Library UNIVERSITY OF CALIFORNIA, DEPARTMENT OF CIVIL ENGINEERING BERKELEY, CALIFORNIA .J& .W -f srii o* iad8 .93t/ -:ot vl^a>; aoiiO bnjs J8KI 3h ESTABLISHED 1845 W. & L. E. GURLEY, TROY, N. Y. MANUFACTURERS OF Civil Engineers' and Surveyors' Instruments All Instruments Sent to the Purchaser Adjusted and Ready for Use. Send for Full Illus- trated Price List and Circular. A MANUAL LAND SURVEYING COMPRISING AN ELEMENTARY COURSE OF PRACTICE WITH INSTRUMENTS AND A TREATISE UPON THE Survey of Public and Private Lands, PREPARED For use of Schools and Surveyors. BY F. HODGMAN, M. S., C. E., Practical Surveyor and Engineer. 'Let things that have to be done be learned by doing them." THE F. HODGMAN CO., CLIMAX, MICHIGAN. 1907. TA55/ Engineering Library Entered according to Act of Congress in the year 1903. BY F. HODGMAN, In the office of the Librarian of Congress, at Washington PREFACE. This addition to the already numerous treatises on land surveymg was caused by the demand of the surveyors of Michigan for a treatise which would deal with the prac- tical questions which meet the surveyor in his every day work in the field. Several admirable treatises were already in existence which dealt amply with the mathe- matical and instrumental part of surveymg. But the perplexing questions which meet the surveyor are not questions of mathematical calculation or of the use of instruments. On the contrary they are, for the most part, questions of how to apply the principles of common law and statutory enactment to the location of boundary lines. These are the controlling considerations in all re- surveys; a class which comprises probably nine-tenths of all the land surveys which are made. Scarcely an allusion to these principles was to be found in any of the works on surveying extant. In 1880 the Michigan Association of Surveyors and Civil Engineers appointed a committee on manual, to prepare a work which would give authori- tative answers to the many questions of practice which came up before them. The committee spent their spare time for five years in an exhaustive research of the laws and the decisions of the highest courts in the land. The chairman attended the meetings of various surveyors' associations and collected their reports. From the great mass of material thus collected, the leading points in the laws of the United States and the decisions of the courts of last resort were selected, covering, as nearly as possible, all the points relative to surveys and boundary lines which arise in the land surveyor's practice The legal decisions quoted are a part of the Common iav? ol the whole country and apply wherever the Common law prevails, whetf-d England, or the United IV PREFACE. States. It should be remembered, however, that differ- ent courts do not always expound the law alike, and sometimes a court reverses its own decisions. When- ever there appears to be a conflict of authorities, the Surveyor should follow the latest decisions in his own State if there be any. It seemed to the committee to be important that the student in land surveying should be taught these things ; that they were as necessary for the beginner to know as for the older practitioner, and hence might properly be incorporated in the text book. Having this in view, it was decided to extend the scope of the manual by including such mathe- matical work as would make it equally adapted to the use of the student as a text book and the practical sur- veyor as a book of reference. In preparing this portion of the work, the leading idea has been that, so far as possible, the student should be taught by actual practice in the field, as well as in the class room; that he should learn to survey by surveying. The solution of a problem in surveying in actual practice is always worked out upon the ground, hence suggestions are made to the student how problems may be solved, instead of giving any formal solution. It is pre-supposed that every successful teacher will have methods of his own for conveying instruction, and will use these suggestions or make different ones as may seem best to him. Doubtless things have been omitted which some would regard as important to have introduced. Such omissions will be supplied by teachers at their pleasure and convenience. We acknowledge our indebtedness to the authors of many treatises which have been consulted in the prepara- tion of this volume, especially to the works of Davies, Oillespie, Hawes and Dunn, also to Messrs. W. & L. E. Gurley for many favors received, and to the officers and members of the Surveyors' Associations of Michigan, Ohio, Indiana, Illinois and Missouri for many valuable suggestions, sympathy and assistance. F. HODGMAN. Climax, Mich., TABLE OF CONTENTS. CHAPTER I. DEFINITIONS. PAGE. .INSTRUMENTS FOR MEASURING DISTANCES. The Chain 2 The Steel Tape 3 Marking Pins 4 Measuring 4 MEASURES OF LENGTH AND AREA, English 9 Old Spanish 11 Old French 13 Standard Measures 14 THE PICKET, To Run Line with 16 To Pass Obstacles- 18 CHAPTER II. INSTRUMENTS. THE SURVEYOR'S COMPASS, Description of 19 Adjustments of 22 Electricity 24 To Run Lines with 24 To Pass Obstacles 25 THE MAGNETIC NEEDLE, Changes in direction of 26 Local Attraction 27 Difference in Instruments 27 Things to be Observed ___ 27 Marking Lines 28 How to find a True Meridian 28 THE TRANSIT, Description and Adjustments 52 How to use - -- 59 Assistants and their Duties 60 The Color Pole 61 Projecting the Line 62 [v] VI TABLE OF CONTENTS. CHAPTER III. INSTRUMENTS, CONTINUED. THE SOLAR COMPASS, Description and Adjustments 65 How to use - 73 SOLAR ATTACHMENT TO TRANSIT. Description and Adjustments 80 How to use 86 CHAPTER IV. MEASUREMENT OF ANGLES. TO MEASURE ANGLES, With Tape and Tins 88 With the Compass 91 With the Transit 100 Verniers 102 TO CORRECT RANDOM LINES, Of one course 93 Of several courses 96 CHAPTER V. PASSING OBSTACLES AND MEASURING INACCESSIBLE DISTANCES. PASSING OBSTACLES, By Parallel Lines 107 By 60 Angles 107 TO MEASURE INACCESSIBLE DISTANCES, By triangles.- 10S Stadia Measures 110 The Gradienter 115 CHAPTER VI. PLATTING AND COMPUTING AREAS. PLATTING, Instruments used 119 COMPUTING AREAS, Triangles 122 Quadrangles : Rectangles, Trapezoids and Trapeziums 124 Irregular Polygons 125 Offsets 120 Rectangular Coordinates 128 Application to Area 129 The Traverse Table 134 Meridian Distances 136 Supplying Omissions 143 Reducing Irregular Polygons 148 Division and Partition of Land ^- 153 Method by Approximations 163 Field Notes 164 Abridging Field Notes 168 TABLE OF CONTENTS. TIT CHAPTER VII. CURYELINEAR SURVEYING. Preliminary Propositions 170 To run Curves with Picket and Tape 171 Field Notes of Transit Lanes 173 To run a Curve with the Transit, Different Methods 171 To locate a Curve from its Middle Point 178 To locate a Curve from some Intermediate Point 179 To locate a Curve from Point of Intersection jgO Passing Obstructions in Line of Curve 181 Compound Curves 1&J Useful Formula 184 CHAPTER VIII. ORIGINAL SURVEYS. Surveys, Classified 186 Original Surveys, Government and Private 186 PUBLIC DOMAIN, How and when Acquired 1S8 Amount of - - 189 Origin of Systems of Surveys of 189 Laws relating to Survey of, where found 193 U. S. LAWS RELATING TO SURVEYS OF PUBLIC LANDS. Appointment of Surveyor General 193 Qualifications of 1H3 Term of Office 194 When Records and Field Notes to be turned over to the State 194 Discontinuance of Office 194 When Authority to vest in Com. of Gen. Land Office 193 Free Access to Field Notes and Records If5 Surveyor General to Employ Deputies 183 To cause Survey of Base and Meridian Lines 196 To cause Survey of Private Land Claims . 177-195 To inspect Surveys in Person or by Agent 197 Pay of Agent 197 Deputy Surveyor to Give Bond 197 Deputy Surveyor to make Oath to Field Notes 198 Penalty for Fraudulent Survey 198 PUBLIC LANDS, How Divided into Townships 198 Township Lines, how marked 199 Townships, how subdivided into Sections 199 Sections, how numbered 171-199 Section Corners, how marked 199 Excess or Deficiency over six miles 199 Lines, how marked and measured 200 VII! TABLE OF CONTENTS. what Surveyors to note in Field Books 300 Disposition of Field Books and making of Plats, ... 200 SECTIONS AND SUBDIVISIONS OF SECTIONS, How Boundaries and Contents are found , 200 U. 8, Survey Corners the true ones '. , , , 201 Corners of X and ^ Sections not set by Government Survey. . 201 Boundary Linee of U. 8. Survey the true ones. .... 201 Those not run, how found 201 True Contents of Sections returned 201 True Contents of % and ^ Sections which are not Returned * , 201 Fractional Sections, how divided 202 When ordinary Course may be departed from. ..... 202 Surveys in Nevada, Oregon, and California 203 When Rectangular System may be departed from . . 203 Instructions, a part of Contract 20i Survey of Mining Claims and Rights of Owners. ... 201 Appointment of Mineral Surveyors 206 Plats and Field Notes of Mining Surveys . , 207 Contracts to be approved by Com. Gen. Land Office 207 Commissioner to fix Prices for Surveys, etc 207 Extra Price in Oregon, Washington, and California 208 Penalty for Interference with Surveys 209 Surveyors appointed to select Timber Lands 210 Duty of Director of Geological Survey 210 FIELD WORK AND CHANGES THAT HAVE BEEN MADE. Two Mile Blocks, Act of 1796 211 Subdivisions into half Sections, Act of 1800 211 Changes in manner of Subdividing, Double Corners, etc , . 211 How Area of Fractions is Calculated 213 INSTRUCTIONS OF 1902. System of rectangular Surveying 218 Establishment of Meridians, Base Lines, and Parallels 219 Division into Townships and Sections. 220 Excess or deficiency in measurement 220 How Townships and Sections are numbered , . . 220 Instruments to be used 221 Tests and Adjustments of , . , . . 221 Chains and Tally pins . 222 Process of chaining 223 Leveling chain and Plumbing phio 223 Marking lines 224 Marking random lines 225 Insuperable objects in line 226 Witness Points where made. 226 Establishing Corners ' 26 Marking Tools 227 Surveying Monuments 227 Descriptions of Corners 227 TABLE OF CONTENTS. IX Abbreviations 228 Standard Township Corners, how marked 230 Witness Corners, how marked 233 Witness Corners in Roads 233 Witness points, how marked 234 Corners on rock 234 Location of Mounds 234 Mounds of Stone '234 Bearing Trees 235 Stones for Corners 237 When Lines to be discontinued at Corners 237 Marks to be Cut 237 Orientation of Corners 237 Size of Posts, Mounds, etc Corner Materials 237 Initial Points ., 238 Base Line 238 Principal Meridians 240 Standard Parallels 040 Guide Meridians 241 Township Exteriors 241 Township exteriors where impassable objects occur. o 4 o Method of Subdividing 243 Method of Subdividing, Exceptions 247 Meandering Streams 249 Meandering Lakes 251 Objects to be noted 253 Prescribed Limits for Closings and Lengths of Lines 255 Field Notes Blank Books furnished 256 What Original Field Notea are 256 SURVEYING BASE LINES AND STANDARD PAR ALLELS BY OFFSETS FROM STRAIGHT LINES. Secant Method and Tables 257 Tangent Method and Tables 264 - CHAPTER IX. SUBDIVISION OF SECTIONS. SUBDIVISION OF SECTIONS 267 Four Different Cases 269 Quarter Sections 270 Half-Quarter Sections 270 Fractional Sections 270 .... X TABLE OF CONTENTS. Section Six 271 Sections made Fractional by Waters 272 Irregular Subdivision of Sections made Fractional by Waters 273 Exceptional Cases 274 Sample Resurvey and Subdivision of a Section 274 Private Surveys 283 Higbway Surveys 284 Surveys for Town Plats 284 What Plats in Michigan must contain 285 What Record of Plats in Michigan must contain 285 Monuments ... 287 CHAPTER X. RESURVEYS. RESUEVEYS. Authority of Surveyor 2^'J What the Surveyor is called on to do 289. DECISIONS OF SUPREME COURTS, Giving Rules for construing Descriptions of Land 289 Adverse Possession 305 Rules of Construction when Land borders on Waters 303 How to Locate Corners and Boundary Lines 317 General Rules 317 Alluvium ., 337 Rules Applicable to U. S. Survey. 339 Mineral Surveys 350 How to Write Descriptions for Deeds 351 CHAPTER XI. RE-LOCATING LOST CORNERS. General Rule 358 Lost Corners of U. S. Survey in Base Lines, etc 357 Lost Closing Section Corners 357 Lost Interior Section Corners 358 Lost Township Corners . 358 Lost Quarter-Section Corners 353 Lost Meander Corners. '. 353 Exceptional Methods 359 HOW TO FIND LOST CORNERS, Evidences of Original Posts 360 Bearing Trees .___ , 361 Fences _ 362 Distant Corners 363 Persons 364 TABLE OF CONTENTS. XI CHAPTER XII. MISCELLANEOUS. Miscellaneous Questions 365 RIGHTS, DUTIES, ETC., OF SURVEYORS 377 To fix Lines by Consent of Parties 377 Have no Authority of their own for that purpose 377 Or to determine where Corners and Lines are 377 Old Boundaries not to be disturbed 378 County Surveyor's Certificate not Admissible in evi- dence in Michigan 378 Surveyor Liable for Damages for Unskillful Work 378 Judicial Functions of Surveyors 379 CHAPTER LEVELING AND DRAINAGE SURVEYING. Definitions 895 Difference between True and Apparent Level 395 Instruments for Leveling 390 The Wye Level and its Adjustments 397 Leveling Rods, Target and Speaking 401 To find Difference in Level of Different Points 403 Drawing Profile 408 Drainage Surveying 409 TABLES. Suggestions to Young Surveyors i-iv Trigonometrical Fonnulae I rv-vi Table of Logarithms 1-16 Natural Sines and Cosines 18- 26 Natural Tangents 28- 39 Logarithmic Sines and Tangents 40-84 Traverse Table 86- 91 Departures _. 92 Natural Secants 93- 94 Azimuth of Polaris at Elongation 94 Gradienter Tables 95 Mean Refractions 96 Acreage of Open Drains 97 Acreage of Tile Drains and Capacity of Tile 98 XII TABLE OP CONTENTS. Azimuths of Tangent Offsets from Tangent 100 Minutes in Decimals of a Degree 101 Inches in Decimals of a Foot 101 Radii and Deflections 101 Tangents and Externals of a 1 Curve 102-105 Curve Formulae 113 Stadia Reductions for Reading 100 106-112 OF LAND SURVEYING. CHAPTEE I. I. DEFINITIONS. FIELD WORK, &c. 1. Land Surveying is the art of measuring distances and running lines on the earth's surface to determine the boundaries or to ascertain the areas of tracts of land. The lines run are not mathematical lines, but are repre- sentations of them, traced upon the earth's surface by means of various instruments, and marked to the eye by chops and notches cut upon trees, or rocks, or by stakes or stones set in the ground, or any other means to render them visible. 2. Original Surveys are the surveys which are first made for the purpose of locating upon the ground the boundaries of tracts of land, and marking them by visible objects. This work is called the Field Work. A full description of what is done is kept by the sur- veyor and is called the field notes. The field notes furnish the data from which to make a map of the land and calculate the area. They also furnish the evidence from which to again find and identify the boundaries upon the ground. 3. Resurveys are those which are made for the pur- pose of finding the boundaries which were marked when the original survey was made. 2 A MANUAL OF LAND SURVEYING. 4. The instruments most commonly used in land surveying are the Chain and Tape for measuring dis- tances, and the Picket, Compass, Solar Compass and Transit for running lines. II. INSTRUMENTS FOR MEASURING DISTANCES AND THEIR USE. 1. The Chain. The word chain is used to represent a distance of 66 feet and also an instrument used for measuring distances. The chain in most general use for land surveying is that invented by Gunter, and known as the Gunter chain. It is 66 feet long and divided into 100 equal parts, called links. The chain is made of wire, in links somewhat less than eight inches long. These are joined by two small, round or oval rings at each joint. The length of one of these longer links, with the two rings or short links taken together, make the distance known as a link. The best surveyor's chains are made of steel wire, having the links brazed to prevent stretch- ing by opening of the joints. Chains have every tenth link marked with a brass tag. The tags at the end of the tenth link from each end have one point; those at the twentieth links have two points; those at the thirtieth links have three points; those at the fortieth links have four points; while that in the centre or fiftieth link is rounded and has no point. Heavy chains of iron wire, with open joints, are of little value. It is very difficult to measure correctly with them, over rough ground, owing to their weight. They stretch rapidly by wear and by the opening of the joints. Chains fifty links long are used to measure over rough ground. 2. Chains Stretch by use, chiefly from wear in the joints. The best steel brazed chains, when in constant use on gritty ground, will stretch six inches or more in a year from this cause alone. They may be corrected in several ways. They may be shortened a limited amount INSTRUMENTS FOR MEASURING DISTANCES. O by turning up the nuts or burrs which hold the handles in place. They may be shortened by taking out short links or rings. The better way is to distribute the correction evenly throughout the chain, by putting each link in a vise and striking lightly on the end with a hammer, shortening it in that way. The links in the chain get bent by use. When many of them are bent, the chain becomes elastic and will elongate from one to two inches when pulled. Chains should be examined before using and the links straight- ened. They should be frequently compared with a stand- ard, that their length may be known, and they should be kept near the true length. 3, Steel Tapes are made for the use of land sur- veyors. They are light, so that they may be readily lev- eled up in measuring over rough ground or on a slope. They do not stre'tch. There are -no links to get kinked and thus cause a false measure. They are in every way more accurate and convenient than the chain. The best tapes for general use are made of the best quality of steel ribbon, polished and blued, from ^ to % of an inch wide, and No. 30 to 32 thick. The wider thinner tapes are nearly useless for field work. Tapes are made of any length and graduated to suit the work for which they are designed. A tape 66 feet long, graduated to links, is best adapted to country use. Tapes 50 or 100 feet long, graduated to feet and hundredths, are better adapted for use in many cities. Tapes from 200 to 400 feet long or even longer are made for special uses. With them long lines may be rapidly measured with an accuracy fairly comparable with the best work of the coast survey. Two precautions need to be observed with steel tapes. When in use they should be kept out at full length and never be doubled on themselves. If doubled they are easily kinked and broken. When done up, they should be wiped clean and wound on open reels to prevent rusting. 4 A MANUAL OF LAND SURVEYING. 4. A light wire is a cheap and handy substitute for the chain or tape. It is necessary to find its length in some way and then for even lengths of the wire it is capable of as accurate work as the best tape. 5. Marking Pins are used with the chain and tape in measuring. They are usually made of heavy wire about 14 inches in length, with one end sharpened to stick in the ground and a ring turned on the other end for convenience in handling. Strips of cloth are tied in the rings so that they can be seen more readily. The marking pins used in the United States surveys have heavy points, for dropping plumb when chaining on slopes. It is convenient to use eleven pins in chain- ing. One of them is stuck at the starting point, the leader takes ten, and then there is always one to start from, when the tallies are kept in even tens. 6. Measuring or chaining. Two men are required for this, and a third man can be of great assistance when chaining on slopes and accurate work is to be done. The care and accuracy required will depend on the interests at stake. The surveyor would mistake his calling who should attempt to measure land worth fifty cents an acre with the same care he would use in meas- uring land worth fifty dollars or more per inch. In making measurements the following things are to be observed, with greater or less care and accuracy of detail, according to the importance of the work in hand. 1st. Chains are not adapted to great accuracy in meas- urements. For the best work use a steel tape, of which the exact length at a given temperature, and the rate of expansion are known. Tapes are usually made to be of standard length at a temperature of about 60, F. The rate of expansion by heat varies with the kind and quality of steel in the tape. It approximates closely to .000007 for each change of a degree in temperature. Thus a tape which is 100 feet long at 60 F. will be 100.014 feet long at 80 F. For very exact measurements INSTRUMENTS FOR MEASURING DISTANCES. 5 take note of the changes in temperature and correct for expansion and contraction. A thermometer is needed for this. 2d. Measure in straight lines. In ordinary work, pick- ets or rods set up along the line, in sufficient numbers for the chainmen to range by, will enable them to secure as great a degree of accuracy as is required in this respect. 3d. Measure on level lines. To do this the tape may be brought to a level line and the successive measures transferred to and from the ground by plumb lines. Use a plumb having a fine, strong line and a long, well balanced, sharp pointed bob. Measure down the slope. The rear chainman should hold the tape steadily and firmly at the mark, bracing his hand against his leg near the ground for a support. The leader brings his end of the tape level and in line. If necessary the follower directs him in doing this. He then applies the line to the point or mark on the tape, with the plumb-bob very nearly touching the ground. When he has the proper tension on the tape, and the plumb hangs perfectly still and true, he depresses the line enough to make a slight mark on the ground with the point of the bob, and sticks his marking pin beside it. Another method of getting the measure on level lines is to drive short stakes or hubs along the line at every change in the slope of the surface. Small headed tacks are driven in the tops of these hubs. The distance between the tackheads is then measured along the sur- face and each measurement recorded. A level is then taken showing the difference in hights of these points. The length of the level line is found by calculation. Between every two hubs we have a right triangle in which we have the hypothenuse given by the tape, and the altitude given by the level, to find the base. By this method the error may be reduced below 1 in 25,000. 6 A MANUAL OF LAND SURVEYING. 4th. The tape must be drawn to the proper tension. Tapes are usually tested under a tension of ten pounds when supported the entire length. They should be further tested to find the amount of additional strain required to overcome the sag, when the tape is not supported between the ends. This varies, in different tapes, from 6 to 12 pounds for a 100 foot tape. The total strain in the unsupported tape in measuring should be from 16 to 22 pounds. The exact amount is to be found for each tape by trial. 7. The following is the general method of procedure in chaining, modified as the circumstances require. We will speak of the chainmen as leader and follower. The leader takes his end of the chain or tape and ten marking pins, and steps briskly in the direction of the line to be measured. One pin is stuck at the starting point. Just before the leader has the chain drawn out at full length, the follower calls "halt," and places his end of the chain in the proper position at the start ing point. The leader shakes out any kinks there may be in the chain, straightens and levels it in the line brings it to the proper tension and sticks his pin, calling "stuck" when he has done so. When the follower hears this signal, and not before, he pulls the marking pin and both move quickly forward, repeating the opera tion until the leader has stuck his last pin or has reached the end of the line. When the leader has stuck his last pin he calls "tally." The follower drops his end of the chain and brings forward the ten pins which he has, and gives them to the leader, who counts them to be sure none have been lost and then proceeds as before. The follower need not return for his end of the chain. The leader will draw it forward to him. When the end of the line is reached the leader holds his end of the chain at that point while the follower drops his end and comes forward and ascertains the distance, if any, between the last pin that was set and the end of the line. INSTRUMENTS FOR MEASURING DISTANCES. 7 When chaining on slopes which are so steep that the whole length of the chain cannot be leveled at once, the leader first draws it forward the whole length and in the line. He then drops the chain and all his marking pins and returns to a point where he can level a part of the chain and measures the distance, sticking one of the fol- lower's marking pins to mark the point, the follower then drops his end of the chain, comes forward and taking the chain at the same point holds it to the mark while the leader measures a second section, and so on in succession till the end of the chain is reached, where the leader sticks one of his own marking pins. It will not often be necessary to take any note of the lengths of the parts of the chain measured. Observe only to measure to and from the same points in the chain, and take care that the count is not lost by getting the marking pins im- properly mixed together. The follower should see that his end of the chain is correctly and firmly held in its position when measuring. He should, when necessary, direct the leader in keeping the true line. The leader should see that his chain is drawn straight, level, in line, and to a uniform tension. To assist him in keeping the line he should observe objects in the range, both front and rear. He should see that his marking pins are set at the exact point. They should either be set plumb or slanting at right angles with the line, so that the measure may be taken from the point. When a plumb line is used, the latter is the better way. Chain men should step quickly between points, and in chaining keep up with a man walking at an ordinary gait of three miles an hour. The follower must not stop the leader by a jerk on the chain. The leader must pull steadily when measuring. No jerking on the chain should be permitted^ If there is a difference in the chainmen the best man should take the lead. The chaining should always be uniform. In jnany surveys uniformity of measure is more important than great exactness. 8 4 MANUAL OF LAND SURVEYING. Tests made by the author have led him to the conclusion, that, in common country surveying with the chain, nothing is gained by level- ing the chain where the ground slopes less than five in a hundred. He finds that in field practice, under the ordinary conditions, more is lost by the sag of the chain than is saved by leveling. In one careful field test, six links was lost in a mile by leveling the chain, that being the net difference in favor of surface measurements for that distance. In that class of work, measurements made along the surface may be corrected on the ground, as follows: When ground slopes 4 in 100 add .1 link per chain 8. The student should practice in the field with the chain and steel tape until he is entirely familiar with their use, and can do accurate and rapid work. He should measure between fixed points over sloping or uneven ground, and repeat the measures until he can secure uniform results. He may be surprised at first to tind that he does not measure twice alike. It is well to drive a small wooden stake at every tally or tenth chain, so that in case a marking pin is lost it will not be necessary to go back farther than to the first stake to remeasure. Beware of errors in counting the links less than a full chain. Count from the right end of the chain or tape. When the chain is used do not mis- take the tag, as 60 instead of 40 or vice versa, or count odd links the wrong way from the tag. Beware of such mis- takes as 64 instead of 56, or 48 instead of 52. The tape is generally numbered the whole length from to 100, Nearly the same care is needed to avoid mistakes in read- ing as with the chain, especially to read the distance from the right end of the tape. Otherwise such mistakes as giving the distance 56 instead of 44 are very liable, to occur. III. MEASURES OF LENGTH AND AREA. 1. The measures in most general use among surveyors are based on the Gunter chain. The surveyor is how- ever frequently required to express his measurements in units of the old linear and square measure. MEASURES OF LENGTH AND AREA. 9 Table of Chain Measure. 7.92 inches or .66 foot=l link. 66 feet 100 links=l chain. 80 chains=l mile. In country surveying the smaller measures are taken in links and parts of a link and distances less than a quarter of a link are not counted In the more exact work in cities, the foot and its subdivisions are in com- mon use, and on account of the greater ease in making computations upon the decimal system, the plan of subdividing the foot decimally is adopted by many surveyors, and is growing in favor 2. Old Linear Measure: 12 inches = 1 foot. 3 feet = 1 yard. 16^ feet = 1 rod. 40 rods =;= 1 rood or furlong. 320 rods = 1 mile. MEASURES FOR AREA 3 Chain Measure: 100,000 square links, or > _ t _. 10 square chains $ - 640 acres = 1 sq mile or section. 36 sections 1 township. In the United States land sys- tem, the square mile is known as the Section. It is subdivided into al- iquot parts, which are described ac- cording to their place in the sec- tion. The manner of naming these subdivisions of a section is indicat- ed in Figure 1. W E 5 T FIG. 1. 10 A MANUAL OF LAND SUKYEYING. When, because of lakes, rivers, reservations, adjacence to township boundaries, or other causes, any of the parts of a section are increased or diminished from their normal amount, they are known and described as Frac- tional. That word is used to indicate that the tract to which it is applied is not one of the regular subdivisions of the section. When a fractional lot is small it is the custom of the United States land department to attach it to, and sell it with, an adjacent larger tract which gives the name to the description of the whole tract. The manner of describing fractional lots is indicated in Fig- ure 2. It is also a custom to number the fractional lots on the plats and describe them by numbers, as for .example, Lot No. 3 of Section 18. The latter method requires a refer- ence to the plat to know the location of the lot, while the former method does not. 4. Old English Land Measure: 144 square inches = 1 square foot. 27234 square feet = 1 square rod. 40 square rods = 1 rood. 160 square rods = 1 acre. Square rods and feet are still in common use as sub- divisions of the acre. The rood and furlong are very nearly if not quite obsolete in the United States. 5. Spanish Measures. In Spanish colonies in America, the Spanish system of land measures was used MEASURES OF LENGTH AND AREA. 11 in describing and measuring the land grants, and has continued in use down to the present time in a large extent of country. The principal unit of measure is the ' vara," which seems to be a somewhat variable one. In a report of the 14th of November, 1851, from the surveyor- general of California, it is stated that all the grants, etc., of lots or lands in California, made either by the Spanish government or that of Mexico, refer to the "vara" of Mexico as the measure of length; that by common con- sent, in California, that measure is considered as exactly equivalent to thirty-three American inches. That officer enclosed a copy of a document he had obtained as being an extract of a treaty made by the Mexican government, from which it would seem that another length is given to the "vara;" and by J. H. Alexander's (of Baltimore) Dic- tionary of Weights and Measures, the Mexican vara is stated to be equal to .92741 of the American yard. The general land office, however, has sanctioned the recog- nition, in California, of the Mexican vara as being equivalent to 33 American inches. Extract of a treaty made with the Mexican government, which accom- panied a report dated November 14, 1851, from the U. S. surveyor- general of California, respecting the ratio of land measures heticeen those employed under the Mexican government and those in use in the United States [From the Mexican ordinance for land and sea.] Article 20th of the agreement entered into between the minister pleni- potentiary of the Mexican government and her agents in London, the 15th of September, 1837, with the holders of Mexican bonds. 20th. In compliance of what is ordered by the seventh article of the preceding law, and in order to carry into effect the stipulation in the preceding agreement in regard to the holders of bonds deferred, it is declared that the act of which mention is made in said agreement answers to 4840 English yards squared, equivalent to 5762.403 Mexican varas square; inasmuch that the "sitio de ganado moyer" contains 4338.464 acres, the Mexican vara having been found by exact measures equal to 837 French millimetres. Reducing the ratio of 4840 square yards and 5762.403 square varas, the vara will be 32.99312 inches Reducing the 4338.464 acres 32.99311 12 A MANUAL OF LAND SURVEYING. 2=3 I 9 Sitio de ganado moyer Square 5,000 5,000 25,000,000 41.023 Criadero de ganado moyer _ _ do. 2,500 2,500 6,250,000 10.255 Sitio de ganado menor do. 3,333 X 3 3,333% 11 111 111 18.232 Criadero de ganado menor do. _ 1 666 2 1,666% 2,777 777g 4.558 Caballeria de tierra Right angled Media caballeria l>n rail 'gram Square 1,104 552 552 552 609,408 304,704 1 Cuarto caballeria o Suerte de tierra Right angled Fenega de sembra- paralPgram 552 276 152,352 ^ duro de maiz do 276 184 50,784 1-12 Sala para casa Square 50 50 2 500 004 Fundo legal para pue- blos _ _ do. 1,200 1,200 1,440,000 3.362 The Mexican vara is the unit of all the measures of length, the pattern and size of which are taken from the Castilian vara of the mark of Burgos, and is the legal vara used in the Mexican republic. Fifty Mexican varas make a measure which is called " cordel," which instru- ment is used in measuring lands. The legal league contains 100 cordels, or 5,000 varas, which is found by multiplying by 100 the 50 varas con- tained in a cordel. The league is divided into two halves and four quarters, this being the only division made of it. Half a league contains 2,500 varas, and a quarter of a league 1,250 varas. Ancientty, the Mexican league was divided into three miles, the mile into a thousand paces of Solomon, and one of these paces into five-thirds of a Mexican vara; consequently, the league had 3,000 paces of Solomon. This division is recognized in legal affairs- but has been a very long time in disuse the same as the pace of Solomon, which in those days was called vara, and was used for measuring lands. The " mark " was equivalent to two varas and seven-eighths that is, eight marks con- MEASURES OF LENGTH AND AREA. 13 taining twenty-three varas and was used for measuring lands, i In Texas the surveys are made on the vara system. A 20- vara chain is used, the area calculated in varas, and when necessary reduced to acres. The field notes contain no system of measurement except varas. Nearly all the old leagues were laid off in rectangular form, and nearly all the subdivisions since have been by lines parallel with the original league lines. The following table of comparisons gives the system of land measures in use in that state: 1 vara = 33^ inches. 1900.8 varas = 1 mile. 25,000,000 sq. varas = 1 league = 4428.4 acres. 1,000,000 " " =1 labor = 177.136 " 5645.376 " = 1 " 1 " " = .000177 " 6. Old French Measures were used in laying off land in the French colonies, and still find a place in some parts of the country. The unit was the "arpent," of which there were different values, varying from three- fourths of an acre to an acre and a half. The " arpent d'ordonnance" or legal arpent equalled 1.262 acres, and contained 100 square perches of 22 "pieds du roi" on a side. The old French linear measures were the old Paris foot called "pied du roi" and its sub-multiples 12 points = 1 ligne. 12 ligne = 1 ponce. 12 ponce = 1 pied du roi = 12.789 inches. 6 pieds du roi = 1 toise, interesting as being the unit employed in the survey of the great French meridian arc, on which the metre was founded. Modern French measures are upon the Metric System. 14: A MANUAL OF LAND SURVEYING. 7t Standard Measures. i The constitution of the United States says that con- gress shall have power to establish a system of weights and measures. It has, however, never done so. In 1832 the secretary of the treasury assumed the authority to adjust and regulate the weights and measures in use in the custom houses, and delegated the construction and adjustment of standards to Mr. Hassler, who was -then superintendent of the coast survey. The standard of length adopted was a yard, as meas- ured between the 27th and 63rd inches of a scale made in London, by Troughton, and brought to this country in 1814. This scale is a copy of the old British Standard, known as the Bird Standard of 1760. At a temperature of 59.62 F. it is equal in length to the Imperial Standard at 62 F. Although Congress never adopted that yard as a standard, it authorized the transmission of copies thereof to the several states. In many of the states these copies have been legally adopted as the standards. Other states have no legal standards. The Michigan standard is a brass yard, of exact length at a temperature of 58.40 F. It is both a line and an end measure. It is doubtful if these standards in the several states are kept in such a manner as to be reliable for purposes of comparison or if they are so kept, whether the officers in charge of them have the skill and the facilities required for making accurate comparisons. Standard rods are sold by dealers but they are more or less discrepant in length. Surveyors who desire to know the true length of their standard measures can send them to the Superintendent of the Coast and Geodetic Survey, at Washington, who will cause them to be compared and the government stamp placed on them, giving their exact length. The examination and test, for which a fee of fifty cents is charged, secures a sufficient degree of accuracy for ordinary purposes of the surveyor. Where an extra degree of accuracy is called for a higher fee is charged. MEASUBES OF LENGTH AND AREA. 15 Although Congress has not adopted a general standard of measure, it has adopted a standard for the measure- ment of the public lands, which so far as the resurvey or subdivision of those lands is concerned is final. In sec- tion 2395 of the revised statutes of the United States, it is enacted that "all lines shall be measured with chains containing two perches of sixteen and one-half feet, each subdivided into twenty five equal links. In section 2396 it is enacted that "All the corners marked in the surveys returned by the Surveyor General shall be established as the proper corners" &c.; and that "the boundary lines actually run and marked in the surveys returned by the Surveyor General, shall be established as the proper boundary lines of the sections and subdivisions for which they were intended, and the length of such lines as re- turned shall be held and considered as the true length thereof.'" This enactment makes an actual standard of measure between every two adjacent corners of the government survey, which is the only legal standard for measures of that line. The surveyor, in resurveying or subdividing the public lands, has thus a standard laid down for him on ever^ line previously run by the government deputy surveyor and has only to adjust his chain to that stand- ard. This is practically done on the ground by apportion- ing any difference between the surveyor's measure of a given line and the length of the line as returned in the field notes pro rata between its different parts. Example. It is required to locate the half-quarter cor- ner on the line described in the field notes as running, "West on corrected line between Sections 11 and 14 39.72, set qr. sec. post," etc. Suppose the surveyor on measuring this line finds the distance between the two corners, as actually marked on the ground, to be by his chain 39.84 chains. Then his chain is too short and its legal length for that line is to its nominal length as 39.72 is to 39.84 and the distance to the half -quarter corner is by the new measure 19.92 chains. 