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 A TREATISE 
 
 ON 
 
 PLANE SURVEYING 
 
 BY 
 
 DANIEL CARHART, C.E. 
 
 .PROFESSOR OP CIVIL ENGINEERING IN THE WESTERN 
 UNIVERSITY OF PENNSYLVANIA 
 
 48504 
 
 GINN & COMPANY 
 
 BOSTON* NEW YORK CHICAGO LONDON
 
 
 
 Entered, according to Act of Congress, in the year 1887, by 
 
 DANIEL CARHART 
 in the Office of the Librarian of Congress, at Washington 
 
 68.5 
 
 JCfje atfcenacutn
 
 PREFACE. 
 
 rr^HIS work, as its name indicates, extends over the field of 
 J- plane surveying. It illustrates and describes the instru- 
 ments employed, their adjustments and uses ; it exemplifies the 
 best methods of solving the common problems occurring in 
 
 Y practice, and furnishes solutions for many special cases which 
 
 ?^ not unfrequently present themselves. An experience of twenty 
 - years in the field and in technical schools confirms the opinion 
 that a work of this kind should be eminently practical ; that 
 the student who desires to become a reliable surveyor needs 
 frequently to manipulate the various surveying instruments in 
 the field, to solve many examples in the class-room, and to 
 exercise good judgment in all these operations. With this in 
 
 ^ view, therefore, the different methods of surveying are treated, 
 ^ directions for using the instruments are given, and these are 
 
 ^ supplemented by numerous examples to be solved, by various 
 field exercises to be performed, and by many queries to be 
 answered. 
 
 Chapter I. is devoted to Chain Surveying, in which direc- 
 tions are given for measuring and ranging out lines, and 
 methods of overcoming obstacles, recording field notes, obtain- 
 ing areas, and plotting a chain survey. 
 
 Chapter II. treats of Compass and Transit Surveying, or 
 when, in addition to the chain, an instrument for measuring 
 angles is employed. In this chapter the compass and transit, 
 the solar attachment, the adjustments of these, and auxiliaries 
 of the transit, such as the stadia wires, gradienter, etc., are 
 fully illustrated and described, and their uses shown. Here 
 the various methods of obtaining the data requisite to deter-
 
 IV PREFACE. 
 
 mine the area, as well as the different methods employed in 
 calculating the contents of laud, are exhibited. Tests of the 
 accuracy of a survey are indicated, numerous methods of over- 
 coming obstacles, supplying omissions, of ascertaining heights 
 and distances, of keeping the field notes, and of plotting a sur- 
 vey are given, while the uses of the solar attachment in deter- 
 mining the latitude of a station and its geographic meridian are 
 exemplified. 
 
 The student now having been taught how to survey land, 
 using a needle instrument, should become acquainted with the 
 declination of the magnetic needle, or variation of the compass, 
 as it is frequently called. This subject is accordingly dis- 
 cussed in Chapter III. Some of the tables and much of the 
 matter is taken from the Reports of the United States Coast 
 and Geodetic Survey. The student will do well to give this 
 chapter a careful inspection, examining the tables and formu- 
 las and the directions for determining the true meridian, thus 
 being prepared with facts, figures, and methods, which will en- 
 able him intelligently to undertake the retracing of old lines, 
 as well as to establish with considerable precision his geo- 
 graphic meridian, and thereby obtain the declination of the 
 needle. 
 
 Chapter IV. is devoted to Laying Out and Dividing Up Land. 
 This subject is of more importance than some suppose, especi- 
 ally to practitioners in the older States of the Union, and is here 
 treated very fully. The principal cases are exemplified, and 
 general directions and suggestions given, so that, it is believed, 
 with a thorough knowledge of this chapter, the student will be 
 enabled, without embarrassment, to meet the requirements of 
 an extensive practice. 
 
 The description, adjustment, and use of the Plane Table form 
 the subject of Chapter V. This instrument is being employed 
 more frequently than formerly in park surveys, in determining 
 positions in harbors, along the lines of proposed highways, in 
 "filling in" large surveys, and 'generally in locating points 
 where extreme accuracy is not required.
 
 PREFACE. V 
 
 In Chapter VI. the system employed by the government in 
 the Survey of the Public Lands is set forth. The description 
 and adjustment of the Solar Compass, which is used quite ex- 
 tensively in these surveys, precede au account of the origin of 
 the system, and the leading points in the " Instructions to Sur- 
 veyors-General " from the commissioner of the land office. A 
 form of recording the notes extracted from the " Instructions" 
 is also given, the chapter closing with formulas and a table for 
 determining the inclination of meridians and deviation of par- 
 allels. 
 
 Chapter VII., on City Surveying, is from the pen of my 
 friend and former colleague, Frederic H. Robinson, C.E., City 
 Engineer of Wilmington, Del. This subject has received but 
 little notice from writers on surveying, although the need of 
 some systematic and practical treatment of it has long been 
 recognized. It therefore affords me much pleasure to acknowl- 
 edge my indebtedness to Professor Robinson for supplying this 
 want, and so enhancing the value of this publication as a text- 
 book. Experience in teaching, and ten years' practice in city- 
 surveys and improvements, eminently qualify him to speak on 
 this important subject with authority and in a manner readily 
 understood by students. 
 
 The special instruments needed in this branch of surveying 
 are illustrated and described ; the adjustment of the Y-level 
 and directions how to level and to record the notes are given ; 
 more refined means of measuring lines are discussed ; tempera- 
 ture, pull, sag, wind, etc., are considered, and corrections indi- 
 cated ; best directions and width of streets, together with the 
 subject of grades, sewers, the establishment of permanent 
 reference points, and adjusting property lines, are fully set 
 forth. 
 
 To my college classmate and esteemed friend, F. Z. Schel- 
 lenberg, C.E., Superintendent of Westmoreland Coal Co., 
 Irwin, Pennsylvania, I am indebted for Chapter VIII., on 
 Mine Surveying. This chapter, though in general explanatory 
 of what is applicable and peculiar to this branch of surveying,
 
 vi PREFACE. 
 
 includes directions for running contours and sketching topog- 
 raphy. It is replete with suggestions that will be valued 
 when, by the aid of the study of mine workings themselves and 
 their ground, illustrations will be afforded which otherwise, as 
 drawings alone, cannot readily be understood. 
 
 The Judicial Functions of Surveyors, as given by Chief Jus- 
 tice Cooley, are set forth in an Appendix. 
 
 Those who are familiar with the elegant tables of logarithms 
 of numbers and of trigonometrical functions prepared by Pro- 
 fessor Wentworth, will likely recognize the use of his electro- 
 plates, from which I have been permitted to print Tables I., 
 III., IV., and VII. To him mv personal acknowledgments 
 are due. The plates from which Tables II., V., VI., VIII., and 
 IX. are printed were prepared expressly for this work. It is 
 thought that the four-place tables of the natural trigonometri- 
 cal functions will be found very useful in connection with sur- 
 veying and engineering operations. They are believed to be 
 correct, having been very carefully compared with others whose 
 accuracy is unquestioned. 
 
 In addition to acknowledgments made elsewhere, I take 
 pleasure in expressing here my thanks to Messrs. W. and L. E. 
 Gurley, of Troy, New York, for the use which I have been per- 
 mitted to make of their valuable catalogue, in the description 
 of certain instruments, and for the loan of several plates for 
 the engraving of instruments ; also to Messrs. Fauth and Co., 
 "Washington, D.C., and to Messrs. Heller and Brightly, and 
 Messrs. Young and Sons, Philadelphia, Pa., for plates which 
 they kindly furnished for the illustration of the subject. 
 
 D. C. 
 
 WESTERN UNIVERSITY OF PENNSYLVANIA, 
 DECEMBER, 1887.
 
 CONTENTS. 
 
 PAGE 
 
 Definitions, and division of the subject 1 
 
 CHAPTER I. 
 CHAIN SURVEYING. 
 
 SECTION I. 
 
 INSTRUMENTS. 
 
 Gunter's chain . 8 
 
 Two-pole or half chain 8 
 
 The engineer's chain 4 
 
 The tape measure 4 
 
 Marking-pins 4 
 
 Straight poles 4 
 
 SECTION II. 
 CHAINING. 
 
 How to chain 4 
 
 Tallying : 5 
 
 Error in chaining 6 
 
 Chaining on sloping ground 8 
 
 Field exercises 10 
 
 Ranging out lines '. 10 
 
 Over a hill ; across a valley 11 
 
 Through a wood 12 
 
 Field exercises 13 
 
 To set off a perpendicular from a line 13, 14 
 
 To let drop a perpendicular from a point to a line 14, 15 
 
 Through a point to run a line parallel to a given line 16, 17 
 
 Obstacles to alignment 17, 18
 
 Viil CONTENTS. 
 
 PAGE 
 
 Obstacles to measurement ... 18-20 
 
 Measurement of heights 20 
 
 Examples and field exercises 21 
 
 SECTION III. 
 RECORDING THE FIELD NOTES. 
 
 By sketch 22 
 
 In columns 23-25 
 
 SECTION IV. 
 MAPPING AND PLOTTING. 
 
 Instruments useful for plotting a chain survey 26 
 
 Drawing-board, T-square, triangles, etc 26, 27 
 
 Scales 28 
 
 Drawing to scale 29 
 
 To ascertain unknown scale 30 
 
 SECTION V. 
 ON AREAS, AND ILLUSTRATIVE EXAMPLES. 
 
 Areas : The area of a triangle 31 
 
 " " rectangle 31 
 
 " " parallelogram 31 
 
 " " trapezoid 31 
 
 " " regular hexagon 31 
 
 " " " octagon 31 
 
 " " " polygon 32 
 
 Table of areas of regular polygons 33 
 
 The area of a circle 33 
 
 " " quadrant 34 
 
 " " sextant 34 
 
 " " circular ring 34 
 
 " " segment 34 
 
 " ellipse 35 
 
 EXAMPLES INDICATING HOW TO SURVEY LAND, TO PLOT THE SURVEY, 
 AND TO COMPUTE THE AREA. 
 
 To survey a triangle 35, 36 
 
 " " rectangle . . 37 
 
 Examples 37
 
 CONTENTS. ix 
 
 PAGE 
 
 To survey a parallelogram 38 
 
 " " trapezoid , 38 
 
 Examples 38, 39 
 
 To survey, plot, and compute the area of a trapezium 39 
 
 Examples 40 
 
 To survey, plot, and compute the area of a polygon, regular or irreg- 
 ular 40, 41 
 
 Examples 41, 42 
 
 To survey, plot, and compute the area of a circle and a circular ring. 43 
 
 " " " sector and a segment 43 
 
 Examples 43 
 
 SECTION VI. 
 OFFSETS AND TIE-LINES. 
 
 Remarks and illustrations 44 
 
 Rectangular co-ordinates and their application to the computation of 
 
 areas 45 
 
 Formula and rule 46 
 
 Examples 47 
 
 Additional formulas and rule 48 
 
 Examples 48-50 
 
 Tie-lines used to survey a lake or pond 51 
 
 Miscellaneous examples 51, 52 
 
 Field exercises 53 
 
 CHAPTER II. 
 COMPASS AND TRANSIT SURVEYING. 
 
 SECTION I. 
 DEFINITIONS AND DESCRIPTION OF INSTRUMENTS. 
 
 A meridian plane; meridian line; the magnetic needle; the magnetic 
 
 meridian 54 
 
 The azimuth, bearing, and meridian distance of a line 56 
 
 Horizontal angle ; vertical angle ; angle of elevation ; angle of depres- 
 sion 55 
 
 The surveyor's compass 55, 66 
 
 To adjust the compass 56 
 
 Engraving of compass 57 
 
 Caution when handling the compass 58
 
 X CONTENTS. 
 
 PAGE 
 
 To re-magnetize the needle 60 
 
 Weight of compass . 60 
 
 The vernier 61 
 
 Formulas for determining the least count 61 
 
 How to space a vernier for a given least count 62 
 
 To read an instrument having a vernier 62 
 
 Exercises on designing verniers 62 
 
 Engraving of surveyor's transit 64 
 
 Description of transit, and section of telescope and cross-wires 65, 66 
 
 Sectional view through the spindle of transit . 67 
 
 The needle, tangent movements, levels, and verniers 68, 69 
 
 The sockets, levelling-plate, and tripod 70, 71 
 
 To adjust the transit ., 71-75 
 
 To use the transit - 76 
 
 Transit attachments and their adjustments 77, 78 
 
 The solar attachment 79, 80 
 
 Engraving of solar attachment (Gurley's) 81 
 
 The adjustment of the solar attachment 83-85 
 
 To find the latitude by means of the solar attachment 85 
 
 To run lines with the solar attachment 87 
 
 Saegmuller's solar attachment and its adjustments 88 
 
 Engraving of Saegmuller's attachment 89 
 
 Latitude by means of Saegmuller's attachment 91 
 
 Latitude by a circumpolar star 91 
 
 Table of refraction 92 
 
 To find the meridian and declination of the needle, using the attach- 
 ment 93 
 
 Time and azimuth with the solar attachment 94 
 
 Error in declination or latitude causing an error in azimuth 94 
 
 Table of errors in azimuth for one-minute error in latitude or declina- 
 tion 95 
 
 The stadia or micrometer 95, 96 
 
 Formula when stadia is perpendicular to level line of sight 97 
 
 Formula when line of sight is not level 99 
 
 Examples 100 
 
 Gradienter 100, 101 
 
 SECTION II. 
 BEARINGS WITH COMPASS, AND ANGLES WITH TRANSIT. 
 
 To obtain the bearing of a line 102 
 
 Reverse bearing ; local attraction 103
 
 CONTENTS. Xi 
 
 PAGE 
 
 Proof -bearings and tests of accuracy 104 
 
 Suggestions and field exercises 105 
 
 To measure angles with the transit, by repetition 106 
 
 " " " " by series ; remark 107 
 
 Deflection angle ; traversing, or surveying by the back angle 108-110 
 
 To traverse a road or a small stream Ill 
 
 Problems on bearings and angles 112-115 
 
 To change the bearings of the sides of a survey 116 
 
 Examples and field exercises 117-119 
 
 SECTION III. 
 PROBLEMS ON PERPENDICULARS AND PARALLELS. 
 
 To let fall a perpendicular from a given point to a line 120, 121 
 
 To prolong a line past an obstacle (various methods) 121-124 
 
 To measure a line past an obstacle (various methods) 124-127 
 
 Examples and field exercises 127, 128 
 
 SECTION IV. 
 HEIGHTS AND DISTANCES. 
 
 Measurement of accessible heights , 129 
 
 Examples 130 
 
 Measurement of inaccessible heights 130-132 
 
 Examples 132, 133 
 
 Inaccessible distances 135 
 
 The three-point problem 136-138 
 
 Miscellaneous problems and field exercises 140,141 
 
 SECTION V. 
 RECORDING THE FIELD NOTES. 
 
 Tabular form 142 
 
 By sketch 142-144 
 
 In columns 147-149 
 
 Large plot of a survey 153 
 
 SECTION VI. 
 LATITUDES AND DEPARTURES. 
 
 Definition of latitude and departure 154 
 
 Formulas and examples on latitudes and departures 155 
 
 The traverse table and explanation of the same 156, 157
 
 Xii CONTENTS. 
 
 PAGB 
 
 The use of the traverse table exemplified 157, 158 
 
 Table of sines and cosines employed ; examples 159 
 
 Testing a survey 160 
 
 Correcting latitudes and departures 161 
 
 " " " " sides weighted 165 
 
 SECTION VII. 
 SUPPLYING OMISSIONS. 
 
 General observations on supplying omissions 166 
 
 Determination of one side 167 
 
 " " the length of two sides 169 
 
 " " bearing of two sides 172 
 
 " " bearing of one side and length of another 174 
 
 SECTION Vin. 
 PLOTTING A COMPASS OR TRANSIT SURVEY. 
 
 Using the protractor 178, 179 
 
 By latitudes and departures 180, 181 
 
 Using cross-section paper 182 
 
 SECTION IX. 
 ON DETERMINING AREAS. 
 
 Determining the area of triangles and parallelograms 183 
 
 " " trapezoids and trapeziums 184 
 
 Examples 185 
 
 Determining the area of polygons 186 
 
 General method for determining the area of any rectilinear figure. 187-190 
 
 Examples 191-193 
 
 Method by total latitudes and departures 193, 194 
 
 Examples 194-196 
 
 Determining areas when offsets are taken ... 196, 197 
 
 Field exercises 197, 198 
 
 CHAPTER III. 
 
 DECLINATION OF THE NEEDLE, OR VARIATION 
 OF THE COMPASS. 
 
 Irregular changes 199 
 
 The diurnal variation . . 200
 
 CONTENTS. Xiii 
 
 PAGE 
 
 Table for reducing observed declinations 201 
 
 The secular variation 201 
 
 Line of no declination, or agonic line .201, 202 
 
 Formulas expressing the magnetic declination 203 
 
 Tables exhibiting annual variation, etc 206 
 
 Effects of the secular change 208 
 
 Rule for obtaining bearings when variation is known 209 
 
 Change determined by old lines 210 
 
 Examples 211, 212 
 
 To obtain the true bearing of a line 212 
 
 To determine the declination, by Polaris 214 
 
 Tables of elongation and azimuth of Polaris 216, 217 
 
 To establish a true meridian with a transit 218 
 
 To obtain approximately the meridian , . . . . 219 
 
 CHAPTER IV. 
 
 LAYING OUT AND DIVIDING LAND. 
 SECTION I. 
 
 LAYING OUT LAND. 
 
 Triangles ; examples 221-223 
 
 Squares 223 
 
 Rectangles ; examples 224, 225 
 
 Parallelograms ; examples 225, 226 
 
 Polygons ; examples 227, 228 
 
 Circles and ellipses ; examples 228-231 
 
 Additional problems 232 
 
 SECTION II. 
 DIVIDING LAND. 
 
 Triangles, various cases ; numerous examples 232-240 
 
 Trapezoids, " " 241-246 
 
 Trapeziums, " " 246-252 
 
 Polygons, " " 253-258 
 
 Irregular boundary 258 
 
 Straightening boundary lines 258 
 
 Miscellaneous examples 259-261
 
 xiv CONTENTS. 
 
 CHAPTER V. 
 PLASE TABLE SURVEYING. 
 
 PAGE 
 
 The plane table ................................................ 262 
 
 Engraving of plane table ........................................ 263 
 
 The alidade and declinator ...................................... 265 
 
 Adjustments ................................................... 266 
 
 Surveying by radiation, progression ... ....................... 267, 268 
 
 " " intersection, resection .............................. 269 
 
 The three-point problem ......................................... 270 
 
 Bessel's method by inscribed quadrilateral ---- . ................... 270 
 
 The two-point problem .............................. ............ 271 
 
 Practical suggestions ............................................ 273 
 
 Field exercises .... ................................ . ......... 274 
 
 CHAPTER VI. 
 THE SURVEY OF THE PUBLIC LANDS. 
 
 The solar compass .............................................. 275 
 
 The latitude and declination arcs ................................. 276 
 
 Engraving of solar compass ...................................... 277 
 
 The hour arc and polar axis ....................................... 279 
 
 Principles of the solar compass ........................ ' ........... 280 
 
 Allowance for declination ............................... ....... 281 
 
 To adjust the solar compass .................................. 283-285 
 
 To use the solar compass ....................................... 285 
 
 To set off the declination ..................................... 285, 286 
 
 Allowance for refraction ........... ............................ 287 
 
 Table of refractions in declination ............................. 288, 289 
 
 Explanation of the table of refractions ........................... 290 
 
 Problems in declination and refraction ........................ 290, 291 
 
 To set off the latitude ........................... ............... 291 
 
 To run lines with the solar compass .............................. 292 
 
 Allowance for the earth's curvature .................... ......... 293 
 
 Time of day by the sun ... ....................................... 294 
 
 Caution as to the false image ................................... 294 
 
 Approximate bearings, and time for using solar compass ............ 295 
 
 Origin of the system for the survey of the public lands ............. 295 
 
 Plan of a township, showing sections ........... , ................. 2%
 
 CONTENTS. XV 
 
 PAGE 
 
 System of rectangular surveying 297 
 
 Conformity to the meridian required 298 
 
 How to survey section lines ; plan of township 299 
 
 An instrument operating independently of needle required 300 
 
 The four-pole chain to be adjusted to 66.06 feet 301 
 
 Process of chaining 302 
 
 Marking lines- 303 
 
 Obstacles in line ; witness points 304 
 
 Establishing corners 304 
 
 Miscellaneous directions 305, 306 
 
 Quarter-section corners ; witness corner 307 
 
 Meandering ., 307 
 
 Base line 308 
 
 Principal meridian ; standard parallels 809 
 
 Auxiliary meridians ; township lines 310 
 
 Method of subdividing townships 311, 313 
 
 Subdivision of sections ." 314-316 
 
 Re-establishment of lost corners 316 
 
 Objects required to be noted 316, 317 
 
 Specimen field notes 318-321 
 
 Inclination of the meridian 322, 323 
 
 Convergency of meridians ; deflection of range-lines 324 
 
 Table of inclination and convergency of meridians 325 
 
 CHAPTER VII. 
 
 CITY SURVEYING. 
 
 Introduction 326, 327 
 
 SECTION I. 
 FIELD INSTRUMENTS, THEIR ADJUSTMENTS AND GENERAL USE. 
 
 Transit, engraving of, showing gradienter, etc .328, 329 
 
 Rods, steel tapes, etc. 331, 332 
 
 Locke's hand-level 333 
 
 Effects of temperature, sag, and wind considered 334 
 
 Corrections for these ; illustrations 335, 336 
 
 Suggestions on measurement 336 
 
 The Y-level 337 
 
 Engravings of Y-level 338,339 
 
 Adjustments of the level 340-343
 
 XVi CONTENTS. 
 
 PAGE 
 
 To use the level 844 
 
 Engravings of New York and Philadelphia levelling-rods 345 
 
 Description of " " " " " 347,348 
 
 The rod-level ; engraving of same 349 
 
 Levelling defined 349, 350 
 
 Curvature and refraction considered 350, 351 
 
 Exercises ; levelling by fore and back sights 352, 353 
 
 Levelling, general method 354, 355 
 
 Form of record ; observations on bench marks 356 
 
 General method of running a grade line 357 
 
 Office instruments 358 
 
 SECTION II. 
 
 A. FIELD WORK. 
 
 Public work ; street lines ; direction and width of streets 358, 359 
 
 The alignment, how made 360 
 
 Preservation of points in line 361 
 
 Street grades ; the profile 362, 363 
 
 llecord of levels for profile 365 
 
 Marking of lines and grades 366 
 
 Private work 366 
 
 Diagram of lot and block 367 
 
 Locating lot corners from descriptions in deed 369 
 
 How to proceed when obstacles intervene 370 
 
 B. OFFICE WORK. 
 
 Public work ; plans, profiles ; exercise 372, 373 
 
 Determination of grades , 374 
 
 Diagram 375 
 
 Drainage 377 
 
 Private work 378 
 
 Conclusion ; books 378, 379 
 
 CHAPTER VIII. 
 
 MINE SURVEYING. 
 
 Its purpose 380 
 
 Instruments employed in making the alignment 381, 382 
 
 Stations, where made, and how designated 382 
 
 Angles ; initial course ; reduced courses , . . . 382
 
 CONTENTS. XVli 
 
 PAGE 
 
 Curves, how laid out 383, 384 
 
 The record 385 
 
 Mapping and plotting ; inside and outside work 385, 386 
 
 The use of the level and rod 387, 388 
 
 The use of the vertical circle 388, 389 
 
 Transference of points from surface to bottom of shaft 389 
 
 Special appliances for taking courses on pitches at high angles 390 
 
 The hanging compass and hanging clinometer 390, 391 
 
 Topography, its use, how taken 391, 393 
 
 Employment of the plane table 393 
 
 Contour lines 393 
 
 Angular cross-sectioning 394 
 
 Transit as first made' in 1831 .. , 396 
 
 APPENDIX. 
 The judicial functions of surveyors 897-411 
 
 TABLES. 
 
 Logarithms of numbers 1-19 
 
 Approximate equation of time 20 
 
 Logarithms of trigonometric functions 21-49 
 
 For determining with greater accuracy than by the preceding 50, 51 
 
 Lengths of degrees of latitude and longitude 52 
 
 Miscellaneous formulas, and equivalents of metres, chains, and feet, 53 
 
 Traverse 54-61 
 
 Natural sines and cosines 62-70 
 
 Natural tangents and cotangents 71-79 
 
 For tables in the body of the book, see " Tables " in Index.
 
 SURVEYING. 
 
 DEFINITIONS, AND DIVISION OP THE SUBJECT. 
 
 1. Surveying is the art of determining and delineating the 
 relative position of points upon the surface of the earth. It 
 consists principally in measuring, laying out, and dividing land ; 
 in establishing lost positions ; in the measurement of heights 
 and distances ; and in the graphical representation of the pecu- 
 liarities of any part of the earth's surface. 
 
 2. It mav be divided into two parts : PLANE SURVEYING and 
 GEODETIC SURVEYING. 
 
 In Plane Surveying the spherical form of the earth is neg- 
 lected ; in other words, the portion of the earth included in the 
 survey is regarded as a horizontal plane. This may be done 
 without sensible error where, as in ordinary land surveying, the 
 operations are limited to surfaces of small extent. 
 
 In Geodetic Surveying the shape of the earth is regarded, 
 since the surfaces under consideration are so extensive, as in 
 the United States Coast and Geodetic Surveys, sensible errors 
 would otherwise arise. 
 
 RKMARK. The spherical excess of a spherical triangle, each 
 of whose sides is one mile, is less than six-thousandths of a 
 second. The excess amounts to only one second for an area of 
 75.5 square miles, each side of the equilateral triangle being 
 then about 13 miles.
 
 2 SURVEYING. 
 
 3. In the following pages Plane Surveying only will be con- 
 sidered, and the subject treated under the following heads : 
 
 CHAIN SURVEYING. 
 
 COMPASS AND TRANSIT SURVEYING. 
 
 PLANE TABLE SURVEYING. 
 
 GOVERNMENT SURVEYING 
 
 CITY SURVEYING. 
 
 MINE SURVEYING. 
 
 In Plane Surveying there are usually three operations : 
 
 1. The Field Work. 
 
 2. The Graphical Representation, or Plot 
 
 3. The Computation.
 
 CHAPTER I. 
 
 CHAIN SUKVEYING, 
 
 SECTION I. 
 
 INSTRUMENTS. 
 
 4. Chain Surveying has chiefly for its object the determina- 
 tion of areas from data obtained by direct measurement of 
 distances between points. The instruments needed are there- 
 fore simply those for measuring lines. 
 
 5. Gunter's Chain, so called from its inventor, is generally 
 used for this purpose. It is made of iron or steel wire, is 66 
 feet in length, and divided into 100 links, so that each link, with 
 half the rings connecting it with the adjoining links, is seven 
 and ninety-two hundredths inches (7.92), or one-hundredth of a 
 chain. Swivels are inserted to keep it from twisting, and every 
 tenth link has a metallic mark attached, so that the number of 
 tens from either end is readily ascertained. Its advantages in 
 surveying farms or fields are apparent : there being 4840 square 
 yards in an acre, and the chain 22 yards long, a square chain 
 will contain one-tenth of an acre ; or, there being 10,000 square 
 links in a square chain, which is one-tenth of an acre, 100,000 
 square links are equivalent to an acre. Hence, if the area of a 
 field is calculated in links, the area is at once shown in acres, 
 by cutting off the last five figures. If the area is found in 
 chains, then since there are ten square chains in an acre, the 
 area is given in acres by cutting off -the last figure. 
 
 6. A Two-Pole, or Half-Chain is sometimes used instead of 
 Gunter's Chain. It is quite convenient for measuring lines 
 where the ground is rough and hilly.
 
 4 PLANE SURVEYING. 
 
 7. The Engineer's Chain is used in surveying railroads and 
 canals, and generally where extensive line surveys are being 
 conducted ; hence not unfrequently it is employed in connection 
 with these surveys, as well as otherwise, in determining areas. 
 It is 100 feet in length, and is divided into 100 links, every 
 tenth link being marked by a piece of brass, as in the four-pole 
 chain. 
 
 8. The Tape Measure is very convenient for taking offsets 
 in a survey, for measuring the boundaries of city lots, cross- 
 sectioning in railroad work, etc. Tapes are "metallic," or 
 steel, and made of various lengths,* 50 feet or 100 feet are 
 commonl}" used, and divided into feet and inches, or feet and 
 tenths of a foot. The latter graduation is preferable for the 
 railroad engineer, and the former for the city engineer. 
 
 9. Eleven Marking-Pins, 12 or 14 inches long, one of which 
 is made of brass, the others of No. 4 iron wire or No. 6 steel, 
 all pointed at one end and formed into a ring at the other, are 
 used in chaining. 
 
 10. Straight Poles about 8 feet long, shod at the bottom with 
 a conical shoe, point down, and painted alternately red and 
 white in foot-width bands, are used to indicate the direction of 
 the line which is being measured, or the position of points to be 
 located.f 
 
 SECTION II. 
 
 A. CHAINING. 
 
 11. Two men are required, a " leader " and a " follower," or 
 head and hind chainman. The chain is first thrown out in the 
 general direction of the line which it is desired to measure, and 
 
 * Steel tapes 1000 feet in length have been frequently used for special 
 purposes. See Mine Surveying, p. 380. 
 t See Article 383.
 
 CHAINING. 5 
 
 examined carefully to see if there are any kinks in it, or bends 
 in the links ; the leader having the inarking-pins in one hand 
 takes hold of the forward end of the chain with the other, and 
 moves on as nearly as he may judge in the direction of the line ; 
 the follower places the rear end of the chain at the station 
 whence the line is to be measured, directs the leader by signals 
 as he approaches the chain's length to get in line, and then calls, 
 "halt"; then the chain must be drawn taut and straight, and 
 the follower having his end of the chain precisely at the starting- 
 point, calls out, "down"; the leader then thrusts one of the 
 iron marking-pins into the ground exactly at the end of the 
 chain and calls out, "down," which is the signal to the follower 
 to advance : proceeding as before until the second length of 
 chain is measured, which is indicated by the follower coming to 
 the pin set in the ground b^y the leader, when the follower cries, 
 "halt," and after placing his end of the chain at the pin, the 
 chain having been drawn taut and straight as before, calls, 
 "down" ; the leader, as before, leaving a pin to mark the end 
 of the chain, repeats, "down" ; the follower then takes up the 
 pin first placed by the leader, and moves on ; thus the party 
 proceeds until the end of the line is reached, the leader placing 
 the pins at his end of the chain, and the follower picking them 
 up at his end. 
 
 If the line ends with less than the length of the chain, the 
 leader places his end at the point which marks the extremity of 
 the line, calls out, "down"; the follower then reads off the 
 number of links between the last pin and the end of the line. 
 The number of whole chain's length of the line is shown by the 
 pins in the hands of the follower, and the number of links 
 counted off added thereto will give the total length in chains 
 and links. 
 
 12. Tally. If the line exceeds eleven chains in length, a 
 transfer of pins from the hind chainman to the head chain man 
 is necessary ; this is called tallying, and is performed in the 
 following manner: At the end of the eleventh chain, the brass
 
 6 PLANE SURVEYING. 
 
 pin the last pin left in the hands of the leader is placed, 
 when he call out "tally"; at this signal the follower drops 
 his end of the chain, advances to the leader, counts over with 
 him the ten iron pins which he has gathered up, and transfers 
 them to the leader, who then withdraws the brass pin, sets an 
 iron one in its place, and the measuring is continued as before.* 
 Each tally should be recorded, especially when chaining very 
 long distances, to avoid error in the final count, t It is obvious 
 that the total length of the line will be equal to the chains and 
 links as indicated above, plus the number of tens shown by the 
 tallies. 
 
 13. The surveyor should guard against error in chaining, by 
 frequently testing his chain, to see that it is of the proper 
 length, if it has been stretched, make a file mark showing its 
 true length, and when in use, see that it is drawn straight, 
 that the forward chainman sticks the pin in line exactly at the 
 end of the chain, or at the mark indicating its true length, and 
 as nearly vertical as possible ; \ and when obtaining the number 
 of links at the end of the line, see that they are not counted 
 
 * Some surveyors use only ten marking-pins, and tally by marking the 
 end of the eleventh chain with a pencil, the finger, or a scratch on the 
 ground, and when the ten pins are transferred to the leader, one of them 
 is thrust in the place thus indicated, and the work is continued as before. 
 
 t In chaining long distances where there are several tallies, the leader 
 and follower may, at each tally, change places, and thereby lessen the 
 liability to error in the final count. See Articles 352, 353. 
 
 } " It has been found by many trials with as good men as can generally 
 be obtained, that with two sets of chainmen instructed alike in the proper 
 manner of keeping their chain level and straight on the line, and of setting 
 the tally pins plumb, as well as holding the ends of the chain to them, 
 a difference has sometimes been made of 36 links, and an average differ- 
 ence of 15 or 16 links to a mile in common timbered land." Hurt, " Gov- 
 ernment Surveying," p. 35. 
 
 The surveyor should have laid down by means of a standard steel tape 
 or otherwise, in a convenient place, and between permanent marks in the 
 ground or on the floor of a large hall, the exact length of a standard chain 
 by which he could test his chain from time to time.
 
 CHAINING. 7 
 
 from the wrong end of the chain, nor the wrong way from the 
 brass mark. 
 
 The pull on the chain, when in use, has a tendency to in- 
 crease its length ; and moreover, since there are a great number 
 of wearing surfaces, if each of these be worn by an extremely 
 small amount, the chain will be considerably elongated. 
 
 In either the surveyor's or engineer's chain there are two 
 small links which connect with the two pieces of wire which 
 form the principal part of what is called the 'link of the chain, 
 thus giving six wearing surfaces to every link ; therefore, if 
 each of these surfaces wears only .005 of an inch, the chain 
 will be increased in length three inches, so that in measuring 
 only a quarter of a mile with a four-pole chain, the error from 
 this cause alone would be Jive feet,* making an error in area 
 of about 4.9 acres in a tract one mile square. This stretching 
 of the chain is partially compensated by the difficulty, and 
 often impracticability, of drawing the chain precisely straight; 
 and so long as the chain is not elongated beyond one-tenth 
 or one-twelfth of one per cent of its length, it may be relied 
 on for accurate work.f 
 
 The true length of a line which has been measured by a chain 
 stretched beyond the standard length may be found from the 
 proportion : 
 
 The length of standard chain : the length of chain used 
 : : the distance measured : the true distance. 
 
 * Tliis error, it is perceived, increases directly with the number of appli- 
 cations of the chain : it is called cumulative. The error arising from erro- 
 neous setting of the pin is termed compensative, that is, it is as likely to be 
 additive as subtractive, and it is shown by the Method of Least Squares, 
 that for this class of errors the square root of the number of errors are 
 probably not compensated. If the error in setting is one inch, in chaining 
 a mile with a Gunter's chain, the probable error would be VbO = about 
 9 inches. 
 
 t To remove the difficulty of drawing the chain perfectly straight, the 
 instructions issued from the United States Land Office, 1880, to Govern 
 ment Surveyors-General, states that the 66 feet chain must be 66.06 feet. 
 See p. 301.
 
 8 PLANE SURVEYING. 
 
 For example, if, with a chain stretched one link over the 
 standard, a line be measured for 2000 feet, we should have 
 
 100 : 101 = 2000 : 2020, the true distance. 
 
 In like manner, for the area of a tract measured with a 
 stretched chain : 
 
 The square of the length of the standard chain 
 : the square of the length of the chain used 
 : : the computed area 
 : the true area. 
 
 If the chain was stretched one link, as in the above example, 
 and the area computed therefrom 20 acres, we should have 
 
 100 2 : 101 2 = 200 sq. chs. : 204.02 sq. chs. for the true area 
 = |$f of the computed area, nearly. 
 
 In general, if A = true area, A l = computed area, L length 
 of chain, and d-L = error in its length (always small). Then 
 A : A, = (L dL) 2 : L\ 
 
 Reducing and rejecting d 2 as inconsiderable, there results 
 A = (l2d)A 1 ; or, the correction to be applied to obtain the 
 true area = 2 dA^. 
 
 This correction is additive when the chain is too long, which 
 is the usual case, and subtractive when the chain is too short. 
 
 14. The surfaces to be measured are in general uneven and 
 broken, not plane ; but however great the inequalities, the area 
 of a tract is considered to be that part of the horizontal plane 
 which is intercepted by vertical planes through its boundaries.* 
 The horizontal distance is therefore required ; hence, when the 
 
 * A vertical line is a line directed to the centre of the earth, or it is a 
 line having a plummet freely suspended to it, and at a state of rest ; a 
 plumb line. 
 
 A vertical plane is a plane embracing a vertical line. 
 
 A horizontal line is a line perpendicular to a vertical line. 
 
 A horizontal plane is a plane perpendicular to a vertical line.
 
 CHAINING. 9 
 
 ground slopes, it is necessary to raise the down-hill end of the 
 chain. If the slope is considerable, only a part of the chain 
 should be used. For example, to measure from L down to JV, 
 the follower holds one end of the chain at L, while the leader, 
 stretching the other towards N, takes as much of it as he can 
 raise to a horizontal position &, and, holding a plummet there, 
 fixes the point c ; the follower, who is now signalled to come for- 
 ward, places at c that point in the chain whence the plummet 
 was suspended to fix c, while the leader advances and, using 
 as much of the chain as possible, locates e, and so on : when 
 the end of the chain is reached, a pin should be transferred 
 
 from the leader to the follower. Where great accuracy is not 
 required, a marking-pin or pebble may be dropped to indicate 
 the points c, e, etc.* To measure up hill from JVto L is less 
 accurate, on account of the difficulty experienced by the fol- 
 lower in holding his end of the chain at the points A,/, d, etc., 
 over their counterparts, i, g, e, etc. 
 
 When chaining steep hills, especially if through a wood or 
 over rough, rocky ground, the work may be greatly facilitated 
 by an extra chain man. He may assist in getting line, straighten- 
 ing the chain, noting the points c, e, etc., marked by the plumb 
 bob, and other duties. f 
 
 * If in connection with the chain a survey is being made with an 
 instrument for measuring angles, vertical and horizontal, the inclina- 
 tion of a slope may be observed, and the length of it measured ; then 
 the horizontal distance required will be equal to the measured distance 
 multiplied by the natural cosine of the angle of inclination. 
 
 t For extreme accuracy in measuring lines, see Chapter VII. Article
 
 10 PLANE SURVEYING. 
 
 EXERCISES. 
 
 1. Set two marks on gently undulating ground and about 
 1000 feet apart, and measure forward and back between these 
 points several times ; the same party once at least each way. 
 
 2. The same between points on hilly and, if possible, bush 
 land. 
 
 3. Chain down a steep hill, and chain up between the same 
 points. 
 
 B. RANGING OUT LINES. 
 
 15. If m chaining any line, as LN, from L toward N, a rod 
 at N can be constantly seen by the rear chainman, he can keep 
 
 the leader in line by ranging him with 
 L N the flagstaff at N. If, however, a 
 
 hill intervenes, a valley, or brush or 
 
 woodland interferring with the alignment, then the line must be 
 first ranged out or points determined in it before the chaining 
 can be performed. 
 
 16. Ranging out a Line. To range out a line requires three 
 persons, each having a rod eight or ten feet long, and a plum- 
 met to indicate when his rod is vertical. Calling these men 
 A, -B, and (7, and supposing A and B in the line, C goes for- 
 ward, and sighting back to A and B, puts his rod in line ; A 
 then advances beyond C and sets his rod in line with C and B ; 
 next B advances and places his rod in line with C and A, and 
 so on the line may be extended any desired length. If, as 
 frequently is the case, one of the party has had more experi- 
 ence or is naturally better qualified for sighting a line, the best 
 results would be obtained by such an one setting all the rods ; 
 for examp\e, C would place his rod in line, then call up A, to 
 whom he would turn over the rod just set, and go forward to 
 line the next ; after which call up B, exchange rods with him, 
 and so on.
 
 RANGING OUT LINES. 
 
 11 
 
 17. Over a Hill. To fix points in a line over a hill, both 
 ends of which are visible from points near the summit, proceed 
 as follows : 
 
 .17 
 
 Place a flagstaff at L, another at N. A man at E 1 signals 
 one at Z>'.in line with L ; D' then directs E' to E" in line with 
 jV; and so on alternately, until the men are at D and E in the 
 line LN. 
 
 18. Across a Valley. To locate points in a line, the ends 
 of which may be seen from each other, but which are separated 
 by a wide, deep valley. 
 
 Fix a point C in line with LN; then a man holding a plumb 
 line at C, and sighting N can direct the setting of the stakes 
 D, E, F, and others.
 
 12 PLANE SURVEYING. 
 
 19. Through a Wood. In chaining through a wood or thick 
 brush land, where the ends cannot be seen from each other, a 
 line * is measured as nearly as may be in the direction of the 
 desired line, and stakes driven every two or three chains, or 
 oftener if necessary. When the end of the line is reached, the 
 distance to the corner is measured, and, b}* proportion, the 
 amount to move each stake to bring it into line is determined. 
 
 For example, let LNbe the true line, and LN 1 the measured 
 line ; c, d, e, etc., points three chains apart. Now, if the length 
 LN' equals 17.40 chains, and NN 1 measured at right angles to 
 LN* = 35 links, LN-\ will equal 
 
 and . LN' (1740 links) :NN' (35 links) 
 
 = Lg (1500 links) : gG (30 links) ; 
 
 or 30 links from g at right angles to LN' will indicate the posi- 
 tion of G, a point in the true line LN. 
 
 1740 : 35 = 1200 : 24, the distance fF, 
 
 1740 : 35 = 900 : 18, the distance eE ; 
 and so on. 
 
 Or, after finding the first distance to set off, either gG or c<7, 
 the others are readily obtained by taking a proportional part 
 of this distance, shown by the several divisions of the line thus : 
 gG represents the fifth division, fF the fourth, eE the third, 
 and so on ; hence, if gG is 30 links, fF will be \ of 30, or 24, 
 
 * Called a random line or trial line. 
 
 t If the distance NN' is a s nail per cent of the total length of the line, 
 the shortest distance between the ends of the lines may be taken for NN', 
 and the length of the measured line for that of the true line. See Article 
 177.
 
 SETTING OFF PERPENDICULARS. 13 
 
 links ; eE, f of 30, or 18 ; (W, f of 30, or 12 ; and c(7, | of 
 30, or 6 links. 
 
 EXERCISES. 
 
 1. Let each student range out a line of several hundred feet, 
 setting all the poles forward, and hack again to the starting- 
 point, and on different kinds of ground, undulating, hilly, and 
 bushy. 
 
 2. Measure a line through a wood or where the ends are not 
 visible from each other. Set stakes, as indicated in Article 19, 
 in the true line 200 feet apart. See how near these stakes are 
 placed in line by ranging. 
 
 C. SETTING OFF PERPENDICULARS. 
 
 20. To erect a perpendicular at a giiven point in a line. 
 
 Let MN be the given line, and P the point at which it is 
 desired to erect a perpen- 
 dicular. Since a triangle 
 formed of the sides 3,4, and 
 5, or any multiple of these, 
 will contain a right angle, 
 we may take parts of a chain 
 representing these distances M- 
 or multiples, having the an- 
 gle made by the shorter sides at P, and set off a perpendicular 
 to a given line, thus: Fasten one end of the chain at K, 30 
 links from P, the end of the ninetieth link at P; then when 
 both parts of the chain are drawn straight by a pull at the 
 fiftieth link, the end of that link will indicate the point which 
 if connected with P will give the perpendicular required. 
 
 21. If the perpendicular is to be of considerable length, then 
 a greater length than PO = 40 links should be used, and the 
 following method would be better : Fasten one end of the chain 
 at P, and with the eightieth link describe an arc be ; measure
 
 14 
 
 PLANE SURVEYING. 
 
 PK= 60 links, and with K as a centre, and with a radius = 100 
 links, the whole length of the chain, describe another arc de ; 
 the intersection of these arcs will give the point required. 
 
 22. Another Method. With the whole length of the chain as 
 a radius, and P as a centre, describe an arc ab ; locate K a 
 chain from P, and with the same radius, and with a centre K^ 
 
 O 
 
 K 
 
 describe an arc cd cutting ab in Q ; extend KQ to 0, so that 
 OQ = QK, then will OP be the perpendicular to the line MN 
 at the point P. Why? 
 
 23. To let drop a perpendicular on a line from a 
 without the line. 
 
 P' 
 
 !;iven point 
 
 First, When the point is accessible. 
 
 Let MN represent the line, and P the point. 
 
 With a length
 
 SETTING OFF PERPENDICULARS. 
 
 15 
 
 of chain somewhat greater than PO, describe an arc cutting 
 MN in the points R and K. With centres R and K, and any 
 radius greater than the half of RK, describe arcs intersecting 
 in Q. A line drawn from P to in the direction of Q will be 
 the perpendicular required. 
 
 If the point is at P 1 at or nearly opposite one end of the line, 
 extend the line if it be possible to N 1 until a sufficient distance 
 is obtained to describe the arcs required. 
 
 24. Or if it is impracticable to prolong the line, as in the 
 figure, where a pond of water prevents, proceed as follows : 
 
 Extend the chain or any convenient portion of it from P to 
 any point R in the line NO. Fix the middle point of RP, as 
 M, and with this as a centre, and a radius 3fP, or its equal MR, 
 describe an arc cutting the given line in 0. Join PO for the 
 perpendicular required.* 
 
 25. Second, When the point is inaccessible. 
 
 Let P be the given point, and LN the line. At any conven- 
 ient point Q in the line LN erect the perpendiculars QO and 
 QR of equal length. Locate Fin the line PO and T in the 
 line RP ; then if a point S be found at the intersection of the 
 
 * The angle ROP is measured by one-half a semi-circumference, and is 
 therefore a right angle. 

 
 16 
 
 PLANE SURVEYING. 
 
 prolongation VR and OT, and a point M be located in ZJVand 
 SP, a line joining M and P will be the perpendicular sought. 
 Why? 
 
 26. Optical Square. To set off perpendiculars from a 
 line, an instrument called the optical square may be used. It 
 is a small cylindrical box containing a mirror, from the upper 
 half of which the silvering is removed. The glass is placed so 
 as to make half a right angle with the line of sight, hence two 
 objects seen in it, the one by direct vision, and the other by 
 reflection, subtend at the point of observation a right angle. 
 
 Or the surveyor's cross, which is simply two pairs of sights 
 set at right angles to each other, and supported upon a staff.* 
 
 D. RUNNING PARALLELS. 
 
 27. Through a given point to run a parallel to a given line, 
 the point and line both being accessible. 
 
 * While these instruments may be employed in chain surveying, 
 neither of them is used in the ordinary practice of a surveyor, as perpen- 
 diculars are expeditiously set off by means of the compass or transit.
 
 RUNNING PARALLELS. 17 
 
 Let LN represent the line, and P the point. Let drop a 
 perpendicular PO, and at some other point K\ erect a perpen- 
 
 L K N 
 
 dicular KR = PO. A line drawn through P and R will be the 
 parallel required. 
 
 28. Otherwise. From any point O in LN run an oblique 
 line to the point P. Through any point R in PO measure a 
 
 N 
 
 Q 
 
 line MQ, so that RQ = ~j~* - A line passing through 
 
 PQ will be the parallel required. 
 
 If R be taken at the middle point of OP, and QR be made 
 equal to MR, the direction of the parallel PQ would be shown 
 at once. 
 
 B. OBSTACLES TO ALIGNMENT. 
 
 29. To prolong a line when an obstacle, as a tree or building, 
 
 prevents direct sighting, we may proceed as follows : 
 
 M PR S 
 
 By Perpendiculars. Let LN be the line which it is desired 
 to prolong past a building B. At two points and N in the
 
 18 PLANE SURVEYING. 
 
 line, set off equal perpendiculars NP and OJf, of such length 
 that a line MP through these may be extended past the 
 obstacle to some point S. At R and S set off perpendiculars 
 to X and F, of the same length as before, at and N, and join 
 XF; it will be the prolongation of LN. 
 
 30. Otherwise : by Equilateral Triangles. On LN, the line 
 to be prolonged, take a distance ON as a base, and construct 
 on it an equilateral triangle NOP; extend the side OP to some 
 
 point Q. Describe an equilateral triangle QRS, and prolong 
 the side QR to F, making QF= QO; finally the construction 
 of the equilateral triangle VXY will give XT the direction 
 sought. 
 
 P. OBSTACLES TO MEASUREMENT. 
 
 31. a. When Both Ends of the Line are Accessible. 
 
 By Perpendiculars. For example, if it is desired to measure 
 one side of a field or farm where a fence, hedge, or bushes 
 prevent chaining on the line, set off perpendiculars, and measure 
 the parallel line. 
 
 Let LN represent a line which, on account of fence and brush, 
 it is impracticable to make the measurement exactly on the line
 
 OBSTACLES TO MEASUREMENT. 
 
 19 
 
 Erect at Z, and N perpendiculars LI and Jtfw, of equal and suffi- 
 cient length so. that a line connecting I and n will clear the 
 obstruction. Measure In ; it will be the length of the required 
 line. 
 
 32. b. When One End is Inaccessible. 
 
 By Symmetrical Triangles. Suppose LP the line, P the 
 inaccessible end, visible, but on the opposite bank of a river. 
 Measure from any 
 point JVnear the river, 
 in a direction diverg- 
 ing from its bank to -r 
 R, making NI=IR. 
 Through any other 
 point Jf, in the line 
 LN, measure through 
 / to K, so MI= IK. 
 If now a point be 
 found in the prolonga- 
 tion of RK, and in 
 line with / and P, RO may be measured and taken for their 
 distance NP.* 
 
 33- Otherwise. Measure from the line the perpendicular 
 LP; erect at P a perpendicular to 
 PN, and extend it to a point M in the 
 prolongation of LN. Measure 
 then the proportion 
 
 ML:LP=LP:LN 
 PL" 
 ML 
 
 gives 
 
 LN: 
 
 " The student will show that ROI and NIP are symmetrical triangles, 
 and NP and RO are homologous.
 
 20 
 
 PLANE SURVEYING. 
 
 34. c. When Both Ends are Inaccessible. 
 
 By Symmetrical Triangles. Let LN be the line, the length of 
 which it is required to determine. Take any point P, measure 
 PO and PM, and find by one of the preceding methods OL, 
 
 MN, and hence, the total length of PL and PN. Now take 
 points R and Q in the lines PL and PN respectively, so that 
 PR : PQ = PL : PN, and measure RQ ; then the required line 
 LN may be calculated by the proportion PQ : PN= RQ : LN. 
 
 G. MEASUREMENT OF HEIGHTS. 
 
 35. To measure the height of a tree or a flag-staff. Let BC 
 represent the height required. At a point D set up a staff of 
 a known height so that, with the eye at -4, 
 C and E will be in line of sight ; measure 
 AD and DB; then the similar triangles 
 ADE and ABC give the proportion 
 
 AD : DE = AB : BC. 
 
 nn _DExAB 
 AD 
 
 Whence
 
 EXERCISES. 21 
 
 EXAMPLES. 
 
 1. If the height of a staff is 4 feet, and the distance from it 
 to a tree = 80 feet, AD being 4^ feet, what is the height of 
 the tree? Ans. 77|| feet. 
 
 QUERIES. If the height of the staff is equal to AD, the 
 length of neither being known, simply the distance AB given, 
 could the height of the tree be ascertained ? 
 
 If the ratio of the height of the staff to AD is known, but 
 not the absolute length, could the required height be found by 
 simply measuring AB1 
 
 Is this method applicable on other than horizontal ground ? 
 
 2. A liberty pole, whose height was 90 feet, standing on a 
 horizontal plane, was broken off, and the extremity of the top 
 struck the ground 28 feet from the bottom of the pole. Re- 
 quired the length of the broken part. 
 
 EXERCISES. 
 
 1. Set a stake 40 feet perpendicularly distant from a given 
 point in a given line. 
 
 2. Through a given point 50 feet from a given line run a 
 parallel 120 feet in length. 
 
 3. Prolong a line beyond a house or other obstacle. 
 
 4. Measure the width of a stream or pond without crossing it. 
 
 5. Run a line to the bank of a stream or lake, and let fall a 
 perpendicular on the line near its extremity from a given point 
 without it. 
 
 6. Measure the height of a tree, flagstaff, or church spire.
 
 22 PLANE SURVEYING. 
 
 SECTION III. 
 
 RECORDING THE FIELD NOTES. 
 
 36. The Field Notes should be kept in a neat, concise, and 
 intelligible manner, exhibiting a complete record of the work 
 done, and the method of doing it, so that a surveyor unac- 
 quainted with the work, and having the record before him, 
 could make a plot of the tract, or go on the field and readily 
 ascertain the position of any point indicated in the notes. 
 
 Either of two methods may be employed, or a combination 
 of them. 
 
 37. Sketch. One is to make a sketch of the tract as the 
 survey progresses, writing the length of each line and indicating 
 the intersection of fences, roads, streams, etc., as shown below. 
 
 For surveying a field or small tract of land, this is a good 
 method, but if the tract is large, many sided, and numerous 
 points to be noted in and 'near the side-lines and diagonals, it 
 would be difficult if not impossible to decipher the sketch on a 
 page of the ordinary field-book, and to make an intelligible 
 record of the work would require a book or sheet inconveniently 
 large to carry about the field.
 
 RECORDING THE FIELD NOTES. 
 
 23 
 
 38. Columns. A method which will answer as well for com- 
 plex as for simple surveys consists in drawing two parallel lines, 
 about an inch apart, extending from top to bottom of the note- 
 book, and near the middle of the left-hand page. Between the 
 lines the distances and stations are to be recorded, commencing 
 at the bottom of the page and proceeding upwards. Roads, 
 fences, streams, etc., should be represented on either or both 
 skies of the column as they naturally appear. The record of 
 the measurements on any line being referred to the beginning 
 of the line. 
 
 The right-hand page may be used for sketching any part of 
 the survey to further elucidate, where necessary, the work done. 
 
 A station is indicated by a triangle (A) or a circle (O) . If the 
 station is at the end of a line it is usual to name it by the letter 
 or number, designating that corner as station A or station 1, and 
 the line extending from A to B is called the line AB^ from 4 to 
 5, the line 4, 5 ; or a line may be designated 
 by its length ; a line that is 3 chains and 
 52 links long would be referred to as the 
 line 352. 
 
 F. S. 
 
 4.78 
 
 4.78 
 
 A false station is a point in a line 
 whence other measurements are to be made 
 either to the right or left, and are desig- 
 nated bv enclosing in a curve its distance 
 from the end of the measured line, or by 
 writing F. S. opposite that distance, as 
 per margin, which shows that there is a 
 false station at a distance of 3.62 chains 
 from A on the line AB. 
 
 A fence, brook, road, etc., intersecting 
 the measured line, should be drawn so as 
 to indicate, as nearly as possible, its inclination thereto, but not 
 as a continuous line ; the ends on each side being directly 
 opposite, as at 4.58 and 5.26, so that if the vertical column 

 
 24 
 
 PLANE SURVEYING. 
 O 
 
 4.58 
 
 5.36 
 
 6.18 
 
 were to vanish by the two lines MN and OP coinciding, the 
 fence or creek would be shown as continuous. 
 
 When the record of a line, as 
 JOT, is complete, and the meas- 
 urement is continued from N, a 
 horizontal line is drawn across the 
 column as shown in the figure. 
 But if the survey closes at the 
 end of a line, as at 0, or if for 
 any reason the work is to proceed 
 from some other point, two lines 
 are drawn across the column. 
 
 3.40 
 
 2.30 
 
 M 
 
 6.80 
 
 Y 
 
 A mark (K) or (l~) placed at 
 the beginning of a line indicates 
 by shape, as well as position, 
 that the line along which it stands 
 bears to the right of the pre- 
 ceding ; the reverse position of 
 the angle (N or ~l) indicates a turn 
 to the left. 
 
 In the figure, MN bears to the 
 right of KM, and NO to the left 
 QtMN.
 
 RECORDING THE FIELD NOTES. 
 
 26 
 
 The record of the survey sketched in Article 37 would be 
 represented by the column method as follows : 
 
 8.35 
 
 9.50 
 D 
 
 D 
 13.00 
 
 C 
 
 c 
 
 11.25 
 
 5.75 
 
 B 
 B 
 
 12.50 
 
 e.oo 
 
 A 
 
 
 D 
 13.50 
 
 
 IS 
 
 A 
 
 
 s 
 
 A 
 
 
 
 15.00 
 
 
 i 
 
 

 
 26 PLANE SURVEYING. 
 
 SECTION IV. 
 
 MAPPING AND PLOTTING. 
 
 39. A Map of a survey is a correct representation or copy 
 of the tract surveyed, exhibiting not only its boundaries, roads, 
 streams, etc., in relative dimensions and positions, but also the 
 irregularities and appearances of its surface. 
 
 A Plot (or Pkc) is an outline map, in which, in general, only 
 the boundaries, roads, streams, and important lines are delin- 
 eated, but no attempt is made to indicate the topography of the 
 tract. The surveyor usually makes a plot of a field or farm 
 survey. The civil engineer makes a map of a proposed railroad. 
 
 INSTRUMENTS USEFUL FOR MAKING A PLOT OF A CHAIN SURVEY. 
 
 40. Drawing-Board, T-Square, Triangles, Dividers, Scale, 
 Drawing Pen and Pencil.* 
 
 A Drawing-Board is a rectangular, smooth board to which 
 the paper that is to contain the drawing is fastened. There 
 are two patterns : one consists of a frame of walnut, or other 
 hard wood, with a detachable centre of soft white pine. The 
 paper, which should be somewhat larger than the detachable 
 centre, being moistened and laid on it, becomes well stretched 
 when the parts of the board are buttoned together and the 
 paper dries. The other is simply a rectangular white pine 
 board made of several pieces of wood laid in different direc- 
 tions to prevent warping. Both patterns are made of various 
 dimensions. 
 
 41. A T -Square, as its name indicates, is a square or ruler 
 with a cross-piece or head at one end, giving it the appearance 
 
 * Other instruments used in drawing are described in Chapter II. 
 Section VIII.
 
 MAPPING AND PLOTTING. 27 
 
 of a letter T. There are two patterns of these, one with a head 
 fixed at right angles to the ruler or blade; the other, in addition 
 to the permanent head, has another head attached to it with a 
 clamp screw, so that by properly setting the movable head, lines 
 of any desired inclination may be drawn. The blade, being long 
 and thin, should be tested occasionally by means of a metallic 
 straight edge or another T-square to see whether or not it is 
 perfectly straight. The correctness of the angles should also 
 be tested ; this may be done as indicated in the next article. 
 
 42. Triangles are made of hard wood, rubber, or metal, and 
 are either solid or have an open centre. The angles are usually 
 30, 60, and 90 degrees, or 45, 45, and 90 degrees, and the 
 longest side rarely exceeds 12 inches. 
 
 The T-square and triangles are frequently employed together 
 to draw parallels, perpendiculars, and many of the oblique lines 
 of a plot.* 
 
 The sides of triangles should be tested occasionally, to see if 
 they are straight, by placing them against the edge of a metallic 
 straight edge. 
 
 The right angle may be tested by placing one of its sides 
 against a straight edge ; mark the direction of the other 
 side, reverse the triangle, but bring the same side against the 
 straight edge, and having the right angle at the same point as 
 before, mark the side again. If the two marks coincide, the 
 angle is right ; otherwise, it is not. 
 
 When correct, the right angle of the triangle may be used to 
 test the correctness of the right angle of the T-square. 
 
 43. Dividers (or Compasses) are made of different sizes and 
 numerous appendages. The surveyor will need at least one 
 with a detachable leg, so that another leg, carrying a pen or 
 
 * The results are tolerably accurate within the limits usually required 
 in a farm survey. It may be well, however, to caution the student not to 
 rely too much upon the accuracy of a point located by means of and near 
 the extremity of a thirty-inch T-square.
 
 28 
 
 PLANE SURVEYING. 
 
 pencil point, may be inserted when necessary. These, it need 
 hardly be said, are used for laying off lines, describing arcs, 
 circles, etc. 
 
 44. Lead-Pencil. Fine quality, hard, used in outlining the 
 work ; and a Drawing-Pen, medium size, for inking in the 
 drawing. 
 
 45. Scales are made of box-wood, metal, ivory, or paper, 
 and are of various kinds. Triangular and diagonal are gener- 
 ally used for plotting chain surveys. The triangular scale for 
 engineers and surveyors is usually 12 inches long, and made of 
 good box-wood, each of the six bevelled faces being graduated 
 with a single scale, viz. : one face contains 10 divisions to the 
 inch, one 20, another 30, another 40, one 50, and one 60 divis- 
 ions ; and generally one inch on each face is subdivided so 
 that an extremely small fraction of an inch may be set off or 
 read. This is a very convenient scale ; not only can very small 
 divisions be readily transferred from it to a drawing, but by 
 simply placing the instrument properly on a line of a drawing, 
 the scale of which is known, its length may be directly deter- 
 mined. 
 
 The Diagonal Scale is usually six inches long, thin and flat, 
 divided transversely into 6 equal parts of one inch each, and 
 longitudinally into ten equal parts. At one end, as AD, one 
 inch is divided by 10 oblique lines, as 8 wi, 6 ?i, etc., into 10 
 equal parts and numbered as shown in the figure. 
 
 P nmA 
 
 Now Fs being .1, the next division between the perpendicular 
 FE and the oblique line sE is .09, the next .08, and the last
 
 MAPPING AND PLOTTING. 29 
 
 division, or one nearest -F, is .01. Hence the scale may be 
 used to measure .01 of an inch, or one hundredth of any divis- 
 ion taken as the unit. For example, to lay off 3.4, place 
 one foot of the dividers at 3 on the line EC and extend the 
 other foot to 4 between DE. To lay off 3.42, place one foot 
 at the intersection of 3, 3, and 2, 2, and the other on the same 
 line 2, 2, at its intersection with 4 p. 
 
 The diagonal scale usually found with a box of drawing in- 
 struments contains various graduations. The simplest are 
 divided to inches, and halves, quarters, tenths, and twelfths 
 of an inch ; each quarter and half subdivided diagonally into 
 tenths, so that a tenth of a quarter can be taken off at once ; 
 and even tenths of these are indicated on the scale besides 
 other divisions of more or less utility. 
 
 Paper scales are frequently employed, and regarding hygro- 
 metric changes are better than the others, for the scale and the 
 paper containing the drawing expand and contract more nearly 
 alike. Generally, however, they are not divided with the same 
 degree of accuracy. 
 
 46. Drawing to a Scale consists in drawing lines whose 
 length shall be some fraction of the length of the line measured. 
 Suppose, for example, a line is 13 chains long, and it is desired 
 to draw it to a scale of 5 chains to an inch ; then 2y 6 ^ inches 
 will evidently be the distance to transfer from the scale to the 
 paper to represent the length of the line. 
 
 A line 10 chains and 50 links in length drawn to a scale of 3 
 chains to an inch will be represented by a line 3| inches long, 
 and so on. The length of the line divided by the number of 
 units chains, yards, feet, etc. to the inch, always giving 
 the distance to be taken off the scale. Obviously the converse 
 of this is true ; that is, the real length of a line may be ascer- 
 tained when the scale is known, by multiplying the units in the 
 length of the line in the drawing bv the number of chains or 
 feet which each unit represents. In the last example the length 
 of the line being found 3.jr inches, and the scale 3 chains to an
 
 30 PLANE SURVEYING. 
 
 inch, the true length = 3.5 x 3 = 10.50 chains. The scale 
 should always be given on the drawing. It may be stated 
 thus: Scale, 3 chains to an inch, 1000 feet to an inch, 2 miles 
 to an inch, or fractionally, and thereby indicating the relative 
 length of the lines in the drawing to those which they repre- 
 sent ; as, 1 : 500, 1 : 2000, etc. 
 
 47. Size of Drawing or Scale to Adopt. In farm surveys 
 of small extent, 1 or 2 chains to an inch may be used ; for 
 medium tracts 3 chains to an inch (1 : 2376) is perhaps the 
 best. The shape of the farm, the length of the shortest and 
 longest sides, as well as the object of the drawing, will, how- 
 ever, influence the surveyor in his decision of the scale. 
 
 48. Scale Unknown. If the area of a tract of land is known 
 but the scale not given, it ma}' be found by measuring the lines 
 of the drawing referred to any convenient scale and comput- 
 ing the area from these determined lengths. Then, since the 
 areas of similar figures are to each other as the squares of their 
 homologous sides, the true scale may be obtained by the pro- 
 portion, 
 
 computed area _ square of assumed scale # 
 known area square of true scale 
 
 SECTION V. 
 ON AREAS, AND ILLUSTRATIVE EXAMPLES. 
 
 A. AREAS. 
 
 49. The following are geometrical truths with which the 
 student is supposed to have an acquaintance, but are given here 
 for convenience of reference. 
 
 *The protractor and other drawing-instruments used in connection with 
 compass and transit surveying are described in Chapter II.
 
 AREAS. 31 
 
 The Area of a Triangle is equal to one-half the product ot its 
 base and altitude. 
 
 In Terms of the Three Sides the area is equd to the square root 
 of the continued product of one-half the sum of the sides, and 
 the half-sum minus each side severally, or in symbols, where 
 A. = area, a, 6, c, the three sides, and s their sum, 
 
 If the triangle is equilateral and s = length of a side, 
 
 50. The Area of a Rectangle is equal to the product of ite 
 
 length and breadth, or A = bl where b = breadth and I = length. 
 
 51. The Area of a Parallelogram is equal to the product of 
 its base and altitude, or A='bh where &= breadth and h= height. 
 
 52. The Area of a Trapezoid is equal to the product of one- 
 half the sum of its parallel sides and the perpendicular distance 
 
 between them, or A=^(m+n) where m and n are the parallel 
 sides, and p the perpendicular distance between them. 
 
 53. The Area of a Regular Hexagon, where s denotes the 
 
 g 
 
 length of one of its sides, is ^4 = -<s 2 - v /3, or it is equal to six 
 
 equal equilateral triangles, having for each side the length of 
 one side of the hexagon. 
 
 54. The Area of a Regular Octagon, each of its sides being 
 unity, may be calculated by the rules of geometry, thus : Let 
 the figure represent the octagon. It is evident that the area of 
 the central square = 1. The sum of the areas of the four 
 triangles m, n, o, _p = l, since their sum equals the square 
 described on db.* Now, the dimensions of each of the four 
 
 * The square described on the diagonal of a square is double the given 
 square.
 
 PLANE SURVEYING. 
 
 remaining figures (rectangles) #, y, z, and M, are 1, and 
 hence the sum of the areas of these four rectangles 
 
 = 4 x i-V2 = % V2 ; 
 adding all the parts, there results 
 
 1 + 1+ 2 V2 = 2 + 2 V2 
 for the area of the octagon. 
 
 55. The Area of a Regular Polygon in terms of the perimeter 
 and apothem, or radius of inscribed circle, is equal to one-half 
 
 the product of the perimeter and apothem, or A = ^ ; p denoting 
 the perimeter, and r the radius of inscribed circle or apothem. 
 
 56. The Area of a Regular Polygon in terms of the number of 
 
 sides and length of one side may 
 be determined as follows : Let 
 r = OP be the radius of the in- 
 scribed circle or apothem, I the 
 length of each side, and n the 
 number of sides, A the area, as 
 before ; then
 
 AREAS. 
 
 33 
 
 and 
 
 A nil , 180 nP , 180 
 
 A = X - cot = cot 
 
 2 2 n 4 n 
 
 If 1=1, and w=8, the area of the polygon (octagon) becomes 
 2 cot 22 30'= 2 + 2V2, as before found. 
 
 57. By the application of the formulas just found, the fol- 
 lowing table may be constructed, showing the apothems and 
 areas of some of the regular polygons, each of whose sides is 
 unity. 
 
 NAMES. 
 
 SIDES. 
 
 APOTHEMS. 
 
 AREAS. 
 
 Triangle . . . . 
 
 3 
 
 0.2886732 
 
 0.4330127 
 
 Square 
 
 4 
 
 0.5000000 
 
 1.0000000 
 
 Pentagon .... 
 
 5 
 
 0.6881910 
 
 1.7204774 
 
 Hexagon .... 
 
 6 
 
 0.8660254 
 
 2.5980762 
 
 Heptagon . . . 
 
 7 
 
 1.0382601 
 
 3.6339124 
 
 Octagon .... 
 
 8 
 
 1.2071069 
 
 4.8284271 
 
 Nonagon .... 
 
 9 
 
 1.3737385 
 
 6.1818242 
 
 Decagon .... 
 
 10 
 
 1.5388418 
 
 7.6942088 
 
 Hendecagon . . 
 
 11 
 
 1.7028439 
 
 9.3656399 
 
 Dodecagon . . . 
 
 12 
 
 1. -8660252 
 
 11.1961524 
 
 Now, since the areas of similar polygons are proportional to 
 the squares on their homologous sides, this table may be used 
 to find the area of any regular polygon named in the table, 
 whatever may be the length of its side. Using the notation 
 above, the principle just enunciated will be expressed as follows : 
 
 I 2 : area in table = P : A, or A. = P X area in table. 
 
 That is, the area of a regular polygon is equal to the square 
 of its side multiplied by the area of a similar polygon each of 
 whose sides is 1 . 
 
 EXAMPLE. The area of a regular pentagon, each side being 30, 
 = 30 2 x 1.7204774=1548.43. 
 
 58. The Area of a Circle is equal to TT multiplied by the 
 square of the radius, or one-half the product of the circumfer-
 
 34 PLANE SURVEYING. 
 
 ence and radius. Let R represent the radius, C the circum- 
 ference, and A the area ; then 
 
 The area of a Quadrant = =- 
 
 59. The Area of a Sextant = - -, and in general, the area 
 
 6 
 
 of any sector of a circle = -^- X irR 2 , in which n denotes the 
 360 
 
 number of degrees in the sector, or A = -^-, in which I denotes 
 the length of the arc. 
 
 60. The Area of a Circular Ring is evidently the difference 
 of the areas of the outer and inner circles ; or, in symbols, if R 
 and r equal the outer and inner radii, A = -^(R 2 r 2 ) . 
 
 61. The Area of a Segment of a circle, as ABC, is evidently 
 equal to the area of the sector AOBC, minus the area of the 
 
