
 the telescope to the approximate second position of the star, 
 that is to say, about the same angle from the meridian, only 
 on the opposite side. The variation of the compass is 
 known roughly all over the world from a map of equal 
 variations such as that in ' Hints to Travellers ' referred 
 to on p. 369, so that the telescope can be placed within 
 a degree or so of its proper place. When the star begins 
 to get near the field, test the vertical arc with the bubble 
 
126 Preliminary Survey 
 
 as before, and if any index error be found, correct it, and 
 clamp the vertical arc again at the same altitude as the first 
 position and free from index error. 
 
 Release the horizontal limb, but on no account touch the 
 vertical limb. Bring the telescope right under or over the 
 star, and when it enters the field clamp the horizontal limb 
 with the star on the central vertical hair, and keep it in that 
 position with the tangent screw of the horizontal limb until 
 it reaches the horizontal axial hair. 
 
 Book the second bearing from the vernier of the hori- 
 zontal limb, and the mean of the two bearings is the mag- 
 netic bearing of the meridian. 
 
 If the observation has been taken to the north, and the 
 bearing of the meridian is greater than zero, or if taken to 
 the south and it is greater than 180, the variation of the 
 compass is west ; and if less than 360 or 1 80, as the case 
 may be, the variation is east. 
 
 Example. Observed Spica at same altitude to the south. 
 
 Mag. bearing in first position . . 155 08' 40" 
 ,, ,, second ,, . 239 39' 40' . 
 
 2)394 48' 20 
 
 Mag. bearing of meridian . . 197 24' 10" 
 
 1 80 o' o" 
 
 Variation of compass \V. . . . 17 24' 10" 
 
 The directrix chosen was a lightning conductor on a 
 large building about 500 feet away, whose magnetic bearing 
 was o , 
 
 Applying the variation found . 17 24' 10' 
 
 True bearing of directrix - . . 216 27' 50" 
 
 Thus when we wish to place the transit in the true 
 meridian we place it in adjustment for collimation and 
 bubble, and clamp the horizontal limb at 216 27' 50". We 
 then release the external axis and adjust the intersection of 
 
Geodetic A stronomy 127 
 
 the cross hairs to the directrix and clamp the external axis. 
 We then release the parallel plate and set the vernier at o 
 or 1 80, when the telescope will be in the true meridian of 
 the place. 
 
 The disadvantage of this method is that it has to be 
 done at night time. In high latitudes there is twilight all 
 through the night during summer, and the stars are hard to 
 see. It takes a long while also, although plotting and other 
 kinds of work may be done during the interval between the 
 two stellar positions. 
 
 MERIDIAN BY CIRCUMPOLAR STARS 
 
 Another method, which may be applied to rough approxi- 
 mations with a plummet string, but which is a very accurate 
 and convenient observation when performed with a transit, 
 is by watching for two circumpolar stars in the same vertical. 
 It requires the knowledge of the latitude, which can be 
 
 a is Dubh ; /3 is Merak ; -y is Phacl ; 8 is Megrez Al Dub ; 
 e is Alioth ; is Mizar al Inak ; rj is Al Kaid. 
 
 obtained at the same time from Polaris with the transit (see 
 P- I 53)> or fr m two P a i rs of stars graphically with a plummet 
 as described on p. 154. 
 
 If the azimuth is taken with a plummet-string, the 
 point of observation is first fixed by a peg with a nail in it, 
 over which the observer stands with his plummet pointing 
 to the nail. An assistant has another stake in hand con- 
 sisting of a piece of 9" x i ' board pointed at the end. When 
 the stars are very nearly in the same vertical the wide stake 
 
128 Preliminary Survey 
 
 is driven, and at the precise moment the assistant puts in a 
 nail to the directions of the observer ; a bullseye lantern is 
 needed to guide the alignment of the nail. The angle of 
 the line connecting the two nails may then be determined 
 by a very simple logarithmic sum, as will be presently 
 described for the accurate method when taken with the 
 transit. 
 
 Adjust the transit as already described (p. 125) with the 
 magnetic north as a starting-point, and when the two stars 
 are in the same vertical book their magnetic bearing. This 
 is all that has to be done with the instrument. 
 
 The azimuth of the stars or angle from the pole is then 
 determined in the following manner for two stars in Ursa 
 Major. 
 
 In Fig. 37 Z represents the zenith, P the pole, HZPR 
 the meridian of the place, and BC the stars ; ZBCD is the 
 great circle passing through the zenith and the stars when 
 the latter are in the same vertical. ZP = 90 PR, i.e. 
 the co-latitude ; hence sin ZP = cos lat. Angle Z in triangle 
 PZC is the azimuth ; PBC is a spherical triangle which 
 alters its form very slowly because the fixed stars BC only 
 move a few seconds a year. In the spherical triangle ZPC 
 we have ] Sin Z (azimuth) : sin b : : sin C : sin ZP ; or in other 
 words sin azim x cos lat = sin b x sin C. This product 
 sin b x sin C is a constant for the whole year within a 
 few seconds of arc. Any two stars can be chosen in the 
 Great Bear, Cassiopea, Draco, or other suitable constella- 
 tion, and the angle C worked out by the equation given in 
 the Appendix. 
 
 The constants for the two well-known stars /3 and e 
 Ursse Majoris are given for years 1890 to 1900 at p. 378 
 of Appendix. This familiar constellation is shown on Fig. 
 37, B as /3 or Merak. It is the farther one of the two 
 * pointers,' and Alioth is the third star from it in the train. 
 
 Alioth is useful for knowing the position of the pole 
 1 See Appendix. Formula 78 Rem. 
 
Geodetic A stronomy 1 29 
 
 star, see ' Alioth,' Glossary. The reader should also study 
 Fig. 122 on p. 395, showing the relative position of all the 
 important circumpolar stars. 
 
 Example. In latitude 51 16' 45" observed ft and e 
 Ursae Majoris in the same vertical and! west of the meridian, 
 the magnetic bearing being 318 27'. 
 
 Tabular constant for 1890 + 10 . . . 1972910 
 Cos latitude 9'79 62 5 
 
 Sin azimuth 58 57' . . . . 9'93 z8 5 
 
 360 
 
 58 57' 
 
 True bearing 301 03' 
 
 Magnetic 318 27' 
 
 Variation of compass . . . . 17 24' W. 
 
 The meridian is found from a previously observed terres- 
 trial object as already described, p. 126. 
 
 The constants for two other stars, not so well known but 
 easily identified from the sidereal chart on p. 393 are given also 
 for years 1890 to 1900 on p. 378 of Appendix. As there is 
 a considerable time interval between their right ascension 
 and that of the Great Bear, they will serve when the other 
 is not obtainable. 
 
 MERIDIAN BY TIME INTERVAL OF CIRCUMPOLAR STARS 
 
 A very delicate adjustment of the instrument in the 
 meridian, particularly suitable for a test when it has been 
 previously adjusted as close as possible by the foregoing, is 
 to watch the culmination of two circumpolar stars differing 
 nearly twelve hours in right ascension, such as /3 Cassiopeae 
 in R. A. 3 min. 21*77 sec -> an( l 7 Ursae Majoris in R. A. 
 TT hours 48 min. 05*84 sec. 
 
 K 
 
130 
 
 Preliminary Survey 
 
 If the instrument is set to the true meridian and moving 
 in a true vertical plane, these stars will transit at a sidereal 
 interval of 15111. 15 -935. sidereal or 15111. 15-683. mean time, 
 but if the line of sight be to one side, 
 as shown by the dotted line, the in- 
 terval will be less or more. In the 
 case shown the error would cause 
 the culmination to be nearly simul- 
 taneous. In the tacheometer de- 
 scribed on p. 307, the micrometer 
 enables the rate of travel in azimuth 
 to be accurately measured, and by 
 a simple proportion the error is 
 eliminated. 
 
 MERIDIAN BY POLE STAR 
 
 Another and extremely useful method is by an observa- 
 tion of the pole star at elongation. This luminary per- 
 forms an apparent orbit of an extremely small radius 'round 
 the pole, but having twenty-four hours to do it in like the 
 rest of the stars. He appears to move straight up and down 
 for a considerable time when at elongation, and to move 
 horizontally when at his upper and lower culmination. 
 He is consequently very suitable for observation at elonga- 
 tion for azimuth, and at culmination for latitude. There is 
 a table given in the N. Aim. for rinding the latitude by an 
 observed altitude of Polaris, out of the meridian. It requires 
 the local mean time to be known at least with some approach 
 to accuracy ; the maximum error produced by an error of 
 a minute of time is about half a minute of latitude. By 
 using this method the latitude and meridian can be deter- 
 mined simultaneously without any mathematics. 
 
 The polar distance of the pole star for January i, 1891, 
 will be i 16' 23'i" with an annual variation of 18 "9". Its 
 
Geodetic Astronomy 131 
 
 azimuth for any latitude when at elongation is given by 
 formula 59, p. 376. 
 
 sin azimuth=* in P Qlar ^stance xrad 
 sin co-latitude 
 
 Example. In latitude 51 16' 45" N. observed Polaris 
 at eastern elongation. Required the azimuth. 
 
 Log sin polar dist 8-3467853 
 
 Rad 10- 
 
 18-3467853 
 Log sin co-latitude ..... 97962456 
 
 Log sin azim 2 2' 9-21" . . . . 8-5505397 
 
 To find the mean time at place when the elongations will 
 take place is a time problem, which subject we must not 
 forestall. To be done exactly, the hour angle corresponding 
 to the azimuth should be calculated by formula 70, p. 376 : 
 
 Sin hour angle = cos azim -=- cos polar dist. 
 
 The polar distance being so small, the hour angle is 
 within an angular minute or two of the complement of the 
 azimuth. Thus in the preceding example the hour angle 
 would be 87 57' 52-9", which only differs by 2-1" from the 
 complement of the azimuth. 
 
 This is a sidereal interval which reduced to mean time 
 by rule on p. 412 gives the mean time from the culmination. 
 The time of culmination is found from Whitaker or N. A. by 
 rule on p. 412. 
 
 The time of elongation ranges according to latitude from 
 5 hours 49 minutes to 5 hours 54 minutes from the cul- 
 mination, which is near enough for ordinary purposes. 
 
 The following short table of logarithmic values of sin 
 polar distance of Polaris will reduce the calculation to the 
 finding and subtracting the log sin co-latitude. 
 
 TABLE XXL Log Sine of Polar Distance of Polaris 
 
 1891 
 1892 
 
 1893 
 1894 
 1895 
 
 8-34678 
 8-34496 
 
 8-343I3 
 8-34130 
 
 8-33947 
 
 1896 
 1897 
 
 1899 
 
 1900 
 
 8-33579 
 8-33395 
 8-33211 
 8-33027 
 
 K 2 
 
132 Preliminary Survey 
 
 Generally speaking the position of the meridian is only 
 required to the nearest minute of azimuth. This can be 
 done in one minute of time by the slide-rule. 
 
 Example as above. By slide-rule with the sine-scale in its 
 initial position we find opposite to sin 38 44' (the co- 
 latitude) the value '626. Extend the sine-scale to the right 
 until the sine polar distance i 16' is under the -626. 
 Then* the right hand i of the rule will be found over the 
 required angle of 2 2'. 
 
 The approximate time of Polaris culmination and other 
 stars is given in ' Hints to Travellers,' p. 153. 
 
 All the foregoing nocturnal observations possess the 
 great advantage of regularity of movement in the celestial 
 bodies observed ; it is often necessary, however, to obtain the 
 true meridian en route, and the writer has found himself 
 compelled to take observations for azimuth more than once 
 in the da} 7 , in order to produce anything like accuracy. 
 The case was one of peculiarly sharp ravines in a densely 
 wooded country, so that the bases were necessarily very short. 
 
 The sun is the great stand-by for such operations as 
 these. There is no mistaking him, and if he were only a 
 little more regular in his movements, he would be all that 
 could be desired. Nevertheless, with a little calculation, 
 the 
 
 SOLAR AZIMUTH 
 
 is the handiest and best observation, all things considered, 
 that the surveyor has at his command. The instrumental 
 work is done in a very few minutes ; the calculation takes 
 about half an hour. The formula is applicable also to the 
 pole star or any other celestial body. 
 
 On Fig. 127 of Glossary the angle at Z in the triangle 
 SPZ is the supplement of the azimuth SZS' taken from 
 the south point. 
 
 Having the three sides, we can by formula 75 at p. 377 of 
 
Geodetic A stronomy 1 3 3 
 
 Appendix (expressed logarithmically, and remembering that 
 
 - =cosec) find the angle Z. 
 sin 
 
 Our data are : First, the latitude of the place approxi- 
 mately. This is usually computed from the latitude and 
 departure of the day's run as a correction to the observed 
 latitude of the day before (see p. 173) ; or if in possession 
 of previously made maps of sufficient accuracy we can scale 
 it from them. 
 
 Secondly, the declination, which we have from the 
 almanack for Greenwich, and which we reduce to local time 
 from the longitude by account. 
 
 Thirdly, the altitude of the sun. The three sides of the 
 triangle SPZ, Fig. 127, are all of them complements of these 
 data. 
 
 PZ=co-latitude 
 
 PS = polar distance 
 
 SZ= co-altitude 
 whence by No. 75, p. 377 
 
 2 log cos \ Z=log sin S + log sin (S-SP) + log cosec 
 ZP + log cosec SZ 20 
 
 Example, At Sevenoaks, April 30, 1890, in latitude by 
 account 51 16' 45 '' N., observed the altitude of and 
 magnetic bearing of JO with transit theodolite. Required 
 the true azimuth, astron. bearing from N. point, and varia- 
 tion of compass, Greenwich time being known by chrono- 
 meter. 
 
 hrs. min. sec. 
 
 Times by watch . < 4 2 2O 
 
 i 4 4 45 
 
 2)8 7 5 
 
 4 3 32-S 
 
 Watch slow on Greenwich mean time . o 2 19 
 
 Greenwich time of observation . . 4 S Si'S 
 
134 Preliminary Survey 
 
 To FIND THE CO-LATITUDE PZ 
 
 Latitude by account . . . .51 16' 45" 
 
 90 
 
 PZ 38 43' IS' 
 
 To FIND THE CO-DECLINATION OR POLAR DISTANCE 
 
 Decl. at noon by ' N. A.' . . 14 50' 42" N. 
 Diff. for 4 hours 5 minutes . . . + o 3' 5-5" 
 
 Decl. at time of observation . . 14 53' 47-5" 
 
 90 
 
 SP ....... 75 06' 12-5" 
 
 Note. If the declination had been south, SP would 
 have been =90 + declination. 
 
 To FIND THE CO-ALTITUDE OR ZENITH DISTANCE 
 
 Altitudes '> 
 
 2)56 5?' 5o". 
 
 28 29' 25" 
 
 Refraction (see Glossary) . . . . o l' 44" 
 Semidiameter (N. A.) . . . . + o 15' 55" 
 Contraction ,, . . . . - o o' i" 
 
 Parallax (see Glossary and N. A. ) . . + o o' 9" 
 
 True altitude of centre . . . .28 43' 44" 
 Whence SZ = 90 -28 43' 44" = 6i 16' 16" 
 
 and S = -=87 32' 517". 
 
 Log sin S . . . = 9-9996021 
 
 + logsin(S-SP) . . = 9333435 1 
 
 log cosec ZP . . =10-2037544 
 
 + log cosec SZ . . =10-0570480 
 
 39-5938396 
 
 20 
 
 Log cos I Z . . ; . = 9-7969198 = 51 12' 28 
 
Geodetic A stronomy 1 3 5 
 
 Whence Z . . . = 102 24' 56" 
 1 80 
 
 Azim. from south point . = 77 35' 04' 
 1 80 
 
 Astron. bearing from north point = 257 35' 04" 
 
 The magnetic bearings of the sun's |Q at the two altitudes 
 were taken at the same time, as described on p. 126. 
 
 1st observation . . . 274 33' 
 2nd observation . . 274 54' 
 
 2)549 27' 
 
 274 43' 30" 
 Semidiameter ... 15' 55" 
 
 Mag. bearing () . 274 59' 25" 
 
 Astron. bearing from N. . 257 35' 04" 
 
 Variation of compass . 17 24' 21" W. 
 
 NOTA BENE 
 
 When the magnetic bearing is in excess of the true 
 bearing reckoned clear round from the north point, the 
 variation is that much west ; and vice versa east. 
 
 The meridian can be also obtained by equal altitudes of 
 the sun, similarly to the method by a fixed star. Owing, 
 however, to the sun's position in the heavens having made 
 a sensible change during the period of observation i.e. that 
 which is due in reality to the earth's orbit, not its rotation 
 a correction termed the equation of equal altitudes has to 
 be calculated at some little length. 
 
 Since the methods given are all of them handier and 
 practicable at any time when the solar equal altitude could 
 be taken, this latter will not be gone into for want of space. 
 The rough and ready method described on pp. 39, 40 with 
 the plane-table is practically this problem, without, however, 
 applying the correction. 
 
136 Preliminary Survey 
 
 For nocturnal observations a good directrix is made 
 from a piece of two-inch board with a circular hole three 
 inches diameter in it, and a wire tacked across 
 it up and down. It is convenient for attach- 
 ing a lantern behind. It should be driven 
 fi rm ly into the ground as far away from the 
 
 FIG. 39 instrument as possible. 
 
 METHOD OF DETERMINING LOCAL MEAN TIME AND 
 LONGITUDE 
 
 The reader should first peruse the following definitions in 
 the Glossary Apparent time, Astronomical time, Civil time, 
 Mean time (which is the same as mean solar time), Sidereal 
 time, Equation of time, Hour angle, and Longitude. 
 
 The subjects of time and longitude are very closely 
 allied. 
 
 If we have Greenwich time by chronometer or other- 
 wise, the difference between it and the local mean time is 
 the longitude in time. This will have become plain from 
 the preliminary remarks on p. 118 of this chapter, 
 
 The tables for converting angles of the earth's rotation 
 into their time-equivalents are all of course based upon 
 twenty-four hours being equal to 360, unless the centesimal 
 system of 400 is used, which facilitates calculation, but is 
 not adopted in this work on account of its novelty. 
 
 The tables are given in Chambers's ' Mathematical 
 Tables ' and Raper. 
 
 BY THE SLIDE-RULE TO REDUCE ARC TO TIME 
 
 Arc Time 
 
 / Degrees give . . minutes 
 
 Multiplying by 4 J Minutes ,, . . . seconds 
 
 (Seconds ,, . . . thirds 
 
 Example. Reduce 157 25' 32" to its time equivalent. 
 Write down the angle on one side as follows, and placing 
 
Geodetic A stronomy 1 3 7 
 
 a i of the slide over a 4 of the rule write the equivalents 
 down opposite. 
 
 Arc Time 
 
 min. hrs. min. sec. 
 
 157. . . =628. . . =10 28 
 
 seconds 
 
 25' . . = 100 . . = o i 40 
 
 thirds 
 T.2" . . . =128. . .=00 2-13 
 
 Answer . . 10 29 42-13 
 
 The hours could have been obtained by dividing the 
 degrees by fifteen. The seconds of arc are generally ex- 
 pressed in seconds and decimals of time, and are therefore 
 also more quickly reduced by dividing by fifteen. 
 
 To REDUCE TIME TO ARC 
 
 Time Arc 
 
 | Hours give . . . degrees 
 
 Multiplying by 15 -l MinUt f" minu ^ 
 
 3 I Seconds ,, . . . seconds 
 
 i Thirds . . . thirds 
 
 Example. Reduce 10 hrs. 29 mins. 42*13 sec. to its 
 arc-equivalent. 
 
 Time Arc 
 
 10 hours x 15 ... . ISO 
 
 29 minutes x 15 = 435' . . . = 7 15' 
 40 seconds x 15 = 600" 10' 
 
 2-133 seconds x 15 = 32" . .= o' 32" 
 
 157 25' 32" 
 
 When we have ascertained the true meridian, the 
 simplest method of determining the local mean time is by 
 the 
 
 CULMINATION OF A FIXED STAR 
 
 Any star within the range of the vertical limb of the 
 instrument will do. Stars of small declination, like those in 
 
138 Preliminary Survey 
 
 Orion, appear to pass the field of view much more quickly 
 than those near the pole. When culminating at Greenwich 
 an error of one minute of arc in determining the meridian 
 would correspond with a time error of three to four seconds, 
 whereas a star close to the pole would take twelve to fifteen 
 seconds to make the same change of azimuth ; the former 
 consequently yields the greater precision. An error of four 
 seconds of time corresponds with one angular minute of 
 longitude. 
 
 Rule. Find the star's right ascension and add twenty- 
 four hours- to it, if necessary. Find the sidereal time at 
 Greenwich mean noon either from Whitaker's Almanack or 
 N. A. ; in the Nautical Almanac it is called the sun's mean 
 right ascension at Greenwich mean noon. 1 Correct the 
 sidereal time thus found for the longitude by account, 
 by rule on p. 142. Subtract this result from the star's 
 right ascension. Remainder will be the interval in sidereal 
 time between mean noon and the culmination. Reduce 
 this interval to mean time by rule p. 412. Final result is 
 local mean time of culmination. 
 
 Example. On December 31, 1890, in longitude by 
 account 157 25' W., find the local mean time of the culmi- 
 nation of Polaris. 
 
 hrs. min. sec. 
 
 Sid. time M. N. Greenwich, N. Aim. . 18 39 27 
 
 Diff. for i hr. 9-86" x Ion. in time 10-49 nrs - = o I 43*4 
 
 Sid. time M. N. at place . . . . 18 37 43-6 
 
 R. A. of Polaris . . . . . .118 52-05 
 
 Add . . 24 o o 
 
 25 1 8 52-05 
 Sid. time M. N. at place . . . . 18 37 43-6 
 
 Sid. interval from mean noon . . .641 08-05 
 Correction subtractive, see p. 412 . . 015-7 
 
 Mean time of culmination . . . . 6 40 02-75 
 1 See ' Sidereal Time,' Glossary. 
 
Geodetic Astronomy 139 
 
 LOCAL MEAN TIME BY SOLAR TRANSIT 
 
 The actual culmination of the sun can be observed 
 when not too near the equator by means of the diagonal eye- 
 piece. The meridian having been determined by a solar 
 observation for azimuth on the same day or by the stars 
 the night before, local mean time can be obtained in a 
 similar way to that just described for a star-transit. If the 
 sun souths too near the zenith we have to take an obser- 
 vation for hour angle in apparent time, as will be presently 
 described, but if we can actually observe the culmination 
 we can save mathematics by merely applying the equation 
 of time as explained on p. 399. The equation of time 
 cannot be perfectly exact unless we know the longitude in 
 time, which is just what we want to know, because the cul- 
 mination is only given in the ephemeris for Greenwich noon ; 
 but its maximum variation is i^ seconds per hour and is 
 sometimes a small fraction of a second per hour. If there- 
 fore we know our position within one hour, that is 15 of 
 longitude, which is over a thousand miles at the equator, it 
 cou'ld not make a greater error than i seconds of time. 
 
 The equation of time approximately follows the time of 
 year, and the following rough guide may be useful to fix upon 
 the memory the positions of maximum and minimum varia- 
 tion of equation of time. 
 
 TABLE XXII. Approximate Equation of Time 
 
 minutes minutes 
 
 January Sun after clock . 3| to 13^ 
 
 February i to n ,, ,, . . . 13^ to 14^ 
 
 February II to 28 ,, ,, . . . 14^ to I2| 
 
 March ,, ,, . . . 12^ to 4 
 
 April i to 15 ,, ,, . . . 4 to o 
 
 April 15 to 30 Sun before clock . o to 3 
 
 May i to 15 ,, . . .3 to 3 50 sec. 
 
 May 15 to 31 ,, ,, . - . .3 50 to 2.\ 
 
 June i to 15 . 2| to o 
 
 June 15 to 30 Sun after clock . . o to 3^ 
 
 July i to 26 ,, ,, . . . 3 to 6 
 
 July 26 to 31 ,, ,, . . . 6^ to 6 
 
140 Preliminary Survey 
 
 minutes minutes 
 
 August Sun before clock . .6 to o 
 
 September ,, ,, . . . o to io 
 
 October ,, ,, . . . ioj to i6 
 
 November ,, ,, . . . 165- to II 
 
 December I to 25 ,, ,, . . . II to o 
 
 December 25 to 31 Sun after clock . o to 3^ 
 
 The surveyor can hardly be anywhere on land nowa- 
 days where he cannot tell his position within TOO miles from 
 some point given on an ordinary atlas. 
 
 The maximum error in equation of time at say a latitude 
 of 70 (which is about as far north as he can go) for a 
 distance error of 100 miles would only be "33 of a second ; 
 we may therefore safely say that the surveyor can always 
 obtain, for the purposes of his calculation, the equation of 
 time, and when he observes apparent noon, by applying the 
 equation of time he has the mean time at place. 
 
 Example. On December 31, 1890, in longitude by 
 account 157 W., observed the culmination of sun's western 
 limbs at nh. 58m. 125. and eastern limb i2h. om. 345. ; what 
 was the error of the watch on local mean time ? 
 
 Longitude in time 10*43 hours behind Greenwich. 
 
 hrs. min. sec. 
 Equation of time at Greenwich, December ) , 
 
 31, mean noon . . . . . ' 
 Diff. for i hour i '19 sec. x 10-43 nrs " . -o o 12*4 
 
 Equation of time to be deducted from ap- 
 parent time 033-6 
 
 Apparent noon is . . . .1200 
 
 Deduct equation at place . . . .033-6 
 
 Mean time of transit, sun's centre . . n 56 56-4 
 
 hrs. min. sec. 
 
 By watch, western limb . . n 58 12 
 eastern limb . . 12 o 34 
 
 transit of centre . . . 1 1 59 23 -o 
 
 Watch fast on local mean time . . . o 2 26-6 
 
Geodetic A stronomy 1 4 1 
 
 When the sun souths too near the zenith, we can obtain 
 the local mean time by a 
 
 SOLAR HOUR-ANGLE 
 X 
 
 This observation may be the same as the one for azi- 
 muth. If the reader will refer back to that problem on 
 p. 132 it will save space to use that description up to the 
 finding the three sides of the triangle SPZ. 
 
 PZ=co-latitude. 
 
 PS=co-declination. 
 
 SZ=co-altitude. 
 
 Angle SPZ, or for shortness P, is the hour angle because 
 it is the angle of rotation at the pole between the position S 
 where the sun is observed and the position S' which it will 
 have on the meridian, and by formula 75 at p. 377 of App. 
 we have, 
 2 log cos \ P=log sin S + log sin (S SZ) + log cosec ZP 
 
 + log cosec SP 20. 
 
 As on p. 138 SZ=6i 16' 16" ; SP=75 06' 12-5" 
 ZP= 3 8 43' 15" 
 S, the half sum, =87 32' 517" 
 and S-SZ=26 16' 357" 
 
 Log sin S 9-9996021 
 
 Log sin (S SZ) ..... 9-6461139 
 
 Log cosec SP 10-0148468 
 
 Log cosec ZP 10-2037544 
 
 39-8643172 
 
 20 
 
 2)19-8643172 
 
 Log cos P, 31 11' 54-2" . . . 9-9321586 
 Whence P = 62 23' 48-4'' 
 
 This hour angle of apparent time is first reduced to its 
 equivalent in apparent time as follows : 
 
142 Preliminary Survey 
 
 hrs. min. sec. 
 
 62 . . . . 248 min. . .480 
 23' . . . .92 sec. . . o i 32 
 
 48 '4" ..003-1 
 
 4 9 35'i 
 
 It is then reduced to mean time by applying the equa- 
 tion of time in N. A. for Greenwich corrected for the assumed 
 longitude. The longitude in time '0135 hr. is applied here 
 merely to show that an error in assumption of 1 2 miles would 
 not make an appreciable difference in the result. 
 
 min. sec. 
 
 Local apparent time as above . . . 4 9 35*1 
 Equation of time at Greenwich, ) _ 
 mean noon, April 30 . . ' 
 
 sec hr. 
 
 Long, in time 48*5 x '0135. . o 0-004 
 
 o 2 54-004 
 
 Mean time at place . . . . .46 41-096 
 Greenwich mean time by chronometer . 4 5 52-8 
 
 Difference of time . . . . . o o 48-3 
 = 12' 5" east longitude. 
 
 hrs. min. sec. 
 
 Time by watch, see p. 137 . . 4 3 3 2 '5 
 
 Local mean time . . . . . 4 6 41-1 
 
 Watch slow on local mean time . . .038-6 
 
 A sidereal hour-angle is obtained in precisely similar 
 manner from a fixed star, only the arc of rotation, reduced 
 to its time equivalent, is an interval of sidereal time and has 
 to be reduced to its mean time equivalent by rule on p. 412. 
 
 The time of apparent noon is also obtained by equal 
 altitudes of the sun by applying the correction of equal 
 altitudes already referred to, or else it may be obtained from 
 the equal altitudes of a fixed star described on p. 1 24 with- 
 out any correction. 
 
 All the previous methods entail the use of a chrono- 
 meter keeping Greenwich time, which, if done properly, is 
 
Geodetic Astronomy 143 
 
 by far the most satisfactory, and indeed the simplest and 
 easiest plan. 
 
 The surveyor cannot, however, as a rule place very much 
 dependence upon his semi-chronometer watch. Where he is 
 able to return to a place at which he has previously fixed the 
 direction of the true meridian and the geographical position, 
 he can after any lapse of time recover the true time. 
 
 This is a most important matter to bear in mind. It 
 will often pay to take several days' journey out of the course 
 to pick up a former station and so readjust the watch, 
 because with a good timekeeper, although it may suddenly 
 change its rate from rough usage, it is generally a gradual 
 acceleration or retardation, and when a cumulative error 
 has been discovered, it may be distributed proportionally 
 over the unchecked back work with very close results. 
 
 The process is simply to set up the transit in the meri- 
 dian, and finding the time at place from a stellar culmi- 
 nation as described on p. 137 determine the error of the 
 watch over a certain interval of time. 
 
 The error of the watch on local mean time can be found 
 anywhere ; but that does not give Greenwich time. The 
 object of going back to the known station is to recover 
 Greenwich time. 
 
 We now come to observations for longitude by what 
 are termed ABSOLUTE METHODS. 
 
 ECLIPSES OF JUPITER'S SATELLITES 
 
 These phenomena, visible through a forty-power tele- 
 scope of one-inch aperture, are timed in the Nautical 
 Almanac for Greenwich mean time. All the observer has 
 to do is to watch both the disappearance and reappearance 
 and reduce the difference of time between the local mean 
 time of its occurrence and its timed occurrence at Green- 
 wich into difference of longitude. 
 
 The configuration of Jupiter's satellites is given for every 
 
144 Preliminary Survey 
 
 day in the Nautical Almanac for north latitude, and must 
 be reversed for south latitude. They are also given for an 
 inverting eyepiece. The first satellite is most rapid in his 
 orbit, and is therefore to be preferred. 
 
 The eclipse being caused by the shadow of the planet 
 falling upon the satellite, the latter may be at some distance 
 from the planet when it takes place, so the observer should 
 be ready a little beforehand . Raper says of this method that 
 though easy and convenient, it is not very accurate ; the 
 eclipse is not instantaneous, and the clearness of the air, and 
 the power of telescope employed, affect considerably the time 
 of the phenomenon. Observers have been found to differ as 
 much as 40 to 50 seconds in the same eclipse. 
 
 ' The observation can only be considered complete when 
 both immersion and emersion of the same satellite have 
 been observed on the same evening, and as nearly as pos- 
 sible under the same circumstances. Thus if the satellite 
 disappear a little sooner than if the air had been clearer, it 
 will emerge a little later from the same cause, and the mean 
 of the two results may be near the truth.' 
 
 LUNAR OBSERVATIONS 
 
 The moon changes her apparent place in the heavens so 
 rapidly that her R. A. and Decl. are given in the Nautical 
 Almanac ephemeris for every hour at Greenwich. Her 
 spherical distances are also given from the sun, certain well- 
 known fixed stars, and any of the planets to which she may 
 be near, for every three hours at Greenwich. Consequently 
 if we observe some recorded phenomena in connection with 
 the moon, or if we time her culmination relatively to that 
 of a fixed star, or measure her distance from one of the 
 registered stars, we have a ready means of finding Green- 
 wich time. If the ephemeris were absolutely correct, these 
 methods would be more valuable than they are, but, unfor- 
 tunately, the moon's apparent motion is so eccentric that it 
 
Geodetic Astronomy 145 
 
 is impossible to register her movements perfectly. In addi- 
 tion to this, the interpolation between the registered times 
 is far from a matter of simple rule of three. To obtain any 
 close results, a lengthy process called Bessel's equation for 
 fourth differences is required, and it is but rarely that the 
 surveyor is able to afford the time for it. There are, how- 
 ever, occasions in which even the approximate computation 
 of the longitude from these lunar phenomena are the only 
 means at the surveyor's disposal. 
 
 LONGITUDE BY A LUNAR OCCULTATION ] 
 
 Occultations of fixed stars by the moon are given in the 
 Nautical Almanac with Greenwich mean time of immersion 
 or disappearance and emersion or reappearance. The longi- 
 tude by account gives a correction for the approximate time 
 of occurrence at place, and the instrument should be set 
 up some time beforehand, as the stars are often of small 
 magnitude and need steady watching. 
 
 At the instant of occultation the apparent R. A. of the 
 moon's limb is the same as that of the star. The calculation 
 is somewhat lengthy, to remove the effect of the moon's 
 parallax ; which being done the true R. A. is deduced and 
 the G. M. T. found. 
 
 If the surveyor only has an ordinary six-inch or five-inch 
 transit-theodolite he will have some difficulty in obtaining a 
 good observation, but with the ' ideal ' tacheometer described 
 at p. 307 he will see it very well 
 
 Any good three-draw telescope attached to a post 
 will do. 
 
 The local mean time has to be first determined by one 
 or other of the methods given on pp. 136-140. 
 
 1 A good illustration of this method is given in Raper's ' Naviga- 
 tion,' p. 305. 
 
146 
 
 Preliminary Survey 
 
 THE LUNAR DISTANCE J 
 
 This observation for longitude has to be taken with the 
 sextant, and will therefore only be explained in principle. 
 
 Let Z be the zenith and M'S' be 
 the observed lunar distance from a 
 star, M'Z and S'Z being the zenith 
 distances or complements of the 
 observed altitudes. If M'S' were 
 the true distance, we could find 
 Greenwich time by interpolating 
 between the registered lunar dis- 
 tances in the Nautical Almanac. 
 But inasmuch as M' and S' are not the true places on 
 account of refraction, parallax, and semi -diameter, the dis- 
 tance is really MS. We can make the corrections MM' and 
 SS' and then making S equal the sum of the true altitudes 
 H + H', and s equal the sum of the apparent altitudes h-\-h', 
 we have by spher. trig. (Chambers's Tract. Math.' 751*?), 
 sin 
 
 \ fo + M'S'). cos 
 
 2 MS=cos 2 \ S - 
 cos H . cos H' . cos 
 
 cos h . cos h 
 
 TERRESTRIAL DIFFERENCE OF LONGITUDE 
 
 When two points are correctly determined as to latitude 
 but the longitude is doubtful, their difference of longitude 
 
 can be computed as follows, 
 providing that they are visible 
 from one another. 
 
 Let A, B be the stations, 
 AP, BP their co-latitudes, the 
 angles A and B their recipro- 
 cal true azimuths, and APB, 
 or P, the required angular 
 FIG 4i difference of longitude. As- 
 
 1 Examples of this method are given in Raper's 'Navigation,' p. 283, 
 Chambers's Math. p. 449, and a graphic method in ' Hints to Travellers. ' 
 
Geodetic Astronomy 147 
 
 suming the earth to be a sphere we have (by formula 81, 
 P- 378)- 
 
 v-'vx v .J ^ . ~ . - .. VWVXA ^ *. * - - -^ / 
 
 sm 1 (#~^) 
 
 LONGITUDE BY MOON-CULMINATING STARS 
 
 The Nautical Almanac furnishes the R. A. of the moon's 
 bright limb for the lower as well as the upper culmination, 
 marked L. C. and U. C. respectively. 
 
 It also gives the variation in R. A. in one hour of longi- 
 tude, that is the variation in her transit over two meridians 
 equidistant from Greenwich and one hour distant from one 
 another. The figures are calculated from the right ascen- 
 sion of the bright limb and include the effect produced by 
 the change of the semi-diameter. 
 
 If we can determine exactly what the right ascension 
 of the moon is at any time and place, we can find, by 
 interpolating between the values for the two nearest hours 
 in the Nautical Almanac, the Greenwich date corresponding 
 to our local time of observation. 
 
 The method adopted is to watch the successive culmina- 
 tions of the moon's bright limb and some fixed star close to 
 her whose right ascension is known. The Nautical Almanac 
 gives a list of such stars which are peculiarly suitable from 
 their position for this purpose. They are generally small 
 however, and need some little practice in star-gazing to 
 identify them. They are chosen with as short a time 
 interval as possible and nearly on the same parallel. 
 
 Very correct results are obtained by this method when 
 the distance between the meridians is not great. 
 
 The best plan is by corresponding observations at two 
 places on the same night. 
 
 Such a case is the following example taken from Pro- 
 fessor Loomis' ' Astronomy : ' Let the right ascension of the 
 moon at the two meridians be A and A', from which we 
 
 L 2 
 
148 Preliminary Survey 
 
 know the moon's motion in R. A. during the interval of the 
 two transits A' A. 
 
 The almanac furnishes the variation of the moon's right 
 ascension corresponding to one hour, which we will repre- 
 sent by V. 
 
 We shall therefore have the proportion 
 
 V : A' A:: i hr. : the difference of longitude. 
 
 Example i. The R. A. of the moon's first limb Sep- 
 tember 6, 1840, was observed at Washington to be 19 h. 21 m. 
 29-905. ; and on the same day at Hudson, Ohio, 19!!. 22m. 
 9-725. Required the difference of longitude of the two 
 places. 
 
 Here A' A=39'82sec. 
 
 That value of V must be taken which corresponds to 
 tlie middle of the interval between the observations, which 
 is found by interpolation to be 135*55 sec - 
 
 i35'55 sec. : 39-82 sec.:: i hr. : 17111. 37-56 sec. . 
 
 An accurate method of determining the longitude upon 
 this principle, both for distant as well as near meridians," 
 involving lengthy calculation, may be found in Professor 
 Chauvenet's 'Spherical Astronomy.' 
 
 METHODS OF DETERMINING THE LATITUDE 
 
 The simplest, most accurate, and most rapid of all the 
 astronomical calculations at the disposal of the surveyor are 
 those for finding the latitude. The observer does not require 
 to know his meridian or his local time, all he needs is a 
 sextant or a transit. In fact, as shown later on, he can even 
 obtain an approximation to the latitude with a plummet- 
 string. 
 
 'In the prefatory remarks to this chapter it was shown 
 that an altitude of the true pole is equal to the latitude of 
 the place. If the pole star were exactly at the pole, we 
 
Geodetic A stronomy 1 49 
 
 should merely have to take its altitude in order to get our 
 latitude. The pole star is not exactly at the celestial pole, 
 so it has to make its own little circle of apparent rotation 
 round the true celestial pole. It will be seen on examining 
 Fig. 127, Glossary, that no star whose polar distance is less 
 than the latitude can set, neither can any star whose polar 
 distance is greater than 90 + co-latitude ever become visible 
 above the horizon. 
 
 The pole star crosses the meridian in the Northern 
 Hemisphere twice in the 24 hours like all the rest of the 
 stars. It is in fact a circumpolar star. When at its upper 
 passage, its altitude will be i 16' 42" greater than, and 
 when on its lower passage as much less than, the latitude. 
 Similarly when we know the declination of any other 
 star we can tell the latitude from a meridional altitude. 
 All we want besides is to know what side of the globe we 
 are on, and in which direction, north or south, the body is 
 culminating. 
 
 In north latitude when the polar distance is equal to the 
 complement of the latitude, the body culminates at the 
 zenith (Z, Fig. 127) because then the altitude =ZP + PR. 
 
 When the polar distance is greater than the co-latitude, 
 it culminates to the south, and when less to the north ; and 
 when equal to the supplement of the latitude, the body 
 only touches the horizon when making its meridional pas- 
 sage. Its diurnal path is represented by HT (Fig. J27). 1 
 
 When the polar distances are as above in south latitudes 
 the culminations are in the opposite direction. 
 
 When the polar distance is 90, or in other words when 
 the body is on the celestial equator, its meridional altitude 
 EH (Fig. 127) will be the complement of the latitude, because 
 HEPR=i8o and EP=9o, therefore HE + PR=9o. 
 
 The polar distance is either the complement of the 
 declination S'P or 90+ the declination UT, and the 
 following rules are deducible from Fig. 127, for obtaining 
 
 1 The paths of the celestial bodies are shown diagrammatically. 
 
150 Preliminary Survey 
 
 the latitude in either hemisphere, when we know the decli- 
 nation and altitude on the meridian. 
 
 Rule i. In the Northern Hemisphere. 
 
 (A) When a body culminates to the south of the 
 observer. 
 
 (1) When the declination is north. Subtract the declina- 
 tion from the altitude, this will be the co-latitude: 
 
 HE=S'H-S'E (Fig. 127, Glossary) 
 
 (2) When the declination is south. Add the declination 
 to the altitude ; this will be the co-latitude : 
 
 HE=EU'4-U'H 
 (-B) When a body culminates to the north. 
 
 (1) When it culminates above the pole. Subtract the 
 co-declination from the altitude. This will be the latitude : 
 
 PR=VR-VP 
 
 (2) When it culminates below the pole. Add the 
 co-declination to the altitude ; result is the latitude : 
 
 PR=V / R + V / P 
 Rule 2. In the Southern Hemisphere. 
 
 (A) When the body culminates to the north o'f the 
 observer. 
 
 (1) When the declination is south. Subtract the decli- 
 nation from the altitude ; this will be the co-latitude : 
 
 QR=S"R-QS" (Fig. 128, Glossary) 
 
 (2) When the declination is north. Add the declina- 
 tion to the altitude ; result will be the co-latitude : 
 
 QR=U // R + U // Q 
 
 (B) When the body culminates to the south of the 
 observer. 
 
 (1) When it culminates above the south pole. Subtract 
 the co-declin. from the altitude ; result will be the latitude : 
 
 OH=V'H-V'O 
 
 (2) When it culminates below the south pole. Add the 
 co-declination to the altitude ; result will be the latitude : 
 
 OH=OV+VH 
 
Geodetic Astronomy 151 
 
 At the equator, all bodies having north declination cul- 
 minate to the north and all bodies having south declina- 
 tion culminate to the south, so that when near 'the line' 
 there can be no mistake as to which rule to adopt. The 
 meridional altitude of a star near either north or south pole, 
 such as a Ursse Majoris or a Crucis, will at once show what 
 side of the line we are on. 
 
 The most accurate method of obtaining the latitude is 
 by a fixed star, but, all things considered, the most useful 
 celestial body is the sun. The moon or one of the planets 
 will all serve the purpose as well as a star ; the only diffe- 
 rence is that the declinations of the bodies of the solar 
 system vary from day to day and we have to know, approxi- 
 mately at least, what the Greenwich time of our observation 
 is in order to get the correct declination from the almanac. 
 The declination of the fixed stars is only very slightly 
 variable from year to year, so that one of them will serve 
 our purpose just as well if we have not the most remote idea 
 of our longitude. But the great value of the sun arises from 
 its being a daylight observation, and to any one not familiar 
 with the stars an unmistakable object. There are but few 
 cases in which we cannot obtain the longitude by account 
 near enough for determining the declination. 
 
 The formidable array of logarithmic figures, which are 
 so associated with astronomical problems, convey the impres- 
 sion of a complexity which does not in fact exist, at least as 
 far as latitude is concerned. There are no trigonometrical 
 calculations at all about obtaining the latitude. The first 
 operation is to free the observed altitude from errors due to 
 dip (if the visible horizon is used), parallax, and refraction, 
 and to correct it for semi-diameter, for all of which terms 
 see explanation in Glossary. The second operation is to 
 reduce the recorded time to Greenwich time from the longi- 
 tude, as far as it is known by account. The sun's maximum 
 variation of declination is only about an angular minute per 
 hour, and an hour corresponds with 15 longitude, so that 
 
1,52 Preliminary Survey 
 
 we must be a very long way out in our reckoning for longi- 
 tude for it to make any serious difference in the calculation. 
 The third operation is to calculate by a simple rule of 
 three sum the declination at the time of observation as re- 
 duced to Greenwich mean time, and finally to apply one of 
 the foregoing rules to the particular case. 
 
 LATITUDE BY CIRCUMPOLAR STARS 
 
 An exceedingly exact though lengthy method of obtaining 
 the latitude is by observing the upper and lower culminations 
 of a circumpolar star. This can be done without knowing 
 the meridian, the declination, or the time, because it is suffi- 
 cient to watch for the maximum and minimum altitudes of the 
 same star. Wfien both observations are corrected for refrac- 
 tion the mean of the two altitudes is the altitude of the 
 
 VR -i- V'R_ 
 celestial pole, that is the latitude of the place : - = 
 
 PR (Fig. 127, Gl.). 
 
 Generally the time is known and the true meridian can 
 be readily ascertained. In this case the operation is 'very 
 speedy, for the altitude is measured at either of the culmina- 
 tions and the declination applied as already explained. 
 
 LATITUDE BY MERIDIONAL ALTITUDE OF A FIXED STAR 
 
 On February 5, 1889, observed the meridional altitude 
 of Sirius in Honolulu, north latitude, star culminating to the 
 south. 
 
 Observed altitude 52 08' 20" 
 
 Refraction o o' 45 
 
 52 07' 35' 
 S. Declination (N. A.) .... 16 33' 52" 
 
 By Rule i. A. 2, Co-latitude . . .68 41' 27" 
 
 90 o' o" 
 
 Latitude 21 18' 33" N. 
 
Geodetic Astronomy 153 
 
 The latitude by a meridional altitude of the sun, which 
 is the sailor's stand-by, is identical in principle, only subject 
 to correction for parallax, which is insensible in the case of 
 a fixed star. 
 
 The following example is taken from Raper's ' Navigation,' 
 illustrating the calculation made on board ship with the 
 sextant. May 3, 1878, longitude 38 W., obs. mer. alt. & 
 56 10' to the southward ; ind. corr. +2' ; height of eye 20 
 feet ; required the latitude. 
 
 Decl. 3rd Table 60 or N. A. . . . 15 43' N. 
 Corr. for 38 W. . '. . + o 2' 
 
 15 45' N. 
 
 Obs. alt. Q . . .56 10' 
 Ind. corr. +2'} o 2' 
 
 Dip. -4'f' 
 
 App. alt. . . . . 56 8' 
 
 Refr. _ - i'i .+ o 15' 
 
 Semidiam. + 16 
 
 True alt 56 23 
 
 Zen. dist 33 37' 33 37' 
 
 Latitude 49 22' N. 
 
 This is a case of Rule i. A., but in Raper the zenith 
 distance is added to the declination for the latitude. An 
 inspection of Fig. 127, Glossary, will show at once thatZE= 
 PR, because PE=9o, and ZR=9o, and ZP is common. 
 
 There are a great variety of methods of determining the 
 latitude. The foregoing are the most common, and as they 
 are always available when the heavens are suitable for 
 observation, we will not take up space with more than an 
 allusion to two others. 
 
 i st. Latitude by an altitude of the sun out of the meri- 
 dian. In the triangle PZS (Fig. 127, Glossary), if we know 
 the time, we have the hour angle SPZ. The co-declination 
 PS we also know, and ZS is the co-altitude. From these 
 
154 Preliminary Survey 
 
 data we can obtain the side PZ, which is the co-latitude. 
 The demonstration of this is given in Chambers's * Mathe- 
 matics,' art. 887. The bare formula is No. 78 in the 
 Appendix. 
 
 2. Latitude by two altitudes of the sun or of a star, 
 and the interval of time between the observations, or the 
 altitudes of two known stars, taken at the same instant, to 
 find the latitude of the place. To those fond of mathe- 
 matical gymnastics of high order this method is recom- 
 mended. It involves a maximum amount of labour with a 
 minimum chance of coming out right at the end. 
 Demonstration in Raper, Chambers's ' Mathematics,' &c. 
 
 In conclusion, as to observations of meridional altitude, 
 the pole star and circumpolar stars have this great 
 advantage that they are longest at culmination, giving good 
 time for a series of altitudes, and the vital necessity of 
 repeating the observation face right and face left, as 
 explained in tacheometry, cannot be too strongly urged. 
 
 The subject of latitude will be concluded with a method 
 of obtaining it without any instrument at all. 
 
 GRAPHIC LATITUDE 
 
 The following simple method of obtaining an approxi- 
 mate latitude is described by Mr. Coles of the Geographical 
 Society, and is given here somewhat in his words with a 
 few additional explanations and practical cautions, but the 
 diagram and example are independently worked out. 
 
 By observing the times when two pairs of stars are 
 vertically above one another the latitude can be obtained 
 by graphic construction in a very short time. All one 
 requires is a plummet-string, and a table of mean positions 
 of fixed stars, such as is given in Whitaker's Almanack. 
 The operation should be repeated once or twice, and the 
 results meaned. It requires a time interval of from four to 
 eight hours, and a timepiece which is at least accurate to a 
 
Geodetic Astronomy 
 
 155 
 
 few seconds for that interval, although it may be quite wrong 
 on local mean time. The positions of the stars are pro- 
 jected stereographically upon the plane of the celestial 
 equator ; the circle marked primitive being the equator to a 
 radius i. P, the centre of the circle, is the projection of the 
 pole. The first point of Aries is marked T, and the other 
 quadrants of right ascension 6 hrs., 12 hrs., and 18 hrs., 
 indicated by their numbers. The projections of the vertical 
 circles through the stars are obtained as will be explained, 
 
 FIG. 42. 
 
 and the intersection of these circles being the zenith, the 
 arc ZP, which (Fig. 127, Gl.) is the co-latitude, gives us the 
 latitude of the place. 
 
 It will be seen that the circles through the stars are 
 much flatter than the primitive, and it is important to have 
 them cut one another as nearly at right angles as possible, 
 so the first pair of stars should be as nearly east, and the 
 second pair as nearly west, as can be conveniently chosen, 
 with as long a time interval as possible. 
 
1 56 Preliminary Survey 
 
 One of the stars may belong to both pairs as in the 
 example, but not necessarily so. 
 
 We will explain the process direct from the example. 
 
 Betelgeuse and Sirius were observed with the plummet- 
 string when bearing about S.E. by S. ; and after nearly six 
 hours, when Sirius was nearly setting, it was observed plumb 
 under Regulus, bearing about W.S.W. The four positions 
 of the stars are marked S and B for the first pair, and S' 
 and R for the second pair. The projections of right 
 ascension are measured by the chords of the angles on the 
 primitive from c v . The projections of polar distance are 
 measured by the tangents of half those arcs from P upon the 
 radial lines drawn to the positions of the stars in R. A. Thus 
 by slide-rule the chord of R.A. of Betelgeuse=2 sin \ angle, 
 and the arc corresponding to 5hrs. 49111. iosec.=87 12' 
 30". Placing the extremity of the sine-scale under the 2, 
 we read for 43 36', 1-379, which is laid off from =y. The 
 declination of Betelgeuse is 7 23' N., therefore polar 
 distance=82 37'. The tangent slide in its initial position 
 gives for 41 i8V '878, and this we lay off from P. The scale 
 of the plot was radius=i decimetre, so that the bevelled 
 edge of the slide-rule served for all the scaling. The dia- 
 gram as printed has been reduced to one-fifth its original 
 size. The first pair of stars being projected, the second 
 pair are treated similarly, except that their right ascension 
 is decreased by the amount of the sidereal time interval 
 between the observations. The projections of the vertical 
 circles are shown on the diagram for both pairs, and their 
 intersection Z, but only the locus C of the second pair is 
 shown, and the auxiliary dotted lines by which it is deter- 
 mined in order to avoid confusion. 
 
 Through R draw RPG, and lay off the perpendicular 
 HPE. Draw ERo and DPF. Draw EFG' cutting RPG pro- 
 duced in G f . Bisect RG' and draw the locus perpendicularly 
 through its centre. Bisect RS' and draw the perpendicular 
 
Geodetic Astronomy 157 
 
 from it to c in the locus, which will be the centre of the 
 projection of the vertical circle RS'. 
 
 Draw circle S'R, and similarly, from a fresh locus, the 
 circle SB cutting S'R in z the zenith. ZP measured by 
 same scale gives the semi-tangent of the co-latitude ; being 
 the projection of a great circle passing through the pole, 
 similarly to the circles of polar distance pS', pR, &c. 
 
 At Kukuihaele, Hawaii, in latitude 20 8' 9" by account, 
 
 hrs. min. sec. 
 
 Observed Betelgeuse and Sirius plumb at . 7 39 15 
 ,, Regulus and Sirius plumb . . 13 26 15 
 
 Mean time interval . . 5 47 o 
 
 Correction by slide-rule (see Sid. Time Glossary) + o o 57 
 
 Sidereal time interval . . 5 47 57 
 
 hrs. min. sec. 
 
 R. A. of Betelgeuse . . . . . 5 49 10 
 ,, ,, Sirius . . . . . 6 40 15 
 
 ,, ,, Regulus, less sid. time interval . 4 14 31 
 ,, ,, Sirius, less sid. time interval . o 52 18 
 Declination of Betelgeuse 7 23' N. and P. D. =82 37' 
 ,, ,, Sirius . 1 6 34' S. and P. D. = 106 34' 
 
 ,, ,, Regulus 1 2 30 34" N.; P. D. = 77 29' 26' 
 
 ZP = semi-tangent of co-latitude = by scale 
 
 of diagram '698 .... =tan. 34 55' 
 
 2 
 
 Co-latitude ...... 69 50' 
 
 90 o' 
 
 Latitude ..... 20 10' 
 
158 Preliminary Survey 
 
 CHAPTER V 
 TACHEOMETRY 
 
 TACHEOMETRY, or rapid measurement, is a term which 
 signifies more than one kind of measurement. It may be 
 defined as telescopic surveying, and has for its aim the pro- 
 duction of a correct map with the least possible assistance and 
 in the least possible time. It performs at one operation the 
 measurement of distance formerly only possible with the 
 chain, and the measurement of elevation formerly made 
 with the level and staff. 
 
 The word is sometimes written tachymetry or takimetry. 
 The French also have a word ' takitechnie ' or ' rapid art,' 
 which includes the use of the slide-rule. The principal dif- 
 ference in the methods of tacheometry is in the mode of 
 mapping. If the plane-table and stadia are used, the map- 
 ping is done in the field, but if the transit and stadia are 
 used, the mapping is done at home from field notes. 
 
 The instruments used are very numerous, some of them 
 highly complicated, and many of them exceedingly ingenious. 
 Their names often affect an originality of principle or an ex- 
 tent of accomplishment which does not really belong to them, 
 whilst they fail to classify them according to their organic 
 type. 
 
 The measurement of distance performed optically is the 
 essential principle of the stadia, and places it in the front 
 rank of all tacheometers. 
 
 There should be a means of distinguishing those instru- 
 ments which measure angles for the calculation of distance 
 
Tacheometry 1 59 
 
 such as the Hadley sextant when used in hydrography ; 
 those which demand the measurement of a base by tape, 
 chain, or pacing, like telemeters and range-finders ; or those 
 which require the use of a rule of three sum or reference to 
 a table, like Eckhold's ' omnimeter,' or the micrometer tele- 
 scope from those in which the distance is practically seen, or 
 at least read at a glance from a graduated rod. This latter 
 achievement, though not by any means new in principle, 
 deserves a special and typical designation in its modern form. 
 The word metroscope, forming a good third with tele- 
 scope and microscope, from metron a measurement and 
 skopeo I behold, would express more satisfactorily the stadia 
 principle. 
 
 Mr. B. H. Brough, in his paper in the ' Min. Proc. List. 
 C. E.' vol. xci., attributes the invention of the stadia to 
 Mr. William Green in 1778. It was used with a simple tube 
 having in its field three horizontal wires, and was used upon 
 the principle that since, rays of light travel in straight lines 
 except as diverted by refraction, the distance of an object is 
 sensibly measured by the extent of the optic retina which is 
 covered by it. 
 
 Thus, supposing we see a man and a boy standing 
 together 400 yards away from us, and the image of the boy 
 upon the eye occupies three-fourths of that of the man ; 
 then, if the boy comes forward so as to be 300 yards from 
 us, he will appear to be exactly the same height as the man. 
 If we arrange the two extreme horizontal wires in a plain 
 tube so that 20 feet away from us a graduated staff will show 
 2 feet subtended by the wires, then at a distance of 40 feet 
 they will subtend 4 feet and so on. 
 
 If we term the distance of the hairs from the eye * hair 
 distance,' and the distance of the staff 'staff distance,' also 
 the distance apart of the hairs ' hair height ' and the space 
 subtended by them on the staff * staff height ,' we have by 
 simple proportion 
 Staff distance : staff height :: hair distance : hair height 
 
160 Preliminary Survey 
 
 from which we can calculate the distance of the staff when 
 we know the other three factors. If, however, we graduate 
 the staff so that one subdivision represents 10 or 100 of hori- 
 zontal measurement, and if we then direct the lower hair to 
 an even figure on the staff, we can simply read the distance 
 from the staff. 
 
 When telescopes are used (and the lens-power is the 
 backbone of stadia measurements) the factor of 'hair 
 distance ' varies with the focus, so that we have no longer a 
 single ratio as a multiplier for determining the distance, and 
 to avoid making a calculation every time, we have to reduce 
 the varying ratio to a single ratio plus a constant, or else an 
 
 FIG. 43. 
 
 optical device is resorted to in the instrument itself to 
 produce the effect described on p. 392. 
 
 Fig. 43 applies to a horizontal sight when the staff is 
 vertical, or to any sight when the staff is held at right 
 angles to the line of sight. 
 
 Let s be the intercepted height on the staff ; /, the height 
 of its image ; a, the distance of the staff from the object- 
 glass ; x, the distance of the image from the object-glass ; 
 f, the focal length ; d, the distance of the axis of the instru- 
 ment in rear of the object-lens ; A, the distance of the staff 
 from the axis of the instrument. 
 
 Then by simple proportion, as before, we have 
 
 t . 
 
Taclieometry 161 
 
 But the general formula for the foci of lenses is 
 
 i ii 
 
 This multiplied by a, becomes 
 
 and substituting for a - on the one side of the equation its 
 value from (i) we have 
 
 " 
 
 or 
 
 /-. is a simple ratio, and f+d is a constant, which ranges in 
 
 different instruments from 12 to 24 inches, i. is generally 
 
 constructed=ioo, that is to say for a fourteen-inch telescope 
 the hairs are placed one seven -hundredths of an inch above 
 and the other seven-hundredths below the axial hair. An 
 amateur will find the best way to do this, to mark the lines 
 on the brass diaphragm with a needle point, as explained 
 on p. 312, then to fix one hair permanently with shellac 
 but to fasten only the extremities of the other hair at the 
 outer edges of the diaphragm ; this will permit of a slight 
 adjustment of the hair to fix it finally by drawing it up or 
 down at the inner edges of the diaphragm with a quill 
 tipped with shellac. If the measurement of the spaces is 
 carefully made, it will only require to be moved by about 
 its own thickness, if at all. This may not leave the hairs 
 perfectly equidistant from the axial hair, but that is of 
 small importance when the two distances are known. Most 
 surveyors who put in their stadia hairs put them in at 
 haphazard, and find their value by experiment, reducing 
 
1 62 Preliminary Survey 
 
 the distance by slide-rule, but by so doing they sacrifice for 
 the sake of a very little extra . trouble the most valuable 
 feature of stadia measurement, and open to themselves a 
 fertile source of error. The diaphragm will hardly ever 
 have to be taken out more than once to finally fix the 
 hairs. 
 
 The constant of f+ d with a fourteen-inch telescope in 
 which the vertical axis was eight inches from the object- 
 glass would be 22 inches or 1-8 feet ; but inasmuch as the 
 readings are only taken to even feet, the constant to be 
 added would be 2 feet, or with a ten-inch telescope hung 
 concentrically it would be 1-25 feet ; but unless at very close 
 distances, only i foot would be deducted. This requires 
 no entry in the book ; it is usually added systematically 
 down the whole row of figures as they are entered by the 
 recorder. 
 
 The principle of all telemetry or tacheometry is triangu- 
 lation, but it may be otherwise styled the determination of 
 parallax. This is a great word amongst astronomers, for by 
 it the whole solar system is plotted to scale, and it serves as 
 a complete illustration of the value of tacheometry. Parallax 
 is the angle at a distant object which subtends a given base. 
 The greater the base, and the more exact the angular 
 measurement, the more correct will the measurement of the 
 distance be. It is possible to obtain a greater degree oj 
 accuracy with a small base and high powers of angular 
 measurement, than with a long base but inaccurate 
 angles. 
 
 For instance, with a range-finder, a base is run out as 
 explained on p. 338 ; it may be 100 or 200 feet, but if the 
 angle is taken with a low power and the base roughly 
 measured, it will not produce results nearly as correct as 
 those of a stadia telescope of high power when forming upon 
 a distant graduated staff a base of only a few feet. 
 
 The base which the astronomers have for determining 
 the distance of the sun is only he diameter of the earth, and 
 
TacJieometry 163 
 
 that is to the distance itself something like a hair's breadth 
 to a distance of six feet. 
 
 Another analogy that should be borne in mind is that 
 exactness is always proportionate to magnitude. It is 
 supposed that the error in the sun's distance does not exceed 
 125,000 miles; this in itself is a large amount, but in 
 93,000,000 it only represents y-^, which, considering the 
 size of the base, is marvellously accurate. Porro, who in 
 1823 introduced the anallatic stadia telescope of high 
 power, claimed never to have exceeded an error of ^-^ up 
 to 660 feet, or -^-^ in distances up to 1,320 feet. This 
 would mean that at a distance of 660 feet, he could read to 
 one-third of a hundredth of a foot on a staff graduated in 
 our way. His telescope must have been an uncommonly 
 good one. 
 
 If a preliminary survey is correct to the limits of the 
 scale, it is all that can be demanded of it, and 100 feet to 
 the inch is about the largest scale used for that purpose, at 
 which less than i foot would be practically unscaleable. 
 Sights should not be taken at distances too great for the 
 instrument to read the graduation distinctly, and this 
 limit varies from 300 feet in small to 1,000 feet in large 
 instruments, beyond which only check sights, or those upon 
 the accuracy of which the survey does not depend, should 
 be taken. 
 
 Even with large telescopes, 300 feet is about the best 
 working limit for the length of sights. In the first place, the 
 reliability of the levels rapidly diminishes beyond that 
 distance just as it does in ordinary levelling, and in the 
 second place, detail is sure to be missed. 
 
 The value of long shots, either within the range of the 
 stadia or beyond it, with the micrometer is very great, both 
 as a check, and also for connecting the survey with useful 
 outlying points, of which the detail is not required, and 
 which cannot be got by intersections. 
 
 M 2 
 
164 Preliminary Survey 
 
 METHODS OF TAKING THE SIGHTS 
 
 In level country it is convenient to set the telescope 
 horizontal like an ordinary level and read off all three hairs, 
 booking the stadia hairs in their column and the axial hair 
 in the column for foresight or intermediate ; there will then 
 be no entry under the vertical limb column, and the level 
 will be reduced by the collimation method. This is only 
 possible where the staff does not travel outside the field. 
 
 When it becomes necessary to use the vertical arc, the 
 lower hair should preferably be directed to an even foot on 
 the staff, and that one which will bring the axial hair as near 
 as possible to the same height above ground at the staff as 
 the line of sight is above ground at the instrument. For 
 instance, if the telescope is 5 feet above ground and the 
 staff is 350 feet away, the stadia being arranged at i in 100, 
 the lower hair should be at 3-00, then the upper will be at 
 6-50, or, if there is an instrumental constant of i foot at 
 6-49, the axial hair being at 4745. 
 
 There are two methods in vogue for holding the staff. In 
 Germany the practice is to hold it at right angles to the line 
 of sight, in America to hold it vertically. It can be easily 
 shown that in both figures, 44 and 45, angle ,*=angle X ; and 
 calling the difference of the two stadia hair readings S and 
 S', and the reading of the axial hair H, we have for Fig. 44, 
 
 A=Sx + </+/ 
 
 or for the case of an instrument whose stadia value is i per 
 100, and constant 1*5 feet, 
 
 AsiooS+i'5 
 
 B=Axsin X 
 
 C=A. cos X + ^=A. cos X + H. sin X 
 In Fig. 45, S=S'. cos X 
 
 ,\A = S'. cos X x f- 
 Or for similar case. 
 
Tacheometry 
 
 165 
 
 A=ioo S'. cos X+ 1-5 
 B=A. sin X=ioo S'. cos X. sin X 
 C=A. cos X=ioo S'. cos 2 X+i'5 
 The writer prefers the first method for the following 
 reasons : 
 
 i. When the staff is held vertically, the accuracy is 
 dependent upon the staff-holder, who has to watch a circular 
 
 FIG. 44 . 
 
 bubble or plumb-bob, but the observer has no means of 
 telling whether the staff is correctly held or not. When the 
 staff is held at right angles, the staff can be furnished with a 
 
 FIG. 45. 
 
 piece of wood whose two faces are each three or rour inches 
 long, one painted white and the other black ; this serves the 
 twofold purpose of enabling the staff-holder to sight square, 
 and as a tell-tale to the observer, who by a wave of the 
 signal-flag makes the staff-holder adjust it with perfect 
 correctness. 
 
1 66 Preliminary Survey 
 
 2. In the first method, the only correction for steep 
 angles in the method of reduction is to add to C the distance 
 /, which, as will be seen, is seldom a scaleable quantity, 
 and from the short table on p. 167 the surveyor will be able 
 to calculate mentally whether the correction will be appre- 
 ciable or not. There is, strictly speaking, a second correc- 
 tion for B, due to H being shorter in Fig. 44 than Fig. 45 
 by H H cos X; but this rarely enters into the calculations. 
 
 In the second method the use of cos 2 X is required 
 which necessitates a specially made slide-rule, or failing a 
 slide-rule a double calculation. For B also a double mul- 
 tiplier is introduced in the shape of sin X x cos X. These 
 formulae have to be applied to all vertical angles. 
 
 The metallic slide-rule made by Mr. J. Kern, of Aarau, 
 Switzerland, is a topographical speciality, which will greatly 
 assist those who prefer to have the staff held vertically, as 
 it gives the horizontal and vertical components for that 
 method. The values of cos 2 are given upon a short 
 slide and sin x cos upon the usual long slide. The 
 metallic division is not more accurate than that of either the 
 boxwood or celluloid rules, and the fit of the slide upon the 
 one examined by the writer was not equal to them. It is 
 more trying to the eye to read, and it is of course much 
 more expensive. It has, on the other hand, the advantage 
 of being more durable and less susceptible to humidity. 
 
 3. If the staff is intended to be kept always in a vertical 
 position, an error on the part of the assistant is much more 
 fatal to accuracy than when it is kept at right angles to the 
 line of sight. 
 
 For instance, let the stadia read one per cent. ; let the 
 line of sight be at an elevation of 45, and let the direct 
 distance be 300 feet. An error of 3 from the vertical 
 position of the staff will produce an error of 10-3 feet in 
 horizontal distance ; whereas the same error from a normal 
 position of the staff will only produce an error of 0*3 foot in 
 horizontal distance. 
 
Tacheometry 
 
 167 
 
 TABLE XXIII. Functions of Angles in per centagc of perpendicular 
 to base. 
 
 is 
 
 II 
 
 T3 | 4) d 
 
 S 8 ! # 1 
 
 ^3 i B'lj 
 
 -s 
 <Sf i |& 2 
 
 ^ a 
 
 j 
 
 . * 
 
 !5 2 
 
 be bJD 
 
 a 
 
 .sag 
 
 lafl u g 
 
 1 &'S 
 
 Percentage 
 perpendicular 
 to base 
 
 9 . 
 
 ' g 
 11 
 
 Angle in 
 degrees and 
 minutes 
 
 ll, 
 
 s'i-s 
 gb 
 
 fc i 
 
 I '00 
 
 i "73 
 
 2'OO 
 2-30 
 
 i o'| 1-7 
 
 *o 9 , ! 2 .' 
 
 1 44 ; 30 
 
 2 o' : 3 '5 
 2 18' ; 4'o 
 
 i 6-00 
 
 i 6-28 
 1 6-85 
 7-00 
 7-42 
 
 6 o' 
 6 17' 
 6 51' 
 
 7 o' 
 
 7 25' 
 
 io'5 
 
 II*O 
 I2'0 
 
 12-3 
 13 o 
 
 IQ'77 
 
 n'oo 
 
 ' 11-32 
 11-87 
 
 12 '00 
 
 10 46' 
 1 6 o' 
 11 19' 
 n 52' 
 
 12 0' 
 
 19-4 
 
 2O"O 
 21 '0 
 81*2 
 
 2 87 
 3'00 
 
 3 '43 
 4 ! 
 5-00 
 
 5'7 2 
 
 3 5 :-' i 
 
 3 26 6'o 
 4 ' 7'o 
 4 35' ! 8-0 
 
 5 o 8 ' 7 
 5 9 9'o 
 5 43' io'o 
 
 7 97 
 8"oo 
 8'53 
 g'oo 
 9*10 
 | 9-6S 
 IO'OQ 
 
 ' IO'2O 
 
 7 5 
 8 o' 
 
 8 3 2| 
 
 9 ^ 
 9 o 39/ , 
 
 10 
 
 10 12' 
 
 14 o 
 
 I 4'5 
 is'o 
 
 i6'o 
 17-0 
 17-6 
 i8'o 
 
 12-95 
 
 1 3 'oo 
 1 3 '50 
 14*00 
 T 4'3 
 I4-58 
 15*00 
 
 ;^; 
 
 13 3 
 14 o' 
 14 2' 
 14 35' 
 
 15 0' 
 
 2 3 
 23-1 
 24-0 
 24*9 
 
 26*0 
 
 26-8 
 
 Example i. Required by first method, B and C. -f 
 
 being 100 ; y+^=i'5 H=5'o ; S=3'oo ; and X=572. 
 This angle corresponds with a slope of i in 10, which is a 
 steeper ascent than is ordinarily met with in roads. 
 A=ioo S + i'5 = 30T'5 
 I3=A xsin X=3oi'5 x '0996 = 30*03 
 /TO per cent, of 5 feet ='50 feet 
 C=A xcos X + /= 301*5 x o'995 + '5 
 = 299-99 feet +-5 
 =300*49 feet 
 
 Note. B' is the height that is actually needed for the re- 
 duction of the levels, so that B would be, strictly speaking, 
 subject to an addition of H H. cos X=o'oi feet, making 
 B' = 3o'O4. 
 
