REESE LIBRARY 
 
 UNIVERSITY OF CALIFORNIA. 
 
 Received _^<fJ^^ ._/< l 
 
 Accessions ^?..^.?^5G?_. Shell 
 
MAKINE SUKVEYING. 
 
A TEEATISE 
 
 ON 
 
 MARINE SURVEYING: 
 
 PREPARED FOR THE USE OF YOUNGER NAVAL OFFICERS, 
 
 WITH QUESTIONS FOR EXAMINATION AND EXERCISES, 
 
 PRINCIPALLY FROM THE PAPERS OF THE 
 ROYAL NAVAL COLLEGE, WITH THE RESULTS. 
 
 $SS?JK 
 
 UJTIVJJJtSITTj 
 
 J. L. ROBINSON, B.A. 
 
 ROYAL NAVAL COLLEGE. 
 
 "There is, I am persuaded, no one amon;,' us .... who thinks so highly <>f 
 himself, as not to believe that he may learn much, and derive much assistance 
 from communication with his brethren, nor so engrossed with his own share 
 of the common work, as not to be desirous of imparting to others whatever 
 has been recommended by his own experience to himself." THIRLWALL. 
 
 Hontron : 
 MACMILLAN AND CO. 
 
 1882 
 
 [The right of Translation and Reproduction is Deserved.] 
 
ier 
 
 PRINTED BY C. J. CLAY, M.A. & SON, 
 AT THE UNIVERSITY PRESS. 
 
PREFACE. 
 
 THE following Treatise on Marine Surveying and its 
 connected subjects is intended to meet an Educational want 
 which has existed for some time past. I mean a small 
 Manual on the subject, arranged on a plan similar to that 
 employed with success in Text-books of Elementary Science, 
 and which combining a sufficient description of the Instru- 
 ments used, and of the various methods pursued, with a care- 
 ful selection of representative Examples, with their solutions, 
 may, it is hoped, serve as a sufficient Introduction to this 
 branch of Science for the use of Younger Naval Officers, and 
 <ilso, if it may be, to bring the subject, at least in its more 
 popular features, before a wider circle than those profession- 
 ally interested in it. 
 
 A glance at the way in which the various subjects are 
 treated, will shew that I have had no intention to write or, 
 if the word be preferred, to compile a Hand-book for the use 
 of the Practical Surveyor. Such an intention might fairly be 
 regarded as an impertinence in one who has never been 
 engaged in the practical work of the Profession, and who 
 must therefore be ignorant of many details which of neces- 
 sity enter largely into any book intended to serve that 
 purpose. 
 
 I have rather had the Examination-room and its require- 
 ments before me in selecting and arranging the materials of 
 my little book. It is well known that the scope of the 
 
VI PREFACE. 
 
 Examination in this subject has been greatly increased since 
 the establishment of the Royal Naval College at Greenwich. 
 Formerly, the only test proposed (irrespective of a vivd voce 
 Examination) was the construction of a very simple Mercator's 
 Chart (vide a typical example, p. 91, No. 2), whereas now in 
 addition to the construction of a chart, involving at times 
 some peculiar difficulties (vide Nos. 11, 14, 16. 18, 22, in 
 Chap. V.), questions are nearly always set which require a 
 sound knowledge of the different methods of Fixing Positions, 
 of the Management of Chronometers, of Meridian Distances, 
 of the Use of Surveying Instruments, and of the varied 
 and important information which is conveyed succinctly 
 but with perfect accuracy by the Symbols employed in the 
 Admiralty charts. It seemed to me that, under these cir- 
 cumstances, a small Manual embodying the necessary infor- 
 mation would be welcomed by young officers who are called 
 on to undergo an Examination in the subject. And if, in 
 addition to meeting the wants of younger students, my other 
 hope should be realized that the Book may succeed in enlist- 
 ing the sympathy of readers for a most interesting and useful 
 branch of study, in exciting a desire in Junior Naval Officers 
 to gain, as opportunity offers, a knowledge of an important 
 branch of their profession, and in laying a firm foundation for 
 the future instruction, theoretical and practical, imparted in 
 the subject in the Royal Naval College and elsewhere, then 
 indeed I shall be rewarded for the many weeks and months 
 of labour which its preparation has required. 
 
 And in this preparation I can conscientiously affirm that 
 no pains have been spared to make it serve its original pur- 
 pose of a Text-book for Young Students. The very best 
 sources of information lay within my reach and have been 
 freely drawn upon : whether these advantages have been 
 utilized in the best way is, of course, a different question. 
 Many kind friends have greatly assisted me by their judg- 
 
PKEFACE. Vll 
 
 ment and technical knowledge. While I feel under obliga- 
 tions to others, I cannot sufficiently say how much I am 
 indebted to Mr Oborn, the Lecturer in Navigation and Nauti- 
 cal Astronomy in the Royal Naval College. In addition to 
 reading through the entire work in manuscript, he carefully 
 corrected the proofs of the first eleven chapters. Many of his 
 suggestions are embodied in the text, and I beg to thank 
 him most sincerely for the great trouble that he has taken to 
 make the little book accurate in its information. Starr- 
 Commander Johnson, the Instructor in Nautical Surveying, 
 in the Royal Naval College, gave me many valuable hints 
 for the improvement of the Chapters on Instruments, Tides, 
 Soundings, and Fixing Positions. These I gladly used either 
 to correct or to supplement my own statements. And, finally, 
 a friend of great experience as a First-class Surveyor read 
 through and corrected several of the more important chapters. 
 The assistance thus given so cheerfully, while of course not 
 removing responsibility from me, cannot but cause me to 
 entertain a greater confidence in submitting my little Book 
 to the public than I should otherwise have been justified in 
 feeling. 
 
 It only remains for me to add, that all possible care has 
 been taken to insure accuracy in the Results appended to 
 the Exercises. I shall, however, be very grateful to have my 
 attention called to errors of any kind which may be found to 
 exist, by those who use the book, as well as for any sugges- 
 tions from Instructors, Students, or Practical Surveyors, which 
 would, in their opinion, tend to make it more generally 
 useful. 
 
 JOHN LOVELL ROBINSON. 
 
 KOYAL NAVAL COLLEGE, 
 July 21, 1882. 
 
TABLE OF CONTENTS. 
 
 CHAPTER I. 
 
 SYMBOLS USED IN CHARTS, &C. 
 
 I. Symbols to denote the quality of the Bottom Examination II. 
 General Abbreviations Examination III. Buoys Colouring Kegula- 
 tions for employing Examination IV. Lights and Lighthouses Expla- 
 nations and Illustrations Illuminating Apparatus Examination V. 
 Conventional Signs Examination VI. Soundings VII. Miscellaneous. 
 Examination VIII. Surveying Symbols Authorities on this subject. 
 
 pp. 119 
 
 CHAPTER II. 
 
 CONSTRUCTION AND USE OF SCALES. 
 
 Definition I. Simply -divided Scales : (a) Scales of Equal Parts 
 Use Mode of constructing Exercises The Natural Scale (/3) Diagonal 
 Scales When employed Exercises in constructing General Exercises in 
 Construction Plain and Diagonal Scales (7) Verniers General prin- 
 ciples Exercises in constructing; Principle of Eeading off To ascertain 
 whether the Limb is correctly graduated Mode of correcting. II. Pro- 
 tracting Scales : Names and Uses of these Construction and Use of a 
 Scale of Chords Scale of Ehumbs; The Sector Sectoral Lines Use of 
 these Marquois Scales Use Exercises To erect a Perpendicular To 
 divide a Line into a number of Equal parts : (i) By Trial (ii) By Propor- 
 tional Compasses (iii) By a Scale of Equal parts (iv) By Marquois 
 Scales pp. 2048 
 
X CONTENTS. 
 
 CHAPTER III. 
 
 LAYING OFF ANGLES. 
 
 Methods of Protracting Angles: (1) By Semicircular Protractor. (2) 
 By Eectangular Protractor. (3) By Circular Protractor. (4) By Scale of 
 Chords. (5) By Computing the Length of a Chord to a given Kadius, 
 Proof of the Kule employed Examples for Exercise . . pp. 49 52 
 
 CHAPTER TV. 
 
 FIXING POSITIONS. 
 
 Necessary Geometrical Theorems To describe a segment capable of 
 containing any given angle : (i) When the Angle is right ; (ii) When it is 
 acute ; (iii) When it is obtuse Exercises Two methods of fixing a posi- 
 tion : (a) The straight line and one angle method. (/3) The Three-point 
 Problem Six different cases of the Three-point Problem : (i) When P is out- 
 side the triangle formed by joining the three points A, B, C, and the central 
 object is on the side of the line joining the other two points, remote from P 
 Two methods Notes on the second method (ii) When P lies outside the 
 triangle, but the central object and P lie on the same side of the line join- 
 ing the other two objects. Note (iii) When P is on one side produced 
 of the triangle formed by joining the three objects A, B and C (iv) When 
 P is on one side of the triangle ABC (v) When the three objects A, B, C 
 are in the same straight line Two methods (vi) When P is inside the 
 triangle ABC Two methods The Indeterminate Case (" on the Circle") 
 Note on this Case Best relative position of the objects when using the 
 Three-point Problem Fixing Positions without describing segments Lines 
 of Position Straight-line method Exercises Danger Angle Useful Prob- 
 lems : (i) To find the length of a line accessible only at the ends (ii) To 
 find the distance of an inaccessible object by the rhombus method (iii) To 
 find the distance of the point where two lines intersect in a river or lake 
 Examination Exercises Answers pp. 53 80 
 
 CHAPTER V. 
 
 CHARTS AND CHART DRAWING. 
 
 Definitions Projection of a point Orthographic Projection Stereo- 
 graphic Projection Central or Gnomonic Projection Mercator's Pro- 
 jection Plane Chart Nautical Mile Advantage and Disadvantage of 
 Mercator's Chart Method of Constructing a Mercator's Chart Erection 
 of the Perpendiculars To lay down the Latitude and Longitude of a 
 Point on a Chart To lay down a Course on a Chart To lay down a 
 Distance on a Chart Two special points noticed more particularly 
 Exercises Answers pp. 81 100 
 
CONTENTS. XI 
 
 CHAPTER VI. 
 
 INSTRUMENTS AND OBSERVING. 
 
 Gunter's Chain Ordinary Surveying Chain Sextant Adjustments 
 To test the Adjustments Method of correcting error in Collimation Pre- 
 caution in observing Theodolite Three principal parts Three principal 
 motions Three Adjustments Method of using To take an "Arc" To 
 repeat an Angle To observe a Vertical Angle The Spirit Level De- 
 scription Three Adjustments The Levelling Staff The Ten-feet Pole 
 Method of Computing the Scale The Station-Pointer Method of using 
 The Barometer Three Methods of correcting the height of the column : 
 (i) By Capacity Correction (ii) By Flexible Base (iii) By Contracted 
 Scale The Marine Barometer Method of Suspending Vernier of the 
 Marine Barometer Method of reading off Method of stowing for car- 
 riage pp. 101120 
 
 CHAPTER VII. 
 
 BASE LINES. 
 
 Use of a Base Base of Verification Example Geodetic Standards of 
 Measurement vary Standards "a traits" and "a bouts" The Knowledge 
 of the exact Length of a Bar at any moment involves three distinct elements. 
 The temperature difficulty is evaded by Three Methods: (i) Borda's (ii) 
 Colby's (iii) Struve's Description of Colby's " Compensation Appara- 
 tus " Note on the U. S. Coast Survey Apparatus Ratio of length of Base 
 to Area of Survey Degree of accuracy required The Base Lines in the 
 Ordnance Survey The Madrid Base Reduction of Base to the Sea Level 
 The Three Elements of the Base Precautions to be observed in selecting 
 the ground Different Methods of Measuring a Base : (i) Masthead Angle 
 (ii) Velocity of Sound Objections to this Method (iii) Patent Log under 
 Steam (iv) Astronomical Observations (v) Direct Measurement Method 
 of proceeding Formulae The Direction of the Base Examination Ex- 
 ercises Answers pp. 121 139 
 
 CHAPTER VIII. 
 
 TRIANGULATION. 
 
 Object sought Personal Equation Method of "piling-up" the Tri- 
 angles Primary and Secondary Triangles Observing Distant Points 
 Assumed Base Spherical Excess Legendre's Theorem Reduction to the 
 Centre Triangulation of England connected with that of the Continent 
 Triangulation of Spain connected with that of Algiers Examina- 
 tion pp. 140- 148 
 
Xll CONTENTS. 
 
 CHAPTER IX. 
 
 LEVELLING. 
 
 Definitions Check Level Simple and Compound Levelling To find 
 difference of Level of two points Method of running a line of Level through 
 a tract of country To lay down a Section Examples Levelling by Theo- 
 dolite By Barometer Use of the Mountain Barometer Button's Rule 
 to compute the difference of Height by heights of the Barometer at the 
 two stations By Thermometer Contours Method of delineating Exa- 
 mination pp. 149 165 
 
 CHAPTER X. 
 
 TIDES AND TIDAL OBSERVATIONS. 
 
 Theories of Bernouilli and Laplace Tidal phenomena Spring and 
 Neap Tides Superior and Inferior Tides Cotidal Lines Head or End of 
 the Tide Difference between Tidal Current and Tidal Stream Flow and 
 Ebb Tide and Half Tide Tide and Quarter Tide Tide and Half Quarter 
 Tide Conj unction Opposition Quadrature Lunation Semi-lunation 
 Perihelion Aphelion Perigee Apogee Circumstances under which the 
 Tides are greatest and least Diurnal inequality Effects of Wind and 
 Barometer on the Tides Eise and Eange of a Tide Mean Level of the 
 Sea Tide Day Priming and Lagging of the Tide Single Day Tides 
 Double Half Day Tides Octant Tides Age of the Tide Eetard Luni- 
 tidal Interval Vulgar Establishment of the Port Corrected Establishment 
 of the Port Semimenstrual Inequality Time and Height of High Water 
 Method of Interpolation First and Second High Water Tidal Constants 
 Methods of Computing the Time of High Water Tidal Observations Ele- 
 ments of a Complete Tidal Table Tide Gauges Precautions to be observed 
 Tidal Evolution Examination pp. 166 199 
 
 CHAPTER XL 
 
 SOUNDINGS. 
 
 Objects sought Sounding Pole Fixing Boat at Starting To keep 
 the Boat on a certain "range" Quality of the bottom to be registered. 
 To fix a sounding Lead Lines must be tested Eeduction of Soundings 
 Method of effecting this reduction Formula for reducing Soundings in a 
 Tidal Eiver Elevations along the course of a Tidal Eiver To discover and 
 fix the position of a hidden Danger Surface Currents Current Log Under- 
 Currents Discharge of a Eiver Definitions To find the Sectional Area To 
 find the Mean Velocity of the Current : (i) By Floats (ii) By the Tachometer 
 Ground Log Buoy and Nipper Examination. . . pp. 200 211 
 
CONTENTS. Xlll 
 
 CHAPTER XII. 
 
 CHRONOMETERS. 
 
 Definitions History Harrison's Watches The Inside of a Chrono- 
 meter Bi-metal Balance Methods of adjusting Secondary Error Airy's 
 Method of Compensation Chronometer Koom in the Eoyal Observatory 
 "Annual Trials" Keport by the Astronomer Eoyal "Trial Numbers 1 ' 
 Chronometers for the Royal Navy History of a Chronometer -while in the 
 Service Packing a Chronometer for transport To take a Going Chrono- 
 meter from the Shore to a Ship Two sources of error to be guarded against 
 Stowage of Chronometers on board Effects of Temperature Three 
 reasons why a Chronometer should be wound at the same hour daily 
 Method of Winding a Chronometer Comparison of Chronometers Bates 
 of Chronometer liable to change To detect a faulty Chronometer when 
 Three are in use Exercises in comparing Chronometers To compare a 
 Chronometer with a Sidereal Clock Examination . . pp. 212234 
 
 CHAPTER XIII. 
 
 MERIDIAN DISTANCES. 
 
 Definition Primary and Secondary Meridians Two Methods of Com- 
 puting the Meridian Distance : (i) Electric Telegraph (ii) Portable 
 Chronometers Precautions to be observed in using the Second Method 
 To find P. M. time of Observations for Equal Altitudes Examination 
 Formulae for Sea Bate Typical Examples worked out as Models General 
 Examination, and Examples for Exercise .... pp. 235 250 
 
 CHAPTER XIV. 
 
 METHOD OF PLOTTING. 
 
 Setting about the Work Projecting the Work from the Field Book. 
 Points to be noted in conducting a Survey Examples of Plotting General 
 Exercises pp. 251289 
 
 Miscellaneous Exercises . . . . . . p. 289 
 
 General Index p. 301 
 
BOOKS CONSULTED. 
 
 The following Works have been carefully consulted in pre- 
 paring the Materials of the following Treatise. 
 
 Admiralty Tide Tables, for the years 1880, 81, 82. 
 
 Admiralty List of Lights on the British Coasts, 1880, 81, 82. 
 
 Airy's " Tides and Waves" in the Encyclopaedia Metropolitan.!. 
 
 Barometer Manual. 
 
 Belcher's Nautical Surveying. 
 
 Brinkley's Elements of Astronomy, edited by Stubbs and Briinnow. 
 
 Burr's Instructions in Practical Surveying. 
 
 Clarke's Geodesy. 
 
 This is a most valuable treatise for the more advanced Mathematical 
 Student. It contains an account of the most recent methods pursued in 
 important Trigonometrical Surveys. 
 
 Compendium of Instructions for Hydrographic Surveyors, 1877. 
 De Kheims. Geometrical Drawing, 1865. 
 Dubois. Cours de Navigation et d'Hydrographie. 
 
 This contains an excellent Chapter on Chronometers and their manage- 
 ment. 
 
 Frome's Outline of the Method of conducting a Trigonometrical Survey. 
 
 Galbraith's (W.), Trigonometrical Surveying. 
 
 Galbraith and Haughton. Tides and Tidal Currents. 
 
 Harbord's Glossary of Navigation. 
 
 Harrison. An Account of the Goings of Mr J. Harrison's Watch, 1767. 
 
 Several very interesting works on this subject, of about the same date, 
 may be seen in the British Museum. 
 
 Haskoll's Land and Marine Surveying. 
 
 Heather's Treatise on Mathematical and Drawing Instruments. 
 
 The original work formed one small volume in Weale's Series. It has 
 lately been enlarged, and is now published in three volumes, containing 
 respectively: " Draiving and Measuring Instruments;" "Optical Instru- 
 ments;" " Surveying and Astronomical Instruments" 
 
BOOKS CONSULTED. XV 
 
 Herschell's Treatise on Astronomy. 
 
 Hull's Nautical Surveying. 
 
 Jeffers' Nautical Surveying. 
 
 Johnson's V. Staff-Corn. Notes on Marine Surveying. 
 
 Laughton's Introduction to Nautical Surveying. 
 
 Ledieu. Les Nouvelles Methodes de Navigation. Paris, 1877. 
 
 The Second Part of this work is devoted exclusively to the Description, 
 Theory, and Use of Chronometers, and ought to be read by all who are 
 interested in Chronometrical Science. 
 
 Manual of Scientific Inquiry. The Articles on Hydrography and the Tides. 
 
 Manual of Surveying for India. 
 
 Pearson's Elementary Treatise on Tides, 1881. 
 
 Philosophical Transactions for the period 1833 1847. 
 
 In tJiese volumes are contained the Original Researches of Dr Whewell 
 and Sir John Lubbock on the Tides. The volumes for 1879 and 1880 con- 
 tain Professor Darwin's Theory of Tidal Evolution. 
 
 Kaper's Practice of Navigation. 3rd Edition. 
 Shadwell's Notes on the Management of Chronometers. 
 
 I beg to acknowledge here my great obligation to this valuable and well- 
 known Treatise. The useful information embodied in my own CJiapters XII 
 and XIII has been obtained to a very considerable extent from Admiral 
 ShadwelVs pages. 
 
 Shortland, Capt. R.N., Papers on Nautical Surveying, published in Naval 
 Science, Vols. H.,IIL, IV. 
 
 Simms. Treatise on Mathematical Instruments. 
 
 Simms. Treatise on Levelling. 
 
 Stevenson's Principles and Practice of Canal and Eiver Engineering. 
 
 Villarceau and De Magnac. Nouvelle Navigation Astronomique. Paris, 
 1877. 
 
 Chapter I. of Part II. is devoted to the subject of Chronometers. M. 
 Villarceau is a recognised authority in Chronometrical Science. 
 
 Note. A very important work on " Hydrographical Surveying" by 
 Capt. Wharton, E.N., has been published while the sheets of the present 
 Book were passing through the press. I should have been very glad to 
 have read it while preparing my own materials. Its practical details and 
 methods of proceeding by an officer of acknowledged capacity and wide 
 experience are of the greatest interest, and its collection of Tables at 
 the end must be of the highest value to all Practical Surveyors. 
 
EXTRACT FROM THE REGULATIONS RELATIVE 
 
 TO EXAMINATION FOR THE RANK OF 
 
 LIEUTENANT. 
 
 NAUTICAL SURVEYING. 
 
 Use of Charts; Rating of Chronometers; Determination of 
 Meridian Distance ; Selection and Measurement of a Base Line ; 
 Determination of Latitude, Longitude, and True Bearing; Tri- 
 angulation; Levelling; Soundings; Fixing Positions; Tide Gauge; 
 Establishment of the Port. 100 marks. 
 
 INSTRUMENTS. 
 
 Construction and Use of Marine Barometer, Sextant, Artifi- 
 cial Horizon, Azimuth Compass, Theodolite and Level. 
 
 40 marks. 
 
3/?7 
 
 A 
 
 'UNIVERSITY, 
 
 MAKINE SUKVEYING. 
 
 CHAPTER I. 
 
 SYMBOLS USED IN CHARTS AND SURVEYING. 
 
 L Symbols used to denote the Quality of the Bottom, n. General Abbre- 
 viations. HI. Buoys. IV. Lights and Lighthouses. V. Conventional 
 Signs. VI. Soundings. VII. Miscellaneous. VHI. Surveying Symbols. 
 
 1. I. QUALITY OF THE BOTTOM: 
 
 Colour b = blue. 
 
 blk = black. 
 
 br = brown, 
 
 d = dark, 
 w = white. 
 
 gn = green, 
 y = yellow. 
 
 gy = grey. 
 spk = speckled. 
 
 Substance cl = clay, 
 m = inud. 
 peb = pebbles, 
 sh = shells. 
 
 crl = coral, 
 oys = oysters. 
 r = rock, 
 st = stones. 
 
 g = gravel. 
 oz = ooze. 
 s = sand, 
 wd = weed. 
 
 Nature brk = broken. 
 
 c = coarse. 
 
 f = fine. 
 
 grd = ground 
 sft = soft. 
 
 h = hard, 
 stf = stiff. 
 
 rot = rotten. 
 
 EXAMINATION. 
 
 Write down the abbreviations for the following Qualities of 
 the Bottom: 
 
 Sand. Coarse gravel. Coral and fine sand. Pebbles mixed 
 with shells. Broken shells. Stiff mud. Sand and pebbles. Grey 
 mud. 
 
 R. M. S. 1 
 
2 MARINE SURVEYING. [CHAP. 
 
 What Qualities of the Bottom do the following abbreviations 
 denote? 
 
 sh; st; r. s. sh; m; sft. cl; crl; stf. cl; c. s; gn. cl.; brk. sh.; 
 wd; spk. sh; s; oz; gy. g. 
 
 2. II. GENERAL ABBREVIATIONS. 
 
 Alt = altitude. Anch Ke = anchorage. B = bay. 
 
 Bar = barometer. Bat y = battery. B k = bank. 
 
 C = cape. C. G. = Coastguard. Cath = Cathedral. 
 
 Ch = church. Chan = channel. Col d = coloured. 
 
 Cr = creek. 
 
 E. D. = a reported Rock or Shoal whose Existence is Doubtful. 
 Fms = fathoms. Ft = feet. G = gulf. 
 
 G* = great. H = hour. H d = head. 
 
 Ho = house. H r = harbour. H. W. = High Water. 
 
 H. W. F. & C. = High Water at Full and Change of the Moon. 
 I = island. I 8 = islands. Kn = knot. 
 
 L = lake. Lat = latitude. Long = longitude. 
 
 L* = light. L. W. = Low Water. Mag z = magazine. 
 Mag c = magnetic. M fc mountain. Np = neaps. 
 
 Obs n Spot + = observation spot. P = port. 
 P. D. = a Danger known to exist but its Position is Doubtful. 
 P k = peak. P* = point. R = river. 
 
 R f = reef. R k = rock. S d = sound. 
 
 Sh = shoal. Sp = springs. Str = strait. 
 
 Tel = telegraph. Therm = thermometer. Var = variation. 
 Vil = village. W. PI = watering place. 
 
 EXAMINATION. 
 
 Write down the abbreviations for the following terms : 
 Harbour, Hour, Head, House, River, Bank, Shoal, Magazine, 
 Magnetic, Creek, Observation Spot. 
 
 What do the following abbreviations represent? 
 C. G.; Vil; L*; M fc ; P. D.; I; H r ; E. D.; G; Mag- S d ; S P ; 
 Tel; Sh. 
 
I.I SYMBOLS USED IN CHARTS AND SURVEYING. 3 
 
 3. III. BUOYS. 
 
 Colours. B. (near a Buoy) = Black. 
 
 Cheq. do. = Chequered. 
 
 H. S. do. = Horizontal Stripes. 
 
 R. do. = Red. 
 
 V. S. do. = Vertical Stripes. 
 
 W. do. = White. 
 
 Shapes. 
 
 Can = *3L Nun = 
 
 * Conical, or 
 
 T = ~^r- Spiral 
 
 Mooring = 
 
 Buoys with Beacons. The following Beacons are used on 
 Buoys : 
 
 Cage = I 7 
 
 Triangle = T 
 
 Diamond = t Globe = I 
 
 The end of a Spit is usually marked thus : 
 
 4. COLOURING BUOYS. The following regulations are in 
 use: 
 
 CHEQUERED. The Buoy is divided into four horizontal and 
 into eight vertical equal parts, to be coloured white and red, or 
 wliite and black alternately. The white squares are then to be 
 reduced by one inch all round. 
 
 VERTICALLY STRIPED. The Buoy is divided into eight equal 
 parts, these being alternately white and red, or white and black. 
 The white stripes are then to be made one-third narrower than 
 the others. 
 
4 MARINE SURVEYING. [CHAP. 
 
 HORIZONTALLY STRIPED. The Buoy is divided into five equal 
 parts, these being coloured alternately white and black, or white 
 and red. The white stripes are then made one-third narrower 
 than the others. 
 
 These various colours are thus represented in charts : 
 White = j~/ (quite plain) Red= _^L (a little shading) 
 
 Black - ^J V.S. = Jffjt H.S. - ^ 
 
 Cheq. = 
 
 5. REGULATIONS FOR THE EMPLOYMENT OF BUOYS. 
 
 (1) The side of a Channel is Starboard or Port with reference 
 to a ship entering a Harbour from seaward. 
 
 (2) The entrance of a Channel, or a turning point, is marked 
 by a conical or spiral buoy. 
 
 (3) The Starboard side of a Channel is marked with Can 
 Buoys of the same colour throughout. Hence the well-known 
 practical rule " whole-coloured buoys on the starboard side." 
 
 (4) The Port side of a Channel is marked with the same 
 shaped buoys, but coloured vertically or chequered. 
 
 (5) If further distinction is required, then conical buoys 
 with globes are on the starboard, and with cages on the port. 
 
 (6) When a shoal, or middle ground, exists in a channel, 
 each end of it is marked by a buoy of that colour in use in that 
 channel, but with annular bands of white. 
 
 If the shoal is of great extent, we must then proceed to use 
 the buoys as though we had two channels to mark out, and the 
 buoys must be placed in accordance with (3) and (4) 
 
 When necessary the outer buoy is distinguished by a staff and 
 diamond, and the inner bv a staff and triangle. 
 
 (7) The position of a wreck is marked by a green Nun 
 Buoy. 
 
I.] SYMBOLS USED IN CHARTS AND SURVEYING. 5 
 
 EXAMINATION. 
 
 (1) Draw a Can Buoy, a Nun Buoy, a Sj)ar Buoy, and a 
 Mooring Buoy. 
 
 (2) Draw a Chequered Nun Buoy, a ./M Caw Buoy, a Ferta'- 
 ca% striped Can Buoy, a Black Spiral Buoy. 
 
 (3) Draw a Can Buoy with a Cage ; a Nun Buoy with a Globe ; 
 and a Buoy used at the end of a Spit. 
 
 (4) You observe in a channel that all the buoys on your port 
 side are black, are you entering or leaving the harbour 1 
 
 (5) On going up a channel a buoy with annular bands of 
 white is reported right ahead, what inference are you to draw 1 
 
 (6) If a shoal in a channel is of great extent, how are the 
 two extreme buoys distinguished from those placed at the sides 1 
 
 (7) How is the position of a wreck marked ? 
 
 (8) What rules are to be observed in painting a Chequered 
 Buoy, a Vertical striped Buoy, and a Horizontal striped Buoy, 
 respectively? 
 
 6. IV. LIGHTS AND LIGHTHOUSES. 
 
 The position of a Lighthouse is marked by a small round 
 black dot, thus ^ 
 
 The following abbreviations are used: 
 L* F = Fixed Light. L* Fl = Flashing Light. 
 
 F and Fl = Fixed and Flashing. Fl g L* = Floating Light. 
 
 L l Int = Intermittent Light. L* Rev = Revolving Light. 
 
 L* Occ = Occulting Light. L l Alt = Alternating Light. 
 
 Min (near a Light) = Minutes. Sec (near a Light) = Seconds. 
 
 Vis (near a Light) = Visible. 
 
 7. EXPLANATIONS AND ILLUSTRATIONS. 
 
 1. F = FIXED OR STEADY. This term sufficiently explains 
 itself. The great majority of the Lights on the British Coasts are 
 of this description. 
 
6 MARINE SURVEYING. [CHAP. 
 
 Examples: (1) Bishops' Rock in Scilly shews a White Fixed 
 Light. 
 
 (2) At the end of the Breakwater in Brixham there is a Red 
 Fixed Light. 
 
 (3) At the Pier Head of the Inner Harbour at Torquay there 
 is a White Fixed Light to Seaward and a Red Fixed Light to the 
 Westward. 
 
 (4) Southsea Castle shews a Fixed Light with Red and Green 
 Sectors. 
 
 (5) Maplin Sands, a Red Fixed Light with a White Sector. 
 The white indicates a particular channel. 
 
 2. FL = FLASHING. There are two kinds : (a) Flashes at short 
 intervals; (6) Groups of Flashes at regular intervals. 
 
 In (b) the appearance presented consists of two or three flashes 
 following each other as quickly as due separation allows, the 
 remainder of the period separating these groups of flashes by a 
 longer interval of eclipse. 
 
 Examples. (1) The Seven Stones Light Vessel shews three 
 flashes in quick succession, followed by 36 seconds darkness. The 
 whole period being one minute *. 
 
 (2) Casquets. Shews three successive flashes of about 2 seconds 
 duration each, with intervals between each flash of about 3 seconds 
 darkness, the third flash being followed by an eclipse of about 18 
 seconds. 
 
 (3) Royal Sovereign Shoal near Eastbourne. Shews three 
 flashes in quick succession every minute; the time thus occupied is 
 about 23 seconds, and then follows an eclipse of about 37 seconds. 
 
 (4) Galley Head (S. of Ireland). Shews six or seven flashes in 
 quick succession every minute ; the duration of this " group " of 
 flashes is about 16 seconds, and then follows an eclipse of 44 
 seconds. 
 
 (5) Holyhead Breakwater. The Light at the end flashes 
 every 7J seconds. 
 
 * The term "flash" must not be taken as denoting an instantaneous 
 appearance, such for instance as that presented by Colomb's Flashing Signals ; 
 it is more deliberate than this, but quicker than the Revolving Light. 
 
I.] SYMBOLS USED IN CHARTS AND SURVEYING. 7 
 
 (6) The Arklow Light Vessel flashes twice in quick suc- 
 cession, followed by 45 seconds darkness. The period being a 
 minute. 
 
 (7) Dungeness shews a white light which flashes every 5 
 seconds, the flashes being of about 2 seconds duration. 
 
 3. F. AND FL. = FIXED AND FLASHING. The appearance pre- 
 sented is a Fixed Light with the addition of White or Coloured 
 Flashes preceded and followed by a short eclipse. 
 
 Example. Craigmore (Firth of Clyde). The period of the 
 Light is about 1 1 seconds, and consists of 5 seconds light followed 
 by three flashes of half a second each, these three flashes being 
 separated by eclipses of 1J, J, 1 J, J seconds. 
 
 4. REV. = REVOLVING. The Light gradually increases to its 
 full power, and then decreases to eclipse. 
 
 Note. At short distances, and in clear weather a faint con- 
 tinuous light may be seen. 
 
 Examples. (1) Wolfs Rock Light revolves every half minute. 
 
 (2) Start Light shews a White Light revolving every minute. 
 
 (3) Hanois Rock Light in Guernsey revolves every 45 seconds. 
 
 Note. It is deemed advisable that the light should not be 
 obscured for too long an interval. There is now no Light in Eng- 
 land withdrawn for more than 105 seconds, the whole period 
 being 120 seconds. Thus the Light on Beachy Head revolves 
 every two minutes, 15 seconds bright, and If minutes dark. 
 
 5. INT. = INTERMITTENT. A Light suddenly and totally 
 eclipsed. The briyldness lasting for more than 30 seconds. 
 
 Examples. (1) Tarbet Ness (E. of Scotland). White Light 
 visible for 2 J minutes, and eclipsed for \ minute. 
 
 (2) Ru Stoer. White Light visible for 1 minute, eclipsed 
 for ^ minute. 
 
 (3) River Ribble (W. of England). White Light visible for 
 3 1 minutes, eclipsed for J minute. 
 
8 MARINE SURVEYING. [CHAP. 
 
 (4) Dundrum Bay (E. of Ireland). Bed Light visible for 45 
 seconds, eclipsed for 15 seconds. 
 
 (5) Rathlin (1ST. of Ireland). White Light with a Red Sector, 
 bright for 50 seconds, eclipsed for 10 seconds. 
 
 6. Occ. = OCCULTING. A Light suddenly and totally eclipsed. 
 The brightness lasting less than 30 seconds. 
 
 Examples. (1) West end of Plymouth Breakwater. The Light 
 suddenly disappears for a space of 3 seconds every half minute. 
 
 (2) North Foreland Light suddenly disappears for the space 
 of 5 seconds every half minute. 
 
 (3) Ardrossan Light, visible for 2 seconds, eclipsed for 2 
 seconds. 
 
 (4) Roche's Point (S. of Ireland), visible for 15 seconds, 
 eclipsed for 5 seconds. 
 
 (5) Wicklow Light, bright for 10 seconds, dark for 3 seconds. 
 
 (6) Loop Head (W. of Ireland). A White Light visible for 
 20 seconds, and eclipsed for 4 seconds. 
 
 Note. The Intermittent and Occulting Lights are in con- 
 trast with Revolving Lights. In the first two the period of 
 brightness is generally long and the period of eclipse is short, 
 whereas in a Revolving Light the period of brightness is short 
 and the period of darkness is long. Moreover in a Revolving 
 Light there is a waxing and waning of the light as the beam 
 packed, as it were, by the action of the annular rings of the lens 
 approaches and recedes from the eye, and the place of the light 
 is indicated (to a near observer) during the intervals of with- 
 drawal by the reflections in the lantern; but in the case of both 
 an Intermittent and Occulting Light the brightness is at its 
 maximum throughout, and the intervening darkness is total. 
 
 7. ALT. = ALTERNATING. Red and White Lights alternately 
 at equal intervals of time without any intervening eclipse. 
 
 Examples. (1) Owers Light Vessel shews alternately a white 
 flash twice and then a red flash. 
 
 (2) Hartland Point (W. of England) shews alternately at 
 intervals of half a minute a white flash twice and then a red flash. 
 
I.] SYMBOLS USED IN CHARTS AND SURVEYING. 9 
 
 8. ILLUMINATING APPARATUS. 
 Two systems are used : 
 
 The Catoptric* (0) or Reflecting system. 
 The Dioptric (D) or Refracting system. 
 
 In the Catoptric the light is usually produced by means of an 
 Argand burner placed in the focus of a silvered metal reflector. 
 In the Dioptric the light comes from one central lamp placed in 
 the focus of a surrounding glass refractor, by which all the best rays 
 of the light are sent horizontally, and to the surface of the sea. 
 
 A combination of the two is sometimes used, and is known as 
 Catodioptric (C. D.), and where the light is very powerful it is 
 possible to convey a portion of it (by reflection) to a lower aper- 
 ture of the Lighthouse to guard against certain local dangers. 
 Both systems admit of lights being shewn either as fixed, inter- 
 mittent, occulting, revolving, flashing, or group flashing, and are 
 classified as of various Orders; the Order being determined in the 
 Catoptric system by the number of lamps and reflectors, and in 
 the Dioptric by the size of the Instrument and of the central 
 flame. Thus in a Dioptric 1st Order Light from the centre of the 
 light to the vane is about 15 feet, in a 6th Order the vertical 
 height is about 6 feet. 
 
 The Catoptric system is preferable where skilled labour is not 
 obtainable, because less care and intelligence are required in the 
 adjustment, it is also better in volcanic districts and in Light 
 Vessels because less liable to be put out of adjustment. 
 
 The illuminant in each system is colza or paraffin oil or gas. 
 The Electric Light is well adapted for the Dioptric system, the 
 light being concentrated at a true focal point. 
 
 9. Coloured Sectors of light are produced by means of coloured 
 glass placed inside the lantern. 
 
 It must be noted that the Bearings of Lights are always Mag- 
 netic and are given as viewed from seaward. 
 
 The Distances at which the Lights are visible are computed on 
 the supposition that the observer's eye is 15 feet above the sea 
 level. 
 
 * Katoptron, a mirror. 
 
10 MARINE SURVEYING. [CHAP. 
 
 Light Vessels have usually the name of the danger they guard 
 against painted distinctly on their sides. These vessels have as 
 many Masts and Balls as they have Lights exhibited at night. 
 When a Light Vessel has drifted from her correct position to one 
 where she is useless as a guide the following method is adopted to 
 intimate the fact. " K fixed red light will be exhibited at each end 
 of the vessel, and a red flare shewn every quarter of an hour." 
 In the daytime the Balls at the mastheads are struck. 
 
 When a Light Vessel marks the position of a wreck, the top- 
 sides are painted green, and in the daytime three Balls are placed 
 on a yard 20 feet above the water, two of these vertically on the 
 safe side, and one Ball on the danger side ; in the night fixed 
 white lights similarly arranged have the same signification. 
 
 When assistance is required from shore, special rockets of 
 little sound but of great brilliancy are fired immediately after 
 a gun. 
 
 When guns are used as fog signals they are fired at intervals 
 of 10 minutes. 
 
 Sirens and horns in Light Vessels are always blown against 
 the wind. 
 
 The Code of Signals in use is the International. 
 
 EXAMINATION. 
 
 (1) What two systems are used in lighting coasts'? Specify 
 the principle of each kind. 
 
 (2) Write down carefully the appearances presented to a 
 mariner by the following descriptions of Lights : L* F. L* Fl. L fc F. 
 andFl. L*Int. L'Occ. L'Rev. L* Alt. 
 
 (3) I find near a Lighthouse marked on a chart the following 
 note : " Rev. 20 sec vis 14 m." Explain it. 
 
 (4) What is a "Group Flashing " Light ? 
 
 (5) What is the chief distinction between an Intermittent 
 and Occulting Light 1 
 
 (6) Contrast these with a Revolving Light. 
 
 (7) How does a Revolving Light differ from an Alternating 
 Light ? 
 
I.] SYMBOLS USED IN CHARTS AND SURVEYING. 11 
 
 (8) How are Coloured Sectors produced 1 ? What is their 
 chief use 1 
 
 (9) In which system is the Electric Light used ? 
 
 (10) Under what circumstances is the Catoptric System 
 preferable to the Dioptric ? 
 
 (11) On approaching land I see that a light is bright for 20 
 seconds and then suddenly disappears for 5 seconds. In what 
 category is this Light to be placed ? 
 
 (12) How are you to know by day and also by night whether 
 a Light Vessel has drifted from her correct position 1 
 
 (13) If a Light Vessel is placed near a wreck, how is she 
 distinguished ? 
 
 (14) How are you to know on which side the danger lies? 
 
 (15) In what way does a Lighthouse signal for help? 
 
 (16) In using a Foghorn what special precaution must be 
 taken '{ 
 
 10. V. CONVENTIONAL SIGNS. 
 
 Note. Small dots form 
 the outside line, and 
 
 Sandy beach dry at L. W. = behind these parallel 
 
 rows of finer dots re- 
 present the sand. 
 
 Small strokes form the outside line, 
 
 Gravelly Beach = VfrYyAvrrf-'" and behind these parallel rows of 
 
 dots represent gravel. 
 
 oooooooooooop Small zeros form the out- 
 Shingle or stony beach = -"^.--=:--rr . , ,. 
 
 side line. 
 
 For examples of the above see the Plate in the Manual of 
 Scientific Enquiry illustrating the Article on Hydrography. 
 
 Sandy shore = -^*^ A shar P line to si ^ nif ^ the ed ^ e of the 
 
 water at High Water. 
 
 Closed curves of small dots 
 to signify the sand with 
 
 Sand with gravel mixed = a f ew irregularly placed 
 
 (dry at L. W.) zeros and heavier dots to 
 
 signify the gravel or coarse stones. 
 
12 
 
 MARINE SURVEYING. [CHAP. 
 
 The same as the last but 
 
 Stone Bank and Beach = 
 
 Sand Bank that dries at L. W. 
 with the gravel or stones omitted. 
 
 Heavier dots in groups or 
 single behind a contin- 
 uous line. 
 Mud Beach 
 (dry at L. W.) 
 
 If Banks, not JBeaches, are to be represented, then the continu- 
 ous line at the edge is to be omitted. 
 
 Sand and Mud mixed - Very light Bhading with 
 
 rows ot dots to represent 
 sand. 
 
 The faintest possible shading. 
 
 (dry at L. W.) 
 
 Coral Reefs = 
 
 Jagged outline with small crosses to de- 
 note rocks 6 feet below the water. 
 
 Rocky Ledges which 
 cover and uncover 
 
 Rocks with less than 6 feet 
 of water on them 
 
 Rocks awash at L. W. 
 
 Narrow rows of irregular 
 shading, the darker parts 
 signifying the sharper 
 and higher ledges. 
 Simply a cross within a lirnit- 
 ''' ing danger line. 
 A small cross with a dot in each 
 quadrant. These dots may be 
 considered as representing the 
 points of rock appearing above 
 the surface at L. W. 
 
 Isolated rocks which do not cover, 
 
 i.e. which are visible at Hierh , + & 
 
 9 *o * 
 
 Water Springs. ^ * 
 
 Rocks of different kinds ( Q <^ J"*\ 
 with limiting danger /' <^>' o + / 
 line. W .* / 
 
I.] 
 
 SYMBOLS USED IN CHARTS AND SURVEYING. 
 
 13 
 
 Cliffy Coast line = 
 
 Shore, steep to = 
 
 Trees. Mangroves 
 
 Palms 
 
 Pines 
 
 Two lines with shading between. 
 Lines radiating to the upper 
 from the summit of a hill behind. 
 Two lines with dark shading between, 
 the wider intervals signify that the 
 height of the cliff increases. 
 
 Generally || W 
 
 3#3L 
 
 Swampy \ 
 
 Marshy v land = "- 
 
 Mossy ) 
 
 Grass 
 
 Meadow 
 
 land = 
 
 Cultivated grounds 
 and gardens. 
 
 Kelp. 
 
 Note. " To economise the labour of the draughtsman, Towns 
 and Buildings may be tinted Red; Sandy Beaches Yellow; Rocks 
 
14 MAKINE SURVEYING. [CHAP. 
 
 shewing at Low Water Brown ; Mud Banks light Black ; Culti- 
 vated land Green*" 
 
 EXAMINATION. 
 
 (1) Represent a Cliffy Coast line, and describe the method of 
 doing so. 
 
 (2) How does a shore steep to differ from a cliffy coast line '? 
 
 (3) What is the distinguishing difference in the modes of 
 representing a Bank and a Beach 1 
 
 (4) Draw the symbol for a Rock with less than 6 feet of water 
 on it. 
 
 (5) Draw the symbol for a Rock awash at L. W. 
 
 (6) Draw a Ledge which covers and uncovers. 
 
 (7) Draw a Rock which does not cover. 
 
 (8) When is a Rock said not to cover ? 
 
 (9) Draw a Coral Reef. 
 
 (10) How would you remember the difference between the 
 symbols for a rock awash at L. W. and a rock with less than 
 6 feet of water on it ? 
 
 (11) Draw a series of rocks with a limiting danger line. 
 
 (12) The limiting danger line is sometimes represented as a 
 continuous line and sometimes by a dotted line, is any distinction 
 sought to be thus conveyed 1 
 
 11. VI. SOUNDINGS. 
 
 Shoal Banks which never uncover, and on which /^ ^ .. 
 the depth of water is known 
 
 1 Fathom line 1 Sometimes when the depth 
 
 2 ,, ,, I lies between 1 fathom and 
 
 3 | 3 fathoms the depth is thus 
 
 4 J marked 
 
 5 ,, Five dots in each series. 
 
 6 ,, Six dots in each series. 
 
 &c. 
 
 * General Instructions for Hydrographic Surveyors, p. 7, note. 
 
I.] SYMBOLS USED IN CHARTS AND SURVEYING. 15 
 
 1 fathom line ----------- Strokes and single dots al- 
 
 ternately. 
 20 ------------- Strokes and two dots al- 
 
 ternately. 
 30 ----------------- Strokes and three dots al- 
 
 ternately. 
 100 ............................ Continuous dots*. 
 
 slfo^ ^ bottom found at the depth expressed. 
 
 12. VII. MISCELLANEOUS. 
 
 ___^ _ 3t=; _-_; Long and short waving 
 
 Breakers, Overfalls lines, such as would be 
 
 and Tide Rips used in representing a 
 
 river rapid. 
 
 Thus the tidal stream in the Irish Sea encounters an extensive 
 projection of the Codling Bank near Wicklow. The outer portion 
 of the stream takes the circuit of the Bank, the inner stream 
 sweeps over it, occasioning an overfall and strong rippling all 
 round the edge by which the Bank may generally be discovered. 
 
 Tide Tables, p. 1 25. 
 
 Anchorage for large vessels a complete anchor. 
 
 Anchorage for small vessels ^ J^ one fluke) or kedge. 
 Churches hjn 
 
 Stone windmill ^^ The building is round. 
 Wooden windmill >*C The building is high. 
 
 A few houses scattered by the sides of 
 Villages ^fiftf the roads, and a few irregularly placed 
 
 ^ S * in the rear. 
 
 * In the Ordnance Maps of Great Britain the boundaries of Counties 
 are represented by the 10 fathom line used in Charts, and those of Town- 
 lands by the 100 fathom line. 
 
16 MARINE SURVEYING. [CHAP. 
 
 - 
 
 Towns The streets are represented by straight lines. 
 
 Currents ** - " An arrow with feathers on both sides. 
 
 ^Yo0e?,tide stream - + Feathers on only one side. 
 
 Note. The feathers are placed on that side on which the 
 stream is inclined to press, if the arrow is thrown out of its line. 
 
 Ebb tide stream - + Plain dart without feathers. 
 
 A current may be considered as the movement of a large body 
 of water. A stream as the movement of surface water to the 
 depth of about 70 fathoms. 
 
 High Water on Full and Change Days, ie. on the days of Full 
 and New Moon, is thus represented, H. W. F. & C. The Hour 
 is always expressed in Roman figures, e.g. vn h . 22 m . 
 
 The Compass on a chart is always drawn on the magnetic 
 meridian, and hence if it is ever required to construct such a com- 
 pass and to subdivide one quadrant, this quadrant must be taken 
 between two cardinal points of the Magnetic Compass and not of 
 the True. 
 
 The underlined figures on a bank signify the depth of water 
 over it at H. W. or else the height of the bank above L. W. 
 
 The method adopted is of course fully explained in the Title of 
 each Chart. 
 
 All Heights are given in feet above H. W. Ordinary Springs, 
 and where there is no tide, then above the sea level. 
 
 All Bearings, as well as the directions of winds and currents, 
 are magnetic. 
 
 In the Admiralty Charts, if changes have been made in the 
 original survey of sufficient importance to require a new plate 
 being engraved, the date of any such change is given at the bottom 
 of the chart. Such a change necessitates the former chart being 
 " destroyed in the presence of the Captain," but minor alterations 
 (e.g. in buoys, lights, &c.), made in red ink by the Navigating 
 Officers, are intimated by the dates in the lower left-hand corner 
 of the chart. The official number of a chart is always found in 
 the right-hand lower corner. 
 
I.] SYMBOLS USED IN CHARTS AND SURVEYING. 17 
 
 The periods of a tide are four in number, viz. 1st, 2nd, 3rd, 4th 
 Quarters, and are denoted by 1st Qr., 2nd Qr., &c. 
 
 The Velocity of the tide is given in knots and fractions of a 
 knot. 
 
 Thus the symbol ^ > signifies a Flood tide stream 
 
 2ndQr. 2Jkn. 
 at the 2nd Qr. running due East at a rate of 2 J knots an hour. 
 
 A Landmark is some particular feature of a coast by means 
 of which the locality may be known by a ship approaching or 
 making the land, or by which she can find her way to an 
 anchorage. 
 
 A Day Hark is a conspicuous object to mark a narrow entrance 
 to a harbour or river, which from the configuration of the coast is 
 difficult to make out from seaward. 
 
 A Leading Hark is generally denoted by two very fine parallel 
 lines close together, and is required in order to keep in a certain 
 channel or passage. 
 
 A Clearing Mark on the contrary is used to enable a vessel to 
 keep clear of a certain danger, and is generally denoted on a chart 
 by a sinyle dotted line. Thus if a ship is beating up a passage 
 wherein a danger exists, she will keep on a certain course until 
 she gets two specified objects in one, or "in transit" as it is called, 
 when she must alter course to avoid the danger ahead. 
 
 Thus we may have this note on a chart or in Sailing Directions : 
 " Lighthouse in one with East Peak clears reefs in 5 fathoms, and 
 kept open (S. b E.) leads through the passage in Mid Channel." 
 
 EXAMINATION. 
 
 (1) Represent a 5-fathom line of soundings. 
 
 (2) Represent the 10-fathom, 20-fathom, and 100-fathom 
 linos. 
 
 (3) " No bottom at 450 fathoms." Draw the representative 
 symbol. 
 
 R. M. S. 2 
 
18 MARINE SURVEYING. [CHAP. 
 
 (4) Draw the symbols used to represent 
 
 An anchorage for large vessels, 
 
 An anchorage for small vessels, 
 
 A church, 
 
 A stone windmill, 
 
 A wooden windmill. 
 
 (5) Draw the symbols which represent 
 
 A current, 
 
 A flood tide stream, 
 
 An ebb tide stream, 
 
 A flood tide stream setting E.S.E. at the 3rd quarter 
 
 5 knots. 
 An ebb tide stream setting N. b W. at the 4th quarter 
 
 2f knots. 
 
 (6) What do the underlined figures on a bank signify ? 
 
 (7) A current sets N.E. ; is this the True or Magnetic 
 Bearing 1 
 
 (8) Distinguish between a Leading Mark and a Clearing Mark. 
 
 (9) How is each represented on a chart ? 
 
 (10) What is a Landmark ] 
 
 13. VIII. SURVEYING SIGNS. 
 Q - Observatory Station. 
 
 y^V = Station where angles are taken either with the sextant 
 or theodolite. 
 
 = R. T. = Right extreme, or Right Tangent. 
 = L. T. = Left extreme, or Left Tangent. 
 
 ^ = B. M. = Bench Mark. This is the well-known mark 
 
 used in the Ordnance Survey to point out the places where the 
 levelling staves were placed in running the lines of levels. We 
 shall have more on this subject below. 
 
fi\ 
 
 I.] SYMBOLS USED IN CHARTS AND SURVEYING. 10 
 
 or p = Two objects in one, or in " transit." 
 -0- = Altitude of the sun's centre. 
 CD = Bearing of the sun's centre. 
 
 GJ = Right limb of the 
 
 sun. 
 
 = Left limb of the sun. 
 
 = Sun's lower limb. 
 75 = Sun's upper limb. 
 
 y = Sun's lower and right limbs, <fec. 
 
 Authorities for this Chapter : 
 
 The Admiralty List of Abbreviations. 
 
 General Instructions for Hydrographic Surveyors, 1877. 
 This contains much information in the Appendices on 
 the subject of Buoyage, Lighthouses, &c. 
 
 "Tide Tables for the British and Irish Ports for the years 
 1881 and 1882." 
 
 "Admiralty List of Lights in the British Islands," 1882. 
 
 Raper in his Navigation has a very good section on the em- 
 ployment of symbols. Vide pages 381 389. 3rd Edition. 
 
 22 
 
CHAPTER II. 
 
 THE CONSTRUCTION AND USE OF SCALES. 
 
 14. DEF. A SCALE is an artificial means of representing 
 any given dimensions, whether angular or linear. E.g. we may 
 suppose that a mile is represented by a straight line 1 inch 
 long, then 2 miles will be represented by 2 inches, and 3^ miles 
 by 3J inches, &c. 
 
 The different kinds of scales may be divided into two classes, 
 viz. (I.) Simply-divided Scales, or Plain Scales; (II.) Protracting 
 Scales. 
 
 15. (I.) SIMPLY-DIVIDED SCALES. 
 
 Of these there are three kinds, (a) Scales of Equal P 
 (/3) Diagonal Scales, (y) Vernier Scales. 
 
 (a). Scales of Equal Parts may be described as follows: 
 
 Assume any convenient length, suppose 6 inches. Draw a 
 straight line 6 inches long, and divide it into 6 equal parts. 
 These six parts are called the Primary divisions. Next divide 
 the first or left-hand primary into 10 equal parts, these smaller 
 parts are known as the secondary divisions. 
 
 Hence we can see that the primary divisions will represent 
 units, if the secondary divisions represent tenths; the priii 
 will represent hundreds if the secondaries are taken as tens, etc. 
 
 These scales of equal parts are generally contained between 
 two fine parallel lines. The two lines of the first primary are 
 divided, one into 10 equal parts, the other into 12 equal parts; one 
 
CHAP. II.] THE CONSTRUCTION AND USE OF SCALES. 21 
 
 of the lines so divided will measure units and decimals, the second 
 will measure feet and inches. 
 
 DEF. If the first primary be an inch long and be divided 
 into 30 equal parts*, the scale is called a scale of 30. Similarly 
 we can have scales of 40, of 50, &c. On a Protractor we have 
 usually plain scales of 30, 35, 40, 45, 50, and 60 on one side, the 
 number which denotes the scale standing on the left of the scale. 
 On the opposite side of the Protractor we have scales of 1, , , 
 lr> ^i f > i> 1? inch* both decimally and duodecimally divided as ex- 
 plained above. 
 
 16. USE OF THESE SCALES. If a secondary division be taken 
 as a unit, the primary will be 10: if the secondary be taken as 10, 
 the primary will be 100; if a secondary be taken as -f-Q, the pri- 
 mary will be a unit, <fec. Hence we can take off from these scales 
 feet and decimals of a foot, yards and decimals of a yard, &c. 
 Thus if the scale of a plan is ^ inch to a mile, and we wish to 
 lay off a distance of 3 -6 miles, we must look for the scale on the 
 Protractor with -J- before it, and then placing one point of the 
 compass on the line with 3 near it we must extend the other point 
 to the 6th secondary division on the decimal line, and this interval 
 will represent 3 '6 miles. 
 
 17. MODE OF CONSTRUCTING THESE SCALES. 
 
 Let it be required to draw a plain scale, 10 units to the inch, 
 and to exhibit 60 units. 
 
 Draw a line 6 inches long, and divide it into 6 equal parts. 
 These are the primary divisions. Divide the first primary into 
 10 equal parts. Next draw a thicker line at a short distance 
 (TU or TT i ncn ) below the first line and draw vertical lines between 
 them from the divisions on the first line as in the diagram. 
 
 Observe the position of the numbers. The zero is placed at 
 the mark between the first and second primary divisions, and then 
 from left to right in order are the numbers 10, 20, &c. This 
 method of numbering is observed in all plain scales. The student 
 
 * Or what amounts to the same, if one-third of the inch be divided into 
 ten equal parts, as on most protractors. 
 
22 MARINE SURVEYING. [CHAP. 
 
 ought to examine the scales on his Protractor to impress this on 
 his mind. In this way lengths are more easily taken off from 
 scales ; e. g. to take 1 5 units from the above scale, we must extend 
 the points of the compass from 10 back to the 5th division in the 
 subdivided primary. 
 
 When very minute divisions are not required we make use 
 of these plain scales, because it is easy to subdivide an inch into 
 10, 15, or 20 equal parts, but if hundredth* of an inch are re- 
 quired recourse must be had to a Diagonal Scale, as we shall 
 see later on. 
 
 EXERCISES IN PLAIN SCALES. 
 
 (1) Construct a scale of 12 feet to 1 inch, and exhibit a 
 length of 70 feet. 
 
 By proportion 12 : 1 :: 70 : x; .'. x = 5-83 inches. 
 
 Now draw a line 5 '83 inches long*. Divide it into 7 equal 
 parts, each of which will represent 10 feet. Subdivide the first 
 primary into 10 equal parts, and each of these subdivisions will 
 represent a single foot. Number as directed above, and print at 
 the beginning " Scale of," and at the end of the scale print the 
 word "feet"; we shall thus have a "Scale of 70 feet." 
 
 Note. Since 1 inch represents 12 feet, or 144 inches of real 
 length, we have the " Natural Scale" = y^. 
 
 DBF. THE NATURAL SCALE is the ratio that the length of 
 a certain unit on the paper or plan bears to the real length of 
 that unit 011 the earth's surface. 
 
 According to the method described above all the Plain Scales 
 are constructed; the secondary divisions varying only with the 
 number of units represented by a primary division; e.g. if we 
 wish to shew feet and inches, a primary will represent 1 foot, and 
 the first primary must be divided into 12 equal parts. If fathoms 
 and feet are to be represented, then each primary w-ill represent 
 a fathom, and the first primary must be divided into 6 equal 
 parts. If furlongs and chains, the first primary must be divided 
 
 * This length must be taken from a Diagonal Scale. 
 
II.] THE CONSTRUCTION AND USE OF SCALES. 23 
 
 into 10 parts (because a chain = 22 yards, and a furlong = 220 
 yards). If miles and furlongs, the first primary will contain. 
 8 equal parts. If miles and cables, the first primary must contain 
 10 equal parts. 
 
 (2) Construct a scale of 20 yards to 1 inch, to shew yards 
 and feet, and exhibit a length of 90 yards. 
 
 (3) Construct a scale of 10 miles to 1 inch, to shew furlongs, 
 and exhibit 50 miles. 
 
 (4) Subdivide into cables a scale on which 1 1 -80 inches are 
 equal to a sea mile, and shew what length will represent 1000 
 feet. (March, 1875.) 
 
 (5) Subdivide into cables a scale 011 which 7 '3 inches are 
 equal to a nautical mile, and shew on the scale a space of 500 
 feet. (April, 1875.) 
 
 (6) Subdivide into cables a scale on which 10 inches are 
 equal to half a nautical mile, and shew on the scale the lengths 
 of 100 feet and 1000 feet. (Dec. 1875.) 
 
 (7) Define the term Natural Scale as used on plans. 
 
 (June, 1876.) 
 
 (8) Subdivide into cables a scale on which 6 inches are equal 
 to a mile of latitude : draw also the corresponding scale of a mile 
 of longitude, the latitude being 50 N. (June, 1876.) 
 
 (9) Draw the corresponding scale in the last question if the 
 latitude is 45 N. (Dec. 1876.) 
 
 (10) Subdivide into cables a scale on which 6 '3 inches are 
 equal to a mile of latitude in latitude 35 N., and draw also the 
 corresponding scale of a mile of longitude. (May, 1877.) 
 
 (1 1) Draw a straight line 4 -75 inches long, and divide it into 
 10 equal parts. (Oct. 1877.) 
 
 (12) How would you test the accuracy of scale of the pub- 
 lished plan of a harbour in which your vessel was lying? 
 
 (Beaufort, March, 1880.) 
 
 (13) The scale of a plan being 5 inches to a nautical mile of 
 6080 feet, calculate the length on the scale for the 10 feet pole 
 
24 MARINE SURVEYING. [CHAP. 
 
 (used in putting in coast line) corresponding to an observed angle 
 of 6' subtended between the cross-bars*. (June, 1880.) 
 
 (14) On a plan the scale is 2 inches to a mile. Calculate 
 the natural scale f. Result 3l ^ 80 . 
 
 (15) Find the natural scale when a mile is represented by 
 an inch. Result 6 ^ ao . 
 
 (16) Find the natural scale wlien 4 inches represent a mile. 
 
 Result 15 40 . 
 
 (17) Find the natural scale when 1 foot represents a mile. 
 
 Result -g^Vrr- 
 
 (18) Given the natural scale 10 ^ 60 , find the scale on the 
 chart. 
 
 Let x = number of inches which represent a mile, 
 x 1 5280x12 
 
 X= 
 
 * 5280 x 12 10560' 10560 
 
 . *. 6 inches represent a mile. 
 
 (19) Given the natural scale 101 ^ TTR7 , find the length which 
 1 inch represents. Result 1 inch =16 miles. 
 
 (20) Given the natural scale ^.^^Q, what length represents 
 a mile? Result J inch. 
 
 (21) Given the natural scale 3-5^577 > what length represents 
 a mile? Result 16 inches. 
 
 (22) Given the natural scale o-sV^ what length represents a 
 mile? Result 25 '34 inches. 
 
 18. Note. In the Ordnance Survey of Great Britain the 
 following scales are adopted: 
 
 (1) Towns = -5^, or 12672 inches to the mile. 
 
 (2) Parishes = ^5^, or 25-34 
 
 In this scale, a square inch represents an acre. 
 
 (3) Counties = T1J i F17 , or 6 inches to the mile. 
 
 (4) The Whole Kingdom = $-5~lFO'> or 1 i ncn * *he mile. 
 
 * See Chapter VI. for a description of this Instrument and the method of 
 computing the Scale. 
 
 t The mile in this and the following questions = 5280 feet. 
 
II.] 
 
 THE CONSTRUCTION" AND USE OF SCALES. 
 
 25 
 
 A natural scale of 50 ^ Qo will permit every hill 100 feet high, 
 a pond 100 feet in diameter, woods 200 or 300 feet across, towns, 
 and large isolated buildings to be delineated. A natural scale 
 of T6 ^ OQ will permit of a complete topographical representation 
 of every object in a country (except fences) in exact proportion 
 to the extent it occupies. 
 
 19. (P). DIAGONAL SCALES. 
 
 In the plain scales as already explained we can subdivide an 
 inch into 10 or even 20 equal parts without difficulty, but when 
 it is necessary to take off a length to two decimal places, i. e. to 
 hundredths, we must make use of a diagonal scale. These scales 
 are constructed as follows : 
 
 Draw eleven straight lines parallel to one another and y 1 ^ of 
 an inch apart. Divide the uppermost of these lines into equal 
 parts, these primary divisions being of any required length such 
 as an inch, a half -inch, &c. Through these points draw perpen- 
 diculars cutting all the parallels, and number these primary divi- 
 sions beginning at the extreme left thus 1, 0, 1, 2, 3,... as already 
 explained in the construction of the plain scale. 
 
 Next subdivide the top and bottom lines of the first primary 
 into 10 equal parts, and number these alternate divisions 2, 4, 6, 8 
 from right to left along the bottom line; and number the alternate 
 parallels 2, 4, 6, 8 from the bottom upwards. Then draw lines as in 
 the diagram, viz. from the zero of the bottom line to the first di- 
 vision of the top, from first division of bottom to the second of the 
 top, (fee., and the scale is completed. 
 
 , B 
 
26 MARINE SURVEYING. [CHAP. 
 
 PROOF. In the triangles AX 1 Y 1 and ABC^ by Euclid vi. 4, 
 we have 
 
 *' x ^ = ^ BC *> but ^ = inch ' 
 
 . '. X l Y l =-- T V . -^ inch = T ^ inch. 
 
 Now if a compass be extended along the second parallel from the 
 bottom, from the perpendicular marked 2 to the point T lt it will 
 take off a distance equal to 2 inch -f 01 inch = 2 -01 inch. 
 
 X Y AX 
 
 S = ~ * 
 
 .\ from perpendicular marked 2 to F 2 =2-02 inches. 
 Hence if we wish to take off I'l inch we measure ZG iy 
 
 1-3 
 
 Similarly, we can see that the interval from the perpendicular 
 marked Z to the point marked P will be 1 '75 inches. 
 
 Note. To take from a diagonal scale the number 
 
 574, we must consider the primary divisions as units, 
 
 46 70, tens, 
 
 253-00, hundreds. 
 
 EXERCISES IN THE CONSTRUCTION OF DIAGONAL SCALES. 
 
 (1) Construct a scale of 120 feet to an inch to measure 700 
 feet, and from which single feet may be taken. 
 
 It is evident that we cannot construct a plain scale to answer 
 the purpose, because we could not divide an inch into 120 equal 
 parts. The scale required must therefore be a diagonal scale, 
 and this is constructed as follows : 
 
 As 120 : 1 :: 700 : x. .'. x =5-83 inches. 
 Draw a straight line 5 -83 inches long; divide this into 7 equal 
 parts ; each part will represent 100 feet. Subdivide the first 
 primary into 10 equal parts, therefore each secondary will repre- 
 
II.] THE CONSTRUCTION AND USE OF SCALES. 27 
 
 sent 10 feet. Now draw 10 lines parallel to the first line and ^ 
 inch apart ; draw the diagonal lines as directed above. Then we 
 can take off distances to single feet. 
 
 (2) Construct a scale of 9 miles to 1-3 inches to measure 
 40 miles, and from which distances to furlongs may be taken. 
 
 As 9 : 1-3 :: 40 : x. .'. x = 5'7 f f inches. 
 
 Divide a line 5 '7 7 inches long into four parts, each part will 
 represent 10 miles. Divide the first primary into 10 parts, each 
 will represent a mile. Next draw 8 lines parallel to the first, or 
 scale line : draw the diagonal lines, and it is evident that we can 
 read to furlongs. 
 
 (3) Construct a scale of 10 feet to 1-5 inches, to measure 
 40 feet, and from which distances to inches may be taken. 
 
 As 10 : 1-5 :: 40 : x. .-. a; = 6 inches. 
 
 Divide a line 6 inches long into four parts, each will represent 
 10 feet. Subdivide the first primary into 10 parts, each will re- 
 present a foot. Next draw 12 lines parallel to the scale line. 
 Draw the diagonal lines as usual, and it is evident that we can 
 take off distances to inches. 
 
 (4) Construct a scale of 100 fathoms, with 18 fathoms re- 
 presented by 1 inch, and from which we can measure feet. 
 
 As 18 : 1 :: 100 : x. .-. x = 5'55 inches. 
 
 Divide a line 5-55 into 10 parts, each will represent 10 fathoms. 
 Subdivide the first primary into 10 parts, each will represent 
 1 fathom. Now draw 6 lines parallel to the scale line, and then 
 we can take off single feet. 
 