 C 
 
 triangle AOB; or, in symbols, since the area of the triangle 
 
 R 2 sin n , , , 
 = - - , and the area of the sector as given above, 
 
 . _ Tr 
 ~~~360~ 2
 
 ILLUSTRATIVE EXAMPLES. 35 
 
 If n is greater than 180, as in the segment A'B'BCA, sin n 
 becomes negative, thereby making the second term of the right- 
 hand member positive, as it should ; since in this case the 
 segment is greater than the sector, and the triangle A'OB' is 
 additive. 
 
 If the lengths of arc and chord are given, denote by 2c the 
 length of chord, the other notation as above ; then 
 
 the minus sign to be used when the segment is less than a semi- 
 circle, and the plus sign when the segment is greater than a 
 semicircle. 
 
 62. The Area of an Ellipse is equal to irAB, in which A and 
 B denote the semi-axes. 
 
 B. ILLUSTRATIVE EXAMPLES. 
 
 EXHIBITING VARIOUS METHODS EMPLOYED TO SURVEY LAND, TO PLOT 
 THE SURVEY, AND TO CALCULATE THE AREA. 
 
 TRIANGLES. 
 
 63. First Method. Measure the perpendicular (7Z), and the 
 segments AD and DB, into 
 
 which it divides the base ; then 
 A _ AB x DC 
 2 
 
 To Make the Plot. Draw AB A. 
 
 according to any convenient 
 
 scale, and locate D ; with the same scale erect at D a perpen- 
 
 dicular = DC. Join CA and CB, and the triangle ABC will 
 
 result. 
 
 tXAMPLES. 
 
 1. Required the area and plot of a triangular field, the per- 
 pendicular of which measures 4.86 chains, and divides the side 
 on which it falls into segments measuring 5.80 chains and 3.16 
 chains, or a total length of 8.96 chains.
 
 86 PLANE SURVEYING. 
 
 Calculation. A = 8 - 96 * 4 - 86 = 21 . 7728 square chains. Divid- 
 ing by 10, since there are 10 square chains in an acre, their 
 
 91 7798 
 results = 2.177 + acres.* (The student will make the 
 
 plot.) 
 
 QUERIES. Could a correct plot of the tract be made if there 
 were given simply the base and altitude ? 
 
 Would there be, usually, any choice of side to take as the 
 base? 
 
 2. A triangular field measures 12.18 chains on one side, and 
 the perpendicular erected at a point 5.10 chains from one end 
 measures 7.54 chains. Calculate the area and make the plot. 
 
 64. Second Method.^ Measure all the sides, and calculate 
 the area by the formula given above for that case. 
 
 EXAMPLES. 
 
 1. The lengths of the sides of a triangle are as follows: 
 AB=40 chains, AC =30 chains, and .8(7=20 chains. Re- 
 quired the area and plot- 
 
 A = V45 x 5 x 15 x 25 = 29.047 acres. 
 
 To Make the Plot. Take 40 chains to any convenient scale 
 in the dividers, and lay it off for the base AB; then, with A 
 as a centre and 30 chains to the same 
 scale in the dividers, describe an arc 
 mm' ; also, with B as a centre and 20 
 chains for radius, describe the arc nn'. 
 
 _ _ The point C connected with A and 
 
 B will give the triangle ABC required. 
 REMARK. It is customary when making a chain survey, to 
 
 * The area is usually expressed in acres and hundredths or thousandths 
 of an acre. 
 
 t Other methods are given in Chapter II. Section IX.
 
 ILLUSTRATIVE EXAMPLES. 37 
 
 measure a proof* line such as CD, and this should always 
 be constructed to test the accuracy of the work. 
 
 2. The three sides of a triangle measure 49, 50.25, and 
 25.69 chains. Find the area. Ans. 61.498 acres. 
 
 3. The sides of a triangular field are 24, 18, and 15 chains. 
 A proof line, 12 chains in length, intersects the longest side or 
 base at a point 10.25 chains from the angle formed by the 
 two longest sides of the field. Required the area and plot. 
 Test accuracy of latter by constructing proof line. 
 
 RECTANGLES. 
 
 65. Measure any two adjacent sides, as AB and BC. The 
 area = A = AB BC. D 
 
 To Plot. Lay off AB to any de- 
 sired scale, and erect a perpendicular 
 with the same scale at the extremities 
 = AD and BC ; connect D and (7, 
 and the required figure will be formed. A. 
 
 EXAMPLES. 
 
 1. The length and breadth of a rectangle are 12.32 and 7.16 
 chains respectively. Required the area. Ans. 8.82 acres. 
 
 2. The length of a rectangle is 1250 feet, and its bread th'840 
 feet. What is its area? Ans. 24.1 acres. 
 
 3. A road running across a farm is ^ of a mile long and 3 
 rods wide. How much laud does it occupy ? Ans. 2\ acres. 
 
 4. The length of a road on a hillside inclined to the horizon 
 at an angle of 20 is 2310 feet, and its width 2| rods. At the 
 rate of $84 per acre, what must be paid to the owner across 
 whose land the road runs? Ans. $189.93. 
 
 * A line to check the measurement. 
 
 48604
 
 38 PLANE SURVEYING. 
 
 PARALLELOGRAMS. 
 
 66. Measure a side, as AB, the perpendicular distance, as 
 j> E c BE, to the opposite side DC, and the 
 
 / i / distance CE. Then A = AB x BE. 
 
 / / To Plot. La\- off the base AB, and 
 
 -4 B at the extremity B erect a perpendicu- 
 
 lar equal BE. Through E draw DC equal to and parallel to 
 AB, making EC its proper length. Join DA and CB, and the 
 parallelogram ABCD will be formed. 
 
 EXAMPLES. 
 
 1. The base of a parallelogram measures 10.54 chains. A 
 perpendicular from one extremity of the base to the opposite 
 side 5.16 chains, and the distance corresponding to EC in the 
 last figure is 1.82 chains. Required the area and plot. 
 
 Ans. 5.439 acres. 
 
 2. A surveyor employed to determine the area of a rhombus, 
 and knowing that the obtuse angles were xiouble the acute, 
 measured the shorter diagonal only, and found it 100 feet. 
 Was the measurement sufficient? If so, give the area. 
 
 QUERIES. Can the area of a rhombus be ascertained if the 
 lengths only of the two diagonals be given ? If either diagonal 
 and a side be given ? 
 
 TRAPEZOIDS. 
 
 67. Measure EC, the perpendicular CD, and BA ; note 
 
 E C where the perpendicular CD meets the 
 
 / ~A base AB. 
 
 CD. 
 
 A D B 
 
 To Plot. Lay off the base AB to the desired scale, and 
 at D erect a perpendicular thereto equal to DC. Through 
 C draw CE of the required length and parallel to AB. Join 
 EA and CB. The figure resulting will be the trapezoid 
 required.
 
 ILLUSTRATIVE EXAMPLES. 39 
 
 EXAMPLES. 
 
 1. The base of a trapezoid measures 12.62 chains, the 
 parallel side 8.14 chains, and the perpendicular 7.44 chains. 
 The distance corresponding to DB in the last figure is 1.12 
 chains. Required the area and plot. Area= 7.723 acres. 
 
 2. A railroad embankment extends 3240 feet perpendicularly 
 across a farm intersecting parallel sides. At one end its base 
 is 96 feet wide, and at the other 60 feet. Supposing the 
 property line is 10 feet from the embankment on each side, 
 how much of the farm is taken for railroad purposes? 
 
 TRAPEZIUMS. 
 
 68. First Method. Measure either diagonal, and the per- 
 pendiculars thereto from the opposite 
 angles, noting the distances AH and 
 EC. 
 
 To Plot. Draw the diagonal AC to AL ^ ^ 
 
 the desired scale, and fix the points H 
 
 and E. At these points erect perpendiculars corresponding to 
 the scale and measurements. Joining DA and DC, and BA 
 and BC, will complete the plot required. 
 
 Second Method. Measure all the sides 
 and a diagonal as shown in the figure, 
 thereby dividing the trapezium into two 
 triangles, all the sides of which are 
 known ; whence the area may be com- 
 puted by the formula for the area of a 
 triungle in terms of the three sides. 
 
 To Plot. Lay off the diagonal AC, and locate the points B 
 and D by methods heretofore given. Connect the points 
 ABC I) A for the plot required.
 
 40 PLANE SURVEYING. 
 
 EXAMPLES. 
 
 1. The diagonal of a trapezium measures 120 rods, and the 
 two perpendiculars 30 and 40 rods ; what is the area ? 
 
 Ans. 26^ acres. 
 
 2. The sides of a trapezium taken in regular order are 
 AB = 5, BC=9, CD = 11, and DA=13 chains, and the 
 diagonal AC 12 chains. Required the area and plot. 
 
 3. The sides of a trapezium are 18.10, 22.14, 28.16, and 
 34.62 chains, and the diagonal from the first to the third corner 
 is 30.76 chains. Determine the area. 
 
 POLYGONS. 
 
 Regular or irregular, five or more sides. 
 69. First Method. Measure all the sides and the diagonals, 
 thus dividing the tract into three or more triangles. The area 
 will equal the sum of the areas 
 of the triangles thus formed. 
 
 To Plot. Draw a line repre- 
 senting the diagonal BE, and 
 construct the triangle ABE on 
 it ; on the other side of BE con- 
 struct BCE; if a pentagon, the 
 plot will be completed by add- 
 ing ODE. 
 
 If a hexagon, there must be 
 measured another diagonal giving four triangles, and generally, 
 for any number of sides n, there will be n 3 diagonals and 
 n 2 triangles, the area of the tract being equal to the sum 
 of the areas of the n 2 triangles. 
 
 If the tract be a regular polygon, the measurement of one 
 side by the aid of the table in (57) will be sufficient to deter- 
 mine the area. 

 
 ILLUSTRATIVE EXAMPLES. 
 
 41 
 
 70. Second Method.* Measure one 
 or more diagonals, and perpendiculars 
 from these to the opposite angles, or 
 corners, thereby dividing the tract into 
 right triangles, or right triangles and 
 trapezoids. The sum of the areas of 
 these figures will equal the area of 
 the polygon. 
 
 EXAMPLES. 
 
 1. The sides of a pentagon taken in regular order are, 6.80, 
 4.20, 5.30, 8.90, and 9.62 chains. The diagonals from the 
 fifth corner to the second and third are each 10 chains. Find 
 the area,t and make a plot. 
 
 2. A side of a regular heptagon measures 4.25 chains. 
 What is the area? 
 
 Given the following field notes to calculate the areas and 
 make the plots. The distances are in chains. 
 
 
 D 
 
 
 D 
 
 
 
 16.75 
 
 
 12.50 
 
 
 C4.50 
 
 13.50 
 
 
 
 
 
 
 
 7.80 
 
 4.60 E 
 
 
 12.90 
 
 4.50 E C 2.80 
 
 5.90 
 
 
 
 9.00 
 
 S' 80 F B 5.50 
 
 3.20 
 
 
 B 4.50 
 
 4.80 
 
 
 2.60 
 
 3.00 F 
 
 
 8.20 
 
 6.25 G 
 
 A 
 
 
 
 A 
 
 
 * Other methods are given in Chapter II. 
 
 t The work may be abridged by using logarithms.
 
 42 
 
 PLANE SURVEYING. 
 
 -Proof Lines 7 - - r - - -> 
 
 5 re 
 
 8 S 
 
 s s
 
 ILLUSTRATIVE EXAMPLES. 43 
 
 CIRCLES AND CIRCULAR RINGS. 
 
 71. Measure the radius or diameter of a circle, and the radii 
 or diameters of a circular ring. 
 
 The area of the former = TT^ = . 
 The area of the latter = *(!? r 2 ). 
 
 EXAMPLES. 
 
 1. The diameter of a circle is 10.16 chains. What is the 
 area? 
 
 2. What is the 'area of a circular ring, the outer and inner 
 radii measuring respectively 20 and 12 rods? 
 
 SECTORS AND SEGMENTS. 
 
 72. Measure the chord AB, and the perpendicular distance 
 or height of arc DE, from the centre of 
 AB to the arc AEB. From these data 
 the radius and the angle at the centre 
 may be found ; and hence the area ob- 
 tained. See (59) and (61). Otherwise, 
 measure the radius BC, and by short 
 chords the arc AEB; whence the area 
 may be computed. (The student will 
 supply the details for both cases.) G 
 
 EXAMPLES. 
 
 1. If the length of the arc of a sector is 500 feet and the 
 radius 1000 feet, how many acres does it contain? Ans. 5.739. 
 
 2. If the chord AB (last figure) =40 feet, and the height of 
 arc DE = 10 feet, what is the area of the segment? 
 
 Ans. 279.558 square feet. 
 
 3. Given the radius, which is bisected by the chord, = 100 
 feet. Required the area of sector and segment.
 
 44 
 
 PLANE SURVEYING. 
 
 SECTION VI. 
 
 OFFSETS AND TIB-LINES. 
 
 73. When any portion of the boundary of a tract of land is 
 irregular, as, for example, when it is a stream or crooked road, 
 the survey along such sides is best effected by 
 measuring a straight line, as LN, and setting 
 off short perpendiculars ra'ra, o'o, and p'p at 
 points m', o', and p', and extending them to 
 the boundary line. Such short perpendiculars 
 are called offsets, and they should be so chosen 
 , that the part of the curve Lm, mo, op, etc., 
 / m intercepted between any two consecutive ones 
 may be considered straight ; whence the area 
 of the part lying between the straight and 
 curved lines may be obtained by adding to- 
 gether the area of the triangles and trapezoids into which it is 
 thus divided. 
 
 If the field notes corresponding to the above figure are as 
 below : 
 
 N 
 9.00 
 7.00 
 2.50 
 1.20 
 
 L 
 
 
 
 1.30 p 
 1.40 o 
 1.00 m 
 
 
 The area between straight line and boundary 
 
 f Area triangle Lmm', 6000 square links, 
 
 I Area trapezoid mm'oo', 15600 " " 
 
 "~ 1 Area trapezoid oo'pp', 60750 " " 
 
 [Area triangle p'pN, 16900 " " 
 
 Their sum = 99250 " " 
 
 cr, .9925 of an acre.
 
 OFFSETS AND TIE-LINES. 
 
 45 
 
 x' 
 
 74. Rectangular Co-ordinates. Let XX' and YT' be two 
 
 straight lines intersecting each other at right angles at 0, and 
 P'P", points in their plane. 
 Then if perpendiculars be drawn 
 thi-ough these points to the lines 
 XX' and YY', the distances 
 cut off on the former are called 
 abscissas, and those on the lat- 
 ter ordinates. The abscissa and ^ 
 ordinate referring to one point, 
 as P, are termed the co-ordi- 
 nates of that point. , 
 
 The lines to which the meas- 
 urements are referred are called the axes; XX' being called 
 the axis of abscissas or axis of X, and YY' the axis of ordi- 
 nates or axis of Y. 
 
 The axes being at right angles, the system is called the 
 rectangular system of co-ordinates. is the origin.* Desig- 
 nating the ordinates measured from the axis of X upward, and 
 the abscissas measured to the right of the axis of Y, as plus, 
 and those downward from the X-axis and to the left of the 
 F-axis, as minus, it is evident that a point can be located in 
 either quadrant very readily by this method. 
 
 If the co-ordinates of P 1 are x = 6 and y = 4, it means 
 simply that Ox' = 6 and Oy' = 4, and the point may be located 
 by drawing the lines as indicated. If x = 5 and y=3, the 
 point is five units to the left of the F-axis, and three units 
 above the X-axis, etc. 
 
 75. Application of Rectangular Co-ordinates to the Computa- 
 tion of Areas. 
 
 Suppose it is required to find the area of any number of 
 trapezoids formed by a broken line, and perpendiculars from its 
 angles upon a straight line as indicated in the figure. XX', the 
 
 * Axes inclined to each other are called oblique.
 
 46 
 
 PLANE SURVEYING. 
 
 straight line, may be taken as the axis of X, and TV the axis 
 of T. Let # x m , x n , etc., y y m , y n , etc., denote respectively 
 the abscissas and ordinates of the points L, M, N, etc. 
 
 I'm 
 
 Y n 
 
 F, 
 
 XI Xm 
 
 The area required 
 
 By expanding and simplifying there results 
 
 (y m - y 
 
 Whence for calculating the area of a tract of land included 
 between a straight line and a broken line, whose angles are 
 given by their co-ordinates upon the straight line as base, we 
 have the following 
 
 RULE. 
 
 Multiply the difference between each ordinate and the second 
 succeeding one by the abscissa of the intervening ordinate. 
 
 Multiply also the sum of the last two ordinates by the last 
 abscissa. 
 
 The half of the algebraic sum of these several products will be 
 the area. 
 
 EXAMPLES. 
 
 Calculate the areas, and make the plots from the following 
 field notes ; the distances are in chains. 
 
 * A similar expression could evidently be found for any number of 
 trapezoids. 

 
 OFFSETS AND TIE-LINES. 
 
 47 
 
 13.60 
 
 .90 
 
 12.40 
 
 1.50 
 
 9.80 
 
 2.10 
 
 5.80 
 
 1.60 
 
 2.80 
 
 1.00 
 
 1.00 
 
 .60 
 
 
 
 
 
 18.90 
 
 1.60 
 
 16.70 
 
 2.00 
 
 12.40 
 
 2.40 
 
 7.40 
 
 1.50 
 
 4.20 
 
 1.00 
 
 1.70 
 
 .20 
 
 
 
 
 
 76. A slight modification of the rule just given will make it 
 applicable to the case where a broken line encloses a tract or 
 forms the boundary of a polygon. 
 
 Let the tract enclosed be represented by the figures, then 
 the area
 
 48 PLANE SURVEYING. 
 
 By expanding, cancelling, and factoring, we may obtain 
 either of the following expressions : 
 
 (y - &,) 
 
 (a;, - x.) ].- (2) 
 
 Whence, for the area of a polygon whose corners are given 
 by their co-ordinates, we have the following 
 
 RULE. 
 
 Take one-half the sum of the products of each { 
 and the difference of its adjacent <{ ^^^^ < l 
 the subtraction in the same direction round the plot.* 
 
 EXAMPLES. 
 
 1. Given the abscissas of the several corners of a field, L, 
 M, N, 0, P, respectively : 
 
 2.00, 5.50, 12.00, 15.00, and 8.60 chains. 
 The corresponding ordinates : 
 
 10.20, 1.80, 4.00, 9.40, and 14.00 chains; 
 to compute the area. 
 
 * The work of computation may be abridged when the abscissas are 
 greater than the ordinates, by making the differences of the a"bscissas the 
 factors with the ordinates ; and when the ordinates are greater than the 
 abscissas, taking the differences of the ordinates with the abscissas. If 
 the axis of ordinates pass through L, the abscissa of that point would 
 vanish. Regard must, in all cases, be had to the resulting signs.
 
 OFFSETS AND TIE-LINES. 
 The form of reduction is as follows : 
 
 49 
 
 CORNERS. 
 
 OBDINATES. 
 
 ABSCISSAS. 
 
 DIFFERENCE BETWEEN 
 ALTERNATE ABSCISSAS. 
 
 DOUBLE AREAS. 
 
 L 
 
 10.20 
 
 2.00 
 
 3.10. 
 
 31.6200 
 
 M 
 
 1.80 
 
 5.50 
 
 - 10.00 
 
 - 18.0000 
 
 N 
 
 4.00 
 
 12.00 
 
 - 9.50 
 
 - 38.0000 
 
 
 
 9.40 
 
 15.00 
 
 3.40 
 
 31.9600 
 
 P 
 
 14.00 
 
 8.60 
 
 13.00 
 
 182.0000 
 
 
 245.5800 
 
 
 - 56. 
 
 v 
 
 2)189.5800 
 
 
 10)94.79 sq. chs. 
 
 
 9.479 acres. 
 
 2. Given the abscissas of the several corners of a field, L t 
 M, N, 0, P, Q, R, respectively : 
 
 0, 6.50, 14.60, 22.80, 20.00, 16.70, 9.90; 
 and the corresponding ordinates : 
 
 13.20, 3.72, 4.40, 3.90, 17.24, 16.90, and 17.30, 
 all in chains ; to determine the area and make a plot. 
 
 3. Given the abscissas of the several corners of a field, A, 
 B, (7, D, E, F, G, H, respectively : 
 
 100, 300, 360, 290, 400, 250, 120, 0; 
 and the corresponding ordinates : 
 
 0, 0, 160, 300, 380, 520, 520, and 330, 
 all in feet ; to determine the area, and make a plot. 
 
 4. Verify Example 3 by a method independent of that given 
 on the preceding page.
 
 50 PLANE SURVEYING. 
 
 5. Required the area and plot from the following field notes : 
 
 t 
 
 
 A 
 
 
 E 
 
 A 
 
 
 
 23.50 
 
 
 41.10 
 
 y 
 
 
 
 F 
 
 Y Diag. DF 
 
 25.80 
 
 
 
 
 F 
 
 
 
 
 
 
 
 21.90 
 
 Diag. CF 
 
 17.65 
 
 
 
 
 
 E 
 
 Y 
 
 B 
 
 V 
 
 
 
 E 
 
 
 F 
 
 A 
 
 
 
 20.50 
 
 
 30.10 
 
 
 
 
 D 
 
 Y 
 
 D 
 
 
 A 
 
 
 D 
 
 
 C 
 
 *e 
 
 
 
 
 17.40 
 
 
 28.90 
 
 ID 
 
 
 1.00 
 
 15.50 
 
 N 
 
 F 
 
 ii 
 I 
 
 T3 
 
 1.60 
 
 13.00 
 
 
 F 
 
 
 OQ fi 
 
 5 
 
 1.75 
 1.00 
 1.20 
 
 8.50 
 6.00 
 3.50 
 
 
 26.75 
 B 
 
 i 
 
 
 
 
 
 C 
 
 K 
 
 
 
 1 
 
 
 C 
 
 On river bank. 
 
 
 
 SB 
 
 
 
 18.00 
 
 
 
 
 o 
 
 
 
 
 
 
 
 2.00 
 
 13.50 
 
 
 
 
 
 3.50 
 
 10.00 
 
 
 
 
 
 2.50 
 
 5.50 
 
 
 
 
 V 
 
 
 
 B 
 
 Y 
 
 
 
 j 
 
 
 B 
 
 
 
 
 ! 
 
 
 18.50 
 
 
 
 * 
 
 i 
 
 
 A 
 
 
 
 
 Other examples containing offsets are given in Chapter II,
 
 OFFSETS AND TIE-LINES. 
 
 77. To find the area of a tract of land when it is impossible 
 to measure the diagonals or perpendiculars, as in the case of a 
 lake or swamp. 
 
 Measure MN and ON, and continue the measurements past 
 their intersection at N, making NH 
 some fractional part of MN, and NK 
 the same part of ON.* Now because 
 of the similarity of the triangles MNO 
 and HKN, MO may be found by 
 measuring a tie-line HK, and divid- 
 ing it by the fraction used. Similarly, 
 LM may be found^ Then OL being , 
 measured, the area of the polygon 
 MNOLR can be computed. In case 
 of a pond or lake, if offsets be taken 
 from the sides of the polygon to the 
 edge of the water, and the sum of the 
 areas thus found included between 
 the sides and the lake be deducted 
 from the area of the polygon, the area 
 of the body of water will be shown. 
 
 MISCELLANEOUS EXAMPLES. 
 
 1. One side of an equilateral triangle measures 18.24 chains. 
 Required the area. 
 
 2. The perpendicular of an equilateral triangular piece of 
 ground measures 160 feet. What is the area? 
 
 Ans. 14780.16 square feet. What part of an acre? 
 
 * Great care should be exercised in the measurements, since the erroi 
 is magnified in the computed lines. If the lines are so taken that KH is 
 one-fourth of MO, an error of one link in measuring KH will make a 
 difference of four links in MO. 
 
 For methods of performing such work more accurately, see Compass 
 and Transit Surveying, Chapter II. Section IX.
 
 62 PLANE SURVEYING. 
 
 3. It is known that the base of an isosceles triangle is f the 
 length of one of its equal sides. The perpendicular measures 
 80 feet. The sides and area are required. 
 
 Ans. Each side, 100 feet; base, 120 feet. 
 Area, 4800 square feet. 
 
 4. Desiring to ascertain the radius of a railroad curve (it 
 being the boundary of a field), a surveyor measured from 
 centre to centre of tracks, a chord of 200 feet ; also the per- 
 pendicular distance from the centre of chord to the middle of 
 tracks, 4 feet. Show that these measurements indicate the 
 radius = 1252 feet. 
 
 QUERY. How should the data obtained in Example 4 be em- 
 ployed to determine the area, assuming that the curve is con- 
 cave to the field ? 
 
 5. The circumference of a circle is 100 rods. How many 
 acres does it contain ? Ans. 4.974. 
 
 QUERY. Can Problem 5 be solved without first finding the 
 radius or diameter? 
 
 6. If the number expressing the 'area of an equilateral tri- 
 angle in square feet is the same as that showing the length of 
 one of its sides in lineal inches, what is its area? 
 
 Ans. 332.55. 
 
 7. The chord of a circle measures 60 feet, and the height of 
 arc, or versed sine, 10 feet. Find in the same circle the versed 
 sine of a chord of 90 feet. Ans. 28.2 feet. 
 
 8. The lengths of two chords lying on the same side of the 
 diameter of a circle are 96 and 60, and their distance apart 26. 
 Required the area between them. 
 
 SUGGESTION. Let x = perpendicular distance from centre of 
 short chord to the nearest point of circumference, and y = per- 
 pendicular distance from centre of long chord to the farthest 
 point of circumference ; that is, measured in the opposite direc- 
 tion from the first
 
 OFFSETS AND TIE-LINES. 58 
 
 Then * (y + 26) = 900. 
 
 y (x + 26) = 2304. 
 
 Whence the diameter is readily determined and thence the 
 area required. 
 
 9. Show that the area of the circumscribed hexagon is to 
 the area of the circumscribed equilateral triangle as 2 is to 3. 
 
 10. Show that the area of a regular inscribed polygon of n 
 
 . , n , . 360 
 
 sides = - r 2 sin 
 
 2 n 
 
 11. Show that the area of a regular circumscribed polygon 
 
 1 &0 
 
 of n sides = w 2 tan 
 
 n 
 
 12. The distance between the centres of two circles, whose 
 diameters are each 50, is equal to 30. What is the area common 
 to the two circles? Ans. 559.15. 
 
 13. Three equal circles being tangent to each other ex- 
 ternally enclose 40 rods. What is the radius of each circle? 
 
 Ans. 15.75 rods. 
 
 EXERCISES. 
 
 1. Survey a polygon, measure all the sides and necessary 
 diagonals, run test-lines, record the notes, make a plot, and 
 compute the area. 
 
 2. Take the boundaries as found above, and complete the 
 survey by measuring one diagonal and perpendicular offsets to 
 the corners. Make record, plot, and computation. 
 
 3. Measure a field partly bounded by a creek or lake, ren- 
 dering it necessary to take offsets thereto. Record the notes, 
 plot, and calculate area. 
 
 4. Survey a pond or small lake by tie-lines and offsets.
 
 CHAPTER H. 
 
 COMPASS AND TEANSIT SUEVEYING. 
 
 SECTION I. 
 
 DEFINITIONS AND DESCRIPTION OP INSTRUMENTS. 
 
 78. The Axis of the earth is the imaginary line about which 
 it rotates. 
 
 The Poles are the points where the axis pierces the earth : one 
 the north pole, the other the south pole. 
 
 79. A Meridian Plane is a plane embracing the earth's axis. 
 
 80. A Meridian Line, or true meridian, is the intersection of 
 a meridian plane with the surface of the earth. 
 
 In plane surveying the meridians passing through the ex- 
 tremities of lines surveyed are considered parallel. 
 
 81. The Magnetic Needle is a thin bar of strongly magnet- 
 ized steel, balanced on a pivot, so that it may turn freely, and 
 always come to rest in the direction of the magnetic meridian. 
 
 82. The Magnetic Meridian is indicated by the direction of 
 a bar magnet, when horizontal, freely suspended and at rest. 
 It does not in general coincide with the geographic meridian. 
 The angle included between them is called the declination of the 
 needle, or variation of the compass.* and the change in this 
 angle is termed the variation of the declination. 
 
 * See Chapter III., on Declination of the Needle.
 
 DESCRIPTION OF INSTRUMENTS. 55 
 
 83 The Azimuth of a Line is the angle which the vertical 
 plaue containing it makes with the plane of the meridian. 
 
 84. The Bearing of a Line, called also the course, is the 
 angle which it forms with the direction of the magnetic needle. 
 
 85. The Meridian Distance of a Point is its perpendicular 
 distance from an assumed meridian. 
 
 86. The Meridian Distance of a Line is the meridian dis- 
 tance of the middle point of that line. 
 
 87. A Horizontal Angle is an angle included between two 
 lines in a horizontal plane. 
 
 A Vertical Angle is an angle included between two lines in 
 a vertical plane. 
 
 88. An Angle of Elevation is a vertical angle, one side of 
 which is horizontal, and the other inclined upward from the 
 angular point. 
 
 89. An Angle of Depression is a vertical angle, one side of 
 which is horizontal, and the other inclined downward from the 
 angular point. 
 
 In Compass and Transit Surveying, in addition to the meas- 
 urement of lines, angles are observed ; hence, besides the 
 instruments previously described, we present the following : 
 
 THE SURVEYOR'S COMPASS. 
 
 90. The Surveyor's * Compass consists essentially of a brass 
 plate carrying a horizontal graduated circle, in the centre of 
 which is suspended, so as to turn freely, a magnetic needle ; 
 and nt the extremities of the plate are attached vertically two 
 flattened pieces of brass, called sights, having fine slits and 
 
 * The Solar Compass is described in Chapter VI.
 
 56 PLANE SURVEYING. 
 
 circular openings in them, by which the instrument is directed 
 upon any object or station. 
 
 In addition to the essentials named, this instrument usually 
 has two small spirit levels set on the plate at right angles to 
 each other, a vernier scale for setting off the declination of the 
 needle, a tangent scale for reading vertical angles, and a brass 
 head for mounting the instrument upon a tripod or a single staff 
 called Jacob's Staff. 
 
 91 The graduated circle is divided into half-degrees, and is 
 figured from to 90 on each side of the centre line of zeros. 
 
 The magnetic needle is from 4 to 6 inches long in the different 
 sizes of compasses, having set in its centre a piece of hardened 
 steel highly polished, which, resting upon the hardened point of 
 the, centre-pin, allows the needle to turn freely, horizontally, 
 and to take its direction in the magnetic meridian. 
 
 92. The needle is lifted from its support by a concealed 
 spring actuated by a screw. The test of the delicacy of a 
 magnetic needle is the number of vibrations which it will make 
 in a certain arc before coming to rest. 
 
 When the compass is not in use, the needle should be screwed 
 up against the glass, and the instrument set so that the north 
 end of the needle points towards the north. 
 
 To ADJUST THE COMPASS. 
 
 93. The Levels. First bring the bubbles into the centre, by 
 the pressure of the hand on different parts of the plate, and 
 then turn the compass half-way around ; should the bubbles 
 run to the end of the tubes, it would indicate that those ends 
 were the highest : lower them by tightening the screws immedi- 
 ately under, and loosening those under the lowest ends until, 
 by estimation, the error is half removed ; level the plate again, 
 and repeat the first operation until the bubbles will remain in 
 the centre during an entire revolution of the compass.
 
 SURVEYOR'S COMPASS. 

 
 DESCRIPTION OF INSTRUMENTS. 59 
 
 94. The Sights may next be tested by observing through the 
 slits a fine hair or thread, made exactly vertical by a plumb. 
 Should the hair appear on one side of the slit, the sight must 
 be adjusted by filing off its under surface on that side which 
 seems the highest. 
 
 95. The Needle is adjusted in the following manner : Having 
 ;he eye nearly in the same plane with the graduated rim of the 
 compass-circle, with a small splinter of wood or a slender iron 
 wire bring one end of the needle in line with any prominent 
 division of the circle, as the zero or ninety-degree mark, and 
 notice if the other, end corresponds with the degree on the 
 opposite side: if it does, the needle is said to "cut" opposite 
 degrees ; if not, bend the centre-pin by applying a small brass 
 wrench, about one-eighth of an inch below the point of the pin, 
 until the ends of the needle are brought into line with the oppo- 
 site degrees. 
 
 Then, holding the needle in the same position, turn the com- 
 pass half-way around, and note whether it now cuts opposite 
 degrees ; if not, correct half the error by bending the needle, 
 and the remainder by bending the centre-pin. 
 
 The operation should be repeated until perfect reversion is 
 secured in the first position. 
 
 This being obtained, it may be tried on another quarter of 
 the circle ; if any error is there manifested, the correction must 
 be made in the centre-pin only, the needle being already 
 straightened by the previous operation. 
 
 96. Electricity. A little caution is necessary in handling the 
 compass, that the glass covering be not excited by the friction 
 of cloth, silk, or the hand, so as to attract the needle to its 
 under surface. 
 
 When, however, the glass becomes electrified, the charge 
 may be removed by breathing upon it, or touching different 
 parts of its surface with the moistened finger.
 
 60 PLANE SURVEYING. 
 
 97. The Needle is remagnetized as follows : 
 
 The operator, being provided with an ordinary permanent 
 magnet, and holding it before him, should pass with a gentle 
 pressure each end of the needle from centre to extremity over 
 the magnetic pole, describing before each pass a circle of about 
 six inches radius, to which the surface of the pole is tangent, 
 drawing the needle towards him, and taking care that the 
 north and the south ends are applied to the opposite poles of 
 the magnet. 
 
 Should the needle be returned in a path near the magnetic 
 pole, the current induced by the contact of the needle and 
 magnet, in the pass just described, would be reversed, and 
 thus the magnetic virtue almost entirely neutralized at each 
 operation. 
 
 When the needle has been passed about twenty-five times 
 in succession, in the manner just described, it may be consid- 
 ered as fully charged. 
 
 A fine brass wire is wound in two or three coils on the south 
 end of the needle, and may be moved back or forth in order to 
 counterpoise the varying weight of the north end. 
 
 98. The Centre-Pin. This should occasionally be examined, 
 and if much dulled, taken out with a brass wrench or with a 
 pair of pliers, and sharpened on a hard oil-stone the operator 
 placing it in the end of a small stem of wood or a pin-vise, 
 and delicately twirling it with the fingers as he moves it back 
 and forth at an angle of about 30 degrees to the surface of the 
 stone. 
 
 When the point is thus made so fine and sharp as to be in- 
 visible to the eye, it should be smoothed by rubbing it on the 
 surface of a soft and clean piece of leather. 
 
 99. Weight. The average weights of the different sizes of 
 compasses, including the brass head of the Jacob-staff, be- 
 ginning wth the smallest, are respectively 5, 7, and 9^ 
 pounds.
 
 DESCRIPTION OF INSTRUMENTS. 61 
 
 THE VERNIER. 
 
 100. A Vernier is an auxiliary scale for measuring smaller 
 divisions than those into which a graduated scale or limb is 
 divided.* The smallest reading of the vernier, or least count, 
 is the difference in length between one division on the gradu- 
 ated scale or limb, and one on the vernier. If the divisions on 
 the vernier are smnller than those on the limb, the vernier is 
 direct; if the reverse, retrograde. 
 
 6 7 8 9 10 11 12 13 14 15 16 17 
 
 Let LM represent any scale divided into tenths, and we wish 
 to measure or read to tenths of these divisions, i.e. to y^ v . 
 Using a direct vernier, we should have 10 spaces on it equal to 
 9 on the scale, and each one of them equal to T 9 of y 1 ^, or y^, 
 of the scale graduation ; giving a least count of y 1 ^ T f 7 = y-^, 
 as desired. To read to twentieths of the divisions on the scale, 
 we should have 20 divisions on the vernier corresponding to 19 
 on the scale, or each space on the vernier equal to if ' iV = '3inr> 
 and giving a least count of -ffo ^fa = -^fa. 
 
 In general, if s = the smallest division of the scale or limb, 
 v = the smallest division of the vernier, 
 n = number of divisions on the vernier, 
 
 we shall have least count = s v = - 
 
 n 
 
 Or, the least count of a vernier is equal to the smallest division 
 of the scale or limb divided by the number of divisions on the 
 vernier.f 
 
 If s = % degree, and n = 30, as ordinarily found on transit 
 
 * It derives its name from Peter Vernier, 1631. 
 
 t It is evidently immaterial whether LM be straight or curved.
 
 62 PLANE SURVEYING. 
 
 plates, the least count will be -$-30 = ^ of a degree = one 
 minute. 
 
 If s = $ degree, and n = 40, oftentimes found on vertical 
 arcs to solar attachments, the smallest reading = \ -H 40 = T | 7 
 of a degree = minute. 
 
 To space a vernier for a given least count, say 10", on a limb 
 
 graduated to 10', we must have n = - = = 60 spaces, 
 
 s-v | 
 covering 59 spaces on the limb. 
 
 101. To read an Instrument having a vernier consists in 
 determining the number of units and fractional parts thereof, 
 into which its scale or limb may be divided, from the zero point 
 on the limb, where the graduation begins, to the zero point of 
 the vernier. 
 
 It is accomplished as follows : Take the reading of the scale, 
 as shown by the last graduation preceding the zero of the ver- 
 nier ; then find a line on the vernier which coincides with a line 
 on the scale. The number of this line, as indicated by the 
 graduation on the vernier, shows how many units of the least 
 count are to be added to the first reading. 
 
 EXERCISES. 
 
 1. A levelling-rod is graduated into feet, tenths, and hun- 
 dredths. It is required to space a direct vernier so that the 
 rod may be read to thousandths of a foot. 
 
 2. An arc is graduated into quarter-degrees, and a vernier 
 of 30 parts covers 29 parts of the arcs ; find the least count. 
 
 3. A scale is divided into inches and tenths of an inch ; plan 
 a direct vernier by means of which the scale may be read to 
 j-l^ of an inch. 
 
 Plan a retrograde vernier to accomplish the same object. 
 
 4. Design a vernier which when applied to a limb graduated 
 into 20' will give a least count of 20". 

 
 SURVEYOR'S TRANSIT. 
 
 NOTE. The principal part of the description of the Compass and Tran- 
 sit, and the plates for the engraving of these instruments, were kindly 
 furnished by Messrs. W. & L. E. Gurley, Troy, X.Y.
 
 DESCRIPTION OF INSTRUMENTS. 
 
 65 
 
 SURVEYOR'S TRANSIT. 
 
 102. The essential parts of the Tran- 
 sit, as shown in the cut, are the telescope 
 with its axis and two supports, the cir- 
 cular plates with their attachments, the 
 sockets upon which the plates revolve, 
 the levelling-head, and the tripod on 
 which the whole instrument stands. 
 
 The telescope is from 10 to 11 inches 
 long, firmly secured to an axis having 
 its bearings nicely fitted in the stand- 
 ards, and thus enabling the telescope 
 to be moved in either direction, or 
 turned completely around if desired. 
 
 The different parts of the telescope 
 are shown in the marginal figure. 
 
 The object-glass is composed of two 
 lenses, so as to show objects without 
 color or distortion, is placed at the end 
 of a slide having two bearings, one at 
 the end of the outer tube, the other in 
 the ring (7(7, suspended within the tube 
 by four screws, only two of which are 
 shown in the cut. 
 
 The object-glass is carried out or in 
 by a pinion working in a rack attached 
 to the slide, and thus adjusted to ob- 
 jects either near or remote as desired. 
 
 The eye-piece is made up of four 
 plano-convex lenses, which, beginning 
 at the eye-end, are called respectively 
 the eye, the field, the amplifying, and 
 the object-lenses, the whole forming a 
 compound microscope having its focus 
 in the plane of the cross-wire ring BB.
 
 66 PLANE SURVEYING. 
 
 The eye-piece is brought to its proper focus usually by twist- 
 ing its milled end, the spiral movement within carrying the eye- 
 tube out or in as desired ; sometimes a pinion, like that which 
 focuses the object-glass, is employed for the same purpose. 
 
 103. The Cross-Wires are two fibres of spider-web or very 
 fine platinum wire, cemented into the cuts on the surface of a 
 
 metal ring, at right angles to each other, so as to divide the 
 open space in the centre into quadrants. 
 
 104. Optical Axis. The intersection of the wires forms a 
 very minute point, which, when the\ r are adjusted, determines 
 the optical axis of the telescope, and enables the surveyor to 
 fix it upon an object with the greatest precision. 
 
 The imaginary line passing through the optical axis of the 
 telescope is termed the "line of collimation," and the opera- 
 tion of bringing the intersection of the wires into the optical 
 axis, is called the "adjustment of the line of collimation." 
 This will be hereafter described. 
 
 105. The Standards of the Transit are firmly attached by 
 their expanded bases to the upper plate, one of them having 
 near the top, as shown in the cut, a little movable box, actu- 
 ated by a screw underneath, by which the telescope axis is made 
 truly horizontal, as will be hereafter described.
 
 DESCRIPTION OF INSTRUMENTS. 
 
 67 
 
 The sectional view here given shows the interior construction 
 of the sockets of the transit, the manner in which it is detached 
 from the spindle, and the means by which it can be taken apart 
 if desired. 
 
 In the figure, the limb BB is attached to the main socket C, 
 which is itself carefully fitted to the conical spindle //, and held 
 in place by the spring catch S. 
 
 The upper plate, AA, carrying the compass-circle, standards, 
 etc., is fastened to the flanges of the socket 7f, which is fitted 
 to the upper conical surface of the main socket (7; the weight 
 of all the parts being supported on the small bearings of the 
 end of the socket, as shown, so as to turn with the least possible 
 friction. 
 
 A small conical centre, in which from below is inserted a 
 strong screw, is brought down firmly upon the upper end of 
 the main socket (7, and thus holds the two plates of the 
 instrument securely together, while at the same time allowing 
 them to move freely around each other in use.
 
 68 PLANE SURVEYING. 
 
 A small disc above the conical centre contains the steel cen- 
 tre-pin upon which rests the needle, as shown ; the disc is fas- 
 tened to the upper plate by two small screws, as represented. 
 
 The main socket with all its parts is of the best bell-metal 
 and is most carefully and thoroughly made, the long bearing of 
 the sockets insuring their firm and easy movement, while at 
 the same time they are entirely out of the reach of dust, or 
 other source -of wear. 
 
 When desii'ed, the whole upper part of the instrument can be 
 taken off from the spindle by pulling out the head of the spring 
 catch at S, and when replaced will be secured by the self-acting 
 spring of the catch. 
 
 The figure also shows the covers of the levelling-screws, the 
 shifting centre of the lower le veiling-plate, and the screw and 
 loop for the attachment of the plummet. 
 
 The compass-box, containing the needle, etc., is covered by 
 a glass to exclude the moisture and air; the circle is silvered, 
 and is divided on its upper surface or rim into degrees and 
 half-degrees, the degree marks being also cut down on its inner 
 edge, and figured from to 90 on each side of the centre or 
 line of zero. 
 
 106. The Magnetic Needle is four to five inches long in the 
 different sizes of transits, its brass cap having inserted in it a 
 little socket or centre of hardened steel, perfectly polished, and 
 this resting upon the hardened and polished point of the centre- 
 pin, allows the needle to play freely in a horizontal direction, 
 and thus take its direction in the magnetic meridian. The 
 needle has its north end designated by a scallop or other mark, 
 and on its south end 'a small coil of fine brass wire, easily 
 moved, so as to bring both ends of the needle to the same 
 level. The needle is lifted from the pin by a concealed spring 
 underneath the upper plate, actuated by a screw shown above, 
 thus raising the button so as to check the vibrations of the 
 needle, or bring it up against the glass when not in use, to 
 avoid the unnecessary wear of the pivot.
 
 DESCRIPTION OF INSTRUMENTS. 69 
 
 107. The Clamp and Tangent Movement, shown in the en- 
 graving, page 64, attached to the plates, serves to fasten the two 
 plates together, so that by the tangent screw they can be slowly 
 moved around each other in either direction, or loosened at will 
 and moved by the hand, thus enabling one to direct the tele- 
 scope rapidly and accurately to the point of sight. 
 
 The Two Levels are shown placed at right angles to each other 
 so as to level the plate in all directions, and adjusted by turn- 
 ing the capstan-head screws at their ends, by a small steel 
 adjusting-pin. The glass vials used in the- levels are ground 
 on their upper interior surface, so as to make the bubble move 
 evenly and with great sensitiveness. 
 
 108. The Lower Plate, or Limb BB, is divided on its upper 
 surface usually into degrees and half-degrees and generally 
 figured in two rows ; viz., from to 360, and from to 90 each 
 way. 
 
 109. The Verniers are double, having on each side of the 
 zero mark thirty equal divisions corresponding precisely with 
 twenty-nine half-degrees of the limb ; they thus read to single 
 minutes, and the number passed over is counted in the same 
 direction in which the vernier is moved. 
 
 The use of two opposite verniers in this and other instru- 
 ments gives the means of " cross-questioning" the graduations, 
 the perfection with which they are centred, and the dependence 
 which can be placed upon the accuracy of the angles indicated. 
 
 Reflectors of silver or celluloid, as in the mountain transit, 
 are often used to throw more light upon the divisions, and more 
 rarely shades of ground glass are employed to give a clear but 
 more subdued light. 
 
 110. The Graduations are made commonly on the brass sur- 
 face of the limb, afterwards filled with black wax, and then 
 finished and silvered. Many instruments, however, have a 
 solid silver plate put over the brass, and the graduations made 
 on the silver itself.
 
 70 PLANE SURVEYING. 
 
 The last is more costly, but insures a finer graduation, with 
 less liability to tarnish or change color. 
 
 111. The Sockets of the transit are compound ; the interior 
 spindle attached to the vernier plate, turning in the exterior 
 socket C when an angle is taken on the limb ; but when the 
 plates are clamped together, the exterior socket itself, and with 
 it the whole instrument, revolves in the socket of the levelling- 
 head. 
 
 The sockets are made with the greatest care, the surfaces 
 being truly concentric with each other, and the bell-metal or 
 composition of which the}" are composed, of different degrees 
 of hardness, so as to cause them to move upon each other easily 
 and with the least possible wear. 
 
 The levelling-head also consists of two plates connected to- 
 gether by a socket, having at its end a hemispherical nut, fitting 
 into a corresponding cavity in the lower plate. 
 
 The plates are 'inclined to each other or made parallel at will 
 by four levelling-screws, of which only two are shown in the 
 section. 
 
 The screws are of bronze or hard composition metal and fitted 
 to long nuts of brass, screwed into the upper parallel plate ; 
 and, as will be noticed, have threads only on the upper ends, 
 the lower part of their steins turning closely in the lower un- 
 threaded part of the nuts. 
 
 By this arrangement dust is excluded from the lower end of 
 the screws, while the brass cover above equally protects the 
 other end. 
 
 The screws rest in little cups or sockets, which are secured 
 to their ends and in which they turn without marring the sur- 
 face of the lower plate, the cups also permitting the screws to 
 be shifted from side to side, or turned around in either direc- 
 tion on the lower plate. 
 
 The clamp and tangent movement of the levelling-head serves 
 to turn the whole instrument upon its sockets, so as to fix the 
 telescope witli precision upon any given point, and when un- 

 
 DESCRIPTION OF INSTRUMENTS. 71 
 
 clamped allowing it to be directed approximately by hand. 
 The tangent screws, as will be seen, press on opposite sides of 
 the clamp-piece, and thus insure a very fine and solid movement 
 of the instrument. 
 
 112. The Lower Levelling-Plate is made in two pieces the 
 upper one, which is screwed fast to the top of the tripod, having 
 a large opening in its centre, in which the smaller lower one is 
 shifted from side to side, or turned completely around. 
 
 By this simple arrangement, termed a shifting centre, the 
 instrument is easily moved over the upper plate, and the plum- 
 met which hangs from the centre P, set precisely over a point, 
 without moving the tripod. 
 
 113. The Levelling-Head of the engineer's transit is attached 
 to the sockets by a screw and washer below ; it can be removed 
 for cleaning, oiling, etc., but should be in place when the in- 
 strument is in use, or packed for transportation. 
 
 114. The Tripod has three mahogany legs, the upper ends 
 of which are pressed firmly on each side of a strong tenon on 
 the solid bronze head by a bolt and nut on opposite sides of the 
 leg ; the nut can also be screwed up at will by a wrench fur- 
 nished for the purpose, and thus kept firm. 
 
 The lower end of the leg has a brass shoe with iron point, 
 securely fastened and riveted to the wood. 
 
 115. To Adjust the Transit. Every instrument should leave 
 the hands of the maker in complete adjustment ; but all are so 
 liable to derangement by accident or careless use, that we deem 
 it necessary to describe particularly those which are most likely 
 to need attention. 
 
 The principal adjustments of the transit are : 
 
 1 . The Levels. 
 
 2. The Line of Collimation. 
 
 3. The Standards.
 
 72 PLANE SURVEYING. 
 
 116. To Adjust the Levels. Set up the instrument upon its 
 tripod as nearly level as may be, and having undamped the 
 plates, bring the two levels above and on a line with the two 
 pairs of levelling-screws ; then, with the thumb and first finger 
 of each hand clasp the heads of two, opposite ; and, turning 
 both thumbs in or out, as may be needed, bring the bubble of 
 the level directly over the screws, exactly to the centre of the 
 opening. Without moving the instrument, proceed in the same 
 manner to bring the other bubble to its centre ; after doing 
 this, the level first corrected may be thrown a little out ; bring 
 it in again ; and when both are in place, turn the instrument 
 half-way around : if the bubbles both come to the centre, they 
 
 would need no correction, but if not, with the adjusting-pin 
 turn the small screws at the end of the levels until the bubbles 
 are moved over half the error ; then bring the bubbles again 
 into the centre by the levelling-screws, and repeat the operation 
 until the bubbles will remain in the centre during a complete 
 revolution of the instrument, and the adjustment will be 
 complete. 
 
 117. To Adjust the Line of Collimation. To make this 
 adjustment, which is, in other words, to bring the inter- 
 section of the wires into the optical axis of the telescope, so 
 that the instrument, when placed in the middle of a straight 
 line, will, by the revolution of the telescope, cut its extremities, 
 proceed as follows : 
 
 Set the instrument firmly on the ground and level it care- 
 fully ; and then, having brought the wires into the focus of the 
 eye-piece, adjust the object-glass on some well-defined point, 
 as the edge of a chimney or other object, at a distance of from 
 200 to 500 feet ; determine if the vertical wire is plumb, by 
 clamping the instrument firmly and applying the wire to the 
 vertical edge of a building, or observing if it will move parallel 
 to a point taken a little to one side : should any deviation be 
 manifested, loosen the cross-wire screws, and by the pressure 
 of the hand on the head outside the tube, move the ring around 
 until the error is corrected. 

 
 DESCRIPTION OF INSTRUMENTS. 73 
 
 The wires being thus made respectively horizontal and 
 vertical, fix their point of intersection on the object selected ; 
 clamp the instrument to the spindle, and having revolved the 
 telescope, find or place some good object in the opposite direc- 
 tion, and at about the same distance from the instrument as 
 the first object assumed. 
 
 Great care should always be taken in turning the telescope, 
 that the position of the instrument upon the spindle is not in 
 the slightest degree disturbed. 
 
 Now, having found or placed an object which the vertical 
 wire bisects, unclamp the instrument, turn it half-way around, 
 and direct the telescope to the first object selected ; having 
 bisected this with the wires, again clamp the instrument, 
 revolve the telescope, and note if the vertical wire bisects the 
 second object observed. 
 
 Should this happen, it will indicate that the wires are in 
 adjustment, and the points bisected are with that of the centre 
 of the instrument, in the same straight line. 
 
 If not, however, the space which separates the wires from 
 the second point observed, will be double the deviation of that 
 point from a true straight line, which may be conceived as 
 drawn through the first point and the centre of the instrument, 
 since the error is the result of two observations, made with the 
 wires when they are out of the optical axis of the telescope. 
 
 For, as in the diagram, let A represent the centre of the 
 instrument, and BC the imaginary straight line, upon the ex- 
 tremities of which the line of collimation is to be adjusted. 
 
 B represents the object first selected, and D the point which 
 the wires bisected, when the telescope was made to revolve. 
 
 When the instrument is turned half around, and the telescope 
 again directed to .B, and once more revolved, the wires will
 
 74 PLANE SURVEYING. 
 
 bisect an object -E, situated as far to one side of the true line 
 as the point D is on the other side. 
 
 The space DE, is therefore the sum of two deviations of the 
 wires from a true straight line, and the error is made very 
 apparent. 
 
 In order to correct it, use the two capstan-head screws on 
 the sides of the telescope, these being the ones which affect the 
 position of the vertical wire. 
 
 Remember that the eye-piece inverts the position of the 
 wires, and therefore, that in loosening one of the screws and 
 tightening the other on the opposite side, the operator must 
 proceed as if to increase the error observed. Having in this 
 manner moved back the vertical wire until, by estimation, one- 
 quarter of the space DE has been passed over, return the 
 instrument to the point jB, revolve the telescope, and if the 
 correction has been carefully made, the wires will now bisect a 
 point C, situated midway between D and E, and in the pro- 
 longation of the imaginary line, passing through the point B 
 and the centre of the instrument. 
 
 To ascertain if such is the case, turn the instrument half 
 around, fix the telescope upon B, clamp to the spindle, and 
 again revolve the telescope towards C. If the wires again 
 bisect it, it will prove that they are in adjustment, and that 
 the points B, A, C, all lie in the same straight line. 
 
 Should the vertical wire strike to one side of (7, the error 
 must be corrected precisely as above described, until it is 
 entirely removed. 
 
 118. To Adjust the Standards. In order that the wires may 
 trace a vertical line as the telescope is moved up or down, it is 
 necessary that both the standards of the telescope should be of 
 precisely the same height. 
 
 To ascertain this and make the correction if needed, proceed 
 as follows : 
 
 Having the line of collimation previously adjusted, set up the 
 instrument in a position where points of observation, such as 

 
 DESCRIPTION OF INSTRUMENTS. 75 
 
 the point and base of a lofty spire, can be selected, giving a 
 long range in a vertical direction. 
 
 Level the instrument, fix the wires on the top of the object, 
 and clamp to the spindle ; then bring the telescope down, until 
 the wires bisect some good point, either found or marked at the 
 base ; turn the instrument half around, fix the wires on the 
 lower point, clamp to the spindle, and raise the telescope to 
 the highest object. 
 
 If the wires bisect it, the vertical adjustment is effected ; if 
 they are thrown to either side, this would prove that the stand- 
 ard opposite that side was the highest, the apparent error being 
 double that actually due to this cause. 
 
 To correct it, one of the bearings of the axis is made mov- 
 able, so that by turning a screw underneath this sliding piece, 
 as well as the screws which hold on the cap of the standard, the 
 adjustment is made with the utmost precision. 
 
 OTHER ADJUSTMENTS OF THE TRANSIT. 
 
 Besides the three adjustments already described which are 
 all that the surveyor will ordinarily have to make there are 
 those of the needle and the object-glass slide which may some- 
 times be required. 
 
 The first is given with the description of the compass ; the 
 last will now be described. 
 
 119. To Adjust the Object-Slide. Having set up and levelled 
 the instrument, the line of collimation being also adjusted for 
 objects from 300 to 500 feet distant, clamp the plates securely, 
 and fix the vertical cross-wire upon an object as distant as may 
 be distinctly seen ; then, without disturbing the instrument, 
 throw out the object-glass, so as to bring the vertical wire upon 
 an object as near as the range of the telescope will allow. 
 Having this clearly in mind, unclamp the limb, turn the instru- 
 ment half-way around, reverse the eye-end of the telescope, 
 clamp the limb, and with the tangent-screw bring the vertical
 
 76 PLANE SURVEYING. 
 
 wire again upon the near object ; then draw in the object-glass 
 slide until the distant object first sighted upon is brought into 
 distinct vision. If the vertical wire strikes the same line as at 
 first, the slide is correct for both near and remote objects ; and, 
 being itself straight, for all distances. 
 
 But if there be an error, proceed as follows : first, with the 
 thumb and forefinger twist off the thin brass tube that covers 
 the screws CC shown in the sectional view of the telescope, 
 p. 65. Next, with the screw-driver, turn the two screws CO 
 on the opposite ides of the telescope, loosening one and tight- 
 ening the other, so as apparently to increase the error, making, 
 by estimation, one-half the correction required. 
 
 Then go over the usual adjustment of the line of collimation , 
 and having it completed, repeat the operation above described ; 
 first sighting upon the distant object, then finding a near one 
 in line, and then reversing, making correction, etc., until the 
 adjustment is complete. 
 
 120. To Use the Transit. The instrument should be set up 
 firmly, the tripod legs being pressed into the ground, so as to 
 bring the plates as nearly level as convenient ; the plates should 
 then be carefully levelled and property clamped, the zeros of the 
 verniers and limb brought into line by the upper tangent-screw, 
 and the telescope directed to the object by the tangent-screws 
 of levelling-head. 
 
 The angles taken are then read off upon the limb, without 
 subtracting from those given by the verniers, in any other 
 position. 
 
 Before an observation is made with the telescope, the eye- 
 piece should be moved in or out, until the wires appear distinct 
 to the eye of the operator ; the object-glass is then adjusted by 
 turning the pinion-head until the object is seen clear and well- 
 defined, and the wires appear as if fastened to its surface. 
 
 The intersection of the wires, being the means by which the 
 optical axis of the telescope is defined, should be brought pre- 
 cisely upon the centre of the object to which the instrument is 
 directed.
 
 DESCRIPTION OF INSTRUMENTS. 77 
 
 The needle is used, as in the compass, to give the bearing of 
 lines, and as a rough check upon the angles obtained by the 
 verniers and limb ; but its employment is only subsidiary to the 
 general purposes of the transit. 
 
 121. Attachments of Transits. The engraving of the Sur- 
 veyor's Transit represents the attachments often applied to the 
 Engineer's Transit, viz. : vertical circle, level on telescope, 
 and clamp and tangent to telescope axis. They are of use 
 where approximate levelling and vertical angles are to be taken 
 in connection with the ordinary use of the transit, and with 
 their adjustments, etc., will now be described. 
 
 122. The Vertical Circle firmly secured to the axis of the 
 telescope is 4^ inches in diameter, plated with silver, divided to 
 hnlf-degrees, and with its vernier enables the surveyor to obtain 
 vertical angles to single minutes. 
 
 123. The Level on Telescope consists of a brass tube about 
 6| inches long, each end of which is held between two capstan- 
 nuts connected with a screw or stem attached to the under side 
 of the telescope tube. 
 
 124. The Clamp and Tangent consists of an arm at one end 
 encircling the telescope axis, and at the other connected with 
 the tangent-screw ; the clamp is fastened at will to the axis by 
 a clamp-screw, inserted at one side of the ring, and then by 
 turning the taugent-screw the telescope is raised or lowered as 
 desired. 
 
 125. To Adjust the Vertical Circle. Having the instrument 
 firmly set up and carefully leveled, bring into line the zeros of 
 the circle and vernier, and with the telescope find or place some 
 well-defined point or line, from 200 to 300 feet distant, which 
 is cut by the horizontal wire. 
 
 Turn the instrument half-way around, revolve the telescope, 
 and fixing the wire upon the same point as before, note if the 
 zeros are again in line.
 
 78 PLANE SURVEYING. 
 
 If not, loosen the capstan-head screws, which fasten the 
 vernier, and move the zero of the vernier over half the error ; * 
 bring the zeros again into coincidence, and proceed precisely as 
 at first, until the error is entirely corrected, when the adjust- 
 ment will be complete. 
 
 This method is not applicable when only an arc of a circle is 
 attached. The adjustment may then be made as follows: 
 Observe successively from each of the two points to the other, 
 and as before use half the error in adjusting the vernier. 
 Verifv b t y repetition. 
 
 A slight error may be most readily removed by putting the 
 zeros in line and then moving the wire itself over half the 
 interval. 
 
 126. The Level is Adjusted by bringing the bubble carefully 
 into the centre by the nuts at each end ; and when there is a 
 vertical circle on the instrument, this should be done when the 
 zeros of circle and vernier are in line and in adjustment ; when 
 there is no vertical circle, proceed as follows : 
 
 127. To Adjust the Level on Telescope. Choose a piece of 
 ground nearly level, and having set the instrument firmly, level 
 the plates carefully, and bring the bubble of the telescope into 
 the centre with the tangent-screw. Measure in any direction 
 from the instrument, from 100 to 300 feet, and drive a stake, 
 and on the stake set a staff, and note the height cut by the 
 horizontal wire ; then take the same distance from the instru- 
 ment in an opposite direction, and drive another slake. 
 
 On that stake set the staff, and note the height cut by the 
 wire when the telescope is turned in that direction. 
 
 The difference of the two observations is evidently the dif- 
 ference of level of the two stakes. 
 
 Set the instrument over the lowest stake, or that upon which 
 
 * Called Index Error. It may be rectified as here shown, or each obser- 
 vation corrected by this amount.
 
 DESCRIPTION OF INSTRUMENTS. 79 
 
 the greatest height was indicated, and bring the levels on the 
 plates and telescope into adjustment as at first. 
 
 Then, with the staff, measure the perpendicular distance from 
 the top of the stake to the centre of one of the horizontal cross- 
 wire screw-heads ; from that distance subtract the difference of 
 level between the two stakes and mark the point on the staff 
 thus found ; place the staff on the other stake, and with the 
 tangent-screw bring the horizontal wire to the mark just found, 
 and the line will be level. 
 
 The telescope now being level, bring the bubble of the level 
 into the centre, by turning the little nuts at the end of the tube, 
 and noting again if the wires cut the point on the staff ; screw 
 up the nuts firmly and the adjustment will be completed. 
 
 128. To Take Apart the Surveyor's Transit. When it is 
 necessary to separate the plates of the transit, proceed as 
 follows : 
 
 (1) Remove the clamp-screw and take off the head of the 
 pinion, both on the north end and outside the compass circle. 
 
 (2) Unscrew the bezel ring containing the glass cover of the 
 compass, remove the needle and button beneath it, and take 
 out the two small screws so as to remove the disc. 
 
 (3) Take the instrument from its spindle, and with a large 
 screw-driver take out the screw from the underside of the coni- 
 cal centre (see figure, p. 67). 
 
 (4) Drive out the centre from below by a round piece of 
 wood, holding the instrument vertical so that the centre will 
 not bruise the circle. 
 
 (5) Set the instrument again upon its spindle, take out the 
 clamp-screw to the tangent movement of the limb, and the work 
 is complete. To put the transit together again, proceed exactly 
 the reverse of the operation thus described. 
 
 129. The Solar Attachment is essentially the solar apparatus 
 of Burt placed upon the cross-bar of the ordinary transit, the 
 polar axis only being directed above instead of below, as in the 
 solar compass.
 
 80 PLANE SURVEYING. 
 
 A little circular disc of an inch and a half diameter, and hav- 
 ing a short, round pivot projecting above its upper surface, is 
 first securely screwed to tho telescope axis. 
 
 Upon this pivot rests the enlarged base of the polar axis, which 
 is also firmly connected with the disc by four capstan-head 
 screws passing from the under side of the disc into the base 
 already named. 
 
 These screws serve to adjust the polar axis, as will be ex- 
 plained hereafter. 
 
 130. The Hour Circle surrounding the base of the polar axis 
 is easily movable about it, and can be fastened at any point 
 desired by two flat-head screws above. It is divided to five 
 minutes of time ; is figured from I. to XII., and is read by a 
 small index fixed to the declination circle, and moving with it. 
 
 A hollow cone, or socket, fitting closely to the polar axis, 
 and made to move snugly upon it, or clamped at any point 
 desired by a milled-head screw on top, furnishes by its two 
 expanded arms below a firm support for the declination arc, 
 which is securely fastened to it by two large screws, as shown. 
 
 131. The Declination Arc is of about 5 inches radius, is 
 divided to quarter degrees, and reads by its vernier to single 
 minutes of arc, the divisions of both vernier and limb being in 
 the same plane. 
 
 The declination arm has the usual lenses and silver plates on 
 the two opposite blocks, made precisely like those of the ordi- 
 nary solar compass, but its vernier is outside the block, and 
 more easily read. 
 
 The declination arm has also a clamp and tangent movement, 
 as shown in the cut. The arc of the declination limb is turned 
 on its axis, and one of the other solar lens used, as the sun is 
 north or south of the equator ; the cut shows its position when 
 it is north. 
 
 The Latitude is set off by means of a large vertical limb hav- 
 ing a radius of 2| inches ; the arc is divided to twenty minutes,
 
 TRANSIT WITH SOLAR ATTACHMENT.
 
 DESCRIPTION OF INSTRUMENTS. 83 
 
 is figured from the centre, each way, up to 80, and is read by 
 its vernier to single minutes. 
 
 It has also a clamp-screw inserted near its centre, by which 
 it can be set fast to the telescope axis in any desired position. 
 
 The vernier of the vertical limb is made movable by the 
 tangent-screw attached, so that its zero and that of the limb 
 are readily made to coincide when, in adjusting the limb to the 
 level of the telescope, the arc is clamped to the axis. 
 
 The usual tangent movement to the telescope axis serves, of 
 course, to bring the vertical limb to the proper elevation, as 
 hereafter described. 
 
 A level on the under side of the telescope, with ground vial 
 and scale, is indispensable in the use of the solar attachment. 
 
 The divided arcs, verniers, and hour circle, are all on silver 
 plate, and are thus easily read and preserved from tarnishing. 
 
 THE ADJUSTMENTS. 
 
 132. The Solar Lenses and Lines are adjusted precisely like 
 those of the ordinary solar, the declination arm being first de- 
 tached by removing the clamp and tangent screws, and the 
 conical centre with its two small screws, by which the arm is 
 attached to the arc. 
 
 The adjuster, which is a short bar f urn, shed with every 
 instrument, is then substituted for the declination arm, the 
 conical centre screwed into its place at one end, and the 
 clamp-screw into the other, being inserted through the hole left 
 by the removal of the tangent-screw, thus securing the adjuster 
 firmly to the arc. 
 
 The arm is then turned to the sun, as described in the article 
 on the Solar Compass, and reversed by the opposite faces of 
 the blocks upon the adjuster, until the image will remain in the 
 centre of the equatorial lines. This adjustment is very rarely 
 needed, as the lenses are cemented in their cells, and the plates 
 securely fastened.
 
 84 PLANE SURVEYING. 
 
 133. The Vernier of the Declination Arc is adjusted by set- 
 ting the vernier at zero, and then raising or lowering the tele- 
 scope by the tangent-screw, until the sun's image appears exactly 
 between the equatorial lines. 
 
 Having the .telescope axis clamped firmly, carefully revolve 
 the arm until the image appears on the other plate. 
 
 If preciseh' between the lines, the adjustment is complete ; 
 if not, move the declination arm by its tangent-screw, until the 
 image will come precisely between the lines on the two opposite 
 plates ; clamp the arm and remove the index error by loosening 
 two flat-head screws on the back, which fasten the movable arc 
 to the declination limb ; place the zero of the limb and vernier 
 in exact coincidence and the adjustment is finished. 
 
 134. To Adjust the Polar Axis. First level the instrument 
 carefully by the long level of the telescope, using in the opera- 
 tion the tangent movement of the telescope axis in connection 
 with the levelling screws of the parallel plates, until the bubble 
 will remain in the centre during a complete revolution of the in- 
 strument upon its axis. 
 
 Place the equatorial sights on the top of the blocks as closely 
 as is practicable with the distinct view of a distant object ; and 
 having previously set the declination arm at zero, sight through 
 the interval between the equatorial sights and the blocks at some 
 definite point or object, the declination arm being placed over 
 either pair of the capstan-head screws on the under side of the 
 disc. 
 
 Keeping the declination arm upon the object with one hand, 
 with the other turn the instrument half around on its axis, and 
 sight upon the same object as before. If the sight strikes 
 either above or below, move the two capstan-head screws imme- 
 diately under the arm, loosening one and tightening the other 
 as may be needed, until half the error is removed. 
 
 Sight again and repeat the operation, if needed, until the 
 sight will strike the same object in both positions of the instru- 
 ment, when the adjustment of the axis in one direction will be 
 complete.
 
 DESCRIPTION OF INSTRUMENTS. 85 
 
 Now turn the instrument at right angles, keeping the sight 
 still upon the same object as before ; if it strikes the same point 
 when sighted through, the axis will be truly vertical in the sec- 
 ond position of the instrument. 
 
 If not, bring the sight upon the same point by the other pair 
 of capstan-head screws now under the declination arc, reverse 
 as before, and continue the operation until the same object will 
 keep in the sight in all positions, when the polar axis will be 
 made precisely at right angles to the level and to the line of 
 collimation of the transit. 
 
 It should here be noted that as this is by far the most delicate 
 and important adjustment of the solar attachment, it should be 
 made with the greatest care, the bubble kept perfectly in the 
 centre and frequently inspected in the course of the operation. 
 
 135. To Adjust the Hour Arc. Whenever the instrument is 
 set in the meridian, as will be hereafter described, the index of 
 the hour arc should read apparent time. 
 
 If not, loosen the two flat-head screws on the top of the hour 
 circle, and with the hand turn the circle around until it does, 
 fasten the screws again, and the adjustment will be complete. 
 
 To obtain mean time, of course the correction of the equa- 
 tion for the given day, as given in the Nautical Almanac, must 
 always be applied. 
 
 136. To Find the Latitude. First level the instrument very 
 carefully, using, as before, the level of the telescope until the 
 bubble will remain in the centre during a complete revolution 
 of the instrument, the tangent movement of the telescope being 
 used in connection with the levelling screws of the parallel plates, 
 and the axis of the telescope firmly clamped. 
 
 Next clamp the vertical arc so that its zero and that of its 
 vernier coincide as near as may be, and then bring them into 
 exact line by the tangent-screw of the vernier. 
 
 Then, having the declination of the sun for 12 o'clock of the 
 given day as affected by the meridional refraction carefully set
 
 86 PLANE SURVEYING. 
 
 off upon the declination arc, note also the equation of time and 
 fifteen or twenty minutes before noon, the telescope being 
 directed to the north, and the object-end lowered until, by 
 moving the instrument upon its spindle and the declination arc 
 from side to side, the sun's image is brought nearly into posi- 
 tion between the equatorial lines. Now bring the declination 
 arc directly in line with the telescope, clamp the axis firmly, 
 and with the tangent-screw bring the image precisely between 
 the lines and keep it there with the tangent-screw, raising it as 
 long as it runs below the lower equatorial line, or, in other 
 words, as long as the sun continues to rise in the heavens. 
 
 When the sun reaches the meridian the image will remain 
 stationary for an instant, and then begin to rise on the plate. 
 
 The moment the image ceases to run below is of course ap- 
 parent noon, when the index of the hour arc should indicate 
 XII, and the latitude be determined by the reading of the ver- 
 tical arc. 
 
 It must be remembered, however, that the angle through 
 which the polar axis has moved in the operation just described 
 is measured from the zenith instead of the horizon, as in the 
 ordinary solar, so that the angle read on the vertical limb is the 
 complement of the latitude. 
 
 The latitude itself is readily found by subtracting this angle 
 from 90 ; thus at Troy, the reading of the limb being found as 
 above directed to be 47 16', the latitude will be 
 
 90 -47 16' = 42 44'. 
 
 It will be noticed that with this apparatus the latitude of any 
 place can be most easily ascertained without any index error, 
 as in the usual solar compass. 
 
 137, To Use the Solar Attachment. From the foregoing de- 
 scription it will be readily understood that good results cannot 
 be obtained from the solar attachment unless the transit is of 
 good construction, furnished with the appliances of a level on 
 telescope, clamp and tangent movement to axis, and vertical
 
 DESCRIPTION OF INSTRUMENTS. 87 
 
 arc with adjustable vernier, and the sockets or centres in such 
 condition that the level of the telescope will remain in the cen- 
 tre when the instrument is revolved upon either socket. 
 
 138. To Bun Lines with the Solar Attachment. Having set 
 off the complement of the latitude of the place on the vertical 
 arc, and the declination for the given day and hour as in the 
 solar, the instrument being also carefully levelled by the tele- 
 scope bubble, set the horizontal limb at zero, and clamp the 
 plates together, loosen the lower clamp so that the transit 
 moves easily upon its lower socket, set the instrument approxi- 
 mately north and south, the object-end of the telescope point- 
 ing to the north, turn the proper solar lens to the sun, and, 
 with one hand on the plates and the other on the revolving arm, 
 move them from side to side, until the sun's image is brought 
 between the equatorial lines on the silver plate. 
 
 The lower clamp of the instrument should now be fastened, 
 and any further lateral movement be made by the tangent- 
 screw of the tripod. The necessary allowance being made for 
 refraction, the telescope will be in the true meridian, and being 
 undamped, may be used like the sights of the ordinary solar 
 compass, but with far greater accuracy and satisfaction in 
 establishing meridian lines. Of course when the upper or 
 vernier plate is undamped from the limb, any angle read by the 
 verniers is an angle from the meridian, and thus parallels of 
 latitude or any other angles from the true meridian may be 
 established as with the solar compass. 
 
 The bearing of the needle, when the telescope is on the meri- 
 dian, will also give the variation of the needle at the point of 
 observation. 
 
 The declination of the needle being set off, and the needle 
 kept then at zero, or " with the sun," lines may be run by the 
 needle alone when the sun is obscured. 
 
 Though when not inconsistent with the remarks following the 
 table on page 95, the sun should be observed for direction at 
 every station.
 
 88 PLANE SURVEYING. 
 
 THE SAEGMULLER ATTACHMENT. 
 
 139. As seen in the engraving on the opposite page, it 
 consists essentially of a small telescope and level, the telescope 
 being mounted in standards, in which it can be elevated or 
 depressed. The standards revolve around an axis, called the 
 polar axis, which is fastened to the telescope axis of the transit 
 instrument. The telescope, called the " Solar Telescope," can 
 thus be moved in altitude and azimuth. Two pointers, attached 
 to the solar telescope to approximately set the instrument, are 
 so adjusted that when the shadow of the one is thrown upon 
 the other the sun will appear in the field of view. 
 
 140. Adjustments. When the apparatus is attached to the 
 transit, which instrument must be in good adjustment, its polar 
 axis should be at right angles both to the horizontal axis of the 
 main telescope and to the line of collimation. 
 
 TEST. Level the transit, and bring the bubble of each tele- 
 scope to the centre of its run. Revolve the solar telescope 
 about its polar axis, and if its bubble remains central, this ad- 
 justment is complete. If not, correct half the movement by 
 the adjusting screws at the base of the polar axis, and the other 
 by revolving the solar telescope on its horizontal axis. 
 
 141. Second. The line of collimation of the solar telescope 
 and the axis of its attached level must be parallel. 
 
 TEST. Bring the telescopes into the same vertical plane, and 
 the large bubble to the middle of its run. Direct then the tran- 
 sit telescope to a mark at a convenient distance away, say 100 
 feet; point also the " solar" to a mark above this equal to the 
 distance between their axes. If now the bubble of the solar 
 telescope is not in the middle of the tube, make it so by the 
 adjusting screws, and the instrument will be in adjustment. 
 
 When the combined instrument is in proper adjustment the 
 bubbles of the telescopes and plates will be in the middle of 
 their tubes, and the lines of collimation parallel.
 
 TRANSIT WITH SOLAR ATTACHMENT, 
 
 AS MADE BY FAUTH & C'<)., \VASUINOTON, D.C.
 
 DESCRIPTION OF INSTRUMENTS. 91 
 
 All the adjustments, including those of the transit, should be 
 frequently examined, and kept as nearly perfect as possible. 
 
 142. The advantages of solar attachments over the ordinar} 7 
 solar compass consist principally in the telescopic sight, and the 
 use of a vertical limb to set off declination and co-latitude. 
 
 LATITUDE. 
 By the Sun. With Saegmuller's Attachment. 
 
 143. Level the transit carefully, point the telescope south, 
 and elevate or depress the object-end, according as the decli- 
 nation of the sun is south or north, an amount equal to the 
 declination.* Bring the solar telescope into the vertical plane 
 of the main telescope, level it carefully, and clamp it. With 
 the solar telescope observe the sun a few minutes before his 
 culmination, bring the horizontal middle wire tangent to the 
 upper limb by moving the transit telescope in altitude and 
 azimuth, and keep it so by the slow-motion screws until the sun 
 ceases to rise. Then take the reading of the vertical arc, cor- 
 rect for index error, if any, for refraction due to altitude,! as 
 per table below ; diminish the result by the sun's semi-diameter, 
 and subtract the result from 90 for the latitude. 
 
 * For declination, consult a nautical almanac. 
 
 t Corrected for index error, the arc reading would be the sum of the 
 co-latitude and refraction. The refraction being due to the meridian alti- 
 tude of the sun, which altitude in the United States is equal to the alge- 
 braic sum of the declination and co-latitude.
 
 92 
 
 PLANE SURVEYING. 
 
 TABLE OF MEAN REFRACTIONS OF CELESTIAL OBJECTS FOR TEMPER- 
 ATURE 50, AND BAROMETER 29.6 INCHES. 
 
 ALTITUDE. 
 
 REFRACTION. 
 
 ALTITUDE. 
 
 REFRACTION. 
 
 10 
 
 5' 15" 
 
 20 
 
 2' 35" 
 
 11 
 
 4 47 
 
 25 
 
 2 02 
 
 12 
 
 4 23 
 
 30 
 
 1 38 
 
 13 
 
 4 03 
 
 35 
 
 1 21 
 
 14 
 
 3 45 
 
 40 
 
 1 08 
 
 15 
 
 3 30 
 
 45 
 
 57 
 
 16 
 
 3 17 
 
 50 
 
 48 
 
 17 
 
 3 04 
 
 60 
 
 33 
 
 18 
 
 2 54 
 
 70 
 
 21 
 
 19 
 
 2 44 
 
 80 
 
 10 
 
 By interpolation, the refraction, due to any altitude within the 
 limits of the table may be found. 
 
 LATITUDE BY CIRCUMPOLAR STAR. 
 
 144. The arc measuring the angle of elevation of the pole at any 
 station indicates the latitude of that station. If, then, the place of the pole 
 were indicated by a heavenly body, its altitude measured and corrected 
 for refraction would give at once the latitude. 
 
 There being no such body, a circumpolar star may be used. Take its 
 altitude at either culmination, subtract refraction due to altitude, and the 
 remainder, increased or diminished by the polar distance according as the 
 lower or upper culmination was observed, will give the latitude. 
 
 Better, when practicable, to observe both culminations, correct for re- 
 fraction, and take the arithmetical mean of the result. Still greater accu- 
 racy would be obtained by taking the mean of observations at upper and 
 lower transit of several eircumpolar stars. 
 
 If A and A' respectively denote the angles measuring, from the north, 
 the altitudes of a circumpolar star at its upper and lower culminations, 
 and r and r' the corresponding refractions, then, 
 
 latitude = 
 
 (r + r')].
 
 DESCRIPTION OF INSTRUMENTS. 93 
 
 To FIND THE MERIDIAN AND DECLINATION OF THE NEEDLE, 
 USING THE ATTACHMENT.* 
 
 145. First. Take the declination of the sun as given in the 
 Nautical Almanac for the given day, and correct it for retrac- 
 tion and hourly change. Incline the transit telescope until this 
 amount is indicated by its vertical arc. If the declination of 
 the sun is north, depress the object-end ; if south, elevate it. 
 Without disturbing the position of the transit telescope, bring 
 the solar telescope into the same vertical plane, and make it 
 horizontal by means of its level. The two telescopes will then 
 form an angle which equals the amount of the declination, and 
 the inclination of the solar telescope to its polar axis will be 
 equal to the polar distance of the sun. 
 
 Second. Without disturbing the relative positions of the two 
 telescopes, incline them and set the vernier to the co-latitude of 
 the place. 
 
 By moving the transit and the solar attachment around their 
 respective vertical axes, the image of the sun will be brought 
 into the field of the solar telescope, and after accurately bisecting 
 it the transit telescope must be in the meridian, and the compass- 
 needle indicates its deviation at that place. 
 
 The vertical axis of the solar attachment will then point to 
 the pole, the apparatus being in fact a small equatorial. Re- 
 volve the main telescope on its horizontal axis, and set a mark 
 at a convenient distance, 1000 feet if practicable. 
 
 Make a reverse observation as follows : Turn the transit 180* 
 in azimuth, and set off the declination, elevating or depressing 
 now the eye-end, according as the declination is south or north; 
 bring the object-end of the solar telescope to point in the direc- 
 tion of the eye-end of that of the main instrument, and level it. 
 Set the vertical arc to the co-latitude of the place, and complete 
 the observation as before. Reverse the large telescope on its 
 
 * For other methods, see Chapter III., p. 218, and Chapter VI., Solar 
 Compass.
 
 94 PLANE SURVEYING. 
 
 horizontal axis, and see if it points to the mark set by the direct 
 observation ; if it do not, take the mean of the two pointings 
 for the meridian. 
 
 If greater accuracy is required, make other observations at 
 different hours of the day, under different conditions of the 
 atmosphere, and compare results with those given in Chapters 
 III. and VI. 
 
 146. Time and azimuth are calculated from 
 an observed altitude of the sun by solving the 
 spherical triangle formed by the sun, the pole, 
 and the zenith of the place. The three sides, 
 SP, PZ, ZS. complements respectively of the 
 declination, latitude, and altitude are given, 
 and we hence deduce SPZ, the hour angle, 
 from apparent noon, and PZS the azimuth 
 of the sun.* 
 
 The " Solar Attachment " solves the same 
 spherical triangle by construction, for the 
 second process brings the vertical axis of the solar telescope to 
 the required distance ZP from the zenith, while the first brings 
 it to the required distance SP from the sun. 
 
 If the two telescopes, both being in position one in the 
 meridian, and the other pointing to the sun are now turned 
 on their horizontal axes, the vertical remaining undisturbed, 
 until each is level, the angle between their directions found 
 by sighting on a distant object is SPZ, the time from appar- 
 ent noon. 
 
 This gives an easy observation for correction of time-piece. 
 
 147. An error either in the declination or latitude will cause 
 an. error in the azimuth. 
 
 These errors in azimuth corresponding to one-minute error in 
 declination or latitude, for various hours and half-hours of the 
 
 * A Table of Equation of Time is given at the end of this book which 
 will be useful in solving analytically the spherical triangle PZS for time.
 
 DESCRIPTION OF INSTRUMENTS. 95 
 
 day, and for different latitudes, have been computed and 
 tabulated.* 
 
 THE SAEGMULLEB ATTACHMENT. 
 
 The table indicates the best time to observe the sun for merid- 
 ian, or to determine the true bearing of a line, to be soon after 
 sunrise or just before sunset. 
 
 However, on account of refraction at these times being great 
 and very uncertain, it is best in general not to make the obser- 
 vation when the sun is nearer the horizon than about 15 degrees. 
 Moreover, the solar apparatus should not be relied on for very 
 accurate work between 10 A.M. and 2 P.M. 
 
 An error in latitude does not cause an error in azimuth when 
 the sun is in the pole of the meridian. 
 
 148. The Stadia, or Micrometer, is a compound cross-wire 
 ring or diaphragm, shown below, having three horizontal wires, 
 of which the middle one is cemented to the ring as usual, while 
 the others, bb and cc, are fastened to small slides, held apart by 
 
 * By Professor H. T. Stewart, C. E., of the Western University of 
 Pennsylvania.
 
 96 
 
 PLANE SURVEYING. 
 
 a slender brass spring hoop, and actuated by independent screws 
 del, by which the distance between the two movable wires can be 
 adjusted to include a given space ; as, 1 foot on a rod 100 feet 
 distant. These wires will in the same manner include 2 feet on 
 
 a rod 200 feet distant, or half a foot at a distance of 50 feet, 
 and so on in the same proportion ; thus furnishing a means of 
 measuring distances especially over broken ground much 
 more easily, and even more accurately, than with a tape or 
 chain. 
 
 149. Its principles may be explained more fully as follows : 
 
 Let the above figure represent a section of a common tele- 
 scope with but two lenses, between which the diaphragm with 
 the stadia wires is placed, and assume that 
 
 / = the focal distance of the object-glass ; 
 
 p = the distance of the stadia wires a and b from each other ; 
 
 d = the horizontal distance of the object-glass to the stadia .
 
 DESCRIPTION OF INSTRUMENTS. 97 
 
 a = stadia reading (BA) ; 
 
 D = horizontal distance from middle of instrument to stadia. 
 
 The telescope is levelled and sighted to a levelling or stadia 
 rod, which is held vertically, hence at right angles with the line 
 of sight. According to a principle of optics, rays parallel to 
 the axis of the lens meet, after being refracted, in the focus of 
 the lens. Suppose the two stadia wires are the sources of those 
 rays, we have, from the similarity of the two triangles a'b'F 
 and FAB, the proportion 
 
 d -f:a=f: p. 
 
 The quotient f:p is, or at least can be made, constant, and 
 may be designated by k ; hence we may write 
 
 d-f=FC = ka. * 
 
 To get the distance from the centre N of the instrument there 
 must be added to FC the value 
 
 c = OF+ON. 
 
 ON is mostly equal to half the focal length of the object- 
 glass ; hence, 
 
 c=1.5/. 
 
 Therefore the formula for the distance of the stadia from the 
 centre of instrument, when that stadia is at right angles to the 
 level line of sight, is 
 
 D = ka + c. (1) 
 
 150. When the line of sight is not level, it is impracticable, 
 especially in long distances, to hold the rod in a vertical plane, 
 and at the same time perpendicular to the line of sight ; hence 
 it is customary to hold the rod vertical, as in the preceding case, 
 and obtain the true distance by applying a correction depending 
 upon the angle of inclination of the sight. 
 
 This correction is deduced as follows : 
 Let A GB = 2 w ; 
 
 n = the angle of inclination ;
 
 98 
 
 PLANE SURVEYING. 
 
 MF= c + GF = c + k x CD = D' ; 
 
 CD must be expressed by AB ; 
 
 MP= the horizontal distance = D' cos 
 
 -p 
 
 Now the angle 
 
 or, 
 
 Hence, 4? 
 
 sin m 
 
 AF = 
 
 sin[90 + (n- 
 GF sin m 
 
 and Z* 
 
 But 
 
 and 
 
 cos (w m) 
 BF sin m 
 
 sin [90 (w 
 (rF sin m 
 
 cos (n -f m) 
 
 r I 
 
 = GFs\nm\ |- 
 
 [_cos (n m) 
 
 _ CD _ CD cos m 
 2tanm~ 2sinm.
 
 DESCRIPTION OF INSTRUMENTS. 99 
 
 Substituting this value of GF in the equation above, we 
 obtain 
 
 ^_CD cos m [cos (n + m)+ cos (n m)] 
 
 2 cos (n + m) cos (n m) 
 ^r, cos 2 n cos 2 w sin 2 ?* sin 2 m 
 
 Ol , Ui/ = U - - 7 
 
 cos n cos" w 
 and Z>' = c + &o c 088 M 
 
 . 
 
 cosw cos 2 m 
 Whence, 
 
 D c cos ?i -f- fra cos 2 w fca sin 2 w tan 2 ra. 
 
 The third term of second member of this equation may be 
 neglected, as it is very small, even for long distances and large 
 angles of elevation (for 1500', n = 45 and k = 100, it is but 
 0.07') ; therefore the final formula for distances, with a stadia 
 kept vertical, and with wires equidistant from the centre wire, 
 is the following: 
 
 D = c cos n + ok cos 2 w. (2) 
 
 The value of ccosw is usually neglected, as it amounts to but 
 1 or 1.5 feet ; it is exact enough to add always 1.25' to the dis- 
 tance as derived from the formula 
 
 (2o)* 
 
 151. The focal length /of the object-glass may be found by 
 focussing the instrument upon some distant object, say a 
 heavenly body, and measuring then the distance between the 
 plane of the cross-wires and that of the objective. ON, being 
 equal to the distance between the objective and the intersection 
 of a plumb-line with the horizontal axis of the telescope, may 
 be obtained by direct measurement. 
 
 The distance p, between the stadia wires, may be determined 
 as follows : 
 
 Set up the instrument on level ground, or nearly so, and 
 measure forward from the plumb-line a distance equal to c, and 
 
 * The above explanation of the stadia is substantially that given by 
 Mr. G. J. Specht, published by Van Nostrand, 1884, though corrected and 
 simplified.
 
 100 / PLANE SURVEYING. 
 
 mark the point ; measure onward from the mark any convenient 
 distance d, 400 or 500 feet, as a base. The telescope being 
 level, observe carefully the space a intercepted by the stadia 
 wires on a levelling-rod held vertically at the farther extremity 
 of the base. 
 
 Then from the proportion d f: a =f : p the required dis- 
 tance p may be obtained. 
 
 EXAMPLES. 
 
 1. Given /= 8 inches, base = 500 feet, and a = 5. 25 feet. 
 Find p=. 084 inches. 
 
 2. At what fractional part of the focal length must the 
 stadia wires be separated so that one foot on the rod will cor- 
 respond to 100 feet base? State also the distance between the 
 wires in terms of the focal length, when one foot on rod cor- 
 responds to 66 feet base. 
 
 3. Measure with a stadia one or more sides of a field, also 
 the distance across a valley, or from one ridge to another, and 
 compare the results with chain measurement between the same 
 points. 
 
 4. Measure with the stadia up or down a hillside, and chain 
 between the same points. Compare results. 
 
 GRADIENTER. 
 
 152. This attachment, as shown on next page, is often used 
 with transits for fixing grades, determining distances, etc. 
 
 It consists mainly of a screw attached to the semicircular 
 expanded arm of the ordinary clamp of the telescope axis ; the 
 screw is accurately cut to a given number of threads, and pass- 
 ing through a nut in one side of the arm, presses against a little 
 stud A fixed to the inside surface of the right-hand standard. 
 
 In the other side of the semicircular arm is inserted a hollow 
 cylinder containing a pin actuated by a strong spiral spring, the 
 end of the pin pressing against the side of the stud opposite 
 that in contact with the screw.
 
 DESCRIPTION OF INSTRUMENTS. 101 
 
 Near the other end of the screw, and turning with it, is a 
 wheel, or micrometer, the rim of which is plated with silver, 
 and divided into one hundred equal parts. 
 
 A small silver scale, attached to the arm and just above the 
 micrometer wheel, is divided into spaces, each of which is just 
 equal to one revolution of the screw ; so that by comparing the 
 edge of the wheel with the divisions of the scale, the number of 
 complete revolutions of the screw can be easily counted. 
 
 It will be seen that when the clamp is made fast to the axis 
 by the clamp-screw, and the gradienter-screw turned, it will 
 move the telescope vertically, precisely like the tangent-screw 
 ordinarily used. 
 
 And as the value of a thread is such that a complete revolu- 
 tion of the screw will move the horizontal cross-wire of the 
 telescope over a space of one foot on a rod at a distance of 
 one hundred feet, it is clear that when the screw is turned 
 through fifty spaces on the graduated head, the wire will pass 
 over fifty one-hundredths, or one-half a foot on the rod, and so 
 on in the same proportion.
 
 102 
 
 PLANE SURVEYING. 
 
 In this way the gradienter can be used in the measurement 
 of distances, precisely like the stadia just described. 
 
 Grades can also be established, with great facility, as fol- 
 lows : First, level the instrument ; bring the telescope level to 
 its centre by the clamp and gradienter screw ; move the gradu- 
 ated head until its zero is brought to the edge of the scale ; and 
 then turn off as many spaces on the head as there are hun- 
 dredths of feet to the hundred in the grade to be established. 
 
 SECTION II. 
 
 A. BEARINGS WITH COMPASS. 
 
 153. To Obtain the Bearing of a Line. At one end of the 
 line, or at any other point in it, set up and level the compass, 
 loosen the needle, and direct the sights toward the other end. 
 The degree on which the needle comes to rest will indicate the 
 angle between the magnetic meridian and the direction of the 
 line, or the bearing. 
 
 For example, if the line lies between the north and east points, 
 as OP, and the angle NOP being, say 42 degrees, the bearing 
 of the line OP\s written, N. 42 E., 
 and read, "north forty-two degrees 
 east." If, as OP', it lies between 
 south and east, and the angle SOP' 
 is, say 74 degrees, it is written, 
 S. 74 E., and read, "south sev- 
 enty-four degrees east" ; in like 
 manner for lines in other quadrants. 
 It will be observed that the bear- 
 ing of a line does not exceed 90. 
 A line which might be read " N. 
 
 90 W." or " S. 90 W." is recorded as west. The bearing can 
 be read most accurately by placing the eye over one end of the 
 needle and taking the reading from the other end.
 
 BEARINGS WITH COMPASS. 
 
 103 
 
 Since the graduations are usually made to half -degrees, the 
 bearing can be taken quite accurately to quarter-degrees, and 
 by practice, even closer, without the use of the vernier. In 
 fact, the principal use of the vernier on a compass is to facili- 
 tate the running of lines from old deeds, where, when the 
 declination is ascertained, it is turned off on the vernier, and 
 the surveyor may use then the bearings as given in the deed by 
 which he is surveying the tract, without making a calculation 
 for the bearing of each line. The vernier cannot be relied on 
 to read bearings to minutes, on account of the difficulty of 
 accurately manipulating it. 
 
 154. Reverse Bearings. Since in plane surveying the meri- 
 dians passing through the extremities of a line are considered 
 parallel, the direct and reverse bearings should indicate the same 
 angle. That is to say, a line, as LM, the bearing of which, taken 
 ati, called also fore-sight, is N. 40 E., when 
 
 taken at M, back-sight, should be S. 40 W. ; 
 the degrees being the same, the letters indi- 
 cating the opposite cardinal points. 
 
 When surveying a tract of land with the 
 compass, the instrument should be set up at 
 every corner, and the bearing and reverse , 
 bearing of every line taken, as a check on the 
 observer's reading and the working of the 
 needle, since a disagreement in the angle thus 
 measured would be evidence sufficient to war- 
 rant a review of the work. 
 
 155. Local Attraction. If the readings of the needle of the 
 fore-sight and back-sight have been correctly made, and there 
 is found a disagreement, local attraction exists. It is usually 
 caused by the presence of ferruginous matter. It may exist at 
 both stations or at only one of them. 
 
 Assuming that the direct and reverse bearings of the preced- 
 ing line agree, then the difference in the reading at the two ends
 
 104 PLANE SURVEYING. 
 
 of the line, when the attraction exists, will show the local vari- 
 ation at the last station, and this correction must be applied to 
 the reading of the needle for the bearing of the next line. If, 
 however, the needle will not reverse on the first line of a survey, 
 then it will be necessary to set up at some other point of the 
 tract ; or, if this is impracticable, select one or more stations 
 near the suspected points, and by taking the bearings of these 
 from the stations, and also the reverse bearings, the intensity 
 and position of the attraction may be determined. 
 
 156. Proof Bearings and Tests of Accuracy. In any 
 important compass survey it is well to check the work by 
 sighting to distant prominent objects, such as buildings, trees, 
 etc., and noting the readings. Since two bearings are required 
 to locate each object, and until it is located it cannot serve as 
 a check, it will be necessary to take at least three bearings to 
 each. If, then, when plotting, the three lines intersect in a 
 point, a proof is given of the correctness of the measurements 
 thus connected. The lengths and bearings of diagonals of the 
 tract may likewise be taken as checks on the accuracy of the 
 work ; also, when in plotting, if the last bearing and distance 
 close the survey, it is considered a proof of the work. The 
 best test, however, of the accuracy of the survey is by Latitudes 
 and Departures, which is explained in Section VI. Articles 207 
 and 208. 
 
 It may be well to caution the student against the fallacy of 
 a test sometimes given, that if the sum of the interior angles, 
 determined from the bearings, equals twice as many right 
 angles, less four, as the figure has sides, it proves the work. 
 This "test," while it furnishes proof for a transit survey in 
 which the interior angles have been measured, will not show 
 that the bearings of a tract have been correctly taken. The 
 student will readily perceive the truth of this statement if he 
 makes or imagines a plot of a field with a certain side the 
 meridian, then conceives the whole plot turned around so that 
 another side comes to the meridian, it will be evident that
 
 BEARINGS WITH COMPASS. 105 
 
 though the bearings are changed, the sum of the interior angles 
 is unaffected. The so-called test would prove the work in either 
 case. 
 
 157. Suggestions. Test frequently to see that the instru- 
 ment is in proper adjustment. Keep the same end ahead. 
 Read from the same end of needle. Sight as low on the flag- 
 staff as possible. Make the line of sight as nearly horizontal 
 as practicable. When reading near the cardinal points, be care- 
 ful that the bearing is not read in the wrong quadrant, also that 
 the common error of reading 56 for 44 is not committed. See 
 that the instrument is set precisely over 
 the station from which the measurements 
 are to be made ; that the flagstaff is exactly 
 on the proper point, and that it is held 
 plumb. Level the instrument carefully ; 
 especially see that it is level across the line of sight. Take 
 the bearing and measure the distance on the true line when 
 practicable ; when not, because of a high fence, bushes, etc., 
 set off the least perpendicular distance therefrom at both ends 
 which will afford a clear view, and take the bearing and dis- 
 tance of the extremities of these perpendiculars. 
 
 EXERCISES. 
 
 1. With a surveyor's compass, by a constant and direct bear- 
 ing only, run a line, say 40 chains in length, over hilly ground, 
 and part of it, if possible, through brush ; then return, using 
 the reverse bearing only. 
 
 2. With the same instrument run another line equally diffi- 
 cult, using both direct and reverse bearings forward and back. 
 
 3. Make a survey of a lot one side of which is near to 
 a railroad track. If local attraction is found, determine its 
 intensity. 
 
 4. Determine the magnetic bearing of eacli part of a broken 
 line of several turns along a railroad track, or where local 
 attraction is known to exist.
 
 106 PLANE SURVEYING. 
 
 B. ANGLES WITH TRANSIT. 
 
 158. With the Transit the survey of a line or the measure- 
 ment of an angle can be made with greater accuracy than with 
 the compass, since the reading of the plates to minutes sup- 
 plants the reading of the needle to quarter or half-quarter 
 degrees, and the pointing power of the transit greatly exceeds 
 that of the compass. 
 
 159. To measure a horizontal angle, as MON. Set up 
 the instrument precisely at ; level it and direct the intersec- 
 tion of the wires to either point, say N. 
 Clamp the instrument firmly to the spindle, 
 note the reading of the vernier, then loosen 
 the vernier plate and bring the telescope 
 quite near the other line so that its ex- 
 tremity M is in the field of view. Clamp 
 the plate, and with its tangent or slow- 
 motion screw bring the line of collimation 
 precisely on M. Again take the reading. 
 
 The difference of the two readings will be the angle required. 
 It is more convenient to make the first sight, ON, with the zero 
 of the limb and plate coincident, since then the reading of the 
 plates after observing M gives at once the angle. If at each 
 observation but one vernier is read, it is best to read every 
 time from the same one ; it is better at each observation, 
 though, to read both verniers and take the mean of these, 
 thereby eliminating eccentricit}*. If, however, great accuracy 
 is required, the measurement of the angles should be taken 
 more than once, by the method of repetition or by series. 
 
 160. By Repetition. Make an observation upon any point, 
 and read both verniers ; clamp the lower plate tc the spindle, 
 direct the telescope to another point, and, as a check, again 
 read the verniers. 
 
 Now, keeping the index at the last reading, turn both plates
 
 ANGLES WITH TRANSIT. 107 
 
 back, and observe again on the first point ; clamp, as before, 
 the lower plate, and turn the upper one so as to sight on the 
 second point. It is perceived that by this operation the angle 
 has been measured twice, but on different parts of the limb. 
 An angle may obviously be repeated any number of times : the 
 mean of the several readings gives more nearly than a single 
 measurement the true angle. The reading at each observation 
 serves as a check on the work. An angle may be repeated by 
 simply noting the reading at the first and last observation, 
 taking their difference, and dividing by the number of repeti- 
 tions. It must be footed, however, how often, if at all, the 
 360 point is passed. Now, if the telescope is plunged, the 
 plates turned 180 in azimuth, and repetitions of the angle 
 again be made, beginning at the second point, the mean of the 
 two sets of readings will give still more nearly the true angle, 
 since the errors of adjustment and twist of station are thus 
 lessened and those of observation reduced. 
 
 161. By Series. Observe as before upon any point, and read 
 the verniers, clamp the lower plate, turn the vernier plate until 
 the telescope may be fixed upon another point, and again 
 read ; thus continue to make observations upon each point 
 desired in their order, sweeping round the horizon, and make 
 the last observation upon the first point. The last reading 
 should be the same as the first. Plunge the telescope, move 
 the plates in azimuth, and observe on the points again, pro- 
 ceeding in the contrary direction. Several series of observa- 
 tions may thus be made, as in the method by repetition. The 
 magnitude of each angle is obtained from the mean of its 
 reading. 
 
 RKMAEiK. Care should be exercised to have the instrument 
 properly centred, that is, set precisely over the centre of the 
 station, especially if the object sighted is near the observer. 
 The error arising from an eccentric setting is inversely as the 
 distance of the object sighted ; an eccentric setting of one inch 
 producing an error of nearly three (3') minutes of arc in sight-
 
 108 PLANE SURVEYING. 
 
 ing 100 feet, while the error arising from a sight of 900 feet 
 is less than oue-tln'rd (') of a minute. 
 
 Read both verniers to eliminate eccentricity. See that the 
 reading is not made from the wrong end of the vernier, and that 
 a half-degree is not omitted, calling the reading, say, 36 15', 
 instead of 36 45'. If great accuracy is required when running 
 a straight or broken line, lessen errors of adjustment by re- 
 versing the instrument in altitude and azimuth, making two 
 sets of observations at each station, and take the mean of their 
 readings. See Article 157. 
 
 If it is desired to locate the lines surveyed with reference to 
 the meridian, the bearing of one of them should be taken b} T 
 the needle of the instrument ; the bearings of the others may 
 be deduced therefrom. See Article 167. 
 
 162. Angle of Deflection. The amount of divergence which 
 a line makes with the preceding is called the deflection, and the 
 angle which measures it is termed the deflection angle. 
 L~ 
 
 In the figure POM is the deflection angle : it is evidently the 
 supplement of LOP. To measure it, set the transit at 0, sight 
 to L, clamp the limb to the spindle and the plates together, 
 then plunge the telescope : it will point to M. Take the reading, 
 unclamp the vernier-plate and move it until the wires intersect 
 P. The difference between the reading now and the first read- 
 ing is the deflection angle. If, when making the first observa- 
 tion, the vernier was at zero, the reading, after sighting P, 
 would indicate at once the angle. 
 
 163. Traversing, or surveying by the back angle, is a method
 
 ANGLES WITH TRANSIT. 
 
 109 
 
 of surveying by which the direction of each line of a survey is 
 compared with the first as a meridian or reference line. It is 
 
 effected as follows : 
 
 ^*" 
 P 
 
 Let it be required to traverse the broken line LMNOPQ. 
 Setup the instrument at M, clamp the vernier at zero, for con- 
 venience, and, with the lower motion, sight L, clamp below, 
 transit the telescope, loosen above and observe N: the reading 
 will show the angle M'MN which the line MN forms with LM. 
 Clamp the plates, move to N, plunge the telescope, and, with 
 the lower motion, sight Jf, the index remaining as at M ; then 
 clamp below, loosen above, transit the telescope, and direct it to 
 O : the index will show the angle which the line NO makes with 
 LM. And so continue until the end of the line. 
 
 To guard against mistakes in reading, and to avoid recording 
 whether the deflection is right or left, it is well to assume all 
 angles measured in the same direction.' In the figure the 
 readings are all to the right, or clockwise, as indicated by 
 the circular arcs, and the record is as follows : 
 
 STATIONS. 
 
 AZIMUTHS 
 
 WITH LM. 
 
 BEARINGS 
 
 WITH LM. 
 
 MAGNETIC BEARINGS 
 ASSUMING BEARING OP 
 LM N. 50 E. 
 
 L 
 
 
 
 North. 
 
 N. 50 E. 
 
 M 
 
 18 
 
 N. 18 E. 
 
 N. 68 E. 
 
 N 
 
 340 
 
 N. 20 W. 
 
 N. 30 E. 
 
 
 
 360 or 
 
 North. 
 
 N. 60 E. 
 
 P 
 
 90 
 
 East. 
 
 S. 4.0 E. 
 
 From the nature of the operation it may be perceived that, 
 algebraically, the azimuth of any line is equal to its deflection
 
 HO PLANE SURVEYING. 
 
 plus the azimuth of the preceding line. This method is partic- 
 ularly adapted to surveying roads, streets, water courses, etc., 
 and even in farm surveying it possesses an advantage over the 
 survey by interior angles, on account of the readiness it affords 
 in obtaining the bearings from the azimuths, and the greater 
 rapidity with which the work may be plotted, since the angle 
 which each line makes with the assumed meridian, or reference 
 line, is taken at once from the field notes. 
 
 Suppose LM in the figure to be the meridian of the survey, 
 and the azimuths of the several lines as recorded in the table. 
 Now, assuming the direction of LM to be north, it is evident 
 that MN will be in the northeast quadrant 18 from the north 
 point, or N. 18 E ; NO will be 20 to the wost of north, or N. 
 20 W. ; OP, making no angle with the meridian, will have a 
 bearing north, and PQ east. 
 
 So that, in general, 
 
 When the azimuth is less than 90, it equals the bearing, and 
 the line is in the northeast quadrant. 
 
 When the azimuth is between 90 and 180, the bearing is 
 southeast, and is the supplement of the azimuth. 
 
 When the azimuth is between 180 and 270, the bearing is 
 southwesterly, and may be found by subtracting 180 from the 
 azimuth. 
 
 When the azimuth is between 270 and 360, the bearing 
 is northwesterly, and is the difference between 360 and the 
 azimuth. 
 
 When the azimuth is 90, the bearing is due east. 
 
 When the azimuth is 180, the bearing is due south. 
 ' When the azimuth is 270, the bearing is due west. 
 
 When the azimuth is 360, the bearing is due north. 
 
 If it is required to find the magnetic or true bearing of any or 
 all the lines, take the magnetic or true bearing of the meridian of 
 the survey and apply it, by addition or subtraction, according as 
 the bearing of the assumed meridian, or standard line, is north- 
 east or southwest. In the example given, suppose the bearing 
 of the assumed meridian LM to be N. 50 E. : then the bearing
 
 ANGLES WITH TRANSIT. 
 
 ill 
 
 of the second line MN will be recorded 18 to the east of the 
 reference line, or N. 68 E. ; the line NO, having a deflection of 
 20 to the left of the reference line will be recorded N. 30 E. ; 
 and OP, N. 50 E. Thus the fourth column is added to the table. 
 
 164. To Traverse a Road, as LMNO. Proceed as indicated 
 in the last article, and in addition measure the lines LM, 
 NO, and perpendicular offsets thereto, at proper distances. 
 
 N . 
 
 If the road deviates much from a straight line, it will be 
 necessary, in order to obtain more correctly the area, to take 
 two offsets at M, one perpendicular to LM, the other to MN\ ' 
 and also two at N, one perpendicular to MN, and the other 
 perpendicular to NO.* 
 
 Likewise to Survey a Small Stream. Traverse and measure 
 the distances between assumed stations, as L, M, N, O, P, so 
 chosen as to make no more of them than is consistent with few 
 and short offsets to the various bends of the stream. If the 
 
 stream is small, not exceeding 10 feet in width, or even wider if 
 shallow, and it is desired to survey it between X and Y, a good 
 plan is to run a straight line between these points and measure 
 offsets therefrom to the stream ; or, if such a line will make the 
 offsets rather long, run RQ, and measure offsets from it to X 
 and Y"and intermediate points. If, however, the stream is wide 
 
 * Article 234.
 
 112 
 
 PLANE SURVEYING. 
 
 and the crossing difficult, it will probably be better to use more 
 stations, as shown in the figure. If a compass is used, the 
 bearings may be taken instead of the angles. 
 
 If a river of considerable width is to be surveyed, it will be 
 necessary, in addition to the measurement of broken lines on 
 each side from which offsets are taken, to make a series of an- 
 gular measurements connecting the lines on one side with those 
 on the other, and thence by trigonometrical calculations deter- 
 mine their relative positions, and ultimately the surface of the 
 river. 
 
 C. PROBLEMS ON ANGLES AND BEARINGS. 
 
 165. Angles between Lines. To determine the angle be- 
 tween two lines, meeting at a point, given by their bearings. 
 
 E W- 
 
 1. If the lines run between the same cardinal points, that is, 
 in the same quadrant, take the difference of their bearings. 
 
 Suppose the bearing of OP is N. 32 W. and that of OQ 
 N. 60 W. ; the angle between them is obviously NOQ - NOP-, 
 or, 60 - 32 = 28. 
 
 2. When the lines run in different quadrants and both above 
 or both below the horizontal or E. and W. line, take the sum 
 of tlreir bearings. If OP bears N. 60 E. and OL N. 20 W., 
 the angle POL = PON+ NOL = 60 + 20 = 80.
 
 PROBLEMS ON ANGLES AND BEARINGS. 
 
 113 
 
 3. If the lines run in diagonally opposite quadrants, subtract 
 the difference of the bearings from 180. Assuming the bear- 
 ing of OP N. 28 E. and of OL S. 58 W., the angle 
 
 POL = 180 - LOM = 180 - (58 - 28) = 150. 
 
 4. When the lines are in different quadrants, and both to the 
 right or both to the left of the vertical or N. and S. line, sub- 
 tract the sum of the bearings from 180. If OP bears N. 65 
 E. and OL S. 42 E., the angle 
 
 POL = 180 - (NOP + SOL) = 180 - (65 + 42) = 73. 
 
 ADDITIONAL EXAMPLES. 
 
 1. A line OP bears N. 40 W. and OL N. 40 E.,; required 
 the angle POL. 
 
 2. Find the angle POL, when OP bears S. 50 E. and OL 
 
 N. 89 E. 
 
 3. Required the angle at 0, when OP bears N. 80 W. and 
 OL S. 79 E. 
 
 4. What is the angle O, if OP runs S. 89| W. and OL 
 N. 89 E. ? 
 
 5. A line OP runs S. 70 W. and OL S. 45 W. Find the 
 angle 0.
 
 114 
 
 PLANE SURVEYING. 
 
 166. There may be given the bearing of a line, as MO, and 
 the deflection angle LOP, to the right or left of the direction 
 of MO, to find the bearing of OP ; or, the bearings of MO and 
 OP may be given to determine the magnitude of the deflection 
 angle LOP. 
 
 a. Given the bearing of a line and the deflection of the next, 
 to find its bearing. 
 
 Suppose MO bears N. 32 W., and the deflection of OP= 20 
 to the left ; the bearing of OP is evidently 20 farther towards 
 the west than MO or its prolongation OL. It is therefore 
 N. 52 W. Again, assuming RO bears N. 60 E. and the de- 
 flection of OQ 40 to the right, it is evident that OQ is in the 
 southeast quadrant, 10 from the east point ; or, its bearing 
 is S. 80 E. 
 
 6. When the bearings of the lines are given, to determine the 
 deflection. 
 
 Suppose LO (p. 115) bears N. 20 E. and OM N. 70 E. ; the 
 deflection of 03/from LO, or its prolongation OP, is evidently 
 70 _ 20 = 50 to the right. Again, the bearing of LO re- 
 maining the same, .and that of OQ N. 30 W., then it is readily 
 seen that the deflection angle is 20 + 30 = 50 to the left.
 
 PROBLEMS ON ANGLES AND BEARINGS. 
 
 JT 
 
 ,P 
 
 115 
 
 General rules' might be given for the cases under the above 
 heads, corresponding to those in the preceding article, but they 
 are deemed unnecessary, as a little reflection will enable the 
 student to determine the required bearing, or angle, in any 
 given case. 
 
 167. Given the angle between two lines, and the bearing of 
 one line, to find the bearing of the other. 
 
 The solution of this problem is ordinarily required in transit 
 surveying, for, when surveying with that instrument, it is com- 
 mon to take the bearing of only one line, and deduce the 
 courses of the others from that bearing and the measured 
 angles. Suppose LO bears N. 24 
 W. and the angle LOP= 82, to 
 find the bearing of OP. It is evi- 
 dent that the bearing of OP or the 
 angle NOP, which gives the degrees 
 in the bearing, 
 
 = 180- (24 + 82) = 74. 
 Hence the bearing of OP is N. 74 E. 
 
 Assume the angle POM= 100, 
 and the bearing of OP as found 
 above; then, since there are 100 
 74, or 26, more in the angle than lies between OP and the
 
 116 PLANE SURVEYING. 
 
 north point, the position of OM is to the west of north 26, or 
 its bearing is N. 26 W. 
 
 Some simple combinations, as indicated in the illustrations 
 given, will enable the student, unencumbered with rules, to 
 readily solve any of the problems coming under this head. 
 
 EXAMPLES. 
 
 1. A line bears S. 89 15' W. What is the bearing of a line 
 perpendicular to it? Also, the bearing of a line making an angle 
 of 135 with it? Is there more than one answer to the last? 
 
 2. If OP bears S. 36 W., and the angle 0^=68, what 
 is the bearing of PL? Ans. N. 32 W. 
 
 M SUGGESTION. Pass a meridian 
 
 ? O through the angle, and consider the 
 given bearing reversed. 
 
 3. The angles L, M, 0, P, of the 
 trapezium are respectively 62, 130, 
 80, and 88, and the bearing of LM 
 N. 70 E. ; find the other bearings.* 
 
 168. To Change the Bearings of the Sides of a Survey. It is 
 sometimes desirable to change the bearings of a survey so that 
 a particular side shall become a meridian. The whole plat is 
 conceived to revolve through an angle sufficient to make the 
 desired side the meridian ; the relative position of the sides 
 remains unaltered. The following rule is substantially that 
 given by Gummere, who states that the method was communi- 
 cated to him by Prof. Robert Patterson, late of Philadelphia. 
 
 RULE. 
 
 Subtract the bearing of the side that is to be made a meridian 
 from those bearings that are between the same points that it is, 
 
 * The calculation may be tested, after having deduced the bearings of 
 all the sides, by taking the last bearing found, as PL, applying the angle 
 L, and observing if it gives the proper bearing of LM.
 
 PROBLEMS ON ANGLES AND BEARINGS. 
 
 117 
 
 and also from those that are between points directly opposite to 
 them. If it is greater than any of those bearings, take the differ- 
 ence, and change west to east or east to west. 
 
 Add the bearing of the side tvhich is to be made a meridian to 
 those bearings tvhich are neither between the same points that it is 
 nor between the points directly opposite to them. If either of the 
 sums exceed 90, take the supplement, and change south to north 
 or north to south,. 
 
 The accompanying diagram of full and dotted lines exhibits 
 the positions of the sides of the following described farm, be- 
 fore and after turning through 16^ to the right: 
 
 (1) N. 16^ W., 24.63 chains ; (3) S. W., 34.28 chains ; 
 
 (2) S. 79 W., 27.00 chains ; (4) .N. 65 E., 37.20 chains, 
 to the place of beginning. The bearings are changed so as to 
 make the first side a meridian. 
 
 EXAMPLES. 
 
 1. Given the bearings of a tract of land : 
 (1) S. 10 E. ; (2) S. 30 W. ; (3) N. 60 W. ; 
 (4) N. 20 W. ; (5) N. 80 E.,
 
 118 PLANE SURVEYING. 
 
 to the place of beginning. Required the changed bearings that 
 the fourth side may be a meridian. 
 
 (1) S. 10 E. (4) North. 
 
 20 
 
 Changed bearing, S. 10 W. 
 
 (2) S. 30 W. (5) N. 80 E. 
 
 20 20 
 
 Changed bearing, S. 50 W. 100 
 
 (3) N. 60 W. 
 
 20 Changed bearing, S. 80 E. 
 
 Changed bearing, N. 40 W. 
 
 The student who avails himself of the hints and methods 
 referring to the manipulation of angles and bearings as given 
 in the preceding articles, will have no difficulty in determining 
 the changed bearings direct from the data, without the use of 
 rules. Thus in the example above it will be observed that each 
 line is turned through 20 to the right; that is, the fourth 
 course is made due north. The next side to it going round to 
 the right, N. 80 E., will be turned the same number of de- 
 grees (20), which places it 10 from the east point in the south- 
 east quarter, or its bearing is S. 80 7 E. ; the first side turning 
 through the same angle (20) will be thrown 10 west of the 
 south point, or S. 10 W. ; the second course will be 20 farther 
 to the southwest, or S. 50 W. ; and the third course turned 
 toward the north point 20 will be N. 40 W. 
 
 2. Find the bearings of all the sides of the following de- 
 scribed tract of land when the second side is made a meridian : 
 
 (1) N. 681 E., 8.42 chains ; (3) S. 78f W., 4.90 chains; 
 
 (2) N. 27 W., 10.25 chains ; (4) S. 1 E., 4.40 chains ; 
 
 (5) S. 12 E., 7.04 chains, 
 to the place of beginning.
 
 PROBLEMS ON ANGLES AND BEARINGS. 119 
 
 3. Given the bearings of a tract of land as follows : 
 
 (1) S. 39i W. ; (3) N. 15 W. ; (5) N. 2 E. ; 
 
 (2) East ; (4) N. 79 E. ; (6) S. 73f W., 
 
 to find the bearings of all the sides when the first becomes a 
 meridian. 
 
 4. Given the bearings of a tract of land as follows : 
 
 (1) S. 79 W. ; (3) N. 89LE. ; (5) S. 80f E. ; 
 
 (2) S. W. ; (4) N. IfE. ; (6) S. 581 E. ; 
 
 (7) N. 39 E. ; (8) N. 16}- W., 
 to find the bearings when the eighth side becomes a meridian. 
 
 EXERCISES. 
 
 1. With a transit, using back and fore sights, run a tangent 
 forward and back over hilly and brush land requiring six or 
 eight settings of the instrument. The last two points set for- 
 ward will give the direction back. Note the distance, if any, 
 between the corresponding positions occupied by the instru- 
 ment. 
 
 2. Traverse, or survey by the back angle, a broken line of 
 six stations, using the first line as the meridian, or reference 
 line, of the survey. Record the notes, indicating the azimuthal 
 angles and bearings. 
 
 3. Measure the three angles of a triangular piece of land, 
 the cornei-s being visible from each other ; see how much, if 
 any, their sum differs from two right angles. 
 
 4. Traverse a pentagonal field, the index at the beginning 
 being sot nt zero, and see if, when finally sighting on the station 
 first occupied, the reading is zero.
 
 120 PLANE SURVEYING. 
 
 SECTION III. 
 
 OBSTACLES. 
 A. PROBLEMS ON PERPENDICULARS AND PARALLELS. 
 
 169. The Obstacles which occur in field work are more easily 
 and expedittously overcome with the compass, or transit, and 
 chain, than with the chain alone. Methods for the latter were 
 given and illustrated in Chapter I. Section II., Chain Survey- 
 ing. 
 
 To erect a perpendicular to a line at any given point. Set up 
 the instrument over the point ; if a compass is used, take the 
 bearing of the line, and then move the instrument in azimuth 
 until a bearing differing 90 from the first is observed. The 
 line of sights will then indicate the direction of the required 
 perpendicular. If a transit is employed, centre on the point, 
 sight to a point in the line, clamp to spindle, and turn the ver- 
 nier plate 90 either way ; then the line of collimation will show 
 the direction of the perpendicular sought. Of course b}- the 
 methods explained above, a line can be run with either instru- 
 ment from any given point and making any given angle thereat 
 with a line. 
 
 170. To let fall a perpendicular from a given point to a line. 
 Let P be the point, and LN the line. If the compass is used, 
 
 take the bearing of LN, remove the 
 instrument to P, and with a bearing 
 differing 90 from the first, run PO 
 for the required perpendicular. With 
 N a transit centre on L, measure the 
 angle OLP, remove to P, and make 
 
 the angle LPO equal to the complement of L ; the line of sight 
 of the instrument will then be in the direction of the required 
 perpendicular.
 
 PERPENDICULARS AND PARALLELS. 
 
 121 
 
 171. To let fall a perpendicular to a line from an inaccessible 
 point. Measure the distance between any two points, as L and 
 N, in the line ; also the angles 
 PLN and LNP. Then in the 
 triangle PLN we have given the 
 side LN and the angles to find 
 PL or PN. Computing PL, 
 the distance 
 
 LO=PLcosPLO. 
 Or we may deduce an expression for LO in terms of the meas- 
 ured line and the observed angles, thus : 
 
 LO=POcotPLO. 
 
 NO = POcotPNO. 
 
 LO:NO = cot PLO : cot PNO, 
 
 LO:LO + NO= cot PLO : cot PLO + cot PNO ; 
 
 jy 
 
 LN cot PLO 
 
 Hence 
 and 
 
 but 
 
 therefore 
 
 cot PLO + cot PNO 
 
 QUERY. Could a line be run not perpendicular as above 
 through an inaccessible point, making any angle with the given 
 line? 
 
 172. To run a line through a given point parallel to a given 
 line. With the compass obtain the bearing of the line, and 
 then from the given point run a 
 
 line with the same bearing. With -<c N 
 
 the transit, LN being the line and 
 
 P the point, centre on L, measure 
 
 the angle NLP, remove to P, and R 
 
 make the angle LPR equal to NLP; the line of collimation 
 
 will then be in the required parallel. 
 
 B. PROBLEMS ON ALIGNMENT. 
 
 173. To prolong a line, as LN, beyond a tree, a building, or 
 any obstacle.
 
 122 
 
 PLANE SURVEYING. 
 
 First Method. By Deflection Angles. Set up the instrument 
 at any point of the line, as N, and deflect, sufficient to pass the 
 obstacle, to any point P. Measure NP, remove to P, deflect 
 to 0, making the angle QPO double the angle at N. 
 
 Measure PO = PN, place the instrument at 0, observe P, 
 plunge the telescope and deflect to R, so that SOR = \ OPQ ; 
 the telescope will then be in the prolongation of LN. 
 
 174. Second Method. By Equilateral Triangle. Deflect 60 
 from the direction of the line at N; measure to P a distance 
 
 sufficient that PO, making an angle of 60 with P^, will clear 
 the obstacle. Measure PO = PN, and turn the telescope in the 
 direction of OR, the prolongation of LN, by deflecting 60 from 
 the direction of PO. 
 
 175. Third Method. By Isosceles Triangle. Deflect at N 
 45 to M, measure NM, make NMO a right angle, and MO 
 
 H 
 
 = MN; at turn into OR by deflecting from the direction of 
 OM 45.
 
 PROBLEMS ON ALIGNMENT. 
 
 123 
 
 176. Fourth Method. By Perpendiculars. Erect a perpen- 
 dicular NK of sufficient length that a line passing through 
 
 K parallel to LN will clear the obstacle ; run KM ; lay off 
 MO = NK, and a right angle turned from MO will indicate the 
 direction of LN, or its prolongation OR. 
 
 177. Random Line. When brush, wood, or any obstruction 
 prevents N being seen from L, run a line LP as nearly as may 
 