16 A MANUAL OF LAND SURVEYING. IV. INSTRUMENTS FOR RUNNING LINES AND THEIR USE. 1. The instruments most commonly used in running lines are the picket, the compass and the transit. There are various modifications of the compass and transit. The methods of running lines with these instruments will be treated of in connection with the description of them. 2. The Picket or Rod is the simplest device for ranging lines. It is simply a straight rod an inch or two in diameter and having a sharp point to stick in the ground. The author prefers to have them sharpened to a long slim point at the top also, and that the pickets shall be of such a length as to be the height of the eye when firmly planted in the ground. Where timber is plenty they may be cut from small straight saplings, or split from body wood as they are wanted, and left standing where they are used, as a guide to the chainmen. 3. To range a line with pickets. Set the first picket at the starting point and a second a short distance away in the direction in which the line is to run. Then go ahead and set picket after picket at such distances apart that at least three of them can be distinctly seen at the same time. Set the pickets plumb and align them by sighting over the sharpened points at the top. A plumb line will be of assistance in rang- ing lines over uneven ground. Set short stakes in the, line at uniform distances apart. Then if the line was intended to strike a particular point and missed, it may be corrected by measuring the perpendicular dis- tance from the line to the point, and then moving each intermediate stake its proportional part of that dis- tance according to the distance it is from the starting point. Example 1. Commencing at the southwest corner of Mr. B.'s farm, I ran north, setting stakes on the trial line every ten chains. At 40.00 chains, my line inter- INSTRUMENTS AND THEIR USE. 17 sected the north line of his farm 32 links east of his northwest corner. What correction must be made for each stake ? Solution. The first stake being set at ^ the distance between points must be corrected % of 32 = 8 links, and as the trial line came out to the east of the corner, the stakes on that line must be moved to the west. The 2d stake being at % the distance between points must be moved west ^ of 32 16 links. Similarly the 3d stake must be moved west 24 links. NOTE. Sections of the United States survey are tracts of one mile square. Monuments are set at each corner called Section Corners. Others are placed midway between them on the section lines called quarter posts or quarter section corners. Some sections greater or less than these are called Fractional Sections. Example 2 Commencing at a point 12 links west of the quarter post in the south side of Section 20, I ran north, setting stakes on the trial line every ten chains. At 80 chains my line intersected the north line of the section, 36 links west of the quarter post. What cor- rection must be made to place the intermediate stakes in the true line between the quarter posts, known as the quarter line ? Answer Commencing with the first ten chain stake tney must be set east, 15, 18, 21, 24, 27, 30, and 33 links respectively. Example 3. Commencing at a point 24 links west of the southwest corner of section 16, I ran a trial line north, setting stakes every ten chains. At 80.36 chains, the line intersected the north line of the section, 32 links east of the section corner. What is the correction to be made at each stake to place it in the true section line and at the equidistant points y Answer to be found by the student. NOTE. This solution requires corrections both for line and meas- ure. It is a cardinal principle of land law that the original measure- ments and monuments which were made in the survey in accordance with which the land was sold are in law the true measures and monu- ments. All subsequent measures for the purpose of locating bound- aries must be made to conform with the original measures. 3 18 A MANUAL OF LAND SURVEYING. Trial or random lines, as they -are usually called, are often run one side of the true line, purposely to avoid obstacles, like fences and hedge rows. The surveyor, by a judicious selection of ground for the random line can often save a great deal of labor and time of the party, by avoiding obstacles which would otherwise have to be removed or offset around. Eandoms from which the true line is to be found should be run with as great care as any line. The student should practice running and measuring trial lines between points until familiar with the pro- cesses. He should run various randoms to find the line between the same points and see how they agree when corrected for true line. 4. To range a true line between points that can not be seen from each other but can both be seen from some inter- mediate point, as a hill. Set up flags at the two points. Two persons then take pickets and station themselves, a short distance apart, at the intermediate position from which the flags can be seen. They face each other and each in turn aligns the other between himself and the flag toward which he faces, until the true line is reached, when the pickets are set in the line. 5. To pass obstacles in the line. From the last two pickets preceding the obstacles, set two other pickets on a line parallel with (the true line and at a sufficient distance to pass the obstacle. Prolong the parallel line far enough to set two pickets beyond the obstacle and then regain the original line by meas- uring back from these two pickets. 6. The methods of running lines with the compass and transit will be given in connection with the descrip- tions of these instruments. DESCRIPTION OF INSTRUMENTS. 19 . CHAPTER II. DESCRIPTION OF INSTRUMENTS. 1. The Surveyor's Compass. The essential fea- tures of the surveyor's compass are a magnetic needle for finding a meridian line, a circle graduated to half degrees .known as the limb, for laying off angles from the meridian, and sights attached for use in prolonging lines on the ground. When the limb and sights are on separate plates move- able upon each other around a common center through an arc of 15 or 20, and a vernier is attached, the instru- ment is known as the Vernier Compass. The use of the vernier is chiefly for setting the sights of the instrument so that they will be in the true north and south line when the magnetic needle points to zero on the limb. There is only a small portion of the earth's surface in which the needle points to the true north. A lino passing through those places where the needle points truly north is called the agonic line or line of no variation. This line runs in a northerly course and is constantly changing its position. At all places outside the line of no variation, the needle points to the east or west of true north. This difference between the direction of the needle and the true meridian is spoken of as the variation, or, more correctly, the declination of the needle. The vernier is used to measure the angle between these two lines. A MANUAL OF LAND SURVEYING. FIG. 3. VERNIER COMPASS-6-INCH NEEDLE. Sometimes there is added a divided circle or limb with ^erniers by which angles can be taken throughout the entire circle independently of the needle. The instrument in this form is called the railroad compass. The addition of leveling screws and a revolving telescope in place of the plain sights makes a surveyor's transit of it. ADJUSTMENTS OF THE COMPASS. 21 The Plain Compass consists of a circular box of brass, usually about six inches in diameter, resting upon an arm of the same metal about fourteen inches in length- At the extremities of the arm are vertical attachments through which are fine slits, terminated at intervals by circular apertures, which serve as sights in directing the instrument upon any point. At the centre of the box is a small vertical pin upon which is balanced a slender magnetized bar of steel, called the Needle. Turning with a free horizontal motion, the pointed ends of the needle traverse the graduated circumference of the circle. The plane of the sights passes through the center of the circle and cuts the circumference in two points marked N and S, otherwise distinguished as the north and the south points of the instrument. From ' these points the graduation of the circle runs 90 in each direction to the points marked E and W. A circle of plate-glass forms the cover of the box. Two small spirit levels are placed at right angles to each other upon the arm, to aid in rendering the plane of the instrument horizontal. The compass is mounted upon a three-legged support called a Tripod, or upon a single staff called a Jacob Staff, with which it is so connected as to admit of being turned in any desired direction. In using the compass, the* surveyor should keep the south end toward his per- son, and read the bearings from the north end of the needle. He will observe that the letters E and W on the face of the compass are reversed from their natural position, to correspond with the line of the sights, in order that the direction may be correctly read. II. ADJUSTMENTS OF THE COMPASS. The Sights of the compass should be truly at right angles with the plate, so that when set up and leveled ready for use the line of sight will be in a vertical plane. 22 A MANUAL OF LAND SURVEYING. The needle should cut opposite degrees in any part of the circle, and should have its ends in line with the centre. The levels should be parallel to the plane of the plate. To adjust the compass to these conditions begin with 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 edge of the tubes, it would indicate that those ends were the highest; lower them by tight- ening the screws immediately under, and loosening those under the lowest ends until, by estimation, the error is half remo . ed ; level the plate again, and repeat the first operation until the bubbles will remain in the centre, during an entire revolution of the compass. 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. The Needle is adjusted in the following manner: Having the eye nearly in the same plane with the grad- uated 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 the small brass wrench, furnished with the compass, about one-eighth of an inch below the point of the pin, until the ends of the needle are brought into line with the opposite degrees. Then, holding the needle in the same position, turn the compass half-way around, and note whether it now cuts opposite degrees ; if not, correct half the error by bend- ing the needle, and the remainder by bending the centre pin. ADJUSTMENTS OF THE COMPASS. 23 The operation should be repeated until perfect rever- sion 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 correc- tion must be made in the centre-pin only, the needle being already straightened by the previous operation. When again made to cut^it should be tried on the other quarters of the circle, and corrections made in the same manner until the error is entirely removed, and the needle will reverse in every point of the divided surface. If the needle has lost its polarity, and needs to be remagnetized, this is effected in the following manner : The operator being provided with an ordinary perma- nent 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 suriace of the pole is tangent, drawing the needle towards him and taking care that the north and south ends are applied to the opposite poles of the magnet. Should the needle be returned in a path near the mag- netic 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 De considered 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. The Centre- Pin. This should occasionally be ex- amined, and if much dulled, taken out with the brass wrench, already spoken of, 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 f and deli 24 A MANUAL OF LAND SURVEYING. cately 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 invisible to the eye, it should be smoothed by rubbing it on the surface of a soft clean piece of leather. 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 electric, the fluid may be removed by breathing upon it, or touching differ- ent parts of its surface with the moistened finger. III. To RUN A LINE WITH THE COMPASS. Set up the instrument at the point from which the line Is to run ; level the plate ; turn the sights in the direction in which the line is to run, which may be ascertained by the needle or otherwise, as is most convenient. An assist- ant, known as the rodman or flagman, goes ahead with a sharp pointed rod or flag pole to such a distance as is convenient, and, guided by the signals of the compass- man, sets his rod in line. When the ground is uneven, the rodman should select his point at the summit of rising ground, when possible to do so, in order to save unneces- sary setting of the compass. He should always select the point most favorable for setting up the instrument, both to get a clear spot for the instrument and to get the best point for taking the next sight. When setting his rod he should face the compass, hold- ing the rod plumb and directly in front of him. He should move steadily in the direction indicated by the signals and not stick the rod down until he receives the signal to do so. After sticking it he should look for further signals, lest a change in its position might be required. After the rod is set the com passman should examine his instrument to see that it is in position, cor- TO PASS OBSTACLES IN THE LINE. 25 recting it and resetting the rod when necessary. He then sets up a picket in line near his instrument, to be used for a back sight, and moves his compass forward in the line to the point marked by the rodman, sets it up in the line, with the sights ranging back to the backsight, and con- tinues the line as far as desirable. The needle may or may not be used, according to circumstances. At the beginning of the line the direction will usually be obtain- ed from the needle. If used afterwards on the same line, care should be taken to have it in proper condition and working freely. When being carried the needle should be raised off the pivot, otherwise the point of the pivot will become dulled and the needle will not traverse freely. IV. To PASS OBSTACLES IN THE LINE. 1. When the obstacle is a tree, and no great degree of accuracy is required, make a mark on the tree where the line strikes it and set the compass up on the opposite side of the tree, putting it in line by taking a backsight on the tree, and finding the direction of the line by the needle. 2. Make an offset far enough to pass the obstacle on a parallel line, the same as when running a picket line. When it is found that the line strikes a tree too large to be removed, set the rod in line near the tree, and then before moving the compass, set the picket for back- sight at one side of it, a sufficient distance to" pass the tree. Then move the compass ahead and set it up the same distance, and direction from the rod that the back- sight picket was set from the compass. Get the direction of the line by ranging to the backsight. Prolong the parallel line beyond the obstacle and regain the true line in a similar manner. Other methods of passing obstacles in line will be given further on. Y. THE MAGNETIC NEEDLE. 1. The compass, because of its being so convenient for use has been for many years the principal instrument used 26 A MANUAL OF LAND SURVEYING. in Land Surveying. It is now very generally superseded by other instruments in surveys where accuracy is re- quired. So far as the direction of lines is concerned, all compass surveying is based on the tendency of the magnetic needle to adjust itself to the magnetic meridian when free to do so, in other words to point north and south. It is however constantly changing its direction. 2. Secular Change. The line of no variation, as it is commonly called, otherwise known as the agonic line seems to have a periodical motion, back and forth, to the east and west, like the swinging of the pendulum. The length of the period is unknown but probably covers sev- eral centuries. In the United States, so far back as known, its motion was to the eastward until the beginning of the present century, since which time it has been moving westward. In Michigan the secular change has been between 3' and 4' per year to the westward for the past sixty years. The agonic line was, iii 1890, in the vicinity of Lansing. 3. Diurnal Change. The needle when undisturbed and free to move, swings back and forth each day through an arc varying from 5' to 2(K or more in amount. In the northern hemisphere the north end of the needle moves westward from about 8 A. M. until about 1 :30 p. M., then returning and reaching its former position at about 8 P. M. The amount of this motion is not uniform from day to day, being least on cloudy days ; nor from month to month, being least in winter. Nor is it the same in different localities. The effect of the diurnal variation is such that if a surveyor were to start a line in the morning and continue running it all day in the same direction, as shown by the needle, he would run a line like a letter S. 4. Irregular Changes. The needle is subject to sudden and violent changes in its direction, sometimes coinci- dent with a thunderstorm or an Aurora Borealis, often without any apparent cause. The writer has observed a THE MAGNETIC NEEDLE. 27 Change of half a degree in less than ten seconds of time, for which there was no apparent or discoverable cause. It was supposed to have been occasioned by a magnetic storm. 5. Local Attraction. Iron ore in the earth, or iron or steel in the vicinity of the needle will deflect it from its normal direction. High mountains or running streams are also said to deflect the needle more or less. Pocket knives and steel watch chains are prolific sources of error as well as chains and axes. 6. Difference in Instruments. It is found by obser- vation that different instruments do not indicate the same declination of the needle when observed at the same time and place. A difference of 15' is not uncom- mon. Eight needles of three types made at Gurley's from the same sheet of steel and tested by an expert for a month on the same center pin, differed in direc- tion, and the difference varied with the time of day. 7. , Things to be Observed in Running Compass Lines. For these reasons it is practically impossible to run a true line and repeat it, relying on the needle alone for direc- 1 ion. Hence in ail original surveys, made with the com- pass, tte field notes of the survey should give the date, and state whether the directions of the lines are given accord- ing to the magnetic meridian. If not, state what the angle is between the magnetic meridian and the meridian adopted for the survey, or in other words state the decli- nation of the needle, estimated or allowed for in the sur- vey. The meridian adopted will usually be as nearly coincident with the true meridian as known. Back- sights should be used whenever the line is prolonged beyond a single sight, both to secure accuracy in the line, and as a check against local disturbances of the needle. They also save time, as a compass can be pointed to a backsight in much less time than it takes a good needle to settle. 28 A MANUAL OF LAND SURVEYING. 8. Marking Lines. It is a cardinal principle of com- mon law, as well as the statute law of the United States with reference to the public lands, that the original surveys as marked on the ground, in accordance with which the land was sold, are conclusive as to the corners and boundary lines. When the land is once sold, no change can be made in the marked boundaries without disturbing the vested rights of the owners. Resurveys are made to find the location on the ground of the original survey. The compass is a useful assistant in pointing out where to look for the more certain evidences, such as marked trees, stakes or corner stones, and, in the absence of anything better, may be used to determine the location of the line. A marked tree of the original survey is, however, better evidence of the location of the line than any line afterward run by a compass. It is possible that the line might be exactly retraced by the compass, but it could not be known to be so without the aid of other evidence. Hence the marks on the ground which define boundary lines cannot be made and kept too plain and permanent. The field notes and records which describe these marks should be full, clear and concise. VI. TRUE MERIDIANS AND HOW TO FIND THEM WITH THE COMPASS. In a country that has had the first surveys made and boundary lines marked, and subsequent surveys are based on these lines, it is very rarely of any consequence to the surveyor to know where the true meridian is. The original boundary lines are unchangeable, and it is no help to the surveyor to know where the true meridian is unless he also knows that the original surveys were in conformity with it, and that the causes of error hereto- fore mentioned can be eliminated. That is very rarely the case. His main concern is to know where the lines were and not where they ought to have been. The writer in nearly a quarter century of active practice as a surveyor has never had occasion, except as a matter of curiosity, to know where the true meridian was. In making the surveys of a country with a compass, it is well to TO FIND A TRUE MERIDIAN. 29 know the position of the true meridian, in order that the lines may be run as nearly in conformity with it as the limitations of the instrument will permit, or that the divergence may be known. Subsequently, a knowledge of the changes in the declination of the needle is all that serves any practical purpose. This can be learned by observations on any line between two permanent points. To find a true north and south line by means of the north star. The north star appears to describe a small circle about the true north point or pole as a center. The radius of this circle is called the Polar Distance of the star. This polar distance is not a constant quantity, but be- comes about ^ of a minute of arc less every year. On the first of January, 1890 it was about 1 16' 41". When in its revolution, the star is farthest from the meridian, it is said to be at its greatest eastern or western elongation. The times of the elongations as given by a correct clock, for latitude from 38 N to 60 N and for the year 1890, are approximately as shown in the following tables: EASTERN ELONGATIONS. Day. Apr. May. June. July. Aug. Sept. H. M. H. M. H. M. H. M. H. M. H. M. 1 7 13 19 25 6 37 A.M. 614 " 550 " 526 " 503 " 439A.M. 416 " 352 " 328 " 305 " 237A.M. 214 " 150 " 126 " 103 " 1239A.M. 121fr " li 52 P.M. 1*29 " 1105 " 10 37 P.M. 1014 " 950 " 927 " 903 " 8 36 P.M. 812 " 748 * 725 " 701 " WESTERN ELONGATIONS. Day. Oct. Nov. I Dec. Jan. Feb. Mar. II. M. H. M. H. M. H. M. H. M. H. M. 1 13 19 25 -fi A.M. 6W " 640 " 617 " 453 " 425A.M. 402 " 338 " 315 " 251 " 228A.M. 204 " 140 " 117 " 1253 " 12 26 A.M. 1202 " 11 39 P.M. 11 15 " 10 51 " 10 24 P.M. 1000 " 936 " 913 " 849 " 8 30 P.M. 806 " 743 " 719 " 655 " 30 A MANUAL OF LAND SURVEYING. To find the meridian of a place by means of an elonga- tion of the north star requires the arrangement of the following preliminaries. Set two posts firmly in the ground about three feet apart east and west, and saw them off to a level about three feet from the ground. Lay upon the posts a plank 3 or 4 feet long and 6 or 8 inches wide, planed smooth on the upper surface, and nail or pin it securely to the supports, forming a sort of table. To the north of the table at a distance of 10 or 12 feet set in the ground a stiff pole 12 or 15 feet high, having a cross bar nailed to its top, in an east and west direction, from which to suspend a plumb-line nearly reaching the ground, and having a bob weighing 1 or 2 pounds, which may be caused to hang in a pail of water, to insure stead- iness. Provide also a block or piece of plank 8 or 10 inches long, and smooth on the under side. Let one of the com- pass sights be fastened at right angles with the upper surface of the block and even with the side which is to be toward the south. Everything being in readiness, the observer, a few minutes before the time of an elongation as given in the above Table, should be at his post and begin moving the block, even with the south edge of the table, keeping the plumb-line and star, as seen through the vertical slit, constantly in range with each other. A light will generally be needed near the plumb-line, to render it visible. As the star approaches its elongation, it will appear to move nearly vertical for several minutes, so as to be seen without moving the sight. When it is certain that the star has reached its elongation, confine the block carefully, by sticking a few tacks along its edges. Pro- ject the vertical slit to the ground by means of a plumb- line and mark the point by setting a substantial stake with its top a little below the surface of the ground. TO FIND A TRUE MERIDIAN. 31 Being still careful not to move the block, let an assist- ant take one of the iron-pointed rods, or a stake, with a light, and go a hundred feet or more toward the star, and having found the point as directed by the observer, in range with the plumb-line as seen through the slit, let him mark it by driving a stake. Having now two stakes in range of the elongation, the remainder of the operation may be deferred till morning. To find the angle which the line as above determined makes with the meridian of the point of observation, requires a trigonometrical computation. Let A be the point of ob- servation, Z, the zenith of that point, HO, an arc of the northern horizon, N, the north point of that arc, /S, the north star at its eastern elongation, PS, the polar distance of the star, A N, the meridian of the point of observation, and AE, the line of the two stakes. The angle sought is NAE = angle PZS = arc NE. ' Now, in the spherical triangle PZS, PZ is the co-latitude of the point A, which must be known. Solving this tri- sin PS sin polar dist. angle, we have sin Z = , or sin Z sin ZP cos lat. From this, the angle Z becomes known, and, accord- ingly, it may be formed on the west side of the line AE, and thus the direction of the meridian AN deter- mined. Ou AN, thus found, let a substantial stake be set a hun- dred yards or more from A t and we have a permanent meridian with which we irtay compare the magnetic meridian at any time, and thus determine the declination of the needle. 32 A MANUAL OF LAND SURVEYING. The declination of the needle is the angle which the magnetic meridian makes with the astronomical merid- ian. For the purpose, simply, of finding the declination of the needle, it is sufficient to lay out on the ground the line of direction of the star at one of its elongations, and then, knowing the bearing of this line as shown by the needle, and the corresponding azimuth of the star, the declination of the needle is readily computed. Thus, let a = azimuth, =b 6 = bearing, and =fc d = declination, accordingly as they are east or west. Then d a (=b 6). RULE. Subtract the bearing from the azimuth. In applying the Rule, due regard is to be had to the algebraic signs. A near approximation to a true meridian may be had by observing the pole star while it is in the same vertical plane with the Bear. star Delta, in the constellation Cas- siopeia. When both are behind the plumb-line together, they are very nearly in the true meridian. When Delta Cassiopeia passes the meridian above the pole, it is too high in the heavens to serve this purpose. It passes the meridian below the pole at midnight April 10th, and may be used for two months before and after that date. Six months later the star Zeta, the last but one in the tail of the Great Bear, takes its * * * place. Fig. 5 shows the relative po- a * sition of these stars and the pole, FIG, 5. Pal* VII. OTHER METHODS FOB FINDING A TRTTB MERIDIAN. There are various other methods for finding a true meridian, several of which are here given. The Method for the determination of the azimuth of Polaris and true meridian at any hour> the star being visible, and the correct local mean time known is from the U. S. Surveying Instructions. IN this article it is proposed to present a method,with two new and compact tables adapted to common clock time, with such plain directions for use that any person of ordinary intelligence can understand and apply them. As the surveyor should have a perfectly clear idea of what is meant by Astronomical Time (used to simplify computations), and the Hour Angle of .Polaris, these terms will now be explained. The Civil Day, according to the customs of society, commences at midnight and comprises twenty-four hours from one midnight to the next following. The hours are counted from 12 to 12 from midnight to noon, after which they are again reckoned from 12 to 12 from noon to midnight. Thus the day is divided into two periods of 12 hours each; the first of which is marked a. m., the last p. m. The Astronomical Day commences at noon on the civil day of the same date. It also comprises twenty-four hours; but they are reckoned from to 24, and from the noon of one day to that of the uext following. The civil day begins twelve hours before the astro- nomical day; therefore the first period of the civil day answers to the last part of the preceding astronomical day, and the last part of the civil day corresponds to the first part of the astronomical day. Thus, January 9, 2 o'clock p. m., civil time, is also January 9, 2 h , astro- nomical time; and January 9, 2 o'clock a. m., civil time, is January 8, 14 h , astronomical time. Ttie rule, then, for the transformation of civil time Into astronomical time is this: If the civil time is marked 107 o4 A MANUAL OF LAND SURVEYING. p. m., take away the designation p. ra., and the astronomical time is haa without furtlwr change ; if the civil time is marked a m., take one from the day and add twelve to the hours, re- move the initials a. m., and the result is the astronomical time wanted. The substance of the above rule may be otherwise stated, as follows: When the surveyor takes an observa- tion during p. m. hours, civil time, he can say: the as- tronomical time is the hours and minutes passed since the noon of this day, and when observing in the a. m. hours, he can say the astronomical time is the hours and minutes elapsed since the noon of yesterday, in either > / Q . O x ** I - V A /} VJ- h\ .u r ji I & '^P \ \ \ I i i \ 1 \ \ \ \ \ \ \ \ \ \ i 1 I I \ r. O ^ > ^O Mfcff UALt #P iAJK D SURVEYING. it is said to cul- minate; above the pole (at S), the passage is called the Upper Culmination, in contradistinction to the Lower Culmination (at S'). In the diagram, which the surveyor may better un- derstand by holding it up perpendicular to the line of sight when he looks toward the pole, Polaris is sup- posed to be on the meridian, where it will be about noon on April 10th of each year. The star appears to revolve around the pole in the direction of the arrows, once in every 23 h 56 m 4 s . 09 of mean solar time ; it consequently comes to and crosses the meridian, or culminates, nearly four minutes earlier each successive day. The apparent mo- tion of the star being uniform, one quarter of the circle will (omitting fractions) be described in 5 h 59 m , one half in ll h 58 m , and three quarters in 17 h 57 m . For the posi- tions S 1} s 2 , s 8 , etc., the angles SPs n SPs a , SPs 3 , etc., are called Hour Angles of Polaris for the instant the star is at SD S 2 , or s 3 , etc., and they are measured by the arcs Ss 15 Ss 2 ., Ss 3 , etc., expressed (in these instructions) in mean solar (common clock) time, and are always counted from the upper meridian (at S), to the west, around the circle from O h O m to 23 h 56 m .l, and may have any value between the limits named. The hour angles, measured by the arcs Ss lf Ss 2 , Ss 3 , Ss 4 , Ss 5 , and Ss 6 , are approximately i* 8 m , 5 h 55 m , 9 h 4 m , 14 h 52% 18 h Ol m , and 22 h 48 m respect- ively; their extent is also indicated, graphically, by broken fractional circles about the pole. The hour an- gle, 5 h 55 m and 18 h Ol m are those at west and east elonga- tion, respectively, in latitude 40 N. Suppose the star observed (e. g.) at the point S 3 ; the time it was at S (the time of upper culmination), taken from the whole circle, 23 h 56 m .l, will leave the arc Ssj, S 2 , s 3 , or the hour angle at the instant of observation; similar relations will obtain when the star is observed in any other position; therefore, in general: Subtract the time of Upper Culmination from the correct local mean time of observation; the remainder will be the Hour Angle of Polaris. The observation will be made as heretofore directed, TO FIND DEPARTMENT OF CIVIL KMClMCCMINC modified as follows : TheT^Wff^fe#*oC^fctfiftW*%i( star to reach elongation; the observation may be made at any instant when Polaris is visible, the exact time being carefully noted. TABLE V. This table gives, in " Part I," the local mean time of the upper culmination of Polaris, on the 1st and loth of each month, for the years 1901 to 1910, inclusive. The times decrease, in each year, to April 10, when they be- come zero; then, commencing at 23 h 56 m .l, the times again decrease until the following April, and so on, con- tinuously. The quantity in the column marked " Diff. for 1 day "is the decrease per day during the interval of time against which it stands, and answers for att the years marked in the table. For any intermediate date, the "Diff. for 1 day" will be multiplied by the days elapsed since the preceding tabular date, and the prod- uct subtracted from the corresponding time, to obtain the required time of upper culmination for the date under consideration. The table answers directly for 108 west longitude. The results of using it for other longitudes will contain an amount of error hardly ap- preciable, as the correction for longitude cannot exceed one-tenth of a minute of time for each 9 degrees of longitude. A few examples will illustrate the use of the table. 1. Required the time of upper culmination of Polaris for a station in longitude 116 west, for March 3, 1904. h. m. Astron. time, U. C. of Polaris, 1904, March 15 151.9 Red. for 12 days is 3.94xi2=47-.3, add. 47.3 Local mean time, U. C. of Polaris, 1892, March 3 2 39.2 The required time may also be obtained by using the table in the opposite direction; by taking the time for March 15, and adding the reduction, as follows: h. m. Astron. time, U. C. of Polaris, 1904,MaT"hl 2 47.0 Red. for 2 days is 3m.94x2=7.9 (Part H) Snbtract 7.9 Local mean time, U. C. of Polaris, 1904, March 3 2 89.1 QS A MANUAL OF LAND SURVEYING. d5 In' this case the two results are identical. If the computation is made both ways, the results will check each other. Part II has been inserted to save the surveyor the lit- tie trouble of making multiplications ; thus, for the above example, look in Part II, under the proper tabu- lar difference, 3 m .94, and opposite the 3d or 17th day of the month in left-hand column is the correction 7 m .9. Computing from a preceding date, for days between April 11 and 15 of any year, the reduction in Part II will be greater than the tabulated time of culmination, in which case 23 h 56 m .l will be added, to make the sub- traction possible. 2. Required, for a station in long. 90 west, the time of U. C. of Polaris for April 14, 1906 h. m. Astrou. time, U. C. of Polaris, 1906, April 1. (Part I) 47.9 Add 2356.1 Sum 2444.0 Reduction to April 14 (Part II;, subtract 51.1 Local mean time, U. C. of Polaris, April 14 23 52.9 Working from the following date, for days between the 9th and 15th of April, the sum will exceed 23 h 56 m .l, and when this occurs subtract 23 h 56. 1 from the sum, and the remainder will be the required time. 3. Required, for a station in long. 90 west, the time of U. C. of Polaris for April 10, 1903 h. m. Astron. time, U. C. of Polaris, 1903, April 15 (Part I) 23 48.5 Reduction for 5 days (Part H), add 19.6 Sum 2408.1 Subtract ,28 56.1 Local mean time, U. C. of Polaris, 1903, April 10 12.0 TO KIND A TRUE MERIDIAN. 39 V. Local mean (astronomical) time of the upper culmination of Poiarit, computed for longitude 108" (7ft. 12m.) west of Greenwich. Part I. Dau. 1801. 1902. 1903. 1904. 1906. 1906. 1907. 1908. Diff. for 1 day. Jan. 1 h. m. 6 39.5 4. m. 6 41.0 6 42. 4 6 43.9 641.4 642.8 644.3 6 46.7 3.96- 16 6 44.2 6 45.7 6 47.1 6 48.6 6 46.1 647.6 649.0 6 50.4 3.96 Fab. 1 4 37.1 4 38.6 4 40.0 4 41.5 4 39.0 4 40.4 4 4L9 443.3 3.96 15 3 41.3 3 43.4 3 44.8 346.3 3 43.8 3 46.2 3 46.7 3 48.1 3.96 UK. I 246.6 248.1 2 49.5 8 47.0 8 48.5 2 49.9 2 51.4 2 48.9 3.H 1 51.5 1 63.0 1 64.4 1 61.9 1 63.4 1 64.8 1 66.3 1 53.8 3.94 Apr. 1 44.6 46.1 47.6 45.0 046-.5 047.9 49.4 46.8 3.94 21 45.6 23 47.1 33 48.6 23 46.0 23 47.5 8348.9 23 60.4 2347.8 3.93 May 1 22 42.8 22 44.3 22 46.7 22 43.2 22 44.7 88 46.1 22 47.6 2846.1 3.93 r 21 47.9 21 48.3 21 50.7 81 48.2 21 49.7 81 61. 