 (5) Construct a scale of 76 miles to 1*3 inches to read to 
 single miles, and to exhibit 500 miles. 
 
 As 76 : 1-3 :: 500 : x. .-. x=S-55. 
 
 Divide 8 '5 5 inches into 5 parts, each will be equal to 100 miles. 
 Divide the first primary into 10 parts, each will represent 10 miles. 
 Draw 10 lines parallel. 
 
28 MARINE SURVEYING. [CHAP. 
 
 EXERCISES IN THE CONSTRUCTION OF SCALES. 
 
 (6) On a map 18 miles are represented by 37 inches. Com- 
 plete the scale to 30 miles. 
 
 As 18 : 3-7 :: 30 : x. .'. a? = 6-1 7 inches. 
 Divide 6*17 inches into 3 parts, each of which will be equal to 
 10 miles. Subdivide the first primary into 10 parts, each of these 
 will represent a mile. 
 
 (7) On a map 30 miles are represented by 18 -6 inches, draw 
 a scale to exhibit 10 miles, and which will measure furlongs. 
 
 As 30 : 18-6 :: 10 : x. .: x=6'2 inches. 
 Divide 6'2 inches into 10 parts, each will represent a mile; 
 subdivide the first primary into 8 parts, each will represent a fur- 
 long. 
 
 (8) A length of 4 ft. 6 in. is represented by a length of 2-5 
 inches. Complete the scale to 10 feet, such that we may measure 
 inches. 
 
 4-5 : 2-5 :: 10 : x. .'. x= 5-55 inches. 
 
 Divide 5-55 inches into 10 parts, each will represent a foot. 
 Subdivide the first primary into 12 parts, each will represent an 
 inch. 
 
 (9) If 3 inches represent 9 fathoms, complete the scale to 
 20 fathoms, such that single feet may be measured. 
 
 (10) Construct a scale of yards to shew 30 yards, the natural 
 scale being y^y. 
 
 Here 1 inch = 187 inches = 15-58 feet = 5-19 yards. 
 As 5-2 : 1 : : 30 : x. .'. x = 5*78 nearly. 
 
 Divide a line 578 inches long into 3 parts, each will represent 
 10 yards. Subdivide the first primary into 10 parts, each of 
 which will represent a yard. 
 
 (11) Construct a scale of 60 miles to shew miles, the natural 
 scale being u^Vr*- 
 
 Here 1 inch = 97 miles. Length of scale = 6 '18 inches. 
 
II.] THE CONSTRUCTION AND USE OF SCALES. 29 
 
 (12) Construct a scale of 100 feet to shew feet, the natural 
 scale being T {^. 
 
 Here 3 feet = 500. Length of scale = 7'2 inches. 
 
 (13) Draw a scale shewing feet and inches, where the natural 
 scale is y 1 ^. 
 
 (14) Draw a scale shewing yards and feet, the natural scale 
 being j v . 
 
 (15) Draw a scale shewing miles and furlongs, the natural 
 scale being ^siny 
 
 (16) Draw a scale of 160 fathoms to 9 inches to shew feet. 
 
 (17) Draw a scale of 60 yards to 4*57 inches to shew feet. 
 
 (18) Draw a scale of 87 fathoms to 5-1 inches to shew 
 fathoms. 
 
 (19) Draw a diagonal scale of 1 foot to '87 inch, shewing 
 inches. 
 
 (20) Draw a diagonal scale of 2 fathoms to '55 inch, shewing 
 feet. 
 
 (21) On a map 183 miles occupy a length of 14 -3 inches, 
 construct a scale to shew furlongs. 
 
 20. If it is desired to take from an ordinary 6-inch Pro- 
 tractor a distance of 9 -9 2 inches, we must take the distance 9 '9 2 
 from the i-inch diagonal scale, and then "step" it twice. If we 
 required 1676 inches, take the distance from the J-inch scale, and 
 " step " it four times. 
 
 21. (y). VERNIERS. 
 
 The important scales which we are about to describe are 
 named from their inventor, Peter Vernier, who died about 1637. 
 
 The Vernier is a contrivance for subdividing to any extent 
 the smallest division on a graduated scale. 
 
 Verniers are either straight or curved. We have examples 
 of the former in those attached to Barometers, and of the latter 
 in those attached to the Sextant, Theodolite, Station Pointer, &c. 
 The principle is the same whatever be the form. 
 
30 MARINE SURVEYING. [CHAP. 
 
 22. Suppose a scale is divided into half degrees, i. e. into 30' 
 spaces, and let a length of 29 of these spaces be taken from the 
 scale by means of a pair of compasses, and placed on paper. 
 Then if this space thus transferred be divided into 30 equal parts, 
 
 f)Q X 30' 
 
 it is evident that each of these latter parts = -- -=/> = 29'. 
 
 ou 
 
 Now 29' is less than 30' by 1', i.e. by -^ of 30'. 
 
 23. Now let n divisions of the Vernier = (nI) divisions of 
 the limb. 
 
 Let I = value of one division of the limb, 
 and v = value of one division of the Vernier; 
 
 n-l 
 
 n \ n \ n n 
 
 Hence the difference between a limb division and a Vernier 
 
 division is -th of the value of a limb division. 
 n 
 
 Note. This difference is known as the Least Reading* of the 
 Vernier, and expresses the degree of minuteness to which we are 
 enabled to read by its aid. 
 
 24. We may discuss the method of construction in a 
 general manner as follows : 
 
 Let I value of a limb division, 
 and n - number of divisions on the Vernier. 
 
 Let (pn 1) limb divisions be taken as the length of the 
 Vernier, and let this be divided into n equal parts. 
 
 Then we have nv = (pn -1)1; 
 
 pn-l j 
 .: v=~ -.1; 
 n 
 
 * Other terms to express this difference are "Accuracy of Beading," 
 and "Least Count." 
 
II.] THE CONSTRUCTION AND USE OF SCALES. 31 
 
 n 
 
 pnl - pnl + I 
 n 
 
 ^l_ 
 n' 
 
 Now it is evident that the value of p does not affect the value 
 of the Least Reading (which under all circumstances is the n th part 
 of 1), but by taking p as 2 or 3 we gain the advantage of being 
 able to distinguish without difficulty the marks in exact coincidence. 
 
 In the case of the sextant, n is generally 60 and I is 10', 
 hence we must have the Least Reading ^ = 10". From the 
 
 above discussion it is evident that we can take as the length 
 of the Vernier (2 x 60 - 1) limb divisions. This, therefore, explains 
 why 60 Vernier divisions are equal to 119 limb divisions. 
 
 25. It is usual, but by no means necessary, to have the 
 divisions of the Vernier smaller than those of the limb. Let the 
 Vernier divisions be the greater. Then 
 
 n divisions of Vernier = (n + 1) divisions of limb; 
 .'. nv=(n+l)l; 
 
 71+1 . 
 
 .-. v = ---- .1; 
 
 n 
 
 7 n + l , , jfn+l n \ . (n + 1 - n\ 1 . 
 
 .-. v-i= .i-i=i( -- i\ =11- ) = -i. 
 
 71 \ n J \ n J n 
 
 Hence the Least Reading is as before -th of a limb division. 
 
 n 
 
 The only practical difference which this will make is that in 
 ordinary instruments where the graduations proceed from right to 
 left, we shall have to count from left to right, i. e. the zero of the 
 Vernier must be considered as being situated at the extreme left 
 instead of at the extreme right. 
 
32 MARINE SURVEYING. [CHAP. 
 
 EXERCISES. 
 
 (1) If a sextant limb be divided into 10' spaces, and 119 of 
 these be taken for the length of the Vernier, and the Vernier 
 contain 120 parts, find the Least Reading. 
 
 Result -4 of 10' = 5". 
 
 (2) If a limb be divided to 10', and 59 of these be taken 
 for the length of the Vernier, and this space on the Vernier be 
 divided into 60 parts, to what minuteness may be read off 1 ? 
 
 Result ^of 10' = 10". 
 
 (3) Suppose that an inch is divided into 10 equal parts, and 
 a Vernier be constructed equal in length -9 inch, and this Vernier 
 space be divided into 10 parts, find the Least Reading. 
 
 Result 1 of 1 inch = inch. 
 
 (4) If 29 spaces of 10' be divided into 30 parts, find the 
 Least Reading. Result ^ of 10' = 20". 
 
 (5) If 29 spaces of 20' be divided into 30 parts, find the 
 Least Reading. Result = 40". 
 
 (6) If 29 spaces of 30' be divided into 30 parts, find the 
 Least Reading. .Result =1'. 
 
 (7) If 39 spaces of 30' be divided into 40 parts, find the 
 Least Reading. Result = 45". 
 
 (8) If 19 spaces of 1 be divided into 20 parts, find the 
 Least Reading. Result ^ of 1 = 3'. 
 
 26. Given the Least Reading, and the graduations of the limb, 
 to construct the Vernier. 
 
 Here we have evidently to find n, i. e. the number of spaces on 
 the Vernier, and then the number of spaces on the limb which 
 inust be taken, is one less or one more than any multiple of n. 
 
 We know from what has gone before that 
 
 ~ 
 
 I value of a limb division 
 
 l-v " Least Reading 
 
II.] THE CONSTRUCTION AND USE OF SCALES. 33 
 
 Of course I, and I v must be expressed in the same unit, minutes, 
 inches, feet, &c. 
 
 EXERCISES. 
 
 (1) Construct a vernier which shall enable us to read to 10", 
 the limb being divided to 10'. 
 
 1 10 x 60 _ n 
 
 Here n = - 7 = z-r = 60. 
 
 I- v 10 
 
 Hence we must take 59 or 61 divisions of the limb as the 
 length of the vernier, and then this space must be divided into 
 60 parts. 
 
 (2) Construct a vernier which shall enable us to read to *01 
 inch, the inch containing 10 parts, 
 
 w = ^ = 10. 
 
 1T5T7 
 
 Hence we must take -9 or 1-1 inch and divide it into 10 parts. 
 
 (3) Suppose a limb is divided into 20' spaces, construct a 
 vernier which shall enable us to read to minutes. 
 
 Result. 20 spaces on vernier = 19 or 21 on limb. 
 
 (4) Suppose we wish to read to 30" in the last example. 
 
 Result. 40 on vernier = 39 or 41 on limb. 
 
 (5) Suppose we wish to read to 3" on a reflecting circle 
 which is graduated to 5'. 
 
 Result. 100 on vernier 99 or 101 on limb. 
 
 (6) The arc of a sextant is divided to 10'. If 119 of these 
 arc divisions be taken for the length of the vernier; into how 
 many divisions must the vernier be cut to give readings to 5" 
 and 10"? Result. 120 and 60. 
 
 (7) If the arc of a sextant be divided to 10', and 79 of these 
 divisions are taken, and this space is divided on the vernier into 
 40 equal parts ; to what extent may readings be obtained? 
 
 Result. 15". 
 
 (8) Make a vernier to read 15" on a sextant which is divided 
 to 15'. Result. 60 vernier = 59 or 61 limb. 
 
 (9) Make a vernier to read to 20" on a sextant divided to 15'. 
 
 Result. 45 vernier = 44 or 46 limb. 
 R. M. S. 3 
 
34 MAKINE SURVEYING. [CHAP. 
 
 (10) The difference between two arc spaces and one vernier 
 space is 15", and the vernier is divided into 60 equal parts. What 
 is the value of a limb division? Result. 15'. 
 
 (11) If m divisions on limb are equal to 1, if n of these 
 are taken to form a vernier, and the vernier is divided into p 
 
 equal parts, find the Least Reading. (May, '80). 
 
 3600" 
 Result. 
 
 mp 
 
 (12) The limb is divided to 10', and m divisions on the limb 
 are equal to n divisions on the vernier ; find the Least Beading 
 
 ( June >' 80 >- Result. 6 -. 
 
 n 
 
 (13) If the degrees on the limb* are divided into 5m equal 
 parts, find the Least Reading, when the length of the vernier is 
 equal to 2x of the limb divisions. , 360" 
 
 mx 
 
 MMIj 
 
 (14) If a limb division is - degrees, and pn 1 of these 
 
 are taken as the length of the vernier, and this length is divided 
 into pn parts; find the Least Reading. , 3600"wa; 
 
 pny 
 
 27. PRINCIPLE OF READING OFF. 
 
 Let the index of the vernier fall between two of the marks 
 on the limb, between, ag., x and x+ 1, reckoned from the zero 
 of the limb, and let I = value of a division on the limb. 
 
 Therefore whole reading = xl + fraction from x to the index of 
 the vernier. 
 
 To discover the value of this fraction we note that the m ih 
 division of the vernier coincides with a division on the limb : 
 
 therefore the fraction = m(l v) = m.-l = I. 
 
 n n 
 
 Example. If the^vernier be divided into 10 parts occupying 
 a space of 9 divisions on the limb, and if the 4th division of the 
 vernier coincides with a division of the limb, the whole reading 
 
II.] THE CONSTRUCTION AND USE OF SCALES. 35 
 
 Suppose =10', and x is the 74th division from the zero of the 
 limb, then the reading = 74 x 10' + ^ x 10' 
 
 In practice we can always discover the value of xl from the 
 numbers engraved on the limb, and the value of I from the 
 
 numbers on the vernier. 
 
 Suppose then that in an ordinary sextant, the index of the 
 vernier falls between 56 10' and 56 20', and that the 5th stroke 
 to the left of 3 on the vernier coincides with a stroke on the Limb, 
 we have evidently m = 23, therefore the fraction 
 
 .: whole reading^ 56 10' 0" + 3' 50" = 56 13' 50". 
 
 Note. As already explained, the value of this fraction can be 
 easily obtained by inspection ; we find in this last case that the 
 5th stroke to the left of the 3rd long line on the vernier is the 
 coincident division, and each smaller stroke measures 10", .*. the 
 fraction = 3' 50". 
 
 28. Generally, we may reason thus: Let 29 divisions on the 
 limb, each of half a degree, be equal in length to 30 on the ver- 
 nier, then each vernier division is less than a limb division 
 by 3^. of a limb division, i.e. by 1'. Now, if the index of the 
 vernier coincide with the zero of the limb, we shall have the 
 30th stroke on the vernier coincident with the 29th division on 
 the limb. We shall have also the 1st division of the vernier 1' 
 to the right of the 1st division of the limb*, the 2nd division of 
 the vernier 2' to the right of the 2nd division of the limb, &c., 
 and the 30th division of the vernier 30' to the right of the 30th 
 division of the limb, and therefore coinciding with the 29th 
 division of the limb, as it ought. Now, let the vernier be moved 
 to the left until its 1st division (not its Index) coincides with the 
 1st division of the limb, then from what has been said, it is 
 evident that the index of the vernier has moved I/; again, if the 
 
 * The graduation is supposed to proceed from right to left as in a sextant. 
 
 32 
 
36 
 
 MARINE SURVEYING. 
 
 [CHAP. 
 
 2nd vernier division coincide with the 2nd limb division, the 
 index has moved 2', because originally this 2nd vernier division 
 was 2' distant from the 2nd limb division, &c. 
 
 DIAGRAMS SHEWING READINGS ON A SEXTANT. 
 
 No.l. 
 
 3 
 
 5 
 
 
 
 
 , 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Limb 
 
 
 
 
 
 
 
 
 
 
 / Vernier 
 
 
 
 
 I f 
 
 Reading is 35 4l' lo" 
 
 No.2. 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 , 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Limb 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Vernier 
 
 
 
 
 
 
 
 Reading is 
 
 201 l' 20" 
 
 
 No.3. 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Limb 
 
 
 
 
 
 
 
 
 
 
 
 Vernier 
 
 
 
 
 
 . 
 
 The Index is between 320 20' and32o 30'. 
 
 The Reading by the vernier is 90 1CX, .-. the whole angle observed is 32<> 29' 10". 
 
 29. TO DETERMINE WHETHER THE LlMB IS CORRECTLY DIVIDED, 
 
 we make the index of the vernier coincide with some stroke on 
 the limb, then if the last vernier stroke coincide with a stroke of 
 the limb, and this occurs at all parts of the limb, we may assume 
 that the graduations are accurate. 
 
 Suppose that the coincidence occurs at x divisions (reckoned on 
 the vernier) from the last, then the vernier is too long or too 
 
II.] THE CONSTKUCTION AND USE OF SCALES. 37 
 
 short by x times the least reading ; this should be ascertained in 
 
 several places and the mean result taken : let the mean error = e, 
 
 I e 
 
 then (n 1) I nv + e. .: n(l v) = l + e. .'. I v = - + - . 
 
 n n 
 
 Hence a reading in which the fraction is m(l v) now becomes 
 
 m, m 
 
 1+ - e. 
 n n 
 
 The correction is + m - if the vernier is too short by e. and 
 n 
 
 m if the vernier is too long by e, 
 
 e.g. If the limb is divided to 10', and the vernier gives 10" as 
 the least reading (in which case n = 60), and the vernier is 
 
 5" 
 found too short by 5", then the correction is m . ^ : but every 6th 
 
 division on the vernier gives 1', we must therefore add 0'5" for 
 every minute read on the vernier. 
 
 30. II. PROTRACTING SCALES. 
 
 Of these there are several kinds. We shall first give the 
 names of these scales, and the abbreviations by which they are 
 usually denoted on Protractors and Sectors, then intimate the uses 
 to which they are put, and finally enter into more detail as regards 
 the construction and method of using the scales which are of most 
 importance in connection with our subject. 
 
 The chief Protracting Scales are as follows: 
 
 (1) Chords denoted by CHO. and C. 
 
 (2) Rhumbs RHU. 
 
 (3) Latitudes LAT. 
 
 (4) Longitudes LON. 
 
 (5) Sines SIN. 
 
 (6) Secants SEC. 
 
 (7) Tangents TAN. 
 
 (8) Semi- tangents S. T. 
 
 (9) Hours Hou. 
 (10) Polygons POL. 
 
38 MARINE SURVEYING. [CHAP. 
 
 31. USES OF THESE SCALES. 
 
 The scales of " Latitudes " and " Hours " are used in the con- 
 struction of sun-dials. 
 
 The scales of " Sines," " Secants," " Tangents," and " Semi- 
 tangents " are used in the various projections of the sphere. 
 
 The scale or line of "Polygons" is useful in inscribing a 
 regular polygon in a circle, or describing a regular polygon on 
 a given straight line. 
 
 The scale of " Longitudes " shews the number of equatorial 
 miles in a degree of longitude at the parallel of latitude indi- 
 cated by any degree on the scale of chords which is usually placed 
 next to it. E.g. In latitude 60, a ship sails E. 79 miles, find the 
 D. Long, made good. Opposite 60 on the scale of chords is 30 
 on the scale of longitudes. This is the number of equatorial miles 
 in a degree of longitude at that latitude ; 
 
 .-. as 30 : 79 :: 60 : x; 
 .: x 158 miles, the required D. Long. 
 
 The scale of "Chords" enables us to lay off an angle of any 
 exact number of degrees, and to measure to any exact degree an 
 angle already protracted. 
 
 The scale of "Rhumbs" is merely a scale of chords of the 
 points and quarter-points of one quadrant of the compass, and is 
 used in laying down the ship's course expressed in points, and in 
 measuring a course already projected. 
 
 32. CONSTRUCTION AND METHOD OF USING A SCALE OF CHORDS. 
 The object of this scale, as already explained, is to lay down 
 
 an angle from a given point in a given straight line, and to mea- 
 sure an angle already laid down. 
 
 The principle of its construction depends on the fact that the 
 side of a regular hexagon inscribed in a given circle is equal to the 
 radius of the circle (Euclid, B. iv. 15). 
 
 Draw two lines OA, OB at right angles; describe the arc AB 
 with centre and radius OA. 
 
 With centres A and B respectively and radii AO BO, de- 
 scribe circles cutting the arc AB in the points marked 6 and 3 
 
II.] 
 
 THE CONSTRUCTION AND USE OF SCALES. 
 
 39 
 
 respectively. Trisect the arcs -43, 36, QJB, and mark the points 
 1, 2, 3, 8. Draw the chord AB, and then with centre A 
 
 transfer to the chord AB, the distances Al, -42, -43, -48. 
 
 Number these points 10, 20, 80. Then AB thus divided 
 
 will be a scale of chords. 
 
 This scale is only divided to 10 degrees. If the quadrant be 
 divided to degrees, and the distances from A be transferred as 
 explained, we shall have a scale of chords divided to degrees, &c. 
 
 33. USE OF THE SCALE OF CHORDS. 
 
 Construct an angle of 45 by the scale of chords. Draw 
 a straight line AB. From the scale of chords take the length 
 from to 60 and lay this distance off along AB ; then describe an 
 arc BD with A as centre, and this length AB as radius. Next 
 take off from the same scale the length from to 45, and describe 
 the arc BC cutting the arc BD in the point G. Join AC, and the 
 angle BA C -45. 
 
 To measure an angle by the scale of chords. 
 
 Let XA Y be an angle, and it is required to find its value. 
 
 With A as centre and distance AB (0 to 60 from the scale) as 
 radius describe an arc BC, cutting AX in B and AY in C. Take 
 off the length of the chord BC from the paper, and placing one 
 point of the compass at the zero of the scale of chords notice to 
 what division on the scale the other point reaches, suppose 38, 
 then the angle JT^Fis an angle of 38. 
 
40 MARINE SURVEYING. [CHAP. 
 
 It is easy to estimate by the eye to a quarter of a degree. If 
 the angle is greater than 90 degrees ; divide it by 2, and step off 
 the distance twice ; or, lay off 90 first, and then the excess of the 
 angle over 90. 
 
 Examples. Lay off by means of the scale of chords the 
 following angles: 40; 73; 107; 136; 163|, and test the 
 accuracy of your work by the graduated edge of your protractor. 
 
 The scale of "Rhumbs" is merely a scale of chords of the 
 angles of deviation from the meridian denoted by the points and 
 quarter-points of the compass. By its aid we can lay down a 
 ship's course expressed in points, or measure a course already laid 
 down. 
 
 E.g. Lay off a course N. N. E. J E. by the scale of rhumbs. 
 
 Let A be the ship's position on any meridian. With AB 
 (=60 from the scale of chords attached) as radius describe an 
 arc BD. Take off BO = 2f from the scale of rhumbs. Measure 
 off EC on the arc BD as already explained. Join AC, then 
 BAG = 2f points. By the converse process a course can be 
 measured. 
 
 34. THE SECTOR. 
 
 It seems advisable to describe this useful instrument in 
 some detail, because in the hands of one who is well acquainted 
 with its use, it is capable of giving many satisfactory results. 
 
 It is composed of two pieces of ivory, 6 inches long and f inch 
 wide, joined by a hinge at one end ; this hinge allows the two 
 limbs to be opened to any required extent. 
 
 On examining the instrument when opened fully, we find the 
 following scales engraved on it. On the back, the length of a 
 foot is divided into 10 primary parts, and these parts are again 
 subdivided into 10, so that we have a foot divided into 100 equal 
 parts. Close to one edge of the instrument we have 12 inches, 
 and these inches decimally divided, so that we have the foot 
 divided into 120 equal parts. 
 
II.] THE CONSTRUCTION AND USE OF SCALES. 41 
 
 On one face of the instrument we have 3 scales, marked respec- 
 tively T, S, N. These are known as "Gunter's Lines," and are 
 the scales of the logarithms of tangents, sines and numbers. It 
 seems unnecessary to dwell more on these scales here. The 
 student is referred for further information to Heather's Treatise 
 on Drawing and Measuring Instruments*. 
 
 On closing the sector we find on each face of it, pairs of lines 
 radiating from the centre of the circular metal hinge. These 
 lines are known as the " sectoral lines " one line of each pair on 
 either leg of the sector. 
 
 On one side we find three scales marked S, T, T. S is a line 
 of sines graduated to 80; T, T are lines of tangents, one extending 
 from to 45, and the other computed to a less radius, extending 
 from 45 to 75. 
 
 On the other face of the instrument we find the sectoral lines 
 marked L, S, C, Pol. 
 
 L is a line of lines, divided on each leg into 100 equal parts; 
 S is a line of secants divided to 75; is a line of chords divided 
 to 60; Pol. is a line of polygons divided from 4 to 12. 
 
 It will be observed that these various scales are contained 
 within three parallel straight Jines, and that the extremity of the 
 innermost of these lines is marked by a small brass nail. All 
 distances from the centre are to be measured along the line thus 
 distinguished, as it is the only one of the three which passes 
 through the centre. 
 
 DEF. A distance taken along a sectoral line beginning at 
 the centre is called a LATERAL DISTANCE. 
 
 DBF. A distance taken from any point in one leg to the 
 corresponding point in the other leg is called a TRANSVERSE DIS- 
 TANCE. 
 
 35. USE OF THE SECTORAL LINES. 
 
 THE LINE OF LINES (L). 
 
 These lines are divided into 10 equal parts, and each primary 
 
 * p. 98. Vol. 168 in Weale's Rudimentary Series. 
 
42 MARINE SURVEYING. [CHAP. 
 
 part contains 10 equal subdivisions, hence the line is divided into 
 100 equal parts. 
 
 Example (1). Divide a straight line 2 inches long, into 10 
 equal parts. 
 
 Take 2 inches as a lateral distance between the points of the 
 compass, and then open the sector until the transverse distance 10 
 10 is 2 inches, then it follows (by Euclid vi. 4) that the trans- 
 verse distances 9 9, 88, 7 7, &c., are '9, -8, 7, &c. of the two 
 inch base 10 10, and these lengths being laid off' from the same 
 extremity of the line to be divided will give the points of section 
 required. 
 
 Example (2). Find -73 of 2 inches. 
 
 Take the lateral distance 2 inches, open the sector until the 
 transverse distance 100 100 = 2 inches. Then take the transverse 
 distance 73 73, and this will evidently be '73 of 2 inches. 
 
 Example (3). On a map, a distance of 1-6 inches represents 
 80 miles, complete the scale to 100. 
 
 Make a transverse distance 80 80 equal to 1*6 inches: then 
 take the transverse distance 100 100. Divide this into 10 equal 
 parts, and subdivide the first primary into 10 parts. 
 
 Example (4). Find of 1 7 inches. 
 
 Make the transverse distance 9 9 equal to 1 '7 inches ; then 
 take the transverse distance 7 7, which will be the length re- 
 quired. 
 
 Example (5). Find || of -8 of 4 inches. 
 
 Make the transverse distance 10 10 equal to 4 inches, and 
 take the transverse distance 8 8. Then make the transverse 
 distance 8585 equal to this distance 88, and take 7272. 
 
 Example (6). Find ff of | of -87 of of 5-3 inches. 
 
 Example (7). On a map, 73 miles = 4 -2 inches, complete the 
 scale to 100 miles. 
 
 Example (8). Construct a scale of equal parts, on which 7 
 inches will represent 1 \ inches. 
 
II.] THE CONSTRUCTION AND USE OF SCALES. 43 
 
 36. LINE OF CHORDS (C). 
 
 This is divided on each leg 'into 6 parts of 10 degrees each 
 and the degrees are further divided into two parts of 30' each. 
 These lines are graduated by the method already explained (p. 38). 
 
 These double lines of chords on the sector have this advantage 
 over the single scale on the Protractor, that in the former we may 
 use any radius less than the widest transverse distance 60 60 
 on the sector, whereas in the latter the radius must be equal to 
 the length 60 on the scale. 
 
 Example (1). To lay off an angle of 50. 
 
 Draw a straight line AB, and describe an arc EG. Open the 
 sector until the transverse distance 60 60 is equal to AB; then 
 without changing the sector, take off the transverse distance 50 
 50, transfer this distance to ED on arc C, and the angle 
 
 Example (2). To lay off an angle of 107. 
 
 Divide by 2, then lay off chord of 53| found as in Example (1) 
 twice. 
 
 Examples for Exercise. Lay off angles of 
 38; 4930 / ; 56 45'; 79 15'; 117 20'; 145 40'; 169. 
 
 From a circle whose radius is 4 inches, cut off a segment 
 capable of containing an angle of 43, and find the length of the 
 chord. Result. 2-95 inches. 
 
 37. LINE OF POLYGONS (Pol.). 
 
 This scale lies nearest to the inner edges of the instrument. 
 Its divisions are marked 4, 5, 6, ...... 12. It has been so gra- 
 
 duated that the transverse distance 5 5 will be contained as 
 a chord five times in the circle whose radius is the transverse 
 distance 6 6; the transverse distance 7 7, will be contained 
 seven times, &c. 
 
 Example. In a given circle whose radius is 4 inches, inscribe 
 a regular pentagon. 
 
 Make the transverse distance 6 6 equal to 4 inches. Take 
 the transverse distance 5 5, and step it round the circle 5 times. 
 Join the adjacent points, and a regular pentagon will be inscribed. 
 
44 MARINE SURVEYING. [CHAP. 
 
 Note. These constructions require much precision, otherwise 
 the polygons will not " close " exactly. The circle ought to be 
 drawn with a very fine pencil or steel bow, the dots marking the 
 points ought to be placed in the middle of the circumference. 
 
 If it is required to describe a regular pentagon on a given 
 straight line, we must proceed thus : Make the transverse distance 
 equal to the given straight line, and then take off the transverse 
 distance 6 6. From each extremity of the given line as centre 
 describe a circle with this radius thus found. The intersection of 
 these arcs will be the centre of the circle, in which the polygon 
 may be inscribed, whose sides are equal to the given straight 
 line. 
 
 Examples for Exercise. 
 
 (1) On a given straight line 1 J inches long, describe a regular 
 heptagon. 
 
 (2) On a given line 1 '4 inches long, construct a regular pen- 
 tagon. 
 
 (3) On a line 1 '5 inches long, construct a regular decagon. 
 
 38. We will conclude this part of our Chapter 011 Scales, &c. 
 by drawing attention to the instruments known as Marquois Scales. 
 These are named after their Inventor. 
 
 They consist of two box-wood rulers, 12 inches long, and a 
 right-angled triangle of the same material. The triangle has its 
 hypotenuse three times the length of tJie shorter side. 
 
 Near the edges on each face of the rulers will be found a pair 
 of scales, the one nearer to the edge is called the " artificial scale," 
 and that immediately behind it is known as the " natural scale." 
 
 These scales are divided into lengths of 10 units, the artificial 
 scale along its entire length, but only the first division of the 
 natural scale is so divided. In all cases however one division of 
 10 units on the artificial scale in equal to three divisions of 10 units 
 each on the natural scale, and hence three natural units are equal 
 to one artificial unit. 
 
 The zero of each scale is at its middle point. The numbering of 
 
II.] THE CONSTRUCTION AND USE OF SCALES. 45 
 
 the artificial scale proceeds right and left from the zero; and that 
 of the natural scale from the extreme left of the scale to the 
 extreme right. 
 
 Beneath the zero of each scale stands a number (15, 20, 40, 
 50, or 60) which denotes the number of units into which an inch 
 is divided. 
 
 39. USE OF THESE SCALES. 
 
 Let it be required to draw two parallel straight lines at a 
 distance of ^ of an inch from each other. Draw one straight 
 line, and place the longer side of the triangle coincident therewith : 
 then place the ruler with the zero of the scale 60 in exact coinci- 
 dence with the index at the middle point of the hypotenuse of the 
 triangle. Keep the ruler firm and slide the triangle along two 
 artificial units, and draw the line parallel to the former; these 
 lines will be separated by ^j- inch. The triangle must be moved 
 to the left if the line is to be drawn above the former, and to the 
 right if it is to be drawn below. 
 
 Examples for Exercise. 
 
 (1) Draw 2 parallel lines 4 inches long and separated by 
 
 -V inch - 
 
 (2) Draw 3 parallel straight lines 3 inches long, the first two 
 -^ of an inch from each other, and the third ^ from the second. 
 
 (3) Draw 9 parallel straight lines 6 inches long and separated 
 by T 2 -g. inch from each other. 
 
 (4) Draw 5 parallel straight lines 2J inches long and distant 
 from each other $ inch. 
 
 (5) Draw 4 parallel straight lines, the first and last pair dis- 
 tant |- inch, and the other pair f inch. 
 
 40. We shall introduce here two important Problems in 
 Geometrical Drawing, which are of frequent occurrence, and are 
 particularly required in chart drawing. These problems are 
 
 (I.) To erect a perpendicular to a given straight line at any 
 required point. 
 
MARINE SURVEYING. 
 
 [CHAP. 
 
 (II.) To divide a given straight line into a required number 
 of equal parts. 
 
 41. I. To ERECT A PERPENDICULAR TO A GIVEN STRAIGHT 
 LINE. 
 
 Let AB be a straight line, and it is required to erect a perpen- 
 dicular at the point B. 
 
 Assume any point above the line AB (the point ought not 
 to be very near the line), and describe a circle with as centre 
 
 and OB as radius : let the circle cut the line in the point C. Join 
 GO, and produce the line CO to cut the circle in the point D, 
 then if D and B be joined, the line BD is perpendicular to AB. 
 
 By Euclid in. 31, the angle in a semicircle is a right angle, 
 .. the angle CBD = W. 
 
 Note. In practice it is necessary to draw only two small por- 
 tions of the circle CBD, viz. the part which cuts the given line 
 at C and the part about D, so that the line CO produced will 
 intersect it. 
 
 42. The student is strongly recommended to use a pricking- 
 point in all these constructions, as by the use of this instrument 
 
II.] THE CONSTRUCTION AND USE OF SCALES. 47 
 
 the points of intersection are more clearly denned than by any 
 other method. 
 
 It may also be remarked that the longer the radius of the 
 circle is taken, the less error will be produced in the required per- 
 pendicular through any slight mistake in the exact position of the 
 point D. This is so self-evident that nothing further need be 
 said. 
 
 43- II. TO DIVIDE A STRAIGHT LINE INTO A NUMBER OF 
 EQUAL PARTS. 
 
 (i.) BY TRIAL. 
 
 Bisect the whole line, then bisect each part, &c. 
 
 But this method enables us only to divide into n parts, where 
 n is some multiple (2. 4. 8...) of 2. It is tedious, and unless 
 great care is exercised, not accurate. 
 
 (ii.) BY PROPORTIONAL COMPASSES. 
 
 Set the index to the number required on the scale marked 
 " Lines." Take the length of the line to be divided between the 
 two points of the compass ; reverse the instrument, and step ofi 
 the distances. 
 
 In chart drawing this is certainly the most expeditious method 
 if the student is in possession of this instrument, 
 
 (iii.) BY A SCALE OP EQUAL PARTS. 
 
 Let AB be a line to be divided, suppose, into 13 equal parts. 
 
 Draw EC making any angle with AB. 
 
 Take any scale of equal parts (for short lines the scale on the 
 edge of the protractor seems well adapted) and placing its zero at 
 A cause its 13th division to coincide with the line BC. Then 
 
 with a pricking-point carefully make a mark at A 1 A g A 3 , and 
 
 either by means of parallel rulers, or by a Marquois triangle and 
 ruler, draw lines parallel to BC. These lines will divide the lino 
 AB into 13 equal parts. 
 
 (iv.) BY MARQUOIS SCALES. 
 
 Select one of the natural scales, a certain number of units on 
 
 
48 MARINE SURVEYING. 
 
 which will be equal to the line which is to be divided; then 
 placing the triangle with its long side over one end of the line, 
 place the zero of the scale selected coincident with the index, and 
 keeping the ruler quite steady slide the triangle along the arti- 
 ficial scale making the index coincide with the necessary strokes : 
 as each stroke is reached make a dot on the given line with 
 a pricking- point ; e.g. divide a line one inch long into 6 parts. 
 Select the scale 60. Make the index coincide with every 10th 
 stroke on the artificial scale. 
 
CHAPTER III. 
 
 LAYING OFF ANGLES. 
 
 44.' AN angle may be protracted by any of the following 
 methods. 
 
 (1) By semicircular Protractor. 
 
 This instrument is graduated to degrees, and hence by estima- 
 tion we can draw an angle within J or J of a degree. 
 
 (2) By rectangular Protractor. 
 
 This is made of various sizes. The usual size in cases of in- 
 struments is 6 inches long, and is graduated to degrees. We can 
 estimate to within ^ or J of a degree, and in larger sizes to within 
 J of a degree. 
 
 (3) By the. circular Protractor. 
 
 By the aid of the attached verniers we can protract to minutes. 
 
 (4) By tJie scale of chords on the Protractor, or the lines of 
 chords on the Sector. 
 
 By estimation to within J, ^, or \ degree. 
 
 (5) By Construction, we can lay off accurately an angle of any 
 given dimensions as follows. 
 
 Note. This last method is called " PROJECTION BY CHORDS." 
 
 45. Let AOB represent any angle a. Take OA equal to any 
 given radius. Describe the arc AB. Bisect OA in C, and describe 
 the arc CD. Draw the chords AB, CD. Bisect the arc AB, then 
 
 line joining the centre with this point of bisection will 
 the chords in E and F respectively and also the angle a. 
 
 R. M. S. 4 
 
 tne ar 
 the li 
 bisect 
 
50 MARINE SURVEYING. [CHAP. 
 
 O^ c A 
 
 By Euclid vi. 4, ^ = ^ = J, .-. CD=\AB = EB. 
 B t ^?- ' - - ' - 
 
 Hence the practical rule : 
 
 " Multiply the natural sine of half the given angle by twice 
 the length of the given radius, and then with centre and 
 a line (called the effective radius) equal to the given radius, 
 describe a circle. Lay off from a scale of equal parts the chord 
 CD equal to the length thus found; join OD, and the angle DOC 
 will be the required angle." 
 
 46. But how is the natural sine computed ? 
 In two ways: (1) Take out the tabular log sine, and subtract 
 10, then take out the natural number corresponding to the result. 
 e.g. Compute the natural sine of 48 26'. 
 
 Tab. log sine 48 26'= 9-874008 
 
 Subtract 10 
 
 1 874008 
 3960 = 
 48 
 
 7481 
 8 
 
 46 
 20 ... 
 
 3 
 
 17 
 
 30 
 
 5 
 
 
 
 nat. sine 48 26'= -7481835. 
 
III.] LAYING OFF ANGLES. 51 
 
 (2) Since vers (90 + a) = 1 - cos (90 + a) = 1 -f sin a, 
 .-. sin a = vers (90 + a) - 1 
 
 Hence the natural sine is most expeditiously computed as follows : 
 Increase the given angle by 90 and take out the tabular versine 
 of this increased angle. Divide the tabular versine by 1000000 
 (because the tabular versines are a million times the natural ver- 
 sines) by placing a decimal point after the left figure; subtract 1, 
 and the result will be the natural sine of the given angle. 
 
 e.g. Compute the natural sine of 48 26', 
 
 48 26' 
 Add 90 
 
 13826 
 
 Tab. vers 138 26'- 1748184 
 Divided by 1000000= 1-748184' 
 Subtract 1 
 
 Natural sine 48 26'= -748184 
 the same result as above. 
 
 Example. Protract an angle of 32 18' accurately with radius 
 4 inches. 
 
 We evidently require the natural sine of 16 9', 
 
 Chord required = 2-225224 
 
 With radius 4 inches long describe a circle. Lay off the chord 
 of 2-225 inches or 4-45 half-inches. Join the centre and the other 
 extremity of the chord, and the angle between the two radii is 
 32 18' as required. 
 
 16 9' 
 Add 90 
 
 1069 
 
 Natural versine = 1-278153 
 
 .-. natural sine= -278153 
 
 Twice radius = 8 
 
 
 42 
 
52 MARINE SURVEYING. [CHAP. III. 
 
 EXAMPLES FOR EXERCISE. 
 
 (1) Protract an angle of 37 40' accurately with radius equal 
 to 5 inches. (Oct. 1875.) 
 
 (2) Protract an angle of 48 30' with radius 5 inches. 
 
 (Nov. 1876.) 
 
 (3) Protract an angle of 74 40' with radius 5 inches. 
 
 (June, 1877.) 
 
 (4) Erect a perpendicular to a straight line using the scale of 
 chords, and also by means of the compass and ruler only, marking 
 and describing the construction in each case. (May, 1878.) 
 
 (5) Divide a straight line 5-8 inches in length into 10 equal 
 parts; at one end erect a perpendicular, and from the other end 
 draw a line which shall make with the given line an angle of 
 65 30'. (Nov. 1878.) 
 
 (6) By means of the scale of chords, protract the following 
 angles: 39; 65; 118. Upon what theory does the application 
 of this scale to angular measurement depend? (Aug. 1879.) 
 
 (7) Protract an angle of 117 25' accurately with radius of 
 4 '5 inches. 
 

 
 
 CHAPTER IV. 
 
 FIXING POSITIONS BY ANGLES. 
 
 47 WE must in the first place investigate a method of de- 
 scribing a segment of a circle capable of containing an angle of 
 a given size. In Euc. in. 33 we have a geometrical method of 
 construction, but in practice it is much more easily accomplished. 
 
 The following geometrical theorems must be known : 
 
 (1) The angles at the base of an isosceles triangle are equal. 
 
 (2) The three angles of a triangle are together equal to two 
 right angles. 
 
 (3) The angle at the centre of a circle is double the angle 
 at the circumference standing on an equal arc. 
 
 (4) The opposite angles of a quadrilateral inscribed in a circle 
 are together equal to two right angles. 
 
 48. There are three cases to be considered, viz. when the 
 angle is right, acute, or obtuse. 
 
 By Euc. in. 31, the angle in a semicircle is right; the angle 
 in a segment greater than a semicircle is acute; and in a segment 
 less than a semicircle the angle is obtuse. 
 
 _CASE L When the given angle is right. 
 
 On the given line as base describe a semicircle, and any angle 
 in this is 90. 
 
54 MARINE SURVEYING. 
 
 CASE II. When the given angle is acute. 
 
 [CHAP. 
 
 Let BAC = a, and EC the given line; 
 L OBC = L OCB- 
 
 .'. 20BC + BOC = 2 right angles. 
 But BOC=2BAC (Euc. m. 20); 
 
 .-. 20BC + 2a=180; 
 .-. OBC + a -90; 
 
 .-. OBC = 90 -a = OCB. 
 
 Also, a being less than 90, the segment BAG is greater than a 
 semicircle, and therefore the centre of the circle and the angle a 
 are on the same side of BC. 
 
 Hence the rule "Aj^each extremity of the given line BC 
 lay off an angle CBO, BCO equal to the complement of the given 
 angle; then with centre and distance OB describe a circle BAG] 
 BAG shall be a segment capable of containing an angle equal 
 to a." 
 
 CASE III. When the given angle is obtuse. 
 
 Let BAC = a, and let D be any point in the conjugate seg- 
 ment BDC. 
 
IV.] 
 
 FIXING POSITIONS BY ANGLES. 
 A 
 
 55 
 
 a=180-Z> (Euc. in. 22), 
 D=\BOG (in. 20) 
 = (180 -2500), 
 .-. a =180 --|(180 -25(70) 
 = 180-90 + 5(70 
 = 90 + 5(70. 
 .-. 500 = a- 90. 
 
 Also, BAC being obtuse, the segment 5-40 is less than a semi- 
 circle, and therefore the centre of the circle is on the side of 50 
 remote from a. 
 
 Hence the rule : " At 5 and the extremities of the given 
 line lay off angles 050, 500 equal to the excess of a above 90. 
 With centre and distance 05, or 00 describe a circle, and the 
 smaller segment shall be capable of containing an angle equal 
 to a." 
 
 EXAMPLES FOB EXERCISE. 
 
 (1) On a straight line 4'35 inches long describe a segment 
 capable of containing an angle of 39, of 78, and of 122. 
 
 (2) On a straight line 2J inches long describe a segment 
 capable of containing an angle of 18 12', of 37 25', and 129 57'. 
 
 (3) A and 5 are two points 5500 yards apart ; construct on 
 the line AB segments which shall contain angles of 72 30'; 141; 
 155 30'. Scale J inch = 1000 yards. 
 
56 MARINE SURVEYING. [CHAP. 
 
 49- There are two principal methods of fixing a position in 
 the survey of a harbour: the first is known as the "straight line 
 and one angle method," and is chiefly used in running lines of 
 soundings; the second is the well-known " Three-point Problem," 
 with its various modifications. 
 
 50. (1) THE STRAIGHT LINE AND ONE ANGLE METHOD. 
 
 Suppose it is required to "fix" the point D situated on the 
 line passing through two known objects A and B. 
 
 Observe the angle BDP between B and some well-defined 
 object P, which must not be too far away. At any point C in 
 the line AB produced make the angle ACE equal to the angle just 
 observed. Then a line through P, a known point in the survey, 
 parallel to JZC, will cut the line at D. 
 
 If a segment capable of containing an angle equal to the 
 observed angle be described on BP as chord or base, it will 
 intersect the line at the point D. 
 
 Note. In using this method it is well to observe, if possible, 
 an angle on both sides to check the accuracy of the work. 
 
 51. (2)1 THE THREE-POINT PROBLEM^ 
 
 This consists in observing at a station the angle between two 
 points A and B, also between two points B and C, and then de- 
 scribing on the lines AB, BC respectively segments of circles 
 capable of containing angles equal to those just observed. The 
 segments will intersect at the stations required. 
 
 52. There are Six CASES of this problem. 
 
 Let the station at which the angles are observed be denoted 
 by the letter P. 
 
FIXING POSITIONS BY ANGLES. 
 
 57 
 
 CASE U When P is outside the triangle formed by joining the 
 three points A, B, G, and tJie central object is on the side of the line 
 joining the other two remote from P. 
 
 1st method. Let APE = a, CPB = ft. 
 
 By the methods already explained describe on AS, BC, seg- 
 ments containing angles a, ft respectively, these segments will 
 intersect in P, the position of observation. 
 
 When the point P is fixed by the intersection of two circles, 
 the method is called the " TWO CIRCLE METHOD" of projection. 
 
 Zng.metliod. A, , C are the three points. Make the angle 
 
 CAD = observed angle ft, and the angle ACD = observed angle a. 
 
 TL 
 
58 
 
 MARINE SURVEYING. 
 
 [CHAP. 
 
 Describe a circle about the triangle ACD ; join DB, and produce 
 it to cut the circle in P. 
 
 Then, by Euc. in. 21, the angle CPD = angle CAD = ft as 
 observed, and also the angle APD = &nglQ ACD = a as observed. 
 Hence P is the position of observation. 
 
 Note 1. When the points D and B are very close together, 
 the 1st method is advisable. 
 
 Note 2. When this 2nd method is practicable it is known 
 aj3 the " gjjRAiGiiT LINE AND CIRCLE METHOD^ of projection. 
 
 C CASE IlJ When P lies outside the triangle, but the central 
 object observed lies on the same side as P. 
 
 Observe the angle APE - a, and the angle CPB = 
 
 On BC and AB describe' segments which shall contain angles 
 equal to ft and a respectively. These segments will intersect in 
 P the position of observation. 
 
 Note. It is undesirable to have the central object near and 
 the other objects far away, because on describing the segments we 
 shall find that their " cut" is not clear, in fact the two circles may 
 approach th condition of external contact. 
 
 "l 
 
 1 1 LI When the point P is on one of the sides of the tri- 
 
 produced. 
 Observe the angle APE - a. 
 
IV.] 
 
 FIXING POSITIONS BY ANGLES. 
 
 59 
 
 The points A, B, C being known points in the survey, the 
 angle BAG is known. 
 
 P* 
 
 Then, Euc. i. 32, GAB = a + ABP- } 
 
 Hence ABP is known. 
 
 At point B lay off this angle thus found, and BP will intersect 
 CA produced in the position of observation. 
 
 Note. Comparing this case with the straight line and one 
 angle method (p. 56) we see that the latter is really a special 
 caseof the three-point problem. 
 
 ^ASE 1^ WJien the point of observation is on one of the sides 
 of the triangle ABG. 
 
 B 
 
 Observe the angle APB - a. 
 
 At C make the angle ACD equal to a. Then a line drawn 
 through B parallel to GD will intersect AG in the point of obser- 
 vation P. 
 
MARINE SURVEYING. 
 
 [CHAP. 
 
 CASE Vj When the three objects A, , C are in the same 
 straight fine. 
 
 1st method. Observe the angle APE a, and the angle 
 
 Then on AB, CB respectively describe 
 containing angles a, /3 respectively. These 
 in P the position of observation. 
 
 2nd method. Lay off A CD = a, and CA D = 
 
 .D 
 
 capable of 
 will intersect 
 
 About the triangle ADC describe a circle. Join D13, and 
 produce it to intersect the circle in the point P, the position of 
 observation. 
 
 Join PA, PC. Then 
 
 APD = ACD = a, 
 
 and CPD=CA=p. 
 
 Hence P is the position of observation. 
 
FIXING POSITIONS BY ANGLES. 
 
 61 
 
 VI. When the point P is within the triangle formed ly 
 joining the given objects. 
 
 1st method. Observe the angle APE = a, and the angle 
 BPC = ft. On AS, EC respectively, describe segments of circles 
 
 containing angles a and (3 respectively. These segments will 
 intersect in P, the position of observation. 
 
 2nd t method. Observe the angles APE = a, BPC = /?, and 
 ^y (as a check). 
 
 At C make ACD = 180 - a, and at A make CAD = 180 - 
 These lines will intersect in D. 
 
62 MARINE SURVEYING. [CHAP. 
 
 About the triangle ADC describe a circle. Join BD, and this 
 line will intersect the circle in P the position of observation. 
 
 By Euc. in. 21, 
 
 angle APD = angle ACD = 180 - a ; 
 therefore EPA = a, as observed. 
 
 Similarly, CPD = CAD = 180 - ft- 
 
 therefore BPC = ft, as observed. 
 
 Hence P is the position of observation. 
 
 This second method is also an example of the " straight line 
 and one circle method" of projection. Vide Case I., Method 2. 
 
 53. The Indeterminate Case. 
 " Observe the angle APB = a and BPC = ft. 
 
 Now if a is equal to the angle C of the triangle ABC, and if 
 ft is equal to the angle A, it is evident that the angle APC is the 
 
 supplement of ABC, and hence, by Euc. in. 22, the four points 
 A, B, C, P are on a circle, and therefore since P may be at any 
 point of the arc APC its position is indeterminate. 
 This case is also known as being "on the circle." 
 Whenever a + ft is the supplement of the known angle ABC 
 of the triangle ABC, the four points A, B, C, P will lie on the 
 circumference of the circle. 
 
 If however no other points are available for observation, then 
 the compass must be resorted to, and one or more bearings taken. 
 
IV.] FIXING POSITIONS BY ANGLES. 63 
 
 Note. In the last figure we have the two segments coincident, 
 and therefore the centres of the two circles coincide. We see 
 then that if the centres of the two circles are very near each other 
 the segments will not give a clear " cut," and therefore the point 
 of intersection will not be sharply defined. The best " cuts" of 
 course are those which most nearly approach a right angle 
 
 In using the three-point problem some little experience is 
 necessary in the selection of suitable objects. The observed 
 angles should be as near 90 as possible, ought not to differ much 
 from each other; the objects ought to be nearly equidistant, not 
 too far distant, and finally no angle less than 25 or 30 ought 
 to be admitted, unless the central object is very distant, when a 
 small angle between it and one of the others is considered good. 
 
 54. The best relative positions of the points in fixing a 
 position by means of the Three -point Problem are : 
 
 (1) If the three objects are in the same straight line. 
 
 (2) If the middle object and the position of observation are 
 on the same side of the line which joins the other two objects. 
 
 (3) If the position of observation is within the triangle 
 formed by the three objects. 
 
 Under these circumstances the Indeterminate Case, or its 
 approximations can never occur. 
 
 If it be required to fix a point by an observed angle of 70 
 between A and B, and an angle of 85 between B and G, we use 
 the notation "^70 B 85 C." 
 
 Hence "hut 56 tree 68 rock" means that the angle ob- 
 served between the hut and tree was 56 and between the tree and 
 rock was 68. The hut, tree, and rock are therefore the " three 
 points," or objects. 
 
 55. The following methods of fixing a position by angles 
 observed between three points without actually describing the 
 segments which contain those angles seem to be worthy of notice 
 by the reader: 
 
 Let the observed angles be AaBj3C. 
 
 Suppose the circles to have been described (Case II., above) and 
 
MARINE SURVEYING. 
 
 [CHAP. 
 
 the point P has been fixed by their intersection, then we obtain 
 the following analysis. Draw BD, BD' the diameters of the 
 circles. Join CD, AD', and draw PA, PB, and PC. 
 
 C 
 
 By Euc. in. 31, the angle ^C r Z> = 90, and by Euc. in. 21, the 
 angle CDB = CPB = the observed angle /3. Hence the angle 
 CBD = 90 - f3. Similarly we have the angle ABD' = 90 - a. By 
 ill. 31, the angles BPD and BPD' are right angles, and therefore 
 DD f is a straight line (i. 14). 
 
 Example. A bears from B WNW. one mile, C bears from B 
 
 NE b E. 1-5 mile. From P we observe A 43 B 55 C. 
 position of P. Scale 1 inch to a mile. 
 
 Fix the 
 
IV.] 
 
 FIXING POSITIONS BY ANGLES. 
 
 65 
 
 Draw BA, BC according to the given scale. Make CBD = 35 
 (the complement of 55) and draw CD at right angles to CB. 
 Make ABD' = 47 (the complement of 43), and draw AD' at right 
 angles to AB. Join DD' and let fall BP perpendicular to DD'. 
 P will be the position of observation. 
 
 The following seems to be a satisfactory method of protraction 
 without drawing the circles. 
 
 Let the notation be A a B /3 C. 
 
 Let P be fixed by the intersection of the two circles in the 
 
 usual way. Join PA, PB, and PC. Produce AP and CP to meet 
 the circles again in D and D' respectively. 
 
 JoinDB, DC, D'A.D'B. 
 
 Then we have by Euc. HI. 21, the angle CBD=CPD 
 ISO - (a + fi) = supplement of the sum of the observed angles. 
 
 Again CBPD is a quadrilateral inscribed in a circle, therefore 
 by Euc. in. 22, we have the angle BCD = BPA = a. Similarly 
 we get ABD' = 180 - (a + /?) and BAD'= /?. 
 
 "We may therefore fix the position of P as follows. 
 Let the notation be A 80 B 44 C. 
 The supplement of the sum of the observed angles = 56. 
 R. M. S. 5 
 
66 
 
 MARINE SURVEYING. 
 
 [CHAP. 
 
 Make CBD = 5S and BCD = 80 (the angle subtended by the 
 points A, B). Let the lines meet in D. Make ABD' = 56 and 
 A C 
 
 BAD' = H (the angle subtended by the points B, C). Let the 
 lines meet in D'. 
 
 Join CD', AD. These lines will intersect in P the position of 
 observation. 
 
 Note. The lines AD, CD' may be called Lines of Position. 
 
 Sometimes the lines must be produced to meet in the observer's 
 position, as in the following example. 
 
 70' 
 
 Let the notation be A 51 B 37 C. 
 
IV.] 
 
 FIXING POSITIONS BY ANGLES. 
 
 67 
 
 The supplement of the sum of the observed angles = 92. Make 
 = 92 and BCD = 51. Make ABD' = 9'2 and A*Z)' = 37. 
 Join AD, CD' and produce them to meet in P, the observer's 
 position. 
 
 The reader may draw the general figure, and prove that this 
 will be the case. 
 
 Note. This method of fixing a position is known as the 
 " STRAIGHT LINE METHOD." 
 
 The following example of the straight line and circle method 
 is very good. 
 
 Let the notation be Aa Bfi C. 
 
 Lay off ACD = a, and CAD = /3. Describe a circle about the 
 triangle ACD. Join JJ, or B lt or 3 , or B z (the middle object) 
 
 
 with D, and produce the line indefinitely to cut the circle again in 
 P. Then it is evident by Euclid in. 21, that APD = ACD = a, 
 and CPD = CAD = p. And also ACP = ADP, and CAP=CDP. 
 
 Hence we have the following method of fixing the position 
 of P without describing the circle. 
 
 Let A, B, C be the points, and let the notation be A 30 B 62 C. 
 
 52 
 
68 
 
 MARINE SURVEYING. 
 
 [CHAP. 
 
 Make ACD = 3Q, and CAD = 62. Join DB and produce it 
 indefinitely. Make ACP = ADB (-58), and CP will meet DB 
 produced in P, the position of observation. 
 
 Finally, let us take the case where the middle (E) object is at 
 a considerable distance from the other two objects, and where one 
 
IV.] FIXING POSITIONS BY ANGLES. 69 
 
 of the angles observed is very small. In this case the radii are 
 so long that it is impracticable to describe the circle. 
 
 Let the notation be A 10 B 46 n C. 
 
 Make CAD = 46, AGD = 10. 
 
 Join D, and produce it indefinitely. Lay off ACP = ADP, 
 which will be found to be 37, and CP will intersect ED in P, 
 the position of observation. 
 
 Before attempting the general exercises at the end of this 
 Chapter the student will find it useful to project each of the 
 following examples by the One-Circle Method, the Two-Circle 
 Method, and the Straight-Line Method. 
 
 (1) When both the observed angles are less than 90 
 Let the Notation be A 43 B 70 C. 
 
 AB = 4000 feet ; BC = 3600 feet ; CA = 5800 feet. 
 Scale Jin. = 1000 feet. 
 
 (2) When both the observed angles are greater than 90 
 Let the Notation be A 110 B 125 C. 
 
 AB = 520Q feet; BC = 3900 feet; CA = 5500 feet. 
 Scale J in. = 1000 feet. 
 
 (3) When one of the observed angles is greater tJian 90, and 
 the other angle is less than 90 
 
 Let the Notation be A 115 B 74 C. 
 AB = 6200 feet ; BC = 5000 feet ; CA = 7200 feet. 
 Scale Jin. =1000 feet. 
 
 DANGER ANGLE. 
 
 56- DBF. The DANGER ANGLE is the angle subtended at a 
 shoal or other hidden danger by two well-defined permanent 
 objects. 
 
 By Euclid in. 21, all angles in the same segment of a circle 
 are equal ; hence if on observing the angle between these two 
 known permanent objects, the angle is found to be equal to that 
 laid down on the chart as observed from the shoal, it follows that 
 the ship is somewhere on the segment and therefore in danger; 
 if the angle is greater than the chart angle the ship is inside the 
 segment, and therefore probably in danger; but if the angle is 
 
70 
 
 MARINE SURVEYING. 
 
 [CHAP. 
 
 less than the given angle the ship is outside the segment and in a 
 position of safety, and will continue so as long as the observed 
 angle is less than the given chart angle. 
 
 The annexed diagram will make this clear. 
 
 A and B are the permanent objects, and S the position of a 
 shoal. 
 
 The angle ASB = a. is observed, and on AB a segment is de- 
 scribed capable of containing an angle equal to a. This angle a 
 is called the "danger angle," the segment is known as the "danger 
 segment," and it is evident that as long as the observed angle 
 taken on board between A and B is less than the angle a, the 
 position of the ship is outside the danger segment. 
 
 57' The following problems are added as being sometimes of 
 use when instruments of the usual kind are not available. 
 
 Problem I. To find the length of a line accessible only at the 
 ends. 
 
 if- 
 
 Let the distance AB be required; select a point C, and set up 
 a mark. Pace BC, and then pace CD equal to BC and in the same 
 direction. Similarly pace CE = AC. ThenDE = AB. (Euc. I. 4.) 
 
IV.] 
 
 FIXING POSITIONS BY ANGLES. 
 
 71 
 
 Problem II. To find the distance of an inaccessible object by 
 the rhombus method. 
 
 Let P be the point on the opposite side of a river, and let the 
 distance BP be required. 
 
 Take a known length of string and tying a knot at its middle, 
 fix the knot at JS, and fasten one end of the string to a peg at A 
 in the direction of P and the other end to a peg situated in any 
 convenient direction C : then removing the knot to the position 
 Z>, the string being kept taut, mark 2), and place a peg at 
 where OP cuts AD. 
 
 The triangles ODC, CEP are similar, 
 BP DC 
 
 BC.CD 
 
 and these quantities on the right-hand side of the equation are 
 known, hence BP may be found. 
 
 E.g. suppose the side of the rhombus was 100 feet, and the 
 distance DO was 11 feet, then 
 
 BP 
 
 100x100 10000 
 
 -- pr -- = - =909 feet, approximately. 
 
 Problem III. To find the distance of the point where two 
 lines intersect in a river or lake. 
 
72 
 
 MARINE SURVEYING. 
 
 [CHAP. 
 
 Let AF and CO meet in P, it is required to determine the 
 length of the line FP, or OP. 
 
 Join FO and make OL = J OF. Join any other point E in AP 
 with 0, and make OK=\OE. Join Z/L" and produce it to cut 
 00 in //. Then the triangles LOH, FOP are similar (Euc. vi. 6; 
 vi. 4), 
 
 .-. PO :IIO = FO : OL = 2 : 1. 
 
 HO can be measured, and hence OP is found. 
 
 EXAMINATION. 
 
 (1) Mention the two principal methods of fixing a position 
 in marine surveying. 
 
 (2) What conditions ought to be looked for in selecting the 
 objects in the three-point problem? 
 
 (3) You are passing along a coast, and take with a sextant 
 two angles between three known points : how would you fix the 
 ship's position by the station pointer ? 
 
 (4) Specify the propositions in Euclid's Third Book on which 
 the theory of the three-point problem depends. 
 
 (5) Specify the various possible cases of this well-known 
 problem. 
 
 (6) Mention those cases in which no ambiguity is possible. 
 
 (7) Define the danger angle. Draw a figure to illustrate 
 your answer. 
 
 (8) What do you understand by the expression " on the 
 circle"? 
 
IV.] FIXING POSITIONS BY ANGLES. 73 
 
 (9) How would you proceed to find the approximate width 
 of a river whose banks are straight and parallel, if unprovided 
 with the usual instruments 1 
 
 (10) Suppose the banks are irregular how would you proceed? 
 
 (11) Two objects at the ends of a given line are accessible 
 but the line itself cannot be measured directly, how would you 
 set about obtaining the approximate distance ? 
 
 (12) Define and illustrate the methods of projection known as 
 "the two circle method," "the straight line and one angle me- 
 thod," "the straight line and one circle method," and the "straight 
 line method." 
 
 (13) How may the ambiguity implied in question 8 be 
 remedied ? 
 
 EXAMPLES FOR EXERCISE. 
 
 (1) Project the following on the scale of 1 inch = a mile 
 
 bluff 67 jetty 106 hut. 
 
 From bluff, jetty bore N. 7 VV. 2 -9 miles and hut bore N. 3730' E. 
 5 '4 miles. Determine by projection the bearing and distance of 
 the point of observation from the jetty. (Dec. 1874.) 
 
 (2) From mosque, palm bears N. 82 E. (true) 1-9 miles, and a 
 shoal S. 63 E. (true) 1'45 miles. Protract on a scale of 2 inches = a 
 rnile. 
 
 Assuming it to be unsafe, standing in on a northerly course 
 with the shoal ahead, to approach nearer than 3 cables, state the 
 " danger angle " which the shore-points would then subtend. 
 
 (Feb. 1877.) 
 
 (3) From flag, spire bears N. 15E. (true) 1-2 miles, and 
 mound bears S. 36 E. 1-33 miles. Project on a scale of 3 inches -a 
 mile. Project also the following line of soundings, reduction being 
 3 feet. 
 
 At 4 h 10 m , spire 3930', flag 3020', mound; 20 feet; 19ft.; 
 18ft.; 18ft. 
 
 At 4 h 24 m , spire 46" 40', flag 34 50', mound ; 17 ft 
 
 (April, 1877.) 
 
 
74 MARINE SURVEYING. [CHAP. 
 
 (4) From flag, mound bears K 34 E. (true) 6990 feet, and the 
 angle mound (on the left) to tree is 58 30'. 
 
 At mound, tree 66 45' nag, 
 At tree, flag 54 45' mound. 
 
 Project the position of the tree by calculated sides; the scale 
 being 3 inches to a mile of 6082 feet. (May, 1877.) 
 
 (5) From A on jibboom at end of a frigate the angle between 
 B on starboard quarter and battery flagstaff was 80 10', and 
 simultaneously the angle observed at B between A and the flag- 
 staff was 79 5'. The horizontal distance between A and midship 
 gun was 100 feet, and the distance between A and B was 300 feet 
 Protract on a scale of 2 inches to 300 feet, and ascertain the 
 distance of the flagstaff from the gun. (Nov. 1877.) 
 
 (6) Two torpedoes are submerged in line on a bearing E b N. 
 and 200 yards apart. The western torpedo bears N. 600 yards 
 from a lighthouse, the eastern bears NW. 400 yards from a fort. 
 The coast line (of sand) runs straight between the lighthouse and 
 fort. The bearings are true. Protract the figure on a scale of 
 1 inch = 400 yards, and shew the true course a vessel from the 
 westward must steer in order to pass in mid-channel between the 
 torpedoes and the sandy coast (March, 1878.) 
 
 (7) Stations A,B,C are in a line on a NW b N. bearing. From 
 B, D bears K 17 15' E. distant one mile. At D the angle between 
 C and 2? = 51, between C and A =7 7 30'. Protract these posi- 
 tions on a scale of 3 inches = a mile. 
 
 Fix (1). Fix (2). 
 
 B<t>D8Q A, 5QB35'A. 
 
 Place these " fixes " on the plan, and find the distance between C 
 and Fix (2). (April, 1878.) 
 
 (8) At A the angle between B and C = 1 20. At B the angle 
 between C and A = 30. The distance from A to B = 1 -5 miles. 
 At D, the angles C 63 A 65 B were observed. Protract on a scale 
 of 4 inches = a mile of 6050 feet Place D on the plan, and calcu- 
 late the natural scale. (May, 1878.) 
 
IV.] FIXING POSITIONS BY ANGLES. 75 
 
 (9) From fort, cliff bears SSE. 21 miles, and point bears SW. 
 2 miles. The bearings are true. At wreck, point 75 fort 89 cliff. 
 Protract on a scale of 1 inch = a mile, and mark the position of the 
 wreck by two methods. (Sept. 1878.) 
 
 (10) At hill, lighthouse 90 cliff; at cliff, hill 60 lighthouse. 
 The hill bears north from cliff distant 1 '5 miles. 
 
 A ship to the southward of cliff and lighthouse observes a rock 
 awash between her and the shore, and in passing takes the follow- 
 ing bearings : 
 
 Rock < Lighthouse K 14 K; rock < hill N. 50 W. ; rock < 
 cliff N. 80 W. 
 
 Bearings are magnetic throughout, variation = 11 E. Protract 
 the above on the scale of 2 inches = a mile 6050 feet, and place the 
 rock awash correctly on the plan. (Oct. 1878.) 
 
 (11) Three buoys, A,fl,C, moored on the outskirts of a sand- 
 bank are equidistant from each other 3000 yards. A bears from 
 B N. 33 E. (Mag.)} C is to the eastward of A and B. 
 
 At A. At B. 
 
 sandbank 8 30' B. A 7 30' sandbank. 
 
 sandbank 47 30' B. A 51 15' sandbank. 
 
 At C. 
 
 B 8 45'- sandbank. 
 
 B 49 30' * sandbank. 
 
 Variation^ 17 W. Protract on a scale of 1-8 inches = 2000 
 yards. 
 
 Rule the True and Mag. Meridians through B, and dot in the 
 sandbank. (Nov. 1878.) 
 
 (12) From ship, a rock bore K 8 E., and after steering W b S. 
 1 -5 miles it bore N. 63 E. Protract on scale of 3 inches = a mile, 
 and give the distances of the ship from the rock at the time 
 of each observation. Bearings magnetic. Variation = 11 W. 
 
 (March, 1879). 
 