 be judged in the direction of LN: when opposite N, as at P, 
 measure the shortest distance from P to N, call it d ; then the 
 
 57.3 xd 
 LP 
 
 angle PLN in degrees = 
 
 Setting up again at L, and applying the correction thus 
 found in a proper manner to the angle or bearing before used, 
 the line LN may be traced. 
 
 N 
 
 Demonstration. When the distance PN does not exceed 5 
 per cent of the length of PL, LN and PL may be regarded as 
 radii of a circle, and PN coincident with the arc which subtends 
 the angle PLN', then 

 
 124 PLANE SURVEYING. 
 
 27rP:360 = PN: PLN, 
 
 360 x PN 57.3 x PN 
 
 PLN. 
 
 LP 
 
 When PN exceeds the limit stated, the angle PLN should be 
 found by measuring PN perpendicularly from PL, and dividing 
 this by the length LP for the tangent of the angle PLN. 
 
 EXAMPLES. 
 
 1. A random line was run N. 41 15' E. 18.34 chains, when 
 the nearest distance to the desired corner, which was to the left, 
 was found to be 16 links. Required the correction and the bear- 
 ing of the true line. Ans. Cor. 30' ; bearing of line, N. 40 45' E. 
 
 2. A random line was run S. 89 45' W. 24.80 chains, when 
 the corner was found 22 links to the right. Find the correction 
 and the bearing of the line. 
 
 3. The length of a random line is 16.64 chains, and a per- 
 pendicular from its extremity to the desired point equals 96 
 links. What correction is needed ? 
 
 -P 
 
 N 
 
 M 
 
 4. A random line LP, 25.12 chains long, run by transit, 
 makes an angle of 27 with LM, and the point P is 18 links to 
 the left of N', LN being the true line. Determine the proper 
 angle to turn off at L with which to trace LN. 
 
 C. PROBLEMS ON MEASUREMENT. 
 
 178. a. When the Ends of the Line are Accessible and 
 Visible from Each Other.
 
 PROBLEMS ON MEASUREMENT. 
 
 125 
 
 The methods indicated in Problems on Alignment will be 
 found useful in many instances for the determination of the 
 lengths of lines, the direct measurements of which are imprac- 
 ticable. Thus, in the figure in Article 176, the distance NO 
 will be found by measuring KM. 
 
 In figure accompanying Article 174 the measurement of either 
 NP or PO will give the side NO. 
 
 Otherwise (Article 175). Measure NM, and multiply it by 
 V^2, or extract the square root of twice the square of NM for 
 the required length NO. 
 
 By random line, as in Article 177, when the shortest distance 
 PN\s taken, the'length of the true line will equal the measured 
 or random line. 
 
 If the perpendicular from P is used, then the length of the 
 true line will equal the square root of the sum of the squares of 
 LP and PN-, that is, 
 
 To ascertain the horizontal measurement of a hillside, take 
 the angle of its slope, measure up or down it (preferably down), 
 and the product of this distance and the cosine of the angle will 
 be the horizontal distance required.* 
 
 179. By Triangulation. Measure LP and the angles L and 
 P ; the sine proportion may then be em- 
 ployed to determine 
 
 180. Otherwise. Measure LP, PN, 
 and the angle P. Then having two sides 
 and the included angle of the' triangle, 
 the third side LN may be computed. 
 
 * When measuring the angle of elevation, the surveyor should sight to 
 a point on the rod a distance above ground equal to the height of the line 
 of collimation of his instrument.
 
 126 PLANE SURVEYING. 
 
 181. b. When One End of the Line is Inaccessible. Let N 
 
 represent the inaccessible but 
 visible end of the line LN, the 
 length of which is desired. Meas- 
 ure LP of such length, if possible, 
 that none of the angles will be less 
 than 30 ; the nearer LNP is equi- 
 
 lateral, the better. Observe the angles L and P. Then, by the 
 
 sine proportion, 
 
 sin (L + P) 
 
 182. When the Points are not Visible from Each Other. In 
 
 the figure let N represent the invisible point in the line LN, 
 the length of which is required. 
 Measure a line in any convenient 
 direction through L, as MP, not- 
 'ing the distances ML and LP, of 
 such a length that the point .2V 
 may be seen from each extrem- 
 ity. Observe the angles P and 
 
 M. In the triangle PMN, find, 
 f ii 
 
 -kj by the sine proportion, the length 
 
 of PN. Then in PNL are known 
 
 two sides and the included angle, 
 
 with which may be found LN. 
 
 It will be observed that the problem requires the measure- 
 
 ment of the distance between two points, L and N, invisible 
 
 from each other, and direction unknown. If it were simply to 
 
 determine the distance from L to an invisible point in the pro- 
 
 longation of OL, we should measure perpendicularly from OL 
 
 to a point P, from which the point N could be seen, observe 
 
 the angle LPN; then LN= PL x tan LPN. 
 
 QUERY. What would be the best method of solving the prob- 
 lem under the last supposition, if it were impracticable to 
 measure a perpendicular from OL?
 
 PROBLEMS ON MEASUREMENT. 
 
 127 
 
 Let it 
 
 183. c. When the Ends of the Line are Inaccessible, 
 be required to determine the length 
 
 of the inaccessible line LN. Meas- 
 ure OP, and observe the angles LON, 
 NOP, OPL, and LPN; then in the 
 triangle LOP compute LO, and in 
 NOP, ON. There will then be given 
 two sides and the included angle of 
 the triangle LON to find LN. 
 
 184. The same general method would apply if the base inter- 
 sected the line the length of which is desired. Suppose it is 
 required to determine the distance be- 
 tween L and N, points on opposite 
 
 sides of two inlets, J^and T. Meas- 
 ure OP and take the angles at the ex- 
 tremities on both sides of the base. 
 There will then be data sufficient to 
 find OL and ON, and finally LN. 
 
 QUERY. Would it be practicable in 
 any case to make OP perpendicular to 
 LN? If so, would it be necessary to 
 measure the distance OP and all the 
 angles, as above? Why? 
 
 N 
 
 EXAMPLES. 
 
 1. To determine the distance between two points L and 2V, 
 on opposite banks of a stream, I measured a base LP = 300 
 feet, and observed the angles which N made with L and P to 
 be 58 45' and 64 50', respectively. Required LN. 
 
 2. If LP in Example 1 were taken at right angles to LN, 
 the angle P being 40 30', what would be the length of LN? 
 
 3. To ascertain the distance LN between two inaccessible 
 points invisible from each other, I measured a line MP through
 
 128 PLANE SURVEYING. 
 
 L, from the extremities of which ^7" could be seen. ML = 160 
 feet ; LP= 200 feet ; angle at M = 65 30' ; angle P= 69 15'. 
 What is the length of LN? 
 
 4. To determine the distance between two points L and N, 
 situated on the side of a river opposite to where I was, a base 
 line OP 400 feet long was measured, and the following angles 
 observed : LON= 68 30' ; NOP = 32 45' ; NPL = 50 30' ; 
 LPO = 40 15'. Required LN. 
 
 EXERCISES. 
 
 1. Prolong a line beyond a house, tree, or other obstruction, 
 using any one of the methods herein given. Return, pass the 
 obstruction by some other method. See how near the starting- 
 point is reached. 
 
 2. Run a trial line of considerable length through a wood, 
 with a view of sighting a stake previously set. Make the 
 proper measurements and calculation to correct the angle and 
 re-run the line. Note the distance, if any, from the stake after 
 the second trial. 
 
 3. Triangulate across a creek or small lake. Use at least 
 two methods. See how near the results agree. 
 
 4. By triangulation determine the distance between two 
 points without going near them. Verify the result by subse- 
 quent measurement. 
 
 5. Measure the distance between two points in a given line, 
 invisible and assumed inaccessible from each other. Compare 
 the results of two methods. Verify subsequently by direct 
 measurement. 
 
 6. Run a trial line between two points which are invisible 
 from each other, on account of an intervening ridge. Correct 
 the angle and re-run the line. If the proper point is not reached, 
 should the angle be again corrected?
 
 ACCESSIBLE HEIGHTS. 129 
 
 SECTION IV. 
 HEIGHTS AND DISTANCES. 
 
 A. ACCESSIBLE HEIGHTS. 
 
 185. Let it be required to determine the height P above a 
 horizontal plane LN. Measure the distance LN and the angle 
 of elevation L. Then 
 
 If the ground is level, or nearly so, the 
 telescope cannot be placed at L, in the / 
 
 horizontal plane with JV, but at some point /'/ 
 
 I, and the angle Pin is measured instead of /'/' 
 PLN. In such a case Nn must be added l [/~ 
 to the calculated height. 
 
 186. Let it be required to find the height of an object stand- 
 ing on an inclined plane ON. Measure 
 the distances NL and LO, and the angles 
 NLP and NOP. In the triangle OLP, 
 by the sine proportion, find PL. Then 
 in the triangle PLN, having two sides 
 and the included angle, PN may be 
 determined. 
 
 187. Otherwise. Measure NL, and at 
 
 L the angles of elevation of N and P. Then the projection of 
 LNon the horizontal plane 
 
 = LM=LNcosNLM, 
 and MN= LN sin NLM ; 
 
 PM=LMt&nPLM; 
 whence PN= PM NM; or, expressed in a single equation, 
 
 PN= LN X cos NLM x tan PLM-LN X sin NLM.
 
 130 PLANE SURVEYING. 
 
 EXAMPLES. 
 
 1. At 120 feet distance from the centre of the foot of a 
 liberty pole, the angle of elevation of its top was 38 40'. 
 Required its height. 
 
 2. The distance LN (see Article 185) measures 90 feet, the 
 angle of elevation I is 42 30', the telescope being 4.8' above 
 the horizontal plane LN. Determine height of the point P. 
 
 3. To determine the height of an object on an inclined plane, 
 two stations, L and (marginal figure, Article 186), were se- 
 lected, one 50 feet and the other 110 feet, measured on the slope 
 from N. The angle NLP=40 15', and NOP 22 30'. Re- 
 quired the height. 
 
 QUERIES. Practically, is it necessary to know the height of 
 instrument * in such cases ? 
 
 If there was a change of slope at L, would any other 
 measurement be necessary to calculate the required height? 
 
 4. Suppose NL (figure, Article 186) measures 60 feet, and 
 the angles of elevation at JO, of N and P, are respectively 
 12 30' and 59 20'. Determine the height of P above N. 
 
 B. INACCESSIBLE HEIGHTS. 
 
 188. To determine the height of an object situated on an 
 inaccessible hill. 
 
 Measure in the same vertical 
 plane with P a horizontal line 
 LN, and observe at N the angles 
 of elevation of the points and P, 
 and at L the angle of elevation of 
 P. In the triangle LNP, by the 
 sine proportion, calculate PN. 
 
 By the same method, find NO 
 from the triangle PON. Then 
 
 * Height of instrument is the height of the line of sight above the 
 the ground, or any other assumed horizontal plane
 
 INACCESSIBLE HEIGHTS. 131 
 
 PO = PN sin PNM - NO sin ONM. 
 
 The student may show, after finding PN and NO as above, 
 a different method of finding PO than that indicated. 
 
 Ex. At a certain station the angle of elevation of the base 
 of a tower on a hill-top was 38 40', and that of the top 
 50 15' ; 190 feet more remote, the angle to the top was 36 20'. 
 The stations being in the same horizontal plane, required the 
 height of tower and of the hill. 
 
 189. Let PO be an object whose height is required. Measure 
 in the same vertical plane with P a 
 
 horizontal base line LN, and observe 
 the angles of elevation PLN and PNO. 
 Then, by the sine proportion, find PN, 
 and 
 
 PO = PN8\nPNO. 
 
 190. Otherwise. PO cot L = LO, 
 
 LO-NO = LN\ 
 or, PO (cot L - cot N) = LN. 
 
 ,, PO== LN 
 
 cot L cot N 
 
 Ex. If LN= 120 feet, and the angles at L and N respec- 
 tively 27 50' and 45 19', 
 
 PO = =136.6 feet. Ans. 
 
 cot 2 7 50' -cot 45 19' 
 
 191. If it is impracticable to locate the base line in a hori- 
 zontal plane, measure from L in the 
 direction of P any line LN, and at L 
 take the angles of elevation of N and P. 
 Observe also the angle at N. By the 
 sine proportion obtain LP. Then 
 
 PO = LPs\nPLO, 
 and PR =PO-RO = LP sin PLO - Z^sin NLO.
 
 132 PLANE SURVEYING. 
 
 QUERY. May the observed angle at N be either LNP or 
 PNR? 
 
 192. Otherwise. L and N being in different planes, measure 
 the horizontal distance between them. Observe the angle of 
 elevation PLO and the horizontal 
 P angles OLN and ONL. By the 
 
 sine proportion find LO. Then 
 
 or, expressed in a single equation, 
 
 po = LNsm LNO tan PLO 
 
 sin NOL 
 
 which equals the height of P above 
 the horizontal plane through L. 
 
 If it is required to find the height of P above the horizontal 
 plane through JV, proceed as follows : Assuming N to be be- 
 low * L, observe at N the angle of elevation of P ; then find the 
 horizontal distance between N and by the sine proportion, 
 using the triangle NLO ; thus, sin : sin L = LN: fourth term. 
 This fourth term will not be NO. since the measurement of the 
 distance and angles employed in the computation is referred 
 to a horizontal plane, and hence the fourth term will express the 
 horizontal distance between N and 0, which equals NR, R being 
 a point in the prolongation of the vertical PO. Whence, 
 PR = NRtenPNR. 
 
 EXAMPLES. 
 
 1. At a certain station the angle of elevation of the top of 
 an inaccessible object situated on a horizontal plane was 60 50', 
 and 120 feet farther away the angle was 29 10'. Required the 
 height of Jthe object and its distance from the first station. 
 
 2. Suppose LN (figure, Article 191) is 140 feet, the angles of 
 elevation at L, of N and P, are respectively 9 25' and 30 16', 
 
 *It may obviously be above or below L ; the same reasoning will hold
 
 INACCESSIBLE HEIGHTS. 
 
 and the angle PNR = 42. Find the height of P above 
 and R. 
 
 3. In figure, Article 192, suppose 
 
 LN= 1000 feet ; angle PLO = 26 18' ; 
 
 angle OLN= 36 20' ; angle PNR = 55 10'. 
 
 angle ONL = 95 40' ; 
 Find PO and RO. 
 
 193. To determine the perpendicular distance from a given 
 horizontal plane of an inaccessible object situated below it. 
 
 Let P be the point whose perpendicular distance from a hori- 
 zontal plane through L is required. Select 
 two points L and N visible from each other, 
 and from which P can be seen. Measure 
 the horizontal distance between them ; ob- 
 serve also the horizontal angles PLN and 
 PNL, and the angle of depression of the 
 point P, at L. By the sine proportion cal- 
 culate the horizontal distance from L to P; 
 this multiplied by the tangent of the angle 
 of depression observed at L will give the perpendicular dis- 
 tance required. 
 
 If L and N are not in the same horizontal plane, observe at 
 N the angle of depression of P, and calculate as above the per- 
 pendicular distance between the point and the horizontal plane 
 through N. The difference of these perpendicular distances 
 will also give the difference in height of L and N. A check on 
 the work may be had by determining from more direct methods 
 already given the difference in elevation of L and N. 
 
 EXAMPLES. 
 
 1. At L and N (last figure) the horizontal angles measure 
 respectively 67 40' and 43 10'; and sighting P, the angles of 
 depression taken in the same order are 32 18' and 21 42'. The 
 distance between the stations being 1200 feet; required the 
 difference in height of P, L, and N.
 
 134 
 
 PLANE SURVEYING. 
 
 2. To find the height of an object, PO, standing on the edge 
 of a lake and inaccessible to L, a 
 station on the opposite rocky 
 shore, a distance of 500 feet 
 was measured from L up the 
 slope to N, where the angles of 
 depression of L, 0, and P were 
 observed respectively, 39 40', 
 25 20', and 21 32'. Required the height of PO. 
 
 194. To determine the height of an object, and its distance 
 from three observing-stations situated in a 
 straight line and in the horizontal plane 
 through the foot of the object. 
 
 Let PO represent the required height ; L, 
 R, and N" the stations ; the angles of eleva- 
 tion of P taken at each and in the order 
 named a, ft, and 0. The distance LR = a, 
 RN= b, and the unknown height = x. It is 
 evident that the triangles POL, POR, and 
 PON are right-angled at 0, and therefore 
 
 OL = x x cot a. 
 OR = x x cot ft. 
 ON= x x cot 6. 
 
 Again, drawing OM perpendicular to LN, we shall have from 
 the acute-angled triangle LOR, 
 
 and from the obtuse-angled triangle NOR, 
 
 RM; 
 
 or, substituting the proper values for the lines represented, we 
 
 shall have 
 
 a? cot 2 a = a? cot 2 (3 + a 2 - 2 a MR, 
 a? cot 2 = x- cot 2 ft + & 2 + 2 b MR.
 
 INACCESSIBLE DISTANCES. 
 
 135 
 
 Eliminating MR by multiplying the first by 6, the second by a, 
 adding and factoring, we obtain 
 
 Whence 
 
 y? cot 2 /? (a + &) + ab(a + 6) . 
 
 ib cot 2 a. + acot 2 cot 2 /3(a + 6) 
 If the stations are equidistant, the formula reduces to 
 
 2a 2 
 
 Or, 
 
 Having obtained the height of P above the plane, the hori- 
 zontal distance from the object to either station may be deter- 
 mined by multiplying this height by the cotangent of the angle 
 of elevation at the station. The oblique distance from either 
 station to P is given by the product of PO and the cosecant of 
 the angle of elevation at the station. 
 
 INACCESSIBLE DISTANCES. 
 
 195. The distance apart of three objects, L, and N, in- 
 accessible from P are known, viz. : 
 LO = 2000 feet, ON= 1800 feet, and 
 .LJV=2400 feet. At P, situated in 
 the prolongation of ON, the observed 
 angle =21 48'; how far is it from 
 station P to each object? 
 
 First calculate angle ; then in the 
 triangle POL there will be known all 
 the angles and one side, whence the re- 
 quired distances may be readily found. 
 
 Usually the station P cannot be chosen so as to fall in ON 
 or OL produced ; then the measurement of two angles will 
 generally be sufficient, with the known distances to locate the
 
 136 
 
 PLANE SURVEYING. 
 
 point of observation. For example, suppose the distances and 
 angles are as follows : 
 
 NO = I = 3000 feet ; 
 
 OL = n = 3600 feet ; 
 
 LN= o = 4800 feet ; 
 angle NPO = a = 23 40' ; 
 angle LPO = (3=22 50'. 
 
 By construction, the point P may be found as follows : Sub- 
 tract from 180 2 LPO, and from LO lay off at L and the 
 angles LOM and OLM, each equal to half the remainder. 
 From the point M thus determined as a centre, and with a 
 
 radius LM, describe the circumference OLP. The angle LPO 
 will then be contained in the segment LPO, and the point P 
 must be somewhere in the circumference OLP. In like man- 
 ner, by means of the angle OPN, find another circumference 
 ONP, in which the point P must be situated. The intersec- 
 tion of these circumferences indicates its position. 
 
 The angle at the circumference being half that at the centre, 
 the angle LMO, subtended by the same chord as LPO, will be 
 2 LPO, and the angles OLM and LOM being equal and to- 
 gether the supplement of LMO, each angle will
 
 INACCESSIBLE DISTANCES. 
 
 137 
 
 Otherwise. Construct an angle NLR equal to OPN\ also 
 LNR equal to OPL, and describe a circumference through the 
 points L, It, and N. The point P must lie in the circumfer- 
 ence, and also in the line drawn from 
 through R. Their point of intersection 
 therefore will indicate its position. 
 
 The student may give the reason. 
 
 196. By Calculation. Pass a circle 
 through the points L, N, P, and join L 
 and N with R, thus forming a triangle 
 in which the angles RLN and RNL are 
 equal, respectively, to the observed an- 
 gles RPN and RPL, and these, with 
 the known side LN, furnish data sufficient to compute the sides 
 LR and RN. Next calculate the angle ONL, whence, by sub- 
 traction, the angle ONR is found. Now, in the triangle NOR 
 there are given two sides and the included angle to find NOR 
 and ORN, or its supplement PRN, and by means of the sine 
 proportion and the triangles PON and POL the distances PN, 
 PO, and PL may be obtained. 
 
 Otherwise. After finding the angle 0, obtain an expression 
 for either OLP or ONP, and then, by the sine proportion, the 
 required distances. 
 
 Denote the angle OLP by <, ONP by i/r, and the other parts 
 as before ; then 
 
 sin B : sin <k = n : OP, or QP= w 
 
 sin/3 
 
 Whence 
 
 sin : sin ^ = I : OP, or OP = 
 
 sin /8 
 
 sin a 
 
 and 
 
 n sin a
 
 138 PLANE SURVEYING. 
 
 Again, < = 360 a /? O f; 
 or, putting 360 a /3 = 0, 
 
 <}>=*0 fa in which 6 is known ; 
 
 and 
 
 Developing the left-hand member, dividing through by cos ^, 
 and simplifying, there results 
 
 tan * = _ ; 
 
 I sin + n sin a cos 6 
 
 or, cot./r = . ^ + oot0. 
 
 n sin a sin 
 
 There are therefore but three steps in the solution : 
 
 1. Calculate the angle 0, and thence obtain 0. 
 
 2. Find tan fa or cot fa 
 
 3. By sine proportion, calculate PN, P0, and PL. 
 
 In the example given, since the sides are in the proportion 
 5:6:8, the angle may be readily found from the well-known 
 formula for the cosine of an angle, 
 
 cos = 25+36-64 _ _ Q5 _ ^ ^ 
 
 uU 
 
 and 
 
 0=213 38'; 
 
 
 whence 
 
 *=10953', 
 
 
 
 < = 103 45'. 
 
 
 
 sin 23 40' 
 
 Ar. co. = 0.396406 
 
 
 : sin 109 53' 
 
 = 9.973307 
 
 
 : : 3000 
 
 = 3.477121 
 
 
 : PO = 7028 
 
 = 3.846834 
 
 * Regard must be given to the signs of the trigonometrical functions.
 
 INACCESSIBLE DISTANCES. 
 
 139 
 
 sin 23 40' 
 sin 46 27' 
 3000 
 
 Ar. co. = 0.396406 
 = 9.860202 
 = 3.477121 
 
 sin 29 50' 
 sin 46 25' 
 3600 
 
 PL = 5242 
 
 = 3.733729 
 
 Ar. co. = 0.303225 
 = 9.859962 
 = 3.556303 
 
 = 3.719490 
 
 If the supplement of the observed angles at P equals the angle 
 at 0, the circle will pass through the three points L, N, and 0, 
 and P may be anywhere on the circumference, and hence its 
 distance is indeterminate by the first method given above ; and, 
 substituting in the formula the proper values to find coti/r by the 
 second method, the numerator of the fraction will become in- 
 finite, as also the cot0 ; hence, such an observation will fail in 
 both cases to locate the point P. 
 
 EXAMPLE. 
 
 Suppose ^=960 rods, NO 576 rods, LO 640 rods, the 
 angle LPO=19, and NPO = 25. Find the distances PO, 
 PN, and PL. 
 
 Ans. PL = 758 rods ; PO = 1310 rods ; PN= 1350 rods. 
 
 197. From the top of a mountain m miles high the angle of 
 depression of a line tangent to the earth's 
 surface is a degrees ; it is required thence 
 to find an expression for the radius of 
 the earth, assuming it to be a sphere. 
 
 Let O represent the centre of the 
 earth ; N the mountain top ; P the point 
 of tan gen cy ; OP and OK radii of the 
 earth; /22V" the height of mountain and 
 prolongation of OR.
 
 140 PLANE SURVEYING. 
 
 Draw NL perpendicular to ON, and denote the radius of the 
 earth by r ; then, since NL and NP are respectively perpendic- 
 ular to NO and OP, the angle NOP = the angle of depression 
 LNP=a. 
 
 Hence (r + m) cos a = r. 
 
 m cos a 
 
 1 cos a 
 
 Ans. 
 
 MISCELLANEOUS PROBLEMS. 
 
 1. Determine the height of a hill, knowing that the angle of 
 elevation of its top from a certain station = 50, and at a 
 station 800 feet more remote the angle of elevation = 36 20'. 
 
 2. The angle of depression, taken from a balloon to a station 
 whose horizontal distance is known = 18 40'. Find the height 
 of the balloon. 
 
 3. Two war vessels, desiring to ascertain their distances 
 from a fort, remove from each other 2000 feet, anil measure the 
 angle between each other and the fort ; the angles being 
 79 40' and 82 20', what were their distances? 
 
 4. Two observers on the same horizontal plane, 1500 feet 
 apart, and in a vertical plane with a balloon, observe its angles 
 of elevation to be 62 40' and 71 10'. Required the height of 
 the balloon. 
 
 5. The passage between two objects L and N being ob- 
 structed by a swamp, the lines P=420 feet, and PN= 540 
 feet, were measured, and the angle LPN observed = 86 42'. 
 Find the distance LN. 
 
 6. What distance can a person whose eye is 5| feet above 
 the ocean see its surface? Assume radius = 3960 miles. 
 
 7. If the sun subtend an angle of 32' 2", and his distance from 
 the earth is 93,000,000 miles, what is his diameter? 
 
 8. What is the altitude of the sun when the shadow of a staff 
 cast on a horizontal plane is to the height of the staff as 7 to 5 ?
 
 MISCELLANEOUS PROBLEMS. 141 
 
 9. If the horizontal parallax * of the moon be 56' 50" and 
 the diameter of the earth 7920 miles, what is the distance of 
 the moon from the earth? 
 
 10. If the moon subtend an angle of 31' 14", when its dis- 
 tance is 240,000 miles, what is its diameter? 
 
 11. When the meridian altitude of the sun is 50, the shadow 
 cast by the peak of a mountain reaches a certain point on a 
 horizontal plain ; but when his meridian altitude is 60, the 
 shadow strikes a point 2000 feet nearer the base of the moun- 
 tain. Determine the height of the mountain above the plain. 
 
 QUERIES. If on the same day two observations were made 
 on the sun for altitude, one or both when he was not on the 
 meridian, and the length of the shadow measured as in Ex. 11, 
 would sufficient data be thus obtained to determine the height 
 of the mountain ? 
 
 Would it be possible with data obtained, as in the first query, 
 to ascertain the height of the mountain if the sun was vertical 
 over the mountain at noon ? 
 
 12. If the height of a mountain is m miles and its top is visi- 
 ble d miles, find an expression for the diameter of the earth, 
 assuming it to be a sphere. 
 
 13. The angle of depression taken on the top of Peak of 
 Teneriffe, which is two and a half miles high, to the farthest 
 visible point was 2 2'. It is required to determine the observed 
 distance and the diameter of the earth, assuming it to be n 
 sphere. Dist., 140,876 miles; Diam., 7936 miles. Ans. 
 
 EXERCISES. 
 
 1. Measure the height of a flagstaff or church spire above 
 the street. 
 
 * The angle at the moon, or other heavenly body, subtended by the 
 semi-diameter of the earth.
 
 142 
 
 PLANE SURVEYING. 
 
 2. Measure the height of a monument, tower, or some other 
 prominent building upon a hill, without obtaining the distance 
 to the foot of the object. Also, if practicable, measure the 
 distance to the foot of the object and the proper angles. Com- 
 pute and compare results with each other, and with the actual 
 height, if it can be ascertained. 
 
 SECTION V. 
 
 RECORDING THE FIELD NOTES. 
 
 198. The Field Notes may be recorded in various ways, de- 
 pending upon the instrument used, and the extent and intricacy 
 of the survey. 
 
 First Method. If the compass is employed, the bearings 
 simply to be taken, distances measured, and the tract bounded 
 by straight lines (no offsets) , the simplest, most compact, and 
 also most convenient form for the subsequent calculation of the 
 area is to write the stations, bearings, and distances in three 
 columns, thus : 
 
 STATIONS. 
 
 BEARINGS. 
 
 DISTANCES. 
 
 REMARKS. 
 
 1 
 
 S. 20 53' E. 
 
 13.11 
 
 To a maple. 
 
 2 
 
 N. 48 10' E. 
 
 13.62 
 
 " birch. 
 
 3 
 
 N. 43 40' W. 
 
 4.73 
 
 " stake and stones. 
 
 4 
 
 N. 45 08' W. 
 
 4.75 
 
 " white oak. 
 
 5 
 
 S. 51J W. 
 
 2.53 
 
 " sandstone. 
 
 6 
 
 S. 72 J W. 
 
 6.56 
 
 " red oak, beginning. 
 
 199. Second Method. If the tract is not large, and there are 
 offsets in addition to the bearings and distances, or if simply 
 the angles and distances are measured, a very good method, 
 especially for a beginner, is to make a rough plat of the survey,
 
 RECORDING THE FIELD NOTES. 
 
 143 
 
 and indicate in their corresponding places on the sketch the 
 bearings, or angles, and the lengths of the lines and offsets, as 
 shown below : 
 
 6.09, 
 
 The above is a sketch of a small field, showing offsets to 
 stream, etc. The following are hasty surveys of boundaries, 
 etc., of land for proposed park in City of Wilmington, Del., 
 July, August, and September, 1885 : 
 Instruments : Transit. Chesterman's 100-foot steel tape. 
 Work: 
 
 Lines run with transit, and carefully measured with steel tape from 
 
 station to station. 
 
 Angles between these lines taken, always from left to right. 
 Magnetic bearings of lines taken. 
 Stations numbered or lettered in regular order. 
 
 Offsets (sometimes angles and distances) taken to locate houses, cor- 
 ners of fences, etc., offsets made at right angles with lines joining 
 stations. 
 Notes : 
 
 Taken free-hand in small note-books (size 5;}" X 3J"). 
 
 Sketches made to suit the page and to make the matter clear for 
 
 plotting. 
 
 The usual checks used on field and office work. 
 Explanation of Sketches : 
 
 No. 1. Single page of note-book. Location of fences on boundary of 
 
 land proposed for park. 
 No. 2. Two opposite pages of note-book. Location of road through 
 
 land proposed for park, showing railroad crossing. 
 No. 3. Two opposite pages of note-book. Location of run between two 
 
 adjoining owners of land proposed for park. 
 
 No. 4. Two opposite pages of note-book. Location of houses, etc., in 
 land proposed for park.
 
 144 
 
 PLANE SURVEYING. 
 
 42V 
 
 ^x 
 
 
 
 / 
 
 \ -V*! 
 
 ssyir* 
 
 jt- - 
 
 fvp 
 
 i 
 
 
 s 
 
 \ 
 
 
 
 1 
 
 
 j: 
 "a 
 
 
 
 
 
 
 S 
 
 Static 
 
 iPost 
 
 No. 1.
 
 RECORDING THE FIELD NOTES. 
 
 145 
 
 Sta. R 5 
 
 107V 
 
 No. 2.
 
 146 
 
 PLANE SURVEYING.
 
 KECOKDLN'G THE FIELD NOTES. 
 
 147 
 
 177 2i- 
 
 200. Third Method. 
 The column method, an- 
 alogous to that shown 
 in Article 38, Chain 
 Surveying, is, however, 
 the most general. If 
 the bearings are taken, 
 they may be inserted in 
 the column either verti- 
 cally or diagonally ; if 
 only the angles are ob- 
 served, they should be 
 placed at the stations 
 which indicate where 
 the measurements were 
 made. The objects to 
 which offsets are meas- 
 ured may be designated 
 or delineated on the 
 marginal side of the line 
 as they naturally ap- 
 pear. Where streams, 
 roads, fences, etc., cross 
 the line, representations 
 of them are made, in- 
 dicating approximately 
 their direction ; or, if 
 desirable, their bear- 
 ings, or angular devia- 
 tions from the line, may 
 be taken and recorded. 
 
 The following notes 
 will more fully explain 
 the method under con- 
 sideration : 
 
 No.
 
 148 
 
 PLANE SURVEYING. 
 
 24.28 
 
 21.08 
 20.68 
 
 13.50 
 6.00 
 
 (6) 
 
 11.38 
 
 CO 
 
 (5) 
 
 (4) 
 
 17.54 
 
 M 
 
 a 
 
 CO 
 
 f3) 
 33.10 
 24.75 
 
 (2) 
 11.90 
 
 0) 
 
 Chestnut stump, cor. to J.S.K. in W.D.C. 
 line. 
 
 West 
 
 Stone, 7 links S.E. of a butternut tree 
 Cor. to A.L. & W.D.C. 
 
 Limestone, middle of public road. 
 
 Stake & Stones, cor. taL.R. in A.L.'s line 
 Middle of road. 
 
 White oak stump 10' S.W. stone house, 
 north side public road, cor. to J.V. & J.L.
 
 RECORDING THE FIELD NOTES. 
 
 149 
 
 
 
 (i) 
 
 
 
 H3.3Q 
 
 
 =%=S := SP=5= 
 
 14.52 
 
 g- 5-^=^=^ 
 
 _ -_ 
 
 14.12 
 
 . . _, ^^l^^r-^J'ftSt. 
 
 
 
 
 
 
 fc 
 
 
 
 (12) 
 
 Limestone, cor. to W.V. in C.C.'s line 
 
 
 10.47 
 
 
 , 
 
 
 
 Oi 
 
 
 
 00 
 CO 
 
 
 
 (11) 
 
 Limestone. 
 
 
 8.00 
 
 
 
 W 
 
 
 
 
 fc' 
 
 
 ! 
 
 (10) 
 
 Limestone, cor. to J.S.K. in C.C.'s line 
 
 
 30.60 
 
 
 
 CO 
 
 
 
 (9) 
 
 Limestone. 
 
 
 8.31 
 
 
 
 CO 
 
 
 
 (8) 
 
 Limestone. 
 
 
 4.45 
 
 
 
 8 
 
 
 i 
 
 (7) 

 
 150 
 
 PLANE SURVEYING. 
 
 k 
 
 
 
 
 (4) 
 
 Limestone. Sta. (4) in foregoing 
 
 
 10.49 
 
 description. 
 
 Stone house |p 
 
 9.00 
 
 
 100' from line. 
 
 
 
 
 ri 
 
 
 
 
 
 oo 
 
 02 
 
 
 
 m 
 
 Limestone. 
 
 N. 12 W. 
 
 10.41 
 
 Lane leading to dwelling, S. 11 E. 
 
 Road 
 
 9.00 
 
 Hf Barn. 
 
 | 
 
 i 
 
 6.40 
 
 M 
 
 j 
 
 3 
 
 
 House 60' from line. 
 
 -1 
 
 W 
 
 
 
 
 
 
 
 8 
 
 
 . 
 
 20.38 
 
 
 
 
 
 8 
 
 
 Stone house Wft, 
 
 (i) 
 
 White oak stump. Sta. (1) :u 
 
 f near corner. 
 
 
 foregoing description.
 
 RECORDING THE FIELD NOTES. 
 
 151 
 
 The bearing and distance of proof-line from P to Station (11)= S.62 W. 
 19.10. 
 
 N. 79 10' E. 6.00 
 
 Housi 
 o Spring Run 
 
 and 15.82toEastlineof Survey 
 
 9.20 
 
 5.75 
 
 OD 
 
 (#) 
 
 8.80 
 
 W 
 
 02 
 
 (I/) 
 
 36.20 
 
 22.38 
 
 W 
 
 02 
 
 Limestone at end of lane on 
 north bank of " Big Brook." 
 
 Limestone on bank of Spring 
 Run. 
 
 Point in lane. 
 
 Limestone in middle of public 
 road at end of lane.
 
 152 PLANE "SURVEYING. 
 
 The notes show that the sides of the tract were first surveyed ; 
 which, with their bearings and distances, include also the loca- 
 tion and general direction of road-crossings, streams, etc., a 
 description of the corners, and the names of owners of property 
 adjoining the survey. Next to traversing the bounding lines, 
 the survey of the public road, crossing the farm from east to 
 west, was made. This road enters the tract at station (1) ; 
 at 6.40 chains from (B) it passes a house which is 60 feet 
 to the right; at 9.00 chains a road to the left, the bearing of 
 which is given ; at 10.41 chains is a corner at end of lane lead- 
 ing to dwelling ; near the east end of road a stone house is 
 located, at 100 feet north of the line ; and at 10.49, station (4) 
 of sides survey is reached, at which point the road leaves the 
 farm. The survey of the lane to the dwelling, and thence to 
 the creek, is next recorded. Here are noted the intersection of 
 a line S. 79 10' W., and the distances on this, east and west, 
 to spring runs, as well as the distances to the east and west 
 sides of the tract;* the dwelling and barn are located, and the 
 limestone on the north bank of Big Brook reached. A line was 
 run from this last point to station (11), which, in connection 
 with the survey of the lanes, the public road, and the cross-line 
 from L to F, gave proof of the accuracy of the work. 
 
 * This line was made a boundary in the subsequent division of the land.
 
 KECOKDING THE FIELD NOTES. 
 
 153
 
 154 PLANE SURVEYING. 
 
 SECTION VI. 
 
 LATITUDES AND DEPARTURES. 
 
 201. The Difference of Latitude of the two ends of a line is 
 the perpendicular distance between the parallels of latitude 
 which pass through them, and is reckoned north or south, 
 according as the bearing is northerly or southerly. 
 
 The Difference of Longitude of the two ends of a line is the 
 perpendicular distance between the meridians which pass through 
 them, and is reckoned east or west, according as the bearing is 
 easterly or westerly. 
 
 The difference of latitude of a line is often called briefly the 
 latitude, or northing or southing ; and the difference of longitude, 
 its departure, or easting or westing. 
 
 It will be perceived from the definitions just given that, when 
 a line bears either due north or south, the distance equals the 
 latitude, and the departure is nothing ; but if the bearing is 
 east or west, the distance and departure are equal, and the lati- 
 tude is zero. Furthermore, it will be seen that in all other 
 cases except those just cited, the latitude, departure, and dis- 
 tance form the three sides of a right triangle : the distance 
 being the hypotenuse, and the latitude and departure the sides 
 about the right angle. 
 
 Let LP represent a line given by its bearing and distance ; it 
 is required to determine its latitude and departure. 
 
 Let OL and PM represent parallels 
 of latitude, and LM and OP meridians. 
 The lengths of LM = OP and LO=MP 
 are required. 
 
 The problem stated simply is : Given 
 in a right triangle LMP the hypotenuse 
 LP (distance), the angle L (bearing), 
 to find the side LM (latitude) , and MP 
 (departure).
 
 LATITUDES AND DEPARTURES. 
 From Trigonometry, LM^ LPvosL, 
 
 155 
 
 So it is seen that the latitude of a line is obtained by taking 
 the product of the distance and the cosine of the bearing, and 
 the departure is equal to the product of the distance and sine of 
 the bearing. 
 
 202. The case just treated is the principal one which the 
 surveyor will use, since it is necessary as will subsequently be 
 seen in computing areas, to determine the latitudes and de- 
 partures ; and by Jthese formulas he will generally obtain them, 
 having taken in the field the bearings, or angles, and distances. 
 
 Other cases, however, will occur in practice referring to the 
 triangle LMP, and for convenience they are here subjoined. 
 
 Designating the length of the line, or distance, by s, the bear- 
 ing by 6, the latitude and departure respectively by I and d, 
 then we may write the following formulas : 
 
 CASE. 
 
 GIVEN. 
 
 KEQUIKED. 
 
 FORMULAS. 
 
 1 
 
 b, s. 
 
 /, d. 
 
 / = s cos b, d = s sin b. 
 
 2 
 
 b, I. 
 
 s, d. 
 
 s = = lsecb, d = l tan b. 
 cos b 
 
 3 
 
 b, d. 
 
 s, I. 
 
 sin 6 tan 6 
 
 4 
 5 
 
 s, I. 
 s, d. 
 
 b, d. 
 b, I. 
 
 cos 6 = -, d = Vs' 2 I' 2 , 
 s 
 
 sin 6 =-, /= V* 3 rf. 
 
 s 
 
 6 
 
 I, d. 
 
 b, s. 
 
 tan b = -, s= VP + rf*. 
 
 EXAMPLES. 
 
 1. Given the bearing and distance of a line, N. 23 54' W. 
 18.25 chains ; required its latitude and departure.
 
 156 PLANE SU It V EYING. 
 
 2. Given the bearing of a line Jf . 87 40' E., and the depart- 
 ure 2640 feet ; find its distance and latitude. 
 
 3. Given the length of a line 24.60 chains, and the departure 
 17.40 ; find its bearing and latitude. 
 
 4. Given the latitude 23.76 chains south, and the departure 
 0.94 chains west; required the bearing and distance. 
 
 5. Given the distance 1886 feet, and the latitude 943 ; deter- 
 mine its bearing and departure. 
 
 6. It is required to find the distance and departure of a line, 
 given the bearing S. 30' W., and latitude 10.80 chains. 
 
 203. The Traverse Table. By the use of Formula 1, last 
 article, latitudes and departures have been calculated for every 
 quarter-degree of the quadrant, corresponding to distances from 
 1 to 10, and even from 1 to 100 ; these results tabulated con- 
 stitute the traverse table. Such a table was considered quite 
 indispensable when the compass was the principal surveying 
 instrument, but since the more accurate transit has to a great 
 extent superseded the compass, and surveyors are now reading 
 to minutes instead of quarter-degrees, the common traverse table 
 reading only to quarter-degrees is of little practical value. 
 
 When, therefore, the bearings are read to minutes, the lati- 
 tudes and departures are generally best obtained from a table 
 of natural sines and cosines.* 
 
 However, for the benefit of those engaged in compass survey- 
 ing, and for those who, though reading to minutes, prefer to 
 obtain by interpolation the latitudes and departures from the 
 traverse table, one is given near the end of this volume. 
 
 * A traverse table in which the calculations are made to every minute 
 of bearing for distances from 1 to 10 and extending to five decimal places, 
 would answer the purpose admirably. Such a table is in existence, but it 
 is not common. The common tables of natural sines and cosines are sim- 
 ply tables of latitudes" and departures corresponding to a unit's distance. 
 With a distance 2, the latitude and departure are twice those in the table; 
 when the distance is 3, three times ; when n, n times.
 
 LATITUDES AND DEPARTURES. 157 
 
 Explanation of the Traverse Table. The number of degrees 
 in the bearing if it does not exceed 45 is found in the left-hand 
 column of the page, and the latitudes and departures, as indi- 
 cated at the top, may be taken under the proper distance ; if 
 the number of degrees is greater than 45, it is found in the right- 
 hand column of the page, and the columns of latitudes and 
 departures are indicated at the bottom. For example : 
 
 1. Let it be required to find the latitude and departure cor- 
 responding to a bearing N. 34 30' E. and distance 5 chains. 
 
 We find in the table, opposite 34 30' and under "distance 5," 
 in the column headed "Lat.," 4.121, and in the column headed 
 "Dep.," 2.832. Hence the latitude and departure are respec- 
 tively 4.12 N. and 2.83 E. 
 
 2. Required the latitude and departure of a line bearing N. 
 72i W. 9 chains. 
 
 Looking in the column at the right of the page for 72 15', 
 and under "distance 9," we find, reading at bottom, 
 in the Lat. column, 2.744 ; 
 in the Dep. column, 8.572. 
 
 Hence the latitude is 2.74 chains N., and the departure 8.57 
 chains W. 
 
 204. The table may be used to find the latitude and depart- 
 ure for any distance however great. If, in first example above, 
 we suppose the bearing to remain the same, but the distance to 
 be 50 chains ; then, since for the same bearing the latitudes and 
 departures vary directly as the distances, the latitude, or depart- 
 ure, for 50 chains is 10 times that for 5 ; and, as multiplying 
 by 10 is in effect removing the decimal point one place to the 
 right, we may take directly from the table opposite 5 the lati- 
 tude and departure of 50, or 41.21 N. and 28.32 E. 
 
 If the distance is not a multiple of 10, but made up of units 
 and tens, we may take out of the table the latitude and depart- 
 ure for the units, and for the tens as indicated above. The sum 
 of these will evidently be the latitude and departure required.
 
 158 PLANE SURVEYING. 
 
 3. Let it be required to find the latitude and departure of a 
 line S. 40 E. 34 chains. 
 
 Looking in the table opposite 40 and under " distance 3," 
 take out at ouce, by conceiving the decimal point removed one 
 place to the right. 
 
 For 30 chains, Lat. 22.98 Dep. 19.28 
 Then " 4 " " 3.06 " 2.57 
 
 34 chains, Lat. 26.04 S. Dep. 21.85 E. 
 
 By an extension of the above principle, the table may be used 
 to obtain the latitude and departure when the distance is com- 
 posed of chains and links. 
 
 4. Given the bearing of a line S. 28 45' W. 26.58 chains, to 
 find its latitude and departure. 
 
 For 20 chains, Lat. = 17.53 Dep. = 9.62 
 
 6 
 .5 " 
 .08 
 
 44 = 5.26 
 " = .44 
 44 = .07 
 
 44 = 2.89 
 44 = .24 
 4 ' = .04 
 
 26.58 chains, 
 
 Lat, = 23.308. 
 
 Dep. = 12. 79 W. 
 
 5. Find by the traverse table the latitude and departure of a 
 line bearing N. 41 45' E. 17.29 chains. 
 
 6. Given the bearing of a line S. W., distance 23.48 
 chains, to find its latitude and departure. 
 
 7. What are the latitude and departure of a line bearing S. 
 8530'E. 135.42 chains? 
 
 8. If the bearing and distance are N. 89f W. 20.09 chains, 
 what are the latitude and departure ? 
 
 205. By means of interpolation the traverse table may be 
 used to find the latitude and departure when the bearing is 
 given to minutes. Thus, the bearing being N. 34 20' E. any 
 given distance, take out the latitude and departure corre-
 
 LATITUDES AND DEPARTURES. 159 
 
 spending to 34 15' and the given distance, and add* to that 
 departure T 5 ^, or , of the difference between it and that corre- 
 sponding to 34 30' and the given distance, for the departure 
 required. Likewise obtain T 5 7 of the difference between the 
 latitudes corresponding to 34 15' and 34 30' and the distance, 
 and subtract* from the latitude first found for the latitude 
 required. 
 
 For a bearing 34 23', the fractional part to be taken of the 
 difference between 34 15' and 34 30' would be T % ; the numer- 
 ator being the excess in minutes above the quarter, and the 
 denominator 15. 
 
 206. In the absence of a traverse table calculated to minutes, 
 the table of natural sines and cosines, as before stated, is the 
 best to use when the bearings are given to minutes. 
 
 It is shown in Article 201 that the cosine of the bearing mul- 
 tiplied by the distance gives the latitude, and the product of the 
 distance and sine of bearing gives the departure. 
 
 EXAMPLES. 
 
 1. The bearing and distance of a line are N. 37 43' W. 
 24.29 chains ; required its latitude and departure. 
 
 Four places of decimals from the table will usually be 
 sufficient. 
 
 The cosine of 37 43' true to four places = .7911. 
 The sine of 37 43' true to four places = .6118. 
 
 .7911X24.29 = 19.21 N. Lat. 
 .6118 x 24.29 = 14.86 W. Dep. 
 
 The following contracted form of multiplication, using five 
 decimal places, gives practically the same result : 
 
 Cosine of bearing = .79105 ; sine of bearing = .61176. 
 
 * The departure increases with an increase of the bearing; the latitude 
 diminishes.
 
 160 PLANE SURVEYING. 
 
 chains, Lat. = 15.8210 Dep. = 12.2352 
 
 " = 3.1642 " = 2.4470 
 Distances 
 
 .0712 " = .0551 
 
 24.29 chains, Lat. = 19.21 N. Dep. = 14.86 W. 
 
 2. Find the latitude and departure of a line bearing S. 62 1 7' E. 
 37.18 chains. 
 
 3. Required the latitude and departure of a line N.8857'W. 
 28.97 chains. 
 
 4. Required the latitude and departure of a line bearing 
 S. i E. 2640 feet. 
 
 5. Given the bearings and distances of two lines running 
 from the same point P, as follows: PO, N. 38 37' E. 1760 
 feet, and PL, N. 71 54' E. 1320 feet ; to find by means of lat- 
 itudes and departures the distance OL. 
 
 6. Assuming PO bears N. 48 17' W. 27.42 chains, and PL 
 S. 36 28' W. 19.24 chains, find, as in the last example, the 
 distance OL between the extremities of the lines. 
 
 207. Testing a Survey. It is evident that when a surveyor 
 has passed completely round a tract of land and returned to the 
 place of beginning, he has gone in a northerly direction just as 
 far as he has gone in a southerly direction, and as far easterly 
 as westerly. Hence the sum of the north latitudes should equal 
 the sum of the south latitudes, and the sum of the east depart- 
 ures equal the sum of the west departures.* 
 
 In practice, this degree of accurac}- is seldom attained, for 
 various causes incident to the manipulation of the instruments, 
 their inherent defects, imperfect chaining, etc. 
 
 * If the survey is effected by traversing (Article 163), the reading at 
 the last station should be 360 or 0. If the interior angles are measured, 
 their sum should equal twice as many right angles less four as the figure 
 has sides. If a small error exists, it must be distributed evenly among the 
 angles, unless on account of the difficulty of observing one or more of the. 
 angles, these should have a larger share of the error. See, also, Article 156
 
 LATITUDES AND DEPARTURES. 
 
 161 
 
 On account of the varying conditions in different surveys, it 
 is impracticable to state precisely how great an error should be 
 allowed without a re-survey of the tract. A rule usually fol- 
 lowed by compass surveyors is to allow an error of 1 link for 
 every 5 chains, 1 : 500. 
 
 This is perhaps a fair average for ordinary farm surveying. 
 If the ground is exceptionally clear, and quite level, an error of 
 1 : 1000 is not too great ; if, on the other hand, the ground is 
 uneven, rocky, and brush} 7 , 1 : 300, or even 1 : 200, might be 
 allowed. The error resulting from a transit survey of the same 
 ground should be much less. For the average case given above, 
 instead of 1 : 500 it should not be much less than 1 : 1200. 
 
 The above rules are cited simply as guides to the young sur- 
 veyor to aid him in forming a standard for himself, based on 
 his own experience. 
 
 208. Correcting Latitudes and Departures, or Balancing the 
 Survey. (1) A survey is balanced when the northings equal 
 the southings, and the eastings equal the westings. When these 
 equalities do not exist, the error is distributed among the lines, 
 proportioned to their lengths. This operation is called correct- 
 ing the latitudes and departures. It is best illustrated by an 
 example : 
 
 7. 
 
 BEARINGS. 
 
 DISTS. 
 
 LATITUDES. 
 
 DEPART- 
 URES. 
 
 CORREC- 
 TIONS. 
 
 CORRECTED 
 LATITUDES. 
 
 CORRECTED 
 DEPART'S. 
 
 f. 
 
 S. 
 
 E. 
 
 W. 
 
 j 
 
 * 
 
 N. 
 
 S. 
 
 E. 
 
 W. 
 
 i 
 
 8. 20 53' E. 
 
 13.11 
 
 . . . 
 
 12.25 
 
 4.67 
 
 
 1 
 
 1 
 
 
 12.27 
 
 4.68 
 
 
 I 
 1 
 
 4 
 f 
 
 N. 48 10' E. 
 N. 43 40' W. 
 N. 45 08' W. 
 8. 51 30' W. 
 
 13.62 
 4.73 
 4.75 
 2.53 
 
 9.08 
 3.42 
 3.35 
 
 ... 
 
 10.15 
 
 3.26 
 3.36 
 1.98 
 
 9 
 
 1 
 1 
 
 1 
 1 
 
 9.06 
 3.41 
 3.34 
 
 
 10.16 
 
 3.26 
 3.35 
 1.98 
 
 1.57 
 
 ... 
 
 1.57 
 
 
 I 
 
 S. 72 30' W. 
 
 6.56 
 
 ... 
 
 1.96 
 
 ... 
 
 6.26 
 
 7 
 
 1 
 4 
 
 ... 
 
 1.97 
 
 ... 
 
 6.25 
 
 
 
 45.30 
 
 15.85 
 
 15.78 
 
 14.82 
 
 14.86 
 
 15.81 
 
 15.81 
 
 14.84 
 
 14.84 
 
 
 
 
 15.78 
 
 
 
 14.82 
 
 
 
 
 
 
 
 Error in latitude, 7 links. 4 linkfl. Error in departure.
 
 162 PLANE SUliV EYING. 
 
 In the table the latitudes and departures corresponding to the 
 several bearings and distances are obtained by means of a table 
 of sines and cosines, and placed in their proper columns. 
 
 The first course being between the south and east, the lati- 
 tude found is written in the column headed S., the departure 
 in column E., and so on, the letters of the course indicating the 
 columns in which to place the latitudes and departures. The 
 difference of the sums in the latitude columns is then taken, and' 
 found to be 7 links : this is the error in latitude. 
 
 The error in departure, found in a corresponding manner, is 
 4 links. 
 
 The total distance round the field is showYi by the footing of 
 the distance column to be 45.30 chains. The distribution of the 
 error is effected then by the proportions : 
 
 For the Latitude. 
 
 Sum of the sides : length of any side = error : correction for that side. 
 45.30 : 13.11 = 7 : 2 
 
 45.30 : 13.62 = 7 : 2 
 
 For the Departure. 
 45.30 : 13.11 = 4 : 1 
 
 It is unnecessary usually to make but one proportion each for 
 the latitude and departure correction, since the error for any 
 other side may be found mentally by comparing its length with 
 that of the side used in the proportion. Whole links only are 
 used. The latitude correction for the second side is a little 
 greater than 2, but it is nearer 2 than 3, and is therefore 
 called 2. 
 
 The corrections thus found are written in their proper col- 
 umns, headed "Correction, Lat. Dep.," opposite the sides to 
 which they refer, and are so applied by addition or subtraction 
 as may be required to reduce the errors to zero. The quanti- 
 ties thus obtained are placed in the columns of corrected lati- 
 tudes and departures to the right of the corrections.
 
 LATITUDES AND DEPARTURES. 163 
 
 Since the southings are too small, the correction 2 is added 
 to 12.25, making 12.27, for the first entry in the column of cor- 
 rected latitudes. The eastings being too small, the correction 
 1 is added to 4.67, making 4.68, to be written under E. in the 
 corrected departures ; and so on for the rest. 
 
 If the corrections have been properly applied, the northings 
 will equal the southings, and the eastings the westings, and the 
 survey is balanced. 
 
 In the example just given, the difference of latitude is 7 and 
 the departure 4 links ; hence, the length of a line to close the 
 survey = A /7 2 + 4* = about 8 links ; and as the perimeter of the 
 tract =45. 30 chains, the "error of the survey," or " error of 
 closure, "= 1 link for 5.66 chains, or 1 : 566. 
 
 Some surveyors prefer a more compact table than that given 
 above, and instead of a double set of latitudes and departures, 
 use but one, and write in ink of different colors the corrected 
 latitudes and departures over the first. Others, again, prefer 
 two columns instead of four for the latitudes and departures, 
 using the plus (-f) sign to indicate north latitudes and east 
 departures, and the minus ( ) sign to indicate south latitudes 
 and west departures. 
 
 The form given above is, however, preferable to either, since 
 a mistake in the application of the corrections is in that more 
 easily detected, the footings are more expeditiously and accu- 
 rately obtained, and the subsequent part of the work referring 
 to the area is thereby facilitated. 
 
 If a side of the survey passes over very rough ground, or 
 through a dense wood, or for any reason it is rendered more 
 difficult to measure than any of the others, the surveyor should 
 exercise his judgment in deciding how much more of the error 
 than the rule would indicate should be applied to that side. 
 
 Regard must also be had to the probability of error in the 
 bearings ; hence, when a side of considerable length is aligned 
 through a thicket, or over very uneven ground, and where often- 
 times the observations are made to top of rod, if it is found 
 that a slight change in the bearing will diminish materially the 
 error, the change should be made.
 
 164 PLANE SURVEYING. 
 
 The diurnal variation of the needle is not unfrequently a 
 source of error in compass surveys. A range of 10 minutes is 
 quite common, and even 15 minutes is occasionally noted. This 
 error may be avoided by measuring the angles of the tract, or 
 testing the compass every two or three hours by setting up and 
 sighting on some line as standard. 
 
 Some authors and surveyors affirm that when the bearing of 
 a line is due east or due west, the error in latitude is nothing, 
 and therefore such a line needs no correction. Likewise a due 
 north and south line has no error in departure. The writer does 
 not concur in this view ; for the errors in compass work are not 
 confined to the chaining, and in transit surveying there is fre- 
 quently considerable error in the angles. In the application of 
 the rule these facts are assumed ; indeed, as soon as a correc- 
 tion, made in the usual manner, is applied to any side, a change 
 of bearing results, for the corrected latitudes and departures no 
 longer belong to the original bearing, but to some other. More- 
 over, there is no more reason for supposing a line runs due 
 north because it is so read than that a line runs N. ^ E. or 
 N. 89| E. being so read ; yet no surveyor would hesitate to 
 apply the rule to either of these, thus assuming that an error in 
 bearing as well as in chaining was committed ; and this is the 
 correct assumption on which, without excepting any side, the 
 distribution of the error, except as follows, should be made. 
 
 (2) If, however, a survey is made with a transit in good ad- 
 justment, the angles, either interior or deflection, being carefully 
 observed, and the test hereinbefore mentioned when applied giv- 
 ing the inference that the angles were accurately measured, and 
 the error of closure therefore due to erroneous chaining, then 
 the correction which should be applied is obtained as follows : 
 
 Add up the columns of latitudes, and also those of departures, 
 and say, as the arithmetical sum of all the -{ * } * to 
 
 correction to be applied to that i,
 
 LATITUDES AND DEPARTURES. 
 
 165 
 
 (3) If greater accuracy is required than can be attained by 
 the preceding methods, each side should be weighted ; that is to 
 say, the surveyor determines the relative difficulties in measure- 
 ment and alignment of the boundaries, considering some one 
 side the standard. Calling the error probably made in the side 
 chosen as standard one (1), another side, which in the judg- 
 ment of the surveyor was, per unit, twice as difficult to measure, 
 would be multiplied by 2, or, as it is termed, have a weight of 
 2 ; another multiplied by 3, or 1, etc. Then, instead of tak- 
 ing the perimeter for the divisor, as was done in the first case 
 above, the sum of the sides thus multiplied or weighted is used, 
 and the proportion is as follows : 
 
 As the sum of the multiplied distances is to any particular 
 multiplied distance, so is the error in { l ^ iltwd * j. to the correction 
 
 . ' 
 
 The following illustrates the method of balancing a survey 
 when the sides are weighted : 
 
 STATIONS. 
 BEARINGS. 
 
 DISTANCES. 
 
 WEIGHTS. 
 
 1 MULTIPLIED 
 DISTANCES. 
 
 LATITUDES. 
 
 DEPART- 
 URES. 
 
 
 
 <s ^ 
 
 CORRECTED 
 LATITUDES. 
 
 CORRECTED 
 DEPART'S. 
 
 JV. 
 
 S. 
 
 E. 
 
 W. 
 
 \ 
 
 1 
 
 jr. 
 
 S. 
 
 E. 
 
 W. 
 
 1 N. 9 W. 
 2 If. 31 E. 
 
 15.50 
 
 25.40 
 
 1 
 
 8 
 
 15.50 
 76.20 
 
 15.31 
 21.77 
 
 
 
 2.43 
 
 1 
 
 6 
 
 3 
 9 
 
 15.32 
 21.83 
 
 
 
 2.41 
 
 . 
 
 13.09 
 
 . . . 
 
 13.18 
 
 3 S. 71 E. 
 
 10.00 
 
 8 
 
 30.00 
 
 
 3.17 
 
 9.48 
 
 
 8 
 
 4 
 
 
 3.14 
 
 9.52 
 
 
 4 S. 10iE. 
 
 19.70 
 
 8 
 
 39.40 
 
 
 19.37 
 
 3.59 
 
 
 8 
 
 5 
 
 
 19.34 
 
 3.64 
 
 
 5 8. 10|W. 
 
 14.60 
 
 11 
 
 21.90 
 
 
 14.34 
 
 
 2.72 
 
 2 
 
 2 
 
 
 14.32 
 
 
 2.70 
 
 6 8. 89 W. 
 
 21.25 
 
 1 
 
 21.25 
 
 
 0.37 
 
 37.25 
 
 ... 
 
 21.25 
 26.40 
 
 2 
 
 2 
 
 37.15 
 
 0.35 
 37.15 
 
 26.34 
 
 21.23 
 26.34 
 
 
 
 
 204.00 
 
 37.08 
 
 26.16 
 
 
 
 
 
 
 
 
 37.08 
 
 
 26.16 
 
 
 
 
 
 
 
 Error in latitude, 17 links. 24 
 
 inks, error In departure. 
 
 * Weights could be applied to the correction of the chaining in the 
 second case, by multiplying the latitudes and departures instead of the 
 lengths of the sides.
 
 166 PLANE SURVEYING. 
 
 EXAMPLES. 
 
 Correct the latitudes and departures in the following examples 
 by the first method : 
 
 1. 2. 
 
 (1) S. E. 22.45 chains; (1) South 22s45 chains ; 
 
 (2) N. 89|E. 67.10 " (2) East 67.10 
 
 (3) N. iW. 23.85 " (3) North 23.85 
 
 (4) S. 89f W. 66.30 " (4) West 66.30 " 
 
 (5) S. 21 1 W. 1.30 " (5) S. 22 W. 1.30" 
 
 EXERCISES. 
 
 A few surveys should now be made, and the methods above 
 given employed in balancing. 
 
 SECTION VII. 
 
 SUPPLYING OMISSIONS. 
 
 209. When, for any cause, it is impracticable to obtain the 
 direction or the length, or both, of a side of a tract of land, 
 these may be obtained by calculation. Even the lengths or 
 bearings of two sides may in general be supplied.* 
 
 The determination, however, of these sides or bearings is 
 based upon the measurements of the other bounding lines and 
 angles ; but as these are not usually precisely correct, and as 
 there are no means of testing them in their application to the 
 solution of problems under this head, it is earnestly recom- 
 mended that all measurements, if possible, be made. 
 
 There are four cases. 
 
 * If the two omitted sides are parallel and equal, their bearings cannot 
 be supplied ; or if they are parallel and of equal or unequal lengths, their 
 distances cannot be computed.
 
 SUPPLYING OMISSIONS. 
 
 167 
 
 CASE I. 
 
 210. Given the bearings and distances of all the sides of a 
 tract of land except the bearing and distance of one side, to deter- 
 mine these. 
 
 Find the latitudes and departures of the given sides. The 
 difference of the northings and southings will show the latitude 
 of the line omitted, and the difference of the eastings and west- 
 ings its departure. Then 
 
 Length of line = Vlat. 2 + dep. 2 
 Tan angle of bearing of line = 
 
 The cardinal points between which the line runs are indicated 
 by the deficiency in the latitude and departure columns. 
 
 EXAMPLES. 
 
 1. Given (1) N. 24| E. 23.75 chains; 
 
 (2) S. 81 E. 11.70 " 
 
 (3) S. 1E. 12.64 " 
 
 (4) S. 11| W. 14.50 " 
 
 To find the length and bearing of a. line connecting the ex 
 tremity of the fourth side with the first corner. 
 
 STA- 
 TIONS. 
 
 BEARINGS. 
 
 DISTS. 
 
 N. 
 
 S. 
 
 E. 
 
 W. 
 
 1 
 
 N. 24} E. 
 
 23.75 
 
 21.61 
 
 
 9.85 
 
 . . . 
 
 2 
 
 S. 811 E. 
 
 11.70 
 
 
 1.78 
 
 11.66 
 
 
 3 
 
 S. 1 E. 
 
 12.64 
 
 . . . 
 
 12.64 
 
 .22 
 
 
 4 
 
 S. 11.1 W. 
 
 14.50 
 
 
 14.21 
 
 
 2.89 
 
 
 
 
 21.61 
 
 28.63 
 
 21.63 
 
 2.89 
 
 
 
 
 
 21.61 
 
 2.89 
 
 
 7.02 N. 18.74 W. 
 
 Length of line = V (7. 02)-' + (18. 74) 2 = 20.01 chains.
 
 168 
 
 PLANE SURVEYING. 
 
 Tan bearing = 
 Bearing 
 
 18.74 
 
 7.02* 
 = N. 69 28' W. 
 
 2. Given the bearings and distances of the sides of a tract of 
 land as follows ; it is required to find the length and bearing of 
 the fourth side.* 
 
 (1) N. llfE. 12.69 chains; 
 
 (2) S. 87f W. 8.50 " 
 
 (3) N. 85 W. 11.70 " 
 (5) S. 821 E. 10.53 " 
 
 The foregoing case may be employed to overcome an obstacle 
 in a line, as LN. Thus, surveying LOPN, 
 there will be given all the sides except LN, 
 which can be determined as above. If it 
 is desired to straighten an old road, the 
 length and direction of the new road may 
 be computed from the distances and deflec- 
 tions, or bearings of the old. 
 
 For example, let ABODE be a crooked 
 road which it is desired to replace by a straight one, AE. The 
 bearings and distances being as follows, the length 
 and bearing of AE are required. 
 
 AB, North, 
 J8C, N. 20 E. 
 CD, N. 35 E. 
 DE, N. 10 W. 
 
 12.70 chains; 
 13.25 " 
 12.75 " 
 16.90 " 
 
 Ans. N. 9 41' E. 52.98 chains. 
 
 EXAMPLE 2. Given the following as the bearings 
 and distances of a road, it is desired to straighten, to 
 find the length and bearing of the new road. 
 
 * In practice, the result should be checked by making a plot of the field.
 
 SUPPLYING OMISSIONS. 169 
 
 (1) N. 12 W. 13. 10 chains; 
 
 (2) N. 8 E. 1G.20 " 
 
 (3) N. 21 W. 14.40 " 
 
 (4) N. 40J E. 15.08 " 
 
 (5) N. 601 W. 16.12 " 
 
 EXAMPLE 3. In last figure but one, suppose LO bears N. 
 44 20' W., distance 3.95 chains. Deflection at from OL 
 30, and OP= 6.90 chains. Deflection at P from OL 100, and 
 PN= 5.40 chains. It is required to find the length and bear- 
 ing of NL. Ans. Bearing south. Length, 12.55 chains. 
 
 CASE II. 
 
 211. Given the bearings and distances of all the sides of a 
 tract of land, except the distances of tivo sides not parallel, to 
 determine these. 
 
 By Article 168, change all the bearings so that one of the 
 sides, whose direction only is known, shall become a meridian. 
 Tabulate the latitudes and departures corresponding to the 
 changed position of the sides. The side made meridian will 
 have 110 departure, and the difference of the eastings and west- 
 ings, therefore, will be the departure of the other unknown side. 
 Now with this departure and the changed bearing the distance 
 and difference of latitude of this side may be found, and should 
 be inserted in their proper places in the table. Then the differ- 
 ence between the northings and southings will be the latitude, 
 or length of the side made a meridian.* 
 
 212. Otherwise. If the deficient sides adjoin. 
 
 If a linef be drawn connecting L and N, a figure, LNOPQ, 
 will be shown, in which all the sides are given except LN: 
 the bearing and distance of this side may, therefore, be calcu- 
 lated by the preceding case. This line and the two sides, LM 
 
 *It is immaterial whether or not the deficient sides adjoin, 
 t Called a closing line since it closes the survey LQPON.
 
 170 PLANE SURVEYING. 
 
 and MN, whose bearings only are given, will form a triangle, 
 in which will be known one side and all the angles, whence 
 the unknown distances, LM and JOT", may be computed. 
 
 213. If the sides do not adjoin. 
 
 In the figure suppose that the distances LM and PO are 
 wanting. Draw Ln and no parallel and equal respectively 
 to MN and NO. Then by joining oP, a closed figure will be 
 formed, all the bearings and distances of which are known 
 except the bearing and distance of the closing line, Po, and 
 these may be found by Case I. Po thus determined, there will 
 be known in the triangle PoO all the angles and one side, to 
 find PO, and 00, which is equal to LM. 
 
 EXAMPLES. 
 
 1. Given the following bearings and distances of the sides of 
 a tract of land, to find the length of the 3d and 6th sides. 
 (See last figure.) 
 
 (1) N. 61 W. 9.38 chains; 
 
 (2) N. 65J E. 8.25 " 
 
 (3) S. 39 E. Unknown; 
 
 (4) S. 2 W. 4.45 chains ; 
 
 (5) S. 46 W. 5.00 
 
 (6) N. 88 W. Unknown.
 
 SUPPLYING OMISSIONS. 
 
 171 
 
 STA- 
 TIONS. 
 
 BEARINGS. 
 
 DlSTS. 
 
 N. 
 
 8. 
 
 E. 
 
 W. 
 
 1 
 
 N. 6i W. 
 
 9.38 
 
 9.32 
 
 
 . . . 
 
 1.06 
 
 2 
 
 N. 65 E. 
 
 8.25 
 
 3.42 
 
 
 7.51 
 
 
 3 
 
 S. 39 E. 
 
 
 
 
 
 
 4 
 
 S. 2 W. 
 
 4.45 
 
 
 4.45 
 
 
 0.16 
 
 5 
 
 S. 4G D W. 
 
 5.00 
 
 
 3.47 
 
 
 3.60 
 
 
 
 
 12.74 
 
 7.92 
 
 7.51 
 
 4.82 
 
 
 
 
 7.92 
 
 
 4.82 
 
 
 
 
 
 4.82 
 
 
 2.69 
 
 
 Tan bearing = M, or Po bears S. 29 10' W. 
 
 Length of Po = V(4.82) 2 +(2.69) 2 = 5.52. 
 
 Angle P therefore = 68 10'. 
 Angle " = 49. 
 
 = 62 50'. 
 
 Angle o 
 
 sin. 49 
 : sin. 62 50' 
 : 5.52 
 
 : PO (3d side) = 6.51 
 
 Ar. co. = 0.122220 
 = 9.949235 
 =i 0.741939 
 
 = 0.813394 
 
 sin. 49 Ar. co. = 0.122220 
 
 : sin. 68 10' =9.967674 
 
 ;: 5.52 = 0.741939 
 
 side) =6. 79 =0.831833 
 
 EXAMPLE 2. Given the following data to supply the omissions 
 
 (1) N. 8 E. 9.80 chains; 
 
 (2) N. 81| E. Unknown ; 
 
 (3) S. 70 E. 
 
 (4) S. 5 W. 17,70 chains; 
 
 (5) N. 87 W. 18. 75 chains, to the beginning.
 
 172 
 
 PLANE SURVEYING. 
 
 EXAMPLE 3. In the last example insert the distances found, 
 and suppose the first and fourth sides are wanting ; determine 
 these by either or both methods. 
 
 CASE III. 
 
 214. Given the bearings and distances of all the sides of o- 
 tract of land, except the bearings of two sides, to determine these. 
 Tabulate the latitudes and departures of the sides completely 
 given ; obtain the difference of the northings and southings, and 
 of the eastings and westings. These differences will be the 
 latitude and departure of a closing line. 
 
 The bearing and distance of the closing line may hence be 
 computed ; then in the triangle formed by this line and the two 
 sides whose distances are given, determine the angles ; and 
 thence, with a proper application of them to the bearing of the 
 closing line, the wanting bearings may be found. 
 
 In the figure let PQONML represent a tract of land in which 
 all the bearings and distances are 
 known except the bearings of QO 
 and NM. 
 
 Drawing nN parallel and equal to 
 QO, and joining Qn and nM, a 
 closed figure, PQnMLP, will be 
 formed, in which the bearing and 
 distance of nM, the closing line, 
 may be calculated by Case I. Then 
 in the triangle MnN, having all the 
 sides, the angles are readily found, and by proper applica- 
 tion of these with the bearing of Mn the bearings of NM, and 
 nN= QO may be obtained. 
 
 NOTE. If the sides whose bearings are required adjoin, the reasoning 
 is evident. If they do not adjoin, a transposition of some of the sides may 
 be made, as in the preceding case, without changing the direction or length 
 of any of them, making the unknown sides adjoin, and with the closing 
 line form the triangle referred to in the last paragraph. The rule is, 
 therefore, applicable to either.
 
 SUPPLYING OMISSIONS. 
 
 173 
 
 EXAMPLES. 
 
 1. Given the following data of a survey, to supply the omis- 
 sions. Referring to the last figure : 
 
 e bearing of 
 
 PQ, 
 
 N. 
 
 3 
 
 E. 
 
 dist. 
 
 4 
 
 .57 
 
 chains. 
 
 (4 
 
 u 
 
 QO, 
 
 
 
 
 tt 
 
 6.25 
 
 ii 
 
 II 
 
 II 
 
 ON, 
 
 S. 
 
 23| 
 
 E. 
 
 !( 
 
 5 
 
 .50 
 
 (t 
 
 (( 
 
 11 
 
 NM, 
 
 
 
 
 (( 
 
 4 
 
 .88 
 
 it 
 
 !< 
 
 II 
 
 ML, 
 
 N. 
 
 87 
 
 W. 
 
 II 
 
 2 
 
 .97 
 
 u 
 
 (( 
 
 II 
 
 LP, 
 
 N. 
 
 43 
 
 W. 
 
 U 
 
 3 
 
 .:},'} 
 
 K 
 
 STA- 
 TIONS. 
 
 LINES. 
 
 BEARINGS. 
 
 DlSTS. 
 
 N. 
 
 8. 
 
 E. 
 
 W. 
 
 P 
 
 PQ 
 
 N.3 E. 
 
 4.57 
 
 4.56 
 
 . . . 
 
 0.24 
 
 . . . 
 
 Q 
 
 0,0 
 
 
 6.25 
 
 
 
 
 
 
 
 ON 
 
 S. 23 E. 
 
 5.50 
 
 . . . 
 
 5.05 
 
 2.19 
 
 . . . 
 
 N 
 
 NM 
 
 
 433 
 
 
 
 
 
 M 
 
 ML 
 
 N. 87 W. 
 
 2.97 
 
 0.15 
 
 ... 
 
 ... 
 
 2.97 
 
 L 
 
 LP 
 
 N. 43 W. 
 
 3.33 
 
 2.43 
 
 
 
 2.28 
 
 
 
 
 
 7.14 
 
 5.05 
 
 2.43 
 
 5.25 
 
 
 
 
 
 5.05 
 
 
 
 2.43 
 
 Deficiency, 2.09 S. Deficiency, 2.82 E. 
 
 Tan of bearing of 
 
 ^=-, and bearing = S. 53 28' E. 
 
 Dist. nM= V(2.09) 2 + (2.82) 2 = 3.51. 
 To find the angle of nMN: 
 
 log 4.33 Ar. co. = 9.363512 
 log 3. 51 " = 9.454693 
 
 log 7.045 = 0.847881 
 
 log .795 = 1.900367 
 
 2)19.566453 
 log cosine nMN= 9.783226 
 
 and < = 52 37' 
 2_ 
 
 105 14'
 
 174 PLANE SURVEYING. 
 
 Now, since Mn bears N. 53 28' W., and the angle 
 = 105 14', the line MN is in the northeast quadrant, and 
 makes an angle with the meridian = 105 14' 53 28' = 51 46', 
 or its bearing is N. 51 46' E. ; and hence, reading in the order 
 the measurements were made, the bearing of NM= S. 51 46' W. 
 
 To find the angle nNM, and thence the bearing of QOs 
 
 6.25 Ar. co. = 9.204120 
 
 :3.51 =0.545307 
 
 : : sin 105 14' = 9.984466 
 
 : sin 32 48' nNM) = 9.733893 
 
 Bearing of NM = S. 51 46' W. 
 
 < nNM 32 48' on west side, add 32 48' 
 
 Bearing of Nn = OQ = S. 84 34' W., 
 
 or bearing of QO = N. 84 34' E. 
 
 2. Supply the omissions from the following data : 
 
 (1) N. 34 W. 13. 00 chains; 
 
 (2) S. 41| W. 12.90 " 
 
 (3) S. 50 E. 8.20 " 
 
 (4) 2.56 " 
 
 (5) 6.90 " 
 
 (6) N. 28 E. 9.95 
 
 CASE IV. 
 
 215. Given the bearings and distances of all the sides of a 
 tract of land except two, one of tvhich has only its bearing given, 
 and the other the distance, to supply the omissions. 
 
 Make a meridian the side whose bearing only is given. Tab- 
 ulate the latitudes and departures corresponding to the changed 
 position of the survey. The side made meridian will have no 
 departure, and the difference of the eastings and westings, 
 therefore, will be the departure of the side whose bearing is 
 unknown. With the given distance and this departure the
 
 SUPPLYING OMISSIONS. 
 
 175 
 
 changed bearing and difference of latitude of this side may be 
 found, and should be inserted in their proper places in the 
 table. Then the difference of the northings and southings will 
 be the latitude, or length, of the side made a meridian. 
 
 216. Otherwise. When the deficient sides adjoin. 
 
 Let the bearing of MN and the distance LM be wanting. 
 Calculate by Case I. the direction 
 and length of the closing line LN. 
 A triangle, LMN, may then be formed 
 in which will be,given the lengths of Q 
 LN and MN, and the angle NLM. 
 The distance LM and the angle N 
 may therefore be computed, and the 
 angle N thus found properly applied 
 to the bearing of the closing line will 
 give the bearing of MN. L M 
 
 217. When the deficient sides do not adjoin. 
 
 Referring to the same figure as before, suppose the bearing 
 of LM and the distance OP wanting. Transpose the sides 
 as there shown, and calculate, as in Case I., the direction and 
 length of the closing line Po. Then, as in the preceding article, 
 there will be given a triangle, OPo, in which are known two 
 sides Po and Oo, and the angle P, whence the bearing of Oo, 
 or LM, and the distance PO, may be determined. 
 
 EXAMPLES. 
 1. Given the following notes, to supply the omissions. 
 
 QP. 
 PO. 
 ON. 
 
 N. 10 
 S. 88 
 S. 16 
 
 E. 
 E. 
 E. 
 
 13.71 chains; 
 18.75 " 
 16.50 
 13.00 * 
 
 NM. S. W. 
 
 ML. N. 80 W. ; 
 
 LQ. N. 36 W. 10.00 chains.
 
 176 
 
 PLANE SURVEYING. 
 
 STA- 
 TIONS. 
 
 LINES. 
 
 BEARINGS. 
 
 DIXTS. 
 
 N. 
 
 8. 
 
 E. 
 
 W. 
 
 Q 
 
 QP 
 
 N. 10 E. 
 
 13.71 
 
 13.50 
 
 
 2.38 
 
 
 P 
 
 PO 
 
 S. 88 E. 
 
 18.75 
 
 . . . 
 
 0.49 
 
 18.74 
 
 . . . 
 
 
 
 ON 
 
 S. 16J E. 
 
 16.50 
 
 . . . 
 
 16.82 
 
 4.68 
 
 
 N 
 
 NM 
 
 
 13.00 
 
 
 
 
 
 M 
 
 ML 
 
 N 80 W 
 
 
 
 
 
 
 L 
 
 LQ 
 
 N. 36 W. 
 
 10.00 
 
 8.09 
 
 
 
 5.88 
 
 
 
 
 
 21.59 
 
 16.31 
 
 25.80 
 
 5.88 
 
 
 
 
 
 16.31 
 
 
 5.88 
 
 
 Deficiency, 5.28 S. Def., 19.92 W. 
 
 Tan of bearing of closing line = 
 
 19.92 
 
 5.28' 
 
 or bearing of NL, S 75 09' W. 
 
 Length of NL = V(5.28) 2 +(19.92) 2 = 20.61, 
 
 and angle MLN = 24 51'. 
 To find angle LMNi 
 
 13.00 (NM) Ar. co. = 8.886057 
 
 : sin. 24 51' L) = 9.623502 
 
 :: 20.61 (LN) = 1.314078 
 
 : sin. 41 47' = 9.823637 
 
 Angle LMN= 180 - 41 47'= 138 13' (see note) 
 Angle LMN= 180 - (138 13'+24 51') = 16 56'. 
 
 NOTE. When the side MN, whose length only is given, is longer than 
 the closing line LN, the angle M must be acute ; if shorter, the angle M 
 may be acute or obtuse, depending upon the length of the side LM, the 
 bearing of which only is known. Hence, when this last relation obtains, it 
 is necessary, in the application of this case, to remove ambiguity, that enough 
 be known concerning the length of the side, whose bearing only is given, 
 to indicate whether the angle M is greater or less than a right angle. 
 
 In the example, LM is known to be shorter than NXf, and hence angle 
 M is obtuse. The ambiguity is not removed by employing the method 
 given in Article 215.
 
 PLOTTING A COMPASS OR TRANSIT SURVEY. 177 
 
 The bearing of NM, S. 75 09' W 16 56'= S. 58 13' W. 
 
 To find the length of LM : 
 
 sin. 24 51' Ar. co. = 0.376498 
 
 : sin. 16 56' = 9.464279 
 
 :: 13.00 =1.113943 
 
 : 9.01 (LM) = 0.954720 
 
 The student may verify by the method in Article 215. 
 
 EXAMPLE. As an exercise, from any of the preceding 
 problems strike out from two sides that do not adjoin the bear- 
 ing of one and the distance of another, and compute them. 
 
 SECTION VIII. 
 
 PLOTTING A COMPASS OR TRANSIT SURVEY. 
 
 218. In addition to the drawing-instruments explained in 
 chain surveying, the draughtsman will now find very convenient 
 an instrument for measuring angles, or, 
 
 A Protractor. It is made of metal* or paper, usually in the 
 form of a semi-circle, the arc of which is divided into 180 equal 
 parts, or degrees, subdivided and numbered both ways. 
 
 To draw a line making a given angle with another at a certain 
 point. Bring the diameter of the protractor to coincide with 
 the given line, its centre with the point, and the arch lying in 
 the direction of the desired line ; then with a sharp pencil or fine 
 needle prick off the required number of degrees ; joining the 
 point thus fixed and the given point completes the problem. 
 
 Some plain scales are graduated to degrees on three edges so 
 
 * For more accurate work there is attached a movable arm or ruler, 
 extending beyond the circumference and carrying a vernier. 
 
 12-inch protractors, complete circle, made of heavy paper, on which 
 are printed the divisions to quarter-degrees, are quite reliable.
 
 178 PLANE SURVEYING. 
 
 as to be used like a protractor, but are objectionable on account 
 of the obliquity of the divisions and their varying lengths. 
 
 219. Illustration. To plot a survey the record of which is 
 as follows : 
 
 (1) N. llf E. 13.19 chains; 
 
 (2) S. 87 W. 8.50 " 
 
 (3) S. 2C4 W. 11.75 " 
 
 (4) S. 82 E. 10.03 " 
 
 With a Protractor. First Method. Represent the meridian 
 by drawing on the paper a line so situated that there will be 
 sufficient room on either or both sides of it, as the case may be, 
 to complete the drawing. Fix upon a point in this line to 
 indicate a corner of the tract, usually " the place of beginning." 
 In this particular example the first corner is the easterly boun- 
 dary, and as it runs northerly, we will draw our meridian near 
 the lower right-hand side of the paper, as at A. Prick off the 
 angle 11| from the north end of the protractor-arch to the 
 right, and draw the line 13.19 chains (AE) to any convenient 
 scale, say 2 chains to an inch, or 6.6 inches. Pass another 
 meridian N'S' through B ; and since the bearing is south- 
 westerly, we prick off the degrees, 87 from the south point, and 
 draw the line 8.50 (BC) to the same scale. In a similar man-
 
 PLOTTING A COMPASS OR TRANSIT SURVEY. 179 
 
 ner draw the line CD, and finally DA, which should end at A. 
 Jf it does not end precisely at A, an error in plotting, or 
 inaccuracy in the survey, would thereby be indicated. 
 
 An error in plotting a line by this method would affect the 
 position, but not the direction of the following lines. 
 
 220. Another Method. By laying off the angles from one 
 point, or from one position of a protractor having a complete 
 circle. With the protractor at any convenient point, P, in the 
 meridian NS, prick off the degrees shown by the bearings, 
 and indicate each, and the side to which it belongs, as per 
 figure. Then, by instruments used for drawing parallel lines, 
 transfer them to their proper places, and make the lengths cor- 
 respond to the scale adopted. The point P, from which all the 
 angles were set off, may or may not be one of the corners of 
 the field. The figure shows that it saves one transfer if so 
 taken.
 
 180 
 
 PLANE SURVEYING. 
 
 EXAMPLES. 
 
 1. Plot a triangle, given tw<5 sides and the included 
 angle. 
 
 2. Given two angles and the included side, to plot the 
 triangle. 
 
 3. Given three sides and two included angles, to plot a 
 trapezium. 
 
 QUERY. Can a trapezium be plotted when there are given all 
 the sides and one angle? 
 
 221. By Latitudes and Departures. The survey being bal- 
 anced, this is the most accurate method, and is equally appli- 
 cable to a compass or transit survey. 
 
 Taking the record of the survey in Article 208, and, using 
 the corrected latitudes and departures, let us make a plot 
 of it. 
 
 ,q Draw through the first sta- 
 
 tion* (M) a meridian, and find, 
 by algebraic additions, from the 
 columns of corrected latitudes 
 and departures, the distance 
 each corner is north or south 
 from this station, called total 
 latitude, and east or west from 
 the meridian called total depar- 
 ture. These distances may be 
 ascertained mentally as we pro- 
 ceed with the drawing, but to 
 avoid error it is best to tabulate 
 them, using three columns, as 
 follows : + indicates distance 
 north or east, and , south or 
 west, from the references. 
 
 * Any station will answer, but the one through which the meridian is 
 supposed to pass in calculating the area is preferable.
 
 PLOTTING A COMPASS OK TRANSIT SURVEY. 
 
 181 
 
 STATIONS. 
 
 TOTAL LATITUDES 
 
 FROM 
 
 STATION M. 
 
 TOTAL DEPARTURES 
 
 FROM 
 
 MERIDIAN NS. 
 
 1 
 
 
 
 
 
 2 
 
 -12.27 
 
 + 4.68 
 
 3 
 
 - 3.21 
 
 + 14.84 
 
 4 
 
 + 0.20 
 
 + 11.58 
 
 5 
 
 + 3.54 
 
 + 8.23 
 
 6 
 
 + 1.97 
 
 + 6.25 
 
 1 
 
 
 
 
 
 The total latitude of the last station is the latitude of the last 
 line with its sign changed. The same is true regarding the 
 total departure of last station. A check is thus had on the 
 work. 
 
 From M lay off on the meridian negatively, or to the south, 
 12.27 chains according to the scale adopted, to A ; from A set 
 off perpendicularly to the east, with the same scale, 4.68 chains, 
 to ; connect M and 0, showing the first line. Set off from 
 3/, again to the south, 3.21 chains, to B; thence perpendicu- 
 larly to the right, or east, 14.68 chains, to P. 
 
 OP represents the second line of the survey. Next lay off 
 20 links to the north from 3f, thence 11.58 chains to the east, 
 and join PQ for the third line, and so on, the last line, SM, 
 requiring a distance laid off on the meridian north = 1.97 ; and 
 a perpendicular thereto, = 6.25 east, when drawn closes the sur- 
 vey, thus proving the correctness of the work. 
 
 A variation of the method just given is to draw two lines, one 
 representing the meridian, the other an east and west line. On 
 the first lay off, as before, the latitudes of the sides, and on 
 the second the corresponding departures; then, by means of 
 dividers, obtain the intersection of co-ordinates, and joining 
 these points shows the plot. 
 
 For plots of ordinary farm surveys the method given above, 
 being equally accurate and more expeditious, is recommended ;
 
 182 
 
 PLANE SURVEYING. 
 
 for plots of extraordinary size, extending over a large drawing- 
 board or made on a large table, the variation noted should be 
 adopted. 
 
 222. Using Cross-Section Paper* and the latitudes and de- 
 partures, a tolerably accurate plot may be made with great 
 facility. The vertical and horizontal lines of the paper may 
 
 represent respectively meridians, and east and west lines. As- 
 sume any convenient point as the beginning of the survey, 
 and suppose the latitude of the first line = 4. 00 chains N., the 
 departure = 6.00 chains E. Count from northward four 
 spaces, thence eastward six spaces, to P ; join OP, thus deline- 
 ating the first side. Suppose the latitude and departure of the 
 
 * Note-books may be procured having the alternate pages ruled in 
 small squares, like cross-section paper.
 
 DETERMINING AREAS. 183 
 
 second side = respectively 3.50 chains N. and 2.25 chains E. ; 
 count off, as before (estimating the fractions of chain), three 
 and a half spaces north and two and a quarter east ; connect 
 the points P and Q for the second side, and so on to the place 
 of beginning. 
 
 SECTION IX. 
 ON DETERMINING AKEAS. 
 
 A. PARTICULAR FORMS AND CASES. 
 TRIANGLES.* 
 
 223. First Method. Measure two sides, as m and n, and the 
 included angle 0. Then the 
 
 224. Second Method. Measure two m 
 angles, as and N, and the included side m. Then 
 
 A_ m 2 sin JV'sin O 
 ~ 2s\n(N+0)' 
 
 PARALLELOGRAMS. 
 
 225. Measure two adjacent sides, m and n, and their in- 
 cluded angle, P. Then h denoting 
 
 the altitude, 
 
 A. = mh = m x n sin P. 
 
 * For other methods than those found in this section of surveying tri- 
 angles, quadrilaterals, and other polygons, see Chain Surveying, Articles 
 03 to 70.
 
 184 PLANE SURVEYING. 
 
 EXAMPLES. 
 
 1. Two sides of a triangle measure 756 feet and 1024 feet, 
 and their included angle 42 45' ; determine the area in acres. 
 
 2. Two angles of a triangle are 59 29' and 65 18', and their 
 included side 932 feet. How many acres does it contain? Plot. 
 
 3. Two sides of a triangle measure 15.24 chains and 13.18 
 chains, and the angle opposite the first 54 25'. Find the area. 
 
 4. Two adjacent sides of a parallelogram are 856 feet and 
 1252 feet, and their included angle 75 48'. Compute the area. 
 
 TRAPEZOIDS. 
 
 226. Measure three sides, say PM, MN, and NO, and one 
 of the included angles, as N. From 
 
 ? the data thus obtained compute the al- 
 
 j\ titude, OL = PK, and the parallel side, 
 j \ PO. Then 
 M K L N A = MN+ PO x pK 
 
 Or, instead of measuring the inclined sides, if it is equally 
 convenient measure the parallel sides, and one of the other 
 sides and an angle as before ; then 
 
 TRAPEZIUMS. 
 
 227. Measure all the sides and one angle. With the data 
 calculate the length of a diagonal dividing the tract into two 
 triangles, in one of which two sides and the included angle will 
 be given, and in the other three sides, whence the area may be 
 found.
 
 DETERMINING AREAS. 185 
 
 228. Or, measure three sides, PM, MN, and ON, and the 
 included angles N and PMN. Draw P r 
 
 MO, calculate the area of the triangle 
 MNO, the diagonal MO, and the angle 
 OWN. Subtract OMN from PMN; 
 then, having two sides and the included 
 angle in the triangle PMO, its area may 
 be computed, which added to the area of MNO gives the re- 
 quired content. 
 
 229. Otherwise. Measure two opposite sides, as OL and MN, 
 and three angles,, as 0, L, and M. 
 
 Conceive the sides ON and LM to 
 be prolonged to meet in some point, 
 P. From the data calculate the 
 areas of the triangles POL and 
 PMN. The difference will give the area sought.* 
 
 EXAMPLES. 
 
 1. Given in a trapezoid (see figure, Article 226) PM = 33 
 rods, MN= 68 rods, NO = 30 rods, and the angle N= 70 ; to 
 find the area, and make a plot. 
 
 2. Given in a trapezium PMNO (see figure, Article 228) 
 PM= 7 chains, MN= 7.50 chains, NO = 6 chains, the angle 
 2V=120, and M= 108 ; to find the area, and make a plot. 
 
 3. Given in a trapezium LMNO (see last figure) L0 = 8 
 chains, MN= 5 chains, and the angles L, M, and N respectively 
 87, 70, and 80 ; to find the area, and make a plot. 
 
 4. Given in a trapezium the angle Ma right angle, the sides 
 MN, NO, OP, and PM respectively 20, 12, 30, and 15 rods; 
 also a perpendicular to MN from N extending to P0=10 
 rods ; to find the area. 
 
 QUERY. Could the area be found without NO? 
 
 * If practicable, observe all the angles, and thereby obtain a check on 
 the measurements.
 
 186 
 
 PLANE SURVEYING. 
 
 POLYGONS- 
 
 230. To find the area of an irregular pentagonal field 
 LMNOP, when all the corners can be seen from one corner, 
 
 as 0. Measure the sides ON, OP, the diagonals 
 OL, OM, and the three angles at 0. Then two 
 sides and the included angle of each triangle 
 thus formed will be given, whence their areas 
 may be calculated and, by addition, the area of 
 the required polygon may be obtained. In like 
 manner, a survey of any small irregular polygonal 
 lot, in which all the corners are visible from one 
 corner, may be effected. If there are n sides, 
 measure from one corner two sides and n 3 diagonals, observ- 
 ing from the same point the n 2 angles which are formed by 
 these diagonals and the two sides. Then, as above, the tract will 
 be divided into n 2 triangles, the area of each may be calculated, 
 and the sum of these areas taken for the area of the polygon. 
 
 231. Or, measure from some point within or without the 
 field radial lines to all the corners, and observe at the same 
 point the angle which these lines make with each other. 
 
 There will thus be given two sides and the included angle of 
 a series of triangles, whence the bounding lines and area may 
 be computed. 
 
 232. Othei-wise. Measure a base line within or without the 
 tract, or use a portion or all of one side as a base line, and 
 
 observe from each extremity of this 
 line the angles formed by it and a 
 visual line through each corner of 
 the tract. There will thus be known 
 two angles and the included side of 
 a series of triangles, whence the 
 bounding lines and area may be 
 calculated. 
 
 The marginal figure represents the
 
 DETERMINING AREAS. 
 
 187 
 
 case where the base line BL is taken outside the tract. It will 
 be noticed that it is possible by this method to survey a farm 
 without entering upon it. 
 
 B. GENERAL METHOD. 
 
 233. The methods given in the last three articles are, however, 
 quite limited in their application, since it rarely happens in a 
 tract of considerable magnitude that all the corners are visible 
 from any one corner, or from any point within or without the 
 field. 
 
 The following niethod of determining the area by means of 
 latitudes and departures is applicable to all right-lined figures, 
 and is the most general and accurate. 
 
 Let POLM represent a tract of land, the area of which is 
 desired. Measure all the sides and angles, interior or deflection, 
 with a single bearing, if the transit 
 is used, or take all the bearings with 
 a compass. Distribute the angular 
 error, if any made by transit (see 
 note, Article 207). Obtain the lati- 
 tudes and departures, and balance 
 the survey. 
 
 Let NS represent a meridian pass- 
 ing through P, the most westerly 
 station of the tract, and .B&, C'c, and 
 Dd meridian distances. Now, if 
 perpendiculars be dropped from the 
 angles 0, -L, and M to the meridian, 
 it will readily appear that the area of 
 POLMP= area oOLMmo minus the 
 sum of the areas of the triangles PoO 
 and PmM, or POLMP = trape- 
 zoids (OoLl + LlmM) triangles 
 (PoO + PmM} 
 
 = Ccx OQ + Dd x LR Bb x Po Ee x Pm. 
 
 N 
 
 Ifl
 
 188 PLANE SURVEYING. 
 
 The computation, then, involves the latitudes and depart- 
 ures, and meridian distances ; the former having been already 
 explained, we shall now indicate how the latter may be obtained, 
 or rather how the double meridian distances are found, since in 
 order to lessen fractions the double lengths are used. 
 
 The double meridian distance, or D.M.D., of the side 
 PO = 2Bb = Oo, its departure. 
 
 The D.M.D. of OL = 2 Cc = Oo + Ll= Oo + QI+QL 
 = 2Bb+Oo+QL. 
 
 The D.M.D. of LM= 2 Dd= 2 Cc + QL - MR. 
 
 The D.M.D. of MP = 2Ee = 2Dd- MR - Mm 
 
 = Mm = its departure. . 
 
 It is evident that, in a corresponding manner, the double 
 meridian 'distances of the bounding lines of a tract may be 
 found, no matter what the number of sides or magnitude of the 
 angles. Hence, considering east departures plus ( + ), and 
 west departures minus ( ), the above deductions may be 
 expressed in 
 
 A GENERAL RULE FOR OBTAINING DOUBLE MERIDIAN 
 DISTANCES. 
 
 The double meridian distance of the first side is equal to its 
 departure. 
 
 The double meridian distance of the second side is equal to the 
 double meridian distance of the first side, plus its departure, plus 
 the departure of the second side. 
 
 The double meridian distance of any side is equal to the double 
 meridian distance of the preceding side, plus its departure, plus 
 the departure of the side itself. 
 
 The double meridian distance of the last side deduced by the
 
 DETERMINING AREAS. 
 
 189 
 
 foregoing rule should equal its departure, and ivill serve as a 
 check on this part of the work.* 
 
 234. Continuing now the work of computing areas and 
 referring to the last figure, we may form the following table : 
 
 STATIONS. 
 
 LINES. 
 
 N. LAT. 
 
 S. LAT. 
 
 E. DEP. 
 
 w. 
 
 DEP. 
 
 D.M.D. 
 
 NORTH 
 DOUBLE 
 AREAS. 
 
 SOUTH 
 DOUBLE 
 AREAS. 
 
 1 
 
 PO 
 
 Po 
 
 
 Oo 
 
 . 
 
 2 Bb 
 
 2 Bb X Po 
 
 
 2 
 
 OL 
 
 . . 
 
 OQ 
 
 QL 
 
 
 2 Cc 
 
 .... 
 
 2CcxOQ 
 
 3 
 
 LM 
 
 
 Lli 
 
 
 MR 
 
 2 Dd 
 
 
 2Dclx LR 
 
 4 
 
 MP 
 
 Pm 
 
 
 
 Mm 
 
 2Ee 
 
 2EeXPm 
 
 
 
 The double meridian distances are placed in the column 
 headed D.M.I). In the column headed North Double Areas are 
 placed '2Bb X Po and 2Ee X Pm, the product of the first and 
 fourth double meridian distances and their corresponding lati- 
 tudes. In the south double area column we find 2 Cc x OQ 
 and 2 Dd X Lit, or the product of the double meridian distances 
 of the second and third sides, and their respective latitudes. 
 In other words, the column in which each of the products of 
 double meridian distance and latitude is to be placed is indi- 
 cated bv the latitude employed in the multiplication. 
 
 Now, twice the area of the triangles POo and PMm, or the 
 subtractive portion of the figure oOLMmo, is given in the north 
 double area column, and twice the area of the trapezoids OoLl 
 and LI Mm, which include the triangles named, is given in the 
 column of south double areas. Half the difference, therefore, 
 
 * The position of the meridian (NS) may be assumed to pass through 
 any other corner, or even through a point outside the survey. A slight 
 modification of the rule just given would make it applicable to any of 
 these cases. For convenience, it is generally assumed to pass through 
 the most westerly station. When a survey is made with the transit, and 
 the area only required, it is most convenient to consider one of the sides 
 of the tract the meridian.
 
 190 PLANE SURVEYING. 
 
 between these sums is the area POLMP required. The reason- 
 ing'being general, and independent of the number of sides, we 
 have for finding the area of any rectilineal figure, the bearings 
 and distances of all the sides being known, the following 
 
 RULE. 
 
 1. Prepare a table as exhibited below. 
 
 2. Find, and place in their proper columns, the latitudes and 
 departures of the several sides of the tract. 
 
 3. Balance the survey (if necessary) . 
 
 4. Find the double meridian distances, with reference to a 
 meridian passing through the most westerly * station, and place 
 them in the D.M.D. column. 
 
 5. Multiply each double meridian distance by its corresponding 
 corrected latitude, and place the product in the column of double 
 areas indicated by the latitude. 
 
 6. One-half the difference of the sums of the columns of 
 double areas will be the required area. 
 
 Let us now take the field notes given in Article 198, and com- 
 pute the area of the tract. 
 
 The student will perceive that the meridian is assumed to 
 pass through the most westerly station, that the double meridian 
 distances are found as directed in 233, that each is multiplied 
 by its corresponding latitude, and the resulting double area 
 product placed in the column of the same name as the latitude. 
 
 Lastly, the difference of the two columns of double areas is 
 taken, the remainder divided by two, giving the number of 
 square chains in the tract, and the result divided by 10 = 12,032 
 acres, which is the area sought. 
 
 On account of the meridian passing through the most west- 
 erly station, and because the field is to the left,f the areas of 
 
 * For convenience simply, see note, preceding article. 
 
 t In the last figure, if the bearings are taken or recited in the order 
 PM, ML, LO, etc., the tract is considefed on the left; if this order is 
 reversed, the tract is on the right.
 
 DETERMINING AREAS. 
 
 191 
 
 w 
 
 -4 
 
 
 O5 en tf- co to H-> 
 
 STATIONS. 
 
 
 cc 02 2! fe! tej cr> 
 
 *J Qt .is. rf^ il^. K5 
 
 fc^ H-* ^ CO GO O 
 
 I i | | s s 
 
 ^ ^ ^ ^ w w 
 
 td 
 H 
 1 
 
 
 05 fcO * rf^ CO CO 
 S OS Cn 05 P 
 
 DISTANCES. 
 
 Oi Oi 
 
 ^-q bo 
 
 oo en 
 
 co co jo 
 
 ! .' en fcS CO ! 
 
 S 
 
 . 
 
 LATITUDES. 
 
 nks. Error 4 links. 2)240.6401 
 10)120.32005 
 12.032 acres. 
 
 Cn 
 ~i 
 
 CO 
 
 ,_, l_l to 
 
 O Cn ' ' t? 
 
 &3 
 
 
 bo 
 t* 
 
 H^ 
 
 _ as 
 
 . . . Oi *^J 
 
 bq 
 
 DEPABTUBES. 
 
 bo be 
 
 t* 01 
 
 O5 H-i CO CO 
 
 ^ s g : : 
 
 S 
 
 -a 
 
 h- ] H-" h-> bS tsS 
 
 H 
 
 a. 
 
 COBBEC- 
 
 TIONS. 
 
 i*> 
 
 ,- ; - ; MM 
 
 i 
 
 in 
 
 CC 
 
 co co co 
 
 : : ss fc : 
 
 ^ 
 
 COBBECTED 
 LATITUDES. 
 
 ei 
 
 CO 
 
 M 
 
 HJ h- ' ' ' t 
 
 ^s 25 : : : is 
 
 ?> 
 
 * 
 
 % 
 
 s *> 
 
 '. '. '. '. 00 
 
 ! 
 
 COBBECTED 
 DEPABTUBES. 
 
 is 
 
 o H-> co co 
 
 g s g s : : 
 
 ^ 
 
 
 1^ S 05 S * 
 
 .if, & 2 fe g 
 
 o 
 
 k - 
 b 
 
 ii li 
 
 : : g | | : 
 
 Hi 
 
 f 
 
 K p : ; ; 25 
 
 CO ^J _ _ *. 
 
 ^ CO . . . CO 
 
 Cn OS O5 
 
 Hi
 
 192 
 
 PLANE SURVEYING. 
 
 the trapezoids are north, and those of the triangles south. If 
 we had assumed the meridian to pass through the most easterly 
 corner, the areas of the trapezoids then formed would be south, 
 and those of the triangles north. 
 
 If the bearings of the lines were reversed, or the survey made 
 with the field to the right, the reverse of the preceding state- 
 ment would be true. 
 
 In either case, however, one-half the difference of the sums 
 of the double areas will give the contents. 
 
 As an exercise the student may obtain an expression for the 
 area of POLMP, last figure, assuming the meridian to pass 
 through L, and passing i-ound by MP, etc., that is, keeping the 
 field to the right. He may also, with the meridian through P, and 
 keeping the field to the left, obtain an expression for the area. 
 
 As a further exercise he may verify the result in the last 
 example solved, taking the meridian through the most easterly 
 station. 
 
 Calculate the areas from the following notes ; also make a 
 plot of each : 
 
 1. 
 
 9 W. 
 31 E. 
 69 E. 
 0^ E. 
 Of W. 
 89 W. 
 
 (1) N. 
 
 (2) N. 
 
 (3) S. 
 
 (4) S. 
 
 (5) S. 
 
 (6) N. 
 
 15.50 chains ; 
 25.40 " 
 
 10.00 
 19.70 
 14.60 
 21.00 
 
 2. 
 
 STA- 
 TIONS. 
 
 LINES. 
 
 DlSTS. 
 
 AZIMUTH WITH 
 LM. 
 
 L 
 
 LM 
 
 22.45 
 
 
 
 M 
 
 MN 
 
 1.30 
 
 22 
 
 N 
 
 NO 
 
 66.30 
 
 90 
 
 
 
 OP 
 
 23.85 
 
 180 
 
 P 
 
 PL 
 
 67.10 
 
 270 
 
 L 
 
 LM 
 
 
 360 nr 0"
 
 DETERMINING AREAS. 
 
 193 
 
 3. 
 
 (1) N. llfE. 
 
 (2) S. 87 W. 
 
 (3) S. 201 W. 
 
 (4) S. 82 E. 
 
 13.19 chains; 
 
 8.50 " 
 11.75 " 
 10.03 " 
 AHS. 
 
 acres. 
 
 If in Article 76 we substitute respectively tor abscissa and 
 ordiiiate of a corner of a tract, departure and latitude of the 
 side ending at said corner, the rule there given may be applied 
 to surveys made with an angular instrument. 
 
 To illustrate, take the example given on page 191: 
 
 CORRECTED 
 LATITUDES. 
 
 CORRECTED 
 DEPARTURES. 
 
 TOTAL 
 LATITUDES. 
 
 TOTAL 
 DEPART- 
 URES. 
 
 DIFFER. 
 
 BETWEEN 
 
 ALTERNATE 
 DEPARTS. 
 
 DOUBLE 
 AREAS. 
 
 jr. 
 
 S. 
 
 E. 
 
 W. 
 
 
 1227 
 
 468 
 
 
 000 
 
 
 
 
 9.06 
 
 
 10.16 
 
 
 -12.27 
 
 4.68 
 
 -14.84 
 
 182.0868 
 
 3.41 
 
 . . 
 
 
 3.26 
 
 - 3.21 
 
 14.84 
 
 -6.90 
 
 22.1490 
 
 3.34 
 
 
 . . 
 
 3.35 
 
 .20 
 
 11.58 
 
 6.61 
 
 1.3220 
 
 
 1.57 
 
 . . 
 
 1.98 
 
 3.54 
 
 8.23 
 
 5.33 
 
 18.8682 
 
 
 1.97 
 
 
 6.25 
 
 1.97 
 
 6.25 
 
 8.23 
 
 16.2131 
 
 2)240.6391 
 
 10) 120 .32 sq. ch. 
 
 12.032 acres. 
 
 In this case the axes were taken through the most westerly 
 station, thereby making the total departures all plus, but giving 
 both plus and minus total latitudes. On account of the signs 
 the double areas are all plus. The axis of ordinates passing 
 through the most westerly station makes the total latitude of 
 that station zero, and consequently there is one less multiplica- 
 tion to be performed. The same would be the case if the f 
 axis passed through the most easterly corner.
 
 194 
 
 PLANE SURVEYING. 
 
 Since the difference of the alternate total departures is equal 
 to the sum of the adjacent departures, the rule just given may 
 be stated as follows : 
 
 Multiply the total latitude of each station by the sum of the 
 departures of the adjacent sides, and take half the sum of these 
 products for the area. 
 
 To illustrate, take the following example : 
 
 S 
 
 
 i 
 
 
 
 
 
 J 
 
 ii 
 
 
 | 
 
 BEARINGS. 
 
 
 
 N. 
 
 S. 
 
 E. 
 
 W. 
 
 II 
 
 I 
 
 DOUBLF AREAS. 
 
 
 
 
 5 
 
 
 
 
 
 3 
 
 $ 
 
 
 i 
 
 N. 25 E. 
 
 433 
 
 3PR 
 
 
 183 
 
 
 000 
 
 
 
 2 
 
 N.76'55'E. 
 
 191 
 
 43 
 
 
 186 
 
 
 393 
 
 369 
 
 145017 
 
 3 
 
 S. 6 3 41' W. 
 
 539 
 
 
 535 
 
 . . 
 
 62 
 
 436 
 
 124 
 
 54064 
 
 4 
 
 S. 25 W. 
 
 40 
 
 . . 
 
 36 
 
 . . 
 
 17 
 
 - 99 
 
 - 79 
 
 7821 
 
 5 
 
 N. 65 W. 
 
 320 
 
 135 
 
 
 
 290 
 
 -135 
 
 -307 
 
 41445 
 
 2)248347 
 
 43560) 124173.5 sq.ft. 
 
 2.852 acres. 
 
 The'studeut may verify the preceding example by this method. 
 
 2. Given the bearings and distances of the sides of a field, as 
 follows, to find the area by each of the two preceding methods. 
 Ascertain, also, the error of the survey. 
 
 (1) N. 61 W. 
 
 (2) N. 65. V E. 
 
 (3) S. 39 E. 
 
 (4) S. 2 W. 
 
 (5) S. 46 W. 
 
 (6) N. 88 W. 
 
 9.38 chains; 
 8.25 " 
 6.51 " 
 4.45 " 
 5.00 " 
 6.79 " 
 
 8. Given the boundaries of a tract of land with the corre- 
 sponding weights, as follows, to determine the area by double
 
 DETERMINING AREAS. 195 
 
 meridian distances, using the weights in balancing the survey as 
 indicated in 3, Article 208. Determine, also, the error of the 
 survey. 
 
 ;i) S. 79 10' W., dist. 27.00 chains, weight, 1; 
 
 " 3; 
 
 " H; 
 
 2; 
 2; 
 " 2; 
 
 " i; 
 " i. 
 
 4. The distances and interior angles of a farm, together with 
 the bearing of one line, are given below. The angles were 
 measured very accurately. It is required to calculate the area, 
 by either of the preceding methods, balancing the survey by 
 (2) Second Case, Article 208. Also make a plot. 
 
 Angle L, 90 ; side LM, 28.00 chains. 
 " M, 148|; " MN,25.2Q " 
 
 (1) S. 
 
 79 10' 
 
 W., 
 
 dist. 27.00 
 
 (2) S. 
 
 i 
 
 W., 
 
 " 34.08 
 
 (3) N. 
 
 89| 
 
 E., 
 
 " 10.47 
 
 (4) N. 
 
 155' 
 
 E., 
 
 " 15.30 
 
 (5) S. 
 
 80| 
 
 E., 
 
 " 7.15 
 
 (6) S. 
 
 58 
 
 E., 
 
 " 11.50 
 
 (7) N. 
 
 39 
 
 E., 
 
 " 9.20 
 
 (8) N. 
 
 16| 
 
 W., 
 
 " 24.63 
 
 
 ii 
 
 N, 
 
 81 1 
 
 ; " 
 
 NO, 
 
 11 
 
 .70 
 
 
 ii 
 
 0, 
 
 220 
 
 a. 
 
 OP, 
 
 12 
 
 .IS 
 
 
 tt 
 
 P, 
 
 90 
 
 a 
 
 PQ, 
 
 27 
 
 .DO 
 
 
 ii 
 
 Q, 
 
 90 
 
 a 
 
 > 
 
 QR, 
 
 15 
 
 .i<; 
 
 
 
 
 R, 
 
 270 
 
 ; " 
 
 RS, 
 
 11 
 
 .!)() 
 
 
 ii 
 
 S, 
 
 90 
 
 > " 
 
 SL, 
 
 21 
 
 .(50 
 
 Bearing 
 
 of LM, N. 
 
 10 
 
 E. 
 
 
 
 
 5. The notes of a survey are given below ; it is required to 
 determine the area by double meridian distances after correct- 
 ing the latitudes and departures by a combination of 2 and 3, 
 Article 208. See also note in same article. 
 
 The interior angles were observed. 
 
 Angle L, 91 44'; side LM, 17. 16 chains; weight, 2. 
 
 " M, 168 20'; " MN, 9.48 " " 1. 
 
 " N, 104 49'; " NO, 8.39 " If 
 
 ' f 0, 179 30'; " OP, 15.28 " " 2.
 
 196 PLANE SURVEYING. 
 
 Angle P, 90 19'; side PQ, 16.05 chains; weight, 2. 
 
 Q, 90 05'; " Q.R, 15.68 " " 3. 
 
 " R, 283 49'; " RS, 11.40 " " 1. 
 
 " S, 71 24'; " SL, 13.80 " 1. 
 
 6. Select a tract of land, some of the sides being much more 
 difficult than the others to align and measure, survey it, weight 
 the sides, balance the latitudes and departures according to the 
 weights, and calculate the area. 
 
 7. Let one party of students survey a tract of uneven or hilly 
 land of considerable magnitude, by means of transit and stadia 
 and rectangular co-ordinates ; another party at the same time, 
 or the same party subsequently, survey the same tract in the 
 usual way. Compare results. 
 
 C. WHEN OFFSETS ARE TAKEN. 
 
 235. Let the annexed figure represent the case. The prop- 
 erty lines are NO, OP, PL, and the centre of the creek * RS. 
 Obtain sufficient data to compute the area of the rectilinear 
 
 .0' 
 
 * When a non-navigable stream forms a boundary of a tract of land, 
 the middle of it is considered the property line, unless otherwise specified. 
 In navigable rivers and tidal waters, the boundary is low-water mark.
 
 DETERMINING AREAS. 
 
 197 
 
 figure LNOP, and take offsets from the line LN to the middle 
 of the stream, as directed in Offsets and Tie-Lines, Article 73 ; 
 and in Traversing, Article 164. Calculate the area of LNOP 
 by one of the preceding methods ; to this add * the sum of the 
 areas of the trapezoids, and triangles formed by the offsets 
 from the line LN to the middle of the creek. If the width of 
 the stream is considerable, and especially if great accuracy is 
 required, the surveyor must not ignore the small triangles f 
 formed at L and N. O 
 
 236. To Find the Area of a 
 Pond or Small .Lake, traverse, 
 or take the bearings of the sides 
 LM, MN, NO, etc., and measure 
 them ; also take offsets, at proper 
 points, from these lines to the 
 edge of the water. 
 
 Calculate the area included be- 
 tween the right lines, and sub- 
 tract therefrom the area found by 
 the offsets ; the remainder will be 
 the area required. 
 
 EXERCISES. 
 
 1. Let one party survey a field with compass and chain, 
 taking bearings and distances of all the sides ; another party 
 survey the same field, using transit and chain, and observing 
 
 * If the base line LN is without the tract, as in LNO'P 1 , the area 
 included between the middle of the stream and LN must be subtracted 
 from that of LNO'P'. 
 
 t Other things being equal, the areas of these small triangles depend 
 upon the obliquity of PL and ON. There will be none formed when PL 
 and NO are perpendicular to the base LN. In the case presented, the 
 area of the triangle at L is to be added, and that of Nn'v subtracted from 
 the sum of the areas of the trapezoids, to obtain the correct content 
 between LNund the middle of the creek.
 
 198 PLANE SURVEYING. 
 
 the interior or deflection angles ; a third part}', using the chain 
 only. Each party should use proof lines, make record, plot, 
 and calculate the area. Compare results. 
 
 2. With a transit, survey a field, a part of which is bounded 
 by a creek, lake, or some crooked line requiring offsets to be 
 taken ; make a plot, and compute the area. 
 
 3. Triangulate a portion of a river or small lake ; make a 
 plot, and compute the area. 
 
 4. Make the necessary measurements to write a description, 
 to make a plot, and to compute the area of a portion of a 
 crooked road. 
 
 5. Observe all the bearings and measure all the sides of a 
 polygonal field, except the bearing and distance of one side. 
 Compute the area, and length and bearing of omitted side. 
 Subsequently observe the bearing and distance, and note, if any 
 disagreement, how much the area is affected thereby.
 
 CHAPTER III. 
 
 DECLINATION OF THE MAGNETIC NEEDLE, OE VAKIATION 
 OP THE COMPASS. 
 
 237. It has been already remarked (Article 82) that the 
 magnetic and geographic meridian do not in general coincide. 
 The angle included by the vertical planes containing these lines, 
 or the angle which the direction of the needle makes with the 
 geographic meridian, is the declination of the needle, sometimes 
 called the variation of the compass. It is different at different 
 places, and is a variable quantity at any place. 
 
 The declination is termed east or west, according as the north 
 end of the needle points to the east or west of the geographic, 
 or true meridian. 
 
 The magnetic declinations of a few places for the year 1885 
 are given below : 
 
 Eastport, Me., 19 10' W. Sitka, Alaska, 2850'E. 
 
 Albany, N.Y., 1011'W. Milledgeville, Ga., 2 32' E. 
 
 Pittsburg, Pa.,* 2 52' W. New Orleans, La., 6 11' E. 
 
 Omaha, Neb., 10 06' E. City of Mexico, Mex., 7 24' E. 
 
 San Francisco, Cal., 16 34' E. 
 
 238. Irregular Changes. The magnetic needle is subject to 
 disturbances during a thunder storm, or an exhibition of aurora, 
 solar changes, and sometimes it is considerably agioated with- 
 out any apparent cause, but probably on account of magnetic 
 or electric disturbances more or less remote. 
 
 The changes, however, which especially concern the surveyor, 
 are the diurnal and secular. 
 
 * At this place, September, 1887, the magnetic declination = 8 01' W.
 
 200 PLANE SURVEYING. 
 
 239. The Diurnal Variation. It has been ascertained, by 
 repeated observations at various places, that the magnetic 
 needle is subject to daily changes ; that at a time varying from 
 two to three hours after sunrise the north end of the needle 
 attains its maximum deviation to the east, or, as it is called, its 
 eastern elongation ; from this time it is deflected westward, 
 attaining its western elongation between 1 and 2 o'clock P.M., 
 whence it retrogrades towards the east. There is sometimes 
 an interruption of the motion at night, but generally a small 
 reversed movement is exhibited, the magnetic meridian being 
 crossed a second time between 7 and 9 P.M. The times at 
 which these limits are reached vary with the seasons : during 
 the north declination of the sun the averages for eastern and 
 western elongations, respectively, are about 7.30 A.M. and 1.15 
 P.M. ; for the remainder of the year, about 8.45 A.M. and 1.45 P.M. 
 
 The average daily direction or mean magnetic meridian is 
 reached in summer about 10.15 A.M., and in winter about 
 10.45 A.M., at Philadelphia, and generally within half an hour 
 of these times at other places. 
 
 The angular range between these limits is not constant, but, 
 as may be seen by the table subjoined, it is considerably greater 
 in summer than in winter, amounting at Philadelphia to 10' 30" 
 in August, and only 6' in November, or a yearly average of 8'. 
 while at Key West, Florida, the average for the year is about 
 5' 30" ; in higher magnetic latitudes the average being more 
 than in the lower. It is least in years of minimum sun spots 
 (as in 1878, for instance), and greatest in years of maximum 
 sun spots (as in 1870), the ratio being about as 7 to 13 of the 
 average amount of these years respectively. The daily vari- 
 ation is at times interrupted, at others enfeebled, and frequently 
 in the winter there are days on which it cannot be recognized. 
 On account of the daily movement of the needle, its variable 
 range during the year, and disturbances from atmospheric phe- 
 nomena, it is well, when taking the bearing of any important 
 line, to record the date, time of day, and condition of the 
 atmosphere, using the subjoined table as far as practicable.
 
 VARIATION OF THE COMPASS. 
 
 201 
 
 240. For reducing the direction of the needle observed at 
 other hours to the mean magnetic meridian, the following table 
 (taken from instructions to United States Deputy Surveyors), is 
 furnished. It gives to the nearest minute the variations of the 
 needle from its average position during the day, for each hour 
 in the day, for the four seasons of the year. 
 
 TABLE FOR REDUCING THE OBSERVED DECLINATION TO THE MEAN 
 DECLINATION OF THE DAY. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Hour 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 Spring 
 
 3' 
 
 4' 
 
 fj 
 
 3' 
 
 1' 
 
 V 
 
 4' 
 
 & 
 
 5' 
 
 4' 
 
 3' 
 
 2' 
 
 1' 
 
 Summer 
 
 4' 
 
 5' 
 
 5' 
 
 4' 
 
 1' 
 
 2' 
 
 4' 
 
 & 
 
 5' 
 
 4' 
 
 y 
 
 2' 
 
 V 
 
 Autumn 
 
 2' 
 
 3' 
 
 3' 
 
 2' 
 
 0' 
 
 2' 
 
 3' 
 
 4' 
 
 3' 
 
 2' 
 
 i' 
 
 I' 
 
 0' 
 
 \Vinter 
 
 1' 
 
 If 
 
 2' 
 
 2' 
 
 1' 
 
 0' 
 
 2' 
 
 3' 
 
 3' 
 
 2' 
 
 i' 
 
 1' 
 
 0' 
 
 241. The Secular Variation. Observations extending through 
 many years, at various places, indicate a continual change taking 
 place in the declination of the needle ; that these changes are 
 not continuous in direction nor uniform in intensity ; that in this 
 country the movement which, at the end of the last century, was 
 eastward is now westward at all places east of the Rocky Moun- 
 tains, and that a period of 250 or 300 years may elapse before 
 the needle will again resume the position it now occupies.* 
 
 242. The Line of no Declination,t or Agonic Line, is the locus 
 of all points on the earth where the direction of the needle is 
 
 * The explanation of the secular change must ultimately be referred 
 to forces of a periodic character, acting for centuries with great regularity. 
 So far no approach has yet been made towards the discovery of the cause 
 of the motion. . . . The study of the variation of the declination so. far 
 would seem to indicate a secular change cycle for stations in the United 
 States, extending over, or varying between, the limits of about 220 or 360 
 years. The data, however, are very uncertain. (U. S. C. & G. S., 1879.) 
 
 t Sometimes called the Line of no Variation.
 
 202 PLANE SURVEYING. 
 
 coincident with the geographic meridian. At all places on the 
 American continent situated to the east of this line the declina- 
 tion is west, and at all places to the west of it, the declination 
 is east. 
 
 The line of no declination has been moving westward during 
 the present century. From a chart published by Professor 
 Loomis, in the American Journal of Science, 1840, it appears 
 that the lines of equal declination, or isogonic lines, crossed 
 the United States in a N.N.W. direction j the deflection towards 
 the west being greatest in Maine. The line of no declination 
 at that time entered North Carolina about midway between 
 Newbern and Wilmington, passed through the middle of Vir- 
 ginia, and into Lake Erie at a point nearly equidistant from 
 Erie, Pa., and Cleveland, Ohio. 
 
 In 1885 the Agonic Line entered the United States a little to 
 the east of Beach Inlet, S.C., thence through Greensboro, N.C., 
 Christiansburg, Va., Point Pleasant, W.Va., St. Clairsville, 
 Ohio, a short distance west of Detroit, and a few miles east of 
 Fort Mackinac, Mich. 
 
 In the year 1700 the declination at Philadelphia, Pa., was 8| 
 west. During the next century it diminished, reaching a mini- 
 mum in 1800 of IV west, since which time it has been increas- 
 ing, and is now, January, 1887, at the Philadelphia State House, 
 lat. 39 56' 54", long. 75 09', 6 50', with an annual increase 
 of 5'. 
 
 243. Mr. Charles A. Schott, chief of the computing division 
 of the U. S. C. & G. S., tabulated the declinations observed 
 at various stations, and deduced from them formulas by 
 which the magnetic declination at various places may be com 
 puted.* 
 
 The places are arranged geographically as far as practicable, 
 and are given by latitude and longitude (west of Greenwich). 
 The epoch to which the formulas refer is 1850, or ra = t 1850. 
 
 * U. S. C. & G. S., 1882. App. 12.
 
 VARIATION OF THE COMPASS. 
 
 203 
 
 FORMULAS EXPRESSING THE MAGNETIC DECLINATION AT VARIOUS PLACES 
 IN THE UNITED STATES, AND FOR ANY TIME WITHIN THE LIMITS OF 
 OBSERVATION. 
 
 NAME OF STATION AND 
 LOCATION. 
 
 LATI- 
 
 TUDK. 
 
 LONGI- 
 TUDE. 
 
 EXPRESSION FOR MAGNETIC 
 DECLINATION. 
 
 Portland Me .... 
 
 43 38.8' 
 
 70 16.6' 
 
 D +10?72 + 2?68 Bin(1.33m + 24.1) 
 
 Burlington, Vt 
 
 44 28.2' 
 43 36.5' 
 
 73 12.3' 
 72 55.5' 
 
 Z>= +10.81 + 3.65 sin (1.30 m- 20.5) 
 + 0.18 sin (7.0m + 132) 
 D +10 03 + 3 82 sin(l am 243) 
 
 Portsmouth, N.H. . . . 
 Newburyport, Mass. . . 
 
 43 04.8' 
 42 48.4' 
 42 31. 9' 
 
 70 43.0' 
 70 49.0' 
 70 52.5' 
 
 D= +10.63 + 3.17 sin(1.44m-4.7) 
 D= +10.07 + 3.10 sin(1.4m + 1.9) 
 D-+ 9.80 + 3.61 sin (1.50m 1.0) 
 
 Boston, Mass '. 
 Cambridge, Mass. . . . 
 
 Nantucket, Mass. . . . 
 Providence, R.I 
 
 Hartford, Conn 
 New Haven, Conn. . . . 
 \lbany N Y 
 
 42 21.5' 
 42 22.9' 
 
 41 17.0' 
 41 49.5' 
 
 41 45.9' 
 41 18.5' 
 42 39.2' 
 
 71 03.8' 
 71 07.7' 
 
 70 06.0' 
 71 24.1' 
 
 72 40.4' 
 72 55.7' 
 73 45 8' 
 
 D= + 9.52 + 2.93 sin(1.30 m + 5.0) 
 D=+ 9.58 + 2.69 sin (1.3m + 7.0) 
 + 0.18 ein (3.2 m + 44) 
 D=+ 9.29 + 2.78 sin(1.35m + 5.5) 
 D=+ 9.10 + 2.99 sin (1.45m -3.4) 
 + 0.19 sin (7.2m + 116) 
 D=+ 8.06 + 2.90 sin(1.25m 26.4) 
 Z>=+ 7.78 + 3.11 sin(1.40i 22.1) 
 D-+ 8.17 + 3.02 sin (1. 44 m 8.3) 
 
 Oxford, N.Y 
 Buffalo, N.Y 
 Toronto, Can 
 
 Erie Pa 
 
 42 26.5* 
 42 52.8' 
 43 39.4' 
 
 42 07.8' 
 
 75 40.5' 
 78 53.5' 
 79 23.4' 
 
 80 05.4' 
 
 D = + 6.19 + 3.24sin(1.35m-18.9) 
 D=+ 3.66 + 3.47 sin(1.4m 27.8) 
 J)=+ 3. 60 + 2.82 sin (1.4m -44.7) 
 + 0.09 sin (9.3m + 136) 
 + 0.08 sin(19m + 247) 
 D-+ 2.26 + 2 71 sin (1.55m 297) 
 
 Marietta, Ohio .... 
 Cleveland, Ohio .... 
 Detroit, Mich 
 Sault de St. Marie, Mich. 
 Cincinnati, Ohio .... 
 St. Louis, Mo 
 New York, N.Y. . . . 
 
 Hatborough, Pa 
 Philadelphia, Pa. . . . 
 Harrisburg, Pa 
 
 39 25.0' 
 41 30.3 ' 
 4220.0' 
 46 29.9' 
 39 08.6' 
 38 38.0' 
 40 42.7' 
 
 40 12.0' 
 39 56.9' 
 
 40 15.9' 
 39 17.8' 
 
 81 28.0' 
 81 42.0' 
 83 03.0' 
 84 20.1' 
 84 25.3' 
 90 12.2* 
 74 00.4' 
 
 75 07.0' 
 75 09.0' 
 
 76 52.9' 
 76 37.0' 
 
 D=+ 0.02 + 2.89 sin(1.4m 40.5) 
 D=+ 0.10 + 2.07 sin (1.40m 6.2) 
 Z> = 0.97 + 2.21 sin (1.50m 15.3) 
 D=+ 1.54 + 2.70 sin (1.45m -58.5) 
 D = - 2.40 + 2.62 sin (1.42 m - 39.8) 
 D = - 7.15 + 2.33 sin (1.4m -20.1) 
 D=+ 6.40 + 2.29 sin (1.6m -5.5) 
 + 0.14 ain (6.3m +6.4) 
 D=+ 5.23 + 3.28 sin(1.54m-13.2) 
 + 0.22 sin (4.1m + 157) 
 Z> = + 5.38+ 3.29 sin(1.55m-23.9) 
 + 0.39 sin(4.0nt + 161) 
 D = + 2.93 + 2.98 sin (1.50 m + 0.2) 
 D-+ 3.20 + 2.57 sin(1.45m 21.2) 
 
 Washington, D.C. . . . 
 Cape Henry, Va. . . . 
 Charleston, S.C 
 Sa'-annah, Ga 
 Key West, Fla 
 
 + 38 53.3' 
 + 36 55.5' 
 32 46.6' 
 32 04.9' 
 24 33 5' 
 
 + 77 00.6' 
 + 76 00.5' 
 79 55.8' 
 81 05.5' 
 81 48.5' 
 
 D=+ 2.47 + 2.52 sin (1.40m -14.6) 
 D=+ 2.54 + 2.41 sin (1.50m -35.4) 
 D=- 2.14 + 2.74 sin (1.35m -1.3) 
 7> = 2.54 + 2.32 sin (1.5m 28.6) 
 D 3.90 + 2.93 sin(1.4m 33.5) 
 
 
 23 09.3' 
 
 82 21 .5' 
 
 D 4.52 + 2.00 sin(1.3m- 26.7) 
 
 
 

 
 204 PLANE SURVEYING. 
 
 FORMULAS EXPRESSING THE MAGNETIC DECLINATION. Continued: 
 
 NAME OF STATION AND 
 LOCATION. 
 
 LATI- 
 TUDE. 
 
 LONGI- 
 TUDE. 
 
 EXPRESSION FOB MAGNETIC 
 DECLINATION. 
 
 Kingston, Jamaica . . . 
 
 17 55.9' 
 
 76 50.6' 
 
 D = 4?64 + 2?04sin(1.2m + 15.9) 
 
 Panama, New Granada . 
 
 8 57.1' 
 
 79 32.2' 
 
 D=- 6.80 + 1.82 sin(0.9 m + 10.4) 
 
 Florence, Ala 
 
 34 47.2' 
 
 87 41.5' 
 
 D = - 4.25 + 2.33 sin (1.3 m - 52.8) 
 
 Mobile Ala 
 
 30 41.4' 
 
 88 02.5' 
 
 D= 4.40 + 2.69 sin (1.45 T/i 76.4) 
 
 New Orleans, La. . . . 
 
 29 57. 2' 
 
 90 03.9' 
 
 D = 5.61 + 2.57 sin(1.4 61.9) 
 
 Vera Cruz, Mexico . . . 
 
 19 11.9' 
 
 96 08.8' 
 
 />=- 4.38 + 5.04 sin (1.10 TO -65.0) 
 
 Mexico, Mexico .... 
 
 19 25.9' 
 
 99 06.0' 
 
 D= 4.34 + 4.44 sin(1.0w 79.2) 
 
 Acapulco, Mexico . . . 
 
 16 50.5' 
 
 99 52.3' 
 
 D=- 4.13 + 4.82 sin(1.0j 81.1) 
 
 San Bias, Mexico . . . 
 
 21 32.6' 
 
 105 15.7' 
 
 D=- 6.51 + 2. 74 sin (0.9 - 106.3) 
 
 Magdalena Bay, L. Cal. . 
 
 24 38.4' 
 
 112 08.9' 
 
 /)= - 7.52 + 3.27 sin (1.25 m -140.6) 
 
 San Diego Cal 
 
 32 42.1' 
 
 117 14.3' 
 
 2> = 12.52 + 1.60 ein(l 2wi 179.8) 
 
 Monterey, Cal 
 
 36 36.1' 
 
 121 53.6' 
 
 D = 12.90 + 3.28 sin(1.0 m 142.6) 
 
 San Francisco, Cal. . . . 
 
 37 47. 5' 
 
 122 27.2' 
 
 D= -13.34 + 3.23 sin(1.00 m 130.3) 
 
 Cape Disappointm't, W.T. 
 
 46 16. 7' 
 
 124 02.0' 
 
 D = 20.26 + 2.36 sin (1.25 m 180.0) 
 
 Sitka, Alaska 
 
 57 02.9 ' 
 
 135 19.7' 
 
 D = -26.77 + 2.33 sin (1.4 m - 111.6) 
 
 Unalashka, Alaska . . . 
 
 53 52.6' 
 
 166 31. 5' 
 
 D = -18.34 + 1.45 sin(1.4 m - 67.8) 
 
 Tyrone Pa 
 
 40 40.0' 
 
 78 15.5' 
 
 Z)=+ 3.46 + 0.0550( 1875.5) 
 
 Pittsburg Pa 
 
 40 27.6' 
 
 80 00.8' 
 
 7) + 2.36 + 00566( 1878.7) 
 
 Chicago, 111 
 
 41 50.0' 
 
 87 36.7' 
 
 Z> = - 6.03 + 0.0281 (t 1850) 
 
 
 
 
 + 0.00082(<-1850)* 
 
 Grand Haven, Mich. . . 
 
 43 05.2' 
 
 86 12.6' 
 
 D = - 4.95 + 0.0380(< 1850) 
 
 
 
 
 + 0.00120(<- 1850)2 
 
 Madison, Wis 
 
 43 04.6' 
 
 89 24.2' 
 
 Z>=- 6.43 + 0.0655(^-1880.3) 
 
 Duluth, Minn.; and Supe- 
 
 46 45.5' 
 
 92 04.5' 
 
 D = -10.17 + 0.0868(* - 1875.8) 
 
 rior City, Wis. . . . 
 
 
 
 
 Rio Janeiro, Brazil . 
 
 -22 54.8' 
 
 43 09.5' 
 
 Z>=+ 0.282 + 0.1395(< 1850) 
 
 
 
 
 + 0.00545 (<- 1850) 2 
 
 San Antonio, Tex. . . . 
 
 + 29 25.4' 
 
 98 29.3' 
 
 D= 10.14 + 0.0204(< 1850) 
 
 
 
 
 + .000024(<- 1850)2 
 
 Omaha, Neb. ; and Council 
 
 41 15.7' 
 
 95 56.5' 
 
 D= -11.66 + 0.0439(*-1850) 
 
 Bluffs, Iowa 
 
 
 
 
 Denver Col 
 
 39 45.3' 
 
 104 59 5 ' 
 
 2) 14.79 4- 0.0258( 1872 9) 
 
 Salt Lake City, Utah. . . 
 
 40 46.1' 
 
 111 53.8' 
 
 D = -15.51 - 0.0930(* 1850) 
 
 
 
 
 + 0.00180(<-1850) 2 
 
 To illustrate the use of the table : Suppose it is desired to 
 ascertain the declination of the needle at Harrisburg for the last 
 of September, 1877, or t= 1877.75. 
 
 Take from the table the expression for the declination at 
 Harrisburg ; that is : 
 
 D= +2.93 + 2.98 sin (1.50m + 0.2).
 
 VARIATION OF THE COMPASS. 205 
 
 Find m= 1877.75 -1850 = 27. 75; 
 
 1.50ra + 0.2 = 41.625 + 0.2 = 41.825, 
 and 2.98 x natural sin 41.825 = 2.98 x .66686 = 1.987. 
 
 .-. D = 2.93 + 1.987 = 4.917 = 4 55' west (the result being 
 plus). The observed declination for the same time was 4 53' 5". 
 The difference between the computed and observed declination 
 is seen to be very small. 
 
 In running old lines it may be necessary to determine the 
 declination at a time anterior to 1850 ; then m will be negative. 
 Suppose the declination at Washington, B.C., for the year 1841 
 is desired. The tabular expression is : 
 
 D = 2.47 + 2.52 sin (1.4m - 14.6), 
 m 1841 -1850 = -9, 
 (1. 4m -14. 6) = -27.2, 
 2.52 sin (-27.2) ==-1.15. 
 
 .. D = 2.47 1.15 = 1.32 west (the resulting sign beingpfots), 
 which agrees practically with the observed declination. 
 
 244. The following table is taken from U. S. C. & G. S. Re- 
 port, 1882, App. 12, Mr. Schott's paper on Secular Variation. 
 It exhibits the computed epoch of greatest easterly deflection 
 reached in the secular motion ; i.e., the date when last reached, 
 or the date (in parenthesis) when it is next expected to be in 
 that position ; the amount in degrees and fractions, and direction 
 (+ west, east) at this, the nearest stationary epoch ; and the 
 computed annual changes in tlie declination of the magnetic 
 needle for the years 1870, 1880, and 1885, a plus sign indicating 
 north end of needle moving westward, a minus sign indicating 
 north end of needle moving eastward.
 
 206 
 
 PLANE SURVEYING. 
 
 LOCATION. 
 
 NEAREST 
 STATIONARY EPOCH 
 OF EASTERLY 
 DIGRESSION. 
 
 AMOUNT 
 AT EASTERLY 
 DIGRESSION. 
 
 ANNUAL CHANGE. 
 
 IN 1870. 
 
 IN 1880. 
 
 IN 1885. 
 
 Paris France 
 
 1581 
 
 10.6 
 
 7.0' 
 
 6.1' 
 
 9.5' 
 
 Halifax, Nova Scotia . 
 
 1728 
 
 + 12.4 
 
 + 1.8' 
 
 + 1.0' 
 
 + 0.5' 
 
 Quebec, Canada .... 
 
 1809 
 
 + 12.1 
 
 + 4.2' 
 
 + 1.6' 
 
 + 0.5' 
 
 Montreal, Canada . . . 
 
 1816 
 
 + 7.6 
 
 + 5.1' 
 
 + 3.1' 
 
 + 2.8' 
 
 Eastport, Me. 
 
 1760 
 
 + 12.5 
 
 + 3.3' 
 
 + 2.7' 
 
 + 2.3' 
 
 Portland, Me 
 
 1764 
 
 + 8.0 
 
 + 2.4' 
 
 + 1.6' 
 
 + 1.2' 
 
 Burlington, Vt 
 
 1810 
 
 + 7.2 
 
 + 5.0' 
 
 + 6.0' 
 
 + 5.8' 
 
 Rutland, Vt 
 
 1806 
 
 + 6.2 
 
 + 6.0' 
 
 + 5.6' 
 
 + 5.3' 
 
 Portsmouth, N.H. . . . 
 
 1791 
 
 + 7.5 
 
 + 4.4' 
 
 + 3.7' 
 
 + 3.3' 
 
 Newburyport, Mass. . . 
 
 1784 
 
 + 7.0 
 
 + 3.9' 
 
 + 3.3' 
 
 + 2.9' 
 
 Salem Mass 
 
 1791 
 
 + 6.2 
 
 + 5.0' 
 
 + 4.1' 
 
 + 3.5' 
 
 Boston, Mass 
 
 1777 
 
 + 6.6 
 
 + 3.4' 
 
 + 2.9' 
 
 + 2.5' 
 
 Cambridge, Mass. . . . 
 
 1783 
 
 + 6.9 
 
 + 2.9' 
 
 + 2.1' 
 
 + 1.8' 
 
 Nantucket, Mass. . . . 
 
 1779 
 
 + 6.5 
 
 + 3.3' 
 
 + 2.7' 
 
 + 2.4' 
 
 Providence, R.I 
 
 1780 
 
 + 6.1 
 
 + 3.8' 
 
 
 
 Hartford, Conn 
 
 1799 
 
 + 5.2 
 
 + 3.8' 
 
 + 3.7' 
 
 + 3.6' 
 
 New Haven, Conn. . . . 
 
 1802 
 
 + 4.7 
 
 + 4.6' 
 
 + 4.3' 
 
 + 4.1' 
 
 Albany, N.Y 
 
 1793 
 
 + 5.2 
 
 + 4.3' 
 
 + 3.7' 
 
 + 3.4' 
 
 Oxford, N.Y 
 
 1797 
 
 + 3.0 
 
 + 4.5' 
 
 + 4.3' 
 
 + 4.0' 
 
 Buffalo, N.Y 
 
 1806 
 
 + 0.2 
 
 + 5.1' 
 
 + 5.0' 
 
 + 4.8' 
 
 Toronto, Canada .... 
 
 
 
 + 4.8' 
 
 + 4.5' 
 
 + 2.3' 
 
 Erie, Pa 
 
 1811 
 
 - 0.5 
 
 + 4.4' 
 
 + 4.2' 
 
 + 4.0' 
 
 Marietta, O 
 
 1815 
 
 - 2.9 
 
 + 4.2' 
 
 + 4.2' 
 
 + 4.2' 
 
 Cleveland, O . 
 
 1790 
 
 2.0 
 
 + 2.8' 
 
 + 2.5' 
 
 + 2.2' 
 
 Detroit, Mich 
 
 1800 
 
 - 3.2 
 
 + 3.4' 
 
 + 3.0' 
 
 + 2.8> 
 
 Sault de St. Marie, Mich. 
 
 1828 
 
 - 1.2 
 
 + 3.6' 
 
 + 4.0' 
 
 + 4.1' 
 
 Cincinnati, . . 
 
 1815 
 
 5.0 
 
 + 3.8' 
 
 + 3.9' 
 
 + 3.8' 
 
 St. Louis, Mo 
 
 1800 
 
 - 9.5 
 
 + 3.4' 
 
 + 3.2' 
 
 + 3.0' 
 
 New York, N.Y 
 
 1797 
 
 + 4.0 
 
 + 2.4' 
 
 + 2.5' 
 
 + 2.6' 
 
 Hatborough, Pa 
 
 1797 
 
 + 1.8 
 
 + 4.6' 
 
 + 4.5' 
 
 
 Philadelphia, Pa 
 
 1800 
 
 + 1.9 
 
 + 4.9' 
 
 + 4.9' 
 
 + 5.3' 
 
 Baltimore, Md 
 
 1802 
 
 + 0.6 
 
 + 3.9' 
 
 + 3.6' 
 
 + 3.2' 
 
 Harrisburg, Pa. .... 
 
 1700 
 
 0.0 
 
 + 4.1' 
 
 + 3.3' 
 
 + 2.8' 
 
 Washington, DC. . . . 1796 
 
 0.0 
 
 + 3.5' 
 
 + 3.2' 
 
 + 3.0'
 
 VARIATION OF THE COMPASS. 
 
 207 
 
 LOCATION. 
 
 NEAREST 
 STATIONARY EPOCH 
 OF EASTERLY 
 DIGKESSION. 
 
 AMOUNT 
 AT EASTERLY 
 DIGRESSION. 
 
 ANNUAL CHANGE. 
 
 IN 1870. 
 
 IN 1880. 
 
 IN 1885. 
 
 Cape Henry, Va 
 Charleston, S.C 
 
 1814 
 1784 
 1809 
 1810 
 1801 
 1762 
 1739 
 1821 
 1841 
 1830 
 1827 
 1839 
 1841 
 1868 
 (1890) 
 (1925) 
 (1903) 
 (1890) 
 (1922) 
 1865 
 1834 
 
 + 0.1 
 - 4.9 
 - 4.9 
 - 6.8 
 - 6.5 
 - 6.7 
 - 8.6 
 - 6.6 
 - 7.1 
 - 8.2 
 - 9.4 
 - 8.8 
 - 9.0 
 - 9.3 
 10.8 
 -14.1 
 -16.2 
 - 16.6 
 -22.6 
 -29.1 
 -19.8 
 
 + 3.8' 
 + 3.5' 
 + 3.6' 
 + 4.3' 
 + 2.7' 
 + 2.0' 
 + 1.5' 
 + 2.8' 
 + 2.8' 
 + 3.1' 
 + 4.2' 
 + 2.4' 
 + 2.4' 
 + 0.1' 
 -1.8' 
 -1.8' 
 -1.8' 
 -1.0' 
 -2.8' 
 + 0.4' 
 + 1.6' 
 
 + 3.7' 
 + 3.0' 
 + 3.5' 
 + 4.2' 
 + 2.7' 
 + 1.6' 
 + 1.4' 
 + 3.1' 
 + 3.4' 
 + 3.6' 
 + 4.9' 
 + 3.0' 
 + 3.2' 
 + 0.5' 
 -1.0' 
 -1.6' 
 -1.3' 
 -0.5' 
 -2.5' 
 + 1.2' 
 + 1.9' 
 + 3.3' 
 + 3.4' 
 + 4.6' 
 + 6.6' 
 + 3.9' 
 
 + 5.2' 
 
 +10.3' 
 + 2.1' 
 
 + 2.6' 
 
 + 1.6' 
 + 0.9' 
 
 + 3.6' 
 + 2.7' 
 + 3.3' 
 + 4.1' 
 + 2.6' 
 + 1.4' 
 + 1.3' 
 + 3.2' 
 + 3.7' 
 + 3.7' 
 + 5.2' 
 + 3.3' 
 + 3.6' 
 + 0.7' 
 -0.5' 
 -1.5' 
 -1.0' 
 -0.3' 
 2.2' 
 + 1.6' 
 + 2.0' 
 
 + 5.1' 
 + 7.3' 
 
 ' +10.7 
 + 2.2' 
 
 + 2.0' 
 
 Key West, Fla 
 Havana, Cuba 
 Kingston, Jamaica . . . 
 Panama, New Granada 
 
 Mobile Ala . . . 
 
 New Orleans, La 
 Vera Cruz, Mexico . . 
 Mexico, Mexico .... 
 Acapulco, Mexico . . . 
 San Bias, Mexico . . . 
 Magdalena Bay, L.Cal. 
 San Diego, Cal 
 
 San Francisco, Cal. . . 
 C. Disappointm't, W.T. 
 Sitka Alaska 
 
 Unalashka, Alaska . . . 
 
 
 
 
 
 Chicago 111 
 
 1833 
 1834 
 
 - 6.3 
 - 5.3 
 
 .... 
 
 Grand Haven, Mich. . . 
 Madison Wis 
 
 Duluth, Wis ) 
 
 
 
 
 Superior City, Wis. . > 
 Rio Janeiro, Brazil . . 
 San Antonio, Tex. . . . 
 Omaha, Neb ) 
 
 
 
 
 
 
 
 
 
 
 Council Bluffs, la. . . ) 
 Denver Col . . 
 
 1876 
 
 -16.7 
 
 .... 
 
 Salt Lake City, Utah .
 
 208 PLANE SURVEYING. 
 
 The variability of the change will be noticed. For example, 
 take New York, Philadelphia, and Harrisburg, places compara- 
 tively near together. 
 
 At New York the change in 1870 was only one-half that at 
 Philadelphia ; but, both increasing, this ratio was maintained 
 throughout the 15 years. At Harrisburg, on the contrary, the 
 annual change in 1870 was nearly six-sevenths that at Phila- 
 delphia, but the change constantly increasing at the latter place 
 while diminishing rapidly at the former, the annual variation at 
 Harrisburg in 1885 was only a little more than one-half that 
 at Philadelphia.* 
 
 245. Effects of the Secular Change. It is evident that if a 
 surveyor should ignore this change, in attempting to establish 
 the corners and to trace the boundary lines of a farm from their 
 description in an old deed, it would be possible for him to return 
 to his place of beginning, but probably none of his other corners 
 would coincide with the true corners. 
 
 A line in the vicinity of Philadelphia, which 12 years ago had 
 a bearing N. 19 E., would now bear N. 20 E., and in the 
 same locality a bearing which at that time was recorded N. 19 
 W. would now be N. 18 W. A variation which, if not cor- 
 rected, would indicate the end of a line 15 chains long over 26 
 links from its true position. 
 
 Take, for example, the notes given in Article 208, page 161, 
 and suppose an interval has elapsed sufficient to make the vari- 
 ation two degrees. The accompanying figure shows the true 
 lines and corners ; also those corresponding to a survey made 
 without taking the variation into account. 
 
 The bearings and distances are as follows : 
 
 (1) S. 20 53' E. 13,11 chains; 
 
 (2) N. 48 10' E. 13.62 " 
 
 (3) N. 43 40' W. 4.73 " 
 
 * For extended investigations on magnetic declination, see U. S. C. & 
 G. S. Reports, 1879, 1881, and 1882.
 
 VARIATION OF THE COMPASS. 
 
 (4) N. 45 08' W. 4.75 chains ; 
 
 (5) S. 51 30' W. 2.53 " 
 
 (6) S. 72 30' W. 6.56 " 
 
 ss 
 
 To allow for a variation of two degrees, we should have the 
 following bearings : 
 
 (1) S. 18 53' E. ; 
 
 (2) N. 50 10' E. ; 
 
 (3) N. 41 40' W. ; 
 
 (4) N. 43 08' W. ; 
 
 (5) S. 53 30' W. ; 
 
 (6) S. 74 30' W. 
 
 246. To deduce a general rule for obtaining the magnetic 
 bearings of old lines when the variation is known.
 
 210 
 
 PLANE SURVEYING. 
 
 Let JV$ represent the direction of the magnetic meridian in 
 the vicinity of a survey made 
 several years ago ; N'S', its 
 direction several years later, at 
 the time of re-survey, aud that 
 the north end of the needle 