 Example 2. Required A', B', and C in Fig. 45 the 
 data being the same, except that H will become 5-02 nearly 
 and S 7 = 3 -02 nearly. At that distance only the nearest 
 hundredth can be read to, so the results cannot be made 
 exactly to correspond with example i ; the difference of 0*1 
 in S' makes a difference of *i ft. in B'. 
 
1 68 Preliminary Survey 
 
 A'=ioo S ; . cos 
 
 = 100 xo'995 x 3 -02 + 1 -5 =302 -oo 
 
 (which is practically A+/) 
 B'=(ioo S'. cos X + i'5) x sin X 
 =302 x 0*995 x 0*0996 
 = 30*08 
 C=ioo S'. cos' 2 X+i*5 
 
 =302x0*995x0*995 +i'5 
 =300*5 
 
 It is true that by a specially constructed slide-rule, the 
 multiplication by cos 2 and by sin x cos is as simple as 
 that by cos and sin ; but beyond the limits of 300 feet with 
 angles as steep as that in the example, the slide-rule will 
 not give the result to a tenth of a foot in elevation, and 
 therefore most of the turning sights and many others also 
 where special accuracy is needed have to be reduced 
 numerically, and then the extra labour of the second method 
 is felt. 
 
 Usually a limit of accuracy is required of about one foot 
 of elevation and ten feet of distance. This upon a scale of 4 
 miles horizontal and 100 feet vertical per inch would not be 
 scaleable, whilst on scales of 400 feet horizontal and 40 feet 
 vertical per inch they would represent -025 inch, which is not a 
 large amount, and this accuracy can be attained by beginners. 
 
 The field work performed with the tacheometer alone, 
 apart from the auxiliary work of contouring and plane-tabling 
 detail, consists of surveying and levelling, and though they 
 are performed simultaneously we will consider them sepa- 
 rately ; and, first 
 
 SURVEYING 
 
 The greater part of the measurements are independent 
 rays whose bearings are read from the horizontal limb. Great 
 care is needed to avoid misreadings, as there is but little 
 opportunity for checking them. Some useful checks are 
 practicable which will be presently described, but the chief 
 
TacJieometry 169 
 
 means of keeping the general direction true is the observation 
 for azimuth described on p. 136, and a frequent comparison 
 with the compass-needle. 
 
 When the distance of any point is measured by the stadia 
 and its bearing also taken it can be plotted either by a pro- 
 tractor or by latitude and departure. Before commencing a 
 survey the true meridian is acertained and some convenient 
 reference-mark or 'directrix' chosen. Then the staff is 
 carried fb all the points in succession which are to be shown 
 on the plan, and finally to the new instrument station, the 
 observations to which are taken with extra care, and a check- 
 sight taken on the directrix before shifting the instrument. 
 It is advisable, if the instrument is fitted with two verniers 
 to the horizontal limb, to remove one of the microscope 
 ' readers ' during all the observations so as to avoid getting 
 on to the wrong vernier. 
 
 When adjusted at the new station, the horizontal limb is 
 clamped to the same bearing, entered in the book from the 
 first to the second station, and the telescope is directed by 
 means of the external axis to the staff held at the first station. 
 The reader is then transferred to the opposite vernier so as 
 to give the bearing it 180 and so replace the zero of the 
 horizontal limb in correspondence with the meridian. 
 
 It may be mentioned that in some countries the south 
 point is called 360 and the north point 180. 
 
 If the bases are short it will be necessary to align the 
 telescope from station to station by a plummet, because the 
 staff being three or four inches wide will give a margin of 
 error. 
 
 At the new station the staff is read a second time by the 
 stadia as a backsight, so that all the primary lines of the 
 traverse are measured twice over. // must be borne in mind 
 that the instrument cannot be carried forward as in levelling 
 ahead of the staff to take a backsight. At every fresh instru- 
 mental station the staff and instrument must change places. 
 The station must be clearly marked. A good stout peg should 
 
170 Preliminary Survey 
 
 be driven down within an inch of the ground and a reference 
 stake about three feet long driven a couple of yards away, 
 having the number or letter of the station chalked on it. 
 Where the intermediate staff stations are at denned points, 
 such as corners of fences, it will be sufficient to describe them 
 in the column for remark, but if they are undefined they 
 should be marked by smaller stakes or builder's laths. A 
 convenient notation on a continuous traverse is to name the 
 instrument stations by the capital letters of the alphabet to 
 the end, and then the small letters, after which it will 
 be safe to begin again with the capitals. The staff stations 
 can then be marked with the letter of the station from which 
 their position has been first determined, and their number 
 as A ls B 26 and so on, a separate column being given to 
 each. When a new instrument station is observed from an 
 old one, or a backsight from a new one upon an old one, 
 the reading of the staff is booked thus : 
 
 Inst. station Staff station Bearing 
 
 A B 175 
 
 B A 355 
 
 THE STADIA HAIRS 
 
 As has been already stated, when the vertical arc is used 
 the lower stadia hair should be directed to an even foot so 
 that the eye will not have the effort of endeavouring to read 
 simultaneously three heights upon the staff. There is gene- 
 rally a slight movement of the staff which makes it very 
 difficult to read unless the telescope is adjusted in this man 
 ner ; but by so doing the stadia are read at a glance, and 
 for the intermediate stations it is not necessary to read the 
 axial hair at all ; since the stadia hairs are equidistant from 
 it, the mean of the two readings will be the reading of the 
 axial hair. It is advisable, however, at the turning points to 
 read the axial hair as a check. 
 
 Some writers advise directing the axial hair to the same 
 
Tacheometry 1 7 1 
 
 height on the staff as the height of the instrument above the 
 ground or peg ; the value of B or B' will then be the differ- 
 ence of level of the two stations. This plan loses more time 
 in observation than it saves in calculation. 
 
 AUXILIARY WORK 
 
 Especially with beginners there is a liability to make a 
 misreading of the horizontal limb with the intermediate 
 sights, and where the work is very particular it is advisable to 
 let an assistant tape from point to point. Mistakes are 
 generally in even degrees, which would at once be shown 
 by the check lines. Taping also comes in very handily 
 where instrumental sights are only practicable at long 
 intervals. 
 
 I n a topographical survey of a portion of Windward Hawaii} 
 Sandwich Islands, the author adopted the following plan for 
 the ravines. These great gorges, locally termed gulches, 
 were sometimes a quarter of a mile wide, and the side slopes 
 varied in height from 100 to 400 feet and in angle from a 
 precipice to 20. The growth of running shrubs, matted 
 ferns, and ironwood was such that in one place it took two 
 hours to obtain a single sight. Half an hour was a very 
 frequent delay. 
 
 The instrument was set up on one side of the ravine and 
 the staff-party remained on the other. They consisted of a 
 staff-holder, clinometer man, and two axemen. All four 
 had axes, and the first thing was to clear for a sight. When 
 the first point was fixed, the slope was measured and a sub- 
 sidiary tape-traverse was run just wherever the jungle was 
 penetrable. The lengths varied from 10 to 50 feet and the 
 total tape traverse sometimes extended to several hundred 
 feet. Each tape measurement was aligned by the prismatic 
 compass, levelled by the Abney level, and the slope taken 
 up and down with the clinometer. As soon as the staff- 
 party reached a place practicable for a clearing, all four 
 
172 Preliminary Survey 
 
 would fall to with the axes again and a fresh point would be 
 thus determined from the instrument. The established points 
 were first plotted, and then the tape traverse plotted on 
 tracing paper and squared in with them ; the coincidence 
 was generally very close, except where the magnetic devi- 
 ation was great. 
 
 When the staff-party reached the limit of instrumental 
 range, a fresh station was selected on the same side of the 
 gulch by means of a backsight with a second staff taken to the 
 station just left, and also check sights to any of the stations 
 on the opposite side where the clearing had been sufficient 
 to put in a flagpole visible from the new station. There 
 was in this way a subsidiary triangulation by which to 
 eliminate error, and great accuracy was obtained. 
 
 When one side of the gulch had been surveyed, the 
 staff and instrument changed sides and the latter occupied 
 the stations in the clearings, the staff party repeating their 
 modus operandi on the other side of the gulch. 
 
 When the surveyor is in good practice with the tacheo- 
 meter, he will very rarely make a misreading, but it is always 
 a very useful check to let the staff-holder carry a passometer 
 (not a pedometer) and book it at every station ; this he can do 
 while the sight is being taken without impeding the progress. 
 
 REDUCTION OF THE TRAVERSE 
 
 The method of latitude and departure has this advantage 
 that protractor work is replaced by square scaling, and the 
 position can be determined in the field at any time whether 
 for the purpose of a check measurement to some known 
 object or for the assistance of the observation for azimuth. 
 When no tables or slide-rule are at hand it becomes a very 
 tedious process and is not often resorted to. 
 
 Chambers's Mathematical Tables give difference of lati- 
 tude and departure for every single degree and for each foot 
 of distance up to 300. The actual distances of a traverse 
 
Tacheometry 173 
 
 rarely exceed 600 feet, so that the latitude and departure 
 can be very readily booked from these tables, if close accu- 
 racy is not required, especially as it is generally possible to 
 arrange the base lines so that they come for the most part 
 on even degrees, by waving the staff-holder to the right or 
 left until he comes to the nearest degree. 
 
 With the slide-rule it is done in the following manner : 
 
 First. Reduce the bearing to its equivalent azimuth by 
 Table XVII. (which should also be pasted on to the slide- 
 rule). 
 
 Second. Find the difference of the angle thus found from 
 90, i.e. the complement. 
 
 Third. Place the initial point of the sine-scale under the 
 distance on the rule, and read off the latitude opposite to 
 the complementary angle and the departure opposite to the 
 angle. With small angles it will give the nearest tenth, but 
 with larger angles the nearest foot. 
 
 Example. Find latitude and departure of base line 
 A/f\m fieldbook, p. 182. Bearing 347-03, dist. 258. 
 (The distance must of course be taken as reduced to the 
 horizontal, which is equal to dist. x cos vert, angle or else 
 by Table XXIV.) 
 
 The azimuth will be 13 -9 7 N.W. and its complement 
 76-03. Place the left hand i of the sine-scale under the 
 258 of the rule, and opposite to angle 13-97 will be found 
 62-4 feet and opposite to 76, 250 feet. 
 
 To reduce the difference of longitude and latitude thus 
 found to true longitude and latitude from Greemvich by 
 Tables XXV. and XXVI. 
 
 This is useful for astronomical observations and check 
 points. 
 
 Example 
 
 Started on January i in long. 157-582 W. 
 " lat. 2i- 45 N. 
 
 Total northings for the day's run 33,425 feet. 
 westings i7> 2 95 
 
174 
 
 Preliminary Survey 
 
 Northings in miles by slide-rule 6-33, multiplied by ^ 
 by slide-rule =5 -52' of latitude N. 
 
 Westings in miles by slide-rnle=3'27 ; multiplied by ~^ 
 =3-01' of longitude W. 
 
 Position on January 2 : 
 
 Long. 157 3i'3 / + 3'oi 
 Lat. 21 2 / + 
 
 3'oi / =i57 34'3 I/ 
 
 ' = 2i 32-52' N. 
 
 As the maritime and geographical positions are given in 
 degrees, minutes, and seconds it is more convenient to 
 retain the old notation for that purpose. 
 
 TABLE XXIV. ' Difference between hypotenuse and base in feet per hun- 
 dred feet of hypotenuse, for sloping ground. (Note. This should 
 be marked on the vertical limb of the transit ; on Y theodolites it 
 usually is.) 
 
 Slope in 
 degrees 
 
 Deduct, 
 feet 
 
 Slope in 
 degrees 
 
 Deduct, 
 feet 
 
 Slope in 
 degrees 
 
 Deduct, 
 feet 
 
 Slope in 
 degrees 
 
 Deduct. 
 feet 
 
 | 
 
 ooi 
 
 I 
 
 420 
 
 i 
 
 1-596 
 
 | 
 i 
 
 4 
 
 3-52I 
 
 
 OO4 
 
 i 
 
 460 
 
 1 
 2 
 
 1-675 
 
 | 
 
 3*37 
 
 I 
 
 OOQ 
 015 
 
 6 
 
 548 
 
 3 
 4 
 II 
 
 1755 
 I-837 
 
 i6 4 
 
 3754 
 3'874 
 
 I 
 
 4 
 
 024 
 
 i 
 
 '594 
 
 i 
 4 
 
 1-921 
 
 i 
 
 3"995 
 
 1 
 
 a 
 
 034 
 
 i 
 
 643 
 
 1 
 
 2-008 
 
 i 
 
 4-118 
 
 .1 
 
 4 
 
 047 
 
 4~ 
 
 693 
 
 3 
 4 
 
 2-095 
 
 f 
 
 4-243 
 
 2 
 
 O6 1 
 
 7 i 
 
 745 
 
 12 
 
 2-185 
 
 17 
 
 4370 
 
 * 
 
 077 
 
 
 800 
 
 i 
 
 2-277 
 
 i 
 
 4-498 
 
 1 
 
 095 
 
 i 
 
 856 
 
 i 
 
 2-370 
 
 i 
 
 4-628 , 
 
 f 
 
 "US 
 
 1 
 
 913 
 
 4~ 
 
 2-466 
 
 4" 
 
 476o j 
 
 3 
 
 137 
 
 8 
 
 973 
 
 13 
 
 2-563 
 
 18 
 
 4-894 
 
 i 
 
 4 
 
 161 
 
 i 
 
 4 
 
 i'035 
 
 1 
 
 47 
 
 2-662 
 
 i 
 
 5-030 
 
 
 187 
 
 i 
 
 1-098 
 
 
 2763 
 
 i 
 
 5-168 
 
 4" 
 
 214 
 
 3 
 4 
 
 164 
 
 1 
 
 2-866 
 
 i 
 
 5-307 
 
 4 
 
 244 
 
 9 
 
 231 
 
 H 
 
 2-970 
 
 19 
 
 5-448 
 
 i 
 
 275 
 
 i 
 
 300 
 
 i 
 
 3-077 
 
 i 
 
 5-59I 
 
 i 
 
 308 
 
 l 
 
 37i 
 
 -| 
 
 3-i85 
 
 ^ 
 
 5-736 
 
 f 
 
 343 
 
 f 
 
 444 
 
 4~ 
 
 3-295 
 
 4 
 
 5-882 
 
 5 
 
 381 
 
 10 
 
 5i9 
 
 15 
 
 3-407 
 
 20 
 
 6-031 
 
 Copied from Trautwine's ' Pocket Book.' 
 
TacJieometry 
 
 175 
 
 TABLE XXV. Length oj a mimite of longitude in different latitudes at 
 the level of the sea in stat^^te miles of $,280 feet. 
 
 Degrees 
 of 
 latitude 
 
 i minute 
 longitude 
 in miles 
 
 Degrees 
 of 
 latitude 
 
 i minute (Degrees 
 longitude i 1 of 
 in miles ..latitude 
 
 i minute 
 longitude 
 in miles 
 
 Degrees 
 of 
 latitude 
 
 i minute 
 longitude 
 in miles 
 
 O 
 
 I-I53 
 
 18 
 
 I -097 36 
 
 0-933 
 
 54 
 
 0-679 
 
 2 
 
 I-I52 
 
 20 
 
 I -084 38 
 
 0-909 
 
 56 
 
 0-646 
 
 4 
 
 I-I50 
 
 22 
 
 I -069 40 
 
 0-886 
 
 ss 
 
 0-612 
 
 6 
 
 1-146 
 
 24 
 
 1-053 42 
 
 0-858 
 
 60 
 
 0-578 
 
 8 
 
 142 
 
 26 
 
 1-037 44 
 
 0-831 
 
 62 
 
 0-542 
 
 10 
 
 135 
 
 28 
 
 1-019 46 
 
 0-802 
 
 64 
 
 0-507 
 
 12 
 
 128 
 
 30 
 
 0-999 48 
 
 0773 
 
 66 
 
 0-470 
 
 H 
 
 119 
 
 32 
 
 0-979 50 
 
 0743 
 
 68 
 
 0-433 
 
 16 
 
 108 
 
 34 
 
 o-957 ; 52 
 
 0-711 
 
 70 
 
 0*395 
 
 TABLE XXVI. Length of a minute of latitude in different latitudes 
 at the level of the sea in statute miles of $, 
 
 Degrees 
 
 i minute >j Degrees 
 
 of 
 
 latitude 
 
 of 
 
 latitude 
 
 in miles 
 
 latitude 
 
 o 1-145 
 
 30 
 
 IO 
 
 I-I45 
 
 40 
 
 20 
 
 1-146 
 
 50 
 
 egrees 
 of 
 
 i minute 
 latitude 
 
 i Degrees 
 of 
 
 i minute 
 latitude 
 
 titude 
 
 in miles 
 
 latitude 
 
 in miles 
 
 30 
 
 I-I48 
 
 60 
 
 I-I52 
 
 40 
 
 I-I5O 
 
 70 
 
 I'I54 
 
 50 
 
 I-I5I 
 
 80 
 
 I-I55 
 
 LEVELLING 
 
 The elevations of the staff-stations are obtained by first 
 determining the elevation of the optical axis of the instru- 
 ment from some bench-mark ; in other words the height 
 of the centre of the trunnion of the telescope above any 
 known or assumed datum. From this elevation the eleva- 
 tions of all succeeding intermediate staff-stations and the 
 next instrument station are determined by calculating the 
 vertical component of the direct distance, and adding to or 
 deducting from it the height of the staff from the ground to 
 where it is intercepted by the axial hair of the telescope. 
 
 Whether at the commencement or at a turning-point, 
 the elevation of the optical axis is obtained by a backsight, 
 and all other sights are foresights. The elevation of optical 
 axis is henceforth termed O. A. to distinguish it from the 
 
176 
 
 Preliminary Survey 
 
 height of the instrument itself above the peg over which it 
 stands, which will be called H. I. 
 
 This latter needs a separate column because it is an 
 independent check. The elevation of instrumental station 
 will be signified by E.I.S., and that of the staff station by 
 E.S.S. The direct distance will be marked D., the hori- 
 zontal component H. C, and the vertical component V. C. 1 
 
 THE BACKSIGHT 
 
 The elevation of optical axis, O. A., is obtained by the 
 following formulae : 
 
 a. When the vertical angle is one of elevation, which is 
 termed //KJ, O.A.=E.S.S. +B.S.-V.C. 
 
 FIG. 46. 
 
 FIG. 47. 
 
 b. When the vertical angle is minus, O.A.=E.S.S. + B.S. 
 + V.C. 
 
 The elevation of instrumental station E.I.S., is obtained 
 in either case by deducting H.I. from O.A. It is needed as 
 a check from the next station. 
 
 THE FORESIGHT 
 
 The elevation of any intermediate staff-station, or of 
 the next instrumental station, is obtained by the following 
 formulae : 
 
 1 See also remarks on adjustment of axial hair to sanie height as 
 that of instrument, p. 174 at foot. 
 
Tacheometry 
 
 177 
 
 (a) When the vertical angle is plus, E.S.S.=O.A.+V.C. 
 - F.S. 
 
 (ft) When the vertical angle is minus, E.S.S.=O.A. 
 (V.C. + RS.). 
 
 In these figures the staff is shown vertical, and the instru- 
 ment not anallatic, but the rules apply to whatever method 
 is adopted of obtaining V.C. 
 
 FIG. 48. 
 
 FIG. 
 
 The first operation backsight A to / \is reduced thus 
 
 E. S. S. = 122-12 
 + B.S. 5-28 
 
 -V. C. 
 
 127-40 
 22-34 
 
 O. A. = 105-06 
 The second intermediate sight A to A, thus : 
 
 O. A. = 105-06 1-63 V.C. 
 - ( V. C. + F. S. ) 4 -39 2 -76 + F. S. 
 
 E. S. S. 100-67 
 
 The foresight A to B, O. A. = 105-06 11-96 V. C. 
 -(V.C. -i- F.S.) 17-48 5'52 + F. S. 
 
 K.S.S. 87-58 " 
 
 It should be remembered that there may be a fall with 
 a plus angle when the staff-reading is greater than the V.C., 
 but never a rise with a minus angle. 
 
 The last backsight here agrees exactly with the foresight, 
 
 N 
 
178 
 
 Preliminary Survey 
 
 Remarks 
 
 i!i - 
 
 O 
 
 o 1 & 
 
 CO 
 
 M ^ M 10 
 
 CO 
 
 m 
 
 n s 
 
 ffi 
 
 vO ro 
 
 CO 
 CO 
 
 W 
 
 ^ \O* 0? 2" ^ 
 
 c-i b b w t^ 
 w o o o oo o 
 
 CO 
 
 W 
 
 8 % 
 8 S- 
 
 
 
 CO <O 10 co ON OO 
 
 8 * ~ ;: n. 
 
 
 
 a 
 
 ^ ^ ^ J? S* o" 
 
 M N CO CO CO CO 
 
 p 
 
 O^ ^ rt ro Jt^ f^ 
 
 rt 
 
 1 
 CO 
 
 t^ O c^ O c^ O | 4 O ^o O to O 
 y> O ioo ^hO *-"O OO OO 
 
 "O^j- cow voco ^ M t^^i- vbco 
 
 II 
 
 "M "o %t 
 
 o o o o o o 
 ^ O O O W N 
 
 11 
 
 8 % ^ % 5 5 
 
 o o o o o o 
 t^. ^- vO HH ro co 
 ^t* *^ vO W vO ^* 
 
 CO M w CO 
 
 CO 
 
 co 
 
 < <T 4 < P4 <! 
 
 1 
 
 < PP 
 
Tackeometry 1 79 
 
 but if there is a discrepancy exceeding the working limit of 
 accuracy the sights should be repeated ; if they only differ 
 by a small amount, such as one or two hundredths, for close 
 levelling the mean should be taken for the O.A. of the new 
 station. The accuracy of the levelling never can equal that 
 of an ordinary levelling instrument, inasmuch as there is 
 always a liability to error in the measurement of the distance, 
 otherwise the accuracy would be the same. Error in the 
 vertical angle is more or less present with the intermediate 
 sights ; but, in the back- and foresights, it is eliminated by 
 the precaution of making each observation in duplicate, 
 face right and face left, and taking the mean angle. This 
 is done by releasing the parallel plate, and rotating the 
 telescope 180 ; then, revolving it in its trunnions, the 
 measurement of the vertical angle in the reversed position 
 will reverse the index error if any. This should be very 
 little in a day's use of the instrument, if it has been properly 
 adjusted in the morning as explained on p. 313. 
 
 The importance of this operation for eliminating index 
 error cannot be overstated. It should never be neglected, hoiv- 
 ever rapid the rate of march. 
 
 Thus, if the vertical angle from A to B was 2 13' 20" 
 when the vertical circle faced to the right, and in the 
 reversed position 2 14' 40", the true vertical angle would 
 be 2 14'. 
 
 The value of the tacheometer for long sights and great 
 differences of level is a cardinal point, but it must be re- 
 membered that the accuracy is in opposite ratio both to 
 the distance and the height. The writer has obtained a 
 difference of elevation exceeding a hundred feet in one shot 
 exactly, the oblique distance being between 300 and 400 feet, 
 but that cannot be relied upon with a tacheometer any more 
 than long shots with the level. The surveyor should there- 
 fore keep to short sights for the centre-line of his survey if 
 running a railway traverse, but the outlying points, such as 
 fences, &c., can be put in with longer sights. 
 
 N 2 
 
i8o 
 
 Preliminary Survey 
 
 In working out intermediate sights by slide-rule, or m 
 checking the back- and foresights, the following short table 
 will serve to show where the decimal point will come in the 
 result ; for instance, in the backsight A /\ the angle is 
 nearly 5, and the distance nearly 250, so it will be at once 
 seen that the result on the slide-rule is 22-34, and not 2-23 
 or 223-4. The list of multipliers and other factors on the 
 back of the Mannheim rule is not of much use to surveyors, 
 so it is best to paste them over and write a table such as 
 this one ; the rules for back- and foresights on pp. 180, 181 ; 
 the table of square-roots of small decimals given on p. 250 ; 
 and some others in this book, in the place of them. 
 
 TABLE XXVII. Sines multiplied by various distances. 
 
 Degrees 
 
 Degrees, 
 minutes, and 
 seconds 
 
 Sine angle 
 multipled by 50 
 
 Sine angle Sine angle 
 multiplied by 250 multiplied by 750 
 
 oi o o'36" 
 
 0087 
 
 043 
 
 13 
 
 05 o 3' o" 
 
 0436 
 
 22 
 
 I- 6 I 
 
 10 
 
 6'0" 
 
 0872 
 
 44 
 
 
 20 
 
 o 12' o" -174 
 
 87 
 
 2-62 
 
 30 
 
 o 1 8' o" -262 
 
 1-31 
 
 3-93 
 
 -50 
 
 o 30' o" -436 
 
 2'lS 
 
 6-55 
 
 75 
 
 o 45' o" 
 
 6 5 
 
 3-27 
 
 9-82 
 
 i-oo 
 
 i o'o" 
 
 87 
 
 436 
 
 13-09 
 
 3-00 
 
 7 O'O" 
 
 2-61 
 
 I3-07 
 
 39-22 
 
 5-00 5 o'o" 4-36 
 
 21-80 
 
 6537 
 
 7-00 7 o'o" 6-09 
 
 3047 
 
 91-40 
 
 10-00 10 o'o" 8-68 
 
 43-41 
 
 130-23 
 
 15-00 15 o'o" 12-94 
 
 6471 
 
 194-13 
 
 20-00 
 
 20 O'o" 17-10 
 
 85-50 
 
 256-51 
 
 25 -oo 25 o' o" 
 
 21-13 
 
 I05-65 
 
 
 30-00 
 
 30 o o 25-00 
 
 I25-00 
 
 
 
 35-oo 35 o'o'' 28-68 
 
 14339 
 
 
 
 40-00 40 o' o" 32-14 
 
 160*69 
 
 
 
 45-00 45 o'o" 35-36 
 
 17678 
 
 
 
 50-00 50 o'o" 38-28 
 
 I9I-4I 
 
 ~ 
 
 55'OQ 55 o'o" 40-96 
 
 204-79 
 
 
 
 60-00 j 60 o'o" 
 
 43-30 
 
 216*50 
 
 
 Referring back to the two first sights in fieldbook, p. 182, 
 to multiply sin 4 57' by 259. Place a i of the sine- scale 
 
Tacheometry 1 8 1 
 
 under 259 of the rule, then opposite to 4 57' will be found 
 223, which by reference to Table XXVII. is seen to mean 
 22 '-3. The slide-rule will only give the nearest tenth, but as 
 it is a backsight it ought to be worked out from a table of 
 sines and tangents. If in the office, Crelle's tables, which 
 give at a single inspection the products of three figures by 
 three figures will be found of great assistance, and this is as 
 close as it is necessary to go. For intermediate sight AA 13 to 
 multiply sin 22' by 254. Place the fiducial mark for minutes 
 on the number scale of the slide, under the 22 of the rule. 
 Then opposite to 254 on the slide will be found 163, which 
 by inspection of the table will be at once identified as i'63 
 feet. 
 
 If the elevations are required in metres, the staff must be 
 graduated also in metres, but the operation is the same. 
 
 CONTOURING 
 
 The elevations of suitable points having been determined 
 in the foregoing manner, the intervening ground may be 
 topographically represented by dotted lines of equal ele- 
 vation called contours, in precisely the same manner as 
 described on p. 61. It is well to distinguish contours which 
 are drawn between two fixed points from those which only 
 depend upon a slope taken from one point, the former being 
 naturally more reliable when the distances are not great. 
 
 PROFILE 
 
 The profile, as it is called in America, or section, as 
 termed in England, is produced from the contours, which 
 are drawn sufficiently close to enable the elevation of each 
 100 feet station to be judged to the nearest foot by the eye. 
 The gradients are first fixed approximately. In Fig. 50 for 
 instance, a suitable place was found about a mile up the 
 gulch for a horseshoe curve. Crossing the river at a 
 suitable height would leave a gradient of about i per cent. 
 
1 82 Preliminary Survey 
 
 falling on both sides, so as to meet the rising bottom 01 the 
 gulch. The next operation was then to mark off at say 
 every 500 feet the position where the gradient intersects the 
 surface. Then with railway sweeps a trial line is drawn to 
 pass as nearly as possible through these points, having due 
 regard to curvature. In a case like this, where the line is 
 located on the side of a hill, the whole of the line must be 
 thrown into cutting, except where the slope is flat enough to 
 admit of embankment. 
 
 It is nearly always cheaper to take out a cutting, even in 
 rock, than to build up a retaining wall. Where this latter 
 course becomes necessary, it is usual to bench the hillside 
 and build up cross-logging^ jointed at the ends with axe- 
 cuts, termed in America cribwork. In other places it is 
 found cheaper and otherwise preferable to build a wall 
 of dry-rubble, but, apart from the question of expense, 
 these erections are liable to destruction from wash-outs 
 and decay, and a little extra expense in cutting is more 
 economical in the long run. 
 
 When the location is on tolerably even country, the trial 
 line should aim at equalising the cuttings and embankments, 
 making them all as short as possible. When the line is on 
 level ground subject to floods, it should be placed entirely 
 in embankment, so as to keep it above highest known flood- 
 level, with frequent openings for letting off the flood waters. 
 
 The details of curvature should all be written up on the 
 plan, for which see p. 217. 
 
 In plotting the profile, one man should take the plan, 
 with contours and trial line drawn on, and another man the 
 ruled profile-paper, which dispenses with all scaling. The 
 first then reads out the hundred-foot stations, and if neces- 
 sary, intermediate points with their elevations, and the other 
 plots them. Thus: station 200, elevation 354; station 201, 
 elevation 353; station 202, elevation 350^; station 202 + 10, 
 350, and so on. 
 
 Fig. 50 is a sketch from memory of one of the Hawaiian 
 
Tacheomelry 
 
 183 
 
 gulches, and is merely intended as an illustration of the 
 method. The scales are sufficiently given by the centre line 
 of the location and the profile. The sharp curve of 40 
 
 FIG. 50. 
 
 (146-2' rad.) on the right of the figure was frequently resorted 
 to either to turn the head of a gulch, or to cut through a 
 rock-spur as shown. 
 
 TACHEOMETRIC CURVE-RANGING 
 
 Curve-ranging is performed by tangential angles with 
 the tacheometer on the same principle as by the transit and 
 
 FIG. 51. 
 
 chain, with this exception. Measurements by the chain are 
 necessarily repetitions of a short chord round the arc of a 
 circle, whereas measurements with the tacheometer are each 
 of them an independent chord, as shown on Fig. 51, where 
 
1 84 
 
 Preliminary Survey 
 
 A is the B.C. (beginning of curve) and AB the tangent 
 produced; A 10 A 20 A 30 , &c., are each of them rays from the 
 instrument, which may be either chords forming equidistant 
 points upon the curve, in which case every radius demands 
 a different series of tangential angles, or else they may be in 
 decimal parts of the radius, which is the method shown on 
 Fig. 51. Whatever the curve may be, points upon it can 
 be at once determined from the following table. 
 
 TABLE XXVIII. Tangential angles for chords with radius- 100. 
 For taclieometric curve -ranging. 
 
 
 Tan- 
 
 
 Tan- 
 
 
 Tan- 
 
 
 Tan- 
 
 
 Tan- 
 
 12 
 
 gential 
 
 12 
 
 gential 
 
 fi 
 
 gential 
 
 1 
 
 gential 
 
 5 
 
 gential 
 
 J3 
 
 angle in 
 
 Ji 
 
 angle in 
 
 J3 
 
 angle in 
 
 
 angle in 
 
 j~ 
 
 angle in 
 
 
 
 degrees 
 
 
 
 degrees 
 
 CJ 
 
 degrees 
 
 
 
 degrees 
 
 o 
 
 degrees 
 
 i 
 
 0-28 
 
 21 
 
 6-03 
 
 41 
 
 11-83 
 
 61 
 
 1776 
 
 8 1 23-89 
 
 2 
 
 3 
 
 C-86 
 
 22 
 23 
 
 6*32 
 6-60 
 
 42 
 43 
 
 12-12 
 I2-42 
 
 62 
 63 
 
 1 8 -06 
 18-36 
 
 82 
 83 
 
 24-21 
 24-52 
 
 4 
 
 1*15 
 
 24 
 
 6-89 
 
 44 
 
 I2-7I 
 
 64 
 
 18-67 
 
 84 
 
 24-83 
 
 5 
 
 i-35 
 
 25 
 
 7-18 
 
 45 
 
 I3-00 
 
 65 
 
 18-97 
 
 85 
 
 25-I5 
 
 6 
 
 1-72 
 
 26 
 
 7-47 
 
 46 
 
 I3-30 
 
 66 
 
 19-27 
 
 86 
 
 25-47 
 
 7 
 
 2-OI 
 
 27 
 
 776 
 
 47 
 
 I3-59 
 
 67 
 
 I9-57 
 
 87 
 
 2578 
 
 8 
 
 2-29 
 
 28 
 
 8-05 
 
 48 
 
 13-89 
 
 68 
 
 19-87 
 
 88 
 
 26-13 
 
 9 
 
 2-58 
 
 29 
 
 8-33 
 
 49 
 
 14-28 
 
 69 
 
 20-18 
 
 89 
 
 26-42 
 
 10 
 
 2-87 
 
 30 
 
 8-62 
 
 50 
 
 I4-48 
 
 70 
 
 20-49 
 
 90 
 
 26-74 
 
 ii 
 
 3-I5 
 
 31 
 
 8-92 
 
 51 
 
 I4-77 
 
 71 
 
 20-79 
 
 91 
 
 27-07 
 
 12 
 
 3'44 
 
 32 
 
 9-21 
 
 52 
 
 I5-07 
 
 72 21-10 
 
 92 
 
 27-39 
 
 13 
 
 373 
 
 33 
 
 9-50 
 
 53 
 
 I5-37 
 
 73 
 
 21-41 
 
 93 
 
 2771 
 
 14 
 
 4-02 
 
 34 
 
 978 
 
 54 
 
 15-67 
 
 74 
 
 21-72 
 
 94 
 
 28-03 
 
 15 
 
 4-30 
 
 35 
 
 10-07 
 
 55 
 
 15-96 
 
 75 
 
 22-02 
 
 95 
 
 28-36 
 
 16 
 
 4-59 
 
 36 
 
 10-37 
 
 56 
 
 I6-26 
 
 76 
 
 22-33 
 
 96 
 
 28-68 
 
 17 
 
 4-88 
 
 37 
 
 10-66 
 
 57 
 
 16-56 
 
 77 
 
 22-64 
 
 97 
 
 29-02 
 
 18 
 
 5' 1 7 
 
 38 
 
 10-95 
 
 58 
 
 16-86 
 
 78 
 
 22-96 
 
 98 
 
 29-33 
 
 19 
 
 5-45 
 
 39 
 
 11-24 
 
 59 
 
 17-16 
 
 79 
 
 23-27 
 
 99 
 
 29-67 
 
 20 
 
 5'74 
 
 40 
 
 n'53 
 
 60 
 
 17-46 
 
 80 
 
 23-58 
 
 IOO 
 
 3O-OO 
 
 Let us suppose that the bearing of AB is 360, and we wish 
 to set out a curve of 5, that is of i,i46ft. radius in chords 
 of about looft. If ii4'6ft. is near enough, we will take the 
 tens and begin with 2 -87, but if it be necessary to come 
 closer than that, we should take 9, 18, 27, &c., and begin 
 with 2-58, which would give us for a first chord 103-1 4ft. 
 The one is as simple as the other by the slide-rule. 
 
TacJieometry 185 
 
 Let us suppose that we take the 9. Place the right 
 hand i of the slide under the 1,146 of the rule, and read off 
 the first dist. io3ft. under the 9, and set the instrument at 
 2*58. Put the staff-holder in line, and make him take 37 
 ordinary steps forward. Keeping him in line wave him back 
 or forward by the Morse ' B ' or ' F ' signal until his staff re- 
 cords the distance by the stadia-hairs, and then put a stake 
 in. Clamp the instrument at 5M7, and read off the distance 
 opposite to 1 8 on the slide-rule=2o6ft. The sub-chords 
 A-io, 10-20, 20-30, 30-40, &c., lengthen as the curve is 
 ranged, but that is quite immaterial for a preliminary 
 survey. If A-io were 103, 70-80 would be io9ft, and 
 60-70 would be 107^. 
 
 Any labourer of average sense will learn to step each 
 distance within a yard, and the time occupied in ranging 
 him in line and moving him back or forward is no longer 
 than that of the ordinary curve-ranging. 
 
 This method is quite suitable for ranging the permanent 
 centre-line of a railway where great precision is not 
 required. 
 
 By leaving a mark at A the instrument can be shifted 
 to any one of the points on the curve, and directed to the 
 tangent. For instance, if it were desired to set up the instru- 
 ment on the tangent at point 80, of which the tangential angle 
 is 23 0> 58, we should clamp the parallel plates at 23'58, and, 
 reversing the telescope, direct it by means of the external 
 axis on A. Then, reversing it back again with the external 
 axis clamped, release the parallel plates, and set the vernier 
 at 47-i6, which is twice the tangential angle, or in other 
 words the angle of deflection, and the instrument would 
 then be on the tangent. 
 
 It is even more advantageous to keep the curve-ranging 
 on true astronomical bearings when on preliminary survey 
 than when on construction, but the manner of doing so is 
 considered in the chapter on curve-ranging. 
 
1 86 Preliminary Survey 
 
 METHOD OF RANGING A CURVE TO FINISH AT A GIVEN 
 ALIGNMENT WITHOUT SHIFTING THE INSTRUMENT * 
 
 It frequently happens that the chaining of a trial line 
 in order to obtain the topography sufficiently to locate the 
 best curve would involve a great deal of labour, and 
 would run a very small chance of being the final one. It 
 is possible without shifting the instrument to locate a curve 
 upon the ground which will pass through any given points 
 and terminate on a given alignment without chaining. Let 
 us suppose that a tangent has ended upon the edge of a 
 ravine, and it is desired to lay out a horse- shoe curve which 
 will follow the slope of the hill ; turn the head of the valley 
 with a trestle, and, skirting the opposite hillside, terminate 
 in a parallel to a boundary of some property which could 
 not be interfered with except at considerable cost. Any 
 other circumstances may fix the direction of the final 
 tangent. If there are no obstructions such as the one in 
 the figure, the tangent will be chosen to suit the ground, 
 and in the nearest direction practicable to the objective 
 point of the railway. 
 
 The surveyor-in-chief will leave an assistant at the 
 transit, and proceed to the neighbourhood of B. He will 
 then fix upon a suitable direction for the final tangent BY, 
 and measure its magnetic bearing with the pocket altazimuth 
 described on p. 321, or by an ordinary prismatic compass. 
 Knowing the variation of the compass by previous observa- 
 tions, he then reduces the bearing to an astronomical 
 bearing, and telegraphs it to the transit man by means of a 
 flag. See Flag-signals (p. 93). The transit-man then works 
 out the deflection angle D, and tangential angle T, which 
 is equal to the angle XAB, and setting his instrument to 
 AB, gives the surveyor-in-chief the E.G. point B. It is 
 
 1 Before perusing the following, the student should read the chapter 
 on curve-ranging. 
 
Tacheometry 1 87 
 
 possible that B may fall considerably more to one side than 
 was intended, so that it would be necessary to prolong 
 or shorten the tangent at A. It is possible also that, after 
 locating D, it may not be the best place for the trestle 
 
 FIG. 52. 
 Preliminary Calculations and Formula. 
 
 AB = 700 feet. 
 
 D =149-48= AC B. 
 
 T = 74 - 74 =BAX. 
 
 Rad=AC=^ B =362-8 feet 
 2 sin T ' 
 
 AD =^ = 44 o- 4 feet 
 
 2 COS - 
 2 
 
 AB = 2R sin T 
 
 Neither of these eventualities need necessitate a change in 
 the direction of BY, but merely in the position of A and 
 B, and the radius of the curve. Assuming that B is a 
 suitable point for the E.G., the staff is held there, and the 
 
1 88 Preliminary Survey 
 
 transit-man measures the length of AB by the stadia, and 
 telegraphs it. He then works out the radius and AD by 
 above formulae and telegraphs them. The surveyor-in- 
 chief then considers whether the radius is a suitable one, 
 or whether, by moving his final tangent a little, he can 
 obtain either a longer radius, or one which corresponds to a 
 more simple degree of curve, or one which is on better 
 ground, and, if necessary, telegraphs back the radius which 
 he desires. In the illustration, the radius being 36 2 '8 feet, 
 and a 1 6 curve having a radius of 359*3 feet, he would 
 telegraph back simply 16, and place his staff for fresh 
 alignment about 7 feet nearer to the transit. The transit- 
 man will then work out AB afresh= 2 x 359*3 x sin T= 
 693-3 feet, and AD=436'2. The chief surveyor will then 
 first drive a stake at the E.G., and then, proceeding to D, will 
 get his alignment and distance from the transit-man, and, if 
 suitable, drive another stake at D marked apex. 
 
 It may be advisable to know the difference of level 
 between A and D, to determine the height of the trestle 
 before finally selecting D. This the transit-man will do by 
 a foresight, as described on p. 176, and if necessary the 
 chief surveyor will take one or more trial points above or 
 below D. Since he is possessed of all the data of the curve, 
 he can with the slide-rule plot the curve on a sketchboard 
 and alternative curves above or below D, if he pleases. 
 
 When the E.G. and apex points have been satisfactorily 
 fixed, and stakes driven, the chief surveyor returns to the 
 transit, during which time the transit man has worked out 
 the data of the curve, and is ready to range it in the method 
 described on p. 183. 
 
 The particulars of this fieldbook will be better under- 
 stood after reading the chapter on curve-ranging. The 
 column ' chainage of,' is so marked for simplicity, although 
 the chain is not actually used. The columns of subtangent 
 and apex distance are not filled in because, the curve not 
 being run to intersection, it is needless to calculate them. 
 
TacJieomctry 
 
 189 
 
 Description of Curve. 
 
 
 
 
 
 1 
 
 
 Total 
 
 u 
 
 
 angle of Astronomical bearing 
 deflection 
 
 S 1 
 
 .1 
 
 S 
 
 bo 
 
 1 
 
 c 
 I 
 
 Ufiamage ot 
 
 .si * i 2 s 
 
 y 1 1 1 1 
 
 a 
 
 1 
 
 3 
 C/2 
 
 3 
 
 1 
 < 
 
 u B | 
 
 5 ! J ' ^ M 
 
 < S, cj j fc 3 
 
 
 
 
 
 
 PQ i <! W 
 
 a 
 
 ^ 1 
 
 
 
 1 
 
 deg. deg. j deg. 
 
 deg. cleg.j deg. 
 
 
 feet 
 
 feet 
 
 feet 
 
 feet 
 
 i 
 
 i i 
 j 149 '48 17-32 
 
 1 1 
 
 54*69 92*06 i66'8o 
 
 16 
 
 3S9'3| 
 
 934'3 
 
 
 
 113 117 j 122 
 + 2 4 +91*1 ; +58-3 
 
 But little is said in Chapter IX. about the various types 
 of tacheometer which are before the public, many of them 
 well worth description, and but little has been said here 
 about the method of using the stadia-rod vertically. The 
 formulae for cos 2 and cos x sin are only tabulated for the 
 centesimal degree in a somewhat bulky form, published by 
 Cuartero of Madrid ; they can be obtained through Messrs. 
 Troughton & Sims. The method given in these pages 
 requires nothing more than an ordinary table of sines and 
 cosines, and the writer has proved its accuracy by consider- 
 able use of it. Upon the Hawaiian survey, a section of 
 seventy miles of railway included a little over one hundred 
 ravines of sizes varying from a hundred feet wide by fifty 
 deep, to a quarter of a mile wide by four hundred feet 
 deep. The intermediate survey on level ground between 
 the ravines was made with the transit, chain, and level, by 
 a separate party in the usual manner. The large ravines 
 were all surveyed with the tacheometer, and frequently 
 where two ravines were close together the intervening 
 ground was also surveyed and afterwards checked with the 
 results obtained by the chain party. In many cases the 
 one party used the stakes of the other as reference 
 marks both for distance and elevation, and the accuracy 
 of the stadia work was on the whole decidedly greater 
 than that of the chaining. The space of this work does 
 not admit of dwelling exhaustively upon many different 
 
190 Preliminary Survey 
 
 methods of doing the same thing ; the one given here is the 
 simplest. 
 
 The following are the conclusions of Mr. Lyman, in his 
 paper, read before the Franklin Institute in 1868. on the 
 experience gained by him from the Schuylkill topographical 
 survey, United States of America. 
 
 1. That the additional lenses and complications adopted 
 by Porro, to cause the centre of anallatism to fall in the 
 exact vertical axis of the instrument, were needless, as the 
 inconvenience from adding to every distance observed a con- 
 stant quantity equal to the sum of the focal distance of the 
 object-glass, and the distance of that glass in front of the 
 vertical axis, rarely exceeding i foot, is comparatively trifling. 
 
 2. That three horizontal hairs or wires are sufficient for 
 all purposes, as the reading of the middle wire affords suffi- 
 cient check on the other two ; that fixed are preferable to 
 movable wires with protected adjusting screws ; and that 
 the fixed wires should be so set that the visible height on 
 the staff intercepted between the middle and either outer 
 wire should bear some exact ratio to the distance such as i 
 foot to 100, thus avoiding calculation. 
 
 3. That the staff should be graduated to hundredths of 
 a foot. 
 
 4. That in combination with the above arrangement, a 
 telescope magnifying only twenty times and reading to the 
 2ooth of a foot at a distance of 660 feet, will produce results 
 as correct as those of Porro's larger and more complicated 
 instrument. 
 
 5. That the errors arising from spherical aberration may 
 under these circumstances be neglected in angles of less 
 than 10 on either side of the focal axis. 
 
 The only point in which the writer's views differ from 
 those of Mr. Lyman is in the power of the telescope 
 necessary to produce the degree of accuracy named : -^ f 
 a foot at 660 feet distance. That he maintains is only pos- 
 sible with double the power specified by Mr. Lyman. 
 
Tack eo me try 191 
 
 THE PLANE-TABLE AND STADIA 
 
 The plane-table itself is described at p. 319 et seq., but 
 when used in combination with a distance-measuring tele- 
 scope it becomes an instrument of much greater value. 
 The simple sight-rule is developed into a heavy brass 
 ruler termed an alidade, to which is attached a telescope 
 usually of 10 inches focal length, furnished with stadia hairs. 
 Instead of having to fill in detail with taping, pacing, and 
 sketching, it is all done with the stadia telescope. The 
 main triangulation still forms the basis of the work, and 
 occasionally the stadia work itself is checked by supple- 
 mentary triangulation. 
 
 The levelling is done in the same way as with the 
 tacheometer. The essential difference between the two 
 methods consists in the absence of a graduated horizontal 
 limb in the alidade, so that when traversing, the accuracy 
 depends upon the back and forward ray with the alidade, 
 which are transferred directly to the map. The transit and 
 stadia are superseding the plane-table and stadia in America. 
 The reasons for the writer's preference for the plane-table as 
 an auxiliary and not as a universal instrument have been 
 stated on p. 35, but it may be added here 
 
 1. The plane-table is not equal to the tacheometer in 
 wooded country, because triangulation is often imprac- 
 ticable. 
 
 2. It is not so suitable where only short bases are 
 possible. 
 
 3. It is not suitable to rainy country. 
 
 4. It is not suitable for putting in railway curves. 
 
 5. It is more awkward to handle. 
 
 6. It is not so well adapted to astronomical obser- 
 vations. 
 
 An engineer does not want to over-burden himself with 
 instruments, but seeks a maximum of efficiency, portability, 
 and economy. The tacheometer will do all that the plane- 
 
192 Preliminary Survey 
 
 table used as a universal instrument will do ; it will do 
 most things better, and the peculiar cases in 'which the 
 plane-table is superior are not of sufficiently frequent occur- 
 rence to a man in general practice to make him choose the 
 instrument. 
 
 RATE OF EXECUTION 
 
 The Hawaiian survey previously alluded to was an 
 example of as rough a piece of country as could be found 
 anywhere. The average rate of progress was under half a mile 
 per diem by two parties, one with the tacheometer and the 
 other with transit and chain. The work included a belt of 
 from 500 to 1,500 feet wide, a hundred ravines of more or 
 less size, and seventeen plantation villages ; the whole being 
 mapped to a scale of 100 feet per inch in order to show the 
 contours at every five feet on the steep side-hill. The 
 ravines were densely wooded, and most of the cultivated 
 land was covered with sugar-cane standing eight to ten feet 
 high. 
 
 The greatest amount of work done in one day with the 
 tacheometer alone was two miles. In that case the country 
 was cleared, the cane was short, there were only two short 
 ravines and one small settlement of about a dozen cottages. 
 
 The little survey of Greatness Mill is intended to show 
 the convenient manner of doing an awkward piece of work 
 by the tacheometer. Everything except building detail was 
 put in from the three stations A, B, and C, and in a few hours. 
 The buildings were afterwards plotted on a tracing with the 
 sketchboard plane-table. Every check-line taken tallied 
 exactly. The profile of the road was obtained from the 
 sights taken to survey it. A photograph was taken at the 
 same time somewhat in the direction of AB. In the case 
 of a railway being made through such a spot as that, the 
 information obtained would show 
 
 i. From the survey a depression of ground calling for 
 a crossing by bridge or viaduct. 
 
Tacheometry 
 
 193 
 
 2. The narrowest part of the water for a crossing. 
 
 3. By the photograph the nature of the property to be 
 interfered with. 
 
 Mr. William Bell Dawson surveyed with the tacheometer 
 and one assistant an area of 180 square miles, including one 
 hundred lakes from seven miles long downwards, in 
 five months. The object was to produce a map of the gold 
 
 Profi,lc of Road- 
 
 Scales, /for, 24/cTo 
 VcJ-t/ 
 
 'Matum'Unf '230? abbir (Jnlnaiike Datum 
 A 
 
 Fro. 53. 
 
 fields on the Atlantic coast of Nova Scotia. The traverses 
 generally closed within an area of 20 to 30 feet radius, but 
 closing errors were further eliminated by independent 
 checks. The map was on a scale of two miles to the inch. 
 He used a 6-inch theodolite with stadia hairs and a 
 Rochon micrometer. The total cost was about 3/. \\s. 6d. 
 per square mile. 
 
 A level should always be furnished with stadia hairs, 
 and if it has a compass also contouring and small surveys 
 can be often done with it of sufficient accuracy for the 
 purpose, and so save bringing out the larger instrument. 
 
194 Preliminary Survey 
 
 CHAPTER VI 
 CHAIN-S 'UR VE YING 
 
 THIS department of survey practice will be only treated in a 
 condensed form to serve as a reference for the preliminary 
 surveyor. In exceptional cases it becomes expedient to 
 make a survey, measure an acreage, or range a curve with 
 the chain or tape alone, or at most with the assistance of an 
 optical square or cross-staff. The most frequent of such 
 cases is where the time of the tacheometer party cannot be 
 profitably employed on a solitary piece of detail, and one or 
 two hands are told off to fill it in with the chain only. 
 
 A short description will also be given of the methods, in 
 vogue in America for running trial lines for railway location 
 with the transit and chain, together with a few notes on 
 curve-ranging with the chain only, reserving for the next 
 chapter the subject of curve-ranging with the theodolite. 
 
 CHAINING 
 
 If chaining could be carried out with rigorous exactness, 
 it would be possible to perform the whole survey with no 
 instruments whatever, although it would be a lengthy affair, 
 especially when the afterwork of levelling over the staked 
 survey is considered. But there are also many essential 
 difficulties which tend to produce error, and render chain 
 surveys unsatisfactory. 
 
 It is just because chaining seems so very simple that it 
 
Chain - Surveying 195 
 
 is often relegated to inferior assistants, who do not study a 
 systematic method of performing their work, and errors 
 frequently arise from the following causes : 
 
 First. The liability of a poor chainman to 'drop a chain,' 
 that is, to make a mis-count with the arrows. 
 
 Secondly. The tendency to measure on sloping ground 
 without making the proper deductions. 
 
 Thirdly. The liability of the chain to being stretched by 
 tension, or to expansion and contraction by variations of 
 temperature, all of which require regular attention, not always 
 devoted to them. 
 
 Fourthly. From twists in the map, due to incorrect tie- 
 lines at the angles of the base lines. 
 
 Fifthly. From confusion in the fieldbook. 
 
 On a level piece of ground, where the plot is drawn to a 
 large scale, such as 20 or 30 feet to the inch, the survey 
 should be made with the transit and chain, as the maximum 
 error should not exceed three inches. The tacheometer will 
 not give this exactitude, and the chain alone is almost sure 
 to develop twists. A survey of this kind, such as a town- 
 district or a railway terminal depot, is out of our province to 
 describe. The main distinguishing features about it are : 
 First, a network of triangulation wherever it is possible to 
 chain bases, or, combined with triangulation, a closed 
 traverse. Second, a system of ordi nates or offsets from the 
 bases thus fixed to all the points required to be plotted. 
 
 The same operation can be performed with the surveyor's 
 compass, but the theodolite is much more precise in the 
 angular measurement. 
 
 There are two or three methods of counting the chainage 
 and it is not of much consequence which system is used, 
 but it is most important that the same system should be 
 rigidly adhered to. The following method is recommended. 
 The leader carries a ranging rod, and the ten arrows, with 
 the forward end of the chain. The follower carries the 
 hinder end of the chain and a clinometer ; he also has a tape 
 
 o 2 
 
196 Preliminary Survey 
 
 attached to his button-hole. The ranging rod has a hand- 
 kerchief or piece of paper attached to it at the height of the 
 follower's eye. 
 
 The leader drags forward the chain to the follower's 
 direction, and hauls it taut. 
 
 The follower shakes it up to get rid of kinks and 
 curvature. 
 
 The leader holds up the ranging rod, to get the final 
 direction, and when fixed, puts the iron point through the 
 handle of the chain to give the last stretch, and then puts in 
 an arrow, and goes ahead. When he has put in his last 
 arrow he runs out the next chain without an arrow, aligns 
 it by the rod, and, leaving the iron point through the handle 
 of the chain, signals to the follower to come forward with 
 the ten arrows. 
 
 The follower first ties a knot in his tape to mark the ten 
 chains, and then comes on, leaving the chain to be dragged 
 forward by the leader when he has received his arrows. The 
 leader then puts in one of the ten arrows at the rod, and pro- 
 ceeds as before. 
 
 When the slope exceeds a degree (or, if the greatest 
 accuracy is required, at every chain) the follower books the 
 slope, and at the end of the line sums up the corrections 
 and deducts the result from the chainage. He stands erect 
 and directs the clinometer to the mark on the ranging rod. 
 
 Measuring by short lengths on a side-slope is very un- 
 satisfactory ; it is difficult to keep the chain level, and still 
 more difficult to plumb down exactly from the high end of 
 the chain when rapidity is an object. 
 
 A table is given at p. 174 from which the deductions 
 from the chainage on sloping ground can be readily made 
 for any given angle. 
 
 SETTING OUT A SQUARE 
 
 Let it be intended to lay off BD perpendicular to AC 
 with the chain alone. Measure off AB= 30 feet. Take the 
 
Chain - Surveying 
 
 197 
 
 10 mark on the chain to A and the opposite end of the chain 
 to B. Put an arrow through the 40 mark nearest to B. 
 Draw out the chain and shake it taut, so that the BD line 
 
 which is equal to 40 feet, and the AD which is 50 feet, shall 
 each be perfectly straight ; then because 3 2 4-4 2 =5 2 ABD is 
 a right angle. Links of a Gunter's chain will of course do 
 just as well, and any multiples of 3, 4, and 5, such as, 6, 8, 
 and 10 ; 9, 12, and 15 ; or 45, 60, and 75. 
 
 To MEASURE THE DEFLECTION DISTANCE OF A NEW 
 BASE LINE 
 
 This is a problem of curve-ranging, but is needed also 
 in all cases where closed triangles cannot be formed. 
 
 Let it be required to find the deflection distance of BD 
 from BC. Measure as long a base B/; as possible, align b 
 correctly with AB, set off bd square and measure it. If 
 possible choose 100 feet or 200 feet for B<$, put in a peg 
 
 FIG. 55. 
 
 with a nail at , and measure bd with a steel tape. BD can 
 then be plotted direct from these data, or else the angle can 
 
 be calculated thus : tan CBD = A. . The entry in the 
 
198 
 
 Preliminary Survey 
 
 fieldbook is marked R or L, according as the new base line 
 turns right or left. 
 
 ANOTHER METHOD 
 
 When the angle of deflection is great, it is better to make 
 the tie-line by the chord of the angle. Measure B as long 
 as possible and also ~Bd equal to it. The fieldbook should 
 
 FIG. 56. 
 
 show by a sketch how the tie-line has been taken. The 
 distance bd should be measured with a steel tape, and the 
 base can be plotted from the data, or else 
 
 i /-1-1-,7-x bd 
 sin iCBD = 
 
 2JD<7 
 
 In making a triangulation survey the aim should be to 
 get as long bases as possible, and the triangles ' well-con- 
 
 FlG. 57- 
 
 ditioned,' that is approximating to equilaterals. When the 
 interior of the survey is partly inaccessible this is impossible, 
 
CJ lain- Survey ing 
 
 199 
 
 and recourse has to be had to external ties like the above. 
 When the triangulation can be made internally, a cross-staff 
 is of great assistance. The handiest form is a brass octagon 
 fitted with hair sights in slits. It stands on a short pole 
 with an iron shoe, and only costs from $s. to io.y. The 
 
 FK;. 58. 
 
 advantage of the- cross-staff will be seen by comparing Figs. 
 57 and 58. It enables the offsets to be made up to 100 feet 
 with the tape, and so diminishes the base lines. 
 
 The fieldbook is arranged with a central column for 
 distances measured along the base lines, and on the right 
 and left arc the offsets opposite to the distances at which 
 they are taken, with remarks to identify them. The field- 
 book is commenced at the bottom of the page. 
 
 Left offsets j Base 
 
 Right offsets 
 
 E 
 
 
 18 716 
 
 
 49 ! 6 55 
 
 
 68 
 
 587 
 
 
 80 
 
 505 
 
 
 98 
 
 392 
 
 
 
 382 
 
 330 to C. 
 
 98 312 
 
 
 105 
 
 190 
 
 
 85 
 
 80 | 
 
 
 To river 17 From 
 
 A 
 
 
2OO Preliminary Survey 
 
 ACREAGE 
 
 The area is divided up into triangles and trapezoids. 
 The area of the triangles is obtained by either of the two 
 following formulae : 
 
 Rule i. To find the area of a triangle from the lengths 
 of the three sides, a useful method when no perpendicular 
 can be measured across the triangle. 
 
 Let a, b, c be the sides, and ^=their half sum ; A the 
 
 c 
 FIG. 59. 
 
 area in square measure of the unit adopted, such as square 
 feet area for lineal feet measurement of the sides. 
 
 A= 
 
 or logarithmically 
 A=i[log -r 
 
 Rule 2. To find the area of a triangle when any one of 
 the sides and the perpendicular from it to the opposite angle 
 are given. 
 
 B * P 
 Let Bs=base, and P = perpendicular ; area= 
 
 The area of a trapezoid is obtained as follows : 
 
 _ c 
 
 h 
 
 8 
 
 FIG. 60. 
 
 Rule 3. Multiply the sum of the parallel sides by the 
 
Chain- Surveying 20 1 
 
 perpendicular distance between them, and half the product 
 is the area. 
 
 or logarithmically 
 
 Log 2 A=log(AB + CD) + log h 
 
 To reduce areas in Gunter's chains to acres. 
 
 10 square chains =i acre 
 100,000 square links =i acre 
 
 To reduce square feet to acreage. 
 
 43,560 square feet=i acre 
 i acre =4 roods 
 i rood =40 perches 
 
 PRELIMINARY WORK WITH TRANSIT AND CHAIN 
 
 In running trial lines with the transit and chain, the 
 curves, unless exceptionally sharp, are not put in. The base 
 lines are measured, and staked from intersection to intersec- 
 tion ; the angles measured, and the stakes afterwards levelled 
 over. The vertical arc of the transit is not used unless 
 when on a steep gradient it is desired to follow the required 
 inclination, in which case the vertical arc is clamped to the 
 corresponding angle, and the line run as nearly as possible 
 to it. 
 
 In America the levelling staff is called a rod, and the 
 ranging rod is called a picket. In running a line there are 
 generally two picket men, sometimes only one. The transit- 
 man sends out a picket-man in the direction which he 
 intends to take, aligns the picket, and takes a reference sight 
 to some good fiducial point. 
 
 There are two chain -men, leader and follower, and one or 
 more axe-men, who prepare stakes about three feet long, 
 pointed at one end, and shaped on the other for a chalk mark. 
 
202 
 
 Preliminary Survey 
 
 The transit-man aligns the chaining by the leader's 
 picket, and the axe-men put in a stake at every hundred 
 feet, or at shorter distances where necessary. 
 
 The chief engineer generally accompanies the chaining 
 party in order to select the ground. 
 
 When the end of the base is reached, the picket-man, 
 leaving a peg at the end of the base, returns to the instru- 
 ment, and changes places with the transit-man to enable 
 the latter to range out a new base. 
 
 There are three methods in use for registering the 
 deflection of the one base from the other. 
 
 The first is the most common see Fig. 61. It takes no 
 
 ,5te 000 
 
 FIG. 61, 
 
 account of the astronomical meridian and only uses the 
 compass-needle as an occasional check. 
 
 The field notes will explain themselves with scarcely 
 any description. The instrument is set back to zero every 
 time ; the slopes of the ground, right or left, are either taken 
 with the vertical arc of the transit or by the clinometer. 
 
 Method 2 see Fig. 62. The second method is by 
 working out the magnetic or astronomical azimuths. If the 
 former, it is sufficient to commence by making the zero of 
 the horizontal limb coincide with the position of the needle 
 when at rest. If the latter, the meridian must first be deter- 
 mined by one of the methods mentioned in Chapter IV. 
 
Chain-Surveying 
 
 203 
 
 Almost all transits are graduated from o to 360, so that 
 this method entails the reduction of the angle as explained 
 on p. 59 ; it is, however, very well suited to plotting by 
 latitude and departure. 
 
 Method 3. By bearings from o to 360 see Fig. 63. These 
 are now also commonly termed azimuths, though not, strictly 
 speaking, azimuths unless taken from south as well as north 
 point. Some distinctive term is necessary for these bearings. 
 ' Northerly bearing ' would express the first idea, but then it 
 needs to be defined as to whether it is astronomical or 
 
 FIG. 62. 
 
 magnetic bearing, which would be expressed by N. Ast. 
 Bearing, or N. Mag. Bearing. 
 
 If we are furthermore to be visited with a centesimal 
 graduation in competition with the old graduation, we should 
 require another letter or two of the alphabet to define our 
 modus operandi. If the term * course ' could by universal 
 consent be relegated to the magnetic, and 'bearing' 
 reserved for the astronomical direction, both being kept 
 solely for graduation from o to 360, whilst the term 
 ' azimuth ' was confined to what it strictly means, it would 
 
204 
 
 Preliminary Survey 
 
 simplify nomenclature ; but it is to be feared that the sailors 
 would not fall in with the proposal. 
 
 It is not necessary either in Methods 2 or 3 to measure 
 the angles of deflection separately. They are only inserted 
 here in order to compare one method with the other. The 
 transit is fitted with two verniers, and when making a devia- 
 tion the instrument is reversed and directed to the previous 
 station, one of the verniers being set at the bearing of the 
 line just run. This is called the backsight. The instrument 
 is then again reversed so as to point forwards, and, unclamp- 
 
 + GO 
 
 FIG. 63. 
 
 ing the parallel plate, directed along the new base-line to 
 the picket fixed at the end of it. The bearing is then read 
 by the other vernier, and booked. 
 
 When two ' readers ' are provided, it is advisable to use 
 only one, and at each change of direction shift it over to the 
 other vernier. Before commencing work, it is of course 
 necessary to see that the line of sight is in true corre- 
 spondence with the zero of the horizontal limb as ex- 
 plained in Chapter IX. It is also necessary to have an 
 instrument in which the graduation is correct, so that the 
 
Chain - Surveying 
 
 205 
 
 readings of the two verniers differ by 180, or in the 
 centesimal instrument by 200. When this is so, using 
 the two verniers forms a check upon the readings. 
 
 The slopes are put in by all three methods as shown on 
 Fig. 6 1. 
 
 CIRCULAR CURVE-RANGING WITH THE CHAIX 
 
 Four methods will be briefly dwelt upon. 
 Method i. Krohnke's tangential system. 
 This is a scheme for obtaining equidistant points upon 
 a curve by laying out abscissae BE 7 , BE", &c., along the 
 
 subtangent of the curve and setting off ordinatcs IV //, B" b" , 
 cS:c., to the required points. When the offsets become long 
 the curve is bisected, each section treated in a similar 
 manner. Krohnke's tables of forty-seven pages are published 
 for setting out curves of any radius by this method. 
 
 Method 2. Jackson's six-point equidistant system. As 
 in Krohnke's method, he obtains equidistant points from 
 unequal abscissae, but by reducing the number of points to 
 six he obtains a manageable curve and much shorter tables. 
 They are given in his ' Aids to Survey Practice,' Crosby 
 Lockwood &: Co. 
 
206 
 
 Preliminary Survey 
 
 Method 3. The method given in Kennedy and Hack- 
 wood's ' Curve-tables ' (E. & F. N. Spon) of equal abscissae 
 and unequal distances on the curve. This is much more 
 simple, and although, for purposes of construction, it is a dis- 
 advantage to have points that are not equidistant, for prelimi- 
 nary survey it makes no difference. The deflection distances 
 are given for Gunter's chains in the end column of each 
 table. 
 
 It may be here as well to explain the principle of laying 
 off deflection distances and to show how points may be set 
 off from a tangent to any required equidistant points upon 
 a circular curve. 
 
 One of the fundamental properties of the circle is that 
 equal chords subtend equal angles at the centre. It requires 
 
 no demonstration that in Fig. 63, BB 1 is = sin D x AB, or 
 BB 2 = sin (D + D') x AB, or that BV/= AB - AB. cos I), 
 or B 4 ^"" = AB - AB. cos (D to D'"). 
 
 It will be presently shown in the chapter on curve-ranging 
 that the angles D, D -f D', &c., are each of them double the 
 corresponding tangential angle given in the tables. 
 
 Thus, supposing the radius 40 chains, we have a tan- 
 gential angle for i chain of o 42' 58" = a deflection or 
 central angle of i 25' 56". 
 
 If we multiply the radius, 2,640 feet, by sin i2$' $6", 
 we shall get 65*985 feet as the co-ordinate corresponding to 
 
Chain-Surveying 207 
 
 a chord of i Gunter's chain, and by cos i 25' 56" we shall 
 get 2,639*17 feet, which deducted from the radius gives us 
 the offset -83 feet or 9-96 inches. The table enables us to 
 find the tangential angle due to length of curve of any radius, 
 and from this we can obtain the deflection distance in the 
 manner just described. This will be better understood after 
 reading the next chapter. In order to throw off a new 
 tangent when the offsets become too long, we have only to 
 go back two, three, or four stations and measure off from it 
 the deflection distance due to its number from the last peg. 
 For instance, if we wished to lay off a tangent at b"" we 
 should go back say to b'" and lay off the deflection distance 
 for i, or to ft", and lay off the deflection distance for 2 chains. 
 We should also lay off from b"" the abscissae = BB 1 or BB 2 
 as the case may be, and this would be a true tangent at b"" , 
 which we could produce and treat in the same manner as 
 BB 4 . 
 
 Example. The total curvature A' of the crowning curve 
 on p. 230 is 63 0< 6, and the radius 36*93 feet; it is required 
 to put in equidistant points b', b' ', b'" upon the curve. Sub- 
 divide A' into a suitable number of parts, say 10, then D 
 (Fig. 6 5 ) = 6 - 3 6; (D + I)') = i272 &c. BB^= 3 6- 93 x 
 sin 6'36 = 4-09 feet. BB 2 = 36*93 x sin i2"j2 = 8*13 
 feet, B 1 '=36-93(1 -cos 6 - 3 6)=o-23 feet. B 2 ^= 3 6- 9 3 
 (i cos i272) = o'9i feet, &c. 
 
 Method 4. The oldest and simplest plan for setting out 
 curves with the chain is the only one which can be done 
 without tables (see Fig. 66). 
 
 The first point in the curve is found similarly to the last 
 method, but the succeeding points are fixed by ranging lines 
 B^a 2 ; PbW, and so on, from which the offsets 2 ^ 2 , aPb* 
 will be each of them double the first offset aW. The first 
 offset a>b l is termed the tangential distance, because it is 
 the chord subtending the tangential angle to a radius of i 
 chain. The succeeding offsets a 2 & 2 , a* & , &c., are termed 
 the deflection distances, because they are equal to the chord 
 
208 Preliminary Survey 
 
 subtending the angle made by the intersection of two tan- 
 gents to the curve at the ends of a chain and to a radius of 
 i chain (see more on nomenclature of curve-ranging at 
 p. 214). 
 
 FIG. 66. 
 
 The formula used in the field from which to find the 
 offsets by this method is 
 
 for tangential dist. offset = chord ! 
 
 deflection dist. offset =~- 
 
 R 
 
 The chord may be i Gunter's chain in links, in which 
 case the radius will be expressed also in links and the 
 offset will be in links, or it may be in feet or metres or any 
 other measure. 
 
 This formula is not quite exact for small radii. 
 
 On a curve of 100 feet radius, the tangential distance for 
 100 feet chord would be 5176 feet against 50 feet by the 
 formula. At 500 the formula would be but -013 feet too 
 little. So that for curves of 10 chains and upwards, set out 
 with i chain chords, the formula is practically correct. 
 
 The subject of curve-ranging with transit and chain is 
 reserved for another chapter. 
 
209 
 
 CHAPTER VII 
 CURVE-RANGING WITH TRANSIT AND CHAIN 
 
 THE subject of curve-ranging has been briefly touched 
 upon in the previous chapter as far as it can be done with 
 the chain only, but it was deemed appropriate to devote a 
 whole chapter to the subject of curve-ranging with the in- 
 strument and chain as being par excellence curve -ranging. 
 
 There are still those who prefer the chain only to any 
 instrument for the purpose of curve-ranging. In the hands 
 of experts, very correct results may be obtained in level 
 country, but it would be interesting to know how the chain- 
 curve-ranger would put in track centres on a 40 curve on the 
 top of an embankment 30 feet high, the total deflection angle 
 being 180. 
 
 To take it all round, curve-ranging with the chain only 
 is, as compared with work by better methods, poor, fudgy ,, 
 and muddling. 
 
 NOMENCLATURE 
 
 The principles of curve-ranging are best understood by 
 keeping in one's mind the idea of a ship's course at sea. 
 
 In the previous chapter, it will have probably been 
 noticed how the methods of running base lines with the 
 transit and chain, especially Method 2 , resemble the deter- 
 mination of a ship's course and position by dead reckoning. 
 