76 MARINE SURVEYING. [CHAP. 
 
 (13) From summit of hill A 300 feet above the sea, the point 
 B bears S. 33 W. distant 1*2 miles, and the theodolite angle be- 
 tween B and 3 rocks awash in line to the eastward was 28 ; the 
 angles of depression being respectively 5 42' 40" ; 4 17' 20"; and 
 3 26' 00. Bearings magnetic. Variation 12E. Protract on a 
 scale of 2-5 inches = a mile of 60G3 feet, and give the magnetic 
 bearings of each rock from the point B. (May, 1879.) 
 
 (14) The top of a lighthouse 180 feet above the sea, subtends 
 a vertical angle to the water line, from A of 9, bearing N. 43 E. 
 from B of 11, bearing N. 53 W. A buoy to the southward lies 
 equidistant from A and B 300 yards. If the light be obscured 
 within a horizontal distance of 380 yards, how far from this limit 
 is the position of the buoy ? Var. = 3 E. Bearings magnetic. 
 Protract on a scale of 0-9 in. = 100 yards. (Aug. 1879.) 
 
 (15) A bears from B W b S. distant mile. 
 D B S b E. J 
 
 C ,, A S W. ,, y^y ,, 
 
 Var. = 10 W. Bearings magnetic. 
 
 At E., A 95 B 58 30' D. Protract on a scale of 2-2 inches 
 a mile. Fix the position of E, and from it give the true bear- 
 ings of A, C, D. Rule the true and magnetic meridians through A. 
 
 (Sept 1879.) 
 (16) From ship, magnetic bearings of 
 
 A = N. 31 E. distant 1 -1 mile. 
 =S73E. 1-4 
 (7 = S29\V. 0-8 
 = N24W. 0-7 
 
 Var. = 5 E. Protract on a scale of 1 -2 inches = a mile of 6060 
 feet, and give the two bearings and distances of these points from 
 each other. (Dec. 1879.) 
 
 (17) When sailing along a coast, a headland was observed to 
 bear NE. JN. (true); having run E. JN. (true) 14 miles, the 
 headland bore W b N. J N. (true) : required its distance from the 
 ship at each observation, and protract its position on a scale of 
 0*5 inches = a nautical mile. (May, 1880.) 
 
IV.] FIXING POSITIONS BY ANGLES. 77 
 
 (18) From house, the following true bearings and distances 
 were observed and measured: Tree N. 58 E. 2J miles; church 
 K 5S. 4J miles; light vessel R 22 S. 3 miles. Protract the 
 above on a scale of 1 *5 inches = a mile, and plot the position of the 
 following soundings : 
 
 House 108 Tree 97 Church (2 feet, mud). 
 
 House 112 Church < Lighthouse (3^ fathoms, coral). 
 
 House 41 Light vessel 44 Church (7 fms., ooze). 
 
 (Aug. 1880.) 
 
 (19) From church the following true bearings and distances 
 were observed and measured; Tree N. 75 E. 5 miles; Windmill 
 S. 81 R 6 miles; Lightship S. 68 R 3 miles. Protract on a 
 scale of 1 ! inches = a mile, and mark the position of the following 
 shoals : 
 
 Church 137 30' Tree 77 30' Windmill (31 fms., hard sand). 
 Church 61 30' Light ship 38 50' Windmill (5 fms., mud 
 and ooze). 
 
 (20) From church the following true bearings and distances 
 were observed and measured : Lighthouse N. 83 E. 7 \ miles ; 
 Tree S. 82 R 2f miles. 
 
 From a vessel the following angles between these objects were 
 observed. Church 46 Tree 48 Lighthouse. 
 
 How must the vessel steer so as to pass one mile south of the 
 lighthouse 1 Protract on a scale of 1 inch = a mile. (Nov. 1880.) 
 
 (21) From Lighthouse, Windmill bears S. 65 E. (true) 1J 
 miles, and Church bears S. 85 E. (true) 3 miles. 
 
 A vessel southward of windmill observes a rock awash between 
 her and the shore, and takes the following bearings (true) while 
 passing ; 
 
 Rock < Lighthouse N. 50 W. ; Rock < Windmill N 29 W. ; 
 
 Rock </> Church N. 40 E. 
 
 Protract the position of the Rock on a scale of 2 inches = a mile. 
 
 (Apr. 1881.) 
 
 (22) From Nubble, Fort bears N. 15 W (true) 3900 feet, 
 and the angle 
 
78 MARINE SURVEYING. [CHAP. 
 
 Fort (on the left) to Quoin is 70 00'. 
 At Fort Quoin 62 30' Nubble. 
 
 At Quoin Nubble 47 30' Fort 
 
 Project the position of Quoin by calculated sides. Scale 6 in. 
 = a mile of 6080 feet. (May 1876.) 
 
 (23) From Coastguard, Mound bore N. 77 \V. (true) 045 of a 
 mile, and Mill bore N. 88 E. 0*56 of a mile ; the following stations 
 were taken to fix a shoal on which the sea breaks too heavily to 
 risk the boat near. Mound 60 V C. G. 47 0' mill 
 
 * 
 Centre of shoal 
 
 Mound 55-0 C. G. 57 30' mill. 
 
 # 
 
 Centre of shoal 
 
 Project the positions on a scale of 5 inches = a mile ; giving the 
 centre of the shoal. (Sept 1876.) 
 
 (24) The mean of a set of observations taken at Pile A , 
 Lat. 520'N., was as follows. Zero, Steeple 360 0' Mag. Bear- 
 ing S. 32 W. 5 h 16 m p.m. AT. 85-00 <t>. Sun's decl. 20 30' N. 
 
 Required true bearing of Steeple from Pile, and the variation. 
 
 (June, 1877.) 
 
 (25) The mean of a set of observations taken at Theodolite 
 A A', in Lat 53 14' N., was as follows : 
 
 Sun's decl. = 1 6 38' N. Semi = 1 5' 50". 
 Zero-Camp A 360 0' Mag. N. 45 E. 
 
 6 h O m a.m. AT. 37 25' 
 
 Required true bearing of the Camp from X y and the Variation. 
 
 (April, 1876.) 
 
 (26) The mean of a set of observations taken at Cairn A , 
 in Lat 50 N., was as follows : Sun's decl. = 12 30' N. 
 
 Zero, Wedge A 360 0' Mag. N. 30 0' E 
 6 h O m a.m. A. T. 40 3' <t> 
 

 IV.] FIXING POSITIONS BY ANGLES. 79 
 
 Required the true bearing of the wedge A from the Cairn A , 
 and the Variation. (Oct. 1876.) 
 
 (27) At Theodolite AJT, in. Lat. 54 3' K, the following 
 observations were made : 
 
 Zero, Lighthouse (Southward of X) 360 0'. 
 
 First set with sun's lower and right limbs. 
 
 3 h 58 m p. m. 35 11' alt 104 19'. 
 
 35 6' 104 26'. 
 
 35 1' 104 33'. 
 
 Second set with sun's upper and left limbs. 
 
 4 h O m p. m. 35 30' alt 104 4'. 
 
 35 26' 104 11'. 
 
 35 21' 104 18'. 
 
 Sun'sdecl. = 1936'6'N. 
 Required the true bearing of the Lighthouse from X. (Aug. 1875.) 
 
 ANSWERS. 
 
 Note. The distances in these results are for the most part 
 given in inches, as actually taken off from the paper. The student 
 can then ascertain the correctness of his work without difficulty. 
 
 (1) S. 71 E. 2-4 miles. 
 
 (2) 63. 
 
 (3) Length of line of soundings = '83 in. = '28 mile. 
 
 1st sounding S. 46 J W. from spire. 
 
 (4) Mound to tree 3-65 in. flag to tree 3'92in. 
 
 (5) 5 '5 5 inches. 
 
 (6) N. 75 E. 
 
 (7) Fix 1 to =2'15in.; fix 2 to <? = l'90in. 
 
 (8) Nat. Scale = T ^ TO -. = 5-65 in. ; DC=5'75'm. 
 
 (9) Wreck bears from point S. 15 J W. 2 miles. 
 
 ( 1 0) Distance from lighthouse = 3. 85 in. ; from hill = 5 -72 in. 
 
 (11) If a:, y, z be the points of the bank nearest to A,J3,C 
 respectively, then xij = 1 '37 in. ; ys = 1 '32 ; zx = 1 '47 in. 
 
80 MARINE SURVEYING. 
 
 (12) From 1st position 143 in. = -715 miles. 
 
 2nd 5 -13 in. = 2 -565,, . 
 
 (13) The rocks are 3000, 4000, 5000 feet respectively from A, 
 and bear N. 42 E.; N. 52 J K ; N. 65 E. 
 
 (14) 1 7 yards. Distance from A to B = 4 -2 9 inches. 
 
 (15) EB = 1 -23 miles ; A = N. 64 W. ; C = S. 48 J W. ; D due 
 East. 
 
 (16) From A, JS bears S. 24 E. 1-87'. 
 
 B, G S. 85JW.2-12. 
 
 C,D N. 9JE. 1-61. 
 
 D,A N. 7of E.0-60. 
 
 (17) Distance from 1st position = 2'73 miles; from 2nd posi- 
 tion = 5 *5 miles. 
 
 (18) No. 1, sounding from house = 3 '0 in.; no. 2, from house 
 = 3'5 in.; no. 3, from light vessel = 2*03; from church 3*86 in. 
 
 (19) No. 1, shoal from church = 4 -25 in. ; from tree = 1 -60. 
 No. 2, shoal from light vessel =1*95 in.; from wind- 
 mill - 4-75. 
 
 (20) Ship from church 3-47 in. Course = S. 57J E. 
 
 (21) Rock to lighthouse = 3-42 in. 
 Rock to windmill = 2-16 in. 
 
 (22) Quoin to nubble = 4-8 in. 
 
 fort =5-0 in. 
 
 (23) Centre of shoal S. 1 3 W. 1 -44 in. 
 
 (24) Steeple bears S. 9 39' W. 
 
 T. B. of sun N. 85 21' W. Yar. = 12 21' W. 
 
 (25) T. B. of sun N. 79 52' E. Camp bears from A'. 
 N 42 1 1' E. Variation 2 49' W. 
 
 (26) N. 4150'E. Var. = ll50'E. 
 
 (27) S. 27 16' 50" E. 
 
CHAPTER V. 
 
 CHARTS AND CHART DRAWING. 
 
 58. A MAP is a representation on a plane of a large portion 
 of the earth's surface. 
 
 A CHART is a representation on a plane of a portion of the 
 earth's surface of large or small extent with special reference to 
 the requirements of the seaman. 
 
 A PLAN is a representation of a very limited extent of the 
 earth's surface, drawn on a plane without reference to the latitude 
 or longitude of the positions ; it has respect only to the relative 
 position and distances of points, for which purpose a scale of dis- 
 tances is annexed. Thus we have plans of harbours, anchorages, 
 
 Note. Plans are frequently placed in the corners of charts for 
 more special use, such plans will give more detailed information 
 could be conveyed on the chart itself. 
 
 * 
 
 
 
 - 
 
 59. DEF. A representation of a figure on a surface formed 
 by the intersection of that surface by lines drawn from the 
 . bsi -rver's eye to every visible point of the figure is called a 
 'rejection. 
 
 The plane on which this representation takes place is known 
 as the " Primitive Plane" or simply the " Primitive" or the 
 " Plane of Projection." 
 
 Projections are of two kinds, Natural and Artificial. 
 A Natural Projection is simply a perspective delineation of 
 any object on the Primitive. 
 
 An Artificial Projection is not a perspective representation. 
 A Projection of the Sphere is thus a representation of the 
 rface of the sphere on a plane. 
 
 R. M. S. 6 
 
82 MARINE SURVEYING. [CHAP. 
 
 The Natural Projections of the sphere are representations of 
 the surface of the sphere on a plane as seen from a certain position 
 of the observer. 
 
 60. The principal Natural Projections are the ORTHOGRAPHIC, 
 the STEREOGRAPHIC, and the CENTRAL or GNOMOXIC. 
 
 DBF. The Orthographic can be explained as follows : Suppose 
 the eye placed at an infinite distance, then all lines drawn from 
 the eye to the sphere will be parallel; accordingly, if perpendicu- 
 lars be let fall from every point on the surface of a hemisphere 
 on its diametral plane as Primitive, the representation will be 
 such as it would appear to the eye at an infinite distance. In this 
 case, however, only the central portions are correctly delineated, 
 whereas the portions near the edges are unduly crowded and 
 distorted. This Projection is therefore only of use for the repre- 
 sentation of small portions of the Earth's surface. 
 
 61. DBF. In the Stereographic Projection the eye must be 
 conceived as situated at one extremity of a diameter of the sphere, 
 and as viewing the concave surface of the sphere through the 
 diametral plane as a Primitive. E.g. If the eye is placed at the 
 South Pole and views the Northern hemisphere through the plane 
 of the Equator, then every point on the surface will appear de- 
 lineated on the plane of the Equator. 
 
 In the former Projection the points as they recede from the 
 centre are crowded together, in this Projection on the contrary 
 their projected dimensions seem to be somewhat enlarged. 
 
 Two properties of this Stereographic Projection make it impor- 
 tant : (1) All circles on the hemisphere are represented by circles 
 on the Projection, and (2) all small triangles on the surface of the 
 sphere are represented by triangles similar to them in the projec- 
 tion. This valuable property insures a general similarity of 
 appearance in the map to the reality, and enables a single hemi- 
 sphere to be represented in a single map. 
 
 62. DBF. In the Central, or Gnomonic Projection, the eye 
 is supposed to be situated at the centre of the sphere, and the 
 
V.] CHARTS AND CHART DRAWING. 83 
 
 Primitive is a tangent plane to the sphere. Since the plane of 
 every great circle passes through the eye, in this Projection every 
 great circle will be represented by a straight line on the Primitive, 
 and hence charts on this Projection are well adapted for Great 
 Circle Sailing *. The circumpolar regions are usually represented 
 on this Projection. It is evident that a complete hemisphere 
 cannot be thus represented, because the great circle which bounds 
 it is on a level with the eye, and therefore parallel to the Primitive 
 plane. The Maps in ordinary Atlases are constructed by this 
 Projection. 
 
 63. It may also be observed that if in the Gnomonic Pro- 
 jection the Primitive plane touch the Earth at any point between 
 the Equator and the Pole, the meridians will be projected as 
 straight lines which will meet at the projection of the Pole. The 
 angle contained between two meridians is called their Convergence/. 
 Now from the definition of a True Bearing it follows also that 
 if two points differ in Latitude the T. B. of the point in the 
 lower Latitude exceeds the reverse T. B. by the value of the 
 coin rrgericy of the meridians of the two places. 
 
 64. MERCATOR'S PROJECTION. 
 
 This is known as an Artificial Projection, and is named after 
 the inventor, who was born in Flanders early in the 1 6th century. 
 The mathematical principles of its construction seem to have been 
 first enunciated by Edward Wright, of Cambridge, at the end of 
 the same century. 
 
 Suppose the eye situated at the centre of the earth, regarded 
 as a hollow crystal sphere. On the surface we may conceive the 
 equator, parallels, and meridians to be painted black, and a hollow 
 cylinder, of infinite length, painted white on the inside, and touch- 
 ing the earth at the equator. Now it is evident that the black lines 
 on the sphere will be projected on to the white background, and 
 also that the intervals between the parallels of latitude will in- 
 crease as the eye ranges from the equator towards the pole: at 
 the pole the line of sight coincides with the axis of the earth, and 
 
 * Godfray's "Chart to facilitate the Practice of Great Circle Sailing" is 
 an example ; the Primitive being a tangent plane at the South Pole. 
 
 62 
 
84 MARINE SURVEYING. [CHAP. 
 
 only meets the circumscribing cylinder at infinity. In other words 
 the meridional parts of 90 = infinity. 
 
 Yet another method may be suggested of getting a clear con- 
 ception of a Mercator's chart. Suppose the earth as before to be 
 circumscribed by a cylinder painted white, only now conceive the 
 earth to be made of some material capable of infinite expansion. 
 If all the features on the surface are recently painted, then as the 
 expansion can take place only in the cylinder, we can understand 
 that the features coming into contact with the cylinder will leave 
 traces in paint on the white ground, but the parts about the pole 
 will be at an enormous distance from the equator, and the pole 
 itself at infinity. 
 
 Thus then in Mercator's Projection the various places on the 
 earth are correctly represented as to their form, but the scale 
 varies greatly. In the polar regions, for example, the scale is 
 extravagantly enlarged ; we may have a small island in a high 
 latitude, seemingly as large as India. 
 
 If the cylinder on which the paint from the expanding earth 
 has been transferred be opened or unrolled, we shall have repre- 
 sented a Mercator's chart of the world. 
 
 65. PLANE CHART. 
 
 Since a small portion of the surface of the globe may be con- 
 sidered as a plane without sensible error, it is evident that charts 
 of coasts, harbours, islands, anchorages, rivers, <fec., may be con- 
 structed without reference to their special latitude and longitude. 
 The object of such a chart is to obtain bearings and distances, 
 and these can be easily obtained from such plane charts. 
 
 In places near the equator the degrees of latitude and longi- 
 tude may be considered as equal*, and therefore a plane chart of 
 regions of low latitude may be constructed of considerable extent. 
 If a plane chart were constructed for regions of large extent in 
 high latitudes, then no direction is correctly shewn except due 
 N. and S., E. and W., and no true distance except on the me- ' 
 ridian. 
 
 * To realise this, let the student consult his Table of Meridional parts 
 for the first 10 or 15 degrees of latitude, and he will notice how little they 
 increase. 
 
V.] CHARTS AND CHART DRAWING. 85 
 
 These plane charts of great extent are no longer constructed, 
 Mercator's charts having completely superseded them. 
 
 66. In drawing plans of harbours, &c., it may be noted that 
 the Nautical Mile used is the Minute of Latitude in. As the earth 
 is an oblate spheroid the degree of latitude increases as we advance 
 from the equator towards the pole, and hence its sixtieth part, or 
 the nautical mile, also increases*. 
 
 67. The two great advantages of Mercator's chart are, 
 (1) All Rhumb lines between two places on the chart are straight 
 lines, and (2) the angle at which this line cuts the parallel me- 
 ridians is the Course. 
 
 The great disadvantage is that although all features retain 
 their correct form, yet the scale for different latitudes varies 
 much, and in very high latitudes it becomes so much exaggerated, 
 that a portion of land within' the Arctic circle may appear two or 
 three times the size of an equal portion within the Tropics. 
 
 METHOD OF CONSTRUCTING A MERCATOR'S CHART. 
 
 68. It seems advisable to notice the following point in the 
 lace for the sake of the younger students. The meridional 
 
 parts for 49= 3382-08 and for 50 = 347447, the difference being 
 92-39. Now what does this difference mean? We may explain 
 it thus. On the principle of construction employed in Mercator's 
 projection, the degrees of latitude increase as we go from the 
 Equator towards the Pole. A degree of the Equator = 60 miles; 
 and the above difference of 92-39 miles signifies that a degree of 
 latitude extending from 49 to 50 contains 92 -39 equatorial miles. 
 
 69. The construction of the chart may be divided into the 
 following steps : 
 
 (1) To find the meridional parts. 
 
 (2) To compute the lengths of the meridional difference of 
 latitude on the given scale. 
 
 * The length of a degree of latitude increases as follows: 
 In lat. 12 the length is 362956 feet. 
 
 52 364951 
 
 ...66.., ...365744-.... 
 
86 MARINE SURVEYING. [CHAP. 
 
 (3) To draw the longitude line, and to divide it according to 
 the given scale. 
 
 (4) To erect the perpendiculars at the ends of this longitude 
 line, and to test the accuracy of the work. 
 
 (5) To set off on the meridian thus drawn, the lengths as 
 found in (2). 
 
 (6) To draw the upper parallel. 
 
 (7) To rule in the meridians and parallels. 
 
 These steps constitute the construction of the frame of the 
 chart. 
 
 (8) To draw outside the frame another fine line about j 1 ^ or 
 y 1 ^ inch distant from, and parallel to it. 
 
 (9) To divide the degrees each into six equal parts of 10'. 
 
 (10) To lay down the latitude and longitude from. 
 
 (11) To lay down the true courses and distances. 
 
 (12) To find the latitude and longitude in. 
 
 (13) To find the bearing and distances to or from any known 
 position. 
 
 (14) To finish off the chart according to individual taste. 
 
 70. "We shall illustrate the principal parts of the above in 
 discussing the following chart. 
 
 Construct a Mercator's chart on a scale of 1-38 inches to a 
 degree of longitude, extending from latitude 48 N. to 51N., and 
 from 20 W. to 26 W. 
 
 To compute the mer. parts and the lengths of the mer. diff. 
 latitudes according to the scale, we arrange the work thus : 
 
 51 = 3568-81 50 = 3474-47 49 = 3382-08 
 
 50=3474-47 4'9= 3382-08 48=3291-53 
 
 diff. = 94-34 diff. = 92-39 diff. = 90-55 
 
 v 
 
 We know from above what these differences imply. } 
 
 Now, by hypothesis, 60 miles at the Equator are represented 
 by 1'3S inches; we have therefore to find wha number of inches 
 will represent 94-34 miles, 92-39 miles, and 90*55 miles re- 
 spectively. 
 
V.] CHARTS AND CHART DRAWING. 87 
 
 We have evidently to work three Simple Proportion sums. 
 Let us take the first. 
 
 As 60 miles : 1*38 inches :: 94-34 miles : x inches; 
 = 94-34 x 1-38; 
 
 94-34x 1-38 . 
 .: x= -^- -inches; 
 
 94-34 x 1-38 
 
 or af\ - x 2 = value of x in ^ inches; 
 oO 
 
 94-34 x 1-38 
 
 . -. - = value of x in ^ inches. 
 
 60 
 
 Hence the well-known Rule: "Multiply the difference of the 
 meridional parts by the scale, and divide the result by 30; the 
 result will be the length to be taken from the J-iiich diagonal 
 scale on the Protractor." 
 
 We may notice that the length for 94-34 is greater than that 
 for 92-39, and this in turn is greater than that for 90-55, and this 
 is as it should be, the degrees increasing as we get farther from 
 the Equator. 
 
 Before we proceed further, the accuracy of these different 
 computations ought to be tested as follows : 
 
 Find the mer. diff. between the two extreme latitudes and 
 compute its length on the scale by the usual method. This length 
 ought evidently to be equal to the sum of the separate parts 
 already found. 
 
 We thus know the size of our chart, and can therefore place 
 it suitably on the drawing paper. 
 
 71. ERECTION OF THE PERPENDICULARS. 
 
 We recommend the method of drawing a perpendicular ex- 
 plained in the Second Chapter. The pencil in the compasses ought 
 to have a very fine point, and the points of intersection ought 
 to be marked with the pricking-point. So much depends on 
 these lines being carefully drawn that the student ought to 
 bestow every attention on this part of the work. 
 
88 MARINE SURVEYING. [CHAP. 
 
 The accuracy of the construction may be tested as follows. 
 Open the compass to any convenient length (about 5 or 6 inches), 
 and measure off equal lengths along the perpendiculars from the 
 longitude line; then the distance between these points thus found 
 ought to be equal to the length of the longitude line. 
 
 The fine line drawn outside the frame is best drawn by the 
 aid of the Marquois Scales, otherwise the eye is the sole judge. 
 
 72. The division of the degrees into equal parts seems on 
 the whole to be most expeditiously performed by means of the 
 proportional compasses. For other methods of dividing a line, 
 the student is referred to the Chapter on Scales. We recommend 
 the graduation to be carried all round the chart, and not merely 
 along the longitude line and one meridian. 
 
 In ruling all lines care must be taken that the ruling pen 
 (which ought to be of the very best make, and ought to be guarded 
 from all rough treatment) is well supplied with ink*, that the 
 ruler is held firmly in its place, that the pen is drawn evenly 
 along the edge, and that the motion proceed from the shoulder 
 joint and not from the wrist. Before any line is drawn on the 
 chart with the pen, the pen ought to be tested on paper of the 
 same kind as that on which the chart is being constructed. 
 
 73. To LAY DOWN THE LATITUDE AND LONGITUDE OF A POINT 
 
 ON A CHART. 
 
 Place a ruler along the latitude of the place found from the 
 graduated meridians, and draw a very fine pencil line about 
 J inch in length under the meridian of the place as judged by 
 the eye. Then lay down the ruler to coincide with the longitude 
 as marked on the top and bottom parallels, and draw a similar 
 line, or make a point on the line drawn before; this will mark 
 the position of the place required. 
 
 The converse process will enable us to find the latitude and 
 longitude of a place on the chart. 
 
 * We can confidently recommend Eowney's Fluid Indian Ink, sold in 
 small bottles at one shilling. 
 
V.] CHARTS AND CHART DRAWING. 89 
 
 To LAY DOWN A COURSE ON A CHART. 
 
 Place the index of the Protractor over the point from which 
 the ship sailed, so that the lower edge may be exactly parallel to 
 the nearest parallel of latitude, and then prick off the course 
 from the graduated edge. 
 
 To LAY DOWN A DISTANCE ON A CHART. 
 
 We know the direction it will take; estimate with the eye 
 the approximate middle point of the line, and take the distance 
 from the graduated meridian, one half the distance above and 
 the other half below the approximate latitude of this middle 
 point. Lay the edge of a ruler through the points, marking the 
 place and the course, and lay off the distance now found. 
 
 The converse process will enable us to find the course and 
 distance between two points*. 
 
 74. We have noticed that the two following points present 
 difficulties to some of the younger students, we shall therefore 
 notice them before concluding this part of our subject. 
 
 Suppose the chart is to extend from 53 N. to 55 40' K, and 
 from 29 10' W. to 32 40' W, Where ought the parallel to be 
 drawn 1 
 
 We would compute the meridional parts for the following 
 pairs : 
 
 55 40' 55 00' 54' 00' 
 
 55 00' 54 00' 53 00' 
 
 The space between 55 and 55 40' is only 40 miles. The 
 parallels might be drawn at 54 and 55, but the space between 
 55 and 55 40' must be divided into only four equal parts, the 
 spaces between the other parallels being divided into six parts. 
 
 The meridians may be drawn at every | degree, e.g. 29 10', 
 29 40', 30 10', &c., or at every even degree, the first two meridians 
 being in this case only 10 miles apart, and the last two being 
 10 miles apart. In such cases however it seems best to space 
 
 * The student is advised to read Raper's remarks on the " Properties of 
 Certain Projections " in his Navigation, p. 127. 
 
90 MARINE SURVEYING. [CHAP. 
 
 the chart so that the appearance of the meridians and parallels 
 may be most pleasing to the eye. The principle is not in any 
 way interfered with. 
 
 75. Suppose the scale is given for a middle latitude, how are 
 we to proceed ? 
 
 E.g. In the December Chart (1875) the scale was 2-13 inches 
 to 1 of middle latitude, the chart to extend from 49 to 52 N. 
 and longitude 9 to 16 W. 
 
 The scale for 1* longitude may be found in two Avays. 
 
 First method. Find the middle latitude of the chart; in this 
 case 50 30'. This is the middle of the degree 50 51. Find 
 the meridional difference of latitude, thus : 
 
 mer. parts for 51-^ 3568-81 
 50, = 3474-47 
 
 diffi. = 94-34 
 
 Now we know from what has gone before that if 60 miles 
 at the equator be represented by x inches, we can compute what 
 number of inches will represent 94*34 miles: and conversely, if 
 we have given the number of inches which represent 94*34 miles, 
 we can find what number of inches will represent 60 miles at the 
 equator, and this will be the required scale. 
 
 Hence as 94'34 : 2-13 :: 6t> : x; 
 
 .'. x- 1*35 inches, the required scale. 
 
 Second method. The degrees of latitude in Mercator's chart 
 are increased in the ratio of the secant oft/te latitude, and hence 
 
 scale at the equator = scale at middle latitude 
 
 x cosine of the middle latitude ; 
 .-. 35 = 2-13 x cos 50 30', 
 
 0-328380 
 9-803510 
 
 loga; = 0-131890 
 .'. x - 1-35, as before. 
 
V.] CHARTS AND CHART DRAWING. 91 
 
 EXAMPLES FOR EXERCISE. 
 
 (1) Mercator's chart. Scale 1 -38 inches - 1 longitude. 
 From lat. 48 K to 51 K and long. 20 W. to 26 W. 
 
 Courses E. JK Distances 148'. Variation 30 W. 
 NNE. 73'. 
 
 SSE. 123'. 
 
 55'. 
 
 Ship sailed from lat. 48 7' N. and long. 25 36' W., and made 
 J-point leeway during the last course, the wind being east. 
 
 Lay down the true courses, and find the latitude and longitude 
 in. (Feb. 1875.) 
 
 (2) Mercator's chart, 1 -25 in. = 1 long. 
 
 From lat. 59 to 63 S. and long. 178 E. to 177 W. 
 Ship sailed from lat. 62 50' S. long. 178 50' E. as follows: 
 NE. 100'; W b S. 80'; K 125'; E b N. f N. 25'; SW. JW. 95'; 
 E.JS. 135'. 
 
 Variation If points E. 
 
 Lay down the true courses, and find lat. and long. in. 
 
 (Aug. 1873.) 
 
 (3) Mercator's chart, 1-33 in. = 1 long. 
 From lat. 64 to 67 30' N. and long. to 5 W. 
 Ship sailed from 64 30' N. and 3 30' W. 
 
 NE.JE. 145'; NW b W. 66'; SSW.JW. 53'; SE b E.|E. 83'; 
 SW. 75'; NW. 60'. 
 
 Variation 2J points W. 
 
 Lay down the true courses, and find the lat. and long. in. 
 
 (Feb. 1874.) 
 
 (4) Mercator's chart, 5 in. = 1 long. 
 
 To extend from lat. 49 N. to 50 k and from long. 8 W. to 
 930'W. 
 
92 MARINE SURVEYING. [CHAP. 
 
 Ship sailed from 49 50' K and 9 20' W. 
 
 Comp. co. Dev. Dist. 
 
 S b W.JW. 5W. 30'. 
 
 SE b E.JE. 8E. 35'. Var. = 23" W. 
 
 N b E.JE. 3E. 35'. 
 
 Find the latitude and longitude in. (March, 1881.) 
 
 (5) Mercator's chart, 1 '65 in. = 1 long. 
 
 To extend from lat. 54 to 56 30' N. and from long. 14 to 
 20 W. 
 
 Ship sailed from 5427'K 1449' W., 
 
 N.f E. 87'; SW b W.103'; N b W. 86'; K 24'. 
 
 Variation = 28 W. 
 
 Observations then placed the ship in lat. 5612'N. and long. 
 1813'W. 
 
 Lay down the true courses, and find the direction and distaiuv 
 set by the current. (March, 1S75.) 
 
 (6) Mercator's chart, 3-46 in. = 1 long. 
 
 To extend from lat. 57 to 58* K and from long. 2 to 5 E. 
 
 Ship sailed from 57 7' K, 2 1 1' E. Corr. mag. E. 54'. 
 
 Variation = 20 W. 
 
 Observation then placed her in 57 33' N., 3 27' E. Find set 
 and drift of the current. Speed of ship and the rate of cunvnt 
 remaining the same, find the ship's course to reach lat. 57 53' N. 
 and long. 434'E. (Oct. 1875; May 1876; Feb. 1877.) 
 
 (7) Mercator's chart, 3 -6 in. = 1 long. 
 
 To extend from lat 38 to 3930 / S. and from long. 77 to 80 E. 
 
 Ship left lat. 39 12' S. and long. 77 9' E. 
 
 E.|S.82';E b N.JK 28'. 
 
 Variation = 26 W. 
 
 An island in lat. 3813'S., long. 79 32' E., was then seen 
 bearing NEJN., and the course was altered to E b SJS.; after 
 making good 17*5 miles in that direction, the island bore XNW. 
 The courses and bearings are cor. mag. Required the position of 
 the ship, and the direction and distance set by the current. 
 
 (Aug. 1875.) 
 
V.] CHARTS AND CHART DRAWING. 93 
 
 (8) Mercator's chart, 1 -38 inches = 1 long. 
 
 To extend from lat. 60 to 6225'N. and from long. 19 to 
 25 W. 
 
 Find by Projection the cor. mag. bearing and distance of a 
 port whose lat. is 6223'K and long. 1957'W. from a ship in 
 lat 60 11' 1ST. and long. 24 36' W. 
 
 Variation = 22 W. 
 
 State also the course that must be steered to reach the port, 
 allowing for a set of N. 34 W. (true) 32 miles, and a deviation of 
 5 E. (April, 1875.) 
 
 (9) Mercator's chart, 1-65 inches = 1 long. 
 
 To extend from lat. 58 to 60 N. and from long. to 6 E. 
 
 Ship sailed from lat. 58 9' N. and long. 22' E. 
 
 E.JN.75'; ENE. 61'; E b N. 25'. 
 
 Variation = 22 W. 
 
 A lighthouse in lat. 5917'N. and long. 454'E. then bore 
 SSEJE., and the angle to a point N.23E. (true) 20 miles from 
 the lighthouse was 68. The courses and bearings are cor. mag. 
 Required the direction and distance set by the current. 
 
 (Sept. 1875.) 
 
 (10) Mercator's chart, 2 -13 in. = 1 of middle latitude. 
 
 To extend from lat. 49 to 52 N. and from long. 9 to 16 W. 
 
 Ship sailed from lat. 4925'N. and long. 1536'W. 
 
 E.JN. 125'; E b N.JN. 98'. 
 
 Variation = 1 5 W. 
 
 A point of land in lat. 51 40' K, long. 10 4' W., then bore NR; 
 after running 30 miles farther on the same course E b N. JK, the 
 point bore N b WJW.; the courses and bearings are cor. mag. 
 Required the position of the ship, and set and drift of the current, 
 if any. (Dec. 1875.) 
 
 (11) Mercator's chart, 2-15 in. = 1 of middle latitude. 
 To extend from lat. 49 to 52 N. and long. 8 to 15 W. 
 Ship sailed from lat. 49 20' N., long. 14 15' W. 
 
 E. 100'; E b K65'; E b N.N. 40'. 
 Variation - 25 W. 
 
94 MARINE SURVEYING. [CHAP. 
 
 A point of land in lat. 5140'N., long. 9 24' W., then bore 
 N b E. (mag.), and the angle to a peak E.JS. (true) 23 miles from 
 the point was 60. Find the position of the ship and the current 
 experienced. (Sept. 1876.) 
 
 (12) Mercator's chart, 2-22 in. = 1 middle latitude. 
 
 To extend from lat. 46 to 49 N., and long. 4 to 11 W. 
 
 Ship sailed from lat. 46 30' N., long. 10 10' W. 
 
 K 120'; ENE. 106'. 
 
 Variation = 2 1W. 
 
 Ushant Light in lat. 4828'N., long. 5 8' W., then bore E b S., 
 and after running 32 miles farther on the same course ENE., the 
 light was lost sight of bearing SW b S. Courses and bearings are 
 magnetic. Required the position of the ship, and the current 
 experienced. (June, 1877.) 
 
 (13) Mercator's chart, 2-6 in. = 1 long. 
 
 To extend from lat. 64 to 65 N. and long. 16 to 20 "NV. 
 
 Ship sailed from lat 64 5' N., long. 16 11' W. 
 
 NW b W. 50'; W b N.JN. 30'; W. 12'.. 
 
 Variation = 5 K 
 
 An islet in lat. 6458'N., long. 1937'W., was then seen 
 bearing 4 points on the starboard bow, and after running 9 miles 
 farther on the same course (west) the islet was exactly abeam. 
 Find the position of the ship and the distance from the islet. 
 
 (April, 1876.) 
 
 (Note. A similar chart in every respect appeared in March, 
 1877.) 
 
 (14) Mercator's chart, 2 -6 in. = 1 long. 
 
 To extend from lat. 49 to 50 30' K, long. 3 to 6 W. 
 
 Ship left lat. 49 1 2' N. , long. 5 40' W. 
 
 ENE. 40'; E b N. JN. 41'. 
 
 Variation = 21 W. 
 
 The Eddystone in lat. 5011'N., long. 4 1 6' W., then bore 
 NNW.f W. (mag.), and the angle to the Start which bears N. 85 E. 
 (true) 25 miles from the Eddystone was 112. Required the posi- 
 
V.] CHARTS AND CHART DRAWING. 95 
 
 tion of the ship, and the comp. course to be steered to pass 7 miles 
 off the Start, allowing for a deviation of 5*E. (Oct. 1876.) 
 
 (Note. Same chart given in April, 1879, and very slightly 
 altered in Oct. 1880.) 
 
 (15) Mercator's chart, 34 in. = 1 long. 
 
 To extend from lat. 36 to 37 30' S. and long. 70 to 73 E. 
 
 Ship left lat. 3720 / S., long. 70 15' E. 
 
 E.JN. 60'; E.JS.42'. 
 
 Variation = 23 W. 
 
 An island in lat. 36 18' S., long. 7230'E., then bore NE b E. JE, 
 and after running 12-5 miles farther on the same course E. J S. , 
 the island bore NNE. J E. The courses and bearings are magnetic. 
 Required the position of the ship, and the current experienced. 
 
 (Nov. 1876.) 
 
 ( 1 6) Mercator's chart, 1 -75 in. = 1 long. 
 
 To extend from lat. 36 to 40 S. and long. 20 to 24 W. 
 
 Ship left lat. 36 20' S., long. 20 30' W. 
 
 SW. 60'; W b S. 100'; SE b S. 90'; S h W. 80'. 
 
 Variation = 8 E. 
 
 Required the latitude and longitude arrived at. In this 
 rxample, an island is in lat. 39 20' S. and long. 21 50' W., and 
 a second island bears from the first W. JN. (niag.) distant 20 miles. 
 A sunken rock lies 10 miles to the westward of the second island 
 with the two islands exactly in line. Place the islands and the 
 sunken rock upon the chart, and state how near to the rock the 
 ship passed. (Oct. 1877, and again in March, 1879.) 
 
 (1 7) Mercator's chart, 1 ; 85 in. = 1 long. 
 
 To extend from lat. 2 30' N. to 2 30' S. and long. to 4 W. 
 A ship sailed from lat. 145'S., long. 030'W., as follows: 
 N. 49 W. 80'; N. 33 E. 85'. Variation = 1 7 W. 
 A second ship sailed from lat. 150'N., long. 3 15' W., as follows : 
 S. 13" E. 90' ; S. 48 E. 95'. Variation = 17 W. 
 Protract the true courses, giving the latitude and longitude of 
 the position at which the tracks cross, and the true bearing and 
 distance of this position from the starting-point of each ship. 
 
 (Feb. 1878.) 
 
96 MARINE SURVEYING. [CHAP. 
 
 (18) Plane chart. (Latitude and longitude equal) 
 
 1-75 inches = 1 degree. 
 
 To extend from lat. 1 to 6 S. and long. 19 to 24 W. 
 
 A steamer in lat. 1 50' S., long. 20 45' W., receives informa- 
 tion that 4 days previously a dismasted ship had been seen in 
 lat. 530'S., long. 21 20' W. Variation = 5 W. , current = W b N. 
 15 miles a day. Mark upon the chart the position of the dis- 
 masted ship at the time the steamer received this information, 
 and state what magnetic course the steamer ought to steer to 
 reach her; state also in what lat. and long, the steamer may 
 expect to find her, if she start at once for the vicinity, and steam 
 at the rate of 10 knots an hour. (March, 1878.) 
 
 (19) Mercator's chart. Scale 1 in. = 5 longitude. 
 
 To extend from lat. 20 to 50 S. and from long. 10 to 60 W. 
 Meridians and parallels to be ruled at intervals of 10 degrees. 
 
 Ship sailed from lat. 34 53' S., long. 56 10' W., direct to 
 lat. 37 8' S., long. 12 10' W. 
 
 Where the track crosses the meridian 20 "W. , the variation = 1 5 W. 
 
 30 W. =11W. 
 
 50 W. = 9E. 
 
 Give the magnetic course at each point. (April, 1878.) 
 
 (20) Mercator's chart. Scale 3 in. = a mile of longitude. 
 
 To extend from lat. 40 10' to 40 13' N. and long. 101 10' to 
 101 13' W. 
 
 A rock having less than 6 feet of water on it at low water 
 average Spring Tides is situated in lat. 40 IT 10" N. and long. 
 101 1 1' 50" W. Variation = 1 1 W. 
 
 Place the rock on the chart, and give its magnetic bearings 
 and distances from each of the inside corners of the margin of the 
 chart. (Oct. 1879.) 
 
 (21) Mercator's chart. Scale 1-1 inch = 1 longitude. 
 
 To extend from lat. 42 20' S. to 48 00' S. and long. 80 00' E. 
 to 86 30' E. 
 
 Taking departure from a position with lighthouse in lat. 
 
V.] CHARTS AND CHART DRAWING. 97 
 
 Variation = 8 W. 
 
 43 00' S. and longitude 82 45' E. bearing west (true) distant 25 
 miles, a vessel sails as follows : 
 
 Comp. co. Dev. Dist. 
 
 SW. JW. 7W. 118') 
 
 E b S. 12 E. 155' 1 Variation = 21 E. 
 
 N. JW. 3W. 158') 
 
 How does the lighthouse bear on the completion of the last 
 course? (Sept. 1880.) 
 
 (22) Mercator's chart Scale 272 in. = 1 longitude, 
 To extend from lat. 61 to 62 N. and long. 18 to 22E. 
 A ship sailed from lat 61 10' N., long. 18 13' R 
 Soundings at the end of each distance : 
 
 Mag. co. Dist. Fms. 
 NNE. 40 30 
 
 m. 
 
 SE b S. 52 47 
 
 N b E. 36 105 
 
 SRJS. 48 70 
 
 m. 
 
 Observations then placed the ship in lat. 6130'N., long. 
 21 12' E. Allowing for current, and supposing the speed uniform, 
 correctly place the soundings on the chart. (Nov. 1875.) 
 
 (23) Mercator's chart. Scale 3-04 in. = 1 longitude. 
 
 To extend from lat. 39 to 40 40' N., long. 124 to 127 W. 
 A ship sailed from lat 39 10' N. and long. 126 48' W. as 
 follows. 
 
 Soundings tried for at the end of each distance. 
 Mag. co. Dist. Fms. 
 
 North 68' ^o^i 
 
 SE. i E. 49 
 
 NNE. 52 ^ Vanatum-1/R 
 
 SE b S. 59 ~ J 
 
 Observations placed the ship in lat. 39 24' N. and long. 
 124 20' W. 
 
 Allowing for the current experienced, and assuming a uniform 
 speed, correctly place the soundings. (Aug. 1876.) 
 
 R. M. S. 7 
 
Mag. co. 
 
 Dist. 
 
 Fms. 
 
 North 
 
 30' 
 
 40 i 
 
 SW. i- W. 
 
 35' 
 
 45 
 
 N.JW. 
 
 44' 
 
 47 
 
 SW. 
 
 40' 
 
 47m.J 
 
 98 MARINE SURVEYING. [CHAP. 
 
 (24) Mercator's chart. Scale 3'6 in. = 1 longitude. 
 To extend from lat. 56 to 57 N. and long. to 3 R 
 A ship sailed from lat. 56 15' N. and long. 2 40' E. as 
 follows. 
 
 Soundings at end of each distance : 
 
 V.mtum-20-W. 
 
 Observations then placed the ship in lat. 56 12' N. and long. 
 24' E. 
 
 Allowing for the current, and assuming a uniform speed, place 
 the soundings in their correct positions. (Dec. 1876.) 
 
 (25) Mercator's chart. Scale 3*7 in. = 1 longitude. 
 To extend from lat. 57 N. to 58 N. and long. 1 to 4 E. 
 A ship sailed from lat. 57 12' N. and longitude 1 25' E. as 
 follows. 
 
 Soundings at the end of each distance : 
 Mag. co. Dist. 
 
 N b E. | E. 30' 
 
 SSE. i E. 47 ~~ . . 
 
 NNE. 37 :: Variation = 20* W. 
 
 SSE. 38 
 
 Observations then placed the ship in lat. 57 26' N. and long. 
 3 35' E. 
 
 Assuming a uniform speed, and allowing for the current, 
 place the soundings in their correct positions. (April, 1877.) 
 
 ANSWERS. 
 
 1. Lat. 50 20' N., Long. 20 46' W. 
 2. Lat. 61 28' N., Long. 177 14' W. 
 3. Lat. 65 3' N., Long. 3 42' W. 
 4. Lat. 49 48' N., Long. 8 25' W. 
 
V.] CHARTS AND CHART DRAWING. 99 
 
 5. Lat. 55 46' N., Long. 19 0' W. Current NE. (true) 37'. 
 
 6. Lat. 57 26' N. , Long. 3 44' E. Current N W b W. (true) 1 1 | '. 
 Mag. co. E. J S. 
 
 7. Lat. DR. 38 29' S., Long. DR. 79 16' E.) Current E. } N. 
 Obs. 3827' S., Obs. 79 27' E.J (true) 9'. 
 
 8. True co. NE.; Mag. co. ENE. 186'; Comp. co. E b N.f N. 
 
 9. Lat. DR. 5941'N., Long. DR. 4 36' E. ^Current S. J W. 
 Obs. 5926'N., Obs. 433'E.J (true) 16'. 
 
 10. Lat. DR. 51 6'N., Long. DR. 1022'W.|Current NW b W. 
 Obs.5110'N., Obs. 10 32' W.) (true) 8'. 
 
 11. Lat. DR. 51 9'N., Long. DR. 944'W.|Current NE. JE. 
 Obs. 5120'N., Obs. 917'WJ (true) 21'. 
 
 12. Lat.DR.4825'N., Long. DR. 528'W.|Current W b N.f N. 
 Obs.4826'N., Obs. &WW.f (true) 5'. 
 
 13. Lat. DR. 6449'N., Long. DR. 1915' W.^Within 8J miles of 
 Obs. 6448'N., Obs. 1918'W.J Islet 
 
 14. Lat. DR. 50 3' N., Long. DR. 4 3'W.)_, 
 
 Obs. 50 4'N., Obs. 40 2 , w jComp.co.E.fS. 
 
 15. Lat. DR. 3637'S., Long. DR. 7212'E. )^ 
 
 Ob, 36035' S.; Obs. 72014' E . Current NE. 4'. 
 
 16. Lat DR. 3938'S., Long. 2321'W. 
 
 2nd Id. is in Lat 39 16' S., Long. 22 16' W. Rock in 
 39 14' S., 22 28' W. Ship passed within 30' of the Rock. 
 
 17. Lat. of Point 12' N., Long. 1 39' W. 
 
 From 1st Ship's point of departure NNW. f W. 124'; from 
 
 2nd Ship's do. SE b S. f E. 125'. 
 
 18. When intelligence is received, wreck is in 524'S., 22020' W. 
 It will be found in 5 22' S., 2237' W. Mag. co. SSW. f W. 
 19. In Long. 50 W. Mag. co. E. J N.; in 30 W. E b S.iS.; in 
 20 W., E b S. JS. 
 
 20. From NE. corner S. 60 W. 2'-4; NW. corner S. 26 E. l'-9 ; 
 SW. comer N. 35 E. 2-7; SE. corner N. 26 W. 3'-02. 
 
 72 
 
100 MARINE SURVEYING. 
 
 21. Lt. Ho. bears W. J N. (true) 79'. Lat. in 43 10' S., Long, 
 in 84 33' E. 
 
 22. Lat. DR. 61 15' N., Long. 21 7'E.; S b E. 15'. 
 
 1st Sounding in Lat. 61 53' N.; 2nd in Lat. 61 19' N.; 
 3rd in 61 57' N. ; 4th in 61 30' N. 
 
 23. Lat. DR. 39 18' N., Long. 124 42' W. 
 
 1st S. in 4017'N.; 2nd in 3938'N.; 3rd in 4018'N.; 
 4th in 39 24' N. 
 
 24. Lat. DR. 56 17' K, Long. 39' E. 
 
 1st S. in 56 24' N. ; 2nd in 56 1 1' N.; 3rd in 56 49' N. ; 4th 
 in5612'N. 
 
 25. Lat. DR. 57 19' N., Long. DR. 3 22' E. 
 
 1st S. in 5744'N.; 2nd in 5711'N.; 3rd in 57 51'N.; 
 4th in 57 26' N. 
 
 Note. The above Results have been found by actually draw- 
 ing the Charts, and it is hoped that they are well within the 
 limits of error which may naturally be looked for in work of 
 the. kind. 
 
CHAPTER VI. 
 
 INSTRUMENTS AND OBSERVING. 
 
 76. GUNTER'S CHAIN. 
 
 This is divided into 100 links, each link being 7 '92 inches 
 long ; hence the whole chain is 66 feet, or 22 yards, or 4 perches. 
 .-. 1 square chain = 16 perches = T ^ acre. 
 
 The great advantage of this chain is the facility with which 
 areas may be computed from the measured lengths of the sides. 
 
 77. THE ORDINARY SURVEYING CHAIN is 100 feet in length. 
 It is composed of steel bars and chain links, each bar and the three 
 chain links being equal to one foot. The chain is divided into 
 segments of 10 feet from the middle towards each handle. These 
 segments are marked by small pieces of brass so divided at their 
 edges that the number of 10 feet from each handle can be detected 
 at a glance. Swivel joints at the handles and at the middle pre- 
 vent the chain getting twisted. 
 
 In measuring with a chain every precaution must be taken, as 
 the results are liable to many errors arising from (1) the chain 
 itself, and (2) the method of using it. 
 
 If the chain is stretched too tight the links give, and therefore 
 the measured length is shorter than the real length; if not stretched 
 tight enough the measured length will be too short. If the chain 
 is a new one a comparison ought to be made at the close of the 
 day's work ; if however it has been in use for some time and found 
 
102 MARINE SURVEYING. [CHAP. 
 
 trustworthy, a comparison once every three or four days will 
 suffice. 
 
 The method of using the chain in measuring a line is fully 
 described in the chapter on Base Lines. 
 
 78. SEXTANT. 
 
 There is no necessity to occupy space in the actual description 
 of this instrument. Hadley invented this invaluable aid to navi- 
 gation about the year 1731 ; the principle of the sextant however 
 appears to have been known to Sir Isaac Newton. We shall 
 merely notice the ADJUSTMENTS in their proper order. 
 
 79. I. The index glass must be perpendicular to the plane 
 of the instrument. 
 
 II. The horizon glass must be perpendicular to the plane of 
 the instrument. 
 
 III. When the horizon glass is parallel to the index glass the 
 index error ought to be zero. 
 
 IV. The line of collimation must be parallel to the plane of 
 the instrument. 
 
 80. TO TEST THESE ADJUSTMENTS. 
 
 I. Is the index glass perpendicular to tJie plane of the 
 instrument ? 
 
 Place the moveable radius called the index bar about the 
 middle of the arc, or somewhat nearer to the commencement of 
 the graduated limb. Then look obliquely into the index glass. 
 If the limb and its reflected image appear in the same line the 
 adjustment is correct ; if not, it must be looked to. If the image 
 is too high, the glass leans forward; if too low, the glass leans 
 backward. 
 
 Note. This correction is carefully made before the instru- 
 ment leaves the maker's hands, and is not easily put out of 
 adjustment. 
 
 II. Is tlie horizon glass perpendicular? No. I. adjustment 
 is first made, because the index glass being perpendicular, if in 
 
VI.] INSTRUMENTS AND OBSERVING. 103 
 
 any position it is parallel to the horizon glass, the latter must also 
 be perpendicular to the plane of the instrument. We test the 
 accuracy of No. II. in two ways. 
 
 (a) By making the reflected image of the sun pass over the 
 sun looked at through the telescope. If the two bodies pass on 
 one side of each other the adjustment must be seen to. 
 
 (/?) By the sea horizon. Set the index about zero, and look 
 directly at the horizon through the unsilvered part of the glass. 
 If the reflected image is in line with this horizon and remains so 
 while the sextant is turned through a very large angle, the 
 adjustment is correct ; if not, it must be seen to. 
 
 A little practice will enable most persons to remedy this 
 source of error by means of the screws behind the horizon glass. 
 
 III. Is the index-error zero ? It is not necessary that it should 
 be so, but it is necessary that its amount should be known, and 
 allowed for in all computations. When the adjustments are made 
 in the above order, the position of the index of the vernier with 
 respect to the zero of the limb may be determined in three differ- 
 ent ways, (a) By making the reflected image of a star or sun to 
 coincide exactly with the object looked at directly through the 
 telescope; (/?) by measuring the sun's diameter on and oft* the 
 arc ; (y) by the sea horizon. 
 
 IV. Is the line of collimation correct ? Make a contact at 
 one of the wires of two stars about 100 or 110 apart. Throw 
 the stars on the other wire : if the contact is still maintained the 
 adjustment is correct; if not, it may be adjusted by the following 
 practical rule*. 
 
 " Open the screws of the collar of the telescope on that side on 
 which the separation of the stars takes place." 
 
 Note. If the separation takes place on the wire farthest from 
 the plane of the instrument the object end of the telescope droops 
 towards the instrument, and if the separation takes place at the 
 nearer wire, then the eye-piece droops. 
 
 * Communicated by Staff Commander V. Johnson, E.N. 
 
10 i MARINE SURVEYING. [CHAP. 
 
 If this adjustment is not correctly made, then the middle 
 of the wires is no longer the true line of sight, and the contact 
 observed there will give angles which are too great*. 
 
 To obtain a good observation with the sextant the following 
 points ought to be regarded : (1) the images ought to be sharply 
 defined by careful focussing, (2) when observing the sun, neutral 
 tints ought be used as less fatiguing to the eye, (3) in using the 
 artificial horizon the suns ought to be of the same brightness. 
 
 Note. In observing the angle between the moon and a star, 
 make the edge of the moon to bisect the star's light ; and in the 
 case of the moon and planet, bring the edge of the moon's disc to 
 the estimated centre of the planet. 
 
 81. THEODOLITE. 
 
 The following description of this instrument will be clearly 
 understood if the student keeps the annexed diagram of a Theo- 
 dolite before him for reference. 
 
 The three principal parts are, the Levelling Plates, the Hori- 
 zontal or Azimuth Limb, and the Vertical Arc. We shall notice 
 these in detail. 
 
 I. THE LEVELLING PLATES. These (marked A and B) are 
 held together by a ball and socket joint, and can be set firm 
 and parallel to each other by means of four milled-headed screws 
 (S) : these screws turn in sockets fixed to the lower plate (A) while 
 their heads press against the under side of the upper plate (B). 
 In some instruments this order is reversed. The screws are set 
 in pairs, and when made to act in contrary directions the instru- 
 ment is set up level. 
 
 II. THE HORIZONTAL OR AZIMUTH LIMB consists of two circu- 
 lar plates, the upper (F) called the vernier circle and the lower (C) 
 known as the graduated circle. The latter projects somewhat 
 beyond the former, the two surfaces being conical and nicely fitted 
 
 * Vide Jean's Navigation, Part I. page 113, for a method of computing 
 the correct angle from the observed angle. This source of error is more 
 frequent than is usually suspected. 
 
VI.] 
 
 INSTRUMENTS AND OBSERVING. 
 
 105 
 
 to each other, the upper having a perfectly easy and very steady 
 motion along the lower. Both circles have horizontal motion round 
 the central axis. This axis consists of two parts, an outer and an 
 inner. The vernier circle is attached to the inner, and the gradu- 
 ated circle to the outer. At opposite points of the vernier plate 
 a short space is chamfered, forming with the graduated circle an 
 even surface. These small spaces contain the verniers. The 
 graduated circle is divided into 20' spaces*, and 39 of these are 
 taken for the length of the vernier, which is divided into 40 equal 
 parts : therefore on such an instrument we can read to half 
 minutes. In smaller instruments we can read only to single 
 minutes. 
 
 III. THE VERTICAL ARC. Attached to the vernier circle are 
 two frames (72) which support the pivots of the vertical arc or 
 semicircle (G), on the top of which is placed the telescope (T). 
 
 * This is only one example. The arcs seem to be graduated in many 
 different ways. 
 
108 MARINE SURVEYING. [CHAP. 
 
 On one side an arm is attached to the axis of the vertical arc, and 
 carries at its other end a microscope for reading off the angles of 
 elevation or of depression. The vernier is fixed, and this micro- 
 scope can be moved along its face for the purpose of reading off. 
 
 82. We next come to speak of the several MOTIONS of the 
 Theodolite. These are three in number, viz. 
 
 I. The absolute horizontal motion of the whole instrument 
 about its axis. 
 
 II. The motion of the vernier circle with respect to the gradu- 
 ated circle. 
 
 III. The motion of the vertical arc. 
 Details of these motions : 
 
 I. The absolute horizontal motion. 
 
 The lower (A) of the two levelling plates is screwed down to 
 the legs of the instrument ; the axis of the instrument passes 
 through to the upper plate, and is fixed at the other end to the 
 vernier circle. When the two circles composing the horizontal 
 limb are clamped together (by screw A"), the whole instrument 
 can be made to turn round on this axis, and can be clamped in 
 any required position by a clamping screw (D) situated between 
 the upper levelling plate and the graduated circle. A tangent 
 screw (F) gives the finer adjustments. 
 
 II. TJie motion of the vernier circle in azimuth. 
 
 The graduated circle (C) can be clamped to the axis as above 
 explained. The vernier circle, having independent motion, can 
 then be turned in any required direction, and can be fixed at 
 pleasure by a clamping screw (A") in its upper surface : a tangent 
 screw (M) will then make the more minute adjustments. 
 
 III. The motion of the vertical arc. 
 
 The vertical arc (6?) can be clamped in any position by means 
 of a screw (P) which acts on the horizontal axis ; a tangent screw 
 (N) at the lower part of the arc moves the arc and the attached 
 telescope through very minute intervals, and thus the observation 
 can be made accurately. 
 
VI.] INSTRUMENTS AND OBSERVING. 107 
 
 83. Before speaking of the adjustments and the methods of 
 making them, we may observe that attached to the vernier circle 
 ( V) are two small spirit-levels (I, I), placed at right angles to each 
 other. These are intended to secure the horizontal position of the 
 azimuth limb. Another level (L) is situated immediately below 
 the telescope ; the intention being to ensure the axis of the tele- 
 scope being horizontal if required. A compass (X) is fixed on the 
 vernier circle to enable approximate bearings to be taken. A 
 plummet attached to a hook immediately beneath the axis enables 
 us to place the vertical axis of the instrument exactly over a given 
 point. 
 
 Coloured eye-pieces are also supplied for the observations 
 of the sun. 
 
 84. The ADJUSTMENTS OF THE THEODOLITE are three in 
 number. 
 
 I. The line of collimation must be correct. 
 
 II. The spirit-level beneath the telescope must be parallel to t/ie 
 line of collimation. 
 
 III. The axis of the instrument must be truly vertical. 
 
 First adjustment. The line of collimation must be correct : i. e. 
 the line joining the centre of the object-glass with the intersection 
 of the cross wires must coincide with the axis of the rings ( Y, Y) 
 on which the telescope rests, i.e. with the " line of the Y's." To 
 ascertain whether this is so or not, we must look through the 
 telescope and cause the intersection of the wires to " bisect " 
 some distant and well-defined point; then turn the telescope 
 upside down, so that the spirit-level is now at the top of the 
 telescope, and observe whether the wires still bisect the object. 
 If so, the adjustment is correct; if not, we must move the circular 
 frame carrying the wires * through half the deviation by turning 
 two of the small screws (releasing one before tightening the 
 other) which keep the diaphragm or frame in its position, and the 
 other half must be corrected by elevating or depressing the tele- 
 scope. If necessary this operation must be repeated until the 
 
 * See diagram on p. 111. 
 
108 MARINE SURVEYING. [CHAP. 
 
 adjustment is perfect. A similar operation will correct the other 
 wire if required. The small screws (a, a) of the diaphragm appear 
 on the outside of the telescope. 
 
 Second adjustment. The spirit-level attached to the telescope must 
 be parallel to the line of collimation. The clips (c, c) which retain 
 the telescope in its Y's (as the cylindrical rings on which it rests 
 are called) being opened, and the vertical arc clamped, bring the 
 bubble to the centre of its " run " by means of the tangent screw 
 (N) of the vertical arc. This done, carefully turn the telescope end 
 for end in its Y's, so as not to disturb the vertical arc; if the 
 bubble resumes its position in the centre of its run, the adjust- 
 ment is correct; but if not, then it must be brought back one 
 half of the distance it has moved by means of the screw at one 
 end of tJte level (Z], and the other half by the tangent screw of the 
 arc (N). This operation must be repeated until the adjustment is 
 perfect. 
 
 Third adjustment. The axis of the instrument must be truly 
 vertical, or what amounts to the same thing, the horizontal limb 
 must be truly horizontal. 
 
 Set the instrument as nearly level as possible by the eye. 
 Clamp the graduated circle (by />), but let the vernier circle be 
 free. Move the latter until the telescope is over two of the levelling 
 screws (S, S) ; then bring the bubble of the telescope level (which 
 is the most sensitive in the instrument) to the middle of its "run" 
 by means of the tangent screw ( JV) of the vertical arc ; next turn 
 the vernier circle through 180 degrees, when if the bubble returns 
 to the middle, the limb is horizontal in that direction ; but, if not, 
 half the difference must be corrected by the levelling screws over 
 which the telescope is lying, and half by means of the tangent 
 screw of the vertical arc. Having done this, turn the vernier 
 circle through 90 degrees, so that the telescope may lie over the 
 other pair of parallel screws ; and by their motion make it hori- 
 zontal. Having thus made the azimuth limb horizontal by means 
 of the sensitive telescope-level, the bubbles of the other levels on 
 the vernier circle must be brought to the centres of their " runs " 
 by the screws which fasten them in their places. 
 
VI.] INSTRUMENTS AND OBSERVING. 109 
 
 The vernier of the vertical arc may now be attended to. The 
 index error (i.e. the deviation of the arrow head from zero, when 
 the Theodolite is perfectly adjusted) is best obtained as follows: 
 repeat the observation of an altitude or depression in the reversed 
 positions both of the telescope and the vernier circle. The two 
 readings will have equal and opposite errors ; the index error is 
 half their difference. 
 
 85. METHOD OF USING THE THEODOLITE. 
 
 Adjust the screws (S, S) between the levelling plates so that 
 equal lengths may appear above the upper plate. Extend the legs 
 of the instrument until the bubbles of the levels on the vernier 
 circle are nearly in the middle of their " runs," and the plummet 
 hangs freely over the required point, and press the legs firmly in 
 the ground. Then unclamp (D) the whole instrument so that it 
 may turn freely about its axis, but keep the other motions 
 clamped (K, P). Adjust the instrument so that the azimuth 
 limb may be horizontal. Clamp the whole instrument (D), and 
 unclamping (A") the two circles of the azimuth limb, set the index 
 of the vernier to 360 or zero of the graduated circle and clamp it; 
 xaniine the other vernier, the index of which ought to coincide 
 with 180. This must be done by means of the attached micro- 
 scope. 
 
 Ni-xt loosen the large clamping screw (D) and turn the whole 
 instrument towards the left of the two stations between which the 
 first angle is to be taken ; bisect this object as closely as possible 
 by hand : then firmly clamp the instrument and make the obser- 
 vation exactly by aid of the tangent screw (F). Now as the index 
 points to zero, and the lower circle is graduated from left to right, 
 it is evident that, by separating the vernier circle and turning it 
 round to the right until the second object is bisected,' the angle 
 can be read off between the first and second object. Both 
 verniers must be read and the mean taken. 
 
 In observing angles with a Theodolite the following method is 
 adopted in the Ordnance Survey. 
 
 Let A, , C...K, be the points observed taken in order of 
 azimuth ; then the instrument being in perfect adjustment A is 
 
110 MARINE SURVEYING. [CHAP. 
 
 bisected and the microscope read, then J5, then (7, and the other 
 points in succession ; after observing K, the movement of the 
 telescope is continued in the same direction round to A, which is 
 observed a second time, to ascertain whether the instrument has 
 moved. This complete round is termed " an arc." A more ordi- 
 nary procedure is to observe the points as before in the order 
 A, , (7, ...K, then reversing the direction of motion of the tele- 
 scope to re-observe in the inverted order, K y If, ...C, B t A. Thus 
 each point is observed twice. 
 
 86. To REPEAT AN ANGLE. After the last observation, without 
 detaching the two circles turn the whole instrument round to the 
 first object, and then unclamping the vernier circle turn it round 
 until the second object is "bisected." The difference between 
 this and the first reading will be double the mean anyle. Again. 
 keep the two circles together, turn round as before to the first 
 station, and then the vernier circle on to the second object; the 
 difference between this reading and the first reading will be three 
 times the mean anyle. etc.* 
 
 87. To OBSERVE A VERTICAL ANGLE. Level and adjust the 
 instrument. Bring the bubble of the telescope-level to the centre 
 of its " run." Make the zero of the arc coincide carefully with 
 the index of the vernier. Then "bisect" the object, and the 
 changed position of the broad arrow will mark the angle of eleva- 
 tion or depression. Reverse in azimuth, also reverse the telescope, 
 and take the mean of the two readings if they are different t. 
 
 88. THE SPIRIT-LEVEL. 
 
 This is also called the " Y-level" from the supports on which 
 the telescope rests being shaped somewhat like the letter Y. The 
 most perfect instrument of this kind is called the Improved 
 Dumpy Y level; the term "dumpy" being given because of its 
 short length and large telescope. 
 
 . * The first object observed in the arc, or round of angles, is termed the 
 zero, because where " bisected," the index points to zero on the graduated 
 circle. 
 
 t Or we may apply the Index Error to the observed altitude or depression, 
 ride p. '109; 
 
VI.] 
 
 INSTRUMENTS AND OBSERVING. 
 
 Ill 
 
 Description of the instrument. 
 
 The Spirit-Level consists of a telescope resting on two supports 
 shaped like the letter Y, these supports being known as the Y's. 
 The instrument may be set level exactly in the same way as the 
 azimuth limb of the Theodolite (vide the third adjustment of that 
 instrument). One of the Y's can be moved in the same way that 
 the telescope of the sextant is moved in its socket. The instru- 
 ment is clamped by means of a collar (E in diagram of the Theo- 
 dolite) which surrounds the vertical axis. Beneath the telescope 
 is fixed a sensitive level, attached by a hinge at one end to the 
 telescope, while a capstan -headed screw at the other end tends 
 to raise or lower that end. Two spider lines at right angles give 
 the centre of view. When this point of intersection is on with 
 any object, the object is technically said to be " bisected." 
 
 ef is a brass ring somewhat smaller than the tube of the tele- 
 scope. Two tine lines of spider's web are fixed to this ring at 
 right angles to each other. By loosening a screw at D and 
 tightening the screw at (7, the ring can be moved from right to 
 left, and similarly up or down as required. These screws project 
 beyond the side of the telescope and can be moved without dif- 
 ficulty. 
 
 Before adjusting the focus of the object-glass that of the eye- 
 piece ought to be looked after, both in this instrument and in the 
 
 Theodolite. The eye-piece must be drawn out until the cross 
 wires are clearly defined, and there is no instrumental parallax, 
 i. e. on looking at some distant object and bringing the intersection 
 
112 MARINE SURVEYING. [CHAP. 
 
 of the wires on with it, there may be no displacement of the 
 contact on moving the eye to the left or right. Such a parallax 
 would be caused if the image of the object falls beyond or falls 
 short of the cross wires. Its presence can always be detected by 
 moving the eye about arid noticing if the cross wires change their 
 position, or are fluttering and indistinct. Instrumental parallax 
 must be corrected as follows : 
 
 1st. Adjust the eye-piece until the cross wires are sharply 
 defined against any object. 
 