1 21 62.6 21 60.1 3.92 Jane 1 2041.2 2042.7 1044.1 8041.6 2043.1 2044.5 tO 46.0 20 43.6 3.92 15 19 46.4 19 47.9 19 49.3 1946.8 19 48.3 19 49.7 19 61.2 19 48.7 3.91 July 1 13 43.8 18 46.3 18 46.7 1844.2 18 46.7 18 47. I 18 48.6 18 46.1 3.91 15 17 49.0 17 60.1 17 51.9 17 49. 4 17 60.9 1758.3 17 53.8 17 61.3 3.92 Aug. 1 16 42.4 16 43.9 16 46.3 16 42. 8 16 44.3 16 46.7 16 47. 2 16 44.7 3 92 15 1647.6 15 49.1 1660.6 1548.0 16 49.5 15 60.9 15 52.4 15 49.9 3.92 Sept. 1 1441.0 1442.5 14 43.9 14 41.4 14 42.9 14 44.3 14 46.8 14 43.3 3.92 15 13 46.1 13 47. 6 13 49.0 1346.6 1348.0 1349.4 13 50.9 13 48.4 3.93 Oct. 1 12 43.3 12 44.8 12 46.2 1843.7 12 45.8 1246.6 12 48.1 12 46.6 3.93 15 11 48.3 11 49.8 11 61.2 11 4-7 11 60.2 11 61.6 H 53.1 11 60.6 3.93 Nov. 1 10 41.4 10 42.9 10 44.3 10 41.8 1043.3 10 44.7 1046.2 1043.7 3.3 16 9 46.4 947.9 9 49.3 9 46.8 9 48.3 9 49.7 9 51.2 9 4S.7 3.94 Dx. 1 8 43.3 8 44.8 8 46.2 8 43.7 8 46.2 8 46.6 8 48.1 8 45.6 3.94 15 7 48.1 7 49.6 7 61.0 748.6 7*0.0 7 51.4 7 62.9 7.60.4 3.95 Part I Continued. Part II. Date. 1909. 1310. 1911. Difl. for Seduction of tabular tine* to intermtdioU dolt,. Iday. Subtract the reduction when competing from *\ preceding Jtn 1 ft. m. 6 43 2 h. m. 6 44 7 *. m. 6 45 1 3%5 or add it when working from .fotlowiny date. 15 5 47.9 5 43.' 4 5 Mis a. 05 Bednction. Arg. "Diff. forl day " IVb. 1 4 40.8 4 42.3 4 43.7 3.03 Day of No. of 15 3 46.6 3 47.1 348.6 3.95 the day* M..r. 1 2 60.3 2 C1.8 2 63.2 3.94 month. m. M. m. w> j^ 15 1 65.2 1 63.7 1 58.1 3.94 3.91. 3.92. 3.93. 3.94. 3.96. eiapeeo. Ayr. 1 048.3 49.8 51.2 3.94 15 mr. i 23 49.3 22 46 5 2360.8 22 48 2352.2 22 49 4 8.93 3 93 7 j 21 5L6 21 63! 21 54^4 3.98 or 16 3.9 3.9 3.9 3.9 3.9 Jon* 1 20 44.9 20 46.4 20 47.8 3.98 or 17 7.8 7.8 7.9 7.9 7.9 16 19 60.1 19 61.6 19 63.0 3.91 or 18 1LY 11.8 11.8 11.8 n July 1 18 47.6 18 49.0 18 60.4 3.91 or 19 15.6 16.7 15.7 15.8 16.8 16 17 2.7 1764.2 17 66.6 a. 62 or 20 19.6 19.6 19: 6 19.7 19.7 Aug. 1 16 4S.1 1647.6 16 49. 3.92 or 21 83.6 23.5 83.6 83.6 23.7 15 15 61.3 15 62. 8 1564.2 S.2 or 22 27.4 27.4 27.6 27.6 27.6 Sept. 1 14 44.7 14.46.2 1447.6 3.92 9 or 23 31.3 31.4 31. i . 31.5 31.6 15 13 49. 8 13 51.3 136S.7 3.93 10 or 24 35.2 35. 3 36.4 36.5 S5.-5 Oct 1 12 47.0 12 48.5 1249.9 3.93 11 or 26 39.1 39.2 39.3 39.4 3.6 19 16 11 62.0 11 63.6 11 64.9 3.93 12 or 26 43.0 43.1 43.8 43.3 4B.4 u HOT. 1 10 46.1 10 46.6 10 48.0 3.93 13 or 27 46.9 47.0 47.8 47.2 47.4 IS 15 9 60.1 9 51.6 9 63.0 3.64 14 or 28 60.8 61.0 61.1 61.2 61.3 It Dec. 1 8 47.0 8 48.5 8 49.9 3.94 89 64.7 64.9 66.0 65.2 66.3 14 16 7 61.8 7 63.3 7 64.7 3.95 30 68. t 68.8 68.9 69.1 60.8 15 81 62.6 62.7 62.9 63.0 63.8 It 40 A MANUAL OF LAND SURVEYING. The surveyor should be careful to employ Part II, Table V, correctly. When the table is used in regular order, the "^Reduction" may be taken from Part II with the argument ("Argument," the quantity on which another quantity in a table depends.) "Day of the month "in left hand column, or, " Number of days elapsed " in right hand column, as may be preferred. In example 2, Part II, may be entered in with the argu. ment 13 days elapsed (from 1st to 14th) in right hand column ; then the reduction, 51 m .l, results, as above written; but when working from a, following date (exam- ple 3), the day of the month in left hand column cannot be used. Mistakes are often made by using the wrong column in Part I ; as a matter of course, the time should always be taken out for the current year. Applications of Tables Y and VII. 4. Required the Hour Angle and Azimuth of Polaris, for a station in latitude 46 N., longitude 90 W M at 8 h 24 p. m , November 7, 1910. h. m. Astronomical time of observation, 1910, Nov. 7 8 24.0 Equivalent to time of Nov. 6 3224.0 h. m. A stron. time. U. C. Polaris, Nov. 1 ( Table V, Part I) . . 10 46.6 Reduction to Nov. 6 (Part II), subtract t>19.7 By reference to the above table, the surveyor will observe that the times between Nor. 1 and 15 are greater than 8 h 24m ; consequently, the culmination for one day earlier, Nov. 6, will be used ; sea directions on page 37 ; also, last clause of example 3, page 38. From Part II, Table V, opposite 6th day of month, and under " 3 94 m ." To subtract, take 1 day from Nov. 7, and add its equivalent, 24 h , to 8* 24m, making, Nov. 6, 32& 24 * (which is the time expressed by Nov. 7, 8" 24n) ; then subtract in the usual manner. d See last clause of footnote, page 40. In case the Hour Angle comes out greater than ll h 58", subtract it from 23 b 56. l m ; see example 4, on above. ' The Hour Angle being less tfian ll h 58, the Azimuth is west; tee pr- cepts, top of Table VII. TO FIND A TRUE MERIDIAN. j Astron. time, U. C. Polaris, HOT. 6 10 26.9, subt. 10 28.9 Hour Angle of Polaris, at observation 21 57.1 Subtract from ... *28 56.1 Time Argument for Table VH 1 59.G Azimuth of Polaris, at observation 51' E 5. Required the Hour Angle and Azimuth of Polaris, for a station in Iatitode41 15T N., longitude 94 W., at 6 16- a. m., Nov. 19, 1901. h. m- Astronomical time of observation, 1901, Nov. 18 18 16.O h. m. Astron. time, U. G. Solaris, Nov. 15 (Table V, Part I), 9 46.4 Reduction to Nov. 18 (Part II) subtract 11.8 Aatron. time, U. C. Polaris, Nov. 18 9 34.6, subt. 9 34.8 Hour Angle of Polaris, at observation, and Time Argument for Table VII , 8 41.4 Azimuth of Polaris, at observation (Table VH), 74' or 1 14* W. TABLE VII. This table gives, for various hour angles, expressed in mean solar time, and for even degrees of latitude from 30 to 50 degrees, the Azimuths of Polaris for 11 years, com- puted for average values of the north polar distance of the star the arguments (reference numbers), being the hour angle (or 23 h 56 m .l, minus the hour angle, when the latter exceeds ll h 58 m ), which is termed the Time Argument; and the latitude of the place of observation. The table is so extended that azimuths may be taken out by mere inspection, and all interpolation avoided, except such as can be performed mentally. Tbe vertical diameter SS', Plato I, Fig. 2, divides the apparent path of Polaris into two equal parts, and for the star at any point s 6 on the east side, there is a corresponding point s lf on the uxst side of the meridian, for which the azimuth Nw is equal to the azimuth Ne. The arc SSj.S's,, taken from the entire circle (or 23* 56.l), leaves the arc Ss 6 , and its equal, Ss,, expressed in time, may be used to find, from Table VII, the azimuth Nw, which is equal to Ne. The hour angles entered in Table VII include only thoe of the west half 42 A MANUAL OF LAND SURVEYING. of the circle ending at S', and when an hour angle greater than lib 58 m re suits from observation, it will be subtracted from 23h 56.!, and the re- mainder will be used as the " time argument " for the table. The surveyor should not confound these two quantities. The hour angle itself always decides the direction of the azimuth and defines the place of the star with reference to the pole and meridian, as noted at top of Table VII. See ex- amples following Table V. The hours of the " time arguments " are placed in the columns headed "Hours," on left. of each page The minutes of the time arguments will be found in the col- umns marked " m.," under the years for which they are computed, arid they are included between the same heavy zigzag lines which inclose the hours to which they belong. The time arguments are given to the nearest half minute ; the occurrence of a period after the minutes of any one of them, indicates that its value is 0.5 greater than printed, the table being so arranged to economize space. The tables will be used as follows: Find the hours of the time argument in the left-hand column of either page ; then, between the heavy lines which inclose the hours, find the minutes in the column marked at the top with the current year. On the same horizontal line with the minutes, the azimuth will be found under the given latitude, which is marked at the top of the right- hand half of each page. Thus, for 1904, time argument, O h 43 m , latitude 36 ; find O h on left-hand page and under 1904, find43 m on ninth line from the top, and on same, line with the minutes, under latitude 36, is the azimuth 17. For 1908, time argument. 9. 33i m , lat. 48, the azi- muth is 1 H', found on the 21st line from top of right- hand page. If the exact time argument is not found in the table, the azimuth should be proportioned to the difference between the given and tabular values of said argument- The table has been arranged to give the azimuths, by simple inspection. No written arithmetical work is re- quired, all being performed mentally. It will always be TO FIND A TRUE MERIDIAN. 43 sufficient to take the nearest whole degree of latitude, and use it as above directed except for a few values near the top of either page, where the difference of azimuths, for 2 difference of latitude, amounts to 4 or 5 minutes of arc. The attention of the surveyor is directed to the fact that he should always use one day of twenty-four hours as the unit when he subtracts the time of culmination from the time of observation. See example 4, page 40 . In any case when the time or upper culmination, taken from Table V, for the given date, would be numerically greater than the astronomical time of observation, the former time will be taken out for a date one day earlier than the date of observation. The surveyor will decide when such condition exists by comparing the time given in the table with his astronomical time of observation. See example 4 and explanations following Table V, page 39. The watch time to be used when making observations on Polaris at all times except elongation should be as accurate as can be obtained. Looking at Table VII, near the top of the page, the surveyor will observe, that for a difference of four minutes in the time argument, there is a change of about two minut es in azimuth ; conse- quently, to obtain the azimuth to tue nearest whole minute of arc, the local mean time, upon which all depends, should be known within two minutes. When the surveyor uses a solar instrument, he can readily determine the time for himself during the afternoon before observing Polaris, or in the morning after observation, and, without mov- ing the hands of his watch, apply the necessary correc- tion to his observed watch time. When the surveyor uses standard railroad time, he will correct the same for the difference of longitude between his station and the standard meridian for which the time is given, at the rate of four minutes of time for each degree of the differ- ence in arc. Thus, if the difference of longitude is 6 45', the equivalent in time will be 27 minutes. The difference of longitude may be taken from a good .map. TABLE VJL Azimuth of Polarit [The hour Angles are expressed in man tolar time. The occurrence of a period after STAR AND AZIMUTH. W. of N. when hour angle is leu than ll b 53=. E. ot N. when hour angle is ffreoKr than 11" 68. Time argument, the star's hour angle (or 23' 56" .1 minus the star's hour angle), for the year POLARIS abote THI POLL To determine the true meridian, the azimuth will be laid off to the ut when the hour angle is leu than 11* 68-, and to the wut when greater thai) 11* 68". I 1 8 I i i i i i s Azimuths (or latitude 05 30 82 84 86 88 40 42 44 40 48 60 ' H 19 23. 28. 33 88 42. 47. 57 14 19 23 33 38 42. 47. 52. W. 14 19 23. K! 38 43 48 52. 57. 14. 19 21 23. 33. 38. 43 48 53 6S 14. 19 24 29 33. 38. 43. 48 53 68 14. 19 24 29 M 38. 43. 48. 53. 58 14. 19. 24 29 34 39 44 48. 53. 58. 14. 19. 24. 28 M 39 44 49 M 59 14. 19. 24. 2U. 34. 39 44 49 64 M 14. 19 21 29 34. 44. 49 54 59 15 19 24 !iS M 39 44 49 64 T 10. 12. 14 16 17. 19. 21. 23 25 27 28. 30. 32 35] 37 39 40. 11 12 14 16 18 20 21 23 25 27 29 31 32. 34. 36 38 39. 41. 43 45 47 48. 60. 62 64 66 67. 59. 11 13 14. 16. 18. 20. 22 24 26 27. 29. 31. 33 35 37 38. 40. 42. 44 46 48 46. 61. 63. 66. 67 * 63 64. 66. 68.' 70 72 74 78 77. 79. 81. 83 86 87 11. 13 15 17 19 21 22. 24. 26. 28. 30 32 34 36 38 39. 41. 43. 46. 47 49 61 53 65 66. 58. !-6?T 62. 64. 66 68 70 72 74 76. 77. 79. 81. 83. 86 87 89 11. 13. 16. 17. 19. 21. 23. 26 27 29 31 S3 35 37 39 40. 42. 44. 46. 48. 60. 62. 64. 66. 68. W 62 64 66 68 70 72 74 76 77. 79. 81. 83. 85. 87. 89. 91. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 48 48 60 62 64 66 58 r*o- 62 64 66 68 70 72 74 78 78 80 82 84 86 88 90 92 94 10. 12. 14. 16. 18 20. 22. 26 27 29 31 33 35 37 39. 10. 13 15 17 19. 21. 23. 26 28 30 32 34. 36. 38. 41 11 13. 15. 18 20 22. 24. 27 29 31. 33. 36 38 40 42. 11. 14 16 18. 21 23 25. 28 30 32. 35 37. 39. 42 44 12 14. 17 19 21. 24 26. 29 31. 34 36. 39 41. 43. 46 6 10 ]5 20 26 31 36 42 47. 53 58. 12 17 22 27 82 37 42. 47. 53 58. 12 17 22 27 32. 37. 43 48 53. 59 12. 17. 22. 27. 33 38 43. 48. 64 13 18 23 28 33. 38. 44 49 54. 13 18 23. 28. 33. 39 44. 50 55 13. 18. 23. 29 34 38. 46 50. 55. 14 19 24 29. 34. 40 45. 51 56. 14 19. 24. 35 40. 46 61. 67 14. 19. 25 30 35. 41 M, 62 57. 10 10 M 30. 36 41. 47 62. 68 43. 45. 47. 49. 61. 53. Z 60 62 64 66 68 70 72 74. 76. 78. 80. 82. 84. 86. 88. po. 93 95 97 46 47 49 51. 63. 65. 57. To" 62 64 66 68. 70. 72. 74. 77 79 81 83 85. 87. 89. 91. 94 96 98 00 46. 49 61 63. 65. 57. reT 62 64. 66. 68. 71 73 76 77. 79. 82 84 86 88. 90. 92. 95 97 99. 01. 03. 48. 61 63 58. 57.| w 62 64. 66. 69 71. 73. 76 78 80. 82. 85 87 89. 91. 94 96 98. 01 03 06. "7. 50. 63 65 57. w 62. 64. 67 69. 72 74. 76. 79 81. 84 86 88. 91 08. 95. 3 00. 03 06 07. 1C 12 59. 6 11. 17 23 29 35 41. 48 54. 0. 6. 12 18 24 30 36 42. 49 55. 1. 7, 12. 18. 24. 30. 37 43. 50 56. 2 7. 13. 19> 25. 31. 38 44, 61 67. 2. 8. 14 20 26 32. 38. 46 62 68. 3 9 15 21 27 33 39. 46 63 4 9. 15. 21. 28 34 40. 47 64 4. 10. 16. 22. 28. 35 41. 48 55 42. 44 46 47. 49. 61 63 54. 66. 9 16 21 27 33 39 45 61. 68 10 16 21. 27. 33. 40 46 52. 10. 16. 22. 28. 84. 40. 47 53. 1 1 59 ,0. 7. 14. 22 30 38. 1. 8. 16 23. 31. 39. 2. 10 17 25 33 41 3. 11 18. 26 34. 43 52 60 61. 63. 66 67 69 70. 72. 74 76 77. 79. 81 83 61. 63 65 66. 68. 70. 72 74 76 77. 79. 81. 83 86 1 5. 12. 19. 27. 35. c 13. 21 28. 37 12 19. 27. 35. 44. 63. 13 21 29 37 46 55 14. 22 30. 38. 47. 67 15. 23. 31. 40. 49. 69 17 26 33 42 51 -?r 11. 23. 37. 53. if 24. 38 M is' 27 40. 0. 17. 29. 43. 58 s 19. 32 46. 10 22 34. 60 12. 24 37. 63. 14. 26. 40. 16. 29 43. 19 32 46. 21 34. 60 32 6. 11 16. 1. 29. 5 lo 10 iiU. 42.J T3 !ii for the UM of land surveyors. miante* of time or of an hour angle Indicate* that its value U 0".5 greater than printed.] STAB ASD AZIMUTH. W. of X. when boar angle i* lem than n> 58. I. of N. when hoor angle i* grtaltr tban Ilk *g. Time argument, the t*r' boor angle {or 23* 56 nuiuu tbe itar'i boor angle), for the rear J POLABU MOV TBS POLB. To determine the true meridian, the azimuth will be laid off to the tax when tbe boar angle to leo than Ilk 58, and to the we* wbo ?rK*r than Ilk 6S-. i i N I i i i | i I 2 Azimuth* for latitude" a SO 82 84 84 S3 40 42 44 46 4S cc \ m. 34 56 m. 28 62. m. 20. 48. m. 9. 45 34 83 a. 79. 78 76 74. 72. 71 97 95 93 100 98 &6 103 101 &s 47 &5 107 1C5 1C3 100 5 ? 1U 105 107 104. 102 100 40. 34 56 27 62. 18 48. 8 44 MM 39 83 el 79 87 t5 83 81. 89 87 85 91 39 87 94 9-2 H 26 37. 48. 23. 35. 46. .VI 21 33. 44. M 18. 31. 42. M I 16 29 40. 51 Is 3i>. 49. 69. 10 24 36. 47. 67. 7 21. 34. 45. T 19 32 43. 54 0. 16 2-i. 41. 52. 57 13 27 3S. 91 76 74 72 70 f~' 77. 76 M 72 70. 79 77 76 74 72 70 II 77 M H s 8 16 24 32 6 14. 23 31 13 21. 29. \l. 20 28 c2 *5 91 95 92. 90 83 85. 10 18. 27 e. 17. 25. 7 16 24. 6. 14. 23 i 13 21. 2. 11. 2. 1 10 H 67. 65. 74 72 76 74 M n M S7 84 64 63 59. 62 58. 60. 57. 49. 56. 48. 56. 47. 64. 46. 53. 45. 52. 44. 51. 43 50. SO 63 64 62 60 TT 64 62 |'. 64 66 H 72 70 6? 66 75 73 70 7o 75 73 71 80. 78 76 73. 0. 58. 55. 68 56 61 4r 57. 13. 20 26 32 38 13 19 25. 31. 37. 12 18. 24. 31 37 11 17. 24 30 36 10. 17 23 29. 35. 0. 16 22. 28. 35 8. 15 21. 28 34 7. 14. 20. 27 33. 6. 13. 20 26. 32. 5. 12. 1 25. 32 18 24. 31 5\. 43. 52. 60. 53. 52 55 53 56. 64. 50 TT 56 V 53 11 59 57 55 53 51 4? 62 "rr 57 M 64 61 47 57 If 52 66. 64 111 h~ 67 M. 55. 56 64. 54 59. 53 59 52. 58. 62 57. 61. 57 51 66. 50 56 49. 55. 42. 41 43. 44. 45 46. 49 4:- 47 * 1 0. 12 17. 23 28 33. 38. 43. 11. 17 22. 27. 38 43. 11 18. 22 27. 32, 38 43 10. 16 21. 10 15. 21 15 20. 14. 20 14 19. 13. 19. 13 19 12. Is. 35. 34 36 34. 37 35 38 36 37 41 M 42. 40. 44 42 45 43 47. 45 32 37. 42. 32 37 42. 31. 36. 42 31 36. 41. 30. 36 41. 30 35. 41 30 35 29. 35 40 30 31 31. 32 33 34 32 30 35 33 31 36 34 32 30 37 35. 33 31 3* 34 32 40 38 36. S3 SI 28. 26 23. 21. 19 16. 14 12 26. 27 2S 2s. 29 54 59 53. 5K. 53 58. 53 58 52. 5S 52. 57. 51. 67 51 56. 51 56 23 21. 19. 18 16 14 12. 10. 9 7 5. 3. 2 22 20 18 16. 14. 12. 11 9 I 5. 3. 2 22 20. 18. 13. 15' 13 11 9 7. 5. 3. 2 22. 21 19 17 15 13. 11. 9. 7. A. 4 2 23 21. 19. 17. 15. 13. 11. '24 22 20 18 16 14 12 M 25 22. s IS. 16. 14. 12. 10. 25. 23. 21. 19 17 15 12. 10. m. 24 2-2 17. 15. 13 11 87. 25 23 20. IS 15 13. 11. 9 14 19 24 28. 33. 38 43. 48. 53 58 8. 13. 18. 23. 28. S3. 38. 43. 48. 53 58 8. 13. 18. 23. 28. 33. 38. 43. 48 53 53 13. 18. 23. 29. 33. 38. 43. 48 53 68 13 18 3 13 1* 29 12. 18 23 12. 17. n. 12. 17. 12 17. 22. 12 17 22 33 38 43 4* 53 5s 33 38 43 48 63 J3 33 3* 43 48 53 15 33 38 43 48 53 U 33 38 43 48 53 Li 32. 3e 43 4? 63 32. 37 43 4? 53 Co 6 4 2 6 6. 6. 7 7 2 2 2 1 1 r 1 r ABLE XII. Convergence/ of Meridians six miles long ana miles apart, and other relevant data, to latitude 70" north. Lat- itude. Convergency. Difference of longi- tude per range. Difference of latitude for On the parallel. Angle. In arc. In time. 1 mile in arc. 1 Tp. in arc. Links. / // ' U, Seconds. - 80 41.9 3 6 0.36 24.02 31 43.6 3. 7 6 4.02 24.27 32 45.4 815 6 7,93 24.53 / i871 5'. 225 33;i 47.2 323 6 12.00 24.80 34 49.1 330 6 16.31 25,09 35 50.9 388 620,95 25.40 36 52.7 346 625.60 25.71 37 54.7 355 6 30,59 26.04 0'.87/> 5'. 221 38 56.8 4 4 6 35.81 26.39 39 58.8 413 6 41.34 26.76 40 60.9 422 6 47.13 27.14 41 63.1 4 31 653.22 27.55 42 65.4 441 6 59. 62 27.97 0'.869 5?. 217 43 67.7 451, 7 6.27 28.42 44 70.1 5 1 7 13.44 28.90 45 72.6 5 12 7 20.93 29.39 46 75.2 523 7 28.81 29.92 47 77.8 534 7 37.10 30.47 0',869 5'. 212 48 80.6 546 7 45.79 31.05 49 83.5 569 7 55.12 31.67 60 86.4 6 12 8 4.83 32.32 51 89.6 6 25 * 15. 17 33.01 52 92.8 6 39 8 26. 13 33.74 0'.868 5'. 207 53 96.2 6 54 8 37. 75 34.52 64 99.8 7 9 8 50.07 35.34 55 103.5 725 9 3.18 36.22 56 107.5 7 42 9 17. 12 37.14 67 111.6 8 9 31.97 38.13 0'.867 5', 202 58 116.0 819 9 47.83 39.19 59 1216 8 38 10 4.78 40.32 60 125.6 859 10 22. 94 41.52 61 130.8 9 22 10 42.42 42.83 . 62 136.3 946 11 3.38 44. 22 ova* 6'4fl8 63 142.2 10 11 11 25. 97 45.73 64 148.0 1038 U 50.37 47.36 65 155.0 11 8 12 16.82 49.12 66 162.8 11 39 12 45.55 51.04 67 170.7 12 13 13 16.88 53.12 o'.see 5U95 68 179.3 12 51 13 51.15 55.'41 69 .188.7 13 31 1428.77 57.90 TO 199.1 14 15 1510.26 60,68 P. 866 6M98 TO FIND A TRUE MERIDIAfc. 47 The number of seconds taken from the 5th column of Table XII (opposite the proper latitude), multiplied by the number of ranges, will give the correction for .longitude in seconds of time. The correction will be subtracted from the standard railroad time of observa- tion, when the surveyor's station is west, or added when east of the standard meridian, as the case may require, to obtain local time. It is immaterial where the sur- veyor obtains, the standard time, provided he gets it right, a result which will be determined in the most satisfactory manner by a direct personal comparison at a telegraph office. Table VII enables the surveyor to obtain the hour angle and azimuth of Polaris at any hour and minute from 1901 to 1911 inclusive, in latitudes 30 to 50, thus combining in two pages the essentials which under ordinary methods would require twenty. Mr. A. W. Barber, expert examiner of surveys for the U. S. Land Department, furnishes the following method, which is accurate, and applicable anywhere. It is valu- able for use in Alaska and other high latitudes, where the previous methods are difficult to apply, and liable to many serious errors: " THE METHOD BY EQUAL ALTITUDES depends on the fact that the circumpolar stars describe invariable circular arcs below the true pole; and that when a peg is set on the ground, to mark the course to a certain star when west of north at an altitude of say 35, and another peg is afterward set to mark the posi- tion of that star, when it has passed to the northeast and risen to the same precise elevation, the meridian of the transit will lie exactly midway between the pegs. It may be used by any careful observer having a good common transit with vertical arc. This process requires no reflecting eye-piece; no full vertical circle; no mathematical tables; no calculations of local mean time, standard time, sidereal time, or as- tronomical and civil day. It does not depend on the date nor the hour of the day, nor any reckoning of cul- mination, hour-angle, elongatio'n, or polar azimuth. Neither does the sun's declination or the atmospheric refraction enter at all into the calculation. Should there be unsuspected error in the graduation or setting of the vertical arc, or some defect of collima- 48 A MANUAL OF LAND SURVEYING, tion in the telescope, it would equally affect both parts of the observer's work, and produce no effect in the re- sulting meridian. Proceed as follows in any latitude or time of the year when the stars are sufficiently discern- ible: 1. Choose a suitable station, set transit firmly, and level precisely, by the telescope level turned in all directions. 2. Set the index arm for vertical angles at zero, and keep it tightly clamped for reading elevation angles. 3. Provide suitable illumination for cross-wires, and also for reading angles, horizontal and vertical. 4. Select a conspicuous star, perhaps 30 or 40 from the pole, which appears two hours more or less before its lower culmination, that is, which stands west of north and is rapidly descending. Identify this star be- yond all chance of error, noting it on a diagram for cer- tainty some hours later. 5. Direct the telescope to this point, fixing the star at the intersection of the cross-wires, and clamp the axis so it will retain the altitude shown on the arc. Read and note down the angle of elevation, and read more than once. 6. Unclamp the axis, bring the telescope to the earth, and have an assistant drive a peg in line with the cross- wire, from 3 to 5 chains distant. A candle held there before a white surface, will exhibit the wire and give the exact point for a tack. 7. Repeat the observation once or twice, at intervals of 10 or 15 minutes, for confirmation of results, marking successive pegs A, B, etc., with degrees and minutes of elevation found. 8. Be ready to observe the upward path of the same star, after it has passed east below the pole. Correct the leveling, set the vertical index successively at each altitude previously noted (beginning with the lowest); and when the star (diagonally ascending in the field of the glass) approaches the horizontal wire, bring the ver- tical wire also upon the star at the intersection, using the slow-motion screw of the horizontal plate. 9. Keep the plate at that point, bring the telescope down, and set peg in line as before. Repeat the process for each observation A, B, etc., taken before midnight, marking each peg B, A, etc., with the elevation in figures. 10. Measure the arc between pegs A A denoting equal altitude, and take one half. Lay off this half from either peg, and set a peg and tack for the true meridian. As a test of correctness, the middle point between pegs TO FIND A TEUE MERIDIAN. 49 B B, and between C C should be found to coincide with the one first found. A single pair is sufficient, except for confirmation. BY EQUAL ALTITUDES OF THE SUN. In this operation a reflecting eye-piece with dark glass will be necessary. The sun's large image in the field can not be centered as truly as a star. It is there- fore found best to place the intersection of the wires at the lower limb (apparently the sun's upper edge, as re- versed by the mirror), and at the precise point of tan- gency, when the sun is just leaving the horizontal wire, apparently descending, with the vertical wire bisecting its disk. This is convenient for the forenoon observations, hence at the corresponding times after noon, with proper altitude of telescope, one must be ready at the moment the sun (now apparently ascending) first reaches the level wire, having the vertical one bisect the sun by the point of tangency, as before. The pair of pegs in this case will be set southeast and southwest of the station. The center of bisecting line of the included arc would be the meridian, were it not for the sun's change of declination in the intervening time. This slight change requires a calculation for cor- rection, which the star process avoids. WORKING BY A REFERENCE MARK. Instead of using pegs and tacks for day-work it is easier to use a reference point or mark. On April 17, after careful leveling, I set the hori- zontal plates at zero with the telescope directed at a distant spire for my mark. Clamp the lower plate fast, and direct the telescope to the sun, observing it as above shown. Find by the horizontal angle that the sun's azi- muth to the left or east of the spire is 30 27'; and by the vertical arc I find his altitude 47 09'. (The semi- diameter may be disregarded in each pair of observa- tions, if the same limb of the sun is used each time.) For the corresponding afternoon observation, I have the index of altitude fixed at 47 09', and watch the sun rise (apparently) to the proper position. At the right moment, clamp the plate, use the slow-motion, and when the disk is in position, find from the horizontal plate that the sun is 84 49' west of the spire or mark. The whole arc is 30 27' 4- 84 49' = 115 16'; and the bisecting meridian is 57 38' from either position of the sun. From this one-half, I subtract the first azimuth of the sun from the mark; 57 38' 30 27' = 27 11' as 60 A MANUAL OF LAND SURVEYING. the true bearing of the mark from the uncorrected meridian, and it apparently bears S. 27 11' E. from the transit. THE CORRECTION FOR DECLINATION at or near the times of the solstices, will be merely theoretical, as an hourly difference of declination less than 10 seconds will be quite negligible. ' But during the rest of the year it should be ascertained; for it would amount to as much as a change of 10', were observations taken six hours apart on September 25, in latitude 65. To calculate this correction: Take one half the change in declination between observations at equal alti- tude; divide these minutes of change by the product of the cosine of the latitude by the sine of half the differ- ence in time expressed in degrees (15 per hour); the quotient will be the minutes of arc for the correction. This is to be applied from south to west from June 21 to December 21 (declination decreasing) and from south to east the rest of the year. EXAMPLE. On April 17, in latitude 39, using stand- ard watch time; second and third pairs of observations, the first pair being already noted. B. 10: 01 A. M. Altitude 49 36' Azimuth E. from mark 26 07' C. 10: 16 A. M. Altitude 51 48' Azimuth E. from mark 21 38' C. 1: 59 P. M. Altitude 51 48' Azimuth W. from mark 76 00' B. 2: 14 P. M. Altitude 49 36' Azimuth W. from mark 80 29' The sum of the measured arcs of the B B positions, 26 07' + 80 29' = 106 36'. The middle point for me- ridian is at 53 18'. As the reference mark is 26 07' from the forenoon sun, its arc from the south meridian point must be 53 18' 26 07', and its course from the transit is again found S. 27 11' E. (Uncorrected.) The third pair, C C, gives 21 38' + 76 00' = 97 38'. One half of this, or 48 49', less 21 38', gives the same resulting arc, 27 11'. CORRECTION FOR CHANGE OF DECLINATION One half of the change on that day in 4% hours was 1.9'. The cosine of the latitude 39 is .78; the sine of half the difference in time (2y 8 hours = about 32) is .53; their product is .78 X .53 = .41. Dividing 1.9' by .41, the quotient is 4y 2 minutes of arc, for correction of the south meridian point eastward. This gives the bearing of the spire or mark from the true meridian, S. 27 06' 30" E. THE TRANSIT. 51 FIG 6. 52 A MANUAL OF LAND SURVEYING. VII. The essential parts of the Transit, as shown in the cut, are the telescope with its axis and two supports, the circular plates with their attachments, the sockets upon which the plates revolve, the leveling head, and the tripod on which the whole instrument stands. The telescope is from ten to eleven inches long, firmly se- cured to an axis having its bearings nicely fitted in the standards, and thus enabling the telescope to be moved in either direction, or turned com- pletely around if desired. The different parts of the telescope are shown in Tig. 7. The object-glass, composed of two lenses, so as to show objects without color or dis- tortion, is placed at the end of a slide having two bearings, one at the end of the outer tube, the other in the ring CC, 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 objects either near or remote as de- sired. The eye-piece is made up of four piano convex lenses, which, beginning at the eye- end, are called respectively the THE TRANSIT, 53 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. The eye-piece is brought to its proper focus usually by turning 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 em- ployed for the same purpose. 1. The Cross -Wires, (Fig. 8 ), are two fibres of spider-web or very fine plat- inum 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 center into quadrants. 2. Optical Axis. The intersection of the wires FIG. 8 forms a very minute point, which, when they 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 operation of bringing the intersection of the wires into the optical axis is called Adjusting: the Line of Ool- limation. This will be hereafter described. 3. The Vertical Circle firmly secured to the axis of the telescope is 4J inches diameter, plated with silver, divided to half degrees, and with its vernier enables the surveyor to obtain vertical angles to single minutes. 4. 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. 54 A MANUAL OF LAND SURVEYING. 5. The Magnetic Needle is four to five inches long in the different sizes of transits, its brass cup having in- serted in it a little socket or center of hardened steel, perfectly polished, and this resting upon the hardened and polished point of the center-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 has a 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 vibra- tions of the needle, or bring it up against the glass when not in use, to avoid the unnecessary wear of the pivot. 6. The Lower Plate, called the Limb, is divided on its upper surface usually into degrees and half -degrees and figured in two rows, Viz., from to 360, and from to 90 each way; sometimes but a single series is used, and then the figures run from to 360 or from to 180 on each side. 7. The Verniers, of which there are two placed op- posite each other against the limb, are auxiliary scales used in measuring smaller portions of the limb than are shown by its graduations. Thirty divisions on the ver- nier correspond precisely with twenty-nine half degrees on the limb. Hence one division on the limb exceeds one division on the vernier by one-thirtieth of one-half of a a degree, that is, by one minute. Accordingly, the number of any division of the vernier, on the side toward which the vernier is moved, which co- incides with a division of the limb is tho number of minutes of arc intercepted by the zero of the vernier and the last preceding division of the limb. Thus, by the device of a vernier we are enabled to measure angles to within one minute although the limb of the transit is graduated only to half-degrees. THE TRANSIT. 55 Adjustments. The principal adjustments of the Tran- sit are (1) The Levels. (2) The Line of Collimation. (3) The Standards. 8. To Adjust the Levels. Set up the instrument upon its tripod as nearly level as may be, and having un- damped the plates, bring the two levels above and on a line with the two pairs of leveling 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 leveling screws, and repeat the operation until the bubbles will remain in the center during a complete rev- olution of the instrument, and the adjustment will be correct. 9. To Adjust the Line of Collimation. To make this adjustment which is, in other words, to bring the intersection of the wires into the optical axis of the tel- escope, so that the Instrument, when placed in the middle of a straight line, will, by the revolution o'f the telescope, cut its extremities proceed as follows: Set the instrument firmly on the ground and level it carefully; und 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 two hundred to five hundred 56 A MANUAL OF LAND SURVEYING. feet; determine if the vertical wire is plumb, by clamping the instrument firmly and applying the wire to the verti- cal edge of a building, or observing if it will move par- allel to a point taken a little to one side; should any dev- iation be manifested, loosen the cross-wire screws, and by the pressure of hand on the head outside the tube, move the ring around until the error is corrected. 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 direction, and at about the same distance from the instrument as the first object assumed. Great care should always be taken in turning the teles- cope, 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 ver- tical 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 ver- tical 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 Iine If not, however, the space which separate the wires from the second point observed^will be double the devia- tion 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 C -B FIG. 9. two observations, made with the wires when they are out of the optical axis of the telescope. THE TRANSIT. 57 For, as in the diagram, let A represent the centre of the instrument, and BC the imaginary straight line, upon the extremities of which the line of collimation is to be ad- justed. 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 S, and once more revolved, the wires will 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 B, i evolve the telescope, and if the correction has. been care- luily made, the wires will now bisect a point, C situated midway between D and E, and in the prolongation of the imaginary line, passing through the point B and the cen- tre of the instrument. To ascertain if such is the case, turn the instrument half around, fix the telescope upon B, clamp to the spin- dle, and again revolve the telescope toward C. If the wires again bisect it, it will prove that they are in adjust- ment, and that the points, B, A, C, all lie in the same straight line. Should the vertical wire strike to one side of C, the error must be corrected precisely as above described, until it is entirelv removed. 58 A MANUAL OF LAND SURVEYING. 10. 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 observa- tion, such as 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 standard opposite that side was the highest, the ap- parent error being double that actually due to this cause. To correct it, one of the bearings of the axis is made movable, so that by turning a screw underneath the slid- ing-piece, as well as the screws which hold on the cap of the standard, the adjustment is made with the utmost precision. 11. To Adjust the Vertical Circle. Having the in- strument firmly set up and carefully leveled, bring into line the zeros of the circle and vernier, and with the tel- escope find or place some well-defined point or line, from one hundred to five hundred feet distant, which is cut by the horizontal wire. Turn the instrument half way around, revolve the tel- escope, and fixing the wire upon the same point as before, note if the zeros are again in line. 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 THE TRANSIT. 5 r precisely as at first, until the error is entirely corrected when the adjustment will be complete. It is not always convenient to make this adjustment so as entirely to eliminate the index error. In this case, the error should be noted and the proper correction made in measuring a vertical angle. To find the index error we have the following RULE, Level the instrument and direct the telescope upon some well defined spot. Note the reading of the circle. Reverse the telescope and turn the vernier plate 180. Direct the telescope upon the point and note the reading of the circle. Subtract the first reading from the second, ami divide the remainder by 2. 12. To Run a Line with the Transit, 1. Setting up the Transit. Set the instrument up over the starting point, centreing it by means of the plumb line. While doing so, place it as nearly level as possible, leaving as little as may be, to be done in leveling up the plates by the leveling screws. There is opportunity for the display of a good deal of skill in setting up a transit over a point, quickly, and in proper position. For hill sides, a tripod having adjustable legs, called an extension tripod, is a great convenience. When the legs are not adjustable, set one leg of the tripod down hill and two legs on the upper side of the line. It is important that the instrument should stand firmly on the ground. Some soils are so yielding that it is impossible for the man at the transit to change the weight of his body from one foot to the other, without getting the transit out of posi- tion. One remedy is, to not change the centre of grav- ity of the person, after the transit is in position, until the observation is taken. Another is, to drive stout stakes into the ground, to set the transit legs on. An- other is to make a bridge of planks or poles' for the transit- man to stand on, so as to carry the bearing of his weight GO A MANUAL OF LAND SURVEYING. as far as possible away from the instrument. Sometimes the aid of an assistant will need to be called in, so that the transitman need not move around the transit before sighting. When the transit is set up firmly in place, loosen the lower clamp and turn the instrument on the spindle till the level tubes are each parallel to an opposite paw of the leveling screws. Turn the parallel pair of screws both inward or out- ward until the bubble comes to the centre. Each level being treated in this way, the limb of the instrument Is caused to be parallel to the horizon. Unclamp the vernier plate and set the zero of the ver- nier to -coincide with the zero of the limb. Clamp the plates in this adjustment. The leveling screws should be kept bearing equally against the plates. Do not turn the leveling screws up too tightly. It tends to spring the plate and causes unnecessary wear of the screw threads. Simply bring them to a firm bearing. 