 The angular changes of direction 33 15' and 44 i' in 
 Fig. 62 are the angles of deflection, and this is.the.y.ery,ro.ot- 
 idea of curve-ranging. The internal angle between the tan- 
 
2io Preliminary Survey 
 
 gents of a curve, commonly termed I in English textbooks, 
 forms no part whatever of the theory to be described here. 
 It conveys no idea of the curve beyond the mere fact that it 
 is the supplement of the angle of deflection. 
 
 This angle of deflection may be either represented by its 
 angular value or else by the difference of bearing of the base- 
 lines, as in navigation. 
 
 Curve-ranging may be defined as deflecting by degrees. 
 It is for this reason that American surveyors have adopted 
 the curve-nomenclature by which the quickness or slowness 
 of a curve is expressed by the number of degrees of a circle 
 which the curve, not the chord^ deflects at each chain length. 
 
 Thus a curve in which a loo-foot chord subtends i of 
 curvature is termed a i -degree curve. 
 
 The reason for this is obvious. When the total number of 
 degrees contained in the deflection angle have been turned, 
 the new base line or tangent will have been reached, and 
 the number of chain lengths contained in the curve will be 
 given by simply dividing the total angle of deflection by the 
 * degree of curve.' 
 
 For instance, if the total angle of deflection be 1^22' 
 and it is desired to put in a 2 curve, that is to say, a curve 
 in which, if a chain length is measured along it, the tangents 
 to the curve at either end of the chain length will deflect 
 from one another by 2, the total length of the curve will 
 
 be^^x 100=861 feet if a 100 feet chain is used, or 861 
 
 2 
 
 links if a Gunter's chain. 
 
 Contrast this simple computation with that of English 
 curve-ranging, where it is expressed as follows : 
 
 Let x be half the angle of intersection, and R the radius. 
 
 Length of the curve = '000582 R (5400 :v). Note : x 
 inust.be expressed in minutes. 
 
 The writer has not taken the trouble-to ascertain whether 
 this formula gives the true circular measure of the curve, or 
 the measure in chains and parts of a chain. If, as is pro- 
 
Curve-Ranging with Transit and Chain 2 1 1 
 
 bable, it is the former, it has not even the advantage of being 
 correct, because curves are not set out by true circular 
 segments but by short chords of chains or parts of a chain, 
 and this is the principle of the American formula. 
 
 The English formula is adapted to logarithmic calcula- 
 tion, whereas the American is performed either by slide-rule 
 or even in the head. 
 
 The ordinary graduation of instruments into degrees and 
 minutes necessitates the reduction of the deflection angle to 
 decimals of a degree, as for instance where the angle is 
 29 47' 30'' and the degree of curve is 6 35' it requires some 
 
 FIG. 67. 
 
 little calculation ; but by the author's system of decimal 
 graduation of the ordinary degree, it is done by the slide* 
 rule instantaneously. 
 
 In Fig. 67, it will be observed that there are two angles 
 marked A, and two marked d. The upper A is the angle 
 of deflection of the curve and is equal to the angle at 
 the centre subtending the curve, which is therefore also 
 marked A. Similarly the angle of deflection of a portion 
 of the curve of which the chord is 100 is=</, and is equal 
 to the central angle also marked d. There has been, and is 
 still, considerable confusion of nomenclature in both England 
 and America, from the fact that some writers insist on term- 
 
 r 2 
 
212 Preliminary Survey 
 
 ing the tangential angle the angle of deflection because it 
 is the angle by which the chord deflects from the tangent. 
 This has been termed the tangential angle, at least from the 
 first of Rankine's books on the subject, and is still so termed 
 by the best authorities. There might have been some room 
 for choice in naming this angle, but it could not with propriety 
 be termed the angle of deflection. 
 
 The only apology for spending so much time in this 
 demonstration is that, unfortunately, writers of high standing 
 have taken up this perplexing nomenclature. To put it con- 
 cisely, the deflection of a curve is its actual change of direc- 
 tion^ or curvature^ and those who adopt the wrong designation 
 would have to describe a curve which had turned round 
 a quarter of a circle as having deflected forty-five degrees, 
 because the angle between the chord and the tangent (the 
 tangential angle) would be 45. The term ' tangential angle ' 
 is used either for the angle for i chain of 66 feet or of 100 
 feet or for the total angle, which is vague. The term ' total 
 tangential angle ' should be given to that for the whole curve, 
 and ' single tangential angle ' for that for the unit of measure- 
 ment. The term ' total deflection angle,' and ' single deflec- 
 tion angle,' will similarly apply to those angles, the latter being 
 the ' degree ' of curve. 
 
 One English writer, imperfectly acquainted with American 
 curve-ranging principles, describes their nomenclature as ' a 
 confusion between the angle at the centre and the angle of 
 deflection,' from which it would appear that the writer was 
 not himself aware that these angles were equal to one 
 another. 
 
 The tangential angle T, or , is equal to half the angle 
 of deflection in circular curves. 
 
 In America the straight is termed the tangent, and the 
 prolongation from the springing to the intersection is termed 
 the subtangent. The point of intersection is so- named, or 
 else the- vertex, '-and the midway point of the curve is called 
 the * apex/ or ' crowning point,' or ' summit.' The curve is 
 
Curve-Ranging with Transit and Chain 213 
 
 marked B.C. at beginning of curve, and E.G. at end of 
 curve. Wherever a change of position occurs with the 
 transit, a 'hub,' or short stump, is driven in with a nail in 
 it, and a reference stake about three feet long close to it. All 
 the other points are marked by long stakes having their dis- 
 tance chalked on them in stations of 100 feet and their 
 excess. Thus 1,143 f eet would be marked 11+43. 
 
 Only short curves are run to intersection point. The 
 method of keeping the fieldbook herein described enables 
 the surveyor to range a curve of any radius, alter it, compound 
 it, reverse it, and calculate his position with reference to his 
 starting point without running the curve to intersection. The 
 only difference in this system from the ordinary practice in 
 America is that the lines are kept on their astronomical 
 bearing. It is a little more trouble when using the gradua- 
 tion of minutes and seconds, but no more trouble with the 
 decimal subdivision, and has the advantage of always 
 affording a ready means of denning the position and checking 
 the work. 
 
 GENERAL PROPERTIES OF CIRCULAR CURVES 
 
 The following demonstrations are amongst the most 
 useful problems occurring in practice (see Fig. 68). 
 
 Let the deflection or total curvature A be represented by 
 DIE, the radius BO or OE, IB the subtangent, and IA 
 the apex distance. 
 
 i. Prove that DIE=BOE. 
 
 In the quadrilateral IBOE angles IBO and IEO are 
 right angles. 
 
 But the internal angles of any quadrilateral are together 
 equal to four right angles. 
 
 Therefore angle BIE + angle BOE are equal to two 
 right angles. 
 
 But angles DIE + BIE are also equal to two right 
 angles ; hence, equating and eliminating BIE, angle DIE 
 = angle BOE. 
 
214 
 
 Preliminary Survey 
 DIE 
 
 
 
 2. Prove that angle IBE= ~ or = or=BOI. 
 
 2 2 
 
 BOI=EOI because A is at the middle of the curve. 
 IBE=IEB because IBE is an isosceles triangle. 
 
 The three angles of any triangle being equal to two right 
 angles, and it having been already shown that BIE-fBOE 
 =two right angles, therefore in the triangle BIE angles 
 BIE, IBE, BEI are equal to the angles BIE + BOE. 
 Equating, eliminating BIE, and setting IBE + BEI=2lBE; 
 2 IBE=BOE, or, which is the same thing, IBE=|DIE. 
 
 3. Prove that the subtangent BI=the radius BO mul- 
 tiplied by the tab. tan. of the tangential angle IBE. IB is 
 evidently proportional to the tangent of angle BOI, which 
 angle we have shown to be equal to IBE. 
 
 4. Prove that the apex distance 
 
 IA = 
 
 radius 
 
 - radius 
 
 cos IBE 
 IA == 10 - AO, but by No. 5, p. 373 - 
 
 jg 
 
Curve-Ranging with Transit and Chain 2 1 5 
 
 .-. 10 = -_ and BOI = IBE 
 cos BOI 
 
 whence I A = -= - radius 
 cos IBE 
 
 GENERAL FORMUL/E 
 
 Let A be the total deflection =2, the tangential angle ; 
 d = the degree of curve or curvature in 100 feet chord ; R 
 =the radius ; C = the chord subtending the whole curve ; 
 S.T. =the sub tangent ; AI = the apex distance ; L = 
 length of curve in feet. 
 
 ___ _ __ 
 
 2 sin* 2sin sin^ 
 
 2 2 
 
 A 
 
 d 50 Ioosin 2 xoosin 
 sin V = R = -C" "C" 
 
 S.T.= R. tan ; AI = -- R 
 cos 
 
 TOO 
 
 We have now sufficient data for ranging a curve. 
 Example i. The intersection point occurred at station 
 
 753 + 34- 
 
 The angle of deflection was 37'9i6. It is intended to 
 put in a 5 curve. Required, the length of the subtangent, 
 the apex distance, the length of the curve, and the chainage 
 of the B. C. and E. C. points and the apex. 
 
 The radius of the curve is found from the table p. 379 
 = 1,146 feet. 
 
 Subtan = 1,146 x tan 3 1_9L = 3937 
 
 2 
 
 Apex dist. = - " 4 . Ql6 - 1,146 = 657 feet 
 cos J/ v 
 
216 Preliminary Survey 
 
 Length of curve = $-L-< x TOO = 758-3 feet 
 
 B. C. point = 753 + 34 - 393-7 feet = 749 + 40-3 
 E. C. point = 749 -f 40-3 -f 758-3 feet = 756 + 98-6 
 
 Apex point at 749 + 40-3 + " - feet = 753 + 19-9 
 Lay off angle El A Fig. 68= """ ? and measure off IA 
 
 2 
 
 = 65-7 feet, and put a stake in marked ' Apex, 753 -f 19-9.' 
 Measure the subtangent both ways, and put in pegs 
 
 with nails, aligning them from the transit at the intersection 
 
 point. Put in reference stakes marked B.C. 749 -f 40*3 and 
 
 E.G. 756 + 98-6. 
 
 Shift the transit to the B. C. point and commence to 
 
 range the curve. The first odd distance will be 59-7 feet ; 
 
 the tangential angle for this will be $- x ^JL? = i ^. The 
 
 2 IOO 
 
 instrument is set to read this angle and a stake put in 
 marked 750. The next angle will be 2'5 (2 30') 4- 1-49 
 = 3'99, which is set off and a stake aligned and driven at 
 another hundred feet and marked 751, and so on. After 
 stake 753 is driven, at a tangential angle of 8 -98 the 
 odd distance, 19-9 to the apex point, is set off, the angle 
 
 being 2-5 x - 9 = o'5o, or angle from the beginning 8 0> 98 
 
 + -50 9'48. This will be one-half the total tangential 
 angle, or one-quarter of the angle of deflection. Both the 
 odd distances to the apex and to the E.G. point are given to 
 the chain-men, and if they do not coincide with the stakes 
 already driven the chain-men either signal the transit-man 
 the amount of divergence when small, or, if large, they 
 return and report. If, for instance, they signal, E.G. point 
 six-tenths to the right, the transit-man will signal back to 
 them to come back three stations or four stations, and he 
 will distribute the error over them. If he is constructing 
 an iron trestle or a brick viaduct he will, of course, repeat 
 
Curve-Ranging with Transit and Chain 217 
 
 his work until there is no sensible error, but it would be a 
 waste of time for an embankment or cutting, when the error 
 is only a few inches. With ordinary care on the part of the 
 transit-man the error, if any, is nearly always in the chainage. 
 The following fieldbook is arranged for astronomical 
 bearings. If the curve is run from zero point, the only dif- 
 ference will be that the entries in the bearing column will 
 be dispensed with. The operation at a turning-point is this. 
 Taking as an example the first shift of the transit at Station 
 
 304 30 
 
 FIG. 69. 
 
 755. In the fieldbook on next page a picket-man is sent to 
 the B.C. The instrument is set up at Station 755 over a 
 peg or c hub ' with a nail in it, fixed by the transit from the 
 B.C. With one of the verniers clamped at 2 90 '31 and the 
 telescope upside down, a backsight is taken by means of the 
 external axis upon the B.C. 
 
 Description of Citrvc. 
 
 Total | 
 
 j j 
 
 
 angle of Astronomical bearing 
 deflection 
 
 i ~ 
 
 4) <2 5 
 
 
 c 
 
 rt 
 
 Chainage of 
 
 
 
 a i s 
 
 bfl 
 
 .a 
 
 
 
 
 1 3 
 
 5 
 
 
 
 
 
 1 
 
 * s| 
 
 fVn 
 
 
 
 > c 
 
 a 
 
 
 1 
 
 1-1 
 
 a 
 
 B. C. 
 
 Apex 
 
 E. C. 
 
 5 
 
 11 ; 5 <J P. 
 
 CJ 
 
 a 
 
 
 
 
 
 
 
 
 
 
 i ** ! 
 
 
 
 
 
 
 deg. 
 
 deg.' deg. | deg. 
 
 deg. 
 
 deg. 
 
 1 feet feet J feet feet 
 
 
 
 37*92 
 
 - 276-32 
 
 285-81 
 
 295-28 
 
 3.4-4 
 
 5 
 
 1,146 393-7 ! 758-3 
 
 65-7 
 
 749 + 
 4'3 
 
 753 + 
 19-9 
 
 756+ 
 
 98-6 
 
 The external axis is then clamped, the telescope is reversed 
 right side up, the parallel plate released, and the instru- 
 ment directed to the tangent of the curve at Sta. 755 by 
 reading with the same vernier the angle 304 '30. Comparing 
 Fig. 69 with the fieldbook we shall at once see that as the 
 
218 
 
 Preliminary Survey 
 
 Ficldbook. 
 
 Chainage 
 
 Bear- 
 
 ing 
 
 Length 
 of chord 
 
 t 
 
 T 
 
 Remarks 
 
 749 + 40-3 
 750 
 751 
 
 752 
 
 753 
 
 754 
 755 
 
 
 
 276-32 
 277-81 
 280-31 
 282'8l 
 285*31 
 285-81 
 287-81 
 290-31 
 I3-99 
 
 O 
 
 597 
 
 lOO'O 
 IOO-O 
 
 loo-o 
 
 19-9 
 
 80-1 
 
 100-0 
 
 o 
 1-49 
 2-5 
 2-5 
 
 0'5 
 
 2'O 
 2'5 
 
 _-- 
 
 O 
 
 1-49 
 
 3-99 
 6-49 
 8-99 
 
 9-49 
 11-49 
 
 13-99 
 
 -" 
 
 B. C. 
 
 . 
 
 Apex 
 Turning point 
 
 755 
 756 
 756 + 98-6 
 
 304-30 
 306-80 
 309-27 
 4-97 
 
 
 
 loo-o 
 
 98-6 
 
 
 
 2'47 
 - 
 
 
 
 4-97 
 
 On tangent at 755 
 E. C. 
 
 
 3I4-24 
 
 
 
 I3-99 
 4-97 
 
 18-96 
 
 2 
 
 On new tangent 
 
 (-' 
 
 
 
 
 
 37-92 
 
 Check 
 
 tangential angle from the B.C. to 755 is 13 -99, if we add to 
 this an equal amount we obtain the angle of deflection, 
 which laid off from Sta. 755 throws us on to the tangent. 
 The following fieldbook is for the same curve, only 
 turned to the left instead of to the right. 
 
 Description of Curve. 
 
 
 
 
 I 
 
 
 Total 
 
 
 
 
 
 angle of 
 
 Astronomical bearing 
 
 
 ' 
 
 
 y i Chainage of 
 
 deflection 
 
 
 V 
 
 3 
 
 V 
 
 JS 
 
 
 
 
 
 
 1 
 
 *3 
 
 / 
 
 1 
 
 to 
 
 j 
 
 -3 
 
 
 
 
 
 
 
 ^ C 
 
 
 
 i 
 
 1 
 
 2 
 
 5| 
 
 
 
 
 
 
 
 M 
 
 
 a 
 
 B. C. j Apex 
 
 E.G. 
 
 * 
 
 
 rt 
 
 < 
 
 
 
 2 
 
 
 
 
 
 
 
 
 
 _ 
 
 37*92 
 
 276-32 
 
 266-84 
 
 257*36 
 
 238-40 
 
 5 
 
 1,146 
 
 393*7 
 
 758-3 
 
 65-7 
 
 749 + 
 
 753 + 
 
 756+ 
 
 
 
 
 
 
 
 
 
 
 
 
 40-3 
 
 19-9 
 
 98-6 
 
Curve-Ranging with Transit and Chain 219 
 
 Fieldbook. 
 
 Chainage 
 
 Bear, 
 ing 
 
 Length 
 of chord 
 
 t 
 
 T 
 
 Remarks 
 
 749 + 40-3 
 750 
 751 
 75 2 
 
 753 
 753+ I 9'9 
 
 754 
 755 
 
 276-32 
 
 274-83 
 272-33 
 269-83 
 267-33 
 266-83 
 264-83 
 262-33 
 
 I3-99 
 
 A, 
 
 100-0 
 100 -0 
 
 100-0 
 
 I9-9 
 
 80-1 
 
 lOO'O 
 
 1-49 
 
 2'5 
 2'5 
 2'5 
 
 0-5 
 
 2'O 
 2'5 
 
 O 
 
 1-49 
 3'99 
 6-49 
 8-99 
 
 9 '49 
 11-49 
 
 1 3 '99 
 
 B. C. 
 
 Apex 
 Turning point 
 
 755 
 756 
 756 + 98-6 
 
 248-34 
 245-84 
 
 243-37 
 4'97 
 
 O 
 
 1 00*0 
 
 98-6 
 
 O 
 2'5 
 
 2-47 
 
 o 
 2-5 
 4-97 
 
 On tangent at 755 
 E.C. 
 
 
 238-40 
 
 
 
 
 On new tangent 
 
 In practice some of these entries are dispensed with, 
 especially when in a hurry. It is however very bad policy to 
 keep a fieldbook that nobody else can understand. If the 
 columns are headed beforehand, or printed, very little time 
 is taken up in making all the entries, and a great deal of 
 time is saved to the reviser when the curve has to be picked 
 up again at any point. 
 
 In England, curve-ranging is not done with a fieldbook. 
 Tables like those of Kennedy and Hackwood supply the 
 tangential angles for curves of radii in Gunter's chains. 
 The B.C. and E.G. points are marked usually with two re- 
 ference pegs, one on either side of the centre peg and close 
 to it. Some surveyors also drive two pegs at every ten chains. 
 Long stakes like those driven in America, with chainage 
 chalked on them, would be welcomed for firewood by the 
 villagers in England. 
 
 The American system of curve-nomenclature by degrees 
 of deflection angle could be also used for Gunter's chains. 
 The chord being expressed as 100 links, and the radius also 
 
22O Preliminary Survey 
 
 in links, the degrees could be used just as they are in the 
 tables, i.e. a curve of i deflection angle per chord of 100 
 links has a radius of 57*30 chains. Compare Table LIV. 
 
 Curve-ranging is nearly always performed with a single 
 theodolite. There is another method with two theodolites 
 which dispenses with chaining ; it is rarely resorted to, and 
 will not be further described than by the remark that the 
 location of points upon the curve is determined by the inter- 
 section of two tangential angles from the two instruments. 
 Thus if the two instruments were set up one at the B.C. and 
 the other at the E.C, point in the fieldbook on p. 218 the 
 first point at 750 would be formed by the intersection of an 
 angle of i'49 from the B.C. with one of 17 '4 7 from the 
 E.G., &c. The points are fixed by a picket held by an 
 assistant, first in line with one theodolite and then retreating 
 and advancing along that line until in the line of the other 
 theodolite. If the chain is not used at all, the curves had 
 best be put in telemetrically as described on p. 183; but if it is 
 used at all there is no time saved by dispensing with it on 
 the curves, unless in the case of a trestle across a steep ravine 
 on a very sharp curve, especially if the measurement is im- 
 peded by undergrowth. As such cases are exceptional it is 
 not advisable to have the time of two transit-men taken up 
 at the one spot on that account. 
 
 An ingenious instrument has been lately invented by 
 Mr. Dalrymple-Hay for ranging curves of radii expressed in 
 Gunter's chains, but it could be easily modified for curves of 
 any other radii. The principle is the adoption of an open 
 and clear graduation in terms of tangential angles, by means 
 of an extended horizontal arc. The index of the limb for 
 a curve of say 20 chains is set at i for the tangential angle of 
 i chain, at 2 for 2 chains, c., and thereby simplicity of read- 
 ing is obtained. 
 
 This device can be fitted to any transit theodolite for 
 eight to twelve guineas. 
 
 The objections to it are : 
 
Curve- Ranging with Transit and Chain 221 
 
 First) that in ordinary practice a transit is wanted alter- 
 nately for running tangents and curves, and when on the 
 tangent the curve apparatus is only an awkward appendage. 
 
 Secondly ', it tends to make curve-ranging more mechanical. 
 This cannot be called an advantage, because there is hardly 
 anything that calls for more intelligent skill in surveying 
 than this branch of it, and a system which enables a man to 
 do work without knowing why he does it will not be likely 
 to produce a good workman. 
 
 REVERSE CURVES 
 
 It is always preferable, having regard to the running 
 gear of the rolling stock, to put in a piece of tangent between 
 the ends of a reverse curve, and when this is done the 
 problem is in nothing different from ranging two distinct 
 curves, one to the right and the other to the left, in the 
 manner just described. 
 
 When the main tangents are parallel it is impossible to 
 unite them by any other than a reverse curve ; but when 
 they are inclined to one another the only reason for pre- 
 ferring a reverse to a plain connecting curve is to obtain a 
 more rapid transit, in which case the reverse curve will 
 either intersect or lie wholly on the further side of one of 
 the main tangents. A sub-tangent at the point of contrary 
 flexure can be chosen, which will make any desired angle 
 with the main tangents ; and an endless number of cases 
 can be formed with a pair of curves of equal deflection and 
 unequal radii, or unequal deflection and equal radii. If 
 curves are wanted which will meet without any intervening 
 tangent, the lengths of sub-tangents are the fixed data, and 
 from the formula on p. 215 transposed, we have R=S.T. 
 cot 0, from which we can obtain the radius and other 
 elements. 
 
 The following problem is the commonest amongst true 
 reverse curves, and suitable to turn-outs and cross-over 
 roads between parallel tangents. 
 
222 
 
 Preliminary Survey 
 
 Whatever the radii, the deflections A of the two curves 
 will each of them be equal to twice the inclination of the 
 common tangent to either of the main tangents. When the 
 radii are equal the point of contrary flexure I bisects 
 the common tangent. When unequal they will be propor- 
 tional to the abscissae of the common tangent BI and IE. 
 
 Reverse curve of unequal radii between parallel tangents. 
 
 FIG. 70. 
 
 If the distance D on the square between the tangents 
 be given, and either L the straight reach, or BE the common 
 tangent be given, the other elements can be found from the 
 following expressions: 
 
 tan 0=-?; cos = BE 
 
 R= 
 
 L 
 BI 
 
 >== 
 
 2 sm iy 
 L=D cot ; 
 
 or if one of the radii R be given, and the distances L and 
 D or BE be given, to find the other radius. 
 
 ^BE-BI_BE(BE-BI) 
 
 2 sin 2D 
 
 Turn-outs are now generally curved from the commence- 
 ment of the switch, when the switch is of the split type, 
 such as is always used on passenger railways in England. 
 
Curve- Ranging with Transit and Chain 223 
 
 In America, the old-fashioned stub-switch, having its heel 
 at the point of curvature, is still largely used. The switch, 
 being common to both tracks, is straight, and the curvature 
 to which it is a tangent commences from its further end. 
 
 To lay out a turn-out for a stub-switch, instead of the 
 main tangent, take the line of direction of the switch when 
 directed towards the curve and which forms an angle / with 
 the main tangent of which 
 
 . throw of switch 
 
 sin /= 
 
 length of switch 
 
 Treat the end of the switch as the point of curvature, 
 and calculate from it as if it were B, The two tangents 
 will then not be quite parallel, but the curve can either be 
 run out to a parallel with the switch, by foregoing formula, 
 and produced to a parallel with the main tangent, or else a 
 separate calculation can be made, which want of space 
 precludes our giving here. 
 
 To DIVERT A CURVE TO A PARALLEL TANGENT 
 
 Very frequently the end of a curve comes too near 
 some fixed point, and it is desired to throw it to one side 
 
 FIG. 71. 
 
 or the other by advancing or receding the B.C. point upon 
 the old tangent. 
 
 Let the distance between the two parallel tangents be y, 
 
224 Preliminary Survey 
 
 and the length of advance or retreat of the B.C. point be x, 
 the deflection angle being A. 
 
 Then x -. - or y=x x sin A 
 sin A 
 
 To FIND THE LINEAR ADVANCE OF THE INNER RAIL 
 ON CIRCULAR CURVES 
 
 In consequence of the outer rail being longer than the 
 inner, the joints of the latter gradually forge ahead unless 
 a special rail is put in occasionally, or the rails cut. Let 
 
 LA=difference of length between outer and inner 
 rails. 
 
 A = deflection angle or total curvature. 
 
 G=gauge. 
 
 General Rule 
 
 LA=Gx Ax -01745 (log 8-24188) * 
 Or if LA be in inches and G in feet 
 
 LA=G x A x -20944 (log 9*32106) 
 
 Rule for Standard Gauge 0/4 feet 8^ inches . 
 LA in inches=A x "98611 (log 9*99392) 
 
 Rule for 3 feet 6 inch Gauge 
 LA in inches=A x -73304 (log 9-86513) 
 
 Rule for 3 feet Gauge 
 LA in inches= Ax '62832 (log 979818) 
 
 Rule for Metre Gauge 
 LA in millimetres = A x 17*4533 (log 1-2418774) 
 
 Generally a consignment of rails includes some 
 ' specials,' six inches or twelve inches shorter than the 
 usual length ; in which case we can find the extent of curva- 
 ture 'needed for putting in a 'special' by substituting for 
 1 Ten added to index No. of unity as with log. sines. 
 
Curve- Ranging with Transit and Chain 225 
 
 LA the decrement /, and inverting the equation the general 
 rule becomes A =- x 57*296, and A has following "values in 
 
 Gr 
 
 the several cases : 
 
 Standard gauge, 6 inch decrement . . . . 6 -084 
 3 feet 6 inch gauge ,, ,,.... 8-185 
 
 3 feet gauge ,, ,,.... 9-549 
 
 Metre gauge ; decrement of 2 decimetres . .11 '459 
 
 If the radius and length of the curve are given, find from 
 
 Table LIV. p. 379 the degree of curve, and A=_L. Thus 
 
 100 
 
 if the curve were 1,146 feet radius, and 2,762 feet long, 
 
 from Table LIV. d= 5 ! oo, and A = - '- 5 = 1 38 i . 
 
 100 
 
 TRANSITION CURVES 
 
 The transition or parabolic curve is a means of toning 
 down the abrupt passage from a tangent into a circular curve. 
 It is called the 'Uebergang' by the Germans and constructed 
 as a true parabola. 
 
 The objection to it in this form is that if a curve becomes 
 deformed or requires renewal the tracklayers are apt to put 
 in what one of them termed to a friend of mine ' somethink 
 of a paraboler.' As the man literally knew as much as 
 the proverbial cow about conic sections, the curve was 
 unique. 
 
 Mr. Searles in his little book entitled 'The Railroad 
 Spiral ' has placed the matter on a much more practical basis, 
 using a combination of short circular curves, forming a close 
 approximation to the parabola, in which, with equal chords, 
 the curvature increases by an equal amount at the end of 
 each chord. He adopts ten angular minutes as the basis, 
 and uses chords of from ten to one hundred feet. Whatever 
 its length, the curvature of the first chord is 10, the second 
 20, the third 30, the fourth 40 minutes, and so on. 
 
 Q 
 
226 Preliminary Survey 
 
 Mr. Searles gives exhaustive and lengthy tables and 
 formulae for putting in any kind of spiral or for substituting a 
 spiral for an existing circular curve whilst retaining as nearly 
 as possible the same ground antl the same length of line. 
 
 The object of the parabolic curve is not to dispense en- 
 tirely with the circular arc, but to attain gradually any suit- 
 able radius with which to form a circular connecting or 
 ' crowning curve ' and leave it again by the same gradual 
 transition to join the further tangent. 
 
 The advantages are chiefly : 
 
 First. Less wear upon the running-gear. 
 
 Second. Less discomfort to the passengers. 
 
 Sharp curves and high speed are the factors that make 
 the demand for the transition-curve. The explanations 
 herein will deal firstly with a case suitable to a tramway or 
 workshop siding of about 40 feet radius, because a better 
 illustration can be given where the whole curve is shown. 
 Spirals for street-railways are becoming every year more 
 important. The radii cannot be increased, but the speed is 
 constantly being accelerated up to the utmost limit in 
 response to the public demand for rapid transit. Mo-tors of 
 one kind and another have been invented which meet the 
 requirements of speed combined with safety. The company 
 protects its cars from derailment upon sharp curves by 
 means of guard rails, but cannot prevent its passengers from 
 the unpleasant swing and jolt when the cars turn a square 
 corner, nor can it avoid the wear and tear to the car wheels 
 and axles from the sudden cross-strain. ' C'est le premier 
 pas qui coute.' 
 
 As pointed out at p. 17 in Chapter I., the wear of loco- 
 motive tyres proves itself to depend largely upon the degree 
 of shock which is imparted. Once the running gear has 
 accommodated itself by its structural flexibility to a sharp 
 curve, the extra wear due to pressure against the outer rail 
 is comparatively small. The super-elevation of the outer 
 rail relieves it, and by the transition curve this elevation is, 
 
Curve- Ranging with Transit and Chain 227 
 
 commenced and gradually increased with the curve, so that 
 the unpleasant sensation of being tilted up in one's carriage 
 whilst still on the tangent is avoided. 
 
 The spiral, even on the sharpest curves, can be made to 
 follow very closely the same ground as a circular arc. It will 
 be seen from Fig. 72 how very slightly the two curves differ, 
 and yet the spiral commences with a radius of 573 feet 
 instead of one of 40 feet. 
 
 The spiral adopted in the following pages differs from 
 that of Mr. Searles in having for its base *2 of a degree 
 instead of ten minutes. For the sharper spirals the radii 
 are given in feet, for the others the degree of curvature per 
 i oo feet chord. Sharp short curves are best put in by offsets 
 as shown in Fig. 73; flatter and longer ones by tangential 
 angles as explained at p. 215 &c. 
 
 Table LV. of Appendix gives the general elements of the 
 decimal spiral which are common to all the other tables. The 
 first column, n, gives the number of the chord from the 
 commencement. The second, n. c., gives the curvature in 
 that chord, which is the same whatever its length may be. 
 The length of the short chord c in feet defines the spiral : 
 thus No. 2 spiral is one in which c is 2 feet. 
 
 The third column, s, gives the total curvature of the 
 spiral from the commencement, in other words the angle of 
 deflection formed by a tangent at any point n with the main 
 tangent. 1 The fourth column, k, is the inclination of any 
 chord to the main tangent, in other words the total curvature 
 opposite the middle point of said chord ; it is the basis of 
 the computation of the ordinates x and y. 
 
 The fifth column, /, is the tangential angle formed by the 
 long chord C at the point of spiral S with the main tangent. 
 It is needed for setting out the curve by tangential angles 
 similarly to a circular curve. In the other tables the column r 
 is the radius of curvature ; d the degree or deflection of 100 
 feet chord. The column x is the ordinate to the main tan- 
 
 1 It is called by Mr. Searles the spiral angle, 
 
 Q 2 
 
228 Prelim inary Survey 
 
 gent from any point n, and y is the corresponding abscissa. 
 The column L gives the length of the spiral. The letter R 
 denoting the radius of the crowning curve does not occur in 
 the tables. C is a long chord ; c the short chord. 
 General formulae of the spiral : 
 
 10 10 y 
 
 #=S*. chord xsin k ; y=2. chord xcos k 
 
 C =S __-_Z_ : . r = ___ C 
 
 sin / cos / ' . n. c. 
 
 2 sin --- 
 
 2 
 
 sub-tangent along "1 
 
 main - tangent for >=j x. cot s 
 
 any value of n\ 
 
 corresponding sub-tangent to = x - 
 
 sin s 
 
 It will be seen, on comparing these columns with the 
 corresponding ones in Mr. Searles' tables, how much simpler 
 they become by the use of the decimal degree. 
 
 The angles X and /* indicated in Fig. 72, are formed by 
 the semichord of the crowning curve with a normal to the 
 main tangent and the central radius of the curve respectively. 
 
 They are found as follows (see Fig. 73) : 
 
 X= 9 o-4 (0 + j); /u=9o J (-s) 
 
 and as a check X-f ^4-0=180. 
 
 The point to be aimed at with spirals, as with ordinary 
 curve-ranging, is to obtain the easiest curve possible within 
 the limits prescribed by the situation. With tramway curves, 
 and in many other cases, it is often the crown of the curve 
 or apex which fixes its other elements. The formulae for 
 putting in a spiral to conform to a pre-determined apex 
 distance is somewhat longer than the other, and requires 
 the rinding of angles X and /^. A piece of spiral is first fixed 
 
 * The symbol 2 is used for the summation up to any point of the 
 product of each chord by the corresponding value of k. 
 
Curve- Ranging ivitJi Transit and Cliain 229 
 
 upon, and the distance of the point of spiral S from the 
 point of intersection I, and the radius R of the crowning 
 curve, are found by the following formulae : 
 
 R= AI.cose- - (I) 
 
 2 COS A. COS fj. 
 
 and IS=y+ tan X [(AI. cos ) #] + AI. sin . . (2) 
 
 As will be presently shown, these formulae are very 
 useful for "putting in a spiral upon a tramway ; but it 
 is not always practicable to fix the apex distance in this 
 way, neither is it always important to do so. The data for 
 a crowning curve of any required radius can be found by 
 the following formulae for a spiral of any chord-length and 
 any curvature. 
 
 Let R be the radius of the crowning curve. 
 
 Let be the half total curvature. 
 
 Let s be the total curvature of one spiral. 
 
 Let x be the ordinate to end of spiral. 
 
 Letjy be the abscissa to end of spiral. 
 
 Let IS be the distance of point of spiral from point of 
 intersection. 
 
 Let AI be the apex distance. 
 
 Then IS=jj'+ tan (R. cos s + x) R. sin s . . (3) 
 
 and AI= R ^_ S _R ( 4 ) 
 
 cos 
 
 The radius of an approximately corresponding circular 
 arc, or trial curve, should be first ascertained by rule on 
 p. 215 for the whole curvature, remembering that the 
 spiral will always somewhat sharpen the rate of curvature 
 at the crown. The next operation is to select from one 
 of the tables a portion of spiral which will lead in a con- 
 venient manner into a crowning curve of not much less 
 radius than the trial curve just found. The third step is 
 to work out the data, either from a fixed apex distance by 
 (i) and (2), or from the fixed radius of crowning curve by 
 (3) and (4). 
 
230 Preliminary Survey 
 
 If not limited by a fixed apex distance, a radius of 
 crowning curve will naturally be chosen which forms a 
 harmonious transition from the spiral, but any radius may 
 be put in to join any spiral, provided s is less than 0. When 
 s=, the two spirals meet, and form a continuous parabola, 
 or nearly so. 
 
 It is also possible to have two spirals of different length 
 and different curvature at the ends of a crowning curve, but 
 the discussion of such peculiar cases as that would be out of 
 place in this work. 
 
 TRAMWAY SPIRAL 
 (Nos. 2 and 5 Spiral) Tables LVI. and LVIL 
 
 Fig. 72 represents the very common case of a square 
 street-turning, the streets being only 30 feet wide, with 5 feet 
 footpaths, and the centre-line is placed somewhat to one 
 side in order to give equal clearance at the corner, and on 
 the further side of the street. This, of course, is only a 
 secondary matter, and has nothing to do with the principle 
 of the spiral ; but the figure will serve to show how,, when 
 the limit of apex distance has been fixed by a trial curve, a 
 spiral can be put in which will differ very slightly from it, 
 and have more clearance at the tight corner. 
 
 The data are as follows. 
 
 A=9o ; =45 ; from which a 40 feet radius is selected 
 for a trial curve, and the subtangent ID, and apex distance 
 AT found by rule on p. 215. It is presumed that the limit 
 of curve-radius is 35 feet. AT is found to be equal to 
 16-57 feet. In order that the spiral may approach co- 
 incidence with the trial curve, the crowning curve must lie 
 outside, and the spiral inside of it. It is found that an 
 approximately constant relation exists between the ordinate 
 x of the end of the spiral, and the distance AA' between 
 the trial curve and the crowning curve at the crown, and for 
 curves of this character it will do to diminish AT by one- 
 tenth of x. 
 
Curve- Ranging with Transit .and Chain 231 
 
 To select a spiral, on examination of Table LVI. it will be 
 seen that any of the points n to 14 are of a curvature 
 greater than 40 feet, so that in using them there will be no 
 danger of the crowning curve flattening the spiral. Which- 
 ever point we choose, we first reduce the apex distance by 
 one-tenth of x. Supposing we use point u. Our data are, 
 
 FIG. 72. 
 
 r=52'09 feet ; .r=i3 = "2 ; x=i"j6 feet ; y= 21 '88 feet. De- 
 ducting from A'l one-tenth of or we have AI= 16*40 feet. 
 
 j t / \ -n 1 6 '40 xcos 4S 1*76 , r . 
 and by (i) R= - ^ '- = 36-93 feet 
 
 2 cos 6o'9Xcos 74-i 
 
 and by (2) IS=2r88 feet + 16-40 x sin 45 + tan 6o'9 
 x (16-40 cos 45 i'76)=5i-i6 feet 
 
 The transition at end of spiral will be from ^=52*09, to 
 R=39'93, or a change of 15-16 feet. 
 
 If we were to end the spiral at point 13, we should have 
 
232 Preliminary Survey 
 
 AI= 16-57 -28=16-29, an d adapting X and /(to the altered 
 data, R would become 35-74 feet, and 18=51-36 feet. The 
 transition would then be from ^=44-08 to R=3574= 
 8-34 feet. The change would be more gradual, but the 
 crowning curve would be 1*19 feet sharper. In either case 
 the travelling public would be none the wiser, the rolling 
 stock would not be affected, and it would be difficult to say 
 which of the points of n, 12, and 13, would be preferable. 
 
 If it were not necessary to fix the apex distance absolutely 
 and it was desired to have a crowning curve with a perfectly 
 harmonious transition, we should choose for a value of R 
 that of r next in order to the point we selected as the end of 
 spiral. Thus with ;/= 13 we should take R=40'93 feet: 
 which is the value of r when //=i4. We do not then need 
 X or ^ ; and by (3) 
 
 18 = 2573 + tan 45 (40-93 x cos i8-2+2'84) 
 
 40-93 sin i8'2 
 and by (4) 
 
 AI==4 o-93 cos i8-2 + 2-84 _ 
 cos 45 
 
 or IS =54-67 feet and AI = i8-o7 feet. 
 
 The point of spiral would not be at an inconvenient 
 length, but the apex would be 1-67 feet nearer to the foot- 
 path. 
 
 When finally selected and calculated the curve should 
 be tabulated for reference and a working drawing made to 
 a large scale in the form of Fig. 73. Any practical track- 
 layer can then put it in or replace it with nothing more than 
 a chalk-line, a set-square, and a steel tape, graduated to feet 
 and himdredths. 
 
 When the survey has been made previous to construction, 
 as it always should be if possible, the data of each curve 
 should be worked out and the rails bent at the rolling mills 
 to the required spirals and crowning curves, painted and 
 stamped so as to identify them. When there is no survey 
 
Curve- Ranging ^vit1l Transit and Cliain 233 
 
 and where no facilities for accurate bending are available at 
 the job, a number of spiral rails should be included in the 
 shipment, from 12 to 30 feet long, having holes for fish- 
 bolts at every two feet in all lengths exceeding 1 2 feet, so that 
 they will only need cutting at the most suitable value of n 
 for each particular curve. 
 
 -Alcos.0 
 
 FIG. 73. 
 
 A template will be made for the centre line of the spiral 
 track, and each of the rails adjusted to it ; they will need a 
 little humouring with the 'Jim Crow' 'to bring them to gauge, 
 not being perfectly concentric circular arcs. 
 
 Another and more exact method is to calculate the 
 spiral for the outer rail, and make a second working drawing 
 from it by laying off ordinates at every two feet equal to the 
 
234 
 
 Preliminary Survey 
 
 gauge. The ordinates for the crowning curve can be put in 
 from the tangent by formula on p. 207, or from the chord as 
 shown on Fig. 73 by formula in Trautwine's ' Pocket Book.' 
 
 THE HORSE-SHOE SPIRAL 
 (Nos. 5, 10, 15, and 25) Tables LVII. to LX. 
 When the total curvature approaches a semicircle (see 
 Fig. 74) it is impossible to run the tangents to an intersec- 
 tion. Long before so great a curvature is reached, as shown 
 in the example, it becomes very inconvenient to do so, and 
 recourse should be had to the apex tangent HAK 
 The deflection A of the main tangents can be measured by 
 
 FIG. 74- 
 
 running an auxiliary base-line DG from one tangent to the 
 other. The sum of the interior angles IDG, IGD is equal 
 to A, but if the instrument is kept to the astronomical bear- 
 ing as explained on p. 203, the difference of bearing 
 between the two tangents will give their deflection, even 
 though two or three auxiliary base-lines have been run 
 between them. 
 
 The point A is generally fixed by inspection of the 
 ground and measured ; HAK being set out with angle 
 
 In exceptional cases, such as when A falls in the middle 
 of a torrent, H and K being on the sides of a precipitous 
 
Curve- Ranging with Transit and Chain 235 
 
 ravine, it is inconvenient to measure HK with the chain, 
 and telemetric measurement may be also difficult on account 
 of heavy brushwood ; we can then measure the distance HG 
 from the end of the auxiliary base to the point opposite 
 the intended apex, and calculate AH by the following 
 formula : 
 
 AH=2^M^--HG.cos-DG.sin(-iM. cot . .(5) 
 2 sin 
 
 Although lengthy, the only angles involved are and 
 the two interior angles IDG, IGD, so that the formula is 
 less tedious than it looks. 
 
 When the distance AH has been obtained, the radius 
 of a circular curve is calculated which will join the tangents 
 and pass through A by the formula 
 
 R'=AH. cot (6) 
 
 R' should be from 5 to 10 per cent, greater than the 
 curve-limit of the survey, in order to allow for the diminu- 
 tion due to the spiral. If R' is too small, a fresh point A' 
 must be chosen somewhat further from the intersection 
 point. It is generally possible, even in very rough country, 
 to obtain a sufficient approximation to AH by stepping or 
 even guessing in order to be within the limit of radius. 
 For instance, supposing to be 76*25, and the curve-limit 
 
 250 feet. AH must not be less than 250 x tan "- = 
 
 198-8 feet, therefore before selecting A, HK is roughly 
 measured to be sure that it is over 400 feet. If the object 
 be the staking out of the curve for actual construction, it 
 will not do to put in A with the tacheometer ; it must be 
 fixed to a tenth of a foot by the chain, but for the first 
 approximation to HK it can be measured by the tacheometer, 
 by pacing, or by the aperture of the two-foot rule, explained 
 on p. 72. Flags are fixed at two points on the first main 
 tangent, such as D and S, and the assistant, keeping himself 
 
236 Preliminary Survey 
 
 in line with them, moves forward along the tangent until 
 he is in the line of HK, the direction of which is given by 
 the surveyor at H with either the theodolite or prismatic 
 compass. 
 
 When a point A has been found which appears to suit 
 the ground best and also to be within the curve-limit, it is 
 finally determined either by the exact measurement of AH 
 (=HK) or by calculation with formula (5). 
 
 To select a spiral, examine any of the tables suitable to 
 the class of curve intended and find one which attains 
 a radius r= or somewhat > R' within a suitable length of 
 spiral. For instance, although it is possible to apply No. 2 
 spiral to a i curve on a trunk railroad, it would be no 
 use whatever, because it attains a radius of that degree at a 
 distance of eight feet. 
 
 No. 50 would attain that radius in a length of 200 feet, 
 or No. 100 in 400 feet, so that one of the last three spirals 
 would be chosen. 
 
 At the transition point T or end of spiral, s should be 
 somewhere between one-fourth and one-half of ; one.-third 
 is best. 
 
 When the spiral is chosen, angles X and p are calculated 
 (see p. 228) and the data of point of spiral KS and radius of 
 crowning curve R are given by the following formulae : 
 
 ,, AH. sin x 
 
 k = -r- ........... (7) 
 
 2 COS A. COS f.t 
 
 KS=j' x. cot + R cos 2 /t ..... (8) 
 
 Example (see Fig. 74). Let the curve limit be 250 feet. 
 
 (H)=76 0> 25. AH is measured=2i7 feet. 
 
 Selecting No. 15 spiral at point 15, we have further data, 
 ^=32'-o6 ; jy=22o /> 96 ; s=24'oo. From these we obtain 
 * = 39'S75> P = 6 3'875> R=2i7 x sin 76-25- 32 / 'o6-f- 
 2 cos 39-875. cos 63'875=264 / '43 ; KS=22o''96 32 r -o6. 
 
 cot 7 6-2 5 + 2X26 4''43 cos ' 6 3 -8 7 5 =3i8'-6 9 . 
 sin 6'2 
 
Curve- Ranging' with Transit and Chain 237 
 
 This will give a perfect transition ; r being 2 86 '5 at the 
 transition point. The length of spiral, 225 feet, will be 
 ample for a curve of. this character, being as long as many 
 narrow-gauge trains. 
 
 If we had chosen No. 10 spiral at point 10, we should, 
 have had a crowning curve of 2 74' -3 2 radius, the last value 
 of r being 286''52. The transition would be equally good, 
 but the length of spiral, 100 feet, would not be sufficient to 
 be thoroughly effective. 
 
 Spirals up to No. 15 inclusive are more suitable for 
 setting out by the ordinates of x and jy, although values of /, 
 the tangential angle, are given from point 10 onwards in 
 No. 15 spiral. Supposing a horseshoe of total curvature 170 
 degrees were required with a curve limit of 200 feet radius, 
 it would be' advisable to adopt No. 15 spiral at point 21, 
 rather than No. 10 spiral at point 13, and the ordinate x of 
 8274 would be inconveniently large for tape measurement. 
 
 THE MOUNTAIN SPIRAL 
 Tables LX. and LXI. 
 
 This term has been chosen to distinguish spirals Nos. 25 
 and 50 as suitable for sharp curves in a standard-gauge line 
 or ordinary curves in a narrow-gauge line. 
 
 The ordinates of x and y will not be used in the field, 
 as they become inconveniently large. The curve will be 
 ranged by the tangential angle /, similarly to the ranging of 
 circular curves. 
 
 Example. Having a total curve deflection of 30, it 
 is desired to put in a 5 crowning curve with a uniform 
 transition from a spiral. 
 
 Choosing a No. 25 spiral we find the transition point is 
 at #=6 and the data are : 
 
 0=15-0 ; s=4-2 ^ = 3-97 ; ^=149-91. 
 r= i, 193*6 ; R=i,i46 feet. 
 
238 Preliminary Survey 
 
 By (3)IS - i49'9 x + tan i5'o (1,146 cos 4- 2 + 3-97) 
 1,146 sin 4 0> 2 = 373*28 feet 
 
 ^ / \ T M4 cos 
 By (4) AI = i-_ TJ */ - 1,146 = 41-33 feet 
 
 cos 15 o 
 
 The apex distance will here only differ by 0-9 foot from 
 that of a 5 curve throughout. 
 
 THE TRUNK LINE SPIRAL 
 Tables LXI. to LXIII. 
 
 This term has been chosen to distinguish spirals 75 and 
 100, but No. 50 is also applicable in many cases. 
 
 Example. Having a total curve deflection of 45 be- 
 tween the two main tangents, it is desired to put in a i 
 crowning curve, so as to form a uniform transition from the 
 spiral. 
 
 Choosing a No. 100 spiral, we find the transition point 
 for a i curve to be where #=4, and the data are : 
 
 =c22-5;^=2; x= 5-23 feet :_y=399'94; R = 5, 730 feet 
 By (3) 18=^399-94 + tan 22'5 (5,730 cos 2 + 5'23)~ 
 
 5,73o + sin 2=2,574'i9 feet. 
 By (4) AI = 5.?7J? ^J + 5-23 _ 5?730==474 . 00 feet 
 
 The apex distance will here only differ by two feet from 
 that of a i curve throughout. For practice the reader 
 might take the apex distance, 474 feet, as the fixed quantity 
 and find the radius and distance IS by formulae (i) and 
 (2). The results will agree with the foregoing assumptions 
 within a small fraction of a foot, but inasmuch as AI is 
 small as compared with R, in order to get R correct to two 
 places of decimals, AI should be given to three or four 
 places, which is not necessary for practical curve-ranging. 
 
Curve- Ranging with Transit and CJiain 239 
 
 RANGING THE SPIRAL FROM AN INTERMEDIATE POINT 
 
 Hitherto the spiral has been supposed to be visible from 
 end to end. Such is generally the case, because when the 
 first trial lines are run, the bush is cleared, and it is only 
 when unusual obstructions, such as sharp rock-points, inter- 
 fere with the view that the spiral cannot be ranged con- 
 tinuously. When, however, from any cause, a break or 
 turning point has to be made, the operation is analogous to 
 that described in circular curves on p. 217, except that the 
 tangential angle / is not half the deflection or 'spiral 
 angle s. 
 
 Supposing a No. 25 spiral is being ranged from a main 
 tangent whose bearing is 33 '34, and it is desired to make a 
 turning point, where n=i6, the curve being to the right. 
 Point 1 6 will be ranged from s, with a tangential angle /= 
 9-33; consequently on a bearing of 33-34 + 9'33=4 2 ' 6 7> 
 on shifting the instrument to #, we require for the tangent 
 at that point a bearing of 33*34 + ^, that is 6o'54; we there- 
 fore clamp the vernier at 42 '6 7 for the back sight, clamp 
 the external axis, reverse the telescope to the forward 
 position, and set the vernier to 6o*54, which will bring the 
 line of sight on to the tangent. 
 
 In ranging the remainder of the curve either by ordinates 
 or tangential angles, a separate set of values of /', x', /, 
 and /' have to be used as given in Table LXIV. If the 
 curve is ranged by ordinates, the values of x' and y' have 
 to be calculated by simple proportion from those of No. 
 100 spiral given in the table; for instance a No. 25 spiral 
 will have values of x and ^=25 per cent, of those in the 
 table. They are found by the same equation as those for 
 the primary tables, viz. x' = 2. chord, sin k' , f = ^. chord, cos 
 k'. If the curve is ranged by tangential angles /', no cal- 
 culation is required, as i' is the same for all spirals. 
 
 By keeping the instrument on the astronomical or any 
 continuous bearing, the transition point or end of spiral can 
 
240 Preliminary Survey 
 
 be very conveniently checked, however many turning points 
 may have been taken. For instance, supposing a curve to 
 the right, and calling the bearing of the main tangent B, the 
 bearing of the transition point when viewed from S will be 
 B-H. With one turning point it will be B + /+/' and so 
 on. When the transition point is reached, the direction of 
 the tangent to the crowning curve is found by taking a 
 back sight to the last turning point with the bearing B + z 
 + /' &c., and reversing the telescope, with the external axis 
 clamped, and the vernier released, the line of sight is then 
 set to the bearing ~B + s. It can easily be seen on the 
 ground whether this is a tangent to the last chord of the 
 spiral, and so check the calculation. 
 
 It is also useful as an independent check to lay off and 
 measure the subtangent from the transition point, or if 
 necessary from any turning point, to the main tangent, 
 which is done by the formula on p. 228. 
 
 WYES AND LOOPS 
 
 When a substitute for a turn-table is needed, it is usual 
 in America to put in a pair of curves turning to the right 
 and left on the same side of the main track, and terminating 
 in a common tangent at right angles to the main track ; the 
 engine runs round one curve into the common tangent, and 
 backshunts on to the main track through the other curve, 
 so coming out end for end. As a substitute for a cross- 
 over road on a double track a loop is sometimes made, by 
 which the train, after describing a complete circle, occupies 
 the other track, but in a reverse position. 
 
241 
 
 CHAPTER VIII 
 
 GRAPHIC CALCULATION FOR PRELIMINARY 
 ESTIMATES 
 
 THE surveyor is compelled to form his first estimates of cost 
 without detailed measurements with chain, tape, and rule, 
 but as little as possible by guess-work. 
 
 If the estimate is based upon a walk-over survey and 
 sketches, he must rely upon his experience of similarly con- 
 structed works, and he will judge the cost mile by mile , 
 according to its general character. 
 
 When sufficient time is allowed to produce a topo- 
 graphical map of more or less accuracy, the operations of 
 the preliminary surveyor, although more rapid, are analogous 
 in principle to those of the executive engineer, who succeeds 
 him with more time at his command. 
 
 The earthwork is measured on a profile plotted from the 
 contours ; the trestles scaled for distance and heights from 
 the same profile, as also the bridges and'culverts. The 
 quantities in all cases are usually taken from tables. 
 
 The use of quantity diagrams combined with the slide- 
 rule is much more suitable to this class of work than long 
 tables of figures and elaborate formuke. 
 
 CALCULATION BY SLIDE-RULE 
 
 Before entering upon the subject of estimates, it will be 
 
 necessary to describe somewhat fully the use of the slide-rule. 
 
 The instrument itself is described on p. 361, but it is 
 
 R 
 
242 Preliminary Survey 
 
 desired to give a detailed description of its uses as applied 
 to preliminary survey. 
 
 The printed explanations sold with the slide-rule con- 
 tain directions for its use, but it is preferred here to arrange 
 them in a different form. 
 
 The easiest way to become familiar with the instrument 
 is to look upon it as a 
 
 i. Geometrical proportion or rule of three sum. Place a i 
 of the slide opposite to a 9 of the rule, and give the figures 
 on the right-hand side of the rule ten times their indicated 
 value. Then, beginning with the central i of the slide, it 
 will be found under 9 of the rule, 2 under 18, 3 under 27, 
 4 under 36, n under 9-9, 2-3 under 207, 37 under 33-3, 
 and so on, every value of the rule corresponding with the 
 slide x 9. We may put it in the form of multiplication as 
 9 X 3=27 x i, or in the form of division as f=V> or in the 
 ordinary rule of three form 9:11:27:3. In any case, the 
 figures on the rule are opposite to their proportionals on the 
 slide. 
 
 We may give the figures on the slide ten times their in- 
 dicated value, and those on the rule 100 times, but the pro- 
 portion remains the same. Thus, in the above illustration 
 37 would become 37 on the slide, and it would be under 
 333 on the rule. 
 
 One of the most useful cases of proportion by the slide- 
 rule is the finding of the angular value of odd distances in 
 railway curves by tangential angles, or in getting the loga- 
 rithms of intermediate numbers or angles by interpolation. 
 
 Example. What is the cosecant of 35 n' i3"'3? 
 
 Log. cosec. 35 ll' by table . . . = 10-2394308 
 Tab. difference for 60" = 1,791 ; 
 
 x 1,791 by slide-rule . . = 397 
 
 - 
 
 DO 
 
 Log. cosec. 35 1 1' 1 3" -3 . * = 10-2393911 
 
 This is done by placing the 1,791 of the slide opposite to 
 the 60 on the rule, and looking for the value on the slide 
 
Graphic Calculation for Preliminary Estimates 243 
 
 corresponding with 13-3 on the rule. It is more exactly 
 done by the lower scales of the rule, although it involves 
 two operations. If the upper scales be used we can only 
 read 1,790 instead of 1,791, but there will still be no mis- 
 take about the 397 to an eye which has had a little practice 
 in estimating the value of the subdivisions. We will, how- 
 ever, take the lower scales. 
 
 Giving the slide 1,000 times its indicated value, we place 
 the 1,791 opposite the 6 of the rule, which we call 60, 
 and we see that the 13-3 on the rule has been overshot 
 altogether, the left hand i of the slide being opposite to 
 the 3*35 on the rule. We slide back until the right hand i of 
 the slide corresponds with 3-35 of the rule, and we shall be 
 still in adjustment for giving the proportion of --J-jJ- as 
 before. Opposite to the 1*33 of the rule (which we give the 
 value of 13-3) we find the exact figure of 397 on the slide. 
 
 This operation is again much simplified by decimal 
 graduation. 
 
 The whole of the succeeding examples of sdak: reduc- 
 tion and plotting, weights and measures and \coinage, are 
 based upon this principle of proportion or geometrical 
 ratio. 
 
 2. Multiplication. When, as described on last page, we 
 put the i of the slide opposite the 9 of the rule, we mul- 
 tiply it by 9, and any other figure on the slide is likewise 
 multiplied by 9 on the rule opposite to it ; therefore, if we 
 want to multiply by any number, we place a i of the slide 
 opposite that number, and the slide and rule will be in adjust- 
 ment to read like a column of a multiplication table headed 
 by that number. Thus, if we wish to multiply 57 X35, we 
 place a i of the lower slide- scale over the latter number. 
 Every figure on the slide will then be opposite to 35 times 
 its value on the rule, and 57 will be found opposite to 1,995 
 exactly. 
 
 3. Division. This is the converse of multiplication 
 contained in the same principle of proportion. 
 
 R 2 
 
244 Preliminary Survey 
 
 Thus, in the preceding example, to divide 1,995 by 57 
 we adjust the instrument in that ratio by placing the 57 of 
 the slide over the 1,995 of the rule, or vice versa, and read 
 off the quotient 35 under the corresponding i. 
 
 4. Involution and Evolution. This is done by simple 
 inspection without using the slide. Any of the figures on 
 the lower scale of the rule are the square roots of those on 
 the upper scale, and vice versa. They are made to * teach ' 
 with one another by the brass marker. Thus the 3 on 
 the lower scale is under the 9 of the upper. 
 
 Example a. Find the square root of 718. Direct the 
 brass index to that figure on the upper scale of the rule, 
 and on the lower scale will be found 26-8. 
 
 Example\). Find the square of 718. The answer will 
 obviously have six figures. Placing the marker at 718 on 
 the lower scale, the upper scale is, as near as one can read, 
 516 that is to say, 516,000. The exact answer is 515,524. 
 
 If we wanted it exactly we should have to make a double 
 inspection with the aid of the lower slide. 
 
 Thus, 718 x 700 = 502,6oq 
 
 718 x 18 = 12,924 
 
 515,524 
 
 and in so doing we should scarcely save any time, because 
 directly we begin to have to put down figures we might as 
 well work it out. 
 
 THE SLIDE-RULE AS A UNIVERSAL DECIMAL SCALE 
 
 Nothing can compare with the slide-rule for plotting 
 in the field. In the Mannheim rule one of the edges is 
 bevelled so as to be used on the plot, and is graduated to 
 millimetres. If we set the slide to read with the rule the 
 proportion of millimetres to the given scale of feet, chains, 
 miles, or what not, we can then apply the millimetre scale 
 directly to the paper without any further calculation, and this 
 
Graphic Calculation for Preliminary Estimates 245 
 
 with the most awkward scales imaginable. Example (a). 
 Let it be desired to plot with the slide-rule from a scale which 
 was intended to be 6 inches to the mile, but which by con- 
 traction of the paper has shrunk to 5*95 inches to the mile. 
 On the back of the rule in the table of useful memoranda 
 will be found under mesuresanglaises^\&^ (foot)=M. 0-3048, 
 i.e. 12 inches= 304/8 millimetres. If we then make 304/8 
 on the slide correspond with 12 on the rule, we shall find 
 opposite to 5 '95 on the rule 151 on the slide. This being the 
 scale value of a mile in millimetres, we can make 151 corre- 
 spond with 5,280, to get the number of millimetres correspon- 
 ding with feet, or 151 with 8,000, to find the value for links. 
 Thus, adjusting the 151 of the rule opposite the 80 of the 
 slide, we have for, say, 23*1 1 chains on slide 43 '6 2 on the rule. 
 The second place of decimals, that is, single links, can scarcely 
 be relied upon at this scale with a small Mannheim rule, 
 but as it cannot be estimated by the eye on the millimetre 
 scale, the rule will give the measurement as closely as it can 
 be scaled. To use the lower scale we must first reduce the 
 proportion of -J^jV to its equivalent of -^ in order to keep 
 within the range of the rule. See p. 243. 
 
 The way in which odd scales of paces can be figured off 
 at a glance in feet or links is explained on p. 57 of chapter 
 on Route Surveying, and they can be plotted on the plan 
 without any further reduction. 
 
 If the scale of miles, chains, or feet, to which it is 
 intended to plot with the slide-rule, be given on the plan, 
 the first process of determining the value of a mile, or other 
 English measure of distance, in millimetres to scale, is dis- 
 pensed with by merely applying the millimetre scale to the 
 paper, and then adjusting the slide-rule to the proportion. 
 
 RAILWAY GRADIENTS 
 
 The nomenclature of gradients on English parliamentary 
 maps for roads or railways is the ratio of perpendicular to 
 base, and is expressed as inclination i in 
 
246 Preliminary Survey 
 
 The nomenclature of railway promoters abroad and 
 ordinary business men is the same ratio expressed in feet 
 per mile. 
 
 Engineers have adopted percentage, such as i per cent, 
 instead of i in 100, 2\ per cent, instead of i in 40 &c., as 
 being the most convenient, both for levelling and contouring. 
 Side slopes are named by the ratio of base to perpendicular, 
 as 2 to i, &c., or sometimes in degrees of slope from the 
 horizontal. 
 
 To REDUCE PERCENTAGE TO FEET PER MILE BY THE 
 SLIDE-RULE 
 
 As i per cent. =5 2 '8 feet per mile, place a i of the lower 
 scale of the slide over the 52-8 of the lower scale of the 
 rule, then 5 of the slide will be over 26-4, that is to say, -5 
 per cent, is equal to 26*4 feet per mile, or 5 per cent, equal 
 to 264 feet per mile. 
 
 Example i. What percentage will be a grade of 324 
 feet per mile ? Opposite the 324 on the rule we find 6-13 on 
 the slide. 
 
 The left hand i of the slide gives grades of i to i '9 per 
 cent., or 10 to 19 per cent. 
 
 The right hand i gives grades of i -9 to 10 per cent., and 
 19 to 100 per cent. 
 
 To REDUCE THE INCLINATION, SUCH AS i IN 20, i IN 
 30, &c., TO FEET PER MILE 
 
 The result is obviously 5,280 divided by the ratio. 
 Place a i of the lower slide-scale over the ratio on the rule, 
 and read off the feet per mile on the slide opposite 5, 2 80 on 
 the rule. 
 
 Example 2. How many feet per mile are there in a 
 grade of i in 18*1 ? 
 
 Place the left hand i of the slide opposite the 18*1 on 
 the rule, and opposite 5,280 on the rule will be found 292*1 
 on the slide. 
 
Graphic Calculation for Preliminary Estimates 247 
 
 To REDUCE THE INCLINATION AS ABOVE TO THE 
 PERCENTAGE 
 
 Place the slide as above, but read the percentage on the 
 slide opposite the right hand of the rule. Thus, in the 
 preceding example the 5-52 is found opposite to the right 
 hand i of the rule. 
 
 To FIND THE ANGLE OF SLOPE CORRESPONDING TO ANY 
 GRADIENT 
 
 1. Reduce the grade in whatever form it is given to its 
 equivalent percentage as just explained. 
 
 2. Find the angle from the line of tangents by placing 
 it in its initial position, and reading off the angle on the slide 
 opposite to the percentage on the rule. 
 
 Example 3. What is the angle corresponding t 270 
 feet per mile ? 
 
 Placing the 270 on the slide above the 5,280 on the rule, 
 we have opposite to the i of the rule 5 1 1 . 
 
 Reversing the slide to line of tangents, we have opposite 
 to 5-1 1 the angle 2 55', or 2'92. 
 
 Example 4. What is the angle corresponding to 2 7 feet 
 per mile ? 
 
 Similarly to the preceding example we find the percent- 
 age=*5i, and this is less than the line of tangents will give. 
 Without reversing the slide we bring the mark for single 
 minutes situated at 3*44 on the slide, and indicated by a 
 single stroke i, opposite to the i on the rule, then the slide 
 will give tangents or sines of small angles which are alike 
 proportional to those angles. The following table shows 
 that -51 per cent, lies somewhere between 6' 30" and 
 34' 1 8" of angular value, and we find the exact angle 17-5' 
 under the 5 1 of the rule. 
 
 Example 5. Suitable for flow of rivers. What is the 
 angle of slope corresponding to a fall of six inches to the 
 
248 Preliminary Survey 
 
 mile ? We see from Table XXIX. that the percentage is 
 somewhere between -002 and -01, and the angle between 
 3" and 20". Placing 5 over 5,280 we find the percentage 
 00948, and by the double stroke mark for seconds the 
 angle 19-$". 
 
 Example 6. What Is the angle of a i in 7-32 slope? 
 With a i of the slide over 7-32 we find the percentage to be 
 13*66, and from the tangent scale in its initial position angle 
 = 7 46'. 
 
 TABLE XXIX. Of leading values of slopes in percentage and angle 
 for checking the slide-rule. 
 
 Feet per 
 mile 
 
 S = tan 
 of slope 
 
 Percentage 
 
 Angle 
 (sexagesimal) 
 
 Angle 
 (decimal) 
 
 0-1 
 
 0000189 
 
 00189 
 
 o o' 3-9" 
 
 ooii 
 
 0-528 -oooiooo -oiooo 
 
 0'20'6" 
 
 0057 
 
 i-o -0001894 -01894 
 
 o o'39-o" 
 
 OI08 
 
 5-28 
 
 ooiooo -i oooo 
 
 o 3' 25 -8" 
 
 0571 
 
 10-0 -001894 '1894 
 
 o 6' 30-0" 
 
 1083 
 
 52-8 -oioooo i -oooo 
 
 o 34' 1 8-0" 
 
 5717 
 
 i oo-o -01894 
 
 1-894 
 
 i 5' 14-0" 
 
 I -0833 
 
 528-0 -loooo 10-000 
 
 5 42' 38-0" 
 
 57106 
 
 1000 -0 
 
 18940 
 
 18-940 
 
 10 43' 45-0" 
 
 107292 
 
 All the values of S are also equal to the sines except the 
 last three, which are somewhat larger. 
 
 To interpolate between the values in this table. 
 
 Example 7. Find all particulars as in the table for a 
 grade of 7 feet per mile. S = 7 x -000189 = -00134. 
 Percentage=7 x -0189=* 134. Angle=7 X39 // =4 / 33". 
 
 SQUARES AND SQUARE-ROOTS OF SMALL DECIMALS 
 
 Table XXX. will facilitate the use of the slide-rule in the 
 involution and evolution of small decimal numbers, specially 
 of those used in Kutter's formula. 
 
 Example 8. Find the square of -0029. From the table 
 we see that this must be between -00000625 and -ooooi, and 
 by inspection of the rule find it to be -00000841. 
 
Graphic Calculation for Preliminary Estimates 249 
 
 TABLE XXX. Ofsyttares and cubes for checking the slide-rule. 
 
 Number 
 
 Square 
 
 Cube 
 
 Number 
 
 Square 
 
 Cube 
 
 0025 
 
 '00000625 
 
 
 1*414 
 
 2 '00 
 
 2*818 
 
 '00316 
 
 'ooooi 
 
 
 i '50 
 
 2-25 
 
 3'375 
 
 005 
 
 '000025 
 
 
 i'732 
 
 3 'oo 
 
 4*196 
 
 0075 
 
 '00005625 
 
 'OOOI 
 
 'oooooio 
 
 2*0 
 
 2'i544 
 
 4-00 
 4*6416 
 
 5'359 
 10*0 
 
 025 
 '0316 
 
 '000625 '0000156 , 3' l62 3 
 'ooi ' '0000316 i 4*6416 
 
 lO'OOOO 
 
 21 '5443 
 
 31*6228 
 
 lOO'O 
 
 '05 
 
 '0025 ! '000125 
 
 zo'o loo'o 
 
 zooo'o 
 
 *75 
 
 '005625 '000422 i 21 '5443 464-1587 
 
 1 0000 '0 
 
 
 O I \ 'OOI 31*6228 lOOO'O 
 
 3i622'8 
 
 ' 2 5 
 
 '0625 ': '0156 46'4i6 2154*4 
 
 tooooo'o 
 
 316 'i i '0316 
 
 lOO'O 
 
 TOOOO'O 
 
 loooooo'o 
 
 5 i '2S '125 II 2I5-4435 
 
 46415*87 
 
 lOOOOOOO'O 
 
 '75 
 
 5625 '422 i 316*227 1 100000*0 
 
 31622777*0 
 
 I'O 
 
 i'o j i'o 464-1587 215443*5 
 
 lOOOOOOOO'O 
 
 1*25 
 
 i "563 | i '953 || I000 ' 
 
 1000000*0 
 
 lOOOOOOOOO'O 
 
 Example 9. Find the square root of '00095. From the 
 table we see that this must be between -025 and '0316. 
 From inspection of the rule it is determined as -0308. 
 
 If the student will take the trouble to work out one or 
 two of these sums in the ordinary way, and then endeavour 
 to obtain one or two decimal square roots by the slide-rule 
 alone without the table, he will at once see what an assist- 
 ance it is. 
 
 CUBES AND CUBE-ROOTS OF NUMBERS 
 
 Invert the slide, keeping the numerical scale upwards. 
 Adjust it so that what was the left-hand upper scale of the 
 slide becomes the right-hand lower scale, with the figures 
 upside down. Now place the number to be cubed on the 
 slide over the same number on the rule, and read off the 
 cube on the slide opposite to the left hand i of the rule. 
 
 Example 10. Find the cube of -373. From the table we 
 see it will be somewhere between '0316 #nd -125. Placing 
 the 373 of the slide over the 373 of the rule, we find above 
 the left hand i of the rule 519, which we read "0519. 
 
 To extract the cube-root with the rule as above, place 
 the number on the slide over the left hand i of the rule, 
 and search for a number on the slide which is opposite 
 to the same number on the rule, that is the cube root. 
 
250 Preliminary Survey 
 
 Example n. Extract the cube root of 1,575. We see 
 from the table that it will be somewhere between 21 and 10. 
 Place the left hand i of the rule under the 1,575 of the slide, 
 and it will be found that the coincident number is 11*63. 
 
 Example 12. Extract the cube root of "0954. From the 
 table we see that it will be between -5 and -316, and placing 
 the left hand i of the rule under the 954 of the slide, we find 
 the coincident number to be 457, which we write as -457. 
 
 RAILWAY SLEEPERS 
 
 Method of obtaining by a single adjustment of the rule, 
 when the dimensions and pitch are given, the number, 
 quantity of cubic feet, quantity of cubic yards, 1 weight in 
 tons of 2,240 Ibs., and price in sterling or dollars. Place 
 the pitch in feet and decimals upon the slide opposite 5,280 
 on the rule. 
 
 Read the number per mile on the rule in thousands 
 opposite the i of the slide. 
 