 2nd. Thrust forward the object-glass by means of the screw 
 at the side of the telescope to get a correct focus ( W in diagram). 
 
 3rd. Then readjust the eye-piece if necessary. 
 
 The parallax must be corrected before we look to the collima- 
 tion. 
 
 The three following ADJUSTMENTS are made in the Spirit- 
 Level. 
 
 I. The line of colliniation must be parallel to the axis of the 
 Y's on which the telescope is lying. It is corrected as in the case 
 of the Theodolite ; vide the first adjustment of the Theodolite. 
 
 II. The spirit-level must be parallel to the line of cottimation, 
 
 After the bubble has been brought into the centre of its "run" by 
 the plate screws, the telescope is reversed in its supports (i. e. 
 turned end for end); if the bubble has moved, it must be brought 
 back to the centre through one half of its displacement by the 
 screw at the end of the level, and through the other half by the 
 plate screws. This will require several repetitions, and then the 
 clips are secured in their places. The object of the adjustment 
 is to make certain that the axis of the telescope is truly hori- 
 zontal. 
 
 III. The Y-supports must be exactly on tJie same level, so that 
 when adjustments I. and II. have been made the axis of the 
 telescope may revolve in a plane at right angles to the axis of the 
 instrument. This may be effected as follows : 
 
 Level the telescope when placed over two of the levelling 
 screws, and then reverse the telescope ; if the bubble has moved 
 
VI.] INSTRUMENTS AND OBSERVING. 113 
 
 it must be adjusted by bringing it back through one half of its 
 displacement by turning the capstan-headed screw placed directly 
 below one of the Y's, which can thereby be raised or lowered in 
 its socket, and through the other half by the plate screws. This 
 operation must then be repeated with the other pair of plate 
 screws. After a few trials it will be found that the bubble 
 remains in the centre of its " run " as the telescope revolves com- 
 pletely round the axis of the instrument. 
 
 89. THE LEVELLING STAFF. 
 
 This is usually made in three pieces which slide into one 
 another in the manner of a telescope. When drawn out a spring 
 at the lower end of each holds it in position, and thus a very 
 convenient staff is formed 14- or 17 feet long. The lowest 
 joint is about 4 inches wide, the next about 3J inches, and the 
 top about 2 1 inches. The entire length is divided into feet, and 
 these are again subdivided into tenths and hundredths. These 
 smallest divisions are coloured black and white alternately. 
 The numbers representing the feet are sometimes painted red. 
 These numbers are placed on the left of the graduations, and the 
 tenths on the right. As the divisions are very small every advan- 
 tage is taken to distinguish the marks to facilitate the reading off 
 through a telescope at some distance; e.g. the top and bottom 
 edges of the horizontal stroke of the figure 7 will coincide with 
 divisions, and the lower edge of the long stroke of the same 
 figure will be in a line with a division; while the whole figure will 
 occupy ten of the hundredth divisions, or one-tenth of a foot. All 
 these minutiae are of assistance when reading off at a distance 
 of 300 or 400 feet. When looked at through the telescope of the 
 level the staff appears inverted. It may be noted that when the 
 assistant holds -the staff he sways it gently to and fro in the 
 direction of the level : the least reading will of course be when the 
 staff is held upright. The vertical wire in the telescope will be 
 the criterion that there is no lateral deviation in the position of 
 the staff. 
 
 R. M. S. 8 
 
114 MARINE SURVEYING. [CHAP. 
 
 90. THE TEN-FEET POLE. 
 
 This consists of two vanes about 2 feet by 15 inches, which 
 slide on the ends of a pole about 13 feet long. On the vanes are 
 painted two lines at right angles to the pole. These vanes are 
 fixed to the pole so that the distance between the centres of the 
 lines is exactly 10 feet. This instrument is used for putting in 
 the coast line. The following explanation will shew the method 
 of using it. 
 
 The scale for the plan must be first determined*. A side of 
 any triangle is computed from the measured base line, and the 
 length of this side is taken from the plan. Then the length of 
 the nautical mile for the given latitude being known, we get the 
 following proportion : 
 
 Scale for 1 mile measured distance from the plan 
 
 length of the nautical mile computed length of side 
 
 e.g. Let the distance between two points on a plan be 7 '5 
 inches, and let the computed distance be 9 120 feet; then, if the 
 mile be 6080 feet, we shall have 
 
 Scale for 1 mile _ 7 '5 
 
 6080 ~9T20* 
 
 . *. Scale for 1 mile = 5 inches. 
 
 This plan would therefore have been projected on a scale of 
 5 inches to a mile, or 1 inch to 1216 feet. 
 
 Computed length in feet 
 .-.A distance in inches on the plan = - . 
 
 I 2i I D 
 
 The Scale for the Ten-feet Pole can now be constructed as 
 follows : 
 
 Let AB represent 10 feet, viz. the distance between the cen- 
 tres of the vanes, and let C be the point of observation ; suppose 
 the angle ACB subtended by the length AB to be 6 (Note. In 
 practice this angle is always measured on and off the arc, and the 
 mean angle taken) : 
 
 6 AD 
 
 Then ** n 2 = CT>' 
 
 Vide Notes on Marine Surveying, by Staff Commander V. F. Johnson, 
 E.N. 
 
VI.] 
 
 INSTRUMENTS AND OBSERVING. 115 
 
 
 
 2 tan - 
 2AD 
 
 10 feet 
 . computed distance = -g-; 
 
 distance in inches on the plan 
 
 Computed distance 
 
 scale for 1 inch on the plan 
 Suppose the scale to be 5 inches to a mile of 6080 feet 
 
 Let = 1', 
 
 log 10= 1-000000 
 
 log cot 1'= 13-536274 
 
 14-536274 
 
 log 1216= 3-084934 
 .-. log distance = 1-451340 
 
 .*. distance = 28 -27 inches. 
 
 Hence if the observed angle be one minute, the distance of the pole 
 from the point of observation will be 28-27 inches 011 the plan. 
 Now we can assume, without sensible error, that when the angles 
 are very small, at half the distance the angle will be twice as 
 great, or conversely; hence the scale for the 10-feet pole may be 
 constructed as follows : 
 
 Observed angle. Inches of scale. 
 
 1' 28-27 
 
 2' 14-13 
 
 3' 9-42 
 
 82 
 
116 MARINE SURVEYING. [CHAP. 
 
 Observed angle. Inches of scale. 
 
 4' 7-06 
 
 5' 5-65 
 
 6' 4-71 
 
 &c. 
 
 Examples for Exercise. Construct a scale for the 10-feet pole 
 if the scale of the plan is 8 inches to the mile, and also if the 
 scale is 6 inches to the mile. The mile = 6080 feet. 
 
 91. THE STATION POINTER. 
 
 This instrument is useful in fixing a point by means of the 
 " Three-point problem." We may give the following description 
 of its construction and use. 
 
 The instrument consists of a ring 6 inches in diameter, and 
 about inch wide*. From a central ring proceed three arms, 
 about 12 inches long, and these can be increased to 18 inches by 
 extra pieces supplied for the purpose. The central arm is fixed, 
 its bevelled edge coinciding with the zero of the graduations. 
 This graduation is continued to the right and left from the zero. 
 The other two arms can be moved through any required angle, and 
 can be fixed by a clamping screw. A tangent screw on the side 
 will give the finer adjustments. Each arm carries a vernier, and 
 by its aid we can lay off angles to single minutes. 
 
 One edge only of each arm is bevelled ; if the right-hand edge 
 is bevelled, the instrument is a " Right-hand Station Pointer," 
 and similarly for the left hand. One of these arms can close up 
 nearer to the index radius than the other. Hence if the left-hand 
 angle is smaller than the left radius can measure we must have 
 resort to a right-hand pointer, and vice versd. 
 
 A small stroke at the centre of the inner ring marks the 
 point of observation required. 
 
 92. METHOD OF USING THE INSTRUMENT. The angles observed 
 are set by means of the verniers, and the bevelled edge of the 
 left-hand radius bar is placed over the left-hand object, and then 
 placing the thumb and fore-finger on either side of the bar, the 
 instrument is moved about until the bevelled edge of the middle 
 
 * These instruments are made of all sizes. 
 
VI.] INSTRUMENTS AND OBSERVING. 117 
 
 bar concides with the middle object, when it will be found easy to 
 make the edge of the third bar coincident with the third object. 
 A pricking point will then mark the position required. 
 
 93. THE BAROMETER. 
 
 This instrument is intended to shew the pressure of the air at 
 any moment. In its simplest form it merely consists of a glass 
 tube, somewhat less than 3 feet long, closed at one end, which 
 is filled with pure mercury, and with the other end inserted in 
 a vessel containing mercury. When held vertically, the column 
 of mercury inside the tube will exactly balance the pressure of 
 the air on a section equal to that of the tube. In general terms, 
 when the level of the mercury rises, the air is exerting an in- 
 creased pressure, when it falls, the air is exerting a diminished 
 pressure. It does not fall within the scope of this work to give 
 the methods of interpreting the records of the barometer. These 
 are fully explained in various publications, and more particularly 
 in the Barometer Manual published by the Board of Trade*. 
 From that excellent little work most of the following details are 
 taken. 
 
 Barometers are best mounted in brass, because its coefficient 
 of expansion by heat is well known, and the tables for correcting 
 barometer readings for temperature (founded upon the coefficients 
 of expansion of mercury, glass, and brass) always give with such 
 barometers identical results. Tables have been formed for barome- 
 ters framed in different kinds of wood, but, for accurate results, 
 these instruments cannot be relied on. 
 
 If, however, and this is important, a scale be applied which is 
 quite independent of the frame, then the reduction for temperature 
 will depend upon the material of the scale, and in such a case 
 wood will answer perfectly as a frame. The scale may be of ivory, 
 porcelain, or enamel, and is fixed in its proper position. 
 
 It is evident that if the barometer rises, the increased quantity 
 
 of mercury in the tube must have come from the cistern, and 
 
 therefore the surface of the mercury in the cistern must be lower 
 
 in consequence ; and the opposite result follows if the barometer 
 
 * J. D. Potter, 31, Poultry. Price One Shilling. 
 
118 MARINE SURVEYING. [CHAP. 
 
 is falling. Hence the varying level of the surface of the cistern 
 must be corrected or allowed for. There are three methods of 
 doing this: (1) by capacity correction, (2) by & flexible base to the 
 cistern, and (3) by a contracted scale. 
 
 94. (I.) By capacity correction. 
 
 When the cistern is covered up and the scale is engraved on 
 the frame this method is adopted : A certain height of the column 
 in the tube is correct by the scale, and the position of the mercury 
 at this time is known as the Neutral Point. When the mercury 
 sinks below this point, the level rises in the cistern above the zero 
 of the scale, and hence the reading will be too great. When the 
 mercury rises above this point, the level in the cistern falls below 
 the zero, and hence the reading is too small. On such a barometer 
 the neutral point is marked (N.P.), and also the relative interior 
 sections of the tube and cistern thus, " Capacity 1 to 50." From 
 these elements, the Correction for Capacity is found by taking 
 g^th of the difference between the height read off and that of the 
 N.P., adding the correction to the reading when the column is 
 above, and subtracting it from the reading when it is below the 
 N.P. 
 
 95. (II.) By a flexible base to the cistern. 
 
 This is the principle of Fortin's Barometer. The upper part 
 of the cistern is made of glass, the base is flexible and is acted 
 upon by a lifting screw placed beneath. From the top of the 
 cistern an ivory cone descends, the point of which is the zero of 
 the scale. Before reading off, the level of the mercury in the 
 cistern is brought to this point by means of the lifting screw, and 
 then the height of the column in the tube as read off from the 
 scale will be the true reading. 
 
 This construction is usually adopted in standard barometers, 
 and cannot be used at sea. 
 
 96. (HI.) By contracted scale. 
 
 In this the highest point of the scale is the neutral point, and 
 the inches of the scale are shortened in proportion to the relative 
 size of the sections of the tube and of the cistern. Thus, if the 
 
VI.] INSTRUMENTS AND OBSERVING. 119 
 
 diameter of the tube be 0-25 in. and that of the cistern be 1'25 in. 
 then the inches on the scale are shortened by 0'04 of an inch. 
 
 97. THE MARINE BAROMETER. 
 
 The greater part of the length of the tube of a marine barome- 
 ter must be made with a very fine bore, to prevent oscillations in 
 the column of mercury from the rolling of the ship. When the 
 bore is not sufficiently contracted, the fluctuations which arise 
 from the motion of the ship, are called " pumping." If on the 
 other hand the bore is too contracted, the instrument is sluggish 
 in responding to the varying pressure of the atmosphere, and is 
 therefore not suitable for very accurate observations. 
 
 Whenever a marine barometer ceases to act, and there are no 
 traces of damage to the instrument, it may be surmised that a 
 particle of dirt, or a bubble of air, has lodged in the very line con- 
 traction of the bore. To remedy this defect the instrument should 
 be taken down, the mercury allowed to fill the tube, and put aside 
 in an inverted position for a few hours. When replaced in its 
 proper position the cause of error will generally be found to have 
 disappeared. 
 
 The marine barometer is mounted in a brass frame, but the 
 cistern is of iron. The frame is open in front and rear to expose 
 the " range portion " of the tube ; the scale is protected from 
 dust by a glass shield. The vernier is engraved 011 a piece of 
 silvered brass tubing, and travels firmly by a rack and pinion 
 motion, the parts being kept in position by friction. The inches of 
 the scale are contracted as explained above. 
 
 The cistern is closed. It contains sufficient mercury to cover 
 the open end of the tube when the instrument is laid flat, or when 
 inverted. A small aperture at the top, covered internally -\vith 
 leather, permits the pressure of the external air being exerted, but 
 prevents the mercury escaping. 
 
 98. METHOD OF SUSPENDING. 
 
 The barometer must be hung in gimbals, and the arms of 
 hammered brass by which it is supported ought not to be shorter 
 than 12 inches to allow sufficient spring. The instrument 
 must hang quite vertically by its own weight only, and especial 
 
120 MARINE SURVEYING. [CHAP. VI. 
 
 care must be taken that the reading is not taken when in any 
 other position, as the reading would be erroneous and too great. 
 It ought to be kept in the shade, in a place easily accessible, and 
 where it cannot come into collision with any object during the 
 heaviest motion of the vessel. 
 
 99. VERNIER OP A STANDARD BAROMETER. 
 
 The scale is divided into inches, and each inch is subdivided 
 into ten equal parts, these again being bisected, thus the inch 
 contains twenty equal parts. Twenty four of these are taken as 
 the length of the vernier, and this space is divided into twenty- 
 five equal parts, thus the Least Reading is 
 
 vs of A=5^=' 002inch - 
 
 In the case of the ordinary marine barometer used on board 
 ship, the Least Reading is "01 inch. 
 
 100. METHOD OF READING. 
 
 The lower edge of the vernier must be brought on a level with 
 the slightly convex surface of the mercury, then this edge, the 
 surface of the mercury and the back edge of the vernier (if there 
 be a back edge) will be in one plane. White paper held behind 
 will facilitate the accuracy of the reading. 
 
 101. METHOD OF STOWING FOR CARRIAGE. 
 
 The vernier must be brought down to the bottom of the 
 scale : then the instrument must be lifted out of the sustaining 
 bracket, and held for a few minutes in an inclined position, so 
 that the mercury may flow gently up and fill the top of the glass 
 tube. It must then be placed lengthwise in its box, with soft 
 packing around it, and the lid screwed, not nailed, on. During 
 the carriage it must be kept free from all jarring blows. 
 
 Note. The student is recommended to read carefully Raper's 
 remarks on the precautions to be observed in taking observa- 
 tions. Vide his Practice of Navigation, Chapter-in, pp. 175 to 
 184. Read also Chauvenet's remarks 011 the errors to which 
 observations are liable. Spherical and Practical Astronomy, 
 Vol. ii. p. 471. 
 
CHAPTER VII. 
 
 BASE LINES. 
 
 102. IF in a nv triangle we know the values of the three 
 angles only, we can determine merely the ratios of the three sides, 
 but if we know in addition the absolute value of any one side, 
 then the absolute values of the other two sides may be deter- 
 mined. Hence the importance of determining in every survey 
 the absolute length of one line accurately. The line so determined 
 is known as the Base Line or simply the Base. 
 
 103. Now if in a triangle we know the length of one side 
 and the two adjacent angles, the triangle can be completely solved. 
 We can then use one of these calculated sides as the known side 
 of a new triangle, and thus proceed indefinitely. Hence the 
 necessity of measuring accurately the base line in the first in- 
 stance, because if erroneous, the error goes on increasing with 
 every triangle until finally the calculated sides become quite 
 erroneous. 
 
 104. DEF. When a triangulation has been carried on over 
 a country it is customary to measure one of the computed sides of 
 a triangle, far distant from the starting point, with the same 
 detail as the original base was measured, as a test of the accuracy 
 of the survey. Such a test is the most severe to which the work 
 can be put. This measured side is known as the " BASE OF 
 VERIFICATION." 
 
122 MARINE SURVEYING. [CHAP. 
 
 E. g. The Ordnance Survey of Great Britain and Ireland was 
 carried on from a base measured on Salisbury Plain. When the 
 network of triangles had reached the north-west of Ireland a base 
 of verification was measured along Lough Foyle. The difference 
 between the computed and measured lengths was less than 5 
 inches ! 
 
 In 1793 a base of verification was measured by Mechain, near 
 Perpignan, to test the accuracy of the work connected with the 
 famous measurement of the Arc of the Meridian. The measured 
 and calculated lengths differed by less than a foot, although the 
 original base was distant more than 430 miles. Another instance 
 of accurate work may be cited. When General Roy died in 1790, 
 the English Ordnance Survey was carried on by General Mudge, 
 from the measured base on Hounslow Heath, through Greenwich, 
 to Dunnose in the Isle of Wight, thence on through Devonshire, 
 Dorset and Wiltshire, and connected with the base of verification 
 on Salisbury Plain. The computed and measured lengths were 
 found to differ by scarcely 1 inch ! 
 
 105. The measurement of a base line, although seemingly not 
 difficult, is by far the most tedious and important part of a Trigo- 
 nometrical survey ; and hence in a very important survey, such 
 as that of an entire country, every refinement which mechanical 
 ingenuity can suggest has been lavished upon this operation to 
 secure, as far as possible, mathematical accuracy. 
 
 We will first dwell briefly on the more exact methods of 
 measuring a base used in such a survey, as likely to prove interest- 
 ing to younger readers, and shall then enter more fully on the 
 various methods used by nautical surveyors. 
 
 The geodetic standards of measurement in different countries 
 vary in length, in form, and in the substance of which they are 
 composed. 
 
 They may however be divided into two classes, known respect- 
 ively as standards " a traits," and standards " a bouts" In the 
 former class, the lines or dots defining the exact measure used are 
 engraved on small disks of silver, platinum, or gold let into the 
 bar ; in the latter, the ends of the bar are generally in the form of 
 
VII.] BASE LINES. 1 23 
 
 a small cylinder presenting a circular disk, either plane or convex, 
 of some hard polished metal, or agate, for the purpose of contact 
 in the operation of measuring a distance. 
 
 Thus the standard used in the Ordnance Survey of Great 
 Britain is 10 feet in length, and in section is a rectangle of 
 1 J inches broad by 2J inches deep. It is supported 011 rollers at 
 J and J of its length. The ends of the bar are cut away to half 
 its depth, so that the dots marking the measure of 10 feet are in 
 the neutral axis of the bar. 
 
 106. Standards are generally furnished with thermometers, 
 which either lie in contact with the metal, or else have their bulbs 
 so arranged as to fit into cavities in the upper surface of the bar. 
 The errors of these thermometers must be known with great 
 exactness, because an error of y 1 ^ of a degree in temperature cor- 
 responds to an error of nearly a millionth of its entire length in 
 an iron bar. These thermometers therefore are from time to time 
 compared with the standard thermometers. Such a standard must 
 be " the best workmanship of the best workman," and the residual 
 errors (in the boiling point, freezing point, calebration, and com- 
 parisons of the standards) can be determined only by observation 
 and experiment. Every thermometer has an index error which 
 varies slightly in the course of time, and must be determined 
 occasionally. When examined in the Ordnance Survey Establish- 
 ment at Southampton, these thermometers are held in water on a 
 small platform of perforated zinc with about 7 inches of water 
 over them. There is a mechanism for slightly agitating the water 
 and thus to prevent any local cooling. The instruments when 
 lying horizontally are read from above by means of powerful 
 micrometer microscopes. 
 
 107. The knowledge of the exact length of a bar at any 
 moment involves three distinct matters : 
 
 (1) Its length at some specified temperature, which is known 
 by repeated comparisons with the standard bar. 
 
 (2) Its coefficient of expansion, which must be obtained from 
 special experiments. 
 
124 MARINE SURVEYING. [CHAP. 
 
 (3) The temperature of the bar at the moment of observation, 
 which can only be discovered by means of the attached ther- 
 mometers. 
 
 108. To evade the temperature difficulty, three forms of con- 
 struction have been adopted. 
 
 (1) Horda's. In this, the measuring bar consists of two rods 
 of metal having quite different rates of expansion : the slight 
 expansions from the normal length being marked by verniers and 
 read by attached microscopes. 
 
 (2) Colby's. In this, by a simple mechanical arrangement, 
 two rods of different expansions are made to present two minute 
 points or dots at a certain constant distance. 
 
 (3) Struves. In this, one end of the bar carries a contact 
 lever, the lower arm of this lever terminates in a polished hrmi 
 sphere, while the upper arm traverses a graduated arc. When the 
 index points to a certain division on the graduated arc, the bar is 
 known to be at its normal length, and its length is also known, 
 and can be allowed for, when the index points to other divisions. 
 
 109. We shall now proceed to explain the principle of Gene- 
 ral Colby's beautiful contrivance, known as the " COMPENSATION 
 APPARATUS," which he invented for the Irish Survey, and with 
 which the Lough Foyle base was measured. 
 
 A 
 
 /' '\ 
 
 A 
 
 ] 
 
 / \ 
 
 iP 
 
 \ / 
 
 / 
 
 
 \ 
 
 / 
 
 \d' e/ 
 
 X 
 
 / 
 
 \ i / 
 
 \ 
 
 AB is a brass rod, DE is an iron rod : these are firmly united 
 at their middle points by a transverse bar (7, but their ends are 
 free to expand or contract under changes in temperature. The 
 
VII.] BASE LINES. 125 
 
 lengths of these bars are the same at a certain temperature, but 
 the expansion of the brass being greater than that of the iron, 
 it follows that if the temperature is greater or less than that 
 selected as the normal, the brass bar AB will expand to ab, or 
 contract to a'6', whereas under similar circumstances the iron 
 bar DE will expand or contract to de or d'e. 
 
 Now AP, BQ are two tongues of steel, about 6 inches long, 
 attached to the extremities of the rods in such a way as not to 
 interfere with the expansion or contraction of the two bars. 
 A minute dot of platinum, almost invisible to the naked eye, is 
 placed at P and Q on the tongues. Under all changes of tempera- 
 ture the distance between P and Q is exactly 10 feet. 
 
 At the normal temperature (60 F.) the two bars being of 
 exactly the same length, we must have the tongue AP parallel 
 to the tongue BQ, and perpendicular to the bars. Suppose 
 the temperature to exceed the normal, then the bar AB expands 
 to ab, but DE expands only to de, and therefore the tongue AP 
 assumes the position ax, and BQ assumes the position by. 
 
 If therefore P and Q are taken at such a distance from the 
 
 , PA expansion of AB 
 
 bars that ,. n = , . .,, it is evident that m the new 
 
 ID expansion ot UL 
 
 positions of ax and by the distance between the points P and Q 
 will not change, although their position above the bars will be 
 somewhat lower. Suppose the distance between the two bars to 
 be 2 inches, and let the expansion of brass : expansion of iron 
 = 83 : 53, and let PD = x inches, then we have 
 
 .-. 53* + 106 = 83*, .-.30* =106, 
 X Do 
 
 .'. # = 3J inches nearly. 
 
 110. About five or six sets of these rods are generally required 
 in measuring a base. 
 
 The method of using the instrument is as follows. The rods 
 are carefully levelled, and are placed so that the dot at the end 
 of one bar is always at a fixed distance from the dot at the end of 
 the next. This is secured by means of powerful microscopes 
 known as " Compensation Microscopes," and constructed on the 
 
126 MARINE SURVEYING. [CHAP. 
 
 same principle as the "Apparatus" itself. These dots are brought 
 under the microscopes, and are thus known to be 6 inches 
 apart. 
 
 HI. In 1845 Professor Bach invented for the United States 
 Coast Survey a measuring apparatus combining the principles of 
 Borda's rods, the compensation tongue of Colby's instrument, and 
 the contact lever of Struve. Surprising results, both as regards 
 time and accuracy, have been obtained by this " Compensating Base- 
 Measuring Apparatus." More than a mile has been measured in 
 a day; and Jeffers, in his Nautical Surveying (p. 105), states that 
 by its means " a base of six miles in length has been measured 
 with a probable error on remeasurement of one-tenth of an inch ; 
 which surpasses the accuracy of angular measurement with our 
 present instruments." It may be remarked that the Lough Foyle 
 base was measured at the rate of only 250 feet a day. 
 
 112. As a preliminary operation to the measurement of the 
 base, it is usual after getting an accurate section of the line by the 
 spirit-level to measure the distance in an approximate manner. 
 A more detailed aligning follows. By the aid of the theodolite 
 over the ends and intermediate points, pickets are driven into the 
 ground at regular intervals; each picket carries a fine mark 
 indicating exactly the line of measurement. This aligning re- 
 quires all the care that can be bestowed upon it. 
 
 113. RATIO or THE LENGTH OF THE BASE TO THE EXTENT OF 
 THE SURVEY. 
 
 This ratio has varied much. Six bases have been measured in 
 the United Kingdom, and these vary from five miles to over ten 
 miles in length, the extent of the survey from north to south 
 bein'g over 750 miles. The base line at the Cape of Good Hope is 
 eight miles long, the ground surveyed being about 600 miles from 
 east to west, by 300 from north to south. In Syria about 200 
 miles have been surveyed, the two bases being each about a mile 
 long. 
 
 The degree of accuracy required in a base line must be settled 
 by the extent of the ground to be surveyed, and also by the object 
 of the survey. 
 
VII.] BASE LINES. 127 
 
 As just stated, six bases have been measured in the 
 Ordnance Survey of the United Kingdom. The earlier by Rams- 
 den's steel chains, the two most recent by General Colby's appa- 
 ratus described above. The following table contains the mea- 
 sured lengths of the bases and their lengths in the corrected 
 Triangulation, Le. when regarded as " Bases of Verification." 
 
 Base Lines of the Ordnance Survey. 
 
 Date. Place. *ed C ff d Die County. . 
 
 1. 1791, Hounslow Heath, 27406'19, 27406'36, +0'17, Middlesex. 
 
 2. 1794, Salisbury Plain, 36576*83, 36577*66, + 0*83, Wilts. 
 
 3. 1801, Misterton Carr, 26344*06, 26343*87, -0'19, Lincoln. 
 
 4. 1806, Rhuddlan Marsh, 24516*00, 24517*60, -fl'60, Flint. 
 
 5. 1817, Belhelvie, 26517'53, 26517-77, +0'24, Aberdeen. 
 
 6. 1827, Lough Foyle, 41640-89,41641-10, + 0*21, Londonderry. 
 
 7. 1849, Salisbury Plain, 36577*86, 36577'66, -0'20, Wilts. 
 
 Clarke's Geodesy, p. 244- 
 
 About two miles were subsequently added to the Lough Foyle 
 Base, making it the longest in the kingdom. 
 
 In the above list, numbers 1, 3, 5 were measured by llams- 
 den's steel chains: the differences between the measured and 
 computed lengths in these cases are not greater than in the case 
 of numbers 6 and 7, which were measured by the compensation 
 apparatus. Hence it has been considered by some authorities 
 that bases measured very carefully with steel chains are deserving 
 of most confidence, and they have the further recommendation of 
 being simple, cheap, and portable. 
 
 115. The lengths of bases have varied much in different 
 countries. In East Prussia a line measured by Bessel was little 
 more than a mile long, whereas in France there is one 11-8 miles. 
 lict\veen these as limits we have them of all lengths : Lough Foyle 
 is over 10 miles; in India, except the line at Cape Comorin, 1-7 
 miles; nine others lie between 6 -4 and 7*8 miles. In the Spanish 
 Triangulation now in progress there are several about 1J miles; 
 the principal base near Madrid is 9"! miles, and in the island of 
 Ivi^a is one just a mile in length. 
 
128 
 
 MARINE SURVEYING. 
 
 [CHAP. 
 
 It is sometimes customary to break up a line into segments 
 and verify the lengths of these segments by triangulation, in 
 other words to treat them as bases of verification. Thus, in the 
 case of the Madrid base mentioned above, the total length is com- 
 posed of five segments. Angles were observed at ten different 
 stations, and then on the assumption that one segment is correctly 
 measured, the lengths of the other four were computed. Regarding 
 the middle segment as the basis of calculation, the measured and 
 computed segments stand as follows : 
 
 Segment. 
 
 1 
 2 
 3 
 4 
 5 
 
 Measured 
 Metres. 
 3077-459 
 2216-397 
 2766-604 
 2723-425 
 3879-000 
 
 Computed 
 Metres. 
 
 Differences 
 Metres. 
 
 3077-462 
 
 + 0-003 
 
 2216-399 
 
 + 0-002 
 - 0-003 
 
 2723-422 
 
 3879-002 
 
 + 0-002 
 
 Sum 14662-885 14662-889 +0-004 
 
 Clarke's Geodesy, p. 173. 
 
 116. To REDUCE A BASE TO THE SEA-LEVEL. 
 
 The base ought to be reduced to the mean level of the sea. 
 This may be effected as follows : 
 
 Let AE measured length (A), 
 ab = reduced length (a). 
 
 r = radius of the earth, supposed to be a sphere, 
 r + h = radius at the mean level of the measured base. 
 
VII.] BASE LINES. 129 
 
 Then by Euclid vi. 33, 4 = -*" / 
 A r + h 
 
 Ar ./- h\~ l 
 
 .'. a= . = ^(1+-) 
 
 r + h \ r) 
 
 ,/- h h 2 h 3 \ 
 
 = A(l--+ -s --= +... ) 
 
 \ r r 2 r 3 / 
 
 Ah Ah' Ah* 
 
 .'. a = A H =- + 
 
 r r" ?- 3 
 
 Ah Ah' Ah 3 
 
 . . A a TT + 5 ... 
 
 r r" ?- J 
 
 The first term only is required. 
 
 Example. In 1815 a base was measured in India equal to 
 30809-07 feet. The mean height of the base above the sea level 
 was 1957 feet, and the mean radius of the earth is 20888153 feet. 
 
 Here A = 30809-07 feet, 
 
 A =1957 feet, 
 r = 20888153 feet, 
 
 .-.A-a = ; log ,4 = 4-488687 
 
 log 7* = 3-291591 
 
 7-780278 
 
 log r = 7-319900 
 
 log (A -a) = 0-460378. -.ji-a = 2-89feet 
 . . Reduced length of base = 30806-18 feet*. 
 
 Note. In ordinary marine surveying, such as the survey of a 
 port, the surface of the earth may be considered as plane without 
 sensible error, the arc of a degree not exceeding its chord by more 
 than 25 feet. 
 
 117. Having thus noticed some of the more important 
 methods of measuring a base line, we now pass on to the methods 
 chiefly employed by the Nautical Surveyor. 
 
 * Manual of Surveying for India, p. 471. 
 R. M. S. 9 
 
130 MARINE SURVEYING. [CHAP. 
 
 In every base there are three elements : 
 
 (1) The latitude and longitude of one extremity. 
 
 This point is usually known as the Observatory Station. 
 
 (2) Tlie measured length. 
 
 (3) The direction or bearing of the base. 
 We shall speak of these separately. 
 
 118. (1) THE LATITUDE AND LONGITUDE OF THE OBSERVATORY 
 STATION. 
 
 The Latitude is usually determined by means of the sextant 
 and artificial horizon, (1) By meridian altitude of the sun, 
 
 (2) By meridian altitude of many stars. 
 
 But in more exact surveys a zenith sector is required ; by its aid 
 the latitude can be determined to 1". 
 
 The Longitude must be determined very carefully by chrono- 
 meters. 
 
 119. (2) THE LENGTH OP THE BASE. 
 
 Circumstances must decide the approximate length required : 
 if the survey is limited, perhaps 1000 yards will suffice, but as 
 a rule 2500 or 3000 yards will be deemed long enough. 
 
 120. Nature of the ground. The following conditions ought 
 to be sought for as far as possible ; 
 
 (a) As level as possible. 
 
 (/?) Near the shore, 
 
 (y) Not much above the sea level. 
 
 (8) A good extent of country or shore, or many points in tli<> 
 harbour, to be visible from the proposed extremities of the line, 
 
 (c) The extremities in view from one another. 
 
 () . The line to run on to some well-defined natural object. 
 
 121. DIFFERENT METHODS OF MEASURING A BASE. 
 (a) By masthead angle. 
 
 (/?) By the velocity of sound. 
 
CO 
 
 VII.] . BASE LINES. 131 
 
 (y) By patent log under steam. 
 
 (S) By astronomical observations. 
 
 (e) By direct measurement. 
 
 We will notice these methods one by one. 
 
 122. FIRST METHOD. BY MASTHEAD ANGLE. 
 
 The distance of a ship from an observer on shore may be 
 required, or from a boat, &c. The height of the mast is supposed 
 to be known ; a white mark (if required) may be placed on the 
 side of the ship at the height of the observer's eye, and finally in 
 observing the angle, the precaution must always be taken of 
 measuring the angle " on and off" the arc, to eliminate the index 
 error. 
 
 Then, Length of base = height x cot angle. 
 
 123. SECOND METHOD. BY THE VELOCITY OF SOUND. 
 Note. This method might be used under two circumstances : 
 (a) Where the required base would lie across a marsh, or 
 
 bog, or the mouth of a creek. 
 
 (ft) Where the base must be measured along coarse shingle. 
 Objections to this method. 
 
 (1) An error of T ^ second in the time produces an error 
 of about 100 feet in the length of the base. 
 
 (2) The velocity of sound is affected by several variable 
 nditions, viz. temperature, wind, and state of the atmosphere. 
 
 The mean of many observations will however give very good 
 results. 
 
 Method of proceeding. A small gun is placed at one end of 
 the proposed line, and a flag is hoisted to shew that everything is 
 ready at the firing station. When this is answered from the 
 observing station, the gun is fired, and the interval between the 
 flash and the report is accurately noted. This operation is re- 
 peated several times, and at each station. The mean of each 
 series is then taken, and if there is much difference in the results, 
 the error is probably owing to the wind. If, however, the mean 
 
 92 
 
132 MARINE SURVEYING. [CHAR 
 
 results are very close, their mean may be considered correct. 
 
 The temperature must be also noted at each station. 
 Then the distance is computed from the formula, 
 Distance = time x velocity of sound at the given temperature. 
 
 The velocity of sound is 1090 feet per second at 32 F., and 
 increases 1 feet per second for an increase of 9 F. in tem- 
 perature. 
 
 Thus at 41 the velocity is 1100, 
 
 at 50 1110, 
 
 at 59 1120, &c.* 
 
 If the velocity of the wind must be taken into account, we 
 proceed thus : 
 
 Let D = required distance, 
 
 s = velocity of sound, corrected for temperature, 
 w = estimated velocity of the wind, (the wind is supposed 
 
 to be in the direction of the measured base), 
 t l = time in seconds with the wind, 
 t a = time in seconds against the wind, 
 T=true time in seconds. 
 
 By 1st series of observations D = (s + w)^ , .'. =s + w. 
 2nd = (s-w)t a , .-. - =s-w. 
 
 -. 
 
 but D = sT. 
 
 * The velocity may be more exactly computed from the formula 
 F=1093 ,Jl + -003665$ where 1093= velocity at 0C, 
 
 t = observed temperature C. 
 
VII.] BASE LINES. 133 
 
 Hence the true interval of time is the liarmonic mean between 
 the times with and against the wind. The true time being known, 
 the length of the base can be found as above. 
 
 124. THIRD METHOD. BY PATENT LOG UNDER STEAM. 
 This can of course give but an approximate result ; it is useful 
 
 however in the case of a running survey of a coast, and will give 
 a distance sufficiently correct for such a purpose, provided that, 
 (1) the speed of the ship exceeds about three knots, and (2) the 
 course is free from currents. 
 
 125. FOURTH METHOD. BY ASTRONOMICAL OBSERVATIONS. 
 This method also will only give a fair approximation, and will 
 
 serve only in the absence of other methods, and for a rough 
 survey of a large extent of coast. It is impossible to fix the 
 position exactly by means of a sextant. This consideration will 
 explain why the length of a base line is not determined on shore 
 by finding the latitude and longitude of each extremity, and 
 then computing the distance. In the first place, an error of 
 10" in the latitude would involve an error of 1000 feet in the 
 distance; and in the second place, the labour is very great to 
 determine with great exactness the position on shore by accurate 
 observations, such as would be necessary for the purpose. 
 
 126. FIFTH METHOD. BY DIRECT MEASUREMENT. 
 
 We now come to speak of the usual method resorted to in the 
 case of most nautical surveys. 
 
 (a) A sea base can be measured as follows : When the surface 
 of the water is free from currents and is calm, the distance 
 between two boats, or between two rocks, can be measured by 
 a line ivell wetted stretched between the points and supported by 
 little floats at suitable intervals. The line should be measured at 
 once in its wet state. 
 
134 MARINE SURVEYING. [CHAP. 
 
 (/?) A land base. 
 
 This, being the most important, must.be described in detail. 
 When the ground has been selected combining as many of the advan- 
 tages above mentioned as possible, the ends of the intended line 
 (A B) are marked with easily distinguishable objects. A theodolite 
 is fixed immediately over one extremity A, the stake having been 
 removed, and the alignment is then effected as follows. Staves 
 are placed, at intervals of 300 or 400 feet, in a direct line with 
 station j5, the true line being ascertained by the telescope of the 
 theodolite. When these staves are firmly driven in, a deep-sea 
 sounding line is stretched taut along the ground so as to touch all 
 the staves on the same side. We thus obtain a straight line between 
 A and B. The actual process of measuring is then carried on. 
 
 One man, called the Leader, being furnished with ten iron 
 pickets about 12 or 14 inches long, and sharply pointed, takes 
 the surveying chain (vide description in the preceding chapter) 
 by one handle, and moves off from A towards B. When the 
 chain has been stretched tight along the ground by the side of 
 the lead line, one end is held over the hole at A by a second 
 man, known as the Follower, and the Leader drives in a picket at 
 the further end. The chainmen then move on; the Follower 
 comes to the picket, and holding his end close up to it the 
 Leader drives in a second picket. The Follower then makes 
 an entry in his note-book that one chain has been measured. 
 They proceed thus, the Follower retaining all the pickets he has 
 removed, until the Leader has exhausted his ten pickets, when the 
 Follower transfers the nine pickets in his possession. When this 
 happens ten chains have been measured, and the record in the 
 Follower's book will stand thus p^4J P"H-J' Wh en station 
 B is approached the links and inches are measured from the last 
 picket in the ground, and a note of this interval made in the 
 book *. Thus the total distance from A to B has been measured. 
 The whole process of measuring is then repeated from B to A } and 
 if the results are not very different the mean of the two results 
 may be regarded as correct. If the measurement is carefully 
 
 * Technically known as the Field Book. 
 
VII. 
 
 BASE LINES. 
 
 135 
 
 made the result ought to be less than a yard in error in a distance 
 of 1000 or 1200 yards. The sources of error are twofold: 
 (1) Overriding of links, which is generally due to carelessness. 
 A third observer to walk along the chain when stretched will 
 prevent this occurring. (2) Unevenness of tfie ground. Hence 
 some practical men have recommended long narrow boards to be 
 laid down by the line where the ground is very uneven. 
 
 127. (3) THE DIRECTION OR BEARING OF THE BASE. 
 This is the third and last " element " of the base line. 
 
 (a) From the Observatory Station A at one end of the line, 
 obtain with the sextant or theodolite the horizontal angle between 
 the sun's N. L. and station B. 
 
 (ft) Compute the sun's True Bearing, either by altitude or 
 time azimuth*. 
 
 We can then compute the T. B. of the station B, and hence 
 the direction or bearing of the base. 
 
 Example 1. The horizontal angle between the second ex- 
 tremity of the base and the sun's centre was found to be 
 114 26' 00"; sun to the right of the object: the sun's T. B. was 
 found to be N. 75 58' E. : find the direction of the base. 
 
 N 
 
 T. B. = N. 75- 58' E. = NAX. 
 
 * The Sun's True Bearing at any hour can be found with great facility 
 by means of the " Sun's True Bearing or Azimuth Tables computed for 
 intervals of four minutes." These tables are supplied to ships by the 
 Admiralty. 
 
136 
 
 MARINE SURVEYING. 
 Observed angle = 114-26' = 
 
 [CHAP. 
 
 .-. Direction of the base 
 
 2r3S-2&' W. 
 
 Example 2. The horizontal angle between an object near 
 the horizon to the left of the sun and the sun's centre was com- 
 puted to be 126- 15', the sun's T. B. was computed to be 
 S. 67-49'E., and the angle between the object and station B 
 was found by theodolite to be 12 26', being to the left of B : 
 find the direction of the base. 
 
 X represents the place of the sun. 
 Sun's T. B. = S. 67 - 49' K = SAX. 
 Obs. angle =126-15 = OAX. 
 
 .\58-26 =SAO, 
 12-26 =OAB. 
 
 .-.70-52 =SAB, 
 . : Direction of the base line AB is S. 70 52' W. 
 
 128. Note. The angle between the sun and the object is 
 seldom horizontal; the horizontal angle may however be computed 
 from the formula*. 
 
 Log. Cos Hor. Angle = log Cos obsd. distance to sun's centre 
 log Cos suns app. alt. 
 
 129. In selecting an object to be observed with the sun, the 
 following conditions ought to be sought for as far as possible. 
 The object ought to be (1) well defined, (2) low, (3) near the 
 horizon, and (4) not less than 90 from the sun. 
 
 * Vide Todkunter's Spherical Trigonometry, p. 92. 3rd Edition. 
 
VII.] BASE LINES. 137 
 
 EXAMINATION. 
 
 (1) What is the use of a base line in a survey ? 
 
 (2) Define a " base of verification." 
 
 (3) Specify the various methods by which the length of a 
 base may be determined. 
 
 (4) In finding a base by masthead angle, what precaution 
 must be taken 1 
 
 (5) Distinguish between standards " ct, traits " and standards 
 " cl, bouts." 
 
 (6) "The geodetic standards vary in different countries;" 
 what does this mean 1 
 
 (7) Mention the causes of variation in the lengths of mea- 
 suring rods. 
 
 (8) What contrivance has been found to insure the uni- 
 formity of length between two points in a measuring rod ? Draw 
 a diagram, and fully explain the principle. 
 
 (9) Describe fully the method by which a base line is mea- 
 sured with the surveying chain. 
 
 (10) Specify the most favourable conditions to be sought in 
 the ground for a base line. 
 
 (11) What are the " three elements of the base"? 
 
 (12) How are the Latitude and Longitude of the Obser- 
 vatory Station usually determined in a marine survey 1 
 
 (13) How is the Direction of the base line found? 
 
 (14) Why is the direction of the base important? 
 
 (15) Under what circumstances may a base measured by 
 patent log be of use ? 
 
 (16) Explain the method of finding the Length of a base by 
 the velocity of sound. 
 
138 MARINE SURVEYING. [CHAP. 
 
 (17) In the case of wind blowing in the direction of the base, 
 when the distance is to be measured by sound, investigate a for- 
 mula by which the true distance may be computed. 
 
 (18) If the time with the wind is 16 seconds, and against it 
 18 '5 seconds, calculate the true time; and if the temperature is 
 62 F., compute the length of the base. 
 
 Results. Time = 17'1 6 seconds, 
 Distance = 20462 feet 
 
 (19) The mean of a set of observations taken at Pile A , 
 lat. 52 00' N., was as follows : 
 
 Zero Steeple 360 . 0' Mag. S. 32 W. 
 At 5 h 16 m P.M. app. time 85 00' fo 
 
 Find the true bearing of Steeple from the Pile, and the varia- 
 tion. Sun's decl. 2030'N. (June, 1877.) 
 
 (20) The mean of a set of obs. taken at theodolite A -V, 
 in lat. 53 14' N. was as follows : Sun's decl. = 16 38' N. Sun's 
 semi = 15' 50". 
 
 Zero Camp ^ 360 0' Mag. N. 45 E. 
 At 6 h O m A.M. app. time 37 25' 
 
 Required the true bearing of the Camp A from X, and the 
 variation. (April, 1876.) 
 
 (21) The mean of a set of obs. taken at Cairn /\, in lat. 
 50 N., was as follows : Sun's decl. 12 30' N. 
 
 Zero Wedge A 360 0'. Mag. N. 30 E. 
 6 h O m A.M. app. time 40 3' (b 
 
 Required the true bearing of the Wedge from the Cairn, and 
 also the- variation. (Oct. 1876.) 
 
 (22) At theodolite A X, in lat. 54 3' N., the following 
 observations were taken. 
 
VII.] BASE LINES 139 
 
 Zero Lighthouse (southward of A') 360 0'. 
 
 First set with sun's lower and right limbs. 
 
 3 h 58 m P.M. 35 IT alt 104 19' 
 
 35 6' 10426' 
 
 35 1' 104 33'. 
 
 Second set with sun's upper and left limbs. 
 
 4 h O m P.M. 35 30' alt 104 4' 
 
 35 26' 104 IT 
 
 35 21' 104 18'. 
 
 Sun's declination = 19 36' 6" K 
 
 Required the true bearing of the Lighthouse from X. 
 
 (Aug. 1875.) 
 
 Answers to Examples 19 22. 
 
 (19) True Bearing = S. 9 39' W. 
 
 Variation = 22 21' W. 
 
 (20) True Bearing = N. 42 10' E. 
 
 Variation = 2 50' W. 
 
 (21) True Bearing = N. 41 50' E. 
 
 Variation =11 50' E. 
 
 (22) True Bearing = S. 30 1 7' E. 
 
CHAPTER VIII. 
 
 TRIANGULATION. 
 
 130. WHEN the base line has been measured the triangula- 
 tion of the survey may be proceeded with. It has been already 
 mentioned that if we know the length of one side of a triangle, 
 and the values of two of its angles, the lengths of the other sides 
 may be computed. 
 
 If a conspicuous object, which we will denote by C, be visible 
 from the extremities A and B of the base, the angles B AC, ABC 
 are observed with the theodolite or sextant, and the line AB being 
 known, we can compute the sides AC and BC. 
 
 Note. If at C the angle ACB be also observed, and the sum 
 of the three observed angles is exactly 180, the triangle is tech- 
 nically said to "close"; but this is very seldom the case: tlm 
 error may be due to the imperfection of even the very best instru- 
 ments, or to the " Personal Equation " of the observer. 
 
 131. 1^ is probable that no two observers will ever see the 
 same phenomenon at exactly the same moment of time, e.g. the 
 instant of a star's transit over a wire. One will see the event a 
 very little before or a very little after another. The difference 
 between the time of observation of some one person, who is known 
 as the Standard, and the observer's time, is called the Personal 
 Equation of the observer. Similarly in taking angles with a large 
 theodolite in an important survey, the Personal Equation has to 
 be taken account of. 
 
CHAP. VIII.] TRIANGULATION. 
 
 Thus in the "Report of the Difference of Longitude between the 
 United States Naval Observatory and the Sayre Observatory of 
 Lehigh University" we have this notification: "My Personal 
 Equation as determined by the use of the instrument at the 
 Washington Observatory was 0"'126 ; the transit observed too 
 soon*." 
 
 132. The selection of the stations for the Triangulation 
 requires considerable experience, as much depends on the nature 
 of the district and the object with which the survey is under- 
 taken. 
 
 The triangles should be as nearly equilateral as possible, be- 
 cause when the angles are all equal, an error in one of the 
 observed angles will produce a minimum of error in the computed 
 sides. If the equilateral triangles are not obtainable, then they 
 must be as " well-conditioned" as possible, i.e. the angles must lie 
 between 30 and 75. An "ill-conditioned" triangle may be 
 denned as one in which the two sides are very long when compared 
 with the base, and therefore the base angles are very much larger 
 than the vertical. It is equally advisable to avoid the contrary 
 error, of having too large a vertical angle. 
 
 133. Increasing from the Base. 
 
 The base line being usually very small in an extensive survey, 
 compared with the distances between the principal points of the 
 triangulation to be ultimately derived from it, the sides of these 
 triangles must as quickly as possible be increased, until they arrive 
 at their greatest limit, and this limit is simply the distance at 
 which they are mutually visible. The following diagram will 
 explain the method of increasing the lengths of the " Primary 
 Triangles " directly from the measured base. 
 
 Let AB be the measured base, C and D represent two trigono- 
 metrical points. Observe all the angles, viz. BAC, ABC, AI) y 
 
 * In this Report, which may be seen in the Library of the Royal Naval 
 College, there is a description of a "Personal Equation Apparatus," designed 
 by Prof. Eastman, for ascertaining the Personal Equations of Observers. 
 
142 
 
 MARINE SURVEYING. 
 
 [CHAP, 
 
 ABD, and compute the distances of C and D from A and B ; then 
 in the triangles DAO, DBC, we have two sides and the included 
 
 angle, and hence DC can be computed from each; we thus have a 
 check upon the accuracy. Next, the distances of E and F from C 
 and D are calculated, and, as before, EF is found from two tri- 
 angles. In the same manner the length of KII is found; and this 
 method is carried on until the trigonometrical stations are at the 
 required distance apart. When this has been accomplished, the 
 primary triangles are piled one upon another until the whole 
 country is embraced ; arrangements are also made either by a 
 second chain of triangles, or else by the triangles overlapping, so 
 that independent values may be obtained for the length of the 
 sides. This is known as the Principal Triangulation. When com- 
 pleted, these Primary triangles are divided into smaller triangles, 
 called Secondary triangles ; these are again subdivided into others, 
 designated as Tertiary triangles, and so on, until the sides of the 
 triangles are about 2 miles long, when they may be considered as 
 straight lines without sensible error. 
 
 134. In the Ordnance Survey of Great Britain the largest- 
 sized theodolites, 3 feet in diameter, were used in fixing the 
 principal stations. The angles of the secondary triangles were 
 observed with smaller instruments. In the Primary triangulatiou 
 
VIII.] TRIANGULATION. 143 
 
 there are 218 stations, at 16 of which no observations were made. 
 The number of observed bearings was 1554. The longest side of 
 any primary triangle was upwards of 1 1 1 miles, and connected 
 Slieve Donard in Co. Down with Sea Fell in Cumberland. The 
 sides of the secondary triangles average, about 8 or 10 miles, and 
 those of the tertiary from 1 to 3 miles. Theodolites of 7, 9, and 
 10 inches diameter were used in measuring the angles of these 
 secondary and tertiary triangles. Kamsden was the maker of the 
 great 3-feet theodolite. 
 
 135- In the First Order of triangles only the most remark- 
 able objects in the country are noted. On these objects signals 
 are placed, and the distances are computed from the Measured 
 Base Line. In the Second Order a series of prominent points are 
 noted in each Primary triangle ; these points form triangles, one 
 of which has a side in common with the Primary triangle, and 
 hence is accurately known. In the same way a Tertiary triangle 
 has a side in common with a secondary triangle, and thus a 
 rigorous exactitude is kept up. 
 
 136. The length of the sides of the smaller triangle depends 
 on the minuteness of the survey ; e.g. if the contents of parishes, 
 estates, &c. are to be calculated, then the sides ought not to be 
 more than 1 or 1J miles for an enclosed country like England, 
 and 2 or 3 miles for a more open country. If no contents are 
 required, and the triangulation is merely to correct a topogra- 
 phical survey, then the distances will depend on the scale of the 
 map. In the Ordnance Survey maps constructed on the scale of 
 6 inches to a mile, two points per square mile were fixed during the 
 triangulation, while on the maps constructed on the scale of 60 
 inches to a mile, sixteen points per square mile were determined. 
 
 137. The reason is, that under the most favourable condi- 
 tions for chaining, a distance so measured is likely to have 20 
 times the amount of error it would have if determined by trian- 
 gulation, and hence the larger the scale on which the work is 
 plotted the closer must be the trigonometrical points if perceptible 
 error is to be avoided. 
 
144 MARINE SURVEYING. [CHAP. 
 
 138. Very distant stations are generally observed at night, 
 the atmosphere being then more adapted for delicate observations. 
 The stations are generally observed by means of Drummond's 
 Light* fixed on them. This consists of a ball of lime 1 J inches 
 in diameter placed in the focus of a parabolic reflector and raised 
 to an intense heat by a stream of oxygen gas. directed on it 
 through a flame of alcohol. The light thus produced is 80 times 
 as intense as that given by an Argand burner. An example of its 
 great power was afforded in the Irish Survey, in which an impor- 
 tant station could not be observed from Devis Mountain, near 
 Belfast, until this light was erected, when, notwithstanding most 
 unfavourable conditions in the weather, the light was brilliantly 
 visible at a distance of nearly 70 miles, and could have been 
 observed at a much greater distance. 
 
 139- ASSUMED BASE. 
 
 Sometimes the triangulation may be carried on before the Base 
 is measured, and then the calculated sides corrected when the Base 
 lias been found ; or, if an error has been made in measuring the 
 Base, the sides of the triangles computed from the erroneous 
 length can be corrected afterwards. 
 
 Thus, let A B = erroneous Base 
 AD = true Base. 
 
 Let BC, A C = the distances to the trigonometrical point E 
 calculated from the erroneous Base. 
 E 
 
 D B 
 
 , DE DA 
 
 By Euclid vi. 4, ^ = ^-j . 
 JJL/ JjJL 
 
 - EA - CA 
 and DE, AE are the true distances required. 
 
 * So called from Lieut. Drummond, who employed it in 182G. 
 
VIII.] TRIANGULATION. 145 
 
 140. SPHERICAL EXCESS. All the observed angles are essen- 
 tially the angles of spherical triangles, and the three angles of every 
 spherical triangle are together greater than 180 (vide Todhunter's 
 Spher. Trig. p. 13). The lines containing the observed angles are 
 really tangents to the earth's surface, whereas to obtain the three 
 points considered as the angular points of a plane triangle, the 
 observed angles must be reduced to the angles contained by the 
 chords of the arcs which form the spherical triangle (Todh. p. 73). 
 The correction for the " spherical excess " is much too small to be 
 applied to angles observed with moderate sized instruments, being 
 completely lost in the greater errors of observation. It must how- 
 ever be taken account of in the primary triangulation. Thus in 
 one of the large triangles of the Ordnance Survey the sum of the 
 three angles was 0"-5 less than 180, and the calculated spherical 
 excess amounted to 1"'29, shewing an error of 1"'79 in the obser- 
 vations. The practical rule adopted is to add one-third of the 
 error to each of the observed angles and thus find the angles of 
 the spherical triangle, and then subtract one-third of the spherical 
 excess from each of the corrected angles, and thus obtain the 
 angles of a plane triangle ready for calculation. 
 
 Xote. The Spherical Excess is obtained from the formula 
 
 . 
 
 or, the excess in seconds = - ^ , where S= area of the triangle, 
 
 T Sill 1 
 
 d r = radius of the earth*. 
 
 141. THE REDUCTION TO THE CENTRE. This is the term 
 applied to the correction when the theodolite is placed near, but 
 not exactly over, the point denoting the station. 
 
 * Vide Todhunter's SpJier. Trig. Chapter ix. throughout; and also Snow- 
 ball's Spher. Trig. Chapter v., especially 6273. The General Eule stated 
 above is derived from Legendre's Theorem, viz.: "If each of the angles of 
 a spherical triangle, whose sides are small when compared with the radius 
 of the sphere, be diminished by one-third of the spherical excess, the triangle 
 may be solved as a plane triangle whose sides are equal to the sides of the 
 spherical triangle, and whose angles are those reduced, angles." 
 
 R. M. S. 10 
 
140 MARINE SURVEYING. [CHAP. 
 
 Suppose A to be the station, and the angle CBD to have been 
 observed from the position J8, then the angle CA D is required. 
 Observe CBD, CBA, and measure AB. 
 
 ADB-, 
 
 .-. CAD + ACB = CBD + ADB, 
 .-. CAD--- CBD + ADB - A CB. 
 
 sinBDA AB . **., AB 
 
 But, . . , ., . - ., . . sin BDA = -j-f- . sm DBA 
 ' sm DBA. AD AD 
 
 sin ACB AB . AB . 
 
 also, -r^r, - -FT,, ' sm ACB = . sm ABC. 
 
 sin ABC AC AC 
 
 These angles being very small, we have 
 
 CAD = CBD + 4^ - sin DBA - A .^ . sin ABC * 
 AD AC 
 
 This correction is not often required, as in the primary triangula- 
 tion care is taken that only those stations are selected which can 
 be actually observed from', and in the secondary and tertiary 
 triangles it is not necessary to observe the third angle. 
 
 142. The triangulation of this country was connected in 1861 
 with that of France and Belgium. The necessary operations woiv 
 carried on by English officers acting in concert with officers of 
 the other countries. The work was commenced in June and 
 was finished in the following January. The following instances 
 
 * The values of AD and AC are known very approximately from the side 
 CD and the observed angles CD A, DC A. 
 
VIII.] TRIANGULATIOX. 147 
 
 of accuracy in the work carried on independently are interesting. 
 The sides of certain triangles were computed by the English 
 and Foreign officers, working from their own bases, the English 
 from the British Triangulatioii carried across the Channel, and the 
 Belgians from their measured base at Ostend. 
 
 Distance. Miles. English Result Belgian Result Difference. 
 
 in metres. in metres. 
 
 Hondschootej ^ zm ^ mi ^ _ Q ^ 
 
 Jvemmel ) 
 
 L } 13-3 21415-30 21415-34 +0-04 
 
 HondschooteJ 
 
 143. Colonel Clarke gives the following interesting account of 
 the operations by which the triangulation of Europe was con- 
 nected with that of Algiers and North Africa*. In 18G8 
 M. Perrier (one of the French officers employed in 1861 in con- 
 necting the triangulation of Great Britain with that of the Conti- 
 nent) satisfied himself that it was possible to connect geodetically 
 Algiers with the peaks of the Sierra Nevada, distant 60 leagues 
 away in Andalusia. In 1879 the French and Spanish officers 
 succeeded in accomplishing the work. 
 
 Twenty miles south-east of Grenada is the highest peak in 
 Spain, Mulhacen, 11420 feet. Distant 50 miles ENE. from this 
 is Tetica, 6820 feet. The line joining these points forms one side 
 of a great quadrilateral ; the opposite side being in Algiers. The 
 terminal points of the African side, which is 66 miles long, arc 
 Filhaousan (3720) and M. Sabiha (1920 feet), each of these moun- 
 tains being about 170 miles from Mulhacen. The other two 
 sides and the diagonals of the quadrilateral span the Mediterra- 
 nean. At each station the signal light was produced by a steam - 
 engine of 6 H. P., working a Gramme's, magneto-electric ma- 
 chine I" in connection with the apparatus of M. Serrin. The 
 labour of transporting to such altitudes this machinery, with the 
 requisite water and fuel, in addition to the ordinary geodetic 
 instruments and equipment, and the maintenance of the whole in 
 
 * Geodesy, p. 261. 
 
 t Vide Ganot's Physics, p. 791. 
 
 102 
 
148 MARINE SURVEYING. [CHAP. VIII. 
 
 a state of efficiency for two months, necessitated the formation of 
 a military post at each station. After immense difficulties the 
 whole was ready on August 20. It was not, however, until Sep- 
 tember 9th that the electric light on Tetica was seen in Algiers ; a 
 red round star-like disk visible at times to the naked eye. On 
 the following day Mulhacen was seen, and the observations were 
 carried on until October 18th, with results which leave nothing to 
 be desired in point of precision. Thus a continuous triangulation 
 now extends from the Shetland Isles into Africa. 
 
 EXAMINATION. 
 
 (1) What is meant by the Triangulation of a country? 
 
 (2) Define a Primary, Secondary, and Tertiary, Triangulation. 
 
 (3) What is meant by the expression " the triangle closes "? 
 
 (4) What two causes tend to prevent a triangle closing I 
 
 (5) What is the " Personal Equation " of an observer ? 
 
 (6) What is an " ill-conditioned " triangle ? 
 
 (7) What conditions ought to be sought in selecting a tri- 
 angle for surveying purposes 1 
 
 (8) Define " Spherical Excess ;" and explain how it is allowed 
 for in computing the triangles of a Trigonometrical Survey. 
 
 (9) Write a note on the method of reducing the observed 
 angles of a primary triangle to a form for computation. 
 
 (10) Define "Reduction to the Centre;" investigate tlu- 
 value of the correction; and draw an explanatory diagram. 
 
 (11) Explain the necessity of an accurate base line, and 
 shew how an error in the measured base may be afterwards 
 allowed for. 
 
 (12) In triangulating between distant objects, explain the 
 cause which renders it necessary that a correction should be 
 applied to the observed horizontal angles for spherical excess. 
 
 (Sept. 1879.) 
 
CHAPTER IX. 
 
 LEVELLING. 
 
 Note. Before studying this chapter the student ought to 
 read the description of the Spirit Level in Chapter vi. ( 88). 
 
 144. DEF. Several points are said to be on a level when 
 they are on the same surface concentric with the surface of still 
 water. 
 
 DEF. Every line traced on the surface of the earth, assumed 
 to be a sphere, is called a Line of True Level. 
 
 DEF. Every line in a plane perpendicular to the direction of 
 a plumb line is called a Line of Apparent Level 
 
 DEF. The difference of level of two points is the difference 
 of the radii of the spheroid upon which the points are situated. 
 
 The object of levelling is to ascertain the relative heights of 
 objects. 
 
 DEF. A continuous line passing through these objects, sup- 
 posed to be in the same vertical plane, forms a Section or Profile. 
 
 145. The heights of these objects are estimated from an 
 imaginary line, either above the highest point or below the lowest 
 point (generally the latter), and this line is known as the Datum 
 Line. 
 
 Thus in the Ordnance Survey of Great Britain and Ireland 
 the Datum Line is the level of the mean tide at Liverpool. In 
 Calcutta the levels of the city are all referred to the bottom, or 
 
150 MARINE SURVEYING. [CHAP. 
 
 sill of the stone at the tide gauge in the dockyard, and this 
 point is 8 -83 feet below the mean sea level. The points where 
 the levelling staves stand when the levelling is in process are 
 known as Bench Marks (B.M.), and are usually engraved 
 on some permanent object, such as a mile stone, or curbstone 
 of a bridge*. The symbol used in the Ordnance Survey is 
 
 the well-known broad arrow ( ^ ) . These marks serve for 
 
 (*) 
 
 future verification, and when the work is carried on from day to 
 day. From these permanent positions side lines of levels can be 
 taken at any time. 
 
 DEF. A CJieck Level is generally run to test the accuracy of 
 the more detailed work. If the difference of level between the 
 two extreme points is found to correspond with that previously 
 ascertained, the presumption is that all the intermediate work is 
 correct. 
 
 146. Levelling is of two kinds, Simple and Compound. 
 Simple Levelling consists in finding the difference of height be- 
 tween two points A and B as follows : Suppose the instrument 
 placed at any point and a levelling staff erected at A ; the reading 
 is noted ; the staff is then removed to /?, and the reading noted ; 
 then the difference of these readings will be the difference of the 
 heights of the points A and B. E.g. At A the reading was 
 7 '25 feet, and at B the reading was 13-64 feet, then we infer that 
 the point B is 6-39 feet below A. 
 
 Compound Levelling is the term applied when a series of 
 levels must be taken between two points far apart, and not mu- 
 tually visible. Each of these levels is accomplished as follows : 
 
 147. The Spirit Level is placed midway between two level- 
 ling staves, but not at a greater distance than 250 feet from each. 
 At greater distances the minute divisions on the staves cannot be 
 
 * One of these marks in the Royal Naval College is on the wall of th>-> 
 building near the door by which the room is reached where the Acting Sub- 
 Lieutenants are examined, and another is near the entrance to the Office of 
 the Director of Studies. 
 
IX.] 
 
 LEVELLING. 
 
 151 
 
 accurately distinguished even m the best telescopes (vide 89). This 
 method eliminates the errors arising from the curvature of the earth 
 and from refraction. The difference between ilw two readings will 
 then be equal to the difference of the heights of the two stations. 
 
 Let A and B be the two stations, the difference of whose 
 heights is required. Let AP and BQ be two levelling staves 
 held at A and B respectively. And let L be the Level erected at 
 C midway between A and B. 
 
 Looking back the height AE is read off: looking forward the, 
 height BD is read off Then by Euc. I. 34, we have BD = EF, 
 .'. AE-BD=AE-FE=AF= difference of the heights. 
 
 Suppose the back reading AE to be 10'7 feet, and the fore 
 reading BD to be 2 -9 feet; then AF=7'8 feet, the height of 
 B above A. 
 
 If BD were the back reading, and AE the forward reading, 
 then we would say that the point A was 7 -8 feet below B. 
 
 Hence the rule : When the back reading exceeds the fore 
 reading, the process is being carried on up hill, but when the back 
 reading is less than the fore reading, the work is going on down hill. 
 
 Example. The back reading is 12 '5 2 feet, and the fore reading 
 is 13-26 feet, which is the higher point? 
 
 Jtesult. The back station is 0-74 feet above the fore station. 
 
 148. The following method is generally pursued in running 
 a series of levels through a tract of country. 
 
152 
 
 MARINE SURVEYING. 
 
 [CHAP. 
 
 A staff-holder places his staff on the bench mark from which 
 the levels are to commence. The Spirit-Level is set up in the 
 most favourable place not more than about 250 feet from the 
 staff, and in the direction in which the levelling is to be carried 
 on ; a second staff is held at the same distance in front of the 
 instrument. When the Level has been adjusted according to the 
 directions given in Chapter vi., draw out the eye-piece of the 
 telescope until the cross- wires are distinct, and then, directing the 
 glass to the back staff, turn the milled-headed screw on the side 
 until the smallest graduations are distinct. The back reading 
 is then made with all possible exactness and entered in the Field 
 Book. See that the instrument is still in adjustment, and read a 
 second time to avoid any error. Next turn the telescope towards 
 the forward staff, and see that the bubble is still in the centre of 
 its "run." Read the divisions on the fore staff most carefully, and 
 note the result; then read a second time for verification. This 
 completes the first levels. 
 
 The surveyor passing the forward staff-holder proceeds to some 
 convenient spot to set up the instrument a second time, and this 
 spot, as before remarked, ought not to be at a greater distance 
 from the staff than 80 or 90 yards. The man who held the back 
 staff now proceeds about the same distance in front of the instru- 
 ment. 
 
 The instrument is again adjusted, and the second levels com- 
 pleted. In this way the work proceeds until the second terminal 
 station is reached. A diagram will perhaps make it easier to 
 understand the process. 
 
 Suppose we wish to ascertain the difference of level between 
 
IX.] LEVELLING. 153 
 
 the two extreme points A and Z. The staves ought to be erected 
 at every point where a serious irregularity exists in the ground. 
 Let two staves be erected, one at A and the other at (7. The 
 Level is placed at B midway between them, and the readings on 
 the staves at A and C are noted and entered. The Level is next 
 placed at Z>, and the staff at C being simply reversed, the gradu- 
 ations being now towards Z, the staff at A is taken forwards to E. 
 The readings at C and E are noted and entered as before. The 
 instrument is then taken to 7< T , and the staff at C to the point Z, 
 and the readings are noted. 
 