2. Assistants and their Duties. The Rod- man. A rodman, often called a flagman, using a rod called a color pole, and one or more axemen are needed. The color pole is often carried by the head chainman. The man who carries the color pole, selects places to set up the instrument, and gets the transit points, is a very important factor in running a line. Nearly as much depends upon him for accuracy and speed as upon the transitman. He should be thoroughly drilled in his duty. He should hold the color pole perpendicularly, clasping it lightly between the thumb and forefinger of both hands, and the hands held above the head. The point should be lifted a little above the ground or hub. He must keep it squarely in front of him, and move his body the same* distance that he does the color pole, when getting a point. As soon as the " All Right" signal is given, let go of the pole. It will fall vertically and make the point plain. U the pole is held to one side it is apt to nave some THE TRANSIT. 61 uneven pressure given which will make it incline more or less. A man cannot stand awkwardly and hold a color pole accurately. He must be able to judge of the stability of the ground to set up on. He must select places where the longest sights can be had, and in running through timbered country he should select transit points where the ground begins to ascend or descend. If any deep ravines or gullies are to be crossed, he must select points- to get across them with the least possible chopping, and without having to set up on a steep hillside. He should not select a point on the shaded side of a big tree, but where the most light comes in through the leaves. A small limb cut out of the way will often let in a wonder- ful amount of light, or a white handkerchief spread over the chest, or a light colored straw hat held in the right position, sometimes reflects enough light to show clearly objects which before were indistinct. In fact, he must be a man of gumption and equal to any emergency. But he cannot do good work unless he is provided with a good color pole. 3. The Color Pole. It should be made from a good piece of straight grained timber. White or Norway pine is good. It is fitted at the bottom with a shoe made from gas pipe, with a steel point welded on, and finished by turning down in a machine. The shoe ought to be of sufficient weight to bring the centre of gravity within two feet of the bottom, so that it will have a greater tendency to hang vertically when held up. The sizes of color poles vary according to the places where they are used. If one is dressed down with planes to a six or eight-sided stick, tapering slightly toward the top, it will keep straight much longer than a stick turned in a lathe. The shoe should be made of sufficient size to receive the stick, without dressing it down to go into the socket. When finished it should be thoroughly tested, to see if the point of the shoe has been set in line with the A MANUAL OF LAND SURVEYING. centre of the pole. Suspend a plumb bob from a point in a ceiling, and mark on the floor the point carried down. Fasten a string in the centre of the top of the color pole and suspend it from the same point. If the point of the shoe covers the mark on the floor it is all right. Prying with a color pole should be prohibited. 4. Axeman. The axemen provide pickets for back- sights, clear the line of brush and trees, and drive stakes and hubs for transit points. They should keep close to the line, so that in clearing through woods they do no unnecessary cutting. A clear line two feet wide through the brush is generally all that is needed. Hubs for transit points should be cut square on top and driven firmly into the earth, nearly level with the surface. 5. Projecting the Line. The flagman selects the point and, facing the transitman, holds the color pole directly in front of him, and guided by the transitman, places it in line and makes a mark in the ground. The axeman then drives a hub at the place and the. rodman again holds up his pole and finds the exact point where the line crosses the hub and a tack is driven. Foremost surveys a line within the limits of a tack head is consid ered close enough. The hub for transit point should not be driven near a large tree, in soft ground, as a breeze will cause the tree to sway so as to move the earth for many feet around it. For a backsight it is a good plan to set up a picket, pointed at the top, so that the point shall coincide with the hole in the eye piece of the telescope. Or it may be set far enough from the transit so that the point may be aligned by the instrument. The picket should be set so firmly in the ground that it will retain its place as long as it is needed. A root will sometimes so press against a picket as to throw the point out of line after it is set. It may be necessary to drive the picket with the axe and then insert a wooden point in a cleft in the top of the picket. Several such points set up in the line before the transit is moved help to secure accuracy in the line. THE TRANSIT. 63 When the backsight is set, the transit is taken forward and set up over the tack point in the hub. The lower clamp is loosened, the telescope reversed and sighted to the backsight and the instrument clamped in that posi- tion. The telescope is then righted and the line contin- ued to the next tack point. When two or more backsight points are visible at once, any error in the adjustment of the instrument or in running the line will be readilv detected, and the proper correction may be applied. If the line of collimation is out of adjustment and it is . not desirable to stop and adjust it, the lower clamp is loosened, the instrument turned half way round and clamped on the backsight. The telescope is then reversed on its axis and a second point marked beside the first. (See Fig. 9.) A tack is thea driven in the true line, which is midway between the two. If the instrument is much out of adjustment it may be necessary to drive three hubs for this purpose. The transit is then set up in the true line, and the line continued as far as necessary, in the same manner. Obstacles in line are passed by offsets to parallel lines, in the same manner as when running lines by pickets or compass. Other methods will be con- sidered in connection with Angular Measurements. Examples, to be solved by the student in the field: 1. Run a line half a mile and mark four or more points along the line with hubs and tacks. 2. Retrace it in the opposite direction, testing the points to see how they agree. 3. Run a line over a hill, marking points at the top and bottom and along the slopes. 4. Retrace it in the opposite direction, testing the points. 5. Run a line across a valley, marking points, and re- trace it in the opposite direction, testing tiie points. 64 A MANUAL OF LAND SURVEYING. CHAPTEE III. DESCRIPTION OF INSTRUMENTS, CONTINUED. FIG. 10. THE SOLAR COMPASS. THE SOLAR COMPASS 65 1. The Solar Compass is an instrument for utilizing the sun's rays to determine a true meridian which has for a number of years been in use in the sur- veys 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, except that the sight vares are attached to the under plate or limb, and this revolves around the upper or vernier plate on which the solar ap- paratus 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. 2. 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. 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. 3. The Latitude Arc, a, has its centre of motion in two pivots, one of which is seen at d, the other is con- cealed in the cut. It is moved either up or down within a hollow arc, seen in the cut, by a tangent- screw at f, 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 thirty-five degrees, so as to be adjustable to the latitude of any place in the United States. 4. The Declination Arc, 6, is also graduated to quarter degrees, and has a range of about twenty-eight degrees. 60 A MANUAL OF LAND SURVEYING. Its vernier, v, reading to single minutes, "is fixed to a movable arm, h, having its center of motion at the end of the declination arc, at g; the arm is moved over the sur- face of the declination arc, and its vernier set to any reading by turning the head of the tangent-screw, ft. It is also securely clamped in any position by a screw, con- cealed in the engraving. 5. 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, (Fig. 11,) fastened by screws to the inside of the opposite block. On the surface of the plate are marked two sets of lines, intersect- ing each other at right angles; of these, bb are termed the hour lines, FIG. 11. an a CG th e equatorial lines, as having reference respectively to the hour of the day and the position of the sun in relation to the equator. In Fig. 11 the equatorial lines are those on the lower block, parallel to the surface of the hour arc, c; the hour lines are of course those at right angles to the first. 6. Equatorial Sights. On the top of each of the rectangular blocks is seen a little sighting-piece, termed the equatorial 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 adjust- ing the different parts of the solar apparatus. 7. 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 didsion being marked 12 and 90 on either side of the graduated lines. 8. The Polar Axis. Through the center of the hour arc passes a hollow socket, p containing the spindle of THE SOLAR COMPASS. 67 the declination 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 gradu- ated side of the declination arc. The axis of the declination arc, or indeed the whole socket p, is appropriately termed the polar axis. 9. 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 above described constitute properly the solar apparatus. Beside these, however, are seen the needle-box, n, with its arc and tangent-screw, t, and the spirit levels, for bringing the whole instrument to a horizontal position. 10. The Needle Box has an arc of about 36 in ex- tent, divided to half degrees, and figured from the center or zero mark on either side. The needle, which is made as in other instruments, ex- cept 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 center, 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 gradu- ated arc, attached to the plate of the compass. 11 . The Levels 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 ten de- grees, and numbered, as shown, from to 90 on each side of the line of sight. 68 A MANUAL OF LAND SURVEYING. These graduations are used in connection with a little brass pin, seen in the center of the plate, to obtain ap- proximate bearings of lines, which are not important enough to require a close observation. 12. 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. These are not shown in the cut. 13. Principles of the Solar Compass. The inter- val between two equatorial lines, cc, in Fig. 10, as well as between the hour linos, 66, is just sufficient to include the circular image of the sun as formed by the solar lens on the opposite end of the revolving arm, 7i, Fig. 9. When, therefore, the instrument is made perfectly hori- zontal, the equatorial lines and the opposite lenses being accurately adjusted to each other by a previous operation r 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 7i, set at zero on the declina- tion arc 6, 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 correspond with the motion of the sun in the heavens, on the given day and at the place of obser- vation; so that if the sun's image were 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 toward the sun, THE SOLAS COMPASS. 69 and the silver plate on which his image is thrown directly opposite. As the sun ascends, the arm must be moved around, until when h 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 6, and the lati- tude arc a, will be in. the same plane. As the sun declines from the meridian, the arm li must be moved in the same direction, until at sunset its posi- tion will be the exact reverse of that it occupied in the morning. 14. Allowance for Declination. Let us now sup- pose the observation made when the sun has passed the equinoctial point, and when his position is affected by declination. By referring to the Almanac, and setting off on the arc his declination for the given day and hour, we are still able to determine his position with the same certainty as if he remained on the equator. When the sun's decimation is south, that is, from the 22d of September to the 20ih of March in each year, the arc b is turned toward the plates of the compass, as shown in the engraving, and the solar lens, o, with the silver plate opposite, are made use of in the surveys. The remainder >f 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 aho 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 position will cause the image to pass above or below the lines, and thus discover the error- 70 A MANUAL OF LAND SURVEYING. "We thus, from the position of the sun in the solar sys- tem, 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 con- struction 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 to run the true meridian, or a due east and west course, but also to set off the horizontal angles with minuteness and accuracy from a direction which never changes, and is unaffected by attraction of any kind. 15. Adjustments. The adjustments of this instru- ment, with which the surveyor will have to do, are sim- ple and few in number, and will now be given in order. 1st. To Adjust the Levels. Proceed precisely as di- rected in the account of the other instruments we have described, by bringing the bubbles into the centre of the tubes by the leveling screws of the tripod, and then re- versing 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. THE SOLAR COMPASS. 71 2d. 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 Jc, and also the clamp-screw, and the conical pivot with its small screws by which the arm and declination arc are con- nected. 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 ad- juster, 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 direction of the sun, and raise or lower the adjuster on the declination 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 position by the tangent of the latitude arc or the leveling-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 confine 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, end for end, on the adjuster, and proceed precisely as in the former case, until the same result is attained. 72 A MANUAL OP LAND SURVEYING. 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. 3d. To Adjust the Vernier of the Declination Aro. Having leveled 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 a. k, 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 leveling-screws of the tripod or socket, and without disturbing the instrument, care- fully revolve the arm h, until the opposit, lens and plate are brought in the direction of the sun, and note if the sun's image comes between the lines as before. If it does, there is no index error of the declination arc; if not, with the tangent-screw k, move the arm until the sun's image passes over half the error; again bring the image between the lines, and repeat the operation as before, until the image will occupy the same position on both 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. 4th. 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 ver- niers and horizontal limb. Then remove the clamp-screw and raise the latitude arc until the polar axis is by esti- * THE SOLAR COMPASS. 73 mation 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. 16. To Use the Solar Compass. Before this instru- ment 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 lat- itude of the place where the observation is made. To Set off the Declination. The declination of the sun, given in tne ephemeris of the Nautical Almanac fuom year to year, is calculated for apparent noon at Greenwich, England. To determine it for any other hour at a place in the United States, reference must be had, not only to the dif- ference of time arising from the longitude, but also to the change of declination from day to day. The longitude of the place, and therefore its difference in time, if not given directly in the tables of the Almanac, can be ascertained very nearly by reference to that of other places given, which are situated on, or very nearly on, the same meridian. It is the practice of surveyors in the states east of the Mississippi, to allow a difference of six hours for the dif- ference in the longitude, calling the declination given in the Almanac for 12 M., that of 6 A M., at the place of ob- servation. Beyond the meridian of Santa Fe, the allowance would be about seven hours, and in California, Oregon, and Wash- ington Territory about eight hours. 74 A MANUAL OF LAND SURVEYING. Having thus the difference of time, we very readily ob- tain the declination for a certain hour in the morning, which would be earlier or later as the longitude was greater or less, and the same as that of apparent noon at Greenwich on the given day. Thus, suppose the observa- tion made at a place, say, five hours later than Greenwich, then the declination given in the Almanac for the given day at noon, affected by the refraction, would be the declination at the place of observation for 7 o'clock A.M.; this gives us the starting-point. To obtain the declination for the other hours of the day, take from the Almanac the declination for apparent noon of the given day, and, as the declination is increas- ing or decreasing, add to or subtract from the decimation of the first hour, the difference for one hour as given in the ephemeris, which will give, when affected by the re- fraction, the declination for the succeeding hour; and proceed thus in "making a table of the declination for every hour of the day. 17. 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 alti- tude 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. 18. Allowance for Refraction. The proper allow- ance to be made for refraction in setting off the declina- tion of the sun upon the Solar Compass has long been a source of perplexity to the surveyor. Accordingly, a table has been prepared, (Table XI), by which the amount of refraction for any hour of the day throughout the year may be readily obtained. The manner of using the table is shown in the solution of the following THE SOLAR COMPASS. 75 Example \. To find the declination for the different hours of April 16, 1883, at Troy, N. Y. Solution. Latitude of Troy, about 42 30' N. Longi- tude, 4 hr., 54 min., 40 sec., practically 5 hr. Apparent noon at Greenwich is 7 A. M. at Troy. Decli- nation of sun at Greenwich at noon of April 16, 1883, as given by Nautical Almanac, K 10 6' 2"-f, and hourly change, 53". Refraction in Lat. 42 30', declination 10, time 5 hr. before noon as given by table, 1' 58". Whence the following figures: N.10" 6' 2"+Ref.5hrs. 1' 58" =10 8' 0" Dec. at 7 A. M. Troy, add hr. dif. 53" N. 10 6' 55"+ " 4 " I'll" 10 8' .6 add hr. dif. 53" N. 10 T 48" + " 3 " 0'52"=-10 8' 40" add hr. dif. 53" N. 10 8' 41"+ " 2 " 039" 10' 9' 20" " 10 add hr. dif. 53" N. 10 9' 34"+ " I '* 0' 36" 10 10' 10" add hr. dif. 53" N. 10 10' 27"+ " " 0' 36" 10 11' 03" ' 12 M. add hr. dif. 53" N. 10 11' 20" + " 1 " 0' 36" IT 11' 56" IP. M. add hr. dif. 53" N. 10 12' 13" + ** 2 " 0' 39" 13* 12' 52" add hr. dif. 53" N. 10 13' 06" + " 3 " 0' 52" 10* 13' 58" add hr. dif 53' N. 10 13' 59"+ " 4 " 1' 11 ' 10 15' 10" ~ add hr. dif. 53' N. 10 14' 49" + " 5 " 1' 58" = 10 16' 50" ' 5 " Example. 2. To find the declination for the different hours of Oct. 16, 1883, at Troy, N. Y. Solution. Declination of sun at Greenwich at noon of Oct. 16, 1883, as given by Nautical Almanac S. 851'47".7. hourly change 55". 76 A MANUAL OF LAND SURVEYING. Refraction 5 hr. before noon, Lat. 42 30', Dec. 9, is very nearly 9 ' 24 " , and operates to diminish the declina- tion. Whence the following: S. 8* 51' 47".7 Ref. 5 hr. 9' 24" 8" 42' 23"-= Dec. at 7A.M. at Troy. add hr. diff. 55" S. 8 52' 42" " 4 " 2 49"= 8 49' 53"= add hr. diff. 55" 8.8 53' 37" " 3" 1 49"= 8 51' 48" add hr. diff. 65" S. 8 54' 32" " 2" 1' 26" 8 53' 06"= " 10 addhr. diff. 55" 8.8 55' 27" " 1 " 1' 14"= 8 54' 13"= " 11 add hr. diff. 55" S. 8 56' 22" " " 1' 14"-= 8 55' 08"= " 12 M. addhr. diff. 55" S. 8 57' 17" " 1 * 1' 14"= 8 56' 03"= " 1 P. M. addhr. diff. 55" S. 8 58' 12 ' * 2 ' 1' 26"= 8" 56' 46"= " 2 * add hr. diff. 55" etc. etc. etc. 19. To Set Off the Latitude. Find the declination of the sun for the given day at noon, at the place of ob- servation, 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 instrument 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 THE SOLAR COMPASS. 77 % 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 passing 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 ob- servation and correction of the latitude arc every day when the weather is favorable, there being also nearly an hour at mid-day when the sun is so near the meridian as not to give the direction of lines with the certainty re- quired. 20. To Bun Lines with the Solar Compass. Hav- ing 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 carefully leveled, the plates clamped at zero on the hori- zontal 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. 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. 78 A MANUAL OF LAND SURVEYING. 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 ordin- ary magnetic compass, the bearings of lines are read from the ends of the needle. 21. Use of the Needle. In running lines, the mag- netic 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 opposite 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, i3 read off in degrees and minutes on the arc, by the edge of which the vernier of the needle-box moves. 22. 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 lati- tude 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. In running due east and west, as in tracing the stand- THE SOLAR COMPASS. 70 ard 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, noting the error on the back sight, and setting off one-half of it on the fore sight on the side toward the pole. 23, 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 latitude. The time thus given is that of apparent noon, and can be reduced to mean time by merely applying the equation of time as directed in the Almanac, and adding or sub- tracting 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 re- volving arm, with reference to the graduations of the hour circle, allowing four minutes of time for each de- gree of the arc, and thus obtaining apparent time, which must be corrected by the equation of time as just de- described.' 24. 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 occupied by the true one. It is caused by the reflection of the true image from the sur- face of the arm, and is a fruitful source of error to the 80 A MANUAL OF LAND SURVEYING. inexperienced surveyor. It can, however, be readily dis- tinguished from the real image by being much less bright, and not so clearly denned. 25. 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 eye, the brass center-pin, and the object observed. The bearing of the line is then read off at the point where the pencil is placed. 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. It is best not to take the sun at morning and evening, when it is within half an hour of the horizon, nor, for about the same interval, before and after it passes the meridian. II. THE SOLAR ATTACHMENT. 1. The Solar Attachment is essentially the solar apparatus of Burt placed upon the cross-bar of the or- dinary transit, the polar axis only being directed above instead of below, as in the solar compass. A little circu- lar disk of an inch and a half diameter, and having a short round pivot projecting above its upper surface, is first screwed firmly to the axis ofthe telescope. Upon this pivot rests the enlarged base of the polar axis, which is also firmly connected with the disk by four THE SOLAR ATTACHMENT. 81 capstan-head screws passing from the under side of the disk into the base already named. These screws serve to adjust the polar axis, as will be explained hereafter. 82 A MANUAL OF LAND SURVEYING. 2. 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 dec- lination arc, which is securely fastened to it by two large screws. 3. The declination arc is of about five inches radius, is divided to quarter degrees, and reads by its ver- lier 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 ordinary 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 declina- tion limb is turned on its axis and one or the other- solar lens used, as the sun is north or south of the equator. 4. The latitude is set off by means of a large verti- cal limb having a radius of two and a half inches; the arc is divided to thirty minutes, is figured from the centre, each way, in two rows, viz. from to 80, and from 90 to 10, the first series being intended for reading vertical angles; the last series for setting off the latitude, 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 de- sired position. The vernier of the vertical limb is made movable by the tangent-screw attached, so that its zero and that oi THE SOLAR ATTACHMENT. 83 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, vernier, and hour circle are all on silver plate, and are thus easily read and preserved from tarnishing. 5. Adjustments. These pertain to the solar lenses and lines, the declination arc, the polar axis and hour arc, as follows: (1) The solar lenses and lines are adjusted precisely like those of the ordinary Solar, the decimation arm being first detached 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 furnished 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. (2) The vernier of tha declination arc is adjusted by setting the vernier at zero, and then raising or lower- ing the telescope 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. 84 A MANUAL OF LAND SURVEYING,, Jf precisely between the lines, the adjustment is com- plete; 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 re- move the index error by loosening two screws that fasten the vernier; place the zeros of the vernier and limb in exact coincidence, tighten the screws, and the adjustment is finished. (3) To Adjust the Polar Axis. First level the instru- ment carefully by the long level of the telescope, using in the operation the tangent movement of the telescope axis in connection with the leveling screws of the parallel plates until the bubble will remain in the centre during a complete revolution of the instrument 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 decimation 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 cap- stan-head screws on the under side of the disk. 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 cap- stan-head screws immediately 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 instrument, when the adjustment of the axis in one direction will be complete. 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 second position of the instrument. THE SOLAB ATTACHMENT. 85 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. It should here be noted that, as this is by far the most delicate and important adjustment of the solar attach- ment, it should be made with the greatest care, the bub- ble kept perfectly in the center and frequently inspected in the course of the operation. (4) To Adjust the Hour Arc. Whenever the instru ment 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 th hour circle, and with the hand turn the circle arounO until it does, fasten the screws again, and the adjustment will be complete. To obtain mean time, of course the correction of th< equation for the given day, as given in the Nautical Al- manac, must always be applied. 6. To Find the Latitude. First level the instru- ment very carefully, using, as before, the level of the telescope until the bubble will remain in the center dur- ing a complete revolution of the instrument, the tangent movement of the telescope being used in connection with the leveling 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 meridianal refraction carefully set 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 86 A MANUAL OF LAND SURVEYING. object-end lowered until, by moving the instrument upon its spindle and the decimation arc from side to side, the sun's image is brought nearly into position 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 re- main stationary for an instant and then begin to rise on the plate. The moment the image ceases to run below is of course apparent noon, when the index of the hour arc should indicate XII, and the latitude be determined by the read- ing of the vertical 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 w T ill 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. ? To Run 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 leveled by the telescope bubble, set the horizontal limb at zero and clamp the plates together, loosen the lower clamp so that 'the transit moves easily THE SOLAR ATTACHMENT. 87 upon its lower socket, set the instrument approximately north and south, the object end of the telescope pointing to the north, turn the proper solar lens to the sun, and with one hand on the plates and the other on the revolv- ing arm, move them from side to side until the sun's image is brought between the equatorial lines on the sil- ver plate. The lower clamp of the instrument should now be fast- ened 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 meridian, will also give the declination of the needle at the point of observation. The declination of the needle being set off, the needle kept then at zero, or *' with the sun," lines may be run by the needle alone, when the sun is obscured. The sun, however, must ever be regarded as the most reliable guide, and should, if possible, be taken at every station. 88 A MANUAL OF LAND SURVEYING. CHAPTEE IV. MEASUREMENT OF ANGLES. 1. The instruments already described are used both for running lines and for measuring angles. The transit is used where the greatest degree of accuracy is required and where angles are to be measured within 1' or less. The compass is used when no great degree of accuracy is required and the measurement of an angle within 5' is as close as is ordinarily expected. Professional Surveyors are provided with the compass or transit in some of their various forms. Students and others may or may not have them. In case of necessity the tape may be used to measure angles, and in connection with the picket, sections of the United States Survey may be subdivided, irregular fields meas- ured, and other similar operations performed, with a ra- pidity and accuracy equal to, if not superior to work done with a compass, the picket being used to run the lines and the tape to measure both distances and angles. 2. To Measure Angles with the Tape. This is most conveniently done with the aid of tables of trigonometrical functions with which the student is supposed to be familiar. Prob. 1. To lay off a right angle from a point p in a given line AB. 7TL FIG. 13. MEASUREMENT OF ANGLES: 89 When the sides of a triangle are to each other as 3, 4 and 5, the angle between the smaller sides is a right angle. Hence to lay off a right angle with the tape or chain, stick a marking pin at p and then measure along the line p m = 3 and stick another pin at m. Then from p as a center with a radius 4 and from m as a center with radius 5 strike arcs intersecting at n. Then will mpn be the required angle. If the line pn is to be prolonged as a picket line, it will be better to range from, if longer sides, as 60, 80 and 100 are used. This is the most useful of the many methods of laying off a right angle with the tape, and can be applied where any method can be. The other methods are, for the most part, more curious than useful. The following is one of the best of them: 2d Method. Measure along the line in opposite direc- tions from_p and stick pins in the line at m and w' mak- ing pm = pm'. Then from m and m' as centres with any radius greater than pm strike two arcs Intersecting at n. Mpn is the required angle. FIG. 14. Prob. 2. From a point p in a given line 'AB to run a line making any required angle with the line AB. 1st Method. From p measure p m equal to the cosine of the required angle and stick a pin in the line at m. Then from m as a centre with a radius equal to the sine of the required angle and from p as a centre and radius r strike arcs intersecting at n. Then mpn will be the required angle and^> and n will be points in the required line. If r 100 then the lengths of cosine and sine are used just as taken from the table of natural sines, only 90 A MANUAL OF LAND SURVEYING. changing the decimal point. Otherwise the tabular numbers must first be multiplied by the length adopted for r. FIG. is. 2d Method. In a similar manner we may use the natural tangents and secants. From p and m as centres, with the secant and tangent of the required angle as radii, strike arcs intersecting at n. Secants not given in the table may be found from the table of natural sines 1 by the formula secant cosine. \ FIG. 16. Example 1. Lay off, by the use of sines and cosines,- an angle of 36 28 X . Solution. Let r = 100 = pn. Then mn = 59.44, pm = 80.4. Ex. 2. Lay off by the use of tangent and secant, an angle of 25 20'. Solution. Let r = 100 = pm. Then mn 47.34; pn = 110.64. Ex 4. Lay off by each method, angles of 48 20', 63 15', 26 32', 8 40', 18 23', 37 06', 82 45'. MEASUREMENT OF AKGL^a. 91 3d Method. By chords. From the point p as a, centre, with any radius, preferably 100, strike an arc mx. Find the natural sine of half the angle. Double it for the chord. With this distance as radius, from m as a centre, strike an arc intersecting the arc mx at n. Then p and n are points in the required line and mpn the required angle. Example 1. Having run the line from the east quarter post of section 26 north to the section corner and marked it with a sufficient number of pickets, it is required to locate the centre line of a highway commencing at the quarter post and running north 22^ west. Solution. Measure north in the line from the quarter post the full length of the tape = 100, stick a marking pin m carefully in line, and strike an arc to the left around the quarter post as a centre. Find the sine of half the angle and double it. SL.e 11 15' X 2 = .19509 X 2 = .39018 or correcting the decimal point 39.018. With this dis- tance as a radius, from m as a centre, locate the inter- secting point n which is a point in the required line. The student should now select a level plat of ground, mark out a line upon it with pickets and solve the pre- ceding examples or similar ones, on the ground, each one by the several different methods and compare results, Also set pickets at the angles of a field of three or more sides and measure the sides and angles of the field. 3. To Measure Angles with the Compass. Set the compass up at the intersection of the lines, be- tween which the angle is to be measured. Put the sights in range with one f the lines and note the reading of the 92 A MANUAL OF lAJtfD SUEVEYING. needle. Then put them in. range with the other line and again note the reading of the needle. Read off from the limb, or calculate the number of de grees passed over by the needle between the two readings. In land surveying, a line traced out upon tlu ground is termed a course and the angle which the link makes with a north and south line is called its bearing or course. In compass work the bearings only are taker The angles between the lines of the survey may be com- puted therefrom if necessary. They are seldom required- In reading and writing down the bearings it is customary to state first the direction of the line from which the bearing is taken and then the angl3, to the east 01 west, which the course makes with that line, e. g., North 60 degrees West. South 5 degrees East. Written N c 60 W; S. 5 E. It is customary in Land Surveying to refer all lines to a meridian real or assumed. The cosine of a bearing multiplied by the length of its course is called the Latitude. The sine of the bearing multiplied by the length of the course is called the Departure. When desirable to find the angles between two lines from their bearings, they may be computed as follows* Calling N. and S. meridianal lettsrs, we have for the angle between two lines from the same station, the fol- lowing: PRINCIPLES. 1. When the meridianal letters are alike and the others unlike, the angle is the sum of the bearings. (2) When the meridianal letters are unlike and the others alike, the angle is the supplement of the sum of the bearings. (3) When loth the meridianal and the other letters are alike, the angle is the difference of the bearings. (4) When both the meridianal and the other letters are unlike, the angle is the supplement of the difference of the t-earings. MEASUREMENT OF ANGLES. 93 Observe that the bearings are given in their proper rel- ative direction with each other and none of them are reversed, as S. E. when it should be N. W. Examples. 1. The bearings, of two lines are N. 60 W. and N. 3 E. What is the angle between them? Ans. 63. a. Required the angles between lines having the fol- lowing bearings: N. 37 E. and S. 26 E.; N. 87 E. and S. 86 W.; S. 15 E. and S. 26 E. Ans. 117; 179; 11. 3. Stake out a triangle in the field and take the bear- ings of the sides. Find the angles of the triangle and compare the sum with 180. 4. Stake out fields having 4, 5 and 6 sides. Take the bearings and find the angles between the sides. 