 Read the quantity of cubic feet per mile on the rule in 
 thousands opposite the value of A in Table XXXI. 
 
 Read the quantity of cubic yards per mile on the rule in 
 hundreds opposite the value of B in the table. Read the 
 quantity of tons per mile on the rule in tens or hundreds, op- 
 posite the value of D in the table. Read the price per mile 
 in sterling or dollars, in hundreds or thousands, from E or F. 
 
 Example i. Find the above desiderata for standard-gauge 
 sleepers 9 feet x 10" x6", pitched 2' 9", at 25. apiece. 
 
 Place the pitch 275 on the slide opposite to 5,280 on 
 the rule. 
 
 Then the i of the slide is opposite to 1,920 No. on the rule- 
 375 ( A ) 7> 200 c - ft. 
 i'39 ,, ( B ) 26 7 c. yds. 
 5-86 (D) 112-1 tons 
 ioo (E) ., 1927. 
 
 1 The quantity of cubic yards is required as a deduction from the 
 ballast. 
 
Grapliic Calculation for Preliminary Estimates 251 
 
 Example 2. Find the above particulars for narrow-gauge 
 sleepers, 6' 6" x 7" X4 /; , pitched 2' 3", at 35 cents apiece. 
 
 Place the pitch 2*25 on the slide opposite to 5,280 on 
 the rule. 
 
 Then the i of the slide is opposite to 2,350 No. on the rule. 
 1-27 (A) 2,985 c. ft. 
 4-69 (B) no-i c. yds. 
 1-98 (D) 46-5 tons 
 
 350 
 
 (F) 
 
 )22 
 
 TABLE XXXI. Quantity, weight and cost table of railway 
 sleepers for standard and narrow gauge. 
 
 A. 
 Dimension Cub. feet 
 Ft. xin. xin. in one 
 1 sleeper 
 
 B. 
 
 Cub. yds. 
 in ten 
 sleepers 
 
 C. D. 
 Wt. in Wt. in 
 
 Ibs. of tons of 
 one 100 : 
 sleeper at] sleepers 
 35 Ibs. per at 35 Ibs. 
 c. ft. per c. ft. 
 
 4 64 0-667 
 
 0-247 23-3 
 
 I-O4 
 
 5 74 0-972 
 
 0-360 i 34-0 
 
 I-52 
 
 6 74 1-167 
 
 0-432 40-6 
 
 1-82 
 
 61 7 4 1-267 
 
 0-469 
 
 44'3 
 
 I- 9 8 
 
 7 7 5 i 1705 
 
 0-632 
 
 597 
 
 2-6 7 
 
 8 86 2-667 
 
 0-988 
 
 93-5 
 
 4-17 
 
 8 96 3-000 
 
 no 
 
 I05-0 
 
 4-69 
 
 8 97! 3-500 
 
 296 
 
 122-5 
 
 5-47 
 
 8 10 6 
 
 3-333 
 
 233 
 
 116-5 
 
 5-20 
 
 8 10 7 
 
 
 441 
 
 136-0 
 
 6-07 
 
 8 10 8 
 
 4-444 
 
 645 
 
 155-3 
 
 6-93 i 
 
 8 12 8 
 
 5-333 
 
 976 
 
 1 86 -8 
 
 8-33 
 
 8 8 6 
 
 2-833 
 
 048 
 
 993 
 
 4-42 
 
 8J 9 6 
 
 3-188 
 
 181 
 
 iii-6 
 
 4-98 
 
 8| 9 7 
 
 3719 
 
 '377 
 
 130-1 
 
 5-81 
 
 8 10 6 
 
 3-542 
 
 312 
 
 124-0 
 
 5-53 
 
 8| 10 7 
 
 4-132 
 
 530 
 
 144-8 
 
 6-45 
 
 8^ 10 8 
 
 4-722 
 
 750 
 
 165-0 
 
 
 8j 12 8 
 
 5-667 
 
 095 
 
 198-0 
 
 8-85 
 
 9 8 6 
 
 3-000 
 
 no 
 
 105-0 
 
 4-69 
 
 9 9 6 
 
 3'375 
 
 250 
 
 118-0 
 
 5-27 
 
 9 97 
 
 3-938 
 
 456 
 
 138-0 
 
 6-15 
 
 9 10 6 
 
 3-750 
 
 388 
 
 131-1 
 
 5-86 
 
 9 10 7 
 
 4'375 
 
 1-619 
 
 152-8 
 
 6-84 
 
 9 10 8 
 
 5-000 
 
 1-851 
 
 175-0 
 
 7-81 
 
 9 12 8 
 
 6-000 
 
 2-221 
 
 2IO-I 
 
 9-37 
 
 E, cost of 1,000 
 sleepers in sterling 
 ati/6, 7551/9,87-5 
 2/0, 100 ; 2/3, 112-5 
 2/6, 125; 2/9, 137-5 
 3/0, 150 ; 4/0, 200 
 5/o, 250. 
 
 F, cost of 1,000 
 sleepers in $ at 35c., 
 350; 5oc.,5oo;7oc., 
 700 ; looc., looo. 
 
252 Preliminary Survey 
 
 EARTHWORK 
 
 It is becoming very customary to use diagrams for 
 measuring earthwork for preliminary estimates. They are 
 usually drawn to large scale, and are correct to a fraction of 
 a yard for level cuttings. To avoid folds, they are in a thin 
 quarto or folio book, which becomes quite unwieldy. The 
 little diagram given on Plate V. is quite close enough for 
 a preliminary estimate. It has the advantage over tables- 
 first, that it is all on one page ; secondly, that it is applicable 
 to any base and any slope. It will be observed that for 
 any additional depth of cutting, the quantities for varying 
 width of base increase as the ordinates of a triangle, but the 
 quantities in the side-slopes increase as ordinates to a 
 curve. The line marked base=o is a datum line for slopes 
 only ; ordinates measured upwards from it to the curve give 
 quantities in the two side- slopes for 100 feet length. 
 
 It is also a datum for the central portion alone ; ordinates 
 measured downwards from it to the line marked with the 
 given width of base give the quantities in the central 
 portion. 
 
 When the total quantity, central and sides, is required, 
 the ordinate is measured clear through from the line marked 
 with the given width of base upwards to the curve. This 
 has to be done with the dividers, or with a slip of paper. 
 If with the former, and much work has to be done, a piece 
 of -tracing-cloth should be gummed over the diagram to 
 preserve it. The length of the ordinate is then applied to 
 the vertical scale of cubic yards. When considerable work 
 is to be done from the same width of base a piece of paper 
 can be gummed down so as to cover all below the datum 
 which is being used. 
 
 Although the side-slopes do not vary directly as the depth, 
 they do vary for the same depth directly as the slope. For 
 instance, a 2 to i slope contains twice as much as a i to i 
 
Graphic Calculation for Preliminary Estimates 253 
 
 for the same depth. We can therefore obtain other values 
 by a simple proportion. 
 
 The diagram is calculated from the prismoidal formula, 
 which is a modification of Simpson's rules for land areas 
 and marine displacements, and may be relied upon within 
 0*5 per cent. If used with a paper slip instead of dividers, 
 it will stand a considerable amount of usage. 
 
 The smaller diagram, Fig. 76, is only an enlargement of 
 that portion of the larger one between o and 25 feet of 
 vertical depth, in order to obtain a clearer reading. 
 
 The diagram gives the contents in cubic yards in 100 
 feet length of level cuttings of any base from 6 to 28 feet, 
 and having side-slopes of i to i or \\ to i. For any 
 less distance than 100 feet, such as Gunter's chains of 
 66 feet, or odd distances, the quantities given by scale are 
 obtained by simple proportion. With the slide-rule, by 
 placing the i of the slide opposite the 66 or other fraction 
 of 100 feet on the rule, and reading opposite the full 
 quantity on the slide, the odd quantity on the rule, 
 
 Example. To find the cubic contents of 100 feet length 
 of embankment, base 18 feet, side-slopes i to i, and 27 feet 
 deep. On the larger diagram, tracing the base line marked 
 1 8 to where it intersects the vertical 27, we scale from said 
 intersection up the vertical to the lower curve, and applying 
 the quantity to the end vertical, we find it measures exactly 
 4,500 cubic yards, which is the required quantity in 100 feet 
 length. 
 
 The cubic content of a level cutting of any length and 
 any slope may be obtained from this diagram by a simple 
 rule of three sum, or by direct scaling with proportional 
 compasses. 
 
 Rule. Multiply the quantity in i to i slopes by the 
 given ratio, and add to the product the quantity for the given 
 base. If the slope is given in degrees from the vertical, 
 multiply by the tabular tangent ; if from the horizontal, by 
 the cotangent of the slope. 
 
254 
 
 Preliminary Survey 
 
 Example. Required the cubic content of a piece of 
 level cutting. Base 24 feet, height 27 feet, slopes f to i, 
 length 13 feet. By diagram, the quantities are for the 
 base 2,400, and for the side slopes 2,700. 
 
 2)7 oX3> ) x-^-=575 cubic yards. 
 4 J 100 
 
 When the cuttings are in side-hill they sometimes have 
 to be equalised, even on preliminary survey, for which a 
 graphic method is given on Fig. 77, which represents a section 
 of ' hog-backed ' cutting ABCDHG, in which the crooked 
 portion ABC is replaced by the straight line CFE ; BE is 
 drawn parallel to AC ; to find E. 
 
 FIG. 77. 
 
 The triangles EBC, BEA are equal, being on the same 
 base and between the same parallels ; deducting the common 
 portion EFB, the remainder EFA, which is added by the 
 equalising line, is equal to the remainder FBC, which is 
 subtracted. The crooked portion DCE is replaced by 
 another equalising line in a similar manner. Finally the 
 one sloping surface line is replaced by a horizontal equivalent 
 as follows : 
 
 Let DE, Fig. 78, be the final equalising slope line. 
 Assume a point a in the centre line of the section above 
 DE, and mark , b' by a parallel run up from HG. Find b" 
 by a parallel run up from E to D ; halve the error b'b" in b'", 
 and b"" a! b'" will be the horizontal surface line of an equiva- 
 lent level cutting. This should be checked by a parallel 
 
Graphic Calculation for Preliminary Estimates 255 
 
 run up from b"" E to D, and if not quite correct, the error 
 bisected a second time. 
 
 FIG. 78. 
 
 The diagram, Fig. 79, is a rough approximation curve 
 constructed from a few actually measured areas, and may be 
 used only within the limits given. 
 
 FIG. 79. 
 
 Example. What would be the equivalent level cutting 
 when the slope of the surface of the ground (or its equiva- 
 lent DE, Fig. 78) is 25, and the central depth 18 feet, base 
 21 feet ? The percentage here is 22, therefore the equivalent 
 level cutting=i8 x '22 + 18=21-96 deep. 
 
256 Preliminary Survey 
 
 IRON BRIDGES 
 
 The diagram on Plate VI. has been prepared from the 
 empirical formula of the late Mr. Trautwine, after checking 
 several spans both of plate-girders and trusses. Designs 
 vary greatly, especially in British construction, sometimes 
 from unavoidable circumstances, more often from a desire 
 for novelty on the part of the designer. It is therefore im- 
 possible to give any precise formula which will cover the 
 engineer's ' personal equation ; ' but it is perfectly possible 
 to give a formula and a diagram which will represent a weight 
 which a bridge of moderate span need not exceed, in order to 
 be safe within a certain limit under given permanent and 
 rolling loads, and that condition is fulfilled by the diagram. 
 The standard gauge covers a range from 4 feet 8J inches to 
 5 feet 3 inches, and the narrow gauge 3 feet or metre. 
 
 The formulae are as follows : 
 
 For plate-girders up to 75 feet. 
 
 Weight in pounds per foot run of girders only =5 xspan 
 in feet + 50 v/span in feet. 
 
 For open trusses up to 250 feet. 
 
 Weight in pounds per foot run of trusses only=4'5 x 
 span in feet +22 \'span in feet. 
 
 The weights of the complete bridge are all scaled for 
 either gauge from the bottom to the curve bearing the proper 
 designation ; the weights of the trusses alone are scaled from 
 the line marked platform only ; the weights of the platform 
 only are scaled from the bottom. 
 
 The following comparisons with the weights of actual 
 structures will show to some extent the divergences to be 
 expected from the formula. 
 
 Oak Orchard Viaduct, New York State, 23 spans of 30 feet 
 each, by Mr. Chas. Macdonaid, engineer. They are plate- 
 girders 20" deep, trussed by a centrepost and eyebars carry- 
 ing a standard-gauge single-line railway. Actual weight 3*1 2 
 tons. Weight by diagram 5 'oo tons. 
 
PLATE GIRDERS 
 60 TO to 
 
 Universal Iron Bridge Diagram. 
 
 Note i. The standard-gauge curve is 
 4' 8i", but will do also for 5' 3". The narrow- 
 gauge curve is 3' o'', but will do also for 
 metre. Calculated for a rolling-load of ii tons 
 per foot run. Iron to bear 5 tons per square 
 inch in tension. 
 
 Note 2. If this diagram is in frequent use 
 with dividers, a piece of dull-back tracing 
 cloth, gummed over it by the four corners, 
 will protect it. 
 
 FIG. 
 
Graphic Calculation for Preliminary Estimates 257 
 
 Railway bridge over the Ohio at Cincinnati, Mr. I. H. 
 Linville engineer. Deck span of no feet carrying standard- 
 gauge single line. Braced on the Pratt system ; actual 
 weight 467 tons, weight by diagram 48 tons. 
 
 Double line, standard-gauge railway truss bridge at 
 Harrisburg, Pa. 21 spans of 156' 6" each. Weight of a 
 span by actual measurement 129 tons. Weight by diagram 
 119 tons. 
 
 Jhelum bridge, British India. Mr. Lee Smith engineer-in - 
 chief. Metre gauge. Single line and two footways. Deck 
 system with lattice bracing of single intersection, spans fifty 
 in number, 97' 6" long. Actual weight of one span 42-4 
 tons ; weight by diagram for single line 29 tons, for double 
 line 50 tons. Mean 39-5 tons. 
 
 IRON TRESTLE PIERS 
 
 Such wide differences of design exist in iron trestles that 
 it is almost impossible to form either a table or a diagram 
 of their weights. 
 
 One of the largest of such structures, the New Portage 
 Viaduct, of America, has towers of maximum height 203 feet. 
 It is built for double-line standard gauge, and carries spans 
 on either side of 118 feet. The weight of the trestle, con- 
 sisting of four columns, and bracing weighs about 1,400 Ibs. 
 per foot of vertical height. 
 
 Another celebrated trestle, the Kinzua Viaduct in 
 Pennsylvania, is built for single-line standard gauge. The 
 maximum height of any tower is 278 feet. The tower is 
 capped by girders of 38 feet span, and supports adjacent 
 spans of 6 1 feet. The weights vary from 500 to 700 Ibs. per 
 foot of vertical height. 
 
 On the Oak Orchard Viaduct of twenty-three 3o-feet 
 spans on single trestles, called ' bents,' consisting of a pair 
 of raking posts and bracing, carrying a single-line standard- 
 gauge railway, the weight of the bents per foot of vertical 
 height up to 75 feet was 160 Ibs. 
 
 s 
 
258 Preliminary Survey 
 
 STONE BRIDGES OF COURSED RUBBLE 
 
 The diagrams on Plate VII. for stone bridges are con- 
 structed from Mr. Trautwine's formulae. The arches are 
 semicircular. The depth of keystone is taken from the 
 formula 
 
 T^ , r, c t \/rad. + half span . f 
 
 Depth of keystone in feet= +-2 feet. 
 
 4 
 
 The thickness of the abutment is taken from the 
 formula 
 
 Thickness in feet at springing= rad ^-^ + ^1*-- + 2 ft . 
 
 
 
 The abutment is plumb on the face, and battered on 
 the back from the thickness at the springings found by the 
 formula to a thickness at ground level == the vertical height 
 from ground to springing, which batter is continued down 
 through the ground to the bottom of the foundations, 3 feet 
 below ground level, and footings added. 
 
 The spandrel walls are T 4 of their vertical height where 
 they join the wing walls and 2 feet 6 inches at cope. 
 
 The wing walls are also -^ of their vertical he-ight at 
 base, and diminishing to 2 feet 6 inches at cope. 
 
 BRICK BRIDGES 
 
 The formula for stone bridges will serve for brick bridges 
 by taking half the quantity for the wing-walls. The quan- 
 tities in the diagram agree with American practice for rubble 
 stone, but wing-walls of brick are not nearly so wasteful of 
 material. 
 
 The two following examples are type bridges on the 
 Eastern and Midlands Railway, England, and agree in the 
 main as to quantity with those of most other railways in 
 England for similar height, span, and width. 
 
 Example i. 2 5 -foot span over-bridge for double line ; 
 elliptical arch ; total height 21*5 fqet, width 18 feet; one 
 counterfort to each abutment. 
 
Pi-ATKVIII 
 
 300 
 
 200 
 
 100 
 
 H 
 
 HT: 60 
 
 4-1 
 
 I- -I 
 
 LJ 
 
 f 1- 
 
 H 
 
 M 
 
 t i 
 
 SO 
 
 TIMBER TRESTLES 
 QUANTITIES >N ONE BENT 
 A One, Cube, foot-1Zft$Jf. 
 
 LDl 
 
 r r 
 
 Jt 
 
 
 % 
 
 " 
 
 40 
 
 30 
 
 20 
 
 10 FEET 
 
 Nfffe.lf thb diagram is in frequent use with dividers, a piece of dull-back tracing 
 doth, gummed over it by the four corners, will protect it. 
 
Graphic Calculation for Preliminary Estimates 259 
 
 Actual Diagram (with half 
 measurement, quantity for wings), 
 cubic yards cubic yards 
 
 Arch, abuts., cforts., and parapets . 196-5 180 
 
 4 wings . ..... 147 '9 140 
 
 Spandrels . . . . . 9*3 41 
 
 3537 36i 
 
 Example 2, 1 5-foot span under- bridge ; segmental arch ; 
 total height 20 feet ; two counterforts to each abut ; width 
 23-5 feet. 
 
 Actual Diagram (with half 
 measurement, quantity for wings), 
 cubic yards cubic yards 
 
 Arch, abuts., cforts. , and ppts. . 200-0 194 
 
 4 wings 126-0 123-5 
 
 Spandrel walls . . . . 4*0 21 
 
 330-0 338-5 
 
 TIMBER TRESTLES 
 
 Plate VIII. gives quantities in one bent. Types i, 2, 
 and 3 are for a trestle carrying a single-line standard-gauge 
 railway on a pair of stringers 14" x 14". The bents 12 feet 
 apart centre to centre. The posts, sills, and caps 12" x 12". 
 The sway-braces and wales 12" x6". The quantities include 
 the longitudinal bracing but not the sleepers (cross-ties). 
 
 Type 4 is for one of a pair of bents forming a pier for a 
 Howe truss. Thus for a height of 45 feet the diagram 
 gives 445 cubic feet in one bent, therefore the pier would 
 contain 890 cubic feet. For narrow-gauge railways three- 
 quarters of the quantities may be taken. 
 
 A Howe truss is a composite bridge commonly used in 
 America, resembling in appearance the lattice-girder in 
 England. The top and bottom chords and the diagonal 
 bracing are of timber, and the stresses are distributed through 
 the members to the piers by vertical tension-rods. 
 
 The following table is an extract from Trautwine's 'Pocket 
 Book,' which, in an excellent article upon trusses, fully 
 
 s 2 
 
260 
 
 Preliminary Sun>ey 
 
 TABLE XXXII. Howe Trusses of limber and iron. Weights 
 approximately equal to iron trusses of same span. 
 
 fi 
 
 
 
 An 
 upper- 
 chord 
 
 A lower 
 chord 
 
 An end 
 brace 
 
 A centre 
 brace 
 
 A 
 
 counter 
 
 End 
 rod 
 
 Centre 
 rod 
 
 a i 
 
 rt 
 
 w 
 
 
 to 
 
 
 V 
 
 
 M 
 
 
 n 
 
 
 ' 
 
 
 
 
 
 " I 
 
 g 
 
 "o 
 
 .0 
 
 o.u 
 
 1 
 
 <s 
 
 | 
 
 -oj 
 
 | 
 
 1) 
 O 1* 
 
 | 
 
 ol 
 
 1 
 
 ss 
 
 | 
 
 W 
 
 
 
 
 Q 
 
 O 
 
 '^"y 
 
 1 
 
 II 
 
 o 
 
 u a 
 
 J-s 
 
 s 
 
 I* 
 
 "8 
 
 II 
 
 *8 
 
 Jr 
 
 XTJ 
 
 "s 
 
 |x 
 
 
 
 
 
 
 
 o 
 
 c3 
 
 Q 
 
 
 o 
 
 oS 
 
 O 
 
 JS 
 
 o 
 
 rt 
 
 o 
 
 rt 
 
 
 
 
 J5 
 
 w 
 
 
 1) 
 
 J^ 
 
 1) 
 
 fc 
 
 1> 
 
 fc 
 
 v z 
 
 4) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 ft. 
 
 ft. 
 
 
 
 ins. 
 
 
 ins. 
 
 
 ins. 
 
 
 ins. 
 
 
 ins. 
 
 
 ins. 
 
 
 ins. 
 
 25 
 
 go 
 
 6 
 
 8 
 g 
 
 3 
 
 SX 6 
 6x9 
 
 3 
 3 
 
 5x12 
 6 x 14 
 
 2 
 
 5X 8 
 6x9 
 
 2 
 
 
 I 
 
 5 x 6 
 
 2 
 
 1 
 
 2 
 
 
 7S 
 
 12 
 
 10 
 
 
 6x 12 
 
 
 6x14 
 
 2 
 
 6xn 
 
 2 
 
 6x 8 
 
 I 
 
 6x 8 
 
 '2 
 
 } 
 
 2 
 
 i| 
 
 100 
 
 15 
 
 ii 
 
 3 
 
 6x14 
 
 3 
 
 6x16 
 
 2 
 
 8X12 
 
 2 
 
 6xio 
 
 I 
 
 6 X 10 i 2 
 
 2! 
 
 2 
 
 
 describes how the truss should be built, and gives the theory 
 of static equilibrium in so simple a manner that for this, or 
 for any ordinary structure of the kind, those who have never 
 studied mathematics can easily work out the proper dimen- 
 sions for any intermediate size. Composite bridges are also 
 largely made of the N type, having the uprights of timber 
 and oblique tension-rods. The total quantity of material is 
 somewhat less with this form of bridge, but there is more 
 ironwork in it. High timber viaducts are not much built 
 now for several reasons, but mainly on account of their 
 inflammability. 
 
 Howe trusses up to 50 feet span can be placed upon 
 timber trestle piers, but if the latter are higher than 60 feet 
 they require three bents to a pier, like the old viaduct at 
 Portage, New York State, which was 234 feet high from bed 
 of river to rail and contained 133,000 cubic feet of timber 
 including trusses. The quantity curve increases very rapidly 
 for trestles over the heights in the diagram. The maximum 
 length of bents on the Portage Viaduct was 190 feet. 
 Taking a proportion of x ^ x 595 (the quantity in the 
 diagram for a bent 60 feet high), multiplying by 3, and 
 allowing for timber in the trusses, the actual quantity would 
 
Graphic Calculation for Preliminary Estimates 261 
 
 be about 2 i times more than the diagram. It is so very 
 rare that timber trestles are built above 60 feet nowadays 
 that it would be needless to extend the diagram. 
 
 0) 
 
 a 
 col 
 
 o 
 
 = 
 
 * 1 S I 
 
 i/) in i/i i/} 
 
 e g 
 
 n 
 
 K 
 
 c 
 
 K 
 
 xK 
 
 I' 
 
 v l^ 
 
 a 
 
 1^ 5 
 
 x 
 
 x 
 
 x 
 
 U3 
 
 SERVICE DIAGRAMS 
 
 The service diagrams Figs. 82 and 83, for tramways and 
 railways working single line with crossing stations, are 
 
262 
 
 Preliminary Survey 
 
 IATV01 
 
 (/) IN Vfl 
 
 >""" i -WV9 
 UJ 
 
 *d 
 
 UJ 
 
 NOUV1S< 
 
 
 
 
 
 
 
 
 
 
 x 
 
 
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 2 
 
 
 z 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 \ 
 
 
 
 
 2 
 
 
 / 
 
 
 
 
 
 
 x 
 
 
 
 
 7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 
 
 
 
 
 \ 
 
 
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 ^ 
 
 
 
 
 
 
 
 
 
 \ 
 
 
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 2 
 
 
 
 
 
 
 
 
 
 
 
 /v^> 
 
 
 
 
 
 
 
 
 
 
 
 
 /O^ 
 
 
 
 
 
 
 
 
 
 
 
 
 @y 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 q_ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 ^, 
 
 
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 , 
 
 
 
 
 
 ^.> 
 
 ^ / 
 
 
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 ^ 
 
 
 
 
 
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 s 
 
 
 
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 ^'*/ 
 
 
 
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 ^>^ 
 
 
 
 
 
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 ^/^ 
 
 
 
 / 
 
 
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 \^ 
 
 
 
 'oN 
 
 
 / 
 
 
 2 
 
 
 
 ^2 
 
 
 ^v^ 
 
 
 
 
 
 
 N 
 
 
 
 ^. 
 
 
 
 *'4^ 
 
 ^^ 
 
 
 \ 
 
 \^ 
 
 
 V 
 
 
 
 4^ 
 
 
 
 
 ^V 
 
 
 \ 
 
 "-t-\ 
 
 $. 
 
 
 
 
 
 
 
 
 
 ^*+ 
 
 
 \ 
 
 
 
 NX 
 
 
 
 
 
 
 
 
 \, 
 
 
 / 
 
 
 ^.x 
 
 
 
 
 
 
 
 
 s 
 
 \\^ 
 
 / 
 
 
 / 
 
 
 
 
 
 
 
 
 
 & ^ 
 
 / 
 
 ,x 
 
 
 
 
 
 
 
 
 
 
 S 
 
 // 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 / / 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 </ 
 
 
 
 
 
 
 
 
 
 
 
 V s 
 
 * 2 
 
 IT) 
 
 to 
 
 tn 
 
 
 
 
 
 m 
 
 
 CM 
 
 
 
 o 
 
 
 
 in 
 
 CO 
 
 
 
 b 
 
 u. 
 
 o 
 
 X 
 
 - 
 
 -) 
 
 * 
 
 .J 
 
 2 
 
 Z 
 
 
 
 o 
 
 N 
 
 in 
 
 CO 
 
 
 
 * 
 
 o 
 
 in 
 
 s 
 
 iT) 
 
 in 
 eO 
 
 a 
 
 c 
 
 in 
 
 o 
 
 fO 
 
Graphic Calculation for Preliminary Estimates 263 
 
 necessarily to a small scale. They should be drawn for 
 working purposes on a sheet of double elephant, which will 
 include the 24 hours for railway and the working hours for 
 tramways. The value of the diagrams is that they show at a 
 glance where the trains meet and pass, so that a special train 
 can be interpolated at any time with no danger of mistaking 
 its connections. 
 
 English railways worked upon the block system with the 
 double line, or with the staff or tablet on single line, need 
 nothing of this kind, but pioneer railways are generally 
 worked merely by telegraph or even telephone, and the 
 diagram forms an indispensable adjunct. It also serves the 
 purpose with tramways of showing the number of cars 
 required to work a certain service. 
 
 In diagram Fig. 82 we will explain the first round trip 
 of No. i tramway car. 
 
 Leave depot 6 A.M. ; run through to terminus with 
 line clear ; arrive terminus 6.55 ; leave terminus 7.00 ; 
 cross No. 2 car on No. 1 1 switch ; cross No. 3 car on No. 
 10 switch, No. 4 car on No. 9 switch, and so on ; arrive 
 depot 7.55 ; leave again at 8. 
 
 In diagram Fig. 83 we will follow No. i freight, whose 
 waybill will be as follows. Leave N at 3 A.M. ; switch off 
 at M to allow No. 2 express to pass ; leave M at 4.10 ; 
 switch off at L for No. 3 ordinary ; leave L at 5.15 ; switch 
 at K for No. 100 express ; leave K at 6.05, switch at J for 
 No. x special ; leave J at 6.45 ; switch at I for No. 101 
 freight and No. 102 ordinary ; leave I at 8.10 ; switch at H 
 for shunting from 8.25 to 8.40 ; ditto at G from 9 to 9.12 
 and at F from 9.40 to 9.45. 
 
 CENTRIFUGAL FORCE 
 
 The following rules are prepared on the assumption of 
 gravity being 32-2 feet per second, and R the radius of 
 rotation ; they are sufficiently approximate when the thick- 
 
264 Preliminary Survey 
 
 ness of the body such as the rim of a flywheel is not 
 more than ^ of the radius from out to out ; the radius being 
 measured to the centre of gravity of the rim. 
 
 Let F= centrifugal force in Ibs. per Ib. weight of rotat- 
 ing body. 
 
 Let F' = centrifugal force in Ibs. per ton of rotating 
 body. 
 
 Let R=radius of rotation in feet. 
 
 Let N=number of revolutions per minute. 
 
 Let M=number of revolutions per second. 
 
 F=-ooo 3 4 RN 2 ...... (i) 
 
 F=i-224 RM 2 ...... (2) 
 
 F=o'76i6 RN 2 ..... (3) 
 
 Example. What is the centrifugal force in Ibs. of a 
 body making 120 revolutions per minute, at a radius of 
 5 feet, and weighing 3 Ibs. ? 
 
 F=i'224X5 X2 2 =24'49 Ibs. per Ib. 
 or for 3 Ibs. = 73 -47 Ibs. 
 
 Formulae suitable for side-stresses on viaducts due to 
 centrifugal force : 
 
 Let V=velocity in feet per second. 
 
 Let VV=velocity in miles per hour. 
 
 Let F=centrifugal force in Ibs. per ton of 2,240 Ibs. 
 
 Let FF= centrifugal force in Ibs. per ton of 2,000 Ibs. 
 
 Let R= radius in feet. 
 
 Let RR=radius in Gunters chains of 66 feet. 
 
 F=6 9 -5-^ 2 (4) 
 
 FF=62-iT- . (5) 
 
 VV 2 
 F 68 . .(6) 
 
 F= I49 . 7 - - (7) 
 
G rap] lie Calculation for Preliminary Estimates 265 
 
 FF=2*o25 ... . (8) 
 
 FF=i 33 *6 X^ 2 . . (9) 
 
 Example i. What is the centrifugal force in Ibs. per ton 
 of 2,240 Ibs. on a curve of 23-5 chains at a velocity of 31*3 
 miles per hour ? 
 
 By slide-rule. Place the brass marker opposite 3 1 -3 on the 
 lower scale of the rule, and bring a 23*5 of the upper scale 
 of the slide to coincide with the brass marker so placed. 
 Read off the result 94*5 feet per ton on the upper scale 
 of the rule opposite to the 2*268 of the slide. 
 
 Example 2. What is the centrifugal force in Ibs. per ton 
 of 2,000 Ibs. at 60 miles per hour on a curve of 4 chains, 33 
 links? 
 
 Place the brass marker opposite 60 on the lower scale of 
 the rule. Bring a 4*33 of the upper scale of the slide to 
 coincide with the brass marker so placed. Read off the 
 result, i, 68 1, on the upper scale of the rule opposite to the 
 2*025 of the slide. 
 
 Example 3. What is the centrifugal force in Ibs. per ton 
 of 2,240 Ibs. at 40 miles per hour on a curve of 5 (1,146 feet 
 radius)? 
 
 40 2 
 
 By No. 7, F=i49*7 x .2209 Ibs. 
 1 146 
 
 For table of certain values of F at different curvatures 
 and speeds, see Tables VII. and VIII., p. 15. 
 
 To REDUCE THERMOMETER SCALES 
 Fahrenheit and Centigrade. 
 
 Boiling-point is at 100 Cent., and 212 Fahr. Zero 
 Cent, corresponds with 32 Fahr., whence 180 Fahr.= 
 100 Cent. 
 
 Place the left hand i of the upper scale of the slide 
 
266 Preliminary Survey 
 
 under the i '8 of the rule ; then the slide will give Centigrade 
 and the rule Fahrenheit, when the constant 32 is added to 
 the latter. 
 
 Thus -i Cent. = -18 + 32 = 32-18 Fahr. 
 iCent.= i-8 + 32=33'8 
 5 Cent.= 9-0 f 32=41-0 
 25 Cent.=45-o-f 32=77 
 100 Cent. = i8o + 32 =212 
 250 Cent. = 450 + 3 2 =482 
 35 Fahr. (32 + 3) =i '66 Cent. 
 300 Fahr. (32 + 268) -149 Cent. 
 
 or by rule C= 5 ( F ~3 2 ). 
 
 9 
 
 Below freezing-point, the values Fahr. read on the rule 
 are deducted from 32. 
 
 Thus i Cent.=32- i -8=30-2 Fahr. 
 -5 =32-9 =23 
 
 Fahrenheit and Reaumur. 
 
 Boiling-point is at 80 Re. and 212 Fahr. 
 
 Zero Re. corresponds with 32 Fahr. 
 
 Place the left hand 8 of the upper scale of the slide 
 under the i'8 of the rule, and read Re. on the slide and 
 Fahr. on the rule, adding the constant 32. 
 
 iRe.=32+ 2-25 = 34-25 Fahr. 
 
 25 Re. 
 
 or by rule F=-9 + 32. 
 4 
 
 Centigrade and Reaumur. 
 
 Zero coincides in these scales, so there is no constant. 
 Place the 8 of the slide under a i of the rule, and read 
 Re. on the slide, and Cent, on the rule. 
 
Graphic Calculation for Preliminary Estimates 267 
 
 Thus i Re. = 1-2 5 Cent. 
 2-4 Re. = 3'oo 
 100 Re. = i25 
 
 or by rule C = 5 R. 
 
 THE SLIDE-RULE AS A UNIVERSAL MEASURER 
 
 Circular Measure of Angles. 
 
 Two movements of the slide are all that is needed for 
 the length of any circular arc to any radius. 
 
 An arc of i to a radius of 100=1745, that is to say is 
 i -745 per cent, of the radius whatever that radius may be. 
 An arc of 10 is 17 -45 per cent, and so on. 
 
 (a) To obtain the value of any arc in percentage of the 
 radius. Place the left hand i of the slide opposite to the 
 1,745 of the rule; then opposite to the angle in degrees and 
 decimals on the slide will be given the percentage on the rule. 
 
 Example. What is the percentage of an arc of 22 30', 
 i.e. 22*5? Placing the slide as directed we find opposite 
 the 22 '5 upon it 39-2, which is the required percentage. 
 
 (b) To obtain the linear value of the same arc to any 
 given radius. Leave the brass marker at the percentage 
 found on the rule so as not to lose it, and move the slide 
 until a i upon it coincides with the given radius. Then 
 move the marker to the percentage on the slide and read off 
 the linear value upon the rule. 
 
 Example. With the 39*2 per cent, found as above, re- 
 quired the linear value of the said arc to a radius of 
 234 feet. 
 
 Placing the slide as directed with the left hand i oppo- 
 site to 234 on the rule, move the marker to 39*2 on the slide 
 and read off the result, 91*7, on the rule. 
 
 Circumferences of circles. Multiply 3 1 4 1 6 by the diameter, 
 or 6-2832 by the radius. When the argument is given in 
 twelfths or sixteenths or sixtieths reduce them to decimals by 
 
268 Preliminary Survey 
 
 Tables XXXIV., XXXIX., and XL., respectively, or by 
 the slide-rule as explained further on. 
 
 (d) Areas of circles. Multiply the square of the diameter 
 by 7854, or the square of the radius by 3*142. 
 
 (e) Surface of cylinders. Multiply the circumference 
 found as above by the length. 
 
 (f) Volume of cylinders. Multiply the area found as 
 above by the length. 
 
 (g) Surface of spheres. Multiply the square of the dia- 
 meter by 3'i4i6. 
 
 Volume of spheres=. diameter 3 x '5236. 
 
 (h) To reduce inches to decimals of a foot. Place 1 2 on 
 the slide opposite to 10 on the rule, and read the decimals 
 on the rule opposite the inches on the slide. 
 
 (i) To reduce fractions of inches to decimals of a foot 
 Place the denominator on the slide opposite the 833 on the 
 rule, which will be regarded as "0833, an d then read off the 
 decimals on the rule opposite to the numerator on the 
 slide. Thus, to express f of an inch in decimals of a foot, 
 place the 32 of the slide opposite the 833 on the rule, which 
 represents the decimal for one inch '0833 ; then 2 1 on the 
 slide will be found opposite to '0547 on the rule. 
 
 (j ) When the fraction is a mixed one of inches and fractions 
 of inches. Find the decimal of the integer by the first pro- 
 cess, and keeping the result on record with the brass marker 
 find the decimal of the fraction by the second process, and 
 add the two results together. 
 
 (k) To reduce fractions of inches to decimals of inches. 
 Place the denominator on the slide opposite a i of the rule 
 and read the decimals on the rule opposite the numerators 
 on the slide. 
 
 (1) To reduce feet to metres or vice versa. Place a i of 
 the slide, representing 100 feet, under the 30*48 of the 
 rule and read off feet on the slide and metres on the rule. 
 
 (m) To reduce yards into metres or vice versa. Place a 
 i of the slide, representing 100 yards, opposite to 91*44 on 
 
Grapliic Calculation for Preliminary Estimates 269 
 
 the rule ; read off yards on the slide and metres on the 
 rule. 
 
 (n) To reduce inches and decimals into millimetres or vice 
 versa. Place a i of the slide to represent one inch opposite 
 to 2 5 '4 on the rule ; read inches on the slide and milli- 
 metres on the rule. 
 
 (o) To reduce kilometres into statute miles and vice versa. 
 Place a i of the slide, to represent one statute mile, opposite 
 i '609 on the rule, and read miles on the slide and kilo- 
 metres on the rule. 
 
 (p) To reduce kilometres into geographical miles and vice 
 versa. Place a i of the slide, to represent one geographical 
 mile, opposite 1*853 on the rule, and read geographical miles 
 on the slide and kilometres on the rule. 
 
 (q) To reduce statute miles to geographical miles and vice 
 versa. Place a i of the slide, to represent one geographical 
 mile, opposite to 1*151 on the rule, and read geographical 
 miles on the slide and statute miles on the rule. 
 
 I square centimetre . = O'JSS square inch 
 
 i ,, metre . . . 107641 ,, feet 
 I ,, ,, 1-19601 ,, yard 
 
 i square kilometre . . = 247*11 acres 
 
 ,, ,, . = 0-38611 square miles 
 
 Metric Weights. ' 
 
 i centigramme . . . - 0-15432 grain 
 i gramme . . . . i5'43 2 
 
 i kilogramme . . 2-2046 pounds 
 
 i tonne .... =2204-6 ,, 
 
 ,,.... 0-9842 ton 
 
 Metric Ctibic Measure. 
 
 i decalitre = 0-35316 c. ft = 2 -2009 British gals. 
 
 ,, =0-28378 U. S. struck bushel = 2 -64 1 79 U. S. liquid gals, 
 
 1 See also Specific Gravity, pp. 389, 390. 
 
TABLE XXXIII. For converting Inches into Decimals of a Foot. 
 
 In. 
 
 Feet 
 
 In. 
 
 Feet 
 
 In. 
 
 Feet 
 
 In. 
 
 Feet 
 
 
 
 0000 
 
 3 
 
 2500 
 
 6 
 
 5000 
 
 9 
 
 7500 
 
 1 
 
 Ti! 
 
 0052 
 
 
 2552 
 
 
 5052 
 
 
 7552 
 
 1 
 
 OIO4 
 
 
 
 2604 
 
 s' 
 
 5104 
 
 i 
 
 7604 
 
 
 0156 
 
 
 2656 
 
 1 
 
 5156 
 
 
 7656 
 
 j. 
 
 0208 
 
 i 
 
 4 
 
 2708 
 
 
 5208 
 
 i 
 
 4 
 
 7708 
 
 
 0260 
 
 
 2760 
 
 
 5260 
 
 
 7760 
 
 f 
 
 0313 
 
 | 
 
 2813 
 
 1 
 
 5313 
 
 f 
 
 7813 
 
 8 
 
 0365 
 
 
 2865 
 
 
 5365 
 
 
 7865 
 
 I 
 
 0417 
 
 i 
 
 2917 
 
 a 
 
 5417 
 
 \ 
 
 7917 
 
 
 0469 
 
 
 2969 
 
 
 5469 
 
 
 7969 
 
 | 
 
 0521 
 
 1 
 
 3021 
 
 | 
 
 5521 
 
 1 
 
 8021 
 
 
 0573 
 
 
 3073 
 
 
 5573 
 
 
 8073 
 
 3 
 
 0625 
 
 3. 
 4 
 
 3125 
 
 
 5625 
 
 3. 
 4 
 
 8125 
 
 
 0677 
 
 
 3177 
 
 
 5677 
 
 
 8177 
 
 T_ 
 
 0729 
 
 I 
 
 3229 
 
 1 
 
 8 
 
 5729 
 
 1 
 8 
 
 8229 
 
 8 
 
 0781 
 
 
 3281 
 
 
 578i 
 
 
 8281 
 
 I 
 
 0833 
 
 4 
 
 '3333 
 
 7 
 
 5833 
 
 10 
 
 8333 
 
 
 0885 
 
 
 3385 
 
 
 5885 
 
 
 8385 
 
 1 
 
 0938 
 
 * 
 
 3438 
 
 
 
 '5937 
 
 ^ 
 
 8437 
 
 
 0990 
 
 
 3490 
 
 
 5990 
 
 
 8490 
 
 J, 
 
 1042 
 
 i 
 
 3542 
 
 i 
 
 6042 
 
 1 
 
 4 
 
 8542 
 
 4 
 
 1094 
 
 
 '3594 
 
 
 6094 
 
 
 8594 
 
 3 
 
 1146 
 
 L 
 
 3646 
 
 f 
 
 6146 
 
 jj- 
 
 8646 
 
 8 
 
 1198 
 
 
 3698 
 
 
 6198 
 
 
 8698 
 
 1 
 
 1250 
 
 1 
 
 2 
 
 3750 
 
 i 
 
 6250 
 
 -| 
 
 8750 
 
 2 
 
 1302 
 
 
 3802 
 
 
 6302 
 
 
 8802 
 
 5 
 
 1354 
 
 I 
 
 3854 
 
 1 
 
 6354 
 
 5 
 
 8 
 
 8^54 
 
 8 
 
 1406 
 
 
 3906 
 
 
 6406 
 
 
 8906 
 
 _3_ 
 
 1458 
 
 f 
 
 3958 
 
 3. 
 
 4 
 
 6458 
 
 3 
 4 
 
 8958 
 
 
 1510 
 
 
 4010 
 
 
 6510 
 
 
 9010 
 
 7 
 
 1563 
 
 i 
 
 4063 
 
 | 
 
 6563 
 
 1 
 8 
 
 9063 
 
 
 1615 
 
 
 4115 
 
 
 6615 
 
 
 91 15 
 
 2 
 
 1667 
 
 5 
 
 4167 
 
 8 
 
 6667 
 
 11 
 
 9167 
 
 
 1719 
 
 
 4219 
 
 
 6719 
 
 
 9219 
 
 JL 
 
 1771 
 
 l 
 
 4271 
 
 i 
 
 6771 
 
 ^> 
 
 9271 
 
 8 
 
 1823 
 
 
 4323 
 
 
 6823 
 
 
 93 2 3 
 
 I 
 
 1875 
 
 i 
 
 '4375 
 
 
 
 6875 
 
 5- 
 
 '9375 
 
 
 1927 
 
 
 4427 
 
 
 6927 
 
 
 9427 
 
 f 
 
 1979 
 
 1 
 
 '4479 
 
 f 
 
 6979 
 
 3 
 f 
 
 9479 
 
 
 2031 
 
 
 "453 * 
 
 
 7031 
 
 
 953 1 
 
 1 
 
 2083 
 
 A 
 
 4583 
 
 i 
 2 
 
 7083 
 
 i 
 
 9583 
 
 3 
 
 2135 
 
 
 4635 
 
 
 7135 
 
 
 9635 
 
 | 
 
 2188 
 
 1 
 
 4688 
 
 1 
 
 7187 
 
 5 
 
 8 
 
 9688 
 
 
 2240 
 
 
 4740 
 
 
 7240 
 
 
 "9740 
 
 3 
 
 2292 
 
 f 
 
 4792 
 
 f 
 
 7292 
 
 t 
 
 9792 
 
 4 
 
 2344 
 
 
 4844 
 
 
 7344 
 
 
 9844 
 
 _7 
 
 2396 
 
 1 
 
 4896 
 
 1 
 
 7396 
 
 2- 
 
 9896 
 
 S 
 
 2448 
 
 
 4948 
 
 
 7448 
 
 ' 
 
 9948 
 
 
 
 
 
 
 
 
 
Graphic Calculation for Preliminary Estimates 271 
 
 TABLE XXXIV. Time, Coinage ', and Measurement in Decimals. 
 
 Pounds sterling 
 
 00466 
 00833 
 01250 
 01666 
 02083 
 02500 
 02916 
 
 03333 
 03750 
 04166 
 04583 
 
 TABLE XXXV. Shillings in Decimals of a Pound. 
 
 Pence 
 Months 
 Inches 
 
 Shillings 
 Years 
 Feet 
 
 Yards 
 
 I 
 
 08 33 
 
 0277 
 
 2 
 
 1667 
 
 0554 
 
 3 
 
 2500 
 
 08 33 
 
 4 
 
 3300 
 
 mi 
 
 5 
 
 4167 
 
 I 3 8 9 
 
 6 
 
 5000 
 
 1666 
 
 7 
 
 5833 
 
 1944 
 
 8 
 
 6667 
 
 '2222 
 
 9 
 
 7500 
 
 25OO 
 
 10 
 
 8333 
 
 2777 
 
 n 
 
 9167 
 
 3056 
 
 Shil- 
 lings 
 
 Pounds 
 sterling 
 
 Shil- 
 lings 
 
 Pounds 
 
 sterling 
 
 Shil- 
 lings 
 
 Pounc 
 sterlin 
 
 I 
 
 05 
 
 6 i -30 
 
 II 
 
 55 
 
 2 
 
 io 7 
 
 35 
 
 12 
 
 60 
 
 3 
 
 15 
 
 40 
 
 13 
 
 6 5 
 
 4 
 
 20 
 
 9 
 
 45 
 
 U 
 
 70 
 
 5 
 
 25 
 
 10 
 
 50 
 
 15 
 
 75 
 
 Pounds 
 sterling 
 
 Shil- 
 lings 
 
 Pounds 
 sterling 
 
 55 
 
 16 
 
 80 
 
 60 
 
 17 
 
 85 
 
 65 
 
 18 
 
 90 
 
 70 
 
 19 
 
 '95 
 
 75 
 
 20 
 
 I'OO 
 
 TABLE XXXVI. --Days, Hours, Minutes, and Seconds. 
 
 \ *T- 
 
 Minutes 
 
 Hours 
 
 Days 
 
 Minutes 
 
 Hours 
 
 Days 
 
 Seconds 
 
 Minutes 
 
 
 Seconds 
 
 Minutes 
 
 
 I 
 
 01667 
 
 000694 
 
 6 
 
 10000 
 
 004166 
 
 2 
 
 03333 
 
 001389 
 
 7 
 
 II667 
 
 004861 
 
 3 
 
 O5OOO 
 
 002083 
 
 8 
 
 13333 
 
 005555 
 
 4 
 
 06667 
 
 002777 
 
 9 
 
 15000 
 
 006250 
 
 i 5 
 
 08333 
 
 003472 
 
 10 
 
 16667 
 
 006940 
 
 TABLE XXXVII. Weeks, Months, and Years. 
 
 Weeks 
 
 Months 
 
 Years 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 23077 
 46154 
 69231 
 92308 
 , I-I5380 
 
 01923 
 03846 
 05769 
 07692 
 09615 
 
 Years 
 
 Weeks 
 
 Months Years 
 
 01923 
 
 6 
 
 I -3846 
 
 H538 
 
 03846 7 
 
 1-6154 
 
 13461 
 
 05769 8 
 
 I '8462 
 
 15384 
 
 07692 9 
 
 2-0769 
 
 17307 
 
 09615 
 
 IO 
 
 2-3077 
 
 19230 
 
2/2 
 
 Preliminary Survey 
 
 TABLE XXXVIII. Days, Weeks, Months, and Years. 
 
 Days 
 
 Weeks 
 
 Months Years 
 
 Days 
 
 Weeks Months Years 
 
 ! 
 
 1429 
 
 0329 -00274 
 
 6 
 
 8572 -1973 -01644 
 
 2 
 
 2857 
 
 0657 -00548 j! 7 
 
 i -oooo -2302 -01918 
 
 3 
 
 4286 
 
 0986 : -00822 
 
 8 
 
 1-1429 ! -2630 -02192 
 
 4 
 
 57H 
 
 1315 -01096 9 
 
 1-2857 '2959 -02466 
 
 5 
 
 7H3 
 
 l644 ; -0X370 IO I-429O ^290 ^2740 
 
 TABLE XXXIX. For converting Fractions into Decimals. 
 
 h 
 
 0156 
 
 
 H -2656 I! 1 
 
 5156 H 
 
 7656 
 
 i 
 
 32 
 
 0312 
 
 
 9 
 
 3 '>: 
 
 2812 i i| 
 
 5312 
 
 25 
 
 78l2 
 
 3 
 
 64 
 
 0469 
 
 
 15 
 
 64 
 
 2969 
 
 35 
 64 
 
 5469 
 
 
 7969 
 
 A 
 
 0625 
 
 
 I? 
 
 3125 
 
 9 
 
 5625 
 
 
 8l25 
 
 8 
 
 0781 
 
 
 
 3281 
 
 II 
 
 
 i! 
 
 828l 
 
 ^> 
 
 0937 
 
 
 
 3437 
 
 j 1| 
 
 '5937 
 
 ?_1 
 
 8437 
 
 64 
 
 1094 
 
 
 
 3594 
 
 i || 
 
 6094 
 
 || 
 
 8594 
 
 1 
 "8 
 
 I25O 
 
 
 3 
 
 3750 
 
 5 
 
 6250 
 
 
 
 8750 
 
 
 1406 
 
 
 25 
 R4 
 
 3906 
 
 41 
 
 9406 
 
 57 
 
 8906 
 
 3% 
 
 1562 
 
 
 
 4062 
 
 1 
 
 6562 
 
 II 
 
 9062 
 
 il 
 
 1719 
 
 
 || 
 
 4219 
 
 
 6719 
 
 64 
 
 9219 
 
 7? 
 
 1875 
 
 
 t? 
 
 4375 
 
 ii 
 
 6875 
 
 If 
 
 '9375 
 
 13 
 
 2031 
 
 
 29 
 64 
 
 453i j It 
 
 7031 
 
 |1 
 
 9531 
 
 32 
 
 2l87 
 
 
 if 
 
 4687 
 
 7187 
 
 |i 
 
 9 687 
 
 15 
 64 
 
 2344 
 
 
 II 
 
 4844 i II 
 
 7344 
 
 ll 
 
 9844 
 
 i 
 
 2500 
 
 
 
 5000 
 
 I 
 
 7500 
 
 I 
 
 i-oboo 
 
 TABLE "XL. For converting Minutes into Decimals of a Degree, or 
 Seconds into Decimals of a Minute (^,). 
 
 Min. 
 
 Degree Min. 
 
 Degree j Min. 
 
 Degree 
 
 Min. Degree 
 
 Sec. 
 
 Minute 
 
 Sec. 
 
 Minute 
 
 Sec. 
 
 Minute 
 
 Sec. Minute 
 
 I 
 
 0167 
 
 16 
 
 2667 
 
 31 
 
 5167 
 
 46 -7667 
 
 2 
 
 0333 
 
 17 
 
 2833 
 
 32 
 
 '5333 
 
 47 7833 
 
 3 
 
 O5OO 
 
 18 -3000 
 
 33 
 
 55oo 
 
 48 
 
 8000 
 
 4 
 
 0667 
 
 19 
 
 3167 
 
 34 
 
 5667 
 
 49 
 
 8l67 
 
 5 
 
 0833 
 
 20 
 
 '3333 
 
 35 
 
 5833 
 
 50 
 
 8333 
 
 6 
 
 IOOO 
 
 21 
 
 35oo 
 
 36 -6000 
 
 5i 
 
 8500 
 
 7 
 
 1167 
 
 22 
 
 3667 
 
 37 
 
 6167 
 
 52 
 
 8667 
 
 8 
 
 1333 
 
 23 
 
 3833 
 
 38 
 
 6333 
 
 53 
 
 8833 
 
 9 
 
 1500 
 
 24 
 
 4000 
 
 39 
 
 6500 
 
 54 
 
 9OOO 
 
 i Io 
 
 1667 
 
 25 
 
 4167 
 
 40 
 
 6667 
 
 55 
 
 9167 
 
 ii 
 
 1833 
 
 26 
 
 4333 
 
 4i 
 
 6833 | 56 
 
 '9333 
 
 12 
 
 2OOO 
 
 27 
 
 4500 
 
 42 
 
 7000 | 57 
 
 9500 
 
 13 
 
 2167 
 
 28 
 
 4667 
 
 43 
 
 7167 1 58 
 
 9667 
 
 14 
 
 2333 
 
 29 
 
 4833 
 
 44 
 
 7333 
 
 59 
 
 9833 
 
 15 
 
 2500 
 
 30 
 
 5000 
 
 45 
 
 7500 
 
 60 
 
 I'OOOO 
 
Graphic Calculation for Preliminary Estimates 273 
 
 TABLE XLI. For converting Seconds into Decimals of a Degree^ or 
 Thirds into Decimals of a Minute (3^0) 
 
 Sec. Deg. ' 
 
 Sec. 
 
 Deg. 
 
 Sec. Deg. ! Sec. 
 
 Deg. 
 
 Sec. i Deg. Sec. 
 
 Dee. 
 
 Thds. 
 
 Min. 
 
 Thds. 
 
 Min. 
 
 Thds. 
 
 Min. ! 
 
 Thds. 
 
 Min. ; 
 
 Thds. 1 Min. 
 
 l 
 
 Thds. 
 
 Min. 
 
 
 
 
 
 
 1 
 
 '0086 ' 
 
 
 !. 
 
 2 
 
 "0005 
 
 12 
 
 0033 
 
 22 
 
 0061 !| 32 
 
 0089 
 
 42 '0117 
 
 . 
 
 'oi44 
 
 3 
 
 0008 
 
 13 
 
 '0036 
 
 , 2 3 
 
 '0064 ' 33 
 
 0092 
 
 43 ' OII 9 
 
 53 
 
 0147 
 
 4 
 
 "OOII 
 
 14 
 
 0039 
 
 1 24 
 
 0067 i 34 
 
 0094 ! 
 
 44 '0122 
 
 54 
 
 '0150 
 
 5 
 
 0014 
 
 15 
 
 '0042 
 
 25 
 
 0069 35 
 
 0097 ( 
 
 45 '0125 
 
 55 
 
 '0153 
 
 6 
 
 0017 
 
 16 
 
 0044 
 
 26 "0072 36 
 
 "oioo 
 
 46 "0128 
 
 : 56 
 
 '0156 
 
 7 
 
 "0019 ' 
 
 17 
 
 0047 
 
 27 
 
 0075 
 
 37 
 
 '0103 
 
 47 '0130 
 
 ; 57 
 
 0158 
 
 8 
 
 "0022 
 
 18 
 
 '0050 
 
 28 
 
 0078 ,! 38 
 
 "0106 
 
 48 '0133 
 
 i 58 
 
 '0161 
 
 9 
 
 '0025 
 
 19 
 
 '0053 
 
 29 
 
 0080 
 
 39 
 
 0108 
 
 49 '0136 
 
 ! 59 
 
 0164 
 
 10 
 
 0028 
 
 20 
 
 '0055 
 
 30 | -0083 : 40 
 
 "Oil I 
 
 50 '0139 
 
 60 
 
 '0167 
 
 Example. What is 17 n' 29" 47"' in decimals of a degree? 
 47 "'=0130'= -0002 
 29" = -0080 
 
 ii' = -1833 
 
 17 
 
 TABLE XLII. Decimals of a Degree in Minutes and Seconds. 
 
 01 
 
 o' 36" 
 
 '21 
 
 12' 36" 
 
 i '4* 
 
 24' 36" 
 
 61 
 
 36' 36" 
 
 8 1 
 
 48' 36'- 
 
 '02 
 
 i' 12" 
 
 '22 
 
 13' 12" 
 
 42 
 
 25' 12" 
 
 '62 
 
 37' 12" 
 
 82 
 
 49' J2" 
 
 03 
 
 i' 48" 
 
 ' 2 3 
 
 13' 48" 
 
 '43 
 
 25' 48" 
 
 '63 
 
 37' 48" 
 
 '83 
 
 49' 48" 
 
 04 
 
 2' 24" 
 
 '24 
 
 14' 24" 
 
 1 -44 
 
 26' 24" 
 
 64 
 
 38' 24" 
 
 84 
 
 5' 24" 
 
 '05 
 
 3' o" 
 
 '25 
 
 15' o" 
 
 ! '45 
 
 27' o" i 
 
 '65 
 
 39 
 
 "85 
 
 5i' o" 
 
 06 
 
 3' 36" 
 
 26 
 
 15' 36" 
 
 i '46 
 
 27' 36" 
 
 66 
 
 39' 36" 
 
 86 
 
 Si' 36" 
 
 07 
 
 4' 12" 
 
 '27 
 
 16' 12" 
 
 i '47 
 
 28' 12" 
 
 67 
 
 40' 12" 
 
 87 
 
 52' 12" 
 
 08 
 
 4' 48" 
 
 28 
 
 16' 4 8 ;/ 
 
 i '48 
 
 28' 48" i 
 
 68 
 
 40' 48" 
 
 88 
 
 52; 48" 
 
 09 
 
 5 24" 
 
 '29 
 
 17' 24" 
 
 ! '49 
 
 29*24" ! 
 
 69 
 
 41 24" 
 
 89 
 
 53 > 24 " 
 
 10 
 
 6' o" 
 
 '3 
 
 18' o" 
 
 '50 
 
 30 o j 
 
 '70 
 
 42' o' 
 
 90 
 
 54' '' 
 
 "II 
 
 6' 36" 
 
 
 i8' 3 6" 
 
 
 30' 3 6 '" 
 
 '7 1 
 
 42' 3 6" 
 
 
 54' 36" 
 
 *I2 
 
 
 '3 2 
 
 19' 12" 
 
 '52 
 
 31' 12" 
 
 '7.2 
 
 43 i 2 ' 
 
 92 
 
 55' 12" 
 
 '13 
 
 7 ' 's" 
 
 '33 
 
 19' 48" 
 
 '53 
 
 3i' 48" 
 
 '73 
 
 43' 48" 
 
 '93 
 
 55; 48" 
 
 '14 
 
 8' 24" 
 
 ! 34 
 
 20' 24" 
 
 "54 
 
 3 2 ' 24" 
 
 '74 
 
 44 24 
 
 '94 
 
 56 24" 
 
 '15 
 
 9' o" i 
 
 
 2l' o" 
 
 '55 
 
 33' ' 
 
 "75 
 
 45' o" 
 
 '95 
 
 57' o" 
 
 16 
 
 9' 36" I 
 
 36 
 
 21' 3 6" 
 
 56 
 
 33' 3 6 " 
 
 76 
 
 45' 36'' 
 
 96 
 
 57' 36" 
 
 17 
 
 10' 12" 
 
 '37 
 
 22' 12" 
 
 "57 
 
 34' 12" i 
 
 '77 
 
 4 6' 12" 
 
 '97 
 
 58' 12" 
 
 18 
 
 10' 48'' 
 
 '38 
 
 22' 48" 
 
 '58 
 
 34' 48" 
 
 78 
 
 4 6' 4 8" 
 
 98 
 
 58' 4 8" 
 
 '19 
 
 Il'2 4 " ! 
 
 '39 
 
 23' 24'' 
 
 '59 
 
 35, 24" 
 
 
 47' 24" 
 
 '99 
 
 59 24" 
 
 '20 
 
 12 O i 
 
 40 
 
 24' o' 
 
 60 
 
 36' o" 
 
 
 48' o" 
 
 I '00 
 
 i o' o" 
 
 Thousandths of a Degree in Seconds and Decimals of a Second. 
 
 3 "-6 
 72 
 
 '003 10 
 
 004 I 14'' 
 
 005 
 
 006 
 
 2l"'6 
 
 007 
 008 
 
 2 5 "'2 
 
 28 7 '8 
 
 009 
 'oio 
 
 32 4 
 36 o 
 
274 
 
 Preliminary Survey 
 
 TABLE XLIII. Of Cotangents of a few leading Angles with their 
 corresponding Tangents for -checking the Slide-rule. 
 
 Tan of 
 
 
 
 Cotan of 
 
 Tan of 
 
 
 
 Cotan of 
 
 5 
 
 0-0874 
 
 !! 
 
 85 60 
 
 1-732 
 
 o 
 
 3 
 
 IO 
 
 0-1763 
 
 80 65 
 
 2-144 
 
 2 5 
 
 15 
 
 0-2679 
 
 75 
 
 70 
 
 2-747 
 
 20 
 
 20 
 
 0-364 
 
 70 
 
 75 
 
 3732 
 
 15 
 
 25 
 
 0-466 
 
 65 
 
 80 
 
 5-671 
 
 IO 
 
 30 
 
 Q'577 
 
 60 
 
 85 
 
 1 1 -430 
 
 5 
 
 35 
 
 0700 
 
 55 
 
 86 
 
 14-300 
 
 4 
 
 40 
 
 0-839 
 
 5o 
 
 87 
 
 19-081 
 
 3 
 
 45 
 
 I'OOO 
 
 45 
 
 88 
 
 28-636 
 
 2 
 
 50 
 
 1-192 
 
 40 
 
 89 
 
 57-29 
 
 I 
 
 55 
 
 1-428 
 
 35 
 
 
 
 
 Rule. To find a tangent of a higher angle than is given 
 by the slide. 
 
 tan =_JL_= * 
 
 cot tan (90-) 
 
 Example. Find tan 73 20'. 
 
 The table shows that the decimal punctuation will be 
 between 2-7 and 37. 
 
 tan (9o~73 2o')=tan 16 40' 
 
 Place the scale of tangents in its initial position, and find 
 tan i64o / = < 299. 
 
 Reverse the slide and adjust the scale of numbers with 
 299 over the i of the rule, and opposite to the other i of 
 the slide will be found 3-34 the required tangent. 
 
 TABLE XLIV. Of some Higher Sines than are clearly 
 given on the Slide-rule. 
 
 Sine of 
 
 
 
 Cosine of 
 
 Sine of 
 
 
 
 Cosine of 
 
 89 
 88 
 87 
 86 
 
 9998 
 '9994 
 9986 
 9976 
 
 
 
 2 
 
 3 
 
 4 
 
 85 
 80 
 
 75 
 70 
 
 9962 
 9848 
 9659 
 '9397 
 
 
 
 5 
 
 10 
 
 15 
 
 20 
 
Graphic Calculation for Preliminary Estimates 275 
 
 MEASUREMENT OF TREE TIMBER 
 
 In measuring felled logs, allowance is made first for 
 squaring, secondly for bark. As to the first, instead of multi- 
 plying the area of cross section by the length, \ of the girth 
 squared is taken as the area, and this is the quantity given 
 in square feet in Hurst's * Pocket Book.' It is about 28 per 
 
 cent, less than the actual cross section. When the tree 
 tapers considerably the two ends and middle are girthed 
 and the average taken. A tree is not called timber unless 
 the stem measures 24 inches in circumference. 
 
 Rule. Marketable area=(| girth) 2 ; marketable cubic 
 contents = length x ( girth) 2 . 
 
 By slide-rule. Find the \ girth in decimals of a foot by 
 
276 Preliminary Survey 
 
 Table XXXIII. (unless a decimal tape is used), square by 
 
 Rule 4, p. 244, and multiply by Rule 2. 
 
 If the bark is on the tree, deduct as follows : 
 For oak, old and thick barked . . . 1*0 of girth. 
 For oak, young and thin barked . . . ,, 
 For elm, pine, and fir . . . . . ~ ,, 
 
 For ash and beech . . . . . . -^ ,, 
 
 Comparison of various beams cut fi'om a log 2 feet in diameter. 
 
 sq. ft. 
 
 a, the most serviceable beam . . x = \ diam., area= 1-92 
 
 b, the stiffest beam . . . x = \ diam., area = 175 
 
 c, square timber .... x = \ diam. , area = 2 -oo 
 Marketable measure of log | girth square . . area = 2 -47 
 Gross area of cross section of log . . . = 3'I4 
 
 To obtain a 12" piece of square timber, the tree must 
 be, allowing for bark, 15" diameter, or 4 feet in circumference, 
 at its smallest end. 
 
 RAILWAY TRACK 
 
 Weight per mile of single track, consisting of two rails 
 and fastenings. For rails only : Weight in tons per mile of 
 single track=area of rail in square inches x 157143 ; the 
 weight in pounds per yard being ten times the area of cross- 
 section in inches. 
 
 Molesworth's tables of Indian State railways produce re- 
 sults as follows, including allowance for waste in fastenings : 
 
 $ft. 6 in. gauge. 
 
 Rails only .... weight in tons = area x 15*65 
 Rails, fishplates, fishbolts, and 
 
 spikes .... ,, x 16-65 
 
 Rails &c. as above, and bearing 
 
 plates .... ,, xi;-86 
 
 Metre gauge. 
 
 Rails only .... weight in tons = area x 1 5 '65 
 Rails, fishplates, fishbolts, and 
 
 spikes .... ,, ,, x 16-65 
 
 Rails &c. as above, and bearing 
 
 plates .... ,, x 17-48 
 
 1 Hurst. 
 
Graphic Calculation for Preliminary Estimates 277 
 
 Example i. What will be the weight of iron in a mile 
 of single track of narrow-gauge railway with 40 pound 
 rails and fastenings, but no bearing plates ? 
 
 Place the i of the slide opposite the 4 of the rule, and 
 opposite to 16-65 ( tne weight per mile corresponding to an 
 inch of section) we find 66 - 6 tons. 
 
 Example 2. What section must a rail have so that the 
 rails only, without fastenings, will amount to 100 tons per 
 mile? Adopting the factor 157143, place the 1571 of the 
 slide opposite the i of the rule ; then opposite the i of the 
 slide will be found 6-36, the required sectional area ; the 
 weight per yard would be 63-6 pounds. 
 
 Note. The weights of the fastenings have not been 
 given in detail ; they cover the weight of ordinary fishplates, 
 but not angled or sleeve fishplates. 
 
 Weight of Angle and Tee Iron. 
 
 Where W=weight in pounds per foot run. 
 
 B= breadth of one flange of angle, or clear 
 
 breadth of head of Tee in inches. 
 D= breadth of other flange of angle, or extreme 
 
 depth of Tee in inches. 
 /=thickness of iron in inches. 
 w= coefficient in table. 
 
 TABLE XLV. Multipliers for Weights of Structural Iron. 
 
 t 
 
 1U 
 
 t 
 
 w 
 
 / w 
 
 t 
 
 IV 
 
 f 
 
 208 
 
 416 
 
 t 
 
 I -041 
 1-25 
 
 t 
 
 1-874 
 
 2-082 
 
 13 
 
 V* 
 
 2708 
 2-916 
 
 t 
 
 625 
 833 
 
 1 
 
 2 
 
 1-458 
 
 1-666 
 
 f 
 
 2-29 
 2-50 
 
 i 
 
 3-124 
 
 3'333 
 
 Intermediate values of w, e.g. for 32nds of an inch, or 
 decimals, can be interpolated by slide-rule. 
 
 Example i. What is the weight of a lineal foot of Tee 
 iron 
 
278 Preliminary Survey 
 
 44 + 3J T&= 7 '3 2 inches, and ^=2-29 
 Answer. 7*32 X2'2g=i6"j$ pounds per foot. 
 Weight of channel and H iron may be found similarly, 
 
 when the web and flanges are of the same mean thickness, 
 
 by the formula W=[D + 2 (Rt)]w. 
 
 FIG. 85. 
 
 Example 2. What is the weight of channel iron in pounds 
 per foot run, size 8" x 6" x ^" ? 
 
 8 + 2 (6 ^)= 19 ; 7;=r666. 
 Answer. 19 x 1-666=31-66 pounds. 
 
 Weight of Round and Square Iron. 
 Rule i. Round iron. Place the 2-61 of the upper scale 
 of the slide under the middle i of the rule. Read the 
 weight in pounds per lineal foot on the upper slide-scale, 
 opposite the square of the diameter in inches on the rule. 
 
 Example. What is the weight of 2 " round iron per lineal 
 foot? Adjusting the rule as described, we first find in 
 Table XXXIX. the diameter in decimals 2-375, an ^ by brass 
 marker we find its square 5*64 on the upper scale of the 
 rule. Under this latter figure we find the result 14*8 Ibs. 
 on the slide. 
 
 Rule 2. Square iron. Instead of the number of 2-61 
 use 3 '33, and proceed as before. For flat iron, use the pro- 
 duct of the breadth by thickness instead of the square of the 
 side. 
 
 For round and square cast iron use 2-43 and 3*097 
 respectively. 
 
 For steel, 2-66 and 3-397. 
 copper, 3-00 and 3-83. 
 brass, 2-84 and 3-63. 
 lead, 3-84 and 4*89. 
 , zinc, 2-40 and 3*06. 
 
Graphic Calculation for Preliminary Estimates 279 
 
 THE SLIDE-RULE AS A 'READY RECKONER' 
 TABLE XLVL Wages and Salaries. 
 
 Multiplier 
 Pence per hour into shillings per day of II hours 0-917 
 
 J J 
 
 10 
 
 0-833 
 
 ,, 
 
 9 55 
 
 0-750 
 
 J 5 
 
 
 
 0-667 
 
 J , 
 
 ,, ,, week of 66 ,, 
 
 5'5 
 
 55 
 
 60 
 
 5-o 
 
 ,, 
 
 54 ,, 
 
 4'5 
 
 55 
 
 J 55 55 48 JJ 
 
 4-0 
 
 5 J 
 
 ,, ; per mo. of 26 d. of n hours 
 
 1-192 
 
 ,, 
 
 ,, ,5 10 ,, 
 
 1-083 
 
 J 
 
 5, 5, 9 ,5 
 
 0-975 
 
 55 
 
 JJ 
 
 0-867 
 
 5 J 
 
 ,, an. of 313 d. at ii ,, 
 
 I4-34 
 
 55 
 
 ,, ,, 5, 10 
 
 13-04 
 
 5? 
 
 55 55 55 9 J J 
 
 11-73 
 
 55 
 
 55 55 5 8 ,, 
 
 10-43 
 
 Shillings per 
 
 day into per week of 6 days 
 
 0-3 
 
 ? ? 
 
 ,, ,, month of 4 weeks 
 
 i -20 
 
 ? ? 
 