 points 2 farther west. It is 
 evident that at the time of the 
 re-survey, the line NS will bear 
 N. 2 E., and OP, which accord- 
 ing to the old survey bears N. 
 48 E., will have its bearing in- 
 creased 2 or N. 50 E. ; but 
 the line OM, the bearing of which was N. 42 W., will now bear 
 N. 40 W. A line recorded as east will be traced by a course 
 S. 88 E., and soon. 
 
 Hence the rule : Increase by the change the bearings which 
 are northeasterly or southwesterly, aud diminish by the same 
 amount the bearings which are northwesterly or southeasterly. 
 The foregoing rule is directly applicable now in the United 
 States, except on the Pacific coast, because the variation is west. 
 That is, the north end of the needle is moving west, thereby 
 increasing the readings of bearings in the N. E. and S. W. quar- 
 ters, and diminishing the readings of those in the N. W. and 
 S. E. quarters. When it becomes east, the words "increase" 
 and "diminish" should be interchanged to make it correct. 
 If a vernier compass is used, the variation may be set off and 
 the lines traced by the old bearings. 
 
 247. Change Determined by Old Lines. If the bearing and date 
 of survey of a line are known, and its extremities visible from each 
 other, setting the instrument on one end and sighting the other 
 will give, by comparison with the recorded bearing, the variation. 
 
 NOTE. Care must be taken by the surveyor, when called upon to run 
 out old lines, the corners not being definitely marked, that the time of the 
 former survey be known ; the date of the deed docs not indicate that of 
 the survey. The description of the lines may have been copied, as they 
 frequently are, from an older deed.
 
 VARIATION OF THE COMPASS. 211 
 
 The variation to be applied to correct magnetic bearings is 
 frequently determined in this way. 
 
 If the boundaries of a tract of land are to be traced, whether 
 the date of the previous survey be known or not, the surveyor 
 seeks to find, if possible, two consecutive marked corners ; then, 
 taking the bearing of these and comparing with the record, 
 he obtains the change sought. 
 
 This change, properly applied to each side, should indicate 
 its direction. 
 
 It frequently, and in large tracts generally, happens that 
 though the corners at the end of a line may be established, they 
 cannot be observed from each other. In such case run a line 
 as nearly as possible from one corner towards the other by the 
 bearing given in the deed, or make first an allowance which may 
 seem proper from the data at hand ; measure from the end of the 
 line thus run the distance to the true corner, and by the 57.3, 
 rule, Article 177 ; or, by the tangent method, same article, find 
 the angle to be added or subtracted, as the case may require, to 
 correct the bearing with which to run the line. The difference 
 between the bearing given in the deed and the corrected bearing 
 will be the change in the declination since the survey recorded 
 in the deed. 
 
 EXAMPLES. 
 
 1. A line, said to have been surveyed in 1860, recorded 
 N. 18 30' E., 24.40 chains, was run in 1885 with a bearing 
 N. 19 45' E., the variation being about 3' to the west per year 
 in its locality, and the corner was 7 links to the right (farther 
 easterly) of the end of the line run. The corrected magnetic 
 bearing and variation are required. 
 
 1 15' + 57 ;^ X 7 = 1 15' + 10' = 1 25' = variation. 
 
 Adding the variation to the bearing of the line run, since the 
 true corner was farther to the east, there results N. 19 55' E. 
 as the corrected magnetic bearing of the line.
 
 212 
 
 PLANE SURVEYING. 
 
 2. If in Example 1 the corner had been found 7 links to the 
 left, what would be the correct bearing of the line? 
 
 3. A line which in 1862 ran S. 34 15' W. 18.56 chains, in 
 1886 bore S. 35 35' W. What was the average change in 
 the declination per year? 
 
 4. Give the corrected magnetic bearing for 1886 of a line in 
 the same locality as that in Example 3, which in 1868 ran due 
 east. 
 
 5. In 1876 a line had a bearing S. 89 45' W. 16.80 chains ; 
 in 1886, running by the same bearing, the true corner was 20 
 links to the north. Give the average annual change, and cor- 
 rect the bearing. 
 
 6. If a line 60.00 chains in length were surveyed in the early 
 part of the day, where the needle deviates 5 minutes east of 
 the mean magnetic meridian, and the same line surveyed soon 
 after mid-dav, the needle then pointing 5 minutes west of the 
 mean magnetic meridian, how far apart would the lines be at 
 their ends, and what the area included between them? 
 
 248. To Obtain the True Bearing of a Line, that is, the 
 bearing with respect to the geographical meridian, when the 
 
 W 
 
 declination is west. Assume NS and N'S' (left-hand figure) to 
 represent respectively the true and magnetic meridian. Then it 
 is evident that the bearing of any line between the north and
 
 VARIATION OF THE COMPASS. 213 
 
 east, or south and west, as OP or OP', will be less referred to 
 NS than when referred to X'S' by the amount of the angle 
 NON' = SOS' = the declination. 
 
 A line running between north and west, as OM, or south 
 and east, as OM', will evidently have its bearing increased by 
 the amount of the change. 
 
 The reverse is true where the declination is east, as may be 
 perceived by reference to the right-hand figure. 
 
 Hence, to get the true bearing from the magnetic for all 
 places east of the line of no declination, i.e. where the declina- 
 tion is tcest, subtract the declination from a bearing which is 
 northeasterly or southwesterly, and add the declination to a 
 bearing which is northwesterly or southeasterly. Where the 
 declination is east, as at all places west of the line of no 
 declination, add the declination to a bearing which is north- 
 easterly or southwesterly, and subtract the declination from a 
 bearing which is northwesterly or southeasterly. Where the 
 declination is west, a bearing that reads north, when reduced to 
 the true bearing, will evidently be west of north the amount 
 of the declination ; if the declination is 3, the bearing will be 
 N. 3 W., and supposing the same declination, a line running due 
 east magnetically will be truly N. 87 E. 
 
 The reverse of the last paragraph is true where the declina- 
 tion is east. 
 
 KKMARK. If, when applying the rule, a negative result is obtained, oare 
 must be exercised in the interpretation of it. For example, if the declina- 
 tion is 3 West, and the needle indicates the bearing of a line N. 1 E., 
 there results, by the rule, 2. This shows simply that the true bearing 
 is to the west of north, or N. 2 W. If the bearing is S. 89 E., adding the 
 declination, as the rule requires, gives evidently the reading N. 88 E. 
 
 Reduce to their true bearings the following, the declination 
 being 2 55' W. : 
 
 N. 2 15' E., East, S. 45 E., South ; S. 87 30' W., N. 88 
 15' W., North. 
 
 Also the following, the declination being 3 40' E. : 
 
 N. 88 E., East, S. 2 E., South ; S. 88 30' W., N. 40 W., 
 North.
 
 214 PLANE SURVEYING. 
 
 249. To Ascertain the Declination.* If a geographical merid- 
 ian were traced on the earth convenient to the operations of 
 the surveyor, he would have the means always at hand by which 
 to determine the declination. He could simply set up his in- 
 strument at a point on the meridian, take the bearing of another 
 point in it, and the reading would be the declination. So the 
 problem resolves itself into the determination of a geographic 
 or true meridian. 
 
 250. By Polaris. If there was a celestial object precisely 
 at the point where the prolongation of the earth's axis pierces 
 the celestial sphere, the direction of the meridian could be ob- 
 tained by simply sighting to the object. This, however, is not 
 the case, but Polaris, or Alpha Ursae Minoris, is a star whose 
 polar distance is, January, 1887, 1 17' 38", f and which appar- 
 ently revolves about the north pole in 23 hours 56 minutes. It 
 therefore culminates twice daily, and twice it attains its greatest 
 distance directly east and west of the pole, called respectively 
 its eastern and western elongation. If, therefore, the Pole Star 
 could be observed at the instant of its culmination, the line of 
 sight would be in the meridian plane ; but since in general the 
 local time of transit is not precisely known, and since the star 
 is then moving at right angles to the plane of the meridian 
 respecting which its motion is at that time a maximum, and 
 consequently a small difference in time would introduce a con- 
 siderable error in arc, this method is not as reliable as that 
 by means of Polaris at its eastern or western elongation, as 
 then the star for a few minutes appears to move in the direction 
 of the vertical wire, or compass-slit, thus affording a favorable 
 
 * For other methods, see Chapter II. Section I,, Solar Attachment; 
 and Chapter VI., Art. Solar Compass. 
 
 t Its polar distance is diminishing at the rate of 20" (19.06") per year. 
 This diminution will continue until the star is within half a degree of the 
 pole, when it will recede. 
 
 In 1890 its polar distance will be 1 16' 42". 
 
 In 1900 its polar distance will be 1 13' 33". 
 
 In 1910 its polar distance will be 1 10' 26".
 
 VARIATION OF THE COMPASS. 215 
 
 opportunity for observing it, and the precise time of observation 
 need not be known. 
 
 Conceive a spherical triangle, the vertices of which are, Z, 
 the zenith of the observer ; P, the north pole ; and S, Polaris. 
 This triangle, when the star is at an 
 elongation, will be right-angled at the 
 star. In this right-angled spherical 
 triangle are known the co-latitude of 
 the observer's station, and the co- 
 declination or polar distance of the 
 star, to find the azimuth * and hour 
 angle. | Using natural functions, the 
 formula for the hour angle is cos P = tan PS cot PZ, and for 
 the azimuth, 
 
 5inZ= s[nPS = &[nP<S + 
 sinPZ coslat + 
 
 It may be well to remark, though it has only a theoretical 
 significance, that these formulas are not applicable to all north 
 latitudes. In other words, there will be no hour angle shown 
 by the first formula, nor azimuthal angle by the second, on that 
 parallel of latitude which agrees in arc distance with Polaris 
 from the equator, and for any point between that parallel and 
 the pole the formulas fail. 
 
 This remark is in general applicable to any circumpolar star. 
 
 QUERIES. Is Polaris a longer time passing from eastern to 
 western elongation, than from western to eastern, to an ob- 
 server whose latitude is 40 ? What is the difference in time to 
 an observer whose latitude is 60? 80? Where would this 
 difference be a minimum ? Where a maximum ? 
 
 * The azimuth of a star is the angle between the meridian plane and 
 the vertical plane through the star. 
 
 t The angle SPZ included between the meridian plane PZ and the 
 plane PS passing through the star. 
 
 } The azimuth of Polaris at elongation varies with the latitude and 
 with the year, as may be seen by the table on page 217
 
 216 
 
 PLANE SURVEYING. 
 
 251. Table of mean local time astronomical (from noon) of 
 the elongations and culminations of Polaris for 1885, latitude 
 40, and longitude 6 hours west of Greenwich. 
 
 FIRST DAY OF 
 
 E. E. 
 
 U.C. 
 
 W. E. 
 
 L. C. 
 
 
 h. m. 
 353 
 
 h. m. 
 6 29.9 
 
 h. m. 
 12 24.6 
 
 h. m. 
 
 18 28.0 
 
 February 
 March 
 
 22 29.0 
 20 38.5 
 
 4 27.6 
 2 37 1 
 
 10 22.2 
 8 31.8 
 
 16 25.6 
 14 351 
 
 April 
 May 
 
 18 36.4 
 16 38.6 
 
 35.0 
 22 33.3 
 
 6 29.7 
 4 31.8 
 
 12 33.1 
 10 35.2 
 
 June 
 July 
 
 14 37.0 
 12 39 5 
 
 20 31.7 
 18 342 
 
 2 30.3 
 32 8 
 
 8 33.7 
 6 362 
 
 August 
 
 10 38.1 
 8 366 
 
 16 32.8 
 14 31 3 
 
 22 27.5 
 20 260 
 
 4 34.8 
 2 333 
 
 October 
 
 6 38.9 
 4 37 
 
 12 33.6 
 
 10 31 7 
 
 18 28.2 
 16 264 
 
 35.5 
 
 22 29 7 
 
 December . . . 
 
 2 38.9 
 
 8 33.5 
 
 14 282 
 
 20 316 
 
 
 
 
 
 
 To correct the tabular times so as to apply to any year subse- 
 quent to 1885, add 0.35 minutes for every year. For any year 
 previous to that date, subtract 0.35 minutes for every year. 
 
 For days not given in the table, interpolate, or allow 3.94 
 minutes for each day, the times varying by this amount. 
 
 To allow for difference of latitude between the limits of 30 
 and 50, add 0.14 for every degree south of 40 ; subtract 0.18 
 for every degree north of 40. 
 
 To refer the tabular times to any year in a quadriennium, 
 observe 
 
 For the first year after a leap year the table is perfect ; for 
 the second year after a leap year add 1 minute ; for the third 
 year after a leap year, add 2 minutes ; for a leap year, and 
 before March 1, add 3 minutes; and for the remainder of the 
 year subtract 1 minute.
 
 VARIATION OF THE COMPASS. 
 
 217 
 
 It will be noticed that there occur two eastern elongations 
 on Jan. 9, and two western elongations on July 9. 
 
 AZIMUTH (FROM THE NORTH) OF POLARIS, WHEN AT ELONGATION, BETWEEN 
 THE YEARS 1887-1895, FOR DIFFERENT LATITUDES BETWEEN +26 
 AND +50. 
 
 Lat. 
 
 1887. 
 
 1888. 
 
 1889. 
 
 1890. 
 
 1891. 
 
 1892. 
 
 1893. 
 
 1894. 
 
 1895. 
 
 + 25 
 
 125.7' 
 
 125.3' 
 
 1 C 25.0' 
 
 1 C 24.6' 
 
 1 C 24.3' 
 
 1 C 23.9' 
 
 1 C 23.6' 
 
 1 C 23.2' 
 
 1 C 22.9' 
 
 26 
 
 26.4 
 
 26.0 
 
 25.7 
 
 25.3 
 
 25.0 
 
 24.6 
 
 24.3 
 
 23.9 
 
 23.6 
 
 27 
 
 27.1 
 
 26.8 
 
 26.4 
 
 26.0 
 
 25.7 
 
 25.4 
 
 25.1 
 
 24,7 
 
 24.3 
 
 28 
 
 27.9 
 
 27.6 
 
 27.2 
 
 26.8 
 
 26.5 
 
 26.2 
 
 25.8 
 
 25.4 
 
 25.1 
 
 29 
 
 28.8 
 
 28.4 
 
 28.0 
 
 27.6 
 
 27.3 
 
 27.0 
 
 26.6 
 
 26.3 
 
 25.9 
 
 30 
 
 29.6 
 
 29.3 
 
 28.9 
 
 28.5 
 
 28.2 
 
 27.8 
 
 27.5 
 
 27.1 
 
 26.8 
 
 31 
 
 30.5 
 
 30.2 
 
 29.8 
 
 29.4 
 
 29.1 
 
 28.8 
 
 28.4 
 
 28.0 
 
 27.6 
 
 32 
 
 31.5 
 
 31.2 
 
 30.8 
 
 30.4 
 
 30.1 
 
 29.7 
 
 29.3 
 
 29.0 
 
 28.6 
 
 33 
 
 32.6 
 
 32.2 
 
 31.8 
 
 31.4 
 
 31.1 
 
 30.7 
 
 30.3 
 
 30.0 
 
 29.6 
 
 34 
 
 33.6 
 
 33.3 
 
 32.9 
 
 32.5 
 
 32.1 
 
 31.8 
 
 31.4 
 
 31.0 
 
 30.6 
 
 35 
 
 34.8 
 
 34.4 
 
 34.0 
 
 33.6 
 
 33.2 
 
 32.9 
 
 32.5 
 
 32.1 
 
 31.7 
 
 36 
 
 36.0 
 
 35.6 
 
 35.2 
 
 34.9 
 
 34.4 
 
 34.0 
 
 33.6 
 
 33.2 
 
 32.9 
 
 37 
 
 37.2 
 
 36.8 
 
 36.4 
 
 36.0 
 
 35.6 
 
 35.2 
 
 34.8 
 
 34.5 
 
 34.1 
 
 38 
 
 38.5 
 
 38.1 
 
 37.7 
 
 37.3 
 
 36.9 
 
 36.5 
 
 36.1 
 
 35.7 
 
 35.3 
 
 39 
 
 39.9 
 
 39.5 
 
 39.1 
 
 38.7 
 
 38.3 
 
 37.9 
 
 37.5 
 
 37.1 
 
 36.7 
 
 40 
 
 41.4 
 
 41.0 
 
 40.5 
 
 40.1 
 
 39.7 
 
 39.3 
 
 38.9 
 
 38.5 
 
 38.1 
 
 41 
 
 42.9 
 
 42.5 
 
 42.0 
 
 41.6 
 
 41.2 
 
 40.8 
 
 40.4 
 
 40.0 
 
 39.6 
 
 42 
 
 44.5 
 
 44.1 
 
 43.6 
 
 43.2 
 
 42.8 
 
 42.4 
 
 42.0 
 
 41.5 
 
 41.1 
 
 43 
 
 46.1 
 
 45.7 
 
 45.3 
 
 44.9 
 
 44.4 
 
 44.0 
 
 43.6 
 
 43.2 
 
 42.7 
 
 44 
 
 47.9 
 
 47.5 
 
 47.1 
 
 46.6 
 
 46.2 
 
 45.8 
 
 46.3 
 
 44.9 
 
 44.4 
 
 45 
 
 49.8 
 
 49.4 
 
 48.9 
 
 48.5 
 
 48.1 
 
 47.6 
 
 47.1 
 
 46.7 
 
 46.2 
 
 46 
 
 51.8 
 
 51.3 
 
 50.9 
 
 50.4 
 
 50.0 
 
 49.5 
 
 49.0 
 
 48.6 
 
 48.2 
 
 47 
 
 53.8 
 
 53.4 
 
 52.9 
 
 52.5 
 
 52.0 
 
 51.5 
 
 51.0 
 
 50.6 
 
 60.2 
 
 48 
 
 56.0 
 
 55.6 
 
 55.1 
 
 54.6 
 
 54.2 
 
 53.7 
 
 53.5 
 
 52.8 
 
 52.3 
 
 49 
 
 58.3 
 
 57.9 
 
 57.4 
 
 56.9 
 
 56.5 
 
 56.0 
 
 55.5 
 
 55.0 
 
 64.5 
 
 + 50 
 
 200.8' 
 
 200.3' 
 
 159.8' 
 
 T59.3' 
 
 158.8' 
 
 F68.4' 
 
 F67.9' 
 
 167.4' 
 
 166.9'
 
 218 PLANE SURVEYING. 
 
 252. To Establish a True Meridian with a Transit.* See 
 that the instrument is in good adjustment. Allow sufficient 
 time before an elongation of the star to "set up" the transit 
 in a desirable position, f See that it is planted firmly, levelled 
 carefully, and that the cross-wires are illuminated J and prop- 
 erly focused. For convenience, set the vernier at zero, and 
 unclamp the lower plate. 
 
 Observe the star a few minutes before its elongation, and 
 keep the vertical wire on it by clamping the lower plate and 
 using the slow-motion screws attached to it. When it has 
 attained its greatest elongation, it will appear for a few moments 
 to coincide with the vertical wire, and then retrograde. Un- 
 clamp the vernier plate, and turn off with it the amount of the 
 azimuth corresponding to the time and place as given in the 
 table of the preceding article. The telescope will then point 
 in the direction of the true meridian, and a mark should be set 
 at as long range as practicable. If preferred, a stake may be 
 set in line of sight at elongation, leaving the turning off of 
 azimuth, and setting mark in meridian until the next day. It 
 would be a little more accurate to take the mean of several 
 observations direct and reverse at eastern and western 
 elongations. 
 
 * See Solar Attachment, Chapter II. Section I. ; also Solar Compass, 
 Chapter VI. 
 
 t Twenty to thirty minutes usually, depending upon the observer. 
 
 t Perforated silvered reflectors, for this purpose, can be obtained of 
 instrument makers. Or, cover with white paper a board 12 or 15 inches 
 square, make a perforation through it of 2 or 3 inches' diameter, and nail 
 on a piece of board to hold a candle. This reflector may be attached to 
 a staff, that it can slide up and down, and adjusted to the height of the 
 telescope. It should be placed about a foot from the object-glass, so that 
 the reflection from the paper will render the cross-wires visible, and at 
 such a height that the star can be observed through the opening. 
 
 The meridian will lie to the west or east of the direction of the tele- 
 scope when elongation was observed, according as the elongation was east 
 or west. The azimuth must be turned off accordingly. Since the direc- 
 tion of the line from the observer's station to the star at elongation is 
 known, the declination may be ascertained even before the meridian is 
 established.
 
 VARIATION OF THE COMPASS. 219 
 
 253. The Direction of the Meridian may be found, though 
 less accurately, bv means of a compass-sight and plumb-line. 
 
 Take a smooth plank about 3 feet in length, and fix it 
 firmly level, and nearly east and west, on supports about 2 feet 
 high. Attach a compass-sight to a board 6 or 8 inches square. 
 At 15 or 20 feet n9rth of the plank suspend a plumb-line by 
 artificial supports, from some projecting point on a building or 
 at the end of a staff projecting from a high window. 
 
 At fifteen or twenty minutes before the time of elongation of 
 the star let an assistant hold a light in such position that the 
 plumb-line may be distinctly seen through the compass-sight 
 when placed on the plank. Move the sight until the plumb- 
 line covers the star. Continue to keep the star and line in that 
 relative position until the star begins to retrograde. The direc- 
 tion of the line of sight then corresponds to that observed by 
 the transit as indicated in the preceding article ; and applying 
 the azimuth therein directed, the meridian may be set out.* 
 
 254. To Obtain approximately the Meridian. In old works 
 on surveying it is stated that the north star (Polaris) is very 
 nearly the meridian when it and Alioth f are in the same vertical 
 plane or line. Others add the time that must elapse after one 
 is vertically above the other before the north star makes its 
 transit, and then by sighting the north star at that instant the 
 meridian may be found. 
 
 This interval is, January, 1887, nearly half an hour. Other 
 stars are now used, being more suitable. Zeta, or Mizar, the 
 star next to Alioth in the tail of the Great Bear, comes to the 
 meridian now almost simultaneously with Polaris and at a con- 
 venient time in the autumn and early winter to make the obser- 
 
 * If possible, a night should be chosen when there is no wind. The 
 slightest disturbance in the air causes considerable vibration of the plumb- 
 line. Using a heavy " bob," and allowing it to vibrate in a vessel of water, 
 will tend to the accuracy of the result. 
 
 t Alioth, or Epsilon: the star in the tail of the Great Bear nearest the 
 quadrilateral.
 
 220 PLANE SUKVEYLNG. 
 
 vation. Delta Cassiopeiae, which is on the same side of the 
 pole as Polaris, makes its transit also about the same time with 
 it, and may be used in the spring and early summer when it is 
 not practicable to make use of Zeta. To make either of these 
 observations, use a transit, or a plumb-line and compass-sight, 
 as explained in the preceding articles ; watch the movements 
 of the stars until they coincide with the plumb-line. The 
 direction of the line of sight then will indicate quite closely the 
 meridian.* 
 
 * The vertical plane including Zeta and Polaris is slowly moving east- 
 ward at about the rate of two minutes in six years. At the present time 
 (1887) Polaris is on the meridian about two minutes before Zeta of the 
 Great Bear, but in six years their respective upper and lower transits will 
 coincide. The vertical plane, including Delta Cassiopeia? and Polaris, is 
 moving westward at about the same rate. Polaris now comes to the 
 meridian about one minute before this star.
 
 CHAPTER IV. 
 
 LAYING OUT AND DIVIDING LAND. 
 
 SECTION I. 
 LAYING OUT LAND. 
 
 A. TRIANGLES. 
 
 255. To lay out a given quantity of land in the form of a 
 triangle when the length of the base is given. 
 
 Denote the given area in square chains or square rods by 
 A,* the length of the base (referred to the same unit) by 6, 
 and the unknown altitude by x. Then 
 
 AT 9 A M P' P K 
 
 - = ^L, oro? = Measure the base, ' 
 
 and at any point in it erect a perpen- 
 
 2 A 
 
 dicular equal to Join the ex- 
 b 
 
 tremity of the perpendicular with the extremities of the base, 
 and a triangle fulfilling the conditions of the question will be 
 exhibited. 
 
 256. Wlien the area is given and the base and altitude in a 
 given ratio. 
 
 * Why not let A denote the number of acres ? 
 
 NOTE. The locus of the vertices of the triangles answering the conditions 
 
 O A 
 
 is a line parallel to the given base and at a distance therefrom = 
 
 o
 
 222 PLANE SURVEYING. 
 
 Designate, as before, the ai'ea by A, the base and altitude 
 respectively by x and y, and - = the ratio ; then 
 
 Or, let mx = base and nx = altitude ; then 
 
 whence 
 
 and l ' 2Am 
 
 2 An 
 
 257. Given area, 6ase, and one side, to make a given angle 
 with the base. 
 
 Denote the base and area as above ; 
 then, since 
 
 PN= OP sin 0, 
 
 6 X OP sin 
 
 and 
 
 N R QP = 
 
 2A 
 b sin 
 
 EXAMPLES. 
 
 1. Lay out an isosceles triangle to contain 6 acres, making 
 the base | the altitude. Locate the altitude and find its length. 
 
 2. Lay out a right triangle containing 4 acres, having the 
 base | the altitude. 
 
 3. It is required to lay out 2 acres in the form of a triangle, 
 the base to be 7.50 chains. Find the length of a side of this 
 triangle which shall make an angle of 40 with the base.
 
 LAYING OUT AND DIVIDING LAND. 223 
 
 258. To lay out an equilateral triangle to contain a given area. 
 Let x = the side, and A = the area ; then, since 
 
 .433 
 
 259. Given the area and the two sides, to lay out the triangle. 
 Denote the given sides by b and c, the area by A, and the 
 
 nknown angle by a; then, since 
 
 'I sin a = A, 
 
 = ~bc" 
 
 EXAMPLES. 
 
 1. Find the side of an equilateral triangle containing one 
 acre. 
 
 2. What is the altitude of the triangle in Example 1 ? How 
 far is it trom the foot of the perpendicular to the centre of the 
 figure? How far from either angle to the centre? 
 
 3. Lay out a triangle containing 2 acres, two sides to be 
 8 chains and 6 chains. What must be the included angle? 
 
 B. QUADRILATERALS. 
 
 SQUARES. 
 
 260. To lay out a given quantity of land in the form of a 
 square. 
 
 Denote the required area in square chains or square rods 
 by -A, and one of the sides by x ; then x = -\J A. 
 
 Measure a distance equal to the V2. ; at each extremity of 
 this line erect a perpendicular of the same length ; connect the 
 extremities of the perpendiculars ; the figure will be a square.
 
 224 PLANE SURVEYING. 
 
 RECTANGLES. 
 
 261. To lay out a given quantity of land in the form of a rec- 
 tangle, one side being given. 
 
 Denote, as before, the area by A, the given side by 6, and 
 by x the unknown side ; then 
 
 .=4 
 
 b 
 
 262. Given the area, and the length to the breadth in a given 
 ratio. 
 
 Denote the area as above ; the length and breadth respec- 
 tively by x and y ; m and n their ratio, so that 
 
 x _ m 
 
 y~ n 
 Then, since xy = A, there results, by substitution, 
 
 -V- 
 
 Or, let mx = the length, and nx the breadth ; then 
 mnx 2 = A ; 
 Am 
 
 whence 
 
 n 
 
 and nx =\| . 
 
 * tn. 
 
 An 
 m 
 
 263. Given the area and the sum of the length and breadth. 
 Denote the sum of the sides by S ; the other notation as 
 above ; then 
 
 and x + y = S ; 
 
 whence 
 
 and y =
 
 LAYING OUT AND DIVIDING LAND. 225 
 
 264. Given the area and the difference of the length and 
 breadth. 
 
 Denote the difference of the sides by d ; the other notation as 
 before ; then 
 
 whence 
 
 and 
 
 EXAMPLES. 
 
 1. How many rods in each side of a square lot which con- 
 tains 1 acre ? How many chains ? How many yards ? 
 
 2. Lay out 6 acres in the form of a rectangle, the length of 
 one side to be 10 chains. Find the adjacent side. 
 
 3. Find the sides of a rectangle which shall contain 15 acres, 
 and the length ^ the breadth. 
 
 4. It is required to lay out a rectangle containing 12 acres, 
 so that the sum of two adjacent sides shall equal 26 chains. 
 What must be the length and breadth ? 
 
 5. Find the sides of a rectangle which shall contain 640 
 square rods, and the difference of whose sides is 10 rods. 
 
 PARALLELOGRAMS. 
 
 265. To lay out a given quantity of land in the form of a 
 parallelogram, the base being given. 
 
 Denote the area and base, as above, and the altitude by x ; then 
 A 
 
 T 
 
 From any point in the base erect a perpendicular equal to 
 A -j- 6, and through the extremity of the perpendicular run a line 
 parallel and equal to the base: a parallelogram will thus be 
 formed, fulfilling the conditions of the question.
 
 226 PLANE SURVEYING. 
 
 266. Given the area, one side, and adjacent angle. 
 Denote the area by A, the base by 6, the given angle by a, 
 p M and by x the side adjacent ; then 
 
 bx sin a = A. ; 
 A 
 
 whence 
 
 b sin i 
 
 L N 
 
 Turn off at L and N, the given angle, measure the dis- 
 tances LP and NM, equal x, and connect M and P for the de- 
 sired figure. 
 
 267'. Given the area and two adjacent sides, to find the included 
 angle. 
 
 Denote the sides by b and c, their included angle by a, and 
 the area as above ; then 
 
 be sin a = A ; 
 
 * 
 
 whence sin a = 
 
 be 
 
 QUERIES. What will the figure become when be = A? 
 When b = c? May the product of be be less than A? Can an 
 expression for the sine be obtained for each case ? 
 
 EXAMPLES. 
 
 1. It is required to lay out a parallelogram to contain 200 
 square rods, having a base of 20 rods. What must be the 
 altitude ? 
 
 2. If in Example 1 it is required that the perpendicular shall 
 
 M be erected at the middle of the base, 
 and terminate at the angle P, as 
 per figure, what length must be 
 given LP, and what the magnitude 
 
 -J of the angle L? 
 
 3. It is required to lay out a parallelogram to contain 48 
 square chains, one side to be 8 chains, and the adjacent angle 
 70. What must be the length of the adjacent side?
 
 LAYING OUT AND DIVIDING LAND. 227 
 
 4. It is required to lay out a parallelogram to contain 2.4 
 acres, the base and adjacent side to be respectively 6 and 5 
 chains. Determine the altitude and tell how to lay out the land. 
 
 5. It is required to lay out a rhombus to contain 32 square 
 chains, each side to be 6 chains. Compute the altitude, and 
 state how to set out the tract ; that is, to establish every corner. 
 
 C. POLYGONS. 
 
 268. To lay out a given quantity of land in the form of a 
 regular polygon of any number of sides. 
 
 Denote the area by A, the number of the sides by n, 
 and the length of one of the sides, as PN in the p L ff 
 figure, by #, and ON, the radius of the circum- 
 scribed circle, by y ; then 
 
 "n X OL X LN= A. 
 
 and 
 
 But the angle LON=, OL= cot 
 
 Whence x = 
 
 and y = ^ 
 
 n 
 
 To lay out the tract, find by the above formula the length of one 
 side, as LN, and stake it out. Then with an instrument for meas- 
 uring angles (transit) set up at one end, as N, sight L, plunge 
 
 the telescope, deflect - to M. Measure NM= NL. Remove 
 
 n 
 
 the instrument to M, deflect from the prolongation of MN, as 
 before, , measure MP, and so continue around, locating PQ,
 
 228 PLANE SURVEYING. 
 
 and finally returning to L. The figure will be the polygon 
 required. 
 
 In a small polygon, if the centre 
 is fixed, it will be better to set up 
 on it and measure therefrom a dis- 
 tance y to N, turn off an angle (the 
 
 instrument still at the centre) = , 
 
 and measure the same distance to M, 
 again turning off an angle equal to 
 N L the last, measure the same distance 
 
 to P, and so on. A stake planted at each extremity of the 
 radial lines will indicate the angular points of the tract.* 
 
 EXAMPLES. 
 
 1. Show how to laj* out 1210 square yards in the form of an 
 octagon. The same for a pentagon ; decagon. 
 
 2. Show how by Article 57 the length of a side of a polygon 
 of a give-n area and any number of sides, within the limits of 
 the table, may be found. 
 
 D. CIRCLES AND ELLIPSES. 
 CIRCLES. 
 
 269. To lay out a given quantity of land in the form of a 
 circle. 
 
 Denote the area by A, and the radius by x ; then, since 
 
 rj 
 
 * A small lot, when great accuracy is not required, may be laid out 
 by fastening one end of a tape at 0, and with a length ON mark out a 
 circumference by means of a pin. Then, beginning at any point in the 
 circumference, measure off the distance x, and continue round the curve. 
 driving a stake at the extremity of each side.
 
 LAYING OUT AND DIVIDING LAND. 229 
 
 When great accuracy is not required, and small circles gener- 
 ally may be laid out by fastening one end of a tape at the 
 centre, and with a common marking-pin held firmly and perpen- 
 dicularly along it at x distance, describe and mark out the 
 circumference. 
 
 270. Or, fix the extremities of two diameters run out per- 
 pendicular to each other, connect these with chords, and the 
 versed sine of 45 to the known radius will give at once the 
 perpendicular distance from the centre of each chord to the cir- 
 cumference. If necessary, the points thus located may be 
 connected and others found in a similar manner. Or the per- 
 pendicular distance from any given point in a chord, of known 
 length, to the circumference may be found by simple geomet- 
 rical truths deduced from the right triangle. 
 
 271. If the circle is too large to be laid out as above, it may 
 be accomplished by means of deflec- L 
 
 tion angles as follows : With the 
 known radius find the angle at the 
 centre (7, which is subtended by a 
 chord OM of any length, say 100 
 feet ; then with the instrument at M, 
 deflect from the tangent ML to, an 
 angle LMO = one-half the central 
 angle OCM, and measure the distance 
 3/0= 100 feet. is a point in the 
 curve.* Again deflect an angle OMP = one-half the central 
 angle, and measure OP = 100 feet to locate /*, another point in 
 curve,* and so on to locate the others. If there is a fractional 
 part of the deflection angle at the closing point, the correspond- 
 ing fractional part of 100 feet may be used. 
 
 * The angle formed by the tangent and chord drawn to the point of 
 contact is measured by one-half the intercepted arc. An inscribed angle 
 has the same measure.
 
 230 
 
 PLANE SURVEYING. 
 
 ELLIPSES. 
 
 272. To lay out a given quantity of land in the form of an 
 ellipse, the greater and lesser diameters to bs in a given ratio. 
 
 Denote the area by A, the greater and less diameter (axes) 
 respectively by mx and nx, in which m and n express the given 
 ratio; then,. 
 
 '-> 
 
 and 
 
 273. Given the area and one of the diameters, to find the 
 other diameter. 
 
 Denote the given diameter by (Z, the unknown by x, and the 
 area as before ; then, since 
 
 we have 
 
 
 An ellipse of small size may be laid 
 out as follows : 
 
 Measure AB equal to the greater diameter (transverse axis)_, 
 and from the centre lay off OF= OF', each equal to the 
 square root of the difference of the squares of the semi- 
 diameters OA, OC. Fix the ends of a steel wire or ribbon of 
 the length AB at F and F 1 ', and with a continuous motion of a 
 marking-pin P, held perpendicularly, keeping the wire taut, the 
 required curve will be traced. 
 
 * See any work on General Geometry or Conic Sections for the area of 
 an ellipse.
 
 LAYING OUT AND DIVIDING LAND. 231 
 
 Or, having found the axis as above, P being any point in 
 the curve, and PR perpendicular to AE at R, by setting off 
 any number of points on AB, we may find from the proportion 
 
 TR 2 : RB x AU = OC* : Od 2 , 
 the corresponding values of PR. 
 
 EXAMPLES. 
 
 1. Find the radius of a circle containing 1 acre. 
 
 2. Find the radius of a sector containing 20 square rods, 
 the angle at the centre being 72. 
 
 3. The area of an ellipse is 1 acre, its diameters in the 
 ratio of 3 : 2 ; find their length. 
 
 4. An ellipse contains 80 square rods, its greater diam- 
 eter 12 rods ; find the lesser diameter. 
 
 5. The greater diameter of an elliptical plot of ground en- 
 closed by a wall 1 foot thick is 240 links, and the lesser 160 
 links, inside measurements. What is the area of the plot, and 
 how much land is occupied by the wall ? 
 
 274. Let it be required to lay out a circle circumscribing a 
 triangle, the sides of which are m, ?i, and p. 
 
 Let be the centre of the circle, 
 R the radius, OL a perpendicular to 
 MN, p = MN, and the other sides as in- 
 dicated in the figure. 
 
 Now NL = |, and angle NOL = P. 
 .-. = 12 sin P, 
 
 P 
 
 R = suTP = 2 sin P 
 
 To find an expression for R in terms of the three sides, sub 
 stitute for sin P its value
 
 232 PLANE SURVEYING. 
 
 2 sin *P 
 whence fl = 
 
 s - m) (is - ) (|s -p) 
 in which s represents the sum of the sides of the triangle. 
 
 ADDITIONAL EXAMPLES. 
 
 1. Circumscribe a circle about a triangle the sides of which 
 are 10, 15, and 20 chains. 
 
 2. Find an expression for the radius with which to inscribe 
 a circle in a triangle the sides of which are m, w, and^>. 
 
 Ans. Twice the area of the triangle, divided by the sum of 
 the sides. 
 
 3. Describe a circle in a triangle the sides of which are 
 30, 40, and 50 rods. 
 
 5. A circular walk, 6 feet wide, is to be made inside of a 
 square which contains \ an acre ; required the area of the walk. 
 
 5. The area of a square is 1 acre, and a circular walk is 
 required to be made in it, touching each side at a point, of such 
 a width that it will take up \ the area of the square. Find the 
 width of the walk and the length of its centre line. 
 
 6. The area of a circular sector of d is m rods ; find an 
 expression for the radius. If d = 60 and m = 300, find R. 
 
 SECTION II. 
 DIVIDING LAND. 
 
 A. TRIANGLES. 
 
 275. To divide a given triangle into two parts in the ratio of 
 m : n by a line parallel to one side.
 
 LAYING OUT AND DIVIDING LAND. 233 
 
 To solve the problem fully, and furnish a check on the work, 
 requires the location of the point or 72, and the length of 
 OR. Denote OR by x, OP by ?/, and by p and 7c the sides respec- 
 tively opposite the angles P and K; then 
 p 2 : x 2 = ra + n : ra, 
 
 Again, ft 2 : y 2 = m + n : 
 
 whence 
 
 If the triangle is to be equally divided, then m = n, and there 
 results 
 
 x _p and _k r 
 
 QUERIES. Is it necessary that LK be known to find either 
 PO or PR? Must LKbe given to find OR? 
 
 EXAMPLES. 
 
 1 . Find a general expression for the distance RK (last figure) . 
 
 2. Show how to divide the triangle LKP into four equiva- 
 lent parts by lines parallel to the base. 
 
 276. To divide a given triangle into two parts in the ratio 
 of m: n by a line from a vertex to the opposite side. 
 
 Let PO be the line, x = LO, and p as above. Then, since tri- 
 angles having the same altitude are to each P 
 other as their bases, we have 
 
 p : x = m + n : n ; 
 
 whence x = pn 
 
 m + n 
 
 EXAMPLES. 
 
 1. Locate on the supposition that the triangle is to be 
 divided into two equivalent parts.
 
 234 PLANE SURVEYING. 
 
 2. Find where the lines from P will meet the base dividing 
 the triangle into three equivalent parts. 
 
 3. The same for any number n parts. 
 
 277. To divide a given triangle into two parts in the ratio of 
 m : n by a line through a given point in one of the sides. 
 
 Denoting PL by x, and the other sides in 
 the usual manner, we have 
 
 m + n : m = Jco : px ; 
 
 - , triko 
 
 p- ----- _L whence x = - 
 
 p(m + n) 
 
 If the parts are to be equivalent, m = ?i, and there results 
 
 EXAMPLE. 
 
 Show how the given triangle LKO may be divided into three 
 equivalent parts by lines radiating from a given point R. 
 
 NOTE. The lines may or may not fall on the same side. Examine both 
 cases. 
 
 278. The same conditions as in the last case, except the tri- 
 angle is to be isosceles. 
 
 Using the same notation and figure as in that case, we have 
 the following equality of ratios : 
 
 m + n : m = ko : x 2 ; 
 
 whence x = J*<L. 
 
 \ra + 
 
 If the parts are to be equivalent, m = ?t, and we have 
 
 EXAMPLE. 
 
 Show how to cut off a given area, in the form of an isosceles 
 triangle, from the corner of a field, only the angle being given.
 
 LAYING OUT AND DIVIDING LAND. 235 
 
 279. The bearings of two sides of a field being given, to cut ojff 
 a triangle having a given area by a line running in a given direc- 
 tion and intersecting the given sides. 
 
 a. Suppose the division line is to make a right angle with 
 either side. Let LO and LQ be 
 the sides, the bearings of which are 
 known, and PR the division line per- 
 pendicular to LQ. The angle L be- 
 comes known through the bearings of 
 the sides which include it, and there 
 follows 
 
 p tan L = PR = L 
 
 But \pl = area = A ; 
 
 hence \p* tan L = A, 
 
 and p = 
 
 tan L 
 
 b. Suppose the angle at R is oblique. Denote LR by or, and 
 LP by //, and find from the bearings the angles at P and R. 
 Then from the two equations, 
 
 %xy sin L = A 
 
 x sin P 
 
 and - = -- - 
 
 y sin R 
 
 may be deduced 
 
 "ilsinZ/ sinP 
 
 QUERY. Is it necessary that the bearings of LO and LQ be 
 given if the field is triangular and the lengths of the sides given? 
 
 EXAMPLES. 
 
 1. The bearing of LO (last figure) is N. 50 E., and LQ 
 S. 82 E. It is required to find the lengths of LR and PR 
 perpendicular thereto, so that 3 acres may be contained in the 
 triangle PLR.
 
 236 PLANE SURVEYING. 
 
 2. Suppose LO = 10, LQ = 8, and OQ = 6 chains. Find 
 the position and length of the division line PE, which, with an 
 angle PEL = 84, will cut off a triangle PEL containing 1.5 
 acres. 
 
 3. Show that if three lines be drawn connecting the middle 
 points of the three sides of a triangle, the four triangles thus 
 formed will be equal. 
 
 280. To divide in a given ratio a given triangle by a line pass- 
 ing through a given point within it. 
 
 Let OQE represent the given triangle, and P the point with- 
 in ; DL the required division line, 
 and DEL : LDOQ = m : n. 
 
 The point P may be located by co- 
 ordinates as PFand PE, lines parallel 
 respectively to OE and QE ; or by its 
 bearing and distance from one of the 
 corners, as E ; or by perpendicular 
 distances PF', PE' from the sides. 
 The distances PF and PE may be 
 
 calculated if the direction and distance PE be known. Denote 
 PF by d, PE by b, DE by x, and EL by y ; then 
 
 x:y = d:y b, or xy = bx + dy ; 
 
 mqo 
 and xy : qo = m : m-\-n, or xy = 
 
 mqo 
 hence bx-\-dy = - 
 
 Or, substituting the value of y= m( * from equation above. 
 
 (*+.) 
 we obtain 
 
 , dmqo mqo 
 
 ox -f- i 
 
 (m + n)x m -f- n 
 
 whence, by reducing and completing the square, there results 
 _ mqo Vm 2 g 2 o 2 4 bdmqo(m + n)
 
 LAYING OUT AND DIVIDING LAND. 237 
 
 2bmqo 
 
 mqo Vm 2 g 2 o 2 4 bdmqo (m + n) 
 
 If the question were to cut off from a corner of a tract of land 
 a given area, by a line passing through a given point within, 
 we might proceed more simply, as follows : 
 
 Denote the area to be cut off by A, and the other notation 
 as alove ; then 
 
 xy sin R = 2 A, 
 
 and x:y = d:y b; 
 
 whence there results 
 
 ' x = A_ -VA* - 2 Abd sin R 
 
 y 
 
 b siu R 
 2Ab 
 
 In each of the two preceding problems there are in general 
 two division lines, as indicated by the double sign, fulfilling the 
 conditions of the question. The student will point out when, 
 if ever, one of these results will not practically answer the first 
 case. Would either result answer practically the second? 
 When, if ever, would the result be imaginary? Why? 
 
 If P were located by its distance PR, and the angle PRL or 
 PRD, the lines PF and PE could be calculated, as before re- 
 marked, and the solution above given made applicable ; or we 
 may proceed as follows : 
 
 Denote PR by d, d sin PRD by 6, d sin PRL by c, and the 
 other notation as above ; then 
 
 Substituting the value of y from the first equation in the 
 second, and reducing, there results, 
 
 a a_2Ax_ 2cA 
 
 b ' bs'mR'
 
 PLANE SURVEYING. 
 
 whence 
 
 or. 
 
 jin 5 
 
 2A 
 
 As\nE 
 
 EXAMPLES. 
 
 1 . Given the three sides of a triangular tract of land (see 
 last figure), QR = 17, OQ = 19, and OR = 22 chains, to divide 
 it into two equivalent parts by a line passing through a point P, 
 within the field. PF and PE = respectively 4 and 9.50 
 chains. The location and length of the division line are re- 
 quired. 
 
 2. It is required to cut off from the angle 0, which is 60, a 
 triangular field to contain 10 acres, by a line DL passing through 
 a point P. The distances PF and PE being 4 and 12 chains 
 respectively, the location and length of the division line are 
 required. 
 
 3. Given the angle ORQ=56 (see last figure but one), 
 PRL = 2Q, and PR= 12 chains. It is required to cut off a
 
 LAYING OUT AND DIVIDING LAND. 239 
 
 triangle DRL, containing 8 acres, by a line DL passing through 
 the point P. The location and length of the division line are 
 required. 
 
 4. Divide a triangular piece of land into three equal parts 
 by lines radiating from a point within. 
 
 SUGGESTION. The locus of the vertices of all triangles having 
 the base LN and one-third the area 
 of LMN is a line parallel to LN and 
 at | the height PM. Similarly for 
 any other side. Find point of inter- 
 section. P 
 
 5. Apply the principle employed in Example 4 to divide a 
 triangle into three parts, in the ratio of 1, 2, and 3, by lines 
 radiating from a point within. 
 
 6. Given two sides of a triangle 6 and 8 chains ; it is required 
 to locate a division line which shall cut off from the vertex an 
 isosceles triangle whose area shall be to the area of the given 
 triangle as 3 : 4. 
 
 7. Given the sides of a triangle 8, 10, and 12 chains; it is 
 required to divide it into a triangle and a trapezium, the ratio 
 of the former to the latter as 2 : 3, by a line extending from the 
 middle of the longest side to some point on the medium side. 
 
 The location of this point and the length of the division line 
 are required. 
 
 8. Divide the triangle given in Example 7 into three equiva- 
 lent parts by lines radiating from the middle of the longest 
 side. Locate the extremities of the division lines. 
 
 9. An angle QOP of a field = 42 30' ; it is required to cut 
 off from some point D, in the line OP, by a line DL, making 
 an angle LDO = 78 30', a triangle containing 2 acres. 
 Locate the division line, and determine its length. 
 
 10. The sides of a triangle are 16, 18, and 24 chains; it is 
 required to divide it into two parts in the ratio of 2:3 by a 
 line perpendicular to the longest side. Locate the division line, 
 and determine its length.
 
 240 PLANE SURVEYING. 
 
 281. To divide a given triangle in a given ratio by a line 
 passing through a given point without it. 
 
 Let ORQ represent the triangle, P 
 the point given by the angle POQ and 
 distance OP, DL the line which 
 shall divide the triangle, so that 
 
 ODL : DLRQ = m:n. 
 Denote OP by 6, OL by x, OD by y, 
 * the angle DOL by 0, the angle POD 
 
 by 0', and the -J^ part of the area by A ; then 
 
 m + n 
 
 ^xy sin O= A; (1) 
 
 also }by sin 0' = area POD, (2) 
 
 and i bx sin ( + 0') = area P0. (3) 
 
 ) $bysmO' = A. (4) 
 
 Substituting in the last equation the value of y taken from 
 (1) and reducing, there results, 
 
 or, 
 whence 
 
 / 
 
 a; sin 
 
 2 Ax 2 A sin 0' 
 
 b sin (0 + 0') sin sin (0+0') 
 A 
 
 b sin (0+0') 
 
 x 
 
 2ylsin 0' 
 
 sin sin (0+0') 6 2 sin 2 (0+0') 
 
 y may be found by substituting the value thus obtained for 
 x, and thence the length of the division line DL. 
 
 EXAMPLES. 
 
 Given, in the triangle OQR, OR = 18.40 chains, RQ = 10.20 
 chains, Q0=20.60 chains, OP =9. 50 chains, and the angle
 
 LAYING OUT AND DIVIDING LAND. 
 
 241 
 
 POQ = 28 30', to divide the triangle into two parts so that 
 OLD : DLRQ = 3:4. The position and length of the division 
 line DL are required. 
 
 B. QUADRILATERALS. 
 
 TRAPEZOIDS. 
 
 282. Given the parallel sides of a trapezoid and the perpen- 
 dicular distance between them, to divide it by a line parallel to these 
 sides into tivo parts having a given ratio. 
 
 Q R 
 
 T 
 
 (c) 
 
 Let OPQR (Fig. c) be the trapezoid, the sides OP, RQ, 
 and the perpendicular distance QT between the bases being 
 given. It is required to divide it by a line Z)Z, so that 
 OPLD : DLRQ = m : n ; that is, practically to locate and de- 
 termine the length of the division line DL. 
 
 Denote the lower base by 6, the upper base by &', the perpen- 
 dicular distance between the bases by /*, the perpendicular dis- 
 tance between the upper base and division line by a;, the length 
 of the division line by y, and the area OPQR by A. Draw QV 
 parallel to RP; then the similar triangles give 
 
 OV:DK= QT-.QF.
 
 242 PLANE SURVEYING. 
 
 Or, OP - QR : DL - QR = QT: QF. 
 
 Or, substituting proper values, 
 
 b b' : y b' = h : x ; 
 - 6') h 
 
 whence 
 
 b b 
 
 But the area of DLQR = (y + &') = -^ 
 
 2 m + n 
 
 Representing for convenience the right-hand member of the 
 last equation by A', we may write 
 
 xy + b'x = 2 A', 
 
 and y = -V. (2) 
 
 Substituting the value of x from (1) in (2) and reducing, 
 there results 
 
 and 
 
 b b' 
 Restoring the value of A', we obtain, 
 
 _ vh J-2L- 2 Ah (b - b') + 6' 2 h 2 
 \ m + n 
 
 The student may indicate how he would trace out on the field 
 the division line thus found. 
 
 283. If instead of the perpendicular distance there be given 
 one of the sloping sides, as OQ (Fig. c). 
 
 Denote OQ by cZ, OD by x, and the other notation as above. 
 Produce the sides until they meet in some point E ; then
 
 LAYING OUT AND DIVIDING LAND. 243 
 
 OPE i 
 DLE: QRE = y 2 :b 12 ; 
 
 or, by division, OPQR : QRE = b 2 - b' 2 : 6' 2 , 
 and DLQR:QRE = y 2 -b' 2 :b'*-, 
 
 whence OPQR : DLQR= b 2 - b 12 : y 2 -V*. 
 
 By division OPLD : DL QR = 6 2 - y 2 : f - 6' 2 ; 
 inserting values, m : n = b 2 y' 2 : y 2 b' 2 ; 
 
 whence 
 
 m -{-n 
 The similar triangles OVQ and QDKgive 
 
 b-b':y-b' = d:d-x. 
 _d(b-y) : 
 
 \b 2 n+V 2 m 
 
 I 
 J 
 
 In Figure d, the unknown sides are symmetrical with respect 
 to :i line joining the centres of the parallel sides ; in Figure e, 
 PR is perpendicular to the parallel sides. The student will 
 show what modification, if anv, may be made in the formulas 
 of the two preceding cases for either of these. 
 
 EXAMPLES. 
 
 1. Given OP= 20 chains, QR= 15 chains, QT= 18 chains, 
 to find the length of the division line DL, so that QRLD shall 
 contain two-thirds as much land as OPLD. 
 
 2. In Figure d, whose sides are equally inclined to the bases, 
 OP= 24 chains, QR =10 chains, and the perpendicular distance 
 QT=20 chains; it is required to locate the extremities of the 
 division line DL, and determine its length, so that it shall 
 divide the tract into two equivalent parts.
 
 244 PLANE SURVEYING. 
 
 3. In Figure e, suppose QR : OP : PR = 3 : 4 : 5, and that the 
 area= 1750 rods ; locate and find the length of the division line 
 DL that shall divide the tract, making OPLD : QRLD =3:4. 
 
 284. To divide a given trapezoid into two parts having a 
 given ratio, by a line intersecting the parallel sides. 
 
 Let OPQR represent the trapezoid, and let it be required to 
 divide it into two equal parts. It is 
 evident if the bases be bisected, and 
 , i / \ a line, as DL, be drawn connecting the 
 ~^ points of division, it will be the division 
 line required. 
 
 Similarly, if the ratio is m : n ; denote 
 
 OP by b, and RQ by b' ; then take OL = _J^_ RD =-^-, 
 and join DL for the line required. 
 
 The student will give the reason. 
 
 If the division line is to pass through a given point D', ob- 
 tain DL as above directed, then measure from D to D', and 
 lay off this distance from L to L'. Join D'L' for the division 
 line required. Why? 
 
 To divide a trapezoid by a line perpendicular to the bases, 
 or parallel to one of the non-parallel sides, divide the line join- 
 ing the middle points of the non-pnrallel sides into two parts in 
 the given ratio, and through the point of division run the re- 
 quired line. If m : n is the ratio, and the bases b and &', the 
 
 distance TKm the last figure = w(& + 6>) . 
 2 (m + n) 
 The student will give the reason. 
 
 EXAMPLES. 
 
 1. Divide a given trapezoid into three equivalent parts by 
 lines intersecting the parallel sides. 
 
 2. Divide a given trapezoid into three parts in the ratio of 
 m : n :p, by lines intersecting the parallel sides.
 
 LAYING OUT AND DIVIDING LAND. 245 
 
 3. The bases of a trapezoid are, OP =20 chains, and QR = 
 15 chains. It is required to divide it into two parts in the ratio 
 of 2 : 3. OL' = 8.50 chains ; locate D'. 
 
 4. Show that /, being the centre of the line connecting the 
 middle of the bases of a trapezoid, is the point through which, 
 if any straight line be drawn meeting the parallel sides, it will 
 divide the trapezoid into two equivalent parts. 
 
 285. Given one side and the adjacent angles of a tract of land, 
 to cut off a trapezoid of a gicen area by a line parallel to the 
 given side. 
 
 Let PO be the given base, P and the known angles indi- 
 cating the direction of the p 
 sides PQ and OR. Denote 
 the area OPLD, to be cut 
 off by A ; the given side ..---" 
 OP by s, PD by y, OL by F "" : 
 ce, DL by z, and suppose 
 
 (0-f-P)<180. 
 
 Produce OR and PQ until they meet in F. 
 Then area OPV - area LDV = A ; 
 
 s 2 sin sin P z 2 sin sin P 
 
 2 
 whence z = \ ( 's 2 -- 
 
 M sin sin P 
 
 When (0 + P) > 180, the produced lines meet in a point on 
 the other side of OP, the sin (0 + P) is also negative, and 
 therefore the fraction under the radical becomes positive. 
 Draw LT parallel to VP; then in the triangle LOT, by sine pro- 
 portion, sin L (= sin F) : sin T ( = sin P) = s z : x ; 
 
 whence X=S 
 
 sinF 
 
 c,. ., _, (s z) sin 
 
 Similarly, y = i --
 
 246 
 
 PLANE SURVEYING. 
 
 REMARK. When great accuracy is not required, and espe- 
 cially if the tract is small and the sides nearly parallel, an 
 approximate perpendicular distance between the bases OP and 
 DL may be obtained by dividing the area to be cut off by the 
 given side OP; then measure the perpendicular and a line 
 through its extremity parallel to the base for an approximate 
 division line. Calculate the area thus cut off, divide the differ- 
 ence between it and the required area by the approximate divis- 
 ion line for a new perpendicular, and thence obtain more nearly 
 the division line sought. 
 
 EXAMPLES. 
 
 1. Deduce an expression for DL by another method. 
 
 2. Show by other methods how OL or PD may be determined. 
 
 3. Given OP, N. 16 30' W., 8.40 chains ; PQ, S. 62 15' W ; 
 and OR, S. 82 W., to cut off a trapezoid containing 4 acres, 
 by a line DL parallel to OP. The position and length of the 
 division line are required. 
 
 4. Given a side of a tract of land 20 chains, and the adja- 
 cent angles 105 and 130, to cut off 36 acres by a line parallel 
 to the given side. Required the position and length of the 
 division line. 
 
 TRAPEZIUMS. 
 
 286. Given the area of a trapezium, one of its sides and adja- 
 cent angles, to divide it by a line parallel to the given side into two 
 parts having the ratio m : n. 
 
 Produce the sides PQ and OR to meet in V. Let OP= s, 
 OL = x, PD = y, DL= z.
 
 LAYING OUT AND DIVIDING LAND. 247 
 
 Calculate the area of 
 
 ' 
 
 2 siuF 
 then A'-A" 
 
 j ,, , 2 2 sin sin P , A , A u\ 
 and the formula - - = 2( A 1 A) 
 sinF 
 
 gives )2sinF(.i'-^. 
 
 \ sin sin P 
 
 Having found 3, x and y ma^y be deduced as in the foregoing 
 case. 
 
 (s z) sin P 
 
 sinF 
 
 (s-z)sin 
 sinF 
 
 REMARK. This problem may be solved by Article 285, taking 
 for the given area to be cut off ' 
 
 EXAMPLE. 
 The boundaries of a trapezium are as follows : 
 
 (1) N. 2 E. 8.00 chains; 
 
 (2) N. 58 E. 13.85 " 
 
 (3) S. 3H E. 14.80 k ' 
 
 (4) S. 82^ W. 20.00 " 
 
 It is required to divide it into two equivalent parts by a line 
 parallel to the third side. Locate it, and determine its length. 
 
 287. Given the bearings of three adjacent sides of a tract of 
 land and the length of the middle one. to cut off trapezium 
 having a given Jtrat, by a line running in a given direction.
 
 248 PLANE SURVEYING. 
 
 Produce the sides PQ and OR till they meet at F. As 
 before, denote OP by s, OL by ar, PZ) by ?/, and .LZ> by z. Ob- 
 tain the angles from the bearings, calculate the area of 
 
 PQY = A' = ** sin sin f \ 
 2 sin F 
 
 and find area DL F= ^' - A. 
 
 R 
 
 L 
 
 
 
 Whence the division line DL = z may be found from the 
 formula 2 2 sinZ>siu Al 
 
 -ZL "~~ *}.) 
 
 2siuF 
 
 z = 
 
 sin D sin Zr 
 By the sine proportion 
 
 sin F 
 and FL = 
 
 whence VO-VL=LO = x = n-sn 
 
 sin F 
 and _ ssin zsin L 
 
 sin F 
 
 REMARK. If (0 + P) > 180, ^ f -^4 in the equation for z 
 will become A 1 -f- .4, and in the formulas for x and y the signs 
 in the numerators will be interchanged, or 
 
 z s * a -^ ~ 
 sin F 
 
 and y = zsin ^~' ss 
 
 sin F
 
 LAYING OUT AND DIVIDING LAND. 249 
 
 EXAMPLE. 
 
 Given LO, S. 76 E. ; OP, N. 8 W. 12.40 chains ; PD, S. 72 
 W. ; it is required to cut off 7 acres bv aline hearing N. 23 W. 
 The length of the division line and the distances OL and DP 
 are to be computed. 
 
 288. Given a trapezium, to divide it into two parts having a given 
 ratio, by a line extending from a given point in one of the sides. 
 
 Let OPQR represent the trapezium the area of which is A, 
 m and n the given ratio. Prolong the sides PQ and OR till 
 they meet in F. Let OR = v, the division line DL = z, RL the 
 given distance to the point L= d, and QD = y. Calculate the 
 area of QRV=A', and add it to - A, thereby obtaining 
 areaofZ)F. p 
 
 Find by the sine proportion VR, 
 and add it to RL, thus obtaining 
 VL. 
 
 Putting VD = x, and VL = i>, .- 
 
 V='2( - A+A\ 
 
 Whence x = VD may be found. 
 
 Finally, with the two sides VD and VL and the included 
 angle V, compute the angle L, and the direction and length of 
 the division line DL ; y may be calculated by a preceding method 
 to check the work. 
 
 EXAMPLES. 
 
 1. Given in a trapezium MNOP (no figure) : 
 MN, 13.00 chains ; 
 NO, 7.30 " 
 OP, 10.40 " 
 PM, 11.10 " 
 and diagonal PN, 13.70 " 
 
 It is required to divide it into two equivalent parts by a line 
 running from a point in the side MN, 6 chains from M. Find
 
 250 PLANE SURVEYING. 
 
 the length of the division line and locate the other extremity 
 of it. 
 
 2. Divide the tract described in Example 1 into two parts, in 
 the ratio of 3 : 4, by a line DL running from some point in MN. 
 and falling perpendicularly upon PO. The part PMDL is to be 
 the greater. Locate the line required, and determine its length. 
 
 289. Given a trapezium, to divide it into two parts having a 
 given ratio, by a line passing through a given point within the 
 tract. 
 
 Let OPQR represent the given trapezium T, the point within 
 it, given by its bearing and dis- 
 tance from some angle, as R. 
 Produce the sides OR and PQ to 
 meet in V. Denote the ratio by 
 ra and n, the area OPQR by A, 
 QR by v, DL by z, VL by , and 
 VI) by y. Find by the sine pro- 
 portion 
 
 y R _ vsinQ V Q _ v sin R 
 sin V" sin F' 
 
 and thence the area VQR = A'. Then in the triangle VRT, 
 having two sides and the included angle, compute FT, which 
 call b, and the angle TVR = a. Putting F a = ft, and 
 - A + A' = A", the following equations may be written : 
 
 xysinV=2A", (1) 
 
 and bx sin a + by sin ft= 2 A". (2) 
 
 Substituting in (2) the value of y from (1), and reducing, 
 there results, 
 
 x = A " I A " 2 2 A" sin ft ^ 
 b sin a. \ b~ sin 2 a sin a sin F 
 
 *rtEL=x-VR=-^- J A " 2 _*A'*infl_vunQ 
 b sin a \ b 2 sin 2 a sin a sin F sin F
 
 LAYING OUT AND DIVIDING LAND. 251 
 
 EXAMPLE. 
 Given the boundaries of a trapezium as follows : 
 
 (1) N. 16J W. 24. 63 chains; 
 
 (2) S. 79 W. 27.00 " 
 
 (3) S. iW. 34.28 
 
 (4) N. 65 E. 37.20 " 
 
 To divide it into two equivalent parts by a line extending 
 from the first to the third side, and passing through a point 
 20 chains distant from the first and second corners. Locate 
 the line and find its length. 
 
 290. Given a trapezium, to divide it into two parts having a 
 given ratio, by a line passing through a given point without the 
 tract. 
 
 P 
 
 Let OQRT represent the trapezium given by the bearings 
 and distances of its sides, P the point without, located by its 
 bearing and distance from T, the ratio m : n. Extend the sides 
 RT and QO until they meet in V. Then the problem may be 
 solved in a similar manner to that in Article 281. 
 
 291. Given a trapezium, to divide it into four equivalent 
 parts, by two lines intersecting opposite sides, one of the division 
 lines being parallel to one of the given sides of the tract. 
 
 Let ABCD represent the given trapezium, FE the division 
 line parallel to DC, and GH the other division line. It is re-
 
 252 
 
 PLANE SURVEYING. 
 
 quired to locate both division lines. Prolong the sides AD and 
 BCto meet in P; also DC and AB to Q. Find AE and EF 
 by methods already given. 
 
 Now, any line cutting the parallel sides of a trapezoid and 
 dividing it into two equivalent parts must pass through a point 
 (the middle of the middle line between the bases). See 
 Article 284. Hence MO becomes known = 1 (CD + EF), and 
 
 P Br- 
 
 aise MC=\FC. In the triangle OMC, compute the angle 
 MCO and the line 0(7; add Z MCO to Z MCQ, and having 
 previously calculated QC, find in the triangle QCO the angle 
 CQO and the side QO. Subtract Z CQO from Z CQB and 
 obtain Z OQ-B. Then putting the side QO = a, QH=x, and 
 Q(? = y, we may write the following equations : 
 
 ay sin HQG = 2 area HQG, 
 
 ax sin CQO + ay sin OQG = 2 area #Q#. 
 
 From these equations obtain iy. Subtract it from AQ, found 
 by sine proportion, and the distance from the corner A to the 
 extremity of the division line GH at G will be the result. 
 
 Then in the triangle QGH find QH', whence the length and 
 bearing of GH may be computed.
 
 LAYING OUT AND DIVIDING LAND. 253 
 
 EXAMPLE. 
 
 It is required to divide the farm described in 288 (Example 1 ) 
 into four equivalent parts by two lines intersecting opposite 
 sides ; one of the division lines is to be parallel to the first side. 
 Locate the division lines, and determine their lengths. 
 
 C. POLYGONS. 
 
 292. Given a polygon, to divide it into two parts having a 
 given ratio, or to cut off a given area, by a line through a given 
 point. 
 
 Let OPQRTV 'represent the polygon given by its bear- 
 ings and distances, or angles and sides, v 
 
 and suppose the line be required to 
 run from P, either an angle or any 
 given point in a side. Calculate the 
 
 area of the polygon , and take the 
 
 m + n 
 
 part of it as the area to be cut off to 
 the right of the line extending from P. Q -B 
 
 Run a trial line* from P as PT, calculate the area of PQET, 
 and determine whether the area thus cut off is too small or too 
 large, and how much. Suppose it is too small ; then the extrem- 
 ity T of the division line PT must be moved towards V to 
 some point T 1 . To find this point, denote TT 1 by x, the angle 
 TTP by T, the distance PTby b, and the area of the triangle 
 PTT by a ; then 
 
 bx sin T = a, 
 
 from which we find x = 
 b sin T 
 
 * The bearing and distance of PT may be calculated from the data 
 given without a trial line as in supplying omissions. If, however, 
 this is done, the surveyor should not omit to measure the division line to 
 verify his work. In fact, it is the best practice, no matter what method is 
 adopted to obtain the division line, to always test the computation by 
 measurement.
 
 254 PLANE SURVEYING. 
 
 This distance measured from T to T will locate T', a 
 point which connected with P will give the division line 
 sought. 
 
 293. Given a polygon, to divide it into two parts in a given 
 ratio, or to cut off a given area, by a line through a given point 
 within the tract. 
 
 Let the marginal figure represent the tract, Pthe given point. 
 If the area to be cut off is not di- 
 rectly given, calculate the contents of 
 the tract, and then by the ratio deter- 
 mine the quantity to be cut off, and 
 denote it by A. Run a trial line TT 
 through P, dividing the polygon as 
 nearly as may be judged in the re- 
 quired manner. Measure TP= b, PT' 
 = c, and the angles T and T'. Calcu- 
 late the area of either part of the polygon, and thus ascertain 
 whether T should approach or recede from 0. Suppose the 
 area TNMVT' is calculated and found too small by a quan- 
 tity a, and that DL represents the division line. Put DP= x, 
 PL = y, the angle PTD = T, PT'L = T', and the angle at the 
 point P = P, which is required, since that will indicate the di- 
 rection of the division line. 
 
 Then cy sin P % bx sin P = a (1) 
 
 =-^^W (2) 
 
 (3) 
 
 sin (T' + P) 
 
 Substituting the values of x and y from (2) and (3) in (1), 
 there results 
 
 c 2 sin T' sin P b 2 sin T sin P 
 
 in(7" + /') sin(T+P) 
 
 = 2 a. (4)
 
 LAYING OUT AND DIVIDING LAND. 256 
 
 Expanding the denominators, dividing each fraction, numer- 
 ator and denominator, by its numerator, and writing for t 
 the cot, there results sin 
 
 fS 7,2 
 
 . (5) 
 
 cot P + cot T' cot P + cot T 
 
 Putting cotP = ^>, cot!T=, and cot7" = ', we may write 
 more simply : 
 
 / c*-l>*\ tc* W 
 
 (t + f - -^-) = ^ ~ tf ; 
 
 whence - 
 
 
 2a 
 Restoring values, we have 
 
 cotP= -f cot T+cot T' 
 
 2a ) 
 
 _ cot T cot r+ 
 
 \ 2 a 4\ 2a 
 
 The problem may be simplified when it is practicable to run 
 the trial line at right angles to one of the sides of the polygon. 
 In the tract given, suppose TT' to be run perpendicularly to 
 R F; then cot T' = 0, and Equation (5) may be written 
 
 * v 
 
 cot P cot P + cot T 
 
 & 
 
 
 4V 2o
 
 256 
 
 PLANE SURVEYING. 
 
 294. Given a polygon, to cut off a given area by a line passing 
 through a given point without the tract. 
 
 Let the marginal figure represent the case. 
 P h y M As in the preceding article, run 
 
 a trial line PT' from P, and sup- 
 pose it is made perpendicular to 
 v RV. Calculate, as before, the con- 
 tent of TNMVT', and ascertain 
 the amount to be added to make 
 the required area. Denote, as 
 before, this area by a, PT=b, 
 the angles at P, T, etc., by P, T, 
 
 R 
 
 PT' = c, PD = x, PL = y 
 etc., and DL the division line ; then 
 
 ^ cy sin P | bx sin P = a, 
 6 sin T 
 
 (1) 
 (*) 
 
 Substituting the values of x and y from (2) and (3) in (1), 
 and reducing as in the preceding problem, there results 
 
 cot,P=- 
 
 The student may verify the value found. 
 
 295. Given a polygon, to divide it into three parts having a 
 given ratio, by lines radiating from a given point. 
 
 a. Let the figure represent the polygon, 
 and suppose the point is in one side at 
 P. Calculate the area of the whole tract 
 and ascertain how much each division is 
 to contain ; then, by Article 292, cut off 
 the required areas PVTSL and POQD, 
 and the problem, is solved.
 
 LAYING OUT AND DIVIDING LAND. 
 
 257 
 
 6. If the point is within the tract,* cut off, by Article 293, 
 one required portion DVTSKD by a line DK through P, and 
 by the preceding article divide the remainder by the line PL as 
 required. If PL cuts off a quadrilateral on either side, Article 
 288 may be used. 
 
 c. If the point is without,* proceed, as in Article 294, to cut 
 off the required portion KVTSDK and HOQLH; the remain- 
 der HLRDKHvt\\\ be the third portion. 
 
 It is evident that this principle may be extended to any 
 number of parts. 
 
 296. To cut off from a given polygon a given area by a line 
 running in a given direction. 
 
 Let the figure represent a tract which it is required to divide 
 into two equivalent parts by a line DL parallel to RS. 
 
 Join QT, calculate its length 
 and bearing, and also the content 
 of QRST. Subtract said content 
 from one-half the area of the whole 
 tract, thereby obtaining the area 
 DLTQ. Then, by Article 287, the 
 length and position of the division 
 line may be determined. It is 
 evident that this principle may be extended to any number of 
 subdivisions. 
 
 * Calculate the area, and ascertain, by the ratio, how much eacb 
 division is to contain.
 
 258 
 
 PLANE SURVEYING. 
 
 EXAMPLES. 
 
 1. The student may indicate how he would divide DQRSTL 
 into two equivalent parts by a line perpendicular to DL. 
 
 2. Show how to divide DPONMVL into two equivalent pails 
 by a line extending from the middle of DL. 
 
 3. Divide the farm described in Article 234, Example 4, into 
 two equivalent parts by a line running due east. 
 
 297. From a tract of land of which one or more of the boun- 
 dary lines is irregular, to cut off a given area. 
 
 L n Q Let OPQRT represent the tract 
 
 which it is required to divide into two 
 equal pails by a line DL parallel to PQ. 
 Survey the land, taking offsets along 
 RO, and calculate the area. Then 
 the problem may be solved by Article 
 285. 
 
 298. To Straighten Boundary Lines. It is sometimes re- 
 quired to substitute a straight line for an irregular or crooked 
 one between farms, and to leave the same quantitv of land as 
 before in each tract. Let ORQ be the line which it is required 
 
 to straighten by a line extend- 
 ing from 0, the bearings and 
 
 distances OR, RQ, and the bear- 
 ing of QT being known. Run a 
 trial line OP, noting the dis- 
 tances RK, OK, KP, and PQ, 
 and calculate the areas of the 
 V triangle ROK and PQK. If it 
 happens that the triangle ROK is equivalent to PQK, then OP 
 will represent the line sought. If, as is generally the case, 
 their areas are not equal, take the difference, and suppose in 
 this case PQK the less. The problem, then, is simply this : 
 Given one side, OP, of a triangle, and the direction of another,
 
 LAYING OFT AND DIVIDING LAND. 259 
 
 PT, to cut off a given area by a line OP*, to find the distance 
 PP. The solution is given in Article 257. 
 
 Otherwise, with the given bearings and distances calculate 
 the area of the triangle ORQ and the length and bearing of the 
 closing line OQ. Then, as before, having one side of a triangle, 
 the direction of another, and the area, find QP 1 and the bearing 
 and distance of OP. The work should be verified by actual 
 measurement of angle and distance. 
 
 EXAMPLE. 
 
 Given OR, N, 59 30' E. 10.60 chains; KQ, S. 70 15' E. 
 19.32 chains ; QT, N. 12 W., to find QP' and the bearing and 
 distance of a line OP' which will straighten the boundary. 
 
 MISCELLANEOUS EXAMPLES. 
 
 1. It is required to lay out a lot to contain one acre, and 
 having an equal frontage on two streets which intersect at an 
 angle of 84 40'. Locate the corners of the property. 
 
 2. From a square tract of land OPQR, which originally 
 contained 160 acres, the southwest 
 
 quarter was sold. It is required to find 
 the uniform width of a strip MNL VTS 
 which shall contain 40 acres. How many 
 rods of fencing will the tract require? 
 
 3. A rectangular tract of land 16.20 
 chains long, and 8. 60 chains wide, valued 
 at $200 per acre, is to be divided among 
 
 three persons so that the first shall have $1,000 worth of it ; the 
 second, $900 ; and the third, the remainder. Locate the points 
 of division on the long side. 
 
 4. The bearings of two sides of a triangle are OM, N 60 E., 
 and ON, S. 40 E. It is required to cut off from the corner 
 an isosceles triangle containing 16 acres. Locate and find the 
 length of the division line.
 
 260 PLANE SURVEYING. 
 
 5. There is a farm in the form of a trapezium the area of 
 which is given as 87.78 acres. The description of its boun- 
 daries is very much effaced ; all that is legible is as follows : 
 
 Beginning at the northwest corner, thence (1) S. 76 E. 
 (distance effaced) ; (2) S. 10 E., distance 25 chains; (3) S. 
 62 W. (distance effaced) ; (4) N. 6 W. (distance effaced). 
 
 It is required to perfect the description. 
 
 SUGGESTION. Prolong the second and fourth sides until they 
 meet, and calculate the area of the triangle exterior to the tract. 
 Add it to the given area, whence the length of the first side 
 may be readily computed ; the second and fourth sides may be 
 found easily by either of two methods. 
 
 6. Required the length of a chord which will cut off one-third 
 part of a circle whose radius is 100 feet. 
 
 SUGGESTION. Let 20 denote the central angle, and r the 
 
 radius, for convenience. Then ^-- sin 20 = Whence 
 
 180 2 3 
 
 may be obtained, and hence the chord. The angle will be 
 the same, no matter what the radius may be. 
 
 7. A trapezoidal field, the two parallel sides of which are 16 
 and 10 chains, and the perpendicular distance between them, 12 
 chains, is to be divided into two equivalent parts b3* a line 
 parallel to the given sides. It is required to determine the 
 length of the division line and locate its extremities, the sides 
 being equally inclined to the bases. 
 
 8. Given the sides of a triangle OR, 280 yards ; RQ, 200 
 yards ; OQ, 300 yards ; the distance from to a point P outside 
 the tract, 220 yards ; and the angle POQ, 20. It is required 
 to run the centre line of a straight road through P and across 
 the field, so as to divide the tract into two equal parts. Locate 
 the points where the road will cross the triangle. 
 
 9. Given the sides of an irregular pentagon, and the per- 
 pendicular distance to each from a point within. Show how to 
 divide the tracts into their equivalent parts. Also into three 
 parts, having the ratio m : n :p.
 
 LAYING OUT AND DIVIDING LAND. 261 
 
 10. Given in a trapezoid MNOP (no figure), PM= 38.50 
 chains ; MN, one of the parallel sides, 64.80 chains ; NO, 41 
 chains ; the angle M, 85 30 f ; and N, 75 40'. It is required to 
 divide the tract into two parts in the ratio of 2 : 3, by a line DL 
 parallel to the parallel sides. The part MNDL is to be the 
 greater. Find the length and location of the division line. 
 
 11. Given one side of a triangular field, 120 yards ; the angle 
 opposite, 20; and the ratio of the other two sides, 7:10. 
 Find the area. 
 
 12. Show that the area of a trapezium is equal to one-half 
 the product of its" diagonals, by the sine of the angle of their 
 intersection. 
 
 13. From a point within a triangular field, the sides of which 
 were equal, I measured the distances to the three angles, and 
 found them 12.5, 10, and 7.5 chains respectively; required 
 the area. 
 
 Ana. 12 A. 1 R. 23 P. 
 
 The above problem is given in Guinmere's Surveying, and by 
 some surveyors it is considered difficult. The following is an 
 outline of a solution ; the student will supply what is wanting : 
 
 With the given distances form the triangle ABC. On AB de- 
 scribe an equilateral triangle ABD ; join 
 CD by a right line, and on it describe 
 an equilateral triangle CDE. CDE is 
 the triangle in question, and B the 
 point within. For BC and BD are evi- 
 dently two of the measured distances, 
 and BE, it will be perceived, is the D 
 
 other, through the similarity and equality of the triangles ADC 
 and BDE. To find the area of CDE, compute the angle BAG, 
 whence the angle CAD becomes known ; now with the two 
 sides AC, AD, and the included angle CAD, CD is easily 
 determined, and hence the required area of the triangle CDE.
 
 CHAPTER V. 
 
 PLANE-TABLE SUEVEYING. 
 
 299. The Plane-Table, as its name indicates, is a table or 
 board which, being covered with paper, and having certain 
 appliances for levelling and sighting, enables the surveyor to 
 determine points and lines, and to delineate them on the paper 
 in their relative position. 
 
 It is used in "filling in" the details of topographical work, 
 and generally for the location of points where great accuracy 
 is not required, on account of the rapidity with which surveys 
 by it may be effected. 
 
 300. The Board, which is rectangular in shape, usually 
 24 by 30 inches, is made of pieces of well-seasoned wood joined 
 advantageously together to prevent warping, and is furnished 
 with rollers or clamps, by means of which the paper is kept 
 securely stretched upon it. 
 
 301. The Plumbing- Arm, which is pointed at one end, and 
 from the other a plummet is suspended, is used to determine 
 the point on the ground immediately under its representative 
 on the board, or vice versa. The lower part of it moves upon an 
 axis which has an index at its extremity, by means of which it 
 may be ascertained when the bob and point upon the table are 
 in the same vertical line. 
 
 302. The Tripod and its Head are similar to those of the 
 ordinary transit, though heavier. 
 
 A metallic plate, screwed fast to the table and having a solid 
 conical spindle projecting from its centre, affords the means of 
 attaching the head to the table.
 
 PLANE-TABLE, 
 As MADE BY HELLER & BRIGHTLY, PHILADELPHIA, PA.
 
 CHAPTER V. 
 
 PLANE-TABLE SURVEYING. 
 
 299. The Plane-Table, as its name indicates, is a table or 
 board which, being covered with paper, and having certain 
 appliances for levelling and sighting, enables the surveyor to 
 determine points and lines, and to delineate them on the paper 
 in their relative position. 
 
 It is used in "filling in" the details of topographical work, 
 and generally for the location of points where great accuracy 
 is not required, on account of the rapidity with which surveys 
 by it may be effected. 
 
 300. The Board, which is rectangular in shape, usually 
 24 by 30 inches, is made of pieces of well-seasoned wood joined 
 advantageously together to prevent warping, and is furnished 
 with rollers or clamps, by means of which the paper is kept 
 securely stretched upon it. 
 
 301. The Plumbing- Arm, which is pointed at one end, and 
 from the other a plummet is suspended, is used to determine 
 the point on the ground immediately under its representative 
 on the board, or vice versa. The lower part of it moves upon an 
 axis which has an index at its extremity, by means of which it 
 may be ascertained when the bob and point upon the table are 
 in the same vertical line. 
 
 302. The Tripod and its Head are similar to those of the 
 ordinary transit, though heavier. 
 
 A metallic plate, screwed fast to the table and having a solid 
 conical spindle projecting from its centre, affords the means of 
 attaching the head to the table.
 
 PLANE-TABLE, 
 As MADE BY HELLER & BRIGHTLY, PHILADELPHIA, PA,
 
 THE PLANE-TABLE. 265 
 
 The tripod-head admits of a slight lateral motion to the 
 board, and is provided with levelling-clamp and tangent- 
 screws similar to the common transit. 
 
 303. Of Alidades there are several kinds. One of the 
 best, however, for ordinarv purposes is indicated in the figure. 
 It consists of a brass ruler or straight edge about 22 
 inches long and two inches wide, from which rises a column 
 surmounted by a telescope. The power of the telescope at 
 least equals that of the common transit, and it is provided with 
 stadia wires, has an attached level, vertical arc, with the 
 necessary adjusting movements. It is set on the column so 
 that the line of collimation is in or near the same vertical plane 
 with the bevelled edge of the ruler. 
 
 A parallel ruler allowing a very slight deviation from this 
 plane is sometimes used, and the work is thereby facilitated. 
 A small level is placed on the top of the column, which serves 
 to indicate any unequal settling of the instrument. Two spirit 
 levels at right angles to each other are placed upon the table to 
 indicate when by the levelling-screws it is made horizontal ; or, 
 the levels are attached to the ruler of the alidade, one in the 
 longitudinal direction of the ruler, the other perpendicular 
 to it. 
 
 304. The Declinator is simply a box containing a magnetic 
 needle which has a range of 12 or 15 degrees on each side 
 of the zero. It is used in orienting the table ; that is, to place 
 a given point on the table over that on the ground which 
 it represents, and to cause a line of the paper to lie in the same 
 vertical plane, or parallel thereto, with its counterpart on the 
 ground. 
 
 Before the table is removed from its first position, or at the 
 time of drawing the first line of the survey, the declinator may 
 be placed upon it, and the needle allowed to rest at zero ; then 
 a pencil drawn alongside the box will trace a north and south 
 line, since the sides of the box are made parallel to the line of
 
 2(36 PLANE SURVEYING. 
 
 zeros.* When the table is oriented at any other station, the 
 declinator will give the same reading if placed along the same 
 line. 
 
 ADJUSTMENTS. 
 
 305. From the nature of the service in some sections of the 
 country, the plane-table is often necessarily subjected to rough 
 usage, and there is a constant liability to a disturbance of the 
 adjustments ; still, in careful hands, a well-made instrument 
 may be used under very unfavorable conditions for a long time 
 without being perceptibly affected. One should not fail, how- 
 ever, to make occasional examinations, and while at work, 
 if any difficulty be encountered which cannot otherwise be 
 accounted for, it should lead directly to a scrutiny of the 
 adjustments. 
 
 306. The Fiducial Edge of the Ruler. This should be a 
 true, straight edge. Place the ruler upon a smooth surface, 
 and draw a line along the edge, marking also the lines at the 
 ends of the ruler. Reverse the ruler, and place the opposite 
 ends upon the marked points, and again draw the line. If the 
 two lines coincide, no adjustment is necessary ; if not, the edge 
 must be made true. 
 
 There is one deviation from a straight line which, by a very 
 rare possibility, the edge of the ruler might assume, and yet 
 not be shown by the above test ; it is when a part is convex 
 and a part similarly situated at the other end concave in ex- 
 actly the same degree and proportion. In this case, on reversal, 
 a line drawn along the edge of the ruler would be coincident 
 with the other, though not a true right line ; this can be tested 
 by an exact straight edge. 
 
 307. The Level Attached to the Ruler. Place the instru- 
 ment in the middle of the table, and bring the bubble to the 
 centre by means of the levelling-screws of the table ; draw lines 
 
 * Any other bearing which may be read will answer the purpose.
 
 THE PLANE-TABLE. 
 
 267 
 
 along the edge and ends of the ruler upon the board to show its 
 exact position, then reverse 180. If the bubble remain cen- 
 tral, it is in adjustment; if not, correct it one-half by means 
 of the levelling-screws of the table, and the othei half by the 
 adjusting-screws attached to the level. This should be re- 
 peated until the bubble keeps its central position, whichever 
 way the ruler may be placed upon the table. This presupposes 
 the plane of the board to be true. If two levels are on the 
 rulers, they are examined and adjusted in a like manner. 
 
 Great care should be exercised in manipulation, lest the 
 table be disturbed. 
 
 308. Cause the line of sight to revolve in a vertical plane, 
 make the bubble of the level attached to the telescope read 
 zero when the line of sight is horizontal, and test the vernier 
 arc for index error, each as in the transit. 
 
 METHODS EMPLOYED IN PLANE-TABLE SURVEYING. 
 
 309. Points may be located with respect to one another by 
 either of four methods. In actual practice, however, a combi- 
 nation of some of them is frequently employed. 
 
 310. By Radiation. Suppose it is required to make a plot 
 of a field KLMNO, all the corners of which can be SCCMI from 
 a point P within it. Place 
 
 the instrument at P, level 
 
 and clamp it. Find a point 
 
 p on the paper, directly 
 
 over P on the ground, and, 
 
 keeping the bevelled edge 
 
 of the ruler on p, point the 
 
 telescope to any corner of 
 
 the tract, as K. By means 
 
 of the stadia wires, or chain, obtain the distance PK, and lay 
 
 it off to any desired scale in the direction of the point sighted,
 
 268 PLANE SUEVEYING. 
 
 thus plotting pk. In a similar manner, locate the other corners. 
 Join by straight lines the points thus determined ; and the 
 resulting figure klmno will represent the tract surveyed. It is 
 obvious that the position of objects such as buildings, trees, 
 etc., if visible, may be determined by this method, and that it 
 is immaterial whether the instrument be set up in the field or 
 at one of the angles, providing all the stations can be seen from 
 the point selected. 
 
 311. By Progression. This method requires the instrument 
 to be set up at every station of the tract to be surveyed. Let 
 
 KLMNO represent, as be- 
 fore, the field, and suppose 
 the instrument is first placed 
 at 7T, and that k on the 
 paper designates this point. 
 With the alidade directed 
 towards />, draw along it an 
 indefinite line. Obtain by 
 stadia or chain the distance 
 KL, and lay it off to a desired scale, thus locating L Remove 
 the instrument to L, orient it, and locate m. Continue in the 
 same manner to locate n and o. 
 
 When the table is oriented at any station, as M, the line 
 ML should lie in the vertical plane, with its representative ml 
 on the plot, and, having gone round the tract, the last line 
 should close with the first station k. 
 
 This method, in conjunction with the preceding, may be 
 employed advantageously in the survey of a road, stream, etc. 
 The centre line of the road or bank of the stream may be trav- 
 ersed by the instrument, placing it at each angle or bend, as 
 in the survey of a field by progression, and determine by the 
 method of radiation the position of prominent objects, such as 
 buildings, bridges, trees, -etc. If there be added to the above 
 a sketch of the general features of the ground, a complete map 
 will be had of the belt of country traversed.
 
 METHODS EMPLOYED. 
 
 269 
 
 312. By Intersections. Let it be required to plot the stations 
 M, 0, P. Measure carefully the base line LN, and draw to a 
 convenient scale In on the paper 
 
 to represent it. At the extremities 
 of this base line orient and point the 
 instrument to the several stations. 
 The intersections of the pairs of 
 lines drawn from the base line to 
 these stations will indicate their 
 position on the plot. Their dis- 
 tances from the base line, if desired, 
 may be obtained by applying the scale used in the construction 
 of In. 
 
 If a field or closed tract of land is to be surveyed, a portion 
 or all of one side may be used as a base line, or a base may be 
 chosen outside the tract. 
 
 This method is obviously well adapted to the mapping of 
 harbors, shore lines, and generally to inaccessible points. 
 
 Of course in this, as in all triangulations, well-conditioned 
 triangles give more satisfactory results ; that is to say, avoid, 
 if possible, angles less than 30 or greater than 150. 
 
 313. By Resection. This method requires the measurement 
 of one line and the accessibility of all the stations. 
 
 Let KLMNO represent the 
 points to be plotted. 
 
 Obtain the distance between 
 two of them, as OK, lay it off 
 on the table to a suitable 
 scale, and let ok represent it. 
 Orient the table at k, point 
 the alidade to L, and draw 
 along its fiducial edge an in- 
 definite line. Remove the in- 
 strument to L, and orient it. Then with the alidade centring 
 on o, point it in the direction of 0, and draw a line along its
 
 270 PLANE SURVEYING. 
 
 edge : this line will intersect kL in some point Z, which will 
 locate L on the plot. Through I draw a line towards J/, 
 remove the instrument to Jf, and proceed as before. Objects 
 on either side of the lines may be determined by radiation or 
 by intersection, and further details, if desired, sketched in as 
 the work proceeds. 
 
 314. Determination of Position by Resection on Three 
 Known Points. In this problem three stations, L, N, 0, are 
 plotted, as Z, ?i, o, on the table, and the instrument being set 
 up over a fourth point P, it is required to find the position of 
 this point on the map. This is the three-point problem of 
 which geometrical constructions and analytical solutions are 
 given in Chapter II. Section IV. It may be solved thus : 
 Fasten a sheet of tracing-paper on the board, fix a point p to 
 represent the station at which the instrument is set ; with the 
 alidade centring on p, direct the telescope successively to Z>, 0, 
 and jV, and draw lines of indefinite length along the ruler's 
 edge towards these stations. Then if the tracing-paper be 
 shifted until the three lines thus drawn coincide with the points 
 Z, o, and , the point p will indicate the position of P. 
 
 The position of this point may now be transferred, by pricking, 
 to the map, the tracing-paper removed, and the table oriented. 
 
 315.* Bessel's Method by Inscribed Quadrilateral. A quadri- 
 lateral is constructed with all the angles in the circumference of 
 a circle, one diagonal of wlr.ch passes through the middle one 
 of the three fixed points and the point sought. On this line 
 the alidade is set, the telescope directed to the middle point, 
 and the table is in position. Resection upon the extreme points 
 intersects in this line and determines the position of the point 
 sought. 
 
 Let a, 6, c, be the points on the sheet representing the signals 
 A, B, (7, in the ground. 
 
 The table is set up at the point to be determined (cZ) and 
 
 * Articles 315 and 316 are from the U. S. C. & G. S. Report for 1880.
 
 METHODS EMPLOYED. 
 
 271 
 
 levelled. The alidade is set upon the line ca, and a directed, 
 by revolving the table, to its corresponding signal A, and the 
 table clamped ; then, with the alidade centring on c, the mid- 
 dle signal B is sighted with the 
 telescope, and the line ce drawn 
 along the edge of the ruler. 
 The alidade is then set upon the 
 line oc, and the telescope di- 
 rected to the signal (7, by re- 
 volving the table, and the table 
 clamped. Then, with the alidade 
 centring on a, the .telescope is 
 directed to the middle signal B, 
 and the line ae is drawn along 
 the edge of the ruler. The 
 point e (the intersection of these 
 two lines) will be in the line 
 passing through the middle point and the point sought. Set 
 the alidade upon the line be, direct 6 to the signal B by revolv- 
 ing the table, and the table will be in position. Clamp the 
 table, centre the alidade upon a, direct the telescope to the 
 signal A, and draw along the ruler the line ad. This will inter- 
 sect the line be at the point sought. Resection upon (7, cen- 
 tring the alidade on c in the same manner as upon A, will verify 
 its position. 
 
 The opposite angles of the quadrilateral adce being supple- 
 mentary, angle ace and angle ade are subtended by the same 
 chord ae and cae and cde are subtended by the same chord 
 ce ; consequently, the intersection of ae and ce at e must fall 
 on the line db ; or, the segments of two intersecting chords 
 in a circle being reciprocally proportional, the triangles ad/ and 
 cef are similar, and the triangles cdf and aef are similar, and 
 d, /, and e must be in a right line passing through 6. 
 
 316. Determination of Position by Resection on Two Known 
 Points. This is called the two-point problem, there being given
 
 272 PLANE SURVEYING. 
 
 bv their projections a, b, two points A and B, to put the plane- 
 table in position at a third point C. (The capital letters refer 
 to points on the ground, and the small ones to their correspond 
 ing projections.) 
 
 Select a fourth point Z), such that the intersections from C 
 and D upon A and B make sufficiently large angles for good 
 determinations. Put the table approximately in position at D, 
 by estimation or by compass, and draw the lines Aa, Bb, inter- 
 secting in d ; through d draw a line to C. Then set up at C, 
 and assuming the point c on the line dC at an estimated dis- 
 tance from d, and putting the table in a position parallel to that 
 which is occupied at D, by means of the line cd, draw the lines 
 from c to A, and from c to B. These will intersect the lines cL4, 
 dB, at points a' and b', which form with c and d a quadrilateral 
 similar to the true one, but erroneous in size and position. 
 
 The angles which the lines ab and a'b' make with each other 
 is the error in position. By constructing now through c a line 
 cd', making the same angle with cd as that which ab makes with 
 a'b', and directing this line cd 1 to D, the table will be brought 
 into position, and the true point c can be found by the inter- 
 sections of aA and bB. 
 
 Instead of transferring the angle of error by construction, we 
 may conveniently proceed as follows, observing that the angle 
 which the line a'b' makes with ab is the error in the position of 
 the table. As the table now stands, a'b' is parallel with AB, 
 but we want to turn it so that 06 shall be parallel to the same. 
 If, therefore, we place the alidade on a'b', and set up a mark
 
 METHODS EMPLOYED. 273 
 
 in that direction, then place the alidade on ab, and turn the 
 table until it again points to the mark, then ab will be parallel 
 to AB, and the table is in position. 
 
 317. Practical Suggestions in using the Plane-Table.* The 
 board should be placed so low as to be readily reached, even 
 at the most remote corner, and yet high enough to enable the 
 observer to take sight with com fort. This will bring it a little 
 below the elbow. 
 
 Care must be taken that no part of the body touch or rest 
 against the edge of the board. In using the alidade, steady 
 the standard with the left hand, while the right swings the rear 
 end of the ruler in the proper direction. 
 
 Thumb-tacks and rollers for holding down the sheet are both 
 found objectionable, especially in high winds. The edges may 
 be pasted underneath, or spring clamps may be used to advan- 
 tage. A scale graduated upon the fiducial edge of the alidade 
 is inconvenient, and in some positious impracticable and waste- 
 ful of time. A detached triangular boxwood or metal scale is 
 greatly to be preferred. Umbrellas or shades, whilst a great 
 relief to the eyes, are cumbersome and troublesome, and by 
 blowing over on the table may cause damage or derangement. 
 Colored glasses screening the eyes will be better, and by using 
 tinted paper, as manilla, instead of white, still more relief is 
 given, and the sheet can be kept cleaner. 
 
 Before leaving the station, and at any intervals not otherwise 
 employed, the " check" shots should be tesieu to determine any 
 displacement of the board. 
 
 Use as hard a pencil, and make as few lines, as possible. 
 In locating points of contours, plot the distance at once along 
 the edge of ruler by detached scale, making only a dot at the 
 point which should receive the number of the contour. 
 
 Objects on a straight line may be quickly located by plotting 
 the ends and determining the intermediate points by intersecting 
 shots. 
 
 * From The Topographer, by L. M. Haupt, C.E., Philadelphia. 

 
 274 PLANE SURVEYING. 
 
 EXERCISES WITH THE PLANE-TABLE. 
 
 1. Make a plane-table survey of a field, using one side as a 
 base line. 
 
 2. Make a survey embracing 200 or 300 rods of a road or 
 stream, locating prominent objects on either side. 
 
 3. Locate several points on the table by intersections, and 
 check the work by resection from these points. 
 
 4. Locate a non-plotted point by resection on three known 
 points tirst method; check by Bessel's method.
 
 CHAPTER VI. 
 
 THE SUBVEY OP THE PUBLIC LANDS OF THE UNITED 
 STATES. 
 
 THE SOLAR COMPASS. 
 
 318. A description of the Solar Compass, the instrument 
 that is extensively used in the survey of the public lands, its 
 adjustment and use, will be given before describing the method 
 employed by the government in these surveys. 
 
 This instrument, so ingeniously contrived for readily deter- 
 mining a true meridian or north and south line, was invented 
 by William A. Burt, of Michigan, and patented by him in 1836. 
 
 It has since come into general use in the surveys of United 
 States public lands, the principal lines of which are required to 
 be run with reference to the true meridian. 
 
 The arrangement of its sockets and plates is similar to that 
 of the Surveyor's Transit, as shown in Chapter II. Section I., 
 except that the sight-vanes are attached to the under plate or 
 limb, and this revolves around the upper or vernier plate on 
 which the solar apparatus is placed. 
 
 The limb is divided to half-degrees, is figured in two rows, 
 as usual, and reads by the two opposite verniers to single 
 minutes. 
 
 THE SOLAR APPARATUS. 
 
 319. The Solar Apparatus is seen in the place of the needle, 
 and in fact operates as its substitute in the field. 
 
 It consists mainly of three arcs of circles, by which can be 
 set off the latitude of a place, the declination of the sun, and 
 the hour of the day.
 
 276 PLANE SURVEYING. 
 
 These arcs, designated in the cut by the letters a, 6, and 
 c, are therefore termed the latitude, the declination, and the 
 hour arcs respectively. 
 
 320. The Latitude Arc a has its centre of motion in two 
 pivots, one of which is seen at d ; the other is concealed in the 
 cut. 
 
 It is moved either up or down within a hollow arc, seen in 
 the cut, by a tangent-screw at /, and is securely fastened in 
 any position by a clamp-screw. 
 
 The latitude arc is graduated to quarter-degrees, and reads 
 by its vernier e to single minutes ; it has a range of about 
 35 degrees, so as to be adjustable to the latitude of any place 
 in the United States. 
 
 321. The Declination Arc b is also graduated to quarter- 
 degrees, and has a range of about 28 degrees. 
 
 Its vernier w, reading to single minutes, is fixed to a movable 
 arm ^, having its centre of motion at the end of the declination 
 arc at g ; the arm is moved over the surface of the declination 
 arc, and its vernier set to any reading by turning the head of 
 the tangent-screw k. It is also securely clamped in any posi- 
 tion by a screw, concealed in the engraving. 
 
 322. Solar Lenses and Lines. At each end of the arm h is 
 a rectangular block of brass, in which is set a small convex 
 lens, having its focus on the surface of a little silver plate A 
 (marginal figure) , fastened by screws to the inside of the oppo- 
 site block. 
 
 On the surface of the plate are marked two sets of lines 
 intersecting each other at right angles ; of 
 these bb are termed the hour lines, and cc the 
 ^ equatorial lines, as having reference respec- 
 tively to the hour of the day and the position of 
 the sun in relation to the equator. In the cut the equatorial 
 lines are those on the lower block, parallel to the surface of the
 
 THE SOLAB COMPASS. 279 
 
 hour arc c ; the hour lines are of course those at right angles 
 to the first. 
 
 323* Equatorial Sights. On the top of each of the rec- 
 tangular blocks is seen a little sighting-piece, termed the equa- 
 torial sight, fastened to the block by a small, milled head-screw, 
 so as to be detached at pleasure. 
 
 They are used, as will be explained hereafter, in adjusting 
 the different parts of the solar apparatus. 
 
 324. The Hour Arc c is supported by the two pivots of the 
 latitude arc already spoken of, and is also connected with that 
 arc by a curved arm, as shown in the figure. 
 
 The hour arc has a range of about 120, is divided to half- 
 degrees, and figured in two series, designating both the hours 
 and the degrees, the middle division being marked 12 and 90 
 on either side of the graduated lines. 
 
 325. The Polar Axis. Through the centre of the hour arc 
 passes a hollow socket p containing the spindle of the declina- 
 tion arc, by means of which this arc can be moved from side to 
 side over the surface of the hour arc, or turned completely round, 
 as may be required. 
 
 The hour arc is read by the lower edge of the graduated side 
 of the declination arc. 
 
 The axis of the declination arc, or indeed the whole socket 
 p, is appropriately termed the polar axis. 
 
 326. The Adjuster. Besides the parts shown in the cut, 
 there is also an arm used in the adjustment of the instrument 
 as described hereafter, but laid aside in the box when that is 
 effected. 
 
 The parts just described constitute properly the solar 
 apparatus. 
 
 Besides these, however, are seen the needle-box n with its 
 arc and tangent screw , and the spirit levels, for bringing the 
 whole instrument to a horizontal position.
 
 280 PLANE SURVEYING. 
 
 327. The Needle-Box n has an arc of about 86 degrees in 
 extent, divided to half-degrees, and figured from the centre or 
 zero mark on either side. 
 
 The needle, which is made as in other instruments, except 
 that the arms are of unequal lengths, is raised or lowered by a 
 lever shown in the cut. 
 
 The needle-box is attached by a projecting arm to a tangent- 
 screw t, by which it is moved about its centre, and its needle 
 set to any variation. 
 
 This variation is also read off by the vernier on the end of 
 the projecting arm, reading to three minutes a graduated arc, 
 attached to the plate of the compass. 
 
 328. The Levals seen with the solar apparatus have ground- 
 glass vials, and are adjustable at their ends like those of other 
 instruments. 
 
 The edge of the circular plate on which the solar work is 
 placed is divided and figured at intervals of 10 degrees, and 
 numbered, as shown, from to 90 on each side of the line of 
 sight. 
 
 These graduations are used in connection with a little brass 
 pin, seen in the centre of the plate, to obtain approximate 
 bearings of lines, which are not important enough to require a 
 close observation. 
 
 329. Lines of Refraction. The inside faces of the sights are 
 also graduated and figured, to indicate the amount of refraction 
 to be allowed when the sun is near the horizon. 
 
 PRINCIPLES OF THE SOLAR COMPASS. 
 
 330. The interval between two equatorial lines cc, in figure 
 on page 276, as well as between the hour lines 66, is just suffi- 
 cient to include the circular image of the sun, as formed by the 
 solar lens on the opposite end of the revolving arm ft, figure 
 on page 277.
 
 THE SOLAR COMPASS. 281 
 
 When, therefore, the instrument is made perfectly horizontal, 
 the equatorial lines and the opposite lenses being accurately 
 adjusted to each other by a previous operation, and the sun's 
 image brought within the equatorial lines, his position in the 
 heavens, with reference to the horizon, will be defined with 
 precision. 
 
 Suppose the observation to be made at the time of one of 
 the equinoxes ; the arm /t, set at zero on the declination arc 
 b ; and the polar axis p, placed exactly parallel to the axis of 
 the earth. 
 
 Then the motion of the arm h, if revolved on the spindle of 
 the declination arc around the hour circle c, will exactly corre- 
 spond with the motion of the sun in the heavens, on the given 
 day and at the place of observation ; so that if the sun's image 
 was brought between the lines cc in the morning, it would 
 continue in the same position, passing neither above nor below 
 the lines, as the arm was made to revolve in imitation of the 
 motion of the sun about the earth. 
 
 In the morning, as the sun rises from the horizon, the arm h 
 will be in a position nearly at right angles to that shown in the 
 cut, the lens being turned towards the sun, and the silver plate 
 on which his image is thrown directly opposite. 
 
 As the sun ascends, the arm must be moved around, until 
 when he has reached the meridian, the graduated side of the 
 declination arc will indicate 12 on the hour circle, and the arm 
 h, the declination arc b, and the latitude arc a will be in the 
 same plane. 
 
 As the sun declines from the meridian, the arm h must be 
 moved in the same direction, until at sunset its position will be 
 the exact reverse of that it occupied in the morning. 
 
 331. Allowance for Declination. Let us now suppose the 
 observation made when the sun has passed the equinoctial point, 
 and when his position is affected by declination. 
 
 Bv referring to the almanac, and setting off on the arc his 
 declination for the given day and hour, we are still able to
 
 282 PLANE SURVEYING. 
 
 determine his position with the same certainty as if he remained 
 on the equator. 
 
 When the sun's declination is south, that is, from the 22d 
 of September to the 20th of March in each year, the arc b is 
 turned towards the plates of the compass, as shown in the en- 
 graving, and the solar lens o, with the silver plate opposite, 
 are made use of in the surveys. 
 
 The remainder of the year the arc is turned from the plates, 
 and the other lens and plate employed. 
 
 When the solar compass is accurately adjusted, and its plates 
 made perfectly horizontal, the latitude of the place, and the 
 declination of the sun for the given day and hour, being also 
 set off on the respective arcs, the image of the sun cannot be 
 brought between the equatorial lines until the polar axis is placed 
 in the plane of the meridian of the place, or in a position parallel 
 to the axis of the earth. The slightest deviation from this posi- 
 tion will cause the image to pass above or below the lines, and 
 thus discover the error. 
 
 We thus, from the position of the sun in the solar system, 
 obtain a certain direction absolutely unchangeable, from which 
 to run our lines and measure the horizontal angles required. 
 
 This simple principle is not only the basis of the construction 
 of the solar compass, but the sole cause of its superiority to the 
 ordinary or magnetic instrument. For in a needle instrument 
 the accuracy of the horizontal angles indicated, and therefore 
 of all the observations made, depends upon " the delicacy of 
 the needle, and the constancy with which it assumes a certain 
 direction, termed the magnetic meridian." 
 
 The principal causes of error in the needle, briefly stated, are 
 the dulling of the pivot, the loss of polarity in the needle, the 
 influence of local attraction, and the effect of the sun's rays, 
 producing the diurnal variation. 
 
 From all these imperfections the solar instrument is free. 
 
 The sights and the graduated limb being adjusted to the solar 
 apparatus, and the latitude of the place and the declination of the 
 sun also set off upon the respective arcs, we are able not only
 
 THE SOLAR COMPASS. 283 
 
 to run the true meridian, or a due east and west course, but 
 also to set off the horizontal angles with minuteness and ac- 
 curacy from a direction which never changes, and is unaffected 
 by attraction of anv kind. 
 
 To ADJUST THE SOLAR COMPASS. 
 
 The adjustments of this instrument, with which the surveyor 
 will have to do, are simple and few in number, and will now 
 be given in order. 
 
 332. To Adjust the Levels. Proceed precisely as directed in 
 the account of the other instruments we have described', by 
 bringing the bubbles into the centre of the tubes by the level- 
 ling-screws of the tripod, and then reversing the instrument 
 upon its spindle, and raising or lowering the ends of the tubes, 
 until the bubbles will remain in the centre during a complete 
 revolution of the instrument. 
 
 333. To Adjust the Equatorial Lines and Solar Lenses. 
 First detach the arm h from the declination arc by withdrawing 
 the screws shown in the cut from the ends of the posts of the 
 tangent-screw k, and also the clamp-screw, and the conical 
 pivot with its small screws by which the arm and declination 
 arc are connected. 
 
 The arm h being thus removed, attach the adjuster in its 
 place by replacing the conical pivot and screws, and insert the 
 clamp-screw so as to clamp the adjuster at any point on the 
 declination arc. 
 
 Now level the instrument, place the arm h on the adjuster, 
 with the same side resting against the surface of the declination 
 arc as before it was detached. Turn the instrument on its 
 spindle so as to bring the solar lens to be adjusted in the direc- 
 tion of the sun, and raise or lower the adjuster on the declina- 
 tion arc, until it can be clamped in such a position as to bring 
 the sun's image as near as may be between the equatorial lines 
 on the opposite silver plate, and bring the image precisely into
 
 284 PLANE SURVEYING. 
 
 position by the tangent of the latitude arc or the levelling- 
 screws of the tripod. Then carefully turn the arm half-way 
 over, until it rests upon the adjuster by the opposite faces of 
 the rectangular blocks, and again observe the position of the 
 sun's image. 
 
 If it remains between the lines as before, the lens and plate 
 are in adjustment ; if not, loosen the three screws which con- 
 fine the plate to the block, and move the plate under their 
 heads, until one-half the error in the position of the sun's image 
 is removed. 
 
 Again bring the image between the lines, and repeat the 
 operation until it will remain in the same situation, in both 
 positions of the arm, when the adjustment will be completed. 
 
 To adjust the other lens and plate, reverse the arm eud for 
 end on the adjuster, and proceed precisely as in the former case, 
 until the same result is attained. 
 
 In tightening the screws over the silver plate, care must be 
 taken not to move the plate. 
 
 This adjustment now being complete, the adjuster should be 
 removed, and the arm h with its attachments replaced as 
 before. 
 
 334. To Adjust the Vernier of the Declination Arc. Hav- 
 ing levelled the instrument, and turned its lens in the direction 
 of the sun, clamp to the spindle, and set the vernier v of the 
 declination arc at zero, by means of the tangent-screw at &, 
 and clamp to the arc. 
 
 See that the spindle moves easily and yet truly in the socket, 
 or polar axis, and raise or lower the latitude arc by turning the 
 tangent-screw/, until the sun's image is brought between the 
 equatorial lines on one of the plates. Clamp the latitude arc 
 by the screw, and bring the image precisely into position by 
 the levelling-screws of the tripod or socket, and without dis- 
 turbing the instrument, carefully revolve the arm /, until the 
 opposite lens and plate are brought in the direction of the sun, 
 and note if the sun's image comes between the lines as before.
 
 THE SOLAR COMPASS. 285 
 
 If it does, there is no index error of the declination arc ; if 
 not, with the tangent-screw &, move the arm until the sun's 
 image passes over half the error ; again bring the image be- 
 tween the lines, and repeat the operation as before, until the 
 image will occupy the same position on both the plates. 
 
 We shall now find, however, that the zero marks on the arc 
 and the vernier do not correspond, and to remedy this error, 
 the little flat-head screws above the vernier must be loosened 
 until it can be moved so as to make the zeros coincide, when 
 the operation will be completed. 
 
 335. To Adjust the Solar Apparatus to the Compass Sights. 
 First level the instrument, and with the clamp and tangent screws 
 set the main plate at 90 by the verniers and horizontal limb. 
 Then remove the clamp-screw, and raise the latitude arc until 
 the polar axis is by estimation very nearly horizontal, and if 
 necessary, tighten the screws on the pivots of the arc, so as to 
 retain it in this position. 
 
 Fix the vernier of the declination arc at zero, and direct the 
 equatorial sights to some distant and well-marked object, and 
 observe the same through the compass sights. If the same 
 object is seen through both, and the verniers read to 90 on the 
 limb, the adjustment is complete ; if not, the correction must 
 be made by moving the sights or changing the position of the 
 verniers. 
 
 To USE THE SOLAR COMPASS. 
 
 336. Before this instrument can be used at any given place, 
 it is necessary to set off upon its arcs both the declination of the 
 sun as affected by its refraction for the given day and hour, and 
 the latitude of the place where the observation is made. 
 
 337. To Set off the Declination. The declination of the sun, 
 given in the ephemeris of the Nautical Almanac from year to 
 year, is calculated for apparent noon at Greenwich, England, 
 or Washington, D.C. 
 
 To determine it for any other hour at a place in the United
 
 286 PLANE SUKVEYING. 
 
 States, reference must be had, not only to the difference of time 
 arising from the longitude, but also to the change of declination 
 from day to day. 
 
 By the use of standard time, which is now quite general 
 throughout the United States, it is very easy to obtain the 
 declination required at any place. 
 
 For those using 75th meridian time, a difference of five hours 
 must be allowed for the difference in declination between the 
 place of observation and Greenwich. 
 
 The time-piece referred to the 75th meridian as standard in- 
 dicating 7 A.M. when it is noon at Greenwich. 
 
 Where the 90th meridian is used as standard, six hours must 
 be allowed, etc. 
 
 To obtain the declination for the other hours of the day, take 
 from the almanac the declination for apparent noon of the given 
 clay, and, as the declination is increasing or decreasing, add to 
 or subtract from the declination of the first hour the difference 
 for one hour as given in the ephemeris, which will give, when 
 affected by the refraction, the declination for the succeeding 
 hour ; and proceed thus in making a table of the declination for 
 every hour of -the day. 
 
 338. Refraction. By reason of the increasing density of the 
 atmosphere from its upper regions to the earth's surface, the 
 rays of light from the sun are bent out of their course, so as to 
 make his altitude appear greater than is actually the case. 
 
 The amount of refraction varies according to the altitude of 
 the body observed ; being when it is in the zenith, about one 
 minute when midway from the horizon to the zenith, and almost 
 34' when in the horizon. 
 
 339. Effect of Incidental Refraction. It will be seen by 
 referring to the instrument, that the effect of the ordinary 
 refraction upon the position of the sun's image with reference 
 to the equatorial lines, which, in fact, are the only ones to be 
 regarded in running lines with the solar compass, is continually
 
 THE SOLAR COMPASS. 287 
 
 changing, not only with the change of latitude, but also with 
 that of the sun's declination from hour to hour, and the motion 
 of the revolving arm as it follows the sun in its daily revolution. 
 
 If the equatorial lines were always in the same vertical plane 
 with the sun, as would be the case at the equator at the time of 
 the equinoxes, it is evident that refraction would have no effect 
 upon the position of the image between these lines, and there- 
 fore would not be of any importance to the surveyor. 
 
 But as we proceed further north, and as the sun's declination 
 to the south increases, the refraction also increases, and must 
 now be taken into account. 
 
 Again, the angle which the equatorial lines make with the 
 horizon is continually changing as the arm is made to follow 
 the motion of the sun during the course of a day. 
 
 Thus, in the morning and evening they are more or less 
 inclined to the horizon, while at noon they are exactly parallel 
 to it. 
 
 And thus it follows that the excess of refraction at morning 
 and evening is in some measure balanced by the fact that the 
 position of the sun's image with reference to the equatorial lines 
 is then less affected by it, on account of the greater inclination 
 of the lines to the horizon. 
 
 340. Allowance for Refraction. The proper allowance to be 
 made for refraction in setting off the declination of the sun 
 upon the solar compass for any hour of any day of the year is 
 given in the following table :
 
 288 
 
 PLANE SURVEYING. 
 
 A TABLE OF MEAN REFRACTIONS IN DECLINATION. 
 
 To apply on the declination arc of Solar Attachment of either Compass 
 or Transits.* 
 
 1 
 
 DECLINATIONS. 
 
 i 
 
 FOR LATITUDE 30. 
 
 1 
 
 + 20 
 
 + 15* 
 
 + 10 D 
 
 + 5 
 
 
 
 -5 
 
 -10^ 
 
 -15 3 
 
 -20^ 
 
 Oh. 
 
 10" 
 
 15" 
 
 21" 
 
 27" 
 
 33" 
 
 40" 
 
 48" 
 
 57" 
 
 1'08" 
 
 2 
 
 14 
 
 19 
 
 25 
 
 31 
 
 38 
 
 46 
 
 54 
 
 1'05 
 
 1 18 
 
 3 
 
 20 
 
 26 
 
 32 
 
 39 
 
 47 
 
 55 
 
 1'06 
 
 119 
 
 136 
 
 4 
 
 32 
 
 39 
 
 46 
 
 52 
 
 1'06 
 
 1'19 
 
 135 
 
 157 
 
 229 
 
 5 
 
 I'OO 
 
 I'lO 
 
 1'24 
 
 1'52 
 
 207 
 
 244 
 
 346 
 
 543 
 
 1306 
 
 FOR LATITUDE S2 5 30'. 
 
 Oh. 
 
 13" 
 
 18" 
 
 24" 
 
 30" 
 
 36" 
 
 44" 
 
 52" 
 
 1'02" 
 
 1'14" 
 
 2 
 
 17 
 
 22 
 
 28 
 
 35 
 
 42 
 
 50 
 
 I'OO 
 
 1 11 
 
 126 
 
 3 
 
 23 
 
 29 
 
 35 
 
 43 
 
 51 
 
 I'Ol 
 
 1 13 
 
 128 
 
 147 
 
 4 
 
 35 
 
 43 
 
 51 
 
 I'Ol 
 
 1'IS 
 
 127 
 
 146 
 
 2 13 
 
 254 
 
 5 
 
 roa 
 
 116 
 
 1'Sl 
 
 153 
 
 220 
 
 305 
 
 425 
 
 736 
 
 
 FOB LATITUDE 35. 
 
 Oh. 
 
 15" 
 
 21" 
 
 27" 
 
 33" 
 
 40" 
 
 48" 
 
 57" 
 
 1'08" 
 
 1'21" 
 
 2 
 
 20 
 
 25 
 
 32 
 
 38 
 
 46 
 
 55 
 
 1'05 
 
 1 18 
 
 135 
 
 3 
 
 26 
 
 33 
 
 39 
 
 47 
 
 56 
 
 1'07 
 
 121 
 
 138 
 
 200 
 
 4 
 
 39 
 
 47 
 
 56 
 
 1'07 
 
 1'20 
 
 136 
 
 1 59 
 
 232 
 
 325 
 
 5 
 
 1'07 
 
 1*20 
 
 1'38 
 
 200 
 
 234 
 
 329 
 
 514. 
 
 1016 
 
 
 FOR LATITUDE 37 30'. 
 
 Oh. 
 
 18" 
 
 24" 
 
 30" 
 
 36" 
 
 44" 
 
 52" 
 
 1'02" 
 
 1'14'' 
 
 1'29" 
 
 2 
 
 22 
 
 28 
 
 35 
 
 42 
 
 50 
 
 I'OO 
 
 112 
 
 126 
 
 145 
 
 3 
 
 29 
 
 36 
 
 43 
 
 52 
 
 1'02 
 
 1 14 
 
 129 
 
 149 
 
 216 
 
 4 
 
 43 
 
 51 
 
 I'Ol 
 
 1'13 
 
 127 
 
 149 
 
 2 14 
 
 254 
 
 405 
 
 5 
 
 I'll 
 
 1'26 
 
 145 
 
 2 10 
 
 249 
 
 355 
 
 615 
 
 1458 
 
 
 * Computed by Edward W. Arms, C.E., for W. and L. E. Gurley 
 Troy, N.Y.
 
 THE SOLAR COMPASS. 
 
 289 
 
 HOUR ANGLE. 
 
 DECLINATIONS. 
 
 FOB LATITUDE 40. 
 
 + 20 3 
 
 + 15 
 
 + 10 
 
 + 5 
 
 
 
 -5 
 
 -10 
 
 -15 
 
 -20' 
 
 Oh. 
 
 21" 
 
 27" 
 
 33" 
 
 40" 
 
 48" 
 
 57" 
 
 1'08" 
 
 1'21" 
 
 1'39" 
 
 2 
 
 25 
 
 32 
 
 39 
 
 46 
 
 52 
 
 1'06 
 
 1 19 
 
 135 
 
 1 57 
 
 8 
 
 33 
 
 40 
 
 48 
 
 57 
 
 1'08 
 
 121 
 
 138 
 
 202 
 
 236 
 
 4 
 
 47 
 
 55 
 
 1'06 
 
 1'19 
 
 136 
 
 158 
 
 230 
 
 321 
 
 459 
 
 5 
 
 1'15 
 
 1'31 
 
 1 51 
 
 220 
 
 305 
 
 425 
 
 734 
 
 2518 
 
 
 FOR LATITUDE 42 30'. 
 
 Oh. 
 
 24" 
 
 30" 
 
 36" 
 
 44" 
 
 52" 
 
 1'02" 
 
 1'14" 
 
 1'29" 
 
 1'49" 
 
 2 
 
 28 
 
 35 
 
 39 
 
 50 
 
 I'OO 
 
 1 12 
 
 126 
 
 145 
 
 2 11 
 
 3 
 
 36 
 
 43 
 
 52 
 
 1'02 
 
 1 13 
 
 129 
 
 149 
 
 2 17 
 
 2 59 
 
 4 
 
 50 
 
 I'OO 
 
 I'll 
 
 1 26 
 
 1 44 
 
 2 10 
 
 249 
 
 355 
 
 6 16 
 
 5 
 
 1'16 
 
 136 
 
 158 
 
 230 
 
 322 
 
 500 
 
 924 
 
 
 
 FOB LATITUDE 45. 
 
 Oh. 
 
 27" 
 
 33" 
 
 40" 
 
 48" 
 
 57" 
 
 1'08" 
 
 1'21" 
 
 1'39 
 
 2'02" 
 
 2 
 
 32 
 
 39 
 
 46 
 
 52 
 
 1'06 
 
 1 19 
 
 1 35 
 
 1 57 
 
 229 
 
 3 
 
 40 
 
 47 
 
 56 
 
 1'07 
 
 121 
 
 138 
 
 200 
 
 234 
 
 329 
 
 4 
 
 54 
 
 1'04 
 
 1'16 
 
 133 
 
 1 54 
 
 224 
 
 311 
 
 438 
 
 8 15 
 
 5 
 
 1'23 
 
 141 
 
 205 
 
 241 
 
 340 
 
 540 
 
 1202 
 
 
 
 FOR LATITUDE 4T 30'. 
 
 Oh. 
 
 30" 
 
 36" 
 
 44" 
 
 52" 
 
 1'02" 
 
 1'14" 
 
 1'29" 
 
 1'49" 
 
 2'18" 
 
 2 
 
 35 
 
 42 
 
 50 
 
 I'OO 
 
 1 12 
 
 1 26 
 
 145 
 
 201 
 
 251 
 
 3 
 
 43 
 
 61 
 
 I'Ol 
 
 I 13 
 
 1 28 
 
 147 
 
 2 16 
 
 2 56 
 
 408 
 
 4 
 
 56 
 
 1'09 
 
 123 
 
 140 
 
 205 
 
 240 
 
 339 
 
 637 
 
 1118 
 
 5 
 
 1'27 
 
 146 
 
 212 
 
 252 
 
 401 
 
 630 
 
 1619 
 
 
 
 FOR LATITUDE 50. 
 
 
 Oh. 
 
 33" 
 
 40" 
 
 48" 
 
 57" 
 
 1'08" 
 
 1'21" 
 
 1'39" 
 
 2'02" 
 
 2'36" 
 
 2 
 
 38 
 
 46 
 
 55 
 
 1'06 
 
 1 18 
 
 135 
 
 157 
 
 228 
 
 319 
 
 3 
 
 47 
 
 56 
 
 i'oe 
 
 1 19 
 
 1 36 
 
 229 
 
 231 
 
 323 
 
 502 
 
 4 
 
 1'02 
 
 1'14 
 
 1 29 
 
 1 48 
 
 2 16 
 
 258 
 
 418 
 
 669 
 
 1947 
 
 5 
 
 1 30 
 
 1 51 
 
 2 19 
 
 304 
 
 422 
 
 728 
 
 2410 
 

 
 290 PLANE SURVEYING. 
 
 EXPLANATION OF THE TABLE OF REFRACTIONS.* 
 
 The table is calculated for latitudes between 30 and 50 at 
 intervals of 2^, that being as near as is required. 
 
 The declination ranges from to 20, both north and south, 
 the + declinations being north, and south, and is given for 
 every 5 degrees, that being sufficiently near for all practical pur- 
 poses. 
 
 The hour angle in the first column indicates the distance of 
 the sun from the meridian in hours, the refraction given for 
 hours being that which affects the observed declination of the 
 sun when on the meridian, commonly known as meridional re- 
 fraction ; the refraction for the hours just before and after noon 
 is so nearly that of the meridian, that it may be called and 
 allowed as the same. 
 
 When the table is used, it must be borne in mind that when 
 the declination is north or + in the table, the refraction is to 
 be added ; when the declination is south or the refraction 
 must be subtracted. 
 
 It will be noticed that the refraction in south or declina- 
 tion increases very rapidly as the sun nears the horizon, show- 
 ing that observations should not be taken with the sun when 
 south of the equator, less than one hour from the horizon. 
 
 Thus, suppose it be required to obtain the declination for any 
 hour in the day, April 16, 1887, at Pittsburg, Pa., where 75th 
 meridian time is used. 
 
 The difference in time is 5 hours, so that the declination 
 given in the ephemeris for apparent noon of that day at Green- 
 wich would be that of 7 A.M. at Pittsburg. Proceed as follows : 
 
 Declination at Greenwich, mean noon, April 16, 1887, 
 N. 10 6' 29" 
 
 Add 1' 51"=refract'n for 5 hrs. [lat. Pittsburg 40 28']. 
 
 Or, N. 10 8' 20" =dec. 7 A.M. at Pittsburg. 
 * See also Refraction Table, page 92.
 
 THE SOLAR COMPASS. 291 
 
 To get the declination for 8 o'clock, same day and place, add 
 53", the difference for one hour because the declination is 
 increasing to the declination taken from the almanac, and 
 this increased by the refraction corresponding to 4 hours from 
 noon will give 10 8' 28" for the required declination. 
 
 Again, suppose it be desired to obtain the corrected dec- 
 lination for 8 A.M. Oct. 15, 1887, same place. 
 
 The declination being now south, the refraction is to be sub- 
 tracted, but the hourly difference is to be added because the 
 declination is increasing, as in the first example ; thus : 
 
 Declination at Greenwich, mean noon, Oct. 15, 1887, 
 
 S. 8 30' 20" 
 Add 56"= dec. for 1 hr., and increasing. 
 
 S. 8 31' 16" 
 Subtract 2' 23"= refr. 4 hrs. from noon. 
 
 Or, S. 8 28' 53"= dec. at 8 A.M. ; 
 
 and so on for any hour in the day, obtaining from the declina- 
 tion at Greenwich, by the proper application of the hourly 
 motion, the declination corresponding to the hour required, and 
 correcting this for refraction due to altitude. 
 
 To facilitate operations, the calculation of the declination for 
 the different hours of the day should be made and noted before 
 the surveyor commences his work. 
 
 341. To Set off the Latitude. Find the declination of the 
 sun for the given day at noon, at the place of observation as 
 just described, and with the tangent-screw set it off upon the 
 declination arc, and clamp the arm firmly to the arc. 
 
 Observe in the almanac the equation of time for the given 
 day, in order to know about the time the sun will reach the 
 meridian. 
 
 Then, about fifteen or twenty minutes before this time, set 
 up the instrument, level it carefully, fix the divided surface of 
 the declination arc at 12 on the hour circle, and turn the instru-
 
 292 PLA^E SURVEYING. 
 
 ment upon its spindle until the solar lens is brought into the 
 direction of the sun. 
 
 Loosen the clamp-screw of the latitude arc, and with the 
 tangent-screw raise or lower this arc until the image of the sun 
 is brought precisely between the equatorial lines, and turn the 
 instrument from time to time so as to keep the image also 
 between the hour lines on the plate. 
 
 As the sun ascends, its image will move below the lines, and 
 the arc must be moved to follow it. Continue thus, keeping it 
 between the two sets of lines until its image begins to pass 
 above the equatorial lines, which is also the moment of its pass- 
 ing the meridian. 
 
 Now read off the vernier of the arc, and we have the latitude 
 of the place, which is always to be set off on the arc when the 
 compass is used at the given place. 
 
 It is the practice of surveyors using the solar compass to set 
 off, in the manner just described, the latitude of the point where 
 the survey begins, and to repeat the observation and correction 
 of the latitude arc every day when the weather is favorable, 
 there being also an hour at mid-day when the sun is so near the 
 meridian as not to give the direction of lines with the certainty 
 required. 
 
 342. To Run Lines with the Solar Compass. Having set 
 off in the manner just given the latitude and declination upon 
 their respective arcs, the instrument being also in adjustment, 
 the surveyor is ready to run lines by the sun. 
 
 To do this, the instrument is set over the station and care- 
 fully levelled, the plates clamped at zero on the horizontal limb, 
 and the sights directed north and south, the direction being 
 given, when unknown, approximately by the needle. 
 
 The solar lens is then turned to the sun, and with one hand 
 on the instrument, and the other on the revolving arm, both 
 are moved from side to side, until the sun's image is made to 
 appear on the silver plate ; when, by carefully continuing the 
 operation, it may be brought precisely between the equatorial 
 lines.
 
 THE SOLAR COMPASS. 293 
 
 Allowance being now made for refraction, .the line of sights 
 will indicate the true meridian ; the observation may now be 
 made, and the flag-man put in position. 
 
 When a due east and west line is to be run, the verniers of 
 the horizontal limb are set at 90, and' the sun's image kept 
 between the lines as before. 
 
 The solar compass being so constructed that when the sun's 
 image is in position the limb must be clamped at in order to 
 run a true meridian line, it will be evident that the bearing of 
 any line from the meridian may be read by the verniers of the 
 limb precisely as in the ordinary magnetic compass : the bear- 
 ings of lines are read from the ends of the needle. 
 
 343. Use of the Needle. In running lines, the magnetic 
 needle is always kept with the sun ; that is, the point of the 
 needle is made to indicate on the arc of the compass-box by 
 turning the tangent-screw connected with its arm on the oppo- 
 site side of the plate. By this means the lines can be run by 
 the needle alone in case of the temporary disappearance of the 
 sun ; but, of course, in such cases the surveyor must be sure 
 that no local attraction is exerted. 
 
 The variation of the needle, which is noted at every station, 
 is read off in degrees and minutes on the arc, by the edge of 
 which the vernier of the needle-box moves. 
 
 344. Allowance for the Earth's Curvature. When long lines 
 are run by the solar compass, either by the true meridian, or 
 due east and west, allowance must be made for the curvature 
 of the earth. 
 
 Thus, in running north or south, the latitude changes about 
 one minute for every distance of 92 chains 30 links, and the 
 side of a township requires a change on the latitude arc of 5' 
 12", the township, of course, being six miles square. 
 
 This allowance is of constant use where the surveyor fails 
 to get an observation on the sun at noon, and is a very close 
 approximation to the truth.
 
 294 PLANE SURVEYING. 
 
 In running due east and west, as in tracing the standard 
 parallels of latitude, the sights are set at 90 on the limb, and 
 the line is run at right angles to the meridian. 
 
 If no allowance were made for the earth's curvature, these 
 lines would, if sufficiently produced, reach the equator, to 
 which they are constantly tending. 
 
 Of course, in running short lines either east or west, the 
 variation from the parallel would be so small as to be of no 
 practical importance ; but when long sights are taken, the 
 correction should be made by taking fore and back sights at 
 every station, noticing the error on the back-sight, and setting 