4. To Correct Courses of Random Lines. CASE 1ST: Where the line has but one course. Random lines as they are usually called are simply trial lines run to find the true line between two fixed points which are not visible from each other. These lines are usually started from one of the points and run as nearly in the true direction as can be estimated. If the estimate proves correct, and the line strikes the point aiired for, the random becomes the true line. If not, the perpendicular distance from the line to the point is measured, from which the correction for the course may be computed. FIG. is. If PC is made perpendicular to AB as is generally the case where randoms are run between corners of the CP Tnited States survey then Tan. CAP = whence AP 04 A MANUAL OF LAND SURVEYING. the angle CAP is found, which is the correction to be applied to the bearing. The angle CAP, when it is quite small, may be found by multiplying 57.3 by PC, and dividing by AC. This is called the Fifty-seven and three-tenths rule. The rule depends upon the fact that for small angles, AP differs insensibly from AC, and CP from the arc sub- tending the angle CAP. Whence, angle C^P:360::CP:2X3.1416X^P, CP 360 OPX57.3 CTX&7.3 or angle CAP X = , or AP 6.2832 AP AC The semi-circumference of a circle, with radius AP, is 3.14159265XAP. Whence arc I' = 3.14159265 X AP -5- 10800. If AP = 1 ch., arc 1' = 0.00029088 ch. = 0.029088 1. If AP = 1 mi. = 80 ch., arc V = 0.029088 1. X 80 = 2.327 1. - 2^ 1. When angle PAG = V and AP or AC = 1 mi., the perpendicular PC, without perceptible error, is 2% links. The line PC is called the departure of AC, for the dis- tance AP or AC. Taking 2% 1, as the departure of 80 ch. at an angle of 1', the departure for 40 ch., would be % of 2% 1. = 1 1. = 1 1. + i of 1 1. For quite small angles, the departure varies directly as the angle. Whence, for 40 ch., the following: Dep. for 1' = 1 1. + | of 1 1. " 2 / = 2 1. + | of 2 1. 3' = 3 1. + I of 31. and so on, practically true, to 60' or 1. For any other distance, at the same angle, the depar- ture varies directly as the distance. Accordingly, Given minutes of angle, to find links of departufe, we have the following: KULE. To the number of minutes, add its one-sixth and multiply the sum by the ratio of the distance to 40 ch. (Good to sixty minutes.) MEASUREMENT OF ANGLES. 95 On the following: GENERAL RULE. Multiply 0.0291 by the number of minutes, and multiply the product by the number of chains in the distance. (Good to 240 minutes.) Example. Given angle 30 7 and distance = 23.20 ch., to find the departure. Since for 40 ch., V of angle gives \\ 1. of depar- ture, we may say, without sensible error for a small angle that 1 1. of departure gives f of l x of angle, for the same distance. Or as it may be written, Dep. of 1 i. = V \ of 1'. Similarly, " " 2 1. = 2' } of 2', " 3 1. = 3' 1 of 3', . and so on, practically true to 60 7 or 1. Eor any other distance with the same departure, the angle varies inversely as the distance. Accordingly, / Given links of departure, to find minutes of angle, we have the following: RULE. From the number of links of departure, sub- tract its one-seventh and divide the remainder by the ratio of the distance to 40 ch. (Good to 60 minutes.) GEXERAL RULE. Multiply 0.0291 by the number of chains in the distance, and divide the number of links of departure by the product. (Good to 240 minutes). In the Table of Departures, the value of PC in chains and decimals is given for angles from V to 60', and for the distances most commonly required in making resur- veys and subdivisions of Sections of the United States Survey. To use the Table: Having measured the outing PC on the ground, find the nearest tabular nuwber in the column for the corresponding distance. The angle will be found in the minute column. Example 1. Commencing at the west quarter post of Section 16, and running north, the random line intersected 96 A MANUAL OF LAND SURVEYING. the north line of the section, 15 links east of the corner What is the amount of the correction for course ? Solution. In 40 chain column, nearest number .151, Corresponding number of minutes 13. 2. Commencing at the south quarter post of section 16 with declination of needle estimated at 2 17' E. set off on the vernier, ran north on random and intersected the north line of the section, 42 links east of the quarter post. What is the declination of the needle as referred to the quarter line? Solution. Distance 80 chains, correction 18 X . As the line came out east of the corner, it is evident that the angle between the magnetic meridian and the quarter line was 18' greater than was estimated, = 2 35'. NOTE. The North and South lines of the United States Survey are, in a legal sense all true meridians, whatever they may be astron- omically, and their locations are fixed by the monuments planted for the section corners and quarter posts. Hence it is a custom amon^ Surveyors to refer the declination of the needle or the variation as it is more frequently called, to these lines, and to mark on each line on their plats, the declination for that line. Under that custom the line referred to in Example 2 would be marked Var. 2 35' E. 3. "East on random between Sections 13 and 24. 79.98 chains intersected east boundary 34 links south of post." What is the bearing of the corrected line running west ? Am. S. 89 45' W. CASE 2ND. Where the line is a broken one of several courses. Surveyors are frequently called on to retrace the lines of angling roads to settle the boundaries of adjacent lands, or to locate meander lines, or to find the boundaries of irregular tracts, where several courses have to be run between the nearest known points of the original survey. In such cases random lines are run according to the notes of the original survey, and temporary stakes driven at the angles of the random line. It will generally be found that corrections for course or distance or for both will have to be made to place the stakes in their correct location. MEASUREMENT OF ANGLES. 97 PROBLEM. To correct a random Hue of several courses. In Fig. 19 let A, B, C, D represent the lines and angles of the original survey between the known points A and D. FIG. 19. Let D f represent the terminus of a random run to re- trace these lines, the direction and distance of which from D is known. From A draw the line AD, producing it indefinitely beyond D ; also, from A as a centre, with radius AD, draw an arc through D. ^N"ow, if tlie error in the random was of direction only, then the point D' would be in the arc. If it was an error of the chain only, D f would be in the line AD or AD produced. Hence the position of D / with reference to the arc and the line AD indicates the kind of correction and in what direction it is to be applied. AD is the length of the original chain in terms of the ADf chain used on the random. That portion of the arc which is intercepted between the point D and a line joining AD', measures the angle of correction. In the field we may calculate the course and length, and run a sufficient part of the line D'A, and then trace the arc from D to its intersection with that line, and thus find the relative length of the lines AD and AD' t by which to determine the correction for the chain and also find the chord of tke angular correction; or they may be calculated as shown in the following example: Example 1. The boundaries of a farm between the nearest known monuments are as follows, (See Fig. 19): 1. N. 16, E. 12.00 chains. 2. N. 72, E. 26.00 " 3. S. 22, E. 14.00 " 98 A MANUAL OF LAND SURVEYING. A random was run with var. 2 30' E. and came out N. 28 E. 32 links from the monument. Required the correction for the variation of needle and for the stakes in the angles of the random line. We will first find the total latitudes and departures of each station on the random line, and the direction and distance of a line, AD', which will join the termini. N. Lat. S. Lat. E. Bep. Tot. Lat. Tot. Dep. 1. N. 16 E. 12.00 2. N. 72 E. 26.00 3. S. 22 E. 14.00 11.54 8.03 12.98 3.31 24.73 5.24 11.54 19.57 6.59 3.31 28.04 33.28 If we now divide the total departure of the point I)' by its total latitude we will have the tangent of the bearing of the line D'A. 33.28 . = 5.050 = tan 78 48' or S. 78 48' W. 6.59 The length of the line D'A = j/6.59 2 + 33. 28 2 == 33.927, If we now subtract the bearing of the line D'D from the bearing of the line D'A we shall have the angle DD'A = 78 48' 28 = 50 48'. Let DH be a perpen dicular from D to the line AD' ; then we have the follow ing equations: D H = D'D sin AD'D = . 32 X 77494 = . 24798-f-. D'H = D'D cos AW = . 32 X ' 63203 = . 20225. AH = A D' D'H = 33 . 926 . 20225 = 33 . 7237+. DH = tan DAD' = .24798+ + 33.7237+ --= .00735 =- AH tan 25' = correction for course. . AH AD = y AH' 2 + HD 2 = = 33. 7237 -r- .99997 = cos DAD' 33. 724. When the angle DAD' is small, AD and AH may be considered equal, without sensible error. AD 33.724 = = .99404 = length of original chain in AD' 33.926 terms of the chain used on the random. As the randon MEASUREMENT OF ANGLES. 99 came out to the left of the true line the variation, 2 3(X E., was too great, hence we subtract the 25', giving 2 05' as the variation of the needle from the meridian of the original survey. To find corrections for the stakes it will be better to refer them to the meridian of the random, hence we will now apply the corrections for course and distance to find the courses and distances of the original survey, as they would be according to the meridian and measure of the random. This done, we calculate their total latitudes and departures. The difference between these arid the latitudes and departures of the correspond- ing points of the random is the correction to be applied. N. Lat. S. Lat. E. Dep. Tot. Lat. Tot. Dep. 1. N. 1625'E. 11.928 11.44 3.37 11.44 3.37 2. N. 72 25 E. 25.844 7.81 24.64 19.25 28,01 3. S. 21 35 E. 13.916 12.94 6.12 6.31 33.12 The last course is computed in this table simply as a check on the work, as it was a condition of the problem that the line LD f was N. 28, E. 32 links; from which it is known that the difference between the two points is: latitude 28 Iks., and departure 15 Iks. We will now com- pare the results in the two tables and find the correction at S, C and D. B Lat. Dep. c Lat. Dep. D Lat. Dep. Random Line Original Line- 11.54 3.31 11.44 3.37 19.57 28.04 19.25 28.01 6.59 33.28 6.31 33.13 Correction S. 10 E. 6 | S. 32 W. 3 S. 28 W. 15 Example 2. Description of a highway between two known points: 1. N.62 E. 14. 00 chains. 2. K43^ E.8.00 " 3. K 5 W. 12.00 " K72^ E. 10.25 " S. 12 W. 6.43 " A random run with var. 2 17' E. came out 62 Iks. east of the point. What is the correction for variation of 100 A MANUAL OF LAND SURVEYING. needle, and what change must be made in the position of each stake at the angles of the random ? 5. To Measure Angles with the Transit. 1. Set up the transit at the apex of the angle and set the zero of the vernier to coincide with the zero of the limb. Clamp the plates in this adjustment and with the clamp to the spindle loosened, turn the telescope in the direction of one of the lines. Clamp the spindle and bring the wire exactly to centre the line by the slow motion screw to the spindle clamp. Unclamp the vernier and turn the telescope in the direction of the other line. Clamp the vernier in that position and make the final adjustment of the wire to the line by the use of the upper tangent screw. The angle may then be read from the limb. 2. Instead of first setting the verniers at zero they may be clamped in any position on the limb and then the differ- ence in the two readings will be the angle. When great accuracy is required numerous readings of the angle me taken on various parts of the limb and the mean of the several results taken for the final reading. 3. To find the angle which the parts of a broken line form with any given line. i: FIG. 20. MEASUREMENT OF ANGLES. 101 SUGGESTIONS. Let ABCDEF be a broken line, and suppose it is required to find the angles which the parts BC, CD, DE and EF form with the line AB. Set the transit at B, with the vernier set at zero. Loosen below, reverse the telescope and direct H to 4. Clamp the limb, revolve the telescope in its horizonta"- axis, unclamp the vernier and direct the telescope to C. The reading of the instrument will be the angle bBC the line which BC forms with the line AB. Remove to C; and, leaving the vernier clamped, un- clamp below, reverse the telescope, and direct it to. 2?. The limb remaining securely clamped, revolve the telescope, unclamp the vernier, and direct to D. The reading will now be the angle cCD which the line CD forms with the line Co or its parallel AB. The work goes on in this manner to its close. Let the student further describe it. If the broken line enclose a field, the reading of the instrument when set as at A and directed to B, having gone entirely around the field, should be 360. This con- stitutes a check against errors occu_Ting anywhere in the work. 4. To measure an angle of elevation or depression. c A FIG. 21. SUGGESTIONS. Set the instrument at the vertex of the angle and level the horizontal limb. 102 A MANUAL OF LAND SURVEYING. Revolve the telescope upward or downward as the case may require, and adjust the line of sight to the inclined sivie of* the angle. Take the reading of the vertical circle, applying the proper correction for index errcr- Otherwise, take the reading of the circle, repeat the observation with the telescope and vernier plate reversed, and find the mean of the two readings for the angle sought. 6. Verniers are auxiliary scales for measuring smaller portions of space than those into which the main scale is divided. They are movable beside the main scale and are divided into parts which are either a little shorter or a little longer than the parts into which the main scale is divided. This small difference in length is what we are enabled to measure. When the limb of a transit is divided to half degrees it is common to make either 29 or 31 divisions of the Vernier Scale equal to 30 on the limb, making each division on the vernier 31' or 29' in length. The zero of the Vernier Scale is the point to which the reading is to be taken. Suppose the zero line of the vernier to make a straight line with some even division of the limb and each division on the vernier scale is 29 7 in length. Xow if the Vernier be moved ] x , the first line of the Vernier Scale from zero in the direction in which the vernier was moved, will be in a line with the first division on the limb. If moved 2' the second lines will coincide; if 3 X the third lines ; and so on to the end of the scale. Such a vernier is called direct reading. It is the kind most commonly used on surveyors' instru- ments. Suppose however that the spaces on the vernier were 31' long. Then when the vernier was moved forward V the first line back of the zero point would coincide with the line in the limb and so on. Such a vernier is called a retrograde vernier. MEASUREMENT OP ANGLES. 103 To read any vernier. If the zero of the vernier coin cides with any division of the scale, that will be the cor- rect reading. If not, note the nearest next less division on the limb, and then look along the vernier scale till a line is found which coincides with a line on the limb. The number of this line on the vernier tells that so many of the subdivisions which the vernier indicates (usually minutes) are to be added to the reading of the entire divisions on the limb. If several lines appear to coincide equally well, take the middle line. A04 A MANUAL OF LAND SURVEYING. CHAPTER V. PASSING OBSTACLES. MEASURING INACCESSIBLE DISTANCES. Having considered the various methods of running lines and measuring angles we are now prepared to take up some further problems in passing obstacles in the line and measuring inaccessible distances. These problems may be solved in the field by the use of the picket and tape, the compass, or the transit. 1. To pass an obstacle in the line and measure, the distance. 1st, by Parallel Lines. Prom a in the line AB run and measure the line ac in any convenient direction, a sufficient distance. From c run cd parallel with AB. FIG. 22. Prom d, run and measure db equal to and parallel with ac. Then ab = cd and b is a point in the line AB. When running through heavy forests or towns it will often be necessary to run several parallel lines before returning to the original line. 2. By 6O Angles. From a run and measure ac making the angle Bac = 60. Run and measure cb ^ ac and the angle acb 60. Then 6 is a point, in the line PASSING OBSTACLES. 105 AS and the angle abc 60, whence the line may be con- tinued; ab will equal at: FIG. 23. 2. To Measure Inaccessible Distances. CASE IST. When the points are visible from each other as ovei' a stream or pond. FIG. 24. I. By Similar Triangles. From a point a in the line AS, required the distance ab across the stream. At a erect a perpendicular ac to the line AB. From c run a perpendicular to cb intersecting AS at d. Measure ac and ad. Then as the triangles cad and bed are similar, ac 2 ad : ac = ac : ab, whence ab = . ad There are numerous other devices for obtaining the distance ab by similar triangles on the ground. Let the student work out some of them in the field. 106 A MANUAL OF LAND SURVEYING. 2. Method by Tangents. Erect a perpendic- ular to AB at a and run it a sufficient distance ac. Meas- ure the angle acb t Then ab = ac X tan acb. If *c is made 100 or 1,000, a& may be read directly from the table of natural tangents, observing to put the decimal point in the proper place. If acb = 45 then ab = ac. 3. Method by Sines. From a run a line ac as most conven- ient. Measure the angles acb and cab and the side ac. Com- pute the angle abc. Then sin abc : sin acb ab ab = ac sin acb FIG. 26. 4. Method by Cosines. From a run a line ac to the point c in a line perpendicular to AB at b. Measure the angle cab and the line ac. Then a& = ae X cos 5. Method by Secants. Run ac as be- fore, to a point c from which a per- pendicular to ac will strike the the point b. Meas- ure ac and the an- FIG. 28. INACCESSIBLE DISTANCES. 107 gle bac. Then ab = ac X secant bac. Ifac= 100 or 1,000 the distance ab is taken directly from the table. 6. By 5 43' Angle. From a lay off the angle bac = 5 43', making be perpen- dicular to ab. Meas- ure be. Then ab = FIG. 29. lObc. This method gives results too large by 1.07 in 1,000. CASE 2ND. Where the points are' irivisibJe from each other. 1. If visible and accessible from a common point c outside the line. Measure the lines ac and be and the angle acb. Sub- FIG. 30. tract this angle from 180 and we have the sum of the remaining angles of the triangle, to find the difference. abc -f bac abc bae tan : tan Then ac + be : ae 6c abe~-\- bac abe bae And 1 2 2 abc -f bac abc bac = abc. Also = bac. 2 2 ab = acX cos bac + bcX cos abc. If a and 6 are inaccessible from c, the sides ac and be may be measured by any of the preceding methods. 2. If instead of two lines ae and bc t we have a broken line of any num. ber of courses, as abcdef, thebear- FlG. 31. 108 A MANUAL OF LAND SURVEYING. ings of which are referred to the line of as a meridian then the algebraic sum of the products of the cosines of the several bearings into their respective distances will be equal to of. In the United States Surveys distances across lakes and bends of large streams are frequently computed from the latitudes and departures of the courses around them. Examples. I. In Fig. 24 ac = 100 ad = 27. Required ab. Am. 370.37+ 2. Same Figure, ac = 250, ad = 96. Required ab. Ans. 651.04+ 3. Fig. 25, ac = 100, angle c = 61 20'. Required ab. Ans. 182.9. 4. Same Figure, ac = 250, angle c = 61 10'. Required ab. Ans. 454.1+ 5. Fig. 26, ac = 500, angle a = 48 20', angle c = 118 1(X. Required ab. Ans.lSSS.l 6. Same Figure, ac = 658, a 54 16', c = 88 32'. Required ab. Ans. 1087.9+ 7. Fig. 27, ac = 1,000, angle a = 28 35'. Required ab. Ans. 878.12+ 8. Same Figure, ac = 950, angle a = 18 56'. Required ab. Ans. 898.6. 9. Fig. 28, ac = 100, angle a = 76 40'. Required ab. Ans. 433.6+ 10. Same Figure, ac = 250, angle a = 56 20'. Required ab. Ans. 450.97. 11. Fig. 29, ac = 900, be = 648, angle c= 112. Re- quired ab. Ans. 1291. 12. Given the following courses and distances along a broken line between the points a and b. Required the distance ab. 1. N. 18 E. 6.25 chains. 2. N. 40 E. 8.00 3. N. 5 W. 12.00 " 4. K. 44 W. 8.68 " Ans. 30.26+ chains. INACCESSIBLE DISTANCES. 109 3. The field notes of the meanders of a lake in sec- tions 11 and 12 in the township 1, south, range 10 west, meridian of Michigan, by the government survey, read as follows: Courses X. 58 E. N.11W. X.63W. Chs. Lks. 10.00 20.00 5.16 Began at post in line of sections 11 and 12 on south side of lake : thence in sec. 12. to post in line of sec. 11 and 12, N. side of lake. X.63W. S. GO W. S. 14 E. S. 33 W. S. r>i E. N.73V4 E 5.00 6.00 10.00 15.00 10.00 7.90 in section 11. to place of beginning. Kequired the distance between the posts on the oppo- site sides of the lake. Compute the distance by the mean- ders on each side of the lake. Compare the results to- gether, and also with the distance returned in the field notes which is 27.27 chains. 14. There is a cliff beside a railroad in the Wasatch Mountains known as the Castle Gate. Desiring to know its height above the railroad grade I set up the transit at Station 744 of the railroad survey and took the angle of elevation to the top of the cliff = 38 42'. Elevation of station 744 = 6573.62 ft Height of instrument above station 744 = 4.84 ft. I next went to station 748 in the line with and 400 ft. farther away from the cliff and again took the angle of elevation to the top of the cliff = 26 15'. Elevation of station 748 == 6567.62 ft. Height of instrument above the station, 4.56 ft. Required the height of the Cas- tle Gate above the station 744 and its horizontal dis- tance. Atiswer. Height 501.54. Distance 620. 110 A MANUAL OF LAND SURVEYING. 15. On Christmas 1881 a party of surveyors climbed a mountain peak, erected a monument on its summit and, named it Christmas Peak. Observations from the line of the railroad survey were made as follows, the stakes of that survey being 100 feet apart: From station 933 + 49.6 P. T. Angle of elevation of summit, 23 42'. Angle to right from railroad line ahead, 76 1CK. Elevation of station, 5005.28 ft. Instrument above station, 4.82 ft. From station, 940 + 31.4 P. C. Angle to left from railroad line back = 82 18'. lie- quired the height of the peak and its distance from sta- tion 933 + 49.6. 3. Other Methods of Measuring Distances, 1. To Gross a stream or pond. Set up the transit at a convenient point, a. Set up a rod at & in the line, at a convenient dis- tance, as 100 feet, from a. Set up a second rod in line at c, over the stream* Any plain, straight rods will answer. Leveling rods with targets are conve- nient. They should be set up plumb. Mark points d and e, in line, on the rods where the horizontal wire of the telescope cuts them, liaise or lower the telescope and mark two other points, /and g, in line on the rods where the wire cuts them. Measure df and eg. Then adf and aeg are similar triangles, and df. : of : : eg : ag. If df~ 1 and af= 100, eg = 6.25; then ag = 625. 2. Stadia Measures. 1. Instead of using two rods as described in the last paragraph, two wires are sometimes placed in the dia- INACCESSIBLE DISTANCES. Ill phragm of the telescope and adjusted at such a distance apart that they will cover a specified space on a rod, as 1 foot when the rod is 100, 200 or any other specified dis- tance away. These wires are one on each side of and parallel with the horizontal wire of the telescope. They may be either fixed on the diaphragm or attached to slides by which their distance apart may be adjusted. When the wires are adjusted to cover a certain space, as one foot on a rod placed 100 feet away, they will cover two feet on a rod 200 feet away, or .5 foot on a rod 50 feet away. This proportion is strictly true only when the measures are taken from a point in front of the in- strument at a horizontal distance from the object glass equal to its focal length. The focal length may be found nearly enough by measuring from the plane of the object glass to the capstan-headed screws which carry the dia- phragm. When the telescope is focused on some very distant object, as the moon or a star, the horizontal dis- tance from the plumb line to the point mentioned forms a constant which is to be added to all the distances as taken from the rod. 2. It is more convenient, though less accurate, to adjust the wires so that they will cover the required space on the rod at a specified distance measured from the center of the instrument. This method is usually adopted on the government surveys, where stadia measures are taken, the length of the base being taken at about a mean of the distances which the stadia is intended to measure. For all shorter distances the reading is too small. For longer distances it is too large. The error is neglected as of no consequence in the class of work for which the stadia is used. When the stadia wires are not adjustable the rod is graduated to conform to the wires. A rod is set up at the selected distance from the transit. The space inter- cepted on it by the wires is subdivided decimally, and the stadia rod graduated to that scale. Where the wires are adjusted to cover a foot on a rod 112 A MANUAL OF LAND SURVEYING. 100 or 200 feet away, the ordinary leveling rod answers the purpose of a stadia rod. 3. In case the measures are not on horizontal lines it will be necessary to apply a correction to the stadia read- ings to reduce them to the horizontal. If the rod has been held perpendicular to the line of sight, the horizontal distance is found by multiplying the distance to the rod by the cosine of the angle of elevation or depression. The position of the rod is determined either by a right- angle sight applied to the rod, or by the rodman slowly moving the top of the rod back and forth until the smallest intercept is obtained. On hillsides it will be found quite as easy to hold the rod perpendicular to the line of sight as to hold it plumb. When the rod is held plumb and the base is measured from the point in front of the transit the reduction to horizontal is made as follows: Let/= focal distance of the telescope, r = space intercepted on the rod as held vertically, s = image of the same intercepted by the stadia wires, CO' = line of sight at an angle e with the horizon. Let A'B' = r' be the intercept on the rod as in- clined at an angle e with the vertical; r f and let b' =f be the corresponding base. Let the an- o gle O'CB or O'CA = v. We shall then have : FIG. 34. Angle OCB = e -f- v, and angle OCA = e v, whence angle OBC = 90 (e -f v\ and angle OAC = 90 (e ,t>). The angle / JB / S = 90 + v, and angle WA'A = 90 v. INACCESSIBLE DISTANCES. 113 In the triangle O'B'B we have O'W sin [90 (e + )] r' cos (e + 1>) = or, = (a) O'B sin (90 + t>) WB cos D In the triangle O'A'A we have O'A' sin [90 (e v)] r' cos (e v) = or, = (ft) O*A sin (90 v) 2O'A cos v Adding (a) and (6), we obtain r'r = 2 cos e (c). 20' J? X O'A Multiplying (a) and (6) together, we obtain r' r' cos 2 e cos 2 sin 2 e sin 2 -e = (d) 4 0'.B X O'A cos 2 Dividing (c) by (d), we have, after a little reduction, r cos e - = - , 00 r 7 cos 2 e sin 2 e tan 2 tfhich is an expression of the relation sought. Cor. With the wires adjusted to one foot on the rod for a base of 100 feet, we should have tan v = 0.005 ft., or tan 2 v = 0.000025 ft Thus, tan 2 v = 0, without material error. Whence formula (e) becomes r' = r cos e. To find the distance CO' we have r' CO' = d' =/- +/+ c = Z/ + /+ a. s Whence, CO = d = (V + / + c) cos e. For vertical rod we have, b' 6 cos e. Whence, d = 6 cos 2 e + (/ -f- c) cos e. (/) The height 00' = h = b sin 2e + (/ + c) sin e. &) Example. Given e = 10 3CK, r = 5.36 ft., and / + c = 1 ft., to find d and h. Solution. Suppose the wires adjusted to give 1 ft. on the rod to the 100 ft., whence 6 = 536 ft. 8 114 A MANUAL OF LAND SURVEYING. Cos e = 0.983 and cos 2 e = 0.9668. Whence, d = 536 X 0.9668 -f 0.98 = 519.18 ft. Sin e = 0.182, and | sin 2e = 0.1792. Whence, h = 536 X 0.1792 + 0.18 = 96.23 ft. Formula (/) may be put in the form d = I cos 2 e -f (/+ c) cos 2 e + (/+ c) cos e (1 cos e). Dropping the last term, we have d = (6 -f / -f c) cos 2 e. (7i) Assuming ^ -f c = 1 ft. as a mean value in different instruments, the omission of the term (/ 4- c) cos e (1 cos e) introduces an error for ordinary elevations of less than 0.01 ft. in a base of 1000 ft. Moreover, the use of formula (h) operates to diminish the very minute error introduced by use of formula (/) For slight elevations, as from 1 to 2, the reduction to horizontal may be omitted. For 5 44 X the amount of the reduction is about one per cent. The correction for hori- zontal measurement is sometimes made by omitting to add/-f c to the base. INACCESSIBLE DISTANCES. 115 4. The Gradienter is an attachment to the transit for fixing grades and determining distances. As made by Gurley, it consists 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 passing 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. 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 100 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 di- visions 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 *ixis of 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 revolution 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-hundredth s, or one-half a foot on the rod, and so on in the same propor- tion. In this way, the gradienter can be used in the measure ment of distances, precisely like the stadia. 8 116 A MANUAL OF LAND SURVEYING. Grades can also be established with great facility, as follows: Level the instrument; bring the telescope level to its centre by the clamp and gradienter screw ; move the graduated 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 hundredths of feet to the hundred in the grade to be established. Having a transit with gradienter attachment, let the student solve the following problems in the field : Prob. 1. To find the grade between two points. SUGGESTIONS. Set the instrument over one of the points, level the plates and the telescope, and bring the zero of the screw to the edge of the scale. Set the target of the leveling rod at height of instru- ment. With the rod held upon the other point, note the num- ber of revolutions of the screw required in bringing the cross- wire upon the center of the target. That number, as so many feet, is the grade. Prob. 2. To find the distance between two points. SUGGESTIONS. Set up and adjust the parts of the in- strument as in Prob. 1. On a leveling rod held upon the other point, note the number of feet covered by one revo- lution of the screw, and multiply that number by 100. If, in order to cover r feet on a rod at a distance of d feet, n revolutions of the screw are required, then we should have: d : 100 :: r : n; whence d = 100 r-s- n. Example Given n = 2.30 and r = 5 ft., to find d. Kesult, d = 217.39 ft. On inclined ground the horizontal sight line may be. above or below the rod. In such cases, as in stadia measurement, a formula of reduction to a horizontal is employed, which may be deduced as follows: INACCESSIBLE DISTANCES. 117 Let CO d (Fig. 34), be a horizontal sight line; Angle OC(y = e, the elevation of telescope to foot of rod; Angle O'CB = v, the angle described by n revolutions of the screw; O'W = r', the space on a rod perpendicular to CO*, subtending angle v, and O'B = r, the corresponding space on a vertical rod. We shall then have, [Formula (a)\ r' sin [90 (e + v)] cos e cos v sin e sin v r sin (90 + v) cos v Whence, r' = r (cos e sin e tan ). r' n Let CO' = d'. Then, tan v = = . d' 100 100 r' 100 r ( Whence, d' = = ] cos e sin e X n n ( -1 100) ( 100 cos e V = r } sineL ( n ) or d' = rj sine^. (1) Now, d = d f cos e. (100 ) Whence, d = r ] cos 2 e % sin 2e [ . (2) ( n ) Cor. If n = 1, we have, d'=r (100 cos e sin e\ (3) and d = r (100 cos 2 e \ sin 2e), (4) in which r is the space on a vertical rod included by one revolution of the screw. The numbers by which this value of r must be thus multiplied for various elevations are given in Table IX. Examples. 1. Given e = 15 20', and r = 5.42 for one revolution of the screw, to find d' and d. SOLUTION. We find in Table IX, factor for inclined distance for 15 = 96.33 " " " 15 30' = 96.09 Difference for 30' = 0.24 whence, " " 2(X= 0.16 118 A MANUAL OF LAND SURVEYING. Whence, factor for inclined distance for 15 20 X = 96.17. Accordingly, d = 5.42 X 96.17 = 521.24 ft. Again, in Table IX we have factor for horizontal distance for 15 = 93.05 " " " " 15 30' = 92.59 Difference for 30 / = 0.46 whence, " 20' = 0.31 Whence, factor for horizontal dist. for 15 20' = 92.74. Hence, d = 5.42 X 92.74 502.65 ft. 2. Given e '= 10.35 rev. to foot of rod, and r = 6.25, to find d / and d. SUGGESTION. From Table X find the angle e, and solve as above. When c is an angle of depression, the point (/ is the upper end of the rod. The application of the formula is, however, the same in this case as in the one considered. Stadia and Gradienter Measurements are found very convenient in solving some of the problems in land surveying, but are almost useless in others. They save time and trouble in measuring across streams, bogs and other places inaccessible to the chain or tape. They furnish a quick and easy means of determining how far it is to an object, but a slow one of locating points at any desired distance, such as setting stakes for a town plat, a ditch line, or a railroad. PLATTING. CHAPTER VI. PLATTING AND COMPUTING AREAS. 1. A Plat or Plot is a representation, upon a small scale, of the lines of a survey. Platting is simply sur- veying on paper. The instruments used are analogous to those used in the field. Lines are marked upon the paper with pencil or pen and ink. Generally they will first be drawn lightly in pencil; afterward the permanent lines will be inked, and all erroneous or superfluous lines erased. Pencils hard enough to hold a fine point without breaking are the best for this use. The right line pen is used for drawing straight lines, it is made in various sizes and forms. One of the best is shown at 6, in Figure 36. The scale of equal parts is the counterpart of the chain or tape. A great variety of scales are made. One of the most useful is the triangular scale (Fig. 36, e). It has six different graduations, all brought to the edge, so that the scale may be laid down on the paper and the distance marked off directly from the, scale. The scale in which the inch is divided into 10, 20,30, 40, 50 and 60 equal parts is the one most useful to the surveyor. Paper scales are made on fine Bristol board, with any graduation desired. They are cheap, and as good as any scale as long as they last. The student may make his own scales on paper. The protractor (Fig. 36, a) takes the place of the com- pass or transit. It is simply the whole or part of a grad- uated circle or limb. Protractors are made in a great variety of forms. One of the cheapest and best has the 120 A MANUAL OP LAND SURVEYING. PLATTING. 121 entire circle graduated to quarter degrees. It is made of paper, has the middle part cut out, and fine threads or wires crossing at the centre of the circle. A paper protractor 14 inches in diameter, graduated to quarter degrees, costs from 30 to 40 cents. Dividers, (Fig. 36, /) are used to space off distances on the plat, or transfer distances from the scale to the plat or the reverse. When provided with pen or pencil points they are used to strike circles and arcs. When they are used for the latter purpose they should have a needle point on the stationary leg. Parallel rulers, as the name indicates, are used in drawing parallel lines. When a paper protractor is used in platting, it is found convenient to fasten it at some point outside the plat and transfer the bearing of the lines from the protractor to the plat by means of the parallel rule. The best rule for this purpose moves upon rollers, (Fig. 36, d) The straight-edge ruler and triangle are also used to mark parallel lines, as well as to lay off angles. Many other articles will be found convenient in platting. A drawing board, made of the softest wood, planed smooth and true, and thumb-tacks to fasten the paper to the board, may almost be considered as necessaries. Neither the student nor surveyor needs many instruments for platting, but those he has should be perfect in their kind. It is not deemed necessary at this point to give further details of these instruments and their uses, any sugges- tion which the student may need being left to the teacher to make. EXERCISES. The first seven exercises are the elementary problems of Geometry, and are designed to be solved on paper by use of the dividers and ruler. 2. 1. To draw a straight line equal to a given straight line. 2. To make an angle equal to a given angle. 122 A MANUAL OF LAND SURVEYING. 3. To draw through a given point a line parallel to a given line. 4. To draw through a given point a line perpendicular to a given line. Two cases. 5. To bisect a given line; a given angle. 6. To construct lines proportional to given lines. 7. To construct a polygon similar to a given polygon. 8. Plat the following lines : (1) 8 chains, to scale of 2 chains to the inch. (2) 10 chains, to scale of 5 chains to the inch. (3) 10 chains, to scale of 4 chains to the inch. (4) 17.25 chains, to scale of 3 chains to the inch. (5) 25.40 chains, to scale of 4 chains to the inch. 9. Plat a triangle whose sides are 13.50 eh., 14.25 ch. and 16.20 ch., on a scale of 5 chains to an inch; on a scale of 3 chains to an inch. 10. Plat a rectangle whose adjacent sides are 9.24 ch. and 13.78 ch., on a scale of 4 chains to the inch. 11. Plat a quadrilateral the sides of which are 22.60 ch., 14.35 ch., 12.20 ch. and 9.80 ch., on a scale of 4 chains to the inch, and having one angle of 83 30'. 12. Measure the remaining angles and find their sum. 13. Plat any figure having five equal sides; measure the interior angles and find their sum. 14. Plat a right triangle having a base of 16.25 ch. and a perpendicular of 8.60 ch. Find the remaining side and angles of the triangle. II. COMPUTING AREAS. In land surveying the areas are computed in triangles and quadrangles. If a field has more than four sides, in making the computation it is parted off into triangles and rectangles or trapezoids, the area of which is com- puted and their sum taken. 1. Area of Triangles. 1. To find the area of a right angled triangle. Multiply the base by one half the perpendicular. COMPUTING AREAS. 123 2. To find the area of an oblique angled triangle. CASE IST. When the sides are given. Let A, B, C represent the angles, and a, 6, c the sides opposite them. a-f 6-f- c Let = s. Let x = area. FIG. 37. 2 Then x = ^/X* ) (* 6 ) ) CASE 2^D. Having two sides and the included angle Let a, 6 be the sides. C the given angle, and x = area. From J? drop a perpendicular, d, to the side 6. This divides the triangle into two right triangles, the area of each of which equals its base multiplied by half the perpendicular, d, and the sum of their areas equals the sum of their bases multiplied by half the perpendicular; bd ab sin C that is, x . But d = a sin C. Hence, x = . 