 Now by what has gone before the difference between the read- 
 ings at A and C is the difference in the height of A and C ; simi- 
 larly we can obtain the difference of the height of C and E, and 
 of the height of E and Z, and thus we can find the difference 
 between the heights of A and Z. In fact, the difference between 
 the sums of the back and fore readings will be the difference 
 between the heights of the initial and terminal stations, the 
 higher of the two being determined by the usual rule (p. 151). 
 
 Example to illustrate this : 
 
 Stations. Back Reading. Fore Reading. 
 
 AtwdC 10-46 ft. 11 -20 ft. 
 
 C&ndE 11-33 8-00 
 
 Z 7-42 7-91 
 
 Sums = 29 "21 27oT 
 
 27-11 
 
 Difference = 2-10 
 .-. A is 2-10 feet lower than Z. 
 
 149. TO LAY DOWN A SECTION. 
 
 In running a check level we require to enter only the back 
 and fore readings, and any remarks that may be necessary in 
 a column reserved for the latter purpose in the Field Book. 
 But to lay down a section the following form of the Field Book is 
 necessary to register all the information required. 
 
154 
 
 MARINE SURVEYING. 
 
 [CHAP. 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 6 
 
 7 8 
 
 9 10 
 
 11 
 
 o 
 
 sJ 
 
 I 
 
 8 
 
 2 
 
 i 
 
 g> . 
 
 
 
 From Initial 
 
 
 02 
 
 
 Q' 
 
 K 
 
 .g 
 
 i 
 
 5 
 
 i 
 
 Q 
 
 2 
 
 ! 
 
 1 
 
 _ 
 g 
 
 Sta 
 
 ion. 
 
 d 
 i 
 
 KeinarUs. 
 
 
 
 
 
 1 
 
 H 
 
 1 
 
 t, 
 
 1 
 
 ^ 
 
 
 
 
 fa 
 
 
 
 Ft Dec. 
 
 
 Ch. Lks. 
 
 
 Ch. Lks.i Ft. Dec. Ft Dec. 
 
 Ft Dec. 
 
 Ft Dec. 
 
 Ft Dec. 
 
 
 1 
 
 13-71 
 
 205 
 
 5-19 
 
 25 
 
 7-96 
 
 7-88 
 
 5-83 
 
 
 
 5-83 
 
 
 
 
 2 
 
 9-40 
 
 208 
 
 2-27 
 
 25 
 
 3-08 
 
 16-30 
 
 
 
 6-90 
 
 
 
 1-07 
 
 
 3 
 
 3-87 
 
 207 
 
 5-08 
 
 23 
 
 3-40 
 
 11-71 
 
 
 
 7-84 
 
 
 
 8-91 
 
 
 4 
 
 2-63 
 
 208 
 
 6-59 
 
 28 
 
 4-00 
 
 12-41 
 
 
 
 9-78 
 
 
 
 18-69 
 
 
 6 
 
 14-62 
 
 205 
 
 3-92 
 
 26 
 
 5-20 
 
 0-95 
 
 13-67 
 
 
 
 
 
 5-02 
 
 
 6 
 
 17-00 
 
 208 
 
 4-64 
 
 29 
 
 3-89 
 
 1-45 
 
 15-55 
 
 
 
 10-53 
 
 
 
 
 n of) 
 ack ] 
 
 61-23 
 
 
 
 
 
 50-70 
 
 
 
 
 
 
 n of) 
 
 50-70 
 
 
 
 
 
 
 
 
 
 
 
 ore ] 
 
 
 
 
 
 
 
 
 
 
 
 
 Diff. 
 
 10-53 which corresponds with the result in Col. 10. 
 
 Hence we infer that Station No. 6 is 10-53 feet above Station 
 No. 1. 
 
 In delineating a section on paper, especially if the irregulari- 
 ties are not very marked, it is necessary to exaggerate the vertical 
 scale in order that these irregularities may become apparent. 
 This process of course distorts the true appearance of the ground. 
 The horizontal scale is usually made some exact part of the verti- 
 cal, so that the proportion may be apparent to the eye. Thus for 
 the vertical scale we may have 25, 50, 100, or 150 feet to an itich, 
 according to the amount of detail required, and then for the 
 horizontal scale we may take from |- to -^ of the above, or even 
 less, if the section is of great length, and the ground generally 
 flat. 
 
 150. Example. 
 
 Let the following data be taken from the Field Book. 
 Distances, 650, 700, 750, 670, 600, 650, 500, 750 feet, and tin- 
 Differences, Fall 12-2, Fall 18*32, Fall 14-09, Fall 0-21, Rise 8-31:, 
 Fall 24, Fall 24-44, Fall 37'79 feet, reckoned from the initial 
 
IX] 
 
 LEVELLING. 
 
 155 
 
 1LHLB 
 
 2 4 
 
 8-32 
 
 0-21 
 
 l2~27 
 
 
 W *5-02 
 
 ! 
 
 
 V07 
 
 
 5 83 
 
 c> 
 
 I I 
 
 II 
 
 8 01 
 
 Fs 
 
 1 
 
156 MARINE SURVEYING. [CHAP. 
 
 station. To form a section from these elements we may take a 
 vertical scale of 50 feet to one inch, and a horizontal scale of 
 1000 feet to one inch. 
 
 The annexed diagram (fig. 1, p. 155) exhibits the section ob- 
 tained from the above data *. 
 
 We simply draw a horizontal line, and lay off on it the 
 distances 650, 700, 750 feet, <fec. on the scale of 1000 feet to the 
 inch, and at these points draw perpendiculars above or below the 
 horizontal line according as the difference of height is a rise or 
 fall from the initial station, and on these vertical lines lay off 
 12-2, 18-32, &c. feet, on the scale of 50 feet to the inch. Then, 
 drawing a line through these points thus found, and attending to 
 any remarks that may appear in the column reserved for that 
 purpose in the Field Book, we obtain a section or profile of the 
 ground. 
 
 151. Let it be required to delineate the section of ground 
 from the information contained in the Field Book given above 
 (H9)f. 
 
 Draw a horizontal line AB (fig. 2, p. 155), and lay off on it tin- 
 back and fore distances from columns 4 and 6, on the scale of 
 10 chains to one inch. At the points E, F, 6 ? , H, <fec. erect per- 
 pendiculars above or below AB, according as the ground at these 
 stations is above or below the station A. This information is 
 supplied in columns 10 and 11. On these perpendiculars we lay 
 off the rise or fall thus obtained on the scale of 50 feet to the 
 inch. Join the points thus found, having respect of course to the 
 column for remarks (wherein may appear that the stations are on 
 knolls, necks, fec.), and the outline thus formed will be the 
 section required. 
 
 152. Let it be required to form a section from the following 
 data. 
 
 * Frome, p. 98. 
 
 t Manual of Surveying for India, p. 547. 
 
IX.] 
 
 LEVELLING. 
 
 157 
 
 1 
 
 2 
 
 o 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 
 1 
 
 1 
 
 2 
 
 
 
 
 
 3 
 
 
 ! 
 
 1 
 
 i 
 
 j 
 
 1 
 
 5 
 
 1 
 
 i 
 
 Remarks. 
 
 
 
 I 
 
 h 
 
 
 
 From Initial 
 Station. 
 
 5 
 
 
 1 
 
 12-0 
 
 4-0 
 
 8-0 
 
 
 
 8-0 
 
 
 
 50 
 
 
 2 
 
 11-5 
 
 2-5 
 
 9-0 
 
 ' 
 
 17-0 
 
 
 
 40 
 
 
 3 
 
 10-2 
 
 7-9 
 
 2-3 
 
 
 
 19-3 
 
 
 
 10 
 
 Knoll. 
 
 4 
 
 2-1 
 
 4-8 
 
 
 
 2-7 
 
 16-6 
 
 
 
 52 
 
 Neck. 
 
 5 
 
 7-4 
 
 3-4 
 
 4-0 
 
 
 
 20-6 
 
 
 
 12 
 
 Knoll. 
 
 6 
 
 2-6 
 
 12-9 
 
 
 
 10-3 
 
 10-3 
 
 
 
 36 
 
 
 7 
 
 1-4 
 
 11-7 
 
 
 
 10-3 
 
 
 
 o-o 
 
 50 
 
 
 8 
 
 3-6 
 
 10-6 
 
 
 
 7-0 
 
 
 
 7-0 
 
 46 
 
 Level of water. 
 
 In this case measure off the distances 50, 40, 10, &c. along 
 the horizontal line (fig. 3, p. 155) on the scale of 50 feet to one 
 inch ; erect the perpendiculars at tho points above the line 
 or below it as required, and set off along these perpendiculars 
 the lengths 8'0, 17'0, 19 '3, <kc. from columns 6 or 7, on the scale 
 of 20 feet to one inch. Then, as already explained, the dotted 
 line will represent the section required. 
 
 In addition to the method of finding the difference of height 
 between two points, already explained, three other methods are 
 sometimes employed. 
 
 (a) By TJieodolite. 
 
 (b) By Barometer. 
 
 (c) By T/iermometer. 
 
 153. BY THEODOLITE. 
 
 The instrument is set up at one extremity of the line to be 
 levelled ; and every irregularity of the ground being marked by 
 pickets, a levelling staff, furnished with a vane which can be 
 fixed at any required height, is set up at the first serious change 
 in the ground. The vane being set at the exact height of the 
 
158 MARINE SURVEYING. [CHAP. 
 
 telescope, the angle of depression is taken. The instrument and 
 the staff are then made to change places, and the angle of 
 elevation is observed. The mean result may be considered asi 
 correct, and then the vertical arc being clamped to this angle the 
 cross-wires are made to bisect the vane. 
 
 The annexed diagram will explain the method of proceeding*: 
 
 Let a Theodolite be set up at A, and a levelling staff, 
 furnished with a sliding vane, be held at B. The vane is fixed at 
 the exact height of the axis of the telescope. Between these two 
 stations a series of intermediate positions a, b, c, <fec., at which a' 
 levelling staff is to be erected, will be determined by the irregu- 
 larity of the ground. 
 
 The angle of depression from A to B is observed, and then 
 the cross-wires are made to bisect the vane at B. The vanes on 
 the staves at a, b, c, <tc. are then shifted until the line of sight 
 passes through their centres. On arriving at 7?, after reading 
 the heights of the vanes at a, 6, c, &c., and measuring the 
 distances A a, ab, be, &c., the Theodolite is placed at B, and the 
 process is repeated up the hill to C : and thus the work goes on. 
 
 154. To lay down a section on paper from the data thus 
 obtained, we must first draw a horizontal line, and then lay down 
 the angles of elevation and depression, and the measured dis- 
 tances laid off along these lines : then the respective heights of 
 the vanes on each staff being laid down on vertical lines passing 
 through these points, will give the spots 011 which the staves 
 stood, and thus the outline of the ground can be ascertained. 
 * Manual of Surveying for India, p. 519. 
 
IX.} LEVELLING. 159 
 
 This method is chiefly used in running check levels which, as 
 intimated above, are intended merely as a test of accuracy in 
 more detailed work. 
 
 155. DIFFERENCE OF HEIGHTS BY THE BAROMETER. 
 
 We can of course obtain only approximate results by this 
 method. Three causes tend to affect the work, (a) climate, 
 (6) latitude, (c) season. 
 
 Thus in England the diurnal range of the barometer is 
 scarcely perceptible owing to the wide fluctuations to which the 
 instrument is subject, but in many parts of the world, e.g. in 
 parts of the Mediterranean, the height of the barometer being 
 known at 9 A.M. on two successive days, its height at any inter- 
 mediate hour may be closely approximated to. 
 
 156. A. detailed description of the Barometer has been 
 already given in Chapter vi< ( 93 101). The following points 
 need only be noticed in this place. 
 
 The height of the column is affected by two causes, (1) Ca- 
 pillarity, (2) Capacity, 
 
 The capillarity tends to depress the height*. In a tube of a 
 diameter O'l inch the error due to this cause amounts to 0*07 
 inch, with a diameter 0*5 inch it amounts to 0'003 inch. 
 
 The error of capacity is thus caused ; if the mercury rises in 
 the tube, the surface of mercury in the cistern must be depressed, 
 and vice versa. Hence it is necessary to determine in some way 
 the zero point of the cistern surface* 
 
 The error is allowed for in three vbays ( 9396). 
 
 (1) By making the area of the cistern about 100 times that 
 of the tube, and then the resulting error is so slight that it may 
 be neglected. 
 
 (2) By shortening the inches of the scale in the required 
 proportion. 
 
 * Vide Everett's Text-Book of Physics, p. 83. The top of the mercury 
 ought always to be convex; if it should ever assume the concave form some 
 imperfection exists in the mercury. 
 
160 MARINE SURVEYING. [CHAP. 
 
 (3) By a flexible base, as in Fortin's barometer, the surface 
 of the mercury can be always raised or depressed to the zero 
 indicated by the ivory point. 
 
 It may be noticed also that a change of temperature of 1 F. 
 causes an expansion or contraction of the mercury of ^V^ of its 
 bulk. Hence in all delicate observations the temperature must 
 be taken. 
 
 157. METHOD OF USING THE MOUNTAIN BAROMETER 
 
 In carrying it from place to place the most essential point to 
 be borne in mind is that the instrument must be always kept in an 
 inverted position. In reading we must make the index appear to 
 touch the curved surface of the mercury. The height can be read 
 off to the YOTnr^ 1 f an ^ nc ^ 1 ( 99). The thermometer attached to 
 the instrument, which shews the temperature of the mercury, and 
 the detached thermometer, which shews that of the atmosphere, 
 must always be noted. 
 
 158. The following is Dr Button's rule for computing t In- 
 difference between the heights of two stations at which observa- 
 tions have been made. 
 
 (1) Correct the readings of the barometer, reducing them to 
 the same temperature by increasing the colder, or diminishing 
 the warmer, by y^tf part for every degree of difference as slu-wn 
 by the attached thermometers. 
 
 (2) Take the difference of the logs of the heights of the 
 barometer thus corrected, and take the four figures of the 
 difference to t/te right of the decimal point as integers. 
 
 This will be the approximate height in fathoms. 
 
 (3) Correct the number thus found for the temperature 
 shewn by the detached thermometers as follows : 
 
 To every degree that the mean of the two differs from 31 
 take as many yj^ths of the fathoms found above, and add this 
 quantity if the temperature be above 31, but subtract it if the 
 temperature be under 31; the result is the true altitude in 
 fathoms. 
 
IX.] LEVELLING. 161 
 
 We will work the following example to illustrate the above 
 method *. 
 
 Corrected Attached Detached 
 
 Bar. Therm. Therm. 
 
 At^l 30409 in. 61 58 
 
 MB 30-278,, 60 57 
 
 30-278 
 TTTT- = 0-003 Mean of Detached Therms. 
 
 ' 30-278 =57-5 
 
 30-281 31 
 
 26-5 
 
 Log 30-409 = 1-483002 
 
 Log 30-281 = 1-481170 .-. Cor. for Temp. 
 
 Diff. =0-001832 26-5x18-32 
 
 .-. Approx. Alt. in fms. = 18*32. 
 
 Cor. for Temp. = 1-116 + 
 
 19-436 or in feet = 116-7. 
 
 159. The following simple formula is found to give good 
 results. 
 
 Height in feet = 55000 x j^ . 
 
 Adding or subtracting j-J^th of the result for every degree 
 (Fahrenheit) that the mean temperature is above or below 55 F. ; 
 where E Reading of Barometer at the Lower Station, and B' 
 = Reading at the Upper. 
 
 The above example worked by this formula will give the 
 height as 119 feet. 
 
 The following formula has been recommended in the case 
 where the height does not exceed 2000 feet. 
 
 T- 52500 fl . 
 
 
 
 where B = Height of the Barometer at the Lower Station, 
 
 * Frome, p. 116. 
 R.M.S. 11 
 
1(52 MARINE SURVEYING. [CHAP. 
 
 B' = Height of Barometer at the Upper Station, 
 
 T = Temperature (C.) at the Lower Station, 
 
 2\ = Temperature (C.) at Upper Station, 
 
 X = Height in Feet. 
 
 The Aneroid is very useful in determining the altitude. Its 
 results are considered good for at least 4000 feet; but it is 
 necessary that this instrument should be very frequently com- 
 pared with a Standard Mercurial Barometer. 
 
 160. DETERMINATION OF HEIGHTS BY THERMOMETER. 
 
 The Thermometer has been used to ascertain the height of a 
 station approximately. It is well known that, as the pressure of 
 the atmosphere diminishes, water boils at a lower temperature, 
 and this fact has been pressed into service to discover the alti- 
 tude of the place above the sea level. As a rule it is found that 
 the heights so determined are somewhat less than those by the 
 Barometer. The chief advantages of this method are, the great 
 portability of the necessary apparatus, and its small liability to 
 injury. The following appear to be the more important conclu- 
 sions arrived at by Col. Sykes during his experiments in India : 
 When the Boiling Point is 212 (F.) the height is feet; when 
 208 the height is 2050 feet; when 204 the height is 4130 feet: 
 when 200, the height is 625$ feet. 
 
 The Heights thus found are to be corrected by a multiplier, 
 the value of which depends on the temperature of the air. When 
 the temperature is 32 (F.) this multiplier is 1-000; when it is 45, 
 the multiplier is 1'027, and when it is 60, the value of the mul- 
 tiplier is 1-058. 
 
 Note. If the height is calculated from the sea level it is said 
 to be an Absolute Height, but if it depends on the assumed height 
 of another station, it is known as a Dependent Height. 
 
 CONTOURS. 
 
 161. DEF. Contours are horizontal lines either round a 
 group of isolated features of ground, or over an entire tract of 
 country. 
 
IX.] LEVELLING. 1G3 
 
 Let us suppose that a mountainous tract is covered with water, 
 and that this water is gradually subsiding. Now if the water, at 
 vertical intervals of 10 or 15 feet, could be supposed to leave a 
 permanent trace on the sides of the mountain, these traces would 
 give us a very clear idea of Contour Lines. They would (a) be 
 perfectly horizontal, and (b) would trace out perfectly the sinuo- 
 sities of the ground. Hence we should have marked out exact 
 horizontal sections of the ground. 
 
 A good example of these Contour Lines is afforded by the 
 Lines of Equal Soundings in a Chart. 
 
 162. The following is the method of tracing out Contour 
 Lines round an isolated feature. The ground must first be very 
 carefully surveyed, and those parts (ridges, water courses, &c.) 
 which define the configuration of the surface must be marked out 
 by pickets; the exact position of these pickets can be fixed on the 
 plan. 
 
 A Spirit-Level is then placed so as to command the best view 
 of the line of level, and a staff with a moveable vane is placed at 
 one of the pickets ; the vane is then raised or lowered until it is 
 bisected by the cross wires of the Level. The staff, with the vane 
 kept at this height, is then shifted to another point about the same 
 level, and moved up or down the slope until the vane is again 
 bisected, when another picket is driven in to mark the position. 
 This process is continued until it becomes necessary to shift the 
 position of the Instrument itself. Now manifestly the points 
 where the staff rested are on the same level, and these points being 
 accurately laid down in the Plan, the lines which join these posi- 
 tions will mark out a horizontal section of the ground, and bo 
 therefore Contour Lines. The same operation is necessary to form 
 the Contours above and below that first laid out. "When tin; 
 vertical interval is small, the pickets which, mark several Contours 
 can be fixed without shifting the position of the Spirit Level. 
 We require to have merely a levelling staff of sufficient length. 
 
 This method of delineating ground was introduced by French 
 Engineers. 
 
 Suppose that AJ), CD, EF represent three Contours, then \vo 
 
 112 
 
 6 
 
164 MARINE SURVEYING. [CHAP. 
 
 infer that the ground is steep between A and E, but slopes away 
 gradually in the direction BDF. 
 
 Shading is also used to shew the relative steepness of ground. 
 The darker shading indicating regions where the ground is more 
 abrupt, the lighter intimates slopes of a gentler description. 
 
 163. Mountain ranges are represented by fine lines, drawn 
 nearly parallel at first, commencing at the summit of the hills. 
 where they are usually drawn close together, and then they 
 extend, as it were, to the level of the plains where they become 
 more divergent. These lines are known as hachure lines, and are 
 supposed to represent the courses which would be taken by rills 
 of water trickling from the summit to the base. 
 
 EXAMINATION. 
 
 (1) Define True Level, Apparent Level. 
 
 (2) What do you understand by two points being on a level? 
 
 (3) Explain the difference between Simple and Compound 
 Levelling. 
 
 (4) Mention the different methods of finding the vertical 
 height of one point above another in levelling a district 
 
 (5) Describe the process of conducting a Levelling operation 
 between two distinct points. 
 
 (6) What is a Check Level ? What is the use of such a 
 Level ? How is it usually obtained ? 
 
 (7) Can you give any formula by means of which the differ- 
 ence of height between two points can be found approximately by 
 the Barometer? 
 
 (8) What is a Section ? Its use ? How can it be formed ? 
 
IX.] LEVELLING. 165 
 
 (9) What are Contour Lines? Can you give an example from 
 the Admiralty Charts 1 
 
 (10) Explain the process of finding these lines. 
 
 (11) If a Chart or Plan were placed in your hand how would 
 you know which hills were the steepest ? 
 
 (12) Describe the use of the Level and Levelling Staves as 
 applied to Nautical Surveying. (May, 1879.) 
 
 (13) An Aneroid barometer shewed at the sea-level 30 '0 
 inches; and at the summit of a hill it shewed 29 '35 inches; and 
 finally on returning to the first station it marked 30 '02 inches. 
 If 85 feet of altitude = ~^ inch as shewn by the Aneroid, find the 
 height of the hill. (Sept. 1879.) Result = 561 feet. 
 
 (14) Explain how the height of a hill may be ascertained by 
 employing an Aneroid barometer. (April, 1880.) 
 
 (15) In the same level with, and equidistant from stations 
 A and B, the height by Level was on the back staff at A 2*5 
 feet, and on the forward staff at B 7 '2 feet. Shew by means of a 
 diagram the difference of height of the ground at A and B. 
 
 (Aug. 1876.) B is 4-7 feet below A. 
 
 (16) At N equidistant from A and B, the height by Level 011 
 back staff' at A was 3-2 feet, that on forward staff at B was 6 '8 
 feet ; shew by diagram the difference of the height of the ground 
 at A and B. (Feb. 1877.) B is 3'6 ft. below A. 
 
 (17) Prove that the difference between the true and apparent 
 Jevel on the earth, regarded as a sphere, is always proportional to 
 the square of the distance. 
 
CHAPTER X. 
 
 TIDES AND TIDAL OBSERVATIONS. 
 
 164. BEFORE the time of Sir Isaac Newton, although many 
 wonderful guesses were made as to the cause of the Tides, it may 
 be very safely asserted that the subject was not in any wise 
 understood. Thus we have the great Kepler writing, "The sphere 
 of attractive virtue which is in the moon extends as far as the 
 earth and entices up the waters, but as the moon flies rapidly 
 across the zenith, and the waters cannot follow so quickly, a flow 
 of the waters is occasioned in the torrid zone towards the west- 
 ward*." So difficult indeed was the problem considered that it 
 was proverbially known as " the grave of human curiosity." 
 
 Newton, however, applied his discovery of the Law of Gravi- 
 tation with remarkable skill to the solution of the Tides difficulty, 
 and explained satisfactorily the phenomena of the Neap and 
 Spring Tides, and many other points connected with the subject. 
 
 Two principal theories have engaged the attention of mathe- 
 maticians since Newton's time, viz. those of Bernoulli (about 
 1745) and Laplace (1774). Bernoulli imagines the earth to be 
 a perfect sphere and covered throughout by water; then, he 
 says, the waters would assume the same form, at any moment, 
 which they would assume if the acting forces were invariable in 
 magnitude and direction. In other words, taking account first of 
 the moon's action, " a prolate spheroid of revolution in a state of 
 
 * Elementary Treatise on Tides, by Rev. J. Pearson, p. 4. 
 
CHAP. X.] TIDES AND TIDAL OBSERVATIONS. 167 
 
 instantaneous equilibrium is imagined, its major and minor axes 
 are assigned, and then solid geometry exhibits the mathematical 
 relation between the length of a radius vector at any point, and 
 the angles which fix its position, the constants being determined 
 from observation. The like also is done when the sun is con- 
 sidered. The excesses above the mean radii in each case are 
 superimposed, and thus a formula for the height is obtained. 
 A similar investigation brings out the interval between the time 
 of transit and that of high- water*." This is a statement of the 
 famous Equilibrium Theory. 
 
 Laplace, however, takes a different course. He calculates the 
 attractive forces of the sun and moon upon the ocean, and finds 
 them to contain constant terms and periodical terms. He next 
 states that in consequence of the resistance and friction of the 
 waters they would soon have assumed a form of equilibrium under 
 the forces which are represented by the constant terms, and then 
 that the state of such a system of bodies is periodical when the 
 forces themselves are periodical. In this way Laplace gets an 
 expression for the height of the Tide the same as that obtained in 
 the former theory t. This method is usually known as the Dy- 
 namical Theory. 
 
 165. TIDAL PHENOMENA. 
 
 If any observer stations himself in any tidal harbour, he will 
 perceive the following changes in the state of the water. He will 
 observe that the water rises and falls twice each day ; the rising 
 of the water is the result of the flowing of the tide, the falling of 
 the water is the result of the ebbing of the tide. When the water 
 ceases to rise, the state of the tide is called high water, wheji it 
 ceases to fall, it is called low tvater. The time of high water is 
 about 40 minutes later on each succeeding day; and these times 
 
 * Pearson, p. 26. . 
 
 t Pearson, p. 26. In his valuable work, Mr Pearson gives an interesting 
 account of the labours of other eminent workers in this branch of Science. 
 On page 27 will be found a description of the Harmonic Analysis of the 
 Tides, a method introduced by Sir William Thomson. Vide also Thomson 
 and Tait's Nat. Phil. Vol. i. 479 for a description of a " Tide Predicting 
 Machine." 
 
1C8 MARINE SURVEYING. [CHAP. 
 
 of high water are observed to bear a close relation to the position 
 of the moon, e.g. at Ipswich it is high water when the moon is 
 nearly south ; at London Bridge, when she is about south-west ; 
 and at Bristol when she bears about E.S.E. But these facts apply 
 to only one of the high waters each day. The second high water 
 occurs about 12 h 24 m after the first. 
 
 Again, the interval between the moon's passage over the 
 meridian and the time of high water varies with the moons age. 
 At new moon, full moon, 1st quarter, and 3rd quarter (or rather 
 the day which follows these four phases) the interval is nearly the 
 same; from new moon to the 1st quarter, and from full moon to 
 the 3rd quarter, the time of H. W. occurs earlier than would be 
 inferred from using this same interval, whereas from the 1st 
 quarter to full moon, and from 3rd quarter to new moon it occurs 
 somewhat later. 
 
 During the days following new and full moon the Tide rises 
 higher and falls lower than at any other period of the month, 
 such tides are known as Spring Tides; but during the days 
 following the 1st and 3rd quarters, the tide has the least range 
 (vide definition given below 183), and such tides are known as 
 Neap Tides. 
 
 166. The tides are caused by the inequalities of the attractions 
 of the sun and moon on the two sides of the earth, not to their 
 total attractions. This fact must be very carefully borne in mind. 
 
 The accompanying diagram will shew how this differential 
 action produces its effects. 
 
 >M 
 
 Let M represent the position of the moon. The shaded figure 
 
X.] TIDES AND TIDAL OBSERVATIONS. 169 
 
 represents the earth, the centre of gravity being at C, and the 
 plain envelope HPH'P the waters of the ocean, supposed to 
 cover the earth completely. 
 
 Since the moon's influence diminishes inversely as the square 
 of her distance*, it is plain that the -waters at H are more 
 attracted by the moon at M than the earth is at C, and again, 
 the earth at C more than the waters at H'. Now as the earth 
 must move as one mass, and the waters being fluid are free to 
 obey the forces which act on them, it is evident that the waters 
 at // will rise towards J/, being drawn away, as it were, from 
 the earth, and the waters at //' will also rise, being left behind, as 
 it were, by the earth, which moves from them faster than they 
 can follow. 
 
 167. Thus we can understand that there is high water at the 
 same time on opposite sides of the earth. The tide of the hemisphere 
 Avhich has the moon above the horizon is called the Superior Tide, 
 and the corresponding tide at the same time in the opposite 
 hemisphere is called the Inferior Tide. Laplace supposed the 
 obverse action of the moon to be produced by what he called an 
 Anti-moon. Hence the Inferior Tide is called an Anti-lunar, or 
 Anti-solar tide, according as it is produced by the obverse action 
 of the moon or the sun f. 
 
 Again, as the total quantity of water is the same, it follows 
 that a rise of the waters at Zf, //' must be accompanied by a fall 
 
 * i.e. at double the distance, her power is one-fourth, at three times the 
 distance, her power is one-ninth, &c. 
 
 t If we take the sun's distance as 23142 times the earth's radius, and 
 Sts mass as 314760 times that of the earth, the earth's action on a particle 
 
 of water at its surface being represented by 1, then - J - and ^ \. 2 will 
 
 I 1 1 u *> 1 to 
 
 represent the sun's attraction on a particle on the sides of the earth 
 Adjacent to it and turned away from it respectively. In the case of the 
 
 . -0123 , -0123 
 moon we have similarly - and -;.,v Hence the differential action in 
 
 the case of the moon is much greater than in the case of the sun. 
 
 Lockyer's Elementary Lessons in Astronomy, p. 321. 
 According to Sir Isaac Newton the height of the Tidal wave due to 
 the moon is to that due to the sun as 58 to 23, or about o to 2. 
 
170 MARINE SURVEYING. [CHAP. 
 
 of the waters in other places ; accordingly we find that there is a 
 fall along the circle PP. 
 
 168. In order that the water should attain the shape HPITF 
 in the above figure, the waters must flow away from P, P towards 
 //, H'. Hence if the whole earth were covered with a deep 
 ocean, and the moon were placed at the point M, H being the 
 point vertically under J/, PP being the great circle, whose plane 
 is at right angles to the line J/////', we may say that there is 
 high water at //, //' and low water at P, P'. 
 
 169. If the moon were fixed at M, the earth being at rest, 
 and the influence of the sun neglected, the waters of the sea 
 would eventually settle down into the above position of equili- 
 brium, and all tidal motion would cease*. But, as a matter of 
 fact, not one of these conditions is found to obtain in practice; 
 the moon appears to move round the earth once in about 24 h 48 m 
 (which period is called a Lunar Day), and hence the points of high 
 water will travel with her, and the circle of low water will also 
 move. The moon will pass the meridian twice in a Lunar Day, 
 and we shall have two high waters and two low waters in every 
 24 h 48. 
 
 170. The sun also produces an effect on the waters of the 
 ocean, but though his mass is many times greater than that of the 
 moon, his far greater distance is the cause of his inferior power in 
 raising the waters (vide note on preceding page). When the two 
 bodies are exerting their influence together, as at new moon, or in 
 opposite directions, as at full moon, the tides produced by the 
 coincidence of the tidal waves, are greater than usual, and form 
 the Spring Tides ; when the actions of these bodies tend to neu- 
 tralize each other, as at the first and third quarters, the tides are 
 lower than usual, and form the Neap Tides. 
 
 171. This, then, is a brief explanation of what the chief Tidal 
 Phenomena would be on the twofold supposition: (1) that the 
 
 * The figure thus assumed is called the Tide Spheroid. 
 
X.] TIDES AND TIDAL OBSERVATIONS. 171 
 
 ocean covered the whole earth, and (2) that no friction existed 
 between the waters and the bed of the ocean. 
 
 These conditions, however, as has been said, not existing, the 
 above explanation will be of no use in estimating beforehand 
 the probable course of events. In fact, so many and great are the 
 modifications which exist, that the Tidal Phenomena are often 
 extremely complicated, and, at the present date, notwithstanding 
 the talent, skill, and labour that have been spent on the theory and 
 observations of the tides, it is generally acknowledged that no 
 other branch of physical science is in so unsatisfactory a state. 
 We proceed, however, to lay down in as simple a manner as 
 possible, such parts of the subject as are necessary for our 
 purpose. 
 
 172. When the moon passes a meridian of a place, it is not 
 high water at that place, but at some place about 30 behind the 
 moon ; in fact the time of high water follows the time of the 
 moon's transit by an interval of time, greater or less, according to 
 the retardation experienced by the water in passing from the open 
 sea round islands, through narrow channels, and up rivers. This 
 phenomenon of retardation of the times of high water is very well 
 shewn in the accompanying "Co-tidal" map of the British seas on 
 the days of new and full moon. 
 
 DEF. "Co-tidal Lines" are lines drawn through all the places 
 at which it is high water at the same time. They thus represent 
 the form of the tide wave which carries the crest of high water 
 from one point to another. 
 
 173. The following explanation of the method by which the 
 tides are produced will perhaps prove useful to the student. 
 
 Let M be the position of the moon, E the centre of the earth, 
 P a particle of water. If PM represent in magnitude and di- 
 rection the moon's attraction on / J , and if X be taken in ME 
 
 MX MP* 
 
 so that = , then A M will represent in magnitude and 
 
 direction the moon's attraction on the earth at E. 
 
172 MARINE SURVEYING. [CHAP. 
 
 Now conceive the earth reduced to rest with respect to the 
 moon, by a force equal and opposite to XM applied to the earth, 
 and to each particle of water on its surface, then the particle P 
 will be acted on by a force PM, and by another equal and parallel 
 
 B 
 
 to JOT, the resultant of which will be PX. Hence every particle 
 of water in the hemisphere ACB will tend towards the line 
 CM. 
 
 Again, if P' be a particle in the opposite hemisphere, and MP 
 represents the moon's attraction on it, and JOT' be taken such 
 
 that = tr&a > then, by similar reasoning, the moon's disturb-^ 
 
 ing action on every particle in that hemisphere ADB will be 
 towards X' away from M. 
 
 Resolving the moon's disturbing effect into the normal and 
 tangential components PN and PX respectively, their magnitudes 
 may be realized as follows : Suppose the earth to be surrounded 
 by water to the depth of 10000 fathoms, then the normal compo- 
 nent PN will raise a tide yj^ of an inch, and the tangential com- 
 ponent PX will raise a tide of 58 inches. Hence it is easy to see 
 what an important part the tangential force plays in the formation 
 of the tides*. 
 
 174. Suppose that on the day of change, ie., the day on 
 which the moon is new, she crossed the meridian of Greenwich at 
 noon, then, reckoning Greenwich time, the main features of the 
 * Brinkley's Astronomy, edited by Stubbs aud Briinnow, p. 262. 
 
X.] 
 
 TIDES AND TIDAL OBSERVATIONS. 
 
 173 
 
 VH 
 
 IX 
 
 vi: 
 
174 MARINE SURVEYING. [CHAP. 
 
 tidal wave (which has its origin in the Southern Ocean) coming 
 up from the Atlantic are the following. The tidal wave brings 
 H. W. to the western coasts of Spain and Portugal about 2 P.M. ; 
 to the western coast of France about 3 P.M. and at 4 P.M. we see 
 in our map the 4 o'clock Co-tidal Line approaching the English 
 and Irish coasts. At 5 o'clock it is H. W. on the west coast of 
 Ireland, along the south coast of Ireland, at Scilly, along South 
 Devon, at Ushant and at Brest. 
 
 At 6 o'clock the line of H. W. has been divided by Ireland. 
 
 (a) The northern branch causes H. W. along the shores of 
 Mayo, West Donegal, and is approaching the shores of Scotland. 
 
 (6) The southern branch is divided by Cornwall; one part 
 enters the Irish Sea, and makes H. W. at Wicklow in Ireland, 
 and at Pembroke in Wales, the other part makes H. W. at 
 Plymouth, Axmouth, and at Guernsey. 
 
 In this way we can trace the progress of the tidal wave. It 
 reaches the straits of Dover 11 hours after the transit. The 
 other branch which at 6 o'clock approaches Scotland, is divided 
 by Scotland, one branch enters the Irish Sea from the north, and 
 meets the branch which entered from the southward, between the 
 Isle of Man and Co. Down in Ireland. The line in which the 
 two branches of a tide meet in this way, is known as the head or 
 end of the tide. We see that the branch which runs down the East 
 coast of England meets, off the Thames, the branch which had 
 come through the straits of Dover and derived from the same 
 Atlantic wave about 12 hours previously, and here we have a 
 second head or end of the tide*. 
 
 It is evident from the map that when it is H. W. at the 
 entrance of the Irish Sea it must be L. W. between the Isle of 
 Man and the Irish coast, and vice versa. Thus the level of the 
 waters has, as it were, an oscillating motion, and the middle of 
 this oscillation crosses the Irish Sea at a line called a nodal line. 
 Along this there is little rise or range of tide. 
 
 * The greatest Rise and Fall of a Tide will generally take place along 
 this line. Vide Galbraith and Haughton's Manual of Tides and Tidal 
 Currents. 
 
X.] TIDES AND TIDAL OBSERVATIONS. 175 
 
 175. It is very important to distinguish between the appa- 
 rent progress of the high water and the real progress of the 
 water itself : in other words between the Tidal Wave and the 
 Tidal Current. 
 
 The actual transference of the water itself is called the tidal 
 current or stream, and is caused by the tidal wave being ob- 
 structed by shoals or banks. The Tidal Wave is not a circulating 
 current, but is a broad and nearly flat undulation, the highest part 
 being that which immediately follows in the track of the moon. 
 On approaching shoal water this undulation is checked and a 
 current is created, e.g. in the Irish Sea the current nowhere 
 exceeds five knots an hour, in the Pentland Firth at spring tides 
 it is as much as eight knots, and yet the tide wave itself as 
 estimated by the co-tidal lines appears to move at a minimum 
 rate of 20 knots and in some places on the coasts at 50 and even 
 100 knots. In the open waters of the Indian Ocean and the 
 South Atlantic where its progress is not interfered with, the 
 tidal wave moves with a very great velocity, in fact little short of 
 that of the moon herself*. We infer therefore that the tidal wave 
 may be compared to the undulations in a rope, or the shaking of 
 a sail, or the fluttering of a flag. The movement only is trans- 
 mitted. We can easily see this in the case of water itself by 
 watching a floating cork. We see it at one moment on the 
 vertex of a wave, at the next in a hollow ; the wave has passed 
 onwards, but the water immediately about the cork is left behind. 
 Similarly with a boat in a heavy swell, the wave comes up 
 threatening to hurry the boat before it, but merely passes under 
 and onwards leaving the boat in much the same position with 
 respect to a point on the bottom. 
 
 176. The flow and ebb of the tide are due to the alteration 
 of the level of the water caused by the tide wave. When the 
 water in a channel has been set in motion, in other words when a 
 tidal current has been produced, the motion of the water does not 
 
 * Thus in the- South Atlantic the velocity of the tidal wave is 700 miles 
 an hour, and in Lat. 60 S., it is estimated at 670 miles. 
 
176 MARINE SURVEYING. [CHAP. 
 
 immediately cease when it is either high or low water, but the 
 momentum still continues to produce its effects. We must, there- 
 fore, remember that the direction of the current does not neces- 
 sarily and in all cases change with the tide, but on the contrary 
 will under certain circumstances continue to run for some Iwurs 
 after the time of high or low water. This is a very important 
 point. Dr Whewell states * that great confusion has arisen from 
 not distinguishing the time of high water and the time of slack 
 water; the latter means the time when the current changes its 
 direction. Owing to this point not being attended to, many 
 observations have been rendered doubtful and many valueless. 
 Hence, then, it must be borne in mind that while the water runs 
 in one direction it does not necessarily rise, nor while running in 
 the opposite direction is it necessarily falling. 
 
 Sir George Airy in his Tides and Waves remarks that if an 
 observer stations himself on London Bridge he will see that the 
 water continues to run upwards even after the surface of the 
 river has dropped two feet. 
 
 177. DEF. The period during which the tide is stationary is 
 called the stand of tJie tide\. 
 
 DEF. If the current of the flood continues to run for three 
 hours after the time of high water, the tide is called a " tide and 
 half -tide" ; if it runs for an hour and a half, it is called a "tide 
 and quarter-tide" ] and if it runs for three-quarters of an hour, 
 it is known as a " tide and half-quarter tide" This special 
 feature of a "tide and half tide" is found in the Solent, and 
 at some of the Channel Islands. 
 
 * In his Paper on " A First approximation to a Map of Cotidal Lines," 
 contained in the Phil. Trans, for 1833, p. 157. 
 
 t For the sake of clearness, we will present in one view the definitions of 
 these terms : 
 
 Slack water is the cessation of the current caused by the tide. 
 
 High water is when the level of the water ceases to rise. 
 
 Low water is when the level of the water ceases to fall. 
 
 Stand of the Tide is the duration of High or Low water without 
 apparent change of level. 
 
X.] TIDES AND TIDAL OBSERVATIONS. 177 
 
 178. We shall here introduce some definitions which may 
 be found useful before proceeding farther with the subject. The 
 Moon is said to be in conjunction when she is on the same side 
 of the Earth as the Sun, and when the centres of the three bodies 
 are in the same plane* ; and she is said to be in opposition when 
 she is on the opposite side of the earth to the sun, and the 
 centres of the three bodies are in the same plane. When in 
 conjunction, the moon is said to be new, when in opposition, she is 
 said to be full. When the lines joining the centre of the earth 
 with the centres of the moon and sun are at right angles, the moon 
 is said to be in quadrature. This happens twice during a lunar 
 month : when it takes place between new and full moon, she is 
 in her first quarter, when it occurs between full and new moon, 
 she is in her third quarter. 
 
 The following symbols are used to represent these positions of 
 the moon : 
 
 Conjunction j Quadrature D 
 
 Opposition <f 
 
 When the moon is passing from conjunction to opposition she 
 is said to wax, and when passing from opposition to conjunction 
 she is said to wane. Finally she is said to be crescent in her first 
 and fourth quarters, and to be gibbous in her second and third 
 quarters. 
 
 The interval between two conjunctions, i.e. between two new 
 moons, is called a lunation, and is exactly equal to 29 d 12 h 
 44 m 2 8 -84. The interval between conjunction and opposition, i.e. 
 between new and full moon, is called a semi-lunation. 
 
 The paths of the heavenly bodies are not circles, but ellipses ; 
 the sun being in one of the foci. 
 
 When the earth in its orbit is nearest to the sun (which 
 takes place in January) it is said to be in perihelion, and when 
 
 * Though not necesarily in the same line. If the bodies are in the same 
 line, we shall have an Eclipse Total, Partial or Annular of the Sun, at 
 New Moon, and an Eclipse of the Moon Total or Partial at Full Moon. 
 R. M. S. 12 
 
178 MARINE SURVEYING. [CHAP. 
 
 it is farthest from the sun, which takes place in July, it is said 
 to be in aphelion. 
 
 When the moon in her orbit is nearest to the earth, she is 
 said to be in perigee, and when farthest from the earth she is said 
 to be in apogee. 
 
 When the sun and moon are in conjunction or in opposition, 
 their combined effect is greatest, and the tides thus produced are 
 known as Spring Tides. When the two bodies are in quadrature 
 the tides are least, and are known as Neap Tides (vide 170). 
 
 179. Note. The spring tides do not occur on the days of 
 new and full moon, but generally two, or three days later; 
 similarly for the neap tides. 
 
 The spring tides are highest at the equinoxes (viz. in March 
 and September), when the sun is close to the equator, and when 
 the moon's declination is zero at full and change. They are least 
 at the solstices (viz. in June and December), when the sun has 
 greatest declination and the moon's declination is also a maximum 
 and of opposite name to that of the sun. When the equinoctial 
 spring tides occur, the combined influence of the sun and moon is 
 exerted along the circumference of the greatest circle on the globe, 
 while in the case of the solstitial tides their influence is exerted 
 along parallels of smaller circumference. Other things being 
 equal, it is evident that the tides of the winter solstice are higher 
 than those of the summer solstice. 
 
 180. When the moon's declination is zero the tides are 
 equally high in both parts of the lunar day. If the moon is not 
 in the equinoctial, i.e. if her declination be either north or south, 
 then there is a difference in the height of the A.M. and P.M. tides, 
 and this difference is known as the Diurnal Inequality. 
 
 E.g. If the moon's declination is 20 N., then the summit of 
 the superior tide is in 20 north latitude, and the summit of the 
 inferior tide is in 20 south latitude, and in this case the tide in 
 Lat. 10 or 17 will not be so high as when the moon's declination 
 is 10 or 17. The daily change in the moon's declination being 
 
. 
 
 
 X.] TIDES AND TIDAL OBSERVATIONS. 179 
 
 considerable, a sensible inequality is thereby produced in the 
 heights of the A.M. and P.M. tides. 
 
 The maximum of the Diurnal Inequality corresponds to the 
 moon's greatest declination, although it may not appear until 
 after the time of greatest declination. In like manner it dis- 
 appears with the Moon's declination, but not until some time 
 after she has crossed the equator ; thus, at Liverpool the age of 
 the Diurnal Inequality is six days, and at Singapore a day and 
 a half. 
 
 Again, at Bristol, from the end of March to the end of Sep- 
 tember, the P.M. tides are 15 inches higher than the A.M. tides, but 
 from Michaelmas to the end of March the A.M. tides are the 
 higher. 
 
 " The tides at Kurrachee, Bombay, and probably other parts 
 in India, are subject to a large Diurnal Inequality, which may 
 accelerate or retard the times of H.W. sometimes to the amount 
 of 1J or 2 hours, and increase or diminish the rise by a foot 
 or more*." "The low water at Singapore is affected by a large 
 Diurnal Inequality amounting at times to 6 feett." "The tides 
 on the coast of Tong King (S.E. China) are subject to a large 
 Diurnal Inequality, one high and one low water generally occur- 
 ring in the 24 hours J." At King George's Sound in Australia 
 there is a large Diurnal Inequality of the times which sometimes 
 duces the two daily tides to one." 
 
 181. There is sometimes a large inequality in the times as well 
 as in the heights, of the morning and afternoon tides. Thus near 
 Cape Florida, this was found at its maximum in June, 1835, to 
 amount to 2 hours ||. 
 
 Dr Whewell states that the Diurnal Inequality "affects in the 
 largest degree the time of high water, and the height of low 
 water 11." And Mr Parkes, who has investigated the tides of 
 Bombay and Kurrachee, infers " that when there is no Diurnal 
 Inequality in high water time, there is none in low water height, 
 
 * Tide Tables, p. 172 note. f Do. p. 175 note. J Do. p. 177 note. 
 Do. p. 184 note. The Diurnal Inequality of the time means the differ- 
 ence between the lunitidal intervals of successive tides, vide 191. 
 
 Raper's Navigation, p. 322. H Phil. Trans. 1840, p. 165. 
 
 122 
 
180 MARINE SURVEYING. [CHAP. 
 
 and when there is none in high water Jieight, there is none in low 
 water 
 
 182. OTHER CAUSES WHICH AFFECT THE TIDES. 
 These are two in number, viz. The winds and the state of the 
 barometer. 
 
 These affect both the times and the heights. Thus in the 
 North Sea a strong N.N.W. gale and a low barometer raise the 
 surface 2 or 3 feet higher, and cause the tide to flow all along the 
 coast from Pentland Firth to London, half an hour longer than the 
 times and heights predicted in the Tables. Again E.S.E. and 
 S.W. winds produce opposite effects, which will be felt down the 
 Channel as far as Dungeness; while, on the contrary, at the 
 entrance of the Channel, at Plymouth, and as far as Portland, a 
 S.W. wind with a low barometer will raise the surface, but a N.E. 
 wind and a high barometer will always lower itt. 
 
 M. Daussy and Sir John Lubbock, on comparing the differences 
 between the observed heights of the tides and their computed 
 heights with the direction of the wind, concluded that the effects 
 of the wind are insensible; but all practical men believe that these 
 effects are considerable. In support of these latter, Sir George 
 Airy states in his Tides and Waves, that on January 3rd, 1841, a 
 gale "lowered the tides in the Thames 5 feet, and produced 
 a depression of about 3 feet at Hull and at Dover, and a sensible 
 effect at Bristol. At Dublin and at Glasgow the tides were raised 
 by itj." 
 
 M. Daussy (mentioned above), the eminent French hydrogra- 
 pher, was the first to point out the influence of the atmospheric 
 pressure on the height of the tide ; he stated that a low barometer 
 is accompanied by a high tide. 
 
 Sir John Lubbock remarks that a rise of 1 inch in the baro- 
 meter is attended by a depression of 7 inches in the height of 
 the tide at London, and of 1 1 inches at Liverpool. Mr Bunt, 
 
 * Phil. Trans. 1868, p. 688. 
 
 t Tide Tables, p. 113. 
 
 J Ency. Hetrop. Vol. in. Mixed Sciences, p. 390. 
 
 In Connaissance des Temps for 1834. 
 
X.] TIDES AND TIDAL OBSERVATIONS. 181 
 
 who has paid much attention to the tides of Bristol, noticed 
 a depression, under the same circumstances, of 13 J inches in the 
 tides of that port. 
 
 Finally, from an examination of the most carefully conducted 
 tidal observations ever made, viz. those carried out under General 
 Colby's directions during the Ordnance Survey of Ireland in 1842, 
 the Astronomer Royal was led to infer " that a negative irregu- 
 larity in the height of the barometer is accompanied by a positive 
 irregularity in the height of the sea, 12 or 14 times as great as 
 that of the barometer*." 
 
 183. THE RISE AND RANGE OF A TIDE. 
 
 DEF. THE MEAN LEVEL OF THE SEA is the middle between 
 the levels of high water and low water at Springs f. 
 
 Though the heights of H. W. and L. W. may vary to a con- 
 siderable extent, yet the mean level does not vary much ; e.g. in 
 Singapore there is sometimes a difference of 6 feet in the heights 
 of the tide, yet the mean level does not alter more than a few 
 inches. The wind is the principal cause of this slight variation. 
 The mean level may be found very closely by the observation 
 of four consecutive tides which include the Diurnal Inequality. 
 
 DEF. THE RISE OF A TIDE is "the vertical rise above the mean 
 low water of Ordinary Spring Tides J". 
 
 Note. The height of a tide is the same as its rise, and the 
 datum line is the level of low water of Ordinary Springs. 
 
 DEF. THE RANGE OF A TIDE is always measured from low 
 water of one tide to the high water of the following tide. 
 
 Thus, in the case of a Spring Tide, its rise and range are iden- 
 tical , but not so in ordinary and Neap Tides. In all tides, 
 
 * " On the Laws of the Tides on the Coasts of Ireland," by Sir G. Airy. 
 Phil. Trans. 1845. 
 
 t But see some remarks in Rev. J. Pearson's work, p. 31. 
 
 Ordinary, because at the Equinoxes the Spring Tides are higher than 
 usual at high water, and lower than usual at low water. 
 
 Restricting the name "Spring Tide" only to the highest tide in the 
 semilunation, and the term "Neap Tide" only to the tide which has the 
 least height. 
 
182 
 
 MARINE SURVEYING. 
 
 [CHAP. 
 
 except Spring Tides, the rise is always greater than the range ; 
 and this difference is a maximum at Neap Tides. The following 
 diagram will, it is hoped, make these definitions clear to the 
 reader. 
 
 XXVII 
 
 XXVI 
 
 XXV 
 
 XXV 
 
 xxiv 
 
 XXIII 
 XXII 
 XXI 
 
 XVIU 
 XV 
 XVI 
 
 e h k 
 
 7 8 9 10 11 12 13 14 15 X 
 
 The graduated column to the left represents a tide gauge 
 divided into quarter feet. The figures along the line OX represent 
 the days of a semilunation, being the time from Spring Tides after 
 new moon to the Spring Tides after full moon. 
 
 The lower curved line ABC R represents the curve of low 
 
 water, while the upper curved line abc r represents the curve 
 
 of high water. 
 
 Thus, let the line OX represent the line of low water at ordi- 
 nary Spring Tides, and let the high water of ordinary Spring 
 
X.] TIDES AND TIDAL OBSERVATIONS. 183 
 
 Tides reach the height of 25 feet, marked a. Next day the water 
 does not rise to that height, nor does it fall so low as A, let b mark 
 the highest point reached, and B its lowest point; on the next 
 day let c be the highest and C the lowest point of the tide; and so 
 on, until the 8th day, when h marks the highest point and H its 
 lowest. After this the highest points ascend and the lowest points 
 descend continually until the 15th day, when we have again the 
 phenomena of Spring Tides, and thus on continuously from Spring 
 Tides to Spring Tides. On the 8th day of the semilunation we 
 have the highest and lowest points of the Neap Tides. 
 
 184. The Rise or Height of each Tide is reckoned by its ver- 
 tical rise above the line OX. Thus on the day of Spring Tide the 
 rise is measured by the line la, next day the rise is measured by 
 26, next day by 3c, next day by 4c?, and so on, until, on the 8th day, 
 the rise of the Neap is Sh. After this we see the rise increasing 
 day by day until we obtain on the 15th day the Spring Rise 15r, 
 corresponding to the height la, 14 days before. We thus see that 
 the Neap Rise is the least. 
 
 185. Again, the Range of the Tide on any day is estimated 
 by the vertical distance between the two curved lines; e.g. on the 
 5th day the range is measured by the line Ee, on the 9th day by 
 Kk, on the 13th day by Pp. It is also evident that the Neap 
 range, on the 8th day, viz. Hh is the least. Hence the reader will 
 observe why a distinction is drawn between the Neap rise and 
 Neap range, but not between the Spring rise and Spring range. 
 The diagram will also explain why the height and range both 
 decrease from Springs to Neaps, and both increase from Neaps 
 to Springs. 
 
 The mean level of the sea will be half-way between the curves 
 of high and low water ( 183). 
 
 186. The interval between the times of H. W. and L. W. is 
 usually greater than the interval between the times of L.W. and 
 H.W., and this is more marked at Springs than at Neaps. Thus 
 at Liverpool the Flow occupies 5 h and the Ebb 6f h . 
 
184 MARINE SURVEYING. [CHAP. 
 
 DBF. A TIDE DAY is " the interval between two successive 
 arrivals at the same place of the same vertex of the tide wave" It 
 varies in length as the two waves, due to the separate action 
 of the Sun and Moon, approach to or recede from coincidence. 
 In the first and third quarters of the moon the solar tide is to the 
 westward of the lunar tide; in the second and fourth quarters it 
 is to the eastward. 
 
 DBF. The lengthening and shortening of the Tide Day, thus 
 caused, on its mean length is called the PRIMING and LAGGING OF 
 THE TIDE. 
 
 The priming will evidently happen in the first and third 
 quarters of the moon ; the lagging in the second and fourth 
 quarters. 
 
 187. DBF. SINGLE DAY TIDES are those which happen only 
 once in 24 hours. 
 
 The most remarkable case is that of Tong King in China. 
 Whewell thus describes it* : " The tide rises and falls every day 
 during about 12 hours each way. The time of rise is three- 
 quarters of an hour later each day, so that in 15 days the time 
 of H.W. advances from 1 P.M. to midnight; after which it does 
 not advance to 1 A.M., but falls back 13 hours to noon, and so on 
 perpetually. In this way H.W. is always P.M. during the summer 
 half of the year (March to October) and A.M. during the other 
 half. When the tide time falls back 13 hours, the tides are 
 scarcely perceptible ; they are greatest at the intermediate times. 
 Newton explained this by two opposing tides, one 6 hours longer 
 than the other. When the Moon is on the Equator, the morning 
 and evening tides of each component tide are equal, and the tides 
 obliterate each other by interference, which takes place about the 
 Equinoxes. At other periods the higher tides of each component 
 daily pair are compounded into a tide which takes place at the 
 intermediate times, i.e. once a day; and this will be after noon, or 
 before, according to the time of the year." 
 
 * Phil. Trans. 1833, p. 224. 
 
X.] TIDES AND TIDAL OBSERVATIONS. 185 
 
 188. DBF. DOUBLE HALF DAY TIDES are tides which rise 
 and i&\\four times in the 24 hours. 
 
 Poole in Dorset is an instance. Here, the tide ebbs and flows 
 twice in 12 hours. It is L.W at about 3 h 30 m ; then flows regu- 
 larly until 5 h 20 m , and makes proper H.W. about 8 h 50 m ; it then 
 ebbs for 1J hours, and again flows for 1J hours, and finally ebbs 
 until L.W. 
 
 .This is a local circumstance, and is caused by the alteration of 
 level by the velocity of the Ebb current near the shore ; "and this 
 alteration of level, from the hydrostatical effect of currents, shews 
 itself in the form of a second rise of the surface, after it has begun 
 to descend from the true H.W.*" 
 
 189. The highest Springs and the lowest Neaps in each 
 lunation take place at different intervals after the moon's Syzy- 
 giest (i.e. conjunction and opposition) and Quadratures, at differ- 
 ent places ; the interval varying from a few hours to a few days. 
 In like manner the tides corresponding to the Moon's octants 
 (Le. when her transits take place 3 hours after the Sun's meridian 
 passage), follow the time of the octants at different intervals at 
 different places. Such tides are called Octant Tides by Dr 
 Whewellt 
 
 190. ESTABLISHMENT OP THE PORT. 
 
 One or two preliminary Definitions are necessary. 
 
 DBF. THE RETARD, or, AGE OF THE TIDE, is " the interval 
 between the transit of the Moon at which the tide originates and 
 the appearance of the tide itself" 
 
 E. g. if the moon passes the meridian at 4 P.M. and the time of 
 H.W. is 7 P.M., this tide does not, in general, correspond to the 
 transit which took place 3 hours previously, but to a transit which 
 may have happened a couple of days back : thus, on the west 
 
 * Whewell, Do. p. 226. 
 
 t This term is applied when the conjunction and opposition are spoken 
 of together. 
 
 Phil. Trans, for 1840, p. 260. 
 
186 MARINE SURVEYING. [CHAP. 
 
 coasts of Spain and France the tide is 1 J days old, at London it is 
 2J days old, on the west coast of Ireland 2 days, on the south- 
 west coast of America 1 day 20 hours ; whereas on the Pacific 
 side of the United States it is scarcely half a day old. 
 
 The special transit to which any tide really corresponds is 
 found by carefully examining the observations of several preceding 
 tides; the highest of these, being due to the united influence 
 of the Sun and Moon, must correspond to that transit of the Moon 
 which took place at noon or midnight*. An extended series of 
 very carefully conducted observations is, however, necessary to fix 
 this with accuracy. 
 
 The term "Retard" explains itself: the tide appears to be 
 " retarded " in following the Moon in her diurnal course. Ber- 
 noulli first applied the word in this connexion. 
 
 191. DBF. THE LUNITIDAL INTERVAL is quite a different 
 thing from the Age of the tide, and is defined as " the interval 
 which elapses between tJie Moon's transit each day and the time 
 of H. W. next following." It is a varying quantity from day to 
 day during the semilunation. The irregularities are due to the 
 angular distance of the Moon from the Sun, to the distances of 
 Sun and Moon from the Earth, and to the changes in their decli- 
 nations : the first is the chief cause. 
 
 DBF. THE ESTABLISHMENT OF THE PORT is, it must be re- 
 membered, merely an interval of time, not, in the first instance, an 
 hour of the day by any clock. 
 
 DBF. THE VULGAR ESTABLISHMENT OF THE PORT is " tlw in- 
 terval between the time of the Moon's meridian passage on the day 
 of new or full moon and the following high water" Le. the lunitidal 
 interval on the days of full and change f. 
 
 DBF. THE CORRECTED ESTABLISHMENT OF THE PORT is " the 
 mean of the lunitidal intervals for all the days between new and 
 full moon" 
 
 * Eaper, p. 320. 
 
 t The Vulgar Establishment is registered on the Admiralty Charts. 
 
X.] TIDES AND TIDAL OBSERVATIONS. 187 
 
 Note. If the new moon occurred exactly at noon or midnight 
 of the place, then the Sun and Moon would pass the meridian at 
 the same instant, and in this case the Establishment of the Port 
 might be given as the time from apparent noon, but it is evident 
 that this must be a very rare event. 
 
 When, therefore, it is said that the Change Tide or Establish- 
 ment of the Port is 9 h 20 m , it does not mean that on every day of 
 new and full moon it is H.W. at 9 h 20 m P.M., but that it is H.W. 
 9 h 20 m after tlie Moon's transit on those days. In the event, how- 
 ever, of the moon passing the meridian at apparent noon, then, as 
 has been explained, it will be H.W. at 9 h 20 m P.M. Keeping this 
 in mind, we have given in the Nautical Almanac the following 
 definition of the Establishment of the Port, viz. " The actual time 
 of .high water when the Moon passes the meridian at the same 
 time as the Sun ; or the interval between the time of transit of 
 the Moon and the time of high water on full and change days." 
 
 DBF. The difference between the Corrected Establishment of 
 the Port and the Lunitidal Interval at each transit of the Moon is 
 termed the Semimenstrual Inequality*. 
 
 192. The Corrected or Mean Establishment may be deter- 
 mined by observing the intervals between the times of the moon's 
 meridian passage and the times of the following high waters for a 
 semilunation, and taking the mean of them. To obtain the 
 elements of a complete tide table, the high waters and low waters 
 during an entire lunation ought to be observed. The most impor- 
 tant lunations are those about the times of the Solstices and 
 Equinoxes. 
 
 193. DETERMINATION BY MEANS OP CURVES. 
 
 To compare the time of H.W. with the time of the Moon's tran- 
 sit, we must compute the latter from the Nautical Almanac, and 
 find how much the time of H.W. is" after the Moon's transit at the 
 
 * Or, semi-monthly inequality. This term is due to Sir John Lubbock. 
 Dr Whewell introduced the terms Vulgar and Mean or Corrected Establish- 
 ments. 
 
188 
 
 MARINE SURVEYING. 
 
 [CHAP. 
 
 place. These differences are lunitidal intervals. Let us suppose 
 that we have obtained the observations for H.W., we annex the 
 times of the Moon's transit, and then put in the lunitidal in- 
 tervals computed from these data. We thus obtain the following 
 table*: 
 
 Day. 
 
 A.M. 
 
 or 
 
 P.M. 
 
 Times of 
 lILrh Water. 
 
 Times of 
 Moon's Transit. 
 
 Lunitidal 
 Intervals. 
 
 
 b m 
 
 h m 
 
 h m 
 
 
 A.M. 
 
 1033 
 
 234 
 
 11 
 
 P.M. 1 7 
 
 1057 
 
 232 
 
 
 A.M. 1 29 
 
 1121 
 
 230 
 
 12 
 
 P.M. 1 51 
 
 1145 
 
 226 
 
 
 A.M. 2 11 
 
 9 
 
 220 
 
 13 
 
 P.M. 2 29 
 
 032 
 
 216 
 
 i 
 
 A.M. 2 48 
 
 055 
 
 2 8 
 
 14 
 
 | 
 
 P.M. 
 
 3 3 
 
 1 19 
 
 2 2 
 
 
 A.M. 
 
 3 21 
 
 1 42 
 
 154 
 
 15 j 
 
 P.M. 
 
 336 
 
 2 6 
 
 148 
 
 
 A.M. 
 
 3 54 
 
 229 
 
 1 40 
 
 16 
 
 P.M. 
 
 4 9 
 
 252 
 
 1 34 
 
 
 A.M. 
 
 426 
 
 315 
 
 128 
 
 17 
 
 P.M. 
 
 443 
 
 339 
 
 1 24 
 
 
 A.M. 
 
 53 43 
 
 1 20 
 
 18 
 
 P.M. 
 
 5 23 4 27 
 
 119 
 
 n \ 
 
 A.M. 
 
 5 46 4 51 
 
 1 18 
 
 19 | 
 
 P.M. 
 
 6 9 
 
 5 16 
 
 1 18 
 
 
 A.M. 
 
 634 
 
 541 
 
 120 
 
 nn 
 20 
 
 P.M. 
 
 7 1 
 
 
 
 We first put in the corrected time of the Moon's transit 
 10 h 33 m . We next insert the time of the following high water 
 l h 7 m . Then take the first from the second (increased where 
 necessary by 12 h ), and the lunitidal interval 2 h 34 m is obtained. 
 
 Full Moon occurred between the 12th and 13th, because on 
 the 13th the Moon was 12 hours from the Sun. Now by taking 
 the times of H. W. as abscissae and the lunitidal intervals as ordi- 
 nates, we can ascertain whether a tolerably regular curve is 
 formed, and therefore whether the lunitidal intervals follow a 
 regular law. 
 
 Again, if we set off the times of the Moon's transit as abscissa? 
 and the lunitidal intervals as ordinates we obtain a curve of semi- 
 
 ; ^fallual of Scientific Enquiry, p. 66. 
 
X.] TIDES AND TIDAL OBSERVATIONS. 189 
 
 monthly inequality; and when this curve has been determined by 
 observation for any place, the hour of JELW. at any time at that 
 place may be predicted. 
 
 Once more, if we set off on this curve the ordinate which 
 corresponds to the time when the Moon's transit is O h or 12 h , we 
 obtain a graphic method of determining the Establishment of the 
 Port. 
 
 Finally, if we set off the times of the Moon's transit as ab- 
 scissae, and the observed heights of the tides as ordinates, we 
 obtain the Law of the Heights of H. W. from Springs to Neaps. 
 The maximum ordinate (- spring height) follows the days of new 
 and full moon by 1, 2, or 3 days, and, as the new or full moon 
 is supposed to produce the spring tide, we thus get the Age of the 
 Tide. 
 
 194. FIRST AND SECOND HIGH WATERS. 
 
 This is a purely local matter, and two very good examples are 
 found in the Solent and at Havre. In the Solent " this double 
 high water is probably caused by the tidal stream at Spithead, for, 
 as long as that stream runs strong to the westward, the tide is 
 kept up in Southampton water, and there is no fall of consequence 
 until the stream begins to slack at Spithead; but when the 
 stream makes to the eastward at Spithead the water falls rapidly 
 at Southampton. After low water the tide rises pretty steadily 
 there for 7 hours, which may be regarded as giving the first 
 or proper high water ; it then ebbs for about an hour, and falls 9 
 inches, when it again begins to rise, and in about \\ hours reaches 
 its former level, and sometimes goes higher, this is called the 
 second high water*." 
 
 At Havre, where the Spring Rise is 22 feet and Neap Rise is 
 1 8 feet, the high water remains stationary for about an hour, with 
 a rise and fall of 3 or 4 inches for another hour; and during 
 a total interval of 3 hours the tide only rises and falls 13 inches. 
 This long period of slack water is very valuable for the traffic of 
 
 * Tide Tables for 1881, p. 110 note. 
 
190 MAHINE SURVEYING. [CHAP. 
 
 the port, and it allows fifteen or sixteen vessels to enter or leave 
 the docks on the same tide*." 
 
 195. DBF. THE TIDAL CONSTANTS for any port are certain 
 corrections of time and height, which being applied with their 
 proper signs to the time and height of the tide at the Standard 
 Port, will give the time and height of high water at the port 
 required. 
 
 E.g. Portsmouth is the Standard Port of reference for all 
 places on the south coast of England, from Littlehampton to 
 Portland ; Plymouth is the standard for all harbours from Brid- 
 port to the Scilly Isles ; London for Gravesend, Woolwich, Green- 
 wich, &c. 
 
 Examples. The Tidal Constants for Gravesend are : 
 - O h 48 m , and - 3 ft 3 in . 
 
 The Tidal Constants for Portland are - 4 h 40 m and - 6 ft 3 in . 
 