 55 55 JJ 4jf JJ 
 
 1-299 
 
 , } 
 
 ,, ,, ann. of 313 days 
 
 15-65 
 
 j5 
 
 week into per month of 4 weeks 
 
 0-2 
 
 ,, 
 
 4| jj 
 
 0-216 
 
 ,, 
 
 ,, ,, ann. of 52 weeks 
 
 2-6 
 
 per week into ^ per month of 4 weeks 
 
 4-0 
 
 ,, 
 
 4 " 
 
 4-33 
 
 5 5 
 
 ,, ann. of 52 weeks 
 
 52-0 
 
 , , month 
 
 (lunar) into per annum 
 
 13-0 
 
 55 
 
 (calendar) into per annum 
 
 12-0 
 
 See also tables, pp. 271, 272. 
 
 Example i. How much per annum is an hourly wage of 
 $\d. worth, at the rate of 9 hours per day, working every 
 day except Sundays? Place the left hand i of the slide 
 opposite the multiplier 11*73 on tne ru ^ e > anc ^ opposite to 
 the wage 3*75^. on the slide will be found 44/. 
 
 Example 2. A man has made lyS/. 15^. in the year. 
 What is the equivalent wage in shillings per day, if he had 
 
280 Preliminary Survey 
 
 worked steadily every day but Sundays? Place the left 
 hand i of the slide opposite the multiplier 15*65 on the 
 rule, and opposite to 17875 on the rule will be found 11-42 
 = iis. $d. 
 
 Example 3. A servant gets i7/. 3^. $d. per annum ; what 
 are his wages for 5 months 2 weeks and 5 days ? 
 
 Find from the tables the decimal equivalents of the time 
 and money as follows : 
 
 Years 
 
 Rate in decimals . 17-000 5 months (Table XXXIV.) -417 
 
 3 shillings (Table XLVII.) -150 2 weeks (Table XXXVII.) -038 
 
 5 pence (Table XLVII.) -021 5 days (Table XXXVIII.) -014 
 
 17-171 -469 
 
 By slide-rule. Place the left hand i of the slide over 
 the 469 on the rule, and read the answer 8-o7/. opposite to 
 1,717 on the slide. 
 
 By table, 8 -077. = 87. is. ^\d. 
 
 From Table XLVII. read off -05 as is. and divide the 
 balance -02 by the multiplier for one penny -00417 ; the 
 
 equivalent fraction would be . For division by slide- 
 
 417 
 
 rule see p. 245. The result will be 4-8^. 
 ENGLISH MONEY 
 
 TABLE XLVII. Decimal Multipliers. 
 
 
 
 
 
 .V. 
 
 d. | /. 
 
 One farthing . . 
 One penny . . 
 One shilling . 
 One pound . . 
 
 00104 
 00417 
 
 05 
 i-o 
 
 0208 
 08 33 
 I'O 
 20 
 
 2 5 I 
 
 i 4 
 
 12 48 
 
 240 960 
 
 Example. To reduce 1317. 13$. g\d. to pounds and deci- 
 mals by the slide-rule. 
 
Graphic Calculation for Preliminary Estimates 281 
 
 With the multiplier -00104 we find \d. . . =-0x331 
 -00417 ,, gd. . ='0374 
 
 -05 i$d. . . =-65 
 
 Answer. . . 13 1-6905 
 
 COLONIAL AND FOREIGN 
 
 Where the value is not at par under the gold standard^ or where 
 silver is not at 440!. per ounce, the slide-rule will give the current value 
 by rule I, />. 242. 
 
 ARGENTINE REPUBLIC SEE CHILL 
 
 AUSTRIA. (Par value.) 
 i/. 
 if. 
 id. 
 
 \L , 
 is. , 
 iff. 
 
 CANADA AND UNITED STATES OF AMERICA. 
 Par Value in Sterling. 
 
 I =4*8; dollars 
 
 i shilling = 24-35 cents 
 I penny - 2*03 ,, 
 I farthing 0-51 ,, 
 
 CHILI, COLOMBIA, AND URUGUAY. (Par value.) 
 
 i / = 5 '34 P eso = 534 centavos 
 i-r. . = '267 ,, . = 267 ., 
 
 I" 7 . = '022 ,, . . = 2'2 ,, 
 
 CHINA. 
 
 Intrinsic value with silver at 44^. per oz. troy. 
 i/. . . = 4-28 taels . . = 428 conderin 
 ls - - '214 >, . - 21-4 ,, 
 
 iff. . = -018 ,, =1-8 
 
 = 
 
 10-215 fl- 
 
 . = 1, 02 1 -5 kreuzer 
 
 = 
 
 5107,, . 
 
 = 5 1 '07 
 
 = 
 
 0425 . 
 
 4'2 
 
 
 BRAZIL. 
 
 
 = 
 
 8-925 milreis 
 446 ,, 
 
 . - 8,925 reis 
 
 . = 446 ,, 
 
 = 
 
 037 
 
 37 
 
282 Preliminary Survey 
 
 FRANCE. (Par value.) 
 
 i/. . . 25-220 francs 
 
 is. . . I '261 ,, . . 126'! centimes 
 
 \d. . . -105 ,, . . 10-5 
 
 EMPIRE OF GERMANY. (Par value.) 
 
 I/. . . = 20-420 marks 
 
 is. . . = i'O2i ., . . 102-1 pfennig 
 
 id. . . = -085 8-5 
 
 TABLE XLVIIL Indian Money at par (i Rupee = 2.s.\ 
 
 Rupees 
 
 Annas 
 
 Pie 
 
 Sterling 
 
 I 
 
 16 
 
 I 9 2 
 
 2 shillings 
 
 0625 
 
 i 
 
 12 
 
 \\ pence 
 
 005208 
 
 0833 
 
 I 
 
 0-125 penny 
 
 i lakh = 100,000 rupees, i crore= 10,000,000 rupees, i pice = 3 
 pie or \ anna. In Ceylon the rupee is divided into 100 cents. 
 
 Example a. What is the value of 473/. sterling in rupees 
 at the exchange of is. *]\d. per rupee ? Using the multiplier 
 in Table XLIX., 
 
 10 : 4,730:: 12-15 x 
 
 whence .#=5,747 rupees. For rule of three sum by slide- 
 rule see p. 243. 
 
 Example b. What is the value of 72 pie at the exchange 
 of is. 8d?. sterling per rupee ? 
 
 First operation : To obtain the par value. By Table 
 XLVIIL, 
 
 i pie : "125^. :: 72 pie : x 
 whence x-=gd. 
 
 Second operation : To obtain the current value. By 
 Table XLIX., 
 
 1 1 '56 : 10 :: gd. \ y 
 
 whence y = 7 '8^. , nearly. 
 
Graphic Calculation for Preliminary Estimates 283 
 
 and 
 
 Illl 
 
 * Z S a 
 
 .*3 ^ X '^ 
 
 
 
 2 
 
 n tc** 
 
 K^ * 
 
 I 
 
 ^6M 
 
 a v 
 
 H JS 
 
 s ^ 
 
 
 CO 
 
 Pi S? 
 
 CO 
 
 !> f^ S? 
 
 
 s? 
 
 
 II II 
 
 II II 
 
 
 II II 
 
 * 
 
 vo 
 
 
 to 
 to 
 
 
 
 
 
 
 SN S 
 
 00 
 
 to 
 O to 
 
 VO 
 
 to O 
 to in 
 
 
 ON 
 
 
 O 00 
 
 
 to t^ 
 
 
 
 
 
 
 M 
 
 to o 
 
 " o 
 
 
 N 
 
 
 to 
 
 v! CO 
 
 o 
 
 ^ 
 
 m 
 
 ON to 
 
 |H 
 
 M 
 
 M OO ON 
 
 oo i ^ < 
 
 
 2i ^ 
 
 . r^ o 
 
 
 op p 
 
 . 
 
 r 9 
 
 w b 
 
 1-1 
 
 HH 
 
 M 
 
 to 
 
 HH 
 
 v: o 
 
 v3 ; u-) 
 
 
 M 
 
 . 
 
 to 
 oo 
 
 1-1 ' vO ON 
 vo O 
 
 So 
 
 to rj- 
 r^ in 
 
 O 00 
 
 vcT 
 
 o o 
 to t^ 
 
 ON O 
 
 ~ 
 
 M 
 
 E 
 
 1-1 
 
 HH 
 
 ^j ON 
 
 
 CO 
 m 
 
 
 m 
 
 
 
 in o 
 
 CO 
 
 to i ^- 
 
 vo vo 
 vO OO 
 m O 
 
 3- 
 
 Joo 
 JXi 
 O 
 
 2 
 
 O 
 
 1-1 
 
 E 
 
 M 
 
 N 
 
 
 to 
 
 
 8 
 
 
 VO 
 
 M 
 
 oo oo 
 
 ON 
 
 vo in 
 
 JN. 
 
 
 1-1 
 
 5 
 
 ^ 
 
 N 00 
 ^ P 
 
 * 
 
 to t^ 
 
 vo O 
 
 M 
 
 2 II 
 
 
 
 M 
 
 s 
 
 VO OO 
 Cs| \O 
 
 to O 
 
 'ON 
 
 ^ OO 
 ON OO 
 C< 
 
 I 
 
 in N 
 
 VO OO 
 
 M 
 
 
 
 h- 4 
 
 
 "- 1 
 
 2 
 
 ^ 0\ 
 
 vT 
 
 to 
 
 00 
 
 oo m 
 
 ^ 
 
 a 
 
 HH 04 *>, 
 
 Hl<S 
 
 M ON 
 
 H|C3 
 
 t*s. ** 
 
 M 
 
 5 2 
 
 T 
 
 ^ < 
 
 \" 
 
 o oo 
 
 to O 
 
 M 
 
 o 
 
 - 1 
 
 s 
 
 
 2 
 
 ts 
 
 00 
 in 
 
 tO ON 
 
 ^ 
 
 in 
 
 in vo 
 
 ^ 
 
 ON O^ 
 
 M 
 
 in oo 
 
 O ON 
 
 ON 
 
 to ON 
 
 W|* 
 
 ^ N 
 
 
 i-* O 
 
 . 
 
 O O 
 
 
 ? *% 
 
 - 
 
 O 
 
 1-1 
 
 2 
 
 M 
 
 
 
 X 
 
 
 X 
 
 
 X 
 
 > 
 
 X tfl 
 
 J> 
 
 * CO 
 
 J> 
 
 * 
 
 
 S3 & 
 
 
 S? P^ 
 
 
 
 
 
 
 
 
 
284 Preliminary Survey 
 
 Example c. To reduce 15 annas 7 pie to cents (looths 
 of a rupee). 
 
 1 6 annas : 100 cents:: 15 annas : x cents 
 whence #=937 ; 
 
 192 pie : 100 cents :: 7 pie : y cents 
 whence ^=37 cents, 
 and answer, 937 + 37=97'4 cents. 
 
 JAPAN. 
 
 Intrinsic value with silver at ^\d. per oz. troy. 
 i/. . . 6 '500 yen . . . 650 sen 
 
 is. . . -325 32-5 
 
 id. ' . . -027 ,, . 27,, 
 
 MEXICO. (Par value.) 
 
 i/. . . =^ 6-160 dollars . . = 616-0 cents 
 
 is. . . = -308 ,, . . . = 30-8 ,, 
 
 id. . . = -025 ,, . . . = 2-5 ,, 
 
 NETHERLANDS. 
 
 i/. . . = 12-00 florins . . =1,200 cents 
 is. . . = -60 ,, . . = 60 ,, 
 
 id. . . = -05 . . - 5 ,, 
 
 The par value according to United States Treasury circular is about 
 
 I per cent. more. 
 
 PERSIA. 
 
 I/. . = 2-800 thomans . = 28-0 banabats . =280 shahis 
 
 is. . = -140 ,, . = l'4 ,, = H 
 
 id. . = -012 ,, . = -128 ,, . = 1-28 ,, 
 
 PORTUGAL. (Par value.) 
 
 . = 4 '500 milreis . . = 4,500 reis 
 
 . == -225 ,, . . = 225 ,. 
 
 . = -019 ,, . . = 19 ,, 
 
 RUSSIA. (Paper currency. ) 
 
 Current value in Whi taker's Almanack for 1890. 
 
 i/. . = 6-3000 roubles . . = 630 kopek 
 
 is. . . = -315 = 3 l 'S " 
 
 id. . . = -0265 ,, . . - 2-65 ,, 
 
GrapJiic Calculation for Preliminary Estimates 285 
 
 SPAIN. (Par value.) 
 I/. = 97000 scudo = 25*200 peseta 2,520 centimes 
 
 IS. =: -4850 ,, = I-260 ,, = 1,260 
 
 id. = -0404 ,, = -105 ,, 105 ,, 
 
 SWEDEN. (Par value.) 
 
 i/. . . = 18*200 crowns . . =* 1,820 ore 
 
 is. . . = -910 ,, 91 ,, 
 
 id. . . = -076 ,, . . - 7'6,, 
 
 TURKEY. (Par value.) 
 
 i/. . . = no'O piastres 
 
 is. . . 5 '5 ,, . . = 220 paras 
 
 id. . . = -46 ,, . . = 18-3 ,, 
 
 VENEZUELA. 
 
 i/. . . = 25-220 bolivars . . = 50*44 decimos 
 is. . . = 1-261 ,, . . = 2-522 ,, 
 id. . . = -105 ,, . = -210 ,, 
 
 Example d. How many Chinese taels are contained in 
 10 Mexican dollars, the former at intrinsic value of 42^. 
 per oz., the latter at par value ? 
 
 First operation : To obtain the current value in Chinese 
 taels of i/. sterling. 
 
 42 : 44:14*28 : ;v=4-48 
 
 Second operation : To obtain the value of 10 Mexican 
 dollars in taels. 
 
 6-16 : 4-48:: 10 : 7-11 taels 
 
 Example e. How many Japanese yen at 44^. per oz. 
 should one receive for 105 U.S. dollars at par value? 
 
 4*87 : 6-50 : : 105 : x 
 ^=140 yen 44 sen 
 
286 Preliminary Survey 
 
 CHAPTER IX 
 INSTRUMENTS 
 
 LEVELS AND LEVELLING 
 
 THE necessity for condensation has led to the insertion 
 here of all that can be said within our limits upon the theory 
 of levelling instead of giving it a separate chapter. It will be 
 attempted to place the fundamental principles of levelling 
 and the adjustments of different kinds of levels in a practical 
 and simple light. The abstruse disquisitions upon possible 
 sources of error in levelling to be found in proceedings of 
 learned societies have no place here. The surveyor . ought 
 to be able to put his instrument in adjustment every morning 
 without referring to a book, and considerable space has 
 been required to make the reason of each adjustment plain. 
 Secondly, he should be alive to the limits of error arising 
 from taking long or unequal sights, curvature of the earth, 
 &c., so that on the one hand he should be prepared to 
 adopt extra precautions for special cases, and on the other 
 hand not to waste time upon refinements which are not 
 essential to his object. 
 
 The adjustment of the level is also the foundation of the 
 adjustment of the theodolite and tacheometer. This is 
 shown in Fig. 88, where the error in the line of sight is 
 treated as a vertical angle which has to be eliminated 
 similarly to the index error of a theodolite. 
 
 For these reasons, the lion's share of this chapter has 
 fallen to the subject of levels. 
 
Instru ments 287 
 
 THEORY 
 
 Two points are said to be upon the same level when 
 they are equidistant from the earth's centre : conseqently, a 
 level line cannot be strictly speaking a straight line. It is 
 a parallel to the curvature of the sea. A horizontal line is 
 a parallel to a tangent to the earth's circumference, and 
 therefore a straight line. In common parlance, the words 
 level and horizontal are synonymous, and it would be pedantic 
 to endeavour to keep their application wholly distinct, but 
 the difference is mentioned because it comes into the 
 question of adjustment for ' collimation.' 
 
 A level adjusted horizontally will not represent an object 
 as high as it really is. This is proved by looking at the top 
 of a vessel's masts when just appearing above the horizon. 
 A level would report the mast-head to be at the same 
 elevation as the observer's eye, whereas it would be perhaps 
 fifty or 100 feet further from the earth's centre. 
 
 The object of the leveller is to determine the relative 
 elevations, or heights above sea-level, of points upon the 
 earth's surface, which is the same thing as their difference 
 of distance from the earth's centre. 
 
 Light always travels in a straight line, unless diverted 
 from its path by the medium traversed. The bending of 
 light by the atmosphere is called refraction : see Glossary. 
 
 Levels are either adjusted with the line of sight, hori- 
 zontal, and consequently subject to correction for both 
 curvature and refraction, or else they are adjusted with the 
 line of sight parallel to a chord of the earth's circumference 
 to allow for curvature and refraction, as by Mr. Gravatt's 
 method, described in ' Heather on Instruments.' 
 
 Refraction makes objects appear too high, consequently 
 it counteracts to a small extent the effect of curvature. It 
 varies with the state of the weather, but as regards levelling 
 it is quite near enough to assume it at its average amount 
 
288 Preliminary Survey 
 
 at sea-level. That is -095 of a foot at a distance of a mile, 
 and varying directly as the square of the distance. 
 
 The Curvature of the earth at a mean radius is='66y 
 foot at a distance of a mile, and also varies (within a limit of 
 many miles) directly as the square of the distance. The 
 allowance to be made for refraction and curvature at a distance 
 of a mile is therefore '572 foot, or for any distance in miles 
 up to 100, more exactly, the correction to be added^ '5717 D 2 . 
 
 TABLE L. Correction for Curvature of the Earth. 
 
 Distance in feet 
 
 Correction + 
 
 Distance in miles 
 
 Correction + 
 
 250 
 
 OOI49 
 
 25 
 
 0356 
 
 5OO 
 
 OO5II 
 
 'SO 
 
 143 
 
 750 
 
 0116 
 
 75 
 
 321 
 
 I,OOO 
 
 0205 
 
 I'O 
 
 572 
 
 1,250 
 
 0320 
 
 1-25 
 
 893 
 
 1,500 
 
 0460 
 
 1-50 
 
 1-286 
 
 1,750 
 
 0620 
 
 175 
 
 1750 
 
 2,000 
 
 0820 
 
 2'OO 
 
 2-287 
 
 Whether the line of sight is adjusted tangentially or to 
 a terrestrial chord, if the back-sights and fore-sights are of 
 equal length, there will be no error in the result on account 
 of curvature. The correction to be added in each case is 
 entirely dependent upon the excess in distance of the one 
 sight over the other. Thus with a back-sight of 300 feet, and 
 a fore-sight of 900 feet, the correction of -002 foot for the 
 back-sight would have to be taken from the correction -017 
 for the fore-sight, leaving an additive correction of '015 foot. 
 But if the back-sight is longer than the fore-sight the correc- 
 tion will be subtractive. Now with the ordinary 14" level, at a 
 distance of 500 feet an almost imperceptible movement of the 
 bubble from the centre of its run will produce a difference 
 in the reading of '03 foot, amounting as it does to about one 
 second of vertical arc, or the twentieth part of one of the 
 small subdivisions usually marked upon the level bubble. 
 
 There are not many men who would have the hardi- 
 
Instruments 289 
 
 hood to swear to their levels to one-hundredth at the distance 
 of 500 feet with a 1 4-inch level ; but it will be seen from 
 the table that both instrument and man must be true to half 
 a hundredth every time at a distance of five hundred feet, 
 or else they may as well leave curvature and refraction out 
 of their calculations. It was partly (i) to obtain this minute 
 accuracy, and partly (2) to set the line of sight in the exact 
 optical axis of the telescope, that Mr. Gravatt invented his 
 elaborate ' three peg ' adjustment. To go through a tedious 
 process to obtain the first condition is a waste of time when 
 it can be added for special exactitude from the table, but 
 besides that, the condition is only really fulfilled for the 
 
 d 
 
 FIG. 86. DUMPY LEVEL. 
 
 A. A. Plane of rotation ; a. a. adjusting screws to same. 
 
 B. B. Horizontal bar ; b. adjusting screw to same. 
 
 C. C. Line of sight ; c. c. adjusting screws to same. 
 
 D. D. Bubble tube; d. d. adjusting screws to same. 
 
 length of base used in the adjustment ; it produces an equal 
 and opposite error at a midway point to what is produced 
 by a horizontal adjustment at the further point. This is evi- 
 dently the case from the fact that no line of sight can be 
 made to follow the curvature of the earth. The second 
 condition is not essential to correct levelling, but it can be 
 fulfilled by a perfectly simple and rapid process, as will be 
 presently shown, but a few words are first needed on the 
 organic principle of the level whether Y or dumpy. 
 
 The spirit-level, like the plummet, is a device for utilising 
 the law of gravity to establish a horizontal or perpendicular 
 
 u 
 
290 Preliminary Survey 
 
 line. Either of them can, by means of a square, do the 
 work of the other. 
 
 If we fill a bottle nearly full of water and cork it, the air- 
 space is always at the top because the water is heavier. If 
 we turn it on its side on a level table the air-space will if 
 very small form a bubble which will stop in any position 
 along the side, because, the sides of the bottle being parallel, 
 no one part is higher than the other. 
 
 The level-tubes of good spirit-levels are very carefully 
 ground to a true curve, so that a movement of the tube in a 
 vertical plane is equal to the right or left for equal vertical 
 angles. The worst kind of level is that which has hardly 
 any curvature and is everlastingly getting off the centre with 
 inappreciable vertical movements. A bubble-glass which is 
 not perfectly uniform in curvature requires more care than 
 a perfect one, but correct levelling can be done with it 
 also. 
 
 Suppose now a bubble-tube with legs like the striding 
 bubble of a theodolite standing upon a table. If the bubble 
 be at the middle it does not prove the table level. ,It only 
 proves that either the table is level or else it has a slope 
 which is equal to a corresponding inequality in the length 
 of the legs. For if we reverse the bubble-tube end for end, 
 the bubble may be displaced from the centre. If the bubble 
 remains in the centre when reversed it proves firstly that the 
 table is horizontal ; secondly, and in consequence, that the 
 legs of the tube are equal. We cannot reverse a bubble upon 
 a sloping plane. The first thing therefore in the adjustment 
 of the level is to make the plane of rotation AA (Fig. 86) 
 horizontal, which we can do whether the line of sight is 
 horizontal or not, and it is therefore here mentioned 
 first. 
 
 The correctness of the bubble is the basis of all the 
 adjustments. The common expression, 'correcting' or 
 'adjusting' the bubble, is a misleading one. The only 
 correction suitable to a defective bubble -tube is to break it 
 
Instruments 291 
 
 up, because nothing can correct an imperfect grinding. 
 When the bubble-tube is truly ground, the bubble is always 
 at the top of the curved surface, and just as a plummet 
 gives the vertical, so the bubble gives a true horizontal 
 line. It is the different parts of the instrument, such as 
 the plane of rotation and line of sight, which have to be 
 adjusted to the bubble, and not the bubble to them. The 
 test of accurate grinding is by marking and measuring the 
 travel of the bubble within equal angular movements in 
 opposite directions. When the bubble-tube may be turned 
 completely round without disturbing the bubble from the centre 
 of its run it proves that the plane of rotation is horizontal. 
 It is not necessary for practical levelling that the line of sight 
 should coincide with the optical or focal axis of the telescope 
 tube : it is sufficient for it to be sensibly horizontal within 
 the range of the focussing screw, and therefore parallel to the 
 plane of rotation. 
 
 When both the plane of rotation and line of sight are 
 horizontal, if the telescope tube is not parallel to them, the 
 line of sight cannot be in the optical axis and will be 
 theoretically thrown out of its horizontality by actuating the 
 focussing screw. This fact is met by another, which is that 
 the adjustment is made for the longest distance that the 
 telescope will read correctly, and in that position the move- 
 ment of the focussing screw is not sufficient to produce 
 sensible error. It is only when the distances are very short 
 that the effect would be appreciable, and then the divergence 
 of the line of sight has not sufficient distance in which to 
 accumulate sensible error. 
 
 The travel of the diaphragm in a vertical plane is so small 
 compared with the field that if the hairs are displaced as far 
 as they will go and the instrument adjusted to horizontality 
 by the method described on p. 296, it will give the same 
 difference of level when set up midway between the stakes 
 or close to either of them. 
 
 The method given here will, however, include an almost 
 
 u 2 
 
292 
 
 Preliminary Survey 
 
 perfect coincidence of the line of sight with the optical axis. 
 It makes all four lines, AA, BB, CC, DD, in Fig. 86 
 perfectly horizontal. It takes a quarter of an hour the 
 first time, and five minutes when the pegs are driven and 
 their difference of level known. 
 
 The adjustment of the plane of rotation is analogous to 
 
 8-0 
 
 STRIDING BUBBLE 
 ADJUSTMENTS 
 
 FIG. 87. 
 
 that of the striding bubble on a table alluded to on p. 290, 
 and will be therefore illustrated in that manner. Referring 
 to Figs, a, b^ c, d, e, the line with hatching represents 
 the plane upon which the level stands. In the case of a 
 theodolite it would be the plane passing through the trunnions. 
 In the case of the Y or dumpy it is AA, Fig. 86, the plane of 
 
Instrumen ts 293 
 
 rotation. It is supposed to be inclined to the horizontal by 
 an angle =0. The bubble-legs are drawn of unequal length 
 to represent in an exaggerated manner the error of adjust- 
 ment which may be in these legs or in the little legs d, d, 
 Fig. 86, and for adjustment of which capstan-headed screws 
 are provided. The angle of error by which the feet are 
 supposed to be out of parallel with the axis of the bubble 
 (that is with a tangent to the upper curved surface of the 
 bubble-tube) is t). It does not matter what proportion each 
 error bears to the other ; they can at once be removed. 
 
 In Fig. a the striding bubble is placed so that the two 
 angles augment the divergency of the bubble from the centre 
 of its run, which becomes=0-f c. In Fig. b the level is 
 turned end for end and the error becomes I 6. 
 
 Now if we bring the bubble from its position in Fig. a 
 to the centre of its run, as shown in Fig. c, correcting the 
 total error 3 + 0, half by the screws on the bubble, and half 
 by altering the inclination of the plane (in the transit by the 
 capstan-headed screws under the trunnion), we shall obtain 
 the following equation. 
 
 Placing the striding bubble in the position of augmented 
 
 error, Fig. a, and deducting from I gives \ ($ Q)- 
 
 Deducting from gives H~^) or ~ 2 ( a ~^ 
 
 that is to say, the error of the bubble and the plane in Fig. c 
 will each then be equal and opposite. 
 
 If, now, we reverse the striding bubble into the position 
 Fig. 4 the reduced error of the bubble will be 0, and if 
 we bring it to the centre of its run by equally dividing the 
 error as before, the error will be eliminated, as in Fig. e. 
 
 Example. The error of the plane was +5, and the 
 bubble 7. Placing the latter so as to augment the error, 
 
294 Preliminary Survey 
 
 Reversing, as in Fig. d, combined error= 2, and again 
 levelling with halved correction, 
 
 Probably the correction will not be accurately halved 
 the first time, and if very much out of adjustment it will 
 take three or four trials. 
 
 The ordinary tradesman's level is planed or ground on 
 its base to a true parallelism with the axis of the bubble, 
 and can only be adjusted, if out of order, by re^planing. 
 
 1. To adjust the plane of rotation for either Y or dumpy 
 levels. Bring the bubble to the centre of its run over a pair 
 of parallel-plate screws. Turn the telescope 90, and repeat 
 over the opposite pair of screws. Bring it back to its first 
 position, and retouch the parallel-plate screws. Turn the 
 telescope 180, so as to be over the first pair of screws, but 
 end for end. If the bubble is off the centre, correct half 
 the error by the screws d, d, and half by a, a, Fig. 86. Repeat 
 until the bubble remains central in each position. Turn 
 the telescope 90, and retouch the parallel-plate screws so 
 as to bring the bubble to the centre, after which it ought 
 to revolve completely in a central position. If it will not, 
 it proves that the plane of rotation is not accurately ground, 
 or that it has become worn by sand or what not, so that it 
 is impossible to produce horizontality in all directions over 
 its surface. This, of course, could only be remedied by a 
 maker, or the instrument could only be used by adjusting 
 the parallel-plate screws at every sight. 
 
 2. To remove parallax in Y or dumpy levels. This is an 
 adjustment of the eyepiece to bring the cross hairs into the 
 common focus of the eyepiece and object-glass. Some- 
 times, with very short-sighted people, the little brass tube 
 into which the eyepiece fits has to be ground down. 
 
Instruments 
 
 295 
 
 The adjustment is performed by directing the telescope 
 upon a distant object, and first focussing the object with 
 the focussing screw, then the cross hairs with the eyepiece, 
 until the object is seen perfectly distinctly, and the hairs 
 are clear and do not appear to shift when the eye is 
 moved. 
 
 3. To place the line of sight approximately in the optical 
 axis of the telescope. Dumpy levels. Hold the staff about 
 thirty feet off, with a sheet of white paper against its back. 
 
 r- 
 
 
 
 
 H 
 
 -v?c 
 
 -<-"j 
 
 1A- 
 
 L|_I "~~ 
 
 f\ 
 
 \\ 
 
 i i 
 
 11 \ 
 
 II ; 
 
 \\ 
 
 k"^ 
 
 
 \\ 
 
 DUMPY LEVEL 
 
 Anjli^TMFMTQ 
 
 ; V77??7> 
 
 %- 
 
 FIG. 88. 
 
 Direct the staff-holder to mark with a pencil the top and 
 bottom of the circular field made upon the paper. If 
 possible, let the staff be held against a wall to steady it. 
 Bisect the space between the pencil marks, and mark the 
 centre. Bring the axial hair to coincidence with this 
 mark by the screws c, c, Fig. 86. The hair will be then in the 
 centre of the shutter, and if the shutter is concentric with the 
 tube, which it is in all properly constructed instruments, and 
 if the glass is perfectly ground, it will also be in the optical 
 axis. In any case it will be quite near enough to avoid all 
 
296 Preliminary Survey 
 
 error, since it is not from this adjustment that the line of 
 sight is made horizontal. When this is done, the screws 
 c, c need never be touched unless the hairs are broken. An 
 ivory scale fastened against a wall will enable the foregoing 
 to be done without assistance. 
 
 4. To make the li?ie of sight horizontal. See Fig. 88. 
 Drive two pegs on nearly level ground about a hundred 
 paces apart. Set up the level so close to one of them that, 
 when the levelling staff is held upon it, the eyepiece will be 
 about half an inch away from it. 
 
 Level the instrument very carefully, and look through 
 the object-glass at the staff, swaying the latter gently until 
 it comes into the focus of the eyepiece about half an inch 
 from the face of the staff. Mark the staff at the centre of 
 the field, which will be about one-eighth of an inch in 
 diameter. Book the height, which we will call instrument- 
 height A. Then remove the staff to the distant peg, and 
 with the bubble in the middle, read the height there in the 
 usual way, and call it staff-reading B. 
 
 Next carry the instrument to the distant peg, and set it 
 up, carefully levelled, close to the staff, taking a similar 
 reading through the object-glass, which call instrument- 
 height A'. Remove the staff to the first peg and read it, 
 calling it staff-reading B'. Now referring to Fig. 88, if the 
 line of sight had been parallel to the bubble, it would be 
 represented by the two horizontal lines in the figure, and the 
 difference of the readings in each position would be the 
 same, that is to say A B / =B A'. 
 
 When, however, the line of sight is inclined to the axis 
 of the bubble, as shown on the figure, it makes a vertical 
 angle of elevation or depression =0. In the case illustrated, 
 being an angle of elevation, the first difference of readings 
 is augmented, and the second difference of readings is 
 diminished by that angle. 
 
 When, therefore, the difference of readings at A, that is 
 A B', is less than the difference of readings at B, that is 
 
Instruments 297 
 
 B A', the line of sight 'throws' upwards, and when more 
 
 it ' throws ' downwards. 
 
 We can express this more simply by the following : 
 Rule. When the sum of the instrument-heights A + A' is 
 
 lesshan 
 
 Qf the staff . readings B + B', the line of 
 more than 
 
 sight throws - and the ha]f difference is to be 
 
 downwards 
 
 deducted from ^ second staff . readi 
 added to 
 
 This last clause needs a little explanation. The dif- 
 ference of the two differences (B A') (A B') clearly 
 measures double the angle 0. The above may be written 
 B + B' (A' + A), and this is twice the error. In the illus- 
 tration, as the line of sight threw upwards, it has to be 
 deducted. 
 
 The crucial test of this method is that wherever the 
 level is placed when the adjustment is complete, whether 
 touching either of the staves or midway between them, the 
 difference of level recorded is precisely the same. 
 
 The angle is caused by the line of sight not being 
 parallel to the plane of rotation, and having determined its 
 amount, we eliminate the whole of it by the screw , Fig. 86, 
 by screwing it up or down until the reading on the staff is 
 corrected by the amount which measures the angle 8 on the 
 staff. This will of course disturb the bubble, and we bring 
 the bubble back to the centre of its run entirely by the 
 screws d^ d. It will be noticed that we have kept the plane 
 of rotation horizontal all the time, and we have now all four 
 lines horizontal. 
 
 The correctness of this adjustment can be seen by 
 treating the Y level in the same manner, for it will be found 
 when completed that the Y's can be thrown open, and the 
 telescope reversed end for end without disturbing the 
 centrality of the bubble. 
 
 We could have produced horizontally in the line of 
 
298 Preliminary Survey 
 
 sight by correcting the final error by the screws c, c, and 
 neglecting the third adjustment altogether, but the method 
 as given is the best. 
 
 When, however, there is no screw b, Fig. 86, under the 
 horizontal bar, as is frequently the case with the smaller 
 dumpy levels, the parallelism of BE with A A can only be 
 corrected by an instrument-maker. We cannot then use the 
 third adjustment on p. 295. We make the first and second 
 adjustments and then proceed directly to the fourth, making 
 the final correction with the collimation screws c, c, Fig. 86. 
 If the level is properly made, the parallelism of BB with 
 A A will be quite near enough to bring the line of sight 
 sufficiently close to the focal axis for all practical 
 purposes. 
 
 Example i. (Line of sight throwing upwards.) 
 
 Readings at Station A. 
 
 Ht. of inst. at A. . . . (A) 5 -23 
 Staff-read, at B 5-15 (B) 
 
 Readings at Station B. 
 
 Ht. of inst. at B. . . . (A') 4-63 
 
 Staff-read, at A 5-03 (B') 
 
 Sum of staff readings . . . lO'iS (B +- B') 
 
 Sum of inst. heights .... . """9 '86 (A + A') 
 
 Difference ..... -32 
 
 Half difference . . . . 'i6 
 
 Reading of true horizontal line from B=5*o3 '16= 
 4 '8 7. To which reading the cross hairs are to be brought 
 by the screw b. 
 
 Example 2. (Line of sight throwing downwards.) 
 
 Readings at Station A. 
 
 lit. of inst. at A. . .5 -08 (A) 
 
 Staff-reading at B 5'i6 (B) 
 
Instruments 2 99 
 
 Readings at Station B. 
 
 lit. ofinst. atB. . . . 4*95 ( A/ ) 
 
 Staff-reading at A 4 '55 ( B ') 
 
 Sum of inst. heights . . . io-O3(A + A')_^ 
 Sum of staff-readings . . . 97 1 -- ""(B + B') 
 
 Difference . . . -32 
 Half difference . . -16 
 
 Reading of the true horizontal line of sight from B= 
 4*55 1- -16=471. 
 
 The difference being the same in these two examples is 
 a mere coincidence. 
 
 The difference of level between the two pegs might have 
 been determined by setting up the instrument midway 
 between them, and taking readings alternately on one and 
 the other ; but the method as given takes no longer, and 
 needs no measurement. 
 
 When, however, the difference of level is known, the 
 instrument need only be set up beside one of them. 
 
 When the adjustments are completed, particular care 
 should be given that all the screws are tight ; if not the 
 adjustment will last but a very short time, but if carefully 
 made it will probably not need touching after a month's 
 steady work. Every day before starting, five or ten 
 minutes at the peg will suffice to show that all the adjust- 
 ments are in order. 
 
 ADJUSTMENTS OF THE Y LEVEL 
 
 1. To adjust the plane of rotation horizontal, see 
 294. 
 
 2. To remove parallax, see p. 294. 
 
 3. To place the line of sight in the optical axis. 
 Direct the telescope on some clearly defined point, and 
 
300 Preliminary Survey 
 
 intersect it with the cross-hairs, revolve the telescope half 
 round in its Y's (being careful not to rotate it on its plane). 
 If the object is not still intersected, correct one half by the 
 screws c, c, Fig. 86, and the other half by the parallel-plate 
 screws (the horizontality of the plane of rotation has nothing 
 to do with this adjustment). Repeat until the intersection 
 is the same in all positions. 
 
 4. To make the line of sight horizontal repeat No. i 
 adjustment and open the Y's. Reverse the telescope end 
 for end. If the bubble is not still central, correct half the 
 error by the screws d, d, and half by b. Repeat until the 
 telescope can be reversed without disturbing the bubble. 
 DD and BB will then be both horizontal. 
 
 5. To place the bubble axis in the same vertical plane 
 with the axis of the telescope. 
 
 An error in this respect is detected by the fact of the 
 bubble not retaining its central position when the telescope 
 is turned a little way in the Y's. The two ends of the 
 bubble are not quite in line, consequently as the telescope 
 is turned, one end rises a little before the other, and this 
 error is corrected by the capstan -headed screw at the side 
 of the screws which correspond to d, but are only found in 
 Y levels and not shown on the figure. 
 
 The Y level has the advantage of requiring no peg- 
 adjustment for collimation. The dumpy level is handier 
 for small sizes. English surveyors prefer it on the ground 
 of its supposed superiority to the Y level in retaining its 
 adjustment. This of course is a consideration when they 
 are in the habit of sending their level to the makers for 
 adjustment. The writer does not hold that view, but 
 believes the Y level will retain its adjustment just as well, 
 last longer without re-grinding of the axis, and is adjusted 
 in the field in a few minutes. The peg adjustment of the 
 dumpy in the usual form of the text-books is a great bug- 
 bear to young engineers, so it brings much grist to the 
 instrument-makers' mill. 
 
Instruments 30 1 
 
 The Y level is almost exclusively the type adopted in 
 America. 
 
 A modification of Y level has been recently introduced 
 by Messrs. Cooke and Sons, of York. The principal dif- 
 ference is that the telescope is contained in a shorter ex- 
 ternal tube terminating in sockets into which the telescope 
 fits very exactly ; it is reversed by withdrawing it most care- 
 fully, and inserting it end for end to obtain the adjustment 
 No. 4, on p. 296. Never having used this instrument, the 
 writer does not wish to speak decidedly about it. Coming 
 from that firm, the workmanship would no doubt be excel- 
 lent, and that always means an instrument which will retain 
 its adjustment well. On the one hand it must be less liable 
 to wear in the bearings, and on the other hand, more 
 awkward to reverse it when it needs adjusting. 
 
 The wear of ordinary Y's can be corrected by the screw 
 b, Fig. 86. 
 
 LEVEL-STAVES 
 
 Two types of these are used : those in which the sight is 
 taken on a sliding vane or target, painted black and white 
 so as to obtain very precise intersection with the cross hairs. 
 On the target is a vernier which reads with graduations on 
 the staff. This type is still somewhat used in America, but 
 hardly at all in England. It can be read at a greater 
 distance than the graduated rod, but requires an assistant 
 who can be depended upon to book the readings correctly. 
 
 Graduated rods are usually marked by lines across at 
 every hundredth of a foot, the spaces being alternately 
 black and white. 
 
 When the staff is used for telemetry, it should have a 
 device for ensuring that it is held at right angles to the line 
 of sight, or else plumb according to the manner of working. 
 For colliery work, an illuminated staff has been successfully 
 employed having the figures painted on glass, and a lamp 
 carried in a thin casing at the back of them. 
 
302 
 
 Preliminary Survey 
 
 DIFFERENT METHODS OF KEEPING THE FIELDBOOK 
 The 'rise and fair Method 
 
 This is the most rigorous plan. The reduction of inter- 
 mediate sights forms a check upon the turning-points. The 
 form of fieldbook is as under : 
 
 Back-sight 
 
 Intermediate 
 
 Foresight 
 
 Rise 
 
 Fall 
 
 Red. level of 
 intermediate 
 
 o-g 
 
 1\ 
 
 ~ $ 
 1 "S'l 
 1 f*5 
 
 8 
 | 
 
 Q 
 
 Total distance 
 
 Remarks 
 
 4-65 
 
 
 
 
 
 
 100*00 
 
 
 
 B. M./ \ on 
 mile- 1 /I\ 1 ston e 
 
 
 2-23 
 
 
 2-42 
 
 
 102*42 
 
 
 
 
 
 
 11-74 
 
 
 
 9'5t 
 
 92-91 
 
 
 
 
 
 
 9 '37 
 
 
 2'37 
 
 
 95-28 
 
 
 
 
 
 8-43 
 
 
 11-13 
 
 
 1-76 
 
 
 93'52 
 
 
 
 Turning-point on 
 peg 
 
 
 5'22 
 
 
 3-21 
 
 
 96-73 
 
 
 
 
 
 
 i'i3 
 
 
 4-09 
 
 
 IOO"82 
 
 
 
 
 
 
 
 7-84 
 
 
 6-71 
 
 
 94-11 
 
 
 
 
 13-08 
 
 
 18-97 
 
 
 
 
 
 
 
 
 
 \ 
 
 13-08 
 
 
 
 
 5 '89 
 
 
 
 
 
 
 
 
 
 
 
 TOO'OO 
 
 
 
 Check 
 
Instruments 303 
 
 By keeping the reduced levels of the intermediate 
 separated from the turning-points, the independent check 
 of back-sights and fore-sights is more conveniently applied 
 at any time, and it should be done at the end of every 
 page. 
 
 The Collimation Method 
 
 This is almost exclusively used in North America, and 
 to a large extent in the Colonies. It is much quicker, and 
 will receive a little more explanation because it is the 
 foundation of the tacheometer fieldbook. It has the dis- 
 advantage of greater liability to error in the reduction of the 
 intermediate sights, each calculation being independent ; 
 but as it enables the railway surveyor to put in his gradient 
 stakes without first sitting down to reduce his levels, its 
 advantages are deemed to outweigh the one disadvantage. 
 
 The back-sight by this method becomes a plus sight, and 
 the only one. All the intermediate sights and the fore-sight 
 are minus sights. 
 
 In Fig. 89 the operation commences with a back-sight 
 
 to determine the elevation of the line of sight, commonly 
 called ' collimation ; ' the staff- reading of 9*00 feet being 
 added to the known or assumed elevation of the fiducial 
 point on which the staff is held at starting. The collimation 
 being known, the elevation of any other point which can be 
 seen is equal to the collimation height less the reading on 
 the staff, and the reduction of the levels is relieved of the 
 column of rise and fall. 
 
304 
 
 Preliminary Survey 
 Form of Fieldbook. 
 
 + 
 
 - 
 
 -T. P. 
 
 Collimation 
 
 Red. level of 
 intermediate 
 
 Red. level of 
 turning-points 
 
 Distance 
 
 Total distance 
 
 Remarks 
 
 9'oo 
 
 
 
 109*00 
 
 
 lOO'OO 
 
 
 
 B.M. on mile- 
 stone 
 
 
 4-10 
 
 
 
 105 'oo 
 
 
 
 
 
 
 lO'OO 
 
 
 
 99 "oo 
 
 
 
 
 
 II'OO 
 
 
 2 '00 
 
 iiS'oo 
 
 
 107*00 
 
 
 
 
 
 
 7'00 
 
 
 
 III'OO 
 
 
 
 
 20 'oo 
 
 . 
 
 9*OO 
 
 
 
 II'OO 
 
 
 
 
 g'oo 
 
 ^ 
 
 
 -- 
 
 ^^ 
 
 i oo 'oo 
 
 
 
 Check 
 
 II'OO 
 
 _ ^ 
 
 __^ ---"' 
 
 
 
 
 
 
 
 The best rules for precise levelling are 
 
 First. To adjust the line of sight truly horizontal, /. e. 
 tangential to the earth's circumference. 
 
 Second. Not to allow any sight to exceed 250 feet 
 distance with a 1 4-inch level or less with a shorter one. 
 
 Third. To keep the back-sights and fore-sights as nearly 
 as possible at equal distances. 
 
 Fourth. When back-sight and fore-sight are at unequal 
 distances to correct by table by adding to every sight the 
 tabular amount where it is not less than 'ooi. It will be quite 
 near enough to measure the distances with a passometer. 
 
 THEODOLITES 
 
 The plain theodolite for ranging base lines and curves 
 is the foundation of most tacheometers. The two types in 
 use are the Y and the transit. For ordinary work the 
 transit is very much to be preferred on account of the 
 liability of the line of sight to be shaken off its position when 
 
Instruments 305 
 
 reversing the telescope in its Y's ; the time taken in rever- 
 sing ; and the danger of leaving the clips loose and dropping 
 the telescope out when shouldering the instrument. 
 
 When telemetric work is the chief use of the instrument 
 the Y type has two advantages which bring it more into 
 competition with the transit. 
 
 i st. Within a considerable range of vertical arc, a tele- 
 scope of twice the focal length usually supplied can be 
 safely and steadily carried in specially constructed Y's. 
 
 2nd. The adjustment for collimation is made more 
 rapidly. 
 
 On the Hawaiian survey, the author used a seven -inch 
 Y theodolite by Elliott Brothers, carrying an eighteen -inch 
 telescope with eyepiece magnifying forty diameters. It was 
 furnished with stadia-hairs reading i per 100, and a 
 movable micrometer hair for long distances. It was a 
 heavy instrument, but, being of such long range, did not 
 want so much shifting. It was carried by one man, over 
 very bad country, and was quite satisfactory. 
 
 The adjustment of the Y theodolite is the same as the 
 Y level for parallax collimation and bubble. The zero is 
 then brought to coincide with the zero of the vernier of the 
 vertical arc when the bubble is at the centre of its run. This 
 is done by means of a small screw fastening the vernier of 
 the vertical limb to the vernier plate over the compass-box. 
 
 The adjustment of the horizontal limb is the same as 
 that for the transit theodolite. 
 
 STANLEY'S NEW PATENT TELEMETRICAL THEODOLITE 
 
 Stanley's New Telemetrical Theodolite, Fig. 90, has the 
 following improvements : A tribrach stage instead of four 
 parallel screws, which permits the instrument being used on 
 a wall without the legs, and prevents all possibility of strain- 
 ing the centre. 
 
 The instrument has a mechanical stage for shifting the 
 centre with exactness over the desired spot. 
 
 x 
 
Preliminary Survey 
 
 The telescope supports and the centre are in one casting, 
 also the size of the bearing on the centre has been increased. 
 
 Instead of webs, which are a constant anxiety to the 
 engineer or surveyor, platinum-iridium points are substi- 
 
 FIG. 90 
 
 tuted. These neither rust nor break, and allow of any dust 
 being brushed off with a camel-hair pencil without inter- 
 fering with the adjustment. 
 
 The eyepiece is also fitted with two vertical adjustable 
 points for measuring distances. 
 
Instruments 307 
 
 A long trough needle is supplied with each instrument 
 in place of the usual small circular compass. 
 
 The head of the tripod is of an improved form, giving 
 greater rigidity. 
 
 The transit-theodolite is the basis of the author's tacheo- 
 meter, and the following description will embrace all that is 
 required for the use and adjustment of the ordinary transit, 
 together with the special points of difference in his parti- 
 cular design. 
 
 CRIBBLE'S ' IDEAL ' TACHEOMETER 
 
 The author has been fortunate enough to be able to ex- 
 periment a good deal in instruments without so much out- 
 lay as most inventors in that line. He has been able to 
 dispose of his theodolites in foreign countries and then get 
 new ones. The ' ideal ' is the fifth instrument specially 
 designed by him for preliminary survey. There is really 
 nothing original about its principle, and yet in its actual 
 form and combination of parts it is unlike any other instru- 
 ment he has met with. 
 
 The principal features of novelty are : 
 
 First. A decimal subdivision of the ordiriary degree of 
 90 to the quadrant. 
 
 By this means all the advantages of the centesimal 
 graduation are obtained whilst retaining correspondence 
 with the published astronomical ephemeris. 
 
 Tables to five places of decimals for the trigonometrical 
 functions, together with logarithms of numbers to five places, 
 can be procured from Messrs. Ascher &Co., Bedford Street, 
 London. They are compiled by Dr. C. Bremiker, and cost 
 is. 6d. bound in cloth. They are the only tables extant for 
 this graduation. Transits can be adapted by merely changing 
 the vernier. 1 
 
 1 It is the original graduation of Briggs in 1633, followed by Roe 
 and Oughtred, in which they endeavoured to get rid of the senseless 
 sexagesimal subdivision, which survives in spite of them. 
 
 x 2 
 
308 
 
 Preliminary Survey 
 
 C adjusting screw to horizontal 
 
 limb 
 
 D compass-box 
 
 E clamping screw to external axis 
 F telescope 
 
 1 parallel-plate screws 
 N frames 
 
 R vertical arc 
 
 W tangent screw 
 
 a striding bubble 
 
 b micrometer head 
 
 c diagonal eyepiece 
 
 d levels of horizontal limb 
 
 c lantern 
 
 f rack-screw 
 
 g adjustment screws to level 
 
 h steel tape 
 
 z plummet 
 
 2 clip-screw and locknut 
 
 \\ 
 
Instruments 309 
 
 Slide-rules graduated decimally can also be obtained, at 
 the same price as for the sexagesimal subdivision, from either 
 Messrs. Tavernier & Gravet at Paris or Messrs. Davis & Sons, 
 Derby. 
 
 Curve-ranging and tacheometry are especially facilitated 
 by this graduation : see Chapter VII. 
 
 Secondly. The advantages of a glass diaphragm are 
 obtained without the usual drawbacks. The loss of light is 
 reduced to a minimum, and the diaphragm can be removed 
 without disturbing the micrometer diaphragm, if it should 
 be found necessary to clean it. Its adjustment does not 
 require the delicate handling which is needed for the micro- 
 meter hair, so that any one who can adjust a transit can 
 adjust this instrument with greater ease and in less time. 
 
 Thirdly. The instrument is a combination of the stadia 
 principle within the limits of legibility of the figures upon 
 the staff, together with the micrometer principle for sights at 
 longer range ; the level staff being adapted for use either 
 on the one principle or the other. 
 
 Fourthly. The telescope is of unusual power ; probably 
 no instrument has yet been constructed of the same lightness 
 and portability with a magnifying power of fifty diameters. 
 It is consequently equally well adapted to astronomical 
 observations and long-range sights of the staff. 
 
 Fifthly. It combines minuteness of levelling power with 
 lightness of construction by making the vertical arc 6 inches, 
 and the horizontal arc 5 inches diameter. Both arcs read 
 with the vernier to 'or degree, which is equal to 36 seconds, 
 but the vertical arc can be estimated by the eye to 18 seconds. 
 In tacheometry it is generally sufficiently accurate to read 
 the horizontal line to the nearest minute, which is more than 
 obtained by the five-inch circle, and the saving in weight is 
 considerable. The trunnion standards also are stayed. 
 
 Sixthly. It has a wide range of efficiency. 
 
 As an astronomical telescope, its power enables a good 
 observation to be taken of the phenomena of Jupiter's 
 
310 Preliminary Survey 
 
 satellites, or the culmination of one of the small stars in the 
 British Association Catalogue, used in conjunction with a 
 lunar transit for determination of the longitude. 
 
 As a micrometer telescope of long range it will determine 
 distances with considerable accuracy at several miles. As a 
 tacheometer it is of superior power to any the writer has yet 
 met with. 
 
 As a handy theodolite for ranging base-lines and railway 
 curves, it has none of the complications of the Wagner- 
 Fennel Tacheometer, the Eckhold Omnimeter, o.r the Porro 
 type of telemeter. It has the appearance of a medium sized 
 simple transit theodolite and can be 
 more easily adjusted. 
 
 The appearance of the diaphragm is 
 not so complicated as the usual tacheo- 
 meter. Only the axial lines are drawn 
 across the glass, the rest are replaced 
 by ticks which cannot possibly lead to 
 FIG. 92. confusion. 
 
 The ticks a and b are the. stadia 
 
 lines. The vertical ticks, or comb as they are called, are 
 spaces of 100 micrometer units, or 1,000 units on the 
 micrometer vernier. 
 
 A three-screw or ' tribrach ' stage has been adopted to 
 avoid strain on the pivot and expedite the adjustment. A 
 centering arrangement was at first included, but has been 
 left out of the last instrument made. Where the chief use 
 is for short bases with the chain, or curve-ranging, the 
 centering clamp is a convenience, but it weighs from 4 to 
 5 pounds, and in tacheometry the bases are long, and the 
 instrument is not so frequently shifted. Any small errors 
 from the width of the staff as a picket, or from inaccurate 
 centering, are eliminated by the observation for azimuth. 
 
 The addition of another lens as first introduced by Porro 
 to make the centre of anallatism (see Glossary) coincide 
 with the vertical axis has been abandoned. The constant 
 
Instruments 3 1 1 
 
 of the instrument is from 175 to 2-0 feet, according to size. 
 Practice has proved that no time need be lost nor any 
 mistake made in adding this constant. It is not worth any 
 extra expense or loss of light in order to eliminate it. 
 
 Instead of the ordinary compass-box, a trough-needle is 
 substituted, which saves the headway under the telescope, 
 and answers the purpose of obtaining the magnetic variation 
 at the commencement of the work equally well. 
 
 A word of caution is needed about the use of the tribrach 
 stage. A very grave defect exists in the form frequently 
 supplied to theodolites, namely, that the clamping plate is 
 not left attached to the parallel-plate screws when the 
 instrument is put away in the box. It forms a separate 
 piece. In addition to the extra trouble in packing away, it 
 is a great danger, inasmuch as there is no warning to the 
 surveyor, who may chance one day to leave his clamp open. 
 With the old-fashioned screw head, his instrument may be 
 loose and turn once or twice round without falling, but one 
 slip of memory with the clamping plate and it would be 
 sudden death to his instrument the moment he shifted it. 
 To remove this danger, a screw is put into the clamping- 
 plate, which has to be removed if the transit has to be set 
 on a wall. Comparatively few surveyors use it thus, and it is 
 generally preferable when taking observations extending over 
 several days to drive very solid pegs 4" x/j." into the ground 
 and cut notches for the feet of the tripod. At any rate, it 
 is so very seldom that the instrument requires to be taken 
 out of the clamping-plate, that it is a most mistaken plan 
 to risk the safety of the instrument in order to make it 
 easily removed. In the * Ideal ' tacheometer, the instrument 
 packs into its case in two pieces. 
 
 For astronomical observation, the usual diagonal eye- 
 piece and lamp for illuminating the axis are provided. 
 
 The plummet is suspended from a short chain, which is 
 a fixture with the instrument. It is preferable to a rigid 
 hook, which is liable to get bent. The hook of the chain 
 
312 
 
 Preliminary Survey 
 
 is strong enough to carry a heavy weight for steadying the 
 instrument in a high wind. The distance of the hook from 
 the centre of the trunnion forms with that from the initial 
 point of a 5 feet steel spring tape 1*50 feet, so as readily 
 to take the height of instrument as an independent check. 
 
 The box of the steel tape forms for ordinary use the 
 plummet, but for special accuracy a pointed plummet is 
 also provided. 
 
 ADJUSTMENTS OF THE 'IDEAL' TACHEOMETER 
 
 Supposing that the micrometer hair has broken, we will 
 commence with 
 
 i. Putting in a spider hair. Prepare the rectangular 
 frame shown in Fig. 93 of copper wire, T ^ diameter or 
 thereabout, soldered together. Catch a field spider and let 
 
 FIG. 93. 
 
 him drop by his web from one end, then wind round the frame 
 till full. Preserve it at the bottom of the level or transit case 
 in a little casing with blocking strips to keep the web from 
 rubbing off. It will then be ready for use at any time. The 
 points where the web touches the frame should be tipped 
 with shellac to fix them. 
 
 Place the diaphragm on the table with a strong light 
 bearing upon it. Tip with shellac the faint lines cut on 
 it to mark where the web should go. Superimpose the 
 web by delicately turning the frame until the web is in its 
 position. Hold it there till the shellac cakes, very gently 
 
Instruments 3 1 3 
 
 pressing the web, so that it shall be quite taut. Remove 
 the frame, and the web will be left in position. 
 
 2. To replace the diaphragms. To put in the glass 
 diaphragm requires no description. To replace the mov- 
 able diaphragm, hold it by a pair of tweezers while insert- 
 ing the capstan-headed screws. Turn the screws until the 
 hair is as nearly vertical and as near the middle of the tube 
 as can be estimated by the eye. Leave the micrometer 
 screw till the last thing. Slide the movable hair to one side 
 so as to be out of the way. 
 
 Insert the eyepiece. 
 
 3. To remove parallax. See p. 294. 
 
 4. To make the line of sight correspond with the zero of the 
 horizontal limb. Set the instrument as nearly level as can be 
 done with the eye, then clamp the lower plate B, and, having 
 undamped the vernier-plate A, direct the telescope on some 
 well-defined object, and bring it into coincidence with the 
 point of intersection of the axial lines on the diaphragm. 
 Take the reading on the horizontal limb AB ; suppose it to 
 be 20 'oo \ then move the vernier-plate A half-round, turn 
 the telescope over, and again intersect the object, taking the 
 reading on the horizontal limb AB suppose it 2oo'04 ; take 
 the difference between this and the first reading +180 
 (which in the present case would be 200, and the differ- 
 ence '04) ; halve this difference, and subtract it from the 
 second reading when it is greater than the first reading + 
 1 80, and add it when it is less ; this is the mean reading 
 (=200 -02). Set and clamp the instrument to this mean 
 reading, and intersect the object by means of the screw 
 which moves the glass diaphragm, sideways. Repeat this 
 operation until the readings taken with the instrument in 
 these two different positions, face right and face left, differ 
 from one another by 180. 
 
 5. To make the line of sight correspond with the zero of 
 the vertical limb. It is not necessary that the line of sight 
 should also be identical with the optical axis. See remarks 
 
314 Preliminary Survey 
 
 on levels, p. 291. Parallelism is sufficient. Level the 
 instrument carefully on the external axis, by means of the 
 levels 4 d on the horizontal limb AB ; next take a pair of 
 verticals i.e. on faces right and left to any well-defined 
 terrestrial object ; set the vertical circle R to the mean 
 of these readings, and clamp it ; now intersect the object, 
 using the two screws z, z which clip the vertical circle 
 R to the stud in the telescope frames N, N, and not 
 the tangent screw W. When the readings on the right 
 face agree with the left face, the index error will be o. 
 The clip-screws z, z are provided with capstan-headed 
 locknuts, which should then be screwed home, as these 
 screws are not used except for adjustment. The leather 
 eaps on the nuts of the clip-screws are to prevent their 
 being moved by mistake instead of the tangent screw. 
 One of them need never be touched. It can be marked, 
 and only the other one used for releasing the telescope, so 
 that the locknut on the untouched clip-screw will bring 
 the vertical limb to its proper position when it is replaced 
 without fresh adjustment. 
 
 Example. With the telescope in its normal position 
 observed the top of church spire. Vertical angle +n'i3. 
 Rotating the horizontal limb and reversing the telescope, 
 vertical angle = + n'i9. Set the vertical limb at n'i6 
 by the tangent screw W, and make the line of sight intersect 
 the object by means of the clip-screw, z. Then reverse the 
 telescope, and rotate the horizontal limb to their original 
 positions, and the vertical arc will still read n'i6. If not 
 quite exact repeat the operation. 
 
 6. To make the plane of rotation of the vernier plate 
 horizontal. Clamp the vertical limb R at o. Tighten 
 the clamp-screw E, unclamp the vernier-plate A, and turn 
 it round until the telescope is immediately over two of 
 the parallel -plate screws I, I ; bring the bubble in the 
 telescope-level P to the middle of its run by the screws 
 I, I ; turn the vernier-plate 1 80 so as to bring the telescope 
 
Instruments 3 1 5 
 
 again over the ,same screws, but with its ends in a reverse 
 position. If the bubble of the telescope level does not 
 remain in the middle of its run, bring it back to that 
 position, half by the parallel-plate screws I, I, and half by 
 the screws g, g. This operation must be repeated until the 
 bubble remains accurately in the centre of its run in 
 both positions of the telescope ; now turn the vernier- 
 plate A until the telescope is directly over the third parallel- 
 plate screw, and bring the bubble to the middle of its run 
 by turning that screw. The bubble .should now retain its 
 position while the vernier-plate is turned completely round, 
 showing that the plane of rotation is truly horizontal or 
 parallel to the axis of the bubble. 
 
 7. To test the accuracy of the external axis. Clamp the 
 vernier-plate to the lower plate by turning the clamp-screw 
 C, and loosen the clamp -screw E ; move the instrument 
 round its external axis, and if the bubble retains its central 
 position during a complete revolution there is no error. If 
 one exists it is only to be remedied by the maker. If, how- 
 ever, adjustment 6 is correct, the work will be correct. 
 
 8. To adjust the levels on the vernier plate. These are 
 only guides for approximate adjustment of the instrument 
 when it is being set up. They are set true when the other 
 adjustments are completed. 
 
 9. Horizontally of the axis of the telescope. This is done 
 with the striding bubble. See full description under levels, 
 p. 292. 
 
 10. To test whether the vertical axial hair has been placed 
 in a vertical line. This adjustment is not required in the 
 ' Ideal ' tacheometer, but is added for the use of those who 
 have ordinary transits. It is not essential to correct work if 
 the intersection of the hairs is always observed, but it is 
 convenient to be able to intersect an object with any part of 
 the vertical line, and it is also useful for telling if a staff &c. 
 is held vertical. 
 
 Fix the intersection of the axial lines on some sharply 
 
Preliminary Survey 
 
 defined terrestrial object, such as the lightning conductor of 
 a building, or, failing any similar point, drive a stake 100 
 paces off and put a nail in. Move the vertical arc up and 
 down and watch whether the intersection point remains on 
 the vertical line ; if not correct it by the diaphragm screws, 
 and check over the former adjustments in case they may 
 have become deranged. 
 
 Other and very delicate adjustments are needed for 
 special purposes, but the above are sufficient to put the 
 instrument in ordinary working order. 
 
 11. To place the micrometer hair in position. Turn the 
 spindle of the screw with the micrometer head removed 
 until the hair coincides with the vertical axial line ; put on 
 the micrometer head very carefully so that the zero is at the 
 zero of the vernier. This needs practice and generally 
 takes a few trials. 
 
 12. To check the stadia lines. From the vertical axis ot 
 the instrument determined by the point of the plummet 
 
 > < 
 
 FIG. 94. 
 
 measure out on the ground the position of the anterior focus, 
 that is a distance composed of the focal length of the object 
 glass + the length from the vertical axis to the object glass. 
 This is the centre of anallatism or point from which heights 
 subtended by the stadia are proportional to the distances of 
 the objects. (See Chapter V .) In the ' Ideal ' /= 1 2 inches 
 and d% inches, therefore /+ d i *6 7 feet. Hold up an 
 
Instruments 317 
 
 ivory scale against a wall or house 80 or 90 feet away, 
 and direct the stadia upon it. Suppose it is a sixty 
 scale and the stadia subtend 662 divisions. Place a 720 
 on the rule of the slide-rule, representing the value of 
 a foot in divisions of a sixty scale opposite a i of the slide, 
 and find the value in feet "9195 opposite to the 662 on the 
 rule. Then with the steel tape measure from the centre of 
 anallatism to the wall. If the stadia are set to i per 100, 
 this ought to measure 91*95 feet. 
 
 Suppose it measures 92-05 feet. Set the 91*95 feet op- 
 posite the 92 '05 feet on the slide-rule, and any distance can 
 then be read off direct, by looking from one scale to the 
 other and adding the constant. Practically the stadia lines 
 are correct, and as it is not required to plot closer than a foot 
 the constant of 2 feet is added to the number of hundreds 
 thus subtended on the staff as the distance from the centre 
 of the instrument. 
 
 Another method when plotting to a large scale is to draw 
 a circle with a radius =/+ Ground the trigonometrical station, 
 and plot the distances (ex. the constant) from the circle. 
 
 A check should be made on the foregoing computation 
 of stadia by measuring a base of about 600 feet on level 
 ground and reading the stadia there also. 
 
 13. To form a table of micrometer values with a ten-foot 
 base. For this a long base on a tangent is needed ; a piece 
 of level railway track is the best. The instrument can be 
 set up a little beyond the B. C. point of the curve so as to be 
 well off the track and out of the vibration of passing trains 
 and yet in line with the tangent. 
 
 Beginning at 500 feet the staff is held in position 
 with the vanes up, and the micrometer value is measured 
 two or three times over. 
 
 At each 100 feet up to about 6,000 feet, fresh readings 
 are taken and booked. They are then plotted as follows. 
 A differential curve is prepared by laying off on a sheet 
 of drawing paper a horizontal base equal by scale to the 
 
318 Preliminary Survey 
 
 total measured length, and at every 100 feet measuring up 
 ordinates equal to the micrometer values. Between the 
 hundreds, the values are interpolated by ordinates to the 
 curve down to the tens, and the units are then interpolated 
 by simple proportion with the slide-rule. 
 
 The curves are put in with ordinary railway sweeps. 
 
 This will be found quite as accurate as more frequent 
 observations in the field ; as a matter of fact, the curves will 
 soften several asperities due to eccentricities of observation. 
 
 Printed profile paper with faint scale lines is convenient 
 for this purpose. 
 
 The ' Ideal ' reads the level staff with the stadia up to 
 1,000 feet ; beyond that the micrometer is used. 
 
 A close approximation to the actual magnifying power 
 of the telescope may be very simply obtained in the follow- 
 ing manner. Set it up at a distance of 20 feet from the 
 anterior focus and cause a levelling-staff to be held there 
 perfectly steady. The stadia will then subtend '2 upon the 
 staff. Direct the lower stadia hair to some convenient 
 figure such as 5-00 feet, then the upper hair will be at 5-20. 
 Open the other eye and look with the one eye through the 
 telescope, and the other unassisted at the staff, until the 
 actual staff appears to be superimposed upon the magnified 
 portion and you seem to see the one through the other. 
 Book the two positions of the stadia hairs on the actual staff. 
 For instance, if the eyepiece magnified 20 diameters, the 
 stadia hairs would appear to cover 2ox'2=4 feet of the 
 natural staff. The lower would then appear to be at 3*10 
 and the upper at yio feet. 
 
 If there are no stadia hairs, take the top and bottom of 
 the field and proceed similarly. 
 
 THE SKETCH-BOARD PLANE-TABLE 
 
 Fig. 95 represents the present improved type of cavalry 
 sketching-board as modified from Col. Richard's original 
 design by Capt. Willoughby Verner, whose field-sketching 
 
Instruments 
 
 319 
 
 FIG. 95. 
 
320 Preliminary Survey 
 
 and reconnaissance in Egypt form an interesting feature of 
 the Soudan Campaign. 
 
 The paper is in a continuous roll, wound round the 
 brass rollers, and turned down under them so as to be 
 always quite tight. It is to a large extent waterproof, being 
 specially prepared, and graduated in quarter-inch squares. 
 
 The line marked ' line of direction ' is that in which the 
 paper is unrolled so that when the line of march does not 
 deviate very much from it, the work will not run off the 
 board. 
 
 The thin line drawn across the compass-box is a cut in 
 the glass, the object of which is to indicate, by turning the 
 compass-box round, the position of the needle, when the 
 board is in the line of direction. The needle, being a loose 
 one, is first observed before starting, the board being set to 
 the line of direction. The compass-box is then turned until 
 the cut in the glass coincides with the needle, and serves 
 ever after to bring back the board into the line of direction. 
 
 This is further explained on p. 41. 
 
 The heads to the brass rollers fit tight in the wood and 
 keep the paper stretched by their friction. In the" cross 
 section the buckle is shown attached to a dovetailed slide 
 and pivot. When used on horseback, the strap fastens the 
 board to the arm, and when fastened to a tripod, the dove- 
 tailed plate runs into a corresponding slide on the tripod. 
 The seven-inch by nine-inch size is more of a sketchboard 
 and the nine-inch by thirteen-inch size is more of a plane- 
 table, but both sizes are used in either way effectually. 
 
 POCKET ALTAZIMUTHS AND COMPASS-CLINOMETERS. 
 
 Strictly speaking an altazimuth is an instrument which 
 gives the altitude and azimuth from one adjustment of its 
 line of sight. 
 
 The first altazimuth was a stationary transit instrument ; 
 what is commonly called now a wall transit. 
 
 Three-screw transit theodolites are now adapted to being 
 
Instruments 
 
 321 
 
 placed on walls ; but the term altazimuth has also become 
 applicable to pocket instruments, in which the azimuth is 
 found by the magnetic bearing and the angle of elevation or 
 depression is read by means of a plummet or a level. 
 
 Another variety might be designed combining the pocket 
 sextant with the measurement of altitudes. Possibly there 
 may be such already. 
 
 Messrs. Casella make an altazimuth in which the mag- 
 netic bearing and altitude by plummet are both read by a 
 microscope and the observation is assisted by a telescope. 
 
 FIG. 96. 
 
 Messrs. Elliott make Colonel O'Grady Haly's compass- 
 clinometer in which the same operations are performed 
 without any lens power. 
 
 All these instruments are practically two instruments in 
 one, to save the time and trouble of taking out first one and 
 then another from separate sling-cases. 
 
 Hand instruments are better without telescopes. Work 
 requiring telescopic accuracy is better done with a tripod. 
 
 A combination of the box sextant with the altitude 
 measurer would have the advantage of greater precision in 
 the horizontal angles and fully as much despatch. On the 
 other hand it would be subject to cumulative error, which 
 
 Y 
 
322 Preliminary Survey 
 
 the compass is not. It would be free, however, from magnetic 
 deviation, which sometimes renders the compass wholly 
 unreliable. 
 
 The plummet makes a slow clinometer and not a sure 
 one either. If the instrument is not held in a vertical 
 plane, the plummet is apt to stick against the side. 
 
 The writer has used nothing of the kind of late years, 
 but confines himself to an arrangement of his own, con- 
 sisting of a combination of Captain Abney's reflecting level 
 with a prismatic compass. 
 
 This is illustrated in Fig. 96. The Abney level is now 
 so well known that it only needs a passing description (see Fig. 
 
 FIG. 97. 
 
 97). Its principle is somewhat that of the vertical arc of a 
 theodolite, only it is in miniature and without lens power. A 
 small telescope has been tried but not with success. A re- 
 flector occupies half the field of the sighting-tube, adjusted 
 at an angle so that when the bubble is at the centre of its 
 run, its reflection is seen in the centre of the tube. 
 