2 2 CASE 3D. Given two angles and the included side. Let A and B be the angles, and c the side given, Find C = 180 (A -f S). Find 6. c sin .B 60 sin J. Sin C : sin B : : c : 6 :. b = x = . sin C 2 CASE 4TH. Given two angles and a side opposite, (A, B and a.) a sin C Find C = 180 (4 -f -B). Find c = . sin A a sin .5 6c sin A Find 6 = . Then x = . sin^L 2 2. Areas of Quadrangles. CASE IST. Squares and rectangles. Multiply the base by the perpendicular. A MANUAL OF LAND SURVEYING. CASE ZNV.Trapezoids. A trapezoid is a figure having four sides, only two of which are parallel. Its area is equal to the half sum of the parallel sides, multi- plied by the perpendicular dis- tance between them. Trapezoid. FIG. 38. CASE 3RD. Trapeziums have no two sides parallel. The area is found by parting off into triangles and comput- ing their areas. 1. Having the sides and Trapezium. FIG. 39. angles given. Let A, B, C, D represent the angles, and a, 6, c, d the sides of the trapezium. Let AC be a diagonal dividing the trapezium into the triangles ABC and ADC. In each of these we have two sides and an included angle given; ab sin B cd sin D hence, x = 1 . 2 2 2. Given the diagonals of a quadrilateral and an angle formed by their intersection, to find the area. Solution. Let ABCD be the quadrilateral, m and n its diagonals, and an angle at which the diagonals intersect. FIG. 40. By Case 2nd, under "Area of Triangles," area AOB= \AOXBO sin " DOC = 1 CO X DO sin " BOG = 1 CO X SO sin 0. Whence, by addition, area ABCD \ (AO + CO) X (BO + DO) sin 0, mn sin or, area ABCD = COMPUTING AREAS. 125 Example. The diagonals of a four-sided field were found to measure 18 ch. and 24 ch. Setting a compass at their intersection, the bearings of two adjacent corners of the field were found to be N. 30 E. and S. 50 E. Required the area of the field. Solution. Applying logarithms in the above formula, having found = 99|, we have i = 18 log 1.255273 n = 24 log 1.380211 = 99 log sin 9.994003 2 ar. co. log 9.698970 Area = 213.03 log 2.32S457 or, Area = 21.303 A. 5. Given three sides, a, 6, d, and the included angles, A *nd D. (See Fig. 39.) Let AC = e, be a diagonal. Let the angle BCA = E, BAG = F, and CAD=G. In the triangle ABC the sides a, b and angle B are known. In the triangle CAD the side d only is known. It is required to find the side e and the angle G. To find G : E + F = 180 B. By tri go- tan E -f F tan E F nometry, a + 6 : a b :: - : - , by 2 2 which we find the sum and the difference of the angles E E+F EF and F. ---- = F, and G = A F. 2 2 6 sin B To find e : Sin F : sin B : : b : e :. e - . 4. This method of finding the area of a trapezium may be applied to polygons of any number of sides, when the sides and angles are given. The polygon is divided into triangles two less in number than the number of sides Each triangle has two sides and the included angle given or readily found. 126 A MANUAL OF LAND SURVEYING. Take for example the irregular polygon of eleven sides shown in Fig. 41, which is divided into nine triangles. In the triangles A, B, C and 7) two sides and the included angle of each are given. From the remaining sides and angles we find two sides and the in eluded angle of the triangles E and F, and so each triangle in turn furnishes the data for computing the adjacent triangle, till all are complete. 3. Offsets. When it is desired to find the area of a field having irregular sides, such as along a stream or lake, it is well to run a straight line where most conve- nient to do so, and then run and measure perpendiculars to the margin of the field. These are called offsets. They divide the space between the straight line. and the margin of the field into triangles and trapezoids, whose areas may be computed separately and the sum taken. If the offsets are equidistant the area may be found by the following RULE. From, the sum of the offsets, subtract the half sum of the extreme ones, and multiply the remainder by the common distance between them. 4. What is the area in acres of the following rigM angled triangles? 1. Base = 23.20 ch., perpendicular = 14.60 eh.? Ans. 16.936 .4 2. Base = 19.46 ch., perpendicular = 12.18 ch. ? What is the area, in acres of the following oblique angled triangles : (See Fig 37.) 3. a = 14.26 ch., 6 = 19.40 ch., c = 12.18 ch. ? Ans. 8.666 A 4. a= 9.43 6 = 11.61 " c= 8.42 " COMPUTING AREAS. 127 5. a= 6.23 " 6 = 14.26 " (7 = 22 40'? Am. 1.11+ A. 6.a = 12.20 " 6 = 20.00 " C=36 15'? 7. ^ = 16 45', .B = 8230', c = 21.16ch.? Am. 6.458+ A. 8. J. = 35, # = 62 42', c = 18.20 " 9. ^1 = 46, 5 = 58 15', a = 26^50 " An*. 40.264 4. 10. A = 37 20', B = 72 40', a = 19.36 tt 11. A square field is 6.25 chains on a side. Required its area. 12. A square field contains 20 acres. What is the length of its sides? Ans. 14.142 ch. 13. What is the area of a rectangle whose sides are 16.41 and 8.26 chains ? 14. A rectangular field containing 16 acres measures 12.50 chains on the base. What is the perpendicular ? Ans. 12.80 ch. 15. Commencing on the margin of a river a line was run across a bend 20.00 chains to the margin. Commenc- ing at the end of the second chain, offsets were taken every two chains, to the margin of the river, as follows: 1.61 ch., 2.27 ch., 3.72 ch., 1.96 ch., 4.23 ch., 2.92 ch. ? 3.26 ch., 2.50 ch. and 1.25 ch. Required the area between the line and the river. Ans. 4.744 acres. 16. Required the area of a field bounded as follows: 1st. North 17.65 ch. 2nd. 8. 36 12 / W. 8.20 ch. 3rd. S. 12 34' W. 7.26 " 4th. S. 58 26' E. 7.53^ " SUGGESTION. First : Change bearings into angles between the lines and compute as two triangles. Second: Take the first line as a base, divide the figure into two right angled triangles and a trapezoid, and com- pute the area. Compare the two methods as to number of figures required for the solution. 17. The sides of a pentagon measure 6.25 chains each. What is its area ? SUGGESTION. Part the figure into three triangles and compute. Also part into five isosceles triangles. Com- pute and compare the two methods. 128 A MANUAL OF LAND SURVEYING. f 5. 1. Rectangular Coordinates. Let XX' and YY' be two lines intersect- ing each other at right angles, as at 0. Let P : , P 2 , P 3 be any p points in the plane of the j I lines. j LetP^, P 2 a 2 PaQa De =; "-or ^ perpendiculars from the points upon the axis XX' , FIG. 43. and P^, P 2 6 2 , P 3 6 3 be perpendiculars from the points upon the axis YY'. The distances Oa :i 2 , Oa 3 are called Abscissas of the points P lt F ? , P 3 ; and the distances Ob^ Ob 2 , Ob 3 are called Ordi nates of the points. The point is called the Origin. The abscissa and ordinate of a point are together called Coordinates of the points. Coordinates at right angles with each other are called Rectangular Coordinates. It is customary to denote abscissas by x and ordinates by y, coordinates of different points in connection with each other being distinguished by use of subscripts. Thus, of the point P lt the coordinates Oa-^ and 06j or a 1 P l may be denoted by tf; t and y l ; of the point P 2 , the coordinates Oa 2 and 06 2 or a 2 P 2 may be denoted by x 2 and2/ 2 ; and so on. It will be seen that the coordinates of a point afford the means of locating it with respect to the axes. The use of longitude and latitude in Geography is an illustration. By use of the signs 4- and , the coordinates of any point in the plane of the axes are readily expressed. COMPUTING AREAS. 129. EXERCISES. 2. 1. Construct the point of which x = 4 and y = 7. 2. Given # = 5 and y = 3, to construct the point. 3. Given x = 3 and y = 6, to construct the point. 4. Given x = 6 and y = 4, to construct the point 5. Given a- = 0, y = 2; or = 5, y = 0; x = 0, y = 0. Required the points, 3. Application to Area. Let it be required to find the area of a series of trapezoids included between perpendiculars from the points of a broken line upon a straight line. Suppose the straight line, as 0J7, to be an axis of abscis- sas, and the first perpen- dicular at the left, as OA, to be an axis of ordinates. FIG. 44. Let x it x 2 ,x a , etc., be the abscissas of the points A, t C, etc., and yi,y 2 ,y a , etc., the corresponding ordinates. Accordingly, the area of the several trapezoids is \ [a (2/i + #2) + (#3 a? a ) fa* + #3) + (a? 4 a? 8 ) (2/3 + y*) H ---- (av, #n-i) &n-i in which n is the number of trapezoids plus one. The above formula may be changed to the form Whence, for the area included between a straight line, as a base, and a broken line whose points are given by their coordinates upon the base, we have the following RULE. From each ordinate subtract the second site- ceeding one and multiply the remainder by the abscissa corresponding to the intervening ordinate. Also, multiply the sum of the last two ordinates by the last abscissa. Divide the algebraic sum of the products by 2. 130 A MANUAL OF LAND SURVEYING. The above formula and Rule have been deduced independently of any supposition as to the relative directions of the parts of the broken line. They are therefore true whatever may be the form of the broken line. That is, whether any part should be perpendicular to the-base, either toward or from it, or whether any part should be turned back- ward respecting the preceding one. SUGGESTION. Let the student verify the rule in a case, for example, like the following, in which BC is represented as being parallel to the base, CD as perpendicular toward it, and FG as being I turned backward from EF. Find how it would be, if one or more of the ordinates were zero; if one or more were negative. 4. 1. Given y also x l = 10, x 2 area. EXERCISES. - 12, */ 2 = 12, y z = 16, */ 4 = 8 and 7/ 5 = 6 t 18, #3 = 24, x = 30 and x 5 = 20, to find Given the following, to find area : (2) (3) (4) 140 435 250 200 320 1000 000 950 812 240 844 725 306 530 500 G40 325 450 415 200 000 000 000 1000 1150 828 650 460 000 200 317 420 305 524 250 5. As a second example of the application of coor- dinates in finding area, let there be taken an ordinary polygon, as APCDEF. (Fig. 46.) Let a?!, x 2 > X 3* e ^c., be the abscissas of the points A, B, C, etc,, and y lt y z , y 3 , etc., tae corresponding ordinates. COMPUTING AREAS. 131 7 d d f 7 P, (I C, I sin b tan 6 Z 4 C,l 6, d cos 6 = d = v"c 2 I 2 . G d 5 c,d b,t sin 6 = 1 i/c 2 d 2 . G d 6 I, d 0, C tan b = c = i/Z 2 -f d' 2 . I The Traverse Table. This table, which is given with others in the back part of the book, shows the lati- tude and departure for any bearing to each quarter degree for any distance from 1 to 10. For other distances, the latitude or departure is found by adding the latitudes or the departures of the partial distances, as shown in the following EXERCISES- 9. 1. Find the latitude and the departure for a bearing of 24, for a distance of 7 ch.; for a distance of 5 eh.; for a distance of 10 cK COMPUTING AREAS. 135 2. Find the latitude and the departure on a bearing of 371, for a distance of 12 ch. OPERATIONS. For 37i, distance 10, lat. = 7.9600 dep. = 6.0529 " " " 2, " =1.5920 " =1.2106 " " 12, " =9.5520; " =7.2635. 3. Find the latitude and departure on a bearing of 40f for a distance of 17.23 ch. OPERATIONS. For 40f, distance 10, lat. = 7.5756, dep.= 6.5276 " " " 7, "< == 5.3030, " = 4.5693 .2 " == 0.15151, = 0.13055 " " " .03" = 0.022727, " = 0.019583 " " " 17.23" =13.053, " =11.247 ANOTHER FORM OF WORK. Bearing. Distances. Latitude*. Departures. 40| 1000 07576 06528 700 53030 45693 20 15151 13055 3 22727 19583 1723 1305.3237 1124.7433 "We take the distance in links, and write the latitude and departure for the first figure of the number, omitting the decimal point ; we write under them the latitude and departure for the second figure, setting them down one place farther toward the right ; under them, the lati- tude and departure for the third figure, setting them one place farther toward the right, and so on. We then add the separate latitudes and separate departures, and point off four figures from the right. The results thus obtained are the latitude and departure sought, as expressed in links. Notice that bearings from 45 upward are found in the right hand column of the table, and the columns of latitude and departure are denoted at the foot of the page. Care needs to be taken here to avoid mistakes of latitudes for departures and departures for latitudes. 136 A MANUAL OF LAND SURVEYING. Find the latitudes and departures for the following bearings and distances : (1) Bearing 52|, Distance 437. (2) Bearing 65, Distance 3669. (3) Bearing 21 f, Distance 2030. (4) Bearing 40, Distance 506. (5) Bearing 81|, Distance 12.34 ch. 1O. Meridian Distance. The distance of a station or any point from the principal meridian is called its Meridian Distance. The meridian distance of a line is the meridian distance of its middle point. If the meridian passing through the extreme easterly or west- erly station of a survey around a tract of land be taken as a base and perpendiculars be drawn from it to each station of the survey, the tract and the space between, it and the meridian will be divided into triangles and trapezoids whose areas are readily computed. Beginning with the station through which the me- ridian passes which we call Sta. 0, then the meridian distance of Sta. 1 will equal the departure of the first course. The meridian distance of any station will equal the alge- braic sum of the departures of all the preceding courses up to that point. The meridian distance of any course or line will equal the half sum of the meridian distances of the stations at the two ends of that course or line. The area of any triangle or trapezoid thus formed will equal the product of the latitude of the line or course on which it is based multiplied by the- meridian dis- tance of that line. The area of the tract is equal to the sum of the areas of all the triangles and trapezoids thus formed minus the sum of the areas of those triangles and trapezoids which lie outside the lines of the survey. The area of the tract is also equal to the difference between the sums of those areas found from latitudes which are northings and of those where they are southings. COMPUTING AREAS. 137 We will now apply the foregoing principles to find the area of the tracts described in the following Field Notes and shown in the figure. On the figure each sta- tion is numbered to correspond with the field notes and each line is also numbered in its order as run. The sev- eral triangles and trapezoids formed by perpendiculars from the stations to the meridian are lettered in their order. Station Bearing Distance N. 26i E. 12.00 ch. 1 X. 59 E. 9.80 " 2 S. 66 E. 19.60 / 3 S. 35 W. 15.68 " 4 S. 66 W. 13.12 " 5 N. 46 W. 11.72 H Bearing 26i Dist. 12.M Lat. 10.74 N. " 59 9.80 " 5.05 N. " 66 " 19.60 7.97 S. " 35 " 15.68 " 12.85 S. " 66 " 13.12 " 5.34 S. " 46 " 14.72 " 10.36 N. Finding from the Traverse Table the latitudes and departures to the nearest link, we have Dep. 5.35 E. " 8.40 E. " 17.91 E. " 8.99 W. " 11.99 W. " 10.59 W. Obviously, in going entirely around a field there should be made the same southing as northing, and the same westing as easting. But from unavoidable lack of pre- cision in the use of instruments, this is practically seldom found to have been, done, according to the figures used. The error, however, can usually be made very small. Finding it large, the entire field work should be reviewed. It is not a settled point among surveyors how great an error of lati- tude or departure may be allowed without resurveying the lot. Some would admit a difference of one link for every three chains in the sum of the distances, others for every five chains, and again others would require it to be within one link for every ten chains. 138 A MANUAL OF LAND SURVEYING. As a check against errors of bearing, a back sight should be taken at every station, and the reverse bearing compared with the corresponding direct bearing of that station. If the two are found to differ considerably, both should be reviewed. Let us now see how small an error of latitude and of departure we have in the present ca.se. Sum of northings = 10.74 + 5.05 + 10.23 = 26.02. " " southings = 7.97 + 12.85+ 5.34 = 26.16. Difference of latitudes = 00.14 = error of latitude. Sum of eastings = 5.35 + 8.40 + 17.91 = 31.66. " westings = 8.99 + 11.99 + 10.59 = 31.57. Difference of departures = 00.09 = error of departure. The above errors may be considered reasonably small for a field of the size of the present one. In practice, some of the courses may have been measured over rough or uneven ground, and, accordingly, such courses should beai a larger proportion of the error. Some of the bearings may have been taken with an indistinct sight, which would dictate the allotment of more than a proportionate amount of the error to them. Distances as measured over uneven ground are liable to be too long. In such cases, the length of a course may be diminished when such change would favor the balancing. Similarly, a doubtful bearing may be changed, if the error should appear to be attributable to it. It is a common mistake to reverse the position of the latitude and departure in the columns. If the bearing is greater than 45 the departure is greater than the latitude, and it is less when the bearing is less than 45. Scan the columns for such errors. 11. Balancing. The next work is to distribute the errors among the several courses in proportion to their lengths, in accordance with the following PRINCIPLE. As the sum of the lengths of all the courses is to the length of each course, so is the total error to the error of that course. This operation is called Balancing. Applying the above principle, we divide the errors by the sum of the lengths of all the courses and multiply the quotients by the length of each course, indicating the products as positive or negative, accordingly as they are to be added or subtracted in making the required correc- tion. COMPUTING AREAS. 139 Thus, 00.14--84.92=00.00165; and 00.09--84.92=OO.OOT06; 00.00165X12=00.0198 or +00.02; and 00.00106X12=00.01272 or 00.01, to the nearest link. In the same manner, by multiplying the above quotients by the lengths of the other courses, the correction for them is readily obtained. Collecting results thus found, we have the following TABLE I. Sta. Latitude. Departure. Cor.L CorD Balanced. N. s. E. W. N. S. E. W. 1 10.74 5.35 +.02 .01 10.76 5.34 2 5.05 8.40 +.02 .01 5.07 8.39 3 7.97 17.91 .03 .02 7.94 17.89 4 12.85 8.99 .03 +.02 12.82 9.01 5 5.34 11.99 .02 +.01 5.32 12.00 6 10.23 10.59 +.02 +.02 10.25 10.61 We next find the Meridian Distance of the several stations. M. D. of Sta. l=Dep. of Course 1=5.34. M. D of Sta. 2=M. D. of Sta. 1+Dep. of C. 2 =5.34+8.39=13.73. M. D. of Sta. 3=M. D. of Sta. 2+Dep. of C. 3 =13.73+17.89=31.62. M. D. of Sta. 4=M. D. of Sta. 3 Dep. of C. 4 =31.629.01=22.61. M. D. of Sta. 5=M. D. of Sta. 4 Dep. of C. 5 =22.6112.00=10.61. M. D. of Sta. 0=M. D. of Sta, 5 Dep. of C. 6 =10.6110.61=0.00 M. D. Of C. 1= M.D.Sta.O+M.D.Sta.l = 0+5.34 = 2= 3= 31. D. Sta. I+M. D. Sta. 2 M. D. Sta. 2+M. D. Sta. 13.73+31.62 = 9.535. .675. 140 A MANUAL OF LAND SURVEYING. M. D. M. D. Sta. 3+M. D. Sta. 4 u r a M. D. Sta. 4+M. D. Sta. 5 " 6 2 M. D. Sta. 5+M. D. Sta. 2 31.62+22.61 _ 9 - ~ . g We may now put the whole matter in compact tabu- lar form as follows. i S. Ml u S3 1y -3 1 1 CO "o PS i I cj OS CC JH -w a*" OJ p go e 02 M P O i-5 p & K 03 N. S. E. W. N. 26iE. 12.00 10.76 5 34 2.67 28.7292 1 N.59 E. 9.80 5.07 8.39 13,73 9.535 48.34245 2 S.66 E. 19.60 7.94 17.89 31.62 22.675 180.0395 a S. 35 W. 15.68 12.82 9.01 22.61 27.115 347.6143 4 S. 66 W. 13.12 5.32 12.00 10 61 16.61 88.3652 5 N.46W. 14.72 10.25 10.61 0.00 5.305 54.37625 26.0826.0831.62 31.62 131-4479 616 0190 131.4479 484.5711 = Acres 48.45711 In this example the area of the tract is evidently equal to the sum of the areas of the trapezoids c d and e based on courses 3, 4, and 5 minus the sum of the areas of the triangles and trapezoid a b and / based on courses 1, 2, and 6. The area of the triangle a equals the M. D. of course or line 1 multiplied by its latitude = 2.67 x 10.76. The area of the trapezoid b equals the M. D. of course 2 multiplied by its latitude = 9.535 X 5.07. In a similar manner we find the area of each triangle and trapezoid. Examples for Solution: The f Rowing exainples are taken from the field notes of the original United States Surveys in Michi- gan and are fair samples of the average work done on the government >and surveys. The meanders of lakes and streams are run for the purpose of finding how much dry or uncovered land is contained in the ad- jacent tract to be paid for by the purchaser. COMPUTING AREAS. 141 Ex. 1. Meanders of a Lake in Section 5. Began at post corner to Sections 4. 5, 8, and 9, thence in Section 5, N. 60 W. 6.50 cli. to S. K Margin of Lake, thence in Sec. 5, N. 25 E. 4.00 ch., thence, N. 51 W. 5.0Q ch.. thence N. 18 W. 7.00 ch., thence X. 3 W. 7.00 ch. r thence N. 63 W. 10.00 ch., thence S. 79 W. 6.00 ch., thence S. 7 W. 13.00 ch., thence S. 20 E. 6.00 ch., thence S. 6 W. 5.00 ch., thence N. 78 E. 14.00 ch., thence S. 27 E. 5.00 ch., thence N. 71 E. 3.87 ch. to place of be- ginning on margin of Lake. Find the area of the lake. Also find the areas of the North and South halves respectively of the quarter sec- tion in which the lake lies, on the supposition that the quarter section is just 40 chains square and that the lines are run with the same variation of the needle as was used in meandering the lake. These areas are given in the official plat as follows: North i, A. 66.18. South i, A. 55.92. 2. Find the area of the lake described in the exam- ple 13, page 109, also the area of each of the quarter- quarter sections adjoining the lake in the south half jf Sections 11 and 12. These areas are marked in the official plat as follows : In Section 11, S. E. i of S. E. i A. 31.50, N. E. i of S. E. A. 20.40. In Section 12. S. W. i of S. W. i A. 37.61. X. W. i of S. W. i A. 27.10. The meander post at the beginning of the survey is 14.00 chains North from the Section Corner. 3. Meander of a Lake in section 2. Began at quarter post in line of Sections -2 and li, thence North 10.00 ch., to S. margin of Lake, thence in Sec. 2, thence S. 57 E. 13.00 ch., thence E. 3.00 ch., thence N. 45 E. 5.00 ch., thence N. 4 W. 6.00 ch., thence N. 70 W. 15.00 ch., thence S. 80 W. 6.00 ch., thence S. 24i E. 7.17 ch., to place of beginning in margin of Lake. Find the area of the Lake also the area of the W. i of S s . E. i' of Section 2 arid of the S. E. i of the S. W. i ol the Section. The first is given on the official plat as A. 62.8S and the latter as A. 38.95. 142 A MAiOJAL OF LAKD SUKVEYING. 13. Problem. Given the bearings of the sides of a field, to find the bearings when the field is supposed to be revolved so as to cause one of the sides to coincide wiih a meridian. EXAMPLES. 1. The bearings of the sides of a field are, 1st, K 12 E., 2d, N. 83 E., 3d, S. 21 W., and 4th, N. 47 W. What will the bearings be, if the field be supposed to be revolved so as to cause the first side to be on a meridian ? Ans.~ 1st, N., 2d, N. 71 E., 3d, S. 9 W., and 4th, N. 59 W. SUGGESTION. Suppose the field to be revolved toward the left, through an angle of 12. Accordingly, each bearing would be changed by that amount. The readings of the new bearings are readily deter- mined by inspection. 2. The bearings of the sides of a field are 1st S. 3| W., 2d N. 86| W., 3d N. 16 E., and 4th E. Required the new bearings when the first side is made to coincide with the meridian. Ans. 1st S., 2d W., 3d X. 13 E., and 4th N. 86| E. 3. The bearings of the sides of a field are 1st S. 20 W., 2d S. 70 W., 3d N. 31 W., 4th N. 45 E., and 5th S. 60 E. Required the new bearings when the third side is made to coincide with the meridian. Ans. 1st S. 51 W., 2d N. 79 W., 3d X., 4th N. 76 E., and 5th S. 29 E. 4. The bearings of the sides of a field are, 1st K 45 E., 2d IS. 30 W., 3d S. 5 E., 4th W., and 5th N. 20 E. What will the bearings become, if the field be revolved so as to bring the third side to the meridian ? Ans. 1st N. 50 E., 2d S. 35 W., 3d S., 4th N. 85 W., 5th N. 25 E. 5. The bearings of the sides of a field are, 1st E., 2d N. 9 E., 3d S. 69 E., 4th S. 66 E., 5th S. 42 W., 6tb S. 75 W., 7th N. 39 W., and 8th N. 42 E. What will the bearings become, if the field be revolved so as to cause the fourth side to coincide with the meridian ? Ans. 1st S. 24 E., 2d N. 75 E., 3d IS. 3 E., 4th S., 5th K. 72 W., etc. Additional exercises may be formed from the above by requiring different sides to be brought to coincide with the meridian. COMPUTING ABEAS. 143 RULE. Change each bearing agreeably with the direc- tion in which the field is mpposed to be r: wived by an amount equal to the bearing of the side which is brought to the meridian, and express the result in accordance with the proper form of denoting bearings. 6. What were the bearings of the sides of a field which are now N. 16| E., E., S. 3 W., and X. 86| W., the vari- ation of the needle having changed 2 toward the west since the former survey ? Supplying Omissions. From inaccessibility of lines and sometimes from accident, omissions may occur in the field notes of a survey. In a closed survey, any two omissions may, in general, be supplied by computa- tion. It is, however, desirable to avoid as far as possible the necessity of supplying omissions in this manner, since it infringes upon the tests which otherwise serve to verify the work. The several cases which may occur are presented in the following problems: 14. Prob. 1. To find an omitted bearing and distance. CASE 1. When the omissions pertain to the same course. In a closed survey, the sum of the northings should equal the sum of the southings; and the sum of the east- ings should equal the sum of the westings. The defect of these equalities in the present case must be on the one hand the latitude and on the other the departure of the omitted course. Example. Sta. Bearing.' Dist. Lat. Dep. A N.31W. 9,40 -B.057 4.841 B N.45E. 9.30 -f-6.576 -f 6.576 C Oniii ted. E S. 20 W. 5.30 4.980 1.813 F S. 70 W. 10.90 -3.728 10.243 144 A MANUAL OF LAND SURVEYING. Solution. Sum of northings = 14.633 of southings = 8.708 Diff. = CG = 5.925 Sum of westings = 16.897 of eastings = 6.576 Diff. = GE = 10.321 FIG. 48. The latitude of the omitted course is thus a southing and its departure, an easting. Its bearing is therefore S. E. To find the bearing or angle GCE, we have GE 10.321 tan GCE = = = 1.74194. CG 5.925 Whence, GCE *== 60 8'; or the required bearing ii . 60 V E. To find the distance CE, we have CE = (5.925 2 + 10.321 2 )* = 12.00. REMARK. It will be noticed that a plat of the field may be made, and the area found without supplying the omissions. CASE 2. When the omissions pertain to different courses. If the field be supposed to be revolved until the side whose length is omitted becomes a meridian, the given bearings being changed accordingly (Art. 13, Prob.), then, since the departure of the side made a meridian is 0, the difference between the sums of the eastings and westings of the other courses is the departure, in its new position, of the side whose bearing is omitted. COMPUTING AREAS. 145 Knowing the length and th departure of this side, its latitude and bearing may be found, (Art. 8). The difference between the sums of the northings and southings of the courses in their new positions, is the length of the side which was made a meridian. Example. Sta. Bearing, g-^ Distance. Lat. Dep. A N. 20 E. North. Omitted. 0.0000 B X.45E. N. 25 E. 8.00 +7.2505 +3.3809 C S. "30 W. S. 10W. 5.00 4.9240 0.8682 D Omitted. 7.20 E West. S. 70 W. 5.92 2.0248 5.5630 Solution. Sum of eastings == 3.3809 " " westings = 6.4312 Difference = 3.0503 (an easting). Latitude of DE = (7.20 2 3.0503 2 )* = 6.5219 (a southing). Sine of changed bearing of DE = 3.0503 -*- 7.20 = 0.42365. Whence " " DE is S. 25 V E. Whence original " " DE was S. 5 V E. Sum of northings = 7.2505 " " southings *-= 13.4707 Difference = 6.22 = length of AB. REMARK. It is sometimes doubtful whether the latitude of the course whose beariug is omitted is a nortliing or a southing. In the present case, the question is determined by a simple inspec- tion of the latitudes, since the sum of the southings is less than the sum of the northings, without considering the northing of the first course. In other cases, there may be two sets of values of the omitted parts. with either of which the problem is satisfied. Practical^, however, the ambiguity is removed by a general knowl- edge wlueh the surveyor has of the directions of the lines. 10 146 A MANUAL OF LAND SURVEYING. 15. Prob. 2. To find the omitted lengths of two courses. CASE 1. When the courses are consecutive. The bearing and length of a line which would close a survey, leaving out the unknown sides, may be found by Prob. 1, Case 1. This line and the unknown sides form a triangle in which the angles, as found from the given bearings, and the length of one side are known. The lengths of the other sides may therefore be computed. The procedure will be readily worked out by the stu- dent, without illustration. CASE 2. When the courses are not consecutive. This case may be treated in the same manner as the preceding. Or, we may suppose the field to be revolved so as to make one of the sides whose length is omitted, a merid- ian, the bearings of the other sides being changed accord- ingly. We may then find the difference of the sums of the eastings and westings, which will be the departure, in its new position, of the other side whose length is wanting. Having the bearing of that side and its departure, its length and latitude may be found. Finding the differ- ence between' the sums of the northings and southings, we obtain the length of the side which was made a meridian. Example. Sta. Bearing. Changed Bearings. Distance. Lat. Dep. A N. 15 E. Nv30 W. 5.00 -f 4.33 2.50 B N. 45 E. North. ' Omitted. 0.00 C S. 55 E. N. 80 E, 10.05 4- 1-75 +0.90 D S. 15 W. S. 30 E. 12.25 ; 10.61 +6.12 E S.75W. S. 30 W. Omitted. F N.33%W. N.785W. 9.96 -f 1.95 9.77 COMPUTING AREAS. N 147 Sum of eastings = 16.02 " " westings = 12.27 Difference = 3.75 = Dist. X sin 30. Whence, length of EF = 3.75 -f- 0.5 = 7.50. Lat. EF = 3.75 + tan 30 *= 6.50. Sum of northings = 8.03 " " southings = 17.11 Difference = 9.08 = length of BC. REMARK. If the sides whose lengths are omitted are parallel, the problem is indeterminate. 16. Prob. 3. To find the omitted bearings of two courses. We find, (Prob. 1, Case 1) the bearing and length of a line which would close a survey, having the lines whose bearings are given as the other sides. The line thus found and the two lines whose bearings are omitted form a triangle. The lengths of the sides of the triangle being known, its angles maybe found; and from the angles and the bearing of one of the sides the bearings of the other sides may be found. The closing line and the triangle are illustrated by the diagram accompanying the following Example. Sta. Bearing. Dist. Lat. pep. A N. 15 E. 5.00 + 4.8296 +1.2941 B Omitted. 9.08 c S. 55 E. 10.05 5.7645 +8.2325 D S. 15 W. 12.25 11.8327 3.1705 E Omitted. 7.50 F N. 33?i \V. 9.96 -f 8.2814 5.5334 I 1(1 148 A MANUAL OF LAND SURVEYING. The side EF, without change of bearing, is represented by CG. BG is the closing line of the field ABGHF, in & , " / ^X^ , which we have Sum of northings = 13.1110 southings = 17.5972 Difference = 4.4862 (a northing). Sum of eastings westings 9.5266 8.7039 Difference = 0.8227 (a westing.) Whence (Prob. 1), bearing BG is K. 10 23' 30" W., and length BG is 4.56. In the triangle BGC, BC = 9.08 and CG = EF '= 7.50. Solving the triangle, we find angle GBC = 55 25' 40", ana angle BGC = 94 31' 49". Whence, bearing BC is N. 45 2' 10" E., and bearing EF is S. 75 V 41" W. REMARK. The problem may possibly have two solutions, accord- ingly as the triangle may fall on either side of the closing line. The ambiguity is, however, practically unimportant. Exercises. To be made by the student in the field. 17. Most of the foregoing problems for finding areas may be simplified and much labor saved in calculation, by reducing the irregular polygons and oblique triangles to right triangles and trapezoids on the plat, and taking their dimensions by direct measurements from the plat, instead of calculating them. If the plat is made on a large enough scale showing not more than four chains COMPUTING AREAS. 149 to the inch and the drafting is carefully done, tne meas- ures on the plat will be very nearly if not quite as good as those taken on the ground, and will give results suffi- ciently close for most purposes. IST METHOD. Draw a diagonal between two distant angles of the figure, and perpendiculars to it from the other angles. FIG. so. 2ND METHOD. Reduce the figure to a single equivalent triangle. FIG. 51. 150 A MANUAL OF LAND SUEVEYING. 1. To reduce the trapezium abed (Fig. 51) to its equiva- lent triangle. Produce the line ab an indefinite distance. With the parallel ruler, or straight edge and triangle, find the point e, where a line through d parallel to ca intersects the line ab. Draw the line ec, intersecting ad at g. Then the triangle ecb is equivalent to the trapezium abed, for the triangles acd and ace, having the same base ac and equal altitudes, are equal; and the triangle aeg being taken from both leaves the triangle eag, which is added to the original figure, equal to the triangle cdg, which is taken from it. The perpendicular may now be drawn from c, and the base eb and altitude fc measured on the plat. 2. By an extension of the same process, any polygon may be reduced to one or more equivalent triangles. It will frequently be found convenient to divide the figure into two or more parts, and reduce the sides separately. The process is indicated in Figure 52. FIG. 52. Let abcdefgh be the polygon to be reduced. Extend one side, as ab, indefinitely for a base. From c draw ci COMPCTTIXG AREAS. 151 parallel to bd. From d draw dk parallel to ei. From e draw el parallel to fk. Having selected / as the vertex of the triangle, we next draw/? for one of its sides. Next, from h draw Jim parallel to ga. From g draw gn parallel to fin. From/ draw fn for the third side of the triangle, and fo, its altitude. The triangle fin is equivalent to the polygon abcdefgh. It is best to draw all these lines lightly on the plat, to avoid errors. If we consider each point, i, k, I, marked in succession on the base as an angle of the polygon, which it is until its successor is located, we have the following GENERAL RULE. Extend one side indefinitely as a base. Commencing at the first angle from the base, draw from it to the base a line parallel to a line joining the two adjacent angles of the polygon. Continue draioing lines to the base from each angle in succession as far as re- quired. Join the last angle from which a parallel was taken, with the last point of intersection on the base, for a side of the final triangle. It is sometimes more convenient not to produce one of the lines of the figure for a base, but to draw a perpen- dicular to it from one end or from the end produced. The same rule applies. 18. The preceding methods of taking measurements from the plat are found very convenient in estimating the area of land benefited by drainage, under the drain laws. Surveyors are frequently called on to make surveys and maps of drainage districts, showing the location of the drains and the location and area of the lands, belong- ing to the various owners, which will be benefited by the drainage. In most, if not all these cases, no man can tell, either before or after the drainage has been executed, just exactly where the dividing line is, between land which is benefited and that which is not benefited. For this rea- son a rapid survey of the approximate line, by stadia 152 A MANUAL OF LAND SURVEYING. measures, is just as good as the most elaborate work with the chain or tape. The one is likely to. get as near the true dividing line as the other. The writer has found the following method to work well in his practice. Suppose a tract of marsh or swamp is to be measured and mapped, having more or less cleared upland around it: Assume some line as a base. A section line or quarter line of the United States Survey answers well for this purpose. From this base run a broken line around the swamp wherever it is most convenient to do so. Set a stake at each angle in the line. Note the length of each course and the angle which it makes with the common base, as described on pagelOl . When the circuit of the swamp has been made, and the transit again set up at the starting point, the work will prove itself. After taking a back sight on the last sta- tion and pointing the telescope along the base line, if the work has all been correctly done, the vernier should give the same reading as it did to start with, showing that just 360 have been passed around. In passing around the swamp an assistant with the stadia rod follows its margin, setting up his rod at every point where it changes its general direction. The transit- man notes down the direction of each point at which the rod is set up, by its angle from the base line and its distance from tl^e transit as read off from the rod. When as many points are iaken as are convenient from one station, the transit is moved up to the next one, and the operation continued till all the desirable points are located. This being done in the field, they are reproduced on the plat on a scale large enough to permit measure- ments on the plat with a reasonable degree of accuracy. The points along the margin of the swamp having been laid down on the plat, are connected by straight lines, and all intersecting farm lines or other points of interest are also laid down. COMPUTING AREAS. 153 We now have a map, showing as correctly as it is pos- sible to do so, the location of the swamp on each man's land. The areas of the several tracts are found by taking the parallel rule and needle point and reducing these irregular polygons to their equivalent triangles and rect- angles, making the necessary measures on the plat and computing the areas from these measures. 