 The Tidal Constants for Scarborough are + O h 49 m and + l ft 5 in . 
 Sunderland being the Standard Port for reference in this last 
 case. 
 
 196. METHODS OF FINDING THE TIME OF H.W. ON ANY 
 
 PARTICULAR DAY. 
 
 There are three methods : 
 
 I. By Stated Rules. 
 
 II. By Admiralty Tide Tables. 
 
 III. By the Tidal Constants. 
 
 197. ! BY RULE as given in treatises on Navigation. 
 
 (a) Find the mean time of the Moon's meridian passage on 
 the given day. 
 
 (/3) Correct this mean time by the equation of time to the 
 nearest minute. 
 
 (y) Take out approximately the Moon's semi-diameter. 
 
 * Tide Tables for 1881, p. Ill note. 
 
X.] TIDES AND TIDAL OBSERVATIONS. 191 
 
 (8) Enter Table (I), page 5, in Inman's Tables, and take out 
 the correction there found with its proper sign. 
 
 (c) Apply this to the mean time of passage. 
 () To this add the Establishment of the Port. 
 
 Then 
 
 i. If the sum < 12 h , the time is the mean time of the 
 afternoon tide on the given day. 
 
 ii. If the sum > 12 h 24 m , or > 24 h 48 m , subtract these sums 
 from it, and the remainder will be the time of H.W. on the after- 
 noon on the given day. 
 
 (rj) When the time of the P.M. tide has thus been found we 
 can find the time of the A.M. tide on the same day by subtracting 
 24 minutes from it; and the time of the A.M. tide on the following 
 day by adding 24 minutes to it. 
 
 This method gives only approximate results, and is now seldom 
 employed. 
 
 198. II. BY ADMIRALTY TIDE TABLES. 
 
 This publication contains the Times and the Heights of the AM. 
 and P.M. tides at twenty-three standard ports in the British Isles, 
 and at Brest, for every day of the year ; and also the times and 
 heights at full and change for the principal places of the globe. 
 These Tables are computed in the Hydrographic Office, and give 
 very interesting and valuable information about the Tides, espe- 
 cially those around our own coasts. In the Tide Tables for 1882 
 are given the tables of Lubbock, Whewell, and Pearson, by means 
 of which the time and height of the tide may be computed. 
 
 The most important Tables are numbered from I. X., and 
 may be briefly noticed. 
 
 Table I. shews the semi-monthly inequality ; or the interval 
 between the Moon's transit, two days preceding a London Tide 
 (and denoted by the letter B), and the time of high water ; the 
 Moon's parallax being 57', her declination 15; the Sun's parallax 
 8 "8" and declination 15. For Portsmouth this constant is about 
 l d 12 h , for London 2 d 3 h , for Hull l d 19 h , for Leith l d 15 h , &c. 
 
192 MARINE SURVEYING. [CHAP. 
 
 Table II. gives the correction for the Moon's parallax. The 
 arguments being the Moon's transit (B) and the H.P. The cor- 
 rection ranges between + 9 m and - 9 m . 
 
 Table III. shews the correction for the Moon's declination. 
 The arguments being the Moon's transit (B) and declination. 
 The limits in this case are + 12 m and - 12 m . 
 
 Table IV. shews the correction for the Sun's declination. The 
 arguments being the Moon's transit (B) and Sun's declination. 
 The limits are + 5 m and - 5 m . 
 
 Table V. shews the correction for the Sun's parallax. The 
 arguments being Moon's transit (B) and Sun's parallax, which 
 ranges between 8-94" and 8-66". 
 
 The limits of the correction are + 3 m and - 3 m . 
 
 These five Tables refer to the time of high water, and a second 
 series of five similar Tables give the corrections to be applied to 
 the height. In using some of these last Tables for other ports 
 than London, the corrections must be multiplied by a certain con- 
 stant which is proportionate to the difference between the semi- 
 monthly inequalities of the two places. 
 
 199. III. BY THE TIDAL CONSTANTS. 
 
 We merely apply to the times of the standard port the tidal 
 constant for the time at the required port; the result is the time 
 of H.W. at the port required. 
 
 Kg. Find the time of H.W. at Greenwich on December 14th, 
 in the afternoon. 
 
 At London Bridge it is H.W at 7 h 37 m P.M. 
 Tidal Constant 15 
 
 .-. P.M. time of H.W. at Greenwich = 7 h 22 m 
 
 200. TIDAL OBSERVATIONS. 
 
 In every survey of a harbour tidal observations form a most 
 important portion of the work. In the next chapter the subject 
 
X.] TIDES AND TIDAL OBSERVATIONS. 193 
 
 of Soundings will be fully explained ; we shall therefore here con- 
 fine our attention to the methods of observing for the Elements of 
 a Tide Table, which are : 
 
 (1) The Establishment of the Port. 
 
 (2) The rise and range of the tide throughout the semi- 
 lunation. 
 
 (3) Any important circumstances which may thus become 
 apparent in the rise and fall, the flow and ebb, of the tide, &c. 
 
 201. A TIDE GAUGE is a long batten, usually coloured black, 
 with divisions painted in white, or cut into the wood. These 
 divisions are feet and inches, or else feet and decimals of a foot. 
 This is fixed firmly and vertically to a post driven into the ground, 
 or fixed to a rock. The zero of the scale must be below the level 
 of the lowest water observed in that spot in the harbour. If 
 the harbour is long and narrow, several such gauges ought to 
 be erected, and their positions carefully noted. A trustworthy 
 observer must be stationed at each when the more important 
 observations are being made. For very delicate observations in 
 somewhat exposed situations, the following seems to be a satisfac- 
 tory form of Tide Gauge. An upright tube, open at the top and 
 closed at the bottom, with two or three small holes in the side near 
 the closed end, is taken of a length somewhat greater than the 
 rise of the Spring Tides at the place. This tube must be firmly 
 fixed to an upright post driven into the ground or otherwise 
 secured, so that the bottom of the tube may be always beneath 
 the lowest water and its top above the highest water. The water 
 is thus enabled to reach the same level inside and outside the tube, 
 but the outside agitation of the waves does not sensibly affect the 
 water inside. An upright rod attached to a float (which must be 
 of nearly the same area as the section of the tube) will thus 
 intimate very clearly the exact height of the water at any 
 moment. 
 
 This may be slightly modified thus: a string from the float 
 may pass over a pulley, and have a weight with an index mark 
 secured to the other end. The index will mark on a graduated 
 E. M. S. 13 
 
194 MARINE SURVEYING. [CHAP. 
 
 sqale the rise arid fall of the float, and therefore of the tide 
 itself* 
 
 202. Observations ought to be made every half hour at least, 
 and every 5 minutes when the time of high and low water draws 
 near. It is well to begin a series of tidal observations by noting 
 the state of the tide every half or quarter of an hour during 
 the whole day and night. In this way any unusual circum- 
 stance at once becomes apparent, such, for instance, as four tides 
 in a lunar day, the rise and fall being checked or accelerated 
 through any local causes, &c. 
 
 203. It is often very difficult to determine the exact time of 
 either high or low water, not merely on account of the waves 
 washing against the gauge (a source of error that can be eliminated 
 in the ways described above), but by reason of the water stopping 
 or hanging near the time of highest or lowest water, or else rising 
 and falling irregularly. Thus Sir G. Airy, in discussing the tidal ob- 
 servations obtained by General Colby during the Irish Survey (182), 
 writes: " The difficulty of fixing on the precise time of high or low 
 water will appear from this statement, that sometimes twenty or 
 twenty-four successive observations (occupying a period of l h 40 m 
 or 2 h ) are registered with the same decimal of a foot for the height. 
 The most perplexing case is that where the change of height, 
 in respect to change of time, follows or may follow different 
 laws before and after the principal phase. Thus at Limerick 
 after L. W. the water sometimes rises as much in 1 minutes as it 
 had previously dropped in 2 hours. It therefore appears right 
 here, if several successive observations about L.W. are registered 
 at the same decimal of a foot, to suppose that the real L.W. is a 
 little before the last of those observations t." 
 
 * Sir G. Airy, in his Treatise already alluded to, when speaking with 
 great approbation of the method pursued in tidal observations at Sheerness, 
 thinks that if the waves outside are very boisterous, the waters might be 
 admitted first into one side of a trough, by a few small holes near the 
 bottom, and then through a second partition into the chamber where the 
 float is placed. Thus all outside disturbance would be eliminated. 
 
 t Phil Trans. 1845, p. 9. 
 
X.] TIDES AND TIDAL OBSERVATIONS. 195 
 
 204. The three following methods of procedure have at dif- 
 ferent times been recommended. 
 
 (1) Note the time when the water ceases to rise, and the 
 time when it begins to fall. The mean of these two times will be 
 the time of high water. 
 
 (2) A more satisfactory method is as follows : At equal 
 intervals of time (e.g. every 5 minutes) for about an hour near the 
 time of H.W. observe the height of the water: either take the 
 means of times and heights for the time and height of H.W., 
 and similarly for L.W. ; or, take the highest point reached for 
 the height and the corresponding time for the moment of H.W. 
 
 (3) A third, and still better, method of dealing with observa- 
 tions made at equal intervals of time, may be adopted by taking 
 the times of observation as abscissae and the heights observed at 
 those times as ordinates, and thus form a curve. 
 
 Example. Find the Time and Height of high water from 
 the following observations : 
 
 Times O h O m , 05, 010, 015, 020, 025, 030, 
 0_35, 040, 045, 050, 055, 060. 
 
 Heights 6 ft - O in -, 66, 66, 69, 610, 611, 70, 
 611, 611, 69, 65, 62, 510. 
 
 Now the selection of the greatest height 7 ft . O in . will give the 
 time of H.W. as O h 30 m , but if transferred to paper*, and the 
 line connecting the ordinates smoothed into a curve, the general 
 run of the height will give the time of H.W. about O h 32 m , or 
 O h 33 m . 
 
 This method of finding the time and height of the H.W. 
 is called Interpolating. 
 
 * Messrs Eden, Fisher and Co., 50, Lombard Street, E.G., sell paper 
 admirably adapted for curve tracing. The square inch is divided by faint 
 blue lines into 64 parts. It is known as " Section-ruled paper," and may be 
 had in different sizes. 
 
 132 
 
196 
 
 MARINE SURVEYING. 
 
 [CHAP. 
 
 The following is the form of the Tide Register supplied to 
 Surveying Ships. 
 
 Register of Tides observed at in the Month of 188... 
 
 Day. 
 
 High Water. 
 
 Moon's 
 Transit. 
 
 Bar 
 at 
 H.W. 
 
 Low Water. 
 
 Moon's 
 Transit 
 
 Bar 
 at 
 L.W. 
 
 II 
 
 "0 
 
 CO 
 
 I 1 
 
 Wind. 
 
 Turn of 
 Stream. 
 
 
 Time 
 H.M. 
 
 Height 
 
 H.M. 
 
 ins. 
 
 Time 
 
 11. .M. 
 
 Height 
 F.I. 
 
 H.M. 
 
 in,. 
 
 F.I. 
 
 D. 
 
 Direct 
 Force. 
 
 Flood 
 H.M. 
 
 Ebb 
 H.M. 
 
 A.M. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 P.M. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (Note. The columns for the moon's transit are to be kept 
 exclusively for the purpose intended, and are not to be otherwise 
 occupied.) 
 
 Observers' Names. 
 
 205. It sometimes happens that there will not be any P.M. 
 tide, because the lunar day is longer than the mean solar day. 
 The mean interval between successive tides is 12 h 24 m 22*. Hence 
 if the A.M. tide is at ll h 50 m , the next high water will not occur 
 until past midnight, and this will be the A.M. tide of the following 
 day. 
 
 Note. It may be well to call attention here to the Four very 
 remarkable Papers 011 Tidal Evolution, by Professor G. H. Dar- 
 win, published in the Philosophical Transactions for the years 
 1879 and 1880. The Mathematical investigations are of the very 
 highest order; but the ordinary reader will find the "Review of 
 the Tidal Theory of Evolution as applied to the Earth and the 
 other members of the Solar System" (pp. 879885, Part n. 1880), 
 in which the Professor gathers up his conclusions, sufficiently 
 popular to give him a deep interest in the important part played 
 by the Tides in the past history of our Earth. The student ought 
 also to read the Phil Trans, for 1879, Part n. pp. 532537, and 
 especially Dr Ball's pamphlet "A Glance through the Corridors of 
 Time " published by Messrs Macmillan and Co. 
 
X.] TIDES AND TIDAL OBSERVATIONS. 197 
 
 EXAMINATION. 
 
 (1) Explain, by means of a diagram, the phenomena of two 
 simultaneous tides in opposite hemispheres. 
 
 (2) Explain the causes of Spring and Neap tides. 
 
 (3) What is an Octant tide ? 
 
 (4) Under what conditions are the spring tides greatest ? 
 
 (5) Distinguish between a tidal wave and a tidal current. 
 
 (6) Explain the following terms : 
 
 Rise and Fall of the tide. 
 Flow and Ebb of the tide. 
 
 (7) Explain the following terms : 
 
 Slack water. 
 Stand of the tide. 
 Head of the tide. 
 
 (8) Define the following terms : 
 
 First of the flood. 
 Last of the flood. 
 First and Second high waters. 
 
 (9) Define the following terms : 
 
 Tide and half tide. 
 
 Tide and quarter tide. 
 
 Tide day. 
 
 Priming and lagging of the tide. 
 
 (10) Define very carefully the following important terms : 
 
 Change tide. 
 
 Vulgar Establishment of the Port. 
 Corrected Establishment of the Port. 
 Lunitidal interval. 
 Semimenstrual inequality. 
 
 (11) Define, and illustrate by a diagram, the following tidal 
 terms: 
 
 Rise of a tide. 
 
 Range of a tide. 
 
 Height of a tide. 
 
 Rise and range of a neap tide. 
 
198 MARINE SURVEYING. [CHAP. 
 
 (12) What are the " elements" of a tide table ? 
 
 (13) What is the diurnal inequality ? What causes it? 
 
 (14) Define the "mean level of the sea." 
 
 (15) What is a lunation ? Give its length. 
 
 (16) What term is applied to the interval between new and 
 full moon? 
 
 (17) When is the moon said to be new, and when is she 
 full? 
 
 (18) Explain the following terms : Quadrature, Perihelion, 
 Aphelion, Perigee, Apogee, Solstice, Equinox. 
 
 (19) Write down the symbols for Conjunction, Opposition, 
 and Quadrature. 
 
 (20) What do you understand by the Superior and Inferior 
 tides ? 
 
 (21) Explain the meaning of the Tidal Constants of a port. 
 
 (22) Describe a tide gauge, and the precautions which ought 
 to be observed in its erection. 
 
 (23) If the observations must be made where waves are 
 likely to prove a source of uncertainty, what method is to be 
 pursued ? 
 
 (24) Explain very fully the method of making tidal observa- 
 tions, and note the seasons at which these observations are most 
 important. 
 
 (25) I find this entry in the official Tide Tables, "Half of the 
 mean Spring Range is 9ft. 4in." Explain this. 
 
 (26) If, 011 a certain date, I find in the Tide Tables that 
 the time of H.W. at a place is 6 h 5 m ; is this mean or apparent 
 time 1 Is it (for England) local or Greenwich time ? 
 
 (27) The height of the tide is 11 ft. 6 in. ; what is the Datum 
 line? 
 
 (28) What are Single Day tides ? 
 
 (29) Explain Double Half Day tides. 
 
X.] TIDES AND TIDAL OBSERVATIONS. 199 
 
 (30) Explain this sentence : " The tides of Avatcha Bay are 
 affected by diurnal inequality." 
 
 (31) Explain this sentence: "The times of H.W. being the 
 corrected and not the vulgar establishment." 
 
 (32) Explain the abbreviation " H.W. F. & C." 
 
 (33) How is the time of H.W. calculated for any particular 
 day when the Change Tide is known ? 
 
 (Exam. Papers, March, 1880.) 
 
 (34) Does the diurnal inequality affect the time or the height 
 of the tide 1 
 
 (35) But other causes may require a different answer. Shew 
 that this is so. 
 
 (36) Does the direction of the tidal current always change at 
 the time of H.W. 1 (Exam. Papers, April, 1880.) 
 
 (37) When do the highest tides occur? (Sept. 1880.) 
 
 (38) State how the time of H.W. by a tide gauge is accu- 
 rately determined, and in what time (apparent or mean) it is 
 expressed. (Dec. 1875.) 
 
 (39) Explain how the mean tide level is ascertained. 
 
 (Sept. 1876.) 
 
 (40) If the tide continues to flow for J-hour after the time of 
 H.W., what term is given to the tide ? 
 
 (41) How would you find the time and height of H.W. at a 
 home port by means of the Tidal Constants 1 
 
 (42) State the effect of a rising and falling barometer respec- 
 tively on the tides. 
 
CHAPTER XL 
 
 SOUNDINGS. 
 
 206. ONE of the most important parts of the survey of any 
 harbour consists in obtaining accurate soundings in its every 
 part. Two objects must be kept in view during the operation, 
 viz. the depth of water, and the nature of the bottom. 
 
 207. I n sounding it is advisable to take casts in straight 
 lines. In shoal water a sounding pole may be used. This in- 
 strument is a long pole graduated to feet and inches, or feet and 
 decimals of a foot. A flat piece of wood at the bottom prevents 
 it sinking into the ground. In deeper water the lead line, care- 
 fully graduated, must be employed. The boat must be kept on a 
 straight line by the method described in the next paragraph, and 
 in this way the presence of shoals, rocks, &c., can be detected. 
 
 208. The boat must be fixed at starting by two angles, in 
 fact by the "three-point problem." Note the direction in which 
 it is intended to sound, and if two natural objects in line in that 
 direction are not to be had, then recourse must be had to staves 
 with flags ( 50), the position of these artificial objects being, how- 
 ever, accurately fixed. And similar marks must be erected along 
 the shore on the lines in which the soundings are to be made. 
 Keeping these marks in transit, the boat is pulled at a uniform 
 speed, and at regular intervals casts are made. If the survey is 
 to be a close one, a cast ought to be made each half minute in 
 water under five fathoms in depth, and each minute in water 
 of a greater depth*. The boat's position ought to be fixed at 
 
 * Some surveyors prefer to space the soundings by the number of strokes 
 of the oars rather than by time intervals. 
 
CHAP. XI.] SOUNDINGS. 201 
 
 given short intervals of time and always fixed at the end of each 
 line of sounding. The boat is then pulled parallel to the shore 
 until the next line is reached, when she sounds in a direction 
 parallel to the former line. In this way the soundings can be 
 fairly spaced on a chart, provided that the speed of the boat is 
 uniform and the casts are made at equal intervals. 
 
 Note. A line must be drawn under all soundings which were 
 made when the tide was up, and which when "reduced" (vide 
 below) to low water would be apparently on a dry bank (p. 16). 
 
 209. In soundings the "quality of the bottom" ought to be 
 always examined, and its nature accurately described on the chart 
 by the side of the figures denoting the depth ( 1). In foggy 
 weather the nature of the bottom is often as much a guide as the 
 depth. A remarkable illustration occurs in nearing the English 
 Channel on the parallel of 4920']Sr. The same depth of 73 
 fathoms occurs at spots 125 miles apart, but in the outer depth 
 the bottom is u fine sand," while in the intier depth it is " ooze," 
 and this distinction in foggy weather is the seaman's only guide*. 
 
 210. In keeping the boat in a line by means of "two 
 objects on with each other," the two objects ought to be sepa- 
 rated by a considerable interval, as, if too close, the plan is not 
 sufficiently sensitive, and the boat might deviate considerably 
 from the required line before the error would be noticed. It is 
 also an important thing to remember that the position of these 
 objects must be accurately protracted on the chart. 
 
 211. To FIX A SOUNDING, we must use either the "three 
 point problem", or the "straight line and one angle method." 
 These two methods have been so fully described in Chapter iv. 
 ( 50 ff.) that nothing further need be said here. 
 
 If a shoal or other danger is discovered, it ought to be at once 
 fixed by the " three-point problem," great care being taken in the 
 selection of the three "points." 
 
 212. During the progress of the sounding operations, the 
 lead lines ought to be frequently tested in their wet state, as two 
 
 * General Instructions for Hydrograpkie Surveyors, 1877, p. 8 note. 
 
202 MARINE SURVEYING. [CHAP. 
 
 causes tend to change their length, viz. (1) constant wetting, and 
 (2) the strain of the lead. The watches supplied to the boats, 
 and to the observers at the tide gauge, ought also to be compared 
 every morning and evening : this precaution will ensure accuracy 
 in the after process of reducing the soundings to the required 
 datum line. 
 
 213. REDUCTION OP SOUNDINGS. All soundings taken during 
 a survey, must be reduced to the level of low water at ord'nun-ii 
 spring tides before they are entered on the chart ; the object of 
 these soundings on the chart being to let the seaman know the 
 least depth of water he can reckon on at any state of the tide. 
 This reduction is thus effected : 
 
 While the soundings are being taken in a boat, an observer is 
 ordered to note carefully the depth of water at the tide gauge, at 
 intervals of 5 or 10 minutes, and to register his observations 
 thus*: 
 
 Time. Depth of Water at the Tide Gauge. 
 
 II. M. FT. IX. 
 
 10 17 6 
 
 10 10 17 9 
 
 10 20 18 
 
 10 30 18 4 
 
 &c. &c. 
 
 The depth of water at each sounding is noted by the officer in 
 the boat, and also the time of each cast. This information is 
 thus registered : 
 
 Time. Depth of Sounding. 
 
 H. M. FT. IN. 
 
 10 63 6 
 
 10 10 73 
 
 10 20 59 8 
 
 10 30 43 3 
 
 <fec. &c. 
 
 * These examples are, of course, only imaginary, and intended merely 
 to explain the method of proceeding. 
 
XI.] SOUNDINGS. 203 
 
 Now, let us suppose that the lowest water observed at any 
 period during the survey, by means of the tide gauge, was 
 5ft. 8 in., then we conclude that the plane of the low water level 
 of ordinary springs intersects the tide gauge at 5 ft. 8 in. above 
 the zero of the scale. Hence, at 10 o'clock, the level of the water 
 in the harbour was 17ft. 6 in. 5ft. 8 in., or lift. 10 in. above 
 the datum line to which the soundings are to be reduced. 
 
 At 10 o'clock a sounding of 63ft. 6 in. was struck. There- 
 fore, to "reduce" this sounding to the given datum line, it is 
 evident that we must subtract lift. 10 in. We thus obtain 
 51 ft. 8 in. as the depth of water to be marked on the chart. The 
 other soundings are reduced in the same way. 
 
 It is usual to make a further reduction of a couple of feet in 
 cases of uncertainty, and where such reduction can be effected 
 without inconveniencing the traffic of a harbour or port. 
 
 214. Perhaps a diagram will render clearer the method of 
 reducing soundings. 
 
 Let AB represent the tide gauge marked to 30 feet. 
 Let CD represent the level of low water at ordinary springs 
 intersecting the tide gauge at 5 ft. 8 inches. Hence, we infer 
 
204 MARINE SURVEYING. [CHAP. 
 
 that under ordinary circumstances, CD will represent the plane 
 of the lowest water in the harbour. 
 
 Let EF mark the height of the water, suppose 17ft. 6 in., 
 when a certain cast has been made ; then the plane of water in 
 the harbour at that instant was higher than the datum line by 
 the vertical distance EC. Now EC -11 ft. 10 in., therefore 
 lift. 10 inches must be subtracted from the sounding made at 
 10 o'clock, whatever it was, in order to reduce it to the datum 
 line. 
 
 215. FORMULA FOR REDUCING SOUNDINGS. 
 Let 8 = depth of a sounding at a certain hour. 
 
 g = depth of water at the tide gauge at the same time. 
 
 d= depth of water at the tide gauge at low water of 
 ordinary springs. 
 
 x = depth of sounding to be registered on the chart*. 
 Then x = s-(g-d), 
 
 = s + d-g. 
 Thus, to reduce the second sounding taken at 10 -10 o'clock, 
 
 f. i. t. i. f. i. t. i. f. i. t. i. 
 
 x = 73 + 5 8- 17 9-78 8-17 9 = 60 11 
 .-. a=60ft. 11 in. 
 
 216. The following rules are recommended by a competent 
 authority as desirable to be observed when sounding in tidal 
 rivers t. 
 
 (1) Soundings, in the immediate vicinity of the tide gauge by 
 which they are to be corrected, are not appreciably affected by 
 deviations from parallelism caused by the river water, and such 
 soundings may be taken at any time and under any circum- 
 stances. 
 
 (2) The farther the soundings are taken from the standard 
 gauge the greater is the probable error arising from non-paral- 
 lelism. 
 
 * Subject to any further reduction that may be deemed advisable. 
 + Stevenson's Canal and River Engineering. 
 
XL] SOUNDINGS. 205 
 
 (3) The soundings ought to be made in. neap rather than in 
 spring tides*. 
 
 (4) The soundings ought to be made in ebb rather than in 
 flood tides. 
 
 (5) Soundings taken in a flood tide, especially during springs, 
 should not be made till within an hour or so of the time of high 
 water. 
 
 Note. The parallelism of a river is affected by two causes ; 
 the rise of the tide may be great, or, the current of the river may 
 be very strong. 
 
 217. Mr Mitchell Henry, of the United States Coast Survey, 
 has proposed the following method of determining the elevations 
 along the course of a tidal river without the aid of a level. 
 
 Set up graduated rods at such distances apart that the 
 "slacks" of the tides may extend from one to another. By 
 simultaneous observations ascertain the difference in the readings 
 of these gauges at the "slack" between the ebb and flood, and 
 again between flood and ebb, then apply the formula : 
 
 Difference of elevation of tJie zeros = one-half the sum of the 
 differences of the readings at the two slack waters. 
 
 218. To fix on a chart the position of a hidden danger 
 which is at a considerable distance from the shore, we must 
 anchor a boat or a buoy over it, and then fix the position of the 
 boat by angles taken to objects on the shore. 
 
 If the danger is so far off that it can be viewed from only 
 two points on shore, then its exact position may be laid down as 
 follows. Anchor a boat or a buoy between the source of danger 
 and the shore, and then the " danger" can be fixed by the " three- 
 point problem". 
 
 219. When an eddy, or still water, intimates danger, the 
 following method of procedure may be adopted if the state of the 
 sea will permit. A buoy is anchored where the danger is 
 
 * Some surveyors adopt the very opposite rule to this. 
 
206 MARINE SURVEYING. [CHAP. 
 
 supposed to exist and the buoy's position is fixed by the " three- 
 point problem." Soundings are then taken round the buoy in 
 ever-enlarging circles until the source of danger is discovered. 
 
 The boat can also be anchored at the head of the eddy and 
 then dropped slowly astern by the boat rope until the point of 
 danger must have been passed. She is then hauled up and again 
 dropped astern on another line, and so on, until the danger is 
 found. When this occurs, the boat anchors to windward, and 
 sufficiently near the danger that on dropping astern by her boat 
 rope she may reach the position necessary to be fixed. 
 
 Or, finally, the danger may be found by the method of 
 " Sweeping." Two boats pull abreast at a certain distance apart. 
 Hanging from each boat's stern is a heavy weight, and a deep- 
 sea lead line connects these at any required depth. Thus it is 
 scarcely possible for any known danger to escape detection. 
 
 220. A SURFACE CURRENT. 
 
 The existence of such a current becomes manifest almost at 
 once. Its rate and direction may be thus determined. Allow a 
 barrico to drop astern by the log line, or else the log ship itself : 
 this determines the rate. Observe the bearing of the barrico by 
 the standard compass, and thus the direction can be determined. 
 
 The rate of the current can also be determined by means of 
 the Current Log. In this instrument the logship is considerably 
 larger than in the ordinary log used at sea. The line is marked 
 at intervals of 10 feet, and is allowed to run out for a certain 
 number of minutes. The following formula will give a very 
 approximate result. 
 
 Let v = rate of the current in knots per hour, 
 f= number of feet run out, 
 t = time in minutes, 
 
 and let the length of the knot be 6000 feet. 
 v x 6000 60 
 
 Then 
 
 60/ / 
 ~ 6000m ~ fOOra ' 
 
XI.] SOUNDINGS. 207 
 
 Hence the practical rule adopted is " Divide the number of 
 feet run out by 100 times the number of minutes; the quotient 
 will be the rate of the current in knots per hour." 
 
 e. g. In 4 minutes 360 feet ran out, 
 360 
 
 221. UNDER-CURRENTS. 
 
 The existence of such currents, as well as their rates and direc- 
 tions, are usually determined in the following way. An " Under- 
 current Float " is made either of two pieces of sheet iron in the 
 form of the letter T, and suspended by cords so that it may 
 remain horizontal, or consists merely of a weighted basket with 
 pieces of sail-cloth fixed on it so as to catch the water*. This 
 float is then suspended by a line of sufficient length to allow it to 
 descend to any required depth, and is suspended from a light 
 block-tin buoy of such a shape as to offer least resistance to the 
 water. This float serves the double purpose, (1) of keeping the 
 underfloat at the same depth, and (2) of indicating the course and 
 strength of the under-current. Smaller floats are thrown out to 
 gauge the upper current, if any, and to ascertain the distance 
 traversed by the buoy in a certain time. 
 
 Note. The under- current ought to be allowed to get up its 
 proper rate in the float before the smaller marks are let go. 
 Under the most favourable conditions we can only hope for 
 approximate results. 
 
 Captain Spratt, in his paper before the Royal Society in 1871, 
 gives an account of observations on the under-currents in the Sea 
 of Marmora ; he found that the best float for the purpose was one 
 made of thin copper or block-tin, suspended so as to remain 
 horizontal, and a buoy anchored with a sinker, served to shew the 
 relative speed of the surface and under-currents. 
 
 * The first kind was used by Stevenson in 1875 in the survey of Cro- 
 marty Firth, the second kind by Carpenter and Jeffrey in 1870, during 
 the cruise of H.M.S. "Porcupine." 
 
208 MARINE SURVEYING. [CHAP. 
 
 222. Mr Mitchell Henry, already alluded to, has used suc- 
 cessfully the following method of discovering the existence of an 
 under-current. A tin cylinder about 40 feet long and 3 inches 
 in diameter is made in separate sections which can be rendered 
 perfectly air-tight. Each section, however, is provided with a 
 stop-cock, so that the instrument may be filled to sink it to any 
 required portion of its length. As the tube, drifts nearly upright 
 in the water, with its top protruding a few inches above the 
 surface, its velocity may be taken as the mean motion of the 
 stream ; if it leans backwards or forwards it shews that its foot 
 rests on a stratum of water which has greater or less motion than 
 the surface drift ; and, finally, if its course differs from that of a 
 surface float, the action of an under-current is recognised, the 
 direction and force of which may be approximately determined. 
 
 223. THE DISCHARGE OF RIVERS. 
 
 DBF. A stream is said to be in its normal condition during 
 ordinary summer weather, i.e. when it is neither dried to its 
 minimum by a long drought, nor swollen to its maximum by 
 heavy rains. 
 
 DEF. By the discharge of a river is meant the quantity of 
 water which passes out of the river in a given time; it is generally 
 expressed in the number of cubic feet per minute. 
 
 This may be computed from the formula : 
 
 Discharge = sectional area x mean velocity. 
 
 We have, therefore, to devise methods for ascertaining the sec-- 
 tional area, and also the mean velocity. 
 
 224. To FIND THE SECTIONAL AREA. 
 
 Select a part of the river where the banks are regular and the 
 stream tranquil, and stretch a graduated cord across the stream. 
 The depth of water can then be found with a sounding pole, or 
 lead line, at every 5 or 10 feet along the cord; thus the mean 
 depth can be ascertained. Then we have 
 
 Sectional area = mean depth x width of the river. 
 
 DEF. The average depth of water at a section is known as the 
 Hydraulic Mean Depth. 
 
XI.] SOUNDINGS. 209 
 
 225. To FIND THE MEAN VELOCITY. 
 
 The most accurate method seems to be to ascertain the surface 
 velocity in the middle of each of the compartments into which the 
 transverse section of the river is divided by the soundings, made 
 as already explained, and then the mean velocity of these may bo 
 taken as representing the mean velocity of the river. 
 
 Two methods are adopted for finding the surface velocity : 
 (1) by Floats, (2) by an instrument called a Tachometer*. 
 
 (1) By means of Floats. A float is let go at such a distance 
 that when it reaches the line of section the stream is exerting its 
 full influence on it. Its time of passage between two points, at a 
 known distance apart, is then noted, and hence the velocity of the 
 stream may be approximately computed. 
 
 (2) By the Tachometer. The principle of this instrument is 
 much the same as that of the Patent Log. The instrument is 
 fixed at any point, and the current impinging on a vane causes 
 it to revolve. The number of revolutions made by the vane being 
 registered on an index, the velocity of the stream is indicated. 
 The instrument is applied at each compartment of the cross sec- 
 tion, and the mean result taken. The great object in view is to 
 ascertain the velocity of the current as it passes the line of 
 section fixed upon. 
 
 226. THE GROUND LOG consists of a logship with a sufficient 
 amount of stray line to permit the logship being anchored at any 
 point. A hand lead line, divided like an ordinary log line, is bent 
 to the logship. The logship remains stationary in the water, and 
 hence the direction and speed of the boat over the bottom, i.e. 
 the course and distance made good, can be found. 
 
 Note. In greater depths than 4 fathoms it is advisable to use 
 the " Buoy and Nipper." In this instrument the stray line runs 
 through a notch until the lead reaches the bottom, when it is 
 caught, and the Buoy remains stationary : then, as before, the 
 speed of the boat can be found. 
 
 * i.e. a speed measurer. 
 R. M. S. 14 
 
210 MARINE SURVEYING. [CHAP. 
 
 The Ground Log is chiefly used in river surveying, and in shoal 
 water, especially where the vessel may be out of sight of land, or 
 where the shore presents no distinct objects by which the position 
 may be fixed. 
 
 EXAMINATION. 
 
 (1) In making a sounding what two objects must be kept in 
 view? 
 
 (2) In sounding over a bay what precautions ought to be 
 taken to insure the detection of all dangers? 
 
 (3) What is meant by " spacing" soundings on a chart? 
 
 (4) How may a boat be kept on a " range " while sounding ? 
 
 (5) How would you fix a sounding on a range ? 
 
 (6) In the case of a very important sounding having been 
 struck, how would you proceed to fix it ? 
 
 (7) What two causes tend to make the lead lines erroneous ? 
 
 (8) To what datum line are soundings reduced in the Admi- 
 ralty charts ? 
 
 (9) Describe fully the method by which this reduction is 
 effected. 
 
 (10) Investigate a formula by means of which this reduction 
 may be computed. 
 
 (11) Suppose a tidal river has to be surveyed, what instruc- 
 tions as regards the soundings would you frame for the guidance 
 of the observers ? 
 
 (12) What do you understand by the parallelism of a river? 
 
 (13) How would you fix the position of a rock visible from 
 only two points on shore ? 
 
 (14) The reported position of a sunken rock in a surveyed 
 harbour is given ; how would you proceed to verify its existence, 
 to fix it accurately, and to find the least water on it, the rise of 
 the tide being considerable? (March, 1875.) 
 
XL] SOUNDINGS. 211 
 
 (15) Explain the principle of the "Ground Log," and state 
 the description of navigation in which it is mostly used. 
 
 (March, 1878.) 
 
 (16) Give any simple methods you are acquainted with for 
 ascertaining the rate and direction of a surface current from 
 a ship at anchor. (March, 1878.) 
 
 (17) A sunken pinnacle rock, of small area, having 13 feet of 
 water upon it, is reported in a vicinity in which the current runs 
 5 knots. Wishing to fix its position accurately, how would you 
 proceed to find the rock 1 (May, 1878.) 
 
 (18) State how you would find approximately the quantity 
 of water passing down a river, through any particular section of 
 the river, in a given tima (Dec. 1879.) 
 
 (19) What methods are adopted to ascertain from a ship at 
 Anchor the velocity and direction of surface and under currents ? 
 
 (Oct. 1880.) 
 
 ^20) How is the sectional area of a river found ? 
 ^21) What is meant by the " hydraulic mean depth"? 
 (22) In what units is the discharge of a river expressed ? 
 
 142 
 
CHAPTER XII. 
 
 CHRONOMETERS. 
 
 227. DEFINITIONS : 
 
 A time-piece simply marks the time. 
 
 A clock shews the time and strikes the hour. 
 
 A watch is a pocket time-piece. 
 
 A repeater is a clock or watch which by mechanism can be 
 made to repeat the hour. 
 
 A chronometer is merely a very perfect watch, or time-piece, in 
 the construction of which no skill is spared; and its mechanism 
 is such that any change of temperature will produce the least 
 possible effect on its performance. 
 
 228. History of the Chronometer'. 
 
 It seems that striking clocks were known in Italy as early as 
 the 13th century, or the beginning of the 14th century. In Eng- 
 land, in 1288, a fine imposed on the Chief Justice of the King's 
 Bench, was devoted to the purchase of a clock for the famous 
 clock-house near Westminster Hall. In 1523, the church of 
 St Mary in Oxford was provided with a clock from the proceeds 
 of fines imposed on the undergraduates. About 1360, Edward III. 
 gave protection to three Dutch clock-makers who were invited by 
 him from Delft. We have some reasons for thinking that clocks 
 were becoming well known in England at the end of the 15th 
 century. Much discussion has taken place about the inventor. 
 
CHAP. XII.] CHRONOMETERS. 213 
 
 It is now generally acknowledged that it was not the product of 
 any single mind, but the result of many inventions. Thus, in the 
 case of our modern chronometers, successive improvements and 
 inventions have brought them to their present state of perfection. 
 
 In 1484, we hear first of a balance-clock being used for astro- 
 nomical purposes : the principle of this instrument was that a 
 finely-adjusted beam balance moving from side to side caught suc- 
 cessively the teeth of a wheel and thus regulated the motion*. 
 
 Such seems to have been the success attending these experi- 
 ments that we find an astronomer named Frisius proposing, about 
 1530, the use of a portable balance-clock for ascertaining the 
 longitude. 
 
 In 1560, Tycho Brahe, the famous Swedish astronomer, pos- 
 sessed four clocks which shewed the hours, minutes, and seconds. 
 The largest had only three wheels, the diameter of one of which 
 was three feet, and had 1200 teeth 011 its rim. Tycho seems to 
 have been the first to notice the effects of temperature, but appa- 
 rently did not know how to explain the facts. 
 
 In 1577, an astronomer named Moestlin had a clock which 
 beat 2528 times in an hour, and by its means determined the 
 Sun's semi-diameter on passing the meridian : he made it to be 
 34' 13". 
 
 Clocks were certainly reduced to a portable size previous to 
 1544, and before this could have taken place the spring must have 
 superseded the heavy weights. This may be considered as the 
 second stage in the development, and prepared the way for the 
 Fusee. Galileo, in watching a chandelier hanging by a long chain 
 in a church at Florence, noticed the isochronism of the pendulum. 
 The invention of the pendulum clock marks the third stage in the 
 history. As usual in cases of this kind the inventor is doubtful. 
 Huyghens applied the invention in a most skilful manner, and hence 
 has been looked upon as the originator. It is now, however, known 
 that in 1641, a London maker, Richard Harris, invented and con- 
 structed a pendulum clock. 
 
 * Vide description and diagram in Godfray's Astronomy, p. 36. 
 
214 MARINE SURVEYING. [CHAP. 
 
 Soon after this invention was made known, Huyghens at- 
 tempted to construct a marine clock, but his success was not 
 great. He noticed that the pendulum beats slower as the latitude 
 is diminished, and thus prepared the way for the correct know- 
 ledge of the Figure of the Earth. 
 
 In 1680, a London watchmaker, named Clement, invented the 
 " anchor" escapement*. This change led to the mode of suspending 
 the pendulum by a thin and flexible spring. The seconds pendu 
 lum, with this escapement, was known as the Royal Pendulum. 
 
 In 1715, George Graham, another London maker of great 
 repute, endeavoured to obviate the effects of temperature by 
 means of his well-known Mercurial Pendulum^. The famous 
 John Harrison, by his Gridiron Pendulum, improved on Graham's 
 invention. Graham then introduced his "dead-beat" escapement 
 as an improvement 011 the anchor or recoil escapement. From the 
 days of Harrison and Graham successive improvements have been 
 introduced into every part of the mechanism. 
 
 229. HARRISON'S WATCHES. 
 
 During the reign of Queen Anne, in 1714, the British 
 Government offered a reward of 20,000 for any method by 
 which the longitude could at all times be determined at sea ; the 
 whole reward would be given if the method, when tested by a 
 voyage to the West Indies, were found true within 30 miles, 
 15,000 if true within 40 miles, and 10,000 if true within 
 60 miles. 
 
 Harrison came to London, in 1728, with drawings of a watch 
 which he deemed would answer the purpose. Dr Halley, the 
 well-known English astronomer, to whom he applied, referred 
 him to Graham, who soon discovered his great ability, and ad- 
 vised him to actually construct his machine before making appli- 
 cation to the Board of Longitude. 
 
 In 1735, he presented his first watch, and next year was sent 
 to Lisbon to test its power in a voyage. He corrected the dead 
 
 * Vide description and diagram in Godfray's Astronomy, p. 41, and also 
 in Ninth Edition of Encyc. Brit. Vol. vi. p. 17. 
 t Vide Godfray's Astronomy, p. 42. 
 
XII.] CHRONOMETERS. 215 
 
 reckoning about a degree and a half*, a success which naturally 
 obtained for him both public and private help. 
 
 In 1739, he finished his second watch, which, though not tried 
 at sea, gained him still further encouragement. In 1749, he pre- 
 sented his third watch f ; it was less complicated and more accurate 
 than the second, as its error was only 3 or 4 seconds a week. 
 Thus encouraged he constructed his fourth and most famous 
 chronometer, and applied for the full reward. The tests were 
 very severe. The chronometer was compared in the Observatory 
 at Greenwich, sealed up, and sent to Portsmouth, where Robert- 
 son, the Master of the Royal Academy, having found its error by 
 equal altitudes, forwarded the observations and results to the 
 Admiralty. The chronometer was then put on board H. M.S. 
 " Deptford," commanded by Captain Digges. It was secured by 
 four separate locks, the keys of which were entrusted to Governor 
 Lyttleton, who was proceeding to Jamaica, to Captain Digges, to 
 the Senior Lieutenant, and to Harrison's son. The ship sailed on 
 Nov. 18th, 1761. During the voyage to Madeira the chrono- 
 meter corrected the dead reckoning, which was sometimes in error 
 to the extent of a degree and a half J. The ship arrived at Ma- 
 deira three days before H.M.S. " Beaver," which had sailed ten 
 days before her, a result, according to Harrison's account pub- 
 lished in 1767, "which was owing to the Beaver being deceived in 
 her reckoning by trusting to the log for want of a more perfect 
 method of finding her longitude." 
 
 In going from Madeira to Jamaica, the chronometer corrected 
 the errors of the longitude, which occasionally amounted to three 
 degrees, and the reckoning of several ships in the convoy varied 
 
 * The Master of H.M.S. " Orford" in the homeward voyage thought 
 that the point of land sighted was the Start, but Harrison, trusting to his 
 timepiece, insisted that it was the Lizard, and he was found to be right. 
 
 t This gained the gold medal of the Royal Society. 
 
 J "In sailing to the Madeiras, Mr Harrison acquainted Capt. Digges 
 with the time when he would see the Island of Porto Santo ; which had they 
 trusted to the ship's reckoning, they could not have seen in that voyage, 
 which would have been a great inconvenience to them, as they were in want 
 of beer"! Vide Account of the goings of Mr J. Harrison's Watch, 1767. 
 
216 MARINE SURVEYING. [CHAP. 
 
 Jive degrees from the correct position ! When the ship arrived at 
 Port Royal, the longitude as found by the chronometer was only 
 five seconds of time in error. The same precautions were taken 
 during the return voyage to Europe, and on arrival at Portsmouth 
 the error in longitude was less than 18 miles. 
 
 In the spring of 1764 the same chronometer was sent on a 
 voyage to Barbados. Before sailing, Harrison drew up a declara- 
 tion as to the rate by which he was content to abide. If the 
 temperature was 42, he said his machine would gain 3 seconds 
 in 24 hours, if 52 the gain would be 2 seconds, if 62 the gain 
 would be 1 second, if 72 there would be neither gain nor loss, 
 and if 82 it would lose one second in 24 hours. In 156 days it 
 was actually found to have gained only 54 seconds, allowing the 
 chronometer to have gained 1 second a day, being the rato by 
 which Harrison would abide; but if allowance be made for the 
 change of temperature according to the above scale, the chrono- 
 meter lost only 15 seconds! a result which shews a "wonderful 
 performance." 
 
 Harrison, after some trouble and anxiety, obtained the whole 
 20,000, and a further sum from the East India Company. 
 
 230. THE INSIDE OF A CHRONOMETER. 
 
 A chronometer derives its power from a coiled spring; the 
 variable force of which is rendered uniform by the fusee, a beau- 
 tiful contrivance, by means of which, on the principle of a variable 
 lever, the main-spring acts through the medium of a chain. On 
 the fusee is cut a curve, into which the chain fits, and which has 
 this peculiar property, that as the chain winds upon it, the dis- 
 tance from the centre of motion of the fusee to the axis of the 
 chain, at the point where it leaves the fusee for the barrel, con- 
 tinually varies; but this variation is such that the product of this 
 distance and the force of the main-spring acting along the chain 
 at that instant is constant, i. e. shall be the same wherever the 
 chain leaves the fusee. Hence, then, we see that the power of the 
 fusee to turn the machinery is always the same, and since the 
 main wheel which communicates motion to all the rest is attached 
 
XII.] 
 
 CHRONOMETERS. 
 
 217 
 
 to the fusee, their centres of motion being coincident, it follows 
 that the power at the teeth of the main wheel is uniform. This 
 power is transmitted through the medium of a train of wheels and 
 pinions till it comes to the Escapement 
 
 Thus far there is not much difference between the works of a 
 chronometer and an ordinary pocket watch. 
 
 231. The distinguishing features of a chronometer are the 
 Escapement and the mode of Compensating the Balance for tem- 
 perature. The latter being the more important for our purposes 
 we shall devote some space to it. 
 
 AA is the balance arm. BB the two segments of the rims 
 attached at one end to the arm and having the other end free. Each 
 segment of the rim is composed of two metals, steel on the inside. 
 
 Flg.2. 
 
 and brass on the outside. W, W, are two weights which can be 
 fixed at any points on the rims. Any increase of temperature 
 diminishes the elastic force of the balance-spring, and hence the 
 chronometer would lose, but, since brass expands more than steel, 
 a curvature of the rims takes place inwards, and therefore the 
 w.-i^hts W, W, approach the centre. The inertia of the balance is 
 thus lessened, and the balance-spring exerts the same influence as 
 it did before the change of temperature. Again, if the tempera- 
 ture falls, the elastic force of the balance increases, and hence the 
 chronometer would gain, but, since brass contracts more than steel, 
 the rims curve outwards, the weights W, W, are therefore removed 
 
218 MARINE SURVEYING. [CHAP. 
 
 to a greater distance from the centre. The inertia of the balance 
 is increased, and the balance-spring has no greater influence than 
 it had before. It is evident that the nearer the weights W, IF, are 
 to the free ends of the rims, the greater will be the space through 
 which they move by any change of temperature, and therefore the 
 greater the variation in the inertia of the balance ; hence it fol- 
 lows that if any change of temperature causes a change of rate 
 the compensation is not sufficiently active. In Fig. 2, the small 
 screws answer the same purpose as the weights in Figs. 1 and 3. 
 In Fig. 3, we have represented the method of making the final 
 adjustment of the "Thermal Compensation," invented in 1875 by 
 the late Astronomer Royal, Sir George Airy, and which is 
 now widely adopted by the best makers. It may be thus 
 described : 
 
 A A is the balance arm turning 011 the staff D. BB the 
 ordinary brass and metal rims carrying the ordinary weights 
 W, W. The new attachment consists of a small arm aa which 
 turns with a stiff friction on the staff D. At each end of this 
 secondary arm, are small springs which keep the little weights 
 iv, w, pressed against the inside of the rims. 
 
 To make the final adjustment, these small weights, w, w, 
 should be placed midway between the large weights W and the 
 ends of the balance arm. If the compensation by these large 
 weights is insufficient the small weights w must be brought nearer 
 to W, and vice versa. In this way no other adjustment is 
 disturbed. 
 
 232. However perfect may be the compensation of a chro- 
 nometer for certain temperatures, it does not hold good for all 
 temperatures. If compensated for great heat and great cold, it 
 gains at medium temperatures; on the contrary, if specially 
 compensated for the latter, then the chronometer is out in either 
 extreme. Chronometer makers at once sought for a remedy, and 
 this remedy, as in Airy's method, is called the " secondary 
 compensation," and the error, which it is intended to obviate, is 
 known as the " secondary error." In the Royal Navy, tho 
 
XII.] CHEONOMETERS. 219 
 
 chronometers are usually compensated for 65 F., with a range of 
 about 15 on either side of the mean. 
 
 To encourage the manufacture of good chronometers, the 
 Admiralty, in the years 1822 1835, presented prizes for the 
 excellence of individual chronometers. Since 1835 the prizes 
 have been withdrawn, but the "annual trials" still continue. In 
 successive years different makers obtained the post of honor, 
 Poole, Frodsham, Hutton, Hewett, Eiffe, Dent. During the five 
 years 1845 1849, no less than 219 chronometers were exhibited, 
 and 79 were bought for the public service; the highest price 
 given being ,62. 
 
 233. CHRONOMETER ROOM IN THE ROYAL OBSERVATORY. 
 
 This is situated in the same building as the Great Equatorial, 
 and is octagonal in form. On the left side of the entrance, are 
 arranged the government chronometers and on the right the 
 " annual trials*." These chronometers are so arranged on double 
 rows of shelves placed back to back, that the superintendent 
 and his assistants can make the daily comparisons at 1 '30 P.M. 
 without moving the instruments. There are usually about 250 
 chronometers in the room. 
 
 A galvano-magnetic clock by Shepherd is fixed to the central 
 pillar. When the mean solar clock of the observatory is cor- 
 rected each day for the 10 o'clock A.M. signal sent to the General 
 Post Office, this clock is also compared, and after 1 o'clock P.M. 
 the chronometers are compared. 
 
 It was formerly the custom to expose the "annual trials" to 
 both a cold and a heat test. The chronometers were exposed in 
 the open air on the coldest days in winter, but now, only a few of 
 the very best are sometimes placed in the outside air, for a short 
 time, to make further trial of their performances. The heat text, 
 however, still continues. 
 
 Occupying one side of the room is the apparatus for this 
 purpose. A zinc or galvanized iron case contains within it 
 
 * No maker can send more than two chronometers to compete in these 
 annual trials. Each maker's instruments are kept together. 
 
220 MARINE SURVEYING. [CHAP. 
 
 13 gas jets, the noxious products of the combustion being carried 
 off by a pipe which communicates with the open air. A wooden 
 case with glass and wood covers is arranged over this stove, and 
 is titted with shelves for the reception of about 80 instruments. 
 The wooden cover being raised, the chronometers can be com- 
 pared through the glass, and thus the temperature is not dis- 
 turbed. 
 
 A thermometer shews the degree of heat, and a vessel of 
 water being also placed inside, the cases of the instruments do 
 not suffer from the excessive dryness of the air. The tem- 
 perature ranges from 95 to 105. 
 
 The "annual trials" remain in the Royal Observatory for six 
 months, viz. from January until July. During this period they 
 are subjected twice, for four weeks on each occasion, to the heat 
 test, generally in March and May ; and, as a rule, it is expected 
 that with a good instrument the " weekly rates" ( = sum of the 
 " daily rates") when in the stove and out of it ought not to differ 
 by more than 10 seconds. Formerly 20 seconds were allowed, 
 but with the recent improvements for compensation the limit has 
 been reduced. 
 
 When the chronometers are placed in the heat case a Chrono- 
 metrical Thermometer (technically known as a " Chron. Therm.") 
 is also enclosed. This instrument differs from an ordinary chro- 
 nometer in' having the two metals of the compensating rims 
 reversed, brass being the inside metal, and steel the outside. 
 The result of this arrangement is that the effect of any change of 
 temperature is enormously increased, e.g. when the temperature 
 in the chronometer room was 37*4 the Chron. Therm, gained in 
 & week 2194 8< 8, when the temperature was 45 it gained 10: 
 and when in the heat case the temperature was 95'9 it lost 
 3056 8 -4. 
 
 234. In the published rates of chronometers, on trial for 
 purchase by the Board of Admiralty, at the Royal Observatory, 
 the results arrived at are arranged in three tables. 
 
XII.] CHRONOMETERS. 221 
 
 (1) The weekly sums of the daily rates in the order of time. 
 
 Note. If the weekly sum exceeds 40 seconds, the compensa- 
 tion of the instrument is looked to by the Officials in charge of 
 the chronometers. 
 
 (2) The weekly sums of the daily rates in the order of 
 temperature, as determined by the Chron. Therm. 
 
 (3) The abstract of the principal changes of rates. 
 
 The chronometers are arranged at the close of the six months 
 in the order of their " Trial Numbers." These trial numbers are 
 computed from the formula 
 
 T. N. twice the greatest difference between any two successive 
 weeks + difference of the greatest and, least weekly sum. 
 
 E.g. In the trials for 1880, Cornell's chronometer had as the 
 greatest weekly difference 3 8 -7, and the difference between the 
 greatest and least weekly sum was 16 '3. 
 
 .-. T. N. = 2x3-7 + 16-3 = 23-7. 
 
 The next in order was Kullberg's. Greatest difference 4-4 ; 
 difference between greatest and least 14 '9. 
 
 .-. T. N. = 2 x 4-4 + 14-9 = 23-7. 
 
 The third was Matheson's. Greatest difference 5 '6, and 
 difference between greatest and least 13-9. 
 
 .-. T. N. = 2x5-6 + 13-9 = 25-1. 
 
 The worst was No. 16494. Greatest difference = 24-1, and 
 difference between greatest and least = 57 '2. 
 
 .-. T. N. = 2 x 24-1 + 57-2 = 105-4. 
 
 The first three or four chronometers on the list have a price 
 put on them by the Officials at the Royal Observatory somewhat 
 in excess of their market value, and these are purchased by the 
 Admiralty. The average cost is about .37 or 38. Sometimes 
 an instrument is bought for .33, but these low-priced chrono- 
 meters are only purchased when they are really wanted. 
 
 The chronometers in the observatory are compared every day 
 at 1 -30 P.M. A skilful observer will compare 200 chronometers 
 in 16 minutes. Two copies of the rates of the government 
 
222 
 
 MARINE SURVEYING. 
 
 [CHAP. 
 
 instruments are kept, one for the information of the Hydro- 
 grapher, the other for use in the Royal Observatory. When the 
 service chronometers are found to be going satisfactorily, they are 
 marked "Ready" in the register, and the Hydrographer selects 
 from his copy those instruments which are to be forwarded to 
 the dockyard dep6ts, or when a ship is commissioned. The 
 "Numbers "of the chronometers thus sent out from the Royal 
 Observatory and their Makers are kept in this form : 
 
 Date. 
 
 Name of Maker 
 and Number. 
 
 . 
 Ship. 
 
 Officer's 
 Name. 
 
 When 
 
 Delivered. 
 
 Remarks. 
 
 
 
 
 
 
 
 E.g. H.M.S. Northampton, which commissioned in 1880, 
 had delivered to her on October 2nd, the following chronometers : 
 Molyneux 2119, Mitchell 621, Dent 2209, Cairns 792, Eifie 3, 
 Pennington (D. W.) 1312. 
 
 The account kept at the Royal Observatory of the repairs of 
 a government chronometer is thus registered : 
 
 Name of Maker 
 and 
 No. of Chron. 
 
 Whether two or. 
 eight Days. 
 
 Date of 
 Last Repairs. 
 
 Maker to whom 
 Notice of Repair 
 is sent. 
 
 1 
 
 ol 
 
 25 
 
 1 
 
 Day of 
 Removal. 
 
 Notes on the 
 Examination. 
 
 IDate of 
 Estimate. 
 
 1 Nature of 
 Repairs. 
 
 1 A mount of 
 
 Estimate. 
 
 IDate of Author- 
 izing Estimate. 
 
 IDate of Report to 
 tin- Admiralty. 
 
 Date of 
 
 Return of the 
 Chronometer. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 235. HISTORY OF A CHRONOMETER WHILE IN USE IN THE 
 ROYAL NAVY. 
 
 From what has gone before, the reader will know by what 
 means a chronometer is admitted into the government service, 
 and how it is selected for use afloat by the Hydrographic depart- 
 ment of the Admiralty. The extracts, given below, from the 
 
XII.] 
 
 CHRONOMETERS. 
 
 223 
 
 Observatory records will shew the "life" a chronometer leads. 
 The first chronometer was returned to the Observatory from 
 H.M.S. Formidable in 1866. 
 
 When a chronometer has been in use for three years, and is 
 on the home station, it is generally sent to be examined by the 
 maker. When its performances are no longer trustworthy, it 
 ceases to be retained in the service, and is usually given to the 
 maker, with a sum of money, in exchange for a new instrument. 
 The old chronometer is often transformed into a Deck Watch. 
 The cost of keeping a chronometer in repair ought not to exceed 
 by much one pound a year. A chronometer, for instance, which 
 was in active service from 1852 until 1875, cost the government 
 ,29 for repairs. 
 
 A new Balance Spring costs about ten pounds. 
 
 To whom 
 Transferred, 
 Maker, Ship, or 
 Depot 
 
 Date of Transfer. 
 
 From whom 
 Received. 
 
 Date of Receipt. 
 
 Dent 
 Serapis 
 Dent 
 
 Sept. 10, 1866 
 July 1, 1867 
 April 18, 1870 
 June 24, 1871 
 Dec. 9, 1872 
 June 17, 1874 
 May 24, 1875 
 June 9, 1876 
 
 Dent 
 
 Oct. 15, 1866 
 April 13, 1870 
 Oct. 31, 1870 
 Nov. 1, 1872 
 Sept. 22, 1873 
 May 13, 1875 
 Jan. 31, 1876 
 
 Portsmouth ... 
 Dent 
 
 Topaze 
 
 Devonport .... 
 Dent 
 
 Dent 
 Tamar 
 Dent ... 
 
 Devonport .... 
 Dent . .. 
 
 Hotspur 
 
 
 EXAMPLE II. 
 
 Dent 
 
 Oct. 1, 1866 
 Feb. 3, 1868 
 
 ren in February 
 Dent 2945. 
 
 Sheerness 
 Dent 
 
 Sept. 26, 1866 
 May 20, 1867 
 
 with 26 in 
 
 Dent 
 
 , 1868, to Dent 
 
 Note. Gn 
 exchange for 
 
224 MARINE SURVEYING. [CHAP. 
 
 236. When a chronometer is being transported the follow- 
 ing precautions are taken*. The brass case containing the chrono- 
 meter is removed from the jimbals, the glass case is unscrewed, 
 and the instrument taken out. The balance is then secured by 
 two thin wedges of cork, and the chronometer replaced, the glass 
 case screwed on, and the whole wrapped up in a sheet of thin 
 paper. The jimbal ring is next removed, and the screws placed 
 in a small packet in the bottom of the case. Some soft stuffing 
 (such as dry oakum, paper shavings, hair, <fcc.) quite free from 
 dust (sawdust or wood shavings are not to be used) is then laid, 
 at the bottom, and the ring placed on it. Over the ring place 
 more stuffing, then the chronometer is laid in carefully, and 
 secured with more stuffing up to the glass lid of the wood box. 
 The case must then be enclosed in a wicker basket, or, if 
 necessary, in a large quantity of stuffing sewed up in stout 
 canvas, so that under no circumstances can the instrument receive 
 a jarring blow. This precaution is of the greatest importance. 
 
 237. METHOD OF TRANSPORTING A GOING CHRONOMETER FROM 
 THE SHORE TO A SHIP. 
 
 The instruments are secured from swinging in their jimbcds 
 by clamps provided for the purpose, and are generally carried 
 between two men in a sheet of canvas. The object of carrying 
 them thus is twofold, (1) to prevent their receiving a j 
 blow, and (2) to keep them free from any circular in<>tin,). 
 Villarceau recommends the following method : a trustworthy man 
 stands within a hoop about a yard in diameter ; he then takes up 
 two chronometers properly secured, one with each hand, and thus 
 safely carries them, free from any circular motion, and also 
 secured from knocking against his person when walking. 
 
 238. STOWAGE ON BOARD. 
 
 When placed on board the following conditions are, as far as 
 possible, sought for in stowing the chronometers. 
 
 (1) Where the traffic is least. (2) Where the tremor from 
 the machinery is least felt. (3) Low down in the ship where the 
 * General Instructions for Hydrographic Surveyors, p. 34. 
 
XII.] CHRONOMETERS. 225 
 
 motion is least. (4) Where damp cannot attack them. (5) Where 
 the ordinary temperature is most uniform. (6) Not near large 
 masses of iron on account of possible magnetic influence*. 
 
 The following negative precautions ought also to be observed : 
 
 (1) The instruments ought never to be removed from their 
 
 position except when the ship is in dry dock, or for extensive 
 
 repairs ; nor (2) ought they to be placed on swinging tables, or 
 
 on a bed during firing, or held in the hand, or placed in drawers. 
 
 The errors arising from these various supposed precautions 
 are, in fact, greater than the errors from those causes which they 
 are supposed to guard against. 
 
 239. In the management of chronometers, the two things to 
 be chiefly guarded against are damp and variable temperature. 
 The former is a prolific source of injury to the instruments ; e.g. 
 here is a remark taken from the " Register of Repairs of Govern- 
 ment Chronometers" in the Royal Observatory, when the chrono- 
 meters are returned from ships paying off, " Chronometer dirty. 
 Pendulum spring rusty." 
 
 240. EFFECT OF TEMPERATURE. 
 
 The general tendency of change of temperature may be thus 
 
 * Villarceau remarks that ships now-a-days are composed almost entirely 
 of iron, and that experience seems to prove that the magnetism of the ship 
 has little influence on the rates, on the supposition that the parts of the 
 chronometer itself are not magnetic. 
 
 If a chronometer does present this phenomenon it is quickly detected by 
 the variations produced by the changes in the direction of the ship's head. 
 On the other hand, Chauvenet states that the " rates of chronometers have 
 been found affected by masses of iron in their vicinity, thus indicating a 
 magnetic polarity of their balances. Such polarity may exist in the balance 
 when first it comes from the maker, or it may be acquired by the chrono- 
 4anding a long time in the same position with respect to the mag- 
 netic meridian. In order to avoid any error that might result from this 
 polarity (whether known or unknown) it will be well to keep the chrono- 
 meters always in the same position. " 
 
 The reader may also consult Shadwell on Chronometers, pp. 14, 15 and 
 notes, for further examples of divergence in opinions on this subject. 
 
 R. M. S. 15 
 
226 MARINE SURVEYING. [CHAP. 
 
 stated : An increase of temperature causes the chronometer to 
 lose, and a decrease of temperature causes it to gain. ( 231.) 
 
 Besides the temperature, the performance of a chronometer 
 depends generally upon the age of tJie oils used in the works. Of 
 course only the very best oil is used, but after a certain time it 
 thickens, and the effect of this is to diminish the amplitude of the 
 vibration of the balance, and thus the chronometer is accelerated ; 
 and this acceleration is found to be almost exactly proportional to 
 the time since it was freshly oiled. There is danger, however, in 
 applying fresh oil, as, when the chronometer is subjected to the 
 heat test, the oil often runs away from the pivot where it is 
 required, and spreads over the plate where it thickens. 
 
 When the chronometers are received on board, they are ar- 
 ranged in a situation combining as many as possible of the above 
 favourable conditions. The dial plates are placed in one direction, 
 and the Standard chronometer is generally placed in the middle. 
 Chronometers are made to run for 1, 2, or 8 days, but most gene- 
 rally for 2 days : thus of 44 chronometers sent for trial in 1880, 
 43 were 2 days, and only a single instrument was made to run 
 for 8 days. An 8-day chronometer is usually wound every 7 
 days, and a 2-day chronometer is wound every day. The winding 
 ought to take place at the same hour every day for the following 
 reasons. 
 
 (1) To insure punctuality and constant habit, so that memory 
 may not prove treacherous. 
 
 (2) To insure an even daily rate, for 24 hours, not 20 hours 
 one day, 30 hours the next. 
 
 (3) To insure the same part of the chain being always in iise, 
 because, if the fusee is not accurately cut, we may obtain a 
 different arc of vibration of the balance. 
 
 241. The winding is performed by a certain number of half- 
 turns of the key, and this number ought to be exactly known and 
 recorded. on a slip of paper attached to the inside of the cover, 
 so that in case of absence or sickness another officer might 
 without fear be entrusted with the duty. In winding, gently 
 
XII.] CHRONOMETERS. 227 
 
 invert the chronometer, draw back the spring-plate at the bottom, 
 and insert the key ; then turn firmly and evenly from right to 
 left, as in an English lever watch, i.e. in the direction contrary 
 to the motion of the hands. As the last half-turn is reached, 
 (the chronometer must always be wound as far as possible), more 
 care ought to be exercised, lest a sudden jerk might cause injury. 
 It may, however, be noted that the resistance to further winding 
 is produced, not by the end of the chain, but by a catch provided 
 to act at the proper moment, and thus to save the chain from 
 being strained or broken. 
 
 When a chronometer has stopped, it does not start again by 
 itself on being wound up. When the winding has been completed, 
 take the case into the hands, and give it a moderately quick and 
 firm circular motion through an arc of about 90 degrees. It is 
 found that if the time of stoppage does not exceed two or 
 three days, the chronometer generally resumes its former daily 
 rate*. 
 
 242. COMPARISON OF THE CHRONOMETERS. 
 
 When a chronometer is sent for service 011 board ship its error 
 on G.M.T. at noon on the day on which it is removed, and its 
 daily rate, are sent with it. Of course its performance on board 
 is soon ascertained, and if its rate has changed under its new con- 
 ditions, this new rate is known as the "harbour rate." When the 
 ship has made a voyage, its error is determined at the port of 
 arrival, either by single or equal altitudes, and the difference 
 between the error at starting and that now determined, when 
 divided by the number of days which have elapsed, gives the "sea 
 rate " or " travelling rate." It seldom happens that the sea and 
 harbour rates are exactly the same. In the rates of chronometers, 
 it is observed that a constant acceleration takes place, i. e. losing 
 rates decrease, and gaining rates increase. This result is due 
 principally to three causes, (1) infiltration of dust, (2) thickening 
 of the oils, (3) wear of the pivots. 
 
 * Here again there seems to be a difference of opinion, but if the instru- 
 ment does not stop for more than the above time, it may be assumed that 
 its rate is the same as before. 
 
 152 
 
228 MARINE SURVEYING. [CHAP. 
 
 243. METHOD OF COMPARING. 
 
 The Standard is generally denoted by the letter Z, and the 
 other chronometers by the letters A , B, C, . . . , or by their numbers, 
 or makers' names. It is found convenient, for the purposes of 
 comparison, to consider the Standard &sfast on all the other instru- 
 ments, adding 12 hours to the Standard's time when necessary. 
 Most chronometers beat half seconds, but some beat 5 times in 
 2 seconds ; hence the former will beat 10 times in 5 seconds, and 
 the latter 10 times in 4 seconds. 
 
 First, to compare the chronometers A, B, C with a deck watch 
 M. Place M where its beats can be distinctly heard, and where 
 its face and those of the chronometers can be seen. 
 