 off one-half of it on the fore-sight on the side towards the pole. 
 
 345. Time of Day by the Sun. The time of day is best 
 ascertained by the solar compass when the sun is on the 
 meridian, as at the time of making the observation for lati- 
 tude. 
 
 The time thus given is that of apparent noon, and can be 
 reduced to mean time, by merel}' applying the equation of time 
 as directed in the almanac, and adding or subtracting as the 
 sun is slow or fast. 
 
 The time, of course, can also be taken before or after noon, 
 by bringing the sun's image between the hour lines, and 
 noticing the position of the divided edge of the revolving 
 arm, with reference to the graduations of the hour circle, 
 allowing four minutes of time for each degree of the arc, and 
 thus -obtaining apparent time, which must be corrected by the 
 equation of time as just described. 
 
 346. Caution as to the False Image. In using the compass 
 upon the sun, if the revolving arm be turned a little one side of 
 its proper position, a false or reflected image of the sun will 
 appear on the silver plate in nearly the same place as that occu- 
 pied by the true one. It is caused by the reflection of the true 
 image from the surface of the arm, and is a fruitful source of 
 error to the inexperienced surveyor. It can, however, be
 
 SURVEY OF THE PUBLIC LANDS. 295 
 
 readily distinguished from the real image by being much less 
 bright, and not so clearly defined. 
 
 347. Approximate Bearings. When the bearings of lines, 
 such as the course of a stream, or the boundaries of a forest, 
 are not desired with the certainty given by the verniers and 
 horizontal limb, a rough approximation of the angle they make 
 with the true meridian is obtained by the divisions on the 
 outside of the circular plate. 
 
 In this operation, a pencil, or thin straight edge of any sort, 
 is held perpendicularly against the circular edge of the plate, 
 and moved around until it is in range with the eve, the brass 
 centre-pin, and the object observed. 
 
 The bearing of the line is then read off at the point where the 
 pencil is placed. 
 
 348. Time for Using the Solar Compass. The solar com- 
 pass, like the ordinary instrument, can be used at all seasons 
 of the year, the most favorable time being, of course, in the 
 summer, when the declination is north, and the days are long, 
 and more generally fair.* 
 
 ORIGIN OP THE SYSTEM FOR THE SURVEY OF THE 
 PUBLIC LANDS.f 
 
 349. The present system of survey of the public lands was 
 inaugurated by a committee appointed by the Continental 
 Congress, of which Thomas Jefferson was chairman. This 
 committee, on May 7, 1784, reported an ordinance requiring 
 public lands to be divided into " hundreds" of ten geographical 
 miles square, and these again subdivided into lots of one mile 
 square, each to be numbered from 1 to 100, commencing in the 
 northwestern corner and continuing from west to east and from 
 
 * See Article 147. 
 
 t The following pages regarding the government surveys are from 
 " Instructions of the General Land Office to the Surveyors-General of the 
 United States relative to the Survey of the Public Lands."
 
 296 
 
 PLANE SUKVEYING. 
 
 east to west consecutively. By subsequent amendment, April 
 26, 1785, the ordinance required the surveyors " to divide the 
 said territory into townships of 7 miles square, by lines running 
 due north and south, and others crossing these at right angles. 
 The plots of the townships, respectively, shall be marked by 
 subdivisions into sections of 1 mile square, or 640 acres in the 
 same direction as the external lines, and numbered from 1 to 
 49, and these sections shall be subdivided into lots of 320 
 acres." This is the first record of the use of the terms " town- 
 ship " and " section." 
 
 This ordinance was subsequently still further amended, and 
 as finally passed on the 20th of May, 1785, provided for town- 
 ships 6 miles square, containing 36 sections of 1 mile square. 
 The first public surveys were made under this ordinance by 
 the direction of the Geographer of the United States. 
 
 6 
 
 5 
 
 4 
 
 5 
 
 2 
 
 1 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 31 
 
 82 
 
 33 
 
 34 
 
 35 
 
 36 
 
 The act of Congress, approved May 18, 1796, provided 
 for the appointment of a surveyor-general, and directed 
 the survey of lands northwest of the Ohio River, and above 
 the mouth of the Kentucky River, " in which the titles of the 
 Indian tribes have been extinguished," and among other pro- 
 visions, that the " sections shall be numbered respectively, 

 
 SURVEY OF THE PUBLIC LAMDS. 297 
 
 beginning with the number one in the northeast section and 
 proceeding west and east, alternately, through the township, 
 with progressive numbers till the thirty-sixth be completed." 
 This method of numbering sections, as shown by the preceding 
 diagram, is still in use. 
 
 The act of Congress, approved Feb. 11, 1805, directs the 
 subdivisions of the public lands into quarter-sections. The 
 act of April 24, 1820, provides for the sale of the public lands 
 in half-quarter-sections, and that in every case of the division 
 of a quarter-section, the division line shall run north and south. 
 April 5, 1832, Congress directed the subdivision of the public 
 lands into quarter-quarters, and requiring the division line to 
 run east and west. 
 
 350. A surveyor-general for each surveying district is 
 appointed by the President, by and with the advice of the 
 Senate. He is required, while in the discharge of the duties 
 of his office, to reside in the district for which he is appointed. 
 His term of office is four years, and he must give bonds, with 
 sufficient security for the penal sum of $30,000, for the faithful 
 disbursement of all public money placed in his hands, and for 
 the faithful performance of the duties of his office. Among 
 other duties prescribed by law and set forth in the manual, the 
 surveyor-general is required to engage a sufficient number of 
 skilful surveyors as his deputies, and shall cause to be sur- 
 veyed, measured, and marked, without delay, all base and 
 meridian lines through such points, and perpetuated by such 
 monuments, and such other correction parallels and meridians, 
 as may be prescribed by law, or by instructions from the 
 General Land Office, in respect to the public lands within his 
 surveying district to which the Indian title has been or may be 
 extinguished. 
 
 351. System of Rectangular Surveying. The public lands 
 of the United States are ordinarily surveyed into rectangular 
 tracts, bounded by lines conforming to the cardinal points.
 
 298 PLANE SURVEYING. 
 
 The public lands shall be laid off, in the first place, into 
 bodies of land 24 miles square, as near as may be. This shall 
 be done by the extension of standard lines from the principal 
 meridian every 24 miles, and by the extension from the base 
 and standard lines, of auxiliary meridians every 24 miles. 
 Thereafter they shall be laid off into bodies of land 6 miles 
 square, as near as may be, called townships, containing, as 
 near as may be, 23,040 acres. The townships shall be sub- 
 divided into 36 tracts, called sections, each containing, as near 
 as may be, 640 acres. Any number or series of contiguous 
 townships, situate north or south of each other, constitute a 
 range. 
 
 (a) The law requires that the lines of the public surveys 
 shall be governed by the true meridian, and that the township 
 shall be six miles square, two things involving in connection 
 a mathematical impossibility. For strictly to conform to the 
 meridian necessarily throws the township out of square, by 
 reason of the convergency of meridians, and hence by adhering 
 to the true meridian results the necessity of departing from the 
 strict requirements of law, as respects the precise area of town- 
 ships and the subdivisions! parts thereof; the township assum- 
 ing something of a trapezoidal form, which inequality develops 
 itself more and more as such, the higher the latitude of the 
 surveys. It is doubtless in view of these circumstances that 
 the law provides (see Section 2 of the act of May 18, 1796) 
 that the section of a mile square shall contain the quantity of 
 640 acres, as nearly as may be; and, morever, provides (see 
 Section 3 of the act of May 10, 1800) in the following words : 
 "And in all cases where the exterior lines of the townships 
 thus to be subdivided into sections or half-sections shall exceed, 
 or shall not extend, 6 miles, the excess or deficiency shall be 
 specially noted, and added to or deducted from the western or 
 northern ranges of sections or half-sections in such township, 
 according as the error may be in running the lines from east to 
 west or from south to north ; the sections and half-sections 
 bounded on the northern and western lines of such townships
 
 SURVEY OF THE PUBLIC LANl)S. 
 
 299 
 
 shall be sold as containing only the quantity expressed in the 
 returns and plats, respectively, and all others as containing the 
 complete legal quantity." 
 
 Sections 5 and 6 of Township No. 6, North, Range No. 34, 
 east, of the principal meridian, Montana, are exhibited below : 
 
 (5) The section lines are surveyed from south to north on 
 true meridians, and from east to west, in order to throw the 
 excesses or deficiencies in measurements on the north and west 
 sides of the township, as required by law. In a case where a 
 township has been partially surveyed, and it is necessary to 
 complete the survey of the same, or where the character of the 
 land is such that only the north or west portions of the town- 
 ship can be surveyed, this rule cannot be strictly adhered to ; 
 but in such cases must be departed from only so far as is 
 absolutely necessary. It will also be necessary to depart from 
 this rule where surveys close upon State or Territorial bound- 
 aries, or upon survej's extending from different meridians. 
 
 (c) The townships are to bear numbers in respect to the 
 base line, either north or south of it ; and the tiers of townships 
 called "ranges" will bear numbers in respect to the meridian 
 line, according to their relative position to it, either on the east 
 or west.
 
 300 PLANE SURVEYING. 
 
 (d) The 36 sections into which a township io subdivided 
 are numbered, commencing with number one at the northeast 
 angle of the township and proceeding west to number 6, and 
 thence proceeding east to number 12, and so on, alternately 
 until the number 36 is in the southeast angle. In all cases 
 of surveys of fractional townships, the sections should bear 
 the same numbers as they would if the township were full. 
 
 (e) Standard parallels shall be established at intervals of 
 every 24 miles, north and south of the base line, and auxiliary 
 meridians at intervals of every 24 miles, east and west of the 
 principal meridian ; the object being to confine the errors result- 
 ing from convergence of meridians and inaccuracies in measure- 
 ments, within the tracts of land bounded by the lines so estab- 
 lished. 
 
 (/) The survey of all principal base and meridian standard 
 parallels, and auxiliary meridian and township lines must be 
 made with an instrument operating independently of the mag- 
 netic needle. Burt's improved solar compass, or other instru- 
 ment of equal utility, must be used of necessity in such cases ; 
 and it is deemed best that such instrument should be used under 
 all circumstances. Where the needle can be relied on, however, 
 the ordinary compass may be used in subdividing and meander- 
 ing. Whenever deputies use instruments with magnetic appa- 
 ratus only, they must test the accuracy of their work and the 
 condition of their instruments by at least three observations 
 upon a circumpolar star, upon different days, between the com- 
 mencement and close of surveying operations in any given 
 township. Deputies using instruments with solar apparatus are 
 not required to make observations of the star Polaris, but they 
 must test their instruments by taking the latitude daily, weather 
 permitting, in running base, standard, meridian, and range 
 lines, and upon three different days, during the execution of 
 subdivisional surveys in each township. They must make com- 
 plete records in their field notes, under proper dates, of the 
 making of all observations in compliance with these instructions, 
 showing the style and condition of the instrument in use, and
 
 SURVEY OF THE PUBLIC LANDS. 301 
 
 the angle formed by comparing the line run with the meridian 
 as determined by observations. 
 
 (g) The construction and adjustments of all surveying in- 
 struments used in the surveying of the public lands of the 
 United States must be tested at least once a year, and oftener 
 if necessary, by comparison with the true meridian, established 
 under the direction of the surveyor-general of the district ; and 
 the instruments must be so modified in construction, or in such 
 a way corrected, as may be necessary to produce the closest 
 possible approximation to accuracy and uniformity in the 
 operation of all such instruments. A record will be made of 
 such examinations, showing the number and style of the instru- 
 ment, name of the maker, the quantity of instrumental error 
 discovered by comparison, in either solar or magnetic apparatus, 
 or both, and means taken for correction. The surveyor-general 
 will allow no surveys to be made until the instruments to be 
 used therefor have been approved by him. 
 
 (ft) The township lines and the subdivision lines will usually 
 be measured by a two-pole chain of 33.03 feet in length, con- 
 sisting of 50 links, and each link being 7.92 inches long. On 
 uniform and level ground, however, the four-pole chain may be 
 used. The measurements will, however, always be represented 
 according to the four-pole chain of 100 links. The four-pole 
 chains must be adjusted to lengths of 66.06 feet, The object 
 in adding six-hundredtlis of a foot to the 66 feet of a four-pole 
 chain is to assure thereby that 66 feet will be set off upon the 
 earth's surface without the application of a greater strain than 
 about 20 pounds by the chainmen, thus providing for loss by 
 vertical curvature of 'the chain, and at the same time avoiding 
 the uncertain results attending the application of strains taxing 
 its elasticity. The deputy surveyor must provide himself with 
 a measure of the standard chain kept at the office of the sur- 
 veyor-general, to be used by him as a field standard. The chain 
 in use must be compared and adjusted with this field standard 
 each working day ; and such field standard must be returned to 
 the surveyor-general's office for examination when his work is 
 'ompleted.
 
 302 PLANE SURVEYING. 
 
 352. Of Tally-Pins. You will use 11 tally-pins made of steel, 
 not exceeding 14 inches in length, weighty enough toward the 
 point to make them drop perpendicularly, and having a ring at 
 the top, in which is to be fixed a piece of red cloth, or some- 
 thing else of conspicuous color, to make them readily seen when 
 stuck in the ground. 
 
 353. Process of Chaining. In measuring lines with a two- 
 pole chain, every five chains are called a tally ; and in measur- 
 ing lines with a four-pole chain, every ten chains are called a 
 tally, because at that distance the last of the 10 tally-pins with 
 which the forward chainman set out will have been stuck. He 
 then cries "tally"; which cry is repeated by the other chain- 
 man, and each registers the distance by slipping a thimble, but- 
 ton, or ring of leather, or something of the kind, on a belt worn 
 for that purpose, or by some other convenient method. The 
 hind chainman then comes up, and having counted in the pres- 
 ence of his fellow the tally-pins which he has taken up, so that 
 both may be assured that none of the pins have been lost, he 
 then takes the forward end of the chain, and proceeds to set 
 the pins. Thus the chainmeu alternately change places, each 
 setting the pins that he has taken up, so that one is forward in 
 all the odd, and the other in all the even, tallies. Such pro- 
 cedure, it is believed, tends to insure accuracy in measurement, 
 facilitates the recollection of the distances to objects on the 
 line, and renders a mis-tally almost impossible. 
 
 354. Levelling the Chain and Plumbing the Pins. The length 
 of every line you run is to be ascertained by precise horizontal 
 measurement, as nearly appi'oximating to an air line as is possi- 
 ble in practice on the earth's surface. This all-important object 
 can only be attained by a rigid adherence to the three following 
 observances : 
 
 Ever keeping the chain stretched to its utmost degree of ten- 
 sion on even ground. 
 
 On uneven ground, keeping the chain not only stretched as
 
 SURVEY OF THE PUBLIC LANDS. 303 
 
 aforesaid, but horizontally levelled. And when ascending or 
 descending steep ground, hills, or mountains, the chain will have 
 to be shortened to one-half its length (and sometimes more), in 
 order accurately to obtain the true horizontal measurement. 
 
 The careful plumbing of the tally-pins, so as to attain pre- 
 cisely the spot where they should be stuck. The more uneven 
 the surface, the greater the caution needed to set the pins. 
 
 355. Marking Lines. All lines on which are to be estab- 
 lished the legal corner boundaries are to be marked after this 
 method, viz. : Those trees which may intercept the line must 
 have two chops or notches on each side of them, without any 
 other marks whatever. These are called sight trees or line trees. 
 A sufficient number of other trees standing within 50 links of 
 the line, on either side of it, are to be blazed on two sides 
 diagonally, or quartering toward the line, in order to render the 
 line conspicuous, and readily to be traced, the blazes to be 
 opposite each other, coinciding in direction with the line where 
 the trees stand very near it, and to approach nearer each other 
 the farther the line passes from the blazed trees. 
 
 Where trees two inches or more in diameter are found, the 
 required blazes must not be omitted. 
 
 Bushes on or near the line should be bent at right angles 
 therewith, and receive a blow of the axe at about the usual 
 height of blazes from the ground sufficient to leave them in a 
 bent position, but not to prevent their growth. 
 
 356. On Trial or Random Lines the trees are not to be 
 blazed, unless occasionally, from indispensable necessity, and 
 then it must be done so guardedly as to prevent the possibility^ 
 of confounding the marks of the trial line with the true. But 
 bushes and limbs of trees may be lopped, and stakes set on the 
 trial or random line, at every ten chains, to enable the surveyor 
 on his return to follow and correct the trial line, and establish 
 therefrom the true line. To prevent confusion, the temporary 
 stakes set on the trial or random lines must be pulled up when 
 the surveyor returns to establish the true line.
 
 304 PLANE SURVEYING. 
 
 357. Insuperable Objects on Line ; Witness Points. Under 
 circumstances where your course is obstructed by impassable 
 obstacles, such as ponds, swamps, marshes, lakes, rivers, 
 creeks, etc., you will prolong the line across such obstacles 
 by means of right-angle offsets ; or, if such be inconvenient, 
 by a traverse or trigonometrical operation, until you regain the 
 line on the opposite side. And in case a north and south, or a 
 true east and west, line is regained in advance of any such 
 obstacle, you will prolong and mark the line back to the ob- 
 stacle so passed, and state all the particulars in relation thereto 
 in your field-book. And at the intersection of lines with both 
 margins of impassable obstacles you will establish a witness 
 point (for the purpose of perpetuating the intersections there- 
 with) , by setting a post, and giving in your field-book the course 
 and distance therefrom to two trees on opposite sides of the 
 line, each of which trees you will mark with a blaze and notch 
 facing the post ; but on the margins of navigable watercourses 
 or navigable lakes you will mark the trees with the proper 
 number of the fractional section, township, and range. 
 
 358. The Best Marking-Tools adapted to the purpose must 
 be provided for marking neatly and distinctly all the letters and 
 figures required to be made at corners, Arabic figures being 
 used exclusively ; and the deputy is always to have at hand the 
 necessary implements for keeping his marking-tools in order. 
 
 359. Establishing Corners. To procure the faithful execu- 
 tion of this portion of a surveyor's dutv is a matter of the 
 utmost importance. After a true coursing and most exact 
 measurement, the establishment of corners is the consummation 
 of the work. If, therefore, the corners be not perpetuated in a 
 permanent and workmanlike manner, the great aim of the sur- 
 veying service will not have been attained. 
 
 The following are the different points for perpetuating cor- 
 ners, viz. : 
 
 (a) For township boundaries, at intervals of every 6 miles.
 
 SURVEY OF THE PUBLIC LANDS. 305 
 
 (6) For section boundaries, at intervals of every mile, or 80 
 chains. 
 
 (c) For quarter-section boundaries, at intervals of every half- 
 mile, or 40 chains. Exceptions, however, occur, as fully set 
 forth hereafter in that portion of the manual showing the man- 
 ner of running township lines and method of subdividing. 
 
 (d) Meander corners are established at all those points where 
 the lines of the public surveys intersect the banks of such 
 rivers, bayous, lakes, or islands, as are by law directed to be 
 meandered. 
 
 360. Miscellaneous. When a rock in place is established for 
 a corner, its dimensions above ground must be given, and a 
 cross ( X ) marked at exact corner point. 
 
 Where mounds of earth are raised " alongside " of corners on 
 N. and S. lines, they must be placed on the W., and on the E. 
 and W. lines on the N. side of corner. In case the character 
 of the land is such that this cannot be done, the deputy will 
 state in his notes instead of " alongside" " S." (on E.). 
 
 In case where pits are practicable, the deputy prefers raising 
 a mound of stone, or stone covered with earth, as more likely 
 to perpetuate the corner ; he will use the form given for mound 
 of stone, omitting the words "pits impracticable," and adding 
 " covered with earth," when so established. 
 
 Where the requisite number of trees can be found within 300 
 links of the corner point, three (3) bearing trees should be 
 established for every standard or closing corner, four (4) for 
 every corner common to four townships or sections, and two (2) 
 for every quarter-section corner or meander corner. In case the 
 requisite number cannot be found within limits, the deputy 
 must state in his field notes, after describing those established, 
 " no other trees within limits," and "dug pits in sees. & ," 
 or " raised a mound of stone alongside." 
 
 Stones 18 inches and less long must be set two-thirds, and 
 over 18 inches long, three-fourths, of their length in the ground. 
 No stones containing less than 504 cubic inches must be used
 
 306 PLANE SURVEYING. 
 
 for corners. Particular attention is called to the " summary of 
 objects and data required to be noted," on pages and of 
 these instructions, and it is expected that the deputy will thor- 
 oughly comply with the same in his work and field notes. 
 
 No mountains, swamp lands, or lands not classed as survey- 
 able, are to be meandered, and all lines approaching such lands 
 must be discontinued at the section or quarter-section corner. 
 
 Where, by reason of impassable objects, the south boundary 
 of a township cannot be established, an east and west line 
 should be run through the township, first random, and then 
 corrected, from one range line to the other, and as far south as 
 possible, and from such line the section lines will be extended 
 in the usual manner, except over any fractions south of said 
 line, which may be surveyed in the opposite direction from the 
 section corners on the auxiliary base thus established. 
 
 When no part of the east or west boundaries can be run, 
 both north and south boundaries will be established as true 
 lines. Allowance for the convergency of meridians must be 
 made whenever necessary.* 
 
 All letters and figures cut in posts or trees must be marked 
 over with red chalk to make them still more plain and durable. 
 Township corners common to four townships, and section cor- 
 ners common to four sections, are to be set diagonally in the 
 earth, with the angles in the direction of the lines. All other 
 corners are to be set square, with the sides facing the direction 
 of the lines. The sizes of wooden posts, mounds, and pits, 
 noted in foregoing descriptions of corners, are to be regarded 
 as minimum, and whenever practicable to increase their dimen- 
 sions, it is desirable to do so. In establishing corners, stones 
 should be used whenever practicable ; then posts ; and lastly, 
 mounds, with stake in pit. 
 
 It is expected that deputy surveyors will carefully read and 
 familiarize themselves with these instructions, and all others 
 
 * See Table of Convergency of Meridians at end of chapter, and 
 explanation of same.
 
 SURVEY OF THE PUBLIC LANDS. 307 
 
 contained in this volume, and will instruct their assistants as to 
 their duties before commencing work. Extra copies will be 
 furnished the deputies for the use of their assistants. 
 
 361. Standard Quarter-Section Corners on standard lines 
 must be established in all respects like other quarter-section 
 corners, with the addition of the letters S.C. ; and if bearing 
 trees are established for such corners, each tree must be marked 
 S.C. \ S.B.T. When a pit is dug at a meander corner, it must 
 be 8 links from the corner on the side opposite the river or lake 
 meandered. 
 
 The letters M.C., for " meander corner," must be marked on 
 the side facing the river or lake meandered. 
 
 362. A Witness Corner, in addition to the marks that would 
 be placed upon the corner for which it is a witness, must have 
 the letters W.C., and be established in all respects like such 
 corner. 
 
 If bearing trees are established for a witness corner, each 
 tree must be marked W.C., in addition to the usual marks. 
 
 363. Meandering. Both banks of navigable rivers are to be 
 meandered by taking the general courses and distances of their 
 sinuosities. 
 
 At those points, when either the township or section lines 
 intersect the banks of a navigable stream, corners are to be 
 established at the time of running these lines. These are 
 called meander corners; and in meandering, you are to 
 commence at one of these corners, coursing the banks, and 
 measuring the distance of each course from your commencing 
 corner to the next meander corner. By the same method, you 
 are to meander the opposite bank of the same river. The 
 crossing distance between meander corners on same line is to be 
 ascertained by triang illation, that the river may be accurately 
 protracted. Rivers not classed under the statute as navigable, 
 but which are well-defined natural arteries of internal communi- 
 cation, will only be meandered on one bank.
 
 308 PLANE SURVEYING. 
 
 All lakes, bayous, and deep ponds which may serve as public 
 highways of commerce must be meandered. 
 
 364. Surveying. Initial points, from which the lines of the 
 public surveys are to be extended, must be established when- 
 ever necessary under special instructions, as may be prescribed 
 in each case by the Commissioner of the General Land Office. 
 The locus of such initial points must be selected with great 
 care and due consideration for their prominence and easy 
 identification, and must be established astronomically. 
 
 The initial point having been established, the lines of the 
 public surveys are to be extended therefrom as follows : 
 
 365. Base Line. The base line shall be extended east and 
 west from the initial point by the use of solar instruments or 
 transits, as may be directed by the surveyor-general in his 
 special written instructions. Where solar instruments are used, 
 the deputy must test said instruments in every 12 miles of line 
 run, by taking the latitude, or by observation on the polar star ; 
 and in all cases where he has reason to suppose that said instru- 
 ment is in error, he must take an observation on the polar star ; 
 and if error be found, must make the necessary corrections 
 before proceeding with his survey. The proper corners shall 
 be established at each 40 and 80 chains, and at the intersection 
 of the line with rivers, lakes, or bayous that should be mean- 
 dered, in accordance with the instructions for the establishment 
 of corners. In order to check errors in measurement, two sets 
 of chainmen, operating independently of each other, must be 
 employed. 
 
 Where transits are used, the line will be run by setting off at 
 the point of departure on the principal meridians a tangent 
 to the parallel of latitude, which will be a line falling at right 
 angles to the said meridian. The survey will be continued on 
 this, line for twelve (12) miles, but the corners will be estab- 
 lished at the proper points by offsets northerly from said line, 
 at the end of each half-mile. In order to offset correctly from
 
 SURVEY OF THE PUBLIC LANDS. 309 
 
 the tangent to the parallel, the deputy will be guided by the table 
 of offsets and azimuths contained in the Manual of Instructions. 
 As the azimuth of the tangent is shown, the angle thence to 
 the true meridian at each mile is readily found, thus indicating 
 the direction of the offset line. The computations are made for 
 a distance of 12 miles, at the end of which observations on the 
 polar star must be taken for the projection of a new tangent. 
 The computations are also upon even degrees of latitude ; off- 
 sets for intervening parallels can be readily determined by 
 interpolation. Where offset distances quarter-section corners 
 exceed 50 links, their direction to the parallel can be deter- 
 mined in like manner by interpolation for azimuth. When said 
 distances are less than 50 links, interpolation for determining 
 the distances will not be required. 
 
 366. Principal Meridian. The principal meridian shall be 
 extended north and south from the initial point, by the use of 
 solar instruments or transits, as may be directed by the sur- 
 veyor-general in his special written instructions. 
 
 Where solar instruments are used, the line will be run in the 
 same manner as prescribed for running the base line by solar 
 instruments. Where transits are used, observations upon the 
 polar star must be taken within each 12 miles of line run. In 
 addition to the above general instructions, it is required that in 
 all cases where the establishment of a new principal meridian 
 seems to be necessary to the surveyor-general, he shall submit 
 the matter, together with his reasons therefor, to the Commis- 
 sioner of the General Land Office, and the survey of such prin- 
 cipal meridian shall not be commenced until written authority, 
 together with such special instructions as he may deem neces- 
 sary, shall have been received from the Commissioner. 
 
 367. Standard Parallels. Standard parallels, which are 
 also called correction lines, shall be extended east and west 
 from the principal meridian, at intervals of every 24 miles 
 north and south of the base line, in the same manner as pre- 
 scribed for running the base line.
 
 310 PLANE SURVEYING. 
 
 Auxiliary Meridians. Auxiliary meridians shall be extended 
 north and south from the base line, at intervals of every 24 
 miles east and west from the principal meridian, in the same 
 manner as prescribed for running the principal meridian. 
 
 It is contemplated that these base, principal meridian, stand- 
 ard, and auxiliary meridian lines shall first be extended over 
 the territory to be surveyed, and that afterwards township and 
 section lines shall be run, where needed, within these tracts of 
 24 miles square, formed by the extension of these principal 
 lines ; and each surveyor-general will therefore cause said prin- 
 cipal lines to be extended as rapidly as practicable. 
 
 368. Exteriors, or Township Lines. The east and west 
 boundaries of townships are always to be run from south to 
 north on a true meridian line ; and the north and south boun- 
 daries are to be run from east to west, or from west to east 
 (according to the relation of the township to be surveyed with 
 reference to prior surveys) , on a random or trial line, and cor- 
 rected back on a true line. The distance north or south of the 
 township corner to be closed upon, from the point of intersec- 
 tion of these random lines with the east or west boundary of 
 the township, must be carefully measured and noted. Should 
 it happen, however, that such random line should fall short, or 
 overrun in length, or intersect the east or west boundary more 
 than three chains' distance from the township corner thereon, as 
 compared with the corresponding boundary on the south (due 
 allowance being made for convergency) the line, and if neces- 
 sary the entire exterior boundaries of the township, must be 
 retraced, so as to discover and correct the error. In running 
 random lines, temporary corners are to be set at each 40 and 
 80 chains, and permanent corners established upon the true 
 line as corrected back, in accordance with instructions, throw- 
 ing the excess or deficiency on the west half-mile, as prescribed 
 by law. Permanent corners are to be established, in accord- 
 ance with instructions, on the east and west township bound- 
 aries at the time they are to be run. Whenever practicable,
 
 SURVEY OF THE PUBLIC LANDS. 311 
 
 the township lines within these tracts of 24 miles square, must 
 be surveyed in regular order from south to north; i.e., the 
 exterior boundaries of the township, in any one range lying 
 immediately north of the south boundary of such tract of 24 
 miles square, must first be surveyed, and the exteriors of the 
 other three townships in said range extended therefrom, in 
 regular order, from south to north ; and it is preferable to sur- 
 vey first the entire range of townships in such tract adjoining 
 the east boundary, or adjoining the west boundary, and the 
 other three ranges in regular sequence. In cases, however, 
 where the character of the land is such that this rule cannot be 
 complied with, the' following will be observed. In extending 
 the south or north boundaries of a township to the west, where 
 the southwest or northeast corners cannot be established in the 
 regular way by running a north and south line, such boundaries 
 will be run west on a true line, allowing for convergency on the 
 west half-mile ; and from the township corner established at 
 the end of such boundary, the west boundary will be ruu north 
 or south, as the case may be. In extending south or north of a 
 township to the east, where the southeast or northeast corner 
 cannot be established in the regular way, the same rule will be 
 observed, except that such boundaries will be run east on a true 
 line, and the ea.si.boundary run north or south, as the case may be. 
 One set of chainmen only is required in running township lines. 
 
 369. Method of Subdividing. The first mile, both on the 
 south and east boundaries of each township you are required to 
 subdivide, is to be carefully traced and measured before you 
 enter upon the subdivision thereof. This will enable you to 
 observe any change that may have taken place in the magnetic 
 variation as it existed at the time of running the township 
 lines, and will also enable you to compare your chaining with 
 that upon the township lines. 
 
 Any discrepancy arising either from a change in the magnetic 
 variation or a difference in measure is to be carefully noted in 
 the field notes.
 
 312 PLANE SURVEYING. 
 
 After adjusting your compass to a variation which you have 
 just found will retrace the eastern boundary of the township, 
 you will commence at the corner to Sections 35 and 36, on the 
 south boundary, and run a line parallel to the range line, 40 
 chains, to the quarter-section corner, which you are to establish 
 between Sections 35 and 36 ; continuing on said course 40 
 chains farther, you will establish the corner to Sections 25, 26, 
 35, and 36. 
 
 From the section corner last named, run a random line, with- 
 out blazing, due east, for the corner of sections 25 and 36, on 
 east boundary, and at 40 chains from the starting-point set a 
 post for temporary quarter-section corner. If you intersect 
 exactly at the corner, you will blaze your random line back, 
 and establish it as the true line ; but if your random line inter- 
 sects the said east boundary either north or south of said corner, 
 you will measure the distance of such intersection, from which 
 you will calculate a course that will run a true line back to the 
 corner from which your random started. You will establish 
 the permanent quarter-section corner at a point equidistant 
 from the two terminations of the true line. 
 
 From the corner of Sections 25, 26, 35, and 36, run due north 
 between Sections 25 and 26, setting the quarter-section post, as 
 before, at 40 chains, and at 80 chains establishing the corner 
 of Sections 23, 24, 25, and 26. Then run a random due east 
 for the corner of Sections 24 and 25 on east boundary ; setting 
 temporary quarter-section post at 40 chains ; correcting back, 
 and establishing permanent quarter-section corner at the equi- 
 distant point on the true line, in the manner directed on the 
 line between Sections 25 and 36. 
 
 In this manner you will proceed with the survey of each suc- 
 cessive section in the first tier until you arrive at the north 
 boundary of the township, which you will reach in running up a 
 random line between Sections 1 and 2. If this random line 
 should not intersect at the corner established for Sections 1 , 2, 
 35, and 36, upon the township line, you will note the distance 
 that you fall east or west of the same, from which distance you
 
 SUKVEY OF THE PUBLIC LANDS. 313 
 
 will calculate a course that will run a true line south to the 
 corner from which your, random started. If the north boundary 
 of a township is a base or standard line, the line between Sections 
 1 and 2 is to be run north as a true line, and the closing corner 
 established at the point of intersection with such base or stand- 
 ard line ; and in such case, the distance from said closing 
 corner to the nearest section or quarter-section corner on such 
 base or standard line must be carefully measured and noted as 
 a " connection line." 
 
 In like manner proceed with the survey of each successive 
 tier of sections until you arrive at the fifth tier ; and from each 
 section corner which you establish upon this tier you are to 
 run random lines to the corresponding corners established upon 
 the range line forming the western boundary of the township ; 
 setting as you proceed each temporary quarter-section corner 
 at 40 chains from the interior section corner, so as to throw 
 the excess or deficiency of measurement on the extreme tier of 
 quarter-sections contiguous to the township boundary ; and on 
 returning establish the true line, and establish thereon the per- 
 manent quarter-section corner. 
 
 It is not required that the deputy shall complete the survey 
 of the first tier of sections from north to south before commenc- 
 ing the survey of the second or any subsequent tier, but the 
 corner on which the random line closes must have been pre- 
 viously established by running the line north on which it is 
 established, except as follows : where it is impracticable to 
 establish such section corner in the regular manner, it may be 
 established by running the east and west line east or west, as 
 the case may be, on a true line, setting the quarter-section 
 corner at 40 chains and the section corner at 80 chains. 
 
 Quarter-section corners, both upon north and south and upon 
 east and west lines, are to be established at a point " equi- 
 distant" from the corresponding section corners, except upon 
 the lines crossing on the north and west boundaries of the town- 
 ship, and in those situations the quarter-section corners will 
 always be established at precisely 40 chains to the north or
 
 314 PLANE SURVEYING. 
 
 west, as the case may be, of the respective section corners 
 from which those lines respectively start, by which procedure 
 the excess or deficiency in the measurements will be thrown, 
 according to law, on the extreme tier of quarter-sections. 
 
 370. Prescribed Limits for Closing, and Length of Lines in 
 Certain Cases. Every north-and-south section line, except those 
 terminating in the north boundary of the township, must be 80 
 chains in length. 
 
 The east-and-west section lines, except those terminating in 
 the west boundary of the township, are to be within 80 links of 
 the actual distance established on the south boundary line of 
 the township for the width of said tier of sections, and must 
 close within 80 links north or south of the section corner. 
 
 The north boundary and south boundary of any one section, 
 except in the extreme western tier, are to be within 80 links 
 of equal length. 
 
 The meanders within each fractional section, or between two 
 meander posts, or of an island in the interior of a section, must 
 close within 1 chain and 50 links. 
 
 In running random township exteriors, if such random lines 
 fall short or overrun in length or intersect the eastern or west- 
 ern boundary, as the case may be, of the township at more than 
 3 chains north or south of the true corner, the lines must be 
 retraced, even if found necessary to measure the meridional 
 boundaries of the township. One set of chain men only is 
 required in subdividing. 
 
 371. Subdivision of Sections. Under the provisions of the 
 act of Congress approved Feb. 11, 1805, the course to be pur- 
 sued in the subdivision of sections is to run straight lines from 
 the established quarter-section corners United States surveys 
 to the opposite corresponding corners, and the point of inter- 
 section of the lines so run will be the corner common to the 
 several quarter-sections ; or, in other words, the legal centre of 
 the section.
 
 SURVEY OF THE PUBLIC LANDS. 315 
 
 In the subdivision of fractional quarter-sections where no 
 opposite corresponding sections have been or can be fixed, the 
 subdivision lines should be ascertained by running from the 
 established corners due north, south, east, or west lines, as 
 the case may be, to the watercourse, Indian boundary line, or 
 other external boundary of such fractional section. The law 
 presupposes the section lines surveyed and marked in the field 
 by the United States deputy surveyors to be due north and 
 south or east and west lines, but in actual experience this is not 
 always the case ; hence, in order to carry out the spirit of the 
 law, it will be necessary in running the subdivisioual lines 
 through fractional sections to adopt mean courses where the 
 section lines are not due lines, or to run the subdivision line 
 parallel to the section line where there is no opposite section 
 line. 
 
 Upon the lines closing on the north and west boundaries of a 
 township the quarter-section corners are established by the 
 United States deputy surveyors at precisely 40 chains to the 
 north or west of the last interior section corners, and the excess 
 or deficiency in the measurement is thrown on the outer tier of 
 lots, as per act of Congress approved May 10, 1800. In the 
 subdivision of quarter-sections, the quarter-quarter corners are 
 to be placed at points equidistant between the section and 
 quarter-section corners, and between the quarter corners and 
 the common centre of the section, except on the last half-mile 
 of the lines closing on the north or west boundaries of a 
 township, where they should be placed at 20 chains, propor- 
 tionate measurement, to the north or west of the quarter- 
 section corner. 
 
 The subdivisional lines of fractional quarter-sections should 
 be run from points on the section lines intermediate between 
 the section and quarter-section corners due north, south, east, 
 or west, to the lake, watercourse, or reservation which renders 
 such tracts fractional. 
 
 When there are double sets of section corners on township 
 and range lines, the quarter corners for the sections south of the
 
 816 FLANE SURVEYING. 
 
 township lines and east of the range lines are not established 
 in the field by the United States surveyors, but in subdividing 
 such sections said quarter corners should be so placed as to suit 
 the calculations of the areas of the quarter-sections adjoining the 
 township boundaries as expressed upon the official plot, adopt- 
 ing proportionate measurements where the present measure- 
 ments of the north or west boundaries of the sections differ 
 from the original measurements. 
 
 372. Re-establishment of Lost Corners. The original cor- 
 ners, when they can be found, must stand as the true corners 
 they were intended to represent, even though not exactly where 
 strict professional care might have placed them in the first 
 instance. 
 
 As has been observed, no existing original corner can be dis- 
 turbed, and it will be plain that any excess or deficiency in 
 measurements between existing corners cannot in any degree 
 affect the distances beyond said existing corners, but must be 
 added or subtracted proportionately to or from the intervals 
 embraced between the corners which are still standing. 
 
 373. Summary of Objects and Data required to be Noted. 
 The precise length of every line run, noting all necessary off- 
 sets therefrom, with the reason and mode thereof. 
 
 The kind and diameter of all bearing trees, with the course 
 and distance of the same from their respective corners, and the 
 precise relative position of ivitness corners to the true corners. 
 
 The kind of materials of which corners are constructed. 
 
 Trees on line. The name, diameter, and distance on line to 
 all trees which it intersects. 
 
 Intersections by line of land objects. The distance at which 
 the line first intersects and then leaves every settler's claim and 
 improvements; prairie, river, creek, or other " bottom "; or 
 swamp, marsh, grove, and windfall, with the course of the 
 same at both points of intersection ; also the distances at which 
 you begin to ascend, arrive at the top, begin to descend, and
 
 SURVEY OF THE PUBLIC LANDS. 31? 
 
 reach the foot of all remarkable hills and ridges, with their 
 courses, and estimated height, in feet, above the level land of 
 the surrounding country, or above the bottom lands, ravines, 
 or waters near which they are situated. 
 
 Intersection by line of water objects. 
 
 All rivers, creeks, and smaller streams of water which the 
 line crosses ; the distances on line at the points of intersection ; 
 and their icidtlis on line. In cases of navigable streams, their 
 width will be ascertained between the meander corners, as set 
 forth under the proper head. 
 
 The land's surface whether level, rolling, broken, or hilly. 
 
 The soil whether first, second, third, or fourth rate. 
 
 Timber the several kinds of timber and undergrowth, in 
 the order in which they predominate. 
 
 Bottom lands to be described as wet or dry ; and if sub- 
 ject to inundation, state to what depth. 
 
 Springs of water whether fresh, saline, or mineral, with 
 the course of the stream flowing from them. 
 
 Lakes and ponds describing their banks and -giving their 
 height, and also depth of water, and whether it be pure or 
 stagnant. 
 
 Improvements towns and villages ; houses or cabins ; fields, 
 or other improvements ; sugar-tree groves, sugar camps, mill 
 seats, forges, and factories. 
 
 Coal bank or beds ; peat or turf grounds ; minerals and ores, 
 with particular description of the same as to quality and extent, 
 and all diggings therefor ; also salt springs and licks. All 
 reliable information you can obtain respecting these objects, 
 whether they be on your immediate line or not, is to appear 
 on the general description to be given at the end of the notes. 
 
 Roads and trails, with their directions whence and whither. 
 
 Rapids, cataracts, cascades, or falls of water, with the esti- 
 mated height of their fall in feet. 
 
 Precipices, caves, sink holes, ravines, stone quarries, ledges 
 of rocks, with the kind of stone they afford. 
 
 Natural curiosities, interesting fossils, petrifactions, organic
 
 318 PLANE SURVEYING. 
 
 remains, etc. ; also all ancient works of art, such as mounds, 
 fortifications, embankments, ditches, or objects of like nature. 
 
 The variation of the needle must be noted at all points or 
 places on the lines where there is found any material change 
 of variation ; and the positions of such points must be perfectly 
 identified in the notes. 
 
 Besides the ordinary notes taken on line (and which must 
 always be written down on the spot, leaving nothing to be sup- 
 plied by memory), the deputy will subjoin, at the conclusion of 
 his book, such further description or information touching any 
 matter or thing connected with the township (or other survey) 
 which he may be able to afford, and may deem useful or neces- 
 sary to be known, with a general description of the township in 
 the aggregate, as respects the face of the country, its soil and 
 geological features, timber, minerals, waters, etc. 
 
 374. Specimen Field Notes of the survey of the Third 
 Standard Parallel North, through Range No. 21 east, of the 
 principal base and meridian in the Territory of Montana, as 
 surveyed by James Page, U. S. Deputy Surveyor. 
 
 On the night of August 22, 1880, I took observation on the 
 star Polaris, in accordance with instructions contained in the 
 " Manual of Surveys," and drove pickets on the line thus estab- 
 lished. 
 
 Survey commenced August 23, 1880, with a Burt's Improved 
 Solar Compass. 
 
 Before commencing this survey, I test my compass on the 
 line established last night, and find it correct. I begin at the 
 standard corner to townships 13 north, ranges 20 and 21 east, 
 which is a post, 4 inches square, marked : 
 
 S.C., T. 13 N., on N. ; R. 21 E., S. 31, on E. ; and R. 
 20 E., S. 36, on W. faces, with 6 notches on N., E., and W. 
 faces, and pits N., E., and W. of post, 6 ft. dist., and mound 
 of earth around post. 
 
 Thence I run
 
 SURVEY OF THE PUBLIC LANDS. 
 
 319 
 
 East, on S. boundary Sec. 31. 
 
 Variation 20 E. 
 Ascend. 
 
 A point about 200 ft. above township cor. top of ridge. 
 Set a sandstone 18 x 8 x 5 ins., 12 ins. in the 
 ground, for standard ^ sec. cor. marked S.C. \ on 
 N. face ; dug pits 18x18x12 ins. E. and W. of 
 stone, 51- ft. dist., and raised a mound of earth 
 1^ ft. high. 31 ft. base alongside ; thence 
 Enter pine timber. 
 
 Set a sandstone 24 x 10 x 7 ins., 18 ins. in the ground 
 for standard cor. to sees. 31 and 32, marked S.C. 
 with 5 notches on E. and 1 notch on W. edges ; 
 from which 
 
 A pine, 12 ins. diam., bears N. 77 E., 41 Iks. dist., 
 marked T. 13 N., R. 21 E., S. 32 B.T. ; 
 
 A pine, 18 ins. diarn., bears N. 50 W., 20 Iks. dist., 
 marked T. 13 N., R. 21 E., S. 31 B.T. ; 
 
 A pine, 7 ins. diam., bears S. 30 W., 119 Iks. dist., 
 marked T. 12 N., R. 21 E., S. 5 B.T. 
 
 Land, high, mountainous, hilly, and rolling. 
 
 Soil, sandy, gravel, and rocky ; 4th rate. 
 
 Timber, pine, 23 chs. ; mostly dead and fallen. 
 
 East on S. boundary Sec. 32. 
 Through timber. 
 Va. 20| E. 
 
 Ravine, course S., about 30 ft. deep. 
 
 Ravine, course S. 20 E., about 20 ft. deep. 
 
 Set a sandstone, 18 x 14 X 5 ins., 12 ins. in the ground, 
 for standard \ sec. cor. marked S.C., \ on N. face, 
 and raised a mound of stone alongside. 
 
 Pits, impracticable. 
 
 Top of ridge, about 100 ft. high. 
 
 Ravine, course S., about 40 feet deep.
 
 320 
 
 PLANE SURVEYING. 
 
 80.00 
 
 Set a post, 4-- ft. long, 4 ins. square, with marked 
 
 stone, 12 ins. in the ground, for standard cor. to 
 
 sees. 32 and 33, marked : 
 
 S.C., T. 13 N., R. 21 E., on N. ; 
 S. 33, on E. ; and 
 S. 32, on W. faces, with 4 notches on E. and 2 notches 
 
 on W. faces, and raised a mound of stone 2 ft. 
 
 high, 4^ ft. base, around post. 
 Land, high and mountainous. 
 Soil, sandy, gravelly, and rocky ; 4th rate. 
 Timber, pine, and fir, 80 chs. ; mostly dead and 
 
 fallen ; some thick undergrowth, same. 
 
 375. Specimen Field Notes of the survey of Township No. 
 6 north, Range No. 34 east, of the principal base and meridian 
 of Montana Territory. 
 
 chains. 
 
 16.40 
 40.00 
 79.96 
 
 39.98 
 
 79.96 
 
 East, on random line, bet. sees. 5 and 8. 
 
 Va. 18 45' E. 
 Over rolling ground. 
 Road to Williamsburg, course S. 
 Set temporary \ sec. cor. 
 Intersected N. and S. line 6 Iks. N. of cor. to sees. 
 
 4, 5, 8, and 9. 
 Thence I run 
 N. 89 56' W. on true line, bet. sees. 5 and 8. with 
 
 same Va. 
 Set a post 3 ft. long, 3 ins. square, with marked stone, 
 
 12 ins. in the ground, for \ sec. cor. marked ^ S. 
 
 on N. face ; dug pits, 18 x 18 x 12 ins. E. and W. 
 
 of post 5 ft. dist., and raised a mound of earth. 
 
 1^ ft. high, 31 ft. base, around post. 
 The cor. to sees. 5, 6, 7, and 8. 
 Land, rolling. 
 Soil, sandy ; 2d rate. 
 No timber.
 
 SUUVKY OF THE PUBLIC LANDS. 
 
 321 
 
 West, on random line, between sees. 6 and 7. 
 
 Over rolling ground. 
 
 Road to Williamsburg, course S. 
 
 Set temporary ^ sec. cor. 
 
 Intersect west boundary of township 15 Iks. S. of cor. 
 
 to sees. 1, 6, 7, and 12, which is a post, 4 ft. long, 
 
 4 ins. square, marked : 
 T. 6 N.S. 6 on N.E. 
 R. 34 E.S. 7 on S.E. 
 R. 33 E.S. 12 on S.W., 
 
 and S.- 1 on N.W. faces, with pits, 18 x 18 x 12 
 
 ins. in each sec., 5^ ft. dist., and mound of earth, 
 
 2 ft. high, 4 ft. base, around post. 
 Thence I run 
 S. 89 54' E. on a true line, bet. sees. 6 and 7, with 
 
 same Va. 
 Set a sandstone, 18 x 14 x 3 ins., 12 ins. in the ground, 
 
 for |- sec. cor., marked \ on N. side; dug pits 
 
 18 x 18 x 12 ins. E. and W. of stone 5J ft. distant, 
 
 and raised a mound of earth, 1 ft. high, 3 base, 
 
 alongside. 
 
 The cor. to sees. 5, 6, 7, and 8. 
 Land, rolling. 
 Soil, sandy ; 2d rate. 
 No timber. 
 
 North, on a random line, bet. sees. 5 and 6. 
 Va. 18 45' E. 
 
 Over rolling ground. 
 
 Set temporary \ sec. cor. 
 
 Intersect N. boundary of township 20 Iks. E. of cor. 
 to sees. 5, 6, 31, and 32, which is a sandstone 
 30 x 12x6 ins., marked with 5 notches on E. and 
 one notch on W. edges, and mound of stone, 2 ft. 
 high, 4 ft. base, alongside. 
 
 Thence I run
 
 PLANE SURVEYING. 
 
 40.05 
 
 80.05 
 
 S. 09' E. on a true line bet. sees. 5 and 6, with 
 same Va. 
 
 Set a sandstone, 16 X 12 x 3 ins. 11 ins. in the ground, 
 for ^ sec. cor. marked \ on W. face ; dug pits, 
 18 x 18 x 12 ins., N. and S. of stone, 5 ft. dist., 
 and raised a mound, of earth, 1^ ft. high, 3^ ft. 
 base, alongside. 
 
 The cor. to sees. 5, 6, 7, and 8. 
 
 Land, rolling. 
 
 Soil, sandy ; 2d rate. 
 
 No timber. 
 
 INCLINATION OF THE MERIDIAN.* 
 
 376. In projecting arcs of a great circle it is of the utmost 
 importance that the surveyor be able to tell the inclination of 
 the meridians for any latitude, and for 
 II any distance of eastings or westings. 
 
 In the following figure, let the two 
 arcs AG and BG be two arcs of a 
 quadrant of the meridian 1 of longi- 
 tude apart. Let AB = the arc of 1 
 F of longitude on the equator = 69.16 
 miles. 
 
 Let DE be an arc of longitude on 
 any parallel of latitude. Also, let EH 
 and DH be the tangents of those me- 
 ridians meeting in the earth's axis 
 produced, and corresponding to the 
 parallel of latitude DE. 
 Then the line EF=DF=cos 7, = cos AD or BE. Also, 
 the angle DFE = 1, and the angle DHE = the inclination of 
 
 * These articles on the inclination and convergency of meridians, and 
 the table calculated in accordance therewith, are substantially those given 
 in the 1886 catalogue of engineers' and surveyors' instruments, by Buff 
 and Berger, Boston, Mass.
 
 SURVEY OF THE PUBLIC LANDS. 323 
 
 the meridians, which is the angle we wish to find, aud which 
 we will represent by X. And because the two triangles FDE 
 and DHE are on the same base ED, and isosceles, their vertical 
 angles vary inversely as their sides ; and we have the equation, 
 1 x EF= X x EH. 
 
 But EF = cos L, and EH = cot ; 
 
 hence X cot L = 1 cos L, 
 
 or X = cos L -T- cot L = sin L. (a) 
 
 That is to say, 
 
 The inclination of the meridians for any difference of longitude 
 varies as the sine of the latitude. 
 
 Since the sine of the latitude is the inclination in decimals 
 of a degree, for one degree of longitude, if we multiply by 3600" 
 we shall have the inclination in seconds of arc. Then, if we 
 divide this by the number of miles in one degree of longitude 
 on that latitude, we shall have the inclination due to one mile 
 on that parallel. Thus, for 
 
 Latitude 43 log. sin = 9.833783 
 
 Multiply by 3600" " = 3.556303 
 
 3.390086 
 
 Divide by 50.66m. = 1 long, on that L. log. = 1.704682 
 48.46" = inclination for one mile of long. 1.685404 
 
 The use of the inclination, as found by the preceding article, 
 is to show the surveyor how much he must deflect a line of 
 survey from the due east or west, to have it meet the parallel 
 at a given distance from the initial point of the survey ; for 
 it will be remembered that a parallel of latitude is a curve 
 having the cosine of the latitude for its radius. And the line 
 due east or west is the tangent of the curve. 
 
 Thus, on latitude 43, it is desired to project a six-mile line 
 west, for the southerly line of a township. 
 
 Remembering that in an isosceles triangle the angle at the 
 base is less than a right angle by half the angle at the vertex, 
 deflect a line towards the pole by the inclination due to three
 
 H24 PLANE SURVEYING. 
 
 miles, or in this case 48.46" x 3 = 2'.25" ; i.e., deflection = 
 \ inclination. 
 
 The table on next page, which was computed from the for- 
 mula (a) above, gives the inclination for one mile, and for six 
 miles on any parallel, from 10 to 60 of latitude; also the 
 convergency for six miles, on any latitude. 
 
 377. The Convergency of the Meridian is readily found for 
 any given distance from the corresponding inclination, by mul- 
 tiplying the sine of the inclination by the given distance. 
 
 Thus, for latitude 43, the inclination for one mile is 48.46"; 
 the sine of which is 0.000235. This, multiplied by the number 
 of links in a mile, which = 8,000, we have the convergency for 
 one mile, = 1.88 links. 
 
 Multiplying this by the number of miles in a township, = 36, 
 and we have the convergenc}* for a township, = 67.68 links. 
 In this manner were the convergeucies of the Table com- 
 puted. 
 
 378. Deflection of Range-Lines from Meridian. The second 
 column of the table shows the surveyor how much he must de- 
 flect the range lines between the several sections of a township 
 from the meridian, in order to make the consecutive ranges 
 of sections in a township of uniform width, for the purpose of 
 throwing the effects of convergency into the most westerly 
 range of quarter-sections, agreeably to law. 
 
 Thus, say between 45 and 55 of latitude, the inclination is 
 practically 1' for every mile of easting or westing. Then, bear- 
 ing in mind that in the United States the surveys are regarded 
 as projected from the east and south to the west and north, the 
 surveyor must project the first range-line between the sections 
 of a township in those latitudes 1' to the left of the meridian. 
 
 The second, 2' ; the third, 3' ; and so on to the fifth, which 
 must be 5' to the left of the meridian on the east side of the 
 township. 
 
 By this means all the convergency of the township is thrown 
 into the sixth, or westerly range of sections, as the law directs.
 
 SURVEY OF THE PUBLIC LANDS. 
 
 325 
 
 The fourth column of the table below shows the amount of 
 this convergence- This column is also useful in subdividing 
 a block of territory embraced by two standard parallels and two 
 guide meridians into townships. Thus, starting a meridian 
 from a standard parallel on latitude 43 N., for the western 
 boundary of a range of township, say the first one west from 
 the guide meridian, and running north, say four townships, 
 the surveyor must make a point that is east of the six-mile point 
 on the northern standard parallel, 4 x 67.7 links = 270. 8 links. 
 The second meridian should fall 8 x 67.7 links to the right of 
 the twelve-mile point. 
 
 TABLE OF INCLINATION AND CONVERGENCY OF THE MERIDIANS. 
 
 13 
 
 2 
 
 Inclination for 
 one mile. 
 
 Inclination for 
 six miles. 
 
 Oonvergency for 
 one township of 
 36 miles. 
 
 
 
 
 
 Inclination for 
 one mile. 
 
 Inclination for 
 six miles. 
 
 Convergency for 
 one township of 
 36 miles. 
 
 Latitude. 
 
 Inclination for 
 one mile. 
 
 1 Inclination for 
 six miles. 
 
 Convergency for 
 one township of 
 36 miles. 
 
 o 
 
 // 
 
 / // 
 
 LINKS. 
 
 
 
 // 
 
 / // 
 
 LINKS. 
 
 
 
 / // 
 
 / // 
 
 LINKS. 
 
 10 
 
 9.18 
 
 55 
 
 13.0 
 
 27 
 
 26.52 
 
 239 
 
 36.9 
 
 11 
 
 50.19 
 
 501 
 
 70.1 
 
 11 
 
 10.13 
 
 101 
 
 14.2 
 
 28 
 
 27.66 
 
 246 
 
 38.6 
 
 46 
 
 52.00 
 
 512 
 
 72.6 
 
 12 
 
 11.07 
 
 106 
 
 15.5 
 
 20 
 
 28.85 
 
 253 
 
 40.2 
 
 46 
 
 53.83 
 
 523 
 
 75.2 
 
 13 
 
 12.02 
 
 112 
 
 16.8 
 
 80 
 
 30.03 
 
 303 
 
 41.9 
 
 47 
 
 55.67 
 
 534 
 
 77.8 
 
 14 
 
 12.98 
 
 118 
 
 18.1 
 
 31 
 
 31.26 
 
 307 
 
 43.6 
 
 48 
 
 57.67 
 
 546 
 
 80.6 
 
 15 
 
 13.96 
 
 124 
 
 19.4 
 
 82 
 
 32.49 
 
 315 
 
 45.4 
 
 49 
 
 59.83 
 
 559 
 
 83.5 
 
 16 
 
 14.93 
 
 130 
 
 20.7 
 
 88 
 
 33.83 
 
 323 
 
 47.2 
 
 50 
 
 1 02.00 
 
 612 
 
 86.5 
 
 17 
 
 15.92 
 
 136 
 
 22.0 
 
 34 
 
 35.17 
 
 331 
 
 49.1 
 
 61 
 
 1 04.17 
 
 625 
 
 89.7 
 
 18 
 
 16.91 
 
 141 
 
 23.4 
 
 86 
 
 36.50 
 
 339 
 
 50.9 
 
 62 
 
 1 06.67 
 
 640 
 
 93.0 
 
 19 
 
 17.93 
 
 147 
 
 24.9 
 
 86 
 
 37.83 
 
 346 
 
 62.7 
 
 68 
 
 1 09.17 
 
 665 
 
 96.4 
 
 20 
 
 18.94 
 
 164 
 
 26.5 
 
 37 
 
 39.17 
 
 355 
 
 54.7 
 
 64 
 
 1 16.67 
 
 710 
 
 100.0 
 
 21 
 
 19.98 
 
 200 
 
 27.8 
 
 38 
 
 40.67 
 
 404 
 
 56.8 
 
 66 
 
 1 14.33 
 
 726 
 
 103.7 
 
 22 
 
 21.02 
 
 206 
 
 29.3 
 
 39 
 
 42.17 
 
 413 
 
 58.8 
 
 56 
 
 1 17.17 
 
 743 
 
 107.6 
 
 23 
 
 22.10 
 
 213 
 
 30.8 
 
 40 
 
 43.67 
 
 422 
 
 60.9 
 
 57 
 
 1 20.00 
 
 800 
 
 111.8 
 
 24 
 
 2.3.17 
 
 219 
 
 32.3 
 
 41 
 
 45.17 
 
 431 
 
 63.1 
 
 58 
 
 1 22.00 
 
 819 
 
 116.2 
 
 25 
 
 24.30 
 
 226 
 
 33.8 
 
 42 
 
 46.85 
 
 441 
 
 65.4 
 
 59 
 
 1 26.66 
 
 840 
 
 120.9 
 
 26 
 
 25.38 
 
 232 
 
 35.4 
 
 1:5 
 
 48.52 
 
 451 
 
 07.7 
 
 60 
 
 130.00 
 
 900 
 
 125.7 
 
 For details of instruction in United States Government Surveying, see 
 Iliiwcs' System of "Rectangular Surveying," Kurt's "Key to Solar Com 
 pass," and Clevenger's "Government Surveying."
 
 CHAPTER VII. 
 
 CITY SUKVEYING. 
 
 INTRODUCTION. 
 
 379. In the broadest sense, the duties of a city engineer in 
 a large city are many and varied. His knowledge and judg- 
 ment are required in the location of the city, the laying out of 
 streets, and the fixing of suitable grades therefor, the establish- 
 ment of a proper water supply, the designing of a suitable sys- 
 tem of sewers, the improvement of the waterways, and the 
 planning of necessary bridges and buildings. Following his 
 judicial functions as a designer are his ministerial functions as 
 a constructor. The field which is thus opened before him, in 
 carrying into execution the plans for the various public works, 
 is a very wide one. 
 
 As the borough grows and expands into the metropolis, its 
 needs in the directions mentioned increase until a division of 
 labor and responsibility becomes expedient and necessary. In 
 securing the best results in engineering practice, as in other 
 work, the tendency is towards specialties ; so that in many cities, 
 in order to secure the services of the best men, and also the 
 best results, the numerous and important duties connected with 
 city engineering have been separated. The province of this 
 work, which is not a treatise on engineering, but on land sur- 
 veying, makes it proper to treat in this chapter, as thoroughly 
 as the intention and limits of the work allow, only what may 
 be classed under the head of surveying, whether it be performed 
 as the special work of the city or town surveyor, or as among 
 the duties of the city engineer, the qualifications of the
 
 CITY SURVEYING. 327 
 
 former by no means fitting a man to perform the varied duties 
 of the latter. 
 
 Although this work is intended for the instruction of the 
 student, not of the experienced surveyor, and hence in many 
 things may go into details which to the latter may seem unim- 
 portant, it is impossible in the limits of a chapter to impart a 
 thorough knowledge of the duties of a citv or town surveyor, 
 indeed, even to mention all his duties and the many operations 
 and methods which only a long and varied practice can impart. 
 General methods will be given and discussed, but any survevor 
 of a practical turn of mind will have his own methods of per- 
 forming much of the routine work pertaining to his situation. 
 
 It is not in harmony with the plan of this work to go into the 
 statement in this chapter of any elaborate theories regarding 
 surveying and the instruments used therein, but to endeavor to 
 give some methods which are found to be applicable in practice 
 and to give good practical results. A thorough knowledge of any 
 one good method of performing a certain work is of much more 
 value to the student than a misty idea of numerous methods. 
 
 Under the two leading heads of this chapter, field instruments 
 and work and office instruments and work, theoretical discus- 
 sions will not be entered into ; not because they do not possess 
 much value, but because we conceive that they are not adapted 
 to the student's present needs and most rapid advancement. 
 Under the former head, in the light of the work which is likely 
 to engage the greater part of the surveyor's time, field instru- 
 ments and methods of using them will be described. Under 
 the latter, the nature of office plans and records will be de- 
 scribed, the instruments and methods used in the work of pro- 
 ducing the plans having been described in other chapters. 
 
 In dividing land and locating the boundaries between parties 
 it is evident that the greater the value or the prospective value 
 of said lands, the more delicate should be the instruments, and 
 the more exact the methods used in the work. The methods 
 and instruments which would for all practical purposes be suffi- 
 ciently exact for the location of a line fence iu the country, 

 
 328 PLANE SURVEYING. 
 
 where land might be purchased for Si 00 per acre, would not at 
 all meet the requirements in locating in a city a line between two 
 parties on land worth $100 per front foot. This fact becomes 
 the more evident when we consider that the structures placed 
 upon party lines in a city are so much more substantial and per- 
 manent in their nature than those thus located in the country. 
 To meet these considerations we shall find that while some of 
 the methods of land surveying previously described in this work, 
 and the instruments used therein, are applicable to the purposes 
 of city surveying, many of the methods will be more exact, and 
 the instruments more numerous and delicate. 
 
 Following the plan heretofore pursued in this work, we will, 
 before discussing the work of the city surveyor, describe the 
 instruments (not described in previous chapters) of most gen- 
 eral use in his work, and explain their adjustments and the 
 general methods of using them. These instruments are the 
 transit and rods, steel tapes, measuring-rods, pocket-thermome- 
 ter, hand-level, spring-balance, plummet, Y-level, le veiling-rods, 
 and rod-levels. 
 
 SECTION I. 
 
 INSTRUMENTS, THEIR ADJUSTMENTS AND GENERAL USES. 
 
 A. FIELD INSTRUMENTS. 
 
 1 
 
 380. The Transit. Full description of the transit, its adjust- 
 ment and uses, may be found in Chapter II. 
 
 381. As precision is the distinguishing feature of city and 
 town surveying, the magnetic needle, which is usually found 
 upon the transits, is in this work of but little use. Angles in 
 carefully made surveys are now taken on the horizontal gradu- 
 ated circle of the transit. The instructions already given in this 
 work regarding the magnetic needle are sufficient reason for the
 
 TRANSIT, 
 
 WITH GRADIENTER, LEVEL TO TELESCOPE, AND VERTICAL ARC, AS MADE BV 
 YOUNG & SONS, PHILADELPHIA, PA.
 
 FIELD INSTRUMENTS. 331 
 
 above. It is, however, desirable that in each city and town 
 the true meridian should be determined and permanently marked. 
 Besides being useful in many other ways which will suggest 
 themselves, it will be of great use as an aid in determining the 
 situation of lines described by their bearings in old deeds, the 
 date of the old survey being known. 
 
 382. The stadia-hairs * and vertical circle for stadia-measure- 
 ments are useful attachments, and the telescope should by all 
 means have a long level-tube attached, as this is of much use 
 in city and town work in running grade lines and in levelling 
 for short distances. . After the level and the manner of using it 
 have been described, the operation of running a grade line will 
 be explained. 
 
 383. Rods. Besides the usual iron-pointed wooden rods, very 
 convenient rods, or pickets, for use with the transit, may be made 
 of gas-pipe about three-quarters of an inch in diameter drawn 
 out on one end to a point, and painted in alternate sections of 
 red and white, red preferred to black because against red the 
 cross-hairs can be seen. 
 
 384. It is b}- no means as easy a matter to run a straight 
 line with a transit as at first thought it may seem to the student. 
 After the selection of suitable weather, reversing at every ex- 
 tension, care in handling the instrument, and with a correspond- 
 ing degree of care on the part of assistants, the results are not 
 always what the most careful would desire. 
 
 385. In marking a line with stakes, it is convenient to have 
 stake-wood which, in cross-section, has one dimension greater 
 than the other. If, in setting the stake, it always be placed 
 with its broader side towards the instrument, its position will 
 afterwards tell one at a glance in which direction the line was 
 run. This is important when several stakes are set on different 
 
 * See Articles 148 to 152, Stadia Measurements.
 
 332 PLANE SURVEYING. 
 
 lines near their intersection, as it will often be the means of 
 avoiding confusion and the resulting errors. 
 
 386. Steel Tapes, etc. Before making an}- important meas- 
 urements for a city or town, it is necessary, in order to avoid 
 subsequent confusion, that a standard of measurement should 
 be adopted. In many parts of an old city or town the intro- 
 duction of a new standard would bring inextricable confusion. 
 If there be a standard, even though it has not been carefully 
 preserved, it should, if possible, be ascertained and regarded. 
 When, however, it is at the option of the surveyor to select his 
 standard, the United States standard should, as tending to uni- 
 formity, be adopted in this country. Standard rods may be 
 procured of the government. With these rods tape lines and 
 other instruments used for a line purpose should be compared, 
 and the variation noted. It is desirable, also, for purposes of 
 comparison, that a standard, 50 feet or 100 feet, at a known 
 temperature, should be carefully laid down with these rods in 
 the corridor of some building, or in some other convenient 
 place. 
 
 Very accurate measuring may be done with graduated wooden 
 rods properly shod with metal ends. These rods are necessarily of 
 but moderate length ; hence, work with them is correspondingly 
 slow. For city work, steel tapes are now in very general use ; 
 and, when properly handled, give very satisfactory results. They 
 are of different lengths and of different widths. For measur- 
 ing full hundreds over tolerably level ground the narrow tape, 
 -^ inch wide and 200 feet long, is very convenient. For general 
 city use the 100-feet tape, f inch in width, is most convenient. 
 
 387. As a rule measurements will be made with the tape in 
 a horizontal position. If not so held, the measurements will 
 afterwai'ds be reduced to the horizontal. In order to determine 
 the horizontal, a hand-level is used to ascertain the difference 
 in elevation of the ground at the two ends of the tape. A cut 
 and description of this convenient little instrument is given 
 below.
 
 FIELD INSTRUMENTS. 333 
 
 Locke's Hand-Level consists of a brass tube about 6 inches 
 long, having, as shown in the figure, a small level on top and 
 near the object end, there being also an opening in the tube 
 beneath, through which the bubble can be seen, as reflected by 
 a glass prism, immediately under the level. Both ends of the 
 tube are closed by plain glass settings to exclude the dust, and 
 there is at the inner end of the sliding or eye tube a semicircu- 
 lar convex lens, which serves to magnify the level bubble, and 
 cross-wire underneath, while it allows the object to be clearly 
 seen through the open half of the tube. 
 
 The cross-wire is fastened to a little frame moving under the 
 level-tube, and adjusted to its place by the small screw shown 
 on the end of the level-case. The level of any object in line 
 with the eye of the observer is determined by sighting upon it 
 through the tube, and bringing the air-bubble of the level into 
 a position where it is bisected by the cross-wire. 
 
 A short telescope is sometimes applied in place of the plain 
 glass ends, enabling levels to be taken at greater distances and 
 with increased accuracy. 
 
 If one or both ends of the tape be held up, the point on the 
 ground vertically under the end of the tape will be determined 
 by means of the plummet, which here needs no description 
 further than to sav that its sides should make such an angle 
 with each other as not to prevent the observer when using it 
 from seeing its point ; neither should it be so long as to be 
 unsteady. 
 
 In all extended and important measurements regard must be 
 had in using the steel tape to standard, temperature, sag, and 
 wind . 
 
 Before using a tape its relation to the standard should be
 
 334 PLANE SURVEYING. 
 
 determined by comparison with the standard, marked as pre- 
 viously described, and the variation noted. 
 
 388. All important measurements, no matter at what tem- 
 perature made, should be reduced to a standard temperature ; 
 for if, at a certain temperature, we determined with a steel taue 
 the distance apart of two points, at a higher temperature that 
 distance on the same tape would be less because the tape is 
 longer ; or, at a lower temperature, greater, because the tape is 
 shorter. The temperature of the air at the time of measurement 
 is ascertained by means of a small thermometer which can be 
 exposed with the tape, and which is so protected that, when not 
 in use, it can be safely carried in the pocket. The standard 
 temperature to which all measurements should be reduced may 
 be taken at pleasure. The correction for expansion and con- 
 traction of the steel tape by heat and cold is 0.000006 per unit 
 per degree F. 
 
 389. When the tape is held suspended, it will always sag in 
 a vertical direction. Hence the horizontal distance between 
 the extreme graduations will be less than if there were no sag. 
 For this reason, when used to measure the distance between 
 two points, it will, without correction, give a result too great; 
 when used without correction to lay down a given distance, it 
 will give it too small. While a formula may be derived by 
 which to make a correction for sag, it will be found quite as 
 satisfactory to determine it by actual trial. The amount of 
 sag will of course depend upon the tension, or pull. This may 
 be regulated by using at one end of the tape a small spring- 
 balance. It is, however, very desirable that on important work 
 the same men at the same ends of the tape should make all 
 measurements. The experience gained in working together 
 will be a most important factor in securing uniform results. 
 
 The effect of wind is in the same direction as that of sag. 
 While much of the work of the surveyor, particularly that in- 
 volving short measurements, must be done regardless of wind,
 
 FIELD INSTRUMENTS. 335 
 
 no good results in long and important measurements can be 
 secured in windy weather. The best correction for wind is to 
 wait for a calm. In windy weather a narrow tape, as it ex- 
 poses less surface to the wind, is useful. 
 
 390. To illustrate what has been said in regard to the correc- 
 tions to be applied to measurements made with the steel tape, 
 let us suppose two examples. 
 
 First. With a steel tape 100 feet long (f inch wide) sus- 
 pended each length at one or both ends, the temperature of 
 the air being 79 F., the distance on the tape between two 
 points is found to, be 550 feet 6| inches. If the tape is 
 inch longer than the standard, and parts of its length propor- 
 tionately longer, the standard temperature, 60 F., and the sag 
 inch in 100 feet, what are the corrections, and what is the 
 actual distance between the points ? 
 
 On account of differing from the standard, as the tape is too 
 long, the distance obtained is too short ; the correction for 
 standard is therefore additive. On account of difference in 
 temperature, the temperature being higher than the standard, 
 as the tape is too long, the distance obtained is too short; 
 the correction for temperature is therefore additive. On 
 account of the sag, as the tape is thereby made too short, the 
 distance obtained is too long ; the correction for sag is there- 
 fore subtractive. 
 
 Correction for standard : 
 
 | in. x 5 = | in. additive. 
 Correction for temperature (79 60 = 19) : 
 
 0.000006 ft. x 550 x 19 = 0.0627 ft. 
 
 0.0627 ft. x 12 = 0.7524 in. = |f in. additive. 
 
 Correction for sag : 
 
 in. x 5 = |f in. subtractive. 
 Total correction : 
 
 + M. in. _f_ i i n . _ II i n . = + T ^ in. additive.
 
 336 PLANE SUE V EYING. 
 
 Actual distance between points : 
 
 550 ft. 6J in. + fa in. = 550 ft. 6ff in. 
 
 Second. Suppose it be required, other things being as 
 before, to locate with the steel tape, when the temperature 
 of the air is 52 F., two points which shall at the standard 
 temperature be 225 feet 4 inches apart. 
 
 What length on the tape must be taken ? 
 
 Correction for standard : 
 
 { in. x 2^ = -2 m - subtractive. 
 Correction for temperature (60 52 = 8) : 
 
 0.000006 ft. x 225 x 8 = 0.0108 ft. 
 
 0.0108 ft, X 12 = 0.1296 in. = fa in. additive. 
 
 Correction for sag : 
 
 \ in. x 2 = ^| in. additive. 
 Total correction : 
 
 fa in. -f u 4 2 in - + it in - = +if in - additive. 
 Length to be taken on tape : 
 
 225 ft. 4| in. + if in. = 225 ft. 4ff in. 
 
 When the tape is not suspended, correction for sag will not 
 be made. 
 
 In short and less important measurements the same attention 
 to corrections is not necessary. 
 
 In practice, the above method has been found to give satis- 
 factory results. 
 
 391. In placing stakes to hold measurements, it is best, and 
 in harmony with the method suggested for placing them on 
 instrument lines, to set them with the greater dimension of 
 cross-section in the direction in which the measurement is 
 being made. 
 
 Measuring is a very important part of the work of the sur- 
 veyor. Even when done with the greatest care, it is difficult to 
 obtain results entirely satisfactory. 

 
 FIELD INSTRUMENTS. 337 
 
 Measurements which are to be directly compared, or are to 
 be used in connection, as in locating parallel lines, should be 
 made under circumstances as nearly as possible identical. 
 Experience and a correct idea of the importance of the work 
 will enable the surveyor to determine the degree of accuracy 
 therein necessary. 
 
 LEV ELLING-lNSTRUMENTS . 
 
 392. The Y-LeveL Of the different varieties of the levelling- 
 instrument, that termed the Y-level has been almost Universally 
 preferred by American engineers, on account of the facility of 
 its adjustment and superior accurac}*. 
 
 The engraving represents a twenty-inch Y-level as made by 
 W. and L. E. Gurley, Troy, N.Y. 
 
 393. The Telescope has at each end a ring of bell-metal, 
 turned very truly, and both of exactly the same diameter ; by 
 these it revolves in the wyes, or can be at pleasure clamped in 
 any position when the clips of the wyes are brought down upon 
 the rings, by pushing in the tapering-pins. 
 
 394. The Level or ground bubble tube is attached to the 
 under side of the telescope, and furnished at the different ends 
 with the usual movements, in both horizontal and vertical 
 directions. 
 
 The aperture of the tube, through which the glass vial 
 appears, is about 5 inches long, being crossed at the centre 
 by a small rib or bridge, which greatly strengthens the tube. 
 
 The level-scale which extends over the whole length is 
 graduated into tenths of an inch, and figured at every fifth 
 division, counting from zero at the centre of the bridge ; the 
 scale is set close to the glass. 
 
 The bubble vial is made of thick glass tube, selected so as to 
 have an even bore from end to end, and finely ground on its 
 upper interior surface, that the run of the air-bubble may be 
 uniform throughout its whole range.

 
 340 PLANE SURVEYING. 
 
 395. The Wyes are made large and strong, of the best bell- 
 metal, and each has two nuts, both being adjustable with the 
 ordinary steel pin. 
 
 The clips are brought down on the rings of the telescope- 
 tube by the Y-pins, which are made tapering, so as to clamp 
 the rings very firmly. 
 
 The clip of one of the wyes has a little pin projecting from 
 it, which, entering a recess filed in the edge of the ring, insures 
 the vertical position of the level and cross-wire. 
 
 396. The Level-Bar is made round, of the best bell-metal, 
 and shaped so as to possess the greatest strength in the parts 
 most STibject to sudden strains. 
 
 Connected with the level-bar is the head of the tripod- 
 socket. 
 
 397. The Tripod-Socket is compound ; the interior spindle 
 Z>, sectional view, upon which the whole instrument is sup- 
 ported, is made of steel, and nicely ground, so as to turn 
 evenlv and firmly in a hollow cylinder of bell-metal ; this, 
 again, has its exterior surface fitted and ground to the main 
 socket EE of the tripod-head. 
 
 The bronze cylinder is held upon the spindle by a washer 
 and screw, the head of the last having a hole in its centre, 
 through which the string of the plumb-bob is passed. 
 
 THE ADJUSTMENTS. 
 
 398. The three adjustments of the level which the surveyor 
 usually has to attend to are the following ; 
 
 1. To adjust the line of collimation, or, in other words, to 
 bring both wires into the optical axis, so that their point of 
 intersection will remain on any given point during an entire 
 revolution of the telescope. 
 
 2. To bring the level-bubble parallel with the bearings of the 
 V-rings, and with the longitudinal axis of the telescope.
 
 FIELD INSTRUMENTS. 341 
 
 3. To adjust the wyes, or to bring the bubble into a position 
 at right angles to the vertical axis of the instrument. 
 
 399. To Adjust the Line of Collimation, set the tripod firmly, 
 remove the Y-pins from the clips, so as to allow the telescope 
 to turn freely, clamp the instrument to the tripod-head, and, by 
 the levelling and tangent screws, bring either of the wires upon 
 a clearly marked edge of some object, distant from 100 to 500 
 feet. 
 
 . Then, with the hand, carefully turn the telescope half-way 
 around, so that the same wire is compared with the object 
 assumed. 
 
 Should it be found above or below, bring it half-way back by 
 moving the capstan-head screws at right angles to it, remem- 
 bering always the inverting property of the eye-piece ; now 
 bring the wire again upon the object, and repeat the first 
 operation, until it will reverse correctlv. 
 
 Proceed in the same manner with the other wire until the 
 adjustment is completed. 
 
 Should both wires be much out, it will be well to bring them 
 nearly correct before either is entirely adjusted. 
 
 When this is effected, unscrew the covering of the eye-piece 
 centring-screws, shown in the sectional view at AA, and move 
 each pair in succession with a small screw-driver, until the wires 
 are brought into the centre of the field of view. 
 
 The inverting property of the eye-piece does not affect this 
 operation, and the screws are moved direct. 
 
 To test the correctness of the centring, revolve the telescope, 
 and observe whether it appears to shift the position of an 
 object. 
 
 Should any movement be perceived, the centring is not 
 perfectly effected. 
 
 It may here be repeated, that in all telescopes the position 
 and adjustment of the line of collimation depends upon that of 
 the object-glass ; and, therefore, that the movement of the eye- 
 piece does not affect the adjustment of the wires in any respect.
 
 342 PLANE SURVEYING. 
 
 When the centring has been once effected, it remains per- 
 manent, the cover being screwed on again to conceal and 
 protect it from derangement at the hands of the curious or 
 inexperienced operator. 
 
 400. To Adjust the Level-Bubble. Clamp the instrument 
 over either pair of le veiling-screws, and bring the bubble into the 
 centre of the tube. 
 
 Now turn the telescope in the wyes, so as to bring the level- 
 tube on either side of the centre of the bar. Should the bubble 
 run to the end, it would show that the vertical plane passing 
 through the centre of the bubble was not parallel to that drawn 
 through the axis of the telescope-rings. 
 
 To correct the error, bring the bubble entirely back, with 
 the capstan-head screws, which are set in either side of the 
 level-holder, placed usually at the object end of the tube. 
 
 Again bring the level-tube over the centre of the bar, and 
 the bubble to the centre ; turn the level to either side, and, if 
 necessary, repeat the correction until the bubble will keep its 
 position, when the tube is turned half an inch or more to either 
 side of the centre of the bar. 
 
 The necessity for this operation arises from the fact that 
 when the telescope is reversed end for end in the wyes in the 
 other and principal adjustment of the bubble, we are not certain 
 of placing the level-tube in the same vertical plane ; and there- 
 fore it would be almost impossible to effect the adjustment 
 without a lateral correction. 
 
 Having now, in great measure, removed the preparatory 
 difficulties, we proceed to make the level-tube parallel with 
 the bearings of the Y-rings. 
 
 To do this, bring the bubble into the centre with the levelling- 
 screws, and then, without jarring the instrument, take the 
 telescope out of the wyes and reverse it end for end. Should 
 the bubble run to either end, lower that end, or, what is equiva- 
 lent, raise the other by turning the small adjusting-nuts, on one 
 end of the level, until by estimation half the correction is made ;
 
 FIELD INSTRUMENTS. 343 
 
 again bring the bubble into the centre, and repeat the whole 
 operation, until the reversion can be made without causing any 
 change in the bubble. 
 
 It would be well to test the lateral adjustment, and make 
 such correction as may be necessary in that, before the hori- 
 zontal adjustment is entirely completed. 
 
 401. To Adjust the Wyes. Having effected the previous 
 adjustments, it remains now to describe that of the wyes, or, 
 more precisely, that which brings the level into position at 
 right angles to the vertical axis, so that the bubble will remain 
 in the centre during 1 an entire revolution of the instrument. 
 
 To do this, bring the level-tube directly over the centre of 
 the bar, and clamp the telescope firmly in the wyes, placing 
 it, as before, over two of the levelling-screws, unclamp the 
 socket, level the bubble, and turn the instrument half-way 
 around, so that the level-bar may occupy the same position 
 with respect to the levelling-screws beneath. 
 
 Should the bubble run to either end, bring it half-way back 
 by the Y-nuts on either end of the bar ; now move the telescope 
 over the other set of levelling-screws, bring the bubble again 
 into the centre, and proceed precisely as above described, 
 changing to each pair of screws, successively, until the adjust- 
 ment is very nearly perfected, when it may be completed over 
 a single pair. 
 
 The object of this approximate adjustment is to bring the 
 upper parallel plate of the tripod-head into a position as nearly 
 horizontal as possible, in order that no essential error may 
 arise, in case the level, when reversed, is not brought precisely 
 to its former situation. When the level has been thus com- 
 pletely adjusted, if the instrument is properly made, and the 
 sockets well fitted to each other and the tripod-head, the bubble 
 will reverse over each pair of screws in any position. 
 
 Should the surveyor be unable to make it perform correctly, 
 he should examine the outside socket carefully to see that it 
 sets securely in the main socket, and also notice that the clamp 
 does not bear upon the ring which it encircles.
 
 344 PLANE SURVEYING. 
 
 When these are correct, and the error is still manifested, it 
 will probably be in the imperfection of the interior spindle. 
 
 After the adjustments of the level have been effected, and 
 the bubble remains in the centre, in any position of the socket, 
 the surveyor should turn the telescope in the wyes until the pin 
 on the clip of the wve will enter the little recess in the ring to 
 which it is fitted, and by which is insured the vertical position 
 of the spirit-level and cross-wire. 
 
 When the pin is in its place, the vertical wire may be applied 
 to the edge of a building ; and in case it should not be parallel 
 with it, two of the cross-wire screws that are at right angles to 
 each other may be loosened, and by the screws outside, the 
 cross-wire ring turned until the wire is vertical ; the line of col- 
 limation must then be corrected again and the adjustments of 
 the level will be complete. 
 
 402. To Use the Level Set the legs firmly into the ground. 
 The bubble should then be brought over each pan- of levelling- 
 screws successively and levelled in each position, any correction 
 that may appear necessary being made in the adjustments. 
 
 Bring the wires precisely in focus and the object distinctly in 
 view, so that all errors of parallax may be avoided. 
 
 This error is seen when the eye of an observer is moved to 
 either side of the centre of the eye-piece of a telescope, in which 
 the foci of the object and eye-glasses are not brought precisely 
 upon the cross-wires and object ; in such a case the wires will 
 appear to move over the surface, and the observation will be 
 liable to inaccuracy. 
 
 In all instances the wires and object should be brought into 
 view so perfectly that the cross-wires will appear to be fastened 
 to the surface, and will remain in that position however the e % ye 
 is moved. 
 
 Care should be exercised during an observation, last the hand 
 touching the instrument inadvertently, or a foot placed near 
 the leg of the tripod, impair the adjustment. 
 
 The weight of a level having a 20-inch telescope, with level-
 
 NEW YORK. P 
 
 LEVELLING-RODS.
 
 FIELD INSTRUMENTS. 347 
 
 ling-head, exclusive of the tripod, is between thirteen and four- 
 teen pounds. 
 
 IiEVELLING-RODS . 
 
 403. The various levelling-rods used by American engineers 
 are made in two or more parts, which slide from each other as 
 they are extended in use. 
 
 404. The New York Rod. This rod, which is shown in the 
 engraving as cut in two, so that the ends may be exhibited, is 
 made of maple, in two pieces, but sliding one from the other, 
 the same end being always held on the ground, and the gradu- 
 ations starting from that point. 
 
 The graduations are made to tenths and hundredths of a foot, 
 the tenth figures being black, and the feet marked with a large 
 red figure. 
 
 The front surface, on which the target moves, reads to 6 
 feet ; when a greater height is required, the horizontal line of 
 the target is fixed at that point, and the upper half of the rod, 
 carrying the target, is moved out of the lower, the reading 
 being now obtained by a vernier on the graduated side, up to 
 an elevation of 12 feet. 
 
 The target is round, made of thick sheet brass, having, to 
 strengthen it still more, a raised rim, which also protects the 
 paint from being defaced. 
 
 The target moves easily on the rod, being kept in any posi- 
 tion by the friction of the two flat plates of brass which are 
 pressed against two alternate sides, by small spiral springs, 
 working in little thimbles attached to the baud which surrounds 
 the rod. 
 
 There is also a clamp-screw on the back, by which it may be 
 securely fastened to any part of the rod. 
 
 The face of the target is divided into quadrants by horizontal 
 and vertical diameters, which are also the boundaries of the 
 alternate colors with which it is painted.
 
 348 PLANE SURVEYING. 
 
 The colors usually preferred are white and red ; sometimes 
 white and black. 
 
 The opening in the face of the target is a little more than a 
 tenth of a foot long, so that in any position a tenth or a foot 
 figure can be seen on the surface of the rod. 
 
 The right edge of the opening is chamfered, and divided into 
 ten equal spaces, corresponding with nine-hundredths on the 
 rod ; the divisions start from the horizontal line which separates 
 the colors of the face. 
 
 The vernier, like that on the side of the rod, reads to thou- 
 sandths of a foot. 
 
 The clamp, which is screwed fast to the lower end of the 
 upper slid ing-piece, has a movable part which can be brought 
 by the clamp-screw firmly against the front surface of the lower 
 half of the rod, and thus the two parts immovably fastened to 
 each other without marring the divided face of the rod. 
 
 405. The Philadelphia Rod. This rod is made of two 
 strips of cherry, each about f inch thick by l inches wide and 
 7 feet long, connected by two metal sleeves, the lower one of 
 which has a clamping-screw for fastening the two parts together 
 when the rod is raised for a higher reading than 7 feet. 
 
 Both sides of the back strip and one side of the front one 
 are planed out -fa inch below the edges ; these depressed sur- 
 faces are painted white, divided into feet, tenths and hundredths 
 of a foot, and the feet and tenths figured. 
 
 The front piece reads from the bottom upward to 7 feet, 
 the foot figures being red and an inch long, the tenth figures 
 black and eight-tenths of an inch long. When the rod is 
 extended to full length, the front surface of the rear half reads 
 from 7 to 13 feet, and the whole front of the rod is figured 
 continuously and becomes a self-reading rod 13 feet long. 
 
 The back surface of the rear half is figured from 7 to 13 
 feet, reading from the top down ; it has a vernier also by 
 which the rod is read to two-hundredths of a foot as it is 
 extended. The target is round and made of sheet-brass, raised
 
 FIELD INSTRUMENTS. 349 
 
 on the perimeter to increase its strength, and is painted in white 
 and red quadrants ; it has also a scale on its chamfered edge, 
 reading to tvvo-hundredths of a foot. 
 
 When a level of less than 7 feet is desired, the target is 
 moved np or down the front surface, the rod being closed 
 together and clamped ; but when a greater height is required, 
 the target is fixed at 7 feet and the rear half slid out, the scale 
 on the back giving the readings like those of the target to two- 
 hundredths of a foot. 
 
 This rod is so graduated that the leveller is enabled to take 
 the reading direct from it, the rodman's duties being simply to 
 hold the rod vertical over the points. It is hence called a self- 
 reading or speaking rod. 
 
 406. The Rod-Level. The figures below represent a level re- 
 cently devised, for the more accurate plumbing of levelling-rods. 
 
 ROD-LEVEL. ROD-LEVEL AS APPLIED TO A ROD. 
 
 The left-hand figure shows it when folded for convenience in 
 carrying. Its convenience and value commend it to general 
 favor. 
 
 407. Levelling is measuring in a vertical direction. In his 
 treatise on levelling, Frederick W. Simms says: "Levelling is 
 the art of tracing a line at the surface of the earth which shall 

 
 350 PLANE SURVEYING. 
 
 cut the directions of gravity everywhere at right angles. . . . 
 The direction of gravity invariably tends towards the centre 
 of the earth, and may be considered as represented by a plumb- 
 line when hanging freely, and suspended beyond the sphere of 
 attraction of the surrounding objects. . . . The operation of 
 levelling may be defined as the art of finding how much higher 
 or lower any one point is than another, or, more properly, the 
 difference of their distances from the centre of the earth." 
 
 A surface like that of still water may be called a level sur- 
 face. The curve formed by the intersection with such a sur- 
 face of a vertical plane is a line of true level; a line tangent to 
 the latter is a line of apparent level. 
 
 Levelling is the art of determining the differences of elevation 
 of two or more points, or of determining how much one point 
 is above or below a line of true level passing through the other 
 point. 
 
 408. From the foregoing it is evident that, on account of the 
 curvature of the earth, a horizontal line is not really through- 
 out its length a level line ; that of two points in the same level 
 line each will have its own horizon. Hence, in levelling, the 
 effect of the curvature of the earth upon the comparative eleva- 
 tions of different points must be taken into consideration. 
 The effect of the curvature is to make objects appear lower 
 than they really are. 
 
 The air nearer the surface of the earth is denser than that 
 farther removed from the surface. This difference in density, 
 causing refraction of light, will affect the elevation of a point 
 as observed through the telescope of a level, so that it also 
 must be taken into consideration. Its effect is to make objects 
 appear higher than they really are. The error caused by refrac- 
 tion is one-seventh as great as that caused by curvature. 
 
 Let us first find an expression for the correction due to the 
 curvature of the earth. That is 
 
 409. To find the deviation from its tangent of a line of true 
 level.
 
 LEVELLING. 
 
 351 
 
 Let represent the centre of the earth, PN a line of true 
 level, and PN' its tangent, or a line of p , 
 
 apparent level. The distance NN' cor- 
 responding to the length of sight PN is 
 required. 
 
 From Geometry, 
 
 PN 1 '^ NN'(2 ON+NN') ; 
 PN~' 2 
 
 20N+NN' 
 
 For ordinary distances, the length of 
 the arc may be regarded as that of the 
 tangent, and NN' as inconsiderable in 
 comparison with 2 ON, the diameter of u 
 the earth. Therefore, calling the length of sight d, the cor- 
 rection c, and the radius of the earth r, we have 
 
 d 2 
 
 and the correction for refraction 
 
 ~7 C ~7 X 2r~14r' 
 
 then the correction due to curvature and refraction, which we 
 will call (7, is 
 
 _!=_.*., 
 
 7 2r Ur 
 
 or, 
 
 This correction must be added to the height of the object as 
 found by the level. 
 
 In practice, the necessity for using the above formula is 
 avoided whenever it is possible to set the level at equal dis- 
 tances from the points whose difference of height is required.
 
 352 
 
 PLANE SURVEYING. 
 
 EXERCISES. 
 
 1. Assuming the diameter of the earth 7,926 miles, show that 
 for a mile sight c = about 8 inches. Find the value of C for the 
 same distance. 
 
 2. What is the correction due to curvature for half a mile? 
 
 3. What is the length of sight when C equals one-tenth of a 
 foot? 
 
 4. Show that, practically, the correction for curvature in feet 
 is equal to two-thirds the square of the distance in miles. 
 
 410. If two points Jf, JV, whose difference of elevation is 
 required, can be observed upon from some point P about equi- 
 distant * from them, not necessarily in their line, set up the 
 level at P, and note the reading of a rod held vertically over 
 each point. The difference of the two readings will indicate 
 the difference of level required. 
 
 411. If the above method is impracticable, set up the instru- 
 ment at some point P either in or out of the line, no matter 
 which from which a rod may be observed on the first station 
 M, and also on another point in the direction of N, about equi- 
 distant with M from the instrument. Remove the level to a 
 
 * Placing the instrument in this position lessens the effects of inaccurate 
 adjustment and renders unnecessary the corrections indicated in Article 
 409.
 
 LEVELLING. 353 
 
 new position P', whence observe again the rod on 0, also the 
 rod reading at N. 
 
 The difference between the readings of the rod at M and 
 shows how much higher the latter is than the former, and in like 
 manner the difference of the readings at and JV gives the differ- 
 ence in elevation of these points, and so on, no matter what the 
 number of stations. The difference in height of M and N 
 
 = Mm Go + Oo' Nn ; 
 or, Mm + Oo' Oo - Nn 
 
 = Mm + Oo' (Oo + Nn) . 
 
 Calling Mm and* Go' back-sights, and the other two, fore- 
 sights, we perceive that the difference of level of two points is 
 shown by subtracting the sum of the fore-sights from the sum 
 of the back-sights. 
 
 412. Again, in levelling, we measure, by means of the rod, 
 how much lower than the line of sight (height of instrument) 
 certain points are. Thus we may determine the relative eleva- 
 tions of the points. Suppose, for example, it be required to 
 determine the difference in elevation of any two points. For 
 reasons already given, set the level equally distant from the 
 points. If this cannot be done, and both observations have to 
 be taken from one of the stations, especially if the distance 
 between them is considerable, correction as previously described 
 must be made. But in this case suppose it is possible ; and 
 suppose that when held on one point, the rod reads 7.255 ; that 
 is, this point may be considered 7.255 below the line of sight, 
 and 4.755 when held on the other; then the first may be con- 
 sidered 7.255 4.755, or 2.500 farther than the second below 
 the line of sight, or lower than the second. 
 
 413. Suppose it be required to determine the difference in 
 elevation between two points, of which one is so much higher 
 than the other that the rod is too short to give a reading on 
 both points for one position of the instrument. In such a case
 
 354 PLANE SURVEYING. 
 
 one or more auxiliary points, called turning-points (T.P.), 
 must be used, and their relative elevations determined. Sup- 
 pose the reading on the first point is 0.824, and on a turning- 
 point is 10.432 ; the latter is then 9.608 below the former. Now 
 the instrument must be moved and set up so as to obtain a 
 reading on the turning-point ; and (we will suppose) on the 
 other of the given points. Suppose that on the former it is 
 1.302, and on the latter 8.634 ; the latter is then 7.332 below the 
 turning-point, or 9.608 + 7.332, or 16.940, below the first of the 
 two given points. 
 
 The first sight taken after setting up the level is called a 
 back-sight, or plus sight ; those taken after this, and before the 
 instrument is moved, are called fore-sights or minus sights. As 
 the difference of the readings of the rod on two points gives 
 their difference of elevation, the difference of the sum of the 
 plus sights, and the sum of the minus sights on T.P.'s and the 
 last point will give the difference in elevation of the extreme 
 points. In the above example 
 
 0.824 10.432 
 
 1.302 8.634 
 
 2.126 19.066 
 
 19.066 - 2.126 = 16.940, as before. 
 
 This is used as a check on level-notes. 
 
 In extended levelling, permanent elevations fixed during the 
 progress of the work for future reference are called bench 
 marks or benches (B.M.). 
 
 414. In levelling, it is customary to refer all elevations to an 
 assumed level plane, called the plane of reference, the datum 
 plane, or simply the datum. Points are then said to be so 
 much above or below the datum. As this plane may be assumed 
 at pleasure, it is generally so taken as to be lower than any 
 point whose elevation is to be determined. In city levelling 
 this plane may be assumed at the height of mean low water.
 
 LEVELLING. 355 
 
 which elevation may be called zero. Then a point which has 
 the elevation 125.37 will be 125.37 above low water. 
 
 If two points have the elevations 125.375 and 105.213 respec- 
 tively, the former is 125.375 105.213, or 20.162 higher than 
 the latter. 
 
 The datum having once been determined, its elevation, or 
 that of a point a known distance above it, should be perma- 
 nently fixed for future reference and comparison. 
 
 415. The levels for profile given under Street Grades, on 
 page 365, show how the field notes in levelling ma}' be kept. 
 The elevation of the bench-mark from which they start is 51.415 
 above the datum. The first plus sight is 7.030, which, added to 
 51.415, gives 58.445, the height of the instrument (H.I.) above 
 the datum. The first minus sight, which is on a turning-point 
 (T.P.), is 0.870, which, subtracted from 58.445, gives 57.575, 
 the height of the T.P. above the datum. The instrument is 
 then moved, set up again in a convenient place, and the work 
 proceeds. 
 
 At one setting of the instrument, the elevations of any points, 
 besides the turning-point, which are not too high or too low to be 
 reached, may be ascertained. It is evident that if any error 
 be made at a T.P., all the following elevations will thereby be 
 affected ; but if made at one of these other points, only the 
 elevation of that point will be affected. Hence the importance 
 of careful observations at T.P's. 
 
 In the above-mentioned form for the keeping of the field 
 notes, all the observations (Obs.) are set in one column. If 
 desired, plus sights and minus sights may be set in different 
 columns ; and of minus sights, those on turning-points may be 
 set in a column by themselves. It will then be easy to apply 
 the check before described. However, the form given is in 
 practice very convenient. 
 
 EXERCISE. 
 
 Tabulate in both of the above forms, also in the form 
 headed
 
 356 
 
 PLANE SURVEYING. 
 
 ELEVATION. REMARKS 
 
 the following level notes : 
 
 Height of B.M 100.000. 
 
 Obs. on B.M 5.132. 
 
 " " Sta. 6.28. 
 
 " " * 1 7.12. 
 
 " '" " 2 8.84. 
 
 " " T.P. 3 9.780. 
 
 From new position of inst. obs. on Sta. 3, 2.160. 
 
 Obs. on Sta. 4 5.89. 
 
 " " " 5 7.92. 
 
 " " " 6 ......... 10.18. 
 
 " T.P. 7 12.020. 
 
 Again on " 7 1.260. 
 
 Obs. on Sta. 8 4.23. 
 
 " M .>*... 9 5.87. 
 
 " " " 10 6.94. 
 
 416. Wind and sunshine affect the accuracy of levelling, as 
 of work with the transit. For very good work it is desirable to 
 have a calm day on which the sun is obscured by clouds. In 
 addition to a proper manipulation of the instrument, the sights 
 should not be longer than from 200 to 300 feet, the rod should 
 be held vertical, and the rod man should select for turning-points 
 good and firm points on stones, pegs, etc., on which the rod 
 may be freely turned or spun around. 
 
 417. Numerous bench-marks should be located in convenient 
 places. In -a city such places are at the intersections of streets, 
 on door-sills of buildings which have become thoroughly settled, 
 on roots of trees, etc. There are many other suitable places 
 which will suggest themselves.
 
 LEVELLING. 
 
 357 
 
 418. In city work, in making a circuit of levels for the 
 establishment of grade elevations and bench-marks, the work 
 should check out with no greater error than 0.01 foot in three 
 miles. 
 
 In levelling, as in all other work, regard must be had to the 
 difference between actual mistakes, the results of carelessness, 
 and the degree of accuracy actually obtainable by the observer. 
 
 We will now describe a general method of running a grade- 
 line with the transit. In the figure the irregular line represents 
 the profile of the ground, and the straight line the grade-line. 
 
 Let it be required to run a grade-line from ^4, elevation 30.29, 
 to B, elevation 28.79 ; elevation of ping or ground at A 33.49, 
 at B 27.26 ; therefore cut at A 3.20 and fill at B 1 .53. 
 
 Set the transit over A ; and, using the long level-tube, take 
 the elevation from a convenient bench. Suppose the H.I. 
 is found to be 38.21 ; then the length of the rod for marking 
 the grade-line (called working height) is 38.2130.29 = 7.92. 
 The rod will then be taken to B and held on the plug. But as 
 the plug is 1.53 below the grade-line at B, the target, when the 
 rod is held for grade on that plug, will be set at 7.92 -f 1.53 = 
 9.45. When thus held, the observer will set the horizontal 
 cross-hair on the middle of the target and clamp the telescope. 
 The line of sight will then be a line parallel with the grade-line 
 and 7.92 above it. Care must be taken to use the rod 7.92, and
 
 358 PLANE SURVEYING, 
 
 not 9.45, as the working height. Measurements may now be 
 made from the line of sight to determine the cut to the grade- 
 line at any intermediate point. 
 
 Suppose at C the rod read 5.97 ; then the cut at that point is 
 7.92-5.97 = 1.95. 
 
 How would you proceed if the instrument were set at B1 
 The cuts or fills to grade at any points may be determined by 
 taking the elevations of the ground at those points and calcu- 
 lating the grade elevations at the same points. The difference 
 of elevation will be the cut or fill required. 
 
 B. OFFICE INSTRUMENTS. 
 
 419. In addition to the various drawing-instruments previ- 
 ously described the student should understand the use of that 
 elegant instrument the polar planimeter. In ascertaining the 
 areas of figures having irregular boundaries it will be found 
 extremely useful. He should also become acquainted with the 
 different methods for the rapid reproduction of drawings. 
 
 SECTION II. 
 
 WORK. 
 
 420. The work of the city surveyor may be divided into two 
 classes : first, public work, or that which he is called upon to 
 perform for the city government ; second, private work, or 
 that which he performs for private citizens. The former is 
 generally connected with the streets ; the latter, with the prop- 
 erty between them. 
 
 Again, all of his work may be classed as field work or office 
 worlr, the former of which we will now consider. 
 
 A. FIELD WORK. 
 
 421. Public Work. There are many and varied natural 
 features and artificial influences affecting the original location
 
 FIELD WORK. 359 
 
 of a town or city. To the thoughtful student many of these 
 will readily suggest themselves. While in the choice of a site 
 the surveyor may have a voice, it is more than probable that 
 his work will commence upon a site already selected. We will 
 now describe some of his more important duties as performed 
 for the town or city government. 
 
 422. Street Lines. The city consists of streets for public 
 use, and of the blocks bounded by them, the land in which is 
 divided and sold to individuals for their private use. Hence 
 we have first to consider the general plan or arrangement of 
 the streets, their widths (the distances between house lines), 
 and their distances apart. There are many general plans which 
 may be adopted, or may be used as the foundation for new 
 ones. When general convenience and the economical division 
 of property are considered, I believe there is none which 
 better meets the requirements than that which is characterized 
 by two systems of parallel streets crossing at right angles. 
 With this general arrangement, and some well-located diagonal 
 avenues, we have the lay-out of a beautiful and convenient 
 city. 
 
 The general directions of the streets should be such that the 
 greatest number may during the day be visited by the sunshine. 
 This will be accomplished if one set of parallel streets runs in a 
 northeasterly and southwesterly direction. 
 
 Every important street should be at least 60 feet wide, while 
 some of the main streets should be at least 100 feet wide, with 
 avenues even wider. The streets will then admit freely air and 
 sunshine, which latter is too often in narrow streets cut off by 
 tall buildings ; while the avenues will be in harmony with their 
 design as elegant thoroughfares. 
 
 Another important consideration which affects the width of 
 streets is the expense of paving and of keeping them in order. 
 
 The distances of the streets from each other will vary very 
 much) according to the purposes for which the included prop- 
 erty is to be used, and how it is to be divided. Thejf may vary
 
 360 PLANE SURVEYING. 
 
 from 300 to 600 feet. The sidewalks will be from one-fifth 
 to one-fourth of the width of the streets. 
 
 In small towns an elaborate design will not be attempted ; 
 but it is alwa3"s best to have in view the possibilities of future 
 growth. 
 
 423. With the transit', the surveyor will run and extend 
 street lines, and will turn off required horizontal angles on the 
 horizontal graduated circle of that instrument. It is convenient 
 to work upon the centre lines of the streets. Two base lines 
 having been carefully located at right angles with each other, 
 the centrelines of the two sets of streets will, with the most 
 reliable measuring-instruments at the disposal of the surveyor, 
 be carefully located parallel with them respectively. If the 
 land is quite level, a 200-foot steel tape is useful. If it 
 be inclined and irregular, a 100-foot tape is better suited 
 to the purpose. In any case, the hand-level, plummet, ther- 
 mometer, etc., should be used. The work, like all work of 
 the surveyor, should be carefully checked by a test of the 
 different angles and distances. All this work should be done 
 with the greatest care. It is desirable, in order to guard 
 against future difficulties in regard to measurements by other 
 pai-ties, to make streets and block distances a little full; 
 that is, greater than they are actually required to be say 
 about one-fourth of an inch in 100 feet. As the work pro- 
 gresses, it will be properly marked with stakes, as before 
 described. After the satisfactory location of the centre lines 
 of the street, the house lines may easily be located therefrom. 
 
 424. The work of the surveyor may be not in laying out and 
 regulating a new town, but in connection with one already laid 
 out. The extensions of the old town may be carried on in 
 harmony with the plan already existing, or they may be on a 
 plan altogether different, and after the manner already described 
 for a new town. He will find that the already built-up portions 
 of the town have been previously regulated, or that they have
 
 FIELD WORK. 361 
 
 not been. If they have been, it is advisable in carrying on 
 the work therein to adhere as closely as possible to established 
 lines, elevations, standard of measurement, etc., lest any altera- 
 tions should lead to expensive and unnecessary legal complica- 
 tions. If the town has never been regulated, the first steps 
 will be to regulate its streets. In doing this a complete survey 
 will be required. Instrument lines will be carefully located 
 with the transit on all streets, and the angles at their intersec- 
 tions determined. These lines will be the basis for the location, 
 by offsets, of all buildings, fences, etc. As the survey goes on, 
 the results will be carefully plotted to a conveniently large 
 scale ; and from the completed plot, an advantageous location of 
 the streets may be determined upon. They will then be located 
 upon the ground to correspond. All important measurements 
 will be made, as before described, with the steel tape, with all the 
 corrections carefully attended to. Offsets to fences, etc., need 
 not be made with so much care, and the corrections will, as a 
 rule, be superfluous. During the progress of the work in an old 
 town, as in a new one, all important lines will be carefully marked 
 with stakes, and upon permanent objects, as houses, etc. 
 
 425. The streets in any 'city or town having been satisfac- 
 torily located according to the general plan, it is necessary, in 
 order to preserve work already done, and to prevent conflict in 
 future work, that the location of the street lines should be pre- 
 served. On account of the perishable nature of wooden stakes, 
 and the fact that they may soon be disturbed, it is necessary 
 to use something more permanent. This is generally found in 
 stones. Mere stones, or monuments used for permanently hold- 
 ing the lines of streets, are differently located and are of differ- 
 ent sizes, depending upon then- location. Sometimes they are 
 placed in the sidewalks 5 feet from the house lines. Then 
 they need not be more than 4 or 5 inches square and 2 feet 
 in length. The line is determined by a small hole drilled in the 
 top of the stone. Sometimes the top of the stone is placed 
 below the surface of the pavement ; sometimes it is placed flush
 
 362 PLANE SURVEYING. 
 
 therewith. Larger stones set in the intersections of the streets, 
 where their centre lines cross, are very conveniently situated 
 for use, and afford a very satisfactory means of marking street 
 lines. On account of their more exposed position, they must 
 be larger than those previously described, and should be set 
 with the greatest care, the materials around them being well 
 packed and rammed. They should be paved about and well 
 protected from danger from traffic. The stones should be 
 square in cross-section about 3 feet long, about 8 inches square 
 on the top, and about 1 foot square on the bottom, the top and 
 bottom being at right angles with the axis of the stone. The 
 line is determined as before by a hole drilled in the top of the 
 stone. From their situation we call these stones centre stones. 
 It is well also to mark substantial buildings standing at the 
 corners of streets with their distances from the house lines of 
 the streets, these distances having been carefully determined 
 by measurements. In general, a line having once been deter- 
 mined upon as satisfactory, every available means should be 
 employed to preserve its location, as any change would ob- 
 viously be attended with inconvenience and danger. 
 
 426. Street Grades. In the selection of a site for a town, 
 and in the location of the streets of a town or city, a topo- 
 graphical map will be of much service. This map will show at 
 a glance the shape of the ground under consideration. If the 
 surface of the earth were cut by horizontal planes 5, 10, 20, or 
 more feet apart, and the curves in which these planes intersect 
 the surface were projected upon a horizoual plane, the resulting 
 lines would be called contour lines or contours. These curves 
 would represent points of the same elevation. Their distances 
 apart would represent relative inclination in the ground, the 
 curves being nearer as the ground is steeper. The determina- 
 tion of these contours is an important feature in topographical 
 surveying. In addition to its other uses, such a map would be 
 of service in locating sewers, also in fixing proper elevations 
 and grades for streets. The field work necessary in the prep-
 
 FIELD WORK. 363 
 
 aratiou of topographical maps, which we will briefly notice, may 
 bo done as follows : Two sets of parallel lines having been 
 located at right angles with each other by means of the transit 
 and tape, the level will be set up, and a number of points at 
 any one elevation above the datum found with the level and 
 the rod, and their locations with reference to the two sets of 
 lines determined. Another set of points as far above or below 
 the former as the planes are apart will in like manner be deter- 
 mined and located, and so on until the entire ground has been 
 gone over. The above method of topographical surveying in 
 determining contours is not a very rapid one. The stadia 
 method is more rapid, and is well adapted to large areas. In 
 addition to the usual horizontal cross-hair in the transit, two 
 others are introduced, one above and one below the former. 
 The instrument has also a vertical circle. The stadia-hairs are 
 so arranged that when the level rod is held at a certain distance 
 from the transit, a certain number of feet on the rod is included 
 between them. The distance of any point from the instrument 
 can be determined, as it varies with the number of feet inter- 
 cepted on the rod. The line of sight must be at right angles 
 to the rod ; if it is not, a calculation must be made to deter- 
 mine the distance. By this distance and a horizontal angle the 
 point is located horizontally.* The elevation of the point above 
 the station at which the instrument is placed is obtained by 
 observing on the rod a point as much above the ground as the 
 telescope is, and taking the vertical angle. The product of the 
 horizontal distance and the tangent of the angle will give the 
 required difference in elevation. The plane table also has been 
 much used in making topographical surveys. 
 
 Street grades themselves will be determined upon in the 
 office, after the necessary data has been obtained in the field 
 
 427. A very convenient method of obtaining the data neces- 
 sary for the determination of elevations and grades for the 
 streets is to obtain a continuous profile of the ground on the 
 
 * See Chapter II., Stadia Measurements, Articles 148-152.
 
 304 PLANE SURVEYING. 
 
 centre line of each street. The work is done in the following 
 manner : The level having been set up, and the height of 
 instrument determined from a convenient bench-mark, an 
 elevation will be taken on a level plug set at the intersection 
 of the centre lines of two streets. Elevations will then be 
 taken at stations, say 50 feet apart, about on the centre line, 
 ' measurements with the tape being commenced at the inter- 
 section before mentioned, and made carefully enough to avoid 
 any error that might affect the work. In addition to the eleva- 
 tions at the stations, elevations should be taken at any interme- 
 diate points where the shape of the ground abruptly changes ; 
 and the points should be located by measurement. These 
 intermediate points are called pluses. When the next inter- 
 section is reached, measurements will be commenced anew, and 
 the levelling continued in the same manner. Elevations on 
 level plugs at intersections, on turning-points, and on benches, 
 which, if not previously established should be established as the 
 work progresses, should be carefully taken with the target. 
 The elevations for the profile should be read without the target 
 to the nearest hundredth. Such circuits should be made in 
 levelling for profiles, and the levelling on the cross-streets 
 should be so carried on as to check the work in every wa} 7 . 
 The level notes, taken as described for the profile of the centre 
 line of a street, are shown below. They are from actual prac- 
 tice. The datum is mean low water in the River, the 
 
 elevation of which is taken as zero. The manner of plotting 
 these notes, and of determining grade lines is given under the 
 head Office Work. 
 
 428. In order to avoid errors in giving grade lines, the 
 grade elevations at the intersections of streets should be per- 
 manently marked. This may be done by placing the centre 
 stones before described so that their tops shall be at the grade 
 elevation. In order to preserve these elevations in case of the 
 removal or disturbance of the stones, bench-marks should be 
 established on convenient door-sills, and in other safe and con-
 
 FIELD WOltK. 
 
 365 
 
 LEVELS ON FIFTH AVENUE, SOUTHERLY FROM MARY- 
 LAND AVENUE. 
 
 FOR PROFILE. 
 
 Nov. 21, 1880, A.M. 
 
 STA. 
 
 OBS. 
 
 H.I. 
 
 EL. 
 
 
 REMARKS. 
 
 B.M. 
 
 
 
 51.415 
 
 
 On west end of door-sill, etc. 
 
 + 
 
 7.030 
 
 ^58.445 
 
 
 
 
 
 s p - 
 
 0.870 
 
 
 57.575 
 
 
 
 1+ 
 
 10.005 
 
 67.580 
 
 ... 
 
 
 
 B.M. &(P. 
 
 1.300 
 
 
 66.280 
 
 
 ( On highest point of red 
 | rock, etc. 
 
 i+ 
 
 0.900 
 
 67.180 
 
 
 
 
 Sta. 0. 
 
 0.000 
 
 
 67.180 
 
 
 ( Plug middle of 5th and 
 \ Md. Aves. 
 
 + 25. 
 
 1.55 
 
 
 
 65.63 
 
 
 
 + 35. 
 
 0.28 
 
 
 66.90 
 
 
 
 1. 
 
 1.50 
 
 
 
 65.68 
 
 
 
 ( 50-ft. Sta. meas. south 
 | from mid. of Md. Ave. 
 
 2. 
 
 3.91 
 
 
 63.27 
 
 
 
 3. 
 
 0.20 
 
 
 
 60.98 
 
 
 
 4. 
 
 8.83 
 
 
 58.35 
 
 
 
 5. 
 
 11.80 
 
 
 
 55.38 
 
 
 
 6. 
 
 13.20 
 
 
 53.98 
 
 
 
 7. 
 
 
 
 
 
 
 
 Plug & ( P. 
 