19. Division and Partition of Land. The sur- veyor is sometimes called on to divide areas into portions having a specified relation to each other, or to part off from a field a given number of acres by a line fulfilling some specified condition with respect to the field divided. / w %? There is a great variety of these problems, most of which occur very rarely in the surveyor's practice. A few of those which occur most frequently are given. Prob. 1. To divide a triangle into parts having a given ratio. CASE \By lines from an angle. Solution. Let ABC be any triangle, and suppose it is required to divide it by a line from B, into two parts having the ratio of m to n. Let BD be the line of division, so c that ABD : DBG :: m : n (1) But ABD : DBG :: AD : DO (2) Combining (1) and (2), we have AD : DC :: m : n, whence, AD : AC :: m : m+n, whence, AD ^= - . Similarly, DC = wi + n m + n Measure the distance AD thus found, and run the line BD. 154 A MANUAL OF LAND SURVEYING. If the triangle were to be divided into three parts in the ratio of m : n : p, we should have mXAC nXAC AD = and DE = - . m + n -\- p m 4- n + p Cor. To part off by a line, as BD, a given area a, we a X AC have AD : AC : : a : area ABC, whence AD . area ABC Examples. I. Find the measurements required to di- vide a trianglar field by lines from an angle to a side whose length is 12.30 -ch., into parts to each other as 2, 3 and 4. 2. Find the measurement required to part off 3.5 acres from a triangular field a side of which is 18.50 ch., and a perpendicular thereupon from the opposite angle is 10.40 ch. CASE 2.~ By lines parallel to a side. Solution. Let D be the point in the side AB from which a line parallel to BC shall divide ABC so that ADE : DECS :: m :n. Then /- Sj ADE : ABC : : m : m + n. <& <" But ADE : ABC :: AD 2 : AB*, FIG. 54. whence, AD 2 : AB 2 : : m : m + n, ( m giving AD = AB ] (m -f n Measure the distance AD thus found, and run DE parallel to BC. If the triangle is required to be divided into three parts in the ratio of m : n : p, we should have m ) x ( m -\- n f tm&AF = AB m-\-n-\-p Cor. 1. To part off a triangle, as ADE, of given area a rOMFUTING AREAS. we have AD = AE r ^ . j Cor. 2. To part off a quadrilateral, as DECB, of given area, a', we may find by Cor. 1 the distance AD required to part off a triangle of the area ABC a' and measure BD = BA AD. Examples. 1. Find the measurement for dividing a triangular field of 12 A. into parts in the ratio of 4 to 5 by a parallel run from a point in a side whose length is 10.35 ch. 2. Find measurements for dividing by parallels, the above field into three equivalent parts. 3. Find measurement for parting off from the same field by a parallel, a triangle of 5 A.; a quadrilateral of CASE Z.By lines perpendicular to a side. Solution. Let ABC be a triangle required to be divided by a perpendicular to AC, into parts having the ratio of ra to n. Let EF be the line of division, so that AEF : EBCF : : m : n, or AEF : ABC : : m : m -f n. (1) Let BD De a perpendicular upon A C Then AEF:ABC::AFXEF:ACXBD::m:m + n. (2) From similar triangles, AF : EF : : AD : BD, AFX 3D whence, EF = - . AD Substituting this value of EF in (2), we have : AC X BD :: m : m + w, AD or AF* :ACXAD ::m:m + n whence, AF = Find AD and then AF. Measure the distance AF and run FE perpendicular to AC. 156 A MANUAL OF LAND SURVEYING. Similarly, may be found the distances to perpendiculars dividing the triangle into three or more parts having a given ratio. Oor. To part ofE a triangle, as AEF, of given area, a, XAD X we have AF = area, A BCD a)* [ . ) The distance AF to a perpendicular which shall part off a triangle AEF = a, may be found otherwise, as follows: triangle AEF = %AF X EF = a, and EF = AF X tan A. Whence, AF = ' tan A Examples. 1. The bearings and lengths of two sides of a triangular field from the same corner are N. 20 E., 15 ch., and N 50 E., 20 ch. Required the measurement from that corner to a perpendicular upon the longer side which shall divide the field into two parts having the ra- tio of 2 to 3. 2. Required the measurement to a perpendicular which shall divide the above field into two equivalent parts; into three equivalent parts. 3. Required the measurement to a perpendicular which shall part off from the same field a triangle of 4 A.; a quadrilateral of 5 A. 2O. Prob. 2. To divide a trapezoid into parts having a given ratio. CASE I. By lines dividing the bases proportionally. Solution. Let ABCD be any trapezoid required to be divided into parts having the ratio Qfm\n:p. This is done in the easiest manner by dividing each base into parts having the ratio to each other as m, n and p, and join- ing the corresponding points of division. The measurements nec- essary to find.the points of division FIG. 56. are: COMPUTING AREAS. 157 m X BC nXBC m + n+p m + n-\-p m-\-n+p and FH = Cor. To part off a given area a by a line, as EF, which shall divide the bases proportionately, we have aXBC aXAD BE = and AF = . area ABCD area ABCD Examples. -1. Given AD, N. 80 E., '12.60 ch., AB, N. 10| E., 8.12 ch., and BC, X. 80 E., 10.34 ch., to find the measurements required in dividing the field into parts having the ratio of 4 to 7, by a line dividing the parallel sides proportionally. 2. Find the measurements for parting off from the above field an area of 5 A., by a line dividing the parallel sides proportionally. CASE 2. By lines parallel to the 'bases. A Solution. Let ABCD be a trape- zoid to be divided into parts in the ratio of m to n, by a line parallel to v ** 1^ \ Suppose EF to be the required ^ line of division, so that 1 :c - 57 - EBCF : AEFD ::m:n. Regarding the sides AB and DC as prolonged to meet at O, we have OAD : OBC : : AD 2 : BC 2 , whence, OAD OBC, or ABCD: OBC'.: AD* BC 2 : BC 2 . (1) Similarly, we have EBCF: OBC:: EF 2 BC Z : BC 2 . (2) Combining (1) and (2), ABCD: EBCF:: 158 A MANUAL OF LAND SURVEYING. Cm X whence EF = \- (a) I m -\-n ) Supposing BH to be parallel to CD, the triangles ABH and EBG give AB : AH :: EB : EG, or AB : AD BC :: EB : EF BC. AB(EF BC} Whence, EB = - - . (6). AD BC Thus, first finding EB by formula (a), we can then find EB by formula (6), and measuring that distance from B, we may run EF parallel to BC, dividing the trapezoid as required. Similarly, a trapezoid may be divided in three or more parts having a given ratio. Indeed, the above formulas may be directly applied to that purpose by making a simple substitution. Cor. To part off a trapezoid of given area a, adjoining BC, we obtain from formula (a) ( a X AD 2 -f (area ABCD a) BO 2 ) K EF = \ - I area ABCD ) '.The distance BE is then found from formula (6). Examples. 1. Given a trapezoidal field ABCD in which AB is an east and west line, 9 ch., BC a north and south line. 5.19 ch., and AD a north and south line, 8 ch., it is required to run a north and south line dividing the field so that the parts on BC and AD shall have the ratio of 2 to 3. , 2. Find the measurement from A to part off from the above field by a north and south line an area of 3 A. ad- joining AD. CASE 3. By lines perpendicular to the bases COMPUTING ABEAS. 159- Solution. Let ABCD be a trapezoid to be divided into parts in the ratio of m to. n by a line perpendicular to AD. L \ Let EF be the line joining the middle points of the non-parallel sides AB and CD. We divide EF, FIG. 57. as a ^ i n t o two parts having the ratio of m to n, and through & run HI perpendicular to AD. To find the point G on the ground, we have the forni- m(BC + AD) ula EG = - -- . Whence, measuring from E the 2 (m -f w) distance .## on the bearing of 13 C, we have the point sought. Cor. To part off a given area a, by a line perpendic- a (BC -f AD) ular to the bases, we have EG = ----- , 2 XareaJ^OD Or, denoting the altitude of the trapezoid by A, we a a have J57 E., 55 Iks. distant, burr oak, 16" " N.54 J L., US " In some surveys, such as laying out additions to cities or villages, or any similar work, it is better to make a rough sketch or plat of the work in the field book and mark the dimensions and directions cf lines on the plat. Field bocks which are ruled in small cross sections are best adapted to this use. ABBREVIATIONS. Where the work of the land sur- veyor consists in re-surveys and sub-dividing sections of the United States Surveys, the field notes may be made more concise and liability of error reduced by al- ways using a definite symbol to refer to each corner of the section or sub-division. The symbols should be simple and adopted upon some system by which 168 A MANUAL, OF LAND SURVEYING. 10 they may be easily remembered and located in trie mind. The system shown in the figure has been used many years by surveyors in Michigan and found sat isfactory. All the corners lying in the exterior lines of the section are numbered in a definite order of rotation in accordance with their relative importance. Let- ters are used for the interior corners, the first letters being used for the corners lying in the quarter lines and the others for the centers of the quarter sections. The following is a sample of the manner of using the symbols in keeping notes upon the U. S. System when sub-dividing a section. Began at 7. Found stake in place and both bearing trees stand- . ing. Planted stone 25" X 8" X 6" marked -+- for corner. Thence north on random. Var. 2 3(X E, setting temporary stakes every 10 chains Intersected Section line 26 links west of 5. At 5 found rotten stake at correct point, S. 28 W. 66 Iks from stump of W. Oak bearing tree of U. S. Survey. Drove stake for corner and put broken earthenware and glass around it and marked Wh. Oak 12" d ; N. 66 E. 42 Iks. Wh. OnklS N 34 W. 63 Iks. 9 5 10 < e a t d b \ h t c g 13 ).22 39.92 9.98 19.96 29.94 20.02 40.18 80.04 From 5 ran east on random, setting temporary stakes every 10 chains. Intersected Sec. line 12 Iks. North of 2. Found earthen post in correct position and bearing trees of resurvey standing. Thence West on corrected line. Set stake on true line. At 10 set stake with stones around it and marked Pine 12 N. 46 W. 79 Iks. dist. Red Oak 24 S. 19 W. 72 dist. Set stake on true line. From 10 ran south on random Var. 2 19' E. and set temporary stakes at 20 and 40 chains. Then went to 6. Found post and bearing trees of resurvey stand- ing. Ran thence West on random Var. 2 20' E. Intersected random from North 6 links South of temp, stake. Intersected random 14 line 8 links North of temp, stake. Int. Sec. line 10 links South of 8. Corner post dug out in road. Set iron plow beam for corner S. 29 W. 76 Iks. from bearing tree of U. S. Survey. Thence East Corrected line. At intersection of quarter lines set post. FIELD NOTES. 169 The following method of abridging field notes is used by the land department of the United States. The plat of a township is lettered and numbered as shown in the diagram. Corners in the township boundary are (r s - 9 H J i K A L I *** tffc. 4t ** T 2 *<*>< Y X (( f s 3 2 7 s fi 5 W 11 f 1 2 6 s # i 1 8 * J 7 6 * / s y 1 \ 3 2 W I, ? i 9 j o * f> P c 9 A ? r S $ T referred to by letter; e. g., 3 or k. Interior section corners are referred to by the numbers of the sections ; e. g. y corner of 9, 10, 15, and 16. Interior quarter sec- tion corners are referred to by their position on the lines, e. 0., K to W at 3 or E to G at 6. The descrip- tions of corners thus referred to are writteniout in the margins of the plats, while all other matter contained in the field notes is, as far as possible marked on the plats themselves. The letters along the margin of the diagram are arranged the same as in Plate III, Instruc- tions of 1902. A different arrangement has been used commencing in the upper left hand corner and passing around the plat in the opposite direction. 170 A MANUAL OF LAND SURVEYING. CHAPTER VII. CTJRVELINEAR SURVEYING. 1. As land surveyors have occasion in laying out streets in villages, parks, cemeteries, race courses, draims, etc., sometimes to make use of curved lines, it has been deemed proper to include in this work a short discussion of the manner of locating the simpler curves, and add such tables as are needed for this use. For a more com- plete exposition of the subject, consult the field books of Henck, Trautwine, Shunk, or Searles. The curve most commonly used is the circular curve, simple or compound. The simple circular curve, as its name indicates, is a circle or an arc. When an arc is used to connect two straight lines, these lines, from their relation to the circle, are termed tangents. The compound circular curve is a combination of arcs having different radii. At the point of junction of any two of these arcs their radii lie in the same straight line. Of the several geometrical propositions on which the theory of running curved lines depends, it will not be necessary for our purpose to recall more than the fol- lowing PRELIMINARY PROPOSITIONS. 1. If a circle be drawn touching each of two intersecting lines at but a single point, then the exterior angle made by the intersection of these lines is equal to the angle at the center of the circle which is measured by the arc intercepted by the two lines at their points of tangency. 2. The angle which either line makes with the chord of the intercepted arc equals one-half the angle at the centre of the circle which is subtended by that chord. CCTtVELINEAR SURVEYING. 171 In Fig. 63 CF and TI represent the two lines tangent to the circle at C and f>aiid intersecting at I. The angle FIT=^Of^ and the angle &CT^= Yz COT. FIG. 63. The angle called the angle, and the?aiigle FCT the tangential angle. Curves are named from the angle which is subtended by a chord ICC) feet long. Thus, if the 100 foot chord subtends an angle of 1 degree, the curve is spoken of as a 1 curve; if of 5, as a 5 curve, and so on. Tables have been prepared giving the various functions of a 1 curve, which are of great assistance in running curved lines, saving nearly all the trouble of calculation. The foot is taken as the primary unit of these tables and is most commonly used, but any other unit using the decimal notation, as a link or metre, is just as readily applied. Curves are run on the ground by successive deflections of chords. The amount of each deflection may be meas- ured on the ground with the tape or turned off on the transit. 2 To run a Curve with Pickets and Tape. First, determine the radius of the curve and the length of chord to be used. The latter is usually 100. From these data the amount of deflection for each chord is determined as follows: chord 2 Defl. dist. = - Tangential d:st. radius defl. disk A MANUAL OF LAND SURVEYING. IlG. 64. Example 1. Let ab be the straight line or tangent which is to be continued from 6 by a curve having a radius of 1,433 feet, using chords of ICO feet. Extend the line ab to c, making be = i/bd 2 cd 2 .- Ex- tend the chord bd to e, making de = bd = df. Extend the chord df in a similar manner, cbd is the tangential angle, and cd the amount of the deflection to be meas- ured from the tangent to find the line of the curve, edf is the deflection angle, and ef is the amount of deflection to be measured off from the extension of the chord bd to find the line of the curve. To find the distance ef. The triangles edf and- dof df 1 10G 2 being similar, ef : df :: df do. /. ef= == do 1433 = 6.98 nearly. The tangential deflection being one-half the chord deflection, cd = y^ef = 3.49. The triangle bed is right-angled at c, hence be ybd* cd 2 = /lOO 2 3.49 2 = 99.94. The difference between be and bd is so small that in all curves of large radius it may be neglected on the ground and be be measured oft' = bd. These lines may be run with pickets, the chords meas- ured with the tape, and the deflections when not too large measured off by a graduated rod or a rod cut to the exact length. CTJRVELIXEAR SURVEYING. 1-7 3 Example 2. Lay off on the ground a curve having a radius of 2,640 feet, using chords of 50 feet. \ ? '' Ex. 3 Lay off a curve having a radius of 819 feet and chord of 50 feet. '- ." Ex. 4. Lay off a curve with radius 2,865 feet, chord 100 feet. Ex. 5. Lay off a curve with radius 1,910, chord 100. j Xm . Lay off a curve with radius 882, chord 50r Ex. 7. Lay off a curve with radius 1,042, chord X00. * i 3. Keeping the Field Notes of Transit Lines. The field notes of transit work where long line's are being run, as for railroads, drains, etc., are usually kept in a different manner from those of other surveys. The notes proper are kept on the left-hand page of the field book. The opposite page is used for explanatory matter, sketches of topography along the line, such as road and stream crossings and obstacles in line, in greater or less minuteness of detail according to circumstances. The line is marked by stakes driven at regular intervals, usually 100 feet or 100 links, and numbered from up- wards. The corresponding numbers are kept on the left- hand column of the note book, commencing at the bottom of the page and running upwards. If the topography is sketched on the right-hand page, th$ number of every stake is put* down in its regular order, and the ruling of the book forms a scale by which the sketches are made. A book ruled in cross-sections is very convenient for this work. If the topography is not taken, the important stations are noted down and the intermediates are omitted. The following abbreviations are used: P. I., point of intersection; P. C., point of curve, or point where the curve begins; P. C. C., point of com- pound curve; P. R. C., j>oint of reverse curve; P. T., point tangent, or point where the curve ends; T. P., turning point, indicating where the transit was set up, also indi- cated by O or /\. The direction of the tangents is kept as shown by the magnetic needle. This serves as a check on the angles of deflection, and assists in locating errors. 174 A MANUA-L OF LAND SURVEYING. SPECIMEN OF ABRIDGED NOTES. [LEFT PAGE.] Notes of Line "B," D. & R. G. mile above the Dead Horse satch Co., Utah. [RIGHT PAGE.] W. R. R., commencing about a Crossing of Price River, Wan- Sta. 322(X/ \ 30 20 P. C. 4 C P. T. S. urve r't. 80 E. Def. 32 20' 15 (XX 730 / \ 7 19 6 00' \ 18 4 30' -f 60 Old Spanish \ Trail, S.70W. 17 16 3 (XX 130' \15 15 o P. C. 3 C urvel'ft P.I. at 17 + 51.4. 10 o jf/ 4 3 Indian trail / 2 1 +50 to Wash^^ ^. 20 ft. wide, = 10 ft. deep. Oo S. 65 E. 4. To Run a Curve with the Transit. The transit is set up on the point in the tangent from which the curve is to commence. The limb is clamped with the verniers at zero, the telescope ranged along the line of the tangent, and the instrument clamped in that position. The tangential angle, = % the deflection angle, is then turned off on the limb. The leading chain-man draws out the chain or tape in the desired direction, and when out at full length, places his rod in line as directed by the signals of the transit-man. He then carefully measures the length of the chord, marking the distance with his rod, which is then aligned the second time. A stake is driven to mark the point, and the chain-men go ahead and measure the second chord, being aligned by the transit-man as before, and thus continue as far as neces- sary or convenient. The transit-man turns off equal CUBYELINEAB SURVEYING. 175 angles on the transit for each successive chord as it is measured. At the end of the last chord which is run from any one setting of the transit, a short stake is driven firmly into the ground and a tack driven in the top of the stake, to mark the exact point. If the curve is to be continued, the transit is moved up to this point, and with the limb clamped as it was used at the last observation, the telescope is ranged back to the point from which the observation was taken, and the instru- ment clamped in that position. As the angles have all been turned off from a point in the circumference of the circle, they are only half as great as the angle at the center subtended by the same chords. Hence the transit- man now unclamps the limb and turns off as much more angle as he had previously laid off. This gives him a new line, tangent to the curve, from which he may continue to lay off chords as before. Some transit men, instead of doubling the angle after the back-sight is taken, turn off an equal amount in the opposite direction on the limb before taking the back- sight. Then, after getting the back-sight, the verniers are brought to zero on the limb, when the telescope will give the line of the new tangent, as before. Curves are usually run to connect two straight lines which have been previously located. In such a case, pre- .liminary to running the curve, it is necessary to find 1st. The deflection angle between the lines. 2nd. The radius of the curve to be used. 3d. The P. C. and P. T. 4th. The length of the curve. The manner of procedure in such a case is indicated in the following: Example 1. To join two straight lines having a deflec- tion angle of 48 16', by a curve the middle point (/) of which shall be at a distance of 112 feet from the point of intersection. Assume that the line abn has been marked with stakes 100 feet apart, and that the point of intersection is found to be at stake No. 116, -f 43.7; in other words, that the P. I. is 43.7 feet beyond stake No. 116. 176 A MANUAL OF LAND SURVEYING. PC Pl FIG. 65. The transit is set up over the point of intersection, the verniers clamped at zero, the telescope reversed and. ranged along the line ab, and the instrument clamp 3d in that position. The telescope is then righted, the appsr clamp loosened, the telescope turned and the limb again clamped with the telescope pointing along the line cde, and the angle read = 48 16'. Before proceeding further, it is necessary to determine the degree of curve to be used. By the conditions of the example, the middle point of the curve is to be 112 feet from the P. 1. Turn- ing to the table of functions of a 1 curve, we find its external secant, cf\ to be 548.8 feet for an angle of 48 16'. 548.8 Dividing this by 112, we find = 4.9, or 4 54', to be 112 the degree of curvature to be used. Next we find the distance be = cd, which is to be measured along the lines to fincl the beginning and end of the curve, the P. C. and P. T. Referring again to our table, we find that the tangent of a 1 degree curve for a deflection of 48 16' is 2567.1, which divided by 4.9, the degree of curvature, gives 523.9. We now measure from the P. I. 523.9 feet along the line cde, and set a hub and drive a tack in it for the P. T. In a similar manner we next locate the beginning of the curve, which, subtracting 5 -f 23.9 from 116 -f 43.7, we find to be at Station 111, + 19.8. If the ground be clear and open, -CUKYELINEAR SURVEYING. 177 so that thE wflole curve may be seen at once, the transit may now be set up on the P. T., and the whole curve and as much of the next tangent cle as desired run at one setting of the instrument, at the same time avoiding most of the errors usually made in running the curve from the P. C. If this cannot be done, the transit is set up at the P. C. with verniers at zero and a foresight on the P. I., or back-sight to some point along the line ab. The P. C. being at Sta. Ill, + 19.8, the first deflection will be for the partial chord found by subtracting 19.8 from 100 = 80.2, or .802 of the full deflection. The tangential deflection for a full chord being 2 27', for the partial chord would be .802 of 2 27' = 1 58' for the first deflec- tion. For each subsequent full chord 2 27' additional is turned off on the transit as 'far as the line can be seen. Say that the line cannot be seen farther than Sta. 116; the several deflections up to that point would be, for Sta. 112, 1 58'; Sta. 113, 4 25'; Sta. 114, 6 52'; Sta. 115, 9 19'; Sta. 116, 11 46'. A hub and tack are driven at Sta. 116, and the transit moved up to that point or, what is better, to the P. T., if the station is visible from there. If the transit is set up at Sta. 116, the back-sight is taken on the P. C., with the limb clamped at 11 46', as at the last observation. The telescope is then righted, and an addi- tional 11 C 46' turned off for the new tangent, from which the subsequent deflections are turned off. For Station 117 the deflection would be 11 46' + 11 46' -f 2 27' = 25 59'; for Sta. 118, 28 26'; for Sta. 119, 30 53'; for Sta. 120, 33 20'; for Sta. 121, 35 47'. Before passing this point, we must know the length of the curve. As there are 48 16' total deflection, and each chord cuts off 4 54' of it, it is evident there are as many 100 foot chords as 4 54' is contained in 48 16'. Reducing 48.266 the minutes to decimals and dividing, we have = 4.9 9.85 chords for the length of the curve. This added to 111 -f 19.8 gives us 121 -{- 04.8 for the end of the curve, and 04.8 feet for the last partial chord. We find the 32 178 A MANUAL OF LAND SURVEYING. deflection for this distance to be .07', giving for the last deflection 35 47' -;- 07' = 35 54'. The work should now prove itself, by coming out at the stake which was previously set for the end of the curve, and we may further test it by setting the transit up at the P. T., back-sight to Sta. 116, with the instrument clamped at 35 54', as last used. Unclamp the limb and turn off as much more as has been turned from Sta. 116, 35 54' 23 32' = 12 22', which added to 35 54' = 48 16', the total deflection. If the work has been accurately performed, a back-sight through the telescope should strike the P. I. It is very seldom that curves run in this way will come out just right, hence it is better to never set up the transit at points in the curve between the P. C. and P. T. when it can readily be avoided. Still it is the ordinary and sometimes the only way the curves can be run. Let the student make the necessary calculations to locate curves from the following data : Ex. 2. Total deflection, 26 50'. External (cf, Fig. 65), 120.87 feet. P. C. at Sta. 112, + 40. Transit moved every 550 feet. Ex. 3. Total deflection, 35 15'. External, 126.2 feet. P. I. at Sta. 262, + 07.3. T. P. at Sta. 263. Ex. 4. Total deflection, 18 36'. Curve, 1 2', P. I. at 96, + 42.6. T. P. at Sta. 93 and 100. The starting point of a curve is sometimes so situated that it is not convenient to set up the transit at that point, or to run the line from it if it were, as in streams,, gullies, etc., and it then becomes convenient to set up the transit first at some intermediate point in the curve which has to be found. 5. To Locate a Curve from the Middle Point. Set the transit up at the P. I. Bisect the inte- rior angle bed (Fig. 65 ). Find the external cf of the desired curve and measure it off on the line of bisection. This gives the middle point of the curve. The transit is then set up at this point and a back-sight taken either on the P. C. or P. I., and the curve run in. Let the student make the necessary calculations and give the various CTJRVELItfEAR SURVEYING. 179 deflections which would be used on the transit to locate from the middle point the curve required in Ex. 1, Fig. 133, the first back-sight to be taken from the P. C. Give the same, the back-sight being taken from the P. I. Also, solve the following curves, to be run from the middle points, back-sights from P. C., also from P. I.: Examples. I. Total deflection, 16 24'. Curve, 1 32'. P. I. at 96, + 27. 2- Total deflection, 26 18'. Curve, 2 24'. P. I. at 13, + 2.7. 3. Total deflection, 35 40'. Curve, 3 16'. P. I. at 97, -f 62.6. It is sometimes convenient, from various reasons 6. To Locate the Curve with the Tran- sit at some other Intermediate Point on the Curve than the middle. Such points may be located by ordinates from the tangent. This is usually done to avoid obstacles in the line of the curve. To find approx- imately on the ground at what point the transit may be set up, the following formula may be used: Let x = length of the ordinate, d = distance along the tangent from the P. C., t = nat. tangent of % the deflection angle of the curve, Then x = d?t. Example. To find whether the transit can be set up at a point on a 4 curve opposite a point on the tangent 4CO feet from the P. C. * = nat. tang., 2 = 03.5. d 2 = 16. /. x == 56. A meas- ure of 56 feet from the tangent will show whether the transit can be set up at this point or not. It will be fonnd the most convenient in running the curve to select the point at a regular station at the end of a full chord, which may be located as follows: Example 1. Total deflection, 48 48'. P. 1. at 62, -f 36. Curve, 4 . To find the 4th full station on the line of the curve, and locate the remainder of the curve from that point. 180 A MANUAL OF LAND SURVEYING. FIG. 66. First find the number of the station at the P. C. be = tangt. of 1 2599.2 -4- 4 == = 649.8 or 4 6 + 49.8. This taken from 62 + 36 = 55 + 86.2, which is the num- ber of the station at the P. C. From here to the 4th fuli station there is then a short chord of 13.8 feet and four full chords. The tangential angle cbd is therefore 4.138 X 2 = 8 16 J'; whence the deflection angle = 16 33', the chord of which, bd, = 413.4. In the right-angled triangle bed, we now have the side bd = 413.4, and the angle cbd = 8 16^', to find the sides be and cd, from which we find that be = 409.1 and cd 58.5. The point c may be found by measuring from the P. I. 649.8 409 = 240.7 = ec. Having thus located the point d, which is Station 60 on the curve, the transit is set up at that point, with the vernier clamped at 90 , and a back- sight taken to the point c. The upper clamp is then loosened and the limb brought to 16 33', which gives the tangent from which the remainder of the curve is located. Let the student calculate the following curves: 2. Total deflection, 36 ' 20'. P. I. at 26, -f 44.6. Curve, 2 30', to be located from the 3rd full station on the curve. 3. Total deflection, 61 18'. P. I. at 42, + 28.5. Curve, 4 40', to be located from 6th full station on the curve. 4. Total deflection, 42 50'. P. I. at 112, -f 72. Curve, 3 18', to be located from Station 114 + 50 on the curve. 7 . Short Curves. When the deflections between the lines are but small, and it is not important that any particular degree of curvature be used, it will be found convenient to make the curve an even two or four sta- tions in length. In case this is done, the curve may be marked out before the transit is moved from the P. I., after observing the deflection angle, and it will not be CCRVELLSEAK SUKVEYLSG. 181 necessary to set it up on the curve at all. The middle 6f the curve will be located by laying off the external secant as before directed. The P. C. and P. I. are also located as usual. If four stations are used, the intermediate sta- tions may be determined from the P. I., the same as if the transit were at the P. C. or P. T., the error being so small that it may usually be neglected. 8- Passing Obstructions in the Line. One method of doing this, by offset from the tangent, has already been sufficiently explained. Another method, which is very generally applicable, is by parallel offsets from the curve. An offset is made in any convenient direction far enough to pass the obstruction. The curve is continued from this point till the obstacle is passed, when the true line is regained by an inset equal to and parallel with the offset. If the lines are run in the man- ner indicated on page 82, (3), this will be a very simple matter, as the telescope will always point in the same direction when the verniers mark the same point on the limb. Fig. 67 illustrates this method of passing obstacles, be and de are equal and parallel. FIG. 67. 9. Compound Curves, being a combination of simple curves, have their several components located in the same manner. They are usually run to fit the topog- raphy of the country through which they are laid, in order to get uniform gradients on street or railroad lines, or save labor and expense in construction. 182 A MANUAL OF LAND SURVEYING. * Having the several straight lines determined which are to form the tangents of the curve, it is only necessary to find the degrees of curvature of the several component curves, which are then located in the manner already described. Usually there will be found on the ground special reasons for selecting a particular radius for one of the component curves, which will thus dictate the radii of the rest. rl FIG. 68. Example 1. Let ac, ce, eg, gi and ik represent tangents of the curve, and bed, def, fgJi and hik the angles of deflection. Let ce = 1370, eg = 1200, gi = 1000. Let bed = 92 , def= 36 , fgh = 23 3 15', and hik = 43 30', the corresponding curves of which we will number 1, 2, 3 and 4. Let the tangents ce and eg be united by a 3^ curve. Required the radii or degrees of curvature of the remaining components of the curve, and the length of the curve. SUGGESTIONS. First find the tangent of a 3 curve for an angle of 36. Tangent of 1 curve for 36 = 1861.8; .'. for 3 curve = 620.6. This leaves 1370 620.6 = 749.4, length of tangent of curve No. 1. Tangent of 1 curve for 92 = 5933.2, which divided by 749.4 7.917 or 7 55', the degree of curvature. Radius, 724.3. Length of curve 92 = - - = 1162 feet. We find the tangent of curve No. 3 7.917 CURVELINEAR SURVEYING. 183 by subtracting the tangent of curve No. 2, 620.6, from the length of the line eg, 1200, = 579.4. The tangent of a 1 curve fur a dellection of 23 15' we find from the table to Le 1178.8, which divided by 579.4 gives the degree of curve to be used, 2.034 2" 02'. The calculations for the remainder of the curve are made in a similar manner. It is customary in running long lines for drains, rail- ways, etc., to run preliminary lines by angles, omitting the curves, till the location of the tangents is definitely determined. Stakes are set and numbered the same as on the final location. Both the staking and measuring are sometimes omitted, the lines being run as simple picket lines. In such case, when the final location is made, the line is staked out to the point of intersection of the tangents and afterward, as the curve is run in, the stakes between the P. C. and P. I. are taken up and moved to their proper place in the line of the curve. Examples for solution. 1. Let the student calculate the curves and plat the line from the following notes of a preliminary angle line, making all the calculations that would be required in the field, and giving the corrected numbers of the stations at the several P. C.'s, P. T.'s, P. C. C.'s and P. R. C.'s: 176 165 P.I. Compound Curve. Angle right, 14. 163 + 20 P.I. Reverse Curve. Angle right, 36. 153 P.I. Reverse Curve. A ng!e left. 17 26'. 144 + 26 P.I. 2 Curve. Angle right, 16 3V 129 118 108 98+15 P.I. of Reverse Curve. Angle right, 53 12'. 85 + 60 P.I. Compound Curve. Angle left, 16. 76 + 48 P. I. Compound Curve. Angle left, 7 C 8'. ffT r^ P. I. Cnrvr \nflo loft l^ooo/ Of o 58 o 48 o 41 o P.I. 2 3 16' Curve. Angle right, 14. 30 o 21 o P.I. 3o Curve. Angle left, 26 32'. 14 o o N. 45 E. 184 A MANUAL OF LAND SURVEYING. 2. The following are notes of the north side of a street in Park Beidler. The measures are taken with a 66 foot tape of 100 links. The street is one chain wide. A tier of lots two chains deep is laid out on each side of the street. The lots are one chain wide on the street, and are marked by stakes set and numbered at regular intervals of one chain. The lines for the south side of the street and for the back ends of the two tiers of lots are to be rim with the transit and tape. Required the details . of these lines and the widths of lots at the back end, the lot lines being at light angles with the street and on the radii of the curves. 8 + 50 6 + 20 Intersect west line of Dawn Street. Course M. and S. P. T. P. R. C. 10 Curve right, P. C. C. 8 Curve left. P. C. C. 5 Curve left. P.O. 2 Curve left. P. T. P. C. C. 6 Curve right. P. B. C. 4 Curve right. P. C. C. 4 Curve left. P. B. C. 8 Curve left. P. C. C. 4 30 7 Curve right. P. C. 3 Curve right. East at right angles with Sylvan St. Course N. and S. The following formula has been found very useful in solving many problems in the location of curves. Like theformula(E = d 2 in Art. 6, it is designed to express %e length of an ordinate from the tangent to the curve: Let x = length of the ordinate, n = length of the curve in chords of 100 feet each, d = degree of curvature. Then x = ln 2 ~d. Thus a 6 curve will have diverged from its tangent at the end of 500 feet, | X 5 2 X 6 = 131.25 feet. By making d equal the difference of the degree of curvature of two curves of different radii but having a common origin, x will be their divergence from each other OEIGIXAL SURVEYS. 185 at the end of n stations. This formula is not mathemat- ically exact, and therefore gives only approximate results, but it is sufficiently correct for all ordinary cases. It is easily remembered; it requires no tables; and with its aid, with such modifications as a little ingenuity will suggest, and a table of actual tangents for a 1 curve, the surveyor can solve almost any case that will ordinarily arise in the field. For example: Suppose a 5 curve to the right 8 stations long has been located, and its extremity falls 28 feet too far to the right to throw the tangent on the best ground. Making x = 28, we obtain d = , showing that a 4 SO 7 ctfrve starting from the same origin would pass through the required spot. Again: Suppose that in this same case the new curve is to commence 200 feet back of the first one; then the required divergence from the tan- gent will be | X 8 2 X 5 23 = 252. Substituting this value for x, and making n = 8 + 2, we have d = 2.88 = 2 53'. 186 A MANUAL OF LAND SUKYEYIFG. CHAPTEE VIII. OKIGINAL SURYEYS. 1. In land surveying, the surveyor has two distinct classes of problems to deal with. In the first '. 4, " 55". .'* May 30,1862, " 12, " 86. " March 3,1875, " 18, " 130. " ;*; I 3,1875, ,,_ a . JL9, -* 105. Such portions of the various acts as are now in force are published by the government in a volume entitled " Existing Land Laws." Those Sections which refer di- rectly to the surveys are as follows: 7. United States Laws relating- to Surveys and Surveyors. SEC. 77. There shall be appointed by the President, by and with the advice and consent of the Senate, a surveyor-general for the States and Territories herein named, embracing, respectively, one surveying district, namely: Louisiana, Florida, Minnesota, Kansas, California, Xevada, Oregon, Nebraska and Iowa, Dakota, Colorado, New Mexico, Idaho, Washington, Montana, Utah, Wyoming, Arizona. 3 Stat. 755; 4 id. 492; 9 id. 496; 10 id. 244, 306, 308, 309, 611; 11 id. 212; 12 id. 176, 211, 214; 11 id. 77, 85. 314, 542; 15 id. 91 ; 16 id. 65, 240; 17 id. 76; 18 id. 18, 34, 121,122, 123, 201, 303; 19 id. 126, 2075 R. S. 2207. SEC. 84, Every surveyor-general shall, before entering on the duties of his office, execute and deliver to the Sec- retary of the Interior a bond, with good and sufficient security, for the penal sum of thirty thousand dollars, conditioned for the faithful disbursement, according to law, of all public money placed in his hands, and for the faithful performance of the duties of his office; and the President has discretionary authority to require a new 13 194 A MANUAL OF LAND SURVEYING. bond and additional security, under the direction of the Secretary of the Interior, for the lawful disbursement of public moneys. 