 Write down the hour and minute of M at which the com- 
 parison is to be made, e.g. let M= 3 h 27 m 00 . Then, when the 
 seconds hand of M arrives at 55 s (if M beats J seconds), or at 56* 
 (if M beats 5 times in 2 seconds), count 0, at the following beats 
 count 1, 2, 3,...; and at the 6th or 7th beat, cast the eye quickly 
 on the face of the chronometer, and at the 10th beat, note care- 
 fully whether the chronometer's seconds hand is (1) at an exact 
 second or J second, or (2) passing from the second to the half 
 second, or (3) from the half second to the second, and estimate as 
 follows : 
 
 In the first case write 8- or 8t 5, in the second case write 
 9 -2 or O f -3, in the third case write 0'-7 or 0*-8. 
 
 Supposed marks 6 h 14 m 8'-3, then M-A= 15 h 27 m 8 0-6 h 14 m 8"-3 
 (12 hours added to M) = 9 h 12 m 51 8 -7. 
 
 After a minute, compare B in like manner. Suppose B 
 = 7 h 23 m 9"-7. Subtract 1 minute, and write 7 h 22 m 9 8 '7, then 
 M-B= 15 h 27 m 8 -0- 7 h 22 m 9"-7 = 8 h 4 m 50 8< 3. 
 
 After another minute, compare C. Suppose (7=4 h 37 m 51 8 -3, 
 subtract 2 minutes and write 4 h 35 m 51'-3, then J^-C7=15 h 27 m 8 '0 
 -4 h 35 m 51 8 -3 = 10 h 51 m 8 8 -7. 
 
 In this way all the comparisons are reduced to the same I 
 If Z is the standard, then the others are compared with it. 
 
XII.] 
 
 CHRONOMETERS. 
 
 229 
 
 The method of comparison on board ship is merely to compare 
 the other chronometers with the Standard. This may be done in 
 the following manner : 
 
 Write down the time shewn by the Standard to some exact 
 minute. The Standard most commonly beats half seconds. At 
 55" count 0, then 1, 2, 3, ... at the next successive beats; at the 
 sixth or seventh look at the seconds hand of the chronometer to be 
 compared (suppose, Dent, or Frodsham, or Eiffe, or A, or some 
 number), and at the tenth beat note the part of the decimal accord- 
 ing to the method just explained. Write down these seconds and 
 decimals of a second, next the minutes, and lastly the hour, and 
 take the difference ; the comparison is then effected. 
 
 244. COPY OF A CHRONOMETER JOURNAL (SHADWELL, p. 29). 
 
 Date. 
 
 Z-A 
 
 2nd 
 Diff. 
 
 Z-B 
 
 2nd 
 Diff. 
 
 z-c 
 
 2nd 
 Diff. 
 
 Remarks. 
 
 1st 
 
 3 24 53 
 
 
 6 55 29 
 
 
 146 
 
 
 Heavy Gale. 
 
 2nd 
 
 3 24 54-7 
 
 1-7 
 
 6 55 41-2 
 
 12-2 
 
 1 4 4-5 
 
 1-5 
 
 j Confused 
 (Cross Sea. 
 
 3rd 
 
 3 24 55 
 
 0-3 
 
 6 55 52 
 
 10-8 
 
 1 4 3-7 
 
 0-8 
 
 Less Motion. 
 
 4th 
 
 3 24 55-8 
 
 0-8 
 
 6 56 2 
 
 10-0 
 
 1 4 2-7 
 
 1-0 
 
 (Night 
 | Quarters. 
 
 Remarks on the above extract : 
 
 (1) If the daily rates of Z and A are the same, then Z A is 
 always the same. 
 
 (2) If the daily rates of Z and A are not the same, but if the 
 rates are uniform, then the Second Difference is the same. 
 
 (3) If the Second Difference is found to vary, it is evident 
 that either Z or A is altering. 
 
 To determine a faulty chronometer we must introduce a third 
 instrument. 
 
 Thus we have (Z-A) - (Z - S) = J3-A, or A-B. Hen- Z 
 has been eliminated. Now on comparing the daily values of Z-A, 
 
2.30 MARINE SURVEYING. [CHAP. 
 
 and A B, if it is found that the Second Difference of A B re- 
 mains constant, while that of Z A is irregular, we infer that A 
 and B are going steadily, and Z is altering; but if the Second 
 Difference of Z A and A B are both irregular, while that of 
 Z B is uniform, then A is the guilty one, while Z and B are 
 going satisfactorily*. 
 
 When several chronometers are to be regulated by single or 
 equal altitudes, these observations are made on shore by the aid of 
 a Deck Watch which must be compared with the Standard before 
 going on shore and on returning, and then the other chronometers 
 must be compared with the Standard. The double comparison 
 will eliminate any difference in the rates of the Deck Watch and 
 the Standard. 
 
 Example. The following comparisons were made before and 
 after observations. 
 
 Before. After. 
 
 Standard = 8 17 O'O 8 47 O'O 
 
 D. Watch=10 8 9-5 1038 8-0 
 
 10 8 50-5 10 .8 52-0 
 
 Deck Watch shews 10 h 19 m 13 -3 s when the observations were 
 made, required the chronometer time of the observations. 
 
 The Deck Watch lost 1 8> 5 in 30 minutes, .-. in 11 minutes (viz. 
 the difference between the first comparison and time of observa- 
 tion) it lost J of 1"*5 or 8t 55, .'. difference between Standard and 
 Deck Watch at moment of observation = 10 h 8 m 51'. 
 
 Deck Watch at moment of observation = 10 h 19 m 13 8 -3 
 Difference between Standard and Deck | 
 Watch at this moment. . . ..} 
 
 .'. Standard's time at moment of obs. = 8 28 4 '3 
 
 Remember that the Standard is fast on all the other chro- 
 nometers. 
 
 * Shadwell, p. 31. 
 
XII.] CHRONOMETERS. 231 
 
 EXAMPLES FOR EXERCISE. 
 
 (1) The following comparisons were made between the Stand- 
 ard and a Deck Watch before and after an observation. Before, 
 Standard 9 h 16 m OO, D.W. 2 h 37 m 18"5. After, Standard 9 h 29 m O"0, 
 D.W. 2 h 50 m 20'-5. The time of observation by the D.W. was 
 2 h 46 m 15 8< 3 j find the Standard's time of observation. 
 
 Result, 9 h 24 m 55-4. 
 
 (2) The following comparisons were made before and after an 
 observation. 
 
 Before, Standard 3 h 27 m O B -0, D.W. ll h 14 m 16'-5. 
 After, Standard 3 38 0-0, D.W. 11 25 20. 
 The time by D.W. when the observation was made was 
 
 H b 20 m 56 i -5. 
 Required the Chronometer time of observation. 
 
 Result, 3 h 33 m 37-9. 
 
 (3) The following comparisons were made : 
 
 Before, Standard 5 h 14 m 30 s , D.W. 3 h ll m 16 B . 
 After, Standard 6 24 0-0, D.W. 4 20 48-7. 
 
 Time by D.W. when the observation was made 3 h 56 m 19 B '5. 
 Required time of observation by the Standard. 
 
 Result, 5 h 59 m 31"-8. 
 
 245. In order to estimate correctly the interval s '2 of a 
 second, the following method has been suggested. Stand in front 
 of a wall which returns an echo, and about 110 feet from it: 
 any sharp noise, such as clapping the hands, will have its echo 
 returned in ^ second. A little practice will render most persons 
 quite competent to determine this portion of time by the ear. 
 
 246. COMPARISON OF A CHRONOMETER WITH A SIDEREAL 
 CLOCK. 
 
 When two chronometers beating J seconds are compared, it 
 will seldom happen that their beats are coincident; they differ by 
 a fraction of a second, the amount of which, by the methods 
 
232 MARINE SURVEYING. [CHAP. 
 
 already described, must be estimated by the ear. It is different, 
 however, in the case where a chronometer and a sidereal clock are 
 compared. In this case, it is quite possible to estimate the differ- 
 ence between the times by the two instruments within ^ second, 
 or, by practice, within even a smaller fraction. Since I 8 sidereal 
 time is less than I 8 mean time, the beats of the clock will gain on 
 those of the chronometer, and certain beats will exactly coincide. 
 If the comparison can be made at this instant, of course the error 
 of the one instrument on the other is determined. The only 
 difficulty, therefore, arises from the impossibility of distinguishing 
 this exact instant; but it is found that the ear will detect the non- 
 coincidence of beats as long as the beats differ by -^ second, and 
 hence the comparison may be obtained within that amount. 
 
 Now 1" sidereal time = s> 997 mean time, .*. a sidereal clock 
 gains 8t 003 on the chronometer in 1", and .'. gains J second in 
 something less than 3 minutes. About every three minutes, there- 
 fore, the two instruments will have coincident beats, and when this 
 is about to occur, the observer begins to count the beats of the 
 chronometer while he looks at the clock : when his ear can no 
 longer detect any difference between the beats, he notes the cor- 
 responding seconds of the two instruments, then writes down the 
 minutes and hours, and the comparison is made. 
 
 Example. A chronometer and sidereal clock were compared 
 by coincident beats as follows : 
 
 First obs. Second obs. 
 
 Chron. 4 h 16 m O'O" 4 h 19 m 10' 
 Clock 1 3 11-5 1 6 22 
 
 Diff. 3 12 48-5 3 12 48 
 
 EXAMINATION. 
 
 (1) In what does the excellence of a chronometer over an 
 ordinary time-piece consist ? 
 
 (2) Explain the principle of the " gridiron pendulum." 
 
XII.] CHKONOMETEKS. 233 
 
 (3) Who first introduced with success the method of finding 
 the longitude by means of chronometers ? 
 
 (4) But a previous attempt had been made; and a still earlier 
 suggestion made on the subject? 
 
 (5) Can you give any description of the mechanism which 
 regulates the movement of a chronometer 1 
 
 (6) Explain the principle of the fusee. 
 
 (7) Describe in some detail the balance of a chronometer, 
 specifying the position of the metals, and noting the effect of 
 changes of temperature. 
 
 (8) Mention the two chief methods of " compensating for 
 temperature." 
 
 (9) Draw a diagram to illustrate Airy's compensation. Ex- 
 plain its object, and its special advantage. 
 
 (10) What is the "heat test" to which chronometers are 
 exposed in the Royal Observatory 1 
 
 (11) What grave objection is there to interfering on board 
 ship with the position of the weights or screws of the balance *? 
 
 (12) Explain the construction of a Chronometrical Thermo- 
 meter, and the use of such an instrument. 
 
 (13) What is meant by the "Trial Number" of a chrono- 
 meter 1 ? and write down the formula by which it is obtained. 
 
 (14) If a chronometer is sent by land, what precautions must 
 be taken in the packing 1 
 
 (15) Mention the precautions which ought to be observed in 
 transporting chronometers from shore to ship. 
 
 (16) Specify the most favourable conditions for the situation 
 of a " Chronometer Room." 
 
 (17) What are the two chief evils to be guarded against in 
 the case of chronometers 1 Give reasons for your answer. 
 
 (18) How would you stow the chronometers in their places'? 
 
 (19) WTiat is the general tendency of variation of tempera- 
 ture on the going of a chronometer ? 
 
234 MARINE SURVEYING. [CHAP. XII. 
 
 (20) Specify the two chief sources of error in the rates of a 
 chronometer. 
 
 (21) Notice the general effect of lapse of time on the rate of 
 a chronometer. 
 
 (22) Name the causes of this. 
 
 (23) Give the three reasons why a chronometer ought to be 
 wound at the same hour daily. Which is the most important of 
 the three ? 
 
 (24) Describe the method of winding a chronometer. 
 
 (25) If a chronometer runs down, how is it set going again ? 
 
 (26) Distinguish between the " harbour rate " and the " sea 
 rate." How are they determined 1 
 
 (27) How would you make a comparison between a Deck 
 Watch and a Chronometer, on the supposition that the Chrono- 
 meter beat half-seconds, and the Deck Watch beat five times in 
 two seconds ? 
 
 (28) If you were in charge of the chronometers in a ship 
 describe your method of comparing them daily. 
 
 (29) What is the use of the " Second Difference " column in 
 the Chronometer Journal ? 
 
 (30) You have three chronometers on board, and reasons 
 having arisen for doubting their performances, describe how you 
 would detect the least trustworthy instrument. 
 
 (31) How are a chronometer and sidereal clock compared ? 
 Within what probable error will the comparison be effected ? 
 
CHAPTER XIII. 
 
 MERIDIAN DISTANCES. 
 
 247. DEF. THE Meridian Distance between two places is 
 measured by the exact difference of time at the two places. Hence 
 if a chronometer which shews the exact mean time at a place A 
 be carried to a place , and its error on B be determined, we can 
 compute the difference of the times at A and , and therefore the 
 Meridian Distance of A and B. 
 
 The Meridian Distance is generally expressed in time, the 
 difference of Longitude in arc. 
 
 248. DEF. Meridians are either primary or secondary. A 
 Primary meridian is that from which different nations begin to 
 count their longitude. E. g. England, the United States, Sweden, 
 Norway, Prussia, and Austria reckon from the meridian of Green- 
 wich; France reckons from Paris; Spain from Cadiz; Portugal 
 from Lisbon, &c. A Secondary meridian is the meridian of a 
 place whose longitude from a Primary meridian is known exactly. 
 Thus on the shores of the North and South Atlantic Oceans there 
 are 19 such Secondary meridians recognized: 7 in the Indian 
 Ocean and Red Sea, 7 in the Eastern Seas, 9 in the Southern 
 Ocean, and 8 in the Pacific. 
 
 249. Their use may be thus explained. A ship wishes to 
 run a Meridian Distance with the object of finding the difference 
 of longitude between, we will suppose, St John's, Newfoundland, 
 
236 MARINE SURVEYING. [CHAP. 
 
 and Halifax, Nova Scotia. She finds, by her chronometers, that 
 the difference of time at the two places is O h 43 m 37 8> 5, and that 
 Halifax is to the westward of St John's. Now St John's is a 
 Secondary meridian and its longitude is known to be 52 40' 47" 
 W. ; if therefore to this the above Meridian Distance expressed 
 in arc (= 10 54' 23") be applied we obtain 63 35' 10" W. the 
 longitude of Halifax. 
 
 It may be also noted that the Meridian Distance between two 
 places A and B does not require the knowledge of the longitude 
 of either place. The Meridian Distance applied to the longitude 
 of one of the places afterwards found will give the absolute longi- 
 tude of the other. 
 
 The surveyor therefore confines his attention to a most careful 
 measurement of the Meridian Distance between some Secondary 
 meridian and the place, the longitude of which it is desired to fix 
 absolutely. 
 
 250. This Meridian Distance is measured in two ways : 
 
 (1) By the Electric Telegraphy where possible. 
 
 (2) By Portable Chronometers. 
 
 251. I. BY THE ELECTRIC TELEGRAPH. Of course where 
 telegraphic communication exists between two places, the mean of 
 a number of signals must give a very perfect result*. But there 
 are very many places where, at present, the only possible method 
 of measuring a Meridian Distance is by transporting chronometers, 
 carefully manipulated, from one point to another. We proceed to 
 describe this method. 
 
 252. II. BY PORTABLE CHRONOMETERS. 
 
 In running a Meridian Distance the following precautions 
 ought to be observed : 
 
 * For the details of such a method, the reader is referred to the " Report 
 on the Difference of Longitude between the U.S. Naval Observatory and the 
 Sayre Observatory of Lehigh University," by Prof. Eastman, Washington, 
 1878. This Report is in the Library of the Royal Naval CoUege. 
 
XIII.] MERIDIAN DISTANCES. 237 
 
 (a) The chronometers ought to be stowed on board so that 
 surrounding circumstances may have least influence on their 
 performance. 
 
 ((3) Never to be touched except for winding and comparisons. 
 
 (y) Only one person to have access to them. 
 
 (8) To be wound at the same hour daily. 
 
 (e) The comparisons to be made with the " Standard," back- 
 ward and forward. 
 
 () The error to be determined on mean time at place before 
 starting, at equal intervals of time (5, 7, or 10 days), to ascertain 
 the performance of individual chronometers. 
 
 (17) The error to be determined by equal altitudes in the 
 following manner: 
 
 i. Compare the Deck Watch with Standard before going 
 ashore. 
 
 ii. Observe in the forenoon thus : 
 
 Take the altitude of the 's L.L. 
 
 Take the 0's U.L. 
 
 Set the index 2 on, and take L.L. and U.L. 
 
 Take 6 sets, 3 with each eye. 
 
 iii. On coming on board compare Deck Watch with the 
 Standard. 
 
 iv. Compare again before going ashore for the afternoon 
 observations. 
 
 v. Take plenty of time, and set index at last altitude, and 
 repeat the observations in a reverse order. 
 
 vi Again compare the Deck Watch with the Standard on 
 finally arriving on board. 
 
 Vide Naval Science, Vol. in. p. 95 ft 
 
 253. The following formula will enable us to find the ap- 
 proximate time by the Deck Watch when the P.M. observations 
 ought to be taken. 
 
238 MAEINE SURVEYING. [CHAP. 
 
 Let = time shewn by watch at the last A.M. observation. 
 x = error of watch on S. M. T. (supposed slow). 
 
 e = equation of time, supposed to be subtractive from mean 
 time. 
 
 ^ + ic = mean time of last A.M. observation. 
 
 .. t + x e = apparent time of last A.M. observation. 
 
 .*. 12 (t -r x - e) = time to apparent noon, or the apparent time of 
 the first P.M. observation. 
 
 .. 12 (t +xe) + e = mean time nearly of first P.M. observation. 
 
 .. 12 (t+x e)+e x= time by watch of first P.M. observation. 
 
 .. l2 t-x + e + e x = time by watch required. 
 
 12 -t-2x + 2e = 12 -t- 2 (x-e)=time required. 
 
 E.g. Let watch shew at the last obs. in the forenoon, 5 h 7 m 20*, 
 watch slow 4 h 12 m 25" on S. M. T. Equation of time 8 m additive 
 to mean time. Then we can find the approximate time by watch 
 for the first P.M. observation as follows : 
 
 Time by watch 5 h 7 m 
 Error 4 12 + 
 
 M. time 9 19 
 Eq. time 8 + 
 
 App. time 9 27 
 12 
 
 2 33 = interval to noon 
 
 = P.M. app. time. 
 2 33 
 Eq. time 8 - 
 
 2 .25 = P.M. mean time. 
 Error 4 12- 
 
 10 13 = time by watch for the first P.M. 
 observation. 
 
XIII.] MEKIDIAN DISTANCES. 239 
 
 254. We can see that if the rate of a chronometer does not 
 change in being carried from one place to another, in other words, 
 if the "travelling rate" or "sea rate" is the same as the "harbour 
 rate," it will be very easy to discover the difference of time at the 
 two places. These rates, however, are very seldom the same, and 
 hence the computation is somewhat more tedious. 
 
 Example*. Where the travelling rate is supposed to remain 
 the same. At Greenwich, May 5th, at mean noon, a chronometer 
 shewed ll h 49 m 42"-75, its rate being 2"-671 gaining. On May 
 17th, at a place A in the United States, at mean noon, it shewed 
 4 h 34 m 47 .. 28 . 
 
 Required the longitude of the place A. 
 
 May 4 23 h 49 m 42 -75 
 May 17 4 34 47-28 
 
 Elapsed time = 12 4 45 4-53 = 12-198 days. 
 Accumulated rate = 2-671 x 12-198 = 32-58 seconds. 
 
 Acc d . rate= 32-58- 
 
 4 34 14-70 
 Time at Greenwich 11 49 42-75 
 
 Long, in time 4 44 31-95 = 71 8' 00" W. 
 
 255. It may be remarked, for the sake of the younger readers, 
 that since the Sun, in his daily course, arrives at the meridian of 
 a place A to the eastward of Greenwich sooner than it arrives at 
 the meridian of Greenwich, the time at A is fast on Greenwich 
 time. 
 
 E.g. If a watch which shews the time at Bombay is brought 
 to Malta, it is fast on Malta time ; if brought to Gibraltar, it will 
 be still more fast on mean time at place. 
 
 Again, suppose that a watch which shews Greenwich mean time 
 were taken to Rome, it will be slow on the mean time at place; it 
 will be still more slow on time at Constantinople, &c. 
 
 * From Chauvenet. 
 
240 MARINE SURVEYING. [CHAP. 
 
 Once more, if a watch which keeps Greenwich time is found 
 to be slow at a place A, and still more slow at a place B, we infer 
 that A and B are both in East longitude, and that B is farther 
 East than A. 
 
 Finally, if a watch which shews mean time at A is found to be 
 fast on mean time at B, then it is evident that B is to the West- 
 ward of A. 
 
 The following questions ought to be thoroughly mastered before 
 the solution of the Meridian Distances is attempted. 
 
 EXAMINATION. 
 
 (1) A chronometer is slow 3 h 12 m 17* on A, and is slow 
 3 h 24m 16 s on B . is A east or west of B ? A west 
 
 (2) A chronometer is fast 7 h 16 m 5* on A, and slow l h 14 m 18' 
 on B; is B east or west of A 1 B east of A. 
 
 (3) A chronometer is fast 2 h 14 m 8 s on A, and fast 4 h 17 m 58' 
 on B] is A east or west of B 1 A is east of B. 
 
 (4) A chronometer is slow on A, 5 h 27 m 33 s , and fast on B, 
 2 h 18 m 49 B ; is B east or west of A ? B west of A. 
 
 (5) A chronometer is slow on A, 3 h 15 m 16 s ; ought the chro- 
 nometer to be fast or slow on B, in order that the place whose 
 time the chronometer shews may be midway between A and B '? 
 
 Fast on B. 
 
 (6) The time at B is fast on a chronometer l b 16 m 35 s ; the 
 chronometer is fast on -4, 5 h ll m 51 s . Is A to the east or west 
 oi B1 A is west of "B. 
 
 256. It may be well to give here the usual Formulae by 
 which the " sea rate " of a Chronometer is found, when running 
 a Meridian Distance between two places A and B. 
 
 Let a l error of Chron. at A before sailing, 
 b l = B on arrival, 
 
 b a = B before sailing, 
 
 2 = A on arrival. 
 
XIII.] MERIDIAN DISTANCES. 241 
 
 Then (a l ~ a a ) total error accumulated during the absence 
 of the Ship from A (Harbour and Sea Error). 
 and (b t ~ b a ) = total error accumulated during the stay at B 
 (Harbour Error). 
 
 .-. (i~a 2 )- (&i~& 2 ) = amount of Error accumulated at Sea, in the 
 two runs. 
 
 Let m number of days from A to B, 
 and n = number of days from B to A. 
 
 ... Sea rate = ~ 
 
 + n 
 
 If only one error is determined at B, then we must use the 
 less satisfactory Formula 
 
 Sea rate = -i - * , where x is the number of days the ship is 
 absent from A. 
 
 Then, knowing the sea rate, we can compute at any moment 
 the Error of the Chronometer on the time at A. 
 
 Finally, Meridian Distance = Error of Chron. on time at A 
 ~ Error of Chron. on time at B. 
 
 Meridian Distance worked out : 
 
 (For greater refinements in the methods of computation, the 
 reader may consult Shad well on Chronometers, Ed. 1861, Chap- 
 ter VIII.). 
 
 Example 1*. 
 
 August 1, chron. fast on Cape Passaro time 7 h 54 m 5 8> 7 
 2, Messina 7 52 23'7 
 
 8, Messina 7 52 19'2 
 
 9, Cape Passaro 7 54 0*6 
 
 * Examples 1, 2, 3 are taken from Papers in Naval Science on Nautical 
 Surveying by Capt. Shortland, Vol. in. 'p. 101. See Capt. Shortland's remarks, 
 p. 102. 
 
 K. M. S. 16 
 
242 MARINE SURVEYING. [CHAP. 
 
 Compute the Meridian Distance between Cape Passaro and 
 Messina. 
 
 Aug. 1, 7 h 54 n 5-7 Aug. 2, 7 h 52 m 23'-7 
 
 9, 7 54 0-6 8, 7 52 19-2 
 
 Loss in 8 days 5'1 Loss in 6 days 4-5 
 
 (harbour and sea). (harbour). 
 
 Loss in 8 days 5'1 
 
 (harbour and sea). 
 
 .-. loss during the 2 days at sea O6 
 
 or sea rate = 0'3 
 
 August 1, Chron. fast on Cape Passaro 7 h 54 m 5 8 '7 
 Change in one day at sea 0'3 
 
 .'. August 2, Chron. fast on Cape Passaro 7 54 54 
 
 Messina 7 52 23-7 
 
 .'. Meridian Distance = . 1 41 '7 
 
 Again, Augusts, Chron. fast on Messina 7 52 19*2 
 
 Change in one day at sea 0*3 
 
 .-. August 9, Chron. fast on Messina 7 52 18-9 
 
 Cape Passaro 7 54 0'6 
 
 .'. Meridian. Distance = 1 41-7 
 
 Since chronometer is faster on Passaro than on Messina, it 
 follows (by Ex. 3 in the Examination) that Passaro is to the 
 westward of Messina. 
 
 Example 2. 
 
 Sept. 4th, Chron. fast on Le Have mean time 5 h 35 m 26 ! '-r7 
 5th, Prospect 5 33 1347 
 
 10th, Prospect 5 33 27'25 
 
 llth, Le Have 5 35 45-34 
 
 Compute the Meridian Distance. 
 
 Result, 2 m 15" -3 9 Le Have to the westward of Prospect. 
 
XIII.] MERIDIAN DISTANCES. 243 
 
 Example 3. 
 
 Sept. 5th, chron. fast on Prospect mean time 5 h 33 m 13 8 '47 
 
 6th, Halifax 5 32 34-66 
 
 9th, Halifax 5 32 4270 
 
 10th, Prospect 5 33 27 -25 
 
 Compute the Meridian Distance. 
 
 Result, O m 41'-68 Prospect to the westward of Halifax. 
 
 Example 4. On leaving a port A, in longitude 1 1 8 4' E. a 
 chronometer was slow on S. M. T. 7 h 47 m 9" '6 ; daily rate I 1 -9 
 gaining ; on arriving at B, the chronometer was slow on S. M. T. 
 7 h 55 m 55*'3 ; daily rate 2"'03 gaining; the interval between 
 these observations was 2-99 days. Required longitude of B. 
 
 (Exam. Papers, May, 1876.) 
 
 Mean rate =1-965 gaining. 
 
 Accumulated rate in 2-99 days = 5"-86 gained. 
 
 Chron. slow on A 7 h 47 ra 9'-6 
 Accum. rate 5 '86 gained. 
 
 .-. Chron. slow on A after 2-99 days 7 47 374 
 Chron. slow on B at same time 7 55 55 '30 
 
 .-. Meridian Distance = 8 51-56; and B is to the 
 
 east of A. 
 
 D. long. = 2 12' 53" E. 
 Long, of 4 = 118 4 OE. 
 
 .-. Long, of .#=120 16 53 E. 
 
 Example 5. On leaving Malta, long. 14 31' K, chron. was 
 fast on S.M.T. l h 20 m ll s -5 ; daily rate 7 8< 23 gaining ; seven days 
 afterwards on arriving at Alexandria, long. 29 51' 40" E., the 
 chron. was fast on S.M.T. O h 20 m 27"'55. Find what change had 
 taken place in the daily rate. (Exam. Papers, 'Oct. 1877.) 
 
 10 2 
 
244 MARINE SURVEYING. [CHAP. 
 
 Long. Alexandria = 29 51' 40" E. 
 Malta =14 31 00 E. 
 
 D. long. = 15 20 40 
 In time = l h l m 22"-6 
 Chron. fast on Malta 1 20 11-5 
 
 .-. If chron. had no daily rate, it\ 
 
 .1 i . ( lo 4o*y 
 
 ought to be fast on Alexandria,) 
 
 It actually was fast 20 27 -55 
 
 .-. Accum. rate in 7 days= 1 38-65 gained 
 .-. Daily rate = 14 '09 gaining 
 Former rate = 7 -23 gaining 
 
 .-. change in rate = 6 -8 6 gained 
 
 Example 6. A chronometer was slow on Monte Video mean 
 time l h 14 m 14 s (long. 56 10' W.); daily rate 1''8 losing. Fourteen 
 days afterwards at sea, the summit of Tristan d'Acunha in lat. 
 37 17' S., long. 12 36' W. was observed to bear N. 38 E. (true), 
 distant 5 miles, when the chronometer was found to be slow on 
 S.M.T. 4 h 9 m OO 8 . Find change in the rate. 
 
 (Exam. Papers, June, 1878.) 
 
 First to determine the exact position of the ship. 
 S. 38 W. 5', . . D. lat, = 4' S. Dep. = 3'-l, .'. D. long. 
 
 Lat of island 37 17 S. (=3-1 sec. 38 19^ 
 
 .-.Latin= 37~21S. * 3'57"W. 
 
 Long, of island =12 36 00 W. 
 
 .-. long, in 12 39 57 W. 
 
 Long, of Monte Video = 56 10' 0" W. 
 Ship : 12 39 57 W. 
 
 .-. D. Long, in arc= 43 30 3 
 
 in time = 2 h 54 m OO" ship to the east. 
 
 Chron. slow on Monte Video 1 14 14 
 
XIII.] .MERIDIAN DISTANCES. 245 
 
 .-. If Chron. had no rate it would) >mij.8 
 
 have been slow on S.M.T. / 
 
 It actually was slow 4 9 00 
 
 .-. Accum. rate in 14 days= 46 lost. 
 
 .-. daily rate = 3'-28 losing 
 Former rate = 1 -80 losing 
 
 . '. change in rate = 1 -48 in a losing direction. 
 
 EXAMINATION, AND EXAMPLES FOR EXERCISE. 
 
 (1) What is a Meridian Distance? 
 
 (2) Define the terms Primary and Secondary meridians, and 
 mention the use to which the latter may be put. 
 
 (3) Describe the precautions which ought to be taken in 
 running a Meridian Distance. (Dec. 1877.) 
 
 (4) Mention the two principal methods of measuring a Meri- 
 dian Distance. 
 
 (5) Having found a Meridian Distance how would you ascer- 
 tain in what direction it ought to be applied to fix the longitude 
 of a station ? Write down an example to illustrate your answer. 
 
 (6) When the mean time is 3 h at Greenwich it is 7 h 51 m 12 s 
 at Bombay, and 13 h 4 m 56" at Sydney. What is the Meridian 
 Distance between Greenwich and Bombay, and between Bombay 
 and Sydney? (Exam. Papers, Sept. 1877.) 
 
 Results. (1) 4 h 51 m 12 s ; (2) 5 h 13 m 44". 
 
 (7) On leaving a port in longitude 116 39' E., a chronometer 
 was slow on S.M.T. 8 h 36 ra 00"; what would be its error on S.M.T. 
 in longitude 119 30' E. (rate not considered) ? (Nov. 1875.) 
 
 Result. 8 h 47 m 24' slow. 
 
246 MARINE SURVEYING. [CHAP. 
 
 (8) A foreign chart shews a rock in long. 9 54' west of the 
 meridian of St Petersburgh. The mean time of St Petersburgh is 
 2 h l m 19" fast on Greenwich M.T. What longitude should be 
 assigned to the rock on an. Admiralty chart? (March, 1878.) 
 
 Result. 20 25' 45" E. 
 
 (9) A Time Ball is dropped at an Observatory in lat. 34 S. at 
 1 P.M. local mean time ; a chronometer on board a Ship 6 miles 
 S.E. (true) of the Observatory, shews at the instant l h 29 m 16'. 
 Required the error of the chronometer on S. M. T. 
 
 (April, 1880.) 
 
 Result. D. Long, between Ship and Obs. = 5' E. = 20* in time. 
 .-. When ball dropped S.M.T. = l h O m 20', .-. error of chronometer 
 
 (10) A Time Ball drops at 1 P.M. mean time of Melbourne 
 Observatory. A chronometer shewed l h 30 m 30" at the same in- 
 stant. The Observation Spot in lat. 39 S. bears west (true) 3'4 
 miles from the Observatory. Required the error of the chrono- 
 meter on mean time of the Observation Spot. (Nov. 1877.) 
 
 Result. O h 30 m 47 /^. 
 
 (11) The last A.M. observation of a set of equal altitudes was 
 taken at 12 h 47 m 34 s by a chronometer slow on AT. at place 
 gh 43111 4.g 8< ]?i nc i the approximate time by the chronometer for the 
 first P.M. observation. (April, 1876.) 
 
 Result. 5 h 44 m 50'. 
 
 (12) The last A.M. observation of a set of equal altitudes was 
 taken at l b 51 m 20 s by a chronometer slow on AT. at place 
 7 h 39 m 00" ; what would be the approximate time by the chrono- 
 meter for the first P.M. observation? (Oct. 1876.) 
 
 Result. 6 h 50 m 40'. 
 
 (13) The last A.M. observation of a set of equal altitudes was 
 taken at 6 h 14 m 19" by a deck watch slow on S.M.T. 2 h 21 m 11' ; 
 equation of time 7 m 36 s subtract! ve from mean time; what would 
 be the time by the watch for the first of the P. M. observations ? 
 
 Result. l h 18 m 31'. 
 
XIII.] MERIDIAN DISTANCES. 247 
 
 (14) Compute the Meridian Distance between Trincomalee 
 and Madras from the following errors of chronometers : 
 
 June 23 slow on Trincomalee M.T. 5 h 39 m 52 8 -82 
 
 29 Madras 5 36 39*30 
 
 July 14 Trincomalee 5 42 15-19 
 
 (May, 1881.) 
 Result. Trincomalee O h 3 m 54 8 -38 east of Madras. 
 
 (15) On leaving a port A, in longitude 120 16' E. a chrono- 
 meter was slow on M.T. at place 8 h 7 m 30', daily rate 2 8 '23 gaining; 
 and on arriving at B, chronometer was slow on S.M.T. 7 h 42 m 58 8> 3, 
 daily rate 2 8 '15 gaining; the interval between the observations 
 was 7-02 days. Required the longitude of B. 
 
 (Sept. 1876.) 
 Result. Long. B= 114 11' 55" E. 
 
 (16) Aug. 5th a chron. was fast on Malta M.T. 2 h 39 m 30"-5, 
 
 12th 2 39 45-9. 
 
 Sailed on Aug. 12 for Gibraltar. 
 
 Aug. 20th the chron. was fast on Gib. M.T. 3 59 12-5, 
 
 26th ' 3 59 27-5. 
 
 Compute the Meridian Distance between Malta and Gibraltar. 
 
 (Feb. 1878.) 
 Result. Malta is l h 19 m 7 8> 8 E. of Gibraltar. 
 
 (17) On leaving a port A, in long. 114 11' E. a chronometer 
 was slow on S.M.T. 4 h 35 m 2'-9; daily rate 12 8 '16 gaining. On 
 arriving at a harbour J5, the chron. was slow on S.M.T. 4 h 59 m 7 8< 5; 
 daily rate 12*'88 gaining; the interval between the observations 
 was 14-98 days. Required the longitude of B. 
 
 (March, 1877.) 
 Result. Long, of B= 120 59' E. 
 
 (18) On August 1st a chronometer was fast on Malta M.T. 
 2 h 13 m 13", daily rate gaining l'-56; on August 7th the chron. was 
 fast on Cyprus M.T. O h 57 ra 51", daily rate 1 '85 gaining. Find 
 the Meridian Distance between Malta and Cyprus, using the mean 
 rate-. (Sept. 1878.) 
 
 Result. Cyprus is l h 15 m 32"-23 E. of Malta. 
 
248 MARINE SURVEYING. [CHAP, 
 
 i 
 
 (19) A chronometer was slow on S.M.T. (long. 56 20' W.) 
 l h 14 m 29", daily rate 1 8> 8 losing; 15 days afterwards at sea, an 
 island, in Lat. 37 10' S., Long. 12 39' W. bore N. 25 E. (true) dis- 
 tant 4 miles, when the chronometer was slow on S.M.T. 4 h 9 m 10". 
 What change had occurred in the rate? (Aug. 1880.) 
 
 Result. Long, ship = 12 41' 8" W. . Change 1 s -44 in a gaining 
 direction. 
 
 (20) On leaving Portsmouth, in Long. 1 6' 15" W. at noon, a 
 chronometer was fast on M.T. at place 3 h 5 m 2"; daily rate gaining 
 1 < 28. Find the time by chronometer at 1 P.M. Plymouth M.T. on 
 the succeeding day, the Longitude of Plymouth being 4 10' 15" W. 
 
 (Oct. 1879.) 
 
 Result. 1 P.M. at Ply mouth = l h 12 m 16" at Portsmouth. 
 . *. Rate must be computed for 1 '05 days. 
 .-. Chron. shewed 4 h 17 m 19'. 
 
 (21) Left Sydney on July 10th, a chronometer being fast on 
 S.M.T. 2 h 10 m O s -45, rate 3"-5 gaining. July 19th, the chronometer 
 was fast on S.M.T. at Tanna Island 3 h 22 m 59". July 31st, on 
 returning to Sydney the chronometer was fast on S.M.T. 2 h ll m 3*. 
 Calculate the Meridian Distance between Sydney and Tanna 
 Island. (Oct. 1877.) 
 
 Result. Tanna Island is l h 12 m 31'-7 W. of Sydney. 
 
 Note. The given rate is not required in the solution. 
 
 (22) On March 3rd in Harbour: 
 
 Chron. A was/as^ at noon on S.M.T. O h 4 m 40*, rate 3* -25 gaining. 
 B was slow 7 52 50, rate 0*33 losing. 
 
 On March 10th at Island: 
 
 Chron. A was fast at 110011 oil S.M.T. l h 4 m 4P,rate 3 8 '18 <jni,nng. 
 ,, B was slow ,, 6 52 50, rate 0'15 gal 
 
 Compute the Meridian Distance between the Harbour and the 
 Island by each chronometer, and state which result you would 
 prefer, and your reason. (March, 1879.) 
 
 Result. By A, Island west of Harbour O h 59 m 38 8 '53. 
 B 1" 0'" 0'. 
 
XIII.] MERIDIAN DISTANCES. 240 
 
 The result by A to be preferred, the rate of A being more 
 uniform. 
 
 Additional Questions from the Examination Papers set for 
 Classes JS l and B 2 . 
 
 (23) On leaving A, in Longitude 122 13' E. chronometer X 
 was slow on M.T. at place 7 h 54 m 27 8- 5, daily rate 3 8> 70 gaining; 
 at , X was slow 8 h 14 m 20'-8 ; and at C, X was slow 8 h 24 m 16'-7, 
 daily rate 3' -35 gaining. The interval between the observations 
 at A and B was 5 -9 9 days, and between the observations at B and 
 C 5'99 days. Required the Longitudes of B and G. 
 
 (June, 1876.) 
 
 Results. Long, of B = 127 16' 36" E. ; Long, of C 
 
 = 129 50' 51" E. 
 
 (24) On leaving A, in. Longitude 114 II' E., chronometer was 
 slow on M.T. at place 7 h 16 m 23"-10, daily rate 3 8 '24 gaming; at 
 B, chronometer was slow 7 h 42 m 54"'95, daily rate 2 8- 04 gaming; 
 the interval between the observations was 14-98 days. Required 
 the Longitude of B. (June, 1877.) 
 
 Results. Long, of .#=120 58' 45" E. 
 
 (25) May 8th, chronometer fast on M.T. at Mauritius, 
 3 h 45 m 46", daily rate gaining 2'-85. 
 
 The same chronometer fast on M.T. at Aden 4 h 36 m 2P-75 at 
 noon on May 19th, and at noon of May 26 fast on M.T. at Aden 
 4 h 36 m 46 8 -25, 
 
 Ascertain the Aden rate ; also the Meridian Distance between 
 Mauritius and Aden, using the mean of the rates. 
 
 (June, 1878.) 
 Results. Aden rate = 3" -5 gaining. 
 
 Meridian Distance = O h 50 m 8 -83 Mauritius to the east 
 of Aden. 
 
 (26) June 1st, chron. fast on M.T. Colombo 3 h 48 m 47"-5 
 
 8 Singapore 2 12 59'5 
 
 ,,15 Singapore 2 13 20-8 
 
 29 Hong Kong 1 32 49 -4 
 
250 MARINE SURVEYING. [CHAP. XIII. 
 
 Find the Singapore rate, and by means of it determine the 
 meridian distance between Colombo and Hong Kong. 
 
 (June, 1879.) 
 Results. Singapore rate = 3" '04 gaining. 
 
 Meridian Distance = 2 h 17 m 2 3" -2 Hong Kong to the 
 eastward of Colombo. 
 
 (27) At noon July 6th, a chronometer slow on M.T. Singa- 
 pore 7 h 44 15" '4 ; on proceeding at once to Malacca, the error of 
 the same chronometer on M.T. of that place, July 12th (noon) 
 was 7 h 38 m 40" -9 slow, and after remaining at anchor, the error at 
 noon of July 16th was 7 h 39 m 8 8- 7 slow; the return voyage to 
 Singapore was then made, and on July 21st (noon), at that latter 
 place the error was found to be 7 h 46 m 12"-7 slow; required the 
 Meridian Distance between Singapore and Malacca. 
 
 (June, 1880.) 
 
 Result. Meridian Distance = O h 6 m 2 3' -3 Singapore to the east- 
 ward of Malacca. 
 
CHAPTER XIV. 
 
 METHOD OF PLOTTING A SURVEY. 
 
 257. Having in the preceding chapters described with more 
 or less detail, as seemed necessary, the different methods of con- 
 structing and using scales, laying off angles, fixing positions, 
 measuring base lines, conducting a triangulation, taking tidal 
 observations, running lines of soundings, managing chronometers 
 and computing meridian distances, it seems well in this last 
 chapter to gather into one compass the various steps in a Survey 
 of a harbour. 
 
 258. The operations then may be arranged under two head- 
 ings. 
 
 I. The method of setting about the work, and the data to be 
 obtained. 
 
 II. The method of projecting on paper the details obtained 
 in I. 
 
 Projecting the work on paper is technically known as 
 Plotting. 
 
 259. I. The method of setting about the work. 
 
 (a) Certain remarkable features are fixed on, and suitable 
 names assigned by which they will be known, such as West 
 mound, Hut station, Flag station, Cairn, <kc. 
 
 (b) Select the best position for the Base Line. 
 
 (c) Measure the length of the Base. 
 
 (d) Determine the Lat. and Long, of one extremity. This 
 point is known as the Observatory Station. 
 
252 MAKINE SURVEYING. [CHAP. 
 
 The Latitude will probably be determined by the meridian 
 altitude of the sun and stars, observed in the artificial horizon. 
 
 The Longitude by sun chronometer, the sun's altitude being 
 observed in an artificial horizon. 
 
 (e) The Direction of the base will also be accurately deter- 
 mined. 
 
 (f) From the Observatory Station, angles must be taken 
 with the Theodolite to all the important points in the survey 
 which are visible. Similarly angles must be taken from all the 
 other stations. And a closer series of angles from such stations 
 as are suitable to enable the coast line to be cut in. 
 
 (g) The tidal observations are also to be made, to determine 
 the Establishment of the Port, Rise and Range of tide, &c. 
 
 (h) The Sounding operations are to be carefully made and 
 reduced to the datum line. 
 
 (i) The Levelling work must also be attended to, for the sake 
 of sections, contours, <fec. 
 
 (j) Topographical details, features of the coast, &c. are to be 
 noted. 
 
 260. II. Plotting the work. 
 
 Def. We may define this operation as the construction of a 
 figure similar to the projection of all the remarkable points of 
 the survey on a plane, a tangent to the Earth's surface, at the 
 middle point of the locality. 
 
 The paper ought to be mounted on a drawing-board. The 
 scale ought to be drawn on the paper, and will therefore vary as 
 the paper itself is subject to the changes in the atmosphere. 
 
 If the plotting is to be engraved it is essential that the si-alt 
 should be engraved with it, because the paper just before receiving 
 the impression is damped and consequently expands. In a length 
 of 2 feet the error will be ^ or -jl f an inch, a difference which 
 
XIV.] METHOD OF PLOTTING A SURVEY. 253 
 
 would render a plan almost useless. Indeed it is seldom that any 
 two impressions from the same plate are of exactly the same size, 
 owing to the moisture in the paper at the time of printing. 
 Printing on dry paper is never found to give satisfactory results*. 
 
 It is often convenient in Plotting Work to have the True or 
 Magnetic Bearings of the several sides of the triangles. This 
 information is secured by the process known as "Surveying by the 
 Back Angle" At the First Station, the Bearing of the Zero is found 
 (>$ 1-7), thus all angles observed may be treated as Bearings, and 
 these may be continued throughout the entire chain by setting 
 the Instrument at each station to the angle from the last onef. 
 
 In important surveys the triangulation must be put in by 
 the method described in 45, 46. This method enables us to 
 lay off angles to single seconds. The coast line is generally put 
 in by the aid of a rectangular protractor. 
 
 The following directions for Plotting the Questions in Exami- 
 nation Papers will, it is hoped, be of some use to the Younger 
 Students. 
 
 (1) In a convenient position on the paper make a dot, and 
 inclose it in a small circle; this will represent the Observatory 
 Station. 
 
 (2) From this point draw the Base Line on the proposed 
 scale of the plan and make a clear mark at the other extremity 
 of the Base. 
 
 (3) Lay off the angles observed at the Observatory Station 
 and at the other end of the Base. These " cuts " will " fix " the 
 principal objects as seen from the extremities of the Base Line. 
 
 (4) Then proceed to "fix" the other points or stations by 
 means of the angles taken at the various stations. 
 
 (5) Afterwards cut in the coast line by the angles observed 
 for that purpose. 
 
 * Maxton's Engineering Drawing, p. 233. 
 t Jeffers, p. 122. 
 
254 
 
 MARINE SURVEYING. 
 
 [CHAP. 
 
 (6) Insert all the topographical information and finish off 
 the frame, &c. 
 
 We shall now plot one or two of the Examples taken from 
 the different Examination Papers, and add a copious collection 
 from the same source for exercise. 
 
 261. Suppose then that we are required to plot the Beau- 
 fort Paper for November, 1878 (No. 8 in the following Exercises) : 
 we proceed as follows. 
 
 (1) Compute the Sun's True Bearing from the known lati- 
 tude, declination and altitude of the centre. This is found to be 
 N. 74 30' E. 
 
 tocallcs. 
 
 (2) Next compute the length of the base line from the 
 known angles of depression to the mainmast and the height of 
 the maintruck 130 feet. Thus we find the base line to be 
 6070 feet, which expressed in the given scale will be '2 -01 im-lu>s. 
 
XIV.] METHOD OF PLOTTING A SURVEY. 255 
 
 (3) We place a dot in the middle of the paper and rule the 
 true meridian through it; and then lay off N. 7430'E. This 
 will be the direction of the sun. 
 
 (4) The direction of C will evidently be 28 30' to the left of 
 this line. From this line we can lay off the angles observed at A. 
 
 (5) Lay off the direction of the ship 29 to the left, or 331 
 to the right. 
 
 (6) Lay off 2-01 inches found in (2)*. This gives the position 
 of the ship. 
 
 (7) Knowing the angle at C (vide the data) between the 
 zero A and the mainmast to be 54 40', and also knowing the angle 
 at A between the zero and mainmast 29 0', we can evidently find 
 the third angle of the triangle. This will be 96 20'. At ship 
 lay off this angle, and the line will cut the zero line in the posi- 
 tion of the station C. Hence we have fixed three points in the 
 Survey. 
 
 (8) We next lay off 45 15' to the right of the zero from A. 
 This will pass through B. We then lay off to the left of zero 
 from C an angle of 48 15' (360- 31 IMS'). These lines will inter- 
 sect in the station B. 
 
 (9) G is fixed by the angles from A and C. 
 
 H A ... a 
 
 D A ... B. 
 
 F B ... C. 
 
 Boulder A ... C. 
 
 (10) The angles subtended at ship by the points C and //, 
 A and G respectively, will be found to correspond with those ob- 
 served by the sextant on board. Thus proving that these points 
 are correctly fixed. 
 
 (11) The Topography of the island will depend on the 
 draughtsman's fancy, but all the directions given must be care- 
 fully attended to. A knowledge of the various Symbols and 
 Conventional Signs is requisite in putting in the given details. 
 
 * The plans are reduced owing to the size of the present pages. 
 
256 
 
 MARINE SURVEYING. 
 
 [CHAP. 
 
 (12) We next take the distances AB, EC, CA from the dia- 
 gonal scale in J inches, and thus compute the lengths in miles. 
 
 (13) The Variation is found to be 9 50 E; accordingly the 
 True and Magnetic Meridians can be ruled through A as directed. 
 
 (14) The height of observer's eye at A above the horizontal 
 plane through the maintruck is 181 feet. Hence the total height 
 of A above the sea is 181 + 130 - 5 = 306 feet 
 
 (15) We draw a line two inches long to represent a mile. 
 We divide this into 10 equal parts. Each will represent a cable. 
 We next draw another line a short distance from the first and 
 parallel to it and finish off according to the directions given in the 
 chapter on Scales ( 17). 
 
 262. Suppose that we have to plot the Final Paper set June, 
 1878 (No. 15 in the following Exercises), we proceed as follows: 
 
 (1) Compute the sun's true azimuth. This is found from the 
 latitude, altitude and declination to be N. 74 40' E. 
 
 
 (2) The distance between the Windmill and Church 
 found to be represented by 4*5 inches. 
 
 s 
 
XIV.] METHOD OF PLOTTING A SURVEY. 257 
 
 (3) A spot is selected about 4 inches from the right-hand 
 edge, and some distance from the lower edge. This will be the 
 Windmill station. 
 
 (4) Draw a line to represent the True meridian. Lay off 
 74 40' to the right, and then 50 5 7' to the left of the meridian. 
 This will be the direction of the Church /\. Along this line take 
 off 4*5 inches. This will give the exact position of Church /\. 
 
 (5) From Church with Windmill as Zero, lay off 57 30', this 
 will be the direction of Tree /\. From Windmill with Church as 
 Zero lay off 61 10' (= 360 - 298 50'). These lines will intersect 
 in the position of the Tree /\ . 
 
 (6) We next fix the Minaret by the angles from Windmill 
 and Tree, and test the " fix " by the angle from Church. 
 
 We next fix Tree Point, from Windmill and Church, and 
 test by the angle from Tree /\. 
 
 Then Spur is fixed, from Windmill and Church, and tested 
 by the given angle from Tree. 
 
 <fec. &c. 
 
 As each point is fixed, it is well to put a small dot in ink and 
 write its name in pencil near it. 
 
 When all the points have been plotted, we must read the 
 topographical information very carefully, and sketch lightly in 
 pencil the outlines required, and then with a crow-quill and 
 Indian ink put in the work as neatly as possible. 
 
 The following very valuable example is, by Sir E. J. Reed's 
 permission, taken from Naval Science, Vol. I. p. 342. 
 
 At Observatory ^ (White Rock). 
 South hill 58 0' ship's foremast. 
 
 North hilll , 
 
 </> depth of Sandy bay/ * 
 Hill over Rocky point 31 00' 
 
 South hill 42 20' Coast mound. 
 Ship's foremast 78 00' Flat rock. 
 K. M. S. 17 
 
258 MARINE SURVEYING. [CHAP. 
 
 Flat rock 63 10' out ext. Cliff h d . 
 
 70 00' Cliff ends, small bay begins. 
 P* inside obs y ^ 89 30' hill over Rocky p*. 
 
 ^ Sandy p* 44 00' 
 
 - Cliff beyond,) 
 
 near Sandy bay/ 
 Fisherman's hut (northern) 33 10' 
 
 South hill 15 5' Stake in depth ofdarge bay. 
 22 25' Sandy bay ends. 
 25 5' near part of next Point. 
 28 55' inner extreme of Rocky Ledge. 
 
 090 1(y footer ditto, 
 
 (depth of Sandyjbay beyond <. 
 
 45 20' Coast Mound p*. 
 
 At Ship's Foremast. 
 Obsy ^ 77 10' South hill. 
 South hill 35 10' North hill. 
 
 42 30' Hill over Rky P*. 
 Hill over Rocky p* 34 00' Coast Mound. 
 
 Flat r k 14 15' Obs* ,. 
 Obs. ^ 1 25' Point inside Obs y A 
 
 5 20' * p*. Beach begins. 
 
 9 20' Sandy p*. 
 16 15' Depth of smaU bay. 
 
 28 50' Cliff 
 
 32 30' nearest part of Cliff. 
 
 36 20' r* Cliff. Beach recommences. 
 
 43 14' Southern hut. 
 46 00' Middle 
 48 00' Northern 
 
 Bay runs J cable outside the huts, in a gentle curve to Stake 
 in its depth 93 to the right of Obs y ^ . 
 
 South Hill 31 00' Beach ends. 
 
XIV.] METHOD OF PLOTTING A SUKVEY. 259 
 
 Point looking this way just right of the North Hill, and - 
 of Rocky Ledge < Hill over that point. 
 
 ' xtrem of 
 
 south R , 
 
 (Beach beyond commences. 
 59 50' Depth of northern bay. 
 73 00' Beach ends. 
 Hill over Rocky p* 34 30' Coast Mound. 
 
 37 50' -- - Coast Mound P fc . 
 43 10' Rock awash. 
 
 Angles at South Hill Station. 
 Ship's Foremast 44 50' Obs y ^. 
 
 North Hill 98 15' Ship's foremast. 
 Hill over Rocky p* 59 50' 
 
 Ship's foremast 45 15' extreme of Point inside Obs y /\. 
 47 20' Sandy p*. 
 51 00' Centre of small clift: 
 55 20' Southern hut. 
 
 56 00' Middle 
 
 54 00' Northern 
 
 P 1 inside Rocky p*i 
 
 V I 55 15' Ship's Foremast. 
 orth Hill from ship] 
 
 </> North 
 
 Rocky ledge begins 54 10' 
 Outer extreme of do. 48 20' 
 
 Angles at Hill over Rocky point Station. 
 
 Ship 77 50' South Hill. 
 Coast Mound 100 00' Ship's Foremast. 
 Obs y A 7 30' Depth of small bay, right of 
 
 Obs y A- 
 
 Ship < - Rocky ledge. 
 
 Coast Mound 61 00' - -- Rocky ledge. 
 
 47 15' Rock awash. 
 
 Depth of Northern bay 117 00' Ship's Foremast. 
 + Coast Mound p fc 93 00' 
 
 Sandy bay ends <f> Coast Mound /\ 
 
 172 
 
260 MARINE SURVEYING. [CHAP. 
 
 Angles at Coast Mound Station. 
 Ship 45 00' South Hill 
 46 00' Hill over Rocky point 
 78 00' North Hill. 
 42 00' Rocky patch begins. 
 33 00' Outer extreme of do. 
 
 eoo AW (Sandy bay begins and curves gently 
 
 (round thence to termination. 
 Eock awash 58 00' Hill over Rocky point. 
 
 Angles on Flat rock. 
 
 ^ Cliff head 90 00' Obs y A- 
 
 ^ do. Southern bay begins 99 5' do. 
 
 Depth of Small bay 23 20' do. 
 
 To plot the above work : 
 
 (1) Make a dot on the most convenient part of the paper, 
 and enclose it in a small circle. This will represent the Obs y A. 
 
 (2) Through this Obs y A draw a true N - and S. line. 
 
 (3) Lay off the T.B. of the ship at the other end of the 
 Base, which in this case is N. 26 50' E.; and the length of the 
 Base = 2*05 miles, determined by Sound and Masthead angle. 
 
 (4) Lay off 58 50' the angle at Obs y A between Ship and 
 South Hill, to the left of the ship. Then 79 10' at the Ship be- 
 tween South Hill and the Obs y A- This fixes South Hill. Now 
 as the angle between Obs y /\ and the Ship observed at South Hill 
 is 44 50', the work is correct. 
 
 (5) Project in the same way the triangle formed by Ship, 
 South Hill, and Hill over Rocky point. Then the triangle formed 
 by Ship, Hill over Rocky point, and Coast Mound. All these 
 points were used as Stations. 
 
 (6) But North Hill was not used as a station, and is " fixed " 
 by the angles obtained at the Ship between it and South Hill, and 
 that obtained at South Hill between the Ship and it ; and is 
 proved by a line of direction obtained at the Obs y A between it 
 
XIV.] 
 
 METHOD OF PLOTTING A SURVEY. 
 
 261 
 
 and South Hill passing through the point of intersection of the 
 lines of direction obtained from the Ship and South Hill. 
 
 Vide diagram I. for this triangulation, and diagram n. for 
 the cutting in of the coast line. 
 
 Diagram I. Shewing the method of plotting the principal triangles, 
 and fixing the chief points. 
 
262 
 
 MARINE SURVEYING. 
 
 [CHAP. 
 
 Diagram II. Shewing the method of cutting in the coast line, and 
 representing the various features. 
 
XIV.] METHOD OF PLOTTING A SURVEY. 263 
 
 BEAUFORT PAPERS. 
 
 No. 1. September, 1874. 
 
 In the exploration of a coast, a lofty peak, Mount Columbus, 
 known to be in lat, 5452'N., long. 16018'E., was observed 
 from the ship to be directly in line with a hill near the shore on 
 a true bearing of N. 60 W. At the same time the angle between 
 M fc Columbus and a Peaked island to the south-west was found to 
 be 80. 
 
 The vessel then stood to the south-west until the Peaked 
 island came in line with Mount Columbus on a true bearing of 
 N. 36 W. ; at the same time the angle between the Hill near the 
 shore and the Peaked island was 25, and that between the 
 Peaked island and a Cliff farther to the south-west, known as 
 Cape Drake, was 101. The altitude of Mount Columbus was 
 here observed ; on the arc 23' 50", off 22' 40"; height of eye 
 18 feet. 
 
 The vessel proceeding to the south-west, when off Cape Drake 
 a landing was effected, and Cape Drake found from observation 
 to be in lat. 54 16' 30" N., the true bearing of M* Columbus being 
 N. 16W., and the angle between the Mount and the Peaked 
 island was found to be 32 30'. 
 
 Project these positions on a scale of an inch to a mile, and 
 determine by projection the longitude of Cape Drake, and the 
 true bearing and distance between that Cape and the Hill near 
 the shore ; also calculate the height of Mount Columbus. 
 
 Results. Long. Cape Drake 160 20' 36" E. 
 
 Hill near shore bears from Cape Drake N. 12 E., 
 
 26-6 miles. 
 M* Columbus is distant 32 miles. 
 
 No. 2. April, 1875. 
 
 Arriving at a coral reef, and there being too much swell to 
 attempt a landing, a position was taken up westward of it with 
 Gull rock (on the west side of the reef, and at the northern edge 
 
264 MARINE SURVEYING. [CHAP. 
 
 of an entrance to a shallow lagoon), in line with a single Palm 
 (on a small sand bay near the eastern extreme) bearing N. 69E., 
 and the following angles observed : 
 
 A. KW. extreme of reef 41 0' Gull rock 11 20' lagoon entrance, 
 
 49 0' S. W. extreme, 
 course N. b E. for 0-98 miles, to 
 
 B. North extreme 77 10' Gull rock 80' W. and S.W. extreme 
 
 in line, 
 course N.E. b E. 1-06 miles, to 
 
 0. N.K extreme 810'Gull rock 260'N.W. extreme 
 Palm 70 0' 
 
 course E. b S.S. 1-67 miles to 
 
 D. S.E. extreme 44 0', Palm in line with lagoon entrance 420'N.E. 
 
 extreme, 
 course S. b W. 1-6 miles, to 
 
 E. South extreme 81 0', Palm 13 0' East extreme, 
 
 course W. b S. 2-05 miles, and the vessel then anchored. 
 
 F. West extreme 16 0' Gull rock 100', lagoon entrance, 
 
 28 0', Palm, 
 
 53 0', South extreme. 
 
 Observations placed the ship in latitude 1031'N., longitude 
 115 7'E., and gave the true bearing of the Palm N. 52 E. 
 
 The courses and bearings are cor. mag. and the variation 5 E. 
 
 Project these positions on a scale of 2 inches to a mile, and 
 determine latitude and longitude of the Palm, giving sketch of 
 reef thus roughly surveyed. 
 
 Results. Lat. Palm= 10 32J']Sr. 
 Long. =1159'E. 
 Distance = 4'87 inches, or 2-43 miles. 
 
 No. 3. October, 1875. 
 In the survey of a coast, Bluff ^ was found to bear K 8 30' E. 
 
XIV.] 
 
 METHOD OF PLOTTING A SURVEY. 
 
 265 
 
 (true), 4-1 miles from South point /\ ; the following /\s were 
 observed at: 
 
 (1) South point A- 
 Zero, Bluff A 360 00' 
 
 Rock awash 6 45 
 
 Round point 19 00 
 
 Wharf * trend of j ^ QQ 
 coast linej 
 
 village 37 00 
 
 ^ ditto 40 00 
 
 Rock hill A 52 20 
 
 This point out 267 00 
 
 Direction of Long reef,) 
 
 extending from thisf 332 00 
 
 point 
 
 Bluff reef 343 30 
 
 Bluff 358 00 
 
 (2) Bluff A 
 Zero, South point 360 00' 
 
 South point 310 
 
 * Long reef 55 00 
 
 Direction of Bluff \ 
 
 reef extending from,l 71 30 
 
 and <f> this point out) 
 
 Trend of coast | ^ ^ 
 
 northward J 
 
 Coast line in 296 00 
 
 Village 300 00 
 
 > ditto 303 30 
 
 Round point 306 20 
 
 Wharf 315 45 
 
 Rock hill A 319 20 
 
 Rock awash . 352 50 
 
 (3) Rock hill A- 
 
 Zero, Bluff A 36 ' 
 Wharf 5 20 
 
 Round point 9 10 
 South point A 2 ? 3 00 
 Rock awash 325 20 
 
 > Long reef 330 30 
 
 Bluff reef 343 40 
 
 Both Long and Bluff reefs are coral, about 2 \ cables in width, 
 with patches above water; the coast line (sand backed with low 
 bush) joins between the points that are cut in. Project this 
 work on a scale of 1*52 inches to a mile, giving a sketch of the 
 port. 
 
 Remits. Distance between Rock awash and Rock Hill 2 -94 in. 
 
 South Hill 3-23 
 
 Left of Bluff reef 3-20 
 
 Extreme of Bluff and Long reef 1-38 
 
266 
 
 MARINE SURVEYING. 
 
 [CHAP. 
 
 No. 4. March, 1877. 
 
 In the survey of a port, lat. 230'N., a base of 2000 feet was 
 measured along its western sand beach, and the following AS 
 were observed at : 
 
 (1) South base A- 
 
 Zero, Signal hill 
 Mean of A.M. observa-) 
 
 tions for the T.B. 
 T.A. -e- 30 31' 
 * of Sand and * } 
 
 of Cliff j 
 * of Cliff and - \ 
 
 of Sand j 
 
 Rock on Sand beach 
 Bluff 
 
 Quail islet 
 
 Summit 
 
 >j 
 
 South point 
 Coast line near 
 North base /\ and line| 
 of beach 
 
 (2) North base A- 
 
 360 0' 
 
 Zero, Quail islet ) 36QO Q , 
 summit j 
 
 85 15 (I) 
 
 > do. 11 45 
 South point 28 00 
 
 2 
 
 South base A 48 30 
 Beach near 258 00 
 
 13 
 25 
 
 Signal hill 262 20 
 of Sand, and) 
 of Cliff /- 
 
 39 40 
 56 00 
 
 _ of Cliff, an* 285QO 
 * of Sand j 
 
 68 20 
 
 Rock on beach 297 50 
 
 90 30 
 
 Bluff 314 20 
 
 116 20 . 
 
 -i Quail islet 350 00 
 
 137 00 
 
 
 342 30 
 
 (3) Quail islet summit 
 
 Zero, North base 360 
 
 -of Sand - of Cliff 30 
 
 Signal hill 34 
 
 * of Cliff, and -, of Sand 46 
 
 Rock on beach < North end) 69 on 
 
 this islet j 
 
 Bluff 82 30 
 
 South extreme this islet 237 
 
 South point 265 20 
 
 South base 314 20 
 
XIV.] METHOD OF PLOTTING A SURVEY. 267 
 
 (4) Sextant /\. At Rock on beach. 
 
 Quail islet 5 0' Summit Quail islet 3 50' Quail islet, 
 
 Beach near 50 30' 98 0' Signal hill 
 
 </> beach near. 
 
 The coast line joins between the points that are cut in; Quail 
 islet, the extremes of Bluff and South point, and cliff fronting 
 Signal hill are low and rocky, rest of beach sand: a reef awash 
 about half a cable wide joins South point and the nearest part of 
 Quail islet; sun's decl. 1045'N. Project and give a sketch of 
 this work on a scale of 5 in. = 1 mile = 6055 feet. 
 
 Remits. T.B. of Sun N. 90 51' E. 
 S. P* to Bluff 34 inches. 
 S. A ^ Signal hill 3-29 in. 
 N. A to Quail islet 2-29 in. 
 S. A to Rock on beach 3-54 in. 
 
 No. 5. May, 1877. 
 
 In the survey of a large lagoon, latitude 20 0' S., a base of 
 2300 feet was measured along the ridge of a sand islet forming 
 its south-west side, and the following /\s were observed at: 
 
 (1) South Base A- ( 2 ) North Base A- 
 
 Zero, Flag A 360 0' Zero, Flag A 360 0' 
 
 Mean of obs. for T.B.V i - Tree islet 64 
 
 taken at 6 A.M. A.T. J U T ,. This islet 76 
 
 - Tree islet 56 30 South base A 102 30 
 
 - This islet 66 North point this isletj 
 
 Trend of do. near 90 <j> centre of joining reef ) 
 
 North base 314 30 Sentry rock 330 
 Sentry rock 342 30 
 
 (3) Flag A. North extreme of Tree islet. 
 
 Zero, South base A 360 0' - This islet 300 0' 
 
 North base A 32 Bight in of do. 326 
 
 ~ South-west islet 380 do. 32820 
 
 Sentry rock 113 South-west islet 339 
 
268 MARINE SURVEYING. [CHAP. 
 
 (4) Sextant A. Sentry rock at North extreme of reef. 
 Trend of reef near, ^ 0(y g Bage 21 o 3(r K ext g _ w islet 
 
 then trends to Flag AC (Trendof reef near, 
 
 40 O'-j then trends to 
 
 ' S.-w. islet. 
 (5) Sextant A. South extreme Tree islet. 
 
 Near point, South-west islet 111 0' Flag A 5 0' Bight in this 
 
 islet in 35 - * do. near. 
 
 Tree islet is covered with the cocoa-nut trees, about a cable 
 across near the centre, and is joined to South-west islet, which 
 is half a cable in breadth, by reefs above water of the same width; 
 lagoon is shallow; Sun's declination 23 0' S. Project and give a 
 sketch of this work on a scale of 5 inches = one mile = 6053 feet. 
 Results. Sentry rock to N. base 2 -7 5 inches. 
 Flag to S. base 3 -50 inches. 
 N. pt. this islet to S. pt. Tree islet 3'07 inches. 
 
 No. 6. October, 1877. 
 
 In surveying a reach of river, ship anchored near mid-channel, 
 and following observations were taken, base being measured by 
 Sound. 
 
 (1) At B A, zero, A A, (2) At A A, zero, distant 
 
 bearing K 5 W. (true] and N. peak. 
 