 11.352 
 
 
 55.828 
 
 .... 
 
 ( Plug centre 5th Ave. and 
 | Anchorage St. 
 
 i + 
 
 4.365 
 
 60.193 
 
 
 
 
 B.M. 
 
 5.480 
 
 
 54.713 
 
 
 
 j Temporary on plug near 
 I fence, etc. 
 
 Sta. 1. 
 
 5.13 
 
 
 55.06 
 
 
 ( 50-ft. sta. meas. south from 
 ( middle of Anchorage St. 
 
 2. 
 
 4.65 
 
 
 55.54 
 
 
 
 3. 
 
 4.93 
 
 
 55.26 
 
 
 
 4. 
 
 5.69 
 
 
 54.50 
 
 
 
 6. 
 
 7.26 
 
 
 52.93 
 
 
 
 6. 
 Plug + 34. 
 
 11.00 
 12.224 
 
 .... 
 
 49.19 
 47.969 
 
 
 ( Plug centre 6th Ave. and 
 I Brown St.
 
 366 PLANE SURVEYING. 
 
 venient places. Besides serving as benches for the stones, 
 these bench-marks will be used in doing very close final level- 
 ling, the tops of the stones being too uneven for that purpose. 
 
 429. Marking of Lines and Grades. The lines and grades 
 of the streets having been finally determined, and the means of 
 preserving them having been established, the marking of these 
 lines and grades for any public work, as street extension and 
 grading, curb setting, sewer and water-pipe laying, etc., can 
 be readily done. Street lines will be run with the transit ; 
 and, in the manner previously described, grade lines will be run 
 with the same instrument. The marking of street lines and 
 grades for the purposes mentioned, the giving of lines and ele- 
 vations for other public work, and measurements of various 
 kinds, as of earthwork, constitute the principal part of the 
 field work to be done for the town or city government by the 
 city or town surveyor ; or, as the officer who does this work 
 may have more extended duties, the principal part of the 
 surveying to be done by the city engineer. 
 
 430. Private Work. Continuing the description of the field 
 work of the town or city surveyor, we will notice the second 
 general class in which his work is comprised ; that is, work for 
 individuals, or private work. In general, for other duties 
 in this connection will fall to his lot, such as surveying large 
 tracts according to methods already described, etc., this 
 work will consist in marking property lines and in giving 
 grades and elevations. As a rule, in a town or city more 
 property lines are marked for buildings than for any other 
 purpose. When the survevor is called upon to locate the 
 lines of a lot, his first inquiry will be as to the data by which 
 to locate them. It is of course understood that in this con- 
 nection the only power of the surveyor is to locate lines accord- 
 ing to given data, not, as man}' persons seem to think, to 
 establish of his own volition new lines. So we will inquire 
 what is proper data for locating such lines. In general, the
 
 J.I 
 
 24'6" r 24V 
 
 v r
 
 FIKLD WORK. 369 
 
 party desiring to have the lines of a lot marked will produce 
 his deed for the property. The young surveyor will be inclined 
 to think that the distances given in deeds are, as to the loca- 
 tion of lines, final. This is not always the case. When walls, 
 alleys, stones, and other permanent landmarks are called for, 
 and can be found, they will take precedence of distances in 
 locating lines. When walls, fences, and other holdings prove 
 undisputed possession for a period of years, though they may 
 not be described in the deed, they govern. In such cases it 
 would be superfluous to mark lines. In towns and cities lots 
 are now as a rule located from the streets. Let us take, in 
 marking the lines for a lot, an example from actual practice. 
 The description taken from the deed is definite, and is as 
 follows : 
 
 Beginning at the easterly side of West Street, between 
 Eighth and Ninth Streets, at the distance of 223 feet from the 
 southerly side of Ninth Street; thence easterly, parallel with 
 Ninth Street, 132 feet to a corner; thence southerly, parallel 
 with West Street, 28 feet to a corner ; thence westerly, par- 
 allel with the first-described line and Ninth Street, 132 feet to 
 the aforesaid easterly side of West Street ; and thence thereby, 
 northerly, 28 feet to the place of beginning. The lot is 
 located as shown in the sketch. The owner desired to have 
 marked upon the ground, for use in building, the two lines par- 
 allel with Ninth Street and the line of the easterly side of 
 West Street. In order that they may not be removed in 
 making excavations for cellars, walls, etc., the nail plugs to 
 mark the lines are set 3 or 4 feet outside of the lot. In the 
 sketch, S, S, S, S represent the stone monuments set at the 
 intersections of the centre lines of the streets to mark lines 
 and grade elevations. Each street is 49 feet wide. In mark- 
 ing the lot, points p, p, will be taken in the centre line of Ninth 
 Street. From these points (if there are no obstructions that 
 prevent) measurements will be made parallel with West Street. 
 Twenty-four feet six inches, half the width of Ninth Street, 
 and 223 feet, the distance from the southerly side of Ninth
 
 370 PLANE SURVEYING. 
 
 Street to the northerly side of the lot, will be laid down, and 
 nails placed in nail plugs at a, a, to mark the northerly line 
 of the lot. From these the southerly line will be located. In 
 a similar manner the front and back lines will be located. 
 Lines strained from a to a and from & to 6 will cross at c, 
 giving a corner of the lot, the nail plugs being undisturbed as 
 the work of building progresses. 
 
 If, on account of impassable obstacles, as buildings, walls, 
 etc., a measurement cannot be made from Ninth Street to the 
 place for the nail plug a back o the lot, the marking O f the side 
 lines will be done as follows : The southeast angle at the inter- 
 section of Ninth and West Streets, 89 51', if not known, will 
 be taken. In addition to the points taken in the centre line 
 of West Street for use in locating the front and back lines of 
 the lot, an additional point p' will be taken, and at this point 
 the angle 89 51' will be thrown in, and the random line p'p 11 
 located parallel with Ninth Street. On this random line points 
 for the location of the side lines will be taken. Now, suppose 
 the point p' is found by measurement to be 257 feet and 
 6 inches from the centre of Ninth Street (all corrections having 
 been made) , or 233 feet from the southerly side thereof. Then 
 the northerly side line will be located by measuring northerly 
 from the line p'p" 10 feet, and the southerly side line by 
 measuring southerly from the line p'p" 18 feet. If the sur- 
 veyor is in possession of an instrument thoroughly reliable for 
 use in angular measurements, the latter method of marking 
 side lines is to be preferred. When one measurement is made 
 along a sidewalk where there are no obstructions, and the 
 other through fences and over various obstructions, it is hardly 
 possible to obtain the degree of accuracy that may be obtained 
 by the angular method. Sometimes it may be necessary to 
 turn off an angle from the random line in order to locate the 
 back line of a lot. The location of lines is often marked by 
 nails in fences, measurements to houses, walls, etc., instead 
 of by nails in plugs. 
 
 After the street lines have been located and marked, the
 
 OFFICE WORK. 371 
 
 work in each block should be done independently of the other 
 blocks. 
 
 In the intervals between routine work it is desirable, in connec- 
 tion with gathering other data, to take and record in a suitable 
 . book, for use as described above, the angles at the intersections 
 of the streets, thus saving time in marking the lines of lots. 
 
 The location from the deed of the lines of a lot is not always 
 so easy as in the example given. It is frequently the case 
 that the distances given are indefinite ; sometimes none are 
 given. In such eases, in the absence of established holdings, 
 or other means of determining the location of property lines, 
 the matter must be settled by an arrangement between adjoin- 
 ing owners. 
 
 In some cases a lot is described in whole or part without 
 distances, but as bounded by the property of other owners. 
 In such a case the location of the lines may, if the descriptions 
 in the deeds of these other proprietors are sufficiently definite, 
 be determined by marking the lines of the other lots. 
 
 431. The city or town surveyor will frequently be called 
 upon for surveys to locate new lines with reference to the 
 street lines, or for surveys of tracts of land in or adjoining 
 the city or town. In such cases his manner of working will 
 be based upon the methods of land-surveying already described. 
 
 Private parties will frequently require, for use in building 
 operations, the marking of grade lines. This will be done in 
 the manner previously described. In marking the grade and 
 height of the building line in front of a lot, it will very often 
 be found convenient to mark the tops of the front line plugs 
 as so much above or below grade elevation. 
 
 B. OFFICE WORK. 
 
 432. Like the field work, the office work of the surveyor may 
 be classified as Public Work and Private Work. 
 
 433. Public Work. All field notes should be sufficiently 
 elaborate to be understood by those who may have occasion to
 
 372 PLANE SURVEYING. 
 
 refer to them. They should be carefully arranged and indexed 
 like all other office records for convenient reference. Plots of 
 work should be made whenever they will aid in the preservation 
 and proper understanding of work done in the field. When 
 plans are sent from the office, copies should always be retained. 
 
 434. It is desirable that, besides the necessary general plans 
 of the town or city, the surveyor should have in his office two 
 sets of plans, of a size convenient for handling, representing 
 the city in sections. For these plans a horizontal scale of 100 
 feet to the inch is suitable. 
 
 The first set should represent street lines. On them should 
 be placed all tha street lines, and, in figures, the widths of 
 streets and block distances, also the location of street monu- 
 ments, measurements made from time to time between centres, 
 angles at the intersections of the centre lines of streets, and 
 any other data of a like nature giving information in regard to 
 horizontal measurements, whether of lines or angles. 
 
 The second set should represent street grades. On them 
 should be placed, as on those of the other set, the street lines 
 and, in figures, the widths of streets, block distances, and 
 location of street monuments. In addition, there should be 
 placed upon them the profiles of the centre lines of the streets. 
 These plans will be used in determining grade lines for the 
 streets, which, after they have been determined, will be placed 
 upon the plans, with the grade elevations (G.E.) and surface 
 elevations at the intersections of the centre lines of streets, 
 grade elevations at curb corners, and any other data giving 
 information in regard to vertical measurements. The street 
 lines having been laid down, we will explain, in connection 
 with the accompanying sketch copied from a plan in actual use, 
 how the data given on page 365 would be used in placing upon 
 the plan the profile of the centre line of Fifth Avenue, and 
 then how the plan would be used in determining suitable grades 
 for the streets.
 
 OFFICE WORK. 373 
 
 435. If the points whose elevations have been determined by 
 the level be connected by a line in a vertical plane, such a line 
 is called a profile. The block distance from Maryland Avenue 
 to Anchorage Street is 297 feet and 9 inches, from Anchorage 
 Street to Brown Street is 294 feet, from Cedar Street to Fifth 
 Avenue is 264 feet, and from Fifth Avenue to Sixth Avenue is 
 160 feet. Maryland Avenue is 64 feet and 6 inches wide, 
 Anchorage and Brown Streets each 40 feet wide, and Cedar 
 Street, Fifth Avenue, and Sixth Avenue each 50 feet wide. 
 The sidewalks on 'Cedar Street and on Fifth, Sixth, and Mary- 
 land Avenues are 12 feet and 9 inches wide, and on Anchorage 
 and Brown Streets are 10 feet wide. By the use of the profile 
 of Fifth Avenue we will illustrate how the profiles of the centre 
 lines of the streets are placed upon the plan. The irregular 
 lines represent profiles. The profile is commenced by consider- 
 ing the centre line of Fifth Avenue, as drawn on the plan, to 
 have the elevation 67.180, which is the elevation in the notes 
 for the surface of the ground at the intersection of the centre 
 lines of Fifth and Maryland Avenues. The stations and pluses 
 as given in the notes are then laid down by scale on the centre 
 line of Fifth Avenue, in the order in which they were taken in 
 the field, beginning at the centre of Maryland Avenue. The 
 elevation at each of the points thus located is then plotted, in a 
 perpendicular to the centre line at that point, with reference 
 to the centre line elevation 67.180. In this case the points ob- 
 tained will all fall below the centre line. These points are 
 points in the profile, and, being joined, will give the profile as 
 shown. The profile of Fifth Avenue having been started at 
 the elevation of the ground at the intersection of Fifth and 
 Maryland Avenues, is said to be swung on Maryland Avenue. 
 In the sketch, the profiles of Cedar Street and Sixth Avenue 
 also are swung on Maryland Avenue. Those of Anchorage and 
 Brown Streets are swung on Cedar Street. 
 
 436. A little thought will make it evident to the student that, 
 as the differences of elevation are small as compared with the
 
 374 PLANE SURVEYING. 
 
 horizontal distances, if both were plotted to the same scale, or, 
 as we say, if the vertical and horizontal scales were made equal, 
 the differences in elevation will scarcely be apparent. This is 
 remedied bv conveniently exaggerating the vertical scale. For 
 example, if the horizontal scale be made 100 feet to the inch, 
 the vertical scale might be made 10 feet to the inch. In the 
 sketch the two scales have this ratio. 
 
 EXERCISE. Let the student select scales, and, in the manner 
 described above, prepare a profile from the field notes given on 
 page 356. 
 
 437. Having thus plotted the streets and profiles in a large 
 area, we may, by use of the plan thus made, determine suitable 
 grades for the streets. This will involve careful study of the 
 shape of the ground, location of watercourses, probable loca- 
 tion of sewers, and effect upon property. The effect of a pro- 
 posed grade for one street upon those which it crosses must be 
 particularly noticed. To properly perform this work involves 
 that knowledge and judgment which can only be acquired by 
 long experience. The straight lines drawn in connection with 
 the profiles represent the surface grades of the finished streets. 
 In fixing the grade for Fifth Avenue, those of the other streets 
 having been taken into consideration, it was found best to have 
 a cut of 2 feet at Maryland Avenue, no cut or fill at Anchorage 
 Street, and a cut of 2 feet at Brown Street. The elevations 
 of the surface at the intersections of Fifth Avenue with Mary- 
 land Avenue, Anchorage Street and Brown Street, are respec- 
 tively 67.180 on the ground, 55.828 and 47.969 on plugs flush 
 with the ground. The grade line having been fixed, the grade 
 elevations (G.E.) at the centres are respectively 65.180, 55.828, 
 and 45.969, and the descents 9.35 feet and 9.86 feet, as shown 
 in the sketch. The nature of grades will depend much upon 
 local considerations. Grades should always be steep enough to 
 secure proper drainage. The inclination should not be less 
 than 1 in 100. Considering the accumulations of dirt on many 
 of our city streets, from 1 to 1.5 in 100 is to be preferred.
 
 OFFICE WORK. 377 
 
 438. In streets in which surface water is carried on the 
 streets, some streets will carry the water in gutters across oth- 
 ers. In the sketch such streets are indicated by having arrows 
 drawn in their directions across intersections. In this manner 
 Fifth Avenue carries the water across Brown Street, and An- 
 chorage Street carries it across Fifth Avenue. The water flow- 
 ing on Fifth Avenue, from Maryland Avenue towards Anchorage 
 Street, will turn into Anchorage Street. The opposite side of 
 Anchorage Street, at the house line, will be a knuckle as high as 
 the centre of the street ; and the water will flow from that point 
 
 i towards Brown Street. In fixing grades great care must be 
 taken to so arrange them that one street shall not be overtaxed 
 with water from the others. An outlet for the surface water is 
 formed in the natural watercourses. 
 
 If the grade of Anchorage Street were very heav}', so that if 
 continued across it would make one side of Fifth Avenue much 
 higher than the other, it would be desirable to break the grade 
 of Anchorage Street at the curb lines of Fifth Avenue, giving 
 only sufficient fall to carry the water across the Avenue. 
 
 439. If the section is sewered, and if the sewers are made 
 "large enough to carry the surface water, the gutters across the 
 streets will be dispensed with, and inlets to the sewers placed 
 at the curb corners of the blocks. 
 
 440. It is often convenient and useful to have plotted on 
 separate streets the profile and grades of each street. 
 
 441. Besides making street and grade plans, it will be a part 
 of the office work of the surveyor to plot, in the usual manner 
 of plotting such work, the surveys made in and about the city 
 or town, for both the city and individuals. 
 
 442. In some cities a registry of property is kept. The 
 plotting of lots in suitable record books, and the keeping up of 
 the records, will be a part of the city surveyor's work.
 
 378 PLANE SURVEYING. 
 
 443. Private Work. This includes the preparation of any 
 plans ordered for their own use by parties other than those con- 
 nected with the city government. 
 
 CONCLUSION. 
 
 444. The student must bear in mind that he can never, from 
 books, learn to be an accomplished surveyor. The practice is 
 ever in advance of the books. Though he should store his 
 mind with book knowledge upon the subject, he will yet be 
 wanting in the knowledge and readiness regarding actual work 
 which can only be acquired by a long experience. Many oper- 
 ations which can with difficulty be understood from pages of 
 explanation, will, when their actual performance is seen, be 
 comprehended in a short time. Again, there is that which can 
 never be learned from books ; that is, the judgment which must 
 be constantly exercised in practising the delicate duties of a city 
 surveyor. Among other things, this judgment will teach him 
 to be very cautious about giving voluntary advice, and careful in 
 giving even that which is requested ; to perform his duties con- 
 scientiously, and to keep clear of all entangling alliances. Let 
 him learn everything connected with a complete performance 
 of his work, from the work of the axeman up ; that, when he 
 directs, he may do it with the same grace with which he should 
 ever follow the directions of his superiors. 
 
 The practice of city surveying is a most excellent drill. If 
 conscientiously performed, it will develop careful and thought- 
 ful habits. However, in practice the student will also have to 
 learn to avoid " fussing" over work, and to proportion to the 
 importance of the work in hand the time and care spent upon a 
 particular work. 
 
 BOOKS. 
 
 445. Valuable information regarding the matters treated of 
 in this chapter will be found in the following publications : 
 
 The manuals and catalogues of instrument-makers.
 
 BOOKS. 379 
 
 "A Treatise on the Principles and Practice of Levelling," by 
 Frederick W. Simms ; published by D. Van Nostraud, New 
 York. 
 
 "A Descriptive Treatise on Mathematical Drawing-Instru- 
 ments," by William F. Stanley ; published by E. & F. N. Spon, 
 New York and London. 
 
 " A Manual of Drafting Instruments and Operations," by S. 
 Edward Warren ; published by John Wiley & Son, New York. 
 
 " The Draughtsman's Handbook of Plan and Map Drawing," 
 by George S. Andre" ; published by E. & F. N. 'Spon, New 
 York and London. 
 
 The student of surveying who wishes to extend his studies 
 into the field of city engineering will find information upon 
 that subject in the numerous works upon its special branches, 
 and in the current technical periodicals of that class. Much 
 information regarding present American practice in city engi- 
 neering will be found in the series of papers on " Municipal 
 Engineering" now being published in '-Engineering News." 
 When completed, these in book form will make a very useful 
 v olume.
 
 CHAPTER VIII. 
 
 MINE SUKVEYING. 
 
 446. The survey of underground excavations (mines) to 
 determine their position and extent may be principally for the 
 purpose of projecting the points upon a horizontal plane as in 
 land surveying. 
 
 But in strata of high inclination and in cavernous spaces 
 various vertical projections will be needed to complete the 
 graphical representation of the workings ; and in fissure veins 
 the elevation may be more important than the plan. 
 
 447. Surveys to depict areas underground may be made with 
 surveyors' compass and chain, but generally now the transit or 
 theodolite is used to take the angles, and the steel tape to meas- 
 ure the distances, and in some mines the tape may be with ad- 
 vantage hundreds of feet in length ; but generally 50 feet for 
 the chain or 100 feet for the tape are most convenient lengths. 
 
 448. The surveyor and each assistant, of course, requires a 
 lamp, and "the sights" are ranged with lamp and plummet, 
 the sight from the instrument being taken upon the flame of 
 the miner's lamp (or candle, it may be) suitably held at the 
 plummet line, which is held to depend from a point fixed or to 
 be fixed in the " roof" or over a point in the " bottom." The 
 plummet string itself may be seen within 300 feet. A chain- 
 pin (arrow) can be used to plumb the light over or under a 
 point. It is advised to display the light at a station for sight 
 only, and therefore in moving it, for any reason, other than 
 vertically, in giving the point, it should be hidden from the 
 observer.
 
 MINE SURVEYING. 381 
 
 The point may be marked by a nail in the timber cap or sill, 
 or be a nail in a peg ; the place of the point in smooth roof is 
 to be made conspicuous by a ring of white paint around it, and 
 also as it may be by reference marks at the sides (pillars) of 
 the passage-way. 
 
 It is a refinement to use a lamp which is also a plummet, and 
 further to place an extra lamp on the bottom under it; two 
 tights seen in the vertical line making its place more certain, 
 and helping to decide that the sight is ready to be taken.* 
 
 It may happen that the line of reflection from standing water 
 can be taken for the line of incidence of a light held under a 
 point, when the roof droops between, the passage being " in 
 swamp" there. 
 
 The surveyor's lamp is made entirely of brass or copper, so as 
 not to affect the magnetic needle of the instrument. 
 
 For use in low openings the tripod of the instrument must be 
 one of short legs (an extra set of shifting legs will answer the 
 purpose), or have extension legs. 
 
 It has been suggested to use two extra tripods, one to set 
 up in advance, for keeping the place of fore-sight and for receiv- 
 ing the instrument, alone carried forward to be mounted there 
 at the same exact spot with facility, while the tripod, left 
 standing at the last place of the instrument, marks the point 
 for back-sight with equal certainty : thus each of three tripods 
 taking its turn in being at a place for fore-sight, remaining 
 there for mounting the instrument upon it, and still remaining 
 for back-sight after the instrument is taken for mounting at 
 next station. There are obvious objections to this in the 
 weight of the luggage, and that only some instruments are 
 made for such ready separate handling. 
 
 Some rays of light must be thrown into the telescope at its 
 object end to make visible the cross-hairs therein. This is gen- 
 erally done by the surveyor, while taking a sight, holding his 
 
 * Eckley B. Coxe derised the plummet lamp, and also a form of it with 
 wire-gauze covering, like the Davy Safety Lamp, for use where fire-damp 
 may be expected.
 
 382 PLANE SURVEYING. 
 
 lamp in his left hand at the front, but a little to one side of the 
 object-glass. A reflector mounted at the object end is a help. 
 One is a silvered flat ring, standing bias, about "2 inches forward 
 from a collar which is slipped over the object end of the tele- 
 scope. It reflects light into the instrument as an annular beam. 
 Another one is a diminutive hemisphere which scatters light 
 caught from the lamp into the tube. 
 
 The change to, and the equable temperature of, the mine 
 require the trying and favor the making of the ordinary adjust- 
 ments of the instrument there. 
 
 449. Stations are generally made only at the angle points of 
 survey lines, and are therefore not regularly distanced. They 
 may be numbered, lettered, or designated by the total distance 
 from the zero of the measurements of their line. Intermediate 
 points are made on the line where, opposite to lateral openings, 
 other lines of survey or important short connections by measure- 
 ment merely may start. The corners of chambers along the 
 passage may be noted by distance without making points ; the 
 size and position of parts of chambers being afterwards taken 
 and noted by sketch with dimensions relatively marked thereon, 
 there being mostly a parallelism in the rock measures which sim- 
 plifies the position and shape that chambers take, so that no 
 special survey of directions is regularly required for them. 
 
 450. Angles between vertical planes of sight (in azimuth) 
 are noted for obtaining the courses as reduced courses from the 
 initial course of survey, by the successive additions and sub- 
 tractions to it and from it of the angles as taken, and modified 
 according to the series of 90 in each quadrant of the circle. 
 
 The initial course had better be referred to true meridian, and 
 comparison with bearings made with allowance for the variation 
 (declination) of the needle. But it has always been recognized 
 that the course, in degrees and minutes, of a quadrant and 
 therefore liable to mistakes as to the particular one of four 
 quadrants would be absolute if the full circle be graduated
 
 MINE SURVEYING. 383 
 
 around to 90, 180, 270, and 360, in the successive quadrants. 
 While it is not agreed whether north or south shall be the zero, 
 the direction of graduation with the movement of the hands on 
 the dial of a watch or clock is conventionally fixed. The bear- 
 ings will be a key to which zero was used in the notes. 
 
 451. It is but seldom that in drifts of mines the alignment as 
 well as the grade requires adjustment to the regularity of straight 
 lines and curves similar to surface railroads ; for the tram-cars 
 will run around very, sharp turns, and for them there is there- 
 fore no necessity of expensive improvements in line. But when 
 a locomotive is to be used, or wire-rope haulage is to be intro- 
 duced, there is apt to be a call for regulation of the line, with 
 regard, especially, to minimum radius of curvature. 
 
 Unlike the longer, flat curves of a railroad, designated 
 according to the American system by the even angular deflec- 
 tions from each other of chords of 100 feet, these sharper 
 curves will go by assumed even radii (in length not less than 
 ten times the gauge of track), and the deflection angles for run- 
 ning them in by the instrument upon short chords will have to 
 be calculated. 
 
 One-half the chord divided by the radius will equal the sine of 
 the angle of deflection from tangent, which is half the angle 
 that two such equal chords will make with each other, and also 
 half the angle at the centre of the circle subtended by the chord. 
 From any point on the circular curve as a position of the instru- 
 ment, successive deflections of the angle will fix the ends of 
 consecutive chords as measured in. Shorter chords (like those 
 less than 100 feet in a railroad curve) have deflection angles 
 approximately proportional to their lengths. 
 
 For ranging the line of direction of a passage that is being 
 opened into the solid, two points for placing lights are given at 
 the start, necessarily near together, until the prolongation of 
 open space allows testing the line by the instrument and giving 
 new points of line. From the three points of a curve line that 
 mark the chords of half the arc, obviously, by simple measure-
 
 384 PLANE SURVEYING. 
 
 ments, a like fourth point may be derived as the face (breast) 
 of the working is advanced. In driving a passage-way describ- 
 ing a semicircle to save weakening pillar at foot of shaft a 
 long, curved gas-pipe was used in ranging around. A large- 
 scale working plot showing offsets secures the proper location 
 of curving and branching passages. 
 
 Outside, besides the fixing of projected curves by deflection 
 angles as above, the laying off of points of arc intermediate on 
 the chord is by foot-rule measurement of ordinates at right 
 angles. 
 
 But without strict regard to data, an expedient way of uniting 
 two intersecting straight lines of track 03* a circular curve (as 
 an arc starting from the one straight line at any distance short 
 of the apex of the lines and ending on the other line an equal 
 distance from the apex) is to find points by linear measure- 
 ment merely. Assuming any tangential distance back from 
 apex to P.C. (point of curve), the beginning, and the same to 
 P.T. (point of tangent), the end of curve, we find a third point 
 of the arc, its middle, as a point midway between the middle of 
 the chord of the whole arc and the apex. One-fourth of this 
 versed sine will be the versed sine (middle ordinate) to be erected 
 on each chord of half the arc for points of the arc. And any 
 other middle ordinates will be as the squares of their arcs or 
 chords. 
 
 This principle applies in rounding off intersecting grades 
 into vertical curves, either convex or concave ; by vertical allow- 
 ances and according to horizontal distances, starting with that 
 at the apex and proceeding similarly to the foregoing as to 
 subdivisions. 
 
 The laying off of curves by chords and versed sine so derived 
 does not require knowledge of length of radius or of amplitude 
 of angle. But when the extent of circular arc between two 
 tangents is to be determined by the length of radius, the tan- 
 gential distance from apex will equal radius multiplied by natural 
 tangent of half the angle of intersection ; and between P.C. 
 and P.T. there will be the same measures of chord as there are 
 of chord singles in angle of intersection.
 
 MINE SURVEYING. 385 
 
 452. In the note-book the left-hand page is used for stations, 
 distances, angles, courses (reduced), and bearings (magnetic), 
 and the opposite right-hand page for offset distances marked 
 relative to a perpendicular line dividing the page, together with 
 sketches and remarks. The notes should begin at the bottom 
 of the pages and proceed upwards, to appear as on the plan to 
 which their results are to be transferred, in their proper relation 
 of position and observation forward. 
 
 The plan of underground work is begun with the plotted net- 
 work of the lines of survey, then the outline of parts excavated 
 is drawn in detail, and these are shaded, as the places become 
 closed in and abandoned, to distinguish what is open work at 
 any period. 
 
 The scale of maps showing the workings, etc., of coal mines 
 is now fixed by law in many of the States at 1 : 1200 as the 
 least; that is, at not less than 1 inch for 100 feet; the purpose 
 of the maps being to aid the official inspection and regulation 
 of the mines for securing the health and safety of the miners. 
 The plan will generally require to show the relation of the 
 workings to surface openings, watercourses, and bounding lines, 
 and to improvements, such as buildings, roads, and railroads. 
 
 The line of outcrops (exposure at the surface of the ground of 
 the mineral beds) within its range will appear on the map, but 
 general topographical detail is reserved for the extended small- 
 scale maps of the surface, which will represent what may be 
 learned of mineral indications also ; from which data in 
 advance of the workings may be derived and confirmed 
 by special explorations, as of proof-holes and deep boring. 
 But upon the mine plaii such elevations (heights of surface 
 above datum) as seem most essential, such as principal ones 
 along the outcrops, highest points of hills, and lowest of 
 streams should be mapped. 
 
 The use of the pantograph, for reducing the irregular figures 
 of mine plans with all details from one scale to another, has 
 found much approval ; and the plammeter is liked for labor- 
 saving and accuracy in determining such areas.
 
 886 PLANE SURVEYING. 
 
 453. In veins, the work being deep and narrow, and pursued 
 from levels or galleries (horizons of working) generally about 
 60 feet apart in height, plans of these levels, drawn in different 
 colors to distinguish them, are superimposed on the map of 
 general plan. The}' show the openings, the gangways, the 
 cross-cuts, etc., with the defining lines of the walls of the vein, 
 and may embrace other separations of the mineral. Longi- 
 tudinal elevation and vertical cross-sections will show the 
 shafts and other connections between the levels, together with 
 the chambers, whether open, filled in, or caved. 
 
 Ore bodies occurring detached and of the most varying 
 dimensions, though often resembling each other as lenticular in 
 shape, make the workings appear in plan, elevation, and cross- 
 section, as the results of exploration in patches. Shafts in the 
 vein will be parallel to pitch of one wall, and therefore varying 
 from the vertical. 
 
 A stratified bed that is to be operated upon, opened, and 
 won by mining, may be conceived as a seam of uniform small 
 thickness extending within limits as a plane surface and in 
 relative position defined by the " strike " (the course of all its 
 level lines, which will all be parallel) and its "dip" (the 
 greatest pitch at right angles to the course of the level-line). 
 But upon the large scale the seam occurs of variable thickness, 
 and with lines of level changing in direction and not parallel at 
 different elevations, to the degree that instead of a plane it is a 
 warped surface. 
 
 The arrangement of permanent works upon the surface of the 
 ground with reference to the lay of the bed as well as the topog- 
 raphy and improvements existing or suited to it, the favorable 
 connection of the lines of haulage and drainage inside, with all 
 to govern outside, present to the mind of the mathematical 
 surveyor applications of the theorems of Descriptive Geometry, 
 as included in adaptation to the ends of practical economy. 
 
 454. Location upon the surface of the ground of the plan of 
 inside work, is a repetition of courses and distances outside in the 

 
 MINE SURVEYING. 387 
 
 same vertical planes. Any particular portion of the workings 
 in progress can thus be compared in natural scale upon actual 
 plan of surface of the ground over them. 
 
 Overlaid plans with elevations and cross-sections of workings, 
 such as were described for workings in veins, are required to 
 show the development in high pitching beds. The " lifts " or 
 levels in such of coal are 100 yards apart, measured on line of 
 pitch. 
 
 Overlaid plans of different parallel seams worked through 
 same shaft are also made, but without systematic elevation 
 and cross-section ; the connections (shafts, slopes, or tunnels) 
 between the beds being through barren ground, and limited 
 to the exigencies of hoisting, draining, and ventilating. 
 
 455. Following the determination in azimuth by courses and 
 distances of the passages in the mine is the determination of 
 their changes in level by the spirit levelling-instrument and the 
 level-rod (as a separate operation, even if the transit be a com- 
 bined instrument having a parallel spirit level attached to its 
 telescope), the work being quite similar to such above ground. 
 But the rod must be limited in height to the low spaces where it 
 is to be used, and is preferably marked with red figures for the 
 feet, and white figures for the tenths, upon a black ground. 
 The top of a simple white target is safer to take, however, than 
 the reading from the instrument of the figures themselves. For 
 accuracy, sights, as above ground, should be limited to 300 
 feet in distance from the instrument. 
 
 From the elevations of points taken by levelling, contour 
 lines can be shown on plan as the mineral bed is exploited. 
 
 Blue is the conventional color for these contour lines and the 
 figures marking their elevation above the datum, on a mine 
 plan, and brown suits for the contrasted surface elevations. 
 
 456. Levelling along passage-ways for the purpose of fixing 
 better gradients of hauling-roads, or for fall of water by rectifi- 
 cation of undulating bottom to improve drainage, requires sta- 

 
 388 PLANE SURVEYING. 
 
 tions especially chained in at regular distances of 50 feet or 
 less ; the marks being temporary ones on the sides to serve for 
 taking the levels and to be referred to as to heights in grading, 
 when the variation of level of bottom from the grade of a 
 station governs the cutting or filling of bottom there, or change 
 of the whole cross-section in height, as it may be. For the 
 adoption of suitable gradients along an extended line, a longi- 
 tudinal vertical section is drawn, called a profile, which exhibits 
 the relation of ground-line levels, and allows the fixing of 
 grade with assurance. The profile may include the line of top 
 as well as of bottom, with section of rock measures to be 
 affected by " ripping " of the roof and " cutting " of bottom. 
 
 457. A Drift or passage along with the measures of a bed 
 will make undulating grade, if course be followed ; and if the 
 drainage-rise be allowed to govern, the alignment will be sacri- 
 ficed. 
 
 Tunnelling, however, being arbitrary, across the measures, is 
 mostly upon directed line and grade. Slopes are mostly upon 
 directed course ; but if within the measures of an inclined bed 
 will mostly be variable in grade. So with an adit, driven to 
 give drainage outfall to the surface. For it, shortening of the 
 distance will probably be the governing condition principally. 
 
 458. For the workings at high pitch, the determination of 
 horizontal and vertical components of the distances on the 
 sloph g lines of top and bottom in a bed, and " hanging wall " 
 and "foot wall" in a vein, will bring the vertical arc of the 
 instrument into requisition, for obtaining the vertical angle, 
 which is always taken as the full angle above the horizontal. 
 Vertical sections, besides such longitudinal ones following 
 broken line of passage within a stratum and showing only adja- 
 cent rock measures, may be made of particular places where 
 there is folding, or fault, of the measures, and for geological or 
 more general purposes they may exhibit the lay and thickness 
 of the various rocks up to the surface, which will as a correct
 
 MINE SURVEYING. 389 
 
 margin show the outcroppings in profile. Vertical sections may 
 be projections upon planes that traverse the measures according 
 to various conditions, and may be constructed of related points 
 from the map that were not determined for their relevancy to 
 this purpose. 
 
 It seems that vertical arcs have had versed sines correspond- 
 ing to radius 1 marked around them for the purpose of telling 
 the allowance upon slope measurement to obtain corresponding 
 horizontal distances, the versed sine being the difference 
 between the hypothenuse as the radius and the horizontal base 
 as the cosine of the vertical right-angled triangle formed ; and 
 the slope length for a given horizontal distance would be 
 greater, according to the versed sine of the angle. 
 
 Vertical arcs have had tangents as rises corresponding to the 
 unit of horizontal distance for the different angles marked upon 
 them. 
 
 A method of dividing the arc according to the sines, without 
 the intervention of the equal graduation into degrees neces- 
 sarily, is the subject of a contribution to "Van Nostrand's 
 Engineering Magazine" for July, 1876, and is appended at the 
 end of this chapter. 
 
 459. The measurement down deep borings or shafts is best 
 made by special flat steel wire, with suitable plummet heavy 
 enough to insure its making the wire line taut. 
 
 The transfer of points down a shaft, as of two to determine 
 a base line for connecting surveys below with those on the 
 surface of the ground, is made by very heavy plummets 
 attached to ordinary wire run off of reels. A portable box 
 to contain the reels, their cranks, and the plummets, is con- 
 venient; the best arrangement being that of reels fixed in a 
 frame that stays in the box. The suspended plummets are to 
 be received below each in a bucket of water, or, if hanging from 
 considerable height, in some thicker liquid to settle the wire 
 lines to a steady position for ranged observation by the instru- 
 ment below. And the observation will be easier upon wire that 
 is whitened there by chalk or paint after being placed.
 
 390 PLANE SURVEYING. 
 
 The plummets in the shaft of the Washington Monument, for 
 showing changes in the verticality of the structure, are steadied 
 in vessels containing a mixture of glycerine and molasses. 
 
 460. For taking courses on pitches at high angles an extra 
 telescope on the axis extended to the outside of one of the 
 standards of transit has been used. Another mining transit 
 has for the same purpose the sweep of the telescope to the ver- 
 tical position, made possible by having its standards made 
 inclined to overhang. But the object-prism placed before the 
 object-glass, allowing sighting at true right angles in any plane, 
 seems most simply to fulfil the requirements for sighting up or 
 down, as well as side wise, and is a ready means applicable to 
 the telescope of any ordinary instrument. A transit adapted 
 in any of these ways for taking vertical sights enables the 
 points of base line, as transferred by plummets to the bottom of 
 the shaft, to be tested and compared with the extended line 
 across the pit top, provided the atmosphere be clear in the shaft 
 and obstructions do not intervene. The vertical adjustment 
 of the instrument itself would be tested by this check, the usual 
 test being on high objects, with reversal of standards to oppo- 
 site sides by turning the horizontal plates. 
 
 A heavy, substantial, simple transit, not weighted with 
 "attachments," is the most reliable. 
 
 461. The use of the hanging compass and of the hanging 
 clinometer of the olden time is retained in small and crooked 
 passages of some metalliferous mines. And their subsidiary 
 use in excavations inconvenient of access or footing of the 
 ordinary (the standing instruments) has lately been recom- 
 mended as of wider application, and they have been introduced 
 into this country. Each of the instruments is to hang by its 
 two hooks, turned opposite ways, to the cord that marks the 
 line. The compass-box levels itself by its 'gimbals (double 
 trunnions), like a ship's compass, in the frame of which the 
 flat hooks with long bearings in line are a part. The clinometer
 
 MINE SURVEYING. 391 
 
 bangs as a vertical arc with plummet to give the inclination of 
 the cord from the horizon, while the compass gives the needle 
 course. The cord is stretched from one low stout tripod to 
 another, or in a curving space may be fastened to a gimlet 
 screwed into side timber beyond intersecting point or angle of 
 two cords. The tripod serves as a stool also for the assistant 
 holding cord to the point on it firmly. The distances are accu- 
 rately measured along the cord by applying a graduated rod to 
 it. The horizontal and vertical components of the measure- 
 ments have to be calculated for plotting on plan and section. 
 In the old mining regions of Europe the surface surveys were 
 also carried on with the same appliances. With .care and 
 patience surprisingly good results in locating connections were 
 attained. The old instruments were graduated in hours and 
 minutes, and the English designations of dial and dialling for 
 the mine compass and operations with it seem to refer to the 
 same original division of its circle. It seems strange to learn 
 that the plotting was protracted by the same compass (swung 
 there on horizontal plate used for straight edge), reference 
 being had to a meridian line fixed in the office, and the drawing- 
 table being a smooth and level stone slab resting on foundation 
 independent of the office floor. 
 
 462. Formerly, when topography was used more for the pic- 
 turing of the plan of landscape in mapping the features for the 
 information of the tourist or the military commander, than for 
 the projection of the contour accurately to fit the location of 
 artificial ways of the different kinds to the ground, hachures 
 were used to indicate character of sloping elevations, and 
 they survive in use upon small-scale maps, to indicate moun- 
 tain chains. They are intended to be lines of pitch, drawn 
 close together so as to graduate changes naturally, and they 
 should be broken at the intersection of the successive level 
 planes with the surface to make terraces however narrow, and 
 suggest level stages in measure of elevation. Now we have on 
 topographical plans contour lines to represent the lines of sue-
 
 392 PLANE SURVEYING. 
 
 cessive levels, say 10 feet apart in rise. They are plotted by 
 connecting all points of elevation that may be determined over 
 the area with regard to the requirements of accuracy in noting 
 the changes ; and they may be considered the margins made by 
 a body of water that had successively risen or receded 10 feet 
 in height at a time over the area. They are to be marked by 
 their elevation above the lowest datum plane, preferably over 
 that of mean tide of the ocean. They turn upon themselves 
 where they enclose a peak or a basin according as the next 
 ones indicate them as higher or lower in the series ; they are 
 farther apart in horizontal distance as slopes are flatter, and 
 where two or more coincide for any distance there is a precipice. 
 
 These points of even elevations of the ground are determined 
 from the levels run along the survey lines, and the cross-section 
 profiles taken at the stations of the lines slopes being taken at 
 right angles to the line with straight edge pole and clinometer or 
 plummet slope level applied to it. Each of these angle instru- 
 ments having a vertical graduated arc, the former with arm 
 hinged at centre of arc and carrying a spirit-level to ascertain 
 the vertical angle included between the levelled arm and the 
 slope of the straight edge under it ; the latter, by the departure 
 from the perpendicular of the plummet, showing the equal 
 departure from the horizontal of the straight edge. 
 
 From the profile of each slope sketched in the field-book and 
 marked with distances and degrees of rise and fall across the 
 survey line, the successive even 10-foot points can be laid off on 
 plan, regard being had in starting with elevation of station to 
 the partial changes required for the first even 10-foot point each 
 way. A scale of horizontal distances for each degree of the 
 arc, to gain 10 feet rise, is made by the topographer of Bristol- 
 board to lay off the points derived by sloping at the stations, 
 and saves the plotting of the profile of cross-section. 
 
 The topographer prefers to draw the contours in the field as 
 taken, using demi-sheets of paper that can be joined at their 
 margins, and upon each of which a portion of the line corre- 
 sponding to its number is plotted, the line having dots along it, 
 spacing the successive stations intermediate of the angle points 

 
 MINE SURVEYING. 393 
 
 of line, and having the elevations corresponding in pencil along- 
 side. The sheets are held in a box that is carried by a shoulder- 
 strap, and the side of which is used in the field as a drawing- 
 board, the particular sheet in use at the time being tacked on it. 
 
 463. The topographer will sketch in the streams, buildings, 
 etc., with reference to measurements however, and will have 
 special lines with small compass, etc., run for him to make 
 contour connections. The operations will rise to the scope of 
 plane-table work, if the drawing-board have a socket with 
 clamps, and be mounted and levelled (by applying a loose hand- 
 level) upon a tripod ; the ruler used on it having small compass 
 sights screwed to its ends for sighting to objects and fixing 
 their position on plot by the graphic triangulation of intersected 
 sight-lines from different stations on the survey -line ; the sta- 
 tion on plot when over its place on ground having a needle 
 stuck upright in it, that has a sealing-wax head for convenient 
 handling, for the purpose of resting the ruler against when sight- 
 ing. Interpolation, or resection, is the reverse sighting from 
 without the line over the plot to two or three poles on stations 
 of the line or other previously located objects, to attain posi- 
 tion, it being understood that the plane table stands with plot 
 in proper relative position always. Secondary triangulation will 
 extend the area of topographic sketching, but this should be 
 checked by connections beyond with surveyed lines and levels. 
 
 The Locke level may be used for taking rises by finding all 
 the points in sight that are at a level of the eye, and, in con- 
 nection with the le veiling-rod, the fall of ground may also be 
 determined by this instrument. For gently undulating ground 
 the use of it is better than sloping. 
 
 464. Contour lines are drawn 10 feet apart in elevation on 
 most plans of extended land and other surveys that are meas- 
 ured in detail, but it is obvious that cases occur where for large- 
 scale work they are taken closer in elevation or farther for 
 small-scale mapping. In the fonm-r rase of large-scale work 
 they may be required exactly as elevations directly located by
 
 394 PLANE SURVEYING. 
 
 spirit levelling-instrument, in the latter case as the approxima- 
 tion from altitudes taken in a few places by the barometer. 
 
 The scope of their usefulness on plans for projecting improve- 
 ments it would be difficult to describe exhaustively. They may 
 be for use in locating the drives and walks and terraces, etc., 
 of a park; the shaping of grounds, under-draining, etc., about 
 a residence ; the laying out of streets, etc., in a hilly town ; the 
 leading of streams of water, large or small, for all purposes in 
 partial or wholly artificial channels, for navigation, water 
 power and supply, irrigation, etc. ; the location of roads and 
 railroads with regard to ease of construction and of favorable 
 gradients, as well as the uses in mining directly, and location 
 of all surface erections collateral thereto or elsewhere, collec- 
 tively known as " the Works." 
 
 ANGULAR CROSS-SECTIONING. 
 
 By F. Z. SCHELLENBERG, C.E. 
 
 Written for "Van Nostrand's Engineering Magazine," July, 1876. 
 
 A most direct and expeditious method to get differences in 
 level between points in sight is by the use of a vertical arc grad- 
 uated to the successive sines 1, 2, 3, ... 100, in quadrant, for 
 the radius of arc 100. 
 
 Multiplying the distance measured in hundreds on the slope 
 by the rate per hundred indicated on the arc gives the difference 
 in level in units. In the higher parts of the arc the correspond- 
 ing cosines may be marked for deriving horizontal distances. 
 
 The applicability of this graduation to such purposes, as 
 described under this caption by R. Bell, C.E., in May number, 
 is obvious, as may also be its use for more extended profiles, 
 for geological cross-sections, for road-grading, or wherever 
 between points obtained by the levelling-instrument its accuracy 
 is not indispensable. 
 
 A clinometer thus graduated enables contour lines for topo- 
 graphical work to be most readily determined. The table 
 following gives the 100 points in the quadrant in terms of the 
 common graduation of 90 to the quadrant.
 
 MINE SURVEYING. 
 
 395 
 
 Vertical 
 
 Horizontal 
 
 
 Vertical 
 
 Horizontal 
 
 
 Distance for 
 
 Distance for 
 
 Angle 
 
 Distance for 
 
 Distance for 
 
 Angle 
 
 100 measured 
 
 100 measured 
 
 with Horizon. 
 
 100 measured 
 
 100 measured 
 
 with Horizon. 
 
 on Slope. 
 
 on Slope. 
 
 
 on Slope. 
 
 on Slope. 
 
 
 
 
 
 000' 
 
 51 
 
 
 :;u 'in' 
 
 1 
 
 . ... 
 
 034' 
 
 52 
 
 ... 
 
 31 20' 
 
 2 
 
 
 
 109' 
 
 53 
 
 .. . . 
 
 32 00' 
 
 3 
 
 . .'.. 
 
 143' 
 
 54 
 
 
 32 41' 
 
 4 
 
 
 
 2 18' 
 
 55 
 
 88.6 
 
 33 22' 
 
 6 
 
 99.9 
 
 2 52' 
 
 56 
 
 
 34 03' 
 
 6 
 
 .... 
 
 3 26' 
 
 57 
 
 
 34 45' 
 
 7 
 
 
 
 4 01' 
 
 58 
 
 
 35 27' 
 
 8 
 
 
 
 4 35' 
 
 59 
 
 
 36 09' 
 
 9 
 
 
 5 10' 
 
 60 
 
 80.6 
 
 36 52' 
 
 10 
 
 99.6 
 
 5 44' 
 
 61 
 
 .... 
 
 37 35' 
 
 11 
 
 
 6 19' 
 
 62 
 
 
 38 19' 
 
 12 
 
 
 6 54' 
 
 63 
 
 .... 
 
 39 03' 
 
 13 
 
 
 7 28' 
 
 64 
 
 
 39 48' 
 
 14 
 
 
 8 03' 
 
 65 
 
 76.0 
 
 40 32' 
 
 15 
 
 98.9 
 
 8 38' 
 
 66 
 
 
 
 41 18' 
 
 16 
 
 
 9 12' 
 
 67 
 
 .... 
 
 42 04' 
 
 17 
 
 
 9 47' 
 
 68 
 
 
 42 51' 
 
 18 
 
 
 10 22' 
 
 69 
 
 
 43 38' 
 
 19 
 
 
 10 57' 
 
 70 
 
 71.4 
 
 44 26' 
 
 20 
 
 98.6 
 
 11 32' 
 
 71 
 
 
 45 14' 
 
 21 
 
 
 12 07' 
 
 72 
 
 
 46 03' 
 
 22 
 
 
 12 43' 
 
 73 
 
 
 46 53' 
 
 23 
 
 
 13 18' 
 
 74 
 
 
 47 44' 
 
 24 
 
 
 13 53' 
 
 75 
 
 66.2 
 
 48 35' 
 
 25 
 
 96.8 
 
 14 29' 
 
 76 
 
 
 49 28' 
 
 26 
 
 
 15 04' 
 
 77 
 
 
 50 21' 
 
 27 
 
 
 15 40' 
 
 78 
 
 .... 
 
 61 16' 
 
 28 
 
 
 16 16' 
 
 79 
 
 
 62 11' 
 
 29 
 
 
 16 51' 
 
 80 
 
 60.6 
 
 63 08' 
 
 30 
 
 95.4 
 
 17 27' 
 
 81 
 
 
 64 06' 
 
 31 
 
 
 18 04' 
 
 82 
 
 
 66 05' 
 
 32 
 
 
 18 40' 
 
 83 
 
 
 66 06' 
 
 33 
 
 
 19 16' 
 
 84 
 
 
 67 08' 
 
 34 
 
 
 19 53' 
 
 85 
 
 52.7 
 
 58 13' 
 
 35 
 
 93.7 
 
 20 29' 
 
 86 
 
 
 59 19 7 
 
 36 
 
 
 21 06' 
 
 87 
 
 .... 
 
 60 28' 
 
 37 
 
 
 21 43' 
 
 88 
 
 .... 
 
 61 39' 
 
 38 
 
 
 22 20 7 
 
 89 
 
 
 62 62' 
 
 39 
 
 
 22 57' 
 
 90 
 
 43.6 
 
 (il (I!)' 
 
 40 
 
 91J6 
 
 23 35' 
 
 91 
 
 
 66 30' 
 
 41 
 
 
 24 12' 
 
 92 
 
 
 6(5 66' 
 
 42 
 
 
 24 50' 
 
 93 
 
 .... 
 
 68 26' 
 
 43 
 
 
 25 28'' 
 
 94 
 
 
 70 03' 
 
 44 
 
 
 26 0(5' 
 
 95 
 
 31.2 
 
 71 is' 
 
 45 
 
 89.3 
 
 26 45' 
 
 9(5 
 
 
 73 44' 
 
 46 
 
 
 27 23' 
 
 97 
 
 .... 
 
 7:. :.(' 
 
 47 
 
 
 28 02' 
 
 98 
 
 
 78 81' 
 
 48 
 
 
 28 41' 
 
 99 
 
 
 81 64' 
 
 49 
 
 
 29 20' 
 
 100 
 
 00.6 
 
 90 00' 
 
 50 
 
 86.6 
 
 30 00' 
 
 

 
 TRANSIT, 
 A.S FIRST MADE IN 1831 BI THE INVENTOR, WILLIAM J. YOUNG, PHILADELPHIA, PA
 
 APPENDIX. 
 
 THE JUDICIAL FUNCTIONS OF SUEVEYOES* 
 
 A* 
 
 WHEN a man has had a training in one of the exact sciences, 
 where every problem within its purview is supposed to be sus- 
 ceptible of accurate solution, he is likely to be not a little impa- 
 tient when he is told that, under some circumstances, he must 
 recognize inaccuracies, and govern his actions by facts which 
 lead him away from the results which theoretically he ought to 
 reach. 
 
 Observation warrants us in saying that this remark may 
 frequently be made of surveyors. 
 
 In the State of Michigan, all our lands are supposed to have 
 been surveyed once or more, and permanent monuments fixed 
 to determine the boundaries of those who should become propri- 
 etors. The United States, as original owner, caused them all 
 to be surveyed once by sworn officers, and as the plan of sub- 
 division was simple, and was uniform over a large extent of 
 territory, there should have been, with due care, few or no 
 mistakes : and long rows of monuments should have been 
 perfect guides to the place of any one that chanced to be miss- 
 ing. The truth unfortunately is, that the lines were very care- 
 lessly run, the monuments inaccurately placed ; and, as the 
 recorded witnesses to these were many times wanting in perma- 
 nency, it is often the case that when the monument was not 
 correctly placed, it is impossible to determine by the record, by 
 the aid of anything on the ground, where it was located. The 
 incorrect record of course becomes worse than useless when tin- 
 witnesses it refers to have disappeared. 
 
 It is, perhaps, generally supposed that our town plats were 
 
 By Chief Justice Cooley of tlie Supreme Court of Michigan. 

 
 398 PLANE SURVEYING. 
 
 more accurately surveyed, as indeed they should have been; 
 for in general there can have been no difficulty in making them 
 sufficiently perfect for all practical purposes. Many of them, 
 however, were laid out in the woods ; some of them by proprie- 
 tors themselves, without either chain or compass, and some by 
 imperfectly trained surveyors, who, when land was cheap, did 
 not appreciate the importance of having correct lines to deter- 
 mine boundaries when land should become dear. 
 
 The fact probably is, that town surveys are quite as inaccurate 
 as those made under authority of the general government. It 
 is now upwards of fifty years since a major part of the public 
 survevs, in what is now the State of Michigan, were made under 
 authority of the United States. Of the lands south of Lansing, 
 it is now forty years since the major part were sold and the 
 work of improvement began. A generation has passed away 
 since they were converted into cultivated farms, and few, if any, 
 of the original corner and quarter stakes now remain. 
 
 The corner and quarter stakes were often nothing but green 
 sticks driven into the ground. Stones might be put around or 
 over these if they were handy, but often they were not, and the 
 witness trees must have been relied upon after the stake was 
 gone. Too often the first settlers were careless in fixing their 
 lines with accuracy while monuments remained, and an irregular 
 brush-fence, or something equally untrustworthy, may have been 
 relied upon to keep in mind where the blazed line once was. A 
 fire running through this might sweep it away, and if nothing 
 was substituted in its place, the adjoining proprietors might in 
 a few years be found disputing over their lines, and perhaps 
 rushing into litigation, as soon as they had occasion to cultivate 
 the land along the boundary. If now the disputing parties call 
 in a surveyor, it is not likely that any one summoned would 
 doubt or question that his duty was to find, if possible, the place 
 of the original stakes which determine the boundary line between 
 the proprietors. 
 
 However erroneous may have been the original survey, the 
 monuments that were set must nevertheless govern, even though 

 
 APPENDIX. . 399 
 
 the effect be to make one half-quarter section 90 acres, and the 
 one adjoining 70 ; for parties buy, or are supposed to buy, in 
 reference to these monuments, and are entitled to what is within 
 their lines, and no more, be it more or less. While the witness 
 trees remain, there can generally be no difficulty in determining 
 the locality of the stakes. When the witness trees are gone, so 
 that there is no longer record evidence of the monuments, it is 
 remarkable how many there are who mistake altogether the duty 
 that now devolves upon the surveyor. It is by no means uncom- 
 mon that we find men, whose theoretical education is thought to 
 make them experts, who think that when the monuments are 
 gone, the only thing to be done is to place new monuments 
 where the old ones should have been, and would have been if 
 placed correctly. This is a serious mistake. The problem is 
 now the same that it was before : To ascertain by the best 
 lights of which the case admits where the original lines were. 
 The mistake above referred to is supposed to have found expres- 
 sion in our legislation ; though it is possible that the real intent 
 of the act to which we shall refer is not what is commonly sup- 
 posed. An act passed in 1869 (Compiled Laws, 593), amend- 
 ing the laws respecting the duties and powers of county survey- 
 ors, after providing for the case of corners which can be identi- 
 fied by the original field notes or other unquestionable testimony, 
 directs as follows : 
 
 "Second. Extinct interior section corners must be re-estab- 
 lished at the intersection of two right lines joining the nearest 
 known points on the original section lines east and west and 
 north and south of it. 
 
 " Third. Any extinct quarter-section corner, except on frac- 
 tional lines, must be re-established equidistant and in a right 
 line between the section corners ; in all other cases, at its pro- 
 portionate distance between the nearest original corners on the 
 same line." 
 
 .The corners thus determined, the surveyors are required to 
 perpetuate by noting bearing troes when timber is near. To
 
 400 PLANE SURVEYING. 
 
 estimate properly this legislation, we must start with the ad- 
 mitted and unquestionable fact that each purchaser from gov- 
 ernment bought such land as was within the original boundaries, 
 and unquestionably owned it up to the time when the monuments 
 became extinct. 
 
 If the monument was set for an interior section corner, but 
 did not happen to be "at the intersection of two right lines 
 joining the nearest known points on the original section lines 
 east and west and north and south of it," it nevertheless deter- 
 mined the extent of his possessions, and he gained or lost 
 according as the mistake did or did not favor him. 
 
 It will probably be admitted that no man loses title to his 
 land or any part thereof merely because the evidences become 
 lost or uncertain. It may become more difficult for him to es- 
 tablish it as against an adverse claimant, but theoretically the 
 right remains ; and it remains as a potential fact so long as he 
 can present better evidence than any other person. And it 
 may often happen that notwithstanding the loss of all trace of 
 a section corner or quarter stake, there will still be evidence 
 from which anv surveyor will be able to determine with almost 
 absolute certainty where the original boundary was between the 
 government subdivisions. 
 
 There are two senses in which the word "extinct" may be used 
 in this connection : one, the sense of physical disappearance ; 
 the other, the sense of loss of all reliable evidence. If the 
 statute speaks of extinct corners in the former sense, it is plain 
 that a serious mistake was made in supposing that surveyors 
 could be clothed with authority to establish new corners by an 
 arbitrary rule in such cases. As well might the statute declare 
 that if a man loses his deed, he shall lose his land altogether. 
 But if by extinct corner is meant one in respect to the actual 
 location of which all reliable evidence is lost, then the following 
 remarks are pertinent : 
 
 1. There would undoubtedly be a presumption in such a case 
 that the corner was correctly fixed by the government surveyor 
 where the field notes indicated it to be.
 
 APPENDIX. 401 
 
 2. But this is only a presumption, and may be overcome by 
 any satisfactory evidence showing that in fact it was placed 
 elsewhere. 
 
 3. No statute can confer upon a county surveyor the power 
 to "establish" corners, and thereby bind the parties concerned. 
 Nor is this a question merely of conflict between State and 
 Federal law ; it is a question of property right. The original 
 surveys must govern, and the laws under which they were made 
 must govern, because the land was bought in reference to them ; 
 and any legislation, whether State or Federal, that should have 
 the effect to change these, would be inoperative, because dis- 
 turbing vested rights. 
 
 4. In any case of disputed lines, unless the parties concerned 
 settle the controversy by agreement, the determination of it is 
 necessarily a judicial act, and it must proceed upon evidence, 
 and give full opportunity for a hearing. No arbitrary rules of 
 survey or of evidence can be laid down whereby it can be ad- 
 judged. The general duty of a surveyor in such a case is plain 
 enough. He is not to assume that a monument is lost, until 
 after he has thoroughly sifted the evidence, and found himself 
 unable to trace it. Even then he should hesitate long before 
 doing anything to the disturbance of settled possessions. Occu- 
 pation, especially if long continued, often affords very satis- 
 factory evidence of the original boundary, when no other is 
 attainable ; and the surveyor should inquire when it originated, 
 how and why the lines were then located as they were, and 
 whether a claim of title has always accompanied the possession, 
 and give all the facts due force as evidence. Unfortunately, it is 
 known that surveyors sometimes, in supposed obedience to the 
 State statute, disregard all evidences of occupation and claim 
 of title, and plunge whole neighborhoods into quarrels and liti- 
 gation by assuming to "establish" corners at points with which 
 the previous occupation cannot harmonize. It is often the case 
 that where one or more corners are found to be extinct, all par- 
 ties concerned have acquiesced in lines which were traced by 
 the guidance of some other corner or landmark, which may or
 
 402 PLANE SURVEYING. 
 
 may not have been trustworthy ; but to bring these lines into 
 discredit, when the people concerned do not question them, not 
 only breeds trouble in the neighborhood, but it must often sub- 
 ject the surveyor himself to annoyance, and perhaps discredit, 
 since in a legal controversy the law, as well as common sense, 
 must declare that a supposed boundary line long acquiesced in 
 is better evidence of where the real line should be than any 
 survey made after the original monuments have disappeared. 
 Stewart v. Carleton, 31 Mich. Reports, 270; Diehl v. Zanger, 
 39 Mich. Reports, 601. And county surveyors, no more than 
 any others, can conclude parties by their surveys. 
 
 The mischiefs of overlooking the facts of possession must 
 often appear in cities and villages. In towns the block and lot 
 stakes soon disappear ; there are no witness trees and no monu- 
 ments to govern, except such as have been put in their places, 
 or where their places were supposed to be. The streets are 
 likely to be soon marked off b}* fences, and the lots in a block 
 will be measured off from these without looking farther. 
 
 Now it may perhaps be known in a particular case that a 
 certain monument still remaining was the starting-point in the 
 original survey of the town plat ; or a surveyor settling in the 
 town may take some central point as the point of departure in 
 his surveys, and assuming the original plat to be accurate, he will 
 then undertake to find all streets and all lots by course and dis- 
 tance according to the plat, measuring and estimating from his 
 point of departure. This procedure might unsettle every line 
 and every monument existing by acquiescence in the town ; it 
 would be very likely to change the lines of streets, and raise 
 controversies everywhere. Yet this is what is sometimes 
 done ; the surveyor himself being the first person to raise the 
 disturbing questions. 
 
 Suppose, for example, a particular village street has been 
 located by acquiescence and used for many years, and the pro- 
 prietors in a certain block have laid off their lots in reference 
 to this practical location. Two lot-owners quarrel, and one of 
 them calls in a surveyor that he may be sure that his neighbor
 
 APPENDIX. 403 
 
 shall not get an inch of land from him. This surveyor under- 
 takes to make his survey accurate, whether the original was or 
 not, and the first result is, he notifies the lot-owners that there 
 is error in the street line, and that all fences should be moved, 
 say, one foot to the east. Perhaps he goes on to drive stakes 
 through the block according to this conclusion. Of course if 
 he is right in doing this, all lines in the village will be unsettled ; 
 but we will limit our attention to the single block. It is not 
 likely that the lot-owners will generally allow the new survey to 
 unsettle their possessions, but there is always a probability of 
 finding some one disposed to do so. We shall then have a law- 
 suit; and with what result? It is a common error that lines 
 do not become fixed by acquiescence in a less time than twenty 
 years. In fact, by statute road lines may become conclusively 
 fixed in ten years ; and there is no particular time that shall be 
 required to conclude private owners, where it appears that they 
 have accepted a particular line as their boundary, and all con- 
 cerned have cultivated and claimed up to it. McNamara v. 
 Seaton, 82 111. Reports, 498 ; Bunce v. Bidwell, 43 Mich. Re- 
 ports, 542. Public policy requires that such lines be not lightly 
 disturbed or disturbed at all after the lapse of any considerable 
 time. The litigant, therefore, who in such a case pins his 
 faith on the surveyor, is likely to suffer for his reliance, and 
 the surveyor himself to be mortified by a result that seems to 
 impeach his judgment. 
 
 Of course nothing in what has been said can require a sur- 
 veyor to conceal his own judgment or to report the facts one 
 way when he believes them to be another. He has no right to 
 mislead, and he may rightfully express his opinion that an origi- 
 nal monument was at one place, when at the same time he is 
 satisfied that acquiescence has fixed the rights of parties as if 
 it were at another. But he would do mischief if he were to 
 attempt to " establish" monuments which he knew would tend 
 to disturb settled rights ; the farthest he has a right to go as 
 an officer of the law is to express his opinion where the monu- 
 ment should be at the same time that he imparts the information
 
 404 PLANE SURVEYING. 
 
 to those who employ him, and who might otherwise be misled, 
 that the same authority that makes him an officer, and entrusts 
 him to make surveys, also allows parties to settle their own 
 boundary lines, and considers acquiescence in a particular line 
 or monument for any considerable period as strong, if not con- 
 clusive, evidence of such settlement. The peace of the com- 
 munity absolutely requires this rule. Foyce v. Williams, 26 
 Mich. Reports, 332. It is not long since that in one of the 
 leading cities of the State an attempt was made to move houses 
 two or three rods into a street, on the ground that a survej', 
 under which the street had been located for many years, had 
 been found on a more recent survey to be erroneous. 
 
 From the foregoing it will appear that the duty of the sur- 
 veyor, where boundaries are in dispute, must be varied by the 
 circumstances. (1) He is to search for original monuments, or 
 for the places where they were originally located, and allow 
 these to control if he finds them, unless he has reason to believe 
 that agreements of the parties express or implied have rendered 
 thtm unimportant. By monuments in the case of government 
 surveys we mean, of course, the corner and quarter stakes ; 
 blazed lines or marked trees on the lines are not monuments ; 
 they are merely guides or finger-posts, if we may use the ex- 
 pression, to inform us with more or less accuracy where the 
 monuments may be found. (2) If the original monuments are 
 no longer discoverable, the question of location becomes one of 
 evidence merely. It is merely idle for any State statute to 
 direct a surveyor to locate or ' ' establish " a corner, as the 
 place of the original monument, according to some inflexible 
 rule. The surveyor, on the other hand, must inquire into all 
 the facts, giving due prominence to the acts of parties con- 
 cerned, and always keeping in mind, first, that neither his opin- 
 ion nor his survey can be conclusive upon parties concerned ; 
 and, second, that courts and juries may be required to follow 
 after the surveyor over the same ground, and that it is exceed- 
 ingly desirable that he govern his action by the same lights and 
 same rules that will govern theirs. On town plats if a surplus or
 
 APPENDIX. 405 
 
 deficiency appears in a block when the actual boundaries are 
 compared with the original figures, and there is no evidence to 
 fix the exact location of the stakes which marked the division 
 into lots, the rule of common sense and the law is that the sur- 
 plus or deficiency is to be apportioned between the lots on an 
 assumption that the error extended alike to all parts of the 
 block. O'Brien v. McGrane, 29 Wis. Reports, 446 ; Quinnin v. 
 Reixers, 46 Mich. Reports, 605. 
 
 It is always possible when corners are extinct that the sur- 
 veyor may usefully act as a mediator between parties, and assist 
 in preventing legal controversies by settling doubtful Hues. 
 Unless he is made for this purpose an arbitrator by legal sub- 
 mission, the parties, of course, even if they consent to follow 
 his judgment, cannot, on the basis of mere consent, be com- 
 pelled to do so ; but if he brings about an agreement, and they 
 carry it into effect by actually conforming their occupation to 
 his lines, the action will conclude them. Of course it is desir- 
 able that all such agreements be reduced to writing ; but this is 
 not absolutely indispensable if they are carried into effect with- 
 out. 
 
 Meander Lines. The subject to which allusion will now be 
 made is taken up with some reluctance, because it is believed 
 the general rules are familiar. Nevertheless, it is often found 
 that surveyors misapprehend them, or err in their application ; 
 and as other interesting topics are somewhat connected with 
 this, a little time devoted to it will probably not be altogether 
 lost. The subject is that of meander lines. These are lines 
 traced along the shores of lakes, ponds, and considerable rivers 
 as the measures of quantity when sections are made fractional 
 by such waters. These have determined the price to be paid 
 when government lands were bought, and perhaps the impres- 
 sion still lingers in some minds that meander lines are boundary 
 lines, and all in front of them remains unsold. Of course this 
 is erroneous. There was never any doubt that, except on the 
 large navigable rivers, the boundary of the owners of the banks 
 is the middle line of the river ; and while some courts have held
 
 406 PLANK SURVEYING. 
 
 that this was the rule on all fresh-water streams, large and small, 
 others have held to the doctrine that the title to the bed of the 
 stream below low-water mark is in the State while conceding to 
 the owners of the bank all riparian rights. The practical differ- 
 ence is not very important. In this State the rule that the centre 
 line is the boundary line is applied to all our great rivers, includ- 
 ing the Detroit, varied somewhat by the circumstance of there 
 being a distinct channel for navigation in some cases with the 
 stream in the main shallow, and also sometimes by the existence 
 of islands. 
 
 The troublesome questions for surveyors present themselves 
 when the boundary line between two contiguous estates is to be 
 continued from the meander line to the centre line of the river. 
 Of course the original survey supposes that each purchaser of 
 land on the stream has a water-front of the length shown by the 
 field notes ; and it is presumable that he bought this particular 
 land because of that fact. In many cases it now happens that 
 the meander line is left some distance from the shore by the 
 gradual change of course of the stream or diminution of the flow 
 of water. Now the dividing line between two government sub- 
 divisions might strike the meander line at right angles, or ob- 
 liquely ; and in some cases, if it were continued in the same 
 direction to the centre line of the river, might cut off from tbe 
 water one of the subdivisions entirely, or at least cut it off from 
 any privilege of navigation or other valuable use of the water, 
 while the other might have a water-front much greater than the 
 length of a line crossing it at right angles to its side lines. The 
 effect might be that, of two government subdivisions of equal 
 size and cost, one would be of very great value as water-front 
 property, and the other comparatively valueless. A rule which 
 would produce this result would not be just, and it has not been 
 recognized in the law. 
 
 Nevertheless, it is not easy to determine what ought to be the 
 correct rule for every case. If the river has a straight course, 
 or one nearly so, every man's equities will be preserved by this 
 rule. Extend the line of division between the two parcels from
 
 APPENDIX. 407 
 
 the meander line to the centre line of the river, as nearly as pos- 
 sible at right angles to the general course of the river at that 
 point. This will preserve to each man the water-front which 
 the field notes indicated, except as changes in the water may 
 have affected it, and the only inconvenience will be that the 
 division line between different subdivisions is likely to be more 
 or less deflected where it strikes the meander line. 
 
 This is the legal rule, and it is not limited to government sur- 
 veys, but applies as well to water-lots which appear as such on 
 town plats. Bay City Gas Light Co. v. The Industrial Works, 
 28 Mich. Reports, 182. It often happens, therefore, that the 
 lines of city lots bounded on navigable streams are deflected as 
 they strike the bank, or the line where the bank was when the 
 town was first laid out. When the stream is very crooked, and 
 especially if there are short bends, so that the foregoing rule is 
 incapable of strict application, it is sometimes very difficult to 
 determine what shall be done ; and in many cases the surveyor 
 may be under the necessity of working out a rule for himself. 
 Of course his action cannot be conclusive ; but if he adopts one 
 that follows, as nearly as the circumstances will admit, the gen- 
 eral rule above indicated, so as to divide as near as may be the 
 bed of the stream among the adjoining owners in proportion to 
 their lines upon the shore, his division, being that of an expert, 
 made upon the ground and with all available lights, is likely to 
 be adopted as law for the case. Judicial decisions, into which 
 the surveyor would find it prudent to look under such circum- 
 stances, will throw light upon his duties, and may constitute a 
 sufficient guide when peculiar cases arise. Each riparian lot- 
 owner ought to have a line on the legal boundary, namely, the 
 centre line of the stream, proportioned to the length of his line 
 on the shore ; and the problem in each case is, how this is to be 
 given him. Alluvion, when a river imperceptibly changes its 
 course, will be apportioned by the same rules. 
 
 The existence of islands in a stream, when the middle line 
 constitutes a boundary, will not affect the apportionment unless 
 the islands were surveyed out as government subdivisions iu the
 
 408 PLANE SURVEYING. 
 
 original admeasurement. Wherever that was the case the pur- 
 chaser of the island divides the bed of the stream on each side 
 with the owner of the bank, and his rights also extend above 
 and below the solid ground, and are limited by the peculiarities 
 of the bed and the channel. If an island was not surveyed as a 
 government subdivision previous to the sale of the bank, it is of 
 course impossible to do this for the purposes of government sale 
 afterwards, for the reason that the rights of the bank owners are 
 fixed by their purchase : when making that they have a right to 
 understand that all land between the meander lines, not sepa- 
 rately surveyed and sold, will pass with the shore in the govern- 
 ment sale ; and having this right, anything which their purchase 
 would include under it cannot afterwards be taken from them. 
 It is believed, however, that the Federal courts would not recog- 
 nize the applicability of this rule to large navigable rivers, such 
 as those uniting the Great Lakes. 
 
 On all the little lakes of the State, which are mere expansions 
 near their mouths of the rivers passing through them, such as 
 the Muskegon, Pere Marquette, and Manistee, 7 the same rule of 
 bed ownership has been judicially applied that is applied to the 
 rivers themselves ; and the division lines are extended under 
 the water in the same way. Rice v. Ruddiman, 10 Mich. 125. 
 If such a lake were circular, the lines would converge to the 
 centre ; if oblong or irregular, there might be a line in the middle 
 on which they would terminate, whose course would bear some 
 relation to that of the shore. But it can seldom be important 
 to follow the division line very far under the water, since all 
 private rights are subject to the public rights of navigation and 
 other use, and any private use of the lands inconsistent with 
 these would be a nuisance, and punishable as such. It is some- 
 times important, however, to run the lines out for some consid- 
 erable distance, in order to determine where one may lawfully 
 moor vessels or rafts for the winter, or cut ice. The ice crop 
 that forms over a man's land of course belongs to him. Lormau 
 v. Benson, 8 Mich. 18 ; People's Ice Co. v. Steamer Excelsior, 
 recently decided.
 
 APPENDIX. 409 
 
 What is said above will show how unfounded is the notion, 
 which is sometimes advanced, that a ripai-ian proprietor on a 
 meandered river may lawfully raise the water in the stream 
 without liability to the proprietors above, provided he does not 
 raise it so that it overflows the meander line. The real fact is, 
 that the meander line has nothing to do with such a case, and 
 an action will lie whenever he sets back the water upon the 
 proprietor above, whether the overflow be below the meander 
 lines or above them. As regards the lakes and ponds of the 
 State, one may easily raise questions that it would be impossible 
 for him to settle. Let us suggest a few questions, some of 
 which are easily answered, and some not : (1) To whom belongs 
 the land under these bodies of water, where they are not mere 
 expansions of a stream flowing through them? (2) What 
 public rights exist in them? (3) If there are islands in them 
 which were not surveyed out and sold by the United States, can 
 this be done now? Others will be suggested by the answers 
 given to these. 
 
 It seems obvious that the rules of private ownership which 
 are applied to rivers cannot be applied to the Great Lakes. 
 Perhaps it should be held that the boundary is at low-water 
 mark, but improvements beyond this would only become unlaw- 
 ful when they became nuisances. Islands in the Great Lakes 
 would belong to the United States until sold, and might be 
 surveyed and measured at any time. The right to take fish in 
 the lakes or to- cut ice is public, like the right of navigation, 
 but is to be exercised in such manner as not to interfere with 
 the rights of shore-owners ; but, so far as these public rights 
 can be the subject of ownership, they belong to the State, not 
 the United States ; and so, it is believed, does the bed of a lake 
 also. Pollard v. Hagan, 3 Howard's U. S. Reports. But 
 such rights are not generally considered proper subjects of 
 sale, but, like the right to make use of the public highways, 
 they are held by the State in trust for all the people. What is 
 said of the large lakes may, perhaps, be said also of many of the 
 interior lakes of the State ; such, for example, as Houghton,
 
 410 PLANE SURVEYING. 
 
 Higgins, Cheboygan, Burt's, Mullet, Whitmore, and many 
 others. But there are many little lakes or ponds which are 
 gradually disappearing, and the shore proprietorship advances 
 pari passu as the waters recede. If these are of any consider- 
 able size, say, even a mile across, there may be questions 
 of conflicting rights which no adjudication hitherto made could 
 settle. Let an}' surveyor, for example, take the case of a pond 
 of irregular form, occupying a mile square or more of territory, 
 and undertake to determine the rights of the shore proprietors 
 to its bed when it shall totally disappear, and he will find lie is 
 in the midst of problems such as probably he has never grappled 
 with, or reflected upon, before. But the general rules for the 
 extension of shore lines which have already been laid down 
 should govern such cases, or at least should serve as guides in 
 their settlement. 
 
 Where a pond is so small as to be included within the lines 
 of a private purchase from the government, it is not believed 
 the public have any rights in it whatever. Where it is not so 
 included, it is believed they have rights of fishery, rights to 
 take ice and water, and rights of navigation for business or 
 pleasure. This is the common belief, and probably the just 
 one. Shore rights must not be so exercised as to disturb these, 
 and the States may pass all proper laws for their protection. 
 It would be easy with suitable legislation to preserve these little 
 bodies of water as permanent places of resort for the pleasure 
 and recreation of the people, and there ought to be such legisla- 
 tion. If the State should be recognized as owner of the beds 
 of these small lakes and ponds, it would not be owner for the 
 purpose of selling. It would be owner only as a trustee for the 
 public use; and a sale would be inconsistent with the right of 
 the bank owners to make use of the water in its natural condi- 
 tion in connection with their estates. Some of them might be 
 made salable lands by draining ; but the State could not drain, 
 even for this purpose, against the will of the shore-owners, un- 
 less their rights were appropriated and paid for. Upon many 
 questions that might arise between the State as owner of the
 
 APPENDIX. 4H 
 
 bed of a little lake and the shore-owners, it would be presump- 
 tuous to express an opinion now, and fortunately the occasion 
 does not require it. 
 
 I have thus indicated a few of the questions with which sur- 
 veyors may now and then have occasion to deal, and to which 
 they should bring good sense and ssund judgment. Surveyors 
 are not, and cannot be, judicial officers, but in a great many 
 cases they act in a quasi judicial capacity, with the acquies- 
 cence of parties concerned; and it is important for them to 
 know by what rules they are to be guided in the discharge of 
 their judicial functions. What I have said cannot contribute 
 much to their enlightenment, but I trust will not be wholly 
 without value.
 
 TABLES.
 
 TABLE I. 
 
 THE 
 
 COMMON OR BRIGGS LOGARITHMS 
 
 OF THE 
 
 NATURAL NUMBERS 
 
 From 1 to 10000. 
 
 
 
 1-100 
 
 N 
 
 log 
 
 I 
 
 log 
 
 K 
 
 log 
 
 N 
 
 log 
 
 * 
 
 log 
 
 1 
 
 0.00000 
 
 21 
 
 1. 32 222 
 
 41 
 
 1.61278 
 
 61 
 
 1.78533 
 
 81 
 
 1.90849 
 
 2 
 
 0. 30 103 
 
 22 
 
 1. 34 242 
 
 42 
 
 1. 62 325 
 
 62 
 
 1. 79 239 
 
 82 
 
 1.91381 
 
 3 
 
 0.47712 
 
 23 
 
 1. 36 173 
 
 43 
 
 1. 63 347 
 
 63 
 
 1.79934 
 
 83 
 
 1.91908 
 
 4 
 
 0.60206 
 
 24 
 
 1.38021 
 
 44 
 
 1. 64 345 
 
 64 
 
 1.80618 
 
 84 
 
 1.92428 
 
 5 
 
 0. 69 897 
 
 25 
 
 1. 39 794 
 
 45 
 
 1. 65 321 
 
 65 
 
 1. 81 291 
 
 85 
 
 1.92942 
 
 6 
 
 0. 77 815 
 
 26 
 
 1.41497 
 
 46 
 
 1. 66 276 
 
 66 
 
 1. 81 954 
 
 86 
 
 1.93450 
 
 7 
 
 0. 84 510 
 
 27 
 
 1.43136 
 
 47 
 
 1. 67 210 
 
 67 
 
 1.82607 
 
 87 
 
 1.93952 
 
 8 
 
 0.90309 
 
 28 
 
 1.44716 
 
 48 
 
 1. 68 124 
 
 68 
 
 1. 83 251 
 
 88 
 
 1.94448 
 
 9 
 
 0. 95 424 
 
 29 
 
 1.46240 
 
 49 
 
 1.69020 
 
 69 
 
 1.83885 
 
 89 
 
 1.94939 
 
 10 
 
 1.00000 
 
 30 
 
 1. 47 712 
 
 50 
 
 1.69897 
 
 70 
 
 1.84510 
 
 90 
 
 1.95424 
 
 11 
 
 1. 04 139 
 
 31 
 
 1.49136 
 
 51 
 
 1. 70 757 
 
 71 
 
 1. 85 126 
 
 91 
 
 1.95904 
 
 12 
 
 1.07918 
 
 32 
 
 1.50515 
 
 52 
 
 1.71600 
 
 72 
 
 1. 85 733 
 
 92 
 
 1.96379 
 
 13 
 
 1. 11 394 
 
 33 
 
 1.51851 
 
 63 
 
 1. 72 428 
 
 73 
 
 1.86332 
 
 93 
 
 1.96848 
 
 14 
 
 1.14613 
 
 34 
 
 1. 53 148 
 
 54 
 
 1. 73 239 
 
 74 
 
 1.86923 
 
 94 
 
 1.97313 
 
 15 
 
 1.17609 
 
 35 
 
 1.54407 
 
 55 
 
 1.74036 
 
 75 
 
 1. 87 506 
 
 95 
 
 1. 97 772 
 
 16 
 
 1.20412 
 
 36 
 
 1. 55 630 
 
 56 
 
 1.74819 
 
 76 
 
 1.88081 
 
 96 
 
 1.98227 
 
 17 
 
 1. 23 045 
 
 37 
 
 1.56820 
 
 67 
 
 1. 75 587 
 
 77 
 
 1.88649 
 
 97 
 
 1.98677 
 
 18 
 
 1. 25 527 
 
 38 
 
 1.57978 
 
 58 
 
 1.76343 
 
 78 
 
 1.89209 
 
 98 
 
 1.99123 
 
 19 
 
 1. 27 875 
 
 39 
 
 1.59106 
 
 59 
 
 1. 77 085 
 
 79 
 
 1.89763 
 
 99 
 
 1.99564 
 
 20 
 
 1. 30 103 
 
 40 
 
 1.60206 
 
 60 
 
 1.77815 
 
 80 
 
 1.90309 
 
 100 
 
 2.00000 
 
 I 
 
 log 
 
 
 
 log 
 
 I 
 
 log 
 
 V 
 
 log 
 
 N 
 
 N 
 
 1-100
 
 100-150 
 
 If 
 
 01234 
 
 56789 
 
 100 
 
 101 
 102 
 103 
 104 
 
 00000 00043 00087 00130 00173 
 00432 00475 00518 00561 00604 
 00860 00903 00945 00988 01030 
 01284 01326 01368 01410 01452 
 01703 01745 01787 01828 01870 
 
 00217 00260 00303 00346 00389 
 00647 00689 00732 00775 00817 
 01 072 01 115 01 157 01 199 01 242 
 01494 01536 01578 01620 01662 
 01912 01953 01995 02036 02078 
 
 106 
 106 
 107 
 108 
 109 
 
 02119 02160 02202 02243 02284 
 02531 02572 02612 02653 02694 
 02938 02979 03019 03060 03100 
 03342 03383 03423 03463 03503 
 03743 03782 03822 03862 03902 
 
 02325 02366 02407 02449 02490 
 02735 02776 02816 02857 02898 
 03141 03181 03222 03262 03302 
 03543 03583 03623 03663 03703 
 03941 03981 04021 04060 04100 
 
 110 
 
 111 
 112 
 113 
 
 114 
 
 04139 04179 04218 04258 04297 
 04532 04571 04610 04650 04689 
 04922 04961 04999 05038 05077 
 05 308 05 346 05 385 05 423 05 461 
 05690 05729 05767 05805 05843 
 
 04336 04376 04415 04454 04493 
 04727 04766 04805 04844 04883 
 05115 05154 05192 05231 05269 
 05500 05538 05576 05614 05652 
 05881 05918 05956 05994 06032 
 
 115 
 116 
 
 117 
 118 
 119 
 
 06070 06108 06145 06183 06221 
 06446 06483 06521 06558 06595 
 06819 06856 06893 06930 06967 
 07188 07225 07262 07298 07335 
 07555 07591 07628 07664 07700 
 
 06258 06296 06333 06371 06408 
 06633 06670 06707 06744 06781 
 07004 07041 07078 07115 07151 
 07372 07408 07445 07482 07518 
 07737 07773 07809 07846 07882 
 
 120 
 
 121 
 122 
 123 
 
 124 
 
 07918 07954 07990 08027 08063 
 08279 08314 08350 08386 08422 
 08636 08672 08707 08743 08778 
 08991 09026 09061 09096 09132 
 09342 09377 09412 09447 09482 
 
 08099 08135 08171 08207 08243 
 08458 08493 08529 08565 08600 
 08814 08849 08884 08920 08955 
 09167 09202 09237 09272 09307 
 09517 09552 09587 09621 09656 
 
 125 
 126 
 127 
 128 
 129 
 
 09691 09726 09760 09795 09830 
 10037 10072 10106 10140 10175 
 10380 10415 10449 10483 10517 
 10721 10755 10789 10823 10857 
 11059 11093 11126 11160 11193 
 
 09864 09899 09934 09968 10003 
 10209 10243 10278 10312 10346 
 10551 10585 10619 10653 10687 
 10890 10924 10958 10992 11025 
 11227 11261 11294 11327 11361 
 
 ISO 
 
 131 
 132 
 133 
 134 
 
 11394 11428 11461 11494 11528 
 11727 11760 11793 11826 11860 
 12057 12090 12123 12156 12189 
 12385 12418 12450 12483 12516 
 12710 12743 12775 12808 12840 
 
 11561 11594 11628 11661 11694 
 11893 11926 11959 11992 12024 
 12222 12254 12287 12320 12352 
 12548 12581 12613 12646 12678 
 12872 12905 12937 12969 13001 
 
 135 
 136 
 137 
 138 
 139 
 
 13033 13066 13098 13130 13162 
 13354 13386 13418 13450 13 481 
 13672 13704 13735 13767 13799 
 13988 14019 14051 14082 14114 
 14301 14333 14364 14395 14426 
 
 13194 13226 13258 13290 13322 
 13513 13545 13577 13609 13640 
 13830 13862 13893 13925 13956 
 14145 14176 14208 14239 14270 
 14457 14489 14520 14551 14582 
 
 14O 
 
 141 
 142 
 143 
 144 
 
 14613 14644 14675 14706 14737 
 14922 14953 14983 15014 15045 
 15229 15259 15290 15320 15351 
 15534 15564 15594 15625 15655 
 15836 15866 15897 15927 15957 
 
 14768 14799 14829 14860 14891 
 15076 15106 15137 15168 15198 
 15381 15412 15442 15473 15503 
 15685 15715 15746 15776 15806 ' 
 15 987 16017 16047 16077 16107 
 
 145 
 146 
 147 
 148 
 149 
 
 16137 16167 16197 16227 16256 
 16435 16465 16495 16524 16554 
 16732 16761 16791 16820 16850 
 17026 17056 17085 17114 17143 
 17319 17348 17377 17406 17435 
 
 16286 16316 16346 16376 16406 
 16584 16613 16643 16673 16702 
 16879 16909 16938 16967 16997 
 17173 17202 17231 17260 17289 
 17464 17493 17522 17551 17580 
 
 ISO 
 
 17609 17638 17667 17696 17725 
 
 17754 17782 17811 17840 17S69 
 
 N 
 
 01234 
 
 36789 
 
 100-150
 
 150-200 
 
 N 
 
 O 1 2 3 4 
 
 56789 
 
 ISO 
 
 17609 17638 17667 17696 17725 
 
 17754 17782 17811 17840 17869 
 
 151 
 
 17898 17926 17955 17984 18013 
 
 18041 18070 18099 18127 18156 
 
 152 
 
 18 184 18213 18241 18270 18298 
 
 18327 18355 18384 18412 18441 
 
 153 
 
 18469 18498 18526 18554 18583 
 
 18611 18639 18667 186% 18724 
 
 154 
 
 18752 18780 18808 18837 18 865. 
 
 18893 18921 18949 18977 19005 
 
 155 
 
 19033 19061 19089 19117 19145 
 
 19173 19201 19229 19257 19285 
 
 156 
 
 19312 19340 19368 19396 19424 
 
 19451 19479 19507 19535 19562 
 
 157 
 
 19590 19618 19645 19673 19700 
 
 19728 19756 19783 19811 19838 
 
 158 
 
 19866 19893 19921 19948 19976 
 
 20003 20030 20058 20085 20112 
 
 159 
 
 20140 20167 20194 20222 20249 
 
 20276 20303 20330 20358 20385 
 
 160 
 
 20412 20439 20466 20493 20520 
 
 20548 20575 20602 20629 20656 
 
 161 
 
 20683 20710 20737 20763 20790 
 
 20817 20844 20871 20898 20925 
 
 162 
 
 20952 20978 21005 21032 21059 
 
 21085 21112 21139 21165 21192 
 
 163 
 
 21219 21245 21272 21299 21325 
 
 21352 21378 21405 21431 21458 
 
 164 
 
 21484 21511 21537 21564 21590 
 
 21617 21643 21669 216% 21722 
 
 165 
 
 21748 21775 21801 21827 21854 
 
 21880 21906 21932 21958 21985 
 
 166 
 
 22011 22037 22063 22089 22115 
 
 22141 22167 22194 22220 22246 
 
 167 
 
 22272 22298 22324 22350 22376 
 
 22401 22427 22453 22479 22505 
 
 168 
 
 22531 22557 22583 22608 22634 
 
 22660 22686 22712 22737 22763 
 
 169 
 
 22789 22814 22840 22866 22891 
 
 22917 22943 22968 22994 23019 
 
 17O 
 
 23045 23070 23096 23121 23147 
 
 23172 23198 23223 23249 23274 
 
 171 
 
 23300 23325 23350 23376 23401 
 
 23426 23452 23477 23502 23528 
 
 172 
 
 23553 23578 23603 23629 23654 
 
 23679 23704 23729 23754 23779 
 
 173 
 
 23805 23830 23855 23880 23905 
 
 23930 23955 23980 24005 24030 
 
 174 
 
 24055 24080 24105 24130 24155 
 
 24180 24204 24229 24254 24279 
 
 175 
 
 24304 24329 24353 24378 24403 
 
 24428 24452 24477 24502 24527 
 
 176 
 
 24551 24576 24601 24625 24650 
 
 24674 24699 24724 24748 24773 
 
 177 
 
 24797 24822 24846 24871 24895 
 
 24920 24944 24969 24993 25018 
 
 178 
 
 25042 25066 25091 25115 25139 
 
 25164 25188 25212 25237 25261 
 
 179 
 
 25285 25310 25334 25358 25382 
 
 25406 25431 25455 25479 25503 
 
 180 
 
 25527 25551 25575 25600 25624 
 
 25648 25672 25696 25720 25744 
 
 181 
 
 25768 25792 25816 25840 25864 
 
 25888 25912 25935 25959 25983 
 
 182 
 
 26007 26031 26055 26079 26102 
 
 26126 2615TT26174 26198 26221 
 
 183 
 
 26245 26269 26293 26316 26340 
 
 26364 26387 26411 26435 26458 
 
 184 
 
 26482 26505 26529 26553 26576 
 
 26600 26623 26647 26670 26694 
 
 185 
 
 26717 26741 26764 26788 26811 
 
 26834 26858 26881 26905 26928 
 
 186 
 
 26951 26975 26998 27021 27045 
 
 27068 27091 27114 27138 27161 
 
 187 
 
 27184 27207 27231 27254 27277 
 
 27300 27323 27346 27370 27393 
 
 188 
 
 27416 27439 27462 27485 27508 
 
 27531 27554 27577 27600 27623 
 
 189 
 
 27646 27669 27692 27715 27738 
 
 27761 27784 27807 27830 27852 
 
 190 
 
 27875 27898 27921 27944 27 967 
 
 27989 28012 28035 28058 28081 
 
 191 
 
 28103 28126 28149 28171 28194 
 
 28217 28240 28262 28285 28307 
 
 192 
 
 28330 28353 28375 28398 28421 
 
 28443 28466 28488 28511 28533 
 
 193 
 
 28556 28578 28601 28623 28646 
 
 28668 28691 28713 28735 28758 
 
 194 
 
 28780 28803 28825 28847 28870 
 
 28892 28914 28937 28959 28 981 
 
 195 
 
 29003 29026 29048 29070 29092 
 
 29115 29137 29159 29181 29203 
 
 196 
 
 29226 29248 29270 29292 29314 
 
 29336 29358 29380 29403 29425 
 
 197 
 
 29447 29469 29491 29513 29535 
 
 29557 29579 29601 29623 29645 
 
 198 
 
 29667 29 688 29710 29732 29754 
 
 29776 29798 29820 29842 29863 
 
 199 
 
 29885 29907 29929 29951 29973 
 
 29994 30016 30038 30060 300S1 
 
 200 
 
 30103 30125 30146.30168 30190 
 
 30211 30233 30255 30276 30298 
 
 N 
 
 O 1 2 3 4 
 
 5 6 7 8 i> 
 
 150-200
 
 200-250 
 
 N 
 
 01234 
 
 5 6 78 9 
 
 200 
 201 
 202 
 203 
 204 
 
 30103 30125 30146 30168 30190 
 30320 30341 30363 30384 30406 
 30535 30557 30578 30600 30621 
 30750 30771 30792 30814 30835 
 30963 30984 31006 31027 31048 
 
 30211 30233 30255 30276 30298 
 30428 30449 30471 30492 30514 
 30643 30664 30685 30707 30728 
 30856 30878 30899 30920 30942 
 31069 31091 31112 31133 31154 
 
 205 
 206 
 207 
 208 
 209 
 
 31175 31197 31218 31239 31260 
 31387 31408 31429 31450 31471 
 31597 31618 31639 31660 31681 
 31806 31827 31848 31 869 31890 
 32015 32035 32056 32077 32098 
 
 31281 31302 31323 31345 31366 
 31492 31513 31534 31555 31576 
 31702 31723 31744 31765 31785 
 31911 31931 31952 31973 31994 
 32118 32139 32160 32181 32201 
 
 21O 
 
 211 
 212 
 213 
 214 
 
 32222 32243 32263 32284 32305 
 32428 32449 32469 32490 32510 
 32634 32654 32675 32695 32715 
 32838 32858 32879 32899 32919 
 33041 33062 33082 33102 33122 
 
 32325 32346 32366 32387 32408 
 32531 32552 32572 32593 32613 
 32 736 32 756 32 777 32 797 32 818 
 32940 32960 32 980 33001 33021 
 33143 33163 33183 33203 33224 
 
 215 
 216 
 217 
 218 
 219 
 
 33244 33264 33284 33304 33325 
 33445 33465 33486 33506 33526 
 33646 33666 33686 33706 33726 
 33846 33866 33885 33905 33925 
 34044 34064 34084 34104 34124 
 
 33345 33365 33385 33405 33425 
 33 546 33 566 33 586 33 606 33 626 
 33746 33766 33786 33806 33826 
 33945 33965 33985 34005 34025 
 34143 34163 34183 34203 34223 
 
 220 
 
 221 
 222 
 223 
 224 
 
 34242 34262 34282 34301 34321 
 34439 34459 34479 34498 34518 
 34635 34655 34674 34694 34713 
 34830 34850 34869 34889 34908 
 35025 35044 35064 35083 35102 
 
 34341 34361 34380 34400 34420 
 34537 34557 34577 345% 34616 
 34733 34753 34772 34792 34811 
 34928 34947 34967 34986 35005 
 35122 35141 35160 35180 35199 
 
 226 
 226 
 227 
 228 
 229 
 
 35218 35238 35257 35276 35295 
 35411 35430 35449 35468 35488 
 35603 35622 35641 35660 35679 
 35793 35813 35832 35851 35870 
 35984 36003 36021 36040 36059 
 
 35315 35334 35353 35372 35392 
 35507 35526 35545 35564 35583 
 35 698 35 717 35 736 35 755 35 774 
 35889 35908 35927 35946 35965 
 36078 36097 36116 36135 36154 
 
 230 
 
 231 
 232 
 233 
 234 
 
 36173 36192 36211 36229 36248 
 36361 36380 36399 36418 36436 
 36549 36568 36586 36605 36624 
 36736 36754 36773 36791 36810 
 36922 36940 36959 36977 369% 
 
 36267 36286 36305 36324 36342 
 36455 36474 36493 36511 36530 
 36642 36661 36680 36698 36717 
 36829 36847 36866 36884 36903 
 37014 37033 37051 37070 37088 
 
 235 
 236 
 237 
 238 
 
 239 
 
 37107 37125 37144 37162 37181 
 37291 37310 37328 37346 37365 
 37475 37493 37511 37530 37548 
 37658 37676 37694 37712 37731 
 37840 37858 37876 37894 37912 
 
 37199 37218 37236 37254 37273 
 37383 37401 37420 37438 37457 
 37566 37585 37603 37621 37639 
 37749 37767 37785 37803 37822 
 37931 37 949- 37 %7 37985 38003 
 
 240 
 
 241 
 242 
 243 
 244 
 
 38021 38039 38057 38075 38093 
 38202 38220 38238 38256 38274 
 38382 38399 38417 38435 38453 
 38561 38578 385% 38614 38632 
 38739 38757 38775 38792 38810 
 
 38112 38130 38148 38166 38184 
 38292 38310 38328 38346 38364 
 38471 38489 38507 38525 38543 ' 
 38650 38668 38686 38703 38721 
 38828 38846 38863 38881 38899 , 
 
 245 
 246 
 247 
 248 
 249 
 
 38917 38934 38952 38970 38987 
 39094 39111 39129 39146 39164 
 39270 39287 39305 39322 39340 
 39445 39463 39 480 39498 39515 
 39620 39637 39655 39672 39690 
 
 39005 39023 39041 39058 39 076 j 
 39182 39199 39217 39235 39252 
 39358 39375 39393 39410 39428 
 39533 39550 39568 39585 39602 
 39 707 39 724 39 742 39 759 39 777 
 
 250 
 
 39794 39811 39829 39846 39863 
 
 39881 39898 39915 39933 39950 
 
 X 
 
 01234 
 
 56789 
 
 200-250
 
 250-300 
 
 
 
 8 
 
 25O 
 
 251 
 252 
 253 
 254 
 
 255 
 256 
 257 
 268 
 259 
 
 260 
 
 261 
 262 
 263 
 
 264 
 
 265 
 
 268 
 269 
 
 270 
 
 271 
 272 
 273 
 274 
 
 275 
 276 
 277 
 278 
 279 
 
 283 
 284 
 
 285 
 286 
 287 
 288 
 289 
 
 29O 
 
 291 
 292 
 293 
 294 
 
 295 
 296 
 297 
 298 
 
 39794 39811 39829 
 39967 39985 40002 
 40140 40157 40175 
 40312 40329 40346 
 40483 40500 40518 
 
 40654 40671 40688 
 
 40824 40841 40858 
 
 40993 41010 41027 
 
 41162 41179 41196 
 
 41330 41347 41363 
 
 39846 39863 
 40019 40037 
 40192 40209 
 40364 40381 
 40535 40552 
 
 40705 40722 
 
 40875 40892 
 
 41044 41061 
 
 41212 41229 
 
 41380 41397 
 
 41497 41514 41531 41547 41564 
 
 41664 41681 41697 41714 41731 
 
 41830 41847 41863 41880 418% 
 
 41996 42012 42029 42045 42062 
 
 42160 42177 42193 42210 42226 
 
 42325 42341 42357 42374 42390 
 
 42 488 42504 42521 42537 42553 
 
 42651 4266V 42684 42700 42716 
 
 42813 42830 42846 42862 42878 
 
 42975 42991 43008 43024 43040 
 
 43136 43152 43169 43185 43201 
 
 43297 43313 43329 43345 43361 
 
 43457 43473 43489 43505 43521 
 
 43616 43632 43648 43664 43680 
 
 43775 43791 43807 43823 43838 
 
 43933 43949 43965 43981 439% 
 
 44091 44107 44122 44138 44154 
 
 44248 44264 44279 44295 44311 
 
 44404 44420 44436 44451 44467 
 
 44560 44576 44592 44607 44623 
 
 44716 44731 44747 44762 44778 
 
 44871 44886 44902 44917 44932 
 
 45025 45040 45056 45071 45086 
 
 45 179 45 194 45 209 45 225 45 240 
 
 45332 45347 45362 45378 45393 
 
 45484 45500 45515 45530 45545 
 
 45637 45652 45667 45682 45697 
 
 45788 45803 45818 45834 45849 
 
 45939 45954 45969 45984 46000 
 
 46090 46105 46120 46135 46150 
 
 46240 46255 46270 46285 46300 
 
 46389 46404 46419 46434 46449 
 
 46538 46553 46568 46583 46598 
 
 46687 46702 46716 46731 46746 
 
 46835 46850 46864 46879 46894 
 
 46982 46997 47012 47026 47041 
 
 47129 47144 47159 47173 47188 
 
 47276 47290 47305 47319 47334 
 
 47422 47436 47451 47465 47480 
 
 47567 47582 47596 47611 47625 
 
 47712 47727 47741 47756 47770 
 
 39881 39898 39915 
 
 40054 40071 40088 
 
 40226 40243 40261 
 
 40398 40415 40432 
 
 40569 40586 40603 
 
 40739 40756 40773 
 40909 40926 40943 
 41 078 41 095 41 111 
 41246 41263 41280 
 41414 41430 41447 
 
 39933 39950 
 40106 40123 
 40278 40295 
 40449 40466 
 40620 40637 
 
 40790 40807 
 
 40960 40976 
 
 41128 41 14* 
 
 412% 41313 
 
 41464 41481 
 
 41581 41597 41614 41631 41647 
 41 747 41 764 41 780 41 797 41 814 
 41913 41929 41946 41963 41979 
 42078 42095 42111 42127 42144 
 42243 42259 42275 42292 42308 
 
 42406 42423 42439 42455 42472 
 
 42570 42586 42602 42619 42635 
 
 42732 42749 42765 42781 42797 
 
 42894 42911 42927 42943 42959 
 
 43056 43072 43088 43104 43120 
 
 43 217 .43 233 43 249 43 265 43 281 
 43377 43393 43409 43425 43441 
 43537 43553 43569 43584 43600 
 43696 43712 43727 43743 43759 
 43854 43870 43886 43902 43917 
 
 44012 44028 44044 44059 44075 
 
 44170 44185 44201 44217 44232 
 
 44326 44342 44358 44373 44389 
 
 44483 44498 44514 44529 44545 
 
 44638 44654 44669 44685 44700 
 
 44793 44809 44824 44840 44855 
 
 44948 44963 44979 44994 45010 
 
 45 102 45 117 45 133 45 148 45 163 
 
 45255 45271 45 2S6 45301 45317 
 
 45408 45423 45439 45454 45469 
 
 45 561 45 576 45 591 45 606 45 621 
 
 45712 45728 45743 45758 45773 
 
 45864 45879 45894 45909 45924 
 
 46015 46030 46045 46060 46075 
 
 46165 46180 46195 46210 46225 
 
 46315 46330 46345 46359 46374 
 46464 46479 46494 46509 46523 
 46613 46627 46642 46657 46672 
 46761 46776 46790 46805 46820 
 46909 46923 46938 46953 46%7 
 
 47056 47070 47085 47100 47114 
 47202 47217 47232 47246 47261 
 47349 47363 47378 47392 47407 
 47494 47509 47524 47538 47553 
 47640 47654 47669 47683 47698 
 
 47784 47799 47813 47828 47842 
 
 250-300
 
 300-350 
 
 N 
 
 01234 
 
 56789 
 
 300 
 
 301 
 302 
 303 
 304 
 
 47712 47727 47741 47756 47770 
 47857 47871 47885 47900 47914 
 48001 48015 48029 48044 48058 
 48144 48159 48173 48187 48202 
 48287 48302 48316 48330 48344 
 
 47784 47799 47813 47828 47842 
 47929 47943 47958 47972 47986 
 48073 48087 48101 48116 48130 
 48216 48230 48244 48259 48273 
 48359 48373 48387 48401 48416 
 
 305 
 306 
 307 
 308 
 309 
 
 48430 48444 48458 48473 48487 
 48572 48586 48601 48615 48629 
 48714 48728 48742 48756 48770 
 48855 48869 48883 48897 48911 
 48996 49010 49024 49038 49052 
 
 48501 48515 48530 48544 48558 
 48643 48657 48671 48686 48700 
 48785 48799 48813 48827 48841 
 48926 48940 48954 48968 48982 
 49066 49080 49094 49108 49122 
 
 310 
 
 311 
 312 
 313 
 314 
 
 49136 49150 49164 49178 49192 
 49276 49290 49304 49318 49332 
 49415 49429 49443 49457 49471 
 49554 49568 49582 49596 49610 
 49693 49707 49721 49734 49748 
 
 49206 49220 49234 49248 49262 
 49346 49360 49374 49388 49402 
 49485 49499 49513 49527 49541 
 49624 49638 49651 49665 49679 
 49762 49776 49790 49803 49817 
 
 315 
 316 
 317 
 318 
 
 319 
 
 49831 49845 49859 49872 49886 
 49969 49982 49996 50010 50024 
 50106 50120 50133 50147 50161 
 50243 50256 50270 50284 50297 
 50379 50393 50406 50420 50433 
 
 49900 49914 49927 49941 49955 
 50037 50051 50065 50079 50092 
 50174 50188 50202 50215 50229 
 50311 50325 50338 50352 50365 
 50447 50461 50474 50488 50501 
 
 320 
 
 321 
 322 
 323 
 
 324 
 
 50515 50529 50542 50556 50569 
 50651 50664 50678 50691 50705 
 50786 50799 50813 50826 50840 
 50920 50934 50947 50 961 50974 
 51055 51068 51081 51095 51108 
 
 50583 505% 50610 50623 50637 
 50718 50732 50745 50759 50772 
 50853 50866 50880 50893 50907 
 50987 51001 51014 51028 51041 
 51 121 51 135 51 148 51 162 51 175 
 
 325 
 326 
 327 
 328 
 
 329 
 
 51188 51202 51215 51228 51242 
 51322 51335 51348 51362 51375 
 51455 51468 51481 51495 51508 
 51587 51601 51614 51627 51640 
 51 720 51 733 51 746 51 759 51 772 
 
 51255 51268 51282 51295 51308 
 51388 51402 51415 51428 51441 
 51521 51534 51548 51561 51574 
 51654 51667 51680 51693 51706 
 51786 51799 51812 51825 51838 
 
 33O 
 
 331 
 332 
 333 
 334 
 
 51851 51865 51878 51891 51904 
 51983 519% 52009 52022 52035 
 52114 52127 52140 52153 52166 
 52244 52257 52270 52284 52297 
 52375 52388 52401 52414 52427 
 
 51917 51930 51943 51957 51970 
 52048 52061 52075 52088 52101 
 52179 52192 52205 52218 52231 
 52310 52323 52336 52349 52362 
 52440 52453 52466 52479 52492 
 
 335 
 336 
 337 
 338 
 339 
 
 52504 52517 52530 52543 52556 
 52634 52647 52660 52673 52686 
 52763 52776 52789 52802 52815 
 52892 52905 52917 52930 52943 
 53020 53033 53046 53058 53071 
 
 52569 52582 52595 52608 52621 
 52699 52711 52724 52737 52750 
 52827 52840 52853 52866 52879 
 52956 52969 52982 52994 53007 
 53084 53097 53110 53122 53135 
 
 340 
 
 341 
 
 342 
 343 
 344 
 
 53148 53161 53173 53186 53199 
 53275 53288 53301 53314 53326 
 53403 53415 53428 53441 53453 
 53529 53542 53555 53567 53580 
 53656 53668 53681 53694 53706 
 
 53212 53224 53237 53250 53263 
 53339 53352 53364 53377 53390 
 53466 53479 53491 53504 53517 
 53593 53605 53618 53631 53643 
 53719 53732 53744 53757 53769 
 
 345 
 346 
 347 
 348 
 349 
 
 53782 53794 53807 53820 53832 
 53908 53920 53933 53945 53958 
 54033 54045 54058 54070 54083 
 54158 54170 54183 54195 54208 
 54283 54295 54307 54320 54332 
 
 53845 53857 53870 53882 53895 
 53970 53983 53995 54008 54020 
 54095 54108 54120 54133 54145 
 54220 54233 54245 54258 54270 
 54345 54357 54370 54382 54394 
 
 350 
 
 54407 54419 54432 54444 54456 
 
 54469 54481 54494 54506 54518 
 
 N 
 
 O 1 2 3 4 
 
 56789 
 
 300-350
 
 350-400 
 
 3 
 
 54407 54419 54432 54444 54456 
 
 54531 54543 54555 54568 54580 
 
 54654 54667 54679 54691 54704 
 
 54777 54790 54802 54814 54827 
 
 54900 54913 54925 54937 54949 
 
 55023 55035 55047 55060 55072 
 
 55145 55157 55169 55182 55194 
 
 55267 55279 55291 55303 55315 
 
 55388 55400 55413 55 425. 55437 
 
 55509 55522 55534 55546 55558 
 
 55630 55642 55654 55666 55678 
 
 55751 55763 55775 55787 55799 
 
 55871 55883 55 895 55907 55919 
 
 55991 56003 56015. 56027 56038 
 
 56110 56122 56134 56146 56158 
 
 56229 56241 56253 56265 56277 
 
 56348 56360 56372 56384 563% 
 
 56467 56478 56490 56502 56514 
 
 56585 56597 56608 56620 56632 
 
 56703 56744 56726 56738 56750 
 
 56820 56832 56844 
 
 56937 56949 56961 
 
 57054 57066 57078 
 
 57171 57 183 57194 
 
 57287 57299 57310 
 
 57403 57415 57426 
 
 57519 57530 57542 
 
 57634 57646 57657 
 
 57749 57761 57772 
 
 57864 57875 57887 
 
 56855 56867 
 
 56972 56984 
 
 57 089 57101 
 
 57206 57217 
 
 57322 57334 
 
 57438 57449 
 
 57553 57565 
 
 57669 57680 
 
 57784 57795 
 
 57898 57910 
 
 57978 57990 58001 58013 58024 
 
 58092 58104 58115 58127 58138 
 
 58206 58218 58229 58240 58252 
 
 58320 58331 58343 58354 58365 
 
 58433 58444 58456 58467 58478 
 
 58546 58557 58569 58580 58591 
 58659 58670 58681 58692 58704 
 58771 58782 58794 58805 58816 
 58883 58894 58906 58917 58928 
 58995 59006 59017 59028 59040 
 
 59106 59118 59129 59140 59151 
 
 59218 59229 59240 59251 59262 
 
 59329 59340 59351 59362 59373 
 
 59439 59450 59461 59472 59483 
 
 59550 59561 59572 59583 59594 
 
 59660 59671 59 682 59693 59704 
 
 59770 59780 59791 59802 59813 
 
 59879 59890 59901 59912 59923 
 
 59988 59999 60010 60021 60032 
 
 60097 60108 60119 60130 60141 
 
 60206 60217 60228 60239 60249 
 
 r, 
 
 54481 54494 54506 54518 
 
 54605 54617 54630 54642 
 
 54728 54741 54753 54765 
 
 54851 54864 54876 54888 
 
 54974 54986.54998 55011 
 
 55 0% 55 108 55 121 55 133 
 55218 55230 55242 55255 
 55340 55352 55364 55376 
 55461 55473 55485 55497 
 55582 55594 55606 55618 
 
 54469 
 54593 
 54716 
 54839 
 54962 
 
 55084 
 55206 
 55328 
 55449 
 55570 
 
 55691 55703 55715 55727 55739 
 
 55811 55823 55835 55847 55859 
 
 55931 55943 55955 55967 55979 
 
 56050 56062 56074 56086 56098 
 
 56170 56182 56194 56205 56217 
 
 56289 56301 56312 56324 56336 
 
 56407 56419 56431 56443 56455 
 
 56526 56538 56549 56561 56573 
 
 56644 56656 56667 56679 56691 
 
 56761 56773 56785 56797 56808 
 
 56879 56891 56902 56914 56926 
 56996 57008 57019 57031 57043 
 57113 57124 57136 57148 57159 
 57229 57241 57252 57264 57276 
 57345 57357 57368 57380 57392 
 
 57461 57473 57484 574% 57507 
 
 57576 57588 57600 57611 57623 
 
 57692 57703 57715 57726 57738 
 
 57807 57818 57830 57841 57852 
 
 57921 57933 57944 57955 57 %7 
 
 58035 58047 58058 58070 58081 
 
 58149 58161 58172 58184 58195 
 
 58263 58274 58286 58297 58309 
 
 58377 58388 58399 58410 58422 
 
 58490 58501 58512 58524 58535 
 
 58602 58614 58625 58636 58647 
 
 58715 58726 58737 58749 58760 
 
 58827 58838 58850 58861 58872 
 
 58939 58950 58961 58973 58984 
 
 59051 59062 59073 59084 59095 
 
 59162 59173 59184 59195 59207 
 
 59273 59284 59295 59306 59 318 
 
 59384 59395 59406 59417 59428 
 
 59494 59506 59517 59528 59539 
 
 59605 59616 59627 59638 59649 
 
 59715 59726 59737 59748 59759 
 59824 59835 59846 59857 59S6S 
 59934 59945 59956 59966 59977 
 60043 60054 60065 60076 600S6 
 60152 60163 60173 60184 60195 
 
 60260 60271 602S2 60293 60304 
 
 350-400
 
 400-450 
 
 H 
 
 01234 
 
 56789 
 
 400 
 401 
 402 
 403 
 404 
 
 60206 60217 60228 60239 60249 
 60314 60325 60336 60347 60358 
 60423 60433 60444 60455 60466 
 60531 60541 60552 60563 60574 
 60638 60649 60660 60670 60681 
 
 60260 60271 60282 60293 60304 
 60369 60379 60390 60401 60412 
 60477 60487 60498 60509 60520 
 60584 60595 60606 60617 60627 
 60692 60703 60713 60724 60735 
 
 405 
 406 
 407 
 408 
 409 
 
 60746 60756 60767 60778 60788 
 60853 60863 60874 60 885 60895 
 60959 60970 60981 60991 61002 
 61066 61077 61087 61098 61109 
 61172 61183 61194 61204 61215 
 
 60799 60810 60821 60831 60842 
 60906 60917 60927 60938 60949 
 61013 61023 61034 61045 61055 
 61 119 61 130 61 140 61 151 61 162 
 61225 61236 61247 61257 61268 
 
 410 
 
 411 
 412 
 413 
 414 
 
 61278 61289 61300 61310 61321 
 61384 61395 61405 61416 61426 
 61490 61500 61511 61521 61532 
 61595 61606 61616 61627 61637 
 61700 61711 61721 61731 61742 
 
 61331 61342 61352 61363 61374 
 61437 61448 61458 61469 61479 
 61542 61553 61563 61574 61584 
 61648 61658 61669 61679 61690 
 61752 61763 61773 61784 61794 
 
 415 
 416 
 417 
 418 
 419 
 
 61805 61815 61826 61836 61847 
 61909 61920 61930 61941 61951 
 62014 62024 62034 62045 62055 
 62118 62128 62138 62149 62159 
 62221 62232 62242 62252 62263 
 
 61857 61868 61878 61888 61899 
 61962 61972 61982 61993 62003 
 62066 62076 62086 62097 62107 
 62170 62180 62190 62201 62211 
 62273 62284 62294 62304 62315 
 
 42O 
 
 421 
 422 
 423 
 424 
 
 62325 62335 62346 62356 62366 
 62428 62439 62449 62459 62469 
 62531 62542 62552 62562 62572 
 62634 62644 62655 62665 62675 
 62737 62747 62757 62767 62778 
 
 62377 62387 62397 62408 62418 
 62480 62490 62500 62511 62521 
 62583 62593 62603 62613 62624 
 62685 626% 62706 62716 62726 
 62788 62798 62808 62818 62829 
 
 425 
 426 
 427 
 428 
 429 
 
 62839 62849 62859 62870 62880 
 62941 62951 62961 62972 62982 
 63043 63053 63063 63073 63083 
 63144 63155 63165 63175 63185 
 63246 63256 63266 63276 63286 
 
 62890 62900 62910 62921 62931 
 62992 63002 63012 63022 63033 
 63094 63104 63114 63124 63134 
 63195 63205 63215 63225 63236 
 63296 63306 63317 63327 63337 
 
 43O 
 
 431 
 432 
 433 
 
 434 
 
 63347 63357 63367 63377 63387 
 63448 63458 63468 63478 63488 
 63548 63558 63568 63579 63589 
 63649 63659 63669 63679 63 689 
 63 749 63 759 63 769 63 779 63 789 
 
 63397 63407 63417 63428 63438 
 63498 63508 63518 63528 63538 
 63599 63609 63619 63629 63639 
 63699 63709 63719 63729 63739 
 63799 63809 63819 63829 63839 
 
 435 
 436 
 437 
 438 
 439 
 
 63849 63859 63869 63879 63 889 
 63949 63959 63 969 63979 63988 
 64048 64058 64068 64078 64088 
 64147 64157 64167 64177 64187 
 64246 64256 64266 64276 64286 
 
 63899 63909 63919 63929 63939 
 63998 64008 64018 64028 64038 
 64098 64108 64118 64128 64137 
 64197 64207 64217 64227 64237 
 64296 64306 64316 64326 64335 
 
 44O 
 
 441 
 442 
 443 
 444 
 
 64345 64355 64365 64375 64385 
 64444 64454 64464 64473 64483 
 64542 64552 64562 64572 64582 
 64640 64650 64660 64670 64 680 
 64738 64748 64758 64768 64777 
 
 64395 64404 64414 64424 64434 
 64493 64503 64513 64523 64532 
 64591 64601 64611 64621 64631 
 64 689 64699 64709 64719 64729 
 64787 64797 64807 64816 64826 
 
 445 
 446 
 447 
 448 
 449 
 
 64836 64846 64856 64865 64875 
 64933 64943 64953 64963 64972 
 65031 65040 65050 65060 65070 
 65128 65137 65147 65157 65167 
 65225 65234 65244 65254 65263 
 
 64885 64895 64904 64914 64924 
 64 982 64992 65002 65011 65021 
 65079 65089 65099 65108 65118 
 65 176 65 186 65 196 65 205 65 215 
 65273 65283 65292 65302 65312 
 
 45O 
 
 65321 65331 65341 65350 65360 
 
 65369 65379 65389 65398 65408 
 
 N 
 
 O 1 2 3 4 
 
 56789 
 
 400-450
 
 450-500 
 
 O 
 
 3 
 
 o 
 
 8 
 
 497 
 498 
 
 65321 65331 65341 65350 65360 
 
 65418 65427 65437 65447 65456 
 
 65514 65523 65533 65543 65552 
 
 65610 65619 65629 65639 65648 
 
 65 706 65 715 65 725 65 734 65 744 
 
 65801 65811 65820 65830 65839 
 65896 65906 65916 65925 65935 
 65992 66001 66011 66020 66030 
 66087 66096 66106 66115 66124 
 66181 66191 66200 66210 66219 
 
 66276 66285 66295 66304 66314 
 
 66370 66380 66 389 66398 66408 
 
 66464 66474 664S3 66492 66502 
 
 66558 66567 66577 66586 665% 
 
 66652 66661 66671 66680 66689 
 
 66745 66755 66764 66773 66783 
 
 66839 66848 66857 66867 66876 
 
 66932 66941 66950 66960 66969 
 
 67025 67034 67043 67052 67062 
 
 67117 67127 67136 67145 67154 
 
 67210 67219 67228 67237 67247 
 
 67302 67311 67321 67330 67339 
 
 67394 67403 67413 67422 67431 
 
 67486 67495 67504 67514 67523 
 
 67578 67587 67596 67605 67614 
 
 67669 67679 67688 67697 67706 
 67761 67770 67779 67788 67797 
 67852 67861 67870 67879 67888 
 67943 67952 67 961 67970 67979 
 68034 68043 68052 68061 68070 
 
 68124 68133 68142 68151 68160 
 
 68215 68224 68233 68242 68251 
 
 68305 68314 68323 68332 68341 
 
 68395 68404 68413 68422 68431 
 
 68485 68494 68502 68511 68520 
 
 68574 68583 68592 68601 68610 
 
 68664 68673 68681 68690 68699 
 
 68753 68762 68771 68780 68789 
 
 68842 68851 68860 68869 68878 
 
 68931 68940 68949 68958 68966 
 
 69020 69028 69037 69046 69055 
 
 69108 69117 69126 69135 69144 
 
 69197 69205 69214 69223 69232 
 
 69285 69294 69302 69311 69320 
 
 69373 69381 69390 69399 69408 
 
 69461 69469 
 69548 69557 
 69636 69644 
 69723 69732 
 69810 69819 
 
 69897 69906 
 
 69478 69487 694% 
 
 69566 69574 69583 
 
 69653 69662 69671 
 
 69740 69749 69758 
 
 69827 69836 69845 
 
 69914 69923 69932 
 
 65369 65379 65389 65398 65408 
 
 65466 65475 65485 65495 65504 
 
 65562 65571 65581 65591 65600 
 
 65658 65667 65677 65686 656% 
 
 65753 65763 65772 65782 65792 
 
 65849 65858 65868 65877 65887 
 
 65944 65954 65963 65973 65982 
 
 66039 66049 66058 66068 66077 
 
 66134 66143 66153 66162 66172 
 
 66229 66238 66247 66257 66266 
 
 66323 66332 66342 66351 66361 
 
 66417 66427 66436 66445 66455 
 
 66511 66521 66530 66539 66549 
 
 66605 66614 66624 66633 66642- 
 
 66699 66708 66717 66727 66736 
 
 66792 66801 66811 66820 66829 
 66885 66894 66904 66913 66922 
 66978 66987 66997 67006 67015 
 67071 67080 67089 67099 67108 
 67164 67173 67182 67191 67201 
 
 67256 67265 67274 67284 67293 
 67348 67357 67367 67376 67385 
 67440 67449 67459 67468 67477 
 67532 67541 67550 67560 67569 
 67624 67633 67642 67651 67660 
 
 67715 67724 67733 67742 67752 
 67806 67815 67825 67834 67843 
 67897 67906 67916 67925 67934 
 67988 67997 68006 680i5 68024 
 68079 68088 68097 68106 68115 
 
 68169 68178 68187 681% 68205 
 68260 68269 68278 68287 68296 
 68350 68359 68368 68377 68386 
 68440 68449 68458 68467 68476 
 68529 68538 68547 68556 68565 
 
 68619 68628 68637 68646 68655 
 
 68708 68717 68726 68735 68744 
 
 68797 68806 68815 68824 68833 
 
 68886 68895 68904 68913 68922 
 
 68 975 68984 68993 69002 69011 
 
 69064 69073 69082 69090 69099 
 69152 69161 69170 69179 69188 
 69241 69249 69258 69267 69276 
 69329 69338 69346 69355 69364 
 69417 69425 69434 69443 69452 
 
 69504 69513 69522 69531 69539 
 69592 69601 69609 69618 69627 
 69679 69688 69697 69705 69714 
 69767 69775 69784 69793 69801 
 69854 69862 69871 69880 69888 
 69940 69949 69958 69966 69975 
 
 450 - 500
 
 10 
 
 500-550 
 
 N 
 
 01234 
 
 56789 
 
 5OO 
 
 501 
 502 
 503 
 504 
 
 69897 69906 69914 69923 69932 
 69984 69992 70001 70010 70018 
 70070 70079 70088 70096 70105 
 70157 70165 70174 70183 70191 
 70243 70252 70260 70269 70278 
 
 69940 69949 69958 69966 69975 
 70027 70036 70044 70053 70062 
 70114 70122 70131 70140 70148 
 70200 70209 70217 70226 70234 
 70286 70295 70303 70312 70321 
 
 505 
 506 
 507 
 508 
 509 
 
 70329 70338 70346 70355 70364 
 70415 70424 70432 70441 70449 
 70501 70509 70518 70526 70535 
 70586 70595 70603 70612 70621 
 70672 70680 70689 70697 70706 
 
 70372 70381 70389 70398 70406 
 70458 70467 70475 70484 70492 
 70544 70552 70561 70569 70578 
 70629 70638 70646 70655 70663 
 70714 70723 70731 70740 70749 
 
 510 
 
 511 
 512 
 513 
 514 
 
 70757 70766 70774 70783 70791 
 70842 70851 70859 70868 70876 
 70927 70935 70944 70952 70961 
 71012 71020 71029 71037 71046 
 71096 71105 71113 71122 71130 
 
 70800 70808 70817 70825 70834 
 70885 70893 70902 70910 70919 
 70969 70978 70986 70995 71003 
 71054 71063 71071 71079 71088 
 71 139 71 147 71 155 71 164 71 172 
 
 515 
 516 
 517 
 518 
 519 
 
 71181 71189 71198 71206 71214 
 71265 71273 71282 71290 71299 
 71349 71357 71366 71374 71383 
 71433 71441 71450 71458 71466 
 71517 71525 71533 71542 71550 
 
 71223 71231 71240 71248 71257 
 71307 71315 71324 71332 71341 
 71391 71399 71408 71416 71425 
 71475 71483 71492 71500 71508 
 71559 71567 71575 71584 71592 
 
 520 
 
 521 
 522 
 523 
 624 
 
 71600 71609 71617 71625 71634 
 71684 71692 71700 71709 71717 
 71767 71775 71784 71792 71800 
 71850 71858 71867 71875 71883 
 71933 71941 71950 71958 71966 
 
 71642 71650 71659 71667 71675 
 71 725 71 734 71 742 71 750 71 759 
 71809 71817 71825 71834 71842 
 71892 71900 71908 71917 71925 
 71975 71983 71991 71999 72008 
 
 525 
 526 
 527 
 528 
 529 
 
 72016 72024 72032 72041 72049 
 72099 72107 72115 72123 72132 
 72181 72189 72198 72206 72214 
 72263 72272 72280 72288 72296 
 72346 72354 72362 72370 72378 
 
 72057 72066 72074 72082 72090 
 72140 72148 72156 72165 72173 
 72222 72230 72239 72247 72255 
 72304 72313 72321 72329 72337 
 72387 72395 72403 72411 72419 
 
 530 
 
 531 
 
 532 
 
 533 
 534 
 
 72428 72436 72444 72452 72460 
 72509 72518 72526 72534 72542 
 72591 72599 72607 72616 7.2624 
 72673 72681 72689 72697 72705 
 72754 72762 72770 72779 72787 
 
 72469 72477 72485 72493 72501 
 72550 72558 72567 72575 72583 
 72632 72640 72648 72656 72665 
 72713 72722 72730 72738 72746 
 72795 72803 72811 72819 72827 
 
 535 
 536 
 537 
 538 
 539 
 
 72835 72843 72852 72860 72 868 
 72916 72925 72933 72941 72949 
 72997 73006 73014 73022 73030 
 73078 73086 73094 73102 73111 
 73159 73167 73175 73183 73191 
 
 72876 72884 72892 72900 72908 
 72957 72 965 72973 72981 72989 
 73038 73046 73054 73062 73070 
 73119 73127 73135 73143 73151 
 73199 73207 73215 73223 73231 
 
 540 
 
 541 
 542 
 543 
 544 
 
 73239 73247 73255 73263 73272 
 73320 73328 73336 73344 73352 
 73400 73408 73416 73424 73432 
 73480 73488 734% 73504 73512 
 73560 73568 73576 73584 73592 
 
 73280 73288 73296 73304 73312 
 73360 73368 73376 73384 73392 
 73440 73448 73456 73464 73472 
 73520 73528 73536 73544 73552 
 73600 73608 73616 73624 73632 
 
 545 
 546 
 547 
 548 
 549 
 
 73640 73648 73656 73664 73672 
 73719 73727 73735 73743 73751 
 73799 73807 73815 73823 73830 
 73878 73886 73894 73902 73910 
 73957 73965 73973 73981 73989 
 
 73679 73687 73695 73703 73711 
 73759 73767 73775 73783 73791 
 73838 73846 73854 73862 73870 
 73918 73926 73933 73941 73949 
 73997 74005 74013 74020 74028 
 
 550 
 
 74036 74044 74052 74060 74068 
 
 74076 74084 74092 74099 74107 
 
 N 
 
 01234 
 
 56789 
 
 500-550
 
 550-600 
 
 n 
 
 
 
 74036 74044 74052 74060 74068 
 
 74115 74123 74131 74139 74147 
 
 74194 74202 74210 74218 74225 
 
 74273 74280 74 288 74296 74304 
 
 74351 74359 74367 74374 74382 
 
 74429 74437 74445 74453 74461 
 
 74507 74515 74523 74531 74539 
 
 74586 74593 74601 74609 74617 
 
 74663 74671 74679 74687 74695 
 
 74741 74749 74757 74764 74772 
 
 74819 74827 74834 74842 7485.0 
 74896 74904 74912 74920 74927 
 74974 74981 74 989 74997 75005 
 75051 75059 75066 75074 75082 
 75 128 75136 75143 75151 75159 
 
 75205 75213 75220 75228 75236 
 
 75282 75289 75297 75305 75312 
 
 75358 75366 75374 75381 75389 
 
 75435 75442 75450 75458 75465 
 
 75511 75519 75526 75534 75542 
 
 75587 75595 75603 75610 75618 
 
 75664 75671 75679 75686 75694 
 
 75740 75747 75755 75762 75770 
 
 75815 75823 75831 75838 75846 
 
 75891 75899 75906 75914 75921 
 
 75 967 75974 75982 75989 75997 
 
 76042 76050 76057 76065 76072 
 
 76118 76125 76133 76140 76148 
 
 76193 76200 76208 76215 76223 
 
 76268 76275 76283 76290 76298 
 
 76343 76350 76358 76365 76373 
 
 76418 76425 76433 76440 76448 
 
 76492 76500 76507 76515 76522 
 
 76567 76574 76582 76589 76597 
 
 76641 76649 76656 76664 76671 
 
 76716 76723 76730 76738 76745 
 
 76790 76797 76805 76812 76819 
 
 76864 76871 76879 76886 76893 
 
 76938 76945 76953 76960 76967 
 
 77012 77019 77026 77034 77041 
 
 77085 77093 77100 77107 77115 
 
 77159 77166 77173 77181 77188 
 
 77232 77240 77247 77254 77262 
 
 77305 77313 77320 77327 77335 
 
 77379 77386 77393 77401 77408 
 
 77452 77459 
 
 77525 77532 
 
 77597 77605 
 
 77670 77677 
 
 77743 77750 
 
 77815 77822 
 
 77466 77474 77481 
 
 77539 77546 77554 
 
 77612 77619 77627 
 
 77685 77692 77699 
 
 77757 77764 77772 
 
 77830 77837 77844 
 
 74076 74084 74092 74099 74107 
 
 74155 74162 74170 74178 74186 
 
 74233 74241 74249 74257 74265 
 
 74312 74320 74327 74335 74343 
 
 74390 74398 74406 74414 74421 
 
 74468 74476 74484 74492 74500 
 
 74547 74554 74562 74570 74578 
 
 74624 74632 74640 74648 74656 
 
 74702 74710 74718 74726 74733 
 
 74780 74788 747% 74803 74811 
 
 74858 74865 74873 74881 74889 
 
 74935 74943 74950 74958 74966 
 
 75012 75020 75028 75035 75043 
 
 75089 75097 75105 75113 75120 
 
 75166 75174 75182 75189 75197 
 
 75243 75251 75259 75266 75274 
 
 75320 75328 75335 75343 75351 
 
 75397 75404 75412 75420 75427 
 
 75473 75481 75488 75496 75504 
 
 75549 75557 75565 75572 75580 
 
 75626 75633 75641 75648 75656 
 
 75702 75709 75717 75724 75732 
 
 75778 75785 75793 75800 75808 
 
 75853 75861 75868 75876 75884 
 
 75929 75937 75944 75952 75959 
 
 76005 76012 76020 76027 76035 
 
 76080 76087 76095 76103 76110 
 
 76155 76163 76170 76178 76185 
 
 76230 76238 76245 76253 76260 
 
 76305 76313 76320 76328 76335 
 
 76380 76388 76395 76403 76410 
 
 76455 76462 76470 76477 76485 
 
 76530 76537 76545 76552 76559 
 
 76604 76612 76619 76626 76634 
 
 76678 76686 76693 76701 76708 
 
 76753 76760 76768 76775 76782 
 
 76827 76834 76842 76849 76856 
 
 76901 76908 76916 76923 76930 
 
 76975 76982 76989 76997 77004 
 
 77048 77056 77063 77070 77078 
 
 77122 77129 77137 77144 77151 
 
 77195 77203 77210 77217 77225 
 
 77269 77276 77283 77291 77298 
 