 Whatever angle is given to the line of sight, the bubble can 
 be brought to the centre of its run without removing the eye 
 from the tube, and the little handle which moves the bubble 
 tube is furnished with an arm and vernier, which indicates 
 the angle of altitude upon a graduated semicircle. The 
 range of view of the bubble through the tube is only up to 
 60, but the author uses his instrument also up to 90 for 
 getting the batter of a wall or the side slope of a steep 
 cutting by placing it on a straight edge, and then bringing 
 
Instruments 323 
 
 the bubble to the centre of its run without looking through 
 the tube. He therefore has the circle graduated up to 90, 
 and often uses it as a 'plumb-level.' 
 
 The prismatic compass forms the handle, which turns 
 the bubble, in place of the usual little brass wheel. Its 
 disc is only i \ inch, but it has a graduation from o to 360 
 and also aquadrantal graduation for working by latitude and 
 departure. The prism magnifies so well that the graduation 
 of single degrees can be easily subdivided by estimation 
 to quarter degrees. It is made by Messrs. Elliot Bros., and 
 does them credit. 
 
 The further use of this instrument is explained in ' Route 
 Surveying,' pp. 61, 64, &c. 
 
 PASSOMETERS 
 
 The passometer is a register of the number of paces 
 taken at any time by a walker ; its use has been explained 
 on p. 54, &c. The graduation of passometers is often very 
 clumsy. The arrangement in Fig. 98 is recommended as 
 being easily readable. The right-hand small dial reads up 
 to 20,000, which is over ten miles. The large hand gives 
 the fifties up to 2,000, and the left-hand small dial the units 
 between the fifties. 
 
 The instrument in Fig. 98 is reading 0,558 paces. The 
 hands of passometers are generally set loose, so that they 
 can be adjusted to zero. For surveying this is a mistake, 
 because they soon get out of teaching with one another. 
 A permanent fit is best that is to say, with square bearings 
 instead of conical, like the hands of a watch. 
 
 The passometer should be attached to the centre of the 
 person ; if placed on one leg it will only count half paces. The 
 best place is hanging from a waistcoat button, with the hook 
 well buttoned in to be secure, and just kept from shaking up 
 and down by the edge of the waistcoat. If too free it will 
 occasionally made a double count. When the counting 
 
 Y 2 
 
324 
 
 Preliminary Survey 
 
 has to be suspended for a while, the passometer can be turned 
 upside down. 
 
 The pedometer and passometer can be made to work 
 together, by adjusting the former until it records a quarter 
 
 FIG. 98. 
 
 mile against the average number of paces to that distance 
 by the latter. The ordinary adjustment of the pedometer to 
 length of pace in inches is not always exact. 
 
 SEXTANTS 
 
 Both the Hadley sextant and the box sextant are so 
 well known that only the adjustments will be here given, 
 which are taken from Mr. Heather's excellent little work on 
 * Instruments.' 
 
 To examine the error arising from the imperfection of the 
 dark glasses. View the sun through the dark glass at the 
 end of the telescope, removing the shades ; make a contact 
 with the reflected image of the sun and its direct image 
 seen through the unsilvered part of the horizon glass. Then, 
 removing the dark glass, set up one shade glass after the 
 other, and book any alterations of angle due to each succes- 
 
Instruments 325 
 
 sive combination. No adjustment can be made for this error ; 
 when registered it has to be applied at every observation. 
 
 The adjustments consist in setting the horizon glass 
 perpendicular to the plane of the instrument, and in setting 
 the line of collimation of the telescope parallel to the plane 
 of the instrument. 
 
 To adjust the horizon glass. While looking steadily at 
 any convenient object, sweep the index slowly along the limb, 
 and, if the reflected image do not pass exactly over the direct 
 image, but one projects laterally beyond the other, then the 
 reflectors are not both perpendicular to the face of the limb. 
 Now the index glass is fixed in its place by the maker, and 
 generally remains perpendicular to the plane of the instru- 
 ment ; and, if it be correctly so, the horizon glass is adjusted 
 by turning a small screw at the bottom of the frame in 
 which it is set, till the reflected image passes exactly over 
 the direct image. 
 
 To examine if the index glass be perpendicular to the 
 plane of the instrument. Bring the vernier to indicate about 
 45, and look obliquely into this mirror, so as to view the sharp 
 edge of the limb of the instrument by direct vision to the 
 right hand and by reflection to the left. If, then, the edge 
 and its image appear as one continued arc of a circle, the 
 index glass is correctly perpendicular to the plane of the 
 instrument ; but if the arc appears broken the instrument 
 must be sent to the maker to have the index glass adjusted. 
 
 To adjust the line of collimation. i. Fix the telescope 
 in its place, and turn the eye-tube round, that the wires in 
 the focus of the eye-glass may be parallel to the plane of the 
 instrument. 2. Move the index till two objects, as the sun 
 and the moon or the moon and a star more than 90 dis- 
 tant from each other, are brought into contact at the wire of 
 the diaphragm which is nearest the plane of the instrument. 
 3. Now fix the index, and altering slightly the position of 
 the instrument, cause the objects to appear on the other 
 wire, and if the contact still remain perfect the line of col- 
 
326 Preliminary Survey 
 
 limation is in correct adjustment. If, however, the two 
 objects appear to separate at the wire that is further from 
 the plane of the instrument, the object end of the telescope 
 inclines toward the plane of the instrument ; but if they 
 overlap, then the object end of the telescope declines 
 from the plane of the instrument. In either case the correct 
 adjustment is to be obtained by means of the two screws, 
 which fasten to the up and down piece the collar holding 
 the telescope, tightening one screw and slackening the other, 
 till, after a few trials, the contact remains perfect at both 
 wires. 
 
 The instrument having been found by the preceding 
 methods to be in perfect adjustment, set the index to zero, 
 and if the direct and reflected images of any object do not 
 perfectly coincide, the arc through which the index has to be 
 moved to bring them into perfect coincidence constitutes 
 what is called the index error, which must be applied to all 
 observed angles as a constant correction. 
 
 To determine the index error. The most approved method 
 is to measure the sun's diameter, both on the arc of the 
 instrument properly so called, to the left of the zero of the 
 limb, and on the arc of excess to the right of the zero of the 
 limb. For this purpose, firstly, clamp the index at about 
 30' to the left of zero, and, looking at the sun, bring the 
 reflected image of his upper limb into contact with the direct 
 image of his lower limb, by turning the tangent screw, and 
 set down the minutes and seconds denoted by the vernier ; 
 secondly, clamp the index at about 30' to the right of zero 
 on the arc of excess, and, looking at the sun, bring the 
 reflected image of his lower limb into contact with the direct 
 image of his upper limb by turning the tangent screw, and 
 set down the minutes and seconds denoted by the vernier 
 underneath the reading before set down. 
 
 Then half the sum of these two readings will be the 
 correct diameter of the sun, and half their difference will be 
 the index error. When the reading on the arc of excess is 
 
Instruments 327 
 
 the greater of the two, the index error then found must be 
 added to all the readings of the instrument, and when the 
 reading on the arc of excess is the less, the index error must 
 be subtracted in all cases. 
 
 To obtain the index error with the greatest accuracy, it 
 is best to repeat the above operation several times, obtaining 
 several readings on the arc of the instrument, and the same 
 number on the arc of excess, and the difference of the 
 sums of the readings in the two cases, divided by the whole 
 number of readings, will be the index error ; while the sum 
 of all the readings divided by their number will be the sun's 
 diameter. 
 
 Example. 
 
 Readings on the arc of the instrument Readings on the art of excess 
 
 35' o" 29' 25" V, 
 
 35' 5" 29' 35" 
 
 35' 10" 29' 20" 
 
 105' 15" 88' 20" 
 
 88' 20" 105' 15" 
 
 2' 49" Index error 
 
 6)193' 35" Sum 
 
 32' 15-8" Sun's diameter 
 
 THE ADJUSTMENTS OF THE Box SEXTANT 
 
 On the upper surface of the instrument close to the rack- 
 screw is a little hole with the square head of a screw inside 
 it. This is a screw for adjusting the horizon glass in the 
 plane of the instrument by observing the reflected image of 
 the sun. 
 
 There is also another screw at the side of the instrument 
 for removing index error by taking readings on the arc of 
 the instrument, and on the arc of excess as described in the 
 adjustments of the Hadley sextant, and then by correcting 
 with the screw so as to make both readings the same. 
 
328 Preliminary Survey 
 
 Both screws are turned by a little key, which is removed 
 for the purpose from its position in the arc-plate opposite 
 the rack screw. 
 
 THE SOLAR COMPASS 
 
 The solar compass is an instrument for determining the 
 true bearing of any object, instead of the magnetic. It 
 performs automatically the operation of finding at any time 
 of day and in any latitude the azimuth of the sun. It can- 
 not of course be used in cloudy weather, and should not be 
 used when the sun is less than one hour above the horizon 
 or less than one hour from the meridian. It stands on a 
 tripod of the same size, and is itself somewhat larger than 
 the ordinary surveyor's compass. 
 
 It is furnished with vertical arcs to set it to the latitude 
 of the place, and a pair of sight vanes. 
 
 Unless the adjustments are very carefully attended to 
 errors are likely to arise, greater than those due to local 
 magnetic deviation. 
 
 The price ranges somewhat between that of a surveying 
 compass and a plain transit. 
 
 THE HELIOSTAT AND HELIOGRAPH 
 
 These instruments are both sun-signals ; the principle is 
 the same in both. A mirror reflects a sunbeam from one 
 point to another. The heliostat is only intended to give a 
 continuous ray, whereas the heliograph is provided with a 
 spring to the mirror, by which it is made to give little jumps, 
 causing it to flash short or long flashes corresponding with 
 the dots and dashes of the Morse code. The heliostat is 
 capable of being used for signalling also, by alternately 
 covering and uncovering the mirror. Both instruments are 
 furnished with sighting vanes on jointed arms, commonly 
 called jiggers, to set the mirror in line with the station to 
 be signalled. The heliograph is also furnished with a duplex 
 
Instruments 329 
 
 mirror, so that when the sun is behind the instrument it 
 can still be used. 
 
 The best heliographs are Galton's sun signal, fitted with 
 a telescope, and Mance's heliograph. Both are expensive 
 instruments, and need not enter into the outfit of the 
 preliminary surveyor. 
 
 The sketch on next page is taken from Capt. Wharton's 
 'Hydrography,' } and illustrates the principle of all such in- 
 struments. It is a heliostat made by a ship's blacksmith. 
 The standard is about 2 J- feet high. ' In soft ground the 
 ends of the legs can be pressed into the earth, and on rocky 
 ground stones placed against the legs will hold the instru- 
 ment steady. The arm ;;/, of light iron, is carried separately, 
 and slips over the shaft of the standard, clamping when 
 required with a screw. 
 
 ' Into a circular socket in head of standard shaft the leg 
 of the frame holding the mirror is shipped ; this is also to 
 be tightened by a retaining screw. The mirror, which can 
 be of any size from 2 to 6 inches or more in diameter, 
 revolves on its retaining screws as an ordinary toilet- 
 table glass, and can be held in any position by the 
 screws. 
 
 ' The ring, of flat wood, is made as light as possible, so as 
 to exert less strain in wind. Across it are nailed crossed 
 strips of copper with a white cardboard disc, about an inch 
 in diameter, fastened to their centre. 
 
 ' The rod that carries this ring slips up and down in a 
 hole at the end of the arm, and is clamped by a retaining 
 screw. 
 
 ' In the centre of the back of the mirror a hole of about 
 I inch diameter is scraped in the tinfoil, being careful to 
 leave a sharp edge. A similar hole is cut out of the wooden 
 back of the glass frame. This we shall call the blind 
 spot. 
 
 1 A cheap instrument of this description is made by Potter of 
 London. 
 
330 
 
 Preliminary Survey 
 
 ' To direct the flash to an object, bring the mirror vertical, 
 and looking through the hole in the centre, revolve the arm 
 until in the direction of the object nearly, clamp it, and 
 adjust the disc-rod as nearly as may be for elevation or 
 depression. Then, slightly loosening the screw, clamping 
 the arm, finally adjust the latter so that the object, as 
 regarded through the hole in the mirror, is obscured by the 
 
 FIG. 99, Captain Wharton's ' Hydrography.' 
 
 a, sliding collar carrying arm in, revolving round s ; b, wooden ring, 
 painted black, with iron wires and white cardboard centre, sliding 
 vertically by means of rod through arm m ; c, iron frame to hold 
 mirror, fitting into socket in top of standard j ; s, iron standard with 
 fixed tripod legs ; d, blind spot in mirror ; e, screw for clamping in 
 iron frame ; f, screw for clamping arm ; g; screw for clamping ring 
 rod. 
 
 white cardboard disc in centre of the ring. By turning the 
 mirror so that the dark shade caused by the blind spot is 
 thrown on to the disc, the flash will be truly directed, and 
 must be kept so by slight alterations of the position of the 
 mirror, which should therefore be clamped only sufficiently 
 to hold it steady and yet admit of gentle movement. The 
 shadow of the blind spot should be slightly smaller than 
 
Instruments 331 
 
 the disc, so as to ensure having it truly in the centre of the 
 latter. 
 
 ' The mirror must be of the best glass, with its faces 
 parallel, or the shadow of the blind spot will be very indis- 
 tinct when the mirror is at a large angle, and also the beam 
 of light will be dispersed before it has traversed many 
 miles. 
 
 ' It is well to have the mirror a fair size, say 6 inches 
 square, as in practice it will be found generally necessary, 
 in order to save time, after once adjusting the flash, to leave 
 a man to keep it on while the surveyor is taking his angles ; 
 and although a man will soon pick up the knack, a larger 
 mirror will allow for eccentricities on his part, and also, on 
 a dull day, a faint flash will be detected from a large 
 mirror, where a small one would not carry any distance. 
 
 * On a bright day a flash from a 3 inch by 2 inch mirror 
 has been seen 55 miles and more. 
 
 ' In hazy weather, angles have been got when the place 
 from which the flash was sent was entirely invisible ; and 
 
 FIG. roo. 
 
 a, a, bent wire ; l>, l>, brightened bullets ; /, JT, line of sight to the 
 Tugela. 
 
 thus whole days have been saved by this simple contriv- 
 ance. 
 
 'Only those who have spent hours, or even days, in 
 straining their eyes to see a distant mark can appreciate the 
 value of a heliostat.' 
 
 The following is an extract from an article in ' Science 
 
332 Preliminary Survey 
 
 for All,' by Major C. Cooper King, of the Royal Military- 
 College, Sandhurst, and explains a handy makeshift shown 
 on Fig. 100. 
 
 'At Ekowe, in the campaign in Zululand, the whole 
 apparatus had to be improvised. . . . Two wires with the 
 upper parts bent into the form of a semicircle, and with 
 cross wires uniting the bent end with the upright part, were 
 tried. The sights were composed of brightened bullets in 
 the centre of the cross wires, and when these rods were set 
 in the ground and aligned with the flash from a common 
 looking-glass, no difficulty was experienced in communi- 
 cating.' 
 
 Sun signalling has the great advantage of being indepen- 
 dent of background. 
 
 TELEMETERS AND RANGE-FINDERS 
 
 The term telemeter, which was introduced by surveyors, 
 has been appropriated to some extent by electricians and 
 others ; and it is probable that the better term tacheometer, 
 which has been quite adopted by the French, will soon 
 become general. 
 
 The simplest form of telemeter is the plane-table, which 
 is a graphic triangulation to attain the same end as the 
 optical telemeter. 
 
 In every case the telemeter measures by a kind 01 
 triangulation, varying from the long base, bearing a con- 
 siderable proportion to the distance to be measured, down 
 to the base of a few inches or at most a few feet, from 
 which distances of over a mile are measured. 
 
 Then, again, telemeters vary in size from the Weldon 
 locket range-finder, about -J inch in diameter, to the 6 feet 
 and 7 feet 6 inch telescope-telemeters of Clarke and Struve. 
 
 Organically they may be divided into three classes. 
 
 i . Those in which the measured base forms an integral 
 part of the instrument itself. Such are Adie's 18 inch and 
 3 feet telemeters, described in Heather's ' Instruments ; ' 
 
Instruments 333 
 
 Piazzi Smyth's 5 feet telemeter, Colonel Clarke's 6 feet, 
 and Otto Struve's 7^ feet telemeter. 
 
 The instrument is held square with the line of sight. 
 The object whose distance has to be measured must be 
 sharply defined. One end of the instrument receives a re 
 fleeted image at right angles, and the other end a reflection 
 at an angle forming the complement of the angle subtended 
 by the length of the instrument at the point observed. 
 
 With so great a multiplication as this, it is evident that 
 the base must be very exactly measured in order to graduate 
 the instrument. 
 
 Adie's telemeter is of brass, and covered with leather in 
 order to diminish changes of length under varying tempera- 
 tures ; but all this class suffer from this cause. They are 
 not much used now. 
 
 2. Those in which the measured base is at the point 
 observed, generally consisting either of a graduated staff or 
 a pair of discs connected by a rod ; the naval surveyors call 
 it a 10 feet pole, the Americans a target 
 
 In this class there are two subdivisions. 
 
 a. Those which have a fixed base and varying angle, as 
 the Rochon micrometer, furnished with a reflector by which 
 the images of the discs are made to coincide, as in a sextant, 
 and the angle subtended is read in terms of distance. 
 
 Messrs. Elliott's army distance-measuring telescope is 
 another of this class. The fixed base is the assumed height 
 of infantry, 5 feet n inches, or cavalry 8 feet 10 inches. 
 Two micrometer wires are fixed in the diaphragm, and are 
 actuated by the micrometer head so as to exactly intercept 
 the object. The micrometer head is graduated differentially 
 in terms of distance ; one side for the cavalry base, the 
 other for the infantry. Targets set at those respective dis- 
 tances on a pole would of course give more exact results. 
 
 Eckhold's omnimeter, Fig. 101, also made by Messrs. 
 Elliott Bros., is a combination of this and the first class 
 alluded to. The instrument is a transit theodolite in prin- 
 
334 
 
 Preliminary Survey 
 
 ciple, but it is furnished with a powerful telescope and a 
 long and powerful microscope (a\ reading a graduated 
 baseplate (). At one operation, by observing top and 
 bottom of a 10 feet pole, the distance and elevation are 
 given by a rule-of-three sum. The instrument has given 
 much satisfaction, both in India and the Colonies ; it is, 
 however, more complicated than the stadia principle, takes 
 longer to adjust, and is not more accurate. The author is 
 
 FIG. 101. 
 
 informed by Messrs. Elliott that an improvement upon this 
 instrument is in course of development by Mr. W. N. 
 Bakewell. 
 
 b. Telemeters which have a variable base and a fixed 
 angle. This is the stadia principle, and the commonest 
 instruments of this type are the stadia or telemetric theo- 
 dolite and the tacheometer. The former instrument is 
 illustrated on p. 307 in a new form by Mr. W. F. Stanley. 
 Theodolites of all sizes are now frequently fitted with stadia- 
 
Instruments 335 
 
 lines. The tacheometer of Messrs. Trough ton & Simms 
 is graduated to 400 primary degrees and a decimal sub- 
 division. Tables of the trigonometrical functions are pub- 
 lished for this graduation. Any kind of decimal graduation 
 affords great facility both to astronomical calculations and 
 telemetry. 
 
 For the author's system of graduation, see p. 307. 
 
 Another feature is the obtaining of what is termed anal- 
 latism (see Glossary), or unchangeableness, at the vertical 
 axis. There is very little practical advantage from this device, 
 which involves another lens. A micrometer screw can of 
 course be added if desired, but it does not usually form an 
 integral part of the tacheometer. 
 
 An instrument of this kind with more novelty about it is 
 the Wagner-Fennel tacheometer. 
 
 All the instruments previously described under this class 
 require the calculations of vertical height to be made by 
 tables or by slide-rule, but this instrument gives them direct, 
 without even the rule-of-three sum required by the Eckhold 
 omnimeter. 
 
 The measurement is by stadia hairs, but the movement 
 of the telescope in a vertical plane instead of being recorded 
 upon a graduated circle causes a vernier to slide backwards 
 or forwards along a scale fixed parallel to the line of 
 sight. 
 
 This vernier actuates a pair of other scales, one hori- 
 zontal and the other vertical, thus giving directly the 
 horizontal and vertical components of the inclined distance 
 measured by the stadia. 
 
 The instrument is highly ingenious, but it is not so 
 suitable for the ordinary work of the railway engineer, such 
 as setting out curves, &c., and it is not adapted to astro- 
 nomical observations. 
 
 3. The third class of telemeters is that in which the 
 base is measured on the ground, at the observer's station. 
 The Hadley sextant, "^though not, strictly speaking, a tele- 
 
336 
 
 Preliminary Survey 
 
 meter, is greatly used as such by naval hydrographers, 
 sometimes in conjunction with a Rochon micrometer. 
 
 The Dredge-Steward omnitelemeter, made by J. H. 
 Steward of London, illustrated and described in ' Engineer- 
 ing,' of August 20, 1886, is a good instrument of this class. It 
 is in appearance and in principle very much like a box sextant, 
 stands on a light tripod, and measures the angle formed at 
 the distant point by the pair of rays from the ends of a 
 base run out by steel tape at the observer's station. It has 
 this special feature, that the base need not be run out 
 square, which saves time when there are obstructions. 
 
 FIG. 102. 
 
 FIG. 103. 
 
 Perhaps the most suitable of any of this class of instru- 
 ments to the ordinary surveyor is the Weldon range-finder, 
 the patent of Colonel Weldon, R.A. There is indeed a 
 new range-finder coming out from Woolwich, but the mili- 
 tary authorities are not as yet communicative upon the 
 subject. 
 
 Figs. 102, 103 represent the watch-size ' Weldon ; ' there 
 is also a ' locket ' size. The following description is mainly 
 from the pamphlet written on the subject by Captain Wil- 
 loughby Verner, R.A. : 
 

 FIG. 104. Range-taking with a 
 
 direction point (using first and 
 
 second prisms). 
 A, position of observer. 
 O, object of which the range is 
 
 required. 
 D, direction point (as distant as 
 
 possible). 
 
 A C 
 
 FIG. 105. Range-taking without 
 a direction point (using second 
 prism only). 
 
 A, position of observer marked 
 by first picket. 
 
 O, object of which the range is 
 required. 
 
 B, second picket. 
 
 C, third picket. 
 
 FIG. 106. Method of measuring a lonj 
 by means of the third prism. Base / 
 Range AO. BC=. Base i 
 AO. 
 
 base 
 
33^ Preliminary Survey 
 
 The Weldon range-finder consists of three prisms of 
 crystal, accurately ground to the following angles : 
 
 1. 90. 
 
 2. 88 51' 15". 
 
 3- 74 53' IS"- 
 
 The range of an object, as at O, is taken by observing the 
 angles OAD, OBD, at the base of a right-angled triangle 
 ABO in Fig. 104, the measured base AB of which=^L o f the 
 distance or range AO. In this case the first prism, 90, and 
 second prism, 88 51' 15", are used. 
 
 A second method, and one equally important, is by 
 observing the angles at the base of an isosceles triangle as 
 BCO in Fig. 105, when the measured base BC is ^ of the 
 distance or range AO. In this instance the second prism 
 (88 51' 15") only is used. 
 
 In order to measure the base AB or BC accurately and 
 rapidly the third prism of 74 53' 51" is used ; but this is 
 merely a convenience and not a necessity, except under 
 very exceptional circumstances. 
 
 It will be at once seen by those who have had any 
 experience of range-finding that there are several objections 
 to this apparently simple process. They may be summa- 
 rised as follows : 
 
 1. Difficulty of obtaining any definite mark at right 
 angles to the object to reflect the latter upon, when em- 
 ploying a base of 5 L. 
 
 2. Difficulty of always finding ground suitable for 
 measurement of base as regards view, general configura- 
 tion, and space, whatever base may be employed. 
 
 These difficulties are more or less common to instru- 
 ments of this class, but the Weldon possesses the two great 
 advantages over most of them of exceeding portability and 
 reliable, permanent reflectors. 
 
 The writer possesses one of these useful little instru- 
 ments, but has not obtained results with it equal to 
 those on record. Considerable practice is needed in 
 
Instruments 339 
 
 using them, both in judging a position for measuring a base 
 and in taking the observation. The right-angled prism is 
 practically an optical square. In Fig. 104 the range is 
 taken thus. Choose a good direction point D or else put in 
 a ranging rod at D making its reflection coincide with the 
 object by means of the right-angled prism. Then using the 
 88 prism retreat along AB leaving a mark at A to keep 
 yourself in line ; when the 88 prism shows a coincidence of 
 D with O, B is reached. AB is measured either by paces 
 or tape, and multiplied by 50= the range. 
 
 In the official trials at Aldershot in 1883, the average 
 error was only 34 yards for each range, whilst in India in 
 1885 it was 35 yards. 
 
 The chief use of this class of hand instrument is on re- 
 connaissance, when neither plane-table nor theodolite can be 
 carried. It is more accurate in the hands of a practised 
 man than the sketching- board used on the portable stand, 
 described in * Route Surveying,' p. 47, and occupies no more 
 space than a watch. The sketching-board, however, has 
 counter advantages of permitting the surveyor to make 
 some representation of the country as he goes. Captain 
 Willoughby Verner advocates the use of the range-finder 
 in conjunction with the sketching-board. The writer has 
 not yet been able to get more correct results with it than 
 with the sketching-board used alone, on its light tripod, as 
 a triangulation, measuring bases with the latter not less than 
 -fjr of the distance, but taking them in any convenient direc- 
 tion and any convenient length, which cannot be done with 
 the range-finder. A great drawback to this class of instru- 
 ment, and one which puts it out of competition with 
 telemetric theodolites wherever the latter can be used, is the 
 time occupied in running out the base. For taking ranges 
 in battle military officers are content to step the base. 
 Captain Verner mentions an amusing incident connected 
 with * field-firing ' of a certain corps, in which a portly 
 sergeant was told off to give the range with the range- 
 
 z 2 
 
34O Preliminary Survey 
 
 finder, but did not understand it, neither did he wish to 
 show his ignorance. He was doubling along in rear of his 
 section, and was seen after each ' rush ' to raise his range- 
 finder to his eye, after the manner of a spy-glass, and call 
 out whatever his fancy moved him. He had orders to give 
 ranges constantly and obeyed them. 
 
 A rapid method of using the range-finder with the 
 accuracy needed on route-survey would be to measure 
 the two angles simultaneously by two observers, each hold- 
 ing one end of a 100 feet steel tape. With the 90 prism in 
 the hand of one, and the 88 in the hand of the other, 
 distances up to 5,000 feet could be measured, or, by each 
 holding the 88 prism, up to 2,500 feet. It would require 
 a boy to hold up the sag of the tape. Each observer should 
 hold his handkerchief under the range-finder, for the other 
 observer to make the conjunction of image with the distant 
 object through the prism. 
 
 The Weldon is now reduced in price, and hardly more 
 expensive than an ordinary optical square, the duties of 
 which are well performed by the 90 prism, and it is not at 
 all liable to derangement. 
 
 Mr. Steward also makes a range-finder called ' The 
 Simplex,' which is similar in principle of action with the 
 Weldon, only the angle is taken with a glass mirror, and 
 it requires adjustment each time. It is practically an optical 
 square with an alternative position of the mirror at a lesser 
 angle. 
 
 THE BATE RANGE-FINDER 
 
 This is a binocular combined with two reflectors. The 
 modus operandi'i* the same as with the Weldon and similar 
 instruments. 
 
 It has the decided advantage ot optical power combined 
 with a wide field, so that for a hand- telemeter, used by one 
 man, it is one of the best of this class. 
 
Instruments 
 
 341 
 
 Like the Dredge-Steward, it allows of considerable lati- 
 tude in measuring the base, and instead of the very small 
 
 FIG. 107. The Bate Range-finder, closed. 
 
 telescope of the latter, is furnished with a fine binocular, 
 suitable for general use. 
 
 The angle subtended by the base measured is expressed 
 in multiples by putting in a gauge between two graduated 
 
 FIG. 108. The Bate Range-finder, open. 
 
 limbs, thus measuring their divergence, or, in other words, 
 plotting the angle in the instrument itself. There is no 
 
342 Preliminary Survey 
 
 calculation beyond the multiplication of the measured base 
 by the multiplier indicated by the gauge. 
 
 It costs about ten guineas, and, considering what there 
 is in it, is not an expensive instrument. 
 
 HYPSOMETRIC INSTRUMENTS AND HYPSOMETRY 
 
 The Barometer. This instrument has become so familiar 
 to the public as a weather-glass, and to engineers and 
 travellers as a height-measurer, that it is hardly necessary 
 to show an external view of it. Fig. 109 represents the internal 
 
 FIG. 109. 
 
 mechanism. The vacuum chamber B is of German silver, 
 firmly attached to the base-plate A. The sides of the 
 vacuum are compressed with a force equal to the weight of 
 a three-year-old child when the air is pumped out, and are 
 kept from collapsing by the powerful spring D. The vary- 
 ing pressure of the atmosphere produces tiny pulsations in 
 the vacuum, which are very greatly multiplied by levers G 
 and J, and chain Q. The lever G acts also as a compensator 
 for changes of length in the mechanism due to temperature, 
 but this does not dispense with the corrections to the 
 hypsometric formula due to difference of temperature, which 
 are given on p. 350, or to the actually recorded difference 
 of level by any aneroid. The altitude scale corresponding 
 
Instrtunents 343 
 
 to equal variations of atmospheric pressure is a differential 
 one, like water pressure on a lock gate, illustrated at p. 1 1 1 . 
 By means of a device termed a worm, similar to that used 
 for equalising the tension of the spring in watches, the 
 differential movement of the index is in the surveying 
 aneroid changed into a uniform pne, so as to admit of the 
 use of a vernier scale. The 5 inch aneroid is the only 
 satisfactory type for the surveyor. It reads with the vernier 
 to feet, not that it is reliable to single feet at one reading 
 by any means. It needs repetition of readings, compensa- 
 tion for temperature, and every other precaution, and then 
 is only reliable in proportion to the number of readings 
 taken, and the range between highest and lowest 
 results. 
 
 It is clumsy and heavy, and it would be a great boon if a 
 small aneroid could be obtained, say 2-J>- or 3 inch, which 
 would give equally good results. 
 
 The reason that this has not been attainable is because 
 there must be a large vacuum and a small range of altitude 
 in order to get a sensitive instrument which will not ' hang' 
 at all when rapidly changing from rise to fall and back 
 again. 
 
 Travellers sometimes specify to instrument makers such 
 absurdities as a 2 inch aneroid reading to single feet, and 
 having a range up to 15,000 feet, and absolutely compen- 
 sated. 
 
 A surveyor needs a maximum of exactitude over small 
 differences of level. He should have a 5 inch aneroid with 
 a range of 3,000 to 5,000 feet for all the ordinary work 
 of reconnaissance. 
 
 For mountain-climbing he should have a 2 inch watch 
 aneroid and a boiling-point thermometer. 
 
 The following results are added to show the range of 
 discrepancy in two aneroids, one a 3 inch and the other 
 a 5 inch, both of them eight guinea instruments by the 
 same makers, of the first rank in London. 
 
344 Preliminary Survey 
 
 Distance, A to B, 0*55 mile. Difference of elevation by 
 Ordnance benchmarks, 69-0 feet. Mean temperature (ave- 
 rage of all the times), 68 Fahr. 
 
 Three-inch Aneroid. 
 
 feet 
 
 feet 
 
 First time. Up . ' . 72 Down . 
 
 . 60 
 
 Second.. ,, . -77 >, 
 
 . 63 
 
 Third ,, . 77 
 
 57 
 
 Fourth ,, ,, . .50 ,, 
 
 . 72 
 
 Fifth ,, . . 66 
 
 . 68 
 
 5)342 
 
 5)320 
 
 Average . . .68-4 
 
 . 64 
 
 Corr. for temperature . 2*3 
 
 
 707 
 
 
 Range of error from true level 
 
 + IO '3 
 
 
 -10-3 
 
 Range of error between two successive readings 
 
 22 
 
 Range of final mean of means of ' ups ' from true 
 
 
 elevation ....... 
 
 I'7 
 
 Range of final mean of means of ' downs ' from 
 
 
 true elevation ...... 
 
 27 
 
 Five-inch- Aneroid. 
 
 * _ 
 
 feet 
 
 feet 
 
 First time. Up . . 67 Down . 
 
 6 5 
 
 Second ,, ,, . .78 ,, 
 
 . 68 
 
 Third ,, . 66 
 
 63 
 
 Fourth ,, ,, . -63 
 
 7i 
 
 Fifth ,, . 64 
 
 - 56 
 
 5)338 
 
 5)323 
 
 67-6 
 
 64-6 
 
 Corr. for temperature . 2-2 
 
 
 69-8 
 
 
 Range from true level ..... 
 
 + 11-3 
 
 
 -ii -3 
 
 Range between two successive readings . 
 
 10 
 
 Range of final mean of means of ' ups ' from true 
 
 
 elevation ....... 
 
 0-8 
 
 Range of final mean of means of ' downs ' from 
 
 
 true elevation ...... 
 
 2'2 
 
Instru men ts 345 
 
 Distance A. to D, 1*1 mile. Difference of elevation by 
 Ordnance benchmarks ii3'2. Mean temperature, 68 Fahn 
 
 Three-inch Aneroid. 
 
 feet 
 
 feet 
 
 112 Down 
 
 . 107 
 
 H7 
 
 97 
 
 125 
 
 . 92 
 
 93 
 
 . 123 
 
 103 
 
 . 1 06 
 
 5)550 
 
 5)525 
 
 IIO 
 
 105 
 
 3-6 
 
 
 First time. Up 
 Second ,, ,. 
 Third ,, 
 Fourth ,, ,, 
 Fifth 
 
 Corr. for temperature 
 
 113-6 
 
 Range of error from true level . . . +15-4 
 
 -18-4 
 
 Range between two successive readings . = 33 
 
 Range of final mean of means of ' ups ' from true 
 
 elevation ....... 0-4 
 
 Range of final mean of means of ' downs ' from 
 
 true elevation . . . . . . 4-6 
 
 Five-inch Aneroid. 
 
 
 
 feet 
 
 feet 
 
 First tim< 
 
 ;. Up . 
 
 103 Down 
 
 IIO 
 
 Second ,. 
 
 
 . 117 
 
 113 
 
 Third , 
 
 
 107 
 
 IOO 
 
 Fourth ,, 
 
 
 - 107 
 
 . 1 08 
 
 Fifth 
 
 
 
 107 
 
 IOO 
 
 
 
 5)541 
 
 5)531 
 
 
 
 108-2 
 
 106-2 
 
 Corr. for 
 
 temperature 
 
 3-6 
 
 
 i ii -8 
 
 Range of error from true level . . + 7 '3 
 
 IO-2 
 
 Range between two successive readings . . 7-0 
 
 Range of final mean of means of ' ups ' from true 
 
 elevation ....... 1-4 
 
 Range of final mean of means of ' downs ' from 
 
 true elevation ?-4 
 
346 Preliminary Survey 
 
 Distance, A to C, 0-776 mile. Correction for tempera- 
 ture added to each reading. Difference of elevation from 
 Ordnance benchmarks, 129*8 feet. 
 
 Five-inch Aneroid. 
 
 feet feet 
 
 P"irst time. Down . . 130*2 Up . . 134*4 
 
 Second ,, ,, . . 131-1 ,,. . 132-2 
 
 Third ., 132*3 ,, . . 121-4 
 
 Fourth ,, ,, 128-3 5; I 3 2 '4 
 
 Fifth ,. ,, . . 107-0 ,, . . 128-9 
 
 Sixth . . 124-8 ., . . 124-8 
 
 Seventh,, ,, . . 117-6 ,, . . 136*5 
 
 Eighth ,, ,, . . 133-9 ,, . . 123-6 
 
 8)1,005-2 8)1,034-2 
 
 125-7 129-3 
 
 Range of error from true difference . . .4-6-7 
 
 -22-8 
 
 Range between two successive readings . . 21-9 
 Range of final mean of means of ' ups ' from true 
 
 elevation ....... 0-5 
 
 Range of final mean of means of ' downs ' from 
 
 true elevation . . . . . . 4^1 
 
 The wide readings will surprise those who are not familiar 
 with the subject, and who have been led to believe other- 
 wise by those who are proficient in the sale, less versed in 
 the construction, and least of all conversant with the use of 
 aneroids. // will be noticed that the five-inch in the above 
 tests has from a quarter to one-half the range of error of the 
 three-inch between any two successive readings. This is a 
 much more important point to the surveyor than the 
 accuracy of the final mean of means. In the three-inch 
 the range of error for the half-mile was two-thirds of that 
 for the mile, whereas in the five-inch the range of error for 
 the mile was less than that for the half-mile. It is fre- 
 quently the case that errors of five and ten feet will be regis- 
 tered in going up and down a hillock whereas the height of a 
 mountain may be correctly given within two or three feet. 
 
 it will also be understood from this why stress is laid -in 
 
Instruments 347 
 
 Chapter II. upon using the Abney level with the aneroid for 
 the smaller rises and falls of route-survey, and making as 
 many repetitions as possible of the aneroid readings which 
 determine the maximum and minimum elevations. These 
 repetitions should all be entered in the fieldbook as care- 
 fully as the original ones, and a symbol placed beside the 
 mean of means on the plan to indicate the degree of 
 accuracy which had been obtained. It will be observed, 
 further, that in every case the mean of the ' ups,' whether 
 the journey commenced by going up or by going down, was 
 more correct than the mean of the ' downs.' Having used 
 a number of instruments in different parts of the world, the 
 writer has always found this the case. The pressure of the 
 atmosphere puts a strain upon the spring D. When going 
 uphill, the spring is relieved, and when going downhill it is 
 again depressed. It is natural that the resilience of the spring 
 should be freer in action than its compression, on account of 
 the intermediate mechanism needed to procure the action. 
 
 PRACTICAL SUGGESTIONS IN PROCURING AND USING 
 ANEROIDS. 
 
 1 . Procure a first-class instrument. 
 
 2. Learn its peculiarities and eccentricities. 
 
 3. Have it examined every year. 
 
 4. Register atmospheric changes by a second instrument 
 at camp. 
 
 5. When reading, hold horizontally, and tap several times. 
 
 6. Take as many repetitions as possible. 
 
 7. Where discrepancy, not due to atmospheric disturb- 
 ance in the district, exists between 'ups ' and 'downs,' prefer 
 the ' ups.' 
 
 8. Always apply the temperature correction. 
 
 9. Three ' ups ' will generally give a mean result correct 
 to 3 feet, or two ' ups ' to 6 feet, in difference of level of 5,0 
 to 500 feet. 
 
348 Preliminary Survey 
 
 The Kew test is useful, and should be specified in order- 
 ing valuable instruments, but even that does not prove the 
 instrument in every way. 
 
 A first-class aneroid is as hard to get as a horse without 
 a blemish. A seasoned vacuum and perfect mechanism 
 inside, together with clear open graduation outside, and a 
 sling-case which will open and shut quickly and safely, are 
 the chief points to aim at. Five-inch aneroids are now 
 made in aluminium to save weight, but the price is very 
 high. 
 
 HYPSOMETRY, 
 
 or height-measuring, is a term applied to determination of the 
 level above the sea, by the barometer or boiling-point thermo- 
 meter, as distinguished from levelling with the spirit level. 
 
 Both operations rest upon the same fundamental prin- 
 ciples of the weight and pressure of the atmosphere, com- 
 monly known as Mariotte's and Charles's laws. These are, 
 firstly, that at a uniform temperature the pressure of any 
 gas varies inversely as its volume, and secondly, that at a 
 uniform pressure the expansion produced by a given increase 
 of temperature is the same for all gases. 
 
 The standard mercury barometer is now but little used 
 for hypsometric purposes on account of the inconvenience 
 in transporting it. The aneroid barometer just described 
 has superseded it, first because of its exceeding portability, 
 secondly because of its superior sensitiveness. It is 
 subject, however, to derangement of its delicate mecha- 
 nism, not alone from accident, but even from changes of 
 climate ; and the boiling-point thermometer, which measures 
 the atmospheric pressure without any mechanism, is gene- 
 rally added to the outfit of anyone who wishes to make ex- 
 tended and reliable hypsometric observations. It must not 
 be supposed that the use of any of these methods is sufficient 
 to ascertain elevations with the accuracy of a spirit-level. 
 They do no more than give the difference of atmospheric 
 
Instruments 349 
 
 pressure at two different situations. When the condition of 
 the atmosphere is steady in the district, that is to say when 
 a standard barometer at either of the stations remains sta- 
 tionary during the period of observation, the difference of 
 pressure at the two situations forms a means of correctly as- 
 certaining the difference of elevation. When, however, there 
 is a disturbance of the air-pressure, no reliance can be 
 placed upon the results unless by also recording the move- 
 ments of a stationary barometer at the two stations by inde- 
 pendent observers. 
 
 With all their draw-backs hypsometric instruments are 
 indispensable, because they are the only means of approxi- 
 mately determining differences of elevation en route. The 
 real makers of them are few, and hardly ever put their names 
 on them. It is not sufficient to order them from a good 
 instrument-maker or even to have them tested at Kew. The 
 best proof of their excellence is in their daily use for a few 
 weeks under conditions of varying temperature, elevation, 
 and methods of transportation. All aneroids are marked 
 'compensated,' and instrument makers will often tell their 
 customers that no further correction is needed on account 
 of this compensation -a statement which always shows that 
 they do not understand the principle of the instrument. 
 
 The compensation of the mechanism of the aneroid causes 
 it to record the correct difference of atmospheric pressure at 
 different times or in different places, and it is therefore not 
 subject to the correction which is applied to mercurial 
 barometers to correct them for the expansion or contraction 
 of the column of mercury. 
 
 But the difference of atmospheric pressure between two 
 different elevations is less at high temperatures than it is at 
 low ones. If the barometer recorded 30" pressure at sea 
 level when the thermometer stood at 32 Fahr., at 1,000 
 feet elevation with the same temperature the pressure would 
 be nearly 2 8" '88. But if the mean temperature were 72 
 Fahr., the pressure would be about 2 8" -96. In a table 
 
35 Preliminary Survey 
 
 graduated with altitudes corresponding to differences of 
 atmospheric pressure at 32 Fahr. mean temperature, the 
 pressure of 28" -96 would only give 923 feet, and the altitude 
 in the table would have to be multiplied by i -083 to give 
 the true altitude. Hence it is much more convenient to 
 know the multipliers for temperature applicable to the 
 aneroids as they are actually constructed, for they are not 
 graduated from 32 Fahr., but from 50 Fahr., by Airy's 
 formula, or from 53 Fahr. by the author's mean formula 
 of English and American standards. 
 
 The compensation of an aneroid is nothing more than a 
 device analogous to that in a watch by which the expansion 
 or contraction of the mechanism of the instrument itself 
 due to changes of temperature is compensated. Nothing 
 can however prevent the expansion of the air produced by 
 any increase of temperature from acting differently upon the 
 aneroid under its altered condition, and nothing can prevent 
 the altered specific gravity of the air due to different alti- 
 tudes and latitudes from likewise differently affecting the 
 instrument. English aneroids are graduated with an altitude 
 scale corresponding to a mean temperature of 50 calculated 
 by Airy's formula or 53 by the author's mean formula ; 
 consequently, as shown in Plate IX. Fig. in, the altitudes 
 indicated by the instrument at lower temperatures are too 
 great and should be treated with the multipliers given, and 
 vice versa at higher temperatures. Each of the subdivisions 
 has a value of '002 and reads thus : i.ooo ; 1-002 1*004, an d 
 so on. The temperature corrections given in the textbooks 
 are arranged from 32 Fahr., so that a double calculation 
 is required. This diagram can be used with the slide-rule. 
 
 When the altitudes are calculated from readings of the 
 barometer, or boiling-point thermometer, formulae are used 
 based upon the researches of Guyot and others, but varying 
 considerably in the coefficients adopted by different experi- 
 menters in different countries. 
 
 The tables given by Mr. Francis Galton in the textbook 
 
OTHT 
 
 PLATE IX. 
 
 'irTO 
 
 26 tt 5 24 5 23 5 f s 2 . 5 M 500 
 
 5 ts 5 Z8 5 Z7 
 
 BAROMETRIC PRESSURE IN INCHES AND TENTHS 
 
 FIG. no. 
 
 in. Correction for intermediate air 
 for aneroids scaled from 53 Fahr. 
 
 Note i. The horizontal lines 
 represent '002 Fahr. 
 
 Note 2. -If this diagram is in frequent use 
 with dividers, a piece of dull-back tracing 
 cloth, gummed over it by the four corners 
 will protect it. 
 
Instruments 3 5 i 
 
 of the Royal Geographical Society arc calculated by Loomis, 
 and differ both from those of Airy, from which most English 
 barometers are graduated, and still more from those of 
 Col. Williamson, the American authority. 
 The formula of the latter is as follows : 
 
 o DxT 
 
 where H= difference of height in feet between two stations. 
 D= difference of barometric pressure between the 
 
 two stations. 
 T=tabular number corresponding with mean tem- 
 
 perature. 
 B=mean barometric pressure. 
 
 A short approximate rule is given by Mr. J. H. Belville 
 of Greenwich Observatory, which is nearly correct between 
 temperatures of 50 and 60 Fahr. 
 
 S : D:: 55,000 : H 
 
 where H= difference of height in feet between stations. 
 S=sum of barometric readings. 
 D= difference of barometric readings, 
 
 Examples will be given to show the divergence of the 
 three authorities, and a series of constants, K, for each degree 
 of temperature from zero to 102 Fahr., by which, with a 
 modification of Belville's rule, a mean result will be obtained 
 between those of the English and American formulae and 
 of so simple a kind that it can be worked by the slide-rule. 
 It is as follows : 
 
 H (difference of height) = 5^ 
 
 o 
 
 where K is the coefficient for mean temperature given in 
 Table LI., D=difTerence of barometric pressures, and S 
 their sum. 
 
352 
 
 Preliminary Survey 
 
 TABLE LI. Value of K in Formula H = 
 
 Deg. 
 M. T. 
 Fahr. 
 
 K. 
 
 Log K. 
 
 Deg. 
 M. T. K. 
 Fahr. 
 
 LogK. 
 
 
 
 48753 
 
 4-68800 
 
 52 54785 
 
 4*73866 
 
 i 
 
 48869 
 
 4-68903 
 
 53 5490* 
 
 4'73958 
 
 2 
 
 48985 
 
 4-69006 
 
 54 550*7 
 
 4-74049 
 
 3 
 
 49101 
 
 4-69109 
 
 55 55*33 
 
 4*74*4* 
 
 4 
 
 49217 
 
 4-69211 
 
 56 55249 
 
 4*74232 
 
 5 
 
 49333 
 
 4-69314 
 
 57 55365 
 
 4*74323 
 
 6 
 
 49449 
 
 4-69416 
 
 5* 5548i 
 
 4-74414 
 
 7 
 
 49565 
 
 4*69517 
 
 59 55597 
 
 4'74505 
 
 8 
 
 49681 
 
 69619 
 
 60 55713 
 
 4'74596 
 
 9 
 
 49797 
 
 69720 
 
 61 55829 
 
 4-74686 
 
 10 
 
 499*3 
 
 69821 
 
 62 55945 
 
 74776 
 
 
 50029 
 
 69922 
 
 63 56061 
 
 74866 
 
 12 
 
 50145 
 
 70023 
 
 64 56*77 
 
 74956 
 
 *3 
 
 50261 
 
 70123 
 
 65 56293 
 
 "75045 
 
 I4 
 
 50377 
 
 70223 
 
 66 56409 
 
 '75*35 
 
 15 
 
 50493 
 
 '70323 
 
 67 56525 
 
 "75224 
 
 16 
 
 50609 
 
 
 68 56641 
 
 '753*3 
 
 I 7 
 
 50725 
 
 - '70522 
 
 69 56757 
 
 4'754 2 
 
 18 
 
 50841 
 
 70621 
 
 70 56873 
 
 4*7549* 
 
 *9 
 
 50957 
 
 70720 
 
 71 56989 
 
 4*75579 
 
 20 
 
 5*073 
 
 70819 
 
 72 57*05 
 
 4*75667 
 
 21 
 
 51189 
 
 70918 
 
 73 57221 
 
 4*75755 
 
 1 22 
 
 
 71016 
 
 74 57337 
 
 4*75843 
 
 2 3 
 
 5142* 
 
 71114 
 
 75 57453 
 
 4*7593* 
 
 24 
 
 5*537 
 
 71212 
 
 76 57569 
 
 4-76019 
 
 2 5 
 
 5*653 
 
 i "71309 
 
 77 57685 
 
 4-76106 
 
 26 
 
 
 71407 
 
 78 578oi 
 
 4-76193 
 
 2 7 
 
 51885 
 
 '7*504 
 
 79 579*7 
 
 4-76281 
 
 28 
 
 52001 
 
 71601 
 
 So 58033 
 
 4-76367 
 
 29 
 
 52117 
 
 4-71698 
 
 81 58149 
 
 4 "76454 
 
 3 
 
 52233 
 
 4*7*794 
 
 82 58265 
 
 4-76541 
 
 3 1 
 
 52349 
 
 4-71891 
 
 83 58381 
 
 4-76627 
 
 3 2 
 
 52465 
 
 4-71987 
 
 84 58497 
 
 4;767*3 
 
 33 
 
 52581 
 
 4-72083 
 
 85 58613 
 
 
 34 
 
 52697 
 
 4-72179 
 
 86 58729 
 
 4-76885 
 
 35 
 
 52813 
 
 4*72274 
 
 87 58845 
 
 4-76971 
 
 36 
 
 52929 
 
 4*72369 
 
 88 58961 
 
 4-77056 
 
 37 
 
 53045 
 
 4-72464 
 
 89 5977 
 
 4-77142 
 
 38 
 
 53161 
 
 4"72559 
 
 90 59*93 
 
 4-77227 
 
 39 
 
 53277 
 
 4*72654 
 
 9* 59309 
 
 4*773*2 
 
 4 
 
 53393 
 
 4-72748 
 
 92 59425 
 
 4*77397 
 
 4 1 
 
 53509 
 
 4*72843 
 
 93 5954* 
 
 4'77482 
 
 4 2 
 
 53625 
 
 4*72937 
 
 94 59657 
 
 4*77566 
 
 43 
 
 5374 1 
 
 4*7303 
 
 95 59773 
 
 4-77650 
 
 44 
 
 53857 
 
 4*73*24 
 
 96 59889 
 
 4*77735 
 
 45 
 
 53973 
 
 4*732*8 
 
 97 60005 
 
 4*778i9 
 
 46 
 
 54089 
 
 4'733ii 
 
 98 60121 
 
 4*77903 
 
 47 
 
 54205 
 
 4 '7344 
 
 99 60237 
 
 4-77986 
 
 48 
 
 54321 
 
 4 '73497 
 
 loo 60353 
 
 4-78070 
 
 49 
 So 
 
 54437 
 54553 
 
 4'73589 
 4-73682 
 
 101 60469 
 
 102 60585 
 
 4-78*53 
 4-78236 
 
 5* 
 
 54669 
 
 4*73774 
 
 
 
Instrumen ts 353 
 
 COMPARISON OF AUTHOR'S FORMULA WITH ENGLISH 
 AND AMERICAN AUTHORITIES 
 
 Example i. 
 
 inches. 
 
 Reading of barometer at lower station . . 26*64 
 
 ,, ,, upper ,, . . 20-82 
 
 Thermometer at lower station .... 70 
 
 ,, upper ,, . . . .40 
 
 Bar. sum =26-64 + 20-82 = 47-46 , T + / . .= uo 
 Difference =26-64-20-82= 5-82 ! M. T. .= 55 
 
 j K, see table 55,133 
 
 TT K.D T T , 
 
 g . Log K . . . -4-74141 
 
 + Log 5 -82 . . . =0-76492 
 
 5-50633 
 -Log 47 -46 . . . =1-67633 
 
 Log alt. 6760-9 ft. . =3-83000 
 
 By slide-rule. Place the 47*46 on the slide opposite to 
 the 55,133 on the rule ; find the result 6,760 on the rule 
 opposite to 5 '82 on the slide. 
 
 The same by the American rule, from the standard work 
 by Lieut. -Col. R. S. Williamson of the U. S. Army, as 
 quoted by Trautwine. 
 
 Rule. Height in feet= 
 difference D of barometer x table No. for MT x constant 30 
 
 mean reading of barometer 
 whence height=6748'5 feet. 
 
 The same by the Geographical Society's rule by Francis 
 Galton, F.R.S. See ' Hints to Travellers,' p. 185. 
 
 Part I. for 26-64 inches. .... 24,506*2 
 20-82 ,,..... 18,066 
 
 900 
 
 6,440-2 
 . x (70" + 40" -64) . . 329-1 
 
 6,769-3 
 
 A A 
 
354 Preliminary Survey 
 
 Example 2. 
 
 Reading of barometer at lower station . . 30 '646 
 
 upper ,, 23-66 
 
 Thermometer at lower station . . 77'5 
 
 ,, ,, upper ... 70-3 
 
 Sum. bar. 30 -046 + 23 '66 = 53-706 j T + / 147-8 
 
 Diff. 30-046-23-66= 6-386 i M. T. - 73-9 
 
 i K, see table 57,337 
 
 H=^5_ . LogK ... 47584350 
 
 Log D 6-386 . . 0-8052289 
 
 5-5636639 
 Log S 53-706 . . 1-7300228 
 
 Log alt. 6,817-8 . . 3-8336411 
 
 By slide-rule as before 6,820. 
 
 The same by the American rule is 6,829 feet and by 
 Gallon's rule 6,805-6 feet. 
 
 Example 3. 
 Bar. lower station ...... 
 
 upper 
 
 S 50-00 
 
 D 6-00 
 
 Temp, lower station .... 60 Fahr. 
 upper ,, . . . . 40 ,, 
 
 M. T 50 K = 54,553 
 
 H = _^P- = 5-M^JL^ 6,546-36 
 
 By the American rule ..... 6,527-5 
 
 By Galton's rule 6,552-7 
 
 By Airy's tables 6,571-5 
 
 Examples i and 2 are taken from ' Trautwine ' and 
 * Hints to Travellers.' Example 3 is. formed of barometric 
 pressures intermediate between the other examples, and 
 the result from Airy's tables is taken from a .pamphlet on 
 
Instruments 
 
 355 
 
 the aneroid by Houlston & Sons, lent to the writer by an 
 instrument maker as being the standard used in graduating 
 aneroids. The author's two surveying aneroids graduated 
 i to 10,000 feet, the other to 5,000, have their zero at 
 3 1 inches and are graduated almost exactly in the same way. 
 The former reads 9,325 feet at 22 inches. Airy's tables 
 above alluded to show an altitude of 9,347 '5 at that pressure 
 and mean temperature of 50 Fahr. 
 
 The American rule would give 9,214-4 feet, Mr. Galton's 
 rule 9,3i8'4, the author's rule 9,263'6, an almost exact 
 mean as before between the English and American autho- 
 rities. 
 
 The usual corrections for temperature of intermediate 
 air are given in tables having 32 Fahr. M. T. as the starting 
 point, and consequently are not capable of being applied to 
 ordinary aneroid readings without a double calculation. 
 
 In the diagram Fig. in multipliers are given having 
 53 Fahr. mean temperature as their starting point, that 
 
 Correction for latitude and 
 decrease of gravity. 
 
 l,AT:0 
 
 Note. The positive correction for de- 
 crease of gravity is to be scaled for the 
 required altitude from the left-hand scale. 
 Thus, for 10,000 feet of altitude the cor- 
 rection 29'8 feet is marked so. The cor- 
 rection for latitude is scaled in the same 
 way, but positive or negative. 
 
 FIG. 112. 
 
 being the basis of the graduation of the ordinary aneroid 
 according to the author's mean formula. By using these 
 multipliers, the correction can be made with one calculation, 
 
356 Preliminary Survey 
 
 and in fact for ordinary work by the slide-rule in the field 
 as close as most instruments will read. 
 
 Example 4. At a mean temperature of 70 Fahr. the 
 aneroid showed an altitude of 1,265 feet. The multiplier is 
 1-037. 
 
 Place the left hand i of the slide opposite to 1*037 on 
 the rule and we have opposite to 1,265 on the sn de 1,310 on 
 the rule, or more correctly, by figures 1,311. 
 
 As a remembrancer for every 10 degrees of temperature 
 above add to 
 
 bdtow 53'Fahrenheit ^55^ percent. ^ the regis- 
 
 tered difference of level by the aneroid. 
 
 The diagrams on Fig. 112 for change and decrease of 
 gravity are added more to exhibit at a glance the amount of 
 such corrections than for actual use, because it is rarely of 
 any practical importance to know them. The local atmo- 
 spheric disturbances may be far greater within a small area 
 than the equivalent of these corrections in barometric pres- 
 sure, so that to apply such refinements as these would be a 
 sheer waste of time. 
 
 THE BOILING-POINT THERMOMETER ! 
 
 ' The boiling-point apparatus (Fig. 113) consists of a ther- 
 mometer A graduated from 180 to 215 ; a spirit-lamp B, 
 which fits into the bottom of a brass tube C that supports 
 the boiler D, and a telescopic tube E which fits tightly on 
 to the top of the boiler. The thermometer is passed down 
 the tube E from the top until within a short distance from 
 the water, which it should never touch, and is supported in 
 that position by an india-rubber washer F. The steam 
 passes from the boiler up the tube E, and escapes by the 
 hole G. To pack this instrument for travelling, withdraw 
 the thermometer and put it into a brass tube, lined with 
 
 1 From 'Hints to Travellers' by John Coles, Esq., Royal Geo- 
 graphical Society 
 
Instruments 
 
 357 
 
 india-rubber having a pad of cotton wool at each end ; take 
 off the tube E, shut it up and put the small end into the 
 boiler D, which it fits, then withdraw the spirit lamp B, screw 
 the cover over the wick, and replace it in C. The whole of 
 this apparatus fits into a circular tin A 
 
 case 6 inches long and 2 inches in dia- 
 meter. 
 
 ' To use the boiling-point thermometer. 
 Take the apparatus to pieces, pour 
 some water into the boiler D, the less 
 the better, as it will boil the quicker 
 (about one quarter full is quite suffi- 
 cient) ; then put the instrument together 
 as shown in the drawing, taking care 
 that the thermometer is at least half an 
 inch clear of the water, and light the 
 spirit-lamp ; as soon as the water boils, 
 the steam, ascending through the tube 
 E, will cause the mercury to rise ; wait 
 until the mercury becomes stationary, 
 and then read the thermometer ; at the 
 same time take the temperature of the 
 air in the shade with an ordinary ther- 
 mometer. 
 
 ' When purchasing this instrument 
 be careful to see that the lamp is large 
 enough to hold a good supply of spirit ; 
 it is a common fault to make it too 
 small. A small screw which may be 
 made of tin to fold up is most useful 
 to place on the windward side, and at a 
 very low temperature is almost indispensable, as the heaf is 
 otherwise carried off too rapidly for the water to boil pro- 
 perly.' 
 
 The following rule for finding the height due to temper- 
 ature of the boiling point, has been prepared in a similar 
 
 
358 Preliminary Survey 
 
 way to that for barometric pressure by adopting a coefficient 
 which produces mean results between the English and 
 American authorities. It is for a mean temperature of inter- 
 mediate air=53 Fahr. instead of freezing point, as custom- 
 ary in the textbooks. This has been done to make the 
 coefficient agree with the graduation of ordinary aneroids, 
 and for other temperatures the coefficient 540 has to be 
 multiplied by the multiplier on diagram Fig. 1 1 1 Plate IX. ; 
 thus, at 32 the coefficient becomes -954x540=51:5, or 
 at 82=54o x 1-064=574-6. 
 
 Rule. Let B= temperature of boiling point in degrees 
 
 Fahr. deducted from 212. 
 H= height of station above level of sea. 
 K= 540 for a mean temperature of interme- 
 diate air of 53 and varying as ex- 
 plained above. 
 H=K. 
 
 Example. 
 
 Boiling points . . . . . . ~\\"tf 
 
 ...... 210-14 
 
 Mean temperature ...... 82 Fahr. 
 
 Required the difference of elevation : 
 
 H = 540 x 1-064 x 0-63 + 0-63- . . = 362-37 
 H' = 540 x 1-064 x i"86 + 1-86- . . = 1,072-14 
 
 Ans. diff. in feet . . . . = 70977 
 
 The value of the boiling-point thermometer consists in 
 the fact that it is a perfectly simple machine, there is no 
 wheelwork to get out of order, no vacuum to play off its 
 caprices. Otherwise it performs the same duty as the 
 aneroid. Its results depend upon the assumption that 
 water boils at 212 at sea-level. This is not always 
 true ; water boils at different temperatures in different 
 latitudes and also varies under different conditions of the 
 atmosphere. In fact the boiling-point is nothing more than 
 
Instruments 359 
 
 a register of the atmospheric pressure, like the aneroid. 
 Water can be put under an air pump at sea level, and made 
 to boil nearly at freezing-point. The boiling-point thermo- 
 meter is only suitable for the first pioneer work in very 
 hilly country, such as finding a pass for a railway through a 
 chain of mountains. It is not frequently used even for 
 that purpose. 
 
 The surveyor generally contents himself with two 
 aneroids rather than spend time in boiling-thermometers. 
 It is not, however, such good work, because the best 
 aneroids are fickle under varying conditions. 
 
 OFFICE INSTRUMENTS 
 
 At a pinch the surveyor can get along with hardly any- 
 thing more than a pair of compasses and a straight-edge. 
 
 For a short survey, where impedimenta are a great objec- 
 tion, with a pocket case of instruments and a slide-rule he 
 can do all the protracting, contouring, scale-making, and 
 gradient-drawing that he wants. What he misses most is a 
 box of good railway sweeps. They are bulky, and it is 
 seldom he can afford space for them. 
 
 The author has combined in one box, 19 inches long 
 by ii inches wide by 8| inches deep, a complete repertory 
 of all the drawing instruments he requires, and though 
 quite bulky, he has contrived to make room for it wherever 
 he has gone. A description may be useful. 
 
 At the. bottom is a full set of railway sweeps, i to 240 
 inches radius. A boxwood rolling parallel ruler with the 
 edges graduated as a protractor and loaded in the middle 
 with lead to steady it. 
 
 On the lower tray is a colour-box and water-dish, recep- 
 tacle for liquid ink, liquid carmine, and liquid prussian blue 
 in their own bottles. Small hammer, wire-cutting nippers, 
 ' Yankee notion ' screw-driver-bradawl, store of pencils, 
 rubber, and colour-brushes, needles, copper-wire, brass 
 
360 Preliminary Survey 
 
 screws, paperclips, small sponge, chamois-leather, palette, 
 lancet, surgical needle. 
 
 On the upper tray are^ six-inch German silver circular 
 protractor with short arm and vernier, reading to '01 degree; 
 three ply magnifiers for ditto. Complete set of drawing 
 instruments, including proportional compasses and trammels. 
 
 In the lid wallet are one 15 inch, 60 set square, one 
 12 inch, 45 set square ; two French curves, card of crow- 
 quills, one 9 inch paper protractor. 
 
 On the lid-flap are one set of ivory scales 10 to 60, one 
 universal scale, all 12 inch. They are laid out in a row in 
 elastic loops, so that they can be withdrawn without needing 
 to search for the right one. Also one small 45 and one 
 small 60 set square. 
 
 It is not only a great convenience, but the best plan of 
 keeping one's instruments to have them all together in a box 
 where each one has its own place, and becomes 'conspi- 
 cuous by its absence.' 
 
 A tee-square is much the best means of protracting angles, 
 where the sheets are not longer than can be reached by it. 
 The steel tee-square with bronze head sold by Charles 
 Churchill & Co. of Finsbury, though an expensive instru- 
 ment, is well worth its price. It is a protractor with a 
 36 inch arm, and a tee-head. It has a clamping screw with 
 a large head to it, so that it is adjusted in a moment to any 
 angle. It is used with an ordinary set square. 
 
 Barnett's diagraph, which only costs us., is another and 
 ingenious instrument for rapid plotting, by the same makers. 
 
 Where no very great accuracy is aimed at, a paper 
 protractor pinned down in one corner of the board, an 
 ordinary tee-squaie, and a single jointed two-foot rule with 
 a pretty stiff joint form a very cheap and efficient method of 
 protracting. 
 
 The tee-square is run up to the protractor, and the 2 -foot 
 rule is set to the angle. It is then run down to the station 
 from which it is to be laid off by the tee-square. It is useful 
 
Instru men ts 361 
 
 for laying off bracing of trestles ; equal and opposite battering 
 walls &c., because the 2 -foot rule has only to be reversed to 
 give the opposite angle. 
 
 It is also very useful for making strain-diagrams 
 although that is outside our province because it is not so 
 liable to shift as a parallel ruler. 
 
 The eidograph and pantagraph are occasionally a great 
 help to the surveyor, but they can hardly be called satis- 
 factory instruments. Reduction of plans is much better 
 done by dividing them into squares, and aided with propor- 
 tional compasses does not take much longer than by the 
 eidograph. 
 
 The planimeter is a most valuable instrument to those 
 who have extensive computations of acreage to make. It 
 
 FIG. 114. Amsler's Planimeter. 
 
 does the whole work by recording, on a pair of indicator 
 wheels, the travel of a roller round the periphery of any 
 figure. Stanley's computing scales, which are much cheaper 
 than the planimeter, are an efficient substitute for it and do 
 not take very much longer. 
 
 THE SLIDE-RULE 
 
 This ancient but not antiquated instrument is not nearly 
 as much appreciated in this country as it ought to be. 
 Thoroughly scientific as it is in principle it has until recent 
 years suffered from the imperfection of mechanical science 
 which prevented the attainment of the same accuracy in 
 graduated scales which is possible with figures. 
 
 The increased skill in making dividing-lathes has enabled 
 the mechanician to produce an instrument of precise accuracy 
 
362 . Preliminary Survey 
 
 only limited by its length and the consequent visibility of 
 the graduation. 
 
 The slide-rule has been formerly known as Coggeshall's 
 rule or the carpenter's slide-rule, and has been regarded as a 
 rough though labour-saving makeshift. 
 
 Ever since its invention by Oughtred it has been mis- 
 understood, until recently the French and Italians have 
 brought it into something of the estimation which it deserves. 
 Oughtred is said to have kept the instrument by him many 
 years out of a settled contempt for those who would apply 
 it without knowledge, having ' onely the superficial scumme 
 and froth of instrumental trickes and practices, and wishing 
 to encourage the way of rationall scientialists, not of ground- 
 creeping Methodicks.' T 
 
 A glance at the catalogue of rules of all sorts, sizes, and 
 prices from 6s. up to io/., manufactured by the firm of 
 Tavernier-Gravet, will show that the French have appreciated 
 this instrument more than we. 
 
 A very decided improvement on the cheaper form of 
 boxwood rules has, however, been made by Messrs. Davis 
 & Son of Derby and London, by overlaying celluloid upon 
 hard wood. It looks like ivory and costs no more than the 
 lo-inch Mannheim slide-rule. The graduation is quite 
 equal to that of the French rule. There are some points of 
 advantage in the writer's opinion in favour of the latter, but 
 which might be easily adopted by the English makers if 
 they were so disposed. 
 
 In Preliminary Survey the slide-rule is simply invaluable, 
 and it is astonishing how quickly its manipulation is acquired, 
 especially by those accustomed to read graduation of any 
 kind. 
 
 If it were only for the one operation of reducing the 
 optically measured distance to horizontal and vertical co- 
 ordinates, and the horizontal distance to latitude and de- 
 parture by two shifts of the slide, no surveyor should be 
 1 Rev. W. E. Elliott, The Slide-rule.' 
 
Instruments 363 
 
 without one ; but that is only one of its numerous func- 
 tions, some of which are described in the chapter on Graphic 
 Calculation. 
 
 To do justice to all the applications of the slide-rule to 
 the various branches of technology would require a small 
 volume for that purpose alone. 
 
 The printed directions supplied with the slide-rule only 
 give general principles, and therefore instructions are given 
 in this manual for a considerable number of those opera- 
 tions which are most commonly required upon survey. 
 Repetition is of course unavoidable, but inasmuch as the 
 use of the instrument may not be continuous, if its special 
 application to some particular rule is forgotten it will not 
 be used at all. It was therefore considered advisable to 
 give numerous examples with the various rules. 
 
 The instructions for using the slide-rule are principally 
 contained in Chapter VIII., but they are also necessarily 
 interspersed all over the book. They are partly for sexa- 
 gesimal degrees, although in many places decimal degrees 
 are given along with minutes and seconds ; Tables XL., XLL, 
 and XLIL, moreover, give a ready reduction of minutes and 
 seconds to decimals of a degree. Nothing has been worked 
 out with the centesimal degree, that being so much of 
 a novelty that everything would have had to be in duplicate. 
 
 MM. Tavernier & Gravet, and Messrs. Davis & Son, 
 make slide-rules graduated for the ordinary degree divided 
 decimally, and they can be used for minutes without any 
 reduction, because the subdivisions are merely at 3', 6', 9', 
 12', &c., which are '05, 'i, "15, -20, and may be used 
 either way. Instead of the mark for single minutes, there 
 is one for -01 degree. 
 
 The slide-rule may be used for long rows of figures by 
 placing them in batches of threes ; but its chief value is for 
 all those small calculations up to an accuracy of ^-J^, which 
 only require one adjustment of the slide ; for in these the 
 operation is performed in less time than it would take to 
 
"Xf" 
 
 -*-* 
 
 Preliminary Survey 
 
 look up the first factor of the 
 sum in the table of logarithms, or 
 to put them down on paper for 
 ordinary reckoning. 
 
 The instrument, in its present 
 form, will perform accurately, and 
 without any mental effort what- 
 ever, a whole mass of tiresome, 
 small calculations in less than 
 half the time it would take a very 
 quick man to work them out by 
 any method he might choose. 
 
 In the field every surveyor 
 knows how books and tables with 
 soiled pages and fluttering leaves 
 become lessons of patience and 
 self-control to him, but the slide- 
 rule enables him to reserve these 
 moral exercises for other occa- 
 , sions- All graphic operations are 
 essentially approximate, but it is 
 possible to arrive at as close an 
 approximation as may be needed 
 for the purpose in view when the 
 object and the principle are both 
 clearly understood, and so to 
 give perfectly correct results with- 
 in the limits prescribed. 
 
 The principle of the slide-rule 
 is that of graphic logarithms. 
 
 The organic formula of loga- 
 rithms is Log (AxB)=log A-f 
 log B. We can obviously per- 
 form this operation by scaling as 
 well as by figures. For if we add 
 the tabular logarithm of A to 
 
Instruments 365 
 
 that of B by scale or by figures, the sum of the two loga- 
 rithms will represent the product of the two numbers. 
 Referring to Fig. 115, the slide has been shown withdrawn 
 one primary division to the left ; and it will be noticed 
 that the i of the upper scale of the rule is over the 2 of 
 the slide, the 2 over the 4, the 3 over the 6, and so on. 
 If the slide had been shown in its initial position, all the 
 figures would have been in correspondence, because they are 
 equal logarithmic scales. That is to say, the instrument 
 maker has constructed the space i to 2, by any convenient 
 scale of equal parts, =301 units, because the logarithm of 
 2 is '301. Similarly the space from i to 3 is made equal to 
 477 units. The space from i to 6 is 778 units for the same 
 reason. 
 