 26 W. (Mag:). Beats of watch between 
 
 Beats of chron. between flash and report tired 
 
 flash and report of at ship 29 
 
 gun fired at ship 18 J B A 22 15/ 
 
 House on hill 9 0' Ship 74 15 
 
 East point north shore 61 10 Far point south shore 80 10 
 
 Prom.point south shore 78 58 West point north shore 105 
 
 Far point south shore 271 15 House on hill 250 
 
 West point north shore East point north shore 305 1 2 
 
 < ship 306 Prom, point south sh. 345 
 
 Sand point </> distant peak. 
 
XIV.] METHOD OF PLOTTING A SURVEY. 269 
 
 (3) At ship 0, zero, A A. 
 
 East point north shore 31 45' 
 Prom, point south shore 53 00 
 Sand point 59 30 
 
 B A 74 00 
 
 Far point south shore 191 
 
 Chronometer used at B has 2 beats to a second. Watch 
 used at A A has 5 beats to two seconds. Sound travels 1090 
 feet a second. Project these positions, scale 1 '5 inches to mile 
 = 6069 feet, using jB's measurement of base. State what number 
 of beats of watch A was wrong. Rule true and magnetic meri- 
 dians. South shore of river is low and sandy, extending nearly 
 straight from Far point past B A to Sand point, thence, in a 
 small bay to Prominent point. North shore of river is straight 
 between points named and A A and consists of a low cliff, 
 A A being highest part. House is on a hill rising gradually 
 above A A- Current W. b S. 1-5 knots. Ship is in 9 fathoms, 
 and a rock awash was found with 
 
 Ship 4> West point 84 0' A A. 
 
 Sketch in these details, and place the rock awash on the plan, 
 giving magnetic bearing of house on hill from it. 
 
 Results. Watch at A in error 5 beats too many. 
 
 Mag. bearing of House from Rock awash N. 10E. 
 Distance of A from Prom, point 3-27 inches. 
 
 West point from Far point 2 '86 inches. 
 
 House from East point 3 -5 2 inches. 
 
 No. 7. September, 1878. 
 
 In a running survey of an island, ship steamed on following 
 courses, angles being taken at the position started from, and at 
 the termination of each course: 
 
270 MARINE SURVEYING. [CHAP. 
 
 Mag. Course. Dist. 
 
 (1) N. 78 E. r-25 
 
 (2) N. 2 W. l'-45 
 
 (3) K 55 W. l'-3 
 
 (4) S. 40 W. 2' -6 
 At Starting Position. 
 
 Lat. 40' S., long. 10 E. 
 
 - island 52 S. peak (K 10 E.) elevated) . 
 
 52'23"j 4 
 
 At end of Course (1). 
 
 - island 20 30' S. peak < S. E. point (N.) Q 
 
 62 W.)} 32 
 
 N. peak 12 ditto 
 
 At end of Course (2). 
 
 S.E. point 61 30' N. peak (S. 79 W.) 24 N. point. 
 S. peak 28 30' ditto. 
 
 At end of Course (3). 
 N. point 17" 30' N. peak (south) , __ 
 
 elevated 47 J 
 ^ island 19 ditto. 
 
 Bearings magnetic. Variation 32 W. Projection plane. 
 
 Protract the above on scale of 2 ins. = a mile of 6046 feet, 
 giving the height of N. and S. peaks, and lat. and long, of the 
 former. 
 
 Measure the horizontal angle the island should subtend at the 
 end of Course (4), and the distance round the coast. 
 
 Ground plan of island resembles an isosceles triangle. In 
 approaching from the north or south the island appears of 
 conical shape, but from the east and west is like a saddle, N. and 
 S. peaks representing the horns. Shores consist of shingle beach. 
 
 Sketch in topography and detail. 
 
XIV.] METHOD OF PLOTTING A SUKVEY. 271 
 
 Results: 
 
 Height K peak = 80 '7 feet (dist. from 4th position 5638 feet). 
 S. =78-7 feet ( 1st 5169 feet). 
 Lat. and Long. K peak 37' 18" S. and 9 58' 30" E. 
 Horizontal angle = 40. 
 Distance round coast 4 '6 2 miles. 
 
 No. 8. November, 1878. 
 
 In survey of an island, /\s were made upon three peaks 
 A, B, C base line being measured by height. of a ship's main 
 truck, moored in the offing. 
 
 At A A- Lat. 20 K 
 Height of eye 5 feet. 
 
 Zero, C A 360 ' Ma g- bearing N. 36 E. 
 
 Sun's centre 28 37 (Corrected declination 18 15' 
 
 alt. of 11 20 1 K for AM. true bearing. 
 
 Beach boulder 6 30 
 
 B A 45 15 
 
 Points 95 30 
 
 Point G 241 
 
 Ship's mainmast 331 
 
 main truck depressed 1 42 30") ^ 
 
 \ For base line. 
 water line 2 56 J 
 
 PointZT 353 50 
 
 MB A- 
 
 Zero, C A 360 ' 
 
 Point^ 99 
 
 Points 211 5 
 
 (A A 273 30 
 
 ( </> direction of range towards west 
 
 Ship's mainmast 331 
 
272 MARINE SURVEYING. [CHAP. 
 
 AtC A- 
 
 Zero, A A 360 0' 
 
 Point G 13 20 
 
 Ship's mainmast 54 40 
 Point H 142 
 
 Point F 285 30 
 
 E A 311 45 
 
 Beach boulder 353 
 Sextant angles from ship. 
 Point H 23 30' C A- 
 A A 21 30 P<> in t & 
 Ship's main truck to water line 130 feet 
 
 Protract the above on scale of 2 inches = 6053 feet = nautical 
 mile. Find the length of the sides of the triangle ABC. Rule 
 true and mag. meridians through A A> an( * gi ye its height above 
 the sea surface. 
 
 Range of hills forms an elbow at j#, extending to A and C. 
 Beach boulder is on the coast, at the head of a rocky bay, which 
 curves between points G and H. South portion of island consists 
 of cultivated plain, with sand coast from point G to D ; also from 
 D to F. A tide rip off point D. Low cliff between points F and 
 H. 
 
 Sketch in details, and divide the scale of the plan to cables. 
 
 Results. AB =1-87 miles. 
 G=1'78 
 (74 = 2-48 
 Var. = 9 50' E. 
 A is 306 feet above the water. 
 
 No. 9. February, 1879. 
 
 In a survey of a group of islands a A was made on the sum- 
 mit of High Island, ship mooring to the eastward, the cutter being 
 sent round to take up a position to the westward of the group. 
 
XIV.] METHOD OF PLOTTING A SURVEY. 
 
 Height of eye 5 
 
 High island A- 
 
 t ? E i/*7 
 
 ' *, _ 
 
 Zero, distant peak 360 0' Mag. bearing N.I 7 E. 
 
 True bearing N. 27 E. 
 
 At Ship (Sextant Station). 
 
 - 4 island 24 45 
 
 Summit do. 31 00 
 
 * do. 42 30 
 
 Ship's mainmast at 
 
 water line 
 
 62 
 
 Summit c 
 
 Do. depressed 
 
 2 2 
 
 45" ^ do. 
 
 * B island 
 
 112 
 
 
 Summit do. 
 
 120 
 
 High isl. 
 
 ^do. 
 
 128 30 
 
 
 C rock 
 
 181 30 
 
 Do. 
 
 Cutter Q 
 
 240 30 
 
 Do. 
 
 D island 
 
 275 30 
 
 
 Summit do. 
 
 292 30 
 
 
 ~do. 
 
 312 00 
 
 
 * Highisl. 11 5'HighislandA 
 
 Crock 20 
 *B island 24 55 
 
 29 00 
 
 32 50 
 
 I f High isl. 
 
 10 (K<* of A and 
 
 W islands. 
 
 18 30 A isl. summit. 
 
 37 30 Do. * 
 
 At Cutter (Sextant Station). 
 
 ^ High island 14 00' High island A 
 
 ^ A island 15 30 
 
 *D island 2800 
 Summit do. 38 30 
 
 ^ do. 45 30 
 
 High island A 16 30 * Hi g h island. 
 
 20 30 * B island. 
 
 25 30 Summit do. 
 
 30 00 - do. 
 
 51 10 Crock. 
 
 Distance by micrometer measurement between High island A 
 and ship's water line at mainmast 6068 feet. 
 
 Protract the above on scale of 2 inches = a nautical mile of 
 6060 feet. 
 
 Rule true and magnetic meridians, and give height of High 
 island summit. 
 
 R. M. S. 
 
 18 
 
274? MARINE SURVEYING. [CHAP. 
 
 The islands have steep cliffy coasts. High island is oval- 
 shaped, greatest diameter north and south, with conical summit. 
 C rock is a sharp pinnacle. 
 
 Sketch in detail ; divide scale to cables, and find the natural 
 scale of the plan. 
 
 Results. Var. 10E. 
 
 High island is 194 feet above the sea. 
 Natural scale = ?a ^ ao . 
 AB= 1*31 inches, CD = 1-53 inches. 
 = 1-08 4 = 1-23 
 D and ship = 2 -5 7 
 
 No. 10. September, 1879. 
 
 In constructing a plan of Greenwich, including part of the 
 river Thames, y\s were made at the Observatory, Gasworks, 
 and Chimney on Essex shore, the distance between the two 
 former by calculation being 4137 feet. 
 
 Sun's declination 7 40' K 
 
 At Observatory ^ lat 5128 / N. 
 
 JZero, St Paul's Cross 360 00' 
 { Mag. bearing N. 36 30' W. 
 
 rSun's centre 164 30 
 ( App. time VIII h A. M. for true bearing. 
 
 Ship flagstaff 10 50 
 
 Dogs Ry. Station 22 30 
 
 Obelisk <j> Gymnasium 30 00 
 
 Church 34 30 
 
 Chimney /\ 43 30 
 
 Asylum ' 48 00 
 
 Gasworks ^ 343 40 
 
 Crane 356 20 
 
 At Gasworks /\. 
 
 Zero, St Paul's Cross 360 00' 
 
 Crane 51 30 
 
XIV.] METHOD OF PLOTTING A SURVEY. 275 
 
 Dogs Ry. Station <j> Chimney A 98 30' 
 
 Asylum 120 00 
 
 Ship flagstaff < obelisk 12800 
 
 Gymnasium 146 30 
 
 Observatory A 159 40 
 
 At Chimney /\. 
 
 Zero, Observatory A 360 00' 
 
 Gymnasium 4 20 
 
 Obelisk 13 52 
 
 Ship 33 00 
 Gasworks A < Dogs Ry. Station 59 00 
 
 Crane 72 00 
 
 Church 94 00 
 
 Asylum 352 00 
 
 Protract upon scale of 7 in. = a mile of 6,084 feet, ruling true 
 and mag. meridian through Observatory /\, and find the natural 
 scale. 
 
 The river, 300 yards broad, curves (the south bank) from 
 Gasworks /\ close past Ship and Asylum ; the north bank about 
 parallel. Mark its course, state how and at what time of tide 
 you would sound, and the manner in which the time of high 
 water at full and change of moon should be observed. 
 
 Bearing in mind the features of the district, mark approxi- 
 mately where you would measure the base-line. Should the 
 heights be given above, and soundings below the same level of 
 river? State how you would fill in upon the plan the streets, 
 buildings, and topography. 
 
 jResults. Chimney to Obelisk =2-48 in. 
 Asylum to Crane = 4'17 
 Station to Obser* =4'23 
 Gasworks to Church =4 -2 6 
 
 Variation = 18 28' W. 
 182 
 
276 MARINE SURVEYING. [CHAP. 
 
 No. 11. October, 1879. 
 
 In the survey of the extinct crater of St Paul's island (now 
 a small harbour), a y\ was made on the ridge at the head of the 
 crater, and three 's on the shores thereof. 
 
 At Ridge A- 
 
 rZero, Hut A 360 (X 
 
 \<f> Dip of ridge 
 True bearing of zero S. 50 E. 
 Mag. bearing S. 66 30' E. 
 (Eastern entrance 30 j 
 
 1 depressed 4 30 J 
 
 West entrance point 49 
 
 Red stone 86 30 
 
 End of ridge on west side 91 40 
 
 At East entrance point . 
 
 Hut 
 Ridge A 
 Red stone 
 End of ridge on| 
 west side / 
 West entrance^ 
 point J 
 
 29 30' 
 40 20 
 85 00 
 
 97 10 
 113 20 
 
 Dip of ridge. 
 Hut 
 
 M 
 
 
 
 
 
 At Hut . 
 
 Red stone 48 0' Ridge A- 
 
 End of ridge on) 
 
 , . , > Do U ,. 
 
 west side J 
 
 West entrance pt. 86 
 East 110 5 
 
 At Red Stone . 
 
 East entrance point 15 50' West entrance point 
 
 Dip of ridge 23 35 East 
 
 Hut 33 30 
 
 Ridge A 78 55 
 
 > > l 
 
XIV.] METHOD OF PLOTTING A SURVEY. 277 
 
 The distance between West entrance point and East entrance 
 point , by line stretched across, was found to be 809 feet. 
 
 Protract the above upon scale of 74 in.= a mile of 6,060 feet. 
 
 Give the height of Ridge A, and rule the true and mag- 
 netic meridians through it. 
 
 The crater is circular, Ridge /\ about 100 yards inland, the 
 other on the coast line, the entrance lying between the East 
 and West entrance points. The ridge of the crater follows the 
 curve of the coast, sloping gradually from the Dip of ridge on 
 Eastern, and end of ridge on Western side, to East and West 
 entrance points, otherwise sloping at an angle of 45 down to 
 the water's edge. 
 
 Ridge /\ seen midway between the points of entrance, is 
 the leading mark into the harbour. 
 
 Sketch in these details, and give the magnetic bearing of the 
 leading mark. 
 
 Remits. Height of ridge is 198 feet Bears N. 27 30' W. 
 
 No. 12. May, 1880. 
 
 When making a plan of a harbour, a base 1,200 yards long 
 was measured on a sandy beach on west side ; angles as follows 
 were observed at the north and south extremes of the base, and 
 also from the summit of an island in the harbour : 
 
 At South Base A zero distant Peak. 
 
 N. Base A 15 00' (bearing N. 54 30' W. true). 
 
 Rock awash at H. W. 50 20 
 
 - Island 83 50 
 
 Island A 88 30 
 
 Island * eartj ^ ^ 
 entrance pointj 
 
278 MARINE SURVEYING. [CHAP. 
 
 At North Base A zero S. Base A . 
 
 ^ Island 314 10' 
 
 Island A 312 00 
 
 * Island 309 00 
 
 East entrance point 298 40 
 
 Rock awash 287 30 
 
 At Island A zero N. Base A . 
 
 Rock awash 23 20' (angle of depression 1 2'). 
 
 East entrance point 133 00 
 S. Base A 301 30 
 
 From South Base A the coast rises in cliffs and trends N. 70 E. 
 (true) for 5 cables and then N. 13W. (true) towards east entrance 
 point, which is fringed by a coral reef. 
 
 On subsequently sounding the harbour, a shoal of 3 fathoms, 
 coarse sand and coral, was discovered, and from which the follow- 
 ing sextant angles were observed : 
 
 East entrance point 51 30' Island A 31 00' S. Base A. 
 
 Protract the above on a scale of 6 in. = one mile (6,080 feet) 
 and find height of the island. 
 
 No. 13. June, 1880. 
 
 From a vessel in lat. 4830'K, long. 28 00' W., the following 
 sextant angles were observed: 
 
 island 23 00' - beach 
 
 32 15 Church 
 
 43 30 summit of island < * beach 
 
 47 00 Windmill 
 68 20 island. 
 
 The vessel then steered S. 74 E. (true) for a distance of 
 2^ miles, when the following angles were taken : 
 
 * island 7 00' - beach 
 
 19 00 Church < * beach 
 
XIV.] METHOD OF PLOTTING A SURVEY. 279 
 
 -. island 33 10' summit of island 
 
 43 00 Windmill 
 57 00 *> island 
 
 From this position the vessel was moved N. 21 E. (true) 
 3 miles, when the following angles were observed : 
 
 * island 40 20' North point of island (distant 1 J miles) 
 
 < * island 
 
 28 40 summit of island 
 
 From the first position the * island bore N. 22 E. (mag- 
 netic), the variation being 10 W., and from the second position 
 the same point bore N. 50 30' W. (true). The island is triangular 
 in shape ; the north side is cliffy and nearly straight ; the south- 
 east extreme is surrounded by rocks which uncover at low water. 
 
 The following soundings were also obtained off south side of 
 island : 
 
 island 59 30' Church 57 20' Windmill (5 fms., hard sand). 
 
 Church < * beach 103 00' * island (7 fms., mud and stones). 
 
 Protract the above on scale of 2 in. = a nautical mile. 
 
 Rule true and magnetic meridians through first position of 
 vessel, and find lat. and long, of summit of island. 
 
 Results. Latitude of summit 48 32' 1 1" N. 
 
 Longitude 27 55 10 W. 
 
 North point to south point 2 -7 inches 
 
 Left extreme to S.E. point 4'28 
 
 North point to church 2 '34 
 Westward sounding to windmill 1*95 ,, 
 
 Eastward 1-95 , 
 
 No. 14. April, 1881. 
 
 The following angles and bearings were observed from a ship, 
 when running along a coast extending about 5 miles in a north 
 and south direction : 
 
280 MARINE SURVEYING. [CHAP. 
 
 From Ship anchored at A, in 12 fathoms, sand. 
 
 North islet (bearing N. 6 E. true) 15 30' Yellow cliff. 
 
 29 30 Rock awash <f> head of 
 
 Green bay. 
 35 00 White cliff < flagstaff 
 
 on hill. 
 60 30 Rock at head of White 
 
 bay < Hill one mile 
 
 inland. 
 88 00 South point of White 
 
 bay. 
 
 Leaving the cutter anchored at A, the ship takes up a second 
 position, B, in 9 fathoms, coral, at 3 miles N. 10 W. (true) from 
 A, and observes as follows: 
 
 From Ship anchored at B. 
 
 South point of White bay 22 00' Cutter at A. 
 
 Rock at head of White bay 41 00 
 
 White cliff <f> Hill one mile inland 56 30 
 
 Rock awash 69 30 
 
 Flagstaff on hill 94 00 
 
 Yellow cliff 113 00 
 
 North islet 142 30 
 
 The Cutter then weighs anchor and proceeds to (7, a position 
 2 1 miles north (true) of the Ship at B, and anchors with Flagstaff 
 on hill < North islet. 
 
 From Cutter anchored at C in 7 fathoms, broken shells. 
 
 South point of White bay 19 00' ship at B. 
 
 White cliff 33 00 
 
 Rock awash 38 30 
 
 Head of Green bay 57 00 
 
 YeUow cliff 66 15 
 
 Flagstaff on hill < north islet 6930 
 
 The coast forms two sandy bays (the southern known as 
 
XIV.] METHOD OF PLOTTING A SURVEY. 281 
 
 White bay and the northern as Green bay), separated by White 
 cliff, a perpendicular cliff towards which the land slopes from the 
 Hill one mile inland. 
 
 The hill on which Flagstaff stands falls steeply to the east- 
 ward, but slopes gently to Yellow cliff, the north point of Green 
 bay. 
 
 North islet nearly joins the shore, and between it and Yellow 
 cliff the coast forms a deep sandy bay. 
 
 Project the foregoing on a scale of one inch = one mile. 
 
 Note. The position of A should be taken at 2 inches from the 
 bottom of the paper and 3 inches from the left-hand margin. 
 
 Re&dts. Distance B to N. islet 2-28 inches. 
 5 to W. cliff 2 -50 
 A to N. islet 5-10 
 C to S. point 5-90 
 
 Note. The following papers were set at the examinations of 
 Classes B, and B 2 . 
 
 No. 15. Final, 1878*. 
 
 In survey of a harbour /\ s were selected at Windmill on 
 table land on east side of entrance, at Church on hill about a mile 
 inland from the harbour's head, and at Tree on a projecting head- 
 land on south-west side of entrance. 
 
 Working from a measured base line the distance between 
 Church A and Windmill A was calculated to be 18,159 feet. 
 
 At Windmill A- Lat - 20 N - 
 
 Zero, Church A 360 ' Ma g- bearing N. 45 W. 
 
 Sun's centre 125 37 ("Corrected decl. 18 5' K 
 
 altitude of 11 20 1 for A.M. true bearing. 
 
 Minaret < end of sand in east bay 7 
 Tree point 291 
 
 Tree A 298 50 
 
 Spur in tree range 310 
 
 * See reduced Diagram in 262. 
 
282 MARINE SURVEYING. [CHAP. 
 
 Cone in tree range 316 10' 
 
 South end of reef extending} 
 from Black point J 
 
 Stony peak < Black point 335 30 
 
 At Church A- 
 
 Zero, Windmill 360 0' 
 
 Windmill point 15 
 
 South end of reef 44 
 
 Black point 48 
 
 Tree point 51 
 
 Tree A 57 30 
 
 Bight of West bay 69 
 
 Spur in tree range 75 10 
 
 Cone 89 30 
 
 Stony peak 103 30 
 
 Minaret 347 
 End of sand in east bay 357 50 
 
 At Tree /\. 
 
 Zero, Church A 360 0/ 
 
 Black point 7 
 
 South end of reef 14 30 
 
 Minaret 20 30 
 
 End of sand in east bay 46 
 
 Windmill A 61 20 
 
 Windmill point 72 50 
 
 Tree point 120 
 Spur </> cone in tree range 290 
 
 Stony peak < bight of \ ^9 
 sand in west bay j 
 
 West bay (shore of sand) curves between Tree point and Black 
 point ; East bay (shore of sand) curves between Black point and 
 end of sand. Black point is a rocky shelf dividing East from 
 West bays. Reef of rocks awash extends southward from Black 
 point. Coast between end of sand and Windmill point straight 
 
XIV.] METHOD OF PLOTTING A SURVEY. 283 
 
 and cliffy. Minaret surrounded by scattered houses. Cultivation 
 around the head of the bay. Tree range of hills extends between 
 Church and Tree /\s, decreasing in height towards the latter. 
 
 Project and give a sketch of this work on a scale of 1 '5 inches 
 = a mile of 6,053 feet, ruling the true and magnetic meridians 
 through windmill /\. 
 
 N.B. Windmill /\ should be about 3 inches from the right-hand 
 edge of the paper. 
 
 Results. Var. 5 57' W. 
 
 Tree point to Windmill point 3*02 inches. 
 Church A to Tree A 4*55 
 
 Tree A to Windmill 4 '40 
 
 Tree A to Stony peak 3-33 
 
 Black point to Windmill point 3 '28 
 
 No. 16. March, 1880. 
 
 In making the survey of a bay a base was obtained by calcu- 
 lation between Beacon /\ and Signal hill /\ equal to 7,500 feet, 
 and the following angles taken : 
 
 Latitude of Beacon A 50 K : Sun's declination 20 N. 
 At Beacon /\. 
 Magnet N. 20 E. 
 
 Zero, Signal hill A 360 0' 
 
 Cliff 18 30 
 
 Bight of sand 28 20 
 
 Extreme of sand spit 34 50 
 
 Rock awash 41 10 
 
 Rock on beach 43 20 
 
 Bluff A 60 
 
 * Bluff 63 30 
 
 Extreme of Beacon shoal 78 10 
 Sun's centre | 
 
 Sun's true altitude 30 10' J 
 Beacon point 107 
 
 - Pier 325 
 
284 MARINE SURVEYING. [CHAP. 
 
 - Pier 337 30' 
 
 - Cliff 348 
 
 Zero 360 
 
 At Signal hill A- 
 
 Zero, Bluff A 360 0' 
 
 * Bluff 3 30 
 
 Rock awash 23 
 
 Extreme of Beacon shoal 30 30 
 
 Beacon point 51 40 
 
 Beacon /\ 55 
 
 Pier 76 50 
 
 Pier 90 
 
 Cliff 93 
 
 ^ Cliff 324 10 
 
 Bight of sand 327 50 
 
 Rock on beach 335 20 
 
 Extreme of sand spit 345 30 
 
 Zero 360 
 
 Angle of depression to rock] 
 
 awash J 
 
 Height of theodolite 5 feet. 
 
 At Bluff A- 
 
 Zero, Signal hill A 360 ' 
 
 ^ Cliff 16 
 
 Extreme of sand spit 20 40 
 
 Bight of sand 28 25 
 
 Rock on beach 57 10 
 
 Extreme of Beacon shoal 280 20 
 
 Beacon point 292 
 
 Beacon A 295 
 
 Rock awash 321 5 
 
 Direction of pier 327 20 
 
 Cliff 344 
 
 Zero 360 
 
XIV.] METHOD OF PLOTTING A SURVEY. 285 
 
 From Beacon point to inner end of pier and 011 to Cliff the coast 
 is steep-to, the Cliff is of moderate height : from east end of cliff 
 to the Rock on beach it is sand, and from thence gradually rises to 
 a high cliff at the Bluff. 
 
 Beacon shoal is one cable in breadth, with patches of less than 
 6 feet at low water ordinary spring tides. 
 
 Sketch in the coast line ; draw the true and magnetic meri- 
 dians, and give the height of the Signal hill /\. 
 
 Scale, 1 mile = 6073 feet = 4*37 inches. 
 
 Results. Height of Signal hill 155 feet. 
 Variation 10 W. 
 
 Signal /\ to outer edge of pier = 3*03 inches. 
 Beacon /\ to rock awash = 2'97 ,, 
 
 West end of cliff to sand spit = 3-95 
 
 No. 17. Final, 1880. 
 
 In survey of a harbour, stations were made on Cliff at north 
 entrance point, Hill on west side, and House on south entrance 
 point. Cliff A bore N. 20 30' E. (true) from Hill A and was 
 found by calculation from a measured base to be 18,020 feet 
 distant. 
 
 At Hill A- 
 
 Zero Cliff A 360 00' 
 
 Rock awash 14 40 
 
 Islet (10 feet high) depressed) 
 
 55' / 
 
 Point, south side of harbour 51 00 
 
 House A 64 20 
 
 Windmill < - beach 327 50 
 
 Church 336 20 
 -beach 347 10 
 
286 MARINE SURVEYING. [CHAP. 
 
 At Cliff^. 
 
 Zero, Hill A 360 00' 
 
 Church 38 10 
 
 Windmill 41 10 
 
 Islet 284 00 
 
 House A 299 50 
 
 Rock awash < Point, south ) _ 
 
 . , , , > ooo UU 
 
 side or harbour J 
 
 At House /\. 
 
 Zero Hill A 360 00 
 
 Point, south side of harbour 9 10 
 
 - beach 14 00 
 
 Windmill 31 20 
 
 Church 34 30 
 
 * beach < Rock awash 38 20 
 
 Cliff A 55 30 
 
 Islet 76 30 
 
 Between Cliff /\ and north extreme of beach the coast, com- 
 posed of cliffs, forms two small bights ; the beach is sandy and 
 nearly straight. Hill A ^ na ^ a m ^ e inland, and stands on 
 summit of a hill which slopes gently to northward, but falls 
 steeply to southward. Point, south side of harbour has several 
 detached rocks which do not cover lying off it, and the shore 
 forms a deep bight on either side. 
 
 Protract on scale of one inch = a mile of 6,080 feet; find 
 height of Hill /\ and sketch in details. 
 
 Results. Distance of Islet from Hill = 19,304 feet 
 Height of Hill = 31 9 feet. 
 Windmill to Islet = 3 '11 inches. 
 Rock awash to Point = 1*18 inches. 
 
XIV.] METHOD OF PLOTTING A SURVEY. 287 
 
 No. 18. December, 1880. 
 
 In a running survey of an island the ship steamed on the 
 following courses, angles being taken at the position started from, 
 and at the termination of each course : 
 
 Mag. course. Dist. 
 
 N. 75 E 1-75 miles. 
 
 North 1-5 
 
 N. 60 W 1-35 
 
 S. 43 W 2-75 
 
 At Ship's 1st position. 
 
 Peak N. 12 E. 
 
 S.W. point 30 0' Peak. 
 
 Peak 43 0' S.E. point 
 
 At Ship's 2nd position. 
 
 Peak N. 60 W. 
 
 ^ island 23 0' Peak < S.E. point. 
 
 Peak 24 0' Hill. 
 Do. 30 0' North point. 
 
 At Ship's 3rd position. 
 
 Hill S. 88 W. 
 S.E. point 39 0' Peak. 
 Peak 22 30' HilL 
 Do. 40 0' North point. 
 
 At Ship's 4th position. 
 
 Hill S. 10 E. 
 
 Peak 18 0' S.W. point. 
 
 At Ship's 5th position. 
 
 North point 6 0' Hill. 
 S.W. point </> Peak. 
 
 Elevation of peak 1 34' 20" on the arc. 
 1 38' 20" off 
 
288 MARINE SURVEYING. [CHAP. 
 
 Variation 25 W. Scale 2 inches = 1 mile = 6,060 feet. 
 Courses and bearings are magnetic. 
 
 Between the S.W. and S.E. points the coast curves to the 
 southward, and is a moderately high cliff; from the S.W. and 
 S.E. points the coast trends directly towards the north point, the 
 West coast is sand, and the East coast shingle. 
 
 Sketch in the island and give the true bearing of the peak 
 from the 5th position, and its height. 
 
 N.B. Place the ship's 1st position 4J incJies from the right 
 and 3 inches from the bottom of the paper. 
 
 Results. Peak bears N.E. (true) from ship's 5th position, is 
 
 1-77 miles distant, and is 301 feet high. 
 Island is 1'55 miles from N. to S.W. point. 
 1-50 K toSJE. 
 1-50 S.W. to S.E. 
 
 No. 19. Final, 1881. 
 
 In survey of a group of islands, a theodolite station was made 
 on High island, Ship anchoring eastward of the group, and the 
 Cutter anchoring northwest of them. 
 
 At High island /\. 
 
 Zero, Distant peak (bearing N. 57 W. true) 360 00'. 
 Rock awash 26 20' 
 
 Cutter 35 00 
 
 - Flat island 65 30 
 
 -^ 79 00 
 
 (Ship's mainmast at water-line 145 40 
 | depressed 48 10" 
 
 Breaker 213 00 
 
 - Red rock 235 00 
 
 * 239 00 
 
XIV.] METHOD OF PLOTTING A SURVEY. 289 
 
 At Ship (sextant station). 
 
 -. High island 930' High island A 10 30 ' * Hi g h island - 
 
 Breaker 26 50 15 00 Rock awash. 
 
 ~Red rock 33 50 23 30 - Flat island. 
 
 37 00 32 30 Cutter (seen over 
 
 Flat island). 
 34 30 Flat island. 
 
 At Cutter (sextant station). 
 
 - High island 10 30' High island A 6 00' Rock awash. 
 
 Flat island 23 00 11 00 - High isl. 
 
 Ship (seen over 
 
 Flat island) 36 50 
 
 Horizontal distance between High island A. and ship's main- 
 mast at waterline, 9,010 feet. 
 
 High island is cliffy and circular in shape with, a conical 
 summit. 
 
 Protract the above on a scale of 2 inches = a nautical mile of 
 6,080 feet, and find height of High island, drawing scale and 
 subdividing it into cables. 
 
 Note. High island station sJiould be placed about the centre 
 of the paper. 
 
 Results. Height of High island 126 feet. 
 
 Cutter is distant from High island 2 -69 inches. 
 Breaker Ship 273 
 
 Red rock from High island 2*20 ,, 
 
 MISCELLANEOUS EXAMPLES AND QUESTIONS. 
 
 (1) Explain the different methods employed in ascertaining 
 the length of a base line. (B x and B 3 , June, 1881.) 
 
 (2) Draw a scale of longitude corresponding to latitude 58 20' 
 N., one mile of latitude being represented by 2-2 inches. 
 
 Result. Scale required is 1*16 inches. (Do.) 
 R. M. S. 19 
 
290 MARINE SURVEYING. [CHAP. 
 
 (3) During a voyage from La Guayra to Cartagena, calling at 
 Puerto Cabello and Curac,oa, the following observations were made 
 for errors of chronometer and rates : 
 
 At La Guayra on May 22, chron. fast 4 h 33 m 7 -8 
 
 28, 4 33 02-2 
 
 Puerto Cabello on June 5, 4 37 19-6 
 
 Curasoa on June 12, 4 41 001 
 
 Cartagena on June 25, 5 7 22 '2 
 
 ,, 29, 5 7 18-1 
 
 Required the Meridian Distances of the three latter places 
 from La Guayra. 
 
 O h 4 m 25 .. 16 . O h gm 12 .. 45 . o* 34 m 47'-25. (1881.) 
 
 (4) From Flagstaff a Tree bore N. 75 W. 5,500 yards, and a 
 Church N. 84 E. 6,100 yards. 
 
 The following angles were taken from a ship to fix her po- 
 sition : 
 
 Tree 50 Flagstaff 45 Church. 
 
 Required the bearing and distance of the flagstaff from the 
 ship. 
 
 Scale \ inch = 1,000 yards. 
 
 Project this question by the one-circle, and the two-circle 
 methods. (B, and B 2 , Dec. 1880.) 
 
 (5) From a Beacon, Green point bore S. W. by S. 5,830 yards, 
 and a Cliff S.E. by S. 6,270 yards. 
 
 The following angles were taken to fix the position of a shoal : 
 
 Green point 120 Beacon 92 Cliff 
 Required the bearing and distance of the shoal from Green 
 
 point. 
 
 Scale J inch = 1,000 yards. 
 
 The two-circle, and the straight line inetlwds to be applied. 
 
 (Do.) 
 
 (6) Explain the different methods used in marine surveying 
 for determining the heights of hills. (B t and B a , Final, 1880.) 
 
XIV.] METHOD OF PLOTTING A SURVEY. 291 
 
 (7) Distinguish between the Theoretical and Accidental Errors 
 to which Observations are liable, and give examples of each kind. 
 
 (8) Draw the symbols employed on Admiralty charts to 
 represent, 
 
 1st, A flood stream setting 2 knots at springs; 
 2nd, A rock awash at low Avater. 
 
 (B 1 and B 3 , Final, 1880.) 
 
 (9) The following angles were taken to fix the position of 
 a Shoal : 
 
 West Point 115 North Patch 88 East Point. 
 
 The distance between West Point and East Point was 7,400 
 yards; from West Point to North Patch 6,180 yards; and East 
 Point to North Patch 6,500 yards. 
 
 Required the distance of the Shoal from West Point. 
 Scale J inch = 1,000 yards. 
 
 N.B. Project this question by the straight line method. 
 
 (Do. Dec. 1877.) 
 
 (10) Explain the different methods used for measuring a Base 
 Line in marine surveying. (Do. Final, 1878.) 
 
 (11) Explain the construction of a Current-Log, said state 
 the Rule adopted to find the rate of a Current. 
 
 (12) From summit A, 239 feet above the sea surface, '2? was 
 elevated 5 45' and depressed 2 12'. 
 
 Horizontal distance between A and B 1,892 feet. 
 A and G 1,530 feet. 
 
 Required the heights of B and C above the sea surface. 
 B 429 feet, C 180 feet. 
 
 (Do. Final, 1878.) 
 
 (13) From Gunboat in 3J fathoms, Frigate in 6 fathoms bore 
 S.S.W. (true), distant 2 miles, and Sloop in 5 fathoms bore 
 W. J S. (true), distant 1'5 miles. 
 
 At Launch in 2J fathoms. 
 Gunboat 24 30' Sloop 35 18' Frigate. 
 
 192 
 
292 MARINE SURVEYING. [CHAP. 
 
 Soundings in fathoms at equal distances between Sloop and 
 Launch 4J, 4, 3. Reduction to low water 9 feet. 
 
 Protract on scale of 2 ins. = a mile, and place position of 
 Launch and the reduced soundings on the plan. (Do.) 
 
 (14) From a Beacon, West Point bore S.W. 6, 340 yards, and 
 East Cape S.S.E. J E. 5,925 yards. 
 
 The following angles were taken to fix the position of a ship : 
 West point 122 Beacon 128 East Cape. 
 
 Required the bearing and distance of the ship from the 
 Beacon. 
 
 Scale J inch = 1,000 yards. 
 
 (Do. April, 1879.) 
 
 (15) From Point A , Cairn bears North (true) 3,040 feet, and 
 Bluff bears S. 22 E., 2,810 feet; project these positions on a scale 
 of 4 inches = one mile = 6,080 feet, also the following line of sound- 
 ings, reduction to low water 3 feet : 
 
 4 h O m Cairn 44 30' Point A 32 0' Bluff 7 
 
 7 6f 6J 
 
 4 h 12 m do. 63 0' do. 44 30' do. 6J. 
 
 s 
 (Do. Final, 1879.) 
 
 (16) Explain how a Base Line should be measured by Sound, 
 and the circumstances under which this method may be found 
 useful. (Do. Final, 1879.) 
 
 (17) If m = height of a masthead, a } = depression of the 
 masthead, a 2 = depression of the waterline, and h = height of the 
 position of observation above the sea, prove the formula 
 
 li = m {1 4- cos a 2 . cosec (a a c^). sin aj. 
 
 If the position of observation is below the masthead, and 
 ttj = angle of elevation of the masthead, prove that 
 
 h = m {1 cos a 2 . cosec (a 2 + a,) . sin a, }. 
 
XIV.] METHOD OF PLOTTING A SURVEY. 293 
 
 (18) Forts A, B, C are equidistant from each other 4 miles. 
 A to the northward bears N. 40 W. (mag.) from B. 
 
 At c. At d. 
 
 122 50' A 93 45' B. 80 30' A 82 0' B. 
 
 Variation 19 Easterly. 
 
 Protract on a scale of 1-2 ins. = a mile, and mark the posi- 
 tions of c and d. (Do. Final, 1879.) 
 
 (19) Breadth of river Thames at Greenwich on a true north 
 bearing from landing stage = 350 yards. Soundings in feet taken 
 on true north bearing, 50 yards apart, as follows : 
 
 2, 7, 11, 17, 16, 14, 5, 2. 
 
 Reduction to L.W. average spring tides 2 feet. Vertical 
 scale 1 inch = 10 feet. Horizontal scale 1 inch = 100 yards. 
 
 Draw the section of the river for low water at average 
 spring tides. (Do.) 
 
 (20) From Rock point, Green point bore West 5-65 miles, 
 and a Peak N. by E. 4 -755 inches. 
 
 The following angles were taken from a Ship to fix her 
 position : 
 
 Green point 70 Peak. 
 
 Do. 60 Rock point. 
 Required the bearing and distance of Rock point from the Ship. 
 
 Scale \ inch = 1 mile. 
 N.B. This example is to be projected by the two-circle metliod. 
 
 (Do. Dec. 1877.) 
 
 (21) Explain with the aid of a diagram (scale 1 inch to 4 feet) 
 the terms, Spring rise, Neap rise, and Neap range ; Spring rise is 
 15 feet and the Neap rise proportionate. (Do. Final, 1876.) 
 
 (22) Explain the meaning of " Surveying by means of the 
 Back Angle." 
 
 (23) At X, in the same line with, and equidistant from stations 
 A and B, the height by level on back staff at A read 2 -5 feet, that 
 on forward staff at B 5 '6 feet; in the same line, at F, equidistant 
 
294 MARINE SURVEYING. [CHAP. 
 
 from B and C, the height on back staff at B read 4 '5 feet, that on 
 forward staff at 05-3 feet. Required the difference of height of 
 the ground at A and C. 
 
 Result. C is 3-9 feet below A. (Do. Final, 1876). 
 
 (24) The position of a Shoal was fixed by the following 
 
 angles : 
 
 Flagstaff 75 Tree 120 Beacon. 
 
 From the Tree, Flagstaff bore S.S.W. J W. 5 -87 miles, and 
 Beacon S.E. J S. 8-13 miles. 
 
 Required the bearing and distance of the Shoal from the Tree. 
 
 Scale J inch = 1 mile. 
 The straigM line method of projection is to be here used. 
 
 (Do.) 
 
 (25) From a hill, White rock bore S.W. 5,680 yards, and 
 Green patch S.E. by E. 7,140 yards. 
 
 The following angles were taken from a ship to fix her 
 position : 
 
 White rock 99 Hill 112 Green patch. 
 Required the bearing and distance of the hill from the ship. 
 
 Scale J inch = 1,000 yards. 
 TJie two-circle method of projection is to be used. (Do.) 
 
 (26) From a beacon a spire bore N. 85 W. 4,550 yards ; and 
 a tower N. 10 E. 5,210 yards. 
 
 The following angles were taken to fix the position of a 
 Shoal : 
 
 Spire 56 Tower. 
 
 Spire 42 Beacon. 
 
 Required the bearing and distance of the Shoal from the 
 Beacon. 
 
 Project this question by either method. 
 
 Scale J inch = 1,000 yards. 
 
 (Do. Dec. 1879.) 
 
XIV.] METHOD OF PLOTTING A SURVEY. 295 
 
 (27) From Flagstaff, Windmill bore N. 33 E. (true), 1 -1 miles, 
 and Church S. 34 30' E., 1'44 miles. 
 
 Project these positions on a scale of 2 inches to a mile ; also 
 the following line of soundings, the reduction to low water being 
 2 feet : 
 
 3 h 40 m Windmill 58 30' Church 6J fms. 
 Flagstaff 56 10 do. m. 
 
 6i _6i-5| 
 
 3 45 Windmill 74 50 Church 5 
 Flagstaff 72 30 do. s.m. 
 
 (Do. Final, 1875.) 
 
 (28) From a Hut, West Point bore S. 20 W. 6,200 yards ; 
 and East Point S. 50 E. 6,500 yards. 
 
 The following angles were taken to fix the position of a 
 Rock: 
 
 West Point 85 Hut 118 East Point. 
 
 Required the bearing and distance of the Rock from the Hut. 
 Project the above by the straight line and the two-circle 
 methods. 
 
 Scale inch = 1 ,000 yards. 
 
 (Do. Dec. 1879.) 
 
 (29) From a Cairn a Flagstaff bore N. 75 W. 6,185 yards, 
 and White patch N. 84 E. 5,850 yards. 
 
 The following angles were taken to fix the position of a Ship : 
 
 Flagstaff 41 Cairn 52 White patch. 
 
 Required the bearing and distance of the Ship from the Cairn. 
 Project the above by the one and the two-circle methods. 
 Scale J inch = 1,000 yards. 
 
 (Do. Dec. 1879.) 
 
 (30) From South Bluff, West Point bore W.N.W. 4,890 
 yards, and East Point E.N.E. 4,650 yards. The following angles 
 were taken to fix the position of a Wreck : 
 
 West Point 62 South Bluff 39 East Point. 
 
296 MARINE SURVEYING. [CHAP. 
 
 Required the bearing and distance of the Wreck from the 
 South Bluff. 
 
 Scale J inch = 1,000 yards. 
 
 TJie one and the two-circle methods of projection to be applied. 
 
 (Do. Dec. 1876.) 
 
 (31) From A in latitude 15 10' 30" N., longitude 80 12'0" E., 
 B bears N. 17 E. (true) 22..560 feet; one mile of latitude = 6,050 
 feet. Required the latitude and longitude of B. 
 
 Results. Lat. of B 15 14' 5" N. 
 
 Long, of 80 13' 7" K 
 
 (Do. Final, 1877.) 
 
 (32) From a hill, the angle of depression to the summit of an 
 islet, 30 feet above the sea and distant 2,500 feet, is 2 11'; height 
 of eye 5 feet. Required the height of the hill. 
 
 Residt. 120 feet. (Do.) 
 
 (33) Define the terms, tidal wave and tidal stream. (Do.) 
 
 (34) Required the natural scale of a plan, drawn on a scale of 
 6 inches = one mile = 6,082 feet. Result. = 1& ^ A . 
 
 (Do.) 
 
 (35) From Sand hill, Wedge bears N. 39 W. (true) 4,550 feet, 
 and Bluff bears N. 78 E., 5,990 feet. Project these positions on a 
 scale of 4 inches = one mile = 6,070 feet; also the following line of 
 soundings, reduction to low water 9 feet : 
 
 3 h 30 m Wedge 33 20' Sand hill 41 40' Bluff lO" 
 
 9f_9J 9J 
 3 42 do. 44 20 do. 39 40 do. 9|. 
 
 (Do.) 
 
 (36) Investigate a Formula for computing the Horizontal 
 Angle between the sun and a given object, the latter not being in 
 the horizon. 
 
 (37) Given two positions on a Mercator's chart of the world 
 shew how to find the distance between them. 
 
XIV.] METHOD OF PLOTTING A SURVEY. 297 
 
 (38) Does a straight line on a Mercator's chart of the world 
 ever represent the shortest distance between two positions 1 
 
 (39) Given the cross-bearings and deviation of the compass, 
 place the ship's position on a chart. 
 
 (40) Given a bearing of an object and an angle between it 
 and another object, lay down the ship's position. 
 
 (41) Lay off a course to pass clear of a known danger, allow- 
 ing for variation and deviation. 
 
 (42) What observations would you make for the Latitude, 
 Longitude, and Variation of the Compass, for a given place? 
 
 (43) Explain the term " tangent " as used in surveying. 
 
 (44) What do you understand by the term " angle of danger " ? 
 
 (45) Lay down a position by a " circle and line of direc- 
 tion," knowing the angles between three points. 
 
 (46) Lay down a position by circles only, knowing the angles 
 between three points. 
 
 (47) Define an " ill-conditioned " triangle. 
 
 (48) State the difficulty sometimes experienced of " being on 
 the circle." 
 
 (49) Mention the method of getting out of this difficulty. 
 
 (50) What precaution must be taken when observing a small 
 angle with a sextant ? 
 
 (51) Draw a cluster of rocks, some above water, others awash, 
 with encircling danger line, extending about a mile east and west, 
 and half a mile north and south. Scale 2 inches = a mile. 
 
 (Dec. 1878.) 
 
 (52) Draw the symbols used on Admiralty charts to represent 
 a Church, a Windmill, and an Observation spot. (June, 1880.) 
 
 (53) Draw the symbols to denote an Anchorage for large 
 vessels, and an Ebb-tidal stream. 
 
298 MARINE SURVEYING. [CHAP. 
 
 (54) Draw the symbols used to represent : 
 
 (i) A can buoy painted red and white in horizontal stripes. 
 
 (ii) A rock awash at low water. 
 
 (iii) The position of a lighthouse. (Oct. 1880.) 
 
 (55) Write the abbreviations for sand and broken shells in 
 describing the nature of the bottom. (Dec. 1880.) 
 
 (56) Draw a coral reef of circular shape, half a mile in dia- 
 meter, with a lagoon in the centre, and rocks awash extending 
 half a mile from its northern side. Scale 2 in = a mile. 
 
 (April, 1881.) 
 
 (57) Draw the symbols used to represent a coral reef, a 
 marsh, and an anchorage for small vessels. (June, 1881.) 
 
 (58) Describe any methods you are acquainted with for find- 
 ing the rate and direction of a surface current from a ship at 
 anchor; also of an under current. (Feb. 1879.) 
 
 (59) What steps are necessary in order to determine with 
 accuracy, by Chronometer, the Meridian Distance between two 
 places? (Dec. 1877.) 
 
 (60) Describe the different methods of measuring a Base, with 
 the necessary precautions for accuracy in each case. 
 
 (Feb. 1875.) 
 
 (61) Does a Nautical Mile vary in length in different parts of 
 the earth? (Sept. 1877.) 
 
 (62) Describe the use of the Level and Levelling-Staves as 
 applied to nautical surveying. (May, 1879.) 
 
 (63) Explain and illustrate what is meant by a landmark, 
 leading mark, clearing mark, danger angle, and transit of land 
 objects. (Dec. 1879.) 
 
 (64) Does the direction of the tidal stream always change 
 at the time of high water? (April, 1880.) 
 
 (65) Explain the abbreviations H.W.F. and C. ; at which 
 seasons do the highest tides occur ? (Sept. 1880.) 
 
XIV.] METHOD OF PLOTTING A SURVEY. 299 
 
 (66) Draw a compass as shewn on Admiralty charts ; diame- 
 ter 3 inches, variation 26 W. (May, 1881.) 
 
 (67) Define the term Neap range of tide. Is the level of the 
 water higher or lower at low water neaps than at low water 
 springs? (Aug. 1880.) 
 
 (68) How is the True Bearing of a point of land determined 
 by means of sextant observations ? (Aug. 1881.) 
 
 (69) Draw a compass similar to those usually shewn 011 Ad- 
 miralty charts, subdividing one quadrant. Variation 23 20' E. 
 
 (Aug. 1881.) 
 
 (70) State how a Theodolite may be used in swinging ship 
 for deviation. (Aug. 1876.) 
 
 (71) What do you understand by the Index Error of a Theo- 
 dolite? 
 
 (72) What is a Day Mark ? 
 
 (73) Distinguish between Absolute and Dependent Heights. 
 
 (74) Distinguish between the terms regular and irregular 
 plotting. 
 
 The following may be regarded as typical questions in the case 
 of a Viva Voce Examination in the Subject. 
 
 (75) Mention the best means of finding the Error of Chrono- 
 meter at sea. 
 
 (76) If the A.M. sights of Equal Altitudes are lost, are the 
 P.M. sights completely lost also? What is the disadvantage attend- 
 ing the loss of the Forenoon sights ? 
 
 (77) Mention, in their proper order, the Adjustments of the 
 Sextant. 
 
 (78) What is the use of the Graduations in a Compass ? 
 
 (79) Mention the Advantages and Disadvantages of this 
 Instrument when compared with a Theodolite. 
 
300 MARINE SURVEYING. [CHAP. XIV. 
 
 (80) Explain how you would set up a Theodolite to take 
 an "arc." 
 
 (81) Lay off on a Chart an Observed Bearing, using Parallel- 
 rulers. 
 
 (82) Note the best Headlands in a given Harbour for fixing 
 a position by the Three-point Problem. 
 
 (83) What substitute would you suggest in lieu of a Station 
 Pointer ? 
 
 (84) Read off and explain the following note about a Light 
 on the Coast : 
 
 120 ft. Rev. 1 min. vis. 16 m. 
 
 (85) Can a Light be seen from a point on the Circle described 
 on the Chart to shew the extent of illumination ? 
 
 (86) Distinguish between a Plan and a Chart; how would 
 you know one from another if placed in your hands ? 
 
 (87) Mention the respective advantages of Mercurial and 
 Aneroid Barometers. 
 
 (88) What is the Datum Line to which Soundings are to be 
 reduced ? 
 
 (89) Is there any difference between the several Compasses 
 engraved on a Chart of considerable extent? 
 
INDEX. 
 
 TJie Numbers refer to the Paragraphs. 
 
 Abbreviations, 2 Balance of a Chronometer, 231 
 
 Absolute Height, 160 Bar, length of, involves three ele- 
 
 Admiralty Tide Tables, 198 meuts, 107 
 
 Afternoon Sights in Equal Altitudes, Barometer, Correction of, 93 
 
 253 Contracted Scale of, 96 
 
 Age of a Tide, 190 Description of, 93 
 
 Airy's Method of Compensation, 231 Fortin's, 95 
 
 Alternating Light, 7 (7) Height by, 159 
 
 Aneroid, 151) Marine, 97 
 
 Angle, Methods of protracting, 44 Method of Suspending, 98 
 
 by Scale of Chords on Pro- Mountain, 157 
 
 tractor, 33 Beading off, 100 
 
 by "Versine Method," 45 Packing for Carriage, 101 
 
 to repeat, with a Theodolite, Vernier of, 99 
 
 86 Effects of, on Tides, 182 
 
 to observe a vertical, with Base, Assumed length of, 139 
 
 ditto, 87 - Direction of, 127 
 
 Horizontal, 128 - Elements of, 117 
 
 "Annual Trials," 232; 233 - Examples of accurate, 104 
 
 Anti-Lunar Tide, 167 ^^^Method of Increasing, 133 
 
 Anti-Solar Tide, 167 - Length, by Astronomical Obser- 
 
 Aphelion, 178 vations, 125 
 
 Apogee, 178 by Chaining, 126 
 
 "Arc," to observe, with a Theodolite, by Masthead Angle, 
 
 85 122 
 
 Assumed Base, 139 by Patent Log, 1-24 
 
 by Velocity of Sound, 
 
 Bach's Base Measuring Apparatus, 123 
 
 111 - Method of Measuring, 126 "- 
 
302 
 
 MARINE SURVEYING. 
 
 3, Madrid, 115 
 
 Names of, in Ordnance Survey, 
 
 114 
 
 Nature of the ground for, 120 
 
 Ratio of length to Size of Sur- 
 
 vey, H3; 119 
 
 - Reduced to the Sea level, 116 
 
 Use of, 102 
 
 - of Verincation, 104 
 Bench Mark, 13; 145 
 Bernouilli's Theory of Tides, 164 
 Borda's Method, 108 
 
 Bottom, Symbols to denote nature of, 
 
 1 
 
 Buoy and Nipper, 226 
 Buoys, Colouring, 4 
 
 Names, Shapes, and Colours 
 
 of, 3 
 
 Regulations for use of, 5 
 
 Capacity, Correction for, 156 
 Capillarity, 156 
 Catodioptric System, 8 
 Catoptric System, 8 
 Centre, Reduction to, 141 
 Chain, Gunter's, 76 
 
 Surveying, 77 
 Change Tide, 191 
 Chart, Explanation of, 58 
 
 Mercator's, 64 
 
 Method of Constructing, 68 ff . 
 Official Number of, 12 
 Plane, 65 
 
 Check Level, 145 
 
 Chords, Angle projected by, 33 ; 45 
 
 Line of, 36 
 
 Scale of, 32 
 
 Use of ditto, 33 
 Chronometer, Balance wheel, 231 
 
 Comparison of, 242; 
 
 243 
 Compared with Sidereal 
 
 Chronometer, compared with Deck 
 
 Watch, 244 
 
 Effect of iron, 238, note 
 Faulty, how detected, 
 
 244 
 
 History of, 228 
 Inside of, 230 
 Journal, 244 
 Life of, hi Royal Navy, 
 
 235 
 
 Method of Winding, 
 241 
 
 Setting go- 
 ing, 241 
 Ought to be wound 
 
 regularly, 240 
 Packing of, for trans- 
 port, 236 
 Rating of, 242 
 Room, in Royal obser- 
 vatory, 233 
 Sources of Error, 239 
 Stowing on board, 238 
 Selection of, for Royal 
 
 Navy, 234 
 Temperature, effect of, 
 
 240 
 Transporting from 
 
 Shore to Ship, 237 
 Chronometrical Thermometer, 233 
 Circle, on the, 53 
 Clearing Mark, 12 
 Colby's Apparatus, 109 
 
 Method of using, 110 
 Collimation, To test, 80 
 
 To correct, 80 
 Coloured Sectors, 8 
 Compensation Apparatus, 109 
 
 Microscopes, 110 
 Conjunction, 178 
 Constants, Tidal, 195 
 Contours, 161; 162 
 Conventional Signs, 10 
 
 Clock, 246 
 
 Convergency of Meridians, 63 
 
INDEX. 
 
 303 
 
 Corrected Establishment of the Port, Fixed Light, 7 (1) 
 
 191 
 
 Cotidal Lines, 172; 173 
 Course, on a Chart, 73 
 Current Log, 220 
 
 Surface, 220 
 
 Tidal, 175 
 
 Symbol for, 12 
 
 Under, 221; 222 
 
 of the Flood, 177 
 
 Danger Angle, 56 
 
 Hidden, 218 ; 219 
 
 Datum Line, Def., 145 
 
 in Soundings, 213 
 
 in Ordnance Survey, 145 
 
 Day Mark, 12 
 
 Dependent Height, 160 
 
 Diagonal Scales, 19 
 
 Dioptric System, 8 
 
 Direction of the Base, 127 
 
 Discharge of a Elver, 223 
 
 Distance on a Chart, 73 
 
 Division of a line into parts, 43 
 
 Diurnal Inequality, in Heights, 180 
 in Times, 181 
 
 Double Half Day Tides, 187 
 
 Drummond Light, 138 
 
 Dynamical Theory of Tides, 164 
 
 Ebb of a Tide, 176 
 
 Elements of a Base, 117 
 
 Error, Secondary, of a Chronometer, 
 
 232 
 
 Establishment of the Port, Def., 190; 
 191 
 
 Method of finding, 192 
 Corrected, 191 
 Vulgar, 191 
 Equal Altitude Observations, 252; 
 
 253 
 
 Equilibrium Theory of Tides, 164 
 Evolution, Tidal, 205 
 
 First and Second High Waters, 194 
 
 Fixed and Flashing Light, 7 (3) 
 Flashing Light, 7 (2) 
 Flow of a Tide, 176 
 Fortin's Barometer, 95 
 
 Gauge, Tide, 201 
 Gravel Beach, 10 
 Ground Log, 226 
 Gunter's Chain, 76 
 
 Hachure Lines, 163 
 Harrison's Watches, 229 
 Heat Test of Chronometers, 233 
 Height, hy Aneroid, 159 
 
 by Barometer, 159 
 
 by Spirit-Level, 147 ; 148 
 
 by Theodolite, 153 
 by Thermometer, 160 
 
 Formulae for, by Barometer, 
 
 159 
 of a Tide, 183; 184 
 
 Absolute, 160 
 Dependent, 160 
 
 Hidden Danger, 218 ; 219 
 
 High Water, Def., 165 ; 177 note 
 
 First and Second, 194 
 Time of, 196 ; 203 ; 204 
 
 Horizontal Angle, 128 
 
 Button's Rule, 158 
 
 Illuminating Apparatus, 8 
 Increasing the Base, 133 
 Indeterminate Case, 53 
 Index Error, in Sextant, 80 
 Theodolite, 84 
 
 Inequality, Diurnal, 180; 181 
 Inferior Tide, 167 
 Intermittent Light, 7 (5) 
 
 Journal, Chronometer, 244 
 
 Lagging of a Tide, 186 
 Land Base, 126 
 
304 
 
 MARINE SURVEYING. 
 
 Landmark, 12 
 
 Laplace's Theory of the Tides, 164 
 
 Lateral Distance, 34 
 
 Latitude on a Chart, 73 
 
 Law of the Heights of High Water, 
 
 193 
 
 Lead Lines, 212 
 Leading Mark, 12 
 Least Reading by Vernier, 23 ff. 
 Legendre's Theorem, 140, note 
 Length of Base, 119 ff. 
 Level, Apparent, 144 
 
 - True, 144 
 
 Difference of, 144 
 
 - Check, 145 
 
 - Spirit, 88; 148 
 
 - Adjustments of, 88 
 Levelling, Def., 144 
 
 by Aneroid, 159 
 by Barometer, 155 
 by Theodolite, 153 
 by Thermometer, 160 
 Simple, 146 
 
 Compound, 146 
 Staff, 89 
 
 Precautions in holding 
 
 ditto, 89 
 Light, Drummond, 138 
 
 Alternating, 7 
 
 - Fixed, 7 
 
 - Fixed and Flashing, 7 
 
 - Flashing, 7 
 
 Intermittent, 7 
 
 - Occulting, 7 
 
 Revolving, 7 
 Light Vessels, 9 
 Lights on Coasts, 6 ; 7 
 
 Limb of a Sextant, to ascertain if 
 
 correctly graduated, 29 
 Line of Chords, 36 
 Lines, 35 
 
 Polygons, 37 
 
 to divide a, 43 
 Lines of position, 55 
 
 Lines, Cotidal, 172 ; 173 
 
 Log, Current, 220 
 
 Log, Ground, 226 
 
 Longitude on a Chart, 73 
 
 Low Water, Def., 165 ; 177, note ; 203 
 
 Lunar Day, 169 
 
 Lunation, 178 
 
 Lunitidal Interval, 191 
 
 Madrid Base, 115 
 Map, Def., 58 
 Marquois Scales, 38 
 
 use of, 39 
 
 Masthead Angle, 122 
 Mean level of Sea, 183 
 Mercator's Chart, Advantages and 
 Disadvantages of, 64 
 
 Construction of, 68 ff. 
 
 Explanation of, 64 
 Meridian, Primary and Secondary, 
 
 248 
 
 use of, 249 
 Distance, Def., 247 
 
 - by Telegraph, 251 
 
 by Chronometers, 
 
 252 
 
 Examples of, 256 
 Middle Latitude, Scale of Chart for, 
 
 75 
 
 Mile, Nautical, Def., 66 
 Mountain Barometer, 157 
 
 how delineated on a Chart, 
 163 
 
 Natural Scale, 17 
 Nautical Mile, 66 
 
 varies in length, 66 
 Neap Tides, 165 ; 170 
 Nodal Line, 174 
 
 Observation Spot, 2 
 Observations with Sextant, 80 
 
 Tidal, 200 
 Observatory Station, 13 
 
INDEX. 
 
 305 
 
 Observatory, Latitude and Longitude 
 
 of, 118 
 
 Occulting Light, 7 (6) 
 Octant Tides, 189 
 Official Number of a Chart, 12 
 Opposition, 178 
 Ordnance Survey, Scales used in the 
 
 plans of, 18 
 
 Parallelism of a River, 217 
 Perigee, 178 
 Perihelion, 178 
 Periods of a Tide, 12 
 Perpendicular, to draw a, 41 
 Personal Equation, 131 
 "Piling-up" triangles, 133 
 Plan, Def., 58 
 Plane Chart, 65 
 Plotting, Method of, 260 
 Pole, Ten-feet, 00 
 Polygons, Line of, 37 
 Position, fixed by straight line and 
 one angle method, 50 
 
 by straight line and one 
 
 circle method, 52 ; 55 
 by straight line method, 52 
 without drawing circles, 55 
 
 Lines of, 55 
 Primary Meridian, 248 
 Priming of a Tide, 186 
 Primitive, 59 
 Problems, Useful, 57 
 Projection of a Point, 59 
 
 Artificial and Natural, 59 
 Central, 62 
 Mercator's, 64 
 Orthographic, 60 
 Stereographic, 61 
 by Chords, 44 ; 45 
 Protracting Scales, 30 ; 31 
 Protractor, Scales on, 15 ; 30 
 
 Quadrature, 178 
 R. M. S. 
 
 Range of a Tide, 183; 185 
 Reduction to the Centre, 141 
 of Soundings, 213 
 Retard of a Tide, 190 
 Revolving Light, 7 (4 ; 6) 
 Rhombus Method of finding a dis- 
 tance, 57 
 Rip, Tide, 12 
 Rise of a Tide, 183 ; 184 
 River, Discharge of, 223 
 
 Elevation along a tidal, 217 
 
 Sectional area of, 224 
 
 Velocity of, 225 
 Ruling-pen, Precautions in using, 72 
 
 Sandy Beach, 10 
 Scale, Def., 14 
 
 of Chords, 32 ; 33 
 
 Diagonal, 19 
 
 - of Equal parts, 15 
 
 use of, 16 
 
 construction of, 17 
 
 - Marquois, 38 ff. 
 
 - Natural, 17 
 
 - Plain, 15 
 
 - Protracting, 30 ; 31 
 
 - of Rhumbs, 31 ; 33 
 
 of Secants, 31 
 
 for a plan, how determined, 90 
 
 Contracted, in Barometer, 96 
 Sea Base, 126 
 
 Sea, Mean level of, 183 
 Secondary Meridian, 248 
 
 Error of Chronometer, 232 
 Section, Def., 144 
 
 to lay down a, 149 ; 154 
 
 Examples of, 150 
 Sectional area of a River, 224 
 Sector, 34 ff. 
 
 Sectoral Lines, 35 
 
 Segment of a Circle, to describe a, 48 
 Semi-lunation, 178 
 Semi-menstrual inequality, 191 
 Sextant, Adjustments of. 79 ; 80 
 
 20 
 
306 MARINE SURVEYING. 
 
 Sextant, Limb of, incorrectly gradua- Theodolite, Description of, 81 
 
 ted, 29 Adjustments of, 84 
 
 Shingle Beach, 10 Motions of, 82 
 
 Single Day Tides, 187 Parts of, 81 
 
 Slack Water, 176 Method of using, 85 
 
 Sound, Velocity of, 123 To repeat an Angle with, 
 
 Soundings, object of, 206 86 
 
 to fix, 208 ; 211 To observe a vertical An- 
 
 must be reduced, 213 gle by, 87 
 
 formula for reducing, 215 Levelling by, 153 
 
 in a river, 216 ; 217 Thermometers in Ordnance Survey, 
 
 Symbols for Lines of, 11 how Compared with 
 
 Sounding Pole, 207 the Standard, 106 
 
 Spherical Excess, 140 Height by, 106 
 
 Spirit Level, 88 Three-point Problem, 51 
 
 Adjustments of, 88 Six cases of, 52 
 
 Spring Tides, 165 ; 170 - Selection of the objects, 
 
 Staff, Levelling, 89 53 
 
 Stand of a Tide, 176 ; 177 note - Best position of the 
 
 Standards, Geodetic, 105 " Points," 54 
 
 " a traits, " and " a bouts," Tidal Day, 186 
 
 105 Constants, 195 
 
 in Ordnance Survey, 105 Current, 175 
 
 Station Pointer, 91 Stream, 175 
 
 Method of using, 92 Observations, 200 
 
 Straight Line Method, 55 Evolution, 205 
 
 Straight Line and One Angle Method, - Table, Elements of, 200 
 
 50 Wave, 175 
 
 Stream, Tidal, 175 Tide, Flow and Ebb of, 165 ; 176 
 
 Struve's Method, 108 Superior and Inferior, 167 
 
 Superior Tide, 167 - Spring and Neap, 165 ; 170 
 
 Surface Current, 220 Head, or End of, 174 
 
 Survey, Points to be observed in Priming and Lagging of, 186 
 
 making a, 259 Age of, 190 
 
 Surveying Chain, 77 Retard of, 190 
 
 Suspending a Barometer, 98 Rise and Range of, 183 ; 184 
 
 Sweeping for a hidden danger, 219 Octant, 189 
 
 Symbols, Chap. I. passim and Half Tide, 177 
 
 and Quarter Tide, 177 
 
 Tachometer, 225 - and Half Quarter Tide, 177 
 
 Temperature, to evade effects of, on a Rip, 12 
 
 Measuring Bar, 108 Gauge, 201 
 
 Ten-feet Pole, 90 Tides, Theories of, 164 
 
 Scale for, 90 Caused by difference of attrac- 
 
INDEX. 
 
 307 
 
 tion on opposite sides of the 
 Earth, 166 
 Tides, when Greatest and Least, 179 
 
 - Single-day, 187 
 
 - Double Half Day, 187 
 
 - Effects of Sun and Moon in 
 
 raising, 167 note 
 
 Time of High Water, how to find, 196 
 Topographical details in a Survey, 259 
 Transverse Distance, 34 
 " Trial Numbers," 234 
 Triangle, ill-conditioned, 132 
 
 Well-conditioned, 132 
 Triangulation, Primary, 133 
 Secondary, 133 
 of Britain connected 
 with that of the 
 Continent, 142 
 of Spain connected 
 with that of North 
 Africa, 143 
 
 Under-Current, 221, 222 
 Under-lined figures on a bank, 12 
 
 United States Coast Survey Appara- 
 tus, 111 
 Useful Problems, 57 
 
 Velocity of a Eiver, 225 
 
 of Sound, 123 
 
 of a Tide, 12 
 
 of a Surface Current, 220 
 
 of an Under Current, 221 ; 
 
 222 
 
 Verification, Base of, 104 
 Vernier, 21 ff. 
 
 of a Standard Barometer, 99 
 
 Least Beading by, 23 ff. 
 
 Exercises in, 25 
 
 Vertical Angle by Theodolite, 87 
 Vulgar Establishment of Port, 191 
 Vessels, Light, 9 
 
 Wave, Tidal, 175 
 
 Wind, effect of, on a Tide, 182 
 
 Winding a Chronometer, Method of, 
 
 241 
 Wreck, how Marked, 5 
 
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