 77342 77349 77357 77364 77371 
 
 77415 77422 77430 77437 77444 
 
 77488 
 77561 
 77634 
 77706 
 
 77779 
 
 77495 
 77568 
 77641 
 77714 
 77786 
 
 77503 77510 77517 
 
 77576 77583 77590 
 
 77648 77656 77663 
 
 77721 77728 77735 
 
 77793 77801 77808 
 
 77851 77859 77866 77873 77880 
 
 550-600
 
 12 
 
 600-650 
 
 N 
 
 01234 
 
 56789 
 
 600 
 6C1 
 602 
 603 
 604 
 
 77815 77822 77830 77837 77844 
 77887 77895 77902 77909 77916 
 77960 77967 77974 77981 77988 
 78032 78039 78046 78053 78061 
 78104 78111 78118 78125 78132 
 
 77851 77859 77866 77873 77880 
 77924 77931 77938 77945 77952 
 779% 78003 78010 78017 78025 
 78068 78075 78082 7SOS9 78097 
 78140 78147 78154 78161 78168 
 
 605 
 606 
 607 
 608 
 609 
 
 78176 78183 78190 78197 78204 
 78247 78254 78262 78269 78276 
 78319 78326 78333 78340 78347 
 78390 78398 78405 78412 78419 
 78462 78469 78476 78483 78490 
 
 78211 782J9 78226 78233 78240 
 78283 78290 78297 78305 78312 
 78355 78362 78369 78376 78 383 
 78426 78433 78440 78447 78455 
 78497 78504 78512 78519 78526 
 
 610 
 
 611 
 612 
 
 613 
 614 
 
 78533 78540 78547 78554 78561 
 78604 78611 78618 78625 78633 
 78675 78682 78689 786% 78704 
 78746 78753 78760 78767 78774 
 78817 78824 78831 78838 78845 
 
 78569 78576 78583 78590 78597 
 78640 78647 78654 78661 78 668 
 78711 78718 78725 78732 78739 
 78 781 78789 78 796 78 803 78810 
 78S52 78859 78866 78 873 78880 
 
 615 
 616 
 617 
 618 
 619 
 
 78 888 78895 78902 78909 78916 
 78958 78 965 78972 78 979 78986 
 79029 79036 79043 79050 79057 
 79099 79106 79113 79120 79127 
 79169 79176 79183 79190 79197 
 
 78923 78930 78937 78944 78951 
 78993 79000 79007 79014 79021 
 79064 79071 7907S 79085 79092 
 79134 79141 79148 79155 79162 
 79204 79211 79218 79225 79232 
 
 62O 
 
 621 
 622 
 623 
 624 
 
 79239 79246 79253 79260 79267 
 79309 79316 79323 79330 79337 
 79379 79386 79393 79400 79407 
 79449 79456 79463 79470 79477 
 79518 79525 79532 79539 79546 
 
 79274 79281 79288 79295 79302 
 79344 79351 79358 79365 79372 
 79414 79421 7942S 79435 79442 
 79 484 79491 79498 79505 79511 
 79553 79560 79567 79574 79581 
 
 625 
 626 
 627 
 628 
 629 
 
 79588 79595 79602 79609 79616 
 79657 79664 79671 79678 79685 
 79727 79734 79741 79748 79754 
 797% 79803 79S10 79817 79824 
 79865 79872 79879 79886 79893 
 
 79623 79630 79637 79644 79650 
 79692 79699 79706 79713 79720 
 79761 79768 79775 79 782 79 789 
 79831 79837 79844 79851 79 858 
 79900 79906 79913 79920 79927 
 
 630 
 
 631 
 632 
 633 
 634 
 
 79934 79941 79948 79955 79 %2 
 80003 80010 80017 80024 80030 
 80072 80079 80 085 SO 092 80099 
 80140 80147 80154 80161 80168 
 80209 80216 80223 80229 80236 
 
 79969 79975 79982 79 989 79996 
 80037 80044 80051 80058 SO 065 
 80106 80113 SO 120 SO 127 80134 
 80175 S01S2 S01SS SO 195 80202 
 80243 80250 80257 80264 SO 271; 
 
 635 
 636 
 637 
 638 
 639 
 
 80277 80284 80291 80298 80305 
 80346 80353 SO 359 80366 SO 373 
 SO 414 80421 80428 80434 80441 
 SO 482 804S9 80496 80502 SO 509 
 SO 550 80557 80564 SO 570 SO 577 
 
 SO 312 SO 318 80325 80332 SO 339 
 S03SO SO 387 80393 80400 SO 407 
 S044S 80455 80462 80468 804751 
 80516 SO 523 SO 530 80536 80 543 ; 
 SO 584 80591 80598 80604 8061H 
 
 640 
 
 641 
 642 
 643 
 644 
 
 80618 SO 625 80632 8063S SO 645 
 S06S6 SO 693 SO 699 80706 SO 713 
 80754 80760 SO 767 80774 SO 781 
 80 821 80828 80835 80 841 80S4S 
 S08S9 SO 895 80902 80909 80916 
 
 80652 80659 80665 80672 8067 
 80720 80726 80733 80740 SO 747] 
 80787 80794 SO 801 SO SOS SO 814 
 80855 SOS62 SOS6S SO 875 SOSS2 
 SO 922 80929 SO 936 80943 SO 949 j 
 
 645 
 
 646 
 647 
 648 
 649 
 
 80956 SO 963 80969 SO 976 80983 
 81023 81030 81037 81043 81050 
 81090 SI 097 81104 81111 81117 
 81158 81164 81 171 81 17S 81184 
 81224 81231 81238 81245 81251 
 
 80990 80996 SI 003 81010 SI 017 
 81057 81064 81070 SI 077 S10S4; 
 81124 81131 81137 81144 81 ISM 
 81191 81 198 81204 SI 211 SI 218 
 SI 258 81265 81271 81278 81285. 
 
 630 
 
 81291 S129S SI 305 81 311 SI 318 
 
 SI 32 SI 331 S133S 81345 SI 351 
 
 M 
 
 01234 
 
 o G 7 8 9 
 
 600-650
 
 650-700 
 
 13 
 
 3 
 
 81291 81298 81305 81311 81318 
 
 81358 81365 81371 81378 81385 
 
 81425 81431 81438 81445 81451 
 
 81491 81498 81505 81511 81518 
 
 81558 81564 81571 81578 81584 
 
 81624 81631 81637 81644 81651 
 
 81690 81697 81704 81710 81717 
 
 81757 81763 81770 81776 81783 
 
 81823 81829 81836 81842 81849 
 
 81889 81895 81902 81908 81915 
 
 81954 81 961 81968 81974 81981 
 
 82020 82027 82033 82040 82046 
 
 82086 82092 82099 82105 82112 
 
 82151 82158 82164 82171 82178 
 
 82217 82223 82230 82236 82243 
 
 82282 82289 82295 82302 82308 
 
 82347 82354 82 360 82367 82373 
 
 82413 82419 82426 82432 82439 
 
 82478 82484 82491 82497 82504 
 
 82543 82549 82556 82562 82569 
 
 82607 82614 82620 82627 82633 
 
 82672 82679 82685 82692 82698 
 
 82737 82743 82750 82756 82763 
 
 82802 82808 82814 82821 82827 
 
 82866 82872 82879 82885 82892 
 
 82930 82937 82943 82950 82956 
 
 82995 83001 83008 83014 83020 
 
 83059 83065 83072 83078 83085 
 
 83123 83129 83136 83142 83149 
 
 83187 83193 83200 83206 83213 
 
 83251 83257 83264 83270 83276 
 
 83315 83321 83327 83334 83340 
 
 83378 83385 83391 83398 83404 
 
 83442 83448 83455 83461 83467 
 
 83506 83512 83518 83525 83531 
 
 83569 83575 83582 83588 83594 
 
 83632 83639 83645 83651 83658 
 
 83696 83702 83708 83715 83721 
 
 83759 83765 83771 83778 83784 
 
 83822 83828 83835 83841 83847 
 
 83885 83891 83897 83904 83910 
 
 83948 83954 83960 83967 83973 
 
 84011 84017 84023 84029 84036 
 
 84073 84080 84086 84092 84098 
 
 84136 84142 84148 84155 84161 
 
 84198 84205 84211 
 
 84261 84267 84273 
 
 84323 84330 84336 
 
 84386 84392 84398 
 
 84448 84454 84460 
 
 84217 
 84280 
 84342 
 84404 
 84466 
 
 84223 
 84286 
 84348 
 84410 
 84473 
 
 81325 81331 81338 81345 81351 
 
 81391 81398 81405 81411 81418 
 
 81458 81465 81471 81478 81485 
 
 81525 81531 81538 81544 81551 
 
 81591 81598 81604 81611 81617 
 
 81657 
 81723 
 81790 
 81856 
 81921 
 
 81987 
 82053 
 82119 
 82184 
 82249 
 
 81664 
 81730 
 817% 
 81862 
 81928 
 
 81994 
 82060 
 82125 
 82191 
 82256 
 
 81671 
 81737 
 81803 
 81869 
 81935 
 
 82000 
 82066 
 82132 
 82197 
 82263 
 
 81677 81684 
 81743 81750 
 81809 81816 
 81875 81882 
 81941 81948 
 
 82007 82014 
 82073 82079 
 82138 82145 
 82204 82210 
 82269 82276 
 
 84510 84516 84522 84528 84535 
 
 82315 82321 82328 82334 82341 
 
 82380 82387 82393 82400 82406 
 
 82445 82452 82458 82465 82471 
 
 82510 82517 82523 82530 82536 
 
 82575 82582 82588 82595 82601 
 
 82640 82646 82653 82659 82666 
 
 82705 82711 82718 82724 82730 
 
 82769 82776 82782 82789 82795 
 
 82834 82840 82847 82853 82860 
 
 82898 82905 82911 82918 82924 
 
 82963 82 969 82975 82982 82988 
 
 83027 83033 83040 83046 83052 
 
 83091 83097 83104 83110 83117 
 
 83155 83161 83168 83174 83181 
 
 83219 83225 83232 83238 83245 
 
 83283 83289 83296 83302 83308 
 
 83347 83353 83359 83366 83372 
 
 83410 83417 83423 83429 83436 
 
 83474 83480 83487 83493 83499 
 
 83537 83544 83550 83556 83563 
 
 83601 83607 83613 83620 83626 
 
 83664 83670 83677 83683 83689 
 
 83727 83734 83740 83746 83753 
 
 83790 83797 83803 83809 83816 
 
 83853 83860 83866 83872 83879 
 
 83916 83923 83929 83935 83942 
 
 83979 83985 83992 83998 84004 
 
 84042 84048 84055 84061 84067 
 
 84105 84111 84117 84123 84130, 
 
 84167 84173 84180 84166 84192 
 
 84230 84236 84242 84248 84255 
 
 84292 84298 84305 84311 84317 
 
 84354 84361 84367 84373 84379 
 
 84417 84423 84429 84435 84442 
 
 84479 84485 84491 84497 84504 
 
 84541 84547 84553 84559 84566 
 
 650-700
 
 14 
 
 700-750 
 
 N 
 
 01234 
 
 56789 
 
 TOO 
 701 
 702 
 703 
 704 
 
 84510 84516 84522 84528 84 535. 
 84572 84578 84584 84590 84597 
 84634 84640 84646 84652 84 658 
 846% 84702 84708 84714 84720 
 84757 84763 84770 84776 84782 
 
 84541 84547 84 553 84559 84566 
 84603 84609 84 615 84 621 84628 
 84665 84671 84677 84683 84689 
 84726 84733 84739 84745 84751 
 84788 84794 84800 84807 84813 
 
 705 
 706 
 707 
 708 
 709 
 
 84819 84825 84S31 84837 84844 
 84880 84887 84893 84899 84905 
 84942 84948 84954 84960 84 967 
 85003 85009 85016 85022 85 028 
 85065 85071 85077 85083 85089 
 
 84850 84856 84862 84868 84874 
 84911 84917 84924 84930 84936 
 84973 84979 84985 84991 84997 
 85 034 85 040 85 046 85052 85 058 
 85095 85101 85107 85114 85120 
 
 71O 
 
 711 
 712 
 713 
 714 
 
 85126 85132 85 138 85 144 85150 
 85187 85193 85199 85205 85211 
 85248 85254 85260 85266 85272 
 85309 85315 85321 85327 85333 
 85370 85376 85382 85388 85394 
 
 85156 85163 85169 85175 85181 
 85217 85224 85230 85236 85242 
 85278 85285 85291 85297 85303 
 85339 85345 85352 85358 85364 
 85400 85406 85412 85418 85425 
 
 715 
 716 
 717 
 718 
 719 
 
 85431 85437 85443 85449 85455 
 85491 85497 85503 85509 85516 
 85552 85558 85564 85570 85576 
 85612 85618 85625 85631 85637 
 85673 85679 85685 85691 85697 
 
 85461 85467 85473 85479 85485 
 85522 85528 85534 85540 85546 
 85582 85588 85594 85600 85606 
 85643 85649 85655 85661 85667 
 85703 85709 85715 85721 85727 
 
 720 
 
 721 
 
 722 
 723 
 724 
 
 85733 85739 85745 85751 85757 
 85794 85800 85806 85 812 85818 
 85854 85860 85866 85872 85878 
 85914 85920 85926 85932 85938 
 85974 85980 85986 85992 85998 
 
 85763 85769 85775 85781 85788 
 85824 85830 85836 85842 85848 
 85884 85890 858% 85902 85908 
 85944 85950 85956 85962 85968 
 86004 86010 86016 86022 86028 
 
 725 
 726 
 727 
 728 
 
 729 
 
 86034 86040 86046 86052 86058 
 86094 86100 86106 86112 86118 
 86153 86159 86165 86171 86177 
 86213 86219 86225 86231 86237 
 86273 86279 86285 86291 86297 
 
 86064 86070 86076 86082 86088 
 86124 86130 86 136 86141 86147 ! 
 86183 86189 86 195 86201 86207 
 86243 86249 86255 86261 86267 
 86303 86308 86314 86320 86326 
 
 73O 
 
 731 
 732 
 733 
 
 734 
 
 86332 86338 86344 86350 86356 
 86392 86398 86404 86410 86415 
 86451 86457 86463 86469 86475 
 86510 86516 86522 86528 86 534 
 86570 86576 86581 86587 86593 
 
 86362 86368 86374 86380 86386 
 86421 86 427 86433 86439 86 445 
 86481 86487 86493 86499 86504 
 86540 86546 86 552 86 558 86564 
 86599 86605 86611 86617 86623 
 
 735 
 736 
 737 
 738 
 
 739 
 
 86629 86635 86641 86646 86652 
 86688 86694 86700 86705 86711 
 86747 86753 86759 86764 86770 
 86806 86812 86817 86823 86829 
 86864 86870 86876 86882 86888 
 
 86658 86664 86670 86676 86682 
 86 717 86 723 86 729 86 735 86 741 
 86776 86 782 86 7SS 86794 86SOO 
 86835 86841 86 847 86853 86859 
 86894 86900 86906 86911 869171 
 
 740 
 
 741 
 742 
 743 
 744 
 
 86923 86929 86935 86941 86947 
 86982 86988 86994 86999 87005 
 87040 87046 87052 87058 87064 
 87099 87105 87111 87116 87122 
 87157 87163 87169 87175 87181 
 
 86953 86958 86964 86970 86976 
 87011 87017 87023 87029 87035 
 87070 87075 87081 87087 87093 
 87128 87134 87140 87146 871511 
 87186 87192 87198 87204 87210 
 
 745 
 746 
 747 
 748 
 749 
 
 87216 87221 87227 87233 87239 
 87274 87280 87286 87291 87297 
 87332 87338 87344 87349 87355 
 87390 873% 87402 87408 87413 
 87448 87454 87460 87466 87471 
 
 87245 87251 87256 87262 87268 
 87303 87309 87315 87320 87326 
 87361 87367 87373 87379 87 384 
 87419 87425 87431 87437 87 442 
 87477 87483 87489 87495 87500 : 
 
 750 
 
 87506 87512 87518 87523 87529 
 
 87535 87541 87547 87552 87558 
 
 N 
 
 O 1 2 3 4 
 
 56789 
 
 700-750
 
 750-800 
 
 15 
 
 8 9 
 
 775 
 770 
 777 
 770 
 779 
 
 780 
 781 
 782 
 783 
 
 784 
 
 785 
 786 
 787 
 788 
 789 
 
 79O 
 
 791 
 792 
 
 793 
 794 
 795 
 796 
 797 
 798 
 799 
 
 87506 87512 87518 87523 87529 
 
 87564 87570 87576 87581 87587 
 
 87622 87628 87633 87639 87645 
 
 87679 87685 87691 87697 87703 
 
 87737 87743 87749 87754 87760 
 
 87795 87800 87806 87812 87818 
 
 S7S52 87858 87864 87869 87875 
 
 87910 87915 87921 87927 87933 
 
 87 967 87973 87978 87984 87990 
 88024 88030 88036 88041 88047 
 
 880S1 88 087 88093 88098 88104 
 
 88138 88144 88150 88156 88161 
 
 88195 88 201 88207 88213 88218 
 
 88252 88 258 88264 88270 88275 
 
 88309 88315 88321 88 326 88332 
 
 88366 88372 88377 88383 88389 
 
 88423 88429 88434 88440 88446 
 
 88 480 88485 88491 88497 88502 
 88536 88542 88547 88553 88559 
 88593 88598 88604 88610 88615 
 
 88649 88655 88660 88666 88672 
 
 88705 88711 88717 88722 88728 
 
 88762 88767 88773 88779 88784 
 
 88818 88824 88829 88835 88840 
 
 88874 88880 8SSSS 88891 SS897 
 
 88930 
 88986 
 89 042 
 89 098 
 89154 
 
 89209 
 89265 
 89321 
 89 3 76 
 89432 
 
 88936 88941 
 88992 88997 
 89048 89053 
 89104 89109 
 89159 89165 
 
 89215 89221 
 89271 89276 
 89326 89332 
 89382 89387 
 89437 89443 
 
 88947 88953 
 89003 89009 
 89059 89064 
 89115 89120 
 89170 89176 
 
 89226 89232 
 89282 89287 
 89337 89343 
 89393 89398 
 89448 89454 
 
 89487 89492 89498 89504 89509 
 
 89542 89548 89553 89559 89564 
 
 89597 89603 89609 89614 89620 
 
 89653 89658 89664 89669 896/5 
 
 89708 89713 89719 89724 89730 
 
 89763 89768 89774 89779 89785 
 
 89818 89823 89829 89834 89840 
 
 89873 89878 89883 89889 89894 
 
 89927 89933 89938 89944 89949 
 
 89982 89988 89993 89998 90004 
 
 90037 90042 90048 90053 90059 
 
 90091 90097 90102 90108 90113 
 
 90146 90151 90157 90162 90168 
 
 90200 90206 90211 90217 90222 
 
 90255 90260 90266 90271 90276 
 
 90309 90314 90320 90325 90331 
 
 87535 87541 87547 87552 87558 
 
 87593 87599 87604 87610 87616 
 
 87651 87656 87662 87668 87674 
 
 87708 87714 87720 87726 87731 
 
 87766 87772 87777 87783 87789 
 
 87823 87829 87835 87841 87846 
 
 87881 87887 87892 87898 87904 
 
 87938 87944 87950 87955 87 961 
 
 879% 88001 88007 88013 88018 
 
 88053 88058 88064 88070 88076 
 
 88110 88116 88121 88127 88133 
 88167 88173 88178 88184 88190 
 88224 88230 88235 88241 88247 
 88281 88287 88292 88298 88304 
 88338 88343 88349 88355 88360 
 
 88395 88400 88406 88412 88417 
 88451 88457 88463 88468 88474 
 88508 88513 88519 88525 88530 
 88564 88570 88576 88581 88587 
 88621 88627 88632 88638 88643 
 
 88677 88683 88689 88694 88700 
 
 88734 88739 88745 88750 88756 
 
 88790 88795 88801 88807 88812 
 
 88846 88852 88857 88863 88868 
 
 88902 88908 88913 88919 88925 
 
 88958 88964 88 969 88975 88981 
 
 89014 89020 89025 89031 89037 
 
 89070 89076 89081 89087 89092 
 
 89126 89131 89137 89143 89148 
 
 89182 89187 89193 89198 89204 
 
 89237 89243 89248 89254 89260 
 89293 89298 89304 89310 89315 
 89348 89354 89360 89365 89371 
 89404 89409 89415 89421 89426 
 89459 89465 89470 89476 89481 
 
 89515 89520 89526 89531 89537 
 89570 89575 89581 89586 89592 
 89625 89631 89636 89642 89647 
 89680 89686 89691 89697 89702 
 89735 89741 89746 89752 89757 
 
 89790 897% 89801 89807 89812 
 
 89845 89851 89856 89862 89867 
 
 89900 89905 89911 89916 89922 
 
 89955 89960 89966 89971 89977 
 
 90009 90015 90020 90026 90031 
 
 90064 90069 90075 90080 90086 
 90119 90124 90129 90135 90140 
 90173 90179 90184 90189 90195 
 90227 90233 90238 90244 90249 
 90282 90287 90293 90298 90304 
 90336 90342 90347 90352 90358 
 
 8 
 
 750-800
 
 16 
 
 800-850 
 
 N 
 
 01234 
 
 56789 
 
 8OO 
 
 801 
 802 
 803 
 804 
 
 90309 90314 90320 90325 90331 
 90363 90369 90374 90380 90385 
 90417 90423 90428 90434 90439 
 90472 90477 90482 90488 90493 
 90526 90531 90536 80542 90547 
 
 90336 90342 90347 90352 90358 
 90390 90396 90401 90407 90412 
 90445 90450 90455 90461 90466 
 90499 90504 90509 90515 90520 
 90553 90558 90563 90569 90574 
 
 805 
 806 
 807 
 808 
 809 
 
 90580 90585 90590 90596 90601 
 90634 90639 90644 90650 90655 
 90687 90693 90698 90703 90709 
 90741 90747 90752 90757 90763 
 90795 90800 90806 90811 90816 
 
 90607 90612 90617 90623 90628 
 90660 90666 90671 90677 90682 
 90714 90720 90725 90730 90736 
 90768 90773 90779 90784 90789 
 90822 90827 90832 90838 90843 
 
 81O 
 
 811 
 812 
 813 
 814 
 
 90849 90854 90859 90865 90870 
 90902 90907 90913 90918 90924 
 90956 90961 90966 90972 90977 
 91009 91014 91020 91025 91030 
 91062 91068 91073 91078 91084 
 
 90875 90881 90886 90891 90897 
 90929 90934 90940 90945 90950 
 90982 90988 90993 90998 91004 
 91036 91041 91046 91052 91057 
 91089 91094 91100 91105 91110 
 
 815 
 816 
 817 
 818 
 819 
 
 91116 91121 91126 91132 91137 
 91169 91174 91180 91185 91190 
 91222 91228 91233 91238 91243 
 91275 91281 91286 91291 91297 
 91328 91334 91339 91344 91350 
 
 91142 91148 91153 91158 91164 
 91196 91201 91206 91212 91217 
 91249 91254 91259 91265 91270 
 91302 91307 91312 91318 91323 
 91355 91360 91365 91371 91376 
 
 820 
 
 821 
 822 
 823 
 
 824 
 
 91381 91387 91392 91397 91403 
 91434 91440 91445 91450 91455 
 91487 91492 91498 91503 91508 
 91540 91545 91551 91556 91561 
 91593 91598 91603 91609 91614 
 
 91408 91413 91418 91424 91429 
 91461 91466 91471 91477 91482 
 91514 91519 91524 91529 91535 
 91566 91572 91577 91582 91587 
 91619 91624 91630 91635 91640 
 
 825 
 826 
 827 
 828 
 829 
 
 91645 91651 91656 91661 91666 
 91698 91703 91709 91714 91719 
 91751 91756 91761 91766 91772 
 91803 91808 91814 91819 91824 
 91855 91861 91866 91871 91876 
 
 91672 91677 91682 91687 91693 
 91724 91730 91735 91740 91745 
 91777 91782 91787 91793 91798 
 91829 91834 91840 91845 91850 
 91882 91887 91892 91897 91903 
 
 830 
 
 831 
 832 
 833 
 834 
 
 91908 91913 91918 91924 91929 
 91960 91965 91971 91976 91981 
 92012 92018 92023 92028 92033 
 92065 92070 92075 92080 92085 
 92117 92122 92127 92132 92137 
 
 91934 91939 91944 91950 91955 
 91986 91991 91997 92002 92007 
 92038 92044 92049 92054 92059 
 92091 92096 92101 92106 92111 
 92143 92148 92153 92158 92163 
 
 835 
 836 
 837 
 838 
 839 
 
 92169 92174 92179 92184 92189 
 92221 92226 92231 92236 92241 
 92273 92278 92283 92288 92293 
 92324 92330 92335 92340 92345 
 92376 92381 92387 92392 92397 
 
 92195 92200 92205 92210 92215 
 92247 92252 92257 92262 92267 
 92298 92304 92309 92314 92319 
 92350 92355 92361 92366 92371 
 92402 92407 92412 92418 92423 
 
 840 
 
 841 
 842 
 843 
 844 
 
 92428 92433 92438 92443 92449 
 92480 92485 92490 92495 92500 
 92531 92536 92542 92547 92552 
 92583 92588 92593 92598 92603 
 92634 92639 92645 92650 92655 
 
 92454 92459 92464 92469 92474 
 92505 92511 92516 92521 92526 
 92557 92562 92567 92572 92578 
 92609 92614 92619 92624 92629 
 92660 92665 92670 92675 92681 
 
 845 
 846 
 847 
 848 
 849 
 
 92686 92691 92696 92701 92706 
 92737 92742 92747 92752 92758 
 92788 92793 92799 92804 92809 
 92840 92845 92850 92855 92860 
 92891 92896 92901 92906 92911 
 
 92711 92716 92722 92727 92732. 
 92763 92768 92773 92778 92 783 
 92814 92819 92824 92829 92834 
 92865 92870 92875 92881 92886 
 92916 92921 92927 92932 92937 
 
 850 
 
 92942 92947 92952 92957 92 962 
 
 92 967 92973 92978 92983 92 988 
 
 N 
 
 01234 
 
 56789 
 
 800-850
 
 850-900 
 
 17 
 
 3 
 
 92942 92947 92952 92957 92962 
 
 92993 92998 93003 93 008 93013 
 
 93044 93049 93054 93059 93064 
 
 93095 93100 93105 93110 93115 
 
 93 146 93 151 93 156 93 16! 03 166 
 
 93197 93202 93207 93212 93217 
 
 93247 93252 93258 93263 93268 
 
 93298 93303 93308 93313 93318 
 
 93349 93354 93359 93364 93369 
 
 93399 93404 93409 93414 93420 
 
 934^0 93455 93460 93465 93470 
 
 93500 93505 93510 93515 93520 
 
 93551 93556 93561 93566 93571 
 
 93601 93606 93611 93616 93621 
 
 93651 93656 93661 93666 93671 
 
 93702 93707 93712 93717 93722 
 
 93752 93757 93762 93767 93772 
 
 93802 93807 93 812 93 81 7 93822 
 
 93852 93857 93862 93867 93 872 
 
 93902 93907 93912 93917 93922 
 
 93952 93957 93962 93967 93972 
 
 94002 94007 94012 94017 94022 
 
 94052 94057 94062 94067 94072 
 
 94101 94106 94111 94116 94121 
 
 94151 94156 94161 94166 94171 
 
 94201 94206 94211 94216 94221 
 
 94250 94255 94260 94265 94270 
 
 94300 94305 94310 94315 94320 
 
 94349 94354 94359 94364 94369 
 
 94399 94404 94409 94414 94419 
 
 94448 94453 94458 94463 94468 
 
 94498 94503 94507 94512 94517 
 
 94547 94552 94557 94562 94567 
 
 945% 94601 94606 94611 94616 
 
 94645 94650 94655 94660 94665 
 
 94694 94699 94704 94709 94714 
 
 94743 94748 94753 94758 94763 
 
 94792 94797 94802 94807 94812 
 
 94841 94846 94 851 94856 94861 
 
 94890 94895 94900 94905 94910 
 
 94939 94944 94949 94954 94959 
 
 94 988 949V3 94998 95002 95007 
 95036 95041 v5 046 95051 95056 
 95085 95090 95 005 95100 95105 
 95134 95139 95 1*3 95148 95153 
 
 95182 
 95231 
 95279 
 95328 
 95376 
 
 95187 
 95236 
 95 284 
 95332 
 95381 
 
 95192 95197 95202 
 95240 95245 95250 
 95289 95294 95299 
 95337 95342 95347 
 95386 95390 95395 
 
 92967 92973 92978 929S3 92988 
 
 93018 93024 93029 93034 93039 
 
 93069 93075 93080 93085 93090 
 
 93120 93125 93131 93136 93141 
 
 93171 93176 93181 93186 93192 
 
 93222 
 93273 
 93323 
 93374 
 93425 
 
 93475 
 93526 
 93576 
 93626 
 93676 
 
 93227 
 93278 
 93328 
 93379 
 93430 
 
 93480 
 93531 
 93581 
 93631 
 93682 
 
 93232 93237 93242 
 93283 93288 93293 
 93334 93339 93344 
 93384 93389 93394 
 93435 93440 93445 
 
 93485 93490 93495 
 
 93536 93541 93546 
 
 93586 93591 93596 
 
 93636 93641 93646 
 
 93687 93692 93697 
 
 95424 95429 95434 95439 95444 
 
 93727 93732 93737 93742 93747 
 
 93777 93782 93787 93792 93797 
 
 93827 93832 93837 93842 93847 
 
 93877 93882 93887 93892 93897 
 
 93927 93932 93937 93942 93947 
 
 93977 93982 93987 93992 93997 
 94027 94032 94037 94042 94047 
 94077 94082 94086 94091 94096 
 94126 94131 94136 94141 94146 
 94176 94181 94186 94191 941% 
 
 94226 94231 94236 94240 94245 
 94275 94280 94285 94290 94295 
 94325 94330 94335 94340 94345 
 94374 94379 94384 94389 94394 
 94424 94429 94433 94438 94443 
 
 94473 94478 94483 94488 94493 
 94522 94527 94532 94537 94542 
 94571 94576 94581 94586 94591 
 94621 94626 94630 94635 94640 
 94670 94675 94680 94685 94689 
 
 94719 94724 94729 94734 94738 
 94768 94773 94778 94783 94787 
 94817 94822 94827 94832 94836 
 
 94 866 94 871- 94 876 94 880 94 885 
 94915 94919 94924 94929 94934 
 
 94963 94968 94973 94978 94983 
 95012 95017 95022 95027 95032 
 95061 95066 95071 95075 95 080 
 95109 95114 95119 95124 95129 
 
 95 158 95 163 95 168. 95 173 95 177 
 
 95207 95211 95216 95221 95226 
 95255 95260 95265 95270 95274 
 95303 95308 95313 95318 95323 
 95352 95357 95361 95366 95371 
 95400 95405 95410 95415 95419 
 
 95448 95453 95458 95463 95 468 
 
 8 
 
 859-900
 
 18 
 
 900-950 
 
 N 
 
 O 1 2 3 4 
 
 56789 
 
 900 
 
 901 
 902 
 903 
 904 
 
 95424 95429 95434 95439 95444 
 95472 95477 95482 95487 95492 
 95521 95525 95530 95535 95540 
 95569 95574 95578 95583 95588 
 95617 95622 95626 95631 95636 
 
 95448 95453 95458 95463 95468 
 95497 95501 95506 95511 95516 
 95545 95550 95554 95559 95564 
 95593 95598 95602 95607 95612 
 95641 95646 95650 95655 95660 
 
 905 
 906 
 907 
 908 
 909 
 
 95665 95670 95674 95679 95684 
 95713 95718 95722 95727 95732 
 95761 95766 95770 95775 95780 
 95809 95813 95818 95823 95828 
 95856 95861 95866 95871 95875 
 
 95689 95694 95698 95703 95708 
 95 737 95 742 95 746 95 751 95 756 
 95 785 95 789 95 794 95 799 95 804 
 95832 95837 95842 95847 95852 
 95880 95885 95890 95895 95899 
 
 910 
 
 911 
 912 
 913 
 
 914 
 
 95904 95909 95914 95918 95923 
 95952 95957 95961 95966 95971 
 95999 96004 96009 96014 96019 
 96047 96052 96057 96061 96066 
 96095 96099 96104 96109 96114 
 
 95928 95933 95938 95942 95947 
 95976 95980 95985 95990 95995 
 96023 96028 96033 96038 96042 
 96071 96076 96080 96085 96090 
 96118 96123 96128 96133 96137 
 
 915 
 916 
 917 
 918 
 919 
 
 96142 96147 96152 96156 96161 
 96190 96194 96199 96204 96209 
 96237 96242 96246 96251 96256 
 96284 96289 96294 96298 96303 
 96332 96336 96341 96346 96350 
 
 96166 96171 96175 96180 96185 
 96213 96218 96223 96227 96232 
 96261 96265 96270 96275 96280 
 96308 96313 96317 96322 96327 
 96355 96360 96365 96369 96374 
 
 920 
 
 921 
 922 
 923 
 924 
 
 96379 96384 96388 96393 96398 
 96426 96431 96435 96440 96445 
 96473 96478 96483 96487 96492 
 96520 96525 96530 96534 96539 
 96567 96572 96577 96581 96586 
 
 96402 96407 96412 96417 96421 
 96450 96454 96459 96464 96468 
 96497 96501 96506 96511 96515 
 96544 96548 96553 96558 96562 
 96591 96595 96600 96605 96609 
 
 925 
 926 
 927 
 928 
 929 
 
 96614 96619 96624 96628 96633 
 96661 96666 96670 96675 96680 
 96708 96713 96717 96722 96727 
 96755 96759 96764 % 769 96774 
 96802 96806 96811 96816 96820 
 
 96638 96642 96647 96652 96656 
 96685 96689 96694 96699 96703 
 96731 96736 96741 96745 96750 
 96778 96783 96788 96792 96797 
 96825 96830 96834 96839 96844 
 
 930 
 
 931 
 932 
 933 
 934 
 
 96848 96853 96858 96862 96867 
 96895 96900 96904 96909 96914 
 96942 96946 96951 96956 96960 
 96988 96993 96997 97002 97007 
 97035 97039 97044 97049 97053 
 
 96872 96876 96881 96886 96890 
 96918 96923 96928 96932 9w937 
 96965 96970 96974 96979 96984 
 97011 97016 97021 97025 97030 
 97058 97063 97067 97072 97077 
 
 935 
 936 
 937 
 938 
 939 
 
 97081 97086 97090 97095 97100 
 97128 97132 97137 97142 97146 
 97174 97179 97183 97188 97192 
 97 220 97 225 .97 230 97 234 97 239 
 97267 97,271 97276 97280 97285 
 
 97104 97109 97114 97118 97123 
 97151 97155 97160 97165 97169 
 97197 97202 97206 97211 97216 
 97243 97248 97253 97257 97262 
 97290 97294 97299 97304 97308 
 
 940 
 
 941 
 942 
 943 
 
 944 
 
 97313 97317 97322 97327 97331 
 97359 97364 97368 97373 97377 
 97405 97410 97414 97419 97424 
 97451 97456 97460 97465 97470 
 97497 97502 97506 97511 97516 
 
 97336 97340 97345 97350 97354 
 97382 97387 97391 97396 97400 
 97428 97433 97437 97442 97 M7 
 97474 97479 97483 97488 97493 
 97520 97525 97529 97534 97539 
 
 945 
 946 
 947 
 948 
 949 
 
 97543 97548 97552 97557 97562 
 97589 97594 97598 97603 97607 
 97635 97640 97644 97649 97653 
 97681 97685 97690 97695 97699 
 97727 97731 97736 97740 97745 
 
 97566 97571 97575 97580 97585 
 97612 97617 97621 97626 97630 
 97658 97663 97667 97672 97676 , 
 97704 97708 97713 97717 97722 ; 
 97 749 97 754 97 759 97 763 97 768 , 
 
 950 
 
 97772 97777 97782 97786 97791 
 
 97795 97800 97804 97809 97813 
 
 N 
 
 01234 
 
 56789 
 
 900-950
 
 950-1000 
 
 19 
 
 N 
 
 950 
 
 951 
 952 
 953 
 
 954 
 
 955 
 956 
 957 
 
 O 
 
 97772 97777 97782 97786 97791 
 97818 97823 97827 97832 97836 
 97864 97868 97873 97877 97882 
 97909 97914 97918 97923 97928 
 97955 97959 97964 97968 97973 
 
 98000 98005 98009 98014 98019 
 98046 98050 98055 98059 98064 
 98091 98096 98100 98105 98109 
 98137 98141 98146 98150 98155 
 98182 98186 98191 98195 98200 
 
 98227 98232 98236 98241 98245 
 98272 98277 98 281 98286 98290 
 98318 98322 98327 98331 98336 
 98363 98367 98372 98376 98381 
 98408 98412 98417 98421 98426 
 
 98453 98457 98462 98466 98471 
 
 98498 98502 98507 98511 98516 
 
 98543 98547 98552 98556 98561 
 
 98588 98592 98597 98601 98605 
 
 98632 98637 98641 98646 98650 
 
 98677 98682 98686 98691 98695 
 98722 98726 98731 98735 98740 
 98767 98771 98776 98780 98784 
 98811 98816 98820 98825 98829 
 98856 98860 98865 98869 98874 
 
 98900 98905. 98909 98914 98918 
 
 98 945. 98949 98954 98958 98 963 
 
 98989 98994 98998 99003 99007 
 
 99034 99038 99043 99047 99052 
 
 99078 99083 99087 99092 99096 
 
 99123 99127 99131 99136 99140 
 
 99167 99171 99176 99180 99185 
 
 99211 99216 99220 99224 99229 
 
 99255 99260 99264 99269 99273 
 
 99300 99304 99308 99313 99317 
 
 99344 99348 99352 99357 99361 
 
 99388 99392 99396 99401 99405 
 
 99432 99436 99441 99445 99449 
 
 99476 99480 99484 99489 99493 
 
 99520 99524 99528 99533 99537 
 
 99564 99568 99572 99577 99581 
 99607 99612 99616 99621 99625 
 99651 99656 99660 99664 99669 
 99695 99699 99704 99708 99712 
 99739 99743 99747 99752 99756 
 
 99782 99787 99791 
 
 99826 99830 99835 
 
 99870 99874 99878 
 
 99913 99917 99922 
 
 99957 99961 99 965 
 
 00000 00004 00009 
 
 99795 
 99839 
 99883 
 99926 
 99970 
 
 00013 00017 
 
 99800 
 99843 
 99887 
 99930 
 99974 
 
 8 
 
 9 
 
 97795 97800 97804 97809 97813 
 97841 97845 97850 97855 97859 
 97886 97891 97896 97900 97905 
 97932 97937 97941 97946 97950 
 97978 97982 97987 97991 979% 
 98023 98028 98032 98037 98041 
 98068 98073 98078 98082 98087 
 98114 98118 98123 98127 98132 
 98159 98164 98168 98173 98177 
 98204 98209 98214 98218 98223 
 98250 98254 98259 98263 98268 
 98295 98299 98304 98308 98 3 13 
 98340 98345 98349 98354 98358 
 98385 98390 98394 98399 98403 
 98430 98435 98439 98444 98 448 
 
 98475 98480 98484 98489 98493 
 
 98520 98525 98529 98534 98538 
 
 98565 98570 98574 98579 98583 
 
 98610 98614 98619 98623 98628 
 
 98655 98659 98664 98668 98673 
 
 98700 98704 98709 98713 98717 
 
 98744 98749 98753 98758 98762 
 
 98789 98793 98798 98802 98807 
 
 98834 98838 98843 98847 98851 
 
 98878 98883 98887 98892 988% 
 
 98923 98927 98932 98936 98941 
 
 98967 98972 98976 98981 98985 
 
 99012 99016 99021 99025 99029 
 
 99056 99061 99065 99069 99074 
 
 99100 99105 99109 99114 99118 
 
 99145 99149 99154 99158 99162 
 
 99189 99193 99198 99202 99207 
 
 99233 99238 99242 99247 99251 
 
 99277 99282 99 286 99291 99295 
 
 99322 99326 99330 99335 99339 
 
 99366 99370 99374 99379 99383 
 
 99410 99414 99419 99423 99427 
 
 99454 99458 99463 99467 99471 
 
 99498 99502 99506 99511 99515 
 
 99542 99546 99550 99555 99559 
 
 99585 99590 99594 99599 99603 
 
 99629 99634 99638 99642 99647 
 
 99673 99677 99682 99686 99691 
 
 99717 99721 99726 99730 99734 
 
 99760 99765 99769 99774 99778 
 
 99813 
 99 856 
 99900 
 99944 
 99987 
 
 00022 00026 00030 
 
 99804 
 99848 
 99891 
 99935 
 99978 
 
 99808 
 99852 
 998% 
 99939 
 99983 
 
 99817 
 99861 
 99904 
 99948 
 99991 
 
 00035 
 
 99822 
 99865 
 99909 
 99952 
 999% 
 
 00039 
 
 
 
 950-1000
 
 TABLE II, 
 
 APPROXIMATE EQUATION OF TIME. 
 
 DATE. 
 
 MINUTES. 
 
 DATE. 
 
 MINUTES. 
 
 DATE. 
 
 MINUTES. 
 
 DATE. 
 
 MINUTES. 
 
 Jan. 1 
 
 4 
 
 Apr. 1 
 
 4 jj 
 
 Aug. 9 
 
 5 : 
 
 Oct. 27 
 
 16 ; 
 
 " 3 
 
 5 
 
 4 
 
 3 1 
 
 " 15 
 
 4 ^ 
 
 Nov. 15 
 
 15 
 
 " 5 
 
 6 
 
 " 7 
 
 2 ^ 
 
 " 20 
 
 3 | 
 
 " 20 
 
 14 I 
 
 " 7 
 
 7 
 
 " 11 
 
 1 J 
 
 " 24 
 
 2 | 
 
 " 24 
 
 13 ! 
 
 9 
 
 8 
 
 " 15 
 
 o 3 
 
 " 28 
 
 i ; 
 
 " 27 
 
 12 g 
 
 " 12 
 
 9 
 
 
 . 
 
 " 31 
 
 
 
 " 30 
 
 11 M 
 
 " 15 
 
 10 
 
 " 19 
 
 i ; 
 
 
 
 Dec. 2 
 
 10 | 
 
 " 18 
 
 11 
 
 " 24 
 
 2 g 
 
 Sept. 3 
 
 i ! 
 
 " 5 
 
 9 ^ 
 
 " 21 
 
 12 * 
 
 " 30 
 
 3| 
 
 (( O 
 
 2 : 
 
 < 7 
 
 8 | 
 
 - 25 
 
 13 | 
 
 May 13 
 
 4 M 
 
 9 
 
 3 ; 
 
 " 9 
 
 7 55 
 
 " 31 
 
 14 ~ 
 
 " 29 
 
 3 | 
 
 " 12 
 
 4 ; 
 
 " 11 
 
 6 3 
 
 Feb. 10 
 
 15 | 
 
 June 5 
 
 2 
 
 " 15 
 
 5 ' 
 
 " 13 
 
 5 J 
 
 " 21 
 
 14 <S 
 
 " 10 
 
 i ; 
 
 " 18 
 
 6 1 
 
 " 16 
 
 4 ^ 
 
 " 27 
 
 13 -g 
 
 " 15 
 
 o 
 
 " 21 
 
 7 1 
 
 " 18 
 
 3 ' 
 
 Mar. 4 
 
 125 
 
 
 
 " 24 
 
 8 44 
 
 20 
 
 2 ! 
 
 " 8 
 
 11 
 
 " 20 
 
 i i 
 
 " 27 
 
 9 o 
 
 22 
 
 i ; 
 
 " 12 
 
 10 
 
 " 25 
 
 2| 
 
 " 30 
 
 10" 
 
 " 24 
 
 o 
 
 " 15 
 
 9 
 
 " 29 
 
 3 I 
 
 Oct. 3 
 
 11 
 
 
 , 
 
 " 19 
 
 8 
 
 July 5 
 
 4 ^ 
 
 6 
 
 12 : 
 
 26 
 
 1 I 
 
 " 22 
 
 7 
 
 " 11 
 
 5 J 
 
 " 10 
 
 13 ; 
 
 28 
 
 2 * 
 
 25 
 
 6 
 
 28 
 
 6 ^ 
 
 14 
 
 14 ; 
 
 30 
 
 sl 
 
 " 28 
 
 5 
 
 
 
 
 " 19 
 
 15 
 
 
 1
 
 TABLE III, 
 
 THE LOOA.KITHMS 
 
 OF THE 
 
 TRIGONOMETRIC FUNCTIONS 
 
 Prom to 3', or 89 57' to 90, for every second; 
 Prom to 2, or 88 to 90, for every ten seconds; 
 Prom 1 to 89, for every minute. 
 
 NOTE. To 
 
 log sin 
 
 all the logarithms 10 is to be appended. 
 
 
 
 log tan = log sk 
 log cos =10. 00 000 
 
 it 
 
 0' 
 
 1' 
 
 2' 
 
 t r 
 
 tt 
 
 0' 
 
 1' 
 
 2' 
 
 tt 
 
 o 
 
 
 
 6. 46 373 
 
 6. 76 476 
 
 60 
 
 30 
 
 6. 16 270 
 
 6. 63 982 
 
 6. 86 167 
 
 30 
 
 1 
 
 4. 68 557 
 
 6.47090 
 
 6. 76 836 
 
 59 
 
 31 
 
 6. 17 694 
 
 6. 64 462 
 
 6. 86 455 
 
 29 
 
 2 
 
 4.98660 
 
 6. 47 797 
 
 6. 77 193 
 
 58 
 
 32 
 
 6. 19 072 
 
 6. 64 936 
 
 6. 86 742 
 
 28 
 
 3 
 
 5. 16 270 
 
 6. 48 492 
 
 6. 77 548 
 
 57 
 
 33 
 
 6. 20 409 
 
 6. 65 406 
 
 6. 87 027 
 
 27 
 
 4 
 
 5. 28 763 
 
 6.49175 
 
 6.77900 
 
 56 
 
 34 
 
 6. 21 705 
 
 6. 65 870 
 
 6. 87 310 
 
 26 
 
 5 
 
 5. 38 454 
 
 6.49849 
 
 6. 78 248 
 
 55 
 
 35 
 
 6. 22 964 
 
 6. 66 330 
 
 6. 87 591 
 
 25 
 
 6 
 
 5. 46 373 
 
 6. 50 512 
 
 6. 78 595 
 
 54 
 
 36 
 
 6. 24 188 
 
 6. 66 785 
 
 6. 87 870 
 
 24 
 
 7 
 
 5. 53 067 
 
 6. 51 165 
 
 6. 78 938 
 
 53 
 
 37 
 
 6. 25 378 
 
 6. 67 235 
 
 6. 88 147 
 
 23 
 
 8 
 
 5. 58 866 
 
 6. 51 808 
 
 6. 79 278 
 
 52 
 
 38 
 
 6. 26 536 
 
 6. 67 680 
 
 6. 88 423 
 
 22 
 
 9 
 
 5. 63 982 
 
 6. 52 442 
 
 6. 79 616 
 
 51 
 
 39 
 
 6.27664 
 
 6. 68 121 
 
 6. 88 697 
 
 21 
 
 10 
 
 5. 68 557 
 
 6. 53 067 
 
 6. 79 952 
 
 50 
 
 40 
 
 6. 28 763 
 
 6. 68 557 
 
 6. 88 969 
 
 20 
 
 11 
 
 5. 72 697 
 
 6. 53 683 
 
 6. 80 285 
 
 49 
 
 41 
 
 6. 29 836 
 
 6. 68 990 
 
 6. 89 240 
 
 19 
 
 12 
 
 5. 76 476 
 
 6. 54 291 
 
 6. 80 615 
 
 48 
 
 42 
 
 6. 30 882 
 
 6. 69 418 
 
 6. 89 509 
 
 18 
 
 13 
 
 5. 79 952 
 
 6. 54 890 
 
 6. 80 943 
 
 47 
 
 43 
 
 6.31904 
 
 6. 69 841 
 
 6. 89 776 
 
 17 
 
 14 
 
 5. 83 170 
 
 6. 55 481 
 
 6. 81 268 
 
 46 
 
 44 
 
 6. 32 903 
 
 6. 70 261 
 
 6.90042 
 
 16 
 
 15 
 
 5. 86 167 
 
 6. 56 064 
 
 6. 81 591 
 
 45 
 
 45 
 
 6. 33 879 
 
 6. 70 676 
 
 6.90306 
 
 15 
 
 16 
 
 5. 88 969 
 
 6. 56 639 
 
 6. 81 911 
 
 44 
 
 46 
 
 6. 34 833 
 
 6. 71 088 
 
 6.90568 
 
 14 
 
 17 
 
 5. 91 602 
 
 6. 57 207 
 
 6. 82 230 
 
 43 
 
 47 
 
 6. 35 767 
 
 6. 71 496 
 
 6. 90 829 
 
 13 
 
 18 
 
 5. 94 085 
 
 6. 57 767 
 
 6. 82 545 
 
 42 
 
 48 
 
 6.36682 
 
 6.71900 
 
 6. 91 088 
 
 12 
 
 19 
 
 5. 96 433 
 
 6. 58 320 
 
 6. 82 859 
 
 41 
 
 49 
 
 6. 37 577 
 
 6. 72 300 
 
 6. 91 346 
 
 11 
 
 20 
 
 5.98660 
 
 6. 58 866 
 
 6. 83 170 
 
 40 
 
 50 
 
 6.38454 
 
 6. 72 697 
 
 6. 91 602 
 
 10 
 
 21 
 
 6. 00 779 
 
 6. 59 406 
 
 6. 83 479 
 
 39 
 
 51 
 
 6.39315 
 
 6.73090 
 
 6.91857 
 
 9 
 
 22 
 
 6. 02 800 
 
 6. 59 939 
 
 6. 83 786 
 
 38 
 
 52 
 
 6. 40 158 
 
 6. 73 479 
 
 6. 92 110 
 
 8 
 
 23 
 
 6. 04 730 
 
 6. 60 465 
 
 6. 84 091 
 
 37 
 
 53 
 
 6. 40 985 
 
 6. 73 865 
 
 6. 92 362 
 
 7 
 
 24 
 
 6. 06 579 
 
 6. 60 985 
 
 6.84394 
 
 36 
 
 54 
 
 6. 41 797 
 
 6. 74 248 
 
 6. 92 612 
 
 6 
 
 25 
 
 6. 08 351 
 
 6. 61 499 
 
 6.84694 
 
 35 
 
 55 
 
 6.42594 
 
 6. 74 627 
 
 6. 92 861 
 
 5 
 
 26 
 
 6. 10 055 
 
 6. 62 007 
 
 6. 84 993 
 
 34 
 
 56 
 
 6. 43 376 
 
 6. 75 003 
 
 6. 93 109 
 
 4 
 
 27 
 
 6. 11 694 
 
 6. 62 509 
 
 6. 85 289 
 
 33 
 
 57 
 
 6.44145 
 
 6. 75 376 
 
 6. 93 355 
 
 3 
 
 28 
 
 6. 13 273 
 
 6. 63 006 
 
 6. 85 584 
 
 32 
 
 58 
 
 6.44900 
 
 6. 75 746 
 
 6.93599 
 
 2 
 
 29 
 
 6. 14 797 
 
 6. 63 496 
 
 6. 85 876 
 
 31 
 
 59 
 
 6. 45 643 
 
 6. 76 112 
 
 6.93843 
 
 1 
 
 30 
 
 6. 16 270 
 
 6. 63 982 
 
 6. 86 167 
 
 30 
 
 60 
 
 6. 46 373 
 
 6. 76 476 
 
 6.94085 
 
 
 
 // 
 
 59' 
 
 58' 
 
 57' 
 
 tt 
 
 tt 
 
 59' 
 
 58' 
 
 57' 
 
 f 
 
 log cot = log cos 
 log sin = 10, 00 000 
 
 log cos
 
 22 
 
 t ft 
 
 log sin 
 
 log tan 
 
 log cos 
 
 tf t 
 
 f tt 
 
 log sin 
 
 log tan 
 
 log cos 
 
 1 1 t 
 
 
 
 _ 
 
 
 
 10.00000 
 
 60 
 
 1O 
 
 7. 46 373 
 
 7. 46 373 
 
 10.00000 
 
 50 
 
 10 
 
 5. 68 557 
 
 5.68557 
 
 10.00000 
 
 50 
 
 10 
 
 7. 47 090 
 
 7. 47 091 
 
 10.00000 
 
 50 
 
 20 
 
 5.98660 
 
 5.98660 
 
 10.00000 
 
 40 
 
 20 
 
 7. 47 797 
 
 7. 47 797 
 
 10.00000 
 
 40 
 
 30 
 
 6. 16 270 
 
 6. 16 270 
 
 10.00000 
 
 30 
 
 30 
 
 7. 48 491 
 
 7. 48 492 
 
 10.00000 
 
 30 
 
 40 
 
 6. 28 763 
 
 6. 28 763 
 
 10.00000 
 
 20 
 
 40 
 
 7.49175 
 
 7. 49 176 
 
 10.00000 
 
 20 
 
 50 
 
 6.38454 
 
 6. 38 454 
 
 10.00000 
 
 10 
 
 50 
 
 7. 49 849 
 
 7. 49 849 
 
 10,00000 
 
 10 
 
 1 
 
 6.46373 
 
 6. 46 373 
 
 10.00000 
 
 59 
 
 11 
 
 7.50512 
 
 7.50512 
 
 10.00000 
 
 49 
 
 10 
 
 6. 53 067 
 
 6. 53 067 
 
 10.00000 
 
 50 
 
 10 
 
 7. 51 165 
 
 7. 51 165 
 
 10.00000 
 
 50 
 
 20 
 
 6.58866 
 
 6. 58 866 
 
 10.00000 
 
 40 
 
 20 
 
 7. 51 808 
 
 7. 51 809 
 
 10.00000 
 
 40 
 
 30 
 
 6. 63 982 
 
 6. 63 982 
 
 10.00000 
 
 30 
 
 30 
 
 7. 52 442 
 
 7. 52 443 
 
 10.00000 
 
 30 
 
 40 
 
 6. 68 557 
 
 6. 68 557 
 
 10.00000 
 
 20 
 
 40 
 
 7. 53 067 
 
 7. 53 067 
 
 10.00000 
 
 20 
 
 50 
 
 6. 72 697 
 
 6. 72 697 
 
 10.00000 
 
 10 
 
 50 
 
 7. 53 683 
 
 7. 53 683 
 
 10.00000 
 
 10 
 
 2 
 
 6.76476 
 
 6.76476 
 
 10.00000 
 
 58 
 
 12 
 
 7. 54 291 
 
 7. 54 291 
 
 10.00000 
 
 48 
 
 10 
 
 6. 79 952 
 
 6.79952 
 
 10.00000 
 
 50 
 
 10 
 
 7. 54 890 
 
 7. 54 890 
 
 10.00000 
 
 50 
 
 20 
 
 6. 83 170 
 
 6. 83 170 
 
 10.00000 
 
 40 
 
 20 
 
 7. 55 481 
 
 7. 55 481 
 
 10.00000 
 
 40 
 
 30 
 
 6. 86 167 
 
 6. 86 167 
 
 10.00000 
 
 30 
 
 30 
 
 7. 56 064 
 
 7.56064 
 
 10.00000 
 
 30 
 
 40 
 
 6.88969 
 
 6. 88 969 
 
 10.00000 
 
 20 
 
 40 
 
 7. 56 639 
 
 7. 56 639 
 
 10.00000 
 
 20 
 
 50 
 
 6. 91 602 
 
 6. 91 602 
 
 10.00000 
 
 10 
 
 50 
 
 7. 57 206 
 
 7. 57 207 
 
 10.00000 
 
 10 
 
 3 
 
 6. 94 085 
 
 6.94085 
 
 10.00000 
 
 57 
 
 13 
 
 7. 57 767 
 
 7. 57 767 
 
 10.00000 
 
 47 
 
 10 
 
 6.96433 
 
 6.96433 
 
 10.00000 
 
 50 
 
 10 
 
 7.58320 
 
 7. 58 320 
 
 10.00000 
 
 50 
 
 20 
 
 6.98660 
 
 6. 98 661 
 
 10.00000 
 
 40 
 
 20 
 
 7.58866 
 
 7. 58 867 
 
 10.00000 
 
 40 
 
 30 
 
 7.00779 
 
 7. 00 779 
 
 10.00000 
 
 30 
 
 30 
 
 7.59406 
 
 7.59406 
 
 10.00000 
 
 30 
 
 40 
 
 7. 02 800 
 
 7. 02 800 
 
 10.00000 
 
 20 
 
 40 
 
 7.59939 
 
 7.59939 
 
 10.00000 
 
 20 
 
 50 
 
 7. 04 730 
 
 7. 04 730 
 
 10.00000 
 
 10 
 
 50 
 
 7.60465 
 
 7. 60 466 
 
 10.00000 
 
 10 
 
 4 
 
 7. 06 579 
 
 7.06579 
 
 10.00000 
 
 056 
 
 14 
 
 7.60985 
 
 7. 60 986 
 
 10.00000 
 
 46 
 
 10 
 
 7.08351 
 
 7.08352 
 
 10.00000 
 
 50 
 
 10 
 
 7. 61 499 
 
 7. 61 500 
 
 10.00000 
 
 50 
 
 20 
 
 7.10055 
 
 7.10055 
 
 10.00000 
 
 40 
 
 20 
 
 7. 62 007 
 
 7.62008 
 
 10.00000 
 
 40 
 
 30 
 
 7. 11 694 
 
 7. 11 694 
 
 10.00000 
 
 30 
 
 30 
 
 7. 62 509 
 
 7. 62 510 
 
 10.00000 
 
 30 
 
 40 
 
 7. 13 273 
 
 7. 13 273 
 
 10.00000 
 
 20 
 
 40 
 
 7.63006 
 
 7. 63 006 
 
 10.00000 
 
 20 
 
 50 
 
 7. 14 797 
 
 7. 14 797 
 
 10.00000 
 
 10 
 
 50 
 
 7. 63 496 
 
 7. 63 497 
 
 10.00000 
 
 10 
 
 5 
 
 7. 16 270 
 
 7.16270 
 
 10.00000 
 
 55 
 
 15 
 
 7. 63 982 
 
 7. 63 982 
 
 10.00000 
 
 45 
 
 10 
 
 7.17694 
 
 7. 17 694 
 
 10.00000 
 
 50 
 
 10 
 
 7. 64 461 
 
 7.64462 
 
 10.00000 
 
 50 
 
 20 
 
 7.19072 
 
 7. 19 073 
 
 10.00000 
 
 40 
 
 20 
 
 7. 64 936 
 
 7.64937 
 
 10.00000 
 
 40 
 
 30 
 
 7.20409 
 
 7.20409 
 
 10.00000 
 
 30 
 
 30 
 
 7. 65 406 
 
 7. 65 406 
 
 10.00000 
 
 30 
 
 40 
 
 7. 21 705 
 
 7. 21 705 
 
 10.00000 
 
 20 
 
 40 
 
 7. 65 870 
 
 7. 65 871 
 
 10.00000 
 
 20 
 
 50 
 
 7. 22 964 
 
 7. 22 964 
 
 10.00000 
 
 10 
 
 50 
 
 7. 66 330 
 
 7. 66 330 
 
 10.00000 
 
 10 
 
 6 
 
 7. 24 188 
 
 7. 24 188 
 
 10.00000 
 
 54 
 
 16 
 
 7. 66 784 
 
 7. 66 785 
 
 10.00000 
 
 44 
 
 10 
 
 7. 25 378 
 
 7. 25 378 
 
 10.00000 
 
 50 
 
 10 
 
 7. 67 235 
 
 7. 67 235. 
 
 10.00000 
 
 50 
 
 20 
 
 7. 26 536 
 
 7. 26 536 
 
 10.00000 
 
 40 
 
 20 
 
 7. 67 680 
 
 7. 67 680 
 
 10.00000 
 
 40 
 
 30 
 
 7. 27 664 
 
 7.27664 
 
 10.00000 
 
 30 
 
 30 
 
 7. 68 121 
 
 7. 68 121 
 
 10.00000 
 
 30 
 
 40 
 
 7. 28 763 
 
 7. 28 764 
 
 10.00000 
 
 20 
 
 40 
 
 7.68557 
 
 7. 68 558 
 
 9.99999 
 
 20 
 
 50 
 
 7. 29 836 
 
 7. 29 836 
 
 10.00000 
 
 10 
 
 50 
 
 7. 68 989 
 
 7.68990 
 
 9.99999 
 
 10 
 
 7 
 
 7.30882 
 
 7.30882 
 
 10.00000 
 
 53 
 
 17 
 
 7.69417 
 
 7. 69 418 
 
 9.99999 
 
 43 
 
 10 
 
 7.31904 
 
 7.31904 
 
 10.00000 
 
 50 
 
 10 
 
 7. 69 841 
 
 7. 69 842 
 
 9.99999 
 
 50 
 
 20 
 
 7. 32 903 
 
 7. 32 903 
 
 10.00000 
 
 40 
 
 20 
 
 7. 70 261 
 
 7. 70 261 
 
 9.99999 
 
 40 
 
 30 
 
 7. 33 879 
 
 7. 33 879 
 
 10.00000 
 
 30 
 
 30 
 
 7. 70 676 
 
 7. 70 677 
 
 9.99999 
 
 30 
 
 40 
 
 7. 34 833 
 
 7. 34 833 
 
 10.00000 
 
 20 
 
 40 
 
 7. 71 088 
 
 7. 71 088 
 
 9.99999 
 
 20 
 
 50 
 
 7. 35 767 
 
 7. 35 767 
 
 10.00000 
 
 10 
 
 50 
 
 7. 71 496 
 
 7. 71 496 
 
 9.99999 
 
 10 
 
 8 
 
 7. 36 682 
 
 7.36682 
 
 10.00000 
 
 52 
 
 18 
 
 7. 71 900 
 
 7.71900 
 
 9.99999 
 
 42 
 
 10 
 
 7.37577 
 
 7.37577 
 
 10.00000 
 
 50 
 
 10 
 
 7. 72 300 
 
 7.72301 
 
 9.99999 
 
 50 
 
 20 
 
 7.38454 
 
 7.38455 
 
 10.00000 
 
 40 
 
 20 
 
 7. 72 697 
 
 7.72697 
 
 9.99999 
 
 40 
 
 30 
 
 7.39314 
 
 7.39315 
 
 10.00000 
 
 30 
 
 30 
 
 7.73090 
 
 7.73090 
 
 9.99999 
 
 30 
 
 40 
 
 7.40158 
 
 7. 40 158 
 
 10.00000 
 
 20 
 
 40 
 
 7. 73 479 
 
 7. 73 480 
 
 9.99999 
 
 20 
 
 50 
 
 7. 40 985 
 
 7. 40 985 
 
 10.00000 
 
 10 
 
 50 
 
 7. 73 865 
 
 7. 73 866 
 
 9.99999 
 
 10 
 
 9 
 
 7. 41 797 
 
 7. 41 797 
 
 10.00000 
 
 51 
 
 19 
 
 7. 74 248 
 
 7. 74 248 
 
 9.99999 
 
 41 
 
 10 
 
 7. 42 594 
 
 7. 42 594 
 
 10.00000 
 
 50 
 
 10 
 
 7. 74 627 
 
 7. 74 628 
 
 9.99999 
 
 50 
 
 20 
 
 7. 43 376 
 
 7. 43 376 
 
 10.00000 
 
 40 
 
 20 
 
 7. 75 003 
 
 7.75004 
 
 9.99999 
 
 40 
 
 30 
 
 7. 44 145 
 
 7. 44 145 
 
 10.00000 
 
 30 
 
 30 
 
 7. 75 376 
 
 7. 75 377 
 
 9.99999 
 
 30 
 
 40 
 
 7. 44 900 
 
 7.44900 
 
 10.00000 
 
 20 
 
 40 
 
 7. 75 745 
 
 7. 75 746 
 
 9.99999 
 
 20 
 
 50 
 
 7. 45 643 
 
 7. 45 643 
 
 10.00000 
 
 10 
 
 50 
 
 7. 76 112 
 
 7. 76 113 
 
 9.99999 
 
 10 
 
 100 
 
 7.46373 
 
 7. 46 373 
 
 10.00000 
 
 50 
 
 20 
 
 7.76475 
 
 7.76476 
 
 9.99999 
 
 40 
 
 t ft 
 
 log cos 
 
 log cot 
 
 log sin 
 
 tt f 
 
 t tt 
 
 log cos 
 
 log cot 
 
 log sin 
 
 r t t 
 
 89'
 
 r tt 
 
 log sin. 
 
 log tan 
 
 log cos 
 
 ft t 
 
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 31 
 
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 21 
 
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 30 
 
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 30 
 
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 24 
 
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 4O 
 
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 2O 
 
 50 
 
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 010 
 
 10 
 
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 50 
 
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 19 
 
 51 
 
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 50 
 
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 50 
 
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 40 
 
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 40 
 
 30 
 
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 9.99997 
 
 30 
 
 30 
 
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 30 
 
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 20 
 
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 20 
 
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 10 
 
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 42 
 
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 18 
 
 52 
 
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 8 
 
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 9.99997 
 
 50 
 
 10 
 
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 50 
 
 20 
 
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 40 
 
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 40 
 
 30 
 
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 30 
 
 30 
 
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 30 
 
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 20 
 
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 20 
 
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 10 
 
 50 
 
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 9. 99 995. 
 
 10 
 
 430 
 
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 17 
 
 53 
 
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 9.99995 
 
 7 
 
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 50 
 
 10 
 
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 50 
 
 20 
 
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 40 
 
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 40 
 
 30 
 
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 30 
 
 30 
 
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 30 
 
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 20 
 
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 20 
 
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 10 
 
 50 
 
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 10 
 
 44 
 
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 16 
 
 54 
 
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 6 
 
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 9.99996 
 
 50 
 
 10 
 
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 50 
 
 20 
 
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 40 
 
 20 
 
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 40 
 
 30 
 
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 30 
 
 30 
 
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 30 
 
 40 
 
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 20 
 
 40 
 
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 20 
 
 50 
 
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 10 
 
 50 
 
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 10 
 
 45 
 
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 9.99996 
 
 15 
 
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 5 
 
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 50 
 
 10 
 
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 50 
 
 20 
 
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 8.12017 
 
 9.99996 
 
 40 
 
 20 
 
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 40 
 
 30 
 
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 9.99996 
 
 30 
 
 30 
 
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 30 
 
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 20 
 
 40 
 
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 20 
 
 50 
 
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 10 
 
 50 
 
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 10 
 
 460 
 
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 14 
 
 56 
 
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 4 
 
 10 
 
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 50 
 
 10 
 
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 50 
 
 20 
 
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 40 
 
 20 
 
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 40 
 
 30 
 
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 30 
 
 30 
 
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 9.99994 
 
 30 
 
 40 
 
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 20 
 
 40 
 
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 20 
 
 50 
 
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 9.99996 
 
 10 
 
 50 
 
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 10 
 
 47 
 
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 9.99996 
 
 13 
 
 57 
 
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 3 
 
 10 
 
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 50 
 
 10 
 
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 50 
 
 20 
 
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 40 
 
 20 
 
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 40 
 
 30 
 
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 30 
 
 30 
 
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 30 
 
 40 
 
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 9.99996 
 
 20 
 
 40 
 
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 20 
 
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 10 
 
 50 
 
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 10 
 
 480 
 
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 9.99996 
 
 12 
 
 58 
 
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 2 
 
 10 
 
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 9.99996 
 
 50 
 
 10 
 
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 50 
 
 20 
 
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 8. 14 800 
 
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 40 
 
 20 
 
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 40 
 
 30 
 
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 30 
 
 30 
 
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 30 
 
 40 
 
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 8. 15 099 
 
 9. 99 996 
 
 20 
 
 40 
 
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 20 
 
 50 
 
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 10 
 
 50 
 
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 10 
 
 49 
 
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 11 
 
 59 
 
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 8. 23 462 
 
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 1 
 
 10 
 
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 50 
 
 10 
 
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 8. 23 585 
 
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 50 
 
 20 
 
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 8. 15 690 
 
 9.99996 
 
 40 
 
 20 
 
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 40 
 
 30 
 
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 9.99996 
 
 30 
 
 30 
 
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 30 
 
 40 
 
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 20 
 
 40 
 
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 8. 23 950 
 
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 20 
 
 50 
 
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 9. 99 995 
 
 10 
 
 50 
 
 8. 24 065 
 
 8. 24 071 
 
 9. 99 993 
 
 10 
 
 50 
 
 8. 16 268 
 
 8. 16 273 
 
 9.99995 
 
 1O 
 
 60 
 
 S. 24 186 
 
 8. 24 192 
 
 9. 99 993 
 
 
 
 t tr 
 
 log cos 
 
 log cot 
 
 log sin 
 
 tf f 
 
 t tr 
 
 log cos 
 
 log cot 
 
 log sin 
 
 " f 
 
 89'
 
 25 
 
 r tr 
 
 log sin 
 
 log tan 
 
 log cos 
 
 ft t 
 
 111 
 
 log sin 
 
 log tan 
 
 log cos 
 
 11 i 
 
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 8. 24 1S6 
 
 S. 24 192 
 
 9.99993 
 
 60 
 
 1O 
 
 8. 30 879 
 
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 9.99991 
 
 5O 
 
 10 
 
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 9.99993 
 
 50 
 
 10 
 
 8.30983 
 
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 9.99991 
 
 50 
 
 20 
 
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 40 
 
 20 
 
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 9.99991 
 
 40 
 
 30 
 
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 30 
 
 30 
 
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 9.99991 
 
 30 
 
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 9.99993 
 
 20 
 
 40 
 
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 9.99991 
 
 20 
 
 50 
 
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 9.99993 
 
 10 
 
 50 
 
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 9.99991 
 
 10 
 
 1 
 
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 9.99993 
 
 59 
 
 11 
 
 8. 31 495 
 
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 9.99991 
 
 49 
 
 10 
 
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 8. 25 029 
 
 9.99993 
 
 50 
 
 10 
 
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 9.99991 
 
 50 
 
 20 
 
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 9.99993 
 
 40 
 
 20 
 
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 8. 31 708 
 
 9.99991 
 
 40 
 
 30 
 
 8. 25 258 
 
 8. 25 265 
 
 9. 99 993 
 
 30 
 
 30 
 
 8. 31 800 
 
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 2 
 
 84718 
 
 84826 
 
 15174 
 
 99892 
 
 58 
 
 3 
 
 84897 
 
 85006 
 
 14994 
 
 99891 
 
 57 
 
 4 
 
 85075 
 
 85 185 
 
 14815 
 
 99891 
 
 56 
 
 5 
 
 85252 
 
 85363 
 
 14637 
 
 99890 
 
 55 
 
 6 
 
 85429 
 
 85 540 
 
 14460 
 
 99889 
 
 54 
 
 7 
 
 85605 
 
 85 717 
 
 14283 
 
 99888 
 
 53 
 
 8 
 
 85 780 
 
 85893 
 
 14107 
 
 99887 
 
 52 
 
 9 
 
 85955 
 
 86069 
 
 13931 
 
 99886 
 
 51 
 
 10 
 
 86128 
 
 86243 
 
 13757 
 
 99885 
 
 50 
 
 11 
 
 86301 
 
 86417 
 
 13 583 
 
 99884 
 
 49 
 
 12 
 
 86474 
 
 86591 
 
 13409 
 
 99883 
 
 48 
 
 13 
 
 86645 
 
 86763 
 
 13237 
 
 99882 
 
 47 
 
 14 
 
 86816 
 
 86935 
 
 13065 
 
 99881 
 
 46 
 
 15 
 
 86987 
 
 87106 
 
 12894 
 
 99880 
 
 45 
 
 16 
 
 87156 
 
 87277 
 
 12723 
 
 99879 
 
 44 
 
 17 
 
 87325 
 
 87447 
 
 12 553 
 
 99879 
 
 43 
 
 18 
 
 87494 
 
 87616 
 
 12384 
 
 99878 
 
 42 
 
 19 
 
 87661 
 
 87785 
 
 12215 
 
 99877 
 
 41 
 
 20 
 
 87829 
 
 87953 
 
 12047 
 
 99876 
 
 40 
 
 21 
 
 87995 
 
 88120 
 
 11880 
 
 99875 
 
 39 
 
 22 
 
 88161 
 
 88287 
 
 11713 
 
 99874 
 
 38 
 
 23 
 
 88326 
 
 88 453 
 
 11547 
 
 99873 
 
 37 
 
 24 
 
 88490 
 
 88618 
 
 11382 
 
 99872 
 
 36 
 
 25 
 
 88654 
 
 88783 
 
 11 217 
 
 99871 
 
 35 
 
 26 
 
 88817 
 
 88948 
 
 11052 
 
 99870 
 
 34 
 
 27 
 
 88980 
 
 89111 
 
 10889 
 
 99869 
 
 33 
 
 28 
 
 89142 
 
 89274 
 
 10726 
 
 99868 
 
 32 
 
 29 
 
 89304 
 
 89437 
 
 10563 
 
 99867 
 
 31 
 
 3O 
 
 89464 
 
 89598 
 
 10402 
 
 99866 
 
 3O 
 
 31 
 
 89625 
 
 89760 
 
 10240 
 
 99865 
 
 29 
 
 32 
 
 89784 
 
 89920 
 
 10080 
 
 99864 
 
 28 
 
 33 
 
 89943 
 
 90080 
 
 09920 
 
 99863 
 
 27 
 
 34 
 
 90102 
 
 90240 
 
 09760 
 
 99862 
 
 26 
 
 35 
 
 90260 
 
 90399 
 
 09601 
 
 99861 
 
 25 
 
 36 
 
 90417 
 
 90557 
 
 09443 
 
 99860 
 
 24 
 
 37 
 
 90574 
 
 90715 
 
 09285 
 
 99859 
 
 23 
 
 38 
 
 90730 
 
 90872 
 
 09128 
 
 99858 
 
 22 
 
 39 
 
 90885 
 
 91029 
 
 08971 
 
 99857 
 
 21 
 
 4O 
 
 91040 
 
 91 185 
 
 08815 
 
 99856 
 
 20 
 
 41 
 
 91 195 
 
 91340 
 
 08660 
 
 99 855 
 
 19 
 
 42 
 
 91349 
 
 91495 
 
 08505 
 
 99854 
 
 18 
 
 43 
 
 91 502 
 
 91650 
 
 08350 
 
 99853 
 
 17 
 
 44 
 
 91655 
 
 91803 
 
 08197 
 
 99852 
 
 16 
 
 45 
 
 91 807 
 
 91957 
 
 08043 
 
 99851 
 
 15 
 
 46 
 
 91959 
 
 92 110 
 
 07890 
 
 99850 
 
 14 
 
 47 
 
 92 110 
 
 92262 
 
 07738 
 
 99848 
 
 13 
 
 48 
 
 92261 
 
 92414 
 
 07586 
 
 99847 
 
 12 
 
 49 
 
 92411 
 
 92565 
 
 07435 
 
 99846 
 
 11 
 
 5O 
 
 92561 
 
 92716 
 
 07284 
 
 99 845 
 
 10 
 
 51 
 
 92 710 
 
 92866 
 
 07 134 
 
 99844 
 
 9 
 
 52 
 
 92859 
 
 93016 
 
 06 984 
 
 99843 
 
 8 
 
 53 
 
 93007 
 
 93 165 
 
 06835 
 
 99842 
 
 7 
 
 54 
 
 93154 
 
 93313 
 
 06687 
 
 99841 
 
 6 
 
 55 
 
 93301 
 
 93462 
 
 06538 
 
 99840 
 
 5 
 
 56 
 
 93448 
 
 93609 
 
 06391 
 
 99839 
 
 4 
 
 57 
 
 93 594 
 
 93756 
 
 06244 
 
 99838 
 
 3 
 
 58 
 
 93 740 
 
 93903 
 
 06097 
 
 99837 
 
 2 
 
 59 
 
 93885 
 
 94049 
 
 05 951 
 
 99 836 
 
 1 
 
 60 
 
 94030 
 
 94 195 
 
 05805 
 
 99 834 
 
 
 
 
 
 8 
 
 8 
 
 11 
 
 9 
 
 
 
 f 
 
 log cos 
 
 log cot 
 
 log tan 
 
 log sin 
 
 t 
 
 85
 
 30 
 
 6 
 
 f 
 
 log sin 
 
 log tan 
 
 log cot 
 
 -IT 
 
 log cos 
 
 n 
 
 t 
 
 
 t 
 
 log sin 
 o 
 
 log tan 
 
 A 
 
 log cot 
 
 1 f\ 
 
 log cos 
 ft 
 
 t 
 
 
 
 94030 
 
 94 195 
 
 05805 
 
 99834 
 
 60 
 
 
 
 
 01 923 
 
 02 162 
 
 97838 
 
 99761 
 
 60 
 
 1 
 
 94174 
 
 94340 
 
 05660 
 
 99833 
 
 59 
 
 
 1 
 
 02043 
 
 02283 
 
 97 717 
 
 99760 
 
 59 
 
 2 
 
 94317 
 
 94485 
 
 05 515 
 
 99832 
 
 58 
 
 
 2 
 
 02163 
 
 02404 
 
 97596 
 
 99759 
 
 58 
 
 3 
 
 94461 
 
 94630 
 
 05370 
 
 99831 
 
 57 
 
 
 3 
 
 02283 
 
 02525 
 
 97475 
 
 99757 
 
 57 
 
 4 
 
 94603 
 
 94773 
 
 05227 
 
 99830 
 
 56 
 
 
 4 
 
 02402 
 
 02645 
 
 97355 
 
 99756 
 
 56 
 
 5 
 
 94746 
 
 94917 
 
 05083 
 
 99829 
 
 55 
 
 
 5 
 
 02520 
 
 02 766 
 
 97234 
 
 99755 
 
 55 
 
 6 
 
 94887 
 
 95060 
 
 04940 
 
 99828 
 
 54 
 
 
 6 
 
 02639 
 
 02885 
 
 97 115 
 
 99753 
 
 54 
 
 7 
 
 95 029 
 
 95 202 
 
 04798 
 
 99827 
 
 53 
 
 
 7 
 
 02 757 
 
 03005 
 
 96995 
 
 99752 
 
 53 
 
 8 
 
 95 170 
 
 95344 
 
 04656 
 
 99825 
 
 52 
 
 
 8 
 
 02874 
 
 03124 
 
 96876 
 
 99751 
 
 52 
 
 9 
 
 95310 
 
 95486 
 
 04514 
 
 99824 
 
 51 
 
 
 9 
 
 02992 
 
 03 242 
 
 96758 
 
 99749 
 
 51 
 
 10 
 
 95450 
 
 95627 
 
 04373 
 
 99823 
 
 50 
 
 
 10 
 
 03 109 
 
 03361 
 
 96639 
 
 99 748 
 
 5O 
 
 11 
 
 95 589 
 
 95 767 
 
 04233 
 
 99822 
 
 49 
 
 
 11 
 
 03226 
 
 03479 
 
 96521 
 
 99747 
 
 49 
 
 12 
 
 95 728 
 
 95908 
 
 04092 
 
 99821 
 
 48 
 
 
 12 
 
 03342 
 
 03597 
 
 96403 
 
 99745 
 
 48 
 
 13 
 
 95867 
 
 96047 
 
 03953 
 
 99820 
 
 47 
 
 
 13 
 
 03458 
 
 03714 
 
 96286 
 
 99744 
 
 47 
 
 14 
 
 96005 
 
 96187 
 
 03813 
 
 99819 
 
 46 
 
 
 14 
 
 03574 
 
 03832 
 
 96168 
 
 99742 
 
 46 
 
 15 
 
 96143 
 
 96325 
 
 03675 
 
 99817 
 
 45 
 
 
 15 
 
 03690 
 
 03 948 
 
 96052 
 
 99741 
 
 45 
 
 16 
 
 96280 
 
 96464 
 
 03 536 
 
 99816 
 
 44 
 
 
 16 
 
 03805 
 
 04065 
 
 95935 
 
 99740 
 
 44 
 
 17 
 
 96417 
 
 96602 
 
 03398 
 
 99815 
 
 43 
 
 
 17 
 
 03920 
 
 04181 
 
 95819 
 
 99738 
 
 43 
 
 18 
 
 96553 
 
 96739 
 
 03261 
 
 99814 
 
 42 
 
 
 18 
 
 04034 
 
 04297 
 
 95 703 
 
 99737 
 
 42 
 
 19 
 
 96689 
 
 96877 
 
 03123 
 
 99813 
 
 41 
 
 
 19 
 
 04149 
 
 04413 
 
 95587 
 
 99736 
 
 41 
 
 20 
 
 96825 
 
 97013 
 
 02987 
 
 99812 
 
 4O 
 
 
 2O 
 
 04262 
 
 04528 
 
 95472 
 
 99734 
 
 40 
 
 21 
 
 96960 
 
 97150 
 
 02850 
 
 99810 
 
 39 
 
 
 21 
 
 04376 
 
 04643 
 
 95357 
 
 99733 
 
 39 
 
 22 
 
 97095 
 
 97285 
 
 02 715 
 
 99809 
 
 38 
 
 
 22 
 
 04490 
 
 04758 
 
 95 242 
 
 99731 
 
 38 
 
 23 
 
 97229 
 
 97421 
 
 02579 
 
 99808 
 
 37 
 
 
 23 
 
 04603 
 
 04873 
 
 95 127 
 
 99730 
 
 37 
 
 24 
 
 97363 
 
 97556 
 
 02444 
 
 99807 
 
 36 
 
 
 24 
 
 04 715 
 
 04987 
 
 95013 
 
 99728 
 
 36 
 
 25 
 
 97496 
 
 97691 
 
 02309 
 
 99806 
 
 35 
 
 
 25 
 
 04828 
 
 05 101 
 
 94899 
 
 99727 
 
 35 
 
 26 
 
 97629 
 
 97825 
 
 02 175 
 
 99804 
 
 34 
 
 
 26 
 
 04940 
 
 05 214 
 
 94786 
 
 99726 
 
 34 
 
 27 
 
 97762 
 
 97959 
 
 02041 
 
 99803 
 
 33 
 
 
 27 
 
 05 052 
 
 05328 
 
 94672 
 
 99 724 
 
 33 
 
 28 
 
 97894 
 
 98092 
 
 01 908 
 
 99802 
 
 32 
 
 
 28 
 
 05 164 
 
 05441 
 
 94559 
 
 99723 
 
 32 
 
 29 
 
 98026 
 
 98225 
 
 01 775. 
 
 99801 
 
 31 
 
 
 29 
 
 05 275 
 
 05553 
 
 94447 
 
 99721 
 
 31 
 
 30 
 
 98 157 
 
 98 358 
 
 01642 
 
 99800 
 
 3O 
 
 
 30 
 
 05386 
 
 05 666 
 
 94334 
 
 99720 
 
 30 
 
 31 
 
 98288 
 
 98490 
 
 01510 
 
 99798 
 
 29 
 
 
 31 
 
 05497 
 
 05 778 
 
 94222 
 
 99 718 
 
 29 
 
 32 
 
 98419 
 
 98622 
 
 01378 
 
 99797 
 
 28 
 
 
 32 
 
 05607 
 
 05890 
 
 94110 
 
 99717 
 
 28 
 
 33 
 
 98549 
 
 98753 
 
 01 247 
 
 99 796 
 
 27 
 
 
 33 
 
 05 717 
 
 06002 
 
 93998 
 
 99716 
 
 27 
 
 34 
 
 98679 
 
 98884 
 
 01 116 
 
 99795 
 
 26 
 
 
 34 
 
 05827 
 
 06113 
 
 93887 
 
 99714 
 
 26 
 
 35 
 
 98808 
 
 99015 
 
 00985 
 
 99793 
 
 25 
 
 
 35 
 
 05937 
 
 06224 
 
 93 776 
 
 99713 
 
 25 
 
 36 
 
 98937 
 
 99145 
 
 00855 
 
 99792 
 
 24 
 
 
 36 
 
 06046 
 
 06335 
 
 93665 
 
 99711 
 
 24 
 
 37 
 
 99066 
 
 99275 
 
 00725 
 
 99791 
 
 23 
 
 
 37 
 
 06155 
 
 06445 
 
 93555 
 
 99710 
 
 23 
 
 38 
 
 99194 
 
 99405 
 
 00595 
 
 99790 
 
 22 
 
 
 38 
 
 06264 
 
 06556 
 
 93444 
 
 99708 
 
 22 
 
 39 
 
 99322 
 
 99534 
 
 00466 
 
 99788 
 
 21 
 
 
 39 
 
 06372 
 
 06666 
 
 93334 
 
 99707 
 
 21 
 
 40 
 
 99450 
 
 99662 
 
 00338 
 
 99787 
 
 20 
 
 
 4O 
 
 06481 
 
 06775 
 
 93225 
 
 99705 
 
 20 
 
 41 
 
 99577 
 
 99791 
 
 00209 
 
 99786 
 
 19 
 
 
 41 
 
 06 589 
 
 06885 
 
 93 115 
 
 99704 
 
 19 
 
 42 
 
 99704 
 
 99 919 
 
 00 081 
 
 99785 
 
 18 
 
 
 42 
 
 06696 
 
 06994 
 
 93006 
 
 99702 
 
 18 
 
 43 
 
 99830 
 
 00 046 
 
 
 99783 
 
 17 
 
 
 43 
 
 06804 
 
 07 103 
 
 92897 
 
 99701 
 
 17 
 
 44 
 
 99 956 
 
 00174 
 
 99826 
 
 99782 
 
 16 
 
 
 44 
 
 06911 
 
 07211 
 
 92 789 
 
 99699 
 
 16 
 
 45 
 
 00082 
 
 00301 
 
 99699 
 
 99781 
 
 15 
 
 
 45 
 
 07018 
 
 07320 
 
 92680 
 
 99698 
 
 15 
 
 46 
 
 00207 
 
 00427 
 
 99573 
 
 99780 
 
 14 
 
 
 46 
 
 07 124 
 
 07428 
 
 92572 
 
 99696 
 
 14 
 
 47 
 
 00332 
 
 00553 
 
 99447 
 
 99778 
 
 13 
 
 
 47 
 
 07231 
 
 07536 
 
 92464 
 
 99695 
 
 13 
 
 48 
 
 00456 
 
 00679 
 
 99321 
 
 99777 
 
 12 
 
 
 48 
 
 07337 
 
 07643 
 
 92357 
 
 99693 
 
 12 
 
 49 
 
 00581 
 
 00805 
 
 99195 
 
 99776 
 
 11 
 
 
 49 
 
 07442 
 
 07751 
 
 92249 
 
 99692 
 
 11 
 
 50 
 
 00704 
 
 00930 
 
 99070 
 
 99775 
 
 1O 
 
 
 50 
 
 07548 
 
 07858 
 
 92142 
 
 99690 
 
 1O 
 
 51 
 
 00828 
 
 01055 
 
 98945 
 
 99773 
 
 9 
 
 
 51 
 
 07653 
 
 07964 
 
 92036 
 
 99689 
 
 9 
 
 52 
 
 00951 
 
 01 179 
 
 98821 
 
 99772 
 
 8 
 
 
 52 
 
 07 758 
 
 08071 
 
 91929 
 
 99687 
 
 8 
 
 53 
 
 01074 
 
 01303 
 
 98697 
 
 99771 
 
 7 
 
 
 53 
 
 07863 
 
 08177 
 
 91 823 
 
 99686 
 
 7 
 
 54 
 
 01 196 
 
 01427 
 
 98573 
 
 99769 
 
 6 
 
 
 54 
 
 07968 
 
 08 283 
 
 91717 
 
 99684 
 
 6 
 
 55 
 
 01318 
 
 01 550 
 
 98450 
 
 99768 
 
 5 
 
 
 55 
 
 08072 
 
 08389 
 
 91611 
 
 99683 
 
 5 
 
 56 
 
 01440 
 
 01673 
 
 98327 
 
 99767 
 
 4 
 
 
 56 
 
 OS 176 
 
 08495 
 
 91 505 
 
 99681 
 
 4 
 
 57 
 
 01 561 
 
 01 796 
 
 98204 
 
 99765 
 
 3 
 
 
 57 
 
 08280 
 
 08600 
 
 91400 
 
 99680 
 
 3 
 
 58 
 
 01682 
 
 01918 
 
 98082 
 
 99764 
 
 2 
 
 
 58 
 
 08383 
 
 08705 
 
 91295 
 
 99678 
 
 2 
 
 59 
 
 01803 
 
 02040 
 
 97960 
 
 99763 
 
 1 
 
 
 59 
 
 08486 
 
 08810 
 
 91190 
 
 99677 
 
 1 
 
 60 
 
 01923 
 9 
 
 02162 
 
 97 838 
 i ft 
 
 99761 
 
 O 
 
 
 6O 
 
 08589 
 
 08914 
 
 91086 
 
 1 ft 
 
 99675 
 
 
 
 t 
 
 log cos 
 
 log cot 
 
 JLU 
 
 log tan 
 
 log sin 
 
 t 
 
 
 / 
 
 
 
 log cos 
 
 log cot 
 
 1U 
 
 log tan 
 
 log sin 
 
 t 
 
 84 C 
 
 83
 
 8 
 
 31 
 
 t 
 
 log sin 
 
 A 
 
 log tan 
 
 Q 
 
 log cot 
 
 1 fl 
 
 log COB 
 
 f 
 
 
 
 08589 
 
 08914 
 
 iu 
 91086 
 
 99675 
 
 60 
 
 1 
 
 08692 
 
 09019 
 
 90981 
 
 99674 
 
 59 
 
 2 
 
 08795 
 
 09123 
 
 90877 
 
 99672 
 
 58 
 
 3 
 
 08897 
 
 09227 
 
 90773 
 
 99670 
 
 57 
 
 4 
 
 08999 
 
 09330 
 
 90670 
 
 99669 
 
 56 
 
 5 
 
 09101 
 
 09434 
 
 90566 
 
 99667 
 
 55 
 
 6 
 
 09202 
 
 09537 
 
 90463 
 
 99666 
 
 54 
 
 7 
 
 09304 
 
 09640 
 
 90360 
 
 99664 
 
 53 
 
 8 
 
 09405 
 
 09742 
 
 90258 
 
 99663 
 
 52 
 
 9 
 
 09506 
 
 09845 
 
 90155 
 
 99661 
 
 51 
 
 10 
 
 09606 
 
 09947 
 
 90053 
 
 99659 
 
 50 
 
 11 
 
 09707 
 
 10049 
 
 89951 
 
 99658 
 
 49 
 
 12 
 
 09807 
 
 10150 
 
 89 850 
 
 99656 
 
 48 
 
 13 
 
 09907 
 
 10252 
 
 89748 
 
 99655 
 
 47 
 
 14 
 
 10006 
 
 10353 
 
 89647 
 
 99653 
 
 46 
 
 15 
 
 10 106 
 
 10454 
 
 89 546 
 
 99651 
 
 45 
 
 16 
 
 10205 
 
 10555 
 
 89 445 
 
 99650 
 
 44 
 
 17 
 
 10304 
 
 10656 
 
 89344 
 
 99648 
 
 43 
 
 18 
 
 10402 
 
 10756 
 
 89244 
 
 99647 
 
 42 
 
 19 
 
 10501 
 
 10856 
 
 89144 
 
 99645 
 
 41 
 
 20 
 
 10599 
 
 10956 
 
 89044 
 
 99643 
 
 4O 
 
 21 
 
 10697 
 
 11056 
 
 88944 
 
 99642 
 
 39 
 
 22 
 
 10795 
 
 11 155 
 
 88845 
 
 99640 
 
 38 
 
 23 
 
 10893 
 
 11 254 
 
 88746 
 
 99638 
 
 37 
 
 24 
 
 10990 
 
 11353 
 
 88647 
 
 99637 
 
 36 
 
 25 
 
 11087 
 
 11452 
 
 88548 
 
 99635 
 
 35 
 
 26 
 
 11 184 
 
 11551 
 
 88449 
 
 99633 
 
 34 
 
 27 
 
 11281 
 
 11649 
 
 88351 
 
 99632 
 
 33 
 
 28 
 
 11377 
 
 11 747 
 
 88253 
 
 99630 
 
 32 
 
 29 
 
 11474 
 
 11 845 
 
 88 155 
 
 99629 
 
 31 
 
 30 
 
 11 570 
 
 11943 
 
 88057 
 
 99627 
 
 3O 
 
 31 
 
 11666 
 
 12040 
 
 87960 
 
 99625 
 
 29 
 
 32 
 
 11 761 
 
 12138 
 
 87862 
 
 99624 
 
 28 
 
 33 
 
 11857 
 
 12235 
 
 87765 
 
 99622 
 
 27 
 
 34 
 
 11952 
 
 12332 
 
 87668 
 
 99620 
 
 26 
 
 35 
 
 12047 
 
 12428 
 
 87572 
 
 99618 
 
 25 
 
 36 
 
 12 142 
 
 12525 
 
 87475 
 
 99617 
 
 24 
 
 37 
 
 12236 
 
 12621 
 
 87379 
 
 99615 
 
 23 
 
 38 
 
 12331 
 
 12 717 
 
 87283 
 
 99613 
 
 22 
 
 39 
 
 12425 
 
 12813 
 
 87187 
 
 99612 
 
 21 
 
 4O 
 
 12519 
 
 12909 
 
 87091 
 
 99610 
 
 2O 
 
 41 
 
 12612 
 
 13004 
 
 86996 
 
 99608 
 
 19 
 
 42 
 
 12706 
 
 13099 
 
 86901 
 
 99607 
 
 18 
 
 43 
 
 12 799 
 
 13 194 
 
 86806 
 
 99605 
 
 17 
 
 44 
 
 12 892 
 
 13289 
 
 86711 
 
 99603 
 
 16 
 
 45 
 
 12 985 
 
 13384 
 
 86616 
 
 99601 
 
 15 
 
 46 
 
 13078 
 
 13478 
 
 86522 
 
 99600 
 
 14 
 
 47 
 
 13171 
 
 13573 
 
 86427 
 
 99598 
 
 13 
 
 48 
 
 13263 
 
 13667 
 
 86333 
 
 99596 
 
 12 
 
 49 
 
 13355 
 
 13 761 
 
 86239 
 
 99595 
 
 11 
 
 50 
 
 13447 
 
 13854 
 
 86146 
 
 99593 
 
 1O 
 
 51 
 
 13539 
 
 13948 
 
 86 052 
 
 99591 
 
 9 
 
 52 
 
 13630 
 
 14041 
 
 85 959 
 
 99589 
 
 8 
 
 53 
 
 13 722 
 
 14134 
 
 85 866 
 
 99588 
 
 7 
 
 54 
 
 13813 
 
 14227 
 
 85 773 
 
 99586 
 
 6 
 
 55 
 
 13904 
 
 14320 
 
 85680 
 
 99584 
 
 5 
 
 56 
 
 13 994 
 
 14412 
 
 85588 
 
 99582 
 
 4 
 
 57 
 
 14 085 
 
 14504 
 
 85496 
 
 99581 
 
 3 
 
 58 
 
 14 175 
 
 14597 
 
 85 403 
 
 99579 
 
 2 
 
 59 
 
 14266 
 
 14688 
 
 85312 
 
 99577 
 
 1 
 
 00 
 
 14356 
 
 14780 
 
 _85 220 
 
 1 A 
 
 99575 
 
 A 
 
 
 
 t 
 
 log cos 
 
 log cot 
 
 IU 
 
 log tan 
 
 log sin 
 
 t 
 
 t 
 
 log sin 
 
 Q 
 
 log tan 
 
 log cot 
 
 log cos 
 
 r 
 
 
 
 14356 
 
 14780 
 
 85220 
 
 99575 
 
 6O 
 
 1 
 
 14445 
 
 14872 
 
 85 128 
 
 99574 
 
 59 
 
 2 
 
 14535 
 
 14 963 
 
 85037 
 
 99572 
 
 58 
 
 3 
 
 14624 
 
 15054 
 
 84946 
 
 99570 
 
 57 
 
 4 
 
 14714 
 
 15 145 
 
 84855 
 
 99568 
 
 56 
 
 5 
 
 14803 
 
 15 236 
 
 84764 
 
 99566 
 
 55 
 
 6 
 
 14891 
 
 15327 
 
 84673 
 
 99565 
 
 54 
 
 7 
 
 14980 
 
 15417 
 
 84583 
 
 99563 
 
 53 
 
 8 
 
 15069 
 
 15508 
 
 84492 
 
 99561 
 
 52 
 
 9 
 
 15 157 
 
 15598 
 
 84402 
 
 99559 
 
 51 
 
 10 
 
 15245 
 
 15688 
 
 84312 
 
 99557 
 
 50 
 
 11 
 
 15333 
 
 15 777 
 
 84223 
 
 99 556 
 
 49 
 
 12 
 
 15421 
 
 15867 
 
 SI 1.1? 
 
 99554 
 
 48 
 
 13 
 
 15 508 
 
 15956 
 
 84044 
 
 99552 
 
 47 
 
 14 
 
 15 596 
 
 16046 
 
 83954 
 
 99550 
 
 46 
 
 15 
 
 15683 
 
 16135 
 
 83865 
 
 99548 
 
 45 
 
 16 
 
 15770 
 
 16224 
 
 83776 
 
 99546 
 
 44 
 
 17 
 
 15857 
 
 16312 
 
 83688 
 
 99545 
 
 43 
 
 18 
 
 15944 
 
 16401 
 
 83599 
 
 99543 
 
 42 
 
 19 
 
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 11 
 
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 12 
 
 35860 
 
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 13 
 
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 14 
 
 35968 
 
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 39071 
 
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 59575 
 
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 46 
 
 15 
 
 36022 
 
 37193 
 
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 45 
 
 
 15 
 
 39121 
 
 40478 
 
 59522 
 
 98643 
 
 45 
 
 16 
 
 36075 
 
 37250 
 
 62750 
 
 98825 
 
 44 
 
 
 16 
 
 39170 
 
 40531 
 
 59 469 
 
 98640 
 
 44 
 
 17 
 
 36129 
 
 37306 
 
 62694 
 
 98822 
 
 43 
 
 
 17 
 
 39220 
 
 40584 
 
 59416 
 
 98636 
 
 43 
 
 18 
 
 36182 
 
 37363 
 
 62637 
 
 98819 
 
 42 
 
 
 18 
 
 39270 
 
 40636 
 
 59364 
 
 98633 
 
 42 
 
 19 
 
 36236 
 
 37419 
 
 62581 
 
 98816 
 
 41 
 
 
 19 
 
 39319 
 
 40689 
 
 59311 
 
 98630 
 
 41 
 
 20 
 
 36289 
 
 37476 
 
 62524 
 
 98813 
 
 40 
 
 
 20 
 
 39369 
 
 40742 
 
 59258 
 
 98627 
 
 40 
 
 21 
 
 36342 
 
 37532 
 
 62468 
 
 98810 
 
 39 
 
 
 21 
 
 39418 
 
 40795 
 
 59205 
 
 98623 
 
 39 
 
 22 
 
 36395 
 
 37588 
 
 62412 
 
 98807 
 
 38 
 
 
 22 
 
 39467 
 
 40847 
 
 59153 
 
 98620 
 
 38 
 
 23 
 
 36449 
 
 37644 
 
 62356 
 
 98804 
 
 37 
 
 
 23 
 
 39517 
 
 40900 
 
 59100 
 
 98617 
 
 37 
 
 24 
 
 36502 
 
 37700 
 
 62300 
 
 98801 
 
 36 
 
 
 24 
 
 39566 
 
 40952 
 
 59048 
 
 98614 
 
 36 
 
 25 
 
 36555 
 
 37756 
 
 62244 
 
 98798 
 
 35 
 
 
 25 
 
 39615 
 
 41005 
 
 58995 
 
 98610 
 
 35 
 
 26 
 
 36608 
 
 37812 
 
 62 188 
 
 98795 
 
 34 
 
 
 26 
 
 39664 
 
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 98607 
 
 34 
 
 27 
 
 36660 
 
 37868 
 
 62132 
 
 98792 
 
 33 
 
 
 27 
 
 39713 
 
 41 109 
 
 58 891 
 
 98604 
 
 33 
 
 28 
 
 36713 
 
 37924 
 
 62076 
 
 98789 
 
 32 
 
 
 28 
 
 39762 
 
 41 161 
 
 58839 
 
 98601 
 
 32 
 
 29 
 
 36766 
 
 37980 
 
 62020 
 
 98786 
 
 31 
 
 
 29 
 
 39S11 
 
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 58786 
 
 98597 
 
 31 
 
 30 
 
 36819 
 
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 61965 
 
 98783 
 
 30 
 
 
 30 
 
 39860 
 
 41 266 
 
 58734 
 
 98594 
 
 30 
 
 31 
 
 36871 
 
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 61909 
 
 98780 
 
 29 
 
 
 31 
 
 39909 
 
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 58682 
 
 98591 
 
 29 
 
 32 
 
 36924 
 
 38147 
 
 61 853 
 
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 28 
 
 
 32 
 
 39958 
 
 41370 
 
 58630 
 
 98588 
 
 28 
 
 33 
 
 36976 
 
 38202 
 
 61 798 
 
 98774 
 
 27 
 
 
 33 
 
 40006 
 
 41422 
 
 58578 
 
 98584 
 
 27 
 
 34 
 
 37028 
 
 38257 
 
 61743 
 
 98771 
 
 26 
 
 
 34 
 
 40055 
 
 41474 
 
 58526 
 
 98581 
 
 26 
 
 35 
 
 37081 
 
 38313 
 
 61 687 
 
 98768 
 
 25 
 
 
 35 
 
 40103 
 
 41 526 
 
 58474 
 
 98 578 
 
 25 
 
 36 
 
 37133 
 
 38368 
 
 61 632 
 
 98765 
 
 24 
 
 
 36 
 
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 98574 
 
 24 
 
 37 
 
 37185 
 
 38423 
 
 61577 
 
 98762 
 
 23 
 
 
 37 
 
 40200 
 
 41629 
 
 58371 
 
 98571 
 
 23 
 
 38 
 
 37237 
 
 38479 
 
 61521 
 
 98759 
 
 22 
 
 
 38 
 
 40249 
 
 41681 
 
 58319 
 
 98568 
 
 22 
 
 39 
 
 37289 
 
 38534 
 
 61466 
 
 98756 
 
 21 
 
 
 39 
 
 40297 
 
 41733 
 
 58267 
 
 98565 
 
 21 
 
 40 
 
 37341 
 
 38589 
 
 61411 
 
 98753 
 
 20 
 
 
 40 
 
 40346 
 
 41 784 
 
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 20 
 
 41 
 
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 19 
 
 
 41 
 
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 98558 
 
 19 
 
 42 
 
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 98746 
 
 18 
 
 
 42 
 
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 18 
 
 43 
 
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 61 246 
 
 98743 
 
 17 
 
 
 43 
 
 40490 
 
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 17 
 
 44 
 
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 98740 
 
 16 
 
 
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 45 
 
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 15 
 
 
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 52 
 
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 8 
 
 
 52 
 
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 53 
 
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 39299 
 
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 7 
 
 
 53 
 
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 98518 
 
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 54 
 
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 54 
 
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 55 
 
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 5 
 
 
 55 
 
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 56 
 
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 4 
 
 
 56 
 
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 4 
 
 57 
 
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 57 
 
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 3 
 
 58 
 
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 58 
 
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 59 
 
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 54 203 
 
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 59 
 
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 58 
 
 
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 54 155 
 
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 58 
 
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 57 
 
 
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 57 
 
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 56 
 
 
 4 
 
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 56 
 
 5 
 
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 55 
 
 
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 55 
 
 6 
 
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 54 
 
 
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 53 965 
 
 98262 
 
 54 
 
 7 
 
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 53 
 
 
 7 
 
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 53918 
 
 98259 
 
 53 
 
 8 
 
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 56792 
 
 98467 
 
 52 
 
 
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 52 
 
 9 
 
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 51 
 
 
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 51 
 
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 53 776 
 
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 11 
 
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 49 
 
 
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 53 729 
 
 98244 
 
 49 
 
 12 
 
 41 861 
 
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 56 592 
 
 98453 
 
 48 
 
 
 12 
 
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 53681 
 
 98240 
 
 48 
 
 13 
 
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 56542 
 
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 47 
 
 
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 98237 
 
 47 
 
 14 
 
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 43 508 
 
 56492 
 
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 46 
 
 
 14 
 
 44646 
 
 46413 
 
 53587 
 
 98233 
 
 46 
 
 15 
 
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 43558 
 
 56442 
 
 98443 
 
 45 
 
 
 15 
 
 44689 
 
 46460 
 
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 98229 
 
 45 
 
 16 
 
 42047 
 
 43607 
 
 56393 
 
 98440 
 
 44 
 
 
 16 
 
 44733 
 
 46507 
 
 53493 
 
 98226 
 
 44 
 
 17 
 
 42093 
 
 43657 
 
 56343 
 
 98436 
 
 43 
 
 
 17 
 
 44776 
 
 46554 
 
 53446 
 
 98222 
 
 43 
 
 18 
 
 42140 
 
 43707 
 
 56293 
 
 98433 
 
 42 
 
 
 18 
 
 44819 
 
 46601 
 
 53399 
 
 98218 
 
 42 
 
 19 
 
 42 186 
 
 43756 
 
 56244 
 
 98429 
 
 41 
 
 
 19 
 
 44862 
 
 46648 
 
 53352 
 
 98215 
 
 41 
 
 20 
 
 42232 
 
 43806 
 
 56194 
 
 98426 
 
 40 
 
 
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 44905 
 
 46694 
 
 53306 
 
 98211 
 
 40 
 
 21 
 
 42278 
 
 43855 
 
 56145 
 
 98422 
 
 39 
 
 
 21 
 
 44948 
 
 46741 
 
 53259 
 
 98207 
 
 39 
 
 22 
 
 42324 
 
 43905 
 
 56095 
 
 98419 
 
 38 
 
 
 22 
 
 44992 
 
 46788 
 
 53212 
 
 98204 
 
 38 
 
 23 
 
 42370 
 
 43954 
 
 56046 
 
 98415 
 
 37 
 
 
 23 
 
 45035 
 
 46835 
 
 53165 
 
 98200 
 
 37 
 
 24 
 
 42416 
 
 44004 
 
 55996 
 
 98412 
 
 36 
 
 
 24 
 
 45077 
 
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 53119 
 
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 36 
 
 25 
 
 42461 
 
 44053 
 
 55947 
 
 98409 
 
 35 
 
 
 25 
 
 45 120 
 
 46928 
 
 53072 
 
 98192 
 
 35 
 
 26 
 
 42507 
 
 44102 
 
 55898 
 
 98405 
 
 34 
 
 
 26 
 
 45 163 
 
 46975 
 
 53025 
 
 98189 
 
 34 
 
 27 
 
 42553 
 
 44151 
 
 55849 
 
 98402 
 
 33 
 
 
 27 
 
 45206 
 
 47021 
 
 52979 
 
 98185 
 
 33 
 
 28 
 
 42599 
 
 44201 
 
 55799 
 
 98398 
 
 32 
 
 
 28 
 
 45249 
 
 47068 
 
 52932 
 
 98181 
 
 32 
 
 29 
 
 42644 
 
 442^0 
 
 55750 
 
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 31 
 
 
 29 
 
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 47114 
 
 52886 
 
 98177 
 
 31 
 
 30 
 
 42690 
 
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 55 701 
 
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 45334 
 
 47160 
 
 52840 
 
 98174 
 
 30 
 
 31 
 
 42735 
 
 44348 
 
 55652 
 
 98388 
 
 29 
 
 
 31 
 
 45377 
 
 47207 
 
 52793 
 
 98170 
 
 29 
 
 32 
 
 42781 
 
 44397 
 
 55603 
 
 98384 
 
 28 
 
 
 32 
 
 45419 
 
 47253 
 
 52747 
 
 98166 
 
 28 
 
 33 
 
 42826 
 
 44446 
 
 55 554 
 
 98381 
 
 27 
 
 
 33 
 
 45462 
 
 47299 
 
 52701 
 
 98162 
 
 27 
 
 34 
 
 42872 
 
 44495 
 
 55505 
 
 98377 
 
 26 
 
 
 34 
 
 45504 
 
 47346 
 
 52654 
 
 98159 
 
 26 
 
 35 
 
 42917 
 
 44544 
 
 55456 
 
 98373 
 
 25 
 
 
 35 
 
 45 547 
 
 47392 
 
 52608 
 
 98155 
 
 25 
 
 36 
 
 42962 
 
 44592 
 
 55408 
 
 98370 
 
 24 
 
 
 36 
 
 45 589 
 
 47438 
 
 52562 
 
 98151 
 
 24 
 
 37 
 
 43008 
 
 44641 
 
 55359 
 
 98366 
 
 23 
 
 
 37 
 
 45632 
 
 47484 
 
 52516 
 
 98147 
 
 23 
 
 38 
 
 43053 
 
 44690 
 
 55310 
 
 98363 
 
 22 
 
 
 38 
 
 45674 
 
 47530 
 
 52470 
 
 98144 
 
 22 
 
 39 
 
 43098 
 
 44738 
 
 55262 
 
 98359 
 
 21 
 
 
 39 
 
 45716 
 
 47576 
 
 52424 
 
 98140 
 
 21 
 
 40 
 
 43 143 
 
 44787 
 
 55213 
 
 98356 
 
 20 
 
 
 40 
 
 45 758 
 
 47622 
 
 52378 
 
 98136 
 
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 41 
 
 43 188 
 
 44836 
 
 55164 
 
 98352 
 
 19 
 
 
 41 
 
 45801 
 
 47668 
 
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 19 
 
 42 
 
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 44 884 
 
 55 116 
 
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 18 
 
 
 42 
 
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 52286 
 
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 18 
 
 43 
 
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 44933 
 
 55067 
 
 98345 
 
 17 
 
 
 43 
 
 45885 
 
 47760 
 
 52240 
 
 98125 
 
 17 
 
 44 
 
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 44981 
 
 55019 
 
 98342 
 
 16 
 
 
 44 
 
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 52194 
 
 98121 
 
 16 
 
 45 
 
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 54971 
 
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 15 
 
 
 45 
 
 45969 
 
 47852 
 
 52148 
 
 98117 
 
 15 
 
 46 
 
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 14 
 
 
 46 
 
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 47897 
 
 52103 
 
 98113 
 
 14 
 
 47 
 
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 45 126 
 
 54874 
 
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 13 
 
 
 47 
 
 46053 
 
 47943 
 
 52057 
 
 98110 
 
 13 
 
 48 
 
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 12 
 
 
 48 
 
 46095 
 
 47989 
 
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 98106 
 
 12 
 
 49 
 
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 11 
 
 
 49 
 
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 51 965 
 
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 11 
 
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 43 591 
 
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 51 
 
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 51 
 
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 9 
 
 52 
 
 43680 
 
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 54 633 
 
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 8 
 
 
 52 
 
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 48171 
 
 51829 
 
 98090 
 
 8 
 
 53 
 
 43 724 
 
 45415 
 
 54 585 
 
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 7 
 
 
 53 
 
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 51 783 
 
 98087 
 
 7 
 
 54 
 
 43769 
 
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 54537 
 
 98306 
 
 6 
 
 
 54 
 
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 48262 
 
 51 738 
 
 98083 
 
 6 
 
 55 
 
 43813 
 
 45511 
 
 54489 
 
 98302 
 
 5 
 
 
 55 
 
 46386 
 
 48307 
 
 51693 
 
 98079 
 
 5 
 
 56 
 
 43857 
 
 45 559 
 
 54 441 
 
 98299 
 
 4 
 
 
 56 
 
 46428 
 
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 51647 
 
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 4 
 
 57 
 
 43901 
 
 45606 
 
 54394 
 
 98 295 
 
 3 
 
 
 57 
 
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 58 
 
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 2 
 
 
 58 
 
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 59 
 
 43990 
 
 45702 
 
 54298 
 
 98288 
 
 1 
 
 
 59 
 
 46552 
 
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 98063 
 
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 60 
 
 44034 
 
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 18 
 
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 24 
 
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 26 
 
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 27 
 
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 28 
 
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 29 
 
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 31 
 
 
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 31 
 
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 39 
 
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 log sin 
 
 f 
 
 48 C 
 
 47'
 
 43 
 
 44' 
 
 49 
 
 f 
 
 log sin 
 
 log tan 
 
 log cot 
 
 log cos 
 
 t 
 
 
 
 9 
 
 9 
 
 10 
 
 rt 
 
 . 
 
 o 
 
 83378 
 
 96966 
 
 03034 
 
 86413 
 
 6O 
 
 1 
 
 83392 
 
 96991 
 
 03009 
 
 86401 
 
 59 
 
 2 
 
 S3 405 
 
 97016 
 
 02984 
 
 86389 
 
 58 
 
 3 
 
 83419 
 
 97042 
 
 02958 
 
 86377 
 
 57 
 
 4 
 
 83 432 
 
 97067 
 
 02933 
 
 86366 
 
 56 
 
 5 
 
 83 446 
 
 97092 
 
 02908 
 
 86354 
 
 55 
 
 6 
 
 83459 
 
 97118 
 
 02SS2 
 
 86342 
 
 54 
 
 7 
 
 83473 
 
 97143 
 
 02857 
 
 86330 
 
 53 
 
 8 
 
 83486 
 
 97168 
 
 02 832 
 
 86 318 
 
 52 
 
 9 
 
 83500 
 
 97193 
 
 02807 
 
 86306 
 
 51 
 
 10 
 
 83 513 
 
 97219 
 
 02 781 
 
 86295 
 
 5O 
 
 11 
 
 83 527 
 
 97244 
 
 02 756 
 
 86283 
 
 49 
 
 12 
 
 83540 
 
 97269 
 
 02 731 
 
 86 271 
 
 48 
 
 13 
 
 83 554 
 
 97295 
 
 02 705 
 
 86 259 
 
 47 
 
 14 
 
 83567 
 
 97320 
 
 02680 
 
 86247 
 
 46 
 
 15 
 
 83 581 
 
 97345 
 
 02655 
 
 86235 
 
 45 
 
 16 
 
 S3 594 
 
 97371 
 
 02629 
 
 86 223 
 
 44 
 
 17 
 
 83608 
 
 97396 
 
 02604 
 
 86211 
 
 43 
 
 18 
 
 S3 621 
 
 97421 
 
 02579 
 
 86200 
 
 42 
 
 19 
 
 83634 
 
 97447 
 
 02553 
 
 86188 
 
 41 
 
 20 
 
 S3 648 
 
 97472 
 
 0252S 
 
 86176 
 
 4O 
 
 21 
 
 83661 
 
 97 497 
 
 02503 
 
 86164 
 
 39 
 
 22 
 
 83674 
 
 97523 
 
 02477 
 
 86152 
 
 38 
 
 23 
 
 83688 
 
 97548 
 
 02 452 
 
 86 140 
 
 37 
 
 24 
 
 83 701 
 
 97573 
 
 02427 
 
 86 128 
 
 36 
 
 25 
 
 83 715 
 
 97598 
 
 02402 
 
 86116 
 
 35 
 
 26 
 
 83728 
 
 97624 
 
 02376 
 
 86104 
 
 34 
 
 27 
 
 S3 741 
 
 97649 
 
 02351 
 
 86092 
 
 33 
 
 28 
 
 83755 
 
 97674 
 
 02326 
 
 86080 
 
 32 
 
 29 
 
 83768 
 
 97 700 
 
 02300 
 
 86068 
 
 31 
 
 30 
 
 83 781 
 
 97 725 
 
 02275 
 
 86056 
 
 3O 
 
 31 
 
 83 795 
 
 97 750 
 
 02250 
 
 86044 
 
 29 
 
 32 
 
 83 SOS 
 
 97776 
 
 02224 
 
 86032 
 
 28 
 
 33 
 
 S3 821 
 
 97 SOI 
 
 02 199 
 
 86020 
 
 27 
 
 34 
 
 83834 
 
 97 826 
 
 02174 
 
 86008 
 
 26 
 
 35 
 
 83S4S 
 
 97851 
 
 02 149 
 
 85 996 
 
 25 
 
 36 
 
 83861 
 
 97877 
 
 02 123 
 
 85 984 
 
 24 
 
 37 
 
 83874 
 
 97902 
 
 02098 
 
 85 972 
 
 23 
 
 38 
 
 83887 
 
 97927 
 
 02073 
 
 85960 
 
 22 
 
 39 
 
 83901 
 
 97953 
 
 02047 
 
 85 948 
 
 21 
 
 40 
 
 83914 
 
 97978 
 
 02022 
 
 85936 
 
 2O 
 
 41 
 
 83927 
 
 98003 
 
 01997 
 
 85924 
 
 19 
 
 42 
 
 83940 
 
 98029 
 
 01971 
 
 85912 
 
 18 
 
 43 
 
 S3 954 
 
 98054 
 
 01946 
 
 85900 
 
 17 
 
 44 
 
 83967 
 
 98 079 
 
 01921 
 
 85 888 
 
 16 
 
 45 
 
 83980 
 
 98 104 
 
 01896 
 
 85876 
 
 15 
 
 46 
 
 S3 993 
 
 98130 
 
 01 870 
 
 85864 
 
 14 
 
 47 
 
 84 006 
 
 98155 
 
 01 845 
 
 85851 
 
 13 
 
 48 
 
 84020 
 
 98180 
 
 01820 
 
 85 839 
 
 12 
 
 49 
 
 84033 
 
 98206 
 
 01 794 
 
 85 827 
 
 11 
 
 50 
 
 84046 
 
 98231 
 
 01 769 
 
 85815 
 
 1O 
 
 '61 
 
 84059 
 
 98256 
 
 01 744 
 
 85 803 
 
 9 
 
 62 
 
 84072 
 
 98 281 
 
 01 719 
 
 85 791 
 
 8 
 
 53 
 
 84 085 
 
 98 307 
 
 01693 
 
 85 779 
 
 7 
 
 54 
 
 84098 
 
 98332 
 
 01668 
 
 85 766 
 
 6 
 
 165 
 
 84112 
 
 98357 
 
 01643 
 
 85 754 
 
 5 
 
 1 66 
 
 84 125 
 
 98 383 
 
 01 617 
 
 85 742 
 
 4 
 
 57 
 
 84 138 
 
 98408 
 
 01 592 
 
 85 730 
 
 3 
 
 68 
 
 84151 
 
 98 433 
 
 01 567 
 
 85 718 
 
 2- 
 
 59 
 
 84164 
 
 98 458 
 
 01 542 
 
 85 706 
 
 1 
 
 60 
 
 84 177 
 
 98 484 
 
 01 516 
 
 85693 
 
 
 
 
 9 
 
 
 10 
 
 9 
 
 
 
 t 
 
 log COB 
 
 log cot 
 
 log tan 
 
 log sin 
 
 f 
 
 f 
 
 log sin 
 
 log tan 
 
 log cot 
 
 log cos 
 
 f 
 
 
 
 9 
 
 Q 
 
 10 
 
 
 
 o 
 
 84 177 
 
 98 484 
 
 01516 
 
 85693 
 
 6O 
 
 1 
 
 84190 
 
 98509 
 
 01491 
 
 85681 
 
 59 
 
 2 
 
 84203 
 
 98534 
 
 01466 
 
 85669 
 
 58 
 
 3 
 
 84216 
 
 98560 
 
 01440 
 
 85657 
 
 57 
 
 4 
 
 84229 
 
 98585 
 
 01415 
 
 85645 
 
 56 
 
 5 
 
 84242 
 
 98610 
 
 01390 
 
 85632 
 
 55 
 
 6 
 
 84255 
 
 98635 
 
 01365 
 
 85620 
 
 54 
 
 7 
 
 84269 
 
 98661 
 
 01339 
 
 85608 
 
 53 
 
 8 
 
 84282 
 
 98686 
 
 01314 
 
 855% 
 
 52 
 
 9 
 
 84295 
 
 98711 
 
 01289 
 
 85 583 
 
 51 
 
 10 
 
 84308 
 
 98737 
 
 01 263 
 
 85571 
 
 5O 
 
 11 
 
 84321 
 
 98762 
 
 01 238 
 
 85559 
 
 49 
 
 12 
 
 84334 
 
 98787 
 
 01213 
 
 85 547 
 
 48 
 
 13 
 
 84347 
 
 98812 
 
 01 188 
 
 85 534 
 
 47 
 
 14 
 
 84360 
 
 98838 
 
 01 162 
 
 85 522 
 
 46 
 
 15 
 
 84373 
 
 98863 
 
 01 137 
 
 85 510 
 
 45 
 
 16 
 
 84385 
 
 98888 
 
 01 112 
 
 85497 
 
 44 
 
 17 
 
 84398 
 
 98913 
 
 01 087 
 
 85485 
 
 43 
 
 18 
 
 84411 
 
 98939 
 
 01061 
 
 85473 
 
 42 
 
 19 
 
 84424 
 
 98964 
 
 01036 
 
 85460 
 
 41 
 
 20 
 
 84437 
 
 98989 
 
 01011 
 
 85448 
 
 40 
 
 21 
 
 84450 
 
 99015 
 
 00985 
 
 85436 
 
 39 
 
 22 
 
 84463 
 
 99040 
 
 00960 
 
 85423 
 
 38 
 
 23 
 
 84476 
 
 99065 
 
 00935 
 
 85 411 
 
 37 
 
 24 
 
 84489 
 
 99090 
 
 00910 
 
 85399 
 
 36 
 
 25 
 
 84502 
 
 99116 
 
 00884 
 
 85386 
 
 35 
 
 26 
 
 84515 
 
 99141 
 
 00859 
 
 85374 
 
 34 
 
 27 
 
 84528 
 
 99166 
 
 00834 
 
 85361 
 
 33 
 
 28 
 
 84540 
 
 99191 
 
 00809 
 
 85349 
 
 32 
 
 29 
 
 84553 
 
 99217 
 
 00783 
 
 85337 
 
 31 
 
 30 
 
 84566 
 
 99242 
 
 00758 
 
 85324 
 
 30 
 
 31 
 
 84579 
 
 99267 
 
 00733 
 
 85312 
 
 29 
 
 32 
 
 84592 
 
 99293 
 
 00707 
 
 85299 
 
 28 
 
 33 
 
 84605 
 
 99318 
 
 00682 
 
 85287 
 
 27 
 
 34 
 
 84618 
 
 99343 
 
 00657 
 
 85274 
 
 26 
 
 35 
 
 84630 
 
 99368 
 
 00632 
 
 85262 
 
 25 
 
 36 
 
 84643 
 
 99394 
 
 00606 
 
 85250 
 
 24 
 
 37 
 
 84656 
 
 99419 
 
 00581 
 
 85237 
 
 23 
 
 38 
 
 84 669 
 
 99444 
 
 00556 
 
 85225 
 
 22 
 
 39 
 
 84682 
 
 99469 
 
 00531 
 
 85 212 
 
 21 
 
 4O 
 
 84694 
 
 99495 
 
 00505 
 
 85200 
 
 20 
 
 41 
 
 84707 
 
 99520 
 
 00480 
 
 85 187 
 
 19 
 
 42 
 
 84720 
 
 99545 
 
 00455 
 
 85 175 
 
 18 
 
 43 
 
 84733 
 
 99570 
 
 00430 
 
 85 162 
 
 17 
 
 44 
 
 84745 
 
 99596 
 
 00404 
 
 85 150 
 
 16 
 
 45 
 
 84758 
 
 99621 
 
 00379 
 
 85 137 
 
 15 
 
 46 
 
 84771 
 
 99646 
 
 00354 
 
 85125 
 
 14 
 
 47 
 
 84 784 
 
 99672 
 
 00328 
 
 85 112 
 
 13 
 
 48 
 
 84796 
 
 99697 
 
 00303 
 
 85 100 
 
 12 
 
 49 
 
 84 809 
 
 99722 
 
 00278 
 
 85087 
 
 11 
 
 5O 
 
 84822 
 
 99747 
 
 00253 
 
 85074 
 
 10 
 
 51 
 
 84835 
 
 99773 
 
 00227 
 
 85062 
 
 9 
 
 52 
 
 84847 
 
 99798 
 
 00202 
 
 85049 
 
 8 
 
 63 
 
 84860 
 
 99823 
 
 00177 
 
 85037 
 
 7 
 
 54 
 
 84873 
 
 99848 
 
 00152 
 
 85024 
 
 6 
 
 55 
 
 84 885 
 
 99874 
 
 00 126 
 
 85 012 
 
 5 
 
 56 
 
 84898 
 
 99899 
 
 00101 
 
 84999 
 
 4 
 
 57 
 
 84911 
 
 99924 
 
 00076 
 
 84986 
 
 3 
 
 58 
 
 84923 
 
 99949 
 
 00051 
 
 84974 
 
 2 
 
 59 
 
 84936 
 
 W 975 
 
 00025 
 
 84961 
 
 1 
 
 60 
 
 84949 
 
 00 OCX) 
 
 00000 
 
 84949 
 
 
 
 
 
 10 
 
 10 
 
 g 
 
 
 t 
 
 log cos 
 
 log cot 
 
 log tan 
 
 log sin 
 
 
 
 46 
 
 45
 
 50 
 
 TABLE IV. 
 
 FOB DETERMINING WITH GREATER ACCURACY THAN CAN BE DONE BY 
 MEANS OF TABLE III. : 
 
 1. log sin, log tan, and log cot, when the angle is between and 2 ; 
 
 2. log cos, log tan, and log cot, when the angle is between 88 and 90 ; 
 
 3. The value of the angle when the logarithm of the function does not 
 
 lie between the limits 8. 54 684 and 11. 45 316. 
 
 FORMULAS FOR THE USE OF THE NUMBERS S AND T. 
 I. When the angle a is between and 2 : 
 
 log sin a = log a" + S. 
 log tan a = log a" + T. 
 log cot a = colog tan a. 
 
 log a" = log sin a S, 
 = log tana T, 
 = colog cot a T. 
 
 II. When the angle a is between 88 and 90 : 
 
 log COS a = log (90" -a)" + S. 
 log COt a = log (90-a)"+ r. 
 
 log tan a = colog cot a. 
 
 log (90 a)" = log cos a S, 
 = log COt a T, 
 
 - colog tana-r, 
 and a = 90 -(90 -a). 
 
 TALIIES OF S AIS T D T, 
 
 a" 
 
 3 
 
 log sin a 
 
 
 a" 
 
 T 
 
 log tan a 
 
 a 
 
 T 
 
 log tan a 
 
 
 
 
 _ 
 
 
 
 
 
 _ 
 
 5146 
 
 
 8. 39 713 
 
 
 4.68557 
 
 
 
 
 4.68557 
 
 
 
 4. 68 567 
 
 
 2409 
 
 
 8.06740 
 
 
 200 
 
 
 6,98660 
 
 5424 
 
 
 8.41999 
 
 
 4.68556 
 
 
 
 
 4. 68 558 
 
 
 
 4. 68 568 
 
 
 3417 
 
 
 8. 21 920 
 
 
 1726 
 
 
 7. 92 263 
 
 5689 
 
 
 8. 44 072 
 
 
 4. 68 555 
 
 
 
 
 4. 68 559 
 
 
 
 4. 68 569 
 
 
 3823 
 
 
 8. 26 795 
 
 
 2432 
 
 
 8. 07 156 
 
 5941 
 
 
 S. 45 955 
 
 
 4.68555 
 
 
 
 
 4.68560 
 
 
 
 4. 68 570 
 
 
 4190 
 
 
 8.30776 
 
 
 2976 
 
 
 8. 15 924 
 
 6184 
 
 
 8. 47 697 
 
 
 4. 68 554 
 
 
 
 
 4. 68 561 
 
 
 
 4. 68 571 
 
 
 4840 
 
 
 8. 37 038 
 
 
 3434 
 
 
 8. 22 142 
 
 6417 
 
 
 8. 49 305 
 
 
 4. 68 553 
 
 
 
 
 4. 68 562 
 
 
 
 4. 68 572 
 
 
 5414 
 
 
 8.41904 
 
 
 3838 
 
 
 S. 26 973 
 
 6642 
 
 
 8.50802 
 
 
 4. 68 552 
 
 
 
 
 4. 68 563 
 
 
 
 4. 68 573 
 
 
 5932 
 
 
 8. 45 872 
 
 
 4204 
 
 
 8. 30 930 
 
 6859 
 
 
 8. 52 200 
 
 
 4. 68 551 
 
 
 
 
 4.68564 
 
 
 
 4. 68 574 
 
 
 6408 
 
 
 8. 49 223 
 
 
 4540 
 
 
 8. 34 270 
 
 7070 
 
 
 S. 53516 
 
 
 4. 68 550 
 
 
 
 
 4. 68 565 
 
 
 
 4.68575 
 
 
 6633 
 
 
 8. 50 721 
 
 
 4699 
 
 
 8. 35 766 ! 
 
 7173 
 
 
 8. 54 145 
 
 
 4. 68 550 
 
 
 
 
 4. 68 565 
 
 
 
 4. 68 575 
 
 
 6851 
 
 
 8. 52 125 
 
 
 4853 
 
 
 8. 37 167 
 
 7274 
 
 
 8. 54 753 
 
 
 4. 68 549 
 
 
 
 
 4.68566 
 
 
 
 
 
 7267 
 
 
 8.54684 
 
 
 5146 
 
 
 8. 39 713 
 
 
 
 
 a." 
 
 S 
 
 log sin a 
 
 
 a" 
 
 T 
 
 log tan a 
 
 a 
 
 T 
 
 log tan a
 
 51 
 
 TABLE IV. 
 
 This table (page 50) must be used when great accuracy is desired in 
 working with angles between and 2, or between 88 and 90. 
 
 The values of S and T are such that when the angle a is expressed 
 in seconds, 
 
 8 = log sin a log a", 
 
 T = log tan a log a". 
 
 Hence, follow the formulas given on the page containing the table. 
 The values of S and T are printed with the characteristic 10 too 
 large, and in using them 10 must always be annexed. 
 
 Find log sin 58' 17". 
 58' 17" = 3497." 
 log 3497 = 3.54370 
 
 8 = 4.68555-10 
 
 log sin 58' 17" = 8.22925 - 10 
 
 Find log tan 52' 47.5". 
 
 52' 47.5" = 3167.5." 
 
 log 3167.5 = 3.50072 
 
 T = 4.68561 -10 
 
 log tan 52' 47.5" = 8.18633-10 
 
 Find log cos 88 26' 41.2". 
 90 - 88 26' 41.2" = 1 33' 18.8* 
 
 log 5598.8 = 3.74809 
 
 8 = 4.68552-10 
 
 log cos 88 26' 41.2" = 8.43361-10 
 
 Find log tan 89 54' 37.362". 
 90 - 89 54' 37.362" = 322.638*. 
 log 322.638 = 2.60871 
 
 T= 5.68558 -10 
 
 log cot 89' 54' 37 .362"= 7.19429-10 
 log tan 89 54 37.362 '' = 2.80671. 
 
 Find the angle, if log sin = 6.72306 - 10. 
 
 6.72306 - 10 
 8 = 4.68557-10 
 
 Subtract, 2.03749 = log 10^.015. 
 
 109.015" = 0> 1 49.016''. 
 
 Find the angle for which log cot = 1 .67604. 
 colog cot = 8.32396 - 10 
 T = 4.68564 - 10 
 
 Subtract, 3.63832 = log 4348.3. 
 
 4348.3" = 1 12' 28.3". 
 
 Find the angle for which log tan = 1.55407. 
 colog tan = 8.44593 10 
 T = 4.68569 - 10 
 
 Subtract, 3.76024 = log 6757.6. 
 
 5757.6" = 1 35' 57.6", 
 and 90 - 1 35' 67.6" = 88 24' 2.4" = angle required
 
 52 
 
 TABLE V, 
 
 SHOWING LENGTHS IN NAUTICAL MILES AND STATUTE MILES OF DEGREES 
 OF LATITUDE AND LONGITUDE IN DIFFERENT LATITUDES. 
 
 DEGREE OF THE PARALLEL. 
 
 DEGREE OF THE MERIDIAN. 
 
 Latitude 
 of 
 Parallel. 
 
 Nautical 
 Miles. 
 
 Statute 
 Miles. 
 
 Latitude 
 of Middle 
 Point. 
 
 Nautical 
 Miles. 
 
 Statute 
 Miles. 
 
 20 
 
 56.404 
 
 65.018 
 
 20 
 
 59.664 
 
 68.777 
 
 21 
 
 56.039 
 
 64.598 
 
 
 
 
 22 
 
 55.657 
 
 64.158 
 
 
 
 
 23 
 
 55.258 
 
 63.698 
 
 
 
 
 24 
 
 54.843 
 
 63.219 
 
 
 
 
 25 
 
 54.411 
 
 62.721 
 
 25 
 
 59.706 
 
 68.825 
 
 26 
 
 53.962 
 
 62.204 
 
 
 
 
 27 
 
 53.497 
 
 61.668 
 
 
 
 
 28 
 
 53.016 
 
 61.113 
 
 
 
 
 29 
 
 52.518 
 
 60.540 
 
 
 
 
 30 
 
 52.005 
 
 59.948 
 
 30 
 
 59.749 
 
 68.875 
 
 31 
 
 51.476 
 
 59.338 
 
 
 
 
 32 
 
 50.931 
 
 58.709 
 
 
 
 
 33 
 
 50.370 
 
 58.063 
 
 
 
 
 34 
 
 49.794 
 
 57.399 
 
 
 
 
 35 
 
 49.203 
 
 56.718 
 
 35 
 
 59.7% 
 
 68.929 
 
 36 
 
 48.597 
 
 56.019 
 
 
 
 
 37 
 
 47.976 
 
 55.304 
 
 
 
 
 38 
 
 47.341 
 
 54.571 
 
 
 
 
 39 
 
 46.960 
 
 53.822 
 
 
 
 
 40 
 
 46.026 
 
 53.056 
 
 40 
 
 59.847 
 
 68.987 
 
 41 
 
 45.348 
 
 52.274 
 
 
 
 
 42 
 
 44.654 
 
 51.476 
 
 
 
 
 43 
 
 43.949 
 
 50.662 
 
 
 
 
 44 
 
 43.230 
 
 49.833 
 
 
 
 
 45 
 
 42.497 
 
 48.988 
 
 45 
 
 59.899 
 
 69.048 
 
 46 
 
 41.752 
 
 48.128 
 
 
 
 
 47 
 
 40.993 
 
 47.254 
 
 
 
 
 48 
 
 40.222 
 
 46.365 
 
 
 
 
 490 
 
 39.439 
 
 45.462 
 
 
 
 
 50 
 
 38.643 
 
 44.545 
 
 50 
 
 59.951 
 
 69.108
 
 53 
 
 TABLE VI, 
 
 MISCELLANEOUS FORMULAE, AND EQUIVALENTS OF METRES, CHAINS, 
 AND FEET. 
 
 : 3.14159265 
 : 0.31830989 
 : 9.86960440 
 : 0.10132118 
 
 Logarithm. 
 0.4971499 
 
 9.5028501 10 
 
 0.9942997 
 
 9.0057006-10 
 
 : 1.77245385 
 0.56418958 
 
 1.46459189 
 
 Logarithm. 
 0.2485749 
 9.7514251 - 10 
 
 0.1657166 
 
 Circumference of circle, diameter being unity, 
 
 Area of circle, radius being unity 
 
 Surface of sphere, diameter being unity . . . . 
 Area of a circle, diameter being unity 
 
 Volume of sphere, diameter being unity 
 
 Volume of sphere, radius being unity . . 
 Arc whose length is equal to the radius : 
 Expressed in degrees 
 
 Expressed in minutes . 
 
 Expressed in seconds . . . . , 
 If radius is unity : 
 
 Length of arc for one degree 
 
 Length of arc for one minute 
 
 Length of arc for one second , 
 Sine of one second 
 
 180 
 
 10800 
 
 648000 
 
 IT 
 
 180 
 
 10800 
 
 648000 
 
 Base of Hyperbolic or Napier's System of Logarithms . : 
 Modulus of Common or Briggs' System of Logarithms . : 
 
 Equatorial radius of the earth in feet 
 
 Polar radius of the earth in feet ; 
 
 Length of degree of latitude at the equator, in feet . . : 
 Length of degree of latitude at 46, in feet : 
 
 : 3.14159265 
 : 0.7853982 
 : 0.52359878 
 : 4.1887902 
 
 = 57.2957795 
 = 3437.74677' 
 : 206264.806 
 
 = 0.0174533 
 = 0.0002909 
 
 = 0.00000485 
 
 = 0.00000485 
 ; 2.7182818 
 0.4342945 
 : 20923600 
 20853657 
 362748.33 
 364571.77 
 
 0.4971499 
 9.8950899 - 10 
 9.7189986-10 
 0.6220886 
 
 1.7581226 
 3.5362739 
 5.3144251 
 
 8.2418774 - 10 
 6.4637261 - 10 
 
 4.68557487 - 10 
 
 4.68557487 - 10 
 0.4342945 
 9.6377843-10 
 
 FEET. METRES. CHAINS. METRES 
 
 0.3048 
 0.6096 
 0.9144 
 1.2192 
 1.52-10 
 1.8288 
 2.1336 
 2.4384 
 2.7432 
 3.0480 
 
 0.0151 
 0.0303 
 0.0455 
 0.0606 
 0.0758 
 0.0909 
 0.1061 
 0.1212 
 0.1364 
 0.1515 
 
 FEET. 
 
 3.2809 
 
 6.5617 
 
 9.8426 
 
 13.1235 
 
 16.4044 
 
 19.6852 
 
 22.9661 
 
 26.2470 
 
 29.5278 
 
 32.8087
 
 54 
 
 TABLE VII. TRAVERSE TABLE. . 
 
 This table gives the latitude and departure to three places of deci- 
 mals for distances from 1 to 10, corresponding to bearings from 
 to 90, at intervals of 15'. 
 
 If the bearing does not exceed 45, it is found in the left-hand 
 column, and the designations of the columns under "Distance" are 
 taken from the top of the page ; but if the bearing exceeds 45, it is 
 found in the right-hand column, and the designations of the columns 
 under "Distance" are taken from the bottom of the page. 
 
 The method of using the table will be made plain by the following 
 examples : 
 
 1. Let it be required to find the latitude and departure of a line 
 running N. 35 15' E. 6 chains. 
 
 On page 60, left-hand column, look for 35 15' ; opposite this bearing, in the 
 vertical column headed "Distance 6," are found 4.900 and 3.463, under the head- 
 ings "Latitude" and "Departure" respectively. Hence latitude, or northing, 
 = 4.900 chains, and departure, or easting, = 3.463 chains. 
 
 2. Let it be required to find the latitude and departure of a Hue 
 running S. 87 W. 2 chains. 
 
 As the bearing exceeds 45, we look in the right-hand column on page 55, and 
 opposite 87, in the column marked " Distance 2," we find (taking the designa- 
 tions of the columns from the bottom of the page) latitude = 0.105 chains, and 
 departure = 1.997 chains. Hence latitude, or southing, = 0.105 chains, and depart- 
 ure, or westing, = 1.997 chains. 
 
 3. Let it be required to find the latitude and departure of a line 
 running N. 15 45' W. 27.36 chains. 
 
 In this case, we find the required number for each figure of the distance sepa- 
 rately, arranging the work as in the following table. In practice, only the last 
 columns under " Latitude " and " Departure " are written. 
 
 Distance. 
 
 Latitude. 
 
 Departure. 
 
 20 = 2 X 10 
 
 7 
 0.3 =3-- 10 
 0.06 = 6 -5- 100 
 
 1.925x10 =19.25 
 6.737 
 2.887 + 10 =0.289 
 5.775 -T- 100 = 0.058 
 
 0.543 X 10 = 5.43 
 1.90 
 0.814+ 10 = 0.081 
 1.628 -T- 100 = 0.016 
 
 27.36 
 
 26.334 
 
 7.427 
 
 Hence latitude = 26.334 chains, and departure = 7.427 chains.
 
 55 
 
 A TABLE OF THE ANGLES 
 
 Which every Point and Quarter Point of the Compass makes with the Meridian. 
 
 North. 
 
 Points. 
 
 1 
 
 o 1 II 
 2 48 45 
 5 37 30 
 8 26 15 
 11 15 
 
 Points. 
 
 V 
 1 
 
 South. 
 
 N. by E. 
 
 N. by W. 
 
 S. by E. 
 
 S. by W. 
 
 N.N.E. 
 
 N.N.W. 
 
 2 
 
 14 3 45 
 16 52 30 
 19 41 15 
 22 30 
 
 2 
 
 S.S.E. 
 
 S.S.W. 
 
 N.E. by N. 
 
 N.W. by N. 
 
 2- ! /4 
 
 P 
 
 25 18 45 
 28 7 30 
 30 56 15 
 33 45 
 
 2 ~$ 
 3 
 
 S.E. by S. 
 
 S.W. by S. 
 
 N.E. 
 
 N.W. 
 
 4 * 
 
 36 33 45 
 39 22 30 
 42 11 15 
 45 
 
 3_i 
 4 
 
 S.E. 
 
 S.W. 
 
 N.E.byE. 
 
 N.W. by W. 
 
 l:fc 
 J : < 
 
 47 48 45 
 50 37 80 
 53 26 15 
 56 15 
 
 4 8 i 
 
 5 
 
 S.E. by E. 
 
 S.W. by W. 
 
 E.N.E. 
 
 W.N.W. 
 
 5-^ 
 6 
 
 If 
 
 59 3 45 
 61 62 30 
 64 41 15 
 
 C7 30 
 70 IS 45 
 73 7 30 
 75 56 15 
 78 45 
 
 !:& 
 
 ! : < 
 
 E.S.E. 
 
 W.S.W. 
 
 E. by N. 
 
 W. by N. 
 
 6-V* 
 7 "* 
 
 E. by S. 
 
 W. by S. 
 
 East. 
 
 West. 
 
 1:8 
 
 T-% 
 
 g 
 
 81 33 46 
 84 22 30 
 87 11 15 
 90 
 
 J3 
 
 East. 
 
 West.
 
 56 
 
 TABLE VII. - TRAVERSE TABLE. 
 
 Bearing. 
 
 Distance 1. 
 
 Distance 2. 
 
 Distance 3. 
 
 Distance 4. 
 
 Distance 5. 
 
 Bearing. 
 
 o r 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 / 
 
 015 
 
 1.000 
 
 0.004 
 
 2.000 
 
 0.009 
 
 3.000 
 
 0.013 
 
 4.000 
 
 0.017 
 
 5.000 
 
 0.022 
 
 8945 
 
 30 
 
 1.000 
 
 0.009 
 
 2.000 
 
 0.017 
 
 3.000 
 
 0.026 
 
 4.000 
 
 0.035 
 
 5.000 
 
 0.044 
 
 30 
 
 45 
 
 1.000 
 
 0.013 
 
 2.000 
 
 0.026 
 
 3.000 
 
 0.039 
 
 4.0CO 
 
 0.052 
 
 5.000 
 
 0.065 
 
 15 
 
 1 
 
 1.000 
 
 0.017 
 
 2.000 
 
 0.035 
 
 3.000 
 
 0.052 
 
 3.999 
 
 0.070 
 
 4.999 
 
 0.087 
 
 89 
 
 15 
 
 1.000 
 
 0.022 
 
 2.000 
 
 0.044 
 
 2.999 
 
 0.065 
 
 3.999 
 
 O.OS7 
 
 4.999 
 
 0.109 
 
 45 
 
 30 
 
 1.000 
 
 0.026 
 
 1.999 
 
 0.052 
 
 2.999 
 
 0.079 
 
 3.999 
 
 0.105 
 
 4.998 
 
 0.131 
 
 30 
 
 45 
 
 1.000 
 
 0.031 
 
 1.999 
 
 0.061 
 
 2.999 
 
 0.092 
 
 3.998 
 
 0.122 
 
 4.998 
 
 0.153 
 
 15 
 
 2 
 
 0.999 
 
 0.035 
 
 1.999 
 
 0.070 
 
 2.998 
 
 0.105 
 
 3.998 
 
 0.140 
 
 4.997 
 
 0.174 
 
 88 
 
 15 
 
 0.999 
 
 0.039 
 
 1.998 
 
 0.079 
 
 2.998 
 
 0.118 
 
 3.997 
 
 0.157 
 
 4.996 
 
 0.196 
 
 45 
 
 30 
 
 0.999 
 
 0.044 
 
 1.998 
 
 0.087 
 
 2.997 
 
 0.131 
 
 3.996 
 
 0.174 
 
 4.995 
 
 0.218 
 
 30 
 
 45 
 
 0.999 
 
 0.048 
 
 1.998 
 
 0.096 
 
 2.997 
 
 0.144 
 
 3.995 
 
 0.192 
 
 4.994 
 
 0.240 
 
 15 
 
 3 
 
 0.999 
 
 0.052 
 
 1.997 
 
 0.105 
 
 2.996 
 
 0.157 
 
 3.995 
 
 0.209 
 
 4.993 
 
 0.262 
 
 87 
 
 15 
 
 0.998 
 
 0.057 
 
 1.997 
 
 0.113 
 
 2.995 
 
 0.170 
 
 3.994 
 
 0.227 
 
 4.992 
 
 0.2S3 
 
 45 
 
 30 
 
 0.998 
 
 0.061 
 
 1.996 
 
 0.122 
 
 2.994 
 
 0.183 
 
 3.993 
 
 0.244 
 
 4.991 
 
 0.305 
 
 30 
 
 45 
 
 0.998 
 
 0.065 
 
 1.996 
 
 0.131 
 
 2.994 
 
 0.196 
 
 3.991 
 
 0.262 
 
 4.989 
 
 0.327 
 
 15 
 
 4 
 
 0.998 
 
 0.070 
 
 1.995 
 
 0.140 
 
 2.993 
 
 0.209 
 
 3.990 
 
 0.279 
 
 4.988 
 
 0.349 
 
 86 
 
 15 
 
 0.997 
 
 0.074 
 
 1.995 
 
 0.148 
 
 2.992 
 
 0.222 
 
 3. 989 
 
 0.296 
 
 4.986 
 
 0.371 
 
 45 
 
 30 
 
 0.997 
 
 0.078 
 
 1.994 
 
 0.157 
 
 2.991 
 
 0.235 
 
 3.988 
 
 0.314 
 
 4.985 
 
 0.392 
 
 30 
 
 45 
 
 0.997 
 
 0.083 
 
 1.993 
 
 0.166 
 
 2.990 
 
 0.248 
 
 3.986 
 
 0.331 
 
 4.983 
 
 0.414 
 
 15 
 
 5 
 
 0.996 
 
 0.087 
 
 1.992 
 
 0.174 
 
 2.989 
 
 0.261 
 
 3.985 
 
 0.349 
 
 4.9S1 
 
 0.436 
 
 85 
 
 15 
 
 0.996 
 
 0.092 
 
 1.992 
 
 0,183 
 
 2.987 
 
 0.275 
 
 3.983 
 
 0.366 
 
 4.979 
 
 0.458 
 
 45 
 
 30 
 
 0.995 
 
 0.096 
 
 1.991 
 
 0.192 
 
 2.986 
 
 0.2SS 
 
 3.982 
 
 0.3S3 
 
 4.977 
 
 0.479 
 
 30 
 
 45 
 
 0.995 
 
 0.100 
 
 1.990 
 
 0.200 
 
 2.9S5 
 
 0.301 
 
 3.980 
 
 0.401 
 
 4.975 
 
 0.501 
 
 15 
 
 6 
 
 0.995 
 
 0.105 
 
 1.989 
 
 0.209 
 
 2.984 
 
 0.314 
 
 3.978 
 
 0.418 
 
 4.973 
 
 0.523 
 
 84 
 
 15 
 
 0.994 
 
 0.109 
 
 1.988 
 
 0.218 
 
 2.9S2 
 
 0.327 
 
 3.976 
 
 0.435 
 
 4.970 
 
 0.544 
 
 45 
 
 30 
 
 0.994 
 
 0.113 
 
 1.9S7 
 
 0.226 
 
 2.981 
 
 0.340 
 
 3.974 
 
 0.453 
 
 4.968 
 
 0566 
 
 30 
 
 45 
 
 0.993 
 
 0.118 
 
 1.986 
 
 0.235 
 
 2.979 
 
 0.353 
 
 3.972 
 
 0.470 
 
 4.965 
 
 0.588 
 
 15 
 
 7 
 
 0.993 
 
 0.122 
 
 1.985 
 
 0.244 
 
 2.978 
 
 0.366 
 
 3.970 
 
 0.4S7 
 
 4.963 
 
 0.609 
 
 83 
 
 15 
 
 0.992 
 
 0.126 
 
 1.984 
 
 0.252 
 
 2.976 
 
 0.379 
 
 3.96S 
 
 0.505 
 
 4.960 
 
 0.631 
 
 45 
 
 30 
 
 0.991 
 
 0.131 
 
 1.983 
 
 0.261 
 
 2.974 
 
 0.392 
 
 3.966 
 
 0.522 
 
 4.957 
 
 0.653 
 
 30 
 
 45 
 
 0.991 
 
 0.135 
 
 1.982 
 
 0.270 
 
 2.973 
 
 0.405 
 
 3.963 
 
 0.539 
 
 4.954 
 
 0.674 
 
 15 
 
 8 
 
 0.990 
 
 0.139 
 
 1.981 
 
 0.278 
 
 2.971 
 
 0.418 
 
 3.961 
 
 0.557 
 
 4.951 
 
 0.696 
 
 82 
 
 15 
 
 0.990 
 
 0.143 
 
 1.979 
 
 0.287 
 
 2.969 
 
 0.430 
 
 3.959 
 
 0.574 
 
 4.948 
 
 0.717 
 
 45 
 
 30 
 
 0.989 
 
 0.148 
 
 1.978 
 
 0.296 
 
 2.967 
 
 0.443 
 
 3.956 
 
 0.591 
 
 4.945 
 
 0.739 
 
 30 
 
 45 
 
 0.988 
 
 0.152 
 
 1.977 
 
 0.304 
 
 2.965 
 
 0.456 
 
 3.953 
 
 0.608 
 
 4.942 
 
 0.761 
 
 15 
 
 9 
 
 0.988 
 
 0.156 
 
 1.975 
 
 0.313 
 
 2.963 
 
 0.469 
 
 3.951 
 
 0.626 
 
 4.938 
 
 0.782 
 
 81 
 
 15 
 
 0.987 
 
 0.161 
 
 1.974 
 
 0.321 
 
 2.961 
 
 0.4S2 
 
 3.948 
 
 0.643 
 
 4.935 
 
 0.804 
 
 45 
 
 30 
 
 0.986 
 
 0.165 
 
 1.973 
 
 0.330 
 
 2.959 
 
 0.495 
 
 3.945 
 
 0.660 
 
 4.931 
 
 0.825 
 
 30 
 
 45 
 
 0.986 
 
 0.169 
 
 1.971 
 
 0.339 
 
 2.957 
 
 0.508 
 
 3.942 
 
 0.677 
 
 4.928 
 
 0.847 
 
 15 
 
 1O 
 
 0.985 
 
 0.174 
 
 1.970 
 
 0.347 
 
 2.954 
 
 0.521 
 
 3.939 
 
 0.695 
 
 4.924 
 
 0.868 
 
 80 
 
 15 
 
 0.984 
 
 0.178 
 
 1.968 
 
 0.356 
 
 2.952 
 
 0.534 
 
 3.936 
 
 0.712 
 
 4.920 
 
 0.890 
 
 45 
 
 30 
 
 0.983 
 
 0.182 
 
 1.967 
 
 0.364 
 
 2.950 
 
 0.547 
 
 3.933 
 
 0.729 
 
 4.916 
 
 0.911 
 
 30 
 
 45 
 
 0.982 
 
 0.187 
 
 1.965 
 
 0.373 
 
 2.947 
 
 0.560 
 
 3.930 
 
 0.746 
 
 4.912 
 
 0.933 
 
 15 
 
 11 
 
 0.982 
 
 0.191 
 
 1.963 
 
 0.382 
 
 2.945 
 
 0.572 
 
 3.927 
 
 0.763 
 
 4.908 
 
 0.954 
 
 79 
 
 15 
 
 0.981 
 
 0.195 
 
 1.962 
 
 0.390 
 
 2.942 
 
 0.585 
 
 3.923 
 
 0.780 
 
 4.904 
 
 0.975 
 
 45 
 
 30 
 
 0.980 
 
 0.199 
 
 1.960 
 
 0.399 
 
 2.940 
 
 0.598 
 
 3.920 
 
 0.797 
 
 4.900 
 
 0.997 
 
 30 
 
 45 
 
 0.979 
 
 0.204 
 
 1.958 
 
 0.407 
 
 2.937 
 
 0.611 
 
 3.916 
 
 0.815 
 
 4.895 
 
 1.018 
 
 15 
 
 12 
 
 0.978 
 
 0.208 
 
 1.956 
 
 0.416 
 
 2.934 
 
 0.624 
 
 3.913 
 
 0.832 
 
 4.891 
 
 1.040 
 
 78 
 
 15 
 
 0.977 
 
 0.212 
 
 1.954 
 
 0.424 
 
 2.932 
 
 0.637 
 
 3.909 
 
 0.849 
 
 4.886 
 
 1.061 
 
 45 
 
 30 
 
 0.976 
 
 0.216 
 
 1.953 
 
 0.433 
 
 2.929 
 
 0.649 
 
 3.905 
 
 0.866 
 
 4.881 
 
 1.082 
 
 30 
 
 45 
 
 0.975 
 
 0.221 
 
 1.951 
 
 0.441 
 
 2.926 
 
 0.662 
 
 3.901 
 
 O.S83 
 
 4.877 
 
 1.103 
 
 15 
 
 13 
 
 0.974 
 
 0.225 
 
 1.949 
 
 0.450 
 
 2.923 
 
 0.675 
 
 3.897 
 
 0.900 
 
 4.872 
 
 1.125 
 
 77 
 
 15 
 
 0.973 
 
 0.229 
 
 1.947 
 
 0.458 
 
 2.920 
 
 0.6SS 
 
 3.894 
 
 0.917 
 
 4.867 
 
 1.146 
 
 45 
 
 30 
 
 0.972 
 
 0.233 
 
 1.945 
 
 0.467 
 
 2.917 
 
 0.700 
 
 3.SS9 
 
 0.934 
 
 4.S62 
 
 1.167 
 
 30 
 
 45 
 
 0.971 
 
 0.238 
 
 1.943 
 
 0.475 
 
 2.914 
 
 0.713 
 
 3.SS5 
 
 0.951 
 
 4.857 
 
 1.188 
 
 15 
 
 14 
 
 0.970 
 
 0.242 
 
 1.941 
 
 0.484 
 
 2.911 
 
 0.726 
 
 3.SS1 
 
 0.968 
 
 4.851 
 
 1.210 
 
 76 
 
 15 
 
 0.969 
 
 0.246 
 
 1.938 
 
 0.492 
 
 2.908 
 
 0.73S 
 
 3.877 
 
 0.985 
 
 4.846 
 
 1.231 
 
 45 
 
 30 
 
 0.968 
 
 0.250 
 
 1.936 
 
 0.501 
 
 2.904 
 
 0.751 
 
 3.873 
 
 1.002 
 
 4.841 
 
 1.252 
 
 30 
 
 45 
 
 0.967 
 
 0.255 
 
 1.934 
 
 0.509 
 
 2.901 
 
 0.764 
 
 3.868 
 
 1.018 
 
 4.835 
 
 1.273 
 
 15 
 
 15 
 
 0.966 
 
 0.259 
 
 1.932 
 
 0.518 
 
 2.898 
 
 0.776 
 
 3.864 
 
 1.035 
 
 4.830 
 
 1.294 
 
 75 
 
 f 
 
 Dep. 
 
 Lat, 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 C f 
 
 Bearing 
 
 Distance 1. 
 
 Distance 2. 
 
 Distance 3. 
 
 Distance 4. 
 
 Distance 5. 
 
 Bearing. 
 
 75- 90'
 
 67 
 
 Bearing, 
 
 Distance 6. 
 
 Distance 7. 
 
 Distance 8. 
 
 Distance 9. 
 
 Distance 1O. 
 
 Bearing. 
 
 o t 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep, 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 O f 
 
 015 
 
 6.000 
 
 0.026 
 
 7.000 
 
 0.031 
 
 8.000 
 
 0.035 
 
 9.000 
 
 0.039 
 
 10.000 
 
 0.044 
 
 89 45 
 
 30 
 
 6.000 
 
 0.052 
 
 7.000 
 
 0.061 
 
 8.000 
 
 0.070 
 
 9.000 
 
 0.079 
 
 10.000 
 
 0.087 
 
 30 
 
 45 
 
 5.999 
 
 0.079 
 
 6.999 
 
 0.092 
 
 7.999 
 
 0.105 
 
 8.999 
 
 0.118 
 
 9.999 
 
 0.131 
 
 15 
 
 1 
 
 5.999 
 
 0.105 
 
 6.999 
 
 0.122 
 
 7.999 
 
 0.140 
 
 8.999 
 
 0.157 
 
 9.999 
 
 0.175 
 
 89 
 
 15 
 
 5.999 
 
 0.131 
 
 6.998 
 
 0.153 
 
 7.998 
 
 0.175 
 
 8.998 
 
 0.196 
 
 9.998 
 
 0.218 
 
 45 
 
 30 
 
 5.998 
 
 0.157 
 
 6.998 
 
 0.183 
 
 7.997 
 
 0.209 
 
 8.997 
 
 0.236 
 
 9.997 
 
 0.262 
 
 30 
 
 45 
 
 5.997 
 
 0.183 
 
 6.997 
 
 0.214 
 
 7.996 
 
 0.244 
 
 8.996 
 
 0.275 
 
 9.995 
 
 0.305 
 
 15 
 
 2 
 
 5.996 
 
 0.209 
 
 6.996 
 
 0.244 
 
 7.995 
 
 0.279 
 
 8.995 
 
 0.314 
 
 9.994 
 
 0.349 
 
 88 
 
 15 
 
 5.995 
 
 0.236 
 
 6.995 
 
 0.275 
 
 7.994 
 
 0.314 
 
 8.993 
 
 0.353 
 
 9.992 
 
 0.393 
 
 45 
 
 30 
 
 5.994 
 
 0.262 
 
 6.993 
 
 0.305 
 
 7.992 
 
 0.349 
 
 8.991 
 
 0.393 
 
 9.991 
 
 0.436 
 
 30 
 
 45 
 
 5.993 
 
 0.288 
 
 6.992 
 
 0.336 
 
 7.991 
 
 0.384 
 
 8.990 
 
 0.432 
 
 9.989 
 
 0.480 
 
 15 
 
 3 
 
 5.992 
 
 0.314 
 
 6.990 
 
 0.366 
 
 7.989 
 
 0.419 
 
 8.988 
 
 0.471 
 
 9.986 
 
 0.523 
 
 87 
 
 15 
 
 5.990 
 
 0.340 
 
 6.989 
 
 0.397 
 
 7.987 
 
 0.454 
 
 8.986 
 
 0.510 
 
 9.984 
 
 0.567 
 
 45 
 
 30 
 
 5.989 
 
 0.366 
 
 6.987 
 
 0.427 
 
 7.985 
 
 0.488 
 
 8.983 
 
 0-549 
 
 9.981 
 
 0.611 
 
 30 
 
 45 
 
 5.987 
 
 0.392 
 
 6.985 
 
 0.458 
 
 7.983 
 
 0.523 
 
 8.981 
 
 0.589 
 
 9.979 
 
 0.654 
 
 15 
 
 4 
 
 5.985 
 
 0.419 
 
 6.983 
 
 0.488 
 
 7.981 
 
 0.558 
 
 8.978 
 
 0.628 
 
 9.976 
 
 0.698 
 
 86 
 
 15 
 
 5.984 
 
 0.445 
 
 6.981 
 
 0.519 
 
 7.978 
 
 0.593 
 
 8.975 
 
 0.667 
 
 9.973 
 
 0.741 
 
 45 
 
 30 
 
 5.982 
 
 0.471 
 
 6.978 
 
 0.549 
 
 7.975 
 
 0.628 
 
 8.972 
 
 0.706 
 
 9.969 
 
 0.785 
 
 30 
 
 45 
 
 5.979 
 
 0.497 
 
 6.976 
 
 0.580 
 
 7.973 
 
 0.662 
 
 S.%9 
 
 0.745 
 
 9.966 
 
 0.828 
 
 15 
 
 o 
 
 5.977 
 
 0.523 
 
 6.973 
 
 0.610 
 
 7.970 
 
 0.697 
 
 8.966 
 
 0.784 
 
 9.962 
 
 0.872 
 
 85 
 
 15 
 
 5.975 
 
 0.549 
 
 6.971 
 
 0.641 
 
 7.966 
 
 0.732 
 
 8.962 
 
 0.824 
 
 9.958 
 
 0.915 
 
 45 
 
 30 
 
 5.972 
 
 0.575 
 
 6.968 
 
 0.671 
 
 7.963 
 
 0.767 
 
 8.959 
 
 0.863 
 
 9.954 
 
 0.959 
 
 30 
 
 45 
 
 5.970 
 
 0.601 
 
 6.965 
 
 0.701 
 
 7.960 
 
 0.802 
 
 8.955 
 
 0.902 
 
 9.950 
 
 1.002 
 
 15 
 
 6 
 
 5.967 
 
 0.627 
 
 6.962 
 
 0.732 
 
 7.956 
 
 0.836 
 
 8.951 
 
 0.941 
 
 9.945 
 
 1.045 
 
 84 
 
 15 
 
 5.964 
 
 0.653 
 
 6.958 
 
 0.762 
 
 7.952 
 
 0.871 
 
 8.947 
 
 0.980 
 
 9.941 
 
 1.089 
 
 45 
 
 30 
 
 5.961 
 
 0.679 
 
 6.955 
 
 0.792 
 
 7.949 
 
 0.906 
 
 8.942 
 
 1.019 
 
 9.936 
 
 1.132 
 
 30 
 
 45 
 
 5.958 
 
 0.705 
 
 6.951 
 
 0.823 
 
 7.945 
 
 0.940 
 
 8.938 
 
 1.058 
 
 9.931 
 
 1.175 
 
 15 
 
 7 
 
 5.955 
 
 0.731 
 
 6.948 
 
 0.853 
 
 7.940 
 
 0.975 
 
 8.933 
 
 1.097 
 
 9.926 
 
 1.219 
 
 83 
 
 15 
 
 5.952 
 
 0.757 
 
 6.944 
 
 0.883 
 
 7.936 
 
 1.010 
 
 8.928 
 
 1.136 
 
 9.920 
 
 1.262 
 
 45 
 
 30 
 
 5.949 
 
 0.783 
 
 6.940 
 
 0.914 
 
 7.932 
 
 1.044 
 
 8.923 
 
 1.175 
 
 9.914 
 
 1.305 
 
 30 
 
 45 
 
 5.945 
 
 0.809 
 
 6.936 
 
 0.944 
 
 7.927 
 
 1.079 
 
 8.918 
 
 1.214 
 
 9.909 
 
 1.349 
 
 15 
 
 8 
 
 5.942 
 
 0.835 
 
 6.932 
 
 0.974 
 
 7.922 
 
 1.113 
 
 8.912 
 
 1.253 
 
 9.903 
 
 1.392 
 
 82 
 
 15 
 
 5.938 
 
 0.861 
 
 6.928 
 
 1.004 
 
 7.917 
 
 1.148 
 
 8.907 
 
 1.291 
 
 9.897 
 
 1.435 
 
 45 
 
 30 
 
 5.934 
 
 0.887 
 
 6.923 
 
 1.035 
 
 7.912 
 
 1.182 
 
 8.901 
 
 1.330 
 
 9.890 
 
 1.478 
 
 30 
 
 45 
 
 5.930 
 
 0.913 
 
 6.919 
 
 1.065 
 
 7.907 
 
 1.217 
 
 8.895 
 
 1.369 
 
 9.884 
 
 1.521 
 
 15 
 
 9 
 
 5.926 
 
 0.939 
 
 6.914 
 
 1.095 
 
 7.902 
 
 1.251 
 
 8.889 
 
 1.408 
 
 9.877 
 
 1.564 
 
 81 
 
 15 
 
 5.922 
 
 0.964 
 
 6.909 
 
 1.125 
 
 7.896 
 
 1.286 
 
 8.883 
 
 1.447 
 
 9.870 
 
 1.607 
 
 45 
 
 30 
 
 5.918 
 
 0.990 
 
 6.904 
 
 1.155 
 
 7.890 
 
 1.320 
 
 8.877 
 
 1.485 
 
 9.863 
 
 1.651 
 
 30 
 
 45 
 
 5.913 
 
 1.016 
 
 6.899 
 
 1.185 
 
 7.884 
 
 1.355 
 
 8.870 
 
 1.524 
 
 9.856 
 
 1.694 
 
 16 
 
 10 
 
 5.909 
 
 1.042 
 
 6.894 
 
 1.216 
 
 7.878 
 
 1.389 
 
 8.863 
 
 1.563 
 
 9.848 
 
 1.737 
 
 80 
 
 15 
 
 5.904 
 
 1.068 
 
 6.888 
 
 1.246 
 
 7.872 
 
 1.424 
 
 8.856 
 
 1.601 
 
 9.840 
 
 1.779 
 
 45 
 
 30 
 
 5.900 
 
 1.093 
 
 6.883 
 
 1.276 
 
 7.866 
 
 1.458 
 
 8.849 
 
 1.640 
 
 9.833 
 
 1.822 
 
 30 
 
 45 
 
 5.895 
 
 1.119 
 
 6.877 
 
 1.306 
 
 7.860 
 
 1.492 
 
 8.842 
 
 1.679 
 
 9.825 
 
 1.865 
 
 15 
 
 11 
 
 5.890 
 
 1.145 
 
 6.871 
 
 1.336 
 
 7.853 
 
 1.526 
 
 8.835 
 
 1.717 
 
 9.816 
 
 1.908 
 
 79 
 
 15 
 
 5.885 
 
 1.171 
 
 6.866 
 
 1.366 
 
 7.846 
 
 1.561 
 
 8.827 
 
 1.756 
 
 9.808 
 
 1.951 
 
 45 
 
 30 
 
 5.880 
 
 1.196 
 
 6.859 
 
 1.396 
 
 7.839 
 
 1.595 
 
 8.S19 
 
 1.794 
 
 9.799 
 
 1.994 
 
 30 
 
 45 
 
 5.874 
 
 1.222 
 
 6.853 
 
 1.425 
 
 7.832 
 
 1.629 
 
 8.811 
 
 1.833 
 
 9.791 
 
 2.036 
 
 15 
 
 12 
 
 5.869 
 
 1.247 
 
 6.847 
 
 1.455 
 
 7.825 
 
 1.663 
 
 8.803 
 
 1.871 
 
 9.782 
 
 2.079 
 
 78 
 
 15 
 
 5.863 
 
 1.273 
 
 6.841 
 
 1.485 
 
 7.818 
 
 1.697 
 
 8.795 
 
 1.910 
 
 9.772 
 
 2.122 
 
 45 
 
 30 
 
 5.858 
 
 1.299 
 
 6.834 
 
 1.515 
 
 7.810 
 
 1.732 
 
 8.787 
 
 1.948 
 
 9.763 
 
 2.164 
 
 30 
 
 45 
 
 5.852 
 
 1.324 
 
 6.827 
 
 1.545 
 
 7.803 
 
 1.766 
 
 8.778 
 
 1.986 
 
 9.753 
 
 2.207 
 
 15 
 
 13 
 
 5.846 
 
 1.350 
 
 6.821 
 
 1.575 
 
 7.795 
 
 1.800 
 
 8.769 
 
 2.025 
 
 9.744 
 
 2.250 
 
 77 
 
 15 
 
 5.840 
 
 1.375 
 
 6.814 
 
 1.604 
 
 7.787 
 
 1.834 
 
 8.760 
 
 2.063 
 
 9.734 
 
 2.292 
 
 45 
 
 30 
 
 5.834 
 
 1.401 
 
 6.807 
 
 1.634 
 
 7.779 
 
 1.868 
 
 8.751 
 
 2.101 
 
 9.724 
 
 2.335 
 
 30 
 
 45 
 
 5.828 
 
 1.426 
 
 6.799 
 
 1.664 
 
 7.771 
 
 1.902 
 
 8.742 
 
 2.139 
 
 9.713 
 
 2.377 
 
 15 
 
 14 
 
 5.822 
 
 1.452 
 
 6.792 
 
 1.693 
 
 7.762 
 
 1.935 
 
 8.733 
 
 2.177 
 
 9.703 
 
 2.419 
 
 70 
 
 15 
 
 5.815 
 
 1.477 
 
 6.785 
 
 1.723 
 
 7.754 
 
 1.969 
 
 8.723 
 
 2.215 
 
 9.692 
 
 2.462 
 
 45 
 
 30 
 
 5.809 
 
 1.502 
 
 6.777 
 
 1.753 
 
 7.745 
 
 2.003 
 
 8.713 
 
 2.253 
 
 9.682 
 
 2.504 
 
 30 
 
 45 
 
 5.802 
 
 1.528 
 
 6.769 
 
 1.782 
 
 7.736 
 
 2.037 
 
 8.703 
 
 2.291 
 
 9.671 
 
 2.546 
 
 15 
 
 15 
 
 5.796 
 
 1.553 
 
 6.761 
 
 1.812 
 
 7.727 
 
 2.071 
 
 8.693 
 
 2.329 
 
 9.659 
 
 2.588 
 
 7.-, 
 
 o t 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 o t 
 
 Bearing. 
 
 Distance 6. 
 
 Distance 7. 
 
 Distance 8. 
 
 Distance 9. 
 
 Distance 1O. 
 
 Bearing. 
 
 75- 90
 
 58 
 
 15- 30 C 
 
 Bearing, 
 
 Distance 1. 
 
 Distance 2. 
 
 Distance 3. 
 
 Distance 4. 
 
 Distance 5. 
 
 Searing. 
 
 o t 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 1 Dep. 
 
 f 
 
 1515 
 
 0.965 
 
 0.263 
 
 1.930 
 
 0.526 
 
 2.894 
 
 0.789 
 
 3.859 
 
 1.052 
 
 4.824 
 
 1.315 
 
 7445 
 
 30 
 
 0.964 
 
 0,267 
 
 1.927 
 
 0.534 
 
 2.891 
 
 0.802 
 
 3.855 
 
 1.069 
 
 4.818 
 
 1.336 
 
 30 
 
 45 
 
 0.962 
 
 0.271 
 
 1.925 
 
 0.543 
 
 2.887 
 
 0.814 
 
 3.850 
 
 1.086 
 
 4.812 
 
 1.357 
 
 15 
 
 16 
 
 0.961 
 
 0.276 
 
 1.923 
 
 0.551 
 
 2.884 
 
 0.827 
 
 3.845 
 
 1.103 
 
 4.806 
 
 1.378 
 
 74 
 
 15 
 
 0.960 
 
 0.280 
 
 1.920 
 
 0.560 
 
 2.880 
 
 O.S39 
 
 3.840 
 
 1.119 
 
 4.800 
 
 1.399 
 
 45 
 
 30 
 
 0.959 
 
 0.284 
 
 1.918 
 
 0.568 
 
 2.876 
 
 0.852 
 
 3.835 
 
 1.136 
 
 4.794 
 
 1.420 
 
 30 
 
 45 
 
 0.958 
 
 0.2SS 
 
 1.915 
 
 0.576 
 
 2.873 
 
 0.865 
 
 3.830 
 
 1.153 
 
 4.788 
 
 1.441 
 
 15 
 
 17 
 
 0.956 
 
 0.292 
 
 1.913 
 
 0.585 
 
 2.869 
 
 0.877 
 
 3.825 
 
 1.169 
 
 4.7S2 
 
 1.462 
 
 73 
 
 15 
 
 0.955 
 
 0.297 
 
 1.910 
 
 0.593 
 
 2.865 
 
 O.S90 
 
 3.820 
 
 1.186 
 
 4.775 
 
 1.483 
 
 45 
 
 30 
 
 0.954 
 
 0.301 
 
 1.907 
 
 0.601 
 
 2.861 
 
 0.902 
 
 3.815 
 
 1.203 
 
 4.769 
 
 1.504 
 
 30 
 
 45 
 
 0.952 
 
 0.305 
 
 1.905 
 
 0.610 
 
 2.857 
 
 0.915 
 
 3.810 
 
 1.220 
 
 4.762 
 
 1.524 
 
 15 
 
 18 
 
 0.951 
 
 0.309 
 
 1.902 
 
 0.618 
 
 2.853 
 
 0.927 
 
 3.804 
 
 1.236 
 
 4.755 
 
 1.545 
 
 72 
 
 15 
 
 0.950 
 
 0.313 
 
 1.899 
 
 0.626 
 
 2.849 
 
 0.939 
 
 3.799 
 
 1.253 
 
 4.748 
 
 1.566 
 
 45 
 
 30 
 
 0.948 
 
 0.317 
 
 1.897 
 
 0.635 
 
 2.845 
 
 0.952 
 
 3.793 
 
 1.269 
 
 4.742 
 
 1.587 
 
 30 
 
 45 
 
 0.947 
 
 0.321 
 
 1.894 
 
 0.643 
 
 2.841 
 
 0.964 
 
 3.788 
 
 1.286 
 
 4.735 
 
 r607 
 
 15 
 
 19 
 
 0.946 
 
 0.326 
 
 1.891 
 
 0.651 
 
 2.837 
 
 0.977 
 
 3.782 
 
 1.302 
 
 4.728 
 
 1.628 
 
 71 
 
 15 
 
 0.944 
 
 0.330 
 
 1.888 
 
 0.659 
 
 2.832 
 
 0.989 
 
 3.776 
 
 1.319 
 
 4.720 
 
 1.648 
 
 45 
 
 30 
 
 0.943 
 
 0.334 
 
 1.885 
 
 0.668 
 
 2.828 
 
 1.001 
 
 3.771 
 
 1.335 
 
 4.713 
 
 1.669 
 
 30 
 
 45 
 
 0.941 
 
 0.338 
 
 1.882 
 
 0.676 
 
 2.824 
 
 1.014 
 
 3.765 
 
 1.352 
 
 4.706 
 
 1.690 
 
 15 
 
 2O 
 
 0.940 
 
 0.342 
 
 1.879 
 
 0.684 
 
 2.819 
 
 1.026 
 
 3.759 
 
 1.368 
 
 4.698 
 
 1.710 
 
 70 
 
 15 
 
 0.938 
 
 0.346 
 
 1.876 
 
 0.692 
 
 2.815 
 
 1.038 
 
 3.753 
 
 1.384 
 
 4.691 
 
 1.731 
 
 45 
 
 30 
 
 0.937 
 
 0.350 
 
 1.873 
 
 0.700 
 
 2.810 
 
 1.051 
 
 3.747 
 
 1.401 
 
 4.683 
 
 1.751 
 
 30 
 
 45 
 
 0.935 
 
 0.354 
 
 1.870 
 
 0.709 
 
 2.805 
 
 1.063 
 
 3.741 
 
 1.417 
 
 4.676 
 
 1.771 
 
 15 
 
 21 
 
 0.934 
 
 0.358 
 
 1.867 
 
 0.717 
 
 2.801 
 
 1.075 
 
 3.734 
 
 1.433 
 
 4.668 
 
 1.792 
 
 69 
 
 15 
 
 0.932 
 
 0.362 
 
 1.864 
 
 0.725 
 
 2.796 
 
 1.087 
 
 3.728 
 
 1.450 
 
 4.660 
 
 1.812 
 
 45 
 
 30 
 
 0.930 
 
 0.367 
 
 1.861 
 
 0.733 
 
 2.791 
 
 1.100 
 
 3.722 
 
 1.466 
 
 4.652 
 
 1.833 
 
 30 
 
 45 
 
 0.929 
 
 0.371 
 
 1.858 
 
 0.741 
 
 2.786 
 
 1.112 
 
 3.715 
 
 1.482 
 
 4.644 
 
 1.853 
 
 15 
 
 22 
 
 0.927 
 
 0.375 
 
 1.854 
 
 0.749 
 
 2.782 
 
 1.124 
 
 3.709 
 
 1.498 
 
 4.636 
 
 1.873 
 
 68 
 
 15 
 
 0.926 
 
 0.379 
 
 1.851 
 
 0.757 
 
 2.777 
 
 1.136 
 
 3.702 
 
 1.515 
 
 4.628 
 
 1.893 
 
 45 
 
 30 
 
 0.924 
 
 0.383 
 
 1.848 
 
 0.765 
 
 2.772 
 
 1.148 
 
 3.696 
 
 1.531 
 
 4.619 
 
 1.913 
 
 30 
 
 45 
 
 0.922 
 
 0.387 
 
 1.844 
 
 0.773 
 
 2.767 
 
 1.160 
 
 3.689 
 
 1.547 
 
 4.611 
 
 1.934 
 
 15 
 
 23 
 
 0.921 
 
 0.391 
 
 1.841 
 
 0.781 
 
 2.762 
 
 1.172 
 
 3.682 
 
 1.563 
 
 4.603 
 
 1.954 
 
 67 
 
 15 
 
 0.919 
 
 0.395 
 
 1.838 
 
 0.789 
 
 2.756 
 
 1.184 
 
 3.675 
 
 1.579 
 
 4.594 
 
 1.974 
 
 45 
 
 30 
 
 0.917 
 
 0.399 
 
 1.834 
 
 0.797 
 
 2.751 
 
 1.196 
 
 3.668 
 
 1.595 
 
 4.585 
 
 1.994 
 
 30 
 
 45 
 
 0.915 
 
 0.403 
 
 1.831 
 
 0.805 
 
 2.746 
 
 1.208 
 
 3.661 
 
 1.611 
 
 4.577 
 
 2.014 
 
 15 
 
 24 
 
 0.914 
 
 0.407 
 
 1.827 
 
 0.813 
 
 2.741 
 
 1.220 
 
 3.654 
 
 1.627 
 
 4.568 
 
 2.034 
 
 66 
 
 15 
 
 0.912 
 
 0.411 
 
 1.824 
 
 0.821 
 
 2.735 
 
 1.232 
 
 3.647 
 
 1.643 
 
 4.559 
 
 2.054 
 
 45 
 
 30 
 
 0.910 
 
 0.415 
 
 1.820 
 
 0.829 
 
 2.730 
 
 1.244 
 
 3.640 
 
 1.659 
 
 4.550 
 
 2.073 
 
 30 
 
 45 
 
 0.908 
 
 0.419 
 
 1.816 
 
 0.837 
 
 2.724 
 
 1.256 
 
 3.633 
 
 1.675 
 
 4.541 
 
 2.093 
 
 15 
 
 25 
 
 0.906 
 
 0.423 
 
 1.813 
 
 0.845 
 
 2.719 
 
 1.268 
 
 3.625 
 
 1.690 
 
 4.532 
 
 2.113 
 
 65 
 
 15 
 
 0.904 
 
 0.427 
 
 1.809 
 
 0.853 
 
 2.713 
 
 1.280 
 
 3.618 
 
 1.706 
 
 4.522 
 
 2.133 
 
 45 
 
 30 
 
 0.903 
 
 0.431 
 
 1.805 
 
 0.861 
 
 2.708 
 
 1.292 
 
 3.610 
 
 1.722 
 
 4.513 
 
 2.153 
 
 30 
 
 45 
 
 0.901 
 
 0.434 
 
 1.801 
 
 0.869 
 
 2.702 
 
 1.303 
 
 3.603 
 
 1.738 
 
 4.503 
 
 2.172 
 
 15 
 
 26 
 
 0.899 
 
 0.438 
 
 1.798 
 
 0.877 
 
 2.696 
 
 1.315 
 
 3.595 
 
 1.753 
 
 4.494 
 
 2.192 
 
 64 
 
 15 
 
 0.897 
 
 0.442 
 
 1.794 
 
 0.885 
 
 2.691 
 
 1.327 
 
 3.587 
 
 1.769 
 
 4.484 
 
 2.211 
 
 45 
 
 30 
 
 0.895 
 
 0.446 
 
 1.790 
 
 0.892 
 
 2.685 
 
 1.339 
 
 3.580 
 
 1.785 
 
 4.475 
 
 2.231 
 
 30 
 
 45 
 
 0.893 
 
 0.450 
 
 1.786 
 
 0.900 
 
 2.679 
 
 1.350 
 
 3.572 
 
 1.800 
 
 4.465 
 
 2.250 
 
 15 
 
 27 
 
 0.891 
 
 0.454 
 
 1.782 
 
 0.908 
 
 2.673 
 
 1.362 
 
 3.564 
 
 1.816 
 
 4.455 
 
 2.270 
 
 63 
 
 15 
 
 0.889 
 
 0.458 
 
 1.778 
 
 0.916 
 
 2.667 
 
 1.374 
 
 3.556 
 
 1.831 
 
 4.445 
 
 2.289 
 
 45 
 
 30 
 
 0.887 
 
 0.462 
 
 1.774 
 
 0.923 
 
 2.661 
 
 1.385 
 
 3.548 
 
 1.847 
 
 4.435 
 
 2.309 
 
 30 
 
 45 
 
 0.885 
 
 0.466 
 
 1.770 
 
 0.931 
 
 2.655 
 
 1.397 
 
 3.540 
 
 1.862 
 
 4.425 
 
 2.328 
 
 15 
 
 28 
 
 0.883 
 
 0.469 
 
 1.766 
 
 0.939 
 
 2.649 
 
 1.408 
 
 3.532 
 
 1.878 
 
 4.415 
 
 2.347 
 
 62 
 
 15 
 
 0.881 
 
 0.473 
 
 1.762 
 
 0.947 
 
 2.643 
 
 1.420 
 
 3.524 
 
 1.893 
 
 4.404 
 
 2.367 
 
 45 
 
 30 
 
 0.879 
 
 0.477 
 
 1.758 
 
 0.954 
 
 2.636 
 
 1.431 
 
 3.515 
 
 1.909 
 
 4.394 
 
 2.386 
 
 30 
 
 45 
 
 0.877 
 
 0.481 
 
 1.753 
 
 0.962 
 
 2.630 
 
 1.443 
 
 3.507 
 
 1.924 
 
 4.384 
 
 2.405 
 
 15 
 
 29 
 
 0.875 
 
 0.485 
 
 1.749 
 
 0.970 
 
 2.624 
 
 1.454 
 
 3.498 
 
 1.939 
 
 4.373 
 
 2.424 
 
 61 
 
 15 
 
 0.872 
 
 0.489 
 
 1.745 
 
 0.977 
 
 2.617 
 
 1.466 
 
 3.490 
 
 1.954 
 
 4.362 
 
 2.443 
 
 45 
 
 30 
 
 0.870 
 
 0.492 
 
 1.741 
 
 0.985 
 
 2.611 
 
 1.477 
 
 3.481 
 
 1.970 
 
 4.352 
 
 2.462 
 
 30 
 
 45 
 
 0.868 
 
 0.496 
 
 1.736 
 
 0.992 
 
 2.605 
 
 1.489 
 
 3.473 
 
 1.985 
 
 4.341 
 
 2.481 
 
 15 
 
 30 
 
 0.866 
 
 0.500 
 
 1.732 
 
 1.000 
 
 2.598 
 
 1.500 
 
 3.464 
 
 2.000 
 
 4.330 
 
 2.500 
 
 6O 
 
 f 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 t 
 
 Bearing 
 
 Distance 1. 
 
 Distance 2. 
 
 Distance 3. 
 
 Distance 4. 
 
 Distance 5. 
 
 Bearing. 
 
 60- 75<
 
 15- 30' 
 
 59 
 
 Bearing, 
 
 Distance 6. 
 
 Distance 7. 
 
 Distance 8. 
 
 Distance 9. 
 
 Distance 10. 
 
 Bearing. 
 
 O f 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 o t 
 
 1515 
 
 5.789 
 
 1.578 
 
 6.754 
 
 1.841 
 
 7.718 
 
 2.104 
 
 8.683 
 
 2.367 
 
 9.648 
 
 2.630 
 
 7445 
 
 30 
 
 5.782 
 
 1.603 
 
 6.745 
 
 1.871 
 
 7.709 
 
 2.138 
 
 8.673 
 
 2.405 
 
 9.636 
 
 2.672 
 
 30 
 
 45 
 
 5.775 
 
 1.629 
 
 6.737 
 
 1.900 
 
 7.700 
 
 2.172 
 
 8.662 
 
 2.443 
 
 9.625 
 
 2.714 
 
 15 
 
 16 
 
 5.768 
 
 1.654 
 
 6.729 
 
 1.929 
 
 7.690 
 
 2.205 
 
 8.651 
 
 2.481 
 
 9.613 
 
 2.756 
 
 74 
 
 15 
 
 5.760 
 
 1.679 
 
 6.720 
 
 1.959 
 
 7.680 
 
 2.239 
 
 8.640 
 
 2.518 
 
 9.601 
 
 2.798 
 
 45 
 
 30 
 
 5.753 
 
 1.704 
 
 6.712 
 
 1.988 
 
 7.671 
 
 2.272 
 
 8.629 
 
 2.556 
 
 9.588 
 
 2.840 
 
 30 
 
 45 
 
 5.745 
 
 1.729 
 
 6.703 
 
 2.017 
 
 7.661 
 
 2.306 
 
 8.618 
 
 2.594 
 
 9.576 
 
 2.882 
 
 15 
 
 17 
 
 5.738 
 
 1.754 
 
 6.694 
 
 2.047 
 
 7.650 
 
 2.339 
 
 8.607 
 
 2.631 
 
 9.563 
 
 2.924 
 
 73 
 
 15 
 
 5.730 
 
 1.779 
 
 6.685 
 
 2.076 
 
 7.640 
 
 2.372 
 
 8.595 
 
 2.669 
 
 9.550 
 
 2.965 
 
 45 
 
 30 
 
 5.722 
 
 1.804 
 
 6.676 
 
 2.105 
 
 7.630 
 
 2.406 
 
 8.583 
 
 2.706 
 
 9.537 
 
 3.007 
 
 30 
 
 45 
 
 5.714 
 
 1.829 
 
 6.667 
 
 2.134 
 
 7.619 
 
 2.439 
 
 8.572 
 
 2.744 
 
 9.524 
 
 3.049 
 
 15 
 
 18 
 
 5.706 
 
 1.854 
 
 6.657 
 
 2.163 
 
 7.608 
 
 2.472 
 
 8.560 
 
 2.781 
 
 9.511 
 
 3.090 
 
 72 
 
 15 
 
 5.698 
 
 1.879 
 
 6.648 
 
 2.192 
 
 7.598 
 
 2.505 
 
 8.547 
 
 2.818 
 
 9.497 
 
 3.132 
 
 45 
 
 30 
 
 5.690 
 
 1.904 
 
 6.638 
 
 2.221 
 
 7.587 
 
 2.538 
 
 8.535 
 
 2.856 
 
 9.483 
 
 3.173 
 
 30 
 
 45 
 
 5.682 
 
 1.929 
 
 6.629 
 
 2.250 
 
 7.575 
 
 2.572 
 
 8.522 
 
 2.893 
 
 9.469 
 
 3.214 
 
 15 
 
 19 
 
 5.673 
 
 1.953 
 
 6.619 
 
 2.279 
 
 7.564 
 
 2.605 
 
 8.510 
 
 2.930 
 
 9.455 
 
 3.256 
 
 71 
 
 15 
 
 5.665 
 
 1.978 
 
 6.609 
 
 2.308 
 
 7.553 
 
 2.638 
 
 8.497 
 
 2.967 
 
 9.441 
 
 3.297 
 
 45 
 
 30 
 
 5.656 
 
 2.003 
 
 6.598 
 
 2.337 
 
 7.541 
 
 2.670 
 
 8.484 
 
 3.004 
 
 9.426 
 
 3.338 
 
 30 
 
 45 
 
 5.647 
 
 2.028 
 
 6.588 
 
 2.365 
 
 7.529 
 
 2.703 
 
 8.471 
 
 3.041 
 
 9.412 
 
 3.379 
 
 15 
 
 20 
 
 5.638 
 
 2.052 
 
 6.578 
 
 2.394 
 
 7.518 
 
 2.736 
 
 8.457 
 
 3.078 
 
 9.397 
 
 3.420 
 
 70 
 
 15 
 
 5.629 
 
 2.077 
 
 6.567 
 
 2.423 
 
 7.506 
 
 2.769 
 
 8.444 
 
 3.115 
 
 9.382 
 
 3.461 
 
 45 
 
 30 
 
 5.620 
 
 2.101 
 
 6.557 
 
 2.451 
 
 7.493 
 
 2.802 
 
 8.430 
 
 3.152 
 
 9.367 
 
 3.502 
 
 30 
 
 45 
 
 5.611 
 
 2.126 
 
 6.546 
 
 2.480 
 
 7.481 
 
 2.834 
 
 8.416 
 
 3.189 
 
 9.351 
 
 3.543 
 
 15 
 
 21 
 
 5.601 
 
 2.150 
 
 6.535 
 
 2.509 
 
 7.469 
 
 2.867 
 
 8.402 
 
 3.225 
 
 9.336 
 
 3.584 
 
 69 
 
 15 
 
 5.592 
 
 2.175 
 
 6.524 
 
 2.537 
 
 7.456 
 
 2.900 
 
 8.388 
 
 3.262 
 
 9.320 
 
 3.624 
 
 45 
 
 30 
 
 5.582 
 
 2.199 
 
 6.513 
 
 2.566 
 
 7.443 
 
 2.932 
 
 8.374 
 
 3.299 
 
 9.304 
 
 3.665 
 
 30 
 
 45 
 
 5.573 
 
 2.223 
 
 6.502 
 
 2.594 
 
 7.430 
 
 2.964 
 
 8.359 
 
 3.335 
 
 9.288 
 
 3.706 
 
 15 
 
 22 
 
 5.563 
 
 2.248 
 
 6.490 
 
 2.622 
 
 7.417 
 
 2.997 
 
 8.345 
 
 3.371 
 
 9.272 
 
 3.746 
 
 68 
 
 15 
 
 5.553 
 
 2.272 
 
 6.479 
 
 2.651 
 
 7.404 
 
 3.029 
 
 8.330 
 
 3.408 
 
 9.255 
 
 3.787 
 
 45 
 
 30 
 
 5.543 
 
 2.296 
 
 6.467 
 
 2.679 
 
 7.391 
 
 3.061 
 
 8.315 
 
 3.444 
 
 9.239 
 
 3.827 
 
 30 
 
 45 
 
 5.533 
 
 2.320 
 
 6.455 
 
 2.707 
 
 7.378 
 
 3.094 
 
 8.300 
 
 3.480 
 
 9.222 
 
 3.867 
 
 15 
 
 23 
 
 5.523 
 
 2.344 
 
 6.444 
 
 2.735 
 
 7.364 
 
 3.126 
 
 8.285 
 
 3.517 
 
 9.205 
 
 3.907 
 
 67 
 
 15 
 
 5.513 
 
 2.368 
 
 6.432 
 
 2.763 
 
 7.350 
 
 3.158 
 
 8.269 
 
 3.553 
 
 9.188 
 
 3.947 
 
 45 
 
 30 
 
 5.502 
 
 2.392 
 
 6.419 
 
 2.791 
 
 7.336 
 
 3.190 
 
 8.254 
 
 3.589 
 
 9.171 
 
 3.988 
 
 30 
 
 45 
 
 5.492 
 
 2.416 
 
 6.407 
 
 2.819 
 
 7.322 
 
 3.222 
 
 8.238 
 
 3.625 
 
 9.153 
 
 4.028 
 
 15 
 
 24 
 
 5.481 
 
 2.440 
 
 6.395 
 
 2.847 
 
 7.308 
 
 3.254 
 
 8.222 
 
 3.661 
 
 9.136 
 
 4.067 
 
 66 
 
 15 
 
 5.471 
 
 2.464 
 
 6.382 
 
 2.875 
 
 7.294 
 
 3.286 
 
 8.206 
 
 3.696 
 
 9.118 
 
 4.107 
 
 45 
 
 30 
 
 5.460 
 
 2.488 
 
 6.370 
 
 2.903 
 
 7.280 
 
 3.318 
 
 8.190 
 
 3.732 
 
 9.100 
 
 4.147 
 
 30 
 
 45 
 
 5.449 
 
 2.512 
 
 6.357 
 
 2.931 
 
 7.265 
 
 3.349 
 
 8.173 
 
 3.768 
 
 9.081 
 
 4.187 
 
 15 
 
 25 
 
 5.438 
 
 2.536 
 
 6.344 
 
 2.958 
 
 7.250 
 
 3.381 
 
 8.157 
 
 3.804 
 
 9.063 
 
 4.226 
 
 65 
 
 15 
 
 5.427 
 
 2.559 
 
 6.331 
 
 2.986 
 
 7.236 
 
 3.413 
 
 8.140 
 
 3.839 
 
 9.045 
 
 4.266 
 
 45 
 
 30 
 
 5.416 
 
 2.583 
 
 6.318 
 
 3.014 
 
 7.221 
 
 3.444 
 
 8.123 
 
 3.875 
 
 9.026 
 
 4.305 
 
 30 
 
 45 
 
 5.404 
 
 2.607 
 
 6.305 
 
 3.041 
 
 7.206 
 
 3.476 
 
 8.106 
 
 3.910 
 
 9.007 
 
 4.345 
 
 15 
 
 26 
 
 5.393 
 
 2.630 
 
 6.292 
 
 3.069 
 
 7.190 
 
 3.507 
 
 8.089 
 
 3.945 
 
 8.988 
 
 4.384 
 
 64 
 
 15 
 
 5.3S1 
 
 2.654 
 
 6.278 
 
 3.096 
 
 7.175 
 
 3.538 
 
 8.072 
 
 3.981 
 
 8.969 
 
 4.423 
 
 45 
 
 30 
 
 5.370 
 
 2.677 
 
 6.265 
 
 3.123 
 
 7.160 
 
 3.570 
 
 8.054 
 
 4.016 
 
 8.949 
 
 4.462 
 
 30 
 
 45 
 
 5.358 
 
 2.701 
 
 6.251 
 
 3.151 
 
 7.144 
 
 3.601 
 
 8.037 
 
 4.051 
 
 8.930 
 
 4.501 
 
 15 
 