 By retreating the slide 301 units to the left in the manner 
 shown we add to the whole scale of the rule an amount= 
 log 2, and consequently represent graphically the multipli- 
 cation of every figure on that scale by 2. The 3 of the 
 rule coincides with the 6 of the slide because on the one 
 477 units are added to 301 units on the other, making 778 
 units, which is the log 6. 
 
 The intermediate graduation is made in the same man- 
 ner, each line being ruled off in the instrument maker's 
 dividing lathe at a distance equal to its tabular logarithm. 
 
 The upper scale of the rule is doubled, so that if we give 
 to the left hand i the value of i, the middle T will be 10 
 and the right hand i will be 100. 
 
 The lower scale is only single in the first place because 
 with less range it has a more open graduation so that by it 
 some figuring can be done twice as closely as by the upper. 
 In the second place it will be observed that all its figures 
 are under their doubles on the upper scale. As twice the 
 log means the square of the number all the lower figures 
 are the square roots of the upper and we can perform invo- 
 lution and evolution without moving the slide at all. 
 
 Where the lower figures are given their indicated value 
 
366 Preliminary Survey 
 
 and are under the right hand half of the upper scale, the 
 figures of the latter must be given ten times their indicated 
 value ; thus 6 is under 3 '6, which must be styled 36 and 
 so on. 
 
 The scales of sines and tangents are constructed from the 
 tabulated logarithmic sines and tangents ; but with a radius 
 of 100, of which the logarithm is 2. Tabular values are 
 always figured to a radius whose logarithm is 10, so that the 
 integral number 8 is deducted from all the tabular values. 
 The process of calculation is the same as with numbers, 
 only the sines and tangents cannot be read lower than 
 34' 23" nor the tangents above 45. Smaller angles and higher 
 tangents can be obtained in another way as explained at 
 p. 247, Ex. 4, and p. 274. 
 
 The use of the brass marker or index is both to retain 
 a number found by one process whilst the slide is being 
 shifted to make a second calculation and also to perform 
 involution and evolution without using the slide. 
 
 THE STATION-POINTER 
 
 This instrument is a treasure to hydrographers. By 
 it they locate their position at sea from three fixed points on 
 shore, very rapidly and exactly. A A, BB, and CC are three 
 arms having a common centre O round which two of them 
 can be moved in any direction ; DE is a circle graduated 
 from zero to 180 on each side of the central fixed arm which 
 is permanently set to zero. The two other arms are provided 
 with "verniers, and when angles are taken with the sextant to 
 the three fixed points forming two angles, one on each side 
 of a central point, these two angles are laid off on each side 
 of zero on the station pointer, which is then laid down on 
 the chart so that each of the three arms points to one of the 
 fixed points. The centre of the instrument will then be at the 
 position of the point of observation on the chart, which is 
 then pricked off through the pin-hole left for the purpose. 
 
Instruments 
 
 36; 
 
 This is the same problem as that described in Route Sur- 
 veying, p. 48, performed by the plane-table. 
 
 The only case when the station-pointer will not locate 
 the station from three known points 
 is when the station happens to be 
 in the circumference of a circle de- 
 scribed about the three points. If a 
 tracing is made of the three arms it 
 will be found that the centre of the 
 graduated limb can be moved into 
 any position in the circumference, 
 and the three arms will still go 
 through the three points. 
 
 It is always practicable to choose 
 points which will not fall into this 
 position. 
 
 The same thing can be done by 
 plotting the bearings on tracing 
 paper and superimposing it upon 
 the chart, working them round until 
 all three intersect the points. This 
 
 of course, not so rapid or cor- 
 
 rect. 
 
 FIG. 116- 
 
 BOOKS 
 
 The Nautical Almanac is the most indispensable portion 
 of the surveyor's library. The chapter on astronomy to- 
 gether with the glossary will explain all that is needed for 
 its use on survey. 
 
 Whitaker's Almanack is a substitute which is quite suffi- 
 cient for the more ordinary calculations of the geographical 
 position. A couple of the shilling edition should be taken, 
 so that those leaves can be cut out which supply the data 
 for observations, and carried about without taking up any 
 room. 
 
368 Preliminary Survey 
 
 Raper's Navigation is one of the best works on nautical 
 astronomy in the English language. It is bulky, and con- 
 tains a great many more subjects than the surveyor requires. 
 It has, however, a full supply of mathematical tables, and 
 if the surveyor has not a . copy of Chambers's or Weale's 
 mathematical tables this work is the best he can procure. 
 
 Chambers's Practical Mathematics is a very suitable book 
 for those who do not intend to go deeply into the subject. 
 It abounds in examples and illustrations, and would doubt- 
 less clear up many points that it was impossible in a treatise 
 like the present one to handle thoroughly on the subject 
 of geodetic astronomy. 
 
 Chambers's Mathematical Tables are more comprehensive 
 than those of Weale's series. Either will do quite well for 
 the surveyor's purpose. 1 
 
 Dr. Crelle^s Calculating Tables are a triumph of German 
 patience. They are a multiplication table in 500 pages of 
 quarto. Three figures by three figures are multiplied by 
 simple inspection. Larger sums are performed by dividing 
 them up into threes. Every extensive survey should be pro- 
 vided with them for office use. Ordinary small surveys need 
 nothing more than the slide-rule. The place of Crelle's tables 
 in the economy of calculation comes between the logarithmic 
 tables and the slide-rule, being quicker than the former for 
 three or four figure calculations and closer than the slide- 
 rule. But even when they are well understood they do 
 not approach the rapidity of the slide-rule for small calcula- 
 tions. 
 
 Trautwine's Pocket Book. This wonderful compendium 
 of general engineering knowledge had reached its 2yth 
 thousand three years ago. There is no one book which the 
 writer would place beside this as a vade-mecum for the 
 engineer who travels abroad either to America or the 
 Colonies. 
 
 1 Dr. Bremiker's Mathematical Tables are for ordinary degrees 
 divided decimally. Ascher and Co., Bedford Street. Price u. 6d. 
 
Instru men ts 3 69 
 
 Molesworttis Pocket Book is likewise a remarkable collec- 
 tion of useful tables and formulae. It is not so explanatory 
 as Trautwine but less bulky and expensive. Engineers going 
 to India will find in it special information upon engineering 
 as practised in that country, which Sir Guildford Molesworth 
 has been in the best possible position to collect. 
 
 Spon 's Shilling Pocket Book of Engineering Formula con- 
 tains, along with other most useful information, a table of 
 sines and tangents which are closer than Molesworth's and 
 are sufficient for ordinary curve-ranging. 
 
 Hints to Travellers. This handbook of the Royal 
 Geographical Society is as remarkable for the modesty of 
 its title as the value of its contents. It is mainly devoted 
 to geodetic astronomy and route-surveying, but it has 
 chapters on all the subjects of interest and importance to 
 travellers, including geology, natural history, anthropology, 
 &c., &c. It is the cheapest five-shillings' worth that the 
 traveller can put into his outfit. The portion on surveying 
 is by Mr. John Coles, the accomplished Instructor to the 
 Royal Geographical Society and Curator of the Library. 
 
 Fieldbooks. The form of fieldbook recommended for 
 tacheometry has been given on p. 178. The printed 
 books can be had through Messrs Elliott Bros. They are 
 marked No. on the outside, and have an eyelet hole for 
 holding the pencil. 
 
 Hold-alls. The best way to preserve pencil and rubber 
 is to keep them and a lo-inch slide-rule in a canvas hold- 
 all, buttoned on to the coat, against which it lies quite flat, 
 and each article when used is replaced in its own proper 
 place. A red and a blue pencil should be added for detail- 
 sketching, or else red-ink and blue-ink fountain-pens. 
 
 MS. books. These should contain about 100 pages. 
 Half a dozen should suffice ; well bound to resist 
 damp climates. 
 
 Stationery and drawing paper. A good supply of ruled 
 profile paper will be found most useful, not only for cross- 
 
 B B 
 
37O Preliminary Survey 
 
 sections and profiles, but also for estimates, diagrams, and 
 different kinds of scale-drawings. 
 
 Tracing paper is not of much use in a trying climate. 
 Tracing cloth with one dull side is best, as pencilling can 
 be done upon it as easily as on tracing paper. A tough 
 writing paper suitable for foreign postage and for using with 
 the manifold writer will be found the most generally useful. 
 Ink-pellets soluble in water make a fair substitute for ordi- 
 nary writing fluid, and are no dearer. A gold pen saves its 
 cost over and over again. 
 
 THE AUTHOR'S OUTFIT OF SURVEYING INSTRUMENTS 
 
 The following complete list of a surveying outfit, such as 
 is recommended for an extensive survey, costs from ioo/. to 
 
 I20/. 
 
 'Ideal' tacheometer. 5-inch horizontal and 6-inch vertical limb. 
 Telescope of 1 2-inch focal length and eyepiece magnifying 50 dia- 
 meters, having micrometer screw head fitted on right-hand side of 
 eyepiece. Fixed stadia hairs i/ioo, and vertical moving hair worked 
 by micrometer screw graduated to I 10,000 inch. Perforated axis 
 and lantern. Diagonal eyepiece. 
 
 Level staff. One ordinary 3-draw Sopwith 16 feet staff, with 2 vanes 
 fixed 10 feet apart for micrometer readings at long distances, and 
 portable table for same. 
 
 Field-books. One dozen specially printed fieldbooks, arranged for 
 stadia measurements and space for sketching. 
 
 One Verner's large size military sketch-board-plane-table and metal 
 tripod-stand, the scales being adapted to railway work. 
 
 Six dozen strips of impervious paper for ditto, ruled j-inch with co- 
 ordinate lines. 
 
 One roo-feet steel chain and 2 sets arrows. 
 
 One loo-feet steel tape, divided into feet and hundredths. 
 
 One loo-feet linen ditto, divided into feet and tenths. 
 
 One bill hook, with long handle. 
 
 One small axe. 
 
 Three ash ranging rods. 
 
 All the above are in two iron-bound deal cases, forming 
 one mule's burden. 
 
Instruments 371 
 
 One 12-inch Y level. 
 
 One Weldon range-finder. 
 
 One Abney level, with prismatic compass. 
 
 Two surveying aneroids, one reading to 5,000 feet, the other to 15,000 
 
 feet, fixed altitude scales and verniers. 
 One keyless semi-chronometer watch. 
 One pedometer and one passometer. 
 One large case of drawing instruments as described. 
 One pocket case of drawing instruments. 
 ' Hints to Travellers. 'Stanford. 
 Bremiker's Mathematical Tables. 
 Chambers's Practical Mathematics. 
 Nautical Almanac. 
 Whitaker's Almanack (2). 
 Dr. Crelle's Calculating Tables. 
 MS. Books (2). 
 
 Tracing cloth with dull back, 24 yards by 30 inches. 
 Profile paper, ruled to scale. Drawing pins, pens, pencils, liquid ink, 
 
 liquid blue, liquid carmine. 
 
APPENDIX 
 
 PLANE TRIGONOMETRY 
 
 FUNCTIONS OF RIGHT-ANGLED TRIANGLES 
 
 B 
 
 a 
 
 . a tan A 
 
 sin A = - = - 
 
 c sec A 
 
 cos A 
 
 tan A = 
 
 cot A 
 
 FIG. 117. 
 
 cosec A 
 i 
 
 c cosec A sec A' 
 a _ sin A _ i 
 b cos A cot A 
 
 sec A - -- = 
 b 
 
 sin A 
 cosec A 
 cot A 
 sec A 
 
 -- 
 
 tan A 
 
 i 
 
 cos A 
 i 
 
 sin A 
 
 cosec A - - 
 
 a tan A 
 
 sin - A + cos 2 A = i 
 
 sin A = v/ 1 cosTTA and cos A = ^ r-~sin~ ' 
 
 sec '-' A = i + tan 2 A and cosec 2 A = i + cot 
 
 sin A x cosec A = 
 
 cos A x sec A = 
 
 tan A x cot A = 
 
 sn 
 
 cos 
 
 A = 
 
 sin (A + B) = sin A. cos B + cos A. sin B 
 sin (A - B) = sin A. cos B - cos A. sin B 
 cos (A + B) = cos A. cos B - sin A. sin B 
 cos (A - B} = cos A. cos B + sin A. sin B 
 
 (i) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 ; (5) 
 
 (6) 
 
 (7) 
 A (8) 
 A (9) 
 
 (10) 
 
 (12) 
 (13) 
 (H) 
 
374 Preliminary Survey 
 
 Functions of any two angles A and B. 
 sn (A + B) + sin (A - B) - 2 sin A. cos B 
 sin (A + B) - sin (A - B) = 2 cos A. sin B 
 cos (A B) + cos (A -f B) = 2 cos A. cos B 
 cos (A - B) - cos (A + B) = 2 sin A. sin B 
 sin 2 A = 2 sin A. cos A .... 
 cos 2 A = cos 2 A sin 2 A . 
 i + cos 2 A = 2 cos '- A 
 i cos 2 A = 2 sin 2 A 
 sin 3 A = 3 sin A - 4 sin 3 A 
 cos 3 A = 4 cos 3 A 3 cos A ... 
 tan (A + B) = _tan_A_+ _fem B 
 i - tan A. tan B 
 
 (A-B) 
 
 tan 2 
 
 tan A - tan B 
 i + tan A. tan B 
 
 2 tan A 
 
 tan ^ A = 
 
 i - tan - A 
 
 sin A 
 i + cos A ' 
 
 cot ^ A = - 
 
 sin A 
 
 i - cos 
 
 cot A 
 
 tan i A 
 
 (15) 
 (16) 
 
 (17) 
 (18) 
 
 (19) 
 
 (20) 
 
 (21) 
 (22) 
 (23) 
 (24) 
 
 (25) 
 (26) 
 
 (28) 
 (29) 
 
 Relations between the sides and angles of triangles.' 
 
 cos A 
 
 cos B - 
 
 cos C = 
 
 lab 
 
 2smiA=<g**- g ><* + g -3 
 2bc 
 
 2 cos 2 i A = (^AfH^l_^) 
 
 2 2bc 
 
 (30) 
 (31) 
 (32) 
 
 (33) 
 
Appendix 375 
 
 Right-angled triangles. 
 
 Case i. Given the hypotenuse c, and a side b. 
 
 sec A - R x c+ b . . .' . (35) 
 
 a = tan A x b -4- R ..... (36) 
 
 C = 90 - A . . . . . (37) 
 
 Case 2. Given a side c and one of the oblique angles A. 
 
 C = 9 o-A .... . (38) 
 
 a = sin A x c -f- R . . . - (39) 
 
 b = cos A x c -f- R . . . (40) 
 
 Oblique-angled triangles. 
 
 Case i. When two angles and a side opposite are given. 
 The sides are proportional to the sines of the opposite angles. 
 Let AB and a be given ; then 
 
 C = 180- (A + B) ..... (41) 
 
 b = *? ....... (42) 
 
 sin A 
 
 c - t4- C . ... . (43) 
 
 sm A 
 
 Case 2. When two sides and an angle opposite to one of 
 them are given. 
 
 Let a, b, and A be given. 
 
 sin B = *J!5lA ..... (44) 
 a 
 
 . . . (45) 
 
 B = 180 - (A + C) ..... (46) 
 or C - 180 - (A + B) . . (47) 
 
 b - *'- Sin B . - . (48) 
 
 sm A 
 
 Case 3. Given two sides and the included angle. 
 Let #, by and C be given. 
 
 A + B = 180- C; 
 
 tan MA - B) = < ~ *)-_^?_ * (AB) ; ( 
 
 # + 
 
 A = i (A + B) + i (A - B) . . . (50) 
 
 c 2 = a ~ + b ~ 2 ab. cos C . . . . (52) 
 When C is obtuse the + sign, and when acute the sign 
 to be used. 
 
76 ^Preliminary Survey 
 
 Case 4, When the three sides are given. 
 Let , b, and c be given and s =.$ (a + 6 
 
 . - 
 
 SPHERICAL TRIGONOMETRY 
 
 Right-angled Spherical Triangles. 
 C 
 
 (53) 
 
 i. Given the hypotenuse $ and one of the angles C, to 
 find the other parts. 
 
 tan a = cos C x R -f- cot b . . . . (54) 
 
 sin c = sin b x sin C -=- R . . . . (55) 
 
 cot A = cos x R-^cot C . . . . (56) 
 
 Case 2. Given the hypotenuse b and a side a. 
 
 cos C - cot b x tan a -f- R . . . . (57) " 
 
 cos c = cos x R -f- cos . . . . (58) 
 
 sin A = sin a x R -f- sin ^ . . . . (59) 
 
 Case 3. Given the two sides a and c. 
 
 cos b = cos x cos a -f- R . . (60) 
 
 cot A = sin c x R -f- tan a . . . (61) 
 
 cot C = sin a x R -f- tan c . . . . (62) 
 
 GZ.W 4. Given the two angles A and C. 
 
 cos b = cot A x cot C -f- R . . . (63) 
 
 cos c = R x cos C -I- sin A . . (64) 
 
 cos a = R x cos A *- sin C . . (65) 
 
 Case 5. Given a side a and its adjacent angle C. 
 
 cot b - R x cos C -f- tan a . . . . (66) 
 
 tan c = R x sin a -j- cot C . . . (67) 
 
 cos A = sin C x cos a -j- R . . . (68) 
 
 Case 6. Given a side c and its opposite angle C. 
 
 sin b = R x sin c -i- sin C . . . . (69) 
 
 sin A = R x cos C -4- cos c . (70) 
 
 shirt - cot C x tan c -^ R . (70 
 
Appendix 377 
 
 This last is termed the ' ambiguous ' case. See Chambers's 
 Practical Mathematics,' p. 379. 
 
 R expressed logarithmically is jcxoooooo. 
 
 Oblique-angled Spherical Trigonometry. 
 A 
 
 , FIG. 120. 
 
 JO 
 
 Rule i. Given three sides to find the angles. 
 Let s = (a + b + c]. 
 
 sin (s U] x sin (s c] , N 
 
 L-,.,1 ..-L, . ( 72 ) 
 
 i B = A/- ' (73) 
 
 V sm a - - 
 
 sin i C .= . - - ' . (74) 
 
 sin a x sin b 
 
 cos , 
 
 /3mT^sm(7^) 
 V sin b x sin <: 
 
 and for B and C the formulae are exactly analogous. 
 Rule 2. Given two sides and the included angle to find the 
 other parts. 
 
 Let a and b be the sides and C the given angle. 
 
 sin (a + b} : sin (a ~ b} : - cot C : tan (A ~ B) ; 
 cos %(a + b}\ cos | (a ~ b} : : cot ^ C : tan (A + B) ; 
 whence A - \ (A + B) +~H A ~ B ) (7 6 ) 
 B = HA + B) - HA - B) . . . (77) 
 
 To find c. 
 
 sin \ (A ~ B) : sin (A + B) : : tan (a - ) : tan * ; 
 /. tan A r = sin (A + B) x tan (a ~ ) -f- sin (A ~ B) 
 
 (78) 
 
 / ^z/^^y spherical triangle the sines of the angles are pro- 
 portional to the sines of the opposite sides, 
 
378 Preliminary Survey 
 
 Rule 3. Given two sides and the angle opposite to one of 
 them. 
 
 Let <2, b, and A be given, then B is found by 
 
 sin a : sin b \ \ sin A : sin B . . . . (79) 
 and for c 
 
 tan \ c : tan (a ~ b} : : sin (A + B) : sin (A ~ B) 
 and for C (80) 
 
 cot C : tan (A ~ B) : : sin -i- (a + &} \ sin (a ~ ) 
 
 (81) 
 TABLE LII. 
 
 Values of sin b x sin Cyfrr Fmr.? 1890 /0 1900 for Azimuth 
 by and 5 Draconis in same Vertical. 
 
 log sin Azim = tabular constant log cos Lat. 
 
 1890 ....... 9-48568 
 
 1891 ....... 
 
 1892 . . 
 
 1893 . 9-48540 
 
 1894 . . 9-48531 
 
 1895 9-48521 
 
 1896 ....... 9-48512 
 
 1897 .... . 9*48503 
 
 1898 ....... 9*48494 
 
 1899 ....... 9-48484 
 
 1900 .. . 9-48475 
 
 TABLE LIII. 
 
 Values of sin. b x sin Cfor Years 1890 to igoofor Azimuth by and 
 e Urstz Majoris (Merak and Alioth} when in same Vertical. 
 
 log sin Azim = tabular constant log cos Lat. 
 
 1890 . . 972910 
 
 1891 .... . 9-72917 
 
 1892 ... . 972923 
 
 1893 . 9'7293o 
 
 1894 . . 972936 
 
 1895 .. 972943 
 
 1896 : . . . . - . . 972950 
 
 1897 ..... . 972956 
 
 1898 ....... 972963 
 
 1899 ....... 9-72969 
 
 1900 ....... 9-72976 
 
Appendix 
 
 TAKLE LIV. 
 
 379 
 
 Radii corresponding to Decimals of a De^ 
 per Chord of loofeet. 
 
 \ > j, 7 0>>\ 
 
 '.gi-ee of'C&Vvalitre' ' 
 
 X ^%/4. ^ 
 
 Angle of 
 
 Radius 
 
 Angle of 
 
 Radius 
 
 Angle of 
 
 Radius 
 
 Angle of 
 
 .~j- 
 
 Radius 
 
 deflection 
 
 in feet 
 
 deflection 
 
 in feet 
 
 deflection 
 
 in feet 
 
 deflection 
 
 in feet 
 
 d 
 
 r 
 
 d 
 
 r 
 
 d 
 
 r 
 
 d 
 
 r 
 
 0-05 
 
 II4592 
 
 2'6o 
 
 22O4 
 
 8-20 
 
 699-3 
 
 2O-OO 
 
 287- 9 
 
 o-io 
 
 57296 
 
 270 
 
 2122 
 
 8-40 
 
 682-7 
 
 2I-OO 
 
 274-4 
 
 0-15 
 
 38197 
 
 2-8o 
 
 2046 
 
 8-60 
 
 666-8 
 
 22-OO 
 
 262-O 
 
 O'2O 
 
 28648 
 
 2-90 
 
 1976 
 
 8-80 
 
 6517 
 
 23-00 
 
 250-8 
 
 O-25 
 
 22918 
 
 3-Qo 
 
 I9IO 
 
 9-00 
 
 637-3 
 
 24-00 
 
 240-5 
 
 0-30 
 
 19098 
 
 3'io 
 
 1848 
 
 9-20 
 
 623-4 
 
 25-00 
 
 231-0 
 
 o'35 
 
 16370 
 
 3-20 
 
 1791 
 
 9-40 
 
 610-2 
 
 26-00 
 
 222-3 
 
 0-40 
 
 14324 
 
 330 
 
 1736 
 
 9-60 
 
 597-5 
 
 27-00 
 
 2I4-2 
 
 0'4S 
 
 12732 
 
 3-40 
 
 1685 
 
 9-80 
 
 585-4 
 
 28-00 
 
 206-7 
 
 0-50 
 
 "459 
 
 3-5o 
 
 1637 
 
 10-00 
 
 5737 
 
 29-00 
 
 I99-7 
 
 o'5S 
 
 10417 
 
 3-60 
 
 1592 
 
 IO-2O 
 
 562-5 
 
 3O-OO 
 
 I93-2 
 
 0-60 
 
 9549 
 
 1 370 
 
 *549 
 
 IO-4O 
 
 5517 
 
 | 3* -0 
 
 I8 7 -I 
 
 0-65 
 
 8815 
 
 3-80 
 
 1508 
 
 10-60 
 
 541-3 
 
 32-00 
 
 lSl'4 
 
 070 
 
 8185 
 
 3 -go 
 
 1469 
 
 10-80 
 
 531-3 
 
 : 33-00 
 
 I76-0 
 
 075 
 
 7639 
 
 4-00 
 
 H33 
 
 i 11-00 
 
 521-7 
 
 34-00 
 
 I7I-0 
 
 0-80 
 
 7162 
 
 4-20 
 
 1364 
 
 11-20 
 
 5 I2 '4 
 
 35' 00 
 
 166-3 
 
 0-85 
 
 6741 
 
 4-40 
 
 1302 
 
 11-40 
 
 503-4 
 
 36-00 
 
 161-8 
 
 0-90 
 
 6366 
 
 4-60 
 
 1246 
 
 1 1 -60 
 
 494-8 
 
 37-00 
 
 157-6 
 
 o-95 
 
 6031 
 
 4-80 
 
 1194 
 
 11-80 
 
 486-4 
 
 38-00 
 
 153-6 
 
 oo 
 
 5730 
 
 5 'OP 
 
 1146 
 
 12-00 
 
 478-3 
 
 39-00 
 
 149-8 
 
 10 
 
 5209 
 
 5-20 
 
 IIO2 
 
 I2-50 
 
 459-3 
 
 40-00 
 
 146 -2 ; 
 
 20 
 
 4775 
 
 5'40 
 
 1061 
 
 I3-OO 
 
 441-7 
 
 42-00 
 
 I39-52 
 
 3 
 
 4407 
 
 5-60 
 
 1024 
 
 I3-50 
 
 425-4 
 
 44-00 
 
 I33-47 
 
 40 
 
 4093 
 
 5-80 
 
 988-3 
 
 I4-00 
 
 410-3 | 
 
 46-00 
 
 127-97 
 
 'SO 
 
 3820 
 
 6-00 
 
 955-4 
 
 H-S 
 
 396-2 
 
 48-00 
 
 122-93 
 
 60 
 
 358i 
 
 6-20 
 
 924-6 
 
 I5-OO 
 
 383-1 i 
 
 50-00 
 
 118-31 
 
 70 
 
 3370 
 
 1 6-40 
 
 895-7 
 
 I5-5 
 
 370-8 
 
 55-oo 
 
 108-28 
 
 80 
 
 3183 
 
 I 6-60 
 
 868-6 
 
 1 6 -oo 
 
 359-3 
 
 60-00 
 
 lOO'OO 
 
 90 
 
 3016 
 
 6-80 
 
 843-1 i 
 
 16-50 
 
 348-5 
 
 65-00 
 
 93-06 
 
 2'OO 
 
 2865 
 
 7-00 
 
 819-0 \ 
 
 17-00 
 
 338-3 , 
 
 70-00 
 
 87-17 
 
 2-10 
 
 2729 
 
 7-20 
 
 796-3 i 
 
 17-50 
 
 3287 ; 
 
 75-00 
 
 82-13 
 
 2'20 
 
 2605 
 
 7-40 
 
 774-8 
 
 18-00 
 
 319-6 
 
 80-00 
 
 7778 , 
 
 2-30 
 
 2491 
 
 7'6o 
 
 754-4 i 
 
 18-50 
 
 311-1 
 
 85-00 
 
 74-01 
 
 2-40 
 
 2387 
 
 7-80 
 
 735-1 
 
 19-00 
 
 302-9 
 
 90-00 
 
 70-71 1 
 
 2-50 
 
 2292 
 
 8-00 
 
 716-8 
 
 i 
 
 19-50 
 
 295-3 
 
 
 
3&O Preliminary Survey 
 
 TABLE LV. 
 
 General Elements of the Decimal Spiral. 
 
 
 
 | 
 
 Total curvature 
 
 Tangential 
 
 No. of point 
 n 
 
 Curvature in 
 one chord 
 
 Total 
 curvature 
 up to n 
 
 up to mid-chord 
 or deflection of 
 short chord from 
 
 angle, or angle 
 between long 
 chord and 
 
 
 . f. 
 
 * 
 
 main tangent 
 
 main tangent 
 
 
 
 
 k 
 
 i 
 
 
 degrees 
 
 degrees 
 
 decrees 
 
 degrees 
 
 I 
 
 O-2 
 
 0-2 
 
 o-i 
 
 O-I 
 
 2 
 
 0'4 
 
 0-6 
 
 0-4 
 
 0-25 
 
 3 
 
 0-6 
 
 I '2 
 
 0-9 
 
 0'47 
 
 4 
 
 0-8 
 
 2-O 
 
 1-6 
 
 075 
 
 5 
 
 I-O 
 
 3 -0 
 
 2'5 
 
 no 
 
 6 
 
 I'2 
 
 4'2 
 
 3-6 
 
 1-52 
 
 7 
 
 I'4 
 
 5*6 
 
 4'9 
 
 2-OO 
 
 8 
 
 1-6 
 
 7-2 
 
 6-4 
 
 2'55 
 
 9 
 
 1-8 
 
 9-0 
 
 8-1 
 
 3-17 
 
 10 
 
 2-0 
 
 II'O 
 
 IO'O 
 
 3-85 
 
 ii 
 
 2'2 
 
 13-2 
 
 I2-I 
 
 4-60 
 
 12 
 
 2-4 
 
 15-6 
 
 14-4 
 
 5-41 
 
 13 
 
 2-6 
 
 18-2 
 
 l6'9 
 
 6-30 
 
 14 
 
 2-8 
 
 21'0 
 
 I9-6 
 
 7-24 
 
 15 
 
 3-0 
 
 24-0 
 
 22'5 
 
 8-26 
 
 16 
 
 3-2 
 
 27-2 
 
 25-6 
 
 9-33 
 
 17 
 
 3 '4 
 
 30*6 
 
 28-9 
 
 10-48 
 
 18 
 
 3-6 
 
 34'2 
 
 32-4 
 
 11-68' 
 
 19 
 
 3'8 
 
 38-0 
 
 36-1 
 
 12-95 
 
 20 
 
 4-0 
 
 42-0 
 
 40-0 
 
 14-29 
 
 21 
 
 4-2 
 
 46-2 
 
 44'I 
 
 15-69 
 
 TABLE LVL 
 Elements of No. 2 Spiral for Traimoays. 
 
 n r 
 
 
 
 y 
 
 L 
 
 feet 
 
 feet 
 
 feet 
 
 feet 
 
 I 572-96 
 
 003 
 
 2-00 
 
 2 
 
 2 286-48 
 
 017 
 
 4'OO 
 
 4 
 
 3 I90-99 
 
 049 
 
 6 -00 
 
 6 
 
 4 I43-24 
 
 105 
 
 8-00 
 
 O 
 
 5 "4-59 
 
 I 9 2 
 
 10-00 
 
 10 
 
 6 95-49 
 
 318 
 
 11-99 
 
 12 
 
 7 81-85 
 
 488 
 
 13-98 
 
 14 
 
 8 71-62 
 
 711 
 
 I5-97 
 
 16 
 
 9 63-66 
 
 '993 
 
 I7-95 
 
 18 
 
Appendlv 
 
 T A i? LE LVI. Continued. 
 
 Elements of No. 2 Spiral for Tramways. 
 
 . 
 
 r j 
 
 y 
 
 L 
 
 
 feet feet feet 
 
 feet 
 
 IO 
 
 57-30 1-34 19-92 
 
 2O 
 
 II 
 
 52-09 1-76 21-88 
 
 22 
 
 12 
 
 4775 2-2 
 
 ,6 23-82 
 
 2 4 
 
 13 
 
 44-08 2* 
 
 4 25-73 
 
 26 
 
 
 40-93 3-51 27-61 
 
 28 
 
 15 
 
 38-20 4-27 29-46 
 
 30 
 
 16 
 
 35-81 5-14 31-26 
 
 32 
 
 17 
 
 33-71 6-10 33-02 
 
 34 
 
 18 
 
 31-84 7-18 34-70 
 
 36 
 
 19 
 
 30-16 8-35 36-32 
 
 38 
 
 20 
 
 28-65 9-64 37-85 
 
 40 
 
 21 
 
 27-29 11-03 39-29 
 
 42 
 
 
 TABL 
 
 E LVI I. 
 
 
 Elements of No. 5 Spiral for 
 
 Tramways and Light Railways. 
 
 n 
 
 ds r 
 
 x y 
 
 L 
 
 
 degrees feet ' 
 
 feet feet 
 
 feet 
 
 I 
 
 4'OO 1432*4 
 
 O-OO9 5'OO 
 
 5 
 
 2 
 
 8-Qi 716-2 
 
 0-044 10-00 
 
 IO 
 
 3 
 
 12-02 477-5 
 
 O-I22 I5-OO 
 
 15 
 
 4 
 
 I6-05 358T 
 
 O-262 2O-OO 
 
 20 
 
 5 
 
 20-10 286-5 
 
 0-480 24-99 
 
 25 
 
 6 
 
 24-18 238-7 
 
 0-794 29-98 
 
 30 
 
 7 
 
 28-29 204-6 
 
 1-22 34-96 
 
 35 
 
 8 
 
 32-43 179-0 
 
 I-78 39-93 
 
 40 
 
 9 
 
 36*62 I59-I6 
 
 2-48 44-88 
 
 45 
 
 IO 
 
 40-86 I43'24 
 
 3-35 49^1 
 
 50 
 
 n 
 
 45-16 I30-22 
 
 4-40 54-70 
 
 55 
 
 12 
 
 49-52 119-40 
 
 5-64 59-54 
 
 60 
 
 13 
 
 53-97 IIO-2O 
 
 7'IO 64-32 
 
 65 
 
 H 
 
 58-50 IO2-31 
 
 8-77 69-03 
 
 70 
 
 15 
 
 63T4 95*54 
 
 IO-69 73-65 
 
 75 
 
 16 
 
 67-89 89-53 
 
 I2-85 78-I6 
 
 80 
 
 17 
 
 72-79 84-25 
 
 15-26 82-54 
 
 85 
 
 18 
 
 77-84 79-58 
 
 17-94 86-76 
 
 90 
 
 19 
 
 83-07 75-40 
 
 20-89 90-80 
 
 95 
 
 20 
 
 88-53 7I-63 
 
 24-10 94-63 
 
 100 
 
 21 
 
 94-26 68-21 
 
 27-58 98-22 
 
 105 
 
382 
 
 Preliminary Survey 
 
 TABLE LVIII. 
 
 Elements of No. 10 Spiral for Light Railways. 
 
 TABLE LIX. 
 Elements of No. 15 Spiral for Light Railways. 
 
 n 
 
 ^ J 
 
 ' 
 
 X 
 
 y L 
 
 
 degrees 
 
 feet 
 
 feet 
 
 feet feet 
 
 i 
 
 2-OOO 
 
 2,865-0 
 
 0-017 
 
 10-00 
 
 IO 
 
 2 
 
 4'ODI 
 
 1,432-0 
 
 0-087 
 
 20-00 
 
 20 
 
 3 
 
 6-003 
 
 954-9 
 
 0-244 
 
 30-00 
 
 30 
 
 4 
 
 8-006 
 
 7I6-2 
 
 0-523 
 
 39-99 
 
 40 
 
 5 
 
 10-012 
 
 573-0 
 
 0-960 
 
 49-98 
 
 50 
 
 6 
 
 I2-O22 
 
 477-5 
 
 1-588 
 
 59-96 60 
 
 7 
 
 I4-034 
 
 409-3 
 
 2-442 
 
 69-93 70 
 
 8 
 
 I6-O52 
 
 358-1 
 
 3-556 
 
 79-87 
 
 80 
 
 9 
 
 18-074 
 
 3I8-3 
 
 4-96 
 
 8977 
 
 90 
 
 10 
 
 2O-IO2 
 
 286-5 
 
 6-70 
 
 99-6I 
 
 100 
 
 ii 
 
 22-136 
 
 260-4 
 
 8-80 
 
 10939 
 
 no 
 
 12 
 
 24-197 
 
 238-6 
 
 11-28 
 
 119-08 
 
 1 20 
 
 13 
 
 26-226 
 
 220-4 
 
 14-19 
 
 I28-65 
 
 130 
 
 H 
 
 28-283 
 
 204-6 
 
 I7-55 
 
 138-07 
 
 140 
 
 15 
 
 30-350 
 
 I9I-O 
 
 21-37 
 
 I473I 
 
 150 
 
 16 
 
 32-427 
 
 I79-I 
 
 25-69 
 
 I56-32 
 
 1 60 
 
 17 
 
 34-5I4 
 
 168-5 
 
 30-53 
 
 165-08 
 
 170 
 
 18 36-614 
 
 I59-2 
 
 35-88 
 
 173-53 
 
 1 80 
 
 19 38726 
 
 I50-8 
 
 4178 
 
 181-61 
 
 190 
 
 20 
 
 40-852 
 
 1433 
 
 48-20 
 
 !8 9 -27 
 
 200 
 
 21 
 
 42-992 
 
 136-4 
 
 55-i6 
 
 196-45 
 
 210 
 
 n 
 
 ds 
 
 r 
 
 X 
 
 y 
 
 L 
 
 
 decrees 
 
 feet 
 
 feet 
 
 feet feet 
 
 I 
 
 i-33 
 
 4297-1 
 
 02 
 
 I5-00 15 
 
 2 
 
 2-67 
 
 2148-6 
 
 13 
 
 3O-OO 30 
 
 3 
 
 4-00 
 
 I432-4 
 
 37 
 
 45 -0 45 
 
 4 5 '33 
 
 10743 
 
 78 
 
 59-99 60 
 
 5 
 
 6-67 
 
 859-6 
 
 1-44 
 
 74-98 75 
 
 6 8-oi 
 
 7l6'2 
 
 2-38 
 
 89-95 
 
 90 
 
 7 
 
 9'34 
 
 6I3-9 
 
 3-66 
 
 104-89 
 
 I0 5 
 
 8 
 
 10-68 
 
 537-2 
 
 5-33 
 
 119-80 
 
 1 20 
 
 9 1 2 -02 
 
 477-5 7-45 
 
 I34-65 
 
 i35 
 
 10 !3"36 
 
 429-7 
 
 10-05 
 
 149-42 
 
 150 
 
 IL 14-71 
 
 390-7 
 
 13-20 
 
 164-09 165 
 
 12 
 
 16-05 
 
 358-1 16-93 178-62 180 
 
Appendix 
 
 383 
 
 TABLE LIX. Continued. 
 Elements of No. 15 Spiral for Light Railways. 
 
 11 
 
 ds 
 
 r 
 
 X 
 
 y 
 
 L 
 
 
 decrees 
 
 feet 
 
 feet 
 
 feet 
 
 feet 
 
 13 
 
 17-40 
 
 3306 
 
 21 29 
 
 192-97 
 
 195 
 
 14 
 
 18-75 
 
 307-0 
 
 26-32 
 
 2O7-IO 
 
 210 
 
 15 
 
 2O-IO 
 
 286-5 
 
 32-O6 
 
 220/96 
 
 225 
 
 16 
 
 21-45 
 
 268-6 
 
 38-54 
 
 234 '49 
 
 240 
 
 'I? 
 
 22-81 
 
 252-8 
 
 4579 
 
 247-62 
 
 255 
 
 18 
 
 24-18 
 
 238-8 
 
 53-83 
 
 260-29 
 
 270 
 
 19 
 
 25-54 
 
 226-2 
 
 62-66 
 
 272-41 
 
 285 
 
 20 
 
 26-91 
 
 214-9 
 
 72-31 
 
 283-90 
 
 300 
 
 21 
 
 28-28 
 
 204-7 
 
 8274 
 
 294-67 
 
 315 
 
 TABLE LX. 
 
 Elements of No. 25 Spiral for Narrow- Gartge Railways. 
 
 n \ ds 
 
 r 
 
 X 
 
 y 
 
 L 
 
 degrees 
 
 feet 
 
 feet 
 
 feet 
 
 feet 
 
 i 0-80 
 
 7I6I-9 
 
 0-043 25-00 
 
 25 
 
 2 
 
 I -60 
 
 3580-9 
 
 0-218 
 
 50-00 50 
 
 3 
 
 2-40 
 
 23873 
 
 0-610 75-00 75 
 
 4 
 
 3 -20 
 
 I790-5 
 
 1-31 
 
 99-99 
 
 IOO 
 
 5 4-oo 
 
 I432-4 
 
 2-40 
 
 124-96 
 
 125 
 
 6 4-80 
 
 1193-6 
 
 3-97 
 
 149-91 
 
 150 
 
 7 5-60 
 
 I023-I 
 
 6-10 
 
 174-82 
 
 175 
 
 8 6-40 
 
 895-2 
 
 8-89 
 
 I99-66 200 
 
 9 7-20 
 
 795-8 
 
 12-41 
 
 224-41 
 
 225 
 
 10 8-01 
 
 716-2 
 
 1675 
 
 249-03 
 
 250 
 
 ii 
 
 8-81 
 
 651-1 
 
 21-99 
 
 273-48 
 
 275 
 
 12 
 
 9-61 
 
 596-9 
 
 28-21 
 
 297-69 
 
 300 
 
 J 3 
 
 10-41 
 
 55i-o 
 
 35-48 321-61 
 
 325 
 
 14 
 
 11-22 
 
 511-6 
 
 43-86 
 
 345-I6 350 
 
 15 12-02 
 
 477-5 
 
 53-43 
 
 368-26 375 
 
 16 12-83 
 
 447-7 
 
 64-23 390-81 400 
 
 17 13-63 
 
 421-4 
 
 76-31 412-70 425 
 
 18 
 
 14-44 
 
 397-9 
 
 8971 433-8i 450 
 
 19 15-24 
 
 377-0 
 
 104-44 454'or 475 
 
 20 
 
 I6-05 
 
 358-2 
 
 120-51 
 
 473-16 
 
 500 
 
 21 16-85 
 
 341 'I 
 
 I37-90 
 
 491-11 
 
 525 
 
334 
 
 Preliminary Survey 
 
 TABLE LXI. 
 
 Elements oj No. 50 Spiral for Narrow- Gauge Railways. 
 
 
 
 ds 
 
 X 
 
 y 
 
 
 
 
 \ 
 
 
 degrees 
 
 feet 
 
 feet 
 
 I 
 
 0*4 
 
 0-087 
 
 5O-OO 
 
 2 
 
 0-8 
 
 0-436 
 
 100-00 
 
 3 
 
 I "2 
 
 I -22 1 
 
 I49-99 
 
 4 
 
 1-6 
 
 2-62 
 
 199-97 
 
 5 
 
 2'O 
 
 4-80 
 
 249-92 
 
 6 
 
 2'4 
 
 7-94 
 
 299-82 
 
 7 
 
 2-8 
 
 12-21 
 
 349^4 
 
 8 
 
 3 "2 
 
 1778 
 
 39933 
 
 9 
 
 3-6 
 
 24-83 
 
 448-83 
 
 10 
 
 4-0 
 
 33-5I 
 
 498-07 
 
 ii 
 
 4'4 
 
 43'99 
 
 546-96 
 
 12 
 
 4-8 
 
 56-42 
 
 59539 
 
 13 
 
 5'2 
 
 70-96 
 
 643 -23 
 
 14 
 
 
 8773 
 
 690-33 
 
 15 
 
 6-0 
 
 106-86 
 
 
 16 
 
 6-4 
 
 128-47 
 
 78I-62 
 
 17 
 
 6-8 
 
 I52-63 
 
 825-40 
 
 18 
 
 7-2 
 
 I79-42 
 
 867-61 
 
 19 
 
 7-6 
 
 208-88 
 
 9O8-OI 
 
 20 
 
 8-0 
 
 241-02 
 
 946 -32 
 
 21 
 
 8-41 
 
 275-81 
 
 9 82-22 
 
 feet 
 
 50 
 
 100 
 
 150 
 
 200 
 
 250* 
 
 300 
 
 350 
 400 
 
 450 
 500 
 550 
 600 
 650 
 700 
 
 750 
 800 
 850 
 900 
 950 
 IOOO 
 
 1050 
 
 TABLE LXII. 
 Elements of No. 75 Spiral for Trunk Lines. 
 
 n 
 
 ds x 
 
 y 
 
 L 
 
 
 degrees, feet 
 
 feet 
 
 feet 
 
 I 
 
 0-27 
 
 0-13 
 
 75-00 
 
 75 
 
 2 
 
 3 
 
 0'8o 
 
 0-65 
 1-83 
 
 149-99 
 224-98 
 
 150 
 225 
 
 4 
 
 1-07 
 
 3'93 
 
 299-95 300 
 
 5 
 
 1-33 j 7-20 i 
 
 374-88 375 
 
 6 
 
 i -60 
 
 11-92 
 
 449-73 450 
 
 7 
 
 1-87 
 
 18-31 
 
 524-46 525 
 
 8 
 
 2-13 
 
 26-67 
 
 598-99 
 
 600 
 
 9 
 
 10 
 
 2'39 
 2-67 
 
 37-24 
 50-26 
 
 673-24 
 748-16 
 
 675 
 
 750 
 
Appendix 
 
 385 
 
 TABLE 'LXIl. Continued. 
 Elements of No. 75 Spiral for Trunk Lines. 
 
 ds 
 
 X 
 
 y \ 
 
 L 
 
 II 
 
 i 2-93 
 
 65-99 
 
 820-44 
 
 825 
 
 12 
 
 j . 3-I9 
 
 84-64 893-08 
 
 900 
 
 13 
 
 
 106-44 964-84 
 
 975 
 
 14 
 
 3-72 131-60 1035-49 ! 
 
 1050 
 
 15 
 
 4-00 160-30 1104-79 
 
 1125 
 
 16 
 
 i 4-26 ; 192-70 
 
 II72-43 
 
 1 200 
 
 17 
 
 4-52 228-94 
 
 I238-08 j 
 
 1275 
 
 18 
 
 4-79 269-13 
 
 I30I-4I 
 
 
 19 
 
 5-05 313-32 
 
 I362-OI 
 
 1425 
 
 20 
 21 
 
 5-33 361-53 i4i9'47 
 5-60 413-72 1473-32 
 
 1500 
 
 1575 
 
 TABLE LXIII. 
 Elements of No. 100 Spiral for Trunk Lines. 
 
 I 
 2 
 
 3 
 4 
 
 I 
 
 8 
 
 9 
 
 10 
 ii 
 
 12 
 
 13 
 
 14 
 
 17 
 
 18 
 
 19 
 
 20 
 21 
 
 rfj 
 
 C 
 
 X 
 
 y 
 
 L 
 
 degrees 
 
 feet 
 
 feet 
 
 feet 
 
 feet 
 
 0-2 
 
 100-00 
 
 0-17 
 
 100-00 
 
 100 
 
 0-4 
 
 199-99 
 
 0-87 
 
 199-99 
 
 2OO 
 
 0-6 
 
 299-98 ; 
 
 2-44 
 
 299-98 
 
 300 
 
 0-8 
 
 399-97 
 
 5-23 
 
 399-94 
 
 4OO 
 
 i -o 
 
 499-94 
 
 9-60 
 
 499-85 
 
 500 
 
 I '2 
 
 599-86 
 
 15-88 
 
 599-65 
 
 600 
 
 i '4 
 
 699-70 
 
 24-42 
 
 699-29 
 
 700 
 
 1-6 
 
 799-45 
 
 35-56 
 
 798-66 
 
 800 
 
 1-8 
 
 899-04 
 
 49-65 
 
 897-66 
 
 900 
 
 2-0 
 
 998-39 
 
 67-02 
 
 996-14 
 
 IOOO 
 
 2-2 
 
 1097-44 
 
 87-98 
 
 1093-92 
 
 IIOO 
 
 2'4 
 
 1196-11 
 
 112-85 
 
 1190-78 
 
 1200 
 
 2-6 
 
 1295-71 
 
 141-92 
 
 1286-46 
 
 1300 
 
 2-8 
 
 1391-77 
 
 175-47 
 
 1380-67 
 
 I4OO 
 
 3-0 
 
 1488-52 
 
 21373 
 
 1473-06 
 
 I5OO 
 
 3-2 
 
 1584-15 
 
 256-94 
 
 1563-24 
 
 1600 
 
 3 '4 
 
 1678-85 
 
 305-27 
 
 1650-79 
 
 1700 
 
 3-6 
 
 1772-36 
 
 358-85 
 
 1735-22 
 
 I800 
 
 3'8 
 
 i 1863-42 
 
 41777 
 
 1 816-02 
 
 I9OO 
 
 4-0 
 
 I948-59 
 
 482-05 
 
 1892-63 
 
 2OOO 
 
 4-2 
 
 I 2040-48 
 
 55I-64 
 
 1964-44 
 
 2100 
 
 C C 
 
386 
 
 Preliminary Survey 
 
 TABLE LXIV. 
 For Ranging the Spiral from an Intermediate point. 
 
 The values of , k } and i are common to all spirals : x and y are for No. 100 
 spiral, but for other spirals x and y are obtained by simple percentage. Thus for No. i s 
 spiral take 15 per cent. 
 
 Instrument at i 
 
 Instrument at 2 
 
 n 
 
 k 
 
 x y i 
 
 n k x 
 
 y 
 
 i 
 
 
 deg. 
 
 feet 
 
 feet i deg. 
 
 
 deg. feet 
 
 feet 
 
 deg. 
 
 i 
 
 o 
 
 o 
 
 
 
 o 
 
 2 
 
 o 
 
 o 
 
 o 
 
 
 
 2 
 
 0*2 
 
 *35 
 
 lOO'OO O"2O 
 
 3 
 
 o*3 
 
 0-52 
 
 100*00 
 
 0*30 
 
 3 
 
 0-7 
 
 i*57 
 
 199-99 0.45 
 
 4 
 
 1*0 
 
 2-27 
 
 199*98 
 
 0-65 
 
 4 
 
 i*4 
 
 4*01 
 
 299-96 i 0-76 
 
 5 
 
 1*9 
 
 5*58 
 
 299*93 
 
 1-07 
 
 5 
 
 2*3 
 
 803 
 
 399*88 
 
 1*15 
 
 6 
 
 3*o 
 
 10*82 
 
 399*79 
 
 i*55 
 
 6 
 
 3*4 
 
 1396 
 
 499-70 
 
 1*60 
 
 7 
 
 4*3 
 
 18-32 499-51 
 
 2"IO 
 
 7 
 
 4*7 
 
 22-15 
 
 
 2*12 
 
 8 
 
 5*8 
 
 28*42 599 'o 
 
 2-72 
 
 8 
 
 6-2 
 
 32*95 
 
 698*78 
 
 2-70 
 
 9 
 
 7*5 
 
 4**47 
 
 698*14 
 
 3*40 
 
 9 
 
 7*9 
 
 46*70 
 
 797'83 
 
 3*35 
 
 10 
 
 9*4 57'8i 
 
 796*80 
 
 4 -I 5 
 
 10 
 
 8-8 
 
 61*99 
 
 896*66 
 
 4*05 
 
 ii ; 11-5 77*74 894*79 
 
 4*97 
 
 ii 
 
 1 1 '9 
 
 82*61 
 
 994'5i 
 
 4*75 
 
 12 13*8 101*60 
 
 991*90 
 
 5*85 
 
 i 12 
 
 14-2 
 
 107*14 
 
 1091*45 5*61 
 
 13 16*3 129*66 
 
 1087*89 
 
 6*80 
 
 13 
 
 16-7 
 19-4 
 
 135*88 
 169*10 
 
 1187*23 i 6*53 
 1281*56 ' 7*52 
 
 14 19*0 162*22 
 15 i 21*9 199*52 
 
 1182*44 
 1275*22 
 
 7-81 
 
 8*89 
 
 15 
 
 22-3 
 
 207*04 
 
 1374*08 : 8*57 
 
 16 25*0 241*78 1365*85 10*04 
 
 16 
 
 25*4 249*94 
 
 1464*41 
 
 9*69 
 
 17 \ 28*3 ; 289*19 1453*9 J "'25 
 
 17 
 
 28-7 297*96 
 
 1552*13 10*87 
 
 18 
 
 31*8 341*88 i 1538*89 
 
 12*55 
 
 18 ' 32*2 ' 351*25 
 
 1636*75 12*11 
 
 19 
 
 35*5 399'96 1620-30 
 
 13*87 
 
 19 35*9 409*88 
 
 I 7 I 7"75 i3'42 
 
 20 
 
 39*4 463'43 ! 1697-57 
 
 15*27 
 
 20 ; 39-8 473*89 1794*58 14*79 
 
 21 
 
 43*5 532*26 1770*11 
 
 16-74 
 
 21 ! 43*9 543*23 1866*63 16*23 
 
 
 Instrument at 3 
 
 Instrument at 4 
 
 3 
 
 
 
 o 
 
 o 
 
 o 
 
 4 
 
 
 
 
 
 o 
 
 o 
 
 4 
 
 0*4 
 
 0*70 
 
 100*00 
 
 0-40 
 
 5 
 
 o*5 
 
 0-87 
 
 100*00 
 
 0*50 
 
 5 
 
 
 2-97 
 
 199-97 
 
 0-85 
 
 6 
 
 1*6 
 
 3 "66 1 199-96 
 
 1*05 
 
 6 
 
 2 '4 
 
 
 299*88 
 
 J '37 
 
 j 
 
 2*9 
 
 8-72 299-83 
 
 1-67 
 
 7 
 
 3*7 
 
 13*61 
 
 399*67 
 
 
 8 
 
 4*4 
 
 16-40 399*53 
 
 2*35 
 
 8 
 
 5*2 
 
 22*67 
 
 499*26 
 
 2-60 
 
 9 
 
 6*1 
 
 27*02 498'97 
 
 3*10 
 
 9 
 
 6-Q 
 
 34*68 
 
 598*54 
 
 3*32 
 
 10 
 
 8*0 
 
 40-94 597-99 
 
 3*92 
 
 10 8*8 
 
 49*98 
 
 697*36 
 
 4-10 
 
 ii 
 
 10*1 
 
 58-48 696-44 
 
 4 -8o 
 
 ii 
 
 10*9 
 
 68*89 
 
 795*56 
 
 4*95 
 
 12 
 
 12*4 
 
 79*95 794" 11 
 
 5*75 
 
 12 
 
 13*2 
 
 9i'73 
 
 892*92 
 
 5*86 
 
 jo 
 
 14-9 
 
 105-66 890*75 
 
 6*76 
 
 13 
 
 
 118-79 
 
 989*18 
 
 6-85 
 
 X 4 
 
 17-6 
 
 135*90 986*07 
 
 7*85 
 
 
 18*4 
 
 150-35 1084*07 
 
 7-90 
 
 15 
 
 20-5 
 
 170*92 1 1079*74 
 
 9'oo 
 
 15 
 
 21*3 
 
 186*68 1177*24 
 
 9"oi 
 
 16 
 
 23*6 
 
 210*96 1171*37 
 
 IO"2I 
 
 16 
 1 8 
 
 24*4 
 27-7 
 31-2 
 
 227*99 ' : 1268*31 
 
 274*47 : I356*85 
 326*27 j 1442*38 
 
 11-44 
 12-75 
 
 19 
 
 26-9 
 3*4 
 
 256*20 1260*55 
 306*80 1346*80 
 362*87 : 1429*61 
 
 11-49 
 12-83 
 14*24 
 
 10 
 
 34*9 
 
 383"49 1 I524*4O 
 
 14*12 
 
 20 
 
 38-0 424"43 ISOB'4 1 
 
 15*71 
 
 20 
 
 38*8 
 
 446-15 1602*33 
 
 
 21 
 
 42-1 49 I- 47 
 
 1582-61 
 
 17*25 
 
 21 
 
 43*9 
 
 514-22 | 1675-59 
 
 17-06 j 
 
 
 
 
 
 
Appendix 
 
 387 
 
 TAKLE LXI V. Continued. 
 For Ranging the Spiral from an Intermediate point. 
 
 Instrument at 5 
 
 Instrumental 6 
 
 1 " , * 
 
 r _i' 
 
 / ; H 
 
 k 
 
 *_ 
 
 y \ i 
 
 
 deg. 
 
 feet 
 
 feet 
 
 deg. 1 
 
 deg. 
 
 _ 
 feet 
 
 feet 
 
 deg. 
 
 r 
 
 o 
 
 o 
 
 o 
 
 o 6 
 
 o 
 
 o 
 
 
 
 6 ! 0-6 
 
 i '05 
 
 99'99 
 
 0*60 7 
 
 0-7 
 
 I '22 
 
 99*99 0*70 j 
 
 7 i '9 
 
 4-36 
 
 199-94 
 
 i '25 
 
 8 
 
 2*2 
 
 5-06 
 
 199-92 i '45 1 
 
 8 : 3'4 
 
 I0"29 
 
 299-76 
 
 1*97 
 
 9 
 
 3'9 
 
 11-86 
 
 299*69 2*27 j 
 
 9 ; s' 1 
 
 o j ?"o 
 
 19-18 
 
 3*"37 
 
 399-37 
 498-62 
 
 2'75 
 3*60 
 
 10 
 ii 
 
 5'8 
 7 "9 
 
 21-97 
 35-7 1 
 
 399-17 3-15 ; 
 
 498*23 4*10 
 
 
 9' 1 
 
 47-18 
 
 597-36 
 
 4'52 
 
 12 
 
 IO"2 
 
 53-42 
 
 596*64 
 
 5' 12 
 
 2 
 
 11-4 
 
 66-95 
 
 695-39 
 
 5-50 j 
 
 !3 
 
 I2'7 
 
 75'4 
 
 694*20 
 
 6*20 
 
 3 
 
 13 '9 
 
 90-97 
 
 792-46 
 
 6-55 
 
 14 
 
 15*4 
 
 101*96 
 
 790*61 
 
 7 '35 
 
 4 
 
 16*6 
 
 "9"54 
 
 888-29 
 
 7 '66 
 
 15 
 
 18*3 
 
 i33'36 
 
 885-55 
 
 8-57 
 
 
 Z 9'5 
 
 152-92 
 
 982-56 
 
 8*85 
 
 16 
 
 21*4 
 
 169-85 
 
 978*66 
 
 9-86 
 
 6 
 
 22-6 
 
 i9i'35 
 
 1074-88 
 
 10*09 
 
 J 7 
 
 24*7 
 
 211-63 
 
 1069*51 
 
 11 '20 
 
 7 
 
 25-9 
 
 235 '03 
 
 1164-83 
 
 11*41 
 
 18 
 
 28-2 
 
 258*89 
 
 11 57 "64 
 
 I2*6l 
 
 8 
 
 29-4 
 
 284-12 
 
 1251*96 
 
 12-79 
 
 19 
 
 3i-9 
 
 3ii-73 
 
 1242-53 
 
 14-08 
 
 9 33*i 
 
 338-73 
 
 I 335'73 
 
 J 4" 2 3 
 
 20 
 
 35"8 
 
 37 - 2 3 
 
 1323-64 
 
 15-63 
 
 20 37-0 
 
 398-9! 
 
 I4i5'59 
 
 iS'74 
 
 21 
 
 39'9 
 
 434-37 
 
 1400-36 
 
 !7" 2 3 
 
 21 i 4 1 ' I 
 
 464-65 i49'95 I 7'3 l 
 
 
 
 
 
 Instrument at 7 
 
 
 
 Instrument at 8 
 
 7 
 
 o 
 
 o 
 
 o o 
 
 8 
 
 
 
 | 
 
 8 
 
 0-8 
 
 1*40 
 
 99 '99 
 
 0*80 : 
 
 9 
 
 0-9 
 
 i'57 
 
 99'99 
 
 o'oo 
 
 9 
 
 2'5 
 
 5V6 
 
 199*89 
 
 I'6 5 : 
 
 
 2-8 
 
 6*46 
 
 199-87 
 
 1*85 
 
 o 
 
 4 '4 
 
 i3"43 
 
 299*60 
 
 2'57 
 
 i 
 
 4*9 
 
 15*00 
 
 299-50 
 
 2-87 
 
 i 
 
 6-5 
 
 2 4'75 
 
 398*96 
 
 3'55 j 
 
 2 
 
 7-2 
 
 27-53 
 
 398-71 
 
 3'95 , 
 
 2 
 
 8'8 
 
 40-05 
 
 497-78 
 
 4'6l i 
 
 3 
 
 9*7 
 
 44'3 8 
 
 497-28 
 
 5*10 
 
 3 
 
 "'3 
 
 59' 6 4 
 
 595' 8 4 5'7 2 ! 
 
 4 
 
 12-4 
 
 65-85 
 
 '95 
 
 6-32 i 
 
 4 
 
 14-0 
 
 83-84 
 
 692*87 6*90 
 
 5 
 
 15-3 
 
 92*24 
 
 "41 
 
 7-60 j 
 
 
 16*9 
 
 112*91 
 
 788-55 j 8*15 i 
 
 6 
 
 18-4 
 
 123*80 
 
 " 2 9 
 
 8'95 ! 
 
 6 
 
 20 'o 
 
 147-11 
 
 882*52 ; 9*46! 
 
 7 
 
 21*7 
 
 160*78 
 
 ./2I 
 
 10*36 
 
 17 
 
 2 3 '3 
 
 186-66 
 
 974*37 10*84 ! 
 
 8 
 
 25*2 
 
 203*36 
 
 969-69 
 
 11-84 
 
 18 
 
 26 '8 
 
 831*75 
 
 1063*62 12*28 
 
 i9 
 
 28*9 
 
 251*69 1057-24 
 
 i3'39 i 
 
 i9 
 20 
 
 30-5 
 34 '4 
 
 282-50 
 339-00 
 
 "49'79 !3'8o ' 
 1232-30 15-38 
 
 20 
 
 21 
 
 32-8 
 36-9 
 
 305-86 
 365*90 
 
 1141*29 
 I22I*26 
 
 15*00 i 
 16-68 
 
 21 38-5 
 
 401*25 I 1310*56 ! 17*02 
 
 
 
 
 
 Instrument at 9 
 
 Instrument at 10 
 
 9 
 
 o 
 
 00 
 
 10 
 
 o 
 
 o ] o o 
 
 \ 10 
 
 I'D 
 
 i '74 
 
 99*98 ; i-o 
 
 ii 
 
 l'l 
 
 1*92 . 99*98 j i "io 
 
 ii 
 
 3'i 
 
 7*15 
 
 199*84 : 2*05 ; 
 
 12 
 
 3-4 
 
 7-85 199-80 ! 2-25 
 
 i 12 
 
 5 '4 
 
 16*56 j 299-39 ; 3"i7 
 
 13 
 
 5 "9 
 
 18-13 299*27 3-47 , 
 
 13 
 14 
 
 7 '9 
 10*6 
 
 30*31 ! 398*44 : 4-35 
 48-70 . 496*74 5*60 
 
 14 
 
 15 
 
 8-6 
 *5 
 
 33-08 398-15 
 53-02 496-14 
 
 4 '75 
 6*10 j 
 
 15 i !3'5 
 
 7 2 '5 593 '9 8 6*92 ; 
 
 16 
 
 14-6 
 
 78-23 592-9! 
 
 7 '52 
 
 16 
 
 16*6 
 
 100*06 i 689-81 j 8-30 
 
 17 
 
 17-9 
 
 108-96 688-07 
 
 9*00 ' 
 
 17 
 
 19-9 
 
 134-65 i 783'84 9'75 , 
 
 18 
 
 21-4 
 
 145-45 781-18 
 
 Jo'SS 
 
 18 
 
 2 3 '4 
 
 174*37 8 75-6i 
 
 11-26 
 
 i9 
 
 25*1 
 
 187-87 i 871-74 
 
 12*16 
 
 19 
 
 27-1 
 
 219*92 9 6 4' 6 3 
 
 12*84 ; 
 
 20 
 
 29*0 
 
 2 36'35 959'20 
 
 13*84 i 
 
 20 
 
 21 
 
 31-0 
 35*1 
 
 27* "43 
 328*93 
 
 1050-35 
 1132*16 
 
 14-49 i 
 16*20 : 
 
 21 
 
 33'i 
 
 290*96 1042*97 
 
 i5'59 
 
 C C "2 
 
388 
 
 Preliminary Survey 
 
 TABLE LXI V. Continued. 
 For Ranging the Spiral from an Intermediate point. 
 
 Instrument at n 
 
 Instrument at 12 
 
 n 
 
 k 
 
 X 
 
 y 
 
 t 
 
 n 
 
 k 
 
 X 
 
 y 
 
 i 
 
 
 deg. 
 
 feet 
 
 feet 
 
 deg. 
 
 
 deg. feet 
 
 feet 
 
 deg. 
 
 i 
 
 
 
 o 
 
 o 
 
 o 
 
 12 
 
 o o 
 
 o 
 
 
 
 2 1*2 
 
 2*09 
 8X6 
 
 99-98 ! i "20 
 
 13 1-3 2-27 ' 99*97 
 
 1-30 
 
 2*6^ 
 
 3 
 4 
 
 K 
 
 U 
 
 19-69 
 
 "yy // 
 299 -I 5 
 
 3'77 
 
 15 , 6*9 21-26 299-01 
 
 u^ 
 4*07 
 
 5 
 
 9'3 
 
 35-85 
 
 397-83 
 
 5'i5 
 
 16 io'o 38*62 
 
 397*49 
 
 5'55 
 
 6 
 
 12-4 
 
 57*33 
 
 495-50 
 
 6'6o 
 
 i7 
 
 i3'3 6l "3 
 
 494-80 
 
 7*10 
 
 7 
 
 i5"7 
 
 84'39 
 
 59i'77 
 
 8-12 
 
 18 
 
 16-8 90-53 
 
 590-54 
 
 8-72; 
 
 8 
 
 19-2 
 
 117-27 
 
 686-21 
 
 9-70 
 
 *9 
 
 20-5 125-55 
 
 684*20 
 
 10*40 
 
 *9 
 
 22-9 
 
 156-19 
 
 778-32 
 
 "'35 
 
 1 20 
 
 24-4 
 
 166*86 
 
 775-27 
 
 12*15 
 
 20 
 
 26-8 
 
 201*27 
 
 867-58 
 
 13*06 
 
 21 
 
 28-5 < 214-58 
 
 863-15 
 
 13*96 
 
 21 
 
 30-9 
 
 252-63 
 
 953-39 
 
 14-84 
 
 
 
 
 Instrument at 13 
 
 Instrument at 14 
 
 13 
 
 o 
 
 o 
 
 O 
 
 
 
 4 
 
 o 
 
 o 
 
 O 
 
 o 
 
 4 
 
 i '4 
 
 2-44 
 
 99-97 
 
 1*40 
 
 
 i'5 
 
 2-62 
 
 99"97 
 
 1-50 
 
 
 4 '3 
 
 9'94 
 
 199-69 ! 2*85 . 
 
 6 
 
 4-6 
 
 10*64 
 
 199-64 
 
 3-05 
 
 6 
 
 7 '4 
 
 22*82 
 
 298*86 4-37 
 
 7 
 
 7'9 
 
 24-38 
 
 298-69 
 
 4-67 j 
 
 7 
 8 
 
 io' 7 
 14*2 
 
 41-39 
 65-92 
 
 397-12 
 494-06 
 
 5-95 
 7*60 
 
 8 
 9 
 
 11-4 
 15-1 
 
 44-I5 
 70*20 
 
 396-72 
 493^7 
 
 6-35 
 8*10 
 
 9 
 
 17-9 
 
 96-65 
 
 589-22 
 
 9'32 
 
 20 
 
 19*0 
 
 102*75 
 
 587-82 
 
 9*92 
 
 20 21-8 
 
 i33'79 
 
 682*07 
 
 11*10 ! 
 
 21 
 
 23-1 
 
 141*99 
 
 679-80 
 
 11*80 
 
 21 ' 25-9 i77'47 
 
 772-02 12-95 i 
 
 
 Instrument at 15 
 
 Instrument at 16 
 
 ig o 
 
 00 
 
 16 
 
 o 
 
 o o 
 
 o 
 
 16 
 
 1-6 
 
 2-79 
 
 99-96 
 
 i -60 
 
 T 7 
 
 i "7 
 
 2*97 99-96 
 
 1-70 
 
 *7 
 
 4 '9 
 
 11-33 
 
 199-59 
 
 3-25 
 
 18 
 
 5-2 12-03 199-54 
 
 3 '45 
 
 18 
 J 9 
 
 8-4 
 
 12*1 
 
 25-94 
 46-90 
 
 298-52 
 396-30 
 
 f97 
 6V5 
 
 J 9 
 
 20 
 
 8*9 
 
 12*8 
 
 27-50 298-34 
 49-66 395*85 
 
 5-27 j 
 7'i5 ' 
 
 20 
 
 i6'o 
 
 74-47 
 
 492 '43 
 
 8*60 
 
 21 
 
 16*9 
 
 78-73 491-54 
 
 9*10 
 
 21 
 
 20' I 
 
 108-83 
 
 586-34 
 
 10*52 
 
 
 
 Instrument at 17 
 
 Instrument at 
 
 5'5 
 9'4 
 13-5 
 
 2 O 
 
 6-1 
 
 Q 
 
 3-14 
 I2'73 
 29-06 
 52-40 
 
 O 
 
 99'95 
 199-49 
 298-I5 
 395-38 
 
 O 
 
 I -80 
 
 3-65' 
 
 5-57 
 
 7-55 
 
 9'9 
 
 3-31 
 13-42 
 30*61 
 
 99 '94 ' 
 *99"43 
 297*94 
 
 Instrument at 20 
 
 O 
 
 
 o 
 
 20 
 
 1 
 
 
 
 o 
 
 
 
 3-49 
 
 99*94 
 
 2*00 
 
 21 
 
 2*1 1 
 
 3-66 
 
 99-93 
 
 2*IO 
 
 14*12 ! 199-37 
 
 4"5 
 
 
 i 
 
 ! 
 
 
 
Appendix 
 
 389 
 
 TABLE LXV. Specific Gravity of Stones, Earths, &>c. Water 
 
 
 taken at 62-3 Ibs. per c. ft. 
 
 
 
 G 
 
 G 
 
 Basalt .... 
 
 275 to 2-95 
 
 Glass 
 
 2-50 to 3-00 
 
 Bricks and brick- 
 
 
 Gneiss and granite 
 
 average 2-65 
 
 work .... 
 
 I '6O tO 2'OO 
 
 Limestone and 
 
 
 Cement (American) 
 
 O'So to 0*90 
 
 marble 
 
 2-65 
 
 (Portland) 
 
 T " "7 C tn T * A C 
 
 Lime 
 
 
 Concrete in lime . 
 
 average 1-9 
 
 Masonry of dressed 
 
 
 ,, Port- 
 
 
 ashlar same G as 
 
 
 land cement . . 
 
 ,, 2-2 
 
 the stone 
 
 
 Coal, Newcastle . 
 
 1-25 
 
 Masomy of rough 
 
 
 ,, anthracite 
 
 1*50 
 
 rubble .... 
 
 I '8O to 2'2O 
 
 ,, in bulk for 
 
 
 Mortar . 
 
 i '40 to I -90 
 
 stowage, 48 cubic 
 
 
 Mud 
 
 1-25 to 175 
 
 feet per ton 
 
 
 Peat 
 
 average 0*40 
 
 Coke 
 
 075 
 
 Pitch 
 
 I-I5 
 
 ,, in bulk for 
 
 Sand 
 
 1-5 to I -95 
 
 stowage 80 to 100 
 
 ! Sandstones . . . 
 
 2'IO tO 2'5O 
 
 cubic feet per ton 
 
 \ Sulphur .... 
 
 average 2*00 
 
 Chalk .... 
 
 ,, 2-50 Tallow . . . . 
 