 27 
 
 5.346 
 
 2.724 
 
 6.237 
 
 3.178 
 
 7.128 
 
 3.632 
 
 8.019 
 
 4.086 
 
 8.910 
 
 4.540 
 
 63 
 
 15 
 
 5.334 
 
 2.747 
 
 6.223 
 
 3.205 
 
 7.112 
 
 3.663 
 
 8.001 
 
 4.121 
 
 8.890 
 
 4.579 
 
 45 
 
 30 
 
 5.322 
 
 2.770 
 
 6.209 
 
 3.232 
 
 7.096 
 
 3.694 
 
 7.983 
 
 4.156 
 
 8.870 
 
 4.618 
 
 30 
 
 45 
 
 5.310 
 
 2.794 
 
 6.195 
 
 3.259 
 
 7.0SO 
 
 3.725 
 
 7.965 
 
 4.190 
 
 8.850 
 
 4.656 
 
 15 
 
 28 
 
 5.298 
 
 2.817 
 
 6.181 
 
 3.286 
 
 7.064 
 
 3.756 
 
 7.947 
 
 4.225 
 
 8.829 
 
 4.695 
 
 62 
 
 15 
 
 5.285 
 
 2.840 
 
 6.166 
 
 3.313 
 
 7.047 
 
 3.787 
 
 7.928 
 
 4.260 
 
 8.809 
 
 4.733 
 
 45 
 
 30 
 
 5.273 
 
 2.863 
 
 6.152 
 
 3.340 
 
 7.031 
 
 3.817 
 
 7.909 
 
 4.294 
 
 8.788 
 
 4.772 
 
 30 
 
 45 
 
 5.260 
 
 2.886 
 
 6.137 
 
 3.367 
 
 7.014 
 
 3.848 
 
 7.891 
 
 4.329 
 
 8.767 
 
 4.810 
 
 15 
 
 29 
 
 5.248 
 
 2.909 
 
 6.122 
 
 3.394 
 
 6.997 
 
 3.878 
 
 7.S72 
 
 4.363 
 
 8.746 
 
 4.848 
 
 61 
 
 15 
 
 5.235 
 
 2.932 
 
 6.107 
 
 3.420 
 
 6.9SO 
 
 3.909 
 
 7.852 
 
 4.398 
 
 8.725 
 
 4.886 
 
 45 
 
 30 
 
 5.222 
 
 2.955 
 
 6.093 
 
 3.447 
 
 6.963 
 
 3.939 
 
 7.833 
 
 4.432 
 
 8.704 
 
 4.924 
 
 30 
 
 45 
 
 5.209 
 
 2.977 
 
 6.077 
 
 3.474 
 
 6.946 
 
 3.970 
 
 7.814 
 
 4.466 
 
 8.682 
 
 4.962 
 
 15 
 
 30 
 
 5.196 
 
 3.000 
 
 6.062 
 
 3.500 
 
 6.928 
 
 4.000 
 
 7.794 
 
 4.500 
 
 8.660 
 
 5.000 
 
 60 
 
 f 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 o t 
 
 Bearing. 
 
 Distance 6. 
 
 Distance 7. 
 
 Distance 8. 
 
 Distance 9. 
 
 Distance 1O. 
 
 Bearing. 
 
 60-- 75'
 
 30- 45 C 
 
 Bearing. 
 
 Distance 1. 
 
 Distance 2. 
 
 Distance 3. 
 
 Distance 4. 
 
 Distance 5. 
 
 Bearing. 
 
 o r 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 f 
 
 3015 
 
 0.864 
 
 0.504 
 
 1.728 
 
 1.008 
 
 2.592 
 
 1.511 
 
 3.455 
 
 2.015 
 
 4.319 
 
 2.519 
 
 5945 
 
 30 
 
 0.862 
 
 0.508 
 
 1.723 
 
 1.015 
 
 2.585 
 
 1.523 
 
 3.447 
 
 2.030 
 
 4.308 
 
 2.538 
 
 30 
 
 45 
 
 0.859 
 
 0.511 
 
 1.719 
 
 1.023 
 
 2.578 
 
 1.534 
 
 3.438 
 
 2.045 
 
 4.297 
 
 2.556 
 
 15 
 
 31 
 
 0.857 
 
 0.515 
 
 1.714 
 
 1.030 
 
 2.572 
 
 1.545 
 
 3.429 
 
 2.060 
 
 4.286 
 
 2.575 
 
 59 
 
 15 
 
 0.855 
 
 0.519 
 
 1.710 
 
 1.038 
 
 2.565 
 
 1.556 
 
 3.420 
 
 2.075 
 
 4.275 
 
 2.594 
 
 45 
 
 30 
 
 0.853 
 
 0.522 
 
 1.705 
 
 1.045 
 
 2.558 
 
 1.567 
 
 3.411 
 
 2.090 
 
 4.263 
 
 2.612 
 
 30 
 
 45 
 
 0.850 
 
 0.526 
 
 1.701 
 
 1.052 
 
 2.551 
 
 1.579 
 
 3.401 
 
 2.105 
 
 4.252 
 
 2.631 
 
 15 
 
 32 
 
 0.848 
 
 0.530 
 
 1.696 
 
 1.060 
 
 2.544 
 
 1.590 
 
 3.392 
 
 2.120 
 
 4.240 
 
 2.650 
 
 58 
 
 15 
 
 0.846 
 
 0.534 
 
 1.691 
 
 1.067 
 
 2.537 
 
 1.601 
 
 3.383 
 
 2.134 
 
 4.229 
 
 2.668 
 
 45 
 
 30 
 
 0.843 
 
 0.537 
 
 1.687 
 
 1.075 
 
 2.530 
 
 1.612 
 
 3.374 
 
 2.149 
 
 4.217 
 
 2.686 
 
 30 
 
 45 
 
 0.841 
 
 0.541 
 
 1.682 
 
 1.082 
 
 2.523 
 
 1.623 
 
 3.364 
 
 2.164 
 
 4.205 
 
 2.705 
 
 15 
 
 33 
 
 0.839 
 
 0.545 
 
 1.677 
 
 1.089 
 
 2.516 
 
 1.634 
 
 3.355 
 
 2.179 
 
 4.193 
 
 2.723 
 
 57 
 
 15 
 
 0.836 
 
 0.548 
 
 1.673 
 
 1.097 
 
 2.509 
 
 1.645 
 
 3.345 
 
 2.193 
 
 4.181 
 
 2.741 
 
 45 
 
 30 
 
 0.834 
 
 0.552 
 
 1.668 
 
 1.104 
 
 2.502 
 
 1.656 
 
 3.336 
 
 2.208 
 
 4.169 
 
 2.760 
 
 30 
 
 45 
 
 0.831 
 
 0.556 
 
 1.663 
 
 1.111 
 
 2.494 
 
 1.667 
 
 3.326 
 
 2.222 
 
 4.157 
 
 2.778 
 
 15 
 
 34 
 
 0.829 
 
 0.559 
 
 1.658 
 
 1.118 
 
 2.487 
 
 1.678 
 
 3.316 
 
 2.237 
 
 4.145 
 
 2.796 
 
 56 
 
 15 
 
 0.827 
 
 0.563 
 
 1.653 
 
 1.126 
 
 2.480 
 
 1.688 
 
 3.306 
 
 2.251 
 
 4.133 
 
 2.814 
 
 45 
 
 30 
 
 0.824 
 
 0.566 
 
 1.648 
 
 1.133 
 
 2.472 
 
 1.699 
 
 3.297 
 
 2.266 
 
 4.121 
 
 2.832 
 
 30 
 
 45 
 
 0.822 
 
 0.570 
 
 1.643 
 
 1.140 
 
 2.465 
 
 1.710 
 
 3.287 
 
 2.280 
 
 4.108 
 
 2.850 
 
 15 
 
 35 
 
 0.819 
 
 0.574 
 
 1.638 
 
 1.147 
 
 2.457 
 
 1.721 
 
 3.277 
 
 2.294 
 
 4.096 
 
 2.868 
 
 55 
 
 15 
 
 0.817 
 
 0.577 
 
 1.633 
 
 1.154 
 
 2.450 
 
 1.731 
 
 3.267 
 
 2.309 
 
 4.083 
 
 2.886 
 
 45 
 
 30 
 
 0.814 
 
 0.581 
 
 1.628 
 
 1.161 
 
 2.442 
 
 1.742 
 
 3.257 
 
 2.323 
 
 4.071 
 
 2.904 
 
 30 
 
 45 
 
 0.812 
 
 0.584 
 
 1.623 
 
 1.168 
 
 2.435 
 
 1.753 
 
 3.246 
 
 2.337 
 
 4.058 
 
 2.921 
 
 15 
 
 36 
 
 0.809 
 
 0.588 
 
 1.618 
 
 1.176 
 
 2.427 
 
 1.763 
 
 3.236 
 
 2.351 
 
 4.045 
 
 2.939 
 
 54 
 
 15 
 
 0.806 
 
 0.591 
 
 1.613 
 
 1.183 
 
 2.419 
 
 1.774 
 
 3.226 
 
 2.365 
 
 4.032 
 
 2.957 
 
 45 
 
 30 
 
 0.804 
 
 0.595 
 
 1.608 
 
 1.190 
 
 2.412 
 
 1.784 
 
 3.215 
 
 2.379 
 
 4.019 
 
 2.974 
 
 30 
 
 45 
 
 0.801 
 
 0.598 
 
 1.603 
 
 1.197 
 
 2.404 
 
 1.795 
 
 3.205 
 
 2.393 
 
 4.006 
 
 2.992 
 
 15 
 
 37 
 
 0.799 
 
 0.602 
 
 1.597 
 
 1.204 
 
 2.396 
 
 1.805 
 
 3.195 
 
 ?.407 
 
 3.993 
 
 3.009 
 
 53 
 
 15 
 
 0.796 
 
 0.605 
 
 1.592 
 
 1.211 
 
 2.388 
 
 1.816 
 
 3.184 
 
 2.421 
 
 3.980 
 
 3.026 
 
 45 
 
 30 
 
 0.793 
 
 0.609 
 
 1.587 
 
 1.218 
 
 2.380 
 
 1.826 
 
 3.173 
 
 2.435 
 
 3.967 
 
 3.044 
 
 30 
 
 45 
 
 0.791 
 
 0.612 
 
 1.581 
 
 1.224 
 
 2.372 
 
 1.837 
 
 3.163 
 
 2.449 
 
 3.953 
 
 3.061 
 
 15 
 
 38 
 
 0.788 
 
 0.616 
 
 1.576 
 
 1.231 
 
 2.364 
 
 1.847 
 
 3.152 
 
 2.463 
 
 3.940 
 
 3.078 
 
 52 
 
 15 
 
 0.785 
 
 0.619 
 
 1.571 
 
 1.238 
 
 2.356 
 
 1.857 
 
 3.141 
 
 2.476 
 
 3.927 
 
 3.095 
 
 45 
 
 30 
 
 0.783 
 
 0.623 
 
 1.565 
 
 1.245 
 
 2.348 
 
 1.868 
 
 3.130 
 
 2.490 
 
 3.913 
 
 3.113 
 
 30 
 
 45 
 
 0.780 
 
 0.626 
 
 1.560 
 
 1.252 
 
 2.340 
 
 1.878 
 
 3.120 
 
 2.504 
 
 3.899 
 
 3.130 
 
 15 
 
 39 
 
 0.777 
 
 0.629 
 
 1.554 
 
 1.259 
 
 2.331 
 
 1.888 
 
 3.109 
 
 2.517 
 
 3.8S6 
 
 3.147 
 
 51 
 
 15 
 
 0.774 
 
 0.633 
 
 1.549 
 
 1.265 
 
 2.323 
 
 1.898 
 
 3.098 
 
 2.531 
 
 3.872 
 
 3.164 
 
 45 
 
 30 
 
 0.772 
 
 0.636 
 
 1.543 
 
 1.272 
 
 2.315 
 
 1.908 
 
 3.086- 
 
 2.544 
 
 3.858 
 
 3.180 
 
 30 
 
 45 
 
 0.769 
 
 0.639 
 
 1.538 
 
 1.279 
 
 2.307 
 
 1.918 
 
 3.075 
 
 2.558 
 
 3. 844 
 
 3.197 
 
 15 
 
 40 
 
 0.766 
 
 0.643 
 
 1.532 
 
 1.286 
 
 2.298 
 
 1.928 
 
 3.064 
 
 2.571 
 
 3.830 
 
 3.214 
 
 5O 
 
 15 
 
 0.763 
 
 0.646 
 
 1.526 
 
 1.292 
 
 2.290 
 
 1.938 
 
 3.053 
 
 2.584 
 
 3.816 
 
 3.231 
 
 45 
 
 30 
 
 0.760 
 
 0.649 
 
 1.521 
 
 1.299 
 
 2.281 
 
 1.948 
 
 3.042 
 
 2.598 
 
 3.802 
 
 3.247 
 
 30 
 
 45 
 
 0.758 
 
 0.653 
 
 1.515 
 
 1.306 
 
 2.273 
 
 1.958 
 
 3.030 
 
 2.611 
 
 3. 788 
 
 3.264 
 
 15 
 
 41 
 
 0.755 
 
 0.656 
 
 1.509 
 
 1.312 
 
 2.264 
 
 1.968 
 
 3.019 
 
 2.624 
 
 3.774 
 
 3.280 
 
 49 
 
 15 
 
 0.752 
 
 0.659 
 
 1.504 
 
 1.319 
 
 2.256 
 
 1.978 
 
 3.007 
 
 2.637 
 
 3.759 
 
 3.297 
 
 45 
 
 30 
 
 0.749 
 
 0.663 
 
 1.498 
 
 1.325 
 
 2.247 
 
 1.988 
 
 2.996 
 
 2.650 
 
 3.745 
 
 3.313 
 
 30 
 
 45 
 
 0.746 
 
 0.666 
 
 1.492 
 
 1.332 
 
 2.23S 
 
 1.998 
 
 2.984 
 
 2.664 
 
 3.730 
 
 3.329 
 
 15 
 
 42 
 
 0.743 
 
 0.669 
 
 1.486 
 
 1.338 
 
 2.229 
 
 2.007 
 
 2.973 
 
 2.677 
 
 3.716 
 
 3.346 
 
 48 
 
 15 
 
 0.740 
 
 0.672 
 
 1.480 
 
 1.345 
 
 2.221 
 
 2.017 
 
 2.961 
 
 2.689 
 
 3.701 
 
 3.362 
 
 45 
 
 30 
 
 0.737 
 
 0.676 
 
 1.475 
 
 1.351 
 
 2.212 
 
 2.027 
 
 2.949 
 
 2.702 
 
 3.686 
 
 3.378 
 
 30 
 
 45 
 
 0.734 
 
 0.679 
 
 1.469 
 
 1.358 
 
 2.203 
 
 2.036 
 
 2.937 
 
 2.715 
 
 3.672 
 
 3.394 
 
 15 
 
 43 
 
 0.731 
 
 0.682 
 
 1.463 
 
 1.364 
 
 2.194 
 
 2.046 
 
 2.925 
 
 2.728 
 
 3.657 
 
 3.410 
 
 47 
 
 15 
 
 0.728 
 
 0.685 
 
 1.457 
 
 1.370 
 
 2.185 
 
 2.056 
 
 2.913 
 
 2.741 
 
 3.642 
 
 3.426 
 
 45 
 
 30 
 
 0.725 
 
 0.688 
 
 1.451 
 
 1.377 
 
 2.176 
 
 2.065 
 
 2.901 
 
 2.753 
 
 3.627 
 
 3.442 
 
 30 
 
 45 
 
 0.722 
 
 0.692 
 
 1.445 
 
 1.383 
 
 2.167 
 
 2.075 
 
 2.889 
 
 2.766 
 
 3.612 
 
 3.458 
 
 15 
 
 44 
 
 0.719 
 
 0.695 
 
 1.439 
 
 1.389 
 
 2.158 
 
 2.0S4 
 
 2.877 
 
 2.779 
 
 3.597 
 
 3.473 
 
 46 
 
 15 
 
 0.716 
 
 0.698 
 
 1.433 
 
 1.396 
 
 2.149 
 
 2.093 
 
 2.865 
 
 2.791 
 
 3.582 
 
 3.489 
 
 45 
 
 30 
 
 0.713 
 
 0.701 
 
 1.427 
 
 1.402 
 
 2.140 
 
 2.103 
 
 2.853 
 
 2.804 
 
 3.566 
 
 3.505 
 
 30 
 
 45 
 
 0.710 
 
 0.704 
 
 1.420 
 
 1.408 
 
 2.131 
 
 2.112 
 
 2.841 
 
 2.816 
 
 3.551 
 
 3.520 
 
 15 
 
 45 
 
 0.707 
 
 0.707 
 
 1.414 
 
 1.414 
 
 2.121 
 
 2.121 
 
 2.828 
 
 2.828 
 
 3.536 
 
 3.536 
 
 45 
 
 f 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 o r 
 
 Bearing 
 
 Distance 1. 
 
 Distance 2. 
 
 Distance 3. 
 
 Distance 4. 
 
 Distance 5. 
 
 Bearing. 
 
 45- 60 C
 
 30- 45' 
 
 Bearing. 
 
 Distance 6. 
 
 Distance 7. 
 
 Distance 8. 
 
 Distance 9. 
 
 Distance 1O. 
 
 Bearing. 
 
 o / 
 
 Lat. Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. i Dep. 
 
 o t 
 
 3O15 
 
 5.183 
 
 3.023 
 
 6.047 
 
 3.526 
 
 6.911 
 
 4.030 
 
 7.775 
 
 4.534 
 
 8.638 i 5.038 
 
 5945 
 
 30 
 
 5.170 
 
 3.045 
 
 6.031 
 
 3.553 
 
 6.893 
 
 4.060 
 
 7.755 
 
 4.568 
 
 8.616 5.075 
 
 30 
 
 45 
 
 5.156 
 
 3. 068 
 
 6.016 
 
 3.579 
 
 6.875 
 
 4.090 
 
 7.735 
 
 4.602 
 
 8.594 
 
 5.113 
 
 15 
 
 31 
 
 5.143 
 
 3.090 
 
 6.000 
 
 3.605 
 
 6.857 
 
 4.120 
 
 7.715 
 
 4.635 
 
 8.572 
 
 5.150 
 
 59 
 
 15 
 
 5.129 
 
 3.113 
 
 5.984 
 
 3.631 
 
 6.839 
 
 4.150 
 
 7.694 
 
 4.669 
 
 8.549 
 
 5.188 
 
 45 
 
 30 
 
 5.116 
 
 3.135 
 
 5.968 
 
 3.657 
 
 6.821 
 
 4.180 
 
 7.674 
 
 4.702 
 
 8.526 
 
 5.225 
 
 30 
 
 45 
 
 5.102 
 
 3.157 
 
 5.952 
 
 3.683 
 
 6.803 
 
 4.210 
 
 7.653 
 
 4.736 
 
 8.504 
 
 5.262 
 
 15 
 
 32 
 
 5.088 
 
 3.180 
 
 5.936 
 
 3.709 
 
 6.784 
 
 4.239 
 
 7.632 
 
 4.769 
 
 8.481 
 
 5.299 
 
 58 
 
 15 
 
 5.074 
 
 3.202 
 
 5.920 
 
 3.735 
 
 6.766 
 
 4.269 
 
 7.612 
 
 4.802 
 
 8.457 
 
 5.336 
 
 45 
 
 30 
 
 5.060 
 
 3.224 
 
 5.904 
 
 3.761 
 
 6.747 
 
 4.298 
 
 7.591 
 
 4.836 
 
 8.434 
 
 5.373 
 
 30 
 
 45 
 
 5.046 
 
 3.246 
 
 5.887 
 
 3.787 
 
 6.728 
 
 4.328 
 
 7.569 
 
 4.869 
 
 8.410 
 
 5.410 
 
 15 
 
 33 
 
 5.032 
 
 3.268 
 
 5.871 
 
 3.812 
 
 6.709 
 
 4.357 
 
 7.548 
 
 4.902 
 
 8.387 
 
 5.446 
 
 57 
 
 15 
 
 5.018 
 
 3.290 
 
 5.854 
 
 3.838 
 
 6.690 
 
 4.386 
 
 7.527 
 
 4.935 
 
 8.363 
 
 5. 483 
 
 45 
 
 30 
 
 5.003 
 
 3.312 
 
 5.837 
 
 3.864 
 
 6.671 
 
 4.416 
 
 7.505 
 
 4.967 
 
 8.339 
 
 5.519 
 
 30 
 
 45 
 
 4.989 
 
 3.333 
 
 5.820 
 
 3.889 
 
 6.652 
 
 4.445 
 
 7.483 
 
 5.000 
 
 8.315 
 
 5.556 
 
 15 
 
 34 
 
 4.974 
 
 3.355 
 
 5.803 
 
 3.914 j! 6.632 
 
 4.474 
 
 7.461 
 
 5.033 
 
 8.290 
 
 5.592 
 
 56 
 
 15 
 
 4.960 
 
 3.377 
 
 5.786 
 
 3.940 
 
 6.613 
 
 4.502 
 
 7.439 
 
 5.065 
 
 8.266 
 
 5.628 
 
 45 
 
 30 
 
 4.945 
 
 3.398 
 
 5.769 
 
 3.965 
 
 6.593 
 
 4.531 
 
 7.417 
 
 5.098 
 
 8.241 
 
 5.664 
 
 30 
 
 45 
 
 4.930 
 
 3.420 
 
 5.752 
 
 3.990 
 
 6.573 
 
 4.560 
 
 7.395 
 
 5.130 
 
 8.217 
 
 5.700 
 
 15 
 
 35 
 
 4.915 
 
 3.441 
 
 5.734 
 
 4.015 
 
 6.553 
 
 4.589 
 
 7.372 
 
 5.162 
 
 8.192 
 
 5.736 
 
 55 
 
 15 
 
 4.900 
 
 3.463 
 
 5.716 
 
 4.040 
 
 6.533 
 
 4.617 
 
 7.350 
 
 5.194 
 
 8.166 
 
 5.772 
 
 45 
 
 30 
 
 4.885 
 
 3.484 
 
 5.699 
 
 4.065 
 
 6.513 
 
 4.646 
 
 7.327 
 
 5.226 
 
 8.141 
 
 5.807 
 
 30 
 
 45 
 
 4.869 
 
 3.505 
 
 5.681 
 
 4.090 
 
 6.493 
 
 4.674 
 
 7.304 
 
 5.258 
 
 8.116 
 
 5.843 
 
 15 
 
 36 
 
 4.854 
 
 3.527 
 
 5.663 
 
 4.115 
 
 6-472 
 
 4.702 
 
 7.281 
 
 5.290 
 
 8.090 
 
 5.878 
 
 54 
 
 15 
 
 4.839 
 
 3.548 
 
 5.645 
 
 4.139 
 
 6.452 
 
 4.730 
 
 7.258 
 
 5.322 
 
 8.064 
 
 5.913 
 
 45 
 
 30 
 
 4.823 
 
 3.569 
 
 5.627 
 
 4.164 
 
 6.431 
 
 4.759 
 
 7.235 
 
 5.353 
 
 8.039 
 
 5.948 
 
 30 
 
 45 
 
 4.808 
 
 3.590 
 
 5.609 
 
 4.188 
 
 6.410 
 
 4.787 
 
 7.211 
 
 5.385 
 
 8.013 
 
 5.983 
 
 15 
 
 37 
 
 4.792 
 
 3.611 
 
 5.590 
 
 4.213 
 
 6.389 
 
 4.815 
 
 7.188 
 
 5.416 
 
 7.986 
 
 6.018 
 
 53 
 
 15 
 
 4.776 
 
 3.632 
 
 5.572 
 
 4.237 
 
 6.368 
 
 4.842 
 
 7.164 
 
 5.448 
 
 7.960 
 
 6.053 
 
 45 
 
 30 
 
 4.760 
 
 3.653 
 
 5.554 
 
 4.261 
 
 6.347 
 
 4.870 
 
 7.140 
 
 5.479 
 
 7.934 
 
 6.088 
 
 30 
 
 45 
 
 4.744 
 
 3.673 
 
 5.535 
 
 4.286 
 
 6.326 
 
 4.898 
 
 7.116 
 
 5.510 
 
 7.907 
 
 6.122 
 
 15 
 
 38 
 
 4.728 
 
 3.694 
 
 5.516 
 
 4.310 
 
 6.304 
 
 4.925 
 
 7.092 
 
 5.541 
 
 7.880 
 
 6.157 
 
 52 
 
 15 
 
 4.712 
 
 3.715 
 
 5.497 
 
 4.334 
 
 6.283 
 
 4.953 
 
 7.068 
 
 5.572 
 
 7.853 
 
 6.191 
 
 45 
 
 30 
 
 4.696 
 
 3.735 
 
 5.478 
 
 4.358 
 
 6.261 
 
 4.980 
 
 7.043 
 
 5.603 
 
 7.826 
 
 6.225 
 
 30 
 
 45 
 
 4.679 
 
 3.756 
 
 5.459 
 
 4.381 
 
 6.239 
 
 5.007 
 
 7.019 
 
 5.633 
 
 7.799 
 
 6.259 
 
 15 
 
 39 
 
 4.663 
 
 3.776 
 
 5.440 
 
 4.405 
 
 6.217 
 
 5.035 
 
 6.994 
 
 5.664 
 
 7.772 
 
 6.293 
 
 51 
 
 15 
 
 4.646 
 
 3.796 
 
 5.421 
 
 4.429 
 
 6.195 
 
 5.062 
 
 6.970 
 
 5.694 
 
 7.744 
 
 6.327 
 
 45 
 
 30 
 
 4.630 
 
 3.816 
 
 5.401 
 
 4.453 
 
 6.173 
 
 5.089 
 
 6.945 
 
 5.725 
 
 7.716 
 
 6.361 
 
 30 
 
 45 
 
 4.613 
 
 3.837 
 
 5.382 
 
 4.476 
 
 6.151 
 
 5.116 
 
 6.920 
 
 5.755 
 
 7.688 
 
 6.394 
 
 15 
 
 40 
 
 4.596 
 
 3.857 
 
 5.362 
 
 4.500 
 
 6.128 
 
 5.142 
 
 6.894 
 
 5.785 
 
 7.660 
 
 6.428 
 
 50 
 
 15 
 
 4.579 
 
 3.877 
 
 5.343 
 
 4.523 
 
 6.106 
 
 5.169 
 
 6.869 
 
 5.815 
 
 7.632 
 
 6.461 
 
 45 
 
 30 
 
 4.562 
 
 3.897 
 
 5.323 
 
 4.546 
 
 6.083 
 
 5.196 
 
 6.844 
 
 5.845 
 
 7.604 
 
 6.495 
 
 30 
 
 45 
 
 4.545 
 
 3.917 
 
 5.303 
 
 4.569 
 
 6.061 
 
 5.222 
 
 6.818 
 
 5.875 
 
 7.576 
 
 6.528 
 
 15 
 
 41 
 
 4.528 
 
 3.936 
 
 5.283 
 
 4.592 
 
 6.038 
 
 5.248 
 
 6.792 
 
 5.905 
 
 7.547 
 
 6.561 
 
 49 
 
 15 
 
 4.511 
 
 3.956 
 
 5.263 
 
 4.615 
 
 6.015 
 
 5.275 
 
 6.767 
 
 5.934 
 
 7.518 
 
 6.594 
 
 45 
 
 30 
 
 4.494 
 
 3.976 
 
 5.243 
 
 4.638 
 
 5.992 
 
 5.301 
 
 6.741 
 
 5.964 
 
 7.490 
 
 6.626 
 
 30 
 
 45 
 
 4.476 
 
 3.995 
 
 5.222 
 
 4.661 
 
 5.968 
 
 5.327 
 
 6.715 
 
 5.993 
 
 7.461 
 
 6.659 
 
 15 
 
 42 
 
 4.459 
 
 4.015 
 
 5.202 
 
 4.684 
 
 5.945 
 
 5.353 
 
 6.688 
 
 6.022 
 
 7.431 
 
 6.691 
 
 48 
 
 15 
 
 4.441 
 
 4.034 
 
 5.182 
 
 4.707 
 
 5.922 
 
 5.379 
 
 6.662 
 
 6.051 
 
 7.402 
 
 6.724 
 
 45 
 
 30 
 
 4.424 
 
 4.054 
 
 5.161 
 
 4.729 
 
 5.898 
 
 5.405 
 
 6.635 
 
 6.080 
 
 7.373 
 
 6.756 
 
 30 
 
 45 
 
 4.406 
 
 4.073 
 
 5.140 
 
 4.752 
 
 5.875 
 
 5.430 
 
 6.609 
 
 6.109 
 
 7.343 
 
 6.788 
 
 15 
 
 43 
 
 4.388 
 
 4.092 
 
 5.119 
 
 4.774 
 
 5.851 
 
 5.456 
 
 6.582 
 
 6.138 
 
 7.314 
 
 6.820 
 
 47 
 
 15 
 
 4.370 
 
 4.111 
 
 5.099 
 
 4.7% 
 
 5.827 
 
 5.481 
 
 6.555 
 
 6.167 
 
 7.284 
 
 6.852 
 
 45 
 
 30 
 
 4.352 
 
 4.130 
 
 5.078 
 
 4.818 
 
 5.803 
 
 5.507 
 
 6.528 
 
 6.195 
 
 7.254 
 
 6.884 
 
 30 
 
 45 
 
 4.334 
 
 4.149 
 
 5.057 
 
 4.841 
 
 5.779 
 
 5.532 
 
 6.501 
 
 6.224 
 
 7.224 
 
 6.915 
 
 15 
 
 44 
 
 4.316 
 
 4.168 
 
 5.035 
 
 4.863 
 
 5.755 
 
 5.557 
 
 6.474 
 
 6.252 
 
 7.193 
 
 6.947 
 
 48 
 
 15 
 
 4.298 
 
 4.187 
 
 5.014 
 
 4.885 
 
 5.730 
 
 5.582 
 
 6.447 
 
 6.280 
 
 7.163 
 
 6.978 
 
 45 
 
 30 
 
 4.2SO 
 
 4.206 
 
 4.993 
 
 4.906 
 
 5.706 
 
 5.607 
 
 6.419 
 
 6.308 
 
 7.133 
 
 7.009 
 
 30 
 
 45 
 
 4.261 
 
 4.224 
 
 4.971 
 
 4.928 
 
 5.681 
 
 5.632 
 
 6.392 
 
 6.336 
 
 7.102 
 
 7.040 
 
 15 
 
 45 
 
 4.243 
 
 4.243 
 
 4.950 
 
 4.950 
 
 5.657 
 
 5.657 
 
 6.364 
 
 6.364 
 
 7.071 
 
 7.071 
 
 45 
 
 f 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 o t 
 
 Bearing, 
 
 Distance 6. 
 
 Distance 7. 
 
 Distance 8. 
 
 Distance 9. 
 
 Distance 1O. 
 
 Bearing. 
 
 45- 60'
 
 TABLE VIII, -NATURAL SINES AND COSINES. 
 
 t 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 t 
 
 
 sin cos 
 
 gin cos 
 
 sin cos 
 
 sin cos 
 
 sin cos 
 
 
 o 
 
 0000 one 
 
 0175 9998 
 
 0349 9994 
 
 0523 9986 
 
 0698 9976 
 
 6O 
 
 1 
 
 0003 one 
 
 0177 9998 
 
 0352 9994 
 
 0526 9986 
 
 0700 9975 
 
 59 
 
 2 
 
 0006 one 
 
 0180 9998 
 
 0355 9994 
 
 0529 9986 
 
 0703 9975 
 
 58 
 
 3 
 
 0009 one 
 
 0183 9998 
 
 0358 9994 
 
 0532 9986 
 
 0706 9975 
 
 57 
 
 4 
 
 0012 one 
 
 0186 9998 
 
 0361 9993 
 
 0535 9986 
 
 0709 9975 
 
 56 
 
 5 
 
 0015 one 
 
 0189 9998 
 
 0364 9993 
 
 0538 9986 
 
 0712 9975 
 
 55 
 
 6 
 
 0017 one 
 
 0192 9998 
 
 0366 9993 
 
 0541 9985 
 
 0715 9974 
 
 54 
 
 7 
 
 0020 one 
 
 0195 9998 
 
 0369 9993 
 
 0544 9985 
 
 0718 9974 
 
 53 
 
 8 
 
 0023 one 
 
 0198 9998 
 
 0372 9993 
 
 0547 9985 
 
 0721 9974 
 
 52 
 
 9 
 
 0026 one 
 
 0201 9998 
 
 0375 9993 
 
 0550 9985 
 
 0724 9974 
 
 51 
 
 1O 
 
 0029 one 
 
 0204 9998 
 
 0378 9993 
 
 0552 9985 
 
 0727 9974 
 
 50 
 
 11 
 
 0032 one 
 
 0207 9998 
 
 0381 9993 
 
 0555 9985 
 
 0729 9973 
 
 49 
 
 12 
 
 0035 one 
 
 0209 9998 
 
 0384 9993 
 
 0558 9984 
 
 0732 9973 
 
 48 
 
 13 
 
 0038 one 
 
 0212 9998 
 
 0387 9993 
 
 0561 9984 
 
 0735 9973 
 
 47 
 
 14 
 
 0041 one 
 
 0215 9998 
 
 0390 9992 
 
 0564 9984 
 
 0738 9973 
 
 46 
 
 15 
 
 0044 one 
 
 0218 9998 
 
 0393 9992 
 
 0567 9984 
 
 0741 9973 
 
 45 
 
 16 
 
 0047 one 
 
 0221 9998 
 
 0396 9992 
 
 0570 9984 
 
 0744 9972 
 
 44 
 
 17 
 
 0049 one 
 
 0224 9997 
 
 0398 9992 
 
 0573 9984 
 
 0747 9972 
 
 43 
 
 18 
 
 0052 one 
 
 0227 9997 
 
 0401 9992 
 
 0576 9983 
 
 0750 9972 
 
 42 
 
 19 
 
 0055 one 
 
 0230 9997 
 
 0404 9992 
 
 0579 9983 
 
 0753 9972 
 
 41 
 
 2O 
 
 0058 one 
 
 0233 9997 
 
 0407 9992 
 
 0581 9983 
 
 0756 9971 
 
 40 
 
 21 
 
 0061 one 
 
 0236 9997 
 
 0410 9992 
 
 0584 9983 
 
 0758 9971 
 
 39 
 
 22 
 
 0064 one 
 
 0239 9997 
 
 0413 9991 
 
 0587 9983 
 
 0761 9971 
 
 38 
 
 23 
 
 0067 one 
 
 0241 9997 
 
 0416 9991 
 
 0590 9983 
 
 0764 9971 
 
 37 
 
 24 
 
 0070 one 
 
 0244 9997 
 
 0419 9991 
 
 0593 9982 
 
 0767 9971 
 
 36 
 
 25 
 
 0073 one 
 
 0247 9997 
 
 0422 9991 
 
 05% 9982 
 
 0770 9970 
 
 35 
 
 26 
 
 0076 one 
 
 0250 9997 
 
 0425 9991 
 
 0599 9982 
 
 0773 9970 
 
 34 
 
 27 
 
 0079 one 
 
 0253 9997 
 
 0427 9991 
 
 0602 9982 
 
 0776 9970 
 
 33 
 
 28 
 
 0081 one 
 
 0256 9997 
 
 0430 9991 
 
 0605 9982 
 
 0779 9970 
 
 32 
 
 29 
 
 0084 one 
 
 0259 9997 
 
 0433 9991 
 
 0608 9982 
 
 0782 9969 
 
 31 
 
 30 
 
 0087 one 
 
 0262 9997 
 
 0436 9990 
 
 0610 9981 
 
 0785 9969 
 
 3O 
 
 31 
 
 0090 one 
 
 0265 9996 
 
 0439 9990 
 
 0613 9981 
 
 0787 9969 
 
 29 
 
 32 
 
 0093 one 
 
 0268 9996 
 
 0442 9990 
 
 0616 9981 
 
 0790 9969 
 
 28 
 
 33 
 
 0096 one 
 
 0270 9996 
 
 0445 9990 
 
 0619 9981 
 
 0793 9968 
 
 27 
 
 34 
 
 0099 one 
 
 0273 9996 
 
 0448 9990 
 
 0622 9981 
 
 0796 9968 
 
 26 
 
 35 
 
 0102 9999 
 
 0276 9996 
 
 0451 9990 
 
 0625 9980 
 
 0799 9968 
 
 25 
 
 36 
 
 0105 9999 
 
 0279 9996 
 
 0454 9990 
 
 0628 9980 
 
 OS02 9968 
 
 24 
 
 37 
 
 0108 9999 
 
 0282 9996 
 
 0457 9990 
 
 0631 9980 
 
 0805 9968 
 
 23 
 
 38 
 
 0111 9999 
 
 0285 9996 
 
 0459 9989 
 
 0634 9980 
 
 0808 9967 
 
 22 
 
 39 
 
 0113 9999 
 
 0288 9996 
 
 0462 9989 
 
 0637 9980 
 
 0811 9967 
 
 21 
 
 4O 
 
 0116 9999 
 
 0291 9996 
 
 0465 9989 
 
 0640 9980 
 
 0814 9967 
 
 2O 
 
 41 
 
 0119 9999 
 
 0294 9996 
 
 0468 9989 
 
 0642 9979 
 
 0816 9967 
 
 19 
 
 42 
 
 0122 9999 
 
 0297 9996 
 
 0471 9989 
 
 0645 9979 
 
 0819 9966 
 
 18 
 
 43 
 
 0125 9999 
 
 0300 9996 
 
 0474 9989 
 
 0648 9979 
 
 0822 9966 
 
 17 
 
 44 
 
 0128 9999 
 
 0302 9995 
 
 0477 9989 
 
 0651 9979 
 
 0825 9966 
 
 16 
 
 45 
 
 0131 9999 
 
 0305 9995 
 
 0480 9988 
 
 0654 9979 
 
 0828 9966 
 
 15 
 
 46 
 
 0134 9999 
 
 0308 9995 
 
 0483 9988 
 
 0657 9978 
 
 0831 9965 
 
 14 
 
 47 
 
 0137 9999 
 
 0311 9995 
 
 0486 9988 
 
 0660 9978 
 
 0834 9965 
 
 13 
 
 48 
 
 0140 9999 
 
 0314 9995 
 
 0488 9988 
 
 0663 9978 
 
 0837 9965 
 
 12 
 
 49 
 
 0143 9999 
 
 0317 9995 
 
 0491 9988 
 
 0666 9978 
 
 08-10 9965 
 
 11 
 
 5O 
 
 0145 9999 
 
 0320 9995 
 
 0494 9988 
 
 0669 9978 
 
 0843 9964 
 
 1O 
 
 51 
 
 0148 9999 
 
 0323 9995 
 
 0497 9988 
 
 0671 9977 
 
 0845 9964 
 
 9 
 
 52 
 
 0151 9999 
 
 0326 9995 
 
 0500 9987 
 
 0674 9977 
 
 0848 9964 
 
 8 
 
 53 
 
 0154 9999 
 
 0329 9995 
 
 0503 9987 
 
 0677 9977 
 
 0851 9964 
 
 7 
 
 54 
 
 0157 9999 
 
 0332 9995 
 
 0506 9987 
 
 0680 9977 
 
 0854 9963 
 
 6 
 
 55 
 
 0160 9999 
 
 0334 9994 
 
 0509 9987 
 
 0683 9977 
 
 0857 9963 
 
 5 
 
 56 
 
 0163 9999 
 
 0337 9994 
 
 0512 9987 
 
 0686 9976 
 
 0860 9963 
 
 4 
 
 57 
 
 0166 9999 
 
 0340 9994 
 
 0515 9987 
 
 06S9 9976 
 
 0863 9963 
 
 3 
 
 58 
 
 0169 9999 
 
 0343 9994 
 
 0518 9987 
 
 0692 9976 
 
 0866 9962 
 
 2 
 
 59 
 
 0172 9999 
 
 0346 9994 
 
 0520 9986 
 
 0695 9976 
 
 0869 9962 
 
 1 
 
 60 
 
 0175 9999 
 cos sin 
 
 0349 9994 
 cos sin 
 
 0523 9986 
 
 0698 9976 
 
 0872 9962 
 
 O 
 
 t 
 
 89 
 
 88 
 
 87 
 
 86 
 
 86 
 

 
 NATURAL SINES AND COSINES. 
 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 f 
 
 
 Mil COS 
 
 sin cos 
 
 sin cos 
 
 sin cos 
 
 sin COB 
 
 
 
 
 0872 9962 
 
 1045 9945 
 
 1219 9925 
 
 1392 9903 
 
 1564 9877 
 
 '< 
 
 1 
 
 0874 9962 
 
 1048 9945 
 
 1222 9925 
 
 1395 9902 
 
 1567 9876 
 
 59 
 
 2 
 
 0877 9461 
 
 1051 9945 
 
 1224 9925 
 
 1397 9902 
 
 1570 9876 
 
 58 
 
 3 
 
 0880 9961 
 
 1054 9944 
 
 1227 9924 
 
 1400 9901 
 
 1573 9876 
 
 57 
 
 4 
 
 0883 9961 
 
 1057 9944 
 
 1230 9924 
 
 1403 9901 
 
 1576 9875 
 
 56 
 
 5 
 
 0886 9961 
 
 1060 9944 
 
 1233 9924 
 
 1406 9901 
 
 1579 9875 
 
 55 
 
 6 
 
 0889 9960 
 
 1063 9943 
 
 1236 9923 
 
 1409 9900 
 
 1582 9874 
 
 54 
 
 7 
 
 0892 9960 
 
 1066 9943 
 
 1239 9923 
 
 1412 9900 
 
 1584 9874 
 
 53 
 
 8 
 
 0895 9960 
 
 1068 9943 
 
 1241 9923 
 
 1415 9899 
 
 1587 9873 
 
 52 
 
 9 
 
 0898 9960 
 
 1071 9942. 
 
 1245 9922 
 
 1418 9899 
 
 1590 9S73 
 
 51 
 
 10 
 
 0901 9959 
 
 1074 9942 
 
 1248 9922 
 
 1421 9899 
 
 1593 9872 
 
 5O 
 
 11 
 
 0903 9959 
 
 1077 9942 
 
 1250 9922 
 
 1423 9898 
 
 1596 9872 
 
 49 
 
 12 
 
 0906 9959 
 
 1080 9942 
 
 1253 9921 
 
 1426 9898 
 
 1599 9871 
 
 48 
 
 13 
 
 0909 9959 
 
 1083 9941 
 
 1256 9921 
 
 1429 9897 
 
 1602 9871 
 
 47 
 
 14 
 
 0912 9958 
 
 1086 9941 
 
 1259 9920 
 
 1432 9897 
 
 1605 9870 
 
 46 
 
 15 
 
 09] 5 9958 
 
 1089 9941 
 
 1262 9920 
 
 1435 9897 
 
 1607 9870 
 
 45 
 
 16 
 
 0918 9958 
 
 1092 9940 
 
 1265 9920 
 
 1438 9896 
 
 1610 9869 
 
 44 
 
 17 
 
 0921 9958 
 
 1094 9940 
 
 1268 9919 
 
 1441 9896 
 
 1613 9869 
 
 43 
 
 18 
 
 0924 9957 
 
 1097 9940 
 
 1271 9919 
 
 1444 9895 
 
 1616 9869 
 
 42 
 
 19 
 
 0927 9957 
 
 1100 9939 
 
 1274 9919 
 
 1446 9895 
 
 1619 9868 
 
 41 
 
 20 
 
 0929 9957 
 
 1103 9939 
 
 1276 9918 
 
 1449 9894 
 
 1622 9868 
 
 4O 
 
 21 
 
 0932 9956 
 
 1106 9939 
 
 1279 9918 
 
 1452 9894 
 
 1625 9867 
 
 39 
 
 22 
 
 0935 9956 
 
 1109 9938 
 
 1282 9917 
 
 1455 9894 
 
 1628 9867 
 
 38 
 
 23 
 
 0938 9956 
 
 1112 9938 
 
 1285 9917 
 
 1458 9893 
 
 1630 9866 
 
 37 
 
 24 
 
 0941 9956 
 
 1115 9938 
 
 1288 9917 
 
 1461 9893 
 
 1633 9866 
 
 36 
 
 25 
 
 0944 9955 
 
 1118 9937 
 
 1291 9916 
 
 1464 9892 
 
 1636 9865 
 
 35 
 
 26 
 
 0947 9955 
 
 1120 9937 
 
 1294 9916 
 
 1467 9892 
 
 1639 9865 
 
 34 
 
 27 
 
 0950 9955 
 
 1123 9937 
 
 1297 9916 
 
 1469 9891 
 
 1642 9864 
 
 33 
 
 28 
 
 0953 9955 
 
 1126 9936 
 
 1299 9915 
 
 1472 9891 
 
 1645 9864 
 
 32 
 
 29 
 
 0956 9954 
 
 1129 9936 
 
 1302 9915 
 
 1475 9891 
 
 1648 9863 
 
 31 
 
 30 
 
 0958 9954 
 
 1132 9936 
 
 1305 9914 
 
 1478 9890 
 
 1650 9863 
 
 3O 
 
 31 
 
 0961 9954 
 
 1135 9935 
 
 1308 9914 
 
 1481 9890 
 
 1653 9862 
 
 29 
 
 32 
 
 9964 9953 
 
 1138 9935 
 
 1311 9914 
 
 1484 9889 
 
 1656 9862 
 
 28 
 
 33 
 
 0967 9953 
 
 1141 9935 
 
 1314 9913 
 
 1487 9889 
 
 1659 9861 
 
 27 
 
 34 
 
 0970 9953 
 
 1144 9934 
 
 1317 9913 
 
 1490 9888 
 
 1662 9861 
 
 26 
 
 35 
 
 0973 9553 
 
 1146 9934 
 
 1320 9913 
 
 1492 9888 
 
 1665 9860 
 
 25 
 
 36 
 
 0976 9952 
 
 1149 9934 
 
 1323 9912 
 
 1495 9888 
 
 1668 9860 
 
 24 
 
 37 
 
 0979 9952 
 
 1152 9933 
 
 1325 9912 
 
 1498 9887 
 
 1671 9859 
 
 23 
 
 38 
 
 0982 9952 
 
 1155 9933 
 
 1328 9911 
 
 1501 9887 
 
 1673 9859 
 
 22 
 
 39 
 
 0985 9951 
 
 1158 9933 
 
 1331 9911 
 
 1504 9886 
 
 1676 9859 
 
 21 
 
 4O 
 
 0987 9951 
 
 1161 9932 
 
 1334 9911 
 
 1507 9886 
 
 1679 9858 
 
 2O 
 
 41 
 
 0990 9951 
 
 1164 9932 
 
 1337 9910 
 
 1510 9885 
 
 1682 9858 
 
 19 
 
 42 
 
 0993 9951 
 
 1167 9932 
 
 1340 9910 
 
 1513 9885 
 
 1685 9857 
 
 18 
 
 43 
 
 0996 9950 
 
 1170 9931 
 
 1343 9909 
 
 1515 9884 
 
 1688 9857 
 
 17 
 
 44 
 
 0999 9950 
 
 1172 9931 
 
 1346 9909 
 
 1518 9884 
 
 1691 9856 
 
 16 
 
 45 
 
 1002 9950 
 
 1175 9931 
 
 1349 9909 
 
 1521 9884 
 
 1693 9856 
 
 15 
 
 46 
 
 1005 9949 
 
 1178 9930 
 
 1351 9908 
 
 1524 9883 
 
 1696 9855 
 
 14 
 
 47 
 
 1008 9949 
 
 1181 9930 
 
 1354 9908 
 
 1527 9883 
 
 1699 9855 
 
 13 
 
 48 
 
 1011 9949 
 
 1184 9930 
 
 1357 9907 
 
 1530 9882 
 
 1702 9854 
 
 12 
 
 49 
 
 1013 9949 
 
 1187 9929 
 
 1360 9907 
 
 1533 9882 
 
 1705 9854 
 
 11 
 
 5O 
 
 1016 9948 
 
 1190 9929 
 
 1363 9907 
 
 1536 9881 
 
 1708 9853 
 
 1O 
 
 51 
 
 1019 9948 
 
 1193 9929 
 
 1366 9906 
 
 1538 9881 
 
 1711 9853 
 
 9 
 
 52 
 
 1022 9948 
 
 11% 9928 
 
 1369 9906 
 
 1541 9880 
 
 1714 9852 
 
 8 
 
 53 
 
 1025 9947 
 
 1198 9928 
 
 1372 9905 
 
 1544 9880 
 
 1716 9852 
 
 7 
 
 54 
 
 1028 9947 
 
 1201 9928 
 
 1374 9905 
 
 1547 9880 
 
 1719 9851. 
 
 6 
 
 55 
 
 1031 9947 
 
 1204 9927 
 
 1377 9905 
 
 1550 9879 
 
 1722 9851 
 
 5 
 
 56 
 
 1034 9946 
 
 1207 9927 
 
 1380 9904 
 
 1553 9879 
 
 1725 9850 
 
 4 
 
 57 
 
 1037 9946 
 
 1210 9927 
 
 1383 9904 
 
 1556 9878 
 
 1728 9850 
 
 3 
 
 58 
 
 1039 9946 
 
 1213 9926 
 
 1386 9903 
 
 1559 9878 
 
 1731 9849 
 
 2 
 
 59 
 
 1042 9946 
 
 1216 9926 
 
 1389 9903 
 
 1561 9877 
 
 1734 9849 
 
 1 
 
 60 
 
 1045 9945 
 
 1219 9925 
 
 1392 9903 
 
 1564 9877 
 
 1736 9848 
 
 
 
 
 cos sin 
 
 cos sin 
 
 cos ftin 
 
 cos sin 
 
 COB Bin 
 
 
 f 
 
 84 
 
 83 
 
 82 
 
 81 
 
 80
 
 64 
 
 NATURAL SINES AND COSINES. 
 
 f 
 
 1O 
 
 11 
 
 12 
 
 13 
 
 14 
 
 r 
 
 
 sin cos 
 
 sin cos 
 
 sin cos 
 
 sin cos 
 
 sin cos 
 
 
 o 
 
 1736 9848 
 
 1908 9816 
 
 2079 9781 
 
 2250 9744 
 
 2419 9703 
 
 6O 
 
 1 
 
 1739 9848 
 
 1911 9816 
 
 2082 9781 
 
 2252 9743 
 
 2422 9702 
 
 59 
 
 2 
 
 1742 9847 
 
 1914 9815 
 
 2085 9780 
 
 2255 9742 
 
 2425 9702 
 
 58 
 
 3 
 
 1745 9847 
 
 1917 9815 
 
 2088 9780 
 
 2258 9742 
 
 2428 9701 
 
 57 
 
 4 
 
 1748 9846 
 
 1920 9814 
 
 2090 9779 
 
 2261 9741 
 
 2431 9700 
 
 56 
 
 5 
 
 1751 9846 
 
 1922 9813 
 
 2093 9778 
 
 2264 9740 
 
 2433 9699 
 
 55 
 
 6 
 
 1754 9845 
 
 1925 9813 
 
 2096 9778 
 
 2267 9740 
 
 2436 9699 
 
 54 
 
 7 
 
 1757 9845 
 
 1928 9812 
 
 2099 9777 
 
 2269 9739 
 
 2439 9698 
 
 53 
 
 8 
 
 1759 9844 
 
 1931 9812 
 
 2102 9777 
 
 2272 9738 
 
 2442 9697 
 
 52 
 
 9 
 
 1762 9843 
 
 1934 9811 
 
 2105 9776 
 
 2275 9738 
 
 2445 9697 
 
 51 
 
 1O 
 
 1765 9843 
 
 1937 9811 
 
 2108 9775 
 
 2278 9737 
 
 2447 9696 
 
 50 
 
 11 
 
 1768 9842 
 
 1939 9810 
 
 2110 9775 
 
 2281 9736 
 
 2450 9695 
 
 49 
 
 12 
 
 1771 9842 
 
 1942 9810 
 
 2113 9774 
 
 2284 9736 
 
 2453 9694 
 
 48 
 
 13 
 
 1774 9841 
 
 1945 9809 
 
 2116 9774 
 
 2286 9735 
 
 2456 9694 
 
 47 
 
 14 
 
 1777 9841 
 
 1948 9808 
 
 2119 9773 
 
 2289 9734 
 
 2459 9693 
 
 46 
 
 15 
 
 1779 9840 
 
 1951 9808 
 
 2122 9772 
 
 2292 9734 
 
 2462 9692 
 
 45 
 
 16 
 
 1782 9840 
 
 1954 9S07 
 
 2125 9772 
 
 2295 9733 
 
 2464 9692 
 
 44 
 
 17 
 
 1785 9839 
 
 1957 9807 
 
 2127 9771 
 
 2298 9732 
 
 2467 9691 
 
 43 
 
 18 
 
 1788 9839 
 
 1959 9806 
 
 2130 9770 
 
 2300 9732 
 
 2470 9690 
 
 42 
 
 19 
 
 1791 9838 
 
 1962 9806 
 
 2133 9770 
 
 2303 9731 
 
 2473 9689 
 
 41 
 
 2O 
 
 1794 9838 
 
 1965 9805 
 
 2136 9769 
 
 2306 9730 
 
 2476 9689 
 
 4O 
 
 21 
 
 1797 9837 
 
 1968 9804 
 
 2139 9769 
 
 2309 9730 
 
 2478 9688 
 
 39 
 
 22 
 
 1799 9837 
 
 1971 9804 
 
 2142 9768 
 
 2312 9729 
 
 2481 9687 
 
 38 
 
 23 
 
 1802 9836 
 
 1974 9803 
 
 2145 9767 
 
 2315 9728 
 
 2484 9687 
 
 37 
 
 24 
 
 1805 9836 
 
 1977 9803 
 
 2147 9767 
 
 2317 9728 
 
 2487 9686 
 
 36 
 
 25 
 
 1808 9835 
 
 1979 9802 
 
 2150 9766 
 
 2320 9727 
 
 2490 9685 
 
 35 
 
 26 
 
 1811 9835 
 
 1982 9802 
 
 2153 9765 
 
 2323 9726 
 
 2493 9684 
 
 34 
 
 27 
 
 1814 9S34 
 
 1985 9801 
 
 2156 9765 
 
 2326 9726 
 
 2495 9684 
 
 33 
 
 28 
 
 1817 9834 
 
 1988 9800 
 
 2159 9764 
 
 2329 9725 
 
 2498 9683 
 
 32 
 
 29 
 
 1819 9833 
 
 1991 9800 
 
 2162 9764 
 
 2332 9724 
 
 2501 9682 
 
 31 
 
 30 
 
 1822 9833 
 
 1994 9799 
 
 2164 9763 
 
 2334 9724 
 
 2504 9681 
 
 30 
 
 31 
 
 1825 9832 
 
 1997 9799 
 
 2167 9762 
 
 2337 9723 
 
 2507 9681 
 
 29 
 
 32 
 
 1828 9831 
 
 1999 9798 
 
 2170 9762 
 
 2340 9722 
 
 2509 9680 
 
 28 
 
 33 
 
 1831 9831 
 
 2002 9798 
 
 2173 9761 
 
 3343 9722 
 
 2512 9679 
 
 27 
 
 34 
 
 1834 9830 
 
 2005 9797 
 
 2176 9760 
 
 2346 9721 
 
 2515 9679 
 
 26 
 
 35 
 
 1837 9830 
 
 2008 9796 
 
 2179 9760 
 
 2349 9720 
 
 2518 9678 
 
 25 
 
 36 
 
 1840 9829 
 
 2011 9796 
 
 2181 9759 
 
 2351 9720 
 
 2521 9677 
 
 24 
 
 37 
 
 1842 9829 
 
 2014 9795 
 
 2184 9759 
 
 2354 9719 
 
 2524 9676 
 
 23 
 
 38 
 
 1845 9828 
 
 2016 9795 
 
 2187 9758 
 
 2357 9718 
 
 2526 9676 
 
 22 
 
 39 
 
 1848 9828 
 
 2019 9794 
 
 2190 9757 
 
 2360 9718 
 
 2529 9675 
 
 21 
 
 40 
 
 1851 9827 
 
 2022 9793 
 
 2193 9757 
 
 2363 9717 
 
 2532 9674 
 
 20 
 
 41 
 
 1854 9827 
 
 2025 9793 
 
 2196 9756 
 
 2366 9716 
 
 2535 9673 
 
 19 
 
 42 
 
 1857 9826 
 
 2028 9792 
 
 2198 9755 
 
 2368 9715 
 
 2538 9673 
 
 18 
 
 43 
 
 1860 9826 
 
 2031 9792 
 
 2201 9755 
 
 2371 9715 
 
 2540 9672 
 
 17 
 
 44 
 
 1862 9825 
 
 2034 9791 
 
 2204 9754 
 
 2374 9714 
 
 2543 9671 
 
 16 
 
 45 
 
 1865 9825 
 
 2036 9790 
 
 2207 9753 
 
 2377 9713 
 
 2546 9670 
 
 15 
 
 46 
 
 1868 9824 
 
 2039 9790 
 
 2210 9753 
 
 2380 9713 
 
 2549 9670 
 
 14 
 
 47 
 
 1871 9823 
 
 2042 9789 
 
 2213 9752 
 
 2383 9712 
 
 2552 9669 
 
 13 
 
 48 
 
 1474 9823 
 
 2045 9789 
 
 2215 9751 
 
 2385 9711 
 
 2554 9668 
 
 12 
 
 49 
 
 1877 9822 
 
 2048 9788 
 
 2218 9751 
 
 2388 9711 
 
 2557 9667 
 
 11 
 
 50 
 
 1880 9822 
 
 2051 9787 
 
 2221 9750 
 
 2391 9710 
 
 2560 9667 
 
 10 
 
 51 
 
 1882 9821 
 
 2054 9787 
 
 2224 9750 
 
 2394 9709 
 
 2563 9666 
 
 9 
 
 52 
 
 1885 9821 
 
 2056 9786 
 
 2227 9749 
 
 2397 9709 
 
 2566 9665 
 
 8 
 
 53 
 
 1888 9820 
 
 2059 9786 
 
 2230 9748 
 
 2399 9708 
 
 2569 9665 
 
 7 
 
 54 
 
 1891 9820 
 
 2062 9785 
 
 2233 9748 
 
 2402 9707 
 
 2571 9664 
 
 6 
 
 55 
 
 1894 9819 
 
 2065 9784 
 
 2235 9747 
 
 2405 9706 
 
 2574 9663 
 
 5 
 
 56 
 
 1897 9818 
 
 2068 9784 
 
 2238 9746 
 
 2408 9706 
 
 2577 9662 
 
 4 
 
 57 
 
 1900 9818 
 
 2071 9783 
 
 2241 9746 
 
 2411 9705 
 
 2580 9662 
 
 3 
 
 58 
 
 1902 9817 
 
 2073 9783 
 
 2244 9745 
 
 2414 9704 
 
 2583 9661 
 
 2 
 
 59 
 
 1905 9817 
 
 2076 9782 
 
 2247 9744 
 
 2416 9704 
 
 2585 9660 
 
 1 
 
 60 
 
 1908 9816 
 
 2079 9781 
 
 2250 9744 
 
 2419 9703 
 
 2588 9659 
 
 
 
 
 cos sin 
 
 cos sill 
 
 cos sin 
 
 cos sin 
 
 cos sin 
 
 
 f 
 
 79 
 
 78 
 
 77 
 
 76 
 
 75 
 
 r
 
 NATURAL SINES AND COSINES. 
 
 65 
 
 f 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 t 
 
 
 sin cos 
 
 Bin cos 
 
 sin cos 
 
 sin cos 
 
 sin cos 
 
 
 o 
 
 25S8 9659 
 
 2756 9613 
 
 2924 9563 
 
 3090 9511 
 
 3256 9455 
 
 6O 
 
 1 
 
 2591 9659 
 
 2759 9612 
 
 2926 9562 
 
 3093 9510 
 
 3258 9454 
 
 59 
 
 2 
 
 2594 9658 
 
 2762 9611 
 
 2929 9561 
 
 30% 9509 
 
 3261 9453 
 
 58 
 
 3 
 
 2597 9657 
 
 2765 9610 
 
 2932 9560 
 
 3098 9508 
 
 3264 9452 
 
 57 
 
 4 
 
 2599 9656 
 
 2768 9609 
 
 2935 9560 
 
 3101 9507 
 
 3267 9451 
 
 56 
 
 5 
 
 2602 9655 
 
 2770 9609 
 
 2938 9559 
 
 3104 9506 
 
 3269 9450 
 
 55 
 
 6 
 
 2605 9655 
 
 2773 9608 
 
 2940 9558 
 
 3107 9505 
 
 3272 9449 
 
 54 
 
 7 
 
 2608 9654 
 
 2776 9607 
 
 2943 9557 
 
 3110 9504 
 
 3275 9449 
 
 53 
 
 8 
 
 2611 9653 
 
 2779 9606 
 
 2946 9556 
 
 3112 9503 
 
 3278 9448 
 
 52 
 
 9 
 
 2613 9652 
 
 2782 9605 
 
 2949 9555 
 
 3115 9502 
 
 3280 9447 
 
 51 
 
 10 
 
 2616 9652 
 
 2784 9605 
 
 2952 9555 
 
 3118 9502 
 
 3283 9446 
 
 5O 
 
 11 
 
 2619 9651 
 
 2787 9604 
 
 2954 9554 
 
 3121 9501 
 
 3286 9445 
 
 49 
 
 12 
 
 2622 9650 
 
 2790 9603 
 
 2957 9553 
 
 3123 9500 
 
 3289 9444 
 
 48 
 
 13 
 
 2625 9649 
 
 2793 9602 
 
 2960 9552 
 
 3126 9499 
 
 3291 9443 
 
 47 
 
 14 
 
 2628 9649 
 
 2795 9601 
 
 2963 9551 
 
 3129 9498 
 
 3294 9442 
 
 46 
 
 15 
 
 2630 9648 
 
 2798 9600 
 
 2965 9550 
 
 3132 9497 
 
 3297 9441 
 
 45 
 
 16 
 
 2633 9647 
 
 2801 9600 
 
 2968 9549 
 
 3134 94% 
 
 3300 9440 
 
 44 
 
 17 
 
 2636 9646 
 
 2804 9599 
 
 2971 9548 
 
 3137 9495 
 
 3302 9439 
 
 43 
 
 18 
 
 2639 9646 
 
 2807 9598 
 
 2974 9548 
 
 3140 9494 
 
 3305 9438 
 
 42 
 
 19 
 
 2642 9645 
 
 2809 9597 
 
 2977 9547 
 
 3143 9493 
 
 3308 9437 
 
 41 
 
 2O 
 
 2644 9644 
 
 2812 95% 
 
 2979 9546 
 
 3145 9492 
 
 3311 9436 
 
 JO 
 
 21 
 
 2647 9643 
 
 2815 9596 
 
 2982 9545 
 
 3148 9492 
 
 3313 9435 
 
 39 
 
 22 
 
 2650 9642 
 
 2818 9595 
 
 2985 9544 
 
 3151 9491 
 
 3316 9434 
 
 38 
 
 23 
 
 2653 9642 
 
 2821 9594 
 
 2988 9543 
 
 3154 9490 
 
 3319 9433 
 
 37 
 
 24 
 
 2656 9641 
 
 2823 9593 
 
 2990 9542 
 
 3156 9489 
 
 3322 9432 
 
 36 
 
 25 
 
 2658 9640 
 
 2826 9592 
 
 2993 9542 
 
 3159 9488 
 
 3324 9431 
 
 35 
 
 26 
 
 2661 9639 
 
 2829 9591 
 
 29% 9541 
 
 3162 9487 
 
 3327 9430 
 
 34 
 
 27 
 
 2664 9639 
 
 2832 9591 
 
 2999 9540 
 
 3165 9486 
 
 3330 9429 
 
 33 
 
 28 
 
 2667 9638 
 
 2835 9590 
 
 3002 9539 
 
 3168 9485 
 
 3333 9428 
 
 32 
 
 29 
 
 2670 9637 
 
 2837 9589 
 
 3004 9538 
 
 3170 9484 
 
 3335 9427 
 
 31 
 
 30 
 
 2672 9636 
 
 2840 9588 
 
 3007 9537 
 
 3173 9483 
 
 3338 9426 
 
 3O 
 
 31 
 
 2675 9636 
 
 2843 9587 
 
 3010 9536 
 
 3176 9482 
 
 3341 9425 
 
 29 
 
 32 
 
 2678 9635 
 
 2846 9587 
 
 3013 9535 
 
 3179 9481 
 
 3344 9424 
 
 28 
 
 33 
 
 2681 9634 
 
 2849 9586 
 
 3015 9535 
 
 3181 9480 
 
 3346 9423 
 
 27 
 
 34 
 
 2684 9633 
 
 2851 9585 
 
 3018 9534 
 
 3184 9480 
 
 3349 9423 
 
 26 
 
 35 
 
 2686 9632 
 
 2854 9584 
 
 3021 9533 
 
 3187 9479 
 
 3352 9422 
 
 25 
 
 36 
 
 2689 9632 
 
 2857 9583 
 
 3024 9532 
 
 3190 9478 
 
 3355 9421 
 
 24 
 
 37 
 
 2692 9631 
 
 2860 9582 
 
 3026 9531 
 
 3192 9477 
 
 3357 9420 
 
 23 
 
 38 
 
 2695 9630 
 
 2862 9582 
 
 3029 9530 
 
 3195 9476 
 
 3360 9419 
 
 22 
 
 39 
 
 2698 9629 
 
 2865 9581 
 
 3032 9529 
 
 3198 9475 
 
 3363 9418 
 
 21 
 
 l 
 
 2700 9628 
 
 2868 9580 
 
 3035 9528 
 
 3201 9474 
 
 3365 9417 
 
 20 
 
 41 
 
 2703 9628 
 
 2871 9579 
 
 3038 9527 
 
 3203 9473 
 
 3368 9416 
 
 19 
 
 42 
 
 2706 9627 
 
 2874 9578 
 
 3040 9527 
 
 3206 9472 
 
 3371 9415 
 
 18 
 
 43 
 
 2709 9626 
 
 2876 9577 
 
 3043 9526 
 
 3209 9471 
 
 3374 9414 
 
 17 
 
 44 
 
 2712 9625 
 
 2879 9577 
 
 3046 9525 
 
 3212 9470 
 
 3376 9413 
 
 16 
 
 45 
 
 2714 9625 
 
 2882 9576 
 
 3049 9524 
 
 3214 9469 
 
 3379 9412 
 
 15 
 
 46 
 
 2717 9624 
 
 2885 9575 
 
 3051 9523 
 
 3217 9468 
 
 3382 9411 
 
 14 
 
 47 
 
 2720 9623 
 
 2888 9574 
 
 3054 9522 
 
 3220 9467 
 
 3385 9410 
 
 13 
 
 48 
 
 2723 9622 
 
 2890 9573 
 
 3057 9521 
 
 3223 9466 
 
 3387 9409 
 
 12 
 
 49 
 
 2726 9621 
 
 2893 9572 
 
 3060 9520 
 
 3225 9466 
 
 3390 9408 
 
 11 
 
 5O 
 
 2728 9621 
 
 2896 9572 
 
 3062 9520 
 
 3228 9465 
 
 3393 9407 
 
 1O 
 
 51 
 
 2731 9620 
 
 2899 9571 
 
 3065 9519 
 
 3231 9464 
 
 33% 9406 
 
 9 
 
 52 
 
 2734 9619 
 
 2901 9570 
 
 3068 9518 
 
 3234 9463 
 
 3398 9405 
 
 8 
 
 53 
 
 2737 9618 
 
 2904 9569 
 
 3071 9517 
 
 3236 9462 
 
 3401 9404 
 
 7 
 
 54 
 
 2740 9617 
 
 2907 9568 
 
 3074 9516 
 
 3239 9461 
 
 3404 9403 
 
 6 
 
 55 
 
 2742 9617 
 
 2910 9567 
 
 3076 9515 
 
 3242 9460 
 
 3407 9402 
 
 5 
 
 56 
 
 2745 9616 
 
 2913 9566 
 
 3079 9514 
 
 3245 9459 
 
 3409 9401 
 
 4 
 
 57 
 
 2748 9615 
 
 2915 9566 
 
 3082 9513 
 
 3247 9458 
 
 3412 9400 
 
 3 
 
 58 
 
 2751 9614 
 
 2918 9565 
 
 3085 9512 
 
 3250 9457 
 
 3415 9399 
 
 2 
 
 59 
 
 2754 9613 
 
 2921 9564 
 
 30S7 9511 
 
 3253 9456 
 
 3417 9398 
 
 1 
 
 60 
 
 2756 9613 
 
 2924 9563 
 
 3090 9511 
 
 3256 9455 
 
 3420 9397 
 
 
 
 
 cos Bin 
 
 cos sin 
 
 cos sin 
 
 cos sin 
 
 cos sin 
 
 
 / 
 
 74 o 
 
 73 
 
 72 
 
 71 
 
 7O '
 
 66 NATURAL SINES AND COSINES. 
 
 t 
 
 2O 
 
 21 
 
 22 
 
 23 
 
 24 
 
 f 
 
 
 sin cos 
 
 sin cos 
 
 sin cos 
 
 sin cos 
 
 sin cos 
 
 
 o 
 
 3420 9397 
 
 3584 9336 
 
 3746 9272 
 
 3907 9205 
 
 4067 9135 
 
 6O 
 
 1 
 
 3423 9396 
 
 3586 9335 
 
 3749 9271 
 
 3910 9204 
 
 4070 9134 
 
 59 
 
 2 
 
 3426 9395 
 
 3589 9334 
 
 3751 9270 
 
 3913 9203 
 
 4073 9133 
 
 58 
 
 3 
 
 3428 9394 
 
 3592 9333 
 
 3754 9269 
 
 3915 9202 
 
 4075 9132 
 
 57 
 
 4 
 
 3431 9393 
 
 3595 9332 
 
 3757 9267 
 
 3918 9200 
 
 4078 9131 
 
 56 
 
 5 
 
 3434 9392 
 
 3597 9331 
 
 3760 9266 
 
 3921 9199 
 
 4081 9130 
 
 55 
 
 6 
 
 3437 9391 
 
 3600 9330 
 
 3762 9265 
 
 3923 9198 
 
 4083 9128 | 54 
 
 7 
 
 3439 9390 
 
 3603 9328 
 
 3765 9264 
 
 3926 9197 
 
 4086 9127 
 
 53 
 
 8 
 
 3442 9389 
 
 3605 9327 
 
 3768 9263 
 
 3929 9196 
 
 4089 9126 
 
 52 
 
 9 
 
 3445 9388 
 
 3608 9326 
 
 3770 9262 
 
 3931 9195 
 
 4091 9125 
 
 51 
 
 1O 
 
 3448 9387 
 
 3611 9325 
 
 3773 9261 
 
 3934 9194 
 
 4094 9124 
 
 5O 
 
 11 
 
 3450 9386 
 
 3614 9324 
 
 3776 9260 
 
 3937 9192 
 
 4097 9122 
 
 49 
 
 12 
 
 3453 9385 
 
 3616 9323 
 
 3778 9259 
 
 3939 9191 
 
 4099 9121 
 
 48 
 
 13 
 
 3456 9384 
 
 3619 9322 
 
 3781 9258 
 
 3942 9190 
 
 4102 9120 
 
 47 
 
 14 
 
 3458 9383 
 
 3622 9321 
 
 3784 9257 
 
 3945 9189 
 
 4105 9119 
 
 46 
 
 15 
 
 3461 9382 
 
 3624 9320 
 
 3786 9255 
 
 3947 9188 
 
 4107 9118 
 
 45 
 
 16 
 
 3464 9381 
 
 3627 9319 
 
 3789 9254 
 
 3950 9187 
 
 4110 9116 
 
 44 
 
 17 
 
 3467 9380 
 
 3630 9318 
 
 3792 9253 
 
 3953 9186 
 
 4112 9115 i 43 
 
 18 
 
 3469 9379 
 
 3633 9317 
 
 3795 9252 
 
 3955 9184 
 
 4115 9114 42 
 
 19 
 
 3472 9378 
 
 3635 9316 
 
 3797 9251 
 
 3958 9183 
 
 4118 9113 
 
 41 
 
 2O 
 
 3475 9377 
 
 3638 9315 
 
 3800 9250 
 
 3961 9182 
 
 4120 9112 
 
 4O 
 
 21 
 
 3478 9376 
 
 3641 9314 
 
 3803 9249 
 
 3963 9181 
 
 4123 9110 
 
 39 
 
 22 
 
 3480 9375 
 
 3643 9313 
 
 3805 9248 
 
 3966 9180 
 
 4126 9109 
 
 38 
 
 23 
 
 3483 9374 
 
 3646 9312 
 
 3808 9247 
 
 3969 9179 
 
 4128 9108 
 
 37 
 
 24 
 
 3486 9373 
 
 3649 9311 
 
 3811 9245 
 
 3971 9178 
 
 4131 9107 
 
 36 
 
 25 
 
 3488 9372 
 
 3651 9309 
 
 3813 9244 
 
 3974 9176 
 
 4134 9106 
 
 35 
 
 26 
 
 3491 9371 
 
 3654 9308 
 
 3816 9243 
 
 3977 9175 
 
 4136 9104 
 
 34 
 
 27 
 
 3494 9370 
 
 3657 9307 
 
 3819 9242 
 
 3979 9174 
 
 4139 9103 ! 33 
 
 28 
 
 3497 9369 
 
 3660 9306 
 
 3821 9241 
 
 3982 9173 
 
 4142 9102 
 
 32 
 
 29 
 
 3499 9368 
 
 3662 9305 
 
 3824 9240 
 
 3985 9172 
 
 4144 9101 
 
 31 
 
 30 
 
 3502 9367 
 
 3665 9304 
 
 3827 9239 
 
 3987 9171 
 
 4147 9100 
 
 3O 
 
 31 
 
 3505 9366 
 
 3668 9303 
 
 3830 9238 
 
 3990 9169 
 
 4150 9098 
 
 29 
 
 32 
 
 3508 9365 
 
 3670 9302 
 
 3832 9237 
 
 3993 9168 
 
 4152 9097 
 
 28 
 
 33 
 
 3510 9364 
 
 3673 9301 
 
 3835 9235 
 
 3995 9167 
 
 4155 9096 
 
 27 
 
 34 
 
 3513 9363 
 
 3676 9300 
 
 3838 9234 
 
 3998 9166 
 
 4158 9095 
 
 26 
 
 35 
 
 3516 9362 
 
 3679 9299 
 
 3840 9233 
 
 4001 9165 
 
 4160 9094 
 
 25 
 
 36 
 
 3518 9361 
 
 3681 9298 
 
 3843 9232 
 
 4003 9164 
 
 4163 9092 
 
 24 
 
 37 
 
 3521 9360 
 
 3684 9297 
 
 3846 9231 
 
 4006 9162 
 
 4165 9091 
 
 23 
 
 38 
 
 3524 9359 
 
 3687 9296 
 
 3848 9230 
 
 4009 9161 
 
 4168 9090 
 
 22 
 
 39 
 
 3527 9358 
 
 3689 9295 
 
 3851 9229 
 
 4011 9160 
 
 4171 9089 
 
 21 
 
 40 
 
 3529 9356 
 
 3692 9293 
 
 3854 9228 
 
 4014 9159 
 
 4173 9088 
 
 2O 
 
 41 
 
 3532 9355 
 
 3695 9292 
 
 3856 9227 
 
 4017 9158 
 
 4176 9086 
 
 19 
 
 42 
 
 3535 9354 
 
 3697 9291 
 
 3859 9225 
 
 4019 9157 
 
 4179 9085 
 
 18 
 
 43 
 
 3537 9353 
 
 3700 9290 
 
 3862 9224 
 
 4022 9155 
 
 4181 9084 
 
 17 
 
 44 
 
 3540 9352 
 
 3703 9289 
 
 3864 9223 
 
 4025 9154 
 
 4184 9083 
 
 16 
 
 45 
 
 3543 9351 
 
 3706 9288 
 
 3867 9222 
 
 4027 9153 
 
 4187 9081 
 
 15 
 
 46 
 
 3546 9350 
 
 3708 9287 
 
 3870 9221 
 
 4030 9152 
 
 4189 9080 
 
 14 
 
 47 
 
 3548 9349 
 
 3711 9286 
 
 3872 9220 
 
 4033 9151 
 
 4192 9079 
 
 13 
 
 48 
 
 3551 9348 
 
 3714 9285 
 
 3875 9219 
 
 4035 9150 
 
 4195 9078 
 
 12 
 
 49 
 
 3554 9347 
 
 3716 9284 
 
 3878 9218 
 
 4038 9148 
 
 4197 9077 
 
 11 
 
 50 
 
 3557 9346 
 
 3719 9283 
 
 3881 9216 
 
 4041 9147 
 
 4200 9075 
 
 10 
 
 51 
 
 3559 9345 
 
 3722 9282 
 
 3883 9215 
 
 4043 9146 
 
 4202 9074 
 
 9 
 
 52 
 
 3562 9344 
 
 3724 9281 
 
 3886 9214 
 
 4046 9145 
 
 4205 9073 
 
 8 
 
 53 
 
 3565 9343 
 
 3727 9279 
 
 3889 9213 
 
 4049 9144 
 
 4208 9072 
 
 7 
 
 54 
 
 3567 9342 
 
 3730 9278 
 
 3891 9212 
 
 4051 9143 
 
 4210 9070 
 
 6 
 
 55 
 
 3570 9341 
 
 3733 9277 
 
 3894 9211 
 
 4054 9141 
 
 4213 9069 
 
 5 
 
 56 
 
 3573 9340 
 
 3735 9276 
 
 3897 9210 
 
 4057 9140 
 
 4216 9068 
 
 4 
 
 57 
 
 3576 9339 
 
 3738 9275 
 
 3899 9208 
 
 4059 9139 
 
 4218 9067 
 
 3 
 
 58 
 
 3578 9338 
 
 3741 9274 
 
 3902 9207 
 
 4062 9138 
 
 4221 9066 
 
 2 
 
 59 
 
 3581 9337 
 
 3743 9273 
 
 3905 9206 
 
 4065 9137 
 
 4224 9064 
 
 1 
 
 6O 
 
 3584 9336 
 
 3746 9272 
 
 3907 9205 
 
 4067 9135 
 
 4226 9063 
 
 O 
 
 
 cos gin 
 
 cos sin 
 
 cos sin 
 
 cos sin 
 
 cos sin 
 
 
 r 
 
 69 
 
 68 
 
 67 
 
 66 
 
 65 '
 
 NATURAL SINES AND COSINES. 
 
 
 25 
 
 26 
 
 27 
 
 28 
 
 2i> 
 
 t 
 
 
 sin cos 
 
 sin cos 
 
 sin cos 
 
 sin cos 
 
 sin cog 
 
 
 o 
 
 4226 9063 
 
 4384 8988 
 
 4540 8910 
 
 4695 8829 
 
 4848 8746 
 
 6O 
 
 1 
 
 4229 9062 
 
 4386 8987 
 
 4542 8909 
 
 4697 8828 
 
 4851 8745 
 
 59 
 
 2 
 
 4231 9061 
 
 4389 8985 
 
 4545 8907 
 
 4700 8827 
 
 4853 8743 
 
 58 
 
 3 
 
 4234 9059 
 
 4392 8984 
 
 4548 8906 
 
 4702 8825 
 
 4856 8742 
 
 57 
 
 4 
 
 4237 9058 
 
 4394 8983 
 
 4550 8905 
 
 4705 8824 
 
 4858 8741 
 
 56 
 
 5 
 
 4239 9057 
 
 4397 8982 
 
 4553 8903 
 
 4708 8823 
 
 4861 8739 
 
 55 
 
 6 
 
 4242 9056 
 
 4399 8980 
 
 4555 8902 
 
 4710 8821 
 
 4863 8738 
 
 54 
 
 7 
 
 4245 9054 
 
 4402 8979 
 
 4558 8901 
 
 47J 3 8820 
 
 4866 8736 
 
 53 
 
 8 
 
 4247 9053 
 
 4405 8978 
 
 4561 8899 
 
 4715 8819 
 
 4868 8735 
 
 52 
 
 9 
 
 4250 9052 
 
 4407 8976 
 
 4563 8898 
 
 4718 8817 
 
 4871 8733 
 
 51 
 
 10 
 
 4253 9051 
 
 4410 8975 
 
 4566 8897 
 
 4720 8816 
 
 4874 8732 
 
 5O 
 
 11 
 
 4255 9050 
 
 4412 8974 
 
 4568 8895 
 
 4723 8814 
 
 4876 8731 
 
 49 
 
 12 
 
 4258 9048 
 
 4415 8973 
 
 4571 8894 
 
 4726 8813 
 
 4879 8729 
 
 48 
 
 13 
 
 4260 9047 
 
 4418 8971 
 
 4574 8893 
 
 4728 8812 
 
 4881 8728 
 
 47 
 
 14 
 
 4263 9046 
 
 4420 8970 
 
 4576 8892 
 
 4731 8810 
 
 4884 8726 
 
 46 
 
 15 
 
 4266 9045 
 
 4423 8969 
 
 4579 8890 
 
 4733 8809 
 
 4886 8725 
 
 45 
 
 16 
 
 4268 9043 
 
 4425 8967 
 
 4581 8889 
 
 4736 8808 
 
 4889 8724 
 
 44 
 
 17 
 
 4271 9042 
 
 4428 8966 
 
 4584 8888 
 
 4738 8806 
 
 4891 8722 
 
 43 
 
 18 
 
 4274 9041 
 
 4431 S965 
 
 4586 8886 
 
 4741 8805 
 
 4894 8721 
 
 42 
 
 19 
 
 4276 9040 
 
 4433 8964 
 
 4589 8885 
 
 4743 8803 
 
 4896 8719 
 
 41 
 
 20 
 
 4279 9038 
 
 4436 S962 
 
 4592 8884 
 
 4746 8S02 
 
 4899 8718 
 
 40 
 
 21 
 
 4281 9037 
 
 4439 8961 
 
 4594 8882 
 
 4749 8801 
 
 4901 8716 
 
 39 
 
 22 
 
 4284 9036 
 
 4441 8960 
 
 4597 8881 
 
 4751 8799 
 
 4904 8715 
 
 38 
 
 23 
 
 4287 9035 
 
 4444 8958 
 
 4599 8879 
 
 4754 8798 
 
 4907 8714 
 
 37 
 
 24 
 
 4289 9033 
 
 4446 8957 
 
 4602 8878 
 
 4756 8796 
 
 4909 8712 
 
 36 
 
 25 
 
 4292 9032 
 
 4449 8956 
 
 4605 8877 
 
 4759 8795 
 
 4912 8711 
 
 35 
 
 26 
 
 4295 9031 
 
 4452 8955 
 
 4607 8875 
 
 4761 8794 
 
 4914 8709 
 
 34 
 
 27 
 
 4297 9030 
 
 4454 8953 
 
 4610 8874 
 
 4764 8792 
 
 4917 8708 
 
 33 
 
 28 
 
 4300 9028 
 
 4457 8952 
 
 4612 8873 
 
 4766 8791 
 
 4919 8706 
 
 32 
 
 29 
 
 4302 9027 
 
 4459 8951 
 
 4615 8871 
 
 4769 8790 
 
 4922 8705 
 
 31 
 
 3O 
 
 4305 9026 
 
 4462 8949 
 
 4617 8870 
 
 4772 8788 
 
 4924 8704 
 
 30 
 
 31 
 
 4308 9025 
 
 4465 8948 
 
 4620 8869 
 
 4774 8787 
 
 4927 8702 
 
 29 
 
 32 
 
 4310 9023 
 
 4467 8947 
 
 4623 8867 
 
 4777 8785 
 
 4929 8701 
 
 28 
 
 33 
 
 4313 9022 
 
 4470 8945 
 
 4625 8866 
 
 4779 8784 
 
 4932 8699 
 
 27 
 
 34 
 
 4316 9021 
 
 4472 8944 
 
 4628 8865 
 
 4782 8783 
 
 4934 8698 
 
 26 
 
 35 
 
 4318 9020 
 
 4475 8943 
 
 4630 8863 
 
 4784 8781 
 
 4937 8696 
 
 25 
 
 36 
 
 4321 9018 
 
 4478 8942 
 
 4633 8862 
 
 4787 8780 
 
 4939 8695 
 
 24 
 
 37 
 
 4323 9017 
 
 4480 8940 
 
 4636 8861 
 
 4789 8778 
 
 4942 8694 
 
 23 
 
 38 
 
 4326 9016 
 
 4483 8939 
 
 4638 8859 
 
 4792 8777 
 
 4944 8692 
 
 22 
 
 39 
 
 4329 9015 
 
 4485 8938 
 
 4641 8858 
 
 4795 8776 
 
 4947 8691 
 
 21 
 
 4O 
 
 4331 9013 
 
 4488 8936 
 
 4643 8857 
 
 4797 8774 
 
 4950 8689 
 
 2O 
 
 41 
 
 4334 9012 
 
 4491 8935 
 
 4646 8855 
 
 4800 8773 
 
 4952 8688 
 
 19 
 
 42 
 
 4337 9011 
 
 4493 8934 
 
 4648 8854 
 
 4802 8771 
 
 4955 8686 
 
 18 
 
 43 
 
 4339 9010 
 
 4496 8932 
 
 4651 8853 
 
 4805 8770 
 
 4957 8685 
 
 17 
 
 44 
 
 4342 9008 
 
 4498 8931 
 
 4654 8851 
 
 4807 8769 
 
 4960 8683 
 
 16 
 
 45 
 
 4344 9007 
 
 4501 8930 
 
 4656 8850 
 
 4810 8767 
 
 4962 8682 
 
 15 
 
 46 
 
 4347 9006 
 
 4504 8928 
 
 4659 8849 
 
 4812 8766 
 
 4965 8681 
 
 14 
 
 47 
 
 4350 9004 
 
 4506 8927 
 
 4661 8847 
 
 4815 8764 
 
 4967 8679 
 
 13 
 
 48 
 
 4352 9003 
 
 4509 8926 
 
 4664 8846 
 
 4818 8763 
 
 4970 8678 
 
 12 
 
 49 
 
 4355 9002 
 
 4511 8925 
 
 4666 8844 
 
 4820 8762 
 
 4972 8676 
 
 11 
 
 50 
 
 4358 9001 
 
 4514 8923 
 
 4669 8843 
 
 4823 8760 
 
 4975 8675 
 
 1O 
 
 51 
 
 4360 8999 
 
 4517 8922 
 
 4672 8842 
 
 4825 8759 
 
 4977 8673 
 
 9 
 
 52 
 
 4363 8998 
 
 4519 8921 
 
 4674 8840 
 
 4828 8757 
 
 4980 8672 
 
 8 
 
 53 
 
 4365 8997 
 
 4522 8919 
 
 4677 8839 
 
 4830 8756 
 
 4982 8670 
 
 7 
 
 54 
 
 4368 8996 
 
 4524 8918 
 
 4679 8838 
 
 4833 8755 
 
 4985 8669 
 
 6 
 
 55 
 
 4371 8994 
 
 4527 8917 
 
 4682 8836 
 
 4835 8753 
 
 4987 8668 
 
 5 
 
 56 
 
 4373 8993 
 
 4530 8915 
 
 4684 8835 
 
 4838 8752 
 
 4990 8666 
 
 4 
 
 57 
 
 4376 8992 
 
 4532 8914 
 
 4687 8834 
 
 4840 8750 
 
 4992 8665 
 
 3 
 
 58 
 
 4378 8990 
 
 4535 8913 
 
 4690 8832 
 
 4843 8749 
 
 4995 8663 
 
 2 
 
 59 
 
 4381 8989 
 
 4537 8911 
 
 4692 8831 
 
 4846 8748 
 
 4997 8662 
 
 1 
 
 60 
 
 4384 8988 
 
 4540 8910 
 
 4695 8829 
 
 4848 8746 
 
 5000 8660 
 
 
 
 
 cos sin 
 
 cos sin 
 
 cos sin 
 
 cos sin 
 
 cos sin 
 
 
 f 
 
 64 
 
 63 
 
 62 
 
 01 
 
 OO 
 
 t
 
 68 
 
 NATURAL SINES AND COSINES. 
 
 r 
 
 3O 
 
 31 
 
 32 
 
 33 
 
 34 
 
 t 
 
 
 sin cos 
 
 sin cos 
 
 sin cos 
 
 in cos 
 
 sin cos 
 
 
 O 
 
 5000 8660 
 
 5150 8572 
 
 5299 8480 
 
 5446 8387 
 
 5592 8290 
 
 6O 
 
 1 
 
 5003 8659 
 
 5153 8570 
 
 5302 8479 
 
 5449 8385 
 
 5594 8289 
 
 59 
 
 2 
 
 5005 8657 
 
 5155 8569 
 
 5304 8477 
 
 5451 8384 
 
 5597 8287 
 
 58 
 
 3 
 
 5008 8656 
 
 5158 8567 
 
 5307 8476 
 
 5454 8382 
 
 5599 8285 
 
 57 
 
 4 
 
 5010 8654 
 
 5160 8566 
 
 5309 8474 
 
 5456 8380 
 
 5602 8284 
 
 56 
 
 5 
 
 5013 8653 
 
 5163 8564 
 
 5312 8473 
 
 5459 8379 
 
 5604 8282 
 
 55 
 
 6 
 
 5015 8652 
 
 5165 8563 
 
 5314 8471 
 
 5461 8377 
 
 5606 8281 
 
 54 
 
 7 
 
 5018 8650 
 
 5168 8561 
 
 5316 8470 
 
 5463 8376 
 
 5609 8279 
 
 53 
 
 8 
 
 5020 8649 
 
 5170 8560 
 
 5319 8468 
 
 5466 8374 
 
 5611 8277 
 
 52 
 
 9 
 
 5023 8647 
 
 5173 8558 
 
 5321 8467 
 
 5468 8372 
 
 5614 8276 
 
 51 
 
 1O 
 
 5025 8646 
 
 5175 8557 
 
 5324 8465 
 
 5471 8371 
 
 5616 8274 
 
 5O 
 
 11 
 
 5028 8644 
 
 5178 8555 
 
 5326 8463 
 
 5473 8369 
 
 5618 8272 
 
 49 
 
 12 
 
 5030 8643 
 
 5180 8554 
 
 5329 8462 
 
 5476 8368 
 
 5621 8271 
 
 48 
 
 13 
 
 5033 8641 
 
 5183 8552 
 
 5331 8460 
 
 5478 8366 
 
 5623 8269 
 
 47 
 
 14 
 
 5035 8640 
 
 5185 8551 
 
 5334 8459 
 
 5480 8364 
 
 5626 8268 
 
 46 
 
 15 
 
 5038 8638 
 
 5188 8549 
 
 5336 8457 
 
 5483 8363 
 
 5628 8266 
 
 45 
 
 16 
 
 5040 8637 
 
 5190 8548 
 
 5339 8456 
 
 5485 8361 
 
 5630 8264 
 
 44 
 
 17 
 
 5043 8635 
 
 5193 8546 
 
 5341 8454 
 
 5488 8360 
 
 5633 8263 
 
 43 
 
 18 
 
 5045 8634 
 
 5195 8545 
 
 5344 8453 
 
 5490 8358 
 
 5635 8261 
 
 42 
 
 19 
 
 5048 8632 
 
 5198 8543 
 
 5346 8451 
 
 5493 8356 
 
 5638 8259 
 
 41 
 
 2O 
 
 5050 8631 
 
 5200 8542 
 
 5348 8450 
 
 5495 8355 
 
 5640 8258 
 
 40 
 
 21 
 
 5053 8630 
 
 5203 8540 
 
 5351 8448 
 
 5498 8353 
 
 5642 8256 
 
 39 
 
 22 
 
 5055 8628 
 
 5205 8539 
 
 5353 8446 
 
 5500 8352 
 
 5645 8254 
 
 38 
 
 23 
 
 5058 8627 
 
 5208 8537 
 
 5356 8445 
 
 5502 8350 
 
 5647 8253 
 
 37 
 
 24 
 
 5060 8625 
 
 5210 8536 
 
 5358 8443 
 
 5505 8348 
 
 5650 8251 
 
 36 
 
 25 
 
 5063 8624 
 
 5213 8534 
 
 5361 8442 
 
 5507 8347 
 
 5652 8249 
 
 35 
 
 26 
 
 5065 8622 
 
 5215 8532 
 
 5363 8440 
 
 5510 8345 
 
 5654 8248 
 
 34 
 
 27 
 
 5068 8621 
 
 5218 8531 
 
 5366 8439 
 
 5512 8344 
 
 5657 8246 
 
 33 
 
 28 
 
 5070 8619 
 
 5220 8529 
 
 5368 8437 
 
 5515 8342 
 
 5659 8245 
 
 32 
 
 29 
 
 5073 8618 
 
 5223 8528 
 
 5371 8435 
 
 5517 8340 
 
 5662 8243 
 
 31 
 
 30 
 
 5075 8616 
 
 5225 8526 
 
 5373 8434 
 
 5519 8339 
 
 5664 8241 
 
 3O 
 
 31 
 
 5078 8615 
 
 5227 8525 
 
 5375 8432 
 
 5522 8337 
 
 5666 8240 
 
 29 
 
 32 
 
 5080 8613 
 
 5230 8523 
 
 5378 8431 
 
 5524 8336 
 
 5669 8238 
 
 28 
 
 33 
 
 5083 8612 
 
 5232 8522 
 
 5380 8429 
 
 5527 8334 
 
 5671 8236 
 
 27 
 
 34 
 
 5085 8610 
 
 5235 8520 
 
 5383 8428 
 
 5529 8332 
 
 5674 8235 
 
 26 
 
 35 
 
 5088 8609 
 
 5237 8519 
 
 5385 8426 
 
 5531 8331 
 
 5676 8233 
 
 25 
 
 36 
 
 5090 8607 
 
 5240 8517 
 
 5388 8425 
 
 5534 8329 
 
 5678 8231 
 
 24 
 
 37 
 
 5093 8606 
 
 5242 8516 
 
 5390 8423 
 
 5536 8328 
 
 5681 8230 
 
 23 
 
 38 
 
 5095 8604 
 
 5245 8514 
 
 5393 8421 
 
 5539 8326 
 
 5683 8228 
 
 22 
 
 39 
 
 5098 8603 
 
 5247 8513 
 
 5395 8*20 
 
 5541 8324 
 
 5686 8226 
 
 21 
 
 40 
 
 5100 8601 
 
 5250 8511 
 
 5398 8418 
 
 5544 8323 
 
 5688 8225 
 
 2O 
 
 41 
 
 5103 8600 
 
 5252 8510 
 
 5400 8417 
 
 5546 8321 
 
 5690 8223 
 
 19 
 
 42 
 
 5105 8599 
 
 5255 8508 
 
 5402 8415 
 
 5548 8320 
 
 5693 8221 
 
 18 
 
 43 
 
 5108 8597 
 
 5257 8507 
 
 5405 8414 
 
 5551 8318 
 
 5695 8220 
 
 17 
 
 44 
 
 5110 8596 
 
 5260 8505 
 
 5407 8412 
 
 5553 8316 
 
 5698 8218 
 
 16 
 
 45 
 
 S3 13 8594 
 
 5262 8504 
 
 5410 8410 
 
 5556 8315 
 
 5700 8216 
 
 15 
 
 46 
 
 5115 8i93 
 
 5265 8502 
 
 5412 8409 
 
 5558 8313 
 
 5702 8215 
 
 14 
 
 47 
 
 .5118 8591 
 
 5267 8500 
 
 5415 8407 
 
 5561 8311 
 
 5705 8213 
 
 13 
 
 48 
 
 5120 8590 
 
 5270 8499 
 
 5417 8406 
 
 5563 8310 
 
 5707 8211 
 
 12 
 
 49 
 
 5123 8588 
 
 5272 8497 
 
 5420 8404 
 
 5565 8308 
 
 5710 8210 
 
 11 
 
 50 
 
 5125 8587 
 
 5275 84% 
 
 5422 8403 
 
 5568 8307 
 
 5712 8208 
 
 10 
 
 51 
 
 5128 8585 
 
 5277 8494 
 
 5424 8401 
 
 5570 8305 
 
 5714 8207 
 
 9 
 
 52 
 
 5130 8584 
 
 5279 8493 
 
 5427 8399 
 
 5573 8303 
 
 5717 8205 
 
 8 
 
 53 
 
 5133 8582 
 
 5282 8491 
 
 5429 8398 
 
 5575 8302 
 
 5719 8203 
 
 7 
 
 54 
 
 5135 8581 
 
 5284 8490 
 
 5432 83% 
 
 5577 8300 
 
 5721 8202 
 
 6 
 
 55 
 
 5138 8579 
 
 5287 8488 
 
 5434 8395 
 
 5580 8299 
 
 5724 8200 
 
 5 
 
 56 
 
 5140 8578 
 
 5289 8487 
 
 5437 8393 
 
 5582 8297 
 
 5726 8198 
 
 4 
 
 57 
 
 5143 8576 
 
 5292 8485 
 
 5439 8391 
 
 5585 8295 
 
 5729 8197 
 
 3 
 
 58 
 
 5145 8575 
 
 5294 8484 
 
 5442 8390 
 
 5587 8294 
 
 5731 8195 
 
 2 
 
 59 
 
 5148 8573 
 
 5297 8482 
 
 5444 8388 
 
 5590 8292 
 
 5733 8193 
 
 1 
 
 60 
 
 5150 8572 
 
 5299 8480 
 
 5446 8387 
 
 5592 8290 
 
 5736 8192 
 
 
 
 
 cos sin 
 
 cos sin 
 
 cos sin 
 
 cos sin 
 
 cos sin 
 
 
 t 
 
 59 
 
 58 
 
 57 
 
 56 
 
 55 
 
 t
 
 NATURAL SINES AND COSINES. 69 
 
 t 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 , 
 
 
 sin cos 
 
 sin cos 
 
 sin cos 
 
 sin cos 
 
 gin cos 
 
 
 o 
 
 5736 8192 
 
 5878 8090 
 
 6018 7986 
 
 6157 7880 
 
 6293 7771 
 
 <><> 
 
 1 
 
 5738 8190 
 
 5880 8088 
 
 6020 7985 
 
 6159 7878 
 
 6295 7770 
 
 59 
 
 2 
 
 5741 8188 
 
 58S3 8087 
 
 6023 7983 
 
 6161 7877 
 
 6298 7768 
 
 58 
 
 3 
 
 5743 8187 
 
 5885 8085 
 
 6025 7981 
 
 6163 7875 
 
 6300 7766 
 
 57 
 
 4 
 
 5745 8185 
 
 5887 8083 
 
 6027 7979 
 
 6166 7873 
 
 6302 7764 
 
 56 
 
 5 
 
 5748 8183 
 
 5890 8082 
 
 6030 7978 
 
 6168 7871 
 
 6305 7762 
 
 55 
 
 6 
 
 5750 8181 
 
 5892 8080 
 
 6032 7976 
 
 6170 7869 
 
 6307 7760 
 
 54 
 
 7 
 
 5752 8180 
 
 5894 8078 
 
 6034 7974 
 
 6173 7868 
 
 6309 7759 
 
 53 
 
 8 
 
 5755 8178 
 
 5897 8076 
 
 6037 7972 
 
 6175 7866 
 
 6311 7757 
 
 52 
 
 9 
 
 5757 8176 
 
 5899 8075 
 
 6039 7971 
 
 6177 7864 
 
 6314 7755 
 
 51 
 
 1O 
 
 5760 8175 
 
 5901 8073 
 
 6041 7969 
 
 6180 7862 
 
 6316 7753 
 
 5O 
 
 11 
 
 5762 8173 
 
 5904 8071 
 
 6044 7967 
 
 6182 7860 
 
 6318 7751 
 
 49 
 
 12 
 
 5764 8171 
 
 5906 8070 
 
 6046 7965 
 
 6184 7859 
 
 6320 "7749 
 
 48 
 
 13 
 
 5767 8170 
 
 5908 8068 
 
 6048 7964 
 
 6186 7857 
 
 6323 7748 
 
 47 
 
 14 
 
 5769 8168 
 
 5911 8066 
 
 6051 7962 
 
 6189 7855 
 
 6325 7746 
 
 46 
 
 15 
 
 5771 8166 
 
 5913 8064 
 
 6053 7960 
 
 6191 7853 
 
 6327 7744 
 
 45 
 
 16 
 
 5774 8165 
 
 5915 8063 
 
 6055 7958 
 
 6193 7851 
 
 6329 7742 
 
 44 
 
 17 
 
 5776 8163 
 
 5918 8061 
 
 6058 7956 
 
 6196 7850 
 
 6332 7740 
 
 43 
 
 18 
 
 5779 8161 
 
 5920 8059 
 
 6060 7955 
 
 6198 7848 
 
 6334 7738 
 
 42 
 
 19 
 
 5781 8160 
 
 5922 8058 
 
 6062 7953 
 
 6200 7846 
 
 6336 7737 
 
 41 
 
 2O 
 
 5783 8158 
 
 5925 8056 
 
 6065 7951 
 
 6202 7844 
 
 6338 7735 
 
 40 
 
 21 
 
 5786 8156 
 
 5927 8054 
 
 6067 7950 
 
 6205 7842 
 
 6341 7733 
 
 39 
 
 22 
 
 5788 8155 
 
 5930 8052 
 
 6069 7948 
 
 6207 7841 
 
 6343 7731 
 
 38 
 
 23 
 
 5790 8153 
 
 5932 8051 
 
 6071 7946 
 
 6209 7839 
 
 6345 7729 
 
 37 
 
 24 
 
 5793 8151 
 
 5934 8049 
 
 6074 7944 
 
 6211 7837 
 
 6347 7727 
 
 36 
 
 25 
 
 5795 8150 
 
 5937 8047 
 
 6076 7942 
 
 6214 7835 
 
 6350 7725 
 
 35 
 
 26 
 
 5798 8148 
 
 5939 8045 
 
 6078 7941 
 
 6216 7833 
 
 6352 7724 
 
 34 
 
 27 
 
 5800 8146 
 
 5941 8044 
 
 6081 7939 
 
 6218 7832 
 
 6354 7722 
 
 33 
 
 28 
 
 5802 8145 
 
 5944 8042 
 
 6083 7937 
 
 6221 7830 
 
 6356 7720 
 
 32 
 
 29 
 
 5805 8143 
 
 5946 8040 
 
 6085 7935 
 
 6223 7828 
 
 6359 7718 
 
 31 
 
 30 
 
 5807 8141 
 
 5948 8039 
 
 6088 7934 
 
 6225 7826 
 
 6361 7716 
 
 3O 
 
 31 
 
 5809 8139 
 
 5951 8037 
 
 6090 7932 
 
 6227 7824 
 
 6363 7714 
 
 29 
 
 32 
 
 5812 S138 
 
 5953 8035 
 
 6092 7930 
 
 6230 7822 
 
 6365 7713 
 
 28 
 
 33 
 
 5814 8136 
 
 5955 8033 
 
 6095 7928 
 
 6232 7821 
 
 6368 7711 
 
 27 
 
 34 
 
 5816 8134 
 
 5958 8032 
 
 6097 7926 
 
 6234 7819 
 
 6370 7709 
 
 26 
 
 35 
 
 5819 8133 
 
 5960 8030 
 
 6099 7925 
 
 6237 7817 
 
 6372 7707 
 
 25 
 
 36 
 
 5821 8131 
 
 5962 S02S 
 
 6101 7923 
 
 6239 781 5 
 
 6374 7705 
 
 24 
 
 37 
 
 5824 8129 
 
 5965 8026 
 
 6104 7921 
 
 6241 7813 
 
 6376 7703 
 
 23 
 
 38 
 
 5826 8128 
 
 5967 8025 
 
 6106 7919 
 
 6243 7812 
 
 6379 7701 
 
 22 
 
 39 
 
 5828 8126 
 
 5969 8023 
 
 6108 7918 
 
 6246 7810 
 
 6381 7700 
 
 21 
 
 4O 
 
 5831 8124 
 
 5972 8021 
 
 6111 7916 
 
 6248 7808 
 
 6383 7698 
 
 20 
 
 41 
 
 5833 8123 
 
 5974 8020 
 
 6113 7914 
 
 6250 7S06 
 
 6385 7696 
 
 19 
 
 42 
 
 5835 8121 
 
 5976 8018 
 
 6115 7912 
 
 6252 7804 
 
 6388 7694 
 
 18 
 
 43 
 
 583S 8119 
 
 5979 8016 
 
 6118 7910 
 
 6255 7802 
 
 6390 7692 
 
 17 
 
 44 
 
 5840 8117 
 
 5981 8014 
 
 6120 7909 
 
 6257 7801 
 
 6392 7690 
 
 16 
 
 45 
 
 5842 8116 
 
 5983 8013 
 
 6122 7907 
 
 6259 7799 
 
 6394 7688 
 
 15 
 
 46 
 
 5S45 8114 
 
 5986 8011 
 
 6124 7905 
 
 6262 7797 
 
 6397 7687 
 
 14 
 
 47 
 
 5847 8112 
 
 5988 8009 
 
 6127 7903 
 
 6264 7795 
 
 6399 7685 
 
 13 
 
 48 
 
 5850 8111 
 
 5990 8007 
 
 6129 7902 
 
 6266 7793 
 
 6401 7683 
 
 12 
 
 49 
 
 5852 8109 
 
 5993 8006 
 
 6131 7900 
 
 6268 7792 
 
 6403 7681 
 
 11 
 
 5O 
 
 5854 8107 
 
 5995 8004 
 
 6134 7898 
 
 6271 7790 
 
 6406 7679 
 
 10 
 
 51 
 
 5857 8106 
 
 5997 8002 
 
 6136 7896 
 
 6273 7788 
 
 6408 7677 
 
 9 
 
 52 
 
 5859 8104 
 
 6000 8000 
 
 6138 7894 
 
 6275 7786 
 
 6410 7675 
 
 8 
 
 53 
 
 5861 8102 
 
 6002 7999 
 
 6141 7893 
 
 6277 7784 
 
 6412 7674 
 
 7 
 
 54 
 
 5864 8100 
 
 6004 7997 
 
 6143 7891 
 
 6280 7782 
 
 6414 7672 
 
 6 
 
 55 
 
 5866 8099 
 
 6007 7995 
 
 6145 7889 
 
 6282 7781 
 
 6417 7670 
 
 5 
 
 56 
 
 5868 8097 
 
 6009 7993 
 
 6147 7887 
 
 6284 7779 
 
 6419 7668 
 
 4 
 
 57 
 
 587 1 8095 
 
 6011 7992 
 
 6150 7885 
 
 6286 7777 
 
 6421 7666 
 
 3 
 
 58 
 
 5873 8094 
 
 6014 7990 
 
 6152 7884 
 
 6289 7775 
 
 6423 7664 
 
 2 
 
 59 
 
 5875 8092 
 
 6016 7988 
 
 6154 7882 
 
 6291 7773 
 
 6426 7662 
 
 1 
 
 (JO 
 
 5S7S 8090 
 
 6018 7986 
 
 6157 7880 
 
 6293 7771 
 
 6428 7660 
 
 
 
 cos sin 
 
 cos sin 
 
 cos sin 
 
 cos sin 
 
 COS Hill 
 
 
 ' 54 
 
 r>:$ c 
 
 r>2^ 
 
 5T" 
 
 50 '
 
 70 NATURAL SINES AND COSINES. 
 
 r 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 t 
 
 
 sin cos 
 
 sin cos 
 
 sin cos 
 
 sin cos 
 
 sin cos 
 
 
 O 
 
 6428 7660 
 
 6561 7547 
 
 6691 7431 
 
 6820 7314 
 
 6947 7193 
 
 6O 
 
 1 
 
 6430 7659 
 
 6563 7545 
 
 6693 7430 
 
 6822 7312 
 
 6949 7191 
 
 59 
 
 2 
 
 6432 7657 
 
 6565 7543 
 
 6696 7428 
 
 6824 7310 
 
 6951 7189 
 
 58 
 
 3 
 
 6435 7655 
 
 6567 7541 
 
 6698 7426 
 
 6826 7308 
 
 6953 7187 
 
 57 
 
 4 
 
 6437 7653 
 
 6569 7539 
 
 6700 7424 
 
 6828 7306 
 
 6955 7185 
 
 56 
 
 5 
 
 6439 7651 
 
 6572 7538 
 
 6702 7422 
 
 6831 7304 
 
 6957 7183 
 
 55 
 
 6 
 
 6441 7649 
 
 6574 7536 
 
 6704 7420 
 
 6833 7302 
 
 6959 7181 54 
 
 7 
 
 6443 7647 
 
 6576 7534 
 
 6706 7418 
 
 6835 7300 
 
 6961 7179 
 
 53 
 
 8 
 
 6446 7645 
 
 6578 7532 
 
 6709 7416 
 
 6837 7298 
 
 6963 7177 
 
 52 
 
 9 
 
 6448 7644 
 
 6580 7530 
 
 6711 7414 
 
 6839 7296 
 
 6965 7175 
 
 51 
 
 1O 
 
 6450 7642 
 
 6583 7528 
 
 6713 7412 
 
 6841 7294 
 
 6967 7173 
 
 50 
 
 11 
 
 6452 7640 
 
 6585 7526 
 
 6715 7410 
 
 6843 7292 
 
 6970 7171 
 
 49 
 
 12 
 
 6455 7638 
 
 6587 7524 
 
 6717 7408 
 
 6845 7290 
 
 6972 7169 
 
 48 
 
 13 
 
 6457 7636 
 
 6589 7522 
 
 6719 7406 
 
 6848 7288 
 
 6974 7167 
 
 47 
 
 14 
 
 6459 7634 
 
 6591 7520 
 
 6722 7404 
 
 6850 7286 
 
 6976 7165 
 
 46 
 
 15 
 
 6461 7632 
 
 6593 7518 
 
 6724 7402 
 
 6852 7284 
 
 6978 7163 
 
 45 
 
 16 
 
 "6463 7630 
 
 6596 7516 
 
 6726 7400 
 
 6854 7282 
 
 6980 7161 
 
 44 
 
 17 
 
 6466 7629 
 
 6598 7515 
 
 6728 7398 
 
 6856 7280 
 
 6982 7159 
 
 43 
 
 18 
 
 6468 7627 
 
 6600 7513 
 
 6730 7396 
 
 6858 7278 
 
 6984 7157 
 
 42 
 
 19 
 
 6470 7625 
 
 6602 7511 
 
 6732 7394 
 
 6S60 7276 
 
 6986 7155 
 
 41 
 
 2O 
 
 6472 7623 
 
 6604 7509 
 
 6734 7392 
 
 6862 7274 
 
 6988 7153 
 
 40 
 
 21 
 
 6475 7621 
 
 6607 7507 
 
 6737 7390 
 
 6865 7272 
 
 6990 7151 
 
 39 
 
 22 
 
 6477 7619 
 
 6609 7505 
 
 6739 7388 
 
 6867 7270 
 
 6992 7149 
 
 38 
 
 23 
 
 6479 7617 
 
 6611 7503 
 
 6741 7387 
 
 6869 7268 
 
 6995 7147 
 
 37 
 
 24 
 
 6481 7615 
 
 6613 7501 
 
 6743 7385 
 
 6871 7266 
 
 6997 7145 
 
 36 
 
 25 
 
 6483 7613 
 
 6615 7499 
 
 6745 7383 
 
 6873 7264 
 
 6999 7143 
 
 35 
 
 26 
 
 6486 7612 
 
 6617 7497 
 
 6747 7381 
 
 6875 7262 
 
 7001 7141 
 
 34 
 
 27 
 
 6488 7610 
 
 6620 7495 
 
 6749 7379 
 
 6877 7260 
 
 7003 7139 
 
 33 
 
 28 
 
 6490 7608 
 
 6622 7493 
 
 6752 7377 
 
 6879 7258 
 
 7005 7137 
 
 32 
 
 29 
 
 6492 7606 
 
 6624 7491 
 
 6754 7375 
 
 6881 7256 
 
 7007 7135 
 
 31 
 
 30 
 
 6494 7604 
 
 6626 7490 
 
 6756 7373 
 
 6884 7254 
 
 7009 7133 
 
 3O 
 
 31 
 
 6497 7602 
 
 6628 7488 
 
 6758 7371 
 
 6886 7252 
 
 7011 7130 
 
 29 
 
 32 
 
 6499 7600 
 
 6631 7486 
 
 6760 7369 
 
 6888 7250 
 
 7013 7128 
 
 28 
 
 33 
 
 6501 7598 
 
 6633 7484 
 
 6762 7367 
 
 6890 7248 
 
 7015 7126 
 
 27 
 
 34 
 
 6503 7596 
 
 6635 7482 
 
 6764 7365 
 
 6892 7246 
 
 7017 7124 
 
 26 
 
 35 
 
 6506 7595 
 
 6637 7480 
 
 6767 7363 
 
 6894 7244 
 
 7019 7122 
 
 25 
 
 36 
 
 6508 7593 
 
 6639 7478 
 
 6769 7361 
 
 6896 7242 
 
 7022 7120 
 
 24 
 
 37 
 
 6510 7591 
 
 6641 7476 
 
 6771 7359 
 
 6898 7240 
 
 7024 7118 
 
 23 
 
 38 
 
 6512 7589 
 
 6644 7474 
 
 6773 7357 
 
 6900 7238 
 
 7026 7116 
 
 22 
 
 39 
 
 6514 7587 
 
 6646 7472 
 
 6775 7355 
 
 6903 7236 
 
 7028 7114 
 
 21 
 
 40 
 
 6517 7585 
 
 6648 7470 
 
 6777 7353 
 
 6905 7234 
 
 7030 7112 
 
 2O 
 
 41 
 
 6519 7583 
 
 6650 7468 
 
 6779 7351 
 
 6907 7232 
 
 7032 7110 
 
 19 
 
 42 
 
 6521 7581 
 
 6652 7466 
 
 6782 7349 
 
 6909 7230 
 
 7034 7108 
 
 18 
 
 43 
 
 6523 7579 
 
 6654 7464 
 
 6784 7347 
 
 6911 7228 
 
 7036 7106 
 
 17 
 
 44 
 
 6525 7578 
 
 6657 7463 
 
 6786 7345 
 
 6913 7226 
 
 7038 7104 
 
 16 
 
 45 
 
 6528 7576 
 
 6659 7461 
 
 6788 7343 
 
 6915 7224 
 
 7040 7102 
 
 15 
 
 46 
 
 6530 7574 
 
 6661 7459 
 
 6790 7341 
 
 6917 7222 
 
 7042 7100 
 
 14 
 
 47 
 
 6532 7572 
 
 6663 7457 
 
 6792 7339 
 
 6919 7220 
 
 7044 7098 
 
 13 
 
 48 
 
 6534 7570 
 
 6665 7455 
 
 6794 7337 
 
 6921 7218 
 
 7046 7096 
 
 12 
 
 49 
 
 6536 7568 
 
 6667 7453 
 
 6797 7335 
 
 6924 7216 
 
 7048 7094 
 
 11 
 
 50 
 
 6539 7566 
 
 6670 7451 
 
 6799 7333 
 
 6926 7214 
 
 7050 7092 
 
 10 
 
 51 
 
 6541 7564 
 
 6672 7449 
 
 6801 7331 
 
 6928 7212 
 
 7053 7090 
 
 9 
 
 52 
 
 6543 7562 
 
 6674 7447 
 
 6803 7329 
 
 6930 7210 
 
 7055 7088 
 
 8 
 
 53 
 
 6545 7560 
 
 6676 7445 
 
 6805 7327 
 
 6932 7208 
 
 7057 7085 
 
 7 
 
 54 
 
 6547 7559 
 
 6678 7443 
 
 6807 7325 
 
 6934 7206 
 
 7059 7083 
 
 6 
 
 55 
 
 6550 7557 
 
 6680 7441 
 
 6809 7323 
 
 6936 7203 
 
 7061 7081 
 
 5 
 
 56 
 
 6552 7555 
 
 6683 7439 
 
 6811 7321 
 
 6938 7201 
 
 7063 7079 
 
 4 
 
 57 
 
 6554 7553 
 
 6685 7437 
 
 6814 7319 
 
 6940 7199 
 
 7065 7077 
 
 3 
 
 58 
 
 6556 7551 
 
 6687 7435 
 
 6816 7318 
 
 6942 7197 
 
 7067 7075 
 
 2 
 
 59 
 
 6558 7549 
 
 6689 7433 
 
 6818 7316 
 
 6944 7195 
 
 7069 7073 
 
 1 
 
 6O 
 
 6561 7547 
 
 6691 7431 
 
 6820 7314 
 
 6947 7193 
 
 7071 7071 
 
 O 
 
 
 cos sin 
 
 cos sin 
 
 cos sin 
 
 cos sin 
 
 cos sin 
 
 
 f 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 f
 
 TABLE IX, NATURAL TANGENTS AND COTANGENTS. ' 
 
 
 
 
 
 
 
 2 
 
 
 3 
 
 
 4 
 
 r 
 
 tan 
 
 cot 
 
 tan 
 
 cot 
 
 tan 
 
 cot 
 
 tan 
 
 cot 
 
 tan 
 
 cot 
 
 
 O 10000 
 
 Infinite 
 
 0175 
 
 57.2900 
 
 0349 
 
 28.6363 
 
 0524 
 
 19.0811 
 
 0699 
 
 14.3007 
 
 GO 
 
 1 0003 
 
 3437.75 
 
 0177 
 
 56.3506 
 
 0352 
 
 28.3994 
 
 0527 
 
 18.9755 
 
 0702 
 
 14.2411 
 
 59 
 
 2 i0006 
 
 1718.87 
 
 0180 
 
 55.4415 
 
 0355 
 
 28.1664 
 
 0530 
 
 18.8711 
 
 0705 
 
 14.1821 
 
 58 
 
 3 10009 
 
 1145.92 
 
 0183 
 
 54.5613 
 
 0358 
 
 27.9372 
 
 0533 
 
 18.7678 
 
 0708 
 
 14.1235 
 
 57 
 
 4 0012 
 
 859.436 
 
 0186 
 
 53.7086 
 
 0361 
 
 27.7117 
 
 0536 
 
 18.6656 
 
 0711 
 
 14.0655 
 
 56 
 
 5 0015 
 
 687.549 
 
 0189 
 
 52.8821 
 
 0364 
 
 27.4899 
 
 0539 
 
 18.5645 
 
 0714 
 
 14.0079 
 
 55 
 
 6 J0017 
 
 572.957 
 
 0192 
 
 52.0807 
 
 0367 
 
 27.2715 
 
 0542 
 
 18.4645 
 
 0717 
 
 13.9507 
 
 54 
 
 7 
 
 0020 
 
 491.106 
 
 0195 
 
 51.3032 
 
 0370 
 
 27.0566 
 
 0544 
 
 18.3655 
 
 0720 
 
 13.8940 
 
 53 
 
 8 0023 
 
 429.718 
 
 0198 
 
 50.5485 
 
 0373 
 
 26.8450 
 
 0547 
 
 18.2677 
 
 0723 
 
 13.8378 
 
 52 
 
 9 0026 
 
 381.971 
 
 0201 
 
 49.8157 
 
 0375 
 
 26.6367 
 
 0550 
 
 18.1708 
 
 0726 
 
 13.7821 
 
 51 
 
 1O 0029 
 
 343.774 
 
 0204 
 
 49.1039 
 
 037S 
 
 26.4316 
 
 0553 
 
 18.0750 
 
 0729 
 
 13.7267 
 
 5O 
 
 11 0032 
 
 312.521 
 
 0207 
 
 48.4121 
 
 0381 
 
 26.2296 
 
 0556 
 
 17.9802 
 
 0731 
 
 13.6719 
 
 49 
 
 12 0035 
 
 286.478 
 
 0209 
 
 47.7395 
 
 0384 
 
 26.0307 
 
 0559 
 
 17.8863 
 
 0734 
 
 13.6174 
 
 48 
 
 13 
 
 0038 
 
 264.441 
 
 0212 
 
 47.0853 
 
 0387 
 
 25.8348 
 
 0562 
 
 17.7934 
 
 0737 
 
 13.5634 
 
 47 
 
 14 
 
 0041 
 
 245.552 
 
 0215 
 
 46.4489 
 
 0390 
 
 25.6418 
 
 0565 
 
 17.7015 
 
 0740 
 
 13.5098 46 
 
 15 
 
 0044 
 
 229.182 
 
 0218 
 
 45.8294 
 
 0393 
 
 25.4517 
 
 0568 
 
 17.6106 
 
 0743 
 
 13.4566 45 
 
 16 
 
 0047 
 
 214.858 
 
 0221 
 
 45.2261 
 
 0396 
 
 25.2644 
 
 0571 
 
 17.5205 
 
 0746 
 
 13.4039! 44 
 
 17 
 
 0049 
 
 202.219 
 
 0224 
 
 44.6386 
 
 0399 
 
 25.0798 
 
 0574 
 
 17.4314 
 
 0749 
 
 13.3515 43 
 
 18 
 
 0052 
 
 190.984 
 
 0227 
 
 44.0661 
 
 0402 
 
 24.8978 
 
 0577 
 
 17.3432 
 
 0752 
 
 13.2996 42 
 
 19 
 
 0055 
 
 180.932 
 
 0230 
 
 43.5081 
 
 0405 
 
 24.7185 
 
 0580 
 
 17.2558 
 
 0755 
 
 13.2480 . 41 
 
 20 
 
 0058 
 
 171.885 
 
 0233 
 
 42.9641 
 
 0407 
 
 24.5418 
 
 0582 
 
 17.1693 
 
 0758 
 
 13.1%9 4O 
 
 21 
 
 0061 
 
 163.700 
 
 0236 
 
 42.4335 
 
 0410 
 
 24.3675 
 
 0585 
 
 17.0837 
 
 0761 
 
 13.1461 39 
 
 22 
 
 0064 
 
 156.259 
 
 0239 
 
 41.9158 
 
 0413 
 
 24.1957 
 
 0588 
 
 16.9990 
 
 0764 
 
 13.0958! 38 
 
 23 
 
 0067 
 
 149.465 
 
 0241 
 
 41.4106 
 
 0416 
 
 24.0263 
 
 0591 
 
 16.9150 
 
 0767 
 
 13.0458 37 
 
 24 0070 
 
 143.237 
 
 0244 
 
 40.9174 
 
 0419 
 
 23.8593 
 
 0594 
 
 16.8319 
 
 0769 
 
 12.9962 36 
 
 25 0073 
 
 137.507 
 
 0247 
 
 40.4358 
 
 0422 
 
 23.6945 
 
 0597 
 
 16.74% 
 
 0772 
 
 12.9469 35 
 
 26 0076 
 
 132.219 
 
 0250 
 
 39.9655 
 
 0425 
 
 23.5321 
 
 0600 
 
 16.6681 
 
 0775 
 
 12.8981 j 34 
 
 27 0079 
 
 127.321 
 
 0253 
 
 39.5059 
 
 0428 
 
 23.3718 
 
 0603 
 
 16.5874 
 
 0778 
 
 12.84% | 33 
 
 28 i OOS1 
 
 122.774 
 
 0256 
 
 39.0568 
 
 0431 
 
 23.2137 
 
 0606 
 
 16.5075 
 
 0781 
 
 12.8014 32 
 
 29 
 
 0084 
 
 118.540 
 
 0259 
 
 38.6177 
 
 0434 
 
 23.0577 
 
 0609 
 
 16.4283 
 
 0784 
 
 12.7536 
 
 31 
 
 3O 
 
 0087 
 
 114.589 
 
 0262 
 
 38.1885 
 
 0437 
 
 22.9038 
 
 0612 
 
 16.3499 
 
 0787 
 
 12.7062 
 
 30 
 
 31 
 
 0090 
 
 110.892 
 
 0265 
 
 37.7686 
 
 0440 
 
 22.7519 
 
 0615 
 
 16.2722 
 
 0790 
 
 12.6591 ! 29 
 
 32 
 
 0093 
 
 107.426 
 
 0268 
 
 37.3579 
 
 0442 
 
 22.6020 
 
 0617 
 
 16.1952 
 
 0793 
 
 12.6124 i 28 
 
 33 
 
 0096 
 
 104.171 
 
 0271 
 
 36.9560 
 
 0445 
 
 22.4541 
 
 0620 
 
 16.1190 
 
 07% 
 
 12.5660 j 27 
 
 34 
 
 0099 
 
 101.107 
 
 0274 
 
 36.5627 
 
 0448 
 
 22.3081 
 
 0623 
 
 16.0435 
 
 0799 
 
 12.5199! 26 
 
 35 
 
 0102 
 
 98.2179 
 
 0276 
 
 36.1776 
 
 0451 
 
 22.1640 
 
 0626 
 
 15.9687 
 
 0802 
 
 12.4742 25 
 
 36 
 
 0105 
 
 95.4895 
 
 0279 
 
 35.8006 
 
 0454 
 
 22.0217 
 
 0629 
 
 15.8945 
 
 0805 
 
 12.4288! 24 
 
 37 
 
 0108 
 
 92.9085 
 
 0282 
 
 35.4313 
 
 0457 
 
 21.8813 
 
 0632 
 
 15.8211 
 
 0808 
 
 12.3838 23 
 
 38 
 
 0111 
 
 90.4633 
 
 0285 
 
 35.0695 
 
 0460 
 
 21.7426 
 
 0635 
 
 15.7483 
 
 0810 
 
 12.3390 22 
 
 39 
 
 0113 
 
 88.1436 
 
 0288 
 
 34.7151 
 
 0463 
 
 21.6056 
 
 0638 
 
 15.6762 
 
 0813 
 
 12.29461 21 
 
 
 
 
 
 
 
 
 
 
 
 
 4O 
 
 0116 
 
 85.9398 
 
 0291 
 
 34.3678 
 
 0466 
 
 21.4704 
 
 0641 
 
 15.6048 
 
 0816 
 
 12.2505 2O 
 
 41 
 
 0119 
 
 83.8435 
 
 0294 
 
 34.0273 
 
 0469 
 
 21.3369 
 
 0644 
 
 15.5340 
 
 0819 
 
 12.20671 19 
 
 42 0122 
 
 81.8470 
 
 0297 
 
 33.6935 
 
 0472 
 
 21.2049 
 
 0647 
 
 15.4638 
 
 0822 
 
 12.16321 18 
 
 43 1 Q12S 
 
 79.9434 
 
 0300 
 
 33.3662 
 
 0475 
 
 21.0747 
 
 0650 
 
 15.3943 
 
 0825 
 
 12.1201 17 
 
 44 0128 
 
 78.1263 
 
 0303 
 
 33.0452 
 
 0477 
 
 20.9460 
 
 0653 
 
 15.3254 
 
 0828 
 
 12.0772 16 
 
 45 
 
 0131 
 
 76.3900 
 
 0306 
 
 32.7303 
 
 0480 
 
 20.8188 
 
 0655 
 
 15.2571 
 
 0831 
 
 12.03461 IS 
 
 46 
 
 0134 
 
 74.7292 
 
 0308 
 
 32.4213 
 
 0483 
 
 20.6932 
 
 0658 
 
 15.1893 
 
 0834 
 
 11.9923 14 
 
 47 
 
 0137 
 
 73.1390 
 
 0311 
 
 32.1181 
 
 0486 
 
 20.5691 
 
 0661 
 
 15.1222 
 
 0837 
 
 11.9504 
 
 13 
 
 48 
 
 0140 
 
 71.6151 
 
 0314 
 
 31.8205 
 
 0489 
 
 20.4465 
 
 0664 
 
 15.0557 
 
 0840 
 
 11.9087 12 
 
 49 
 
 0143 
 
 70.1533 
 
 0317 
 
 31.5284 
 
 0492 
 
 20.3253 
 
 0667 
 
 14.9898 
 
 0843 
 
 11.8673 11 
 
 5O 
 
 OJ46 
 
 68.7501 
 
 0320 
 
 31.2416 
 
 0495 
 
 20.2056 
 
 0670 
 
 14.9244 
 
 0846 
 
 11.8262 1O 
 
 51 
 
 0148 
 
 67.4019 
 
 0323 
 
 30.9599 
 
 0498 
 
 20.0872 
 
 0673 
 
 14.85% 
 
 0849 
 
 11.7853 9 
 
 52 0151 
 
 66.1055 
 
 0326 
 
 30.6833 
 
 0501 
 
 19.9702 
 
 0676 
 
 14.7954 
 
 0851 
 
 11.7448 
 
 8 
 
 53 0154 
 
 64.8580 
 
 0329 
 
 30.4116 
 
 0504 
 
 19.8546 
 
 0679 
 
 14.7317 
 
 0854 
 
 11.7045 
 
 7 
 
 54 
 
 0157 
 
 63.6567 
 
 0332 
 
 30.1446 
 
 0507 
 
 19.7403 
 
 0682 
 
 14.6685 
 
 0857 
 
 11.6645 
 
 6 
 
 55 
 
 0160 
 
 62.4992 
 
 0335 
 
 29.8823 
 
 0509 
 
 19.6273 
 
 0685 
 
 14.6059 
 
 0860 
 
 11.6248 
 
 5 
 
 56 
 
 0163 
 
 61.3829 
 
 0338 
 
 29.6245 
 
 0512 
 
 19.5156 
 
 0688 
 
 14.5438 
 
 0863 
 
 11.5853 4 
 
 57 
 
 0166 
 
 60.3058 
 
 0340 
 
 29.3711 
 
 0515 
 
 19.4051 
 
 0690 
 
 14^823 
 
 0866 
 
 11.5461 
 
 3 
 
 58 
 
 0169 
 
 59.2659 
 
 0343 
 
 29.1220 
 
 0518 
 
 19.2959 
 
 0693 
 
 14.4212 
 
 0669 
 
 11.5072 
 
 2 
 
 59 
 
 0172 
 
 58.2612 
 
 0346 
 
 28.8771 
 
 0521 
 
 19.1879 
 
 0696 
 
 14.3607 
 
 0872 
 
 11.4685 
 
 1 
 
 6O 
 
 0175 
 
 57.2900 
 
 0349 
 
 28.6363 
 
 0524 
 
 19.0811 
 
 0699 
 
 14.3007 
 
 0875 
 
 11.4301 
 
 
 
 cot 
 
 tan 
 
 cot 
 
 tan 
 
 cot 
 
 tan 
 
 cot 
 
 tan 
 
 cot 
 
 tan 
 
 
 ' 89 
 
 88 
 
 87 
 
 80 
 
 S.T 
 
 t
 
 72 
 
 NATURAL TANGENTS AND COTANGENTS. 
 
 t 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 o , 
 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 
 
 0875 11.4301 
 
 1051 9.5144 
 
 1228 8.1443 
 
 1405 7.1154 
 
 1584 6.3138 ! GO 
 
 1 
 
 0878 11.3919 
 
 1054 9.4878 
 
 1231 8.1248 
 
 1408 7.1004 
 
 1587 6.3019 59 
 
 2 
 
 0881 11.3540 
 
 1057 9.4614 
 
 1234 8.1054 
 
 1411 7.0855 
 
 1590 6.2901 58 
 
 3 
 
 0884 11.3163 
 
 1060 9.4352 
 
 1237 8.0860 
 
 1414 7.0706 
 
 1593 6.2783 57 
 
 4 
 
 0887 11.2789 
 
 1063 9.4090 
 
 1240 8.0667 
 
 1417 7.0558 
 
 1596 6.2666 56 
 
 5 
 
 0890 11.2417 
 
 1066 9.3831 
 
 1243 8.0476 
 
 1420 7.0410 
 
 1599 6.2549 55 
 
 6 
 
 0892 11.2048 
 
 1069 9.3572 
 
 1246 8.0285 
 
 1423 7.0264 
 
 1602 6.2432 i 54 
 
 7 
 
 OS95 11.1681 
 
 1072 9.3315 
 
 1249 8.0095 
 
 1426 7.0117 
 
 1605 6.2316 53 
 
 8 
 
 0898 11.1316 
 
 1075 9.3060 
 
 1251 7.9906 
 
 1429 6.9972 
 
 1608 6.2200 j 52 
 
 9 
 
 0901 11.0954 
 
 1078 9-2806 
 
 1254 7.9718 
 
 1432 6.9827 
 
 1611 6.2085 51 
 
 1O 
 
 0904 11.0594 
 
 1080 9.2553 
 
 1257 7.9530 
 
 1435 6.9682 
 
 1614 6.1970 SO 
 
 11 
 
 0907 11.0237 
 
 1083 9.2302 
 
 1260 7.9344 
 
 1438 6.9538 
 
 1617 6.1856 49 
 
 12 
 
 0910 10.9882 
 
 1086 9.2052 
 
 1263 7.9158 
 
 1441 6.9395 
 
 1620 6.1742 48 
 
 13 
 
 0913 10.9529 
 
 1089 9.1803 
 
 1266 7.8973 
 
 1444 6.9252 
 
 1623 6.1628 47 
 
 14 
 
 0916 10.9178 
 
 1092 9.1555 
 
 1269 7.8789 
 
 1447 6.9110 
 
 1626 6.1515 46 
 
 15 
 
 0919 10.8829 
 
 1095 9.1309 
 
 1272 7.8606 
 
 1450 6.8969 
 
 1629 6.1402 45 
 
 16 
 
 0922 10.8483 
 
 1098 9.1065 
 
 1275 7.8424 
 
 1453 6.8828 
 
 1632 6.1290 44 
 
 17 
 
 0925 10.8139 
 
 1101 9.0821 
 
 1278 7.8243 
 
 1456 6.8687 
 
 1635 6.1178 43 
 
 18 
 
 0928 10.7797 
 
 1104 9.0579 
 
 1281 7.8062 
 
 1459 6.8548 
 
 1638 6.1066 i 42 
 
 19 
 
 0931 10.7457 
 
 1107 9.0338 
 
 1284 7.7883 
 
 1462 6.8408 
 
 1641 6.0955 i 41 
 
 2O 
 
 0934 10.7119 
 
 1110 9.0098 
 
 1287 7.7704 
 
 1465 6.8269 
 
 1644 6.0844 4O 
 
 21 
 
 0936 10.6783 
 
 1113 8.9860 
 
 1290 7.7525 
 
 1468 6.8131 
 
 1647 6.0734 39 
 
 22 
 
 0939 10.6450 
 
 1116 8.9623 
 
 1293 7.7348 
 
 1471 6.7994 
 
 1650 6.0624 38 
 
 23 
 
 0942 10.6118 
 
 1119 8.9387 
 
 1296 7.7171 
 
 1474 6.7856 
 
 1653 6.0514 ! 37 
 
 24 
 
 0945 10.5789 
 
 1122 8.9152 
 
 1299 7.6996 
 
 1477 6.7720 
 
 1655 6.0405 36 
 
 25 
 
 0948 10.5462 
 
 1125 8.8919 
 
 1302 7.6821 
 
 1480 6.7584 
 
 1658 6.0296 | 35 
 
 26 
 
 0951 10.5136 
 
 1128 8.8686 
 
 1305 7.6647 
 
 1483 6.7448 
 
 1661 6.0188 34 
 
 27 
 
 0954 10.4813 
 
 1131 8.8455 
 
 1308 7.6473 
 
 1486 6.7313 
 
 1664 6.0080 ' 33 
 
 28 
 
 0957 10.4491 
 
 1134 8.8225 
 
 1311 7.6301 
 
 1489 6.7179 
 
 1667 5.9972 ; 32 
 
 29 
 
 0960 10.4172 
 
 1136 8.7996 
 
 1314 7.6129 
 
 1492 6.7045 
 
 1670 5.9865 ' 31 
 
 3O 
 
 0963 10.3854 
 
 1139 8.7769 
 
 1317 7.5958 
 
 1495 6.6912 
 
 1673 5.9758 3O 
 
 31 
 
 0966 10.3538 
 
 1142 8.7542 
 
 1319 7.5787 
 
 1497 6.6779 
 
 1676 5.9651 29 
 
 32 10969 10.3224 
 
 1145 8.7317 
 
 1322 7.5618 
 
 1500 6.6646 
 
 1679 5.9545 28 
 
 33 0972 10.2913 
 
 1148 8.7093 
 
 1325 7.5449 
 
 1503 6.6514 
 
 1682 5.9439 27 
 
 34 
 
 0975 10.2602 
 
 1151 8.6870 
 
 1328 7.5281 
 
 1506 6.6383 
 
 1685 5.9333 26 
 
 35 
 
 0978 10.2294 
 
 1154 8.6648 
 
 1331 7.5113 
 
 1509 6.6252 
 
 1688 5.9228 25 
 
 36 
 
 0981 10.1988 
 
 1157 8.6427 
 
 1334 7.4947 
 
 1512 6.6122 
 
 1691 5.9124 24 
 
 37 
 
 0983 10.1683 
 
 1160 8.6208 
 
 1337 7.4781 
 
 1515 6.5992 
 
 1694 5.9019 23 
 
 38 J0986 10.1381 
 
 1163 8.5989 
 
 1340 7.4615 
 
 1518 6.5863 
 
 1697 5.8915 22 
 
 39 0989 10.1080 
 
 1166 8.5772 
 
 1343 7.4451 
 
 1521 6.5734 
 
 1700 5.8811 21 
 
 4O 0992 10.0780 
 
 1169 8.5555 
 
 1346 7.4287 
 
 1524 6.5606 
 
 1703 5.8708 2O 
 
 41 0995 10.0483 
 
 1172 8.5340 
 
 1349 7.4124 
 
 1527 6.5478 
 
 1706 5.8605 19 
 
 42 0998 10.0187 
 
 1175 8.5126 
 
 1352 7.3962 
 
 1530 6.5350 
 
 1709 5.8502 18 
 
 43 i 1001 9.9893 
 
 1178 8.4913 
 
 1355 7-3800 
 
 1533 6.5223 
 
 1712 5.8400 17 
 
 44 ! 1004 9.9601 
 
 1181 8.4701 
 
 1358 7.3639 
 
 1536 6.5097 
 
 1715 5.8298 16 
 
 
 
 
 
 
 45 ' 1007 9.9310 
 
 1184 8.4490 
 
 1361 7.3479 
 
 1539 6.4971 
 
 1718 5.8197 15 
 
 46 1010 9-9021 
 
 1187 8.4280 
 
 1864 7.3319 
 
 1542 6.4846 
 
 1721 5.8095 14 
 
 47 1013 9.8734 
 
 1189 8.4071 
 
 1367 7.3160 
 
 1545 6.4721 
 
 1724 5.7994 13 
 
 48 1016 9.8448 
 
 1192 8.3863 
 
 1370 7.3002 
 
 1548 6.4596 
 
 1727 5.7894 12 
 
 49 1019 9.8164 
 
 1195 8.3656 
 
 1373 7.2844 
 
 1551 6.4472 
 
 1730 5.7794 11 
 
 5O 1022 9.7882 
 
 1198 8.3450 
 
 1376 7.2687 
 
 1554 6.4348 
 
 1733 5.7694 1O 
 
 51 1025 9.7601 
 
 1201 8.3245 
 
 1379 7.2531 
 
 1557 6.4225 
 
 1736 5.7594 9 
 
 52 1028 9.7322 
 
 1204 8.3041 
 
 1382 7.2375 
 
 1560 6.4103 
 
 1739 5.7495 8 
 
 53 | 1030 9.7044 
 
 1207 8.2838 
 
 1385 7.2220 
 
 1563 6.3980 
 
 1742 5.7396 7 
 
 54 
 
 1033 9.6768 
 
 1210 8.2636 
 
 1388 7.2066 
 
 1566 6.3S.S9 
 
 1745 5.7297 6 
 
 55 
 
 1036 9.6499 
 
 1213 8.2434 
 
 1391 7.1912 
 
 1569 6.3737 
 
 1748 5.7199 5 
 
 56 
 
 1039 9.6220 
 
 1216 8.2234 
 
 1394 7.1759 
 
 1572 6.3617 
 
 1751 5.7101 4 
 
 57 
 
 1042 9.5949 
 
 1219 8.2035 
 
 1397 7.1607 
 
 1575 6.3496 
 
 1754 5.7004 3 
 
 58 1045 9.5679 
 
 1222 8.1837 
 
 1399 7.1455 
 
 1578 6.3376 
 
 1757 5.6906 2 
 
 59 1048 9.5411 
 
 1225 8.1640 
 
 1402 7.1304 
 
 1581 6.3257 
 
 1760 5.6809 1 
 
 6O | 1051 9.5144 
 
 1228 8.1443 
 
 1405 7.1154 
 
 1584 6.3138 
 
 1763 5.6713 O 
 
 cot tan 
 
 cot tan 
 
 cot tan 
 
 cot tan 
 
 cot tan 
 
 ' 84 
 
 83 
 
 82 
 
 81 
 
 80 '
 
 NATURAL TANGENTS AND COTANGENTS. 
 
 73 
 
 
 
 11 
 
 12 
 
 13 
 
 14 
 
 f 
 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 
 
 
 1763 5.6713 
 
 1944 5.1446 
 
 2126 4.7046 
 
 2309 4.3315 
 
 2493 4.0108 
 
 <IO 
 
 1 
 
 1766 5.6617 
 
 1947 5.1366 
 
 2129 4.6979 
 
 2312 4.3257 
 
 24% 4.0058 
 
 59 
 
 2 
 
 1769 5.6521 
 
 1950 5.1286 
 
 2132 4.6912 
 
 2315 4.3200 
 
 2499 4.0009 
 
 58 
 
 3 
 
 1772 5.6425 
 
 1953 5.1207 
 
 2135 4.6845 
 
 2318 4.3143 
 
 2503 3.9959 
 
 57 
 
 4 
 
 1775 5.6330 
 
 1956 5.1128 
 
 2138 4.6779 
 
 2321 4.3086 
 
 2506 3.9910 
 
 56 
 
 5 
 
 1778 5.6234 
 
 1959 5.1049 
 
 2141 4.6712 
 
 2324 4.3029 
 
 2509 3.9861 
 
 55 
 
 6 
 
 1781 5.6140 
 
 1962 5.0970 
 
 2144 4.6646 
 
 2327 4.2972 
 
 2512 3.9812 
 
 54 
 
 7 
 
 1784 5.6045 
 
 1965 5.0892 
 
 2147 4.6580 
 
 2330 4.2916 
 
 2515 3.9763 
 
 53 
 
 8 
 
 1787 5.5951 
 
 1968 5.0814 
 
 2150 4.6514 
 
 2333 4.2859 
 
 2518 3.9714 
 
 52 
 
 9 
 
 1790 5.5857 
 
 1971 5.0736 
 
 2153 4.6448 
 
 2336 4.2803 
 
 252.1 3.9665 
 
 51 
 
 1O 
 
 1793 5.5764 
 
 1974 5.0658 
 
 2156 4.6382 
 
 2339 4.2747 
 
 2524 3.9617 
 
 5O 
 
 11 
 
 1796 5.5671 
 
 1977 5.0581 
 
 2159 4.6317 
 
 2342 4.2691 
 
 2527 3.9568 
 
 49 
 
 12 
 
 1799 5.5578 
 
 1980 5.0504 
 
 2162 4.6252 
 
 2345 4.2635 
 
 2530 3.9520 
 
 48 
 
 13 
 
 1802 5.5485 
 
 1983 5.0427 
 
 2165 4.6187 
 
 2349 4.2580 
 
 2533 3.9471 
 
 47 
 
 14 
 
 1805 5.5393 
 
 1986 5.0350 
 
 2168 4.6122 
 
 2352. 4.2524 
 
 2537 3.9423 
 
 46 
 
 15 
 
 1808 5.5301 
 
 1989 5.0273 
 
 2171 4.6057 
 
 2355 4.2468 
 
 2540 3.9375 
 
 45 
 
 16 
 
 1811 5.5209 
 
 1992 5.0197 
 
 2174 4.5993 
 
 2358 4.2413 
 
 2543 3.9327 
 
 44 
 
 17 
 
 1814 5.5118 
 
 1995 5.0121 
 
 2177 4.5928 
 
 2361 4.2358 
 
 2546 3.9279 
 
 43 
 
 18 
 
 1817 5.5026 
 
 1998 5.0045 
 
 2180 4.5864 
 
 2364 4.2303 
 
 2549 3.9232 
 
 42 
 
 19 
 
 1820 5.4936 
 
 2001 4.9969 
 
 2183 4.5800 
 
 2367 4.2248 
 
 2552 3.9184 
 
 41 
 
 2O 
 
 1823 5.4845 
 
 2004 4.9894 
 
 2186 4.5736 
 
 2370 4.2193 
 
 2555 3.9136 
 
 4O 
 
 21 
 
 1826 5.4755 
 
 2007 4.9819 
 
 2189 4.5673 
 
 2373 4.2139 
 
 2558 3.9089 
 
 39 
 
 22 
 
 1829 5.4665 
 
 2010 4.9744 
 
 2193 4.5609 
 
 2376 4.2084 
 
 2561 3.9042 
 
 38 
 
 23 
 
 1832 5.4575 
 
 2013 4.9669 
 
 21% 4.5546 
 
 2379 4.2030 
 
 2564 3.8995 
 
 37 
 
 24 
 
 1835 5.4486 
 
 2016 4.9594 
 
 2199 4.5483 
 
 2382 4.1976 
 
 2568 3.8947 
 
 36 
 
 25 
 
 1838 5.4397 
 
 2019 4.9520 
 
 2202 4.5420 
 
 2385 4.1922 
 
 2571 3.8900 
 
 35 
 
 26 
 
 1841 5.4308 
 
 2022 4.9446 
 
 2205 4.5357 
 
 2388 4.1868 
 
 2574 3.8854 
 
 34 
 
 27 
 
 1844 5.4219 
 
 2025 4.9372 
 
 2208 4.5294 
 
 2392 4.1814 
 
 2577 3.8807 
 
 33 
 
 28 
 
 1847 5.4131 
 
 2028 4.9298 
 
 2211 4.5232 
 
 2395 4.1760 
 
 2580 3.8760 
 
 32 
 
 29 
 
 1850 5.4043 
 
 2031 4.9225 
 
 2214 4.5169 
 
 2398 4.1706 
 
 2583 3.8714 
 
 31 
 
 30 
 
 1853 5.3955 
 
 2035 4.9152 
 
 2217 4.5107 
 
 2401 4.1653 
 
 2586 3.8667 
 
 3O 
 
 31 
 
 1856 5.3868 
 
 2038 4.9078 
 
 2220 4.5045 
 
 2404 4.1600 
 
 2589 3.8621 
 
 29 
 
 32 
 
 1859 5.3781 
 
 2041 4.9006 
 
 2223 4.4983 
 
 2407 4.1547 
 
 2592 3.8575 
 
 28 
 
 33 
 
 1862 5.3694 
 
 2044 4.8933 
 
 2226 4.4922 
 
 2410 4.1493 
 
 2595 3.8528 
 
 27 
 
 34 
 
 1865 5.3607 
 
 2047 4.8860 
 
 2229 4.4860 
 
 2413 4.1441 
 
 2599 3.8482 
 
 26 
 
 35 
 
 1868 5.3521 
 
 2050 4.8788 
 
 2232 4.4799 
 
 2416 4.1388 
 
 2602 3.8436 
 
 25 
 
 36 
 
 1871 5.3435 
 
 2053 4.8716 
 
 2235 4.4737 
 
 2419 4.1335 
 
 2605 3.8391 
 
 24 
 
 37 
 
 1874 5.3349 
 
 2056 4.8644 
 
 2238 4.4676 
 
 2422 4.1282 
 
 2608 3.8345 
 
 23 
 
 38 
 
 1877 5.3263 
 
 2059 4.8573 
 
 2241 4.4615 
 
 2425 4.1230 
 
 2611 3.8299 
 
 22 
 
 39 
 
 1880 5.3178 
 
 2062 4.8501 
 
 2244 4.4555 
 
 2428 4.1178 
 
 2614 3.8254 
 
 21 
 
 40 
 
 1883 5.3093 
 
 2065 4.8430 
 
 2247 4.4494 
 
 2432 4.1126 
 
 2617 3.8208 
 
 20 
 
 41 
 
 1887 5.3008 
 
 2068 4.8359 
 
 2251 4.4434 
 
 2435 4.1074 
 
 2620 3.8163 
 
 19 
 
 42 
 
 1890 5.2924 
 
 2071 4.8288 
 
 2254 4.4374 
 
 2438 4.1022 
 
 2623 3.8118 
 
 18 
 
 43 
 
 1893 5.2839 
 
 2074 4.8218 
 
 2257 4.4313 
 
 2441 4.0970 
 
 2627 3.8073 
 
 17 
 
 44 
 
 1896 5.2755 
 
 2077 4.8147 
 
 2260 4.4253 
 
 2444 4.0918 
 
 2630 3.8028 
 
 16 
 
 45 
 
 1899 5.2672 
 
 2080 4.8077 
 
 2263 4.4194 
 
 2447 4.0867 
 
 2633 3.7983 
 
 15 
 
 46 
 
 1902 5.2588 
 
 2083 4.8007 
 
 2266 4.4134 
 
 2450 4.0815 
 
 2636 3.7938 
 
 14 
 
 47 
 
 1905 5.2505 
 
 2086 4.7937 
 
 2269 4.4075 
 
 2453 4.0764 
 
 2639 3.7893 
 
 13 
 
 48 
 
 1908 5.2422 
 
 2089 4.7867 
 
 2272 4.4015 
 
 2456 4.0713 
 
 2642 3.7848 
 
 12 
 
 49 
 
 1911 5.2339 
 
 2092 4.7798 
 
 2275 4.3956 
 
 2459 4.0662 
 
 2645 3.7804 
 
 11 
 
 5O 
 
 1914 5.2257 
 
 2095 4.7729 
 
 2278 4.3897 
 
 2462 4.0611 
 
 2648 3.7760 
 
 10 
 
 51 
 
 1917 5.2174 
 
 2098 4.7659 
 
 2281 4.3838 
 
 2465 4.0560 
 
 2651 3.7715 
 
 9 
 
 52 
 
 1920 5.2092 
 
 2101 4.7591 
 
 2284 4.3779 
 
 2469 4.0509 
 
 2655 3.7671 
 
 8 
 
 53 
 
 1923 5.2011 
 
 2104 4.7522 
 
 2287 4.3721 
 
 2472 4.0459 
 
 2658 3.7627 
 
 7 
 
 54 
 
 1926 5.1929 
 
 2107 4.7453 
 
 2290 4.3662 
 
 2475 4.0408 
 
 2661 3.7583 
 
 6 
 
 55 
 
 1929 5.1848 
 
 2110 4.7385 
 
 2293 4.3604 
 
 2478 4.0358 
 
 2664 3.7539 
 
 5 
 
 56 
 
 1932 5.1767 
 
 2113 4.7317 
 
 22% 4.3546 
 
 2481 4.0308 
 
 2667 3.7495 
 
 4 
 
 57 
 
 1935 5.1686 
 
 2116 4.7249 
 
 2299 4.3488 
 
 2484 4.0257 
 
 2670 3.7451 
 
 3 
 
 58 
 
 1938 5.1606 
 
 2119 4.7181 
 
 2303 4.3430 
 
 2487 4.0207 
 
 2673 3.7408 
 
 2 
 
 59 
 
 1941 5.1526 
 
 2123 4.7114 
 
 2306 4.3372 
 
 2490 4.0158 
 
 2676 3.7364 
 
 1 
 
 6O 
 
 1944 5.1446 
 
 2126 4.7046 
 
 2309 4.3315 
 
 2493 4.0108 
 
 2679 3.7321 
 
 
 
 
 cot tan 
 
 cot tan 
 
 cot tan 
 
 cot tan 
 
 cot tan 
 
 
 / 
 
 79 
 
 78 
 
 77 
 
 7<i 
 
 75 
 
 f
 
 74 
 
 NATURAL TANGENTS AND COTANGENTS. 
 
 t 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 t 
 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 
 o 
 
 2679 3.7321 
 
 2867 3.4874 
 
 3057 3.2709 
 
 3249 3.0777 
 
 3443 2.9042 
 
 6O 
 
 1 
 
 2683 3.7277 
 
 2871 3.4836 
 
 3060 3.2675 
 
 3252 3.0746 
 
 3447 2.9015 
 
 59 
 
 2 
 
 2686 3.7234 
 
 2874 3.4798 
 
 3064 3.2641 
 
 3256 3.0716 
 
 3450 2.8987 
 
 58 
 
 3 
 
 2689 3.7191 
 
 2877 3.4760 
 
 3067 3.2607 
 
 3259 3.0686 
 
 3453 2.8960 
 
 57 
 
 4 
 
 2692 3.7148 
 
 2880 3.4722 
 
 3070 3.2573 
 
 3262 3.0655 
 
 3456 2.8933 
 
 56 
 
 5 
 
 2695 3.7105 
 
 2883 3.4684 
 
 3073 3.2539 
 
 3265 3.0625 
 
 3460 2.8905 
 
 55 
 
 6 
 
 2698 3.7062 
 
 2886 3.4646 
 
 3076 3.2506 
 
 3269 3.0595 
 
 3463 2.8878 
 
 54 
 
 7 
 
 2701 3.7019 
 
 2S90 3.4608 
 
 3080 3.2472 
 
 3272 3.0565 
 
 3466 2.8851 
 
 53 
 
 8 
 
 2704 3.6976 
 
 2893 3.4570 
 
 3083 3.2438 
 
 3275 3.0535 
 
 3469 2.8824 
 
 52 
 
 9 
 
 2708 3.6933 
 
 2896 3.4533 
 
 3086 3.2405 
 
 3278 3.0505 
 
 3473 2.8797 
 
 51 
 
 1O 
 
 2711 3.6891 
 
 2899 3.4495 
 
 3089 3.2371 
 
 3281 3.0475 
 
 3476 2.8770 
 
 5O 
 
 11 
 
 2714 3.6848 
 
 2902 3.4458 
 
 3092 3.2338 
 
 3285 3.0445 
 
 3479 2.8743 
 
 49 
 
 12 
 
 2717 3.6806 
 
 2905 3.4420 
 
 3096 3.2305 
 
 3288 3.0415 
 
 3482 2.8716 
 
 48 
 
 13 
 
 2720 3.6764 
 
 2908 3.4383 
 
 3099 3.2272 
 
 3291 3.0385 
 
 3486 2.8689 
 
 47 
 
 14 
 
 2723 3.6722 
 
 2912 3:4346 
 
 3102 3.2238 
 
 3294 3.0356 
 
 3489 2.8662 
 
 46 
 
 15 
 
 2726 3.6680 
 
 2915 3.4308 
 
 3105 3.2205 
 
 3298 3.0326 
 
 3492 2.8636 
 
 45 
 
 16 
 
 2729 3.6638 
 
 2918 3.4271 
 
 3108 3.2172 
 
 3301 3.0296 
 
 3495 2.8609 
 
 44 
 
 17 
 
 2733 3.6596 
 
 2921 3.4234 
 
 3111 3.2139 
 
 3304 3.0267 
 
 3499 2.8582 
 
 43 
 
 18 
 
 2736 3.6554 
 
 2924 3.4197 
 
 3115 3.2106 
 
 3307 3.0237 
 
 3502 2.8556 
 
 42 
 
 19 
 
 2739 3.6512 
 
 2927 3.4160 
 
 3118 3.2073 
 
 3310 3.0208 
 
 3505 2.8529 
 
 41 
 
 2O 
 
 2742 3.6470 
 
 2931 3.4124 
 
 3121 3.2041 
 
 3314 3.0178 
 
 3508 2.8502 
 
 4O 
 
 21 
 
 2745 3.6429 
 
 2934 3.4087 
 
 3124 3.2008 
 
 3317 3.0149 
 
 3512 2.8476 
 
 39 
 
 22 
 
 2748 3.6387 
 
 2937 3.4050 
 
 3127 3.1975 
 
 3320 3.0120 
 
 3515 2.8449 
 
 38 
 
 23 
 
 2751 3.6346 
 
 2940 3.4014 
 
 3131 3.1943 
 
 3323 3.0090 
 
 3518 2.8423 
 
 37 
 
 24 
 
 2754 3.6305 
 
 2943 3.3977 
 
 3134 3.1910 
 
 3327 3.0061 
 
 3522 2.8397 
 
 36 
 
 25 
 
 2758 3.6264 
 
 2946 3.3941 
 
 3137 3.1878 
 
 3330 3.0032 
 
 3525 2.8370 
 
 35 
 
 26 
 
 2761 3.6222 
 
 2949 3.3904 
 
 3140 3.1845 
 
 3333 3.0003 
 
 3528 2.8344 
 
 34 
 
 27 
 
 2764 3.6181 
 
 2953 3.3868 
 
 3143 3.1813 
 
 3336 2.9974 
 
 3531 2.8318 
 
 33 
 
 28 
 
 2767 3.6140 
 
 2956 3.3832 
 
 3147 3.1780 
 
 3339 2.9945 
 
 3535 2.8291 
 
 32 
 
 29 
 
 2770 3.6100 
 
 2959 3.3796 
 
 3150 3.1748 
 
 3343 2.9916 
 
 3538 2.8265 
 
 31 
 
 30 
 
 2773 3.6059 
 
 2962 3.3759 
 
 3153 3.1716 
 
 3346 2.9887 
 
 3541 2.8239 
 
 3O 
 
 31 
 
 2776 3.6018 
 
 2965 3.3723 
 
 3156 3.1684 
 
 3349 2.9858 
 
 3544 2.8213 
 
 29 
 
 32 
 
 2780 3.5978 
 
 2968 3.3687 
 
 3159 3.1652 
 
 3352 2.9829 
 
 3548 2.S187 
 
 28 
 
 33 
 
 2783 3.5937 
 
 2972 3.3652 
 
 3163 3.1620 
 
 3356 2.9800 
 
 3551 2.8161 
 
 27 
 
 34 
 
 2786 3.5897 
 
 2975 3.3616 
 
 3166 3.1588 
 
 3359 2.9772 
 
 3554 2.8135 26 
 
 35 
 
 2789 3.5856 
 
 2978 3.3580 
 
 3169 3.1556 
 
 3362 2.9743 
 
 3558 2.8109 
 
 25 
 
 36 
 
 2792 3.5816 
 
 2981 3.3544 
 
 3172 3.1524 
 
 3365 2.9714 
 
 3561 2.8083 
 
 24 
 
 37 
 
 2795 3.5776 
 
 2984 3.3509 
 
 3175 3.1492 
 
 3369 2.9686 
 
 3564 2.8057 
 
 23 
 
 38 
 
 2798 3.5736 
 
 2987 3.3473 
 
 3179 3.1460 
 
 3372 2.9657 
 
 3567 2.S032 
 
 22 
 
 39 
 
 2801 3.5696 
 
 2991 3.3438 
 
 3182 3.1429 
 
 3375 2.9629 
 
 3571 2.8006 
 
 21 
 
 4O 
 
 2805 3.5656 
 
 2994 3.3402 
 
 3185 3.1397 
 
 3378 2.9600 
 
 3574 2.7980 
 
 2O 
 
 41 
 
 2808 3.5616 
 
 2997 3.3367 
 
 3188 3.1366 
 
 3382 2.9572 
 
 3577 2.7955 
 
 19 
 
 42 
 
 2811 3.5576 
 
 3000 3.3332 
 
 3191 3.1334 
 
 3385 2.9544 
 
 3581 2.7929 
 
 18 
 
 43 
 
 2814 3.5536 
 
 3003 3.3297 
 
 3195 3.1303 
 
 3388 2.9515 
 
 3584 2.7903 
 
 17 
 
 44 
 
 2817 3.5497 
 
 3006 3.3261 
 
 3198 3.1271 
 
 3391 2.9487 
 
 3587 2.7878 
 
 16 
 
 45 
 
 2820 3.5457 
 
 3010 3.3226 
 
 3201 3.1240 
 
 3395 2.9459 
 
 3590 2.7852 
 
 15 
 
 46 
 
 2823 3.5418 
 
 3013 3.3191 
 
 3204 3.1209 
 
 3398 2.9431 
 
 3594 2.7827 
 
 14 
 
 47 
 
 2827 3.5379 
 
 3016 3.3156 
 
 3207 3.1178 
 
 3401 2.9403 
 
 3597 2.7801 
 
 13 
 
 48 
 
 2830 3.5339 
 
 3019 3.3122 
 
 3211 3.1146 
 
 3404 2.9375 
 
 3600 2.7776 
 
 12 
 
 49 
 
 2833 3.5300 
 
 3022 3.3087 
 
 3214 3.1115 
 
 3408 2.9347 
 
 3604 2.7751 
 
 11 
 
 5O 
 
 2836 3.5261 
 
 3026 3.3052 
 
 3217 3.1084 
 
 3411 2.9319 
 
 3607 2.7725 
 
 10 
 
 51 
 
 2839 3.5222 
 
 3029 3.3017 
 
 3220 3.1053 
 
 3414 2.9291 
 
 3610 2.7700 
 
 9 
 
 52 
 
 2842 3.5183 
 
 3032 3.2983 
 
 3223 3.1022 
 
 3417 2.9263 
 
 3613 2.7675 
 
 8 
 
 53 
 
 2845 3.5144 
 
 3035 3.2948 
 
 3227 3.0991 
 
 3421 2.9235 
 
 3617 2.7650 
 
 7 
 
 54 
 
 2849 3.5105 
 
 3038 3.2914 
 
 3230 3.0961 
 
 3424 2.9208 
 
 3620 2.7625 
 
 6 
 
 55 
 
 2852 3.5067 
 
 3041 3.2880 
 
 3233 3.0930 
 
 3427 2.9180 
 
 3623 2.7500 
 
 5 
 
 56 
 
 2855 3.5028 
 
 3045 3.2845 
 
 3236 3.0899 
 
 3430 2.9152 
 
 3627 2.7575 
 
 4 
 
 57 
 
 2858 3.4989 
 
 3048 3.2811 
 
 3240 3.0868 
 
 3434 2.9125 
 
 3630 2.7550 
 
 3 
 
 58 
 
 2861 3.4951 
 
 3051 3.2777 
 
 3243 3.0838 
 
 3437 2.9097 
 
 3633 2.7525 
 
 2 
 
 59 
 
 2864 3.4912 
 
 3054 3.2743 
 
 3246 3.0S07- 
 
 3440 2.9070 
 
 3636 2.7500 
 
 1 
 
 60 
 
 2867 3.4874 
 
 3057 3.2709 
 
 3249 3.0777 
 
 3443 2.9042 
 
 3640 2.7475 O 
 
 
 cot tan 
 
 cot tan 
 
 cot tan 
 
 cot tan 
 
 cot tan 
 
 / 74 
 
 73 
 
 72 
 
 71 
 
 70 '
 
 NATURAL TANGENTS AND COTANGENTS. 
 
 75 
 
 ' 20 
 
 21 
 
 22 
 
 23 
 
 24 1 ' 
 
 
 tan cot 
 
 tail cot 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 o 
 
 3640 2.7475 
 
 3839 2.6051 
 
 4040 2.4751 
 
 4245 2.3559 
 
 4452 2.2460 
 
 6O 
 
 1 
 
 3643 2.7450 
 
 3842 2.6028 
 
 4044 2.4730 
 
 4248 2.3539 
 
 4456 2.2443 
 
 59 
 
 2 
 
 3646 2.7425 
 
 3845 2.6006 
 
 4047 2.4709 
 
 4252 2.3520 
 
 4459 2.2425 
 
 58 
 
 3 
 
 3650 2.7400. 
 
 3849 2.5983 
 
 4050 2.4689 
 
 4255 2.3501 
 
 4463 2.2408 
 
 57 
 
 4 
 
 3653 2.7376 
 
 3852 2.5961 
 
 4054 2.4668 
 
 4258 2.3483 
 
 4466 2.2390 
 
 56 
 
 5 
 
 3656 2.7351 
 
 3855 2.5938 
 
 4057 2.4648 
 
 4262 2.3464 
 
 4470 2.2373 
 
 55 
 
 6 
 
 3659 2.7326 
 
 3859 2.5916 
 
 4061 2.4627 
 
 4265 2.3445 
 
 4473 2.2355 
 
 54 
 
 7 
 
 3663 2.7302 
 
 3862 2.5893 
 
 4064 2.4606 
 
 4269 2.3426 
 
 4477 2.2338 
 
 53 
 
 8 
 
 3666 2.7277 
 
 3865 2.5871 
 
 4067 2.4586 
 
 4272 2.3407 
 
 4480 2.2320 
 
 52 
 
 9 
 
 3669 2.7253 
 
 3869 2.5848 
 
 4071 2.4566 
 
 4276 2.3388 
 
 4484 2.2303 
 
 51 
 
 1O 
 
 3673 2.7228 
 
 3872 2.5826 
 
 4074 2.4545 
 
 4279 2.3369 
 
 4487 2.2286 
 
 5O 
 
 11 
 
 3676 2.7204 
 
 3875 2.5804 
 
 4078 2.4525 
 
 4283 2.3351 
 
 4491 2.2268 
 
 49 
 
 12 
 
 3679 2.7179 
 
 3879 2.5782 
 
 4081 2.4504 
 
 4286 2.3332 
 
 4494 2.2251 
 
 48 
 
 13 
 
 3683 2.7155 
 
 3882 2.5759 
 
 4084 2.4484 
 
 4289 2.3313 
 
 4498 2.2234 
 
 47 
 
 14 
 
 3686 2.7130 
 
 3885 2.5737 
 
 4088 2.4464 
 
 4293 2.3294 
 
 4501 2.2216 
 
 46 
 
 15 
 
 3689 2.7106 
 
 3889 2.5715 
 
 4091 2.4443 
 
 4296 2.3276 
 
 4505 2.2199 
 
 45 
 
 16 
 
 3693 2.7082 
 
 3892 2.5693 
 
 4095 2.4423 
 
 4300 2.3257 
 
 4508 2.2182 
 
 44 
 
 17 
 
 3696 2.7058 
 
 3895 2.5671 
 
 4098 2.4403 
 
 4303 2.3238 
 
 4512 2.2165 
 
 43 
 
 18 
 
 3699 2.7034 
 
 3899 2.5649 
 
 4101 2.4383 
 
 4307 2.3220 
 
 4515 2.2148 
 
 42 
 
 19 
 
 3702 2.7009 
 
 3902 2.5627 
 
 4105 2.4362 
 
 4310 2.3201 
 
 4519 2.2130 
 
 41 
 
 2O 
 
 3706 2.6985 
 
 3906 2.5605 
 
 4108 2.4342 
 
 4314 2.3183 
 
 4522 2.2113 
 
 4O 
 
 21 
 
 3709 2.6961 
 
 3909 2.5583 
 
 4111 2.4322 
 
 4317 2.3164 
 
 4526 2.2096 
 
 39 
 
 22 
 
 3712 2.6937 
 
 3912 2.5561 
 
 4115 2.4302 
 
 4320 2.3146 
 
 4529 2.2079 
 
 38 
 
 23 
 
 3716 2.6913 
 
 3916 2.5539 
 
 4118 2.4282 
 
 4324 2.3127 
 
 4533 2.2062 
 
 37 
 
 24 
 
 3719 2.6S89 
 
 3919 2.5517 
 
 4122 2.4262 
 
 4327 2.3109 
 
 4536 2.2045 
 
 36 
 
 25 
 
 3722 2.6865 
 
 3922 2.5495 
 
 4125 2.4242 
 
 4331 2.3090 
 
 4540 2.2028 
 
 35 
 
 26 
 
 3726 2.6841 
 
 3926 2.5473 
 
 4129 2.4222 
 
 4334 2.3072 
 
 4543 2.2011 
 
 34 
 
 27 
 
 3729 2.6818 
 
 3929 2.5452 
 
 4132 2.4202 
 
 4338 2.3053 
 
 4547 2.1994 
 
 33 
 
 28 
 
 3732 2.6794 
 
 3932 2.5430 
 
 4135 2.4182 
 
 4341 2.3035 
 
 4550 2.1977 
 
 32 
 
 29 
 
 3736 2.6770 
 
 3936 2.5408 
 
 4139 2.4162 
 
 4345 2.3017 
 
 4554 2.1960 
 
 31 
 
 30 
 
 3739 2.6746 
 
 3939 2.53S6 
 
 4142 2.4142 
 
 4348 2.2998 
 
 4557 2.1943 
 
 3O 
 
 31 
 
 3742 2.6723 
 
 3942 2.5365 
 
 4146 2.4122 
 
 4352 2.2980 
 
 4561 2.1926 
 
 29 
 
 32 
 
 3745 2.6699 
 
 3946 2.5343 
 
 4149 2.4102 
 
 4355 2.2962 
 
 4564 2.1909 
 
 28 
 
 33 
 
 3749 2.6675 
 
 3949 2.5322 
 
 4152 2.4083 
 
 4359 2.2944 
 
 4568 2.1892 
 
 27 
 
 34 
 
 3752 2.6652 
 
 3953 2.5300 
 
 4156 2.4063 
 
 4362 2.2925 
 
 4571 2.1876 
 
 26 
 
 35 
 
 3755 2.6628 
 
 3956 2.5279 
 
 4159 2.4043 
 
 4365 2.2907 
 
 4575 2.1859 
 
 25 
 
 36 3759 2.6605 
 
 3959 2.5257 
 
 4163 2.4023 
 
 4369 2.2889 
 
 4578 2.1842 
 
 24 
 
 37 3762 2.6581 
 
 3963 2.5236 
 
 4166 2.4004 
 
 4372 2.2871 
 
 4582 2.1825 
 
 23 
 
 38 ! 3765 2.65:8 
 
 3966 2.5214 
 
 4169 2.3984 
 
 4376 2.2853 
 
 4585 2.1808 
 
 22 
 
 39 3769 2.6534 
 
 3969 2.5193 
 
 4173 2.3964 
 
 4379 2.2835 
 
 4589 2.1792 
 
 21 
 
 4O 
 
 3772 2.6511 
 
 3973 2.5172 
 
 4176 2.3945 
 
 4383 2.2817 
 
 4592 2.1775 
 
 2O 
 
 41 
 
 3775 2.648S 
 
 3976 2.5150 
 
 4180 2.3925 
 
 4386 2.2799 
 
 45% 2.1758 
 
 19 
 
 42 
 
 3779 2.6464 
 
 3979 2.5129 
 
 4183 2.3906 
 
 4390 2.2781 
 
 4599 2.1742 
 
 18 
 
 43 
 
 3782 2.6441 
 
 3983 2.5108 
 
 4187 2.3886 
 
 4393 2.2763 
 
 4603 2.1725 
 
 17 
 
 44 
 
 3785 2.6418 
 
 3986 2.5086 
 
 4190 2.3867 
 
 4397 2.2745 
 
 4607 2.1708 
 
 16 
 
 45 
 
 3789 2.6395 
 
 3990 2.5065 
 
 4193 2.3847 
 
 4400 2.2727 
 
 4610 2.1692 
 
 15 
 
 46 
 
 3792 2.6371 
 
 3993 2.5044 
 
 4197 2.3828 
 
 4404 2.2709 
 
 4614 2.1675 
 
 14 
 
 47 
 
 3795 2.6348 
 
 3996 2.5023 
 
 4200 2.3808 
 
 4407 2.2691 
 
 4617 2.1659 
 
 13 
 
 48 
 
 3799 2.6325 
 
 4000 2.5002 
 
 4204 2.3789 
 
 4411 2.2673 
 
 4621 2.1642 
 
 12 
 
 49 3802 2.6302 
 
 4003 2.4981 
 
 4207 2.3770 
 
 4414 2.2655 
 
 4624 2.1625 
 
 11 
 
 5O 3805 2.6279 
 
 4006 2.4960 
 
 4210 2.3750 
 
 4417 2.2637 
 
 4628 2.1609 
 
 10 
 
 51 3809 2.6256 
 
 4010 2.4939 
 
 4214 2.3731 
 
 4421 2.2620 
 
 4631 2.1592 
 
 9 
 
 52 3812 2.6233 
 
 4013 2.4918 
 
 4217 2.3712 
 
 4424 2.2602 
 
 4635 2.1576 
 
 8 
 
 53 3815 2.6210 
 
 1017 2.4897 
 
 4221 2.3693 
 
 4428 2.2584 
 
 4638 2.1560 
 
 7 
 
 54 
 
 3819 2.6187 
 
 4020 2.4876 
 
 4224 2.3673 
 
 4431 2.2566 
 
 4642 2.1543 
 
 6 
 
 55 
 
 3822 2.6165 
 
 4023 2.4855 
 
 4228 2.3654 
 
 4435 2.2549 
 
 4645 2.1527 
 
 5 
 
 56 
 
 3825 2.6142 
 
 4027 2.4834 
 
 4231 2.3635 
 
 4438 2.2531 
 
 4649 2.1510 
 
 4 
 
 57 
 
 3829 2.6119 
 
 4030 2.4813 
 
 4234 2.3616 
 
 4442 2.2513 
 
 4652 2.1494 
 
 3 
 
 58 
 
 3832 2.6096 
 
 4033 2.4792 
 
 4238 2.3597 
 
 4445 2.24% 
 
 4656 2.1478 
 
 2 
 
 59 
 
 3835 2.6074 
 
 4037 2.4772 
 
 4241 2.3578 
 
 4449 2.2478 
 
 4660 2.1461 
 
 1 
 
 6O 3839 2.6051 
 
 4040 2.4751 
 
 4245 2.3559 
 
 4452 2.2460 
 
 4663 2.1445 
 
 
 
 1 cot tan 
 
 cot tan 
 
 cot tan 
 
 cot tan 
 
 cot tan 
 
 
 ' 69 
 
 68 
 
 67 
 
 66 
 
 5 C 
 
 i
 
 76 
 
 NATURAL TANGENTS AND COTANGENTS. 
 
 f 
 
 25 
 
 26 
 
 27 
 
 28 C 
 
 29 
 
 t 
 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 
 o 
 
 4663 2.1445 
 
 4877 2.0503 
 
 5095 1.9626 
 
 5317 1.8807 
 
 5543 1.8040 
 
 60 
 
 1 
 
 4667 2.1429 
 
 4881 2.0488 
 
 5099 1.9612 
 
 5321 1.8794 
 
 5547 1.8028 
 
 59 
 
 2 
 
 4670 2.1413 
 
 4885 2.0473 
 
 5103 1.9598 
 
 5325 1.8781 
 
 5551 1.8016 
 
 58 
 
 3 
 
 4674 2.1396 
 
 4888 2.0458 
 
 5106 1.9584 
 
 5328 1.8768 
 
 5555 1.8003 
 
 57 
 
 4 
 
 4677 2.1380 
 
 4892 2.0443 
 
 5110 1.9570 
 
 5332 1.8755 
 
 5558 1.7991 
 
 56 
 
 5 
 
 4681 2.1364 
 
 4895 2.0428 
 
 5114 1.9556 
 
 5336 1.8741 
 
 5562 1.7979 
 
 55 
 
 6 
 
 4684 2.1348 
 
 4899 2.0413 
 
 5117 1.9542 
 
 5340 1.8728 
 
 5566 1.7966 
 
 54 
 
 7 
 
 4688 2.1332 
 
 4903 2.0398 
 
 5121 1.9528 
 
 5343 1.8715 
 
 5570 1.7954 
 
 53 
 
 8 
 
 4691 2.1315 
 
 4906 2.0383 
 
 5125 1.9514 
 
 5347 1.8702 
 
 5574 1.7942 
 
 52 
 
 9 
 
 4695 2.1299 
 
 4910 2.0368 
 
 5128 1.9500 
 
 5351 1.8689 
 
 5577 1.7930 
 
 51 
 
 1O 
 
 4699 2.1283 
 
 4913 2.0353 
 
 5132 1.9486 
 
 5354 1.8676 
 
 5581 1.7917 
 
 50 
 
 11 
 
 4702 2.1267 
 
 4917 2.0338 
 
 5136 1.9472 
 
 5358 1.8663 
 
 5585 1.7905 
 
 49 
 
 12 
 
 4706 2.1251 
 
 4921 2.0323 
 
 5139 1.9458 
 
 5362 1.8650 
 
 5589 1.7893 
 
 48 
 
 13 
 
 4709 2.1235 
 
 4924 2.0308 
 
 5143 1.9444 
 
 5366 1.8637 
 
 5593 1.7881 
 
 47 
 
 14 
 
 4713 2.1219 
 
 4928 2.0293 
 
 5147 1.9430 
 
 5369 1.8624 
 
 5596 1.7868 
 
 46 
 
 15 
 
 4716 2.1203 
 
 4931 2.0278 
 
 5150 1.9416 
 
 5373 1.8611 
 
 5600 1.7856 
 
 45 
 
 16 
 
 4720 2.1187 
 
 4935 2.0263 
 
 5154 1.9402 
 
 5377 1.8598 
 
 5604 1.7844 
 
 44 
 
 17 
 
 4723 2.1171 
 
 4939 2.0248 
 
 5158 1.9388 
 
 5381 1.8585 
 
 5608 1.7832 
 
 43 
 
 18 
 
 4727 2.1155 
 
 4942 2.0233 
 
 5161 1.9375 
 
 5384 1.8572 
 
 5612 1.7820 
 
 42 
 
 19 
 
 4731 2.1139 
 
 4946 2.0219 
 
 5165 1.9361 
 
 5388 1.8559 
 
 5616 1.7808 
 
 41 
 
 2O 
 
 4734 2.1123 
 
 4950 2.0204 
 
 5169 1.9347 
 
 5392 1.8546 
 
 5619 1.7796 
 
 4O 
 
 21 
 
 4738 2.1107 
 
 4953 2.0189 
 
 5172 1.9333 
 
 5396 1.8533 
 
 5623 1.7783 
 
 39 
 
 22 
 
 4741 2.1092 
 
 4957 2.0174 
 
 5176 1.9319 
 
 5399 1.8520 
 
 5627 1.7771 
 
 38 
 
 23 
 
 4745 2.1076 
 
 4960 2.0160 
 
 5180 1.9306 
 
 5403 1.8507 
 
 5631 1.7759 
 
 37 
 
 24 
 
 4748 2.1060 
 
 4964 2.0145 
 
 5184 1.9292 
 
 5407 1.8495 
 
 5635 1.7747 
 
 36 
 
 25 
 
 4752 2.1044 
 
 4968 2.0130 
 
 5187 1.9278 
 
 5411 1.8482 
 
 5639 1.7735 
 
 35 
 
 26 
 
 4755 2.1028 
 
 4971 2.0115 
 
 5191 1.9265 
 
 5415 1.8469 
 
 5642 1.7723 : 34 
 
 27 
 
 4759 2.1013 
 
 4975 2.0101 
 
 5195 1.9251 
 
 5418 1.8456 
 
 5646 1.7711 ! 33 
 
 28 
 
 4763 2.0997 
 
 4979 2.0086 
 
 5198 1.9237 
 
 5422 1.8443 
 
 5650 1.7699 
 
 32 
 
 29 
 
 4766 2.0981 
 
 4982 2.0072 
 
 5202 1.9223 
 
 5426 1.8430 
 
 5654 1.7687 
 
 31 
 
 30 
 
 4770 2.0965 
 
 4986 2.0057 
 
 5206 1.9210 
 
 5430-1.8418 
 
 5658 1.7675 
 
 3O 
 
 31 
 
 4773 2.0950 
 
 4989 2.0042 
 
 5209 1.9196 
 
 5433 1.8405 
 
 5662 1.7663 
 
 29 
 
 32 
 
 4777 2.0934 
 
 4993 2.0028 
 
 5213 1.9183 
 
 5437 1.8392 
 
 5665 1.7651 
 
 28 
 
 33 
 
 4780 2.0918 
 
 4997 2.0013 
 
 5217 1.9169 
 
 5441 1.8379 
 
 5669 1.7639 
 
 27 
 
 34 
 
 4784 2.0903 
 
 5000 1.9999 
 
 5220 1.9155 
 
 5445 1.8367 
 
 5673 1.7627 
 
 26 
 
 35 
 
 4788 2.0887 
 
 5004 1.9984 
 
 5224 1.9142 
 
 5448 1.8354 
 
 5677 1.7615 
 
 25 
 
 36 
 
 4791 2.0872 
 
 5008 1.9970 
 
 5228 1.9128 
 
 5452 1.8341 
 
 5681 1.7603 
 
 24 
 
 37 
 
 4795 2.0856 
 
 5011 1.9955 
 
 5232 1.9115 
 
 5456 1.8329 
 
 5685 1.7591 
 
 23 
 
 38 
 
 4798 2.0840 
 
 5015 1.9941 
 
 5235 1.9101 
 
 5460 1.8316 
 
 5688 1.7579 ! 22 
 
 39 
 
 4802 2.0825 
 
 5019 1.9926 
 
 5239 1.9088 
 
 5464 1.8303 
 
 5692 1.7567 21 
 
 40 
 
 4806 2.0809 
 
 5022 1.9912 
 
 5243 1.9074 
 
 5467 1.8291 
 
 5696 1.7556 2O 
 
 41 
 
 4S09 2.0794 
 
 5026 1.9897 
 
 5246 1.9061 
 
 5471 1.8278 
 
 5700 1.7544 19 
 
 42 
 
 4813 2.0778 
 
 5029 1.9883 
 
 5250 1.9047 
 
 5475 1.8265 
 
 5704 1.7532 18 
 
 43 
 
 4816 2.0763 
 
 5033 1.9868 
 
 5254 1.9034 
 
 5479 1.8253 
 
 5708 .7520 | 17 
 
 44 
 
 4820 2.0748 
 
 5037 1.9854 
 
 5258 1.9020 
 
 5482 1.8240 
 
 5712 1.7508 
 
 16 
 
 45 
 
 4823 2.0732 
 
 5040 1.9840 
 
 5261 1.9007 
 
 5486 1.8228 
 
 5715 .7496 
 
 15 
 
 46 
 
 4827 2.0717 
 
 5044 1.9825 
 
 5265 1.8993 
 
 5490 1.8215 
 
 5719 .7485 
 
 14 
 
 47 
 
 4831 2.0701 
 
 5048 1.9811 
 
 5269 1.8980 
 
 5494 1.8202 
 
 5723 .7473 
 
 13 
 
 48 
 
 4834 2.0686 
 
 5051 1.9797 
 
 5272 1.8967 
 
 5498 1.8190 
 
 5727 .7461 
 
 12 
 
 49 
 
 4838 2.0671 
 
 5055 1.9782 
 
 5276 1.8953 
 
 5501 1.8177 
 
 5731 .7449 
 
 11 
 
 5O 
 
 4841 2.0655 
 
 5059 1.9768 
 
 5280 1.8940 
 
 5505 1.8165 
 
 5735 .7437 1O 
 
 51 
 
 4845 2.0640 
 
 5062 1.9754 
 
 5284 1.8927 
 
 5509 1.8152 
 
 5739 .7426 9 
 
 52 
 
 4849 2.0625 
 
 5066 1.9740 
 
 5287 1.8913 
 
 5513 1.8140 
 
 5743 .7414 8 
 
 53 
 
 4852 2.0609 
 
 5070 1.9725 
 
 5291 1.8900 
 
 5517 1.8127 
 
 5746 .7402 
 
 7 
 
 54 
 
 4856 2.0594 
 
 5073 1.9711 
 
 5295 1.8887 
 
 5520 1.8115 
 
 5750 .7391 
 
 6 
 
 55 
 
 4859 2.0579 
 
 5077 1.9697 
 
 5298 1.8873 
 
 5524 1.8103 
 
 5754 .7379 
 
 5 
 
 56 
 
 4863 2.0564 
 
 5081 1.9683 
 
 5302 1.8860 
 
 5528 1.8090 
 
 5758 .7367 
 
 4 
 
 57 
 
 4867 2.0549 
 
 5084 1.9669 
 
 5306 1.8847 
 
 5532 1.8078 
 
 5762 .7355 
 
 3 
 
 58 
 
 4870 2.0533 
 
 5088 1.9654 
 
 5310 1.8834 
 
 5535 1.8065 
 
 5766 .7344 
 
 2 
 
 59 
 
 4874 2.0518 
 
 5092 1.9640 
 
 5313 1.SS20 
 
 5539 1.8053 
 
 5770 .7332 
 
 1 
 
 60 
 
 4877 2.0503 
 
 5095 1.9626 
 
 5317 1.8807 
 
 5543 1.8040 
 
 5774 1.7321 
 
 
 
 
 cot tan 
 
 cot tan 
 
 cot tan 
 
 cot tan 
 
 cot tan 
 
 f 
 
 64 
 
 68 
 
 62 
 
 61 
 
 60 '
 
 NATURAL TANGENTS AND COTANGENTS. 
 
 77 
 
 ' 
 
 3O 
 
 31 
 
 32 
 
 33 C 
 
 34 
 
 r 
 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 
 
 
 5774 1.7321 
 
 6009 1.6643 
 
 6249 1.6003 
 
 6494 1.5399 
 
 6745 1.4826 !O 
 
 1 
 
 5777 1.7309 
 
 6013 1.6632 
 
 6253 1.5993 
 
 6498 1.5389 
 
 6749 1.4816 
 
 59 
 
 2 
 
 5781 1.7297 
 
 6017 1.6621 
 
 6257 1.5983 
 
 6502 1.5379 
 
 6754 1.4807 
 
 58 
 
 3 
 
 5785 1.7286 
 
 6020 1.6610 
 
 6261 1.5972 
 
 6506 1.5369 
 
 6758 1.4798 
 
 57 
 
 4 
 
 5789 1.7274 
 
 6024 1.6599 
 
 6265 1.5962 
 
 6511 1.5359 
 
 6762 1.4788 
 
 56 
 
 5 
 
 5793 1.7262 
 
 6028 1.6588 
 
 6269 1.5952 
 
 6515 1.5350 
 
 6766 1.4779 
 
 55 
 
 6 
 
 5797 1.7251 
 
 6032 1.6577 
 
 6273 1.5941 
 
 6519 1.5340 
 
 6771 1.4770 
 
 54 
 
 7 
 
 5801 1.7239 
 
 6036 1.6566 
 
 6277 1.5931 
 
 6523 1.5330 
 
 6775 1.4761 
 
 53 
 
 8 
 
 5805 .7228 
 
 6040 1.6555 
 
 6281 1.5921 
 
 6527 1.5320 
 
 6779 1.4751 
 
 52 
 
 9 
 
 5808 1.7216 
 
 6044 1.6545 
 
 6285 1.5911 
 
 6531 1.5311 
 
 6783 1.4742 
 
 51 
 
 1O 
 
 5812 .7205 
 
 6048 1.6534 
 
 6289 1.5900 
 
 6536 1.5301 
 
 6787 1.4733 
 
 5O 
 
 11 
 
 5816 1.7193 
 
 6052 1.6523 
 
 6293 1.5890 
 
 6540 1.5291 
 
 6792 1.4724 
 
 49 
 
 12 
 
 5820 .7182 
 
 6056 1.6512 
 
 6297 1.5880 
 
 6544 1.5282 
 
 67% 1.4715 
 
 48 
 
 13 
 
 5824 .7170 
 
 6060 1.6501 
 
 6301 1.5869 
 
 6548 1.5272 
 
 6800 1.4705 
 
 47 
 
 14 
 
 5828 1.7159 
 
 6064 1.6490 
 
 6305 1.5859 
 
 6552 1.5262 
 
 6805 1.46% 
 
 46 
 
 15 
 
 5832 1.7147 
 
 6068 1.6479 
 
 6310 1.5849 
 
 6556 1.5253 
 
 6809 1.4687 
 
 45 
 
 16 
 
 5836 1.7136 
 
 6072 1.6469 
 
 6314 1.5839 
 
 6560 1.5243 
 
 6813 1.4678 
 
 44 
 
 17 
 
 5840 1.7124 
 
 6076 1.6458 
 
 6318 1.5829 
 
 6565 1.5233 
 
 6817 1.4669 
 
 43 
 
 18 
 
 5844 1.7113 
 
 6080 1.6447 
 
 6322 1.5818 
 
 6569 1.5224 
 
 6822 1.4659 
 
 42 
 
 19 
 
 5847 1.7102 
 
 6084 1.6436 
 
 6326 1.5808 
 
 6573 1.5214 
 
 6826 1.4650 
 
 41 
 
 20 
 
 5851 1.7090 
 
 6088 1.6426 
 
 6330 1.5798 
 
 6577 1.5204 
 
 6830 1.4641 
 
 40 
 
 21 
 
 5855 1.7079 
 
 6092 1.6415 
 
 6334 1.5788 
 
 6581 1.5195 
 
 6834 1.4632 
 
 39 
 
 22 
 
 5859 1.7067 
 
 6096 1.6404 
 
 6338 1.5778 
 
 6585 1.5185 
 
 6839 1.4623 
 
 38 
 
 23 
 
 5863 1.7056 
 
 6100 1.6393 
 
 6342 1.5768 
 
 6590 1.5175 
 
 6843 1.4614 
 
 37 
 
 24 
 
 5867 1.7045 
 
 6104 1.6383 
 
 6346 1.5757 
 
 6594 1.5166 
 
 6847 1.4605 
 
 36 
 
 25 
 
 5871 1.7033 
 
 6108 1.6372 
 
 6350 1.5747 
 
 6598 1.5156 
 
 6851 1.45% 
 
 35 
 
 26 
 
 5875 1.7022 
 
 6112 1.6361 
 
 6354 1.5737 
 
 6602 1.5147 
 
 6856 1.4586 
 
 34 
 
 27 
 
 5879 1.7011 
 
 6116 1.6351 
 
 6358 1.5727 
 
 6606 1.5137 
 
 6860 1.4577 
 
 33 
 
 28 
 
 5883 1.6999 
 
 6120 1.6340 
 
 6363 1.5717 
 
 6610 1.5127 
 
 6864 1.4568 
 
 32 
 
 29 
 
 5887 1.6988 
 
 6124 1.6329 
 
 6367 1.5707 
 
 6615 1.5118 
 
 6869 1.4559 
 
 31 
 
 30 
 
 5890 1.6977 
 
 6128 1.6319 
 
 6371 1.5697 
 
 6619 1.5108 
 
 6873 1.4550 
 
 30 
 
 31 
 
 5894 1.6965 
 
 6132 1.6308 
 
 6375 1.5687 
 
 6623 1.5099 
 
 6877 1.4541 
 
 29 
 
 32 
 
 5898 1.6954 
 
 6136 1.6297 
 
 6379 1.5677 
 
 6627 1.5089 
 
 6881 1.4532 
 
 28 
 
 33 
 
 5902 1.6943 
 
 6140 1.6287 
 
 6383 1.5667 
 
 6631 1.5080 
 
 6886 1.4523 
 
 27 
 
 34 
 
 5906 1.6932 
 
 6144 1.6276 
 
 6387 1.5657 
 
 6636 1.5070 
 
 6890 1.4514 
 
 26 
 
 35 
 
 5910 1.6920 
 
 6148 1.6265 
 
 6391 1.5647 
 
 6640 1.5061 
 
 6894 1.4505 
 
 25 
 
 36 
 
 5914 1.6909 
 
 6152 1.6255 
 
 6395 1.5637 
 
 6644 1.5051 
 
 6899 1.44% 
 
 24 
 
 37 
 
 5918 1.6898 
 
 6156 1.6244 
 
 6399 1.5627 
 
 6648 1.5042 
 
 6903 1.4487 
 
 23 
 
 38 
 
 5922 1.6887 
 
 6160 1.6234 
 
 6403 1.5617 
 
 6652 1.5032 
 
 6907 1.4478 
 
 22 
 
 39 
 
 5926 1.6875 
 
 6164 1.6223 
 
 6408 1.5607 
 
 6657 1.5023 
 
 6911 1.4469 
 
 21 
 
 40 
 
 5930 1.6864 
 
 6168 1.6212 
 
 6412 1.5597 
 
 6661 1.5013 
 
 6916 1.4460 
 
 2O 
 
 41 
 
 5934 1.6853 
 
 6172 1.6202 
 
 6416 1.5587 
 
 6665 1.5004 
 
 6920 1.4451 
 
 19 
 
 42 
 
 5938 1.6842 
 
 6176 1.6191 
 
 6420 1.5577 
 
 6669 1.4994 
 
 6924 1.4442 
 
 18 
 
 43 
 
 5942 1.6831 
 
 6180 1.6181 
 
 6424 1.5567 
 
 6673 1.4985 
 
 6929 1.4433 
 
 17 
 
 44 
 
 5945 1.6820 
 
 6184 1.6170 
 
 6428 1.5557 
 
 6678 1.4975 
 
 6933 1.4424 
 
 16 
 
 45 
 
 5949 1.6808 
 
 6188 1.6160 
 
 6432 1.5547 
 
 6682 1.4966 
 
 6937 1.4415 
 
 15 
 
 46 
 
 5953 1.6797 
 
 6192 1.6149 
 
 6436 1.5537 
 
 6686 1.4957 
 
 6942 1.4406 
 
 14 
 
 47 
 
 5957 1.6786 
 
 61% 1.6139 
 
 6440 1.5527 
 
 6690 1.4947 
 
 6946 1.4397 
 
 13 
 
 48 
 
 5961 1.6775 
 
 6200 1.6128 
 
 6445 1.5517 
 
 6694 1.4938 
 
 6950 1.4388 
 
 12 
 
 49 
 
 5965 1.6764 
 
 6204 1.6118 
 
 6449 1.5507 
 
 6699 1.4928 
 
 6954 1.4379 
 
 11 
 
 50 
 
 5969 1.6753 
 
 6208 1.6107 
 
 6453 1.5497 
 
 6703 1.4919 
 
 6959 1.4370 
 
 1O 
 
 51 
 
 5973 1.6742 
 
 6212 1.6097 
 
 6457 1.5487 
 
 6707 1.4910 
 
 6963 1.4361 
 
 9 
 
 52 
 
 5977 1.6731 
 
 6216 1.6087 
 
 6461 1.5477 
 
 6711 1.4900 
 
 6%7 1.4352 
 
 8 
 
 53 
 
 5981 1.6720 
 
 6220 1.6076 
 
 6465 1.5468 
 
 6716 1.4891 
 
 6972 1.4344 
 
 7 
 
 54 
 
 5985 1.6709 
 
 6224 1.6066 
 
 6469 1.5458 
 
 6720 1.4882 
 
 6976 1.4335 
 
 6 
 
 55 
 
 5989 1.6698 
 
 6228 1.6055 
 
 6473 1.5448 
 
 6724 1.4872 
 
 6980 1.4326 
 
 5 
 
 56 
 
 5993 1.6687 
 
 6233 1.6045 
 
 6478 1.5438 
 
 6728 1.4863 
 
 6985 1.4317 
 
 4 
 
 57 
 
 5997 1.6676 
 
 6237 1.6034 
 
 6482 1.5428 
 
 6732 1.4854 
 
 6989 1.4308 
 
 3 
 
 58 
 
 6001 1.6665 
 
 6241 1.6024 
 
 6486 1.5418 
 
 6737 1.4844 
 
 6993 1.4299 
 
 2 
 
 59 
 
 6005 1.6654 
 
 6245 1.6014 
 
 6490 1.5408 
 
 6741 1.4835 
 
 6998 1.4290 
 
 1 
 
 6O 
 
 6009 1.6643 
 
 6249 1.6003 
 
 6494 1.5399 
 
 6745 1.4826 
 
 7002 1.4281 
 
 
 
 
 cot tan 
 
 cot tan 
 
 cot tan 
 
 cot tan 
 
 cot tan 
 
 
 t 
 
 59 
 
 58 
 
 57 
 
 50 
 
 55 
 
 t
 
 78 
 
 NATURAL TANGENTS AND COTANGENTS. 
 
 / 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 9 
 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 tan 
 
 cot 
 
 o 
 
 7002 1.4281 
 
 7265 1.3764 
 
 7536 1.3270 
 
 7813 1.2799 
 
 8098 
 
 1.2349 6O 
 
 1 
 
 7006 1.4273 
 
 7270 1.3755 
 
 7540 1.3262 
 
 7818 1.2792 
 
 8103 
 
 1.2342 59 
 
 2 
 
 7011 1.4264 
 
 7274 1.3747 
 
 7545 1.3254 
 
 7822 1.2784 
 
 8107 
 
 1.2334 58 
 
 3 
 
 7015 1.4255 
 
 7279 1.3739 
 
 7549 1.3246 
 
 7827 1.2776 
 
 8112 
 
 1.2327 57 
 
 4 
 
 7019 1.4246 
 
 7283 1.3730 
 
 7554 1.3238 
 
 7832 1.2769 
 
 8117 
 
 1.2320 56 
 
 5 
 
 7024 1.4237 
 
 7288 1.3722 
 
 7558 1.3230 
 
 7836 1.2761 
 
 8122 
 
 1.2312 55 
 
 6 
 
 7028 1.4229 
 
 7292 1.3713 
 
 7563 1.3222 
 
 7841 1.2753 
 
 8127 
 
 1.2305 54 
 
 7 
 
 7032 1.4220 
 
 7297 1.3705 
 
 7568 1.3214 
 
 7846 1.2746 
 
 8132 
 
 1.2298 53 
 
 8 
 
 7037 1.4211 
 
 7301 1.3697 
 
 7572 1.3206 
 
 7850 1.2738 
 
 8136 
 
 1.2290 52 
 
 9 
 
 7041 1.4202 
 
 7306 1.3688 
 
 7577 1.3198 
 
 7855 1.2731 
 
 8141 
 
 1.2283 51 
 
 1O 
 
 7046 1.4193 
 
 7310 1.36SO 
 
 7581 1.3190 
 
 7860 .2723 
 
 8146 
 
 1.2276 5O 
 
 11 
 
 7050 1.4185 
 
 7314 1.3672 
 
 7586 1.3182 
 
 7865 .2715 
 
 8151 
 
 1.2268 49 
 
 12 
 
 7054 1.4176 
 
 7319 1.3663 
 
 7590 1.3175 
 
 7869 .2708 
 
 8156 
 
 1.2261 48 
 
 13 
 
 7059 1.4167 
 
 7323 1.3655 
 
 7595 1.3167 
 
 7874 .2700 
 
 8161 
 
 1.2254 47 
 
 14 
 
 7063 1.4158 
 
 7328 1.3647 
 
 7600 1.3159 
 
 7879 .2693 
 
 8165 
 
 1.2247 46 
 
 15 
 
 7067 1.4150 
 
 7332 1.3638 
 
 7604 1.3151 
 
 7883 1.2685 
 
 8170 
 
 1.2239 45 
 
 16 
 
 7072 1.4141 
 
 7337 1.3630 
 
 7609 1.3143 
 
 7888 1.2677 
 
 8175 
 
 1.2232 44 
 
 17 
 
 7076 1.4132 
 
 7341 1.3622 
 
 7613 1.3135 
 
 7893 1.2670 
 
 8180 
 
 1.2225 43 
 
 18 
 
 7080 1.4124 
 
 7346 1.3613 
 
 7618 1.3127 
 
 7898 1.2662 
 
 8185 
 
 1.2218 42 
 
 19 
 
 7085 1.4115 
 
 7350 1.3605 
 
 7623 1.3119 
 
 7902 1.2655 
 
 8190 
 
 1.2210 41 
 
 2O 
 
 7089 1.4106 
 
 7355 1.3597 
 
 7627 1.3111 
 
 7907 1.2647 
 
 8195 
 
 1.2203 4O 
 
 21 
 
 7094 1.4097 
 
 7359 1.3588 
 
 7632 1.3103 
 
 7912 1.2640 
 
 8199 
 
 1.2196 39 
 
 22 
 
 7098 1.4089 
 
 7364 1.3580 
 
 7636 1.3095 
 
 7916 1.2632 
 
 8204 
 
 1.2189 38 
 
 23 
 
 7102 1.4080 
 
 7368 1.3572 
 
 7641 1.3087 
 
 7921 1.2624 
 
 8209 
 
 1.2181 37 
 
 24 
 
 7107 1.4071 
 
 7373 1.3564 
 
 7646 1.3079 
 
 7926 1.2617 
 
 8214 
 
 1.2174 36 
 
 25 
 
 7111 1.4063 
 
 7377 1.3555 
 
 7650 1.3072 
 
 7931 1.2609 
 
 8219 
 
 1.2167 35 
 
 26 
 
 7115 1.4054 
 
 7382 1.3547 
 
 7655 1.3064 
 
 7935 1.2602 
 
 8224 
 
 1.2160 34 
 
 27 
 
 7120 1.4045 
 
 7386 1.3539 
 
 7659 1.3056 
 
 7940 1.2594 
 
 8229 
 
 1.2153 33 
 
 28 
 
 7124 1.4037 
 
 7391 1.3531 
 
 7664 1.3048 
 
 7945 1.2587 
 
 8234 
 
 1.2145 32 
 
 29 
 
 7129 1.4028 
 
 7395 1.3522 
 
 7669 1.3040 
 
 7950 1.2579 
 
 8238 
 
 1.2138 31 
 
 3O 
 
 7133 1.4019 
 
 7400 1.3514 
 
 7673 1.3032 
 
 7954 .2572 
 
 8243 
 
 1.2131 3O 
 
 31 
 
 7137 1.4011 
 
 7404 1.3506 
 
 7678 1.3024 
 
 7959 1.2564 
 
 8248 
 
 1.2124 29 
 
 32 
 
 7142 1.4002 
 
 7409 1.3498 
 
 7683 1.3017 
 
 7964 1.2557 
 
 8253 
 
 1.2117 | 28 
 
 33 
 
 7146 1.3994 
 
 7413 1.3490 
 
 7687 1.3009 
 
 7969 1.2549 
 
 8258 
 
 1.2109 27 
 
 34 
 
 7151 1.3985 
 
 7418 1.3481 
 
 7692 1.3001 
 
 7973 1.2542 
 
 8263 
 
 1.2102 i 26 
 
 35 
 
 7155 1.3976 
 
 7422 1.3473 
 
 7696 1.2993 
 
 7978 1.2534 
 
 8268 
 
 1.2095 25 
 
 36 
 
 7159 1.3968 
 
 7427 1.3465 
 
 7701 1.2985 
 
 7983 1.2527 
 
 8273 
 
 1.2088 24 
 
 37 
 
 7164 1.3959 
 
 7431 1.3457 
 
 7706 1.2977 
 
 7988 1.2519 
 
 8278 
 
 1.2081 23 
 
 38 
 
 7168 1.3951 
 
 7436 1.3449 
 
 7710 1.2970 
 
 7992 1.2512 
 
 8283 
 
 1.2074 22 
 
 39 
 
 7173 1.3942 
 
 7440 1.3440 
 
 7715 1.2962 
 
 7997 1.2504 
 
 8287 
 
 1.2066 | 21 
 
 4O 
 
 7177 1.3934 
 
 7445 1.3432 
 
 7720 1.2954 
 
 8002 1.2497 
 
 8292 
 
 1.2059 |2O 
 
 41 
 
 7181 1.3925 
 
 7449 1.3424 
 
 7724 1.2946 
 
 8007 1.2489 
 
 8297 
 
 1.2052 19 
 
 42 
 
 7186 1.3916 
 
 7454 1.3416 
 
 7729 1.2938 
 
 8012 1.2482 
 
 8302 
 
 1.2045 18 
 
 43 
 
 7190 1.3908 
 
 7458 1.3408 
 
 7734 1.2931 
 
 8016 1.2475 
 
 8307 
 
 1.2038 17 
 
 44 
 
 7195 1.3899 
 
 7463 1.3400 
 
 7738 1.2923 
 
 8021 1.2467 
 
 8312 
 
 1.2031 16 
 
 45 
 
 7199 13891 
 
 7467 1.3392 
 
 7743 1.2915 
 
 8026 1.2460 
 
 8317 
 
 1.2024 15 
 
 46 
 
 7203 1.3882 
 
 7472 1.3384 
 
 7747 1.2907 
 
 8031 1.2452 
 
 8322 
 
 1.2017 14 
 
 47 
 
 7208 1.3874 
 
 7476 1.3375 
 
 7752 1.2900 
 
 8035 1.2445 
 
 8327 
 
 1.2009 13 
 
 48 1 7212 1.3865 
 
 7481 1.3367 
 
 7757 1.2892 
 
 8040 1.2437 
 
 8332 
 
 1.2002 12 
 
 49 7217 1.3857 
 
 7485 1.3359 
 
 7761 1.2884 
 
 8045 1.2430 
 
 8337 
 
 1.1995 11 
 
 5O 7221 1.3848 
 
 7490 1.3351 
 
 7766 1.2876 
 
 8050 1.2423 
 
 8342 
 
 1.1988 1O 
 
 51 7226 1.3840 
 
 7495 1.3343 
 
 7771 1.2869 
 
 8055 1.2415 
 
 8346 
 
 1.1981 9 
 
 52 7230 1.3831 
 
 7499 1.3335 
 
 7775 1.2861 
 
 8059 1.2408 
 
 8351 
 
 1.1974 : 8 
 
 53 7234 1.3823 
 
 7504 1.3327 
 
 7780 1.2853 
 
 8064 1.2401 
 
 8356 
 
 1.1967 7 
 
 54 7239 1.3814 
 
 7508 1.3319 
 
 7785 1.2846 
 
 8069 1.2393 
 
 8361 
 
 1.1960 6 
 
 55 ! 7243 1.3806 
 
 7513 1.3311 
 
 7789 1.2838 
 
 8074 1.2386 
 
 8366 
 
 1.1953 5 
 
 56 7248 1.3798 
 
 7517 1.3303 
 
 7794 1.2830 
 
 8079 1.2378 
 
 8371 
 
 1.1946 4 
 
 57 7252 1.3789 
 
 7522 1.3295 
 
 7799 1.2822 
 
 8083 1.2371 
 
 8376 
 
 1.1939 3 
 
 58 7257 1.3781 
 
 7526 1.3287 
 
 7803 1.2815 
 
 8088 1.2364 
 
 8381 
 
 1.1932 2 
 
 59 7261 1.3772 
 
 7531 1.3278 
 
 7808 1.2807 
 
 8093 1.2356 
 
 8386 
 
 1.1925 j 1 
 
 GO 7265 1.3764 
 
 cot tan 
 
 7536 1.3270 
 
 7813 1.2799 
 
 8098 1.2349 
 
 8391 
 
 1.1918 O 
 
 ' 54 
 
 53 
 
 52 
 
 51 C 
 
 5O '
 
 NATURAL TANGENTS AND COTANGENTS. 
 
 79 
 
 t 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 t 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 tan cot 
 
 
 o 
 
 8391 1.1918 
 
 8693 1.1504 
 
 9004 1.1106 
 
 9325 1.0724 
 
 9657 1.0355 
 
 6O 
 
 1 
 
 8396 1.1910 
 
 8698 1.1497 
 
 9009 1.1100 
 
 9331 1.0717 
 
 9663 1.0349 
 
 59 
 
 2 
 
 8401 1.1903 
 
 8703 1.1490 
 
 9015 1.1093 
 
 9336 1.0711 
 
 9668 1.0343 
 
 58 
 
 3 
 
 8406 1.1896 
 
 8708 1.1483 
 
 9020 1.1087 
 
 9341 1.0705 
 
 9674 1.0337 
 
 57 
 
 4 
 
 8411 1.1889 
 
 8713 1.1477 
 
 9025 1.1080 
 
 9347 1.0699 
 
 9679 1.0331 
 
 56 
 
 5 
 
 8416 1.1882 
 
 8718 1.1470 
 
 9030 1.1074 
 
 9352 1.0692 
 
 9685 1.0325 
 
 55 
 
 6 
 
 8421 1.1875 
 
 8724 1.1463 
 
 9036 1.1067 
 
 9358 1.0686 
 
 9691 1.0319 
 
 5! 
 
 7 
 
 8426 1.1868 
 
 8729 1.1456 
 
 9041 1.1061 
 
 9363 1.0680 
 
 9696 1.0313 
 
 5: 
 
 8 
 
 8431 1.1861 
 
 8734 1.1450 
 
 9046 1.1054 
 
 9369 1.0674 
 
 9702 1.0307 
 
 52 
 
 9 
 
 8436 1.1854 
 
 8739 1.1443 
 
 9052 1.1048 
 
 9374 1.0668 
 
 9708 1.0301 
 
 51 
 
 1O 
 
 8441 1.1847 
 
 8744 1.1436 
 
 9057 1.1041 
 
 9380 1.0661 
 
 9713 1.0295 
 
 5O 
 
 11 
 
 8446 1.1840 
 
 8749 1.1430 
 
 9062 1.1035 
 
 9385 1.0655 
 
 9719 1.0289 
 
 49 
 
 12 
 
 8451 1.1833 
 
 8754 1.1423 
 
 9067 1.1028 
 
 9391 1.0649 
 
 9725 1.0283 
 
 48 
 
 13 
 
 8456 1.1826 
 
 8759 1.1416 
 
 9073 1.1022 
 
 9396 1.0643 
 
 9730 1.0277 
 
 47 
 
 14 
 
 8461 1.1819 
 
 8765 1.1410 
 
 9078 1.1016 
 
 9402 1.0637 
 
 9736 1.0271 
 
 46 
 
 15 
 
 8466 1.1812 
 
 8770 1.1403 
 
 9083 1.1009 
 
 9407 1.0630 
 
 9742 1.0265 
 
 45 
 
 16 
 
 8471 1.1806 
 
 8775 1.1396 
 
 9089 1.1003 
 
 9413 1.0624 
 
 9747 1.0259 
 
 44 
 
 17 
 
 8476 1.1799 
 
 8780 1.1389 
 
 9094 1.0996 
 
 9418 1.0618 
 
 9753 1.0253 
 
 43 
 
 18 
 
 8481 1.1792 
 
 8785 1.1383 
 
 9099 1.0990 
 
 9424 1.06] 2 
 
 9759 1.0247 
 
 42 
 
 19 8486 1.1785 
 
 8790 1.1376 
 
 9105 1.0983 
 
 9429 1.0606 
 
 9764 1.0241 
 
 41 
 
 20 
 
 8491 1.1778 
 
 8796 1.1369 
 
 9110 1.0977 
 
 9435 1.0599 
 
 9770 1.0235 
 
 4O 
 
 21 
 
 8496 1.1771 
 
 8801 1.1363 
 
 9115 1.0971 
 
 9440 1.0593 
 
 9776 1.0230 
 
 39 
 
 22 
 
 8501 1.1764 
 
 8806 1.1356 
 
 9121 1.0964 
 
 9446 1.0587 
 
 9781 1.0224 
 
 38 
 
 23 
 
 8506 1.1757 
 
 8811 1.1349 
 
 9126 1.0958 
 
 9451 1.0581 
 
 9787 1.0218 
 
 37 
 
 24 
 
 8511 1.1750 
 
 8816 1.1343 
 
 9131 1.0951 
 
 9457 1.0575 
 
 9793 1.0212 
 
 36 
 
 25 
 
 8516 1.1743 
 
 8821 1.1336 
 
 9137 1.0945 
 
 9462 1.0569 
 
 9798 1.0206 
 
 35 
 
 26 
 
 8521 1.1736 
 
 8827 1.1329 
 
 9142 1.0939 
 
 9468 1.0562 
 
 9804 1.0200 
 
 34 
 
 27 8526 1.1729 
 
 8832 1.1323 
 
 9147 1.0932 
 
 9473 1.0556 
 
 9810 1.0194 
 
 33 
 
 28 8531 1.1722 
 
 8837 1.1316 
 
 9153 1.0926 
 
 9479 1.0550 
 
 9816 1.0188 
 
 32 
 
 29 
 
 8536 1.1715 
 
 8842 1.1310 
 
 9158 1.0919 
 
 9484 1.0544 
 
 9821 1.0182 
 
 31 
 
 30 
 
 8541 1.1708 
 
 8847 1.1303 
 
 9163 1.0913 
 
 9490 1.0538 
 
 9827 1.0176 
 
 3O 
 
 31 
 
 8546 1.1702 
 
 8852 1.1296 
 
 9169 1.0907 
 
 9495 1.0532 
 
 9833 1.0170 
 
 29 
 
 32 
 
 8551 1.1695 
 
 8858 1.129C 
 
 9174 1.0900 
 
 9501 1.0526 
 
 9838 1.0164 
 
 28 
 
 33 
 
 8556 1.1688 
 
 8863 1.1283 
 
 9179 1.0894 
 
 9506 1.0519 
 
 9844 1.0158 
 
 27 
 
 34 
 
 8561 1.1681 
 
 8868 1.1276 
 
 9185 1.0888 
 
 9512 1.0513 
 
 9850 1.0152 
 
 26 
 
 35 
 
 8566 1.1674 
 
 8873 1.1270 
 
 9190 1.0881 
 
 9517 1.0507 
 
 9856 1.0147 
 
 25 
 
 36 
 
 8571 1.1667 
 
 8878 1.1263 
 
 9195 1.0875 
 
 9523 1.0501 
 
 9861 1.0141 
 
 24 
 
 37 
 
 8576 1.1660 
 
 8884 1.1257 
 
 9201 1.0869 
 
 9528 1.0495 
 
 9867 1.0135 
 
 23 
 
 38 
 
 8581 1.1653 
 
 8889 1.1250 
 
 9206 1.0862 
 
 9534 1.0489 
 
 9873 1.0129 
 
 22 
 
 39 
 
 8586 1.1647 
 
 8894 1.1243 
 
 9212 1.0856 
 
 9540 1.0483 
 
 9879 1.0123 
 
 21 
 
 40 
 
 8591 1.1640 
 
 8899 1:1237 
 
 9217 1.0850 
 
 9545 1.0477 
 
 9884 1.0117 
 
 20 
 
 41 
 
 8596 1.1633 
 
 8904 1.1230 
 
 9222 1.0843 
 
 9551 1.0470 
 
 9890 1.0111 
 
 19 
 
 42 
 
 8601 1.1626 
 
 8910 1.1224 
 
 9228 1.0837 
 
 9556 1.0464 
 
 98% 1.0105 
 
 18 
 
 43 
 
 8606 1.1619 
 
 8915 1.1217 
 
 9233 1.0831 
 
 9562 1.0458 
 
 9902 1.0099 17 
 
 44 
 
 8611 1.1612 
 
 8920 1.1211 
 
 9239 1.0824 
 
 9567 1.0452 
 
 9907 1.0094 ! 16 
 
 45 
 
 8617 1.1606 
 
 8925 1.1204 
 
 9244 1.0818 
 
 9573 1.0446 
 
 . 9913 1.0088 
 
 15 
 
 46 
 
 8622 1.1599 
 
 8931 1.1197 
 
 9249 1.0812 
 
 9578 J.0440 
 
 9919 1.0082 
 
 14 
 
 47 
 
 8627 1.1592 
 
 8936 1.1191 
 
 9255 1.0805 
 
 9584 1.0434 
 
 9925 1.0076 
 
 13 
 
 48 
 
 8632 1.1585 
 
 8941 1.1184 
 
 9260 1.0799 
 
 9590 1.0428 
 
 9930 1.0070 
 
 12 
 
 49 
 
 8637 1.1578 
 
 8946 1.1178 
 
 9266 1.0793 
 
 9595 1.0422 
 
 9936 1.0064 
 
 11 
 
 5O 
 
 8642 1.1571 
 
 8952 1.1171 
 
 9271 1.0786 
 
 9601 1.0416 
 
 9942 1.0058 1O 
 
 51 
 
 8647 1.1565 
 
 8957 1.1165 
 
 9276 1.0780 
 
 9606 1.0410 
 
 9948 1.0052 
 
 9 
 
 52 
 
 8652 1.1558 
 
 8962 1.1158 
 
 9282 1.0774 
 
 9612 1.0404 
 
 9954 1.0047 
 
 8 
 
 53 
 
 8657 1.1551 
 
 8967 1.1152 
 
 9287 1.0768 
 
 9618 1.0398 
 
 9959 1.0041 
 
 7 
 
 54 
 
 8662 1.1544 
 
 8972 1.1145 
 
 9293 1.0761 
 
 9623 1.0392 
 
 9965 1.0035 
 
 6 
 
 55 
 
 8667 1.1538 
 
 8978 1.1139 
 
 9298 1.0755 
 
 9629 1.0385 
 
 9971 1.0029 
 
 5 
 
 56 
 
 8672 1.1531 
 
 8983 1.1132 
 
 9303 1.0749 
 
 9634 1.0379 
 
 9977 1.0023 
 
 4 
 
 57 
 
 8678 1.1524 
 
 8988 1.1126 
 
 9309 1.0742 
 
 9640 1.0373 
 
 9983 1.0017 
 
 3 
 
 58 
 
 8683 1.1517 
 
 8994 1.1119 
 
 9314 1.0736 
 
 9646 1.0367 
 
 9988 1.0012 
 
 2 
 
 59 
 
 8688 1.1510 
 
 8999 1.1113 
 
 9320 1.0730 
 
 9651 1.0361 
 
 9994 1.0006 
 
 1 
 
 6O 
 
 8693 1.1504 
 
 9004 1.1106 
 
 9325 1.0724 
 
 9657 1.0355 
 
 1.0000 1.0000 
 
 
 
 
 cot tan 
 
 cot tan 
 
 cot tan 
 
 cot tan 
 
 cot tan 
 
 
 F 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 /
 
 INDEX. 
 
 Agonic line 201 
 
 Alignment in mines 381-384 
 
 obstacles to and problems on 17, 120, 304, 370 
 
 street 360 
 
 Angle of deflection 108 
 
 of depression ; of elevation 56 
 
 horizontal ; vertical 66 
 
 measurement between lines 112 
 
 measurement by repetition 106 
 
 measurement by series 107 
 
 traversing 108 
 
 problems 113 
 
 Areas, determination of 35, 183, 187, 196 
 
 Azimuth, determination by solar attachment 94 
 
 table of errors in 96 
 
 Balancing the survey 161 
 
 weights used 166 
 
 Bearings, compass 102 
 
 reverse 103 
 
 proof 104 
 
 changing 116 
 
 when the variation is known 209 
 
 to obtain true 212 
 
 Chain, engineer's 4 
 
 Gunter's ; two-pole 3 
 
 required length of four-pole 301 
 
 Chaining 4, 302, 334 
 
 Circle, area of 33, 43 
 
 laying out 228-231, 388 884 
 
 vertical 388,389 
 
 Clinometer, the hanging 390, 391 
 
 Compass, surveyor's 56 
 
 adjustment 66
 
 82 INDEX. 
 
 PAGE 
 
 Compass bearings 102 
 
 solar . . '. 276 
 
 principles and adjustments of solar 280-285 
 
 the hanging 390 
 
 the diurnal variation of 200 
 
 the secular variation of 201, 208 
 
 variation determined by old lines 210 
 
 Contour lines 393 
 
 Co-ordinates, application of rectangular 45 
 
 Corners, establishing 304 
 
 lot 369 
 
 quarter-section ; witness 307 
 
 re-establishment of 316 
 
 Course, initial, reduced 382 
 
 special appliances , 390 
 
 Cross-sectioning, angular 394 
 
 Curvature and refraction 350, 351 
 
 Curves, how laid out 229, 230, 383, 384 
 
 Declination, allowance for 281 
 
 formula expressing 203 
 
 line of no 201 
 
 of the needle 199 
 
 by Polaris 214 
 
 problems 290, 291 
 
 setting off 285, 286 
 
 Departure and latitude 154 
 
 correcting 161 
 
 Distances, measurement of inaccessible 135-137 
 
 Dividing land 232 
 
 Drainage 377 
 
 Ellipse, area of 35 
 
 laying out 228-231 
 
 Error in chaining 7, 8 
 
 caused by needle 164 
 
 the survey 163 
 
 Field notes, recording the 22, 41, 142, 147, 318, 356, 385 
 
 Formulas expressing magnetic declination 203 
 
 Grade lines 357 
 
 Grade line marking 366 
 
 Grades, determination of 374 
 
 Gradienter 100, 329 
 
 Graduation .. .69
 
 INDEX. 83 
 
 FAGK 
 
 Heights, measurement of 20, 129-131 
 
 Hexagon, area of regular 31' 
 
 Judicial functions of surveyors 397 
 
 Latitude and departure 154 
 
 correcting 161 
 
 difference of 164 
 
 finding 86 
 
 by circumpolar star 92 
 
 by the sun, with Saegmuller's attachment 91 
 
 setting off 291 
 
 Laying out land 221 
 
 Level, adjustment 340-343 
 
 wye 337 
 
 engraving of wye 338, 339 
 
 use of 343, 387, 388 
 
 Locke's hand 333 
 
 rod 349 
 
 Levelling-rods 345 
 
 Levelling defined 349 
 
 fore and back sights 352, 353 
 
 general method 354, 355 
 
 Line, agonic 201 
 
 azimuth of 65 
 
 bearing of 65 
 
 base 308 
 
 meridian . 64 
 
 meridian distance of 55 
 
 obstacles in 17, 122, 124, 126, 304 
 
 inaccessibility of one end 19, 126 
 
 inaccessibility of both ends .20, 127 
 
 running a grade 357 
 
 Lines, offset and tie 44-61 
 
 parallel 16 
 
 perpendicular 13 
 
 ranging 
 
 in 
 
 marking 303, 366 
 
 preservation of points in 361 
 
 surveying section 296 
 
 surveying township 298, 299, 301 
 
 straightening boundary 268 
 
 variation determined by old 210 
 
 contour 393
 
 84 INDEX. 
 
 PAGE 
 
 Local attraction 103 
 
 Longitude, difference of 154 
 
 Mapping, see Plotting. 
 
 Meandering 307 
 
 Measurement, corrections 335 
 
 suggestions on 336 
 
 obstacles to, problems on 18, 124 
 
 Meridian, the principal 309 
 
 the magnetic , 54 
 
 auxiliary 310 
 
 principal base and , 300 
 
 establishing with transit 218 
 
 establishing with solar attachment 93 
 
 obtaining approximately 219 
 
 convergency of 324 
 
 inclination of 322, 323 
 
 deflection of range lines from 324 
 
 general rule for obtaining the double 188 
 
 Micrometer or stadia wires 95, 96 
 
 Needle, the magnetic 54, 68 
 
 adjustment of 59 
 
 declination of 199 
 
 remagnetization of 60 
 
 Objects to be noted in surveying public land 316 
 
 Obstacles 17, 19, 120-127, 304, 370 
 
 Octagon, area of regular 31 
 
 Omissions, supplying 166 
 
 Parallelogram, area of 31, 38, 183 
 
 laying out 225, 226 
 
 Parallels, standard 300, 309 
 
 Perpendiculars and parallels, problems on 13, 15, 17, 120 
 
 Pins, marking 4, 302 
 
 Plane-table surveying 262 
 
 Plane-table, engraving of 263 
 
 adjustments , 266 
 
 surveying, various methods 267-269 
 
 use in topography 393 
 
 practical suggestions on 273 
 
 exercises 274 
 
 Plans 372 
 
 Plot, farm 153 
 
 Plotting 26, 177, 180, 182, 385, 386
 
 INDEX. 85 
 
 PAOB 
 
 Poles, or rods, for sighting , 4 f 331 
 
 of the earth 55 
 
 Polygon, area of 31-40, 186 
 
 laying out 227, 228 
 
 dividing up 253-258 
 
 Profiles 363, 364, 373 
 
 level notes for 356, 365 
 
 Public lands, survey of 275 
 
 origin of system of survey 275, 295 
 
 Record of levels for profile 365 
 
 Recording notes, sfe Field Notes. 
 
 Rectangle, area of 31, 37 
 
 laying out 224, 225 
 
 Rectangular surveying 297 
 
 Refraction, allowance for 287 
 
 tables 02, 288 
 
 explanation of table of 290 
 
 problems in 290, 291 
 
 curvature and 350 
 
 Ring, area of 37, 43 
 
 Rods or poles 4, 331 
 
 levelling 345, 348 
 
 Scale, drawing to 29 
 
 to adopt 30, 385 
 
 unknown 30 
 
 Scales 28 
 
 Sections 299 
 
 subdividing 314-316 
 
 Segment, area of 34, 43 
 
 Sextant, area of 34 
 
 Shaft, transference of points down 389 
 
 Solar compass 276 
 
 engraving of 277 
 
 hour arc, polar axis 279 
 
 principles 280 
 
 adjustments .283-286 
 
 to use . 286 
 
 running lines with 292 
 
 time for using 296 
 
 time of day by sun with 294 
 
 Solar attachment of the transit (Gurley's) 79 
 
 engraving of 81
 
 86 INDEX. 
 
 PAGE 
 
 Solar attachment, adjustment of 80 
 
 running lines with 87 
 
 Saegmuller's 88 
 
 adjustment of Saegmuller's 88, 89 
 
 time and azimuth with 94 
 
 Squares, laying out 223 
 
 Stadia wires, or micrometer 95, 96 
 
 level line of sight perpendicular to 97 
 
 line of sight inclined to 99 
 
 Streets, direction and width '. 358, 359 
 
 alignment of 360 
 
 grade and profile 362, 363 
 
 preservation of points in ... 361 
 
 Table of areas of regular polygons 33 
 
 errors in azimuth 95 
 
 elongation and azimuth of Polaris 216, 217 
 
 inclination and convergency of meridians 325 
 
 refraction 92 
 
 secular variation . 206 
 
 traverse 156 
 
 TABLES AT END OF BOOK. 
 
 Tables of logarithms of numbers 1-19 
 
 approximate equations of time 20 
 
 logarithms of trigonometric functions . '. 21, 49 
 
 for determining with greater accuracy 50, 51 
 
 lengths of degree of latitude and longitude 52 
 
 miscellaneous formulae, and equivalents of metres, chains, and feet, 53 
 
 traverse 54-61 
 
 natural sines and cosines 62-70 
 
 natural tangents and cotangents 71-79 
 
 Tallying 3 
 
 Tape measures 4, 332, 380 
 
 Telescope, section of 65, 338, 339 
 
 adjusting level on .... 78 
 
 Testing a survey 160 
 
 Time by solar attachment 94 
 
 Topography 362, 363, 391-393 
 
 Township, surveying boundaries of 298, 299, 301 
 
 subdividing,. ...311-313
 
 INDEX, 87 
 
 PAQB 
 
 Township plan 290 
 
 Transit, showing gradienter, vertical arc, etc 329 
 
 surveyor's (full circle) ' 64 
 
 adjustments 71 
 
 measurement of angles 106 
 
 attachments 77 
 
 adjustment of attachments 78 
 
 solar attachment (see Solar Attachment) 79 
 
 as first made 396 
 
 to take apart , 79 
 
 Trapezium, area? of 39, 184 
 
 dividing 246-252 
 
 Trapezoid, area of 31, 38, 184 
 
 dividing 241-246 
 
 Traverse table 166 
 
 Triangle, area of 31, 35, 183 
 
 dividing 232-240 
 
 laying out J 221-223 
 
 Tripod 71 
 
 Tripods, extra 381 
 
 Vernier, the 01 
 
 the transit 69 
 
 determination of least count of 61 
 
 to read an instrument having 62 
 
 exercises on spacing 62 
 
 Vertical circle 77 
 
 to adjust 7V

 
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