 0-90 
 
 Earth .... 
 
 1-50 to 2-00 Tar 
 
 ,, I'OO 
 
 Flint 
 
 average 2'6o Traprock 
 
 2-80 to 3-00 
 
 TABLE LXVI. Metals and Alloys. 
 
 Aluminium . . . 
 
 average 2-60 
 
 Iron (cast) . . 
 
 average 7*23 
 
 Antimony . 
 
 ,, 670 
 
 ,, (wrought) . 
 
 778 
 
 Babbett or white 
 
 
 Lead 
 
 ,, 1 1 '40 
 
 metal .... 
 
 730 
 
 Mercury .... 
 
 13-60 
 
 Bismuth .... 
 
 ,, 9-80 
 
 Platinum . . .21 -50 to 23-00 
 
 Brass 
 
 8-40 
 
 Silver .... 
 
 average 10-50 
 
 car : : : : 
 
 8'8o 
 ,, 19-00 
 
 Steel 
 Tin 
 
 775 to 8 -oo 
 average 7-30 
 
 Gun metal 
 
 8 -co 
 
 7i nr 
 
 7-00 
 
 
 TABLE LXVIL Timber. 
 
 , j WW 
 
 Acacia . . .- . 
 
 070 to 0-80 
 
 Cork 
 
 average 0-24 
 
 
 Ash 
 
 070 to 076 Ebony .... 
 
 1-19 
 
 Beech .... 
 
 070 to O'8o Elm (English) . 
 
 0-56 
 
 Box 
 
 average i '25 ,, (American) . 
 
 0-72 
 
 Cedar (Lebanon) 
 
 0*4.0 Fir 
 
 O* C T 
 
 ,, (American) . 
 
 ,, O'55 Hornbeam . 
 
 076 
 
 (West Indies) 
 
 ,, 070 Iron wood 
 
 1*15 
 
 Chestnut. . . 
 
 ,, O'6i Larch .... 
 
 '55 
 
390 
 
 Preliminary Stirvey 
 
 TABLE LXVIL Timber. (Continued^ 
 
 Lignum vitas . . average I '33 
 
 M ahogany ( H on - 
 
 duras) ... ,, 0-56 
 
 Mahogany (Spa- 
 nish) .... ,, 0-85 
 
 Maple . . . . ,, 0-67 
 
 Oak (American) . ,, 0-80 
 
 (English) . . 078 to 0-93 
 
 Pine (white) . . average 0-40 
 
 ,, (yellow) . . ,, 0-55 
 
 ,, (red) ... 0-57 to 0-65 
 ,, heart of long- 
 leafed southern 
 
 yellow .... i "04 
 
 Teak 0-74 to 0-86 
 
 TABLE LXVIIL Liquids. 
 
 I -06 
 
 Sulphuric acid 
 
 . . . 1-84 
 
 0-80 
 
 Water, distilled at 
 
 
 070 
 
 62 Fahr., bar. 
 
 
 1-20 
 
 30 in., 62-355 
 
 
 1-22 ' Ibs. per cubic 
 
 
 O-QA foot . 
 
 i -oo 
 
 0-92 Water, at 212 
 
 
 0-88 Fahr 
 
 . O'Q 1 ?? 
 
 0-92 Sea .... 
 
 i -026 to i -030 
 
 Acetic acid 
 Alcohol . . . 
 Ether .... 
 Hydrochloric acid 
 Nitric acid 
 Oil (linseed) . . 
 
 ,, (olive) . 
 
 ,, (petroleum) . 
 
 ,, (whale) 
 
 TABLE LXIX. Multipliers for reducing Specific Gravity to Weight oj 
 certain Volumes, Water taken at 62 '3 Ibs. 
 
 Weight of cubic centimetre in grammes . = G 
 
 ,, kilolitre or cubic metre in 
 
 tonnes of 1,000 kilogrammes ... G 
 
 Weight of decalitre in kilogrammes . G x 10 
 
 cubic inch in Ibs. . . . = Gx 0*036 
 
 cubic foot in Ibs. . . . G x 62 '30 
 
 cubic yard in tons . . . G x 0751 
 
 Brit. imp. gallon in Ibs. G x 10 
 
 Brit. imp. bushel in Ibs. = G x 80 
 
 U. S. liquid gal. in Ibs. . G x 8-322 
 
 U. S. struck bushel in Ibs. . = 
 
 No. of cubic yards in one ton . . . = 
 No. of cubic feet in one ton . = 
 
 No. of Brit. imp. gals, in one ton 
 
 55 55 bushels ,, 
 
 ,, U. S. liquid gals. ,, 
 
 struck bushels in one ton = 
 
GLOSSARY 
 
 Account, By, a term used for either longitude or latitude when cal- 
 culated from other data than observations. 
 
 Age of the Tide is the interval between the time of new or full moon 
 and the time of the next spring tide, and varies from i^ to 3 days. 
 
 Alioth, a star otherwise called e Ursa.- Majoris, in the constellation of 
 the. Great Bear (see Fig. 122). 
 
 Altazimuth, an instrument for measuring at one adjustment of the 
 line of sight the angle of altitude and of azimuth (see pp. 320, 322). 
 
 Altitude is the angular elevation of a heavenly body, or, in other 
 words, the arc of a great circle passing through a heavenly body 
 
Preliminary Survey 
 
 measured from it to the true horizon (see Horizon). Let A, Fig. 121 , 
 be the observer's eye, S the object, in altitude. SOH is the true 
 altitude plus refraction, which has to be deducted (see Refraction). 
 IO is drawn parallel to AS. So SOI represents the correction due 
 to parallax in altitude (see Parallax). The sensible horizon is drawn 
 for simplicity at the observer's eye, instead of at the earth's surface, 
 because it only involves an inappreciable error of parallax, due to 
 the increased length of radius of the earth at high elevations (see 
 Parallax, Equatorial). 
 
 Amplitude is the spherical angle at the zenith contained between the 
 plane of the prime vertical at the point of observation and that of 
 the meridian of any celestial body observed on the horizon, rising 
 or setting. It is measured by the arc of the horizon from the east 
 or west point to the body ; it is shown by AW in Fig. 133, or as 
 AC in Fig. 127, and is therefore the complement of the azimuth. 
 It is a very useful method of finding the variation of the compass 
 at sea, but is of no use where there is no horizon. When a star 
 is setting its true place is already under the horizon by about half 
 a degree (see Refraction) and therefore more to the westward than 
 it appears to be. If we take an amplitude when the star is 34' 
 above the horizon we shall be pretty near the truth ; if we take it 
 when actually setting it will appear too much to the west that is, 
 to the right of its true place. The greater the southern declination 
 in the northern hemisphere that is, the natter the arc of the body's 
 transit across the heavens the greater the error will be. In the 
 case of bodies with parallax (see Parallax) the error due to that 
 cause makes them appear too low. The moon's parallax is nearly 
 double the refraction ; so she is still about half a degree above the 
 horizon when she appears to set, and her true place is to the left of 
 her apparent place. 
 
 Anallatism (Greek a privative, and alasso, to alter, unchangeableness). 
 The centre of anallatism is that point in a distance-measuring tele- 
 scope from which the distance of any object is proportional to the 
 height intercepted upon the staff by two horizontal wires in the dia- 
 phragm. In ordinary telescopes it is situated at the anterior focus. 
 In some tacheometers it is made to coincide with the vertical axis 
 by means of an additional lens. 
 
 Aneroid (Greek a privative, and neros, wet), an instrument for measur- 
 ing the pressure of the atmosphere (see p. 342). 
 
 Angle (Latin angulus> a corner) may be plane or spherical (see Spheri- 
 cal Angle). A plane angle is formed by the inclination of two 
 straight lines to one another ; it has been reckoned from o to 360 
 
Glossary 393 
 
 degrees all over the world until lately, when the centesimal system 
 of dividing the circle into 400 degrees, and each degree into 100 
 minutes, has been coming into use for the purpose of simplifying 
 calculations. An angle of 90 is called a right angle, one less than 
 90 an acute, and one greater than 90 an obtuse angle. 
 
 Angle Complement. The complement of an angle is the difference 
 between it and 90. The cosine is the sine of the complement of 
 an angle, and the cotangent and cosecant are tangent and secant 
 of the same complementary angle. 
 
 Angle Supplement, the difference between an angle and 180. 
 
 Aphelion (Greek apo, from, and helios, sun), that point in the orbit of a 
 comet or planet which is farthest from the sun. 
 
 Apogee (Greek apo, from, and ge, the earth), that point in the moon's 
 orbit which is farthest from the earth. 
 
 Apparent (Latin ad, to, and parere, to appear), that which is opposite 
 to the true or real. The apparent position of a celestial object is 
 that which it appears to have to the observer with an instrument 
 before being corrected for refraction, parallax, c. 
 
 Apparent Time is the hour angle of the sun (see Hour Angle) reckoned 
 westward from the meridian. It is the time shown by the sun- 
 dial. The sun's apparent place in the heavens is constantly chan- 
 ging, owing to the earth's orbit, but this being elliptical, the movement 
 is not uniform and is represented in the almanacs by daily changes 
 of right ascension, with rate for one hour. A clock keeping appa- 
 rent time would have to be altered every day, so the expedient of 
 mean or average time is resorted to (see Mean Time). Apparent 
 time is first found from observation and then reduced to mean time 
 by the equation of time (see Equation of Time). 
 
 Arc of Excess, in sextants that part of the graduated arc behind the 
 zero. 
 
 Aries, First Point of (see Right Ascension). 
 
 Argument (Latin argumentum, a thing taken for granted) means any 
 mathematical datum or known quantity from which to determine 
 others. 
 
 Ascension (see Right Ascension). 
 
 Astronomical Time (see Civil Time). 
 
 Augmentation of the moon's semi -diameter is the increase in angular 
 dimension when in altitude above what it appears on the horizon, 
 owing to its approach towards the observer, until, when in the zenith, 
 it is closer by the amount of the earth's radius. The table is given in 
 Chambers's ' Mathematical Tables. ' 
 
 Axis (Greek axon, an axle), an imaginary line joining the north and 
 
394 Preliminary Survey 
 
 south poles of a celestial body upon which it is supposed to rotate. 
 The imaginary line about which the vertical limb of a theodolite 
 rotates. Is of very wide application (see also Optical). 
 
 Azimuth (Arabic samatha, to go towards) is the spherical angle at 
 the zenith contained between the plane of the meridian of the place 
 and that of the great circle of altitude passing through the object 
 observed (see Fig. 133, p. 414). If Z be the zenith and NZS the 
 meridian, Z )) A a great circle of altitude, N A is the azimuth. It is 
 measured upon the horizon from the north or south point, which- 
 ever is nearest. (See also Course and Amplitude, of which the 
 azimuth is the complement.) 
 
 Barometer, an instrument for measuring the pressure of the atmosphere 
 (seep. 342 &c.). 
 
 Binary Stars are double stars which revolve round one another ; when 
 the motion appears to be rectilinear they are merely called double 
 stars. 
 
 Circumpolar Stars, those whose polar distance is less than the latitude 
 of the place, and which therefore do not set but culminate twice. 
 At the North Pole the whole celestial hemisphere is circumpolar ; 
 at the Equator none. See Fig. 122. 
 
 'Civil Time, like astronomical time, is a term having reference more to 
 date than to time. Both are mean time (see Mean Time), but civil 
 time begins its day from midnight, and astronomical time from the 
 succeeding noon, so that January I, 1890, at 6 A.M. by civil time; is 
 December 31, 1889, 18 hours astronomical time. But January I, 
 1890, 2 P.M. civil time, is the same date and time astronomically. 
 
 Co- altitude. See Zenith Distance. 
 
 Collimation, Line of (Lat. cum, with ; limes, a limit), in telescopes 
 is the axis of a pencil of light reaching the eye through the tube ; or, 
 which is the same thing, it is the straight line joining the two foci 
 of the double-convex lens forming the object-glass and the focus of 
 the eye-piece. The line of collimation is defined in levels and 
 theodolites by two intersecting spider hairs or some such device, 
 attached to a brass diaphragm which is placed in the optical axis 
 by adjusting screws. 
 
 Colure (Gr. kolouo, I cut in the middle), two celestial meridians (see 
 Meridian, Celestial) whose planes are at right angles to one another ; 
 whose line of intersection is terminated by the poles, and which cut 
 the celestial sphere into quarters. One of these semicircles bisects 
 the equator at the spring and autumn equinoxes, and the other at 
 the summer and winter solstices. 
 
 Compass, Solar. See p. 328. 
 
Glossary 
 
 395 
 
 Compass, Variation of, is the angle between the astronomical meridian 
 and the direction of the compass needle when at rest under the in- 
 fluence of terrestrial, but undisturbed by local, magnetism. It is 
 
 FIG. 122. 
 
 subject both to a diurnal oscillation and an annual variation as well 
 as to local disturbances. 
 
 Constellation (Lat. cum, together and stella, a star), a portion of 
 the heavens marked on globes and maps by dotted boundary lines 
 
396 Preliminary Survey 
 
 in which the main feature is a well-defined group of stars supposed 
 to resemble some terrestrial object and accordingly designated. 
 Contraction of the semi-diameter of sun and moon are sensibly the same 
 in amount, and arise from unequal refraction in the upper and lower 
 
 limb. The table is given in Chambers's ' Mathematical Tables. ' 
 Co-ordinates, Rectangular^ a pair of straight lines locating any point 
 in a plane by measuring its shortest distance from two fiducial 
 lines at right angles to one another. Thus the rectangular co- 
 
 N 
 
 W 
 
 s 
 
 FIG. 123. 
 
 ordinates x y, x' y', x" y" determine the positions of the points a, 
 If, and c relatively to the lines N.S. and E.W., and if N.S. be the 
 meridian (see ( Meridian, Magnetic,' and ' Meridian, Celestial ') y,y' , 
 andy are the latitude ; x, x', and x" the departure (see Latitude, 
 Difference of). 
 
 Course in navigation and land-surveying is the direction of the line 
 being travelled or measured with reference to the magnetic meridian 
 or true meridian. It is the azimuth of the objective point. It is 
 also called ' bearing ' in land-surveying more frequently than course. 
 It is reckoned in each quadrant separately, from the north east- 
 wards and westwards, and from the south eastwards and west- 
 wards. This method is suitable to traversing with a large field 
 compass, being ready for reduction to latitude and departure. 
 Another method is to reckon clear round from o to 360. Most 
 countries reckon the o from the north point, some from the south. 
 Theodolites are arranged to read from o to 360, as the former 
 method would not be suitable; but when working to latitude and 
 departure the angles must first be reduced to azimuthal form. See 
 Table X VII. , p. 59. 
 
 Conjunction (Lat. cum, together ; jungere, to join). Two bodies are 
 said to be in conjunction when they appear in nearly the same part 
 of the heavens. It is necessary, therefore, that one of them should 
 
Glossary 397 
 
 have an apparent movement through the heavens, such as the moon, 
 which is in conjunction with a star when it has the same longitude 
 or right ascension. 
 
 Culmination (Lat. admen, the top) is the passage of a celestial body 
 across the meridian of a place. In the northern hemisphere the 
 sun and most of the stars used in observation culminate southwards ; 
 consequently the term ' to south ' is used for southern culmination. 
 See also Circumpolar Stars. 
 
 Declination is the distance of a celestial body from the celestial equator 
 measured north or south on the arc of a celestial meridian passing 
 through it. Declination corresponds exactly with terrestrial latitude. 
 In Pig. 127 SD is the north declination of a body S (see also Polar 
 Distance). In some almanacs N. and S. declination are marked 
 + and respectively. 
 
 Degree (Lat. degredior, to go down : from de, down, and gradus, a 
 step), a division of the circle which in the sexagesimal is g| 5 or 
 in the centesimal is ^ of the total circumference. 
 
 Departure in a traverse means the easting or westing from a known point 
 which is taken as the origin of rectangular co-ordinates. It is equal 
 to the distance run multiplied by the sine of the angle, azimuth, 
 or course (see Course, Azimuth). 
 
 Depression or Dip of the Horizon is the angle of depression of the 
 apparent horizon, due to elevation of the eye above the level of the 
 sea. If we direct a levelling instrument in good adjustment towards 
 the horizon from the top of a high cliff we shall at once perceive that 
 there is a depression, and with a large-sized transit theodolite we 
 can measure that angle with sufficient accuracy to know the height 
 of the cliff within ten to twenty feet. Depression arises from the 
 curvature of the earth. The values are given for different elevations 
 of the eye in Table L. , p. 288. These answer for correcting an 
 altitude taken at sea, or for estimating the elevation of a cliff in the 
 manner just described. The depression is always deducted from 
 the observed altitude. A simple way to keep it in memory is ' Dip 
 makes me see too much, and therefore I deduct it. ' 
 
 Diameter (Gr. dia^ through, and metron, a measure) in its ordinary 
 use is limited to the circle and the sphere of which it is double the 
 distance from centre to circumference. It also means the breadth 
 of anything. 
 
 Diaphragm, in telescopes is an annular brass plate fixed in the focus by 
 adjusting screws and forming both a passage to confine the light and 
 a frame to hold the cross hairs. 
 
 Dip of the Horizon. See Depression, 
 
398 
 
 Preliminary Survey 
 
 Diurnal Inequality of Heights is, in irregular tides, the difference 
 between the height of high water of each successive tide. 
 
 Diurnal Inequality of Time is. in irregular tides, the difference between 
 the lunitidal intervals of each successive tide. 
 
 Eclipse (Gr. ekleipsts, a disappearance), the phenomenon occurring in 
 the heavens from the disappearance of one body in the shadow 
 of another. The following diagram illustrates the theory of the 
 various forms of eclipses of sun and moon, from which it will be 
 seen that an eclipse of the sun (which is more correctly an occulta- 
 tion, see Occupation), can only take place when the sun and moon 
 
 'Total, dipse cftheJi 
 FIG. 124. 
 
 FIG. 125. 
 
 Total chpse oftfic-Jfcon 
 
 FIG. 126. 
 
 are in conjunction, or. in other words, at new moon, and an eclipse 
 of the moon, which is really an eclipse, can only take place at oppo- 
 sition or full moon. 
 
 An eclipse of a satellite of Jupiter is due to similar causes. When 
 entering the shadow its immersion is said to take place, and on 
 leaving it is said to emerge. The idea of obtaining the longitude 
 by observing eclipses of Jupiter's satellites originated with Galileo. 
 
 Ecliptic, the great circle of the heavens which the sun appears to 
 describe in the year ; it derives its name from the fact that eclipses 
 can only take place when the moon is also on the ecliptic. It is 
 commonly called the sun's annual path, to distinguish it from the 
 sun's diurnal path, due to the earth's axial rotation. 
 
 Ecliptic, Obliquity of, the inclination of the plane of the ecliptic to 
 that of the celestial equator, producing the phenomena of the 
 seasons. The angle is about 23 27 , and is very gradually dimi- 
 
Glossary 399 
 
 nishing. It is measured by the declination at the solstitial points, 
 June 21 and December 21. 
 
 Elongation (Lat. longe, afar off), the angular distances of the pole star 
 eastward or westward of the true pole ; also the similar distances of 
 a planet from the sun or a satellite from its primary. 
 
 Equation of Time, the daily correction at mean noon to be added to 
 or deducted from the apparent time ascertained by observation, in 
 order to obtain mean time {see Mean Time). It is sometimes ex- 
 pressed as sun 'after clock' or 'before clock,' meaning that the 
 equation is to be added to or deducted from the apparent time. 
 Thus : when the sun is on the meridian it is noon of apparent time ; 
 if in the table the equation is marked ' sun after clock ' 5 rnin. 6 sec. 
 it will be then o hr. 5 min. 6 sec. astronomical, or 12 hrs. 5 min. 
 6 sec. civil time. If the table had it ' sun before clock ' 1 1 min. 
 3 sec., when the sun culminated it would be II hrs. 48 min. 57 sec. 
 civil time, and 23 hrs. 48 min. 57 sec. astronomical time of the 
 previous day's date (see Civil Time, Astronomical Time ; see also 
 Sidereal Time). 
 
 Equator, Celestial, is the intersection of the plane of the terrestrial 
 equator with the celestial sphere (EQ in Fig. 127). 
 
 Equator, Terrestrial, is the great circle of the earth's surface whose 
 plane is midway between the poles and at right angles to the earth's 
 axis. AD in Fig. 129. 
 
 Equatorial instrument, a telescope which is made to move, by hand 
 or by clockwork, in the equator, or, in other words, in which the axis 
 of rotation, instead of being set vertical, as in an ordinary transit 
 theodolite, points to the celestial pole. 
 
 Equinox. See Equinoctial. 
 
 Equinoctial (Lat. cequus, equal ; nox, night), another name for the 
 celestial equator (see Equator), because when the sun is in it the 
 nights are equal all over the world. It will be obvious in looking at 
 a celestial globe that whatever angle the pole makes with the horizon 
 the equator always intersects with the prime vertical (see Prime 
 Vertical) at the horizon ; consequently the equator is in every latitude 
 half above and half below the horizon. If an observer could be sta- 
 tioned precisely at the north or south pole, the equator would then be 
 coincident with the horizon, and he would see a half-sun going clear 
 round the horizon at the equinox. The sun is in the equinoctial on 
 March 21 and September 21 (see Right Ascension). 
 
 Establishment, Vulgar, is the lunitidal interval when the time of moon's 
 meridian passage is o hr. o min. or 12 hrs. It is termed by Raper 
 the tide-hour, and defined, as the apparent time of the first high 
 
4OO Preliminary Survey 
 
 water that takes place in the afternoon of the day of full or change. 
 In German Hafenzeit, or harbour-time. 
 
 Establishment, Mean, is the mean of all the lunitidal intervals in a 
 semi-lunation, and is often less by ten to forty minutes than the 
 vulgar establishment. 
 
 Fiducial, any point, line, or arc which is known, fixed, or may be 
 otherwise relied upon for locating others, such as the meridian of 
 Greenwich ; the datum line of a profile or cross section ; an ordnance 
 benchmark, &c. &c. 
 
 Focal Length, the distance from the ' centre ' of the lens to the focus. 
 The ' centre ' of a double-convex lens is that point in the axis which 
 is midway between the two surfaces. 
 
 Focus, the common meeting-point of all the converging rays passing 
 through a lens. In the double-convex lens of a telescope the focus 
 inside the tube is termed the focus, that outside the tube is termed 
 the anterior focus. In German Brennpunkt, or burning-point. 
 
 Geographical Mile, or Admiralty Knot, is 1-15152 statute mile. It 
 slightly differs from that of the United States, which 151-15157 statute 
 mile and adopted by the Coast Survey as being the linear distance 
 in the arc of I minute of a great circle of a true sphere whose surface 
 area is equal to that of the earth at sea level. It also equals I -85324 
 kilometres. 
 
 Gibbous (Lat. gibbus, convex -bunched), when the moon is rather more 
 than half but less than full. 
 
 Great Circle of a sphere is any circle described about it whose plane 
 passes through its centre, as HZRN or PSDO in Fig. 127. 
 
 Great Circle Sailing is sailing between two points on the earth's surface 
 upon the arc of a great circle (see Spherical Distance). 
 
 Horary Angle. See Hour Angle. 
 
 Horizon. The sensible horizon is a plane parallel to that of the true 
 horizon, but touching the surface of the earth at the point of obser- 
 vation. 
 
 The true horizon \s the intersection of the celestial sphere by a 
 plane GH (Fig. 121) passing through the centre of the earth and 
 at right angles to a diamecer of the earth at the observer's stand- 
 point. 
 
 The visible or apparent horizon is the intersection of a conical 
 surface, of which the apex is the observer's eye, with the sphere 
 (see EF, Fig. 121). The dip of the horizon is equal to the comple- 
 ment of half the angle of the cone. 
 
 Hour Angle is the angle at the pole contained between the meridian of 
 the place and the celestial meridian passing through any celestial 
 
Glossary 
 
 401 
 
 body. SPZ, Fig. 127, is the hour angle ; it is measured by the arc 
 DE of the celestial equator. The calculation of the hour angle is 
 reduced from arc to time by the proportion of the total time of a re- 
 volution to the hour angle ; thus, as 360 : 24 hours : : hour angle in 
 arc : hour angle in time. The tables of log. sines, cosines, &c., in 
 Raper give the horary values of all angles, in addition to which 
 tables for converting arc into time and vice versa are given. For 
 formulae adapted to slide-rule see p. 136. 
 
 The calculation of the hour angle of the sun is the commonest 
 method of obtaining apparent time at place, from which by equa- 
 tion of time (see Equation of Time) and a chronometer registering 
 Greenwich mean time the longitude is easily calculated (see also 
 Apparent Time, Longitude). The hour angle of a star is sidereal 
 time, which can be reduced to mean time by rule (p. 410). 
 
 Hypsometric, 'height-measuring,' is used for observations with the 
 aneroid or boiling-point thermometer to determine the approximate 
 elevation above the sea. 
 
 Integral, consisting of entire numbers, as contrasted with fractions. 
 
 Kilometre, a distance of one thousand metres (see Metre). 
 
 Knot. See Geographical Mile. 
 
 FIG. 127. 
 
 Latitude, Difference of, in traversing called latitude for shortness, is the 
 
 northing or southing of the base line (see Co-ordinates and Traverse). 
 
 Latitude (Terrestrial], the spherical distance (see Spherical Distance) 
 
402 
 
 Preliminary Sutvey 
 
 between the equator and the position of the observer, measured 
 north and south upon a meridian of longitude. It is represented 
 by BC, Fig. 129. 
 
 FIG. 128. 
 
 FIG. 129. 
 
 Lens, in telescopes a circular glass, truly ground to a surface of that 
 curvature which causes the rays of light passing through it to be 
 refracted at angles which meet in a common centre. 
 
Glossary 403 
 
 Limb of a celestial body means the extreme edge of the circumference, 
 upon which the observation is taken. : 
 
 Limb of a theodolite is the vertical or horizontal portion cf the instru- 
 ment (see Chapter IX.), 
 
 Longitude ( Terrestrial) is the spherical angle at the pole (see Spherical 
 Angle) between the plane of the meridian of Greenwich (see Meri- 
 dian, Terrestrial) and that of the meridian of the place of observation. 
 Thus the angle ANC in Fig. 129 is the longitude of C west of 
 Greenwich when AEND is the meridian of Greenwich, and is 
 measured on any of the parallels of latitude in angular measure. 
 Notice in the figure that though the small circles of latitude give the 
 same arc of a circle as the equator, and consequently the difference 
 of longitude, they do not give the true spherical distance that is 
 (see Spherical Distance), the arc of a great circle which is the flattest 
 circle, and consequently the shortest distance between any two points. 
 
 Lunitidal Interval is the time that elapses each day between the 
 transit of the moon over the meridian and high water. 
 
 Mean Distance of a planet from the sun is the mean of the perihelion 
 and aphelion distances, which see. 
 
 Mean Time. Instead of correcting the watch daily to keep apparent 
 time (see Apparent Time) the average length of a solar day throughout 
 the year is calculated and termed a mean solar day. Both civil 
 and astronomical time are kept in this way (see Civil Time and 
 Astronomical Time). The corrections by which to obtain it from 
 observed apparent time are given in the almanacs and called 
 equation of time. Mean time is the only possible method of regulat- 
 ing time by the sun and yet keeping it uniform (see Standard Time). 
 
 Meridian, Celestial. The observer's celestial meridian is the great 
 circle of the celestial sphere passing through the zenith the pole, 
 the north and south points of the horizon, and the south pole. Its 
 plane is therefore at right angles to that of the prime vertical, and is 
 shown by ZHNR in Fig. 127. 
 
 The celestial meridian of a star is a semicircle of the heavens 
 which passes through it. 
 
 Meridian (Distance). See Hour Angle. 
 
 Meridian, Magnetic, is the line of direction of the compass needle (see 
 Compass) produced by the magnetic polarities of the earth. The 
 lines of equal magnetic variation do not even approximate to great 
 circles of the earth's surface, although the magnetic poles approximate 
 in position to the terrestrial poles. 
 
 Meridian, Terrestrial, is a great circle of the earth's surface passing 
 through the north and south poles and the place of observation. 
 
 D D 2 
 
404 Preliminary Survey 
 
 Every place has its meridian, but certain ones are chosen by the 
 several sea-going nations as the basis of their calculations of 
 longitude, such as the meridian of Greenwich, of Paris, of Washing- 
 ton, of Lisbon. A conference was held two years ago with a view 
 to adopting one for the world, but no agreement was come to. 
 
 Metre, the French standard measure of length = one ten-millionth of 
 the distance from the pole to the equator as measured by earlier 
 astronomers = 39*37079 British and American inches. It is about 
 \ inch longer than the seconds pendulum. 
 
 Micrometer (Gr. mikros, small ; metron, a measure), an instrument for 
 measuring small angles or distances (see p. 317, Chapter IX.). 
 
 Mile (see Statute, Geographical). 
 
 Moon-culminating Stars are certain stars which lie close to the moon's 
 path through the heavens, and from this cause furnish a ready 
 means of obtaining, at their culmination, the difference of local time 
 at any two places, and hence the longitude. The interval between 
 the culminations at Greenwich is obtained from the ' Nautical 
 Almanac ' by the difference in right ascension of the moon and star. 
 The interval at place is found by watch, and the difference between 
 the two intervals is proportional to the difference of longitude. 
 
 Nadir, the point in the celestial sphere at the opposite extreme to the 
 zenith. 
 
 Nautical Mile. See Geographical Mile. 
 
 Nodes (Lat. nodus, a knot) are the points of intersection of a planet's 
 orbit with the sun's path or ecliptic. The 'ascending node' is 
 where it crosses the ecliptic from south to north, and the descend- 
 ing node the opposite point. 
 
 Obliquity of the Ecliptic. See Ecliptic. 
 
 Occultation is the disappearance or hiding of a celestial body by the 
 intervention of another. Thus the stars in the moon's path are 
 occulted by her, and the satellites of a planet by the body of the 
 planet. 
 
 Optical Axis in instruments is the line joining the centres of the true 
 spherical surfaces of the lenses. 
 
 Orientation, the general direction of a chain of triangles, or the placing 
 of a plane-table or similar instrument so that it will preserve the 
 same line of direction. 
 
 Parallax (see Fig. 121, where the parallax in altitude is shown 
 as angle SOI. It is the difference of altitude which would exist 
 between two simultaneous observations of the same star by two 
 observers, one stationed at the earth's surface and the other at its 
 centre). All calculations of celestial bodies are reduced to the earth's 
 
Glossary 405 
 
 centre, and therefore parallax is always to be added. A way to keep 
 it in memory is to think of the earth's centre as our true place of 
 observation, so that being raised up too high by an amount equal 
 to the earth's radius we see the body too low. 
 
 When the body is in the zenith the angle of parallax, as will be 
 seen by inspection of Fig. 121, is eliminated. It is at a maximum 
 when the body is on the horizon, where it is termed horizontal 
 parallax, and the table for its values for different bodies and at 
 different seasons is given in the almanacs together with a table for 
 the sun's parallax in altitude. 
 
 Parallax, Eqiiatorial. In the Nautical Almanac ' the equatorial or 
 longest radius of the earth is used for computing parallax ; so when 
 great accuracy is sought the equatorial parallax must be again re- 
 duced by a correction for latitude. Only the sun, moon, and planets 
 have parallax. The fixed stars, being at too vast a distance, have no 
 appreciable parallax except in a few instances, where the parallax is 
 measured by assuming as a base not the earth's radius, but the 
 diameter of the earth's orbit round the sun. 
 
 Parallax in Altitude. Given the horizontal parallax of a celestial body 
 and its altitude, to find its parallax in altitude. The sun's parallax 
 in altitude is given in a table, because the variation is sensibly 
 constant, but the horizontal parallax of the moon is deduced by the 
 following rule : 
 
 Let P = the horizontal parallax 
 P' = the parallax in altitude 
 A = the apparent altitude 
 R : cos A : : sin P : sin P' 
 ... sinFs= cos_A. sinP 
 
 R 
 Or by logarithms : 
 
 log sin P' = log cos A + log sin P - 10. 
 
 Parallax in telescopes, the apparent dancing about of the cross hairs when 
 the eye is shifted about during observation. It arises from the eye- 
 piece not being correctly in focus. The cross hairs should be clearly 
 defined by moving the eyepiece out or in. 
 Perigee (Gr. peri, near ; ge, the earth), the converse of apogee (see 
 
 Apogee). 
 Perihelion (Gk. peri, near ; helios, the sun), the converse of aphelion 
 
 (see Aphelion). 
 
 Pointers, the stars a and j8 in the Great Bear, Ursa Major (see Fig. 122), 
 or familiarly the Waggon and Horses. The former of them is also 
 called Dubhe. The seven principal stars of this constellation are 
 
406 Preliminary Survey 
 
 all that are commonly known as belonging to it, but there are many 
 more. These appear to revolve round the pole without in our 
 latitudes even touching the horizon, the pointers maintaining all the 
 while their direction towards the pole star. 
 
 Polar Distance is the arc of a celestial meridian passing through a 
 celestial body measured from the pole to the body. In the northern 
 hemisphere it is the complement of the declination when that is 
 north, and equals 90 + the declination when it is south. In the 
 southern hemisphere the polar distance, i.e. distance from the south 
 pole, is vice versd. 
 
 Pole (Gr. poled, I turn) (Celestial), the intersection of the earth's axis 
 when produced to the celestial sphere. The apparent unchange- 
 ableness of this point renders it the basis of all astronomical mea- 
 surements. The point does actually change from year to year, but 
 not with sufficient rapidity to enter into daily calculations of latitude 
 and longitude. 
 Pole (Terrestrial), the two points north and south forming the apices 
 
 of the axis of the earth's rotation (N, S, Fig. 129). 
 Pole Star is a star of the second magnitude near the celestial pole in the 
 end of the tail of the Little Bear ; it is called either o Ursse Minoris 
 or Polaris ; its right ascension for 1890 is I hr. 18 min. 29-1 sec., 
 its declination 88 J 43' 18" N. Fifty years ago it was R.A. I hr. 
 2 min. 10-683 sec - and declination 88 27' 2i"'94 N. ; it is therefore 
 travelling (in appearance) slowly towards the celestial pole. 
 Precession of the Equinoxes is a slow retrograde movement of the equi- 
 noctial points, due to the attraction of the sun. and planets (see 
 Aries, First Point of). 
 
 Prime Vertical is the great circle of the celestial sphere passing through 
 
 the zenith, east and west points of the horizon, and the nadir. Its 
 
 plane is therefore at right angles to that of the meridian. It is 
 
 shown as ZCN in Fig. 127. 
 
 Primitive, the great circle upon the plane of which a stereographic 
 
 projection is made. 
 
 Quadrant (Lat. quadrans, a fourth part), the fourth part of a circle. 
 An instrument so named was used in taking altitudes before the 
 introduction of the sextant. 
 
 Radius Vector (Lat. radius, a sunbeam ; vector, a bearer), the 
 shortest distance from the centre of the earth to the centre of the 
 sun at any point of the earth's orbit. Has also the meaning of 
 radius of curvature to any curve other than a circle at any particular 
 point in the curve, such as the distance from the centre of the 
 earth to any point upon its spheroidal surface. 
 
Glossary 
 
 407 
 
 Range oj Tide is the difference between the height of high- and low- 
 water levels of any one tide without any reference to datum. Is 
 also termed height of tide. 
 
 Refraction (Lat. refrangere, to bend) is the bending of the ray of light 
 proceeding from a celestial body when passing at an angle from the 
 rarer ether, or whatever the medium may be, into our denser atmo- 
 sphere. It makes bodies appear higher than they are, and the cor- 
 rection for it is given in Chambers's ' Mathematical Tables,' 
 Whitaker, &c. It is always to be deducted. In consequence of 
 this law all bodies appear to rise earlier and to set later than they 
 really do, with the sole exception of the moon (see Amplitude). If 
 the ray of light were passing from a dense into a rarer medium it 
 would be bent the opposite way. A simple but entirely unphilo- 
 sophical and somewhat grotesque way of remembering the direction in 
 
 FIG. 130. 
 
 which the ray is bent is to imagine the pencil of light as a long thin 
 wand bending down with the weight of the star at the end of it. 
 When in a vertical position it will not deflect either way, but the 
 more the angle of depression the greater the deflection. The 
 direction of the ray as it reaches the eye, or line of sight, is where 
 the eye sees the star ; but its true position is below that, at the end 
 of the curved pencil of light ; so that we must deduct the correction 
 for refraction from the observed altitude. It is just the same for 
 all bodies, at whatsoever distance they may be. It depends entirely 
 upon the angle and density of the medium, and the tables give 
 corrections for difference of barometric pressure and temperature 
 where close calculation is required. It varies from o' when the body 
 is in the zenith to 34' on the horizon. 
 
 I\ight Ascension of a celestial body is analogous to the longitude of a 
 terrestrial position. It is the arc of the celestial equator measured 
 from a meridian passing through the first point of Aries to a 
 meridian passing through the celestial body. The first point of 
 Aries has nothing in particular to mark it in the heavens ; it is the 
 vernal equinoctial point which in the times of the ancient astronomers 
 
408 Preliminary Survey 
 
 was actually situated in Aries, but, owing to the precession of 
 the equinoxes, is now in another constellation altogether. It i? 
 quite an imaginary point upon the celestial equator, chosen, like 
 the observatory of Greenwich in terrestrial calculations, as a 
 fiducial point from which to map the stars. Fortunately for astro- 
 nomers there has not been the same display of national feeling 
 in the selection of a celestial meridian as there has been about the 
 choice of a common terrestrial meridian, so there is but one. The 
 right ascension is reckoned from west to east, and is expressed in 
 hours and minutes, the 360 of the equator being 24 hours (see 
 Sidereal Time). The celestial semicircle which crosses the poles 
 and the first point of Aries is called the vernal equinoctial colure, 
 and its opposite semicircle the autumnal equinoctial colure, because 
 on March 21 and September 21 or thereabouts the sun's path (see 
 Ecliptic) intersects the celestial equator. The sun has then no R. A. 
 and no Decl. , and day and night are equal all over the world (see 
 Equinoctial). 
 
 Rise of a tide is the height of the high-water level above the low spring 
 datum. 
 
 Satellites (Lat. safeties, a companion), the little bodies which revolve 
 round the planets. 
 
 Sea Mile. See Geographical Mile. 
 
 Semi-diameter of sun and moon is half the angle subtended by the 
 diameter of the visible disc ; it varies according to the bodies' 
 distance from the earth, and values are given in the almanacs ; by 
 it observations of the upper or lower limb are reduced to the centre. 
 
 Sextant (Lat. sextans, a sixth part), an angular reflecting instrument for 
 making celestial observations (see Chapter IX. ). 
 
 Sidereal Time. If a star is watched passing any fixed point, such as the 
 line between two perfectly straight vertical rods, on successive 
 evenings by a correct watch, it will be seen to pass them 3 min. 
 56 sec. (more correctly 3 min. 55*91 sec.) earlier each evening. 
 This movement is perfectly regular, and means simply the time of 
 one complete revolution of the earth upon its axis. Sidereal time 
 is needed to find the time when any star will culminate, and to 
 correct watches or chronometers, which may be done by a transit 
 instrument to a fraction of a second. 
 
 It would be of no use as civil time, because the time would keep 
 dropping back. Sidereal time commences when the first point of 
 Aries is on the meridian of the place (see Right Ascension), and is 
 counted through 24 hours until the same point comes round again. 
 It is a shorter measure of time than mean time, 
 
Glossary 409 
 
 24 hrs. of sidereal time = 23 hrs. 56 min. 4-0906 sec. of mean 
 time, and 24 hrs. of mean time = 24 hrs. 3 min. 56-5554 sec. of 
 sidereal time. 
 
 Sidereal time at mean noon is the heading of a column in the 
 Nautical Almanac Ephemeris, and also in Whitaker. 
 
 In the American Nautical Almanac the same thing is termed 
 sidereal time, or right ascension of mean sun. 
 
 In Chambers's ' Mathematics 5 it is called the sun's mean right 
 ascension at mean noon. 
 
 All these names are sufficiently reasonable, but such a difference 
 of nomenclature is confusing to the beginner, who has some difficulty 
 in grasping the thought of a ' mean sun.' 
 
 We will confine ourselves to the first-mentioned expression and 
 endeavour to put it in popular language. 
 
 Everybody knows that the sun appears to go round the heavens 
 once a day and once a year, owing to the earth's daily rotation and 
 annual orbit. 
 
 The sun's yearly path is indicated by different stars appearing 
 at sunset at one time of the year from another. At sunset in March 
 the brilliant constellation of Orion is nearly overhead. In June it 
 is the sun's bedfellow, so we do not see it at all. 
 
 The starting-point of star-measurement (see Right Ascension) is 
 an arbitrary point in the heavens called the first point of Aries, 
 situated in a semicircle which is termed the vernal equinoctial 
 colure, because the sun is always there at spring-time. 
 
 This starting-point is marked 24 hrs. or o on the celestial 
 globes, and it is also the commencement of sidereal time at any 
 place. 
 
 When the first point of Aries is on the meridian of any place it 
 is sidereal noon, just as when the sun is on the meridian it is apparent 
 noon. Hence in the American Nautical Almanac there is a 
 column headed 'Mean Time of Sidereal Noon.' In the British 
 Nautical Almanac it is termed 'Mean Time of Transit of First Point 
 of Aries.' 
 
 Sidereal time is a perfectly regular measure of time like mean time, 
 but it is not the same measure, since we see it gains about 4 minutes 
 a day. 
 
 Why, then, do we use it at all ? Because the sidereal time at mean 
 noon given in the almanac enables us to tell when any star will 
 culminate, as will be presently shown. 
 
 Why do we not use it exclusively ? Because it does not keep 
 with the sun. We should have to put our breakfast hour on 
 
4IO Preliminary Survey 
 
 every morning if we kept sidereal time, and say Monday at 8, 
 Tuesday at 8 '04, &c., or else we should soon be breakfasting at 
 midnight. 
 
 'If a star has a R.A. of 24 hours, like the second star of Cassi- 
 opeia nearly (see Fig. 122), it will culminate at sidereal noon. If its 
 R.A. is I hour it will culminate I sidereal hour afterwards, and 
 so on. Hence we can find the mean time of any star's culmination 
 by adding its right ascension reduced to mean time to the mean time 
 of sidereal noon given in the almanac (see Figs. 131, 132). 
 
 Thus : Find the mean time of the culmination of a star in 
 4 hrs. R.A. on May 18, 1889. 4 hrs. sidereal =-3 hrs. 59 min. 
 20-7 sec. mean time, which is the interval between the passing of 
 the first point of Aries and the star across the meridian. 
 
 But sidereal noon by the almanac was at 20 hrs. 15 min. 10*46 sec. 
 mean time on May 17. Adding the R.A. in mean time, we have 
 o hr. 14 min. 31 -16 sec. as the mean time on the i8th of the star' 5 
 culmination. 
 
 Or we may do it in another way. The sidereal time at mean 
 noon on May 1 8 is given as 3 hrs. 45 min. 26-47 sec - This repre- 
 sents the interval of sidereal time between the culmination of the 
 first point of Aries and mean noon. But the sidereal interval 
 between the first point of Aries and the star is 4 hours. Therefore, 
 if we deduct the one from the other we get the sidereal interval 
 14 min. 33-53 sec. from mean noon. This reduced to mean time is 
 the same as we had before, 14 min. 31 'l6 sec. 
 
 The two rules are therefore as follows : To find the mean time 
 of any star's culmination at any meridian. 
 
 Rule i. To the mean time of sidereal noon on the previous day 
 or the given day add the star's right ascension reduced to mean 
 time. 
 
 (If the two quantities make more than 24 hours take the previous 
 day, if less than 24 hours take the day itself. ) 
 
 Rule 2. From the star's right ascension, increased if necessary 
 by 24 hours, deduct the sidereal time at mean noon : result will be a 
 sidereal interval which reduced to mean time will be the answer. 
 
 When the meridian is not the same as Greenwich the mean 
 time, apparent time, or sidereal time have all to be corrected 
 for the difference of longitude reduced to time, as explained on 
 p. 136. 
 
 It is, no doubt, with the object of making the matter clearer that 
 sidereal time at mean noon is expressed in some books as the mean 
 right ascension of mean sun. One is told to imagine a sun which 
 
Glossary 
 
 411 
 
 keeps mean time in its movements, and whose right ascension will 
 therefore be the sun's right ascension plus or minus the equation of 
 time. 
 
 Since the right ascension of any body is synonymous with the 
 sidereal time of its culmination, the right ascension of this imaginary 
 sun when on the meridian that is, at mean noon is the same thing 
 as sidereal time at mean noon. 
 
 There is not, however, any such thing really as apparent right 
 ascension or mean right ascension ; it is only a hyperbole for con- 
 veying the twofold idea of the real sun keeping apparent time and 
 an imaginary sun keeping mean time. 
 
 It is important not to confound the expression ' the sun's mean 
 right ascension at mean noon ' with that of ' the sun's right ascension 
 at mean noon.' The latter only differs by one or two seconds from 
 its right ascension at apparent noon that is to say, it is the difference 
 of right ascension which the sun has made during the interval re- 
 presented by the equation of time whereas the sun's mean right 
 ascension, or sidereal time, at mean noon is the position of an 
 imaginary sun whose right ascension always differs from that of the 
 true sun by the equation of time itself. 
 
 This explanation has only been added because of the difference of 
 nomenclature. The term * sidereal time at mean noon ' is sufficiently 
 intelligible for our present purpose without introducing the idea of 
 two kinds of right ascension. 
 
 FIG. 131. Position of the Celes- 
 tial Equator at Sidereal Noon, 
 May 17, 20 hrs. 15 min. 10*46 
 sec. Mean Time. 
 
 FIG. 132. Position of the Celestia 
 Equator at Culmination of Star 
 at o hr. 14 min. 31*16 sec. Mean 
 Time, May 18, and 4 hrs. Sidereal 
 Time. 
 
 The ' mean sun ' of the foregoing example is shown in the figure 
 close to the star ; but 14 min. 31-6 sec. ahead. 
 
 Note. The position of the heavens in these two figs, is as they 
 would appear in the southern hemisphere, where the sun's culmination 
 
412 Preliminary Survey 
 
 is to the north. The primitive is the ecliptic. The only reason for this 
 arrangement was that the north point and first point of Aries might 
 both be at the top of the paper, as being perhaps more readily under- 
 stood. 
 
 When the equation of time at mean noon is marked as sun 
 
 u af ! er do( * , mean time is that much- ..J^erthan Ume 
 
 before clock slower than 
 
 and sidereal time at mean noon is, that much ess - the sun's 
 
 more than 
 
 right ascension at mean noon. In the illustration just given the 
 equation of time was 3 min. 46-69 sec. ' before clock,' and the sun's 
 right ascension at mean noon was 3 hrs. 41 min. 3978' sec., which 
 added together make 3 hrs. 45 min. 26-47 sec -> which is the sidereal 
 time at mean noon. 
 
 To reduce sidereal to mean time intervals by slide rule : 
 
 Place theright-hand i of the slide over the 9-83 of the rule, and for 
 hours and decimals of sidereal time on the slide read off seconds 
 and decimals on the rule, which are to be deducted from the sidereal 
 interval to obtain the mean-time interval; thus 2-50 hrs. sidereal 
 time = 2 hrs. 30 min. is opposite to 24-6 seconds to be deducted. 
 
 To redtice mean to sidereal time intervals by the slide rule : 
 
 Place the right-hand I of the slide over the 9*86 of the rule, and 
 for hours and decimals of mean time on the slide read off seconds and 
 decimals on the rule, which are to be added to the mean-time interval ; 
 thus 6-50 hours mean time are opposite to 64-1 sees., or I min. 4-1 
 sees, to be added. 
 
 To rediice sidereal time at mean noon at Greenwich to sidereal 
 time at local mean noon by slide rule : 
 
 Express the difference of longitude in hours and decimals. 
 
 Adjust the rule with a I of the slide over 9-86 on the rule, and 
 for hours of difference of longitude on the slide read the correction in 
 seconds and decimals of sidereal time on the rule. 
 
 Example. What will be the sidereal time at mean noon in New 
 York, Ion. 74 W., it being 16 hrs. 41 min. n sec. at Greenwich? 
 
 The Ion. in time is 74 x 4 = 296 min., or 4-93 hrs., which being 
 W. is the amount behind Greenwich. 
 
 Sid. time, mean noon, Greenwich . i6 h 41 II s 
 4-93 hrs. x 9 -86 sec. . . . o o 48-6 
 
 Sid. time at mean noon, New York . 16 40 22*4 
 Signs of the Zodiac are twelve symbols denoting the constellations suc- 
 cessively traversed by the sun in his apparent annual circuit of the 
 heavens. They are as follows with the sun's position in them ; 
 
Glossary 413 
 
 Aries, the Ram . . r . . March 20 to April 20 
 
 Taurus, the Bull . . . . April 20 to May 21 
 
 Gemini, the Twins . . a . . May 21 to June 21 
 
 Cancer, the Crab . . 25 . . June 21 to July 22 
 
 Leo, the Lion . . . $ . . July 22 to August 23 
 
 Virgo, the Virgin . . i^ . . August 23 to September 23 
 
 Libra, the Scales . . i . . September 23 to October 23 
 
 Scorpio, the Scorpion . m . . October 23 to November 22 
 
 Sagittarius, the Archer . / \ . November 22 to December 21 
 
 Capricornus, the He-Goat, vf . December 21 to January 20 
 
 Aquarius, the Waterman . as . . January 20 to February 1 8 
 
 Pisces, the Fishes . . x . . February 18 to March 20 
 
 The signs of the zodiac are supposed to be Chaldean or Egyptian 
 hieroglyphics, intended to represent some occurrences peculiar to the 
 month in which the sun occupied each of the constellations at that 
 time. Thus the spring signs show productiveness of nature. 
 When the sun is in Libra the autumnal equinox takes place, whence 
 the origin of that title is evident. Explanations more or less likely 
 are given to all the rest of them. 
 
 Small Circles of a sphere are those whose planes do not pass through its 
 centre, such as the parallels of latitude (Fig. 129). 
 
 Solstice (Lat. sol, the sun, and stare, to stand), the two periods, June 21 
 and December 21, when the sun's declination is temporarily con- 
 stant. 
 
 South, To. Sec Culmination. 
 
 Sphere, Celestial, is the apparent vault of the heavens supposed to 
 be viewed by an observer at the centre of the earth, in which the 
 heavenly bodies appear to be situated and upon which their relative 
 positions or movements are determined by measurements of spheri- 
 cal distances taken from arbitrary but fiducial circles and points 
 supposed to be drawn upon the surface of the sphere like the meri- 
 dians of longitude and parallels of latitude upon the terrestrial maps. 
 
 Spherical Angle is that formed at any point upon a sphere by two great 
 circles intersecting there. It is measured by the inclination of their 
 planes or by the angle between the tangents to the circles at the 
 point of intersection. Thus the spherical angle NZA (Fig. 133) may 
 be measured by the angle NOA between the planes, or by the angle 
 TZT between the tangents, or by the arc NA of the great circle 
 whose plane is at right angles to those of the two intersecting planes. 
 
 Spherical Distance is the arc of a great circle (see Great Circle) passing 
 through two points which is intercepted between them, as SD, 
 
Preliminary Survey 
 
 PZ in Fig. 127 or as NA Fig. 133. It is the shortest distance 
 upon the spherical surface between any two points. 
 Standard Time on the continent of America is a form of keeping mean 
 time (see Mean Time) by which there are no variations except those 
 of an even hour at a time, since 1 5 of Ion. correspond "with one 
 hour's difference of time. The time of every fifteenth meridian, be- 
 ginning at New York with the 75th and ending with San Francisco on 
 the I20th, rules the belt for about 7^ on each side of it. The conve- 
 nience of railway systems causes in some cases the overlapping of the 
 
 FIG. 133. 
 
 times, so they are distinguished by the following terms: Intercolonial 
 time, Eastern time, Central time, Western or Mountain time, and 
 Pacific time. The first mentioned is time of the 6oth meridian and 
 is used in Nova Scotia. 
 
 Statute Mile) the British and American standard of long measure ; 
 it is equal to 8 furlongs, or 80 chains (Gunter), or 320 rods, or 
 1,760 yards, or 5,280 feet, or 63,360 inches. It is also equal to 
 0-868719 of a knot (see Geographical Mile) and 1*609315 kilo- 
 metre. 
 
 ' Semi-mensual inequality of heights ' is the difference between the 
 heights of spring and neap tides above mean water-level. 
 
 ' Semi-mensual inequality of time ' is the difference between the greatest 
 and smallest lunitidal interval. 
 
 Supplement, the difference between any angle and a semicircle. 
 
 Tacheometer (Gr. tacheos, swiftly, and metreo, I measure), same as tele- 
 meter, but exclusively applied to instruments furnished with tele- 
 scopes. 
 
 Telemeter (Gr. tele, far off, and metreo, I measure), an instrument 
 for measuring distance without chaining. 
 
Glossary 415 
 
 Time. See Apparent, Astronomical, Civil, Mean, Sidereal. 
 
 Traverse in land-surveying is used in contradistinction to triangu- 
 lation (see Triangulation). It is the method of surveying by mea- 
 suring base lines in length and angular direction continuously for- 
 ward, whereas in triangulation the lengths are computed by trigo- 
 nometry or graphic construction. 
 
 A closed traverse is one in which the base lines box the compass back 
 to the starting-point (see Course). 
 
 ' Working a traverse ' is the reduction of the angular base lines to 
 rectangular co-ordinates of latitude and departure (see Latitude 
 and Departure). The term is also largely used in navigation. 
 
 Triangulation in land-surveying is the determination of points by the 
 intersection of rays taken from the ends of a base of known length. 
 It is the foundation of geodetic operations of large extent as well as 
 the most cursory field-sketching with a sketch-board. It is the root 
 principle of all range-finders and telemeters and of the whole science 
 of surveying. It is hardly ever used without traversing as well. In 
 primary triangulation for geodetic survey the detail is filled in by 
 traversing (see Traverse), and upon a route survey traverse the detail 
 is sketched from triangulation. So the two principles dovetail into 
 one another, taking alternative forms according as accuracy or 
 despatch is the main point aimed at. 
 
 Zenith is that point in the heavens which is directly overhead. It 
 would be the celestial pole if the observer were standing at the earth's 
 pole, and on the celestial equator if he were crossing the 'line.' 
 
 Zenith Distance is the coaltitude or complement of the altitude. In 
 Fig. 127 it is indcated by SZ. 
 
 Zodiac (Gr. zone, a girdle), a belt of the heavens extending 8 on 
 either side of the ecliptic, within which the sun, moon, and the 
 major and many of the minor planets perform their annual revolu- 
 tions (see Signs of the Zodiac). 
 
INDEX 
 
 ABN 
 
 ABNEY'S level, 322 
 
 Acreage, 200 
 
 Adie's telemeter, 332, 333 
 
 Altazimuth, pocket, traverse and 
 
 profile by, 54, 63 
 description of, 320 
 Aneroid barometer, 342 
 Angle iron, weight of, 277 
 Arc reduced to time, 126 
 Astronomy, Chauvenet's, 116 
 
 BARNETT'S diagraph, 360 
 Bate range-finder, 340 
 Boiling-point thermometers, 350 
 Borings, 113 
 Breakwater, Port Said, 113 
 
 Chicago, 114 
 
 Bremiker's Mathematical Tables, 
 
 368 
 Bridges, iron, 256 
 
 stone, 258 
 
 brick, 258 
 
 CHAINING, 194 ; methods of, 195 
 Chambers' s Practical Mathe- 
 matics, 368 
 
 Mathematical Tables, 368 
 Channel iron, weight of, 278 
 Chronometers, 121 
 Cisterns, discharge from, 106 
 Clarke's, Col. , telemeter, 333 
 Climate, affects location, 3 
 Clinometer (compass), 321 
 Coast lining, 82 
 Coggeshall's slide-rule, 36^ 
 Compass-clinometer, 321 
 
 Casella's 321 
 
 O'Grady Haly's, 321 
 
 solar, 328 
 
 Computing scales, Stanley's, 361 
 Concrete, 113 
 
 DRE 
 
 Crelle's Tables, 368 
 Cross-ties, railway, 250 
 
 table of, 251 
 Cuartero's tables, 189 
 Curvature of the earth, 288 
 Curve-ranging, Krohnke's me- 
 thods, 205 
 
 Jackson's method, 205 
 
 Kennedy and Hackwood's 
 method, 206 
 
 by chord and offset, 208 
 
 with transit and chain, 209 
 
 nomenclature, 209 
 
 by astronomical bearings, 217 
 
 tacheometric, 183 
 Curves, resistance of iron, 13 
 
 cost of operating, 17 
 
 reverse, 221 
 
 diversion of, 223 
 
 linear advance of inner rail, 224 
 
 transition, 225 
 
 DAVIS'S Rules, 362 
 
 Dawson, Wm. Bell, his survey of 
 
 Nova Scotia, 193 
 Deflection distance, 197 
 Directrix for night observations, 
 
 136 
 Distance, measurement of, 150 
 
 measured by gunfire, 86 
 Dock, Hull, cost of, 113 
 
 West India, 114 
 
 Antwerp, 114 
 
 Marseilles, 114 
 
 Leith, 114 
 
 Liverpool Graving, 114 
 
 Malta Graving, 115 
 
 Portsmouth, 115 
 
 Honfleur Sluicing Basin, 115 
 Drawing instruments, 359 
 Dredge-Steward omnitelemeter, 
 
 336 
 
Index 
 
 4*7 
 
 DRE 
 
 Dredging, 112 
 plant, 112 
 
 EARTH, curvature of, 288 
 Earthwork, mensuration, 253 
 
 equalisation, 254 
 Eidograph, 361 
 
 Elliott's army telescope, 333 
 
 omnimeter, 333 
 Equation of time, 139 
 
 FIELD BOOK, 369 
 Force, centrifugal, 263 
 
 GLOSSARY, 391 
 
 Gradients, method of overcoming, 
 
 4 
 
 cost of traction on, 6 
 
 railway, 245 
 Graduation of slide-rule, 362 
 Gravity, specific, 389 
 Greatness Mill, survey of, 192 
 Gribble, ' Ideal ' Tacheometer, 
 
 307 
 
 HELIOGRAPH, 328 
 Heliostat, 328 
 Hints to travellers, 369 
 Hold-alls, 369 
 Hydraulics, roo 
 Hydrostatics, no 
 Hypsometry, 348 
 
 IRON, angle, 277 
 
 tin, 277 
 
 channel, 278 
 
 round, 278 
 
 square, 278 
 
 KUTTER'S formula. 
 
 103, 105 
 
 LATITUDE, 117 
 
 general rules, 118 
 
 by circumpolar stars, 152 
 
 by solar altitude out of meri- 
 dian, 152 
 
 double altitude, 154 
 
 NAU 
 
 Latitude by a pair of stars, 154 
 Level, dumpy, 289 
 
 striding bubble, 294 
 
 plane of rotation, 294 
 
 parallax, 294 
 
 adjustment of, 29^ 
 - Y, 299 
 
 staves, 301 
 
 fieldbook, 302 
 Levelling, 286 
 
 theory of, 287 
 Levels, 286 
 Longitude, 1 18 
 
 by culmination of a rived star, 
 
 137 
 
 by solar transit, 139 
 
 by solar hour angle, 141 
 
 by Jupiter's satellites, 143 
 
 by lunar observations, 144 
 
 by lunar occultation, 145 
 
 by lunar distance, 146 
 
 by reciprocal azimuths, 146 
 
 Jby moon-culminating stars, 147 
 Lyman on tacheometrv, 190 
 
 MAKESHIFTS for protracting, 360 
 Mannheim's rules, 362 
 Manuscript books, 369 
 Mapping, 74 
 
 by plane construction, 75 
 
 by conical projection, 77 
 
 by stereographic projection, 79 
 
 by Mercator's projection, 80 
 
 by gnomonic projection, 89 
 
 ' Mathematics,' Chambers's, 116 
 Mensuration by slide-rule, 267 
 Meridian, by sun shadows, 124 
 
 by ordinary watch, 124 
 
 by equal 'altitudes of a star, 
 124 
 
 by circumpolar stars, 127 
 
 by pole star, 130 
 
 by solar azimuth, 132 
 Metric measures, 269 
 Molesworth's Pocket Book, 369 
 Money, calculations of, by slide- 
 rule, 280 
 
 Moore, Lieut. W. N., survey by, 
 
 82 
 Morse code, 92 
 
 NAUTICAL ALMANAC, 3 
 
 K E 
 
Preliminary Survey 
 
 OFF 
 
 OFFICE instruments, 359 
 Olive, W. I., 105 
 Otto Struve's telemeter. 333 
 Oughtred slide-rule, 362 
 Outfit for tacheometric survey, 
 370 
 
 PANTAGRAPH, 361 
 
 Parallax, analogy with t-aeheome- 
 
 try, 162 
 Passometer, 51, 323 
 
 traverse by, 54 
 
 contours by, 61 
 
 profile by, 61 
 Pedometer, 324 
 
 Permanent way, weight of, 277 
 Photography, 33 
 
 Piazzi Smyth's telemeter, 333 
 Pipes, discharge from, 106 
 Plane-table, 318 
 
 triangulation by, 36 
 
 variation of compass by, 39 
 
 traversing by, 42 
 
 as range-finder, 49 
 
 compared with sextant, 35 
 
 with stadia, 19 
 Plane-tabling, accuracy of, 44 
 
 as an auxiliary, 46 
 Plane trigonometry. 373 
 Planimeter, 361 
 Protractor, tee-square, 360 
 
 QUAYS, New York, 114 
 
 RAILWAY track, weight of, 276 
 
 American, management and 
 cost of, ii 
 
 estimates for, 22 
 
 Australian, 24 
 
 Indian, 25 
 
 service diagram for, 262 
 Range-finder, 71 
 Range-finders, 332 
 Raper's ' Navigation,' 368 
 Reconnaissance, 30 
 Report, subject matter of a, 2 
 Road-rollers, cost of, 26 
 Roads, Indian, cost of, 25 
 
 Australian, cost of, 26 
 Rochon's micrometer, 333 
 Round iron, weight of, 278 
 
 TAC 
 
 SCALES, thermometer, 265 
 Searles's ' Railroad Spiral,' 227 
 Sextant, surveying by, 53 
 
 adjustment of, 324 
 
 box, adjustment of, 327 
 Signalling, with heliostat, 93 
 Signals, flag, 95 
 Sketchboard, 318 
 Sketching, 33 
 
 with aid of maps, 43 
 
 with plane-table, 34 
 Sleepers, railway, 250 
 Slide-rule, Kern's metallic, 166 
 
 calculation by, 241 
 
 arithmetic by, 242, 243 
 
 as decimal scale, 244 
 
 for gradients, 245 
 
 for centrifugal force, 265 
 
 for mensuration, 267 
 
 as a ready reckoner, 279 
 
 described, 361 
 Sluicegate, pressure on, in 
 Solar compass, 328 
 Spherical trigonometry, 376 
 Spider hairs, to put in, 313 
 Spiral tramway, 230 
 
 horse-shoe, 234 
 
 mountain, 237 
 
 trunk-line, 238 
 Spon's shilling P. H. , 369 
 Square, setting out a, 196 
 Square iron, weight of, 278 
 Stadia, theory of, 168 
 Stanley's telemetric theodolite, 308 
 Station pointers, 366 
 Stationery, 369 
 
 Steward's simplex range-finder, 
 
 340 
 
 Stone crushers, cost of, 26 
 Survey, with pole and micrometer, 
 
 82 
 
 from the boat, 83 
 
 from the ship, 85 
 
 with transit and chain, 201 
 Surveying, with chain and cross- 
 staff, 198, 199 
 
 Surveyor, qualifications of, i 
 Symbols in hydrography, 192 
 
 TABLES, see 'Index to Tables, 
 
 p. 419 
 
 Tacheometer, Cribble's, 307 
 Troughton & Sims's, 189 
 
Index 
 
 419 
 
 TAC 
 
 Tacheometer, adjustment of, for 
 line of sight, 314 
 
 of micrometer, 316 
 
 registration of micrometer 
 values, 313 
 
 Tacheometric survey, outfit for, 
 
 370 
 Tacheometry, 158 
 
 auxiliary work, 171 
 
 survey of Hawaii, 171 
 
 traverse by, 172 
 
 levelling by, 175 
 
 fieldbook for, 178 
 
 contouring, 181 
 
 profile, 181 
 
 Hawaiian gulch, 182, 183 
 
 curve-ranging, 183 
 Tanks, discharge from, 106 
 Tavernier-Gravet's rules, 362 
 Tee iron, weight of, 277 
 Telemeters, 332 
 Theodolites, 304 
 
 Tide gauges, 96 
 Tides and Currents, 89 
 Timber, tree, measurement of, 276 
 Time, reduced to arc, 137 
 Traffic, estimates of, 9 
 
 WYE 
 
 Tramways, cost of, 27 
 
 service diagram for, 261 
 Trautwine's ' Pocket Book,' 368 
 Trestles, iron, 257 
 
 timber, 259 
 Trigonometry, plane, 373 
 
 spherical, 376 
 
 Troughton & Sims's tacheometer, 
 
 189 
 Two-foot rule as a range-finder, 72 
 
 UNION PACIFIC RAILWAY, 28 
 
 WAGES and salaries, calculations 
 
 of, by slide-rule, 179 
 Wagner-Fennel tacheometer, 335 
 Warehouses of brick, 115 
 Water (falling), horse-power of, 
 
 109 
 
 Webb, Lieut. V.B., 89 
 Weirs, discharge over, 106 
 Weldon range-finder, 336 
 Whitaker's Almanack, 367 
 Workshops, for railway, 29 
 Wyes and loops, 240 
 
 INDEX TO TABLES 
 
 V 
 
 TABLE 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 V. 
 
 VI. 
 
 VII. 
 
 VIII. 
 
 IX. 
 
 X. 
 
 XI. 
 
 XII. 
 
 XIII. 
 
 XIV. 
 
 XV. 
 
 XVI. 
 
 XVII. 
 
 XVIII. 
 
 Traction on grades . 
 
 Assistant engines 
 
 Curve-resistance ..... 
 Curvature and grades on American roads 
 Curve limits for fixed wheel bases 
 Curve limits at different speeds 
 
 Running expenses affected by curves 
 Compensation for curves 
 Running expenses of American railroads 
 Statistics of American railroads . 
 
 Statistics of Australian roadj 
 Reduction of azimuths 
 Morse alphabet . 
 
 i- AGE 
 
 5 
 6 
 6 
 
 13 
 14 
 
 14 
 15 
 J 5 
 18 
 
 . 18 
 
 19 
 19 
 20 
 
 21 
 . 21 
 
 24 
 
 59 
 
 93 
 
420 
 
 Preliminary Svrvey 
 
 TABLE 
 
 XIX. 
 
 XX. 
 
 XXI, 
 
 XXII. 
 
 XXIII. 
 
 XXIV. 
 
 XXV. 
 
 XXVI. 
 
 XXVII. 
 
 XXVIII. 
 
 XXIX. 
 
 XXX. 
 
 XXXI. 
 
 XXXII. 
 
 XXXIII. 
 
 XXXIV. 
 
 XXXV. 
 
 XXXVI. 
 
 XXXVII. 
 
 XXXVIII. 
 
 XXXIX. 
 
 XL. 
 
 XLI. 
 
 XLII. 
 
 XLIII. 
 
 XL IV. 
 
 XLV. 
 
 XLVI. 
 
 XLVII. 
 
 XLVI II. 
 
 XLLX. 
 
 L. 
 
 LI. 
 
 LII. 
 
 bill. 
 
 LIV. 
 
 LV. 
 
 LV1. 
 
 LVI1. 
 
 LV1II. 
 
 LIX. 
 
 LX. 
 
 LXI. 
 
 LXII. 
 
 LXI II. 
 
 LXIV. 
 
 LXV. 
 
 LXVI. 
 
 LXVI1. 
 
 LXVIII. 
 
 LXIX. 
 
 Kutter's coefficient . . , . . .''*.*'. 
 Comparison of Beard more and Kutter ' '_-.:, , . . . 
 Sine of polar distance of Polaris . ..,.'.. 
 Equation of time' (approximate) . . , '. v '. '. r -_ 
 Functions of angles in percentage . ; . . , . . , .\ '. 
 Difference between hypotenuse and base . ..... 
 
 Length of a minute of longitude . . * , f , . . 
 Length of a minute of latitude . . . . 
 
 Sines multiplied by various distances . . 
 Angles for tacheometric curve-ranging. '.',-. 
 Leading values of slopes in percentage . . 
 Squares and cubes . . . . . . 
 
 Railway sleepers (cross ties) ....... 
 
 Howe trusses '. 
 
 Inches into decimals of a foot . .. . . 
 
 Time, coinage, and linear measurement . ./-.., 
 Shillings and pounds . . ." . . 
 Days, hours, minutes, and seconds . ., L . 
 Weeks, months, and years . . . . . 
 Days, weeks, months, and years . ... . . 
 
 Vulgar fractions into decimals '. . . . 
 
 Minutes into decimals of a degree . . . . 
 
 Seconds into decimals of a degree . . . 
 Decimals of a degree into minutes and seconds . . . 
 
 Cotangents . . . * 
 
 Sines .;' . . . . ' .- ; .; 
 
 Multipliers for structural iron . . . . \. 
 
 Wages and salaries . . ..... . .. 
 
 Decimal multipliers for English money . .-; . . 
 
 Indian money at par . 
 
 Sterling currency and rupees at various rates of exchange 
 Curvature of the earth . . ' . ., . ~^' ', 
 
 Multipliers for hypsometry . . . . ' . \. 
 Azimuth by j8 and 5 Draconis . . . . ' v , - 
 Azimuth by y8 and e Ursae majoris . . . , , . ' " 
 Radii of curves in feet and degrees per TOO feet . 
 General elements of spiral . . . . 
 Elements of No. 2 spiral . . ''. . v "'. " i. ' . 
 
 No. 5 
 
 No. 10 
 
 No. 15 
 
 No. 25 
 
 No. 50 
 
 No. 75 
 
 No. lop 
 
 surveying the spiral 
 
 from an intermediate 
 
 Tables for 
 
 point 
 
 Specific gravity of stones, earth, and other minerals . 
 ,, ,, metals and alloys . . '. ' f . .* 
 
 ,, ,, timber . . ' .' v .. 
 
 liquid . . . ' . . 
 
 Multipliers for reducing specific gravity to weight 
 certain volumes 
 
 139 
 167 
 174 
 
 1 80 
 184 
 248 
 249 
 
 251 
 260 
 270 
 271 
 271 
 271 
 271 
 272 
 272 
 272 
 273 
 273 
 274 
 274 
 277 
 279 
 280 
 282 
 283 
 288 
 352 
 378 
 378 
 379 
 380 
 380 
 38i 
 382 
 382 
 383 
 384 
 384 
 385 
 
 386 
 389 
 389 
 389 
 390 
 
 
no