3 Stat. 697 ; ft. S. 2215, 2216, U. S. v. Vanzandt, 11 Wheat, 184; U. S. v. Tingey, 6 Pet. 115; Farrar and Brown v. U. S,, 5 id. 373; U. S. v. Bradley, 10 id. 343; U. S. vs. Linn, 15 id. 290; U. S. v. Prescott,3 How. 578; U. 8. v. Boyd, 5 id. 29; Bryan v. U. S., 1 Black, 140; Bov- den v. United States, 13 Wall. 17; Bevans v. U. S , 13 id. 56; U. 8. v. Thomas, 15 id. 337; U.S. v. Stephenson, l McClean, C. C. 462; U. S. v. Linn, 2 id. 501 ; U. S. v. Ward, 3 id. 179. 8 Op. Att. Gen. 7. Cir. G. L. O., July 1, 1871 ; id. May 14, 1879. Treasury Cir., July 13, 1871 (Copp's L. L. 783; 1 Lester's L. L. 312, 314). SEC. 85. The commission of each surveyor-general shall cease and expire in four years from the date thereof, un- less sooner vacated by death, resignation, or removal from office. 3 Stat. 697; E. S. 2217. Best r. Polk, 18 Wall. 112. Decision Com. G. L. O., Feb. 20, 1858 (1 Lester's L. L. 340). SEC. 8C. Every surveyor-general, except where the Pres- ident sees cause otherwise to determine, is authorized to continue in the uninterrupted discharge of his regular official duties after the day of expiration of his commis- sion and until a new commission is issued to him for the same office, or until the day when a successor enters upon the duties of such office; and the existing official bond of any officer so acting shall be deemed good and sufficient and in force until the date of the approval of a new bond to be given by him, if recom missioned, or otherwise, for the additional time he may so continue officially to act, pursuant to the authority of this section. 10 Stat. 247; 18 id, 62; K. S. 2222. SEC. 87. Whenever the surveys and records of any sur- veying distri ct are com pleted, the surveyor-general thereof shall be required to deliver over to the Secretary of State of the respective states, including such surveys, or to such other officer as may be authorized to receive them, all the field-notes, maps, records, and other papers apper- taining to land titles within the same; and the office of ORIGINAL SURVEYS. 195 surveyor-general in every such district shall thereafter cease and be discontinued. 5 Stat. 384; 19 id. 121 ; R. S. 2218. SEC. 88. In all cases of discontinuance, as provided in the preceding section, the authority, powers, and duties of the surveyor-general in relation to the survey, resur- vey, or subdivision of the lands therein, and all matters and things connected therewith, shall be vested in and devolved upon the Commissioner of the General Land Office. 10 Stat. 152; R.S.2219. SEC. 89. Under the authority and direction of the Com- missioner of the General Land Office, any deputy surveyor or other agent of the United States shall have free access to any such field-notes, maps, records, and other papers for the purpose of taking extracts therefrom or making copies thereof without charge of any kind; but no transfer of such public records shall be made to the authorities of any State until such State has provided by law for the reception and safe-keeping of such public records and for the allowance of free access thereto by the authorities of the United States. 10 Stat. 152; 18 id. 62; R. S. 2220, 2221. SEC. 90. Every surveyor-general shall engage a sufficient number of skillful surveyors as his deputies, to whom he is authorized to administer the necessary oaths upon their appointments. He shall have authority to frame regulations for their direction, not inconsistent with law or the instructions of the General Land Office, and 'to remove them for negligence or misconduct in office. Taylor and Quarlls v. Brown, 5 Cranch, 234; Craig et al. v. Braxford, 3 Wheat, 594; Ellicott et al. v. Pearl, 10 Pet. 412; Brown's Lessee v. Clements, 3 How. 650. Reed v. Con way 20 Mo. 22; same case, 26 id, 13; Hamil v. Carr, 21 Ohio St. 258; Doe v. Hildreth, 2 Irid. 274; McClintock v. Rodgers, 11 Ills. 279. Cir. G. L. O., June 26, 1880. Second. He shall cause to be surveyed, measured, and marked, without delay, all base and meridian lines through 196 A MANUAL OP LAND SURVEYING. 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 sur- veying district, to which the Indian title has been or may be hereafter extinguished. Gazzen v. Phillips' Lessee, 20 How. 372. 3 Op. Att. Gen., 281, 284. Atshire v. Hulse, 1 Ohio, 170; Hastings v. Stevenson, 2 d. 9; Mc- Kinney v. McKinney, 8 id. 423; Eamil v. Carr, 21 Ohio St. 258; Hendrick v. Eno, 42 Iowa 411 ; Saint Louis v. Walker, 40 Mo. 383; Jordan v. Barrett, 13 La. 24; Fowler i>. Duval, 11 id. 5C1; Cox v. Jones, 47 Cal. 412. Cir. G. L. O., June 26, 1880. Third. He shall cause to be surveyed all private land claims within his district after they have been confirmed by authority of Congress, so far as may be necessary to complete the survey of the public lands. Menard's Heirs v. Massey, 8 How. 293; Kissell v. St. Louis Public Schools, 18 id. 19; Stanford v. Taylor, 18 id. 409; Ballance v. For- syth, 24 id. 183; U. S. v. Fossat, 25 id. 445; Carondelet v. St. Louis, 1 Black, 179; U. S. v. Sepulveda, 1 Wall. 104; U. S. v. Halleck, 1 id. 439; U. S. v. Billings, 2 id. 444; Sutler's case, 2 id. 562; U. S. v. Pacheco, 2 id. 587; Fossat case, 2 id. 649; Dehon v. Bernal, 2 id. 774; U. S. v. Armijo, 5 Cd. 444; Higueras v. U. S. 5 id. 827; Maguire v. Tyler, 8 id. 650; Lynch v. Bernal 9 id. 315; Henshaw v. Bissell, 18 id. 255; Shepley et al. v. Cowan et al., 1 Otto, 330; Miller et al. v t Dale et al, 2 id. 473; Van Eeynegand v. Bolton, 6 id. 33; U. S. v'. Throckmorton, 8 id. 61 ; Snyder v. Sickles, 8 id. 203; Scull v. U. S., 8 id, 410. Bissell v. Henshaw, 1 Saw. C. C. 553; Leroy v. Jamison, 3 id. 369. Gibson v. Chouteau, 39 Mo. 536 ; Milburn v. Hardy, 28 id. 514; Funkhouser v. Hantz, 29 id. 540; Dent v. Legesson, 29 id. 489; Carondelet v. St. Louis, 29 id. 527; Maguire v. Tyler, 30 id. 202; Robins v. Eckler, 36 id. 494; Clark v. Heammerle, 36 id. 620; Gib- son v. Chouteau, 39 id. 536; Vasquez v. Ewing, 42 id. 247; Glasgow v. Lindell,50id. 60; Eector v. Gaines, 19 Ark. 70 ; Ashley v.Kector, 20 id. 359; Meaux v. Breaux, 10 Martin (La.) 364; Moon v. Wilkin- son, 13 Cal. 478; Boggs v. Mining Co., 14 id. 279; Mott v. Smith, 16 id. 534 ; Johnson v. Van Dyke, 20 id. 225 ; McGarrahan v. Maxwell, 27 id. 75; Treadway v. Scmple, 28 id. 652; Searle v. Ford, 29 id. 104; Mahoney v. Van Winkle, 33 id. 448; Morrill v. Chapman, 35 id. 85; Yates v. Smith, 38 id. 60; San Diego u. Allison, 46 id. 163. De- cisions Sec. Int., July 1C, 1872; Aug. 8, 1876; Aug. 17, 1876; March 16,1877. Decisions Com. G. L. O., Aug. 18, 1860; Sept. 18, 1874; Nov. 3, 1874; Sept. 18, 1875; Oct. 28, 1875; June 26, 1879. Cir. G. L. O., June 26, 1880. ORIGINAL SURVEYS. 197 Fourth. He shall transmit to the register of the respec- tive land offices within his district general and particular plats of all lands surveyed by him for each land district; and he shall forward copies of such plats to the Commis- sioner of the General Land Office. Barnard v. Ashley, 18 How. 43; Water and Mining Co. v . Bugbee, 6 Otto. 1G5; Hamil v. Carr, 21 Ohio St. 258; Doe v. Hildreth, 2 Ind 274; Pope v. Athearn, 42 Cal. 606; Com. G. L. O. Instructions to Surveyor-General, April 17, 1879. Fifth. He shall, so far as is compatible with the desk duties of his office, occasionally inspect the surveying operations while in progress in the field, sufficiently to satisfy himself of the fidelity of the execution of the work according to contract;and the actual and necessary expenses incurred by him while so engaged shall be allowed ; and where it is incompatible with his other duties for a surveyor-general to devote the time necessary to make a personal inspection of the work in progress, then he is authorized to depute a confidential agent to make such examination, and the actual and necessary expenses of such person shall be allowed and paid for that service, and five dollars a day during the examination in the field; but such examination shall not be protracted beyond thirty days, and in no case longer than is actually neces- sary; and when a surveyor-general, or any person em- ployed in his office at a regular salary, is engaged in such special service he shall receive only his necessary expenses in addition to his regular salary. 1 Stat. 464; 13 id. 325; 4 id. 492; 10 id. 245, 247; 18 id. 34; 19 id. 126; R. 8.2223. Sec. Int. Instructions, July l, 1874; Sept. 21, 1874. Cir. G. L. O., June 26, 1880. SEC. 91. Every deputy surveyor shall enter into a bond, with sufficient security, for the faithful performance of all surveying contracts confided to him: and the penalty of the bond, in each case, shall be double the estimated amount of money accruing under such contracts, at the rate per mile stipulated to be paid therein. The suffici- 198 A MANUAL OF LAND SURVEYING. ency of the sureties to all such bonds shall be approved and certified by the proper surveyor-general. 4 Stat. 493; 10 id. 247; R. S. 2230. U. S. v. Vanzandt, 11 Wheat. 184; U. S. v. Tingey, 5 Pet. 115; Farrar et al. v. U. S., 5 id. 373; U. S. v. Bradley, 10 id. 343; U. S. v. Linn, 15 id. 290. U. S. v. Stephenson, 1 McLean, C C. 462. SEC. 92. The surveyors-general, in addition to the oath now authorized by law to be Administered to deputies on their appointment to office, shall require each of their deputies, on the return of his surveys, to take and sub- scribe an oath that those surveys have been faithfully and correctly executed according to law and the instruc- tions of the surveyor-general. 9 Stat. 79; R. S. 2231. Ellicott and Meredith v. Pearle, 10 Pet. 412; U. S. v. Hanson, 16 id. 196; Bollard et al. v. Dwight et al., 4 Cranch, 421 ; Taylor et al v. Brown, 5 id. 234. Cir. G. L. O., June 26, 1880. SEC. 93. The district attorney of the United States, in whose district any false, erroneous, or fraudulent surveys have been executed, shall, upon the application of the proper surveyor-general, immediately institute suit upon the bond of such deputy, and the institution of such suit shall act as a lien upon any property owned or held by such deputy or his sureties at the time such suit was instituted. 9 Stat. 79; R.S.2232. SEC. 99. The public lands shall be divided by north and south lines run according to the true, meridian, and by others crossing them at right angles, so as to form town- ships of six miles square, unless where the line of an Indian reservation, or of tracts of land heretofore sur- veyed or patented, or the course of navigable rivers, may render this impracticable; and in that case this rule must be departed from no further than such particular circum- stances require. McKinney v, McKinney, 8 Ohio, 423; Hamil v. Carr, 21 Ohio St. 258. Decision Sec. Int , Jan. 24, 1880. Cir. G. L. O , June 26, 1880. Second. The corners of the townships must be marked with progressive numbers from the beginning, each dis- ORIGINAL SURVEYS. 199 tance of a mile between such corners must be also dis- tinctly marked with marks different from those of the corners. Third. The township shall be subdivided into sections, containing, as nearly as may be, six hundred and forty acres each, by running through the same, each way, par- allel lines at the end of every two miles; and by making a corner on each of such lines, at the end of every mile. The sections shall be numbered, respectively, beginning with the number one in the northeast section and pro- ceeding west and east alternately through the township with progressive numbers till the thirty-six be completed. Grogan r. Knight, 27 Ccl. 516. Decision Sec. Int., April 14, 1879. Cir. G. L. O., June 26, 1880. Fourth. The deputy surveyors, respectively, shall cause to be marked on a tree near each corner established in the manner described, and within the section, the number of such section, and over it the number of the township within which such section may be; and the deputy sur- veyors shall carefully note, in their respective field-books, the names of the corner-trees marked and the numbers so made. Cir. G. L. O., June 26, 1880. Fifth. Where the exterior lines of the townships which may be subdivided into sections or half -sections exceed, or do not extend six miles, the excess or deficiency shall be specially noted, and added to or deducted from the western and northern ranges of sections or half-sections in such townships, according as the error may be in run- ning the lines from east to west, or from north to south; the sections and half-sections bounded on the northern and western lines of such townships shall be sold as con- taining only the quantity expressed in the returns and plats respectively, and all others as containing the com- plete legal quantity. Knight v. Elliott, 57 Mo. 317; Vaughn v. Tate, 64 id. 491; Walters v. Commons, 2 Port. (Ala-) 38; Lewen r. Smith, 7 id. 428. Decision Sec. Int., April 14, 1879, Cir. G. L. O., June 26, 1880. 200. A MANUAL OF LAND SURVEYING. Sixth. All lines shall be plainly marked upon trees, and measured with chains, containing two perches of sixteen and one-half feet each, subdivided into twenty-five equal links; and the chain shall be adjusted to a standard to be ' kept for that purpose. Bradley v. Taylor, 5 Crancli, 191 ; Mclvers v. Walker, 9 id. 173; Shipp v. Miller's Heirs, 2 Wheat. 316; Holmes v. Trout, 7 Pet. 171; Brown v. Huger, 21 How. 303; Meron v. Whitney, 5 Otto, 551; Robinson v. Moon, 4 McLean, C. C. 279. Oakley v. Stuart, 52 Cal. 521. Cir. G. L. O., June 26, 1880. Seventh. Every surveyor shall note in his field-book the true situations of all mines, salt licks, salt springs, and and mill-seats which come to his knowledge; all water courses over which the line he runs may pass; and also the quality of the lands. Newsom v. Pryor's Lessee, 7 Wheat. 7; Preston v. Bowman, 6 id. 580; Patterson v Jenks, 2 Pet. 216. Eighth. These field books shall be returned to the sur- veyor-general, who shall cause therefrom a description of the whole lands surveyed to be made out and transmitted to the officers who may superintend the sales. He shall also cause a fair plat to be made of the townships and fractional parts of townships contained in the lands, de- scribing the subdivisions thereof and the marks of the corners. This plat shall be recorded in books to be kept for that purpose; and a copy thereof shall be kept open at the surveyor-general's office for public information, and other copies shall be sent to the places of the sale and to tne General Land Office. 1 Stat. 465; 2 id. 73; 19 id. 348; K. S. 2395. Taylor et al. v. Brown, 5 Crancli, 234; Barnard v. Ashley, 18 How. 43; Water and Mining Co. v. Bugbee, 6 Otto, 165. Eector v. Gaines, 19 Ark. 70; Lewen v, Smith, 5 Port. (Ala.) 428 ; Mptt v. Smith, 16 Cal. 534; Hamil v. Carr, 21 Ohio St. 258; Doe v. Hildreth, 2 Ind. 274; McClintock v. Eod- gers, 11 Ills. 279. Decision Sec. Int., Jan. 15, 1878 Decision Com. G. L. O., April 17, 1879. SEC. 100. The boundaries and contents of the several sections, half-sections, and quarter-sections of the public ORIGINAL SURVEYS. 201 lands shall be ascertained in conformity with the follow- ing principles: First. All the comers marked in the surveys, returned by the surveyor-general, shall be established as the proper corners of sections, or subdivisions of sections, which they were intended to designate; and the corners of half and quarter sections, not marked on the surveys, shall be placed as nearly as possible equidistant from those two corners which stand on the same line. Second. The boundary lines, actually run and marked in the surveys returned by the surveyor-general, shall be established as the proper boundary lines of the sections, or subdivisions, for which they were intended, and the length of such lines, as returned, shall be held and con- sidered as the true length thereof. And the boundary lines which have not been actually run and marked shall be ascertained by running straight lines from the estab- lished corners to the opposite corresponding corners; but in those portions of the fractional townships where no such opposite corresponding corners have been or can be fixed, the boundary lines shall be ascertained by running from the established corners due north and south or east and west lines, as the case may be, to the water-course, Indian boundary line, or other external boundary of such fractional township. Mott v. Smith, 1C Cal. 534; Guin v. Brandon. 2y Ohio St. 656; McCIin- tock r. Rodgers, 11 Ills. 279; Goodman r, Myriek, 5 Oreg. 65. Cir. G. L. O., June 26, 1880. Third. Each section or subdivision of section, the con- tents whereof have been returned by the surveyor-gen- eral, shall be held and considered as containing the exact quantity expressed in such return ; and the half-sections and quarter-sections, the contents whereof shall not have been thus returned, shall be held and considered as con- taining the one-half or the one-fourth part, respectively, 202 A MANUAL OF LAND SURVEYING. of the returned contents of the* section of which they make part. 2 Stat. 313; R. S. 2396. Lindsey v. Hawes, 2 Black, 554; U. S. v. Pa- checo, 2 Wall. 587; Kailway Co. v. Schurmier, 7 id. 272; County of Saint Clair v. Livingston, 23 id. 46; Heidekoper v. Brooms, 1 Wash.C. C. 109; Coon v. Ten, 1 Pet. C. C. 496. 2 Op. Att. Gen. 578. Knight v. Elliott, 57 Mo. 317; Vaughn v. Tate, 64 id. <91; Waters v. Commons, 2 Port. (Ala.) 38 ; Lewen v. Smith, 7 id. 428; Billingsly v. Bates, 30 Ala. 376 ; Doe r. Hildreth, 2 Ind. 274; Gro- gan v. Knight, 27 Cal. 516. Decision Com. G. L. O., May 17, 1875. Cir. G. L. O., June 26, 1880. SEC. 101. In every case of the division of a quarter-sec- tion the line for the division thereof shall run north and south, and the corners and contents of half quarter-sec- tions which may thereafter be sold shall be ascertained in the manner and on the principles directed and pre- scribed by the section preceding, and fractional sections containing one hundred and sixty acres or upwards shall in like manner, as nearly as practicable, be subdivided into half quarter-sections, under such rules and regula- tions as may be prescribed by the Secretary of the Inte- rior, and in every case of a division of a half quarter- section, the line for the division thereof shall run east and west, and the corners and contents of quarter quarter- section, which may thereafter be sold, shall be ascertained, as nearly as may be, in the manner and on the principles directed and prescribed by the section preceding; and fractional sections containing fewer or more than one hundred and sixty acres shall in like manner, as nearly as may bo practicable, be subdivided into quarter quarter- sections, under such rules and regulations as may be pre- scribed by the Secretary of the Interior. 3 Stat. 566 ; 4 id. 503 ; R. S. 2397. Gazzam r. Phillips' Lessee, 20 How. 372 ; Railway Co. v. Schurmier, 7 Wall. 272. Buel v. Tuley, 4 Mc- Lean, C. C. 268. Wharton v. Littlefield, 30 Ala. 245. 3 Op. Att. Gen. 281,284. Decision Sec. Int., April 14,1879. Decision Com. G. L, O., May 17, 1875. Cir. G. L. O., June 26, 1880. SEC. 102. Whenever, in the opinion of the President, a departure from the ordinary method of surveying land ORIGINAL SURVEYS. 203 on any river, lake, bayou, or water-.course would promote the public interest, he may direct the surveyor-general, in whose district such land is situated, and where the change is intended to be made, to cause the lands thus situated to be surveyed in tracts of two acres in width^ fronting oa any river, bayou, lake, or water-course, and running back the depth of forty acres; which tracts of land so surveyed shall be offered for sale entire, instead of in half quarter-sections, and in the usual manner, and on the same terms in all respects as the other public lands of the United States. 4 Stat. 34 ; R. S. 2407. SEC. 103. In extending the surveys of the public lands in the State of Nevada, the Secretary of the Interior may vary the lines of the subdivisions from a rectangular form, to suit the circumstances of the country. 14 Stat. 86 ; R, S. 2408. Heydenfeldt v. Mining Co., 3 Otto, 634. SEC. 104. The Secretary of the Interior, if he deems it advisable, is authorized to continue the surveys in Ore- gon and California, to be made after what is known as the geodetic method, under such regulations and upon such terms as have been or may hereafter be prescribed by the Commissioner of the General Land Office; but none other than township lines shall be run where the land is unfit for cultivation; nor shall any deputy sur- veyor charge for any line except such as may be actually run and marked or for any line not necessary to be run. 9 Stat. 496 ; 10 id. 245 ; K. S. 2409. SEC. 105. "Whenever, in the opinion of the Secretary of the Interior, a departure from the rectangular mode of surveying and subdividing the public lands in California would promote the public interests, he may direct such change to be made in the mode of surveying and desig- nating such lands as he deems proper, with reference to the existence of mountains, mineral deposits, and the ad- vantages derived from timber and water privileges; but such lands shall not be surveyed into less than one hun- 204 A MANUAL OF LAND SURVEYING. dred and sixty acres or subdivided into less than forty acres. 10 Stat. 245 : B. S. 2410. Cir. G. L. O., June 26, 1880. SEC. 106. The public surveys shall extend over all min- eral lands, and all subdividing of surveyed lands into lots less than one hundred and sixty acres may be done by county and local surveyors at the expense of claimants; but nothing contained in this section shall require the survey of waste or useless lands. 10 Stat. 15, 21 ; 16 id. 218 ; E. S. 2406. SEC. 107. The printed manual of instructions relating to tne puolic survey*, prepared at the General Land Office, and bearing date January first, nineteen hundred and three, the instructions of the Commissioner of the General Land Office, and the special instructions of the surveyor-general, when not in conflict with such printed manual or the instructions of the Commissioner, shall be taken and deemed to be a part of every contract for surveying the public lands. 12 Stat. 409 ; K. S. 2399. Cir. G. L. O., June 26, 1880. SEC. 108. Legal subdivisions of forty acres of placer lands may be subdivided into ten-acre lots. 16 Stat. 213 ; R. S. 2330. SEC. 2320. Mining- claims upon veins or lodes of quartz or other rock in place bearing gold, silver, cinnabar, lead, tin, copper, or other valuable deposits, heretofore located, shall be governed as to length along the vein or lode by the customs, regulations, and laws in force at the date of their location. A mining-claim located after the tenth day of May, eighteen hundred and seventy-two, whether located by one or more persons, may equal, but shall not exceed, one thousand five hundred feet in length along the vein or lode; but no location of a mining-claim shall be made until the discovery of the vein or lode within the limits of the claim located. No claim shall extend more than three hundred feet on each side of the middle ORIGINAL SURVEYS. 205 of the vein at the surface, nor shall any claim be limited by any mining regulation to less than twenty-five feet on each side of the middle of the vein at the surface, except where adverse rights existing on the tenth day of May, eighteen hundred and seventy-two, render such limita- tion necessary. The end-lines of each claim shull be parallel to each other. 10 May, 1872, c. 152, S. 2. V. 17, p. 91. SEC. 2322. The locators of all mining locations hereto- fore made or which shall hereafter be made, on any min- eral vein, lode, or ledge, situated on the public domain, their heirs and assigns, where no adverse claim exists on the tenth day of May, eighteen hundred and seventy-two, so long as they comply with the laws of the United States, and with State, Territorial and local regulations not in conflict with the laws of the United States govern- ing their possessory title, shall have the exclusive right of possession and enjoyment of all the surface included within the lines of their locations, and of all veins, lodes, and ledges throughout their entire depth, the top or apex of which lies inside of such surface-lines extended down- ward vertically, although such veins, lodes, or ledges may so far depart from a perpendicular in their course down- ward as to extend outside the vertical side-lines of such surface locations. But their right of possession to such outside parts of such veins or ledges shall be confined to such portions thereof as lie between vertical planes drawn downward as above described, through the end- lines of their locations, so continued in their own direc- tion that such planes will intersect such exterior parts of such veins or ledges. And nothing in this section shall authorize the locator or possessor of a vein or lode which extends in its downward course beyond the vertical lines of his claim to enter upon the surface of a claim owned or possessed by another. 10 May, 1872, C. 152, S. 3, V. 17, p. 91. SEC. 2323. Where a tunnel is run for the development of a vein or lode, or for the discovery of mines, the own- 206 A MANUAL OF LAND SURVEYING. ers of such tunnel shall have the right of possession of all veins or lodes within three thousand feet from the face of such tunnel on the line thereof, not previously known to exist, discovered in such tunnel, to the same extent as if discovered from the surface; and locations on the line of such tunnel of veins or lodes not appearing on the surface, made by other parties after the commence- ment of the tunnel, and while the same is being prose- cuted with reasonable diligence, shall be invalid ; but failure to prosecute the work on the tunnel for six months shall be considered as an abandonment of the right to all undiscovered veins on the line of such tunnel. 10 May, 1872, C. 152, S. 4, V. 17, p. 92. SEC. 2324. The miners of each mining-district may make regulations not in conflict with the laws of the United States, or with the laws of the State or Territory in which the district is situated, governing the location, manner of recording, amount of work necessary to hold possession of a mining-claim, subject to the following requirements: The location must be distinctly marked on the ground so that its boundaries can be readily traced. All records of mining-claims hereafter made shall con- tain the name or names of the locators, the date of the location, and such a description of the claim or claims located by reference to some natural object or permanent monument as will identify the claim. 10 May, 1872, C. 152, S. 5, V. 17, p. 92. SEC. 109. The surveyor-general of the United States may appoint in each land district containing mineral lands as many competent surveyors as shall apply for ap- pointment to survey mining claims. The expenses of the survey of vein or lode claims, and the survey and sub- division of placer claims into smaller quantities than one hundred and sixty acres, shall be paid by the applicants, and they shall be at liberty to obtain the same at the most reasonable rates, and they shall also be at liberty to employ any United States deputy surveyor to make the ORIGINAL SURVEYS. 207 survey. The Commissioner of the General Lund Office shall have power to establish the maximum charges for such surveys; and to the end that he may be fully in- formed on the subject, each applicant shall file with the register a sworn statem3nt of all charges and fees paid by such applicant for surveys, which statement shall be transmitted to the Commissioner of the General Land Office. 17Stat.95; 19 id. 52; R. S. 2334. Decision Cora. G. L. O., April 20, 1877. SEC. 110. The surveyor-general of the United States shall prepare or cause to be prepared a plat and field-notes of all mining surveys made by authority of law, which shall show accurately the boundaries of such claims; and, when warranted by the facts, he shall give to the claim- ant his certificate that five hundred dollars* worth of labor has been expended or improvements made upon the claim by the claimant or his grantors, and that the plat is correct, with such further description by such refer- ence to natural objects or permanent monuments as shall identify the claim, and furnish an accurate description. to be incorporated in the patent. 17Stat.92 R. S.2325 SEC. 111. Contracts for the survey of the public lands shall not become binding upon the United States until approved by the Commissioner of the General Land Office, except in such cases as the Commissioner may otherwise specially order. 12 Stat. 409 ; R. S. 2398. Maguire v. Tyler, 1 Black, 201 ; Parks v. Ross. 11 How. 362-; Spencer r. Lapsley, 20 id 264. Reed r. Con way, 26 Mo. 13. Decision Sec. lut., Feb. 27, Io78. SEC. 112. The Commissioner of the General Land Office has power, and it shall be his duty, to fix the prices per mile for public surveys, which shall in no case exceed the maximum established by law; and, under instructions to be prepared by the Commissioner, an accurate account shall be kept by each surveyor-general of the cost of sur- 208 A MANUAL OF LAND SURVEYING. veying and platting private land claims, to be reported to the General Land Office, with the map of such claim; and patents shall not issue for any such private claim, nor shall any copy of such survey be furnished, until the cost of survey and platting has been paid into the Treas- ury by the claimant or other party; and before any land granted to any railroad company by the United States shall be conveyed to such company or any persons entitled thereto, under any of the acts incorporating or relating to said company, unless such company is exempted by law from the payment of such cost, there shall first be paid into the Treasury of the United States the cost of sur- veying, selecting, and conveying the same by the said company or persons in interest. 12 Stat. 4C9 ; 18 id. 384 ; 19 id. Ill ; E. S. 2400 Railway Co. v. Prescott, 16 Wall. 6C3; Railway Co. v. McShane, 22 id. 444; Hannewell v. Cass Co., 22 id. 4G4; Colorado Co. v. Commissioners, 5 Otto, 259. Decisions Sec. Int., Dec. 17, 1874; Feb. 27, 1873; Feb. 20, 1879; March 5, J879; April 2, 1879. Decisions Com. G. L. O., April 18, 1867 ; August 18, 1867; Feb. 17, 1869 ; March 26, 1870. Cir. G. L. O., June 2G, 1880. SEC. 113. The Commissioner of the General Land Office may authorize, in his discretion, public lands in Oregon densely covered with forests or thick undergrowth, to be surveyed at augmented rates, not exceeding eighteen dol- lars per mile for standard parallels, fifteen dollars for townships, and twelve dollars for section lines; and under like conditions he may allow augmented rates i;i California, and in Washington Territory, not exceeding eighteen dollars per linear mile for standard parallels, sixteen dollars for township, and fourteen dollars for section lines. 16 Stat. 304, 305 ; 17 id. 358 ; R. S. 2404, 2405. Decision Sec. Int., June 16, 1879. Cir. G. L. O., June 26, 1880. SEC. 114. Whenever the public surveys, or any portion of them, in the States of Oregon and California, are so required to be made as to render it expedient to make, compensation for the surveying thereof by the clay instead ORIGINAL SURVEYS. 209 of by the mile, it shall be lawful for the Commissioner ot the General Land Office, under the direction of the Secre- tary of the Interior, to make such fair and reasonable allowance, as, in his judgment, may be necessary to insure the accurate and faithful execution of the work. lOStat. 247; R. S. 2411. Decision Sec. Int., June 16, 1879. Cir. G. L. O., June 26, 1880. SEC. 118. Each surveyor-general, when thereunto duly authorized by law, shall cause all confirmed private land claims within his district to be accurately surveyed, and shall transmit plats ani field-notes thereof to the Com- missioner of the General Land Office for his approval. When publication of such surveys is authorizsd by law, the proof thereof, together with any objections properly filed and all evidence submitted either in support of or in opposition to the approval of any such survey, shall also be transmitted to said Commissioner. 2 Stat. 326, 352; 3 id. 325 ; 5 id. 740 ; 9 id. 242, 633 ; 10 id. 244, 308, 599; 11 id. 294; 12 id. 172, 209, 369, 409 ; 13 id. 332, 344; 14 id. 218; 16 id. 64, 304 ; 18 id. 305; 19 id. 121, 202 : R. 8. 2447. Bissell v. Penrose, 8 How. 317 ; Villalobus v. TJ. S., 10 id. 541 ; Ledoux v. Black, 18 id. 473 ; U. S. v. Fossat, 20 id. 413; Brown . Huger, 21 id. 305 ; U. S. r. Fossat, 21 id. 445 , Castro r. Hendricks, 23 id. 438; Ballance v. For- syth, 24 id. 183; tJ. S. v. Sepulveda, 1 Wall. 104; U. S. v. Halleck, 1 id, 439; U. S. r. Vallejo, 1 id. 658 ; Sutter's case 2 id. 562 ; Fossat case, 2 id. G49 ; Higueras v. U. , 5 id. 827 ; Alviso v. U. S., 8 id, 337. 12 Op. Att. Gen. 116, 250; 14 id, 74, 601. U. S, v. Garcia, 1 Saw. C.C. 383; Russell v. Henshaw, 1 id. 553; Leroy v. Jamison, 3 id. 369; TJ. S. v. Flint, 4 id. 42. Dent v. Sergerson, 29 Mo. 480 ; Fowler v. Duvall, 11 La. Ann. 5G1 ; Waterman v. Smith, 13 Cal. 373; Moore v Wilkerson, 13 id. 478; Men-it v. Judd, 14 id. CO; Mott v. Smith, 16id. 534 ; Johnson v. Van Dyke, 20 id. 225 ; McGarraghan v. Maxwell, 27 id. 75; Scale v. Ford, 29 id. 104. Cir. G. L. O., June 26, 1880. SEC. 120. Every person who in any manner, by threat or force, interrupts, hinders, or prevents the surveying of the public lands, or of any private land claim which has been or may be confirmed by the United States, by the persons authorized to survey the same, in conformity with the instructions of the Commissioner of the General 14 210 A MANUAL OF LAND SURVEYING. Land Office, shall be fined not less than fifty dollars nor more than three thousand dollars, and be imprisoned not less than one nor more than three years. 4 Stat. 417 ; E. S. 2412. SEC. 121. Whenever the President is satisfied that forci- ble opposition has been offered, or is likely to be offered, to any surveyor or deputy surveyor in the discharge of his duties in surveying the public lands, it may be lawful for the President to order the marshal of the State or district, by himself or deputy, to attend such surveyor or deputy surveyor with sufficient force to protect such officer in the execution of his duty, and to remove force should any be offered. 4 Stat. 417 ; R. S. 2413. SEC. 122. The President is authorized to appoint sur- veyors of public lands, who shall explore such vacant and unappropriated lands of the United States as produce the live-oak and red- cedar timbers, and shall select such tracts or portions thereof, where the principal growth is of either of such timbers, as in the judgment of the Sec- retary of the Navy may be necessary to furnish for the Navy a sufficient supply of the same. Such surveyors shall report to the President the tracts by them selected, with the boundaries ascertained and accurately desig- nated by actual survey or water- courses. 3 Stat. 347 ; E. S. 2459. U. S. v. Briggs, 9 How. 351. SEC. 123. The director of the geological survey shall, under the Interior Department, have the direction of the geological survey and the classification of the public lands and examination of the geological structure, min- eral resources, and products of the national domain. 20 Stat. 394. 8. Manner of Field Work and Changes that have been Made. In accordance with these laws, in- structions have been issued from time to time, by the ORIGINAL SURVEYS. 211 Commissioners of the General Land Office, directing the manner in which the field work should be performed. In the earlier surveys under the act of 1796 (Sec. 2395 B. S. See p. 199, Sec. 99, Third,) the township was sub- divided by parallel lines two miles apart. The mile posts were planted on these lines, but no half mile (or quarter- section) corners set. The act of 1800 provided that the townships west of the Muskingum Kiver should be subdivided into half sections of 320 acres each, as near as may be, by parallel lines run through them from east to west and from north to south at distances of a mile apart. Half-mile posts were to be set on the east and west lines, but not on the lines running north and south. The act of 1805 (Sec. 2396 R. S. P. 200, Sec. 100) covers in its provisions the two classes of surveys above noted, as well as the principles governing all subsequent surveys of the public lands. Since that time, few changes have been made in the manner of carrying on the surveys. The principal changes have been in the instruments used and in the manner of closing the eubdivisional lines on the exterior boundary of the township. In the earlier surveys, the lines were all run by the magnetic needle. Now the direction of all lines must be determined independently of the needle, the use of which for running lines or determining courses is prohibited. In the surveys made previous to 1846, the deputy surveyors were required to close the subdivision lines upon the corners previously set on the east line of the township, but not on those set on the north and west lines. Double corners were thus produced on all the exterior lines of the township. The same system 212 A MANUAL OF LAND SURVEYING. prevailed in some of th'e surveying districts as late as 1854, and perhaps later. It is thus laid down in the instructions of 1815. "Each side of a section must be made one mile in measure by the chain, and quarter-section corners are to be established at every half mile, except when in the closing of a section if the measure of the closing side should vary from 80 chains or one mile, you are in that case to place the quarter- section corners equidistant, or at an average distance from the corners of the section; but in running out the sectional lines on the west or north side of the township, you will establish your quarter- section posts or corners at the distance of half a mile from the last corner, and leave the remaining excess or defect on the west or north tier of quarter-sections, which balance or remainder you will carefully measure and put down in your field-notes in order to calculate the remain- ing or fractional quarter-section on the north and west side of the township: also in running to the western or northern boundary, unless your sectional lines fall in with the posts established there for the corners of sections in the adjacent townships, you must set post and mark bearing trees at the points of intersection of your lines with the town boundaries, and take the distance of your corners from the corners of the sections of the adjacent townships, and note that and the side on which it varies in chains or links, or both. The sections must be made to close by running a ran- dom line from one corner to another, except on the north and west ranges of sections, and the true line between them is to be established by means of offsets." Under the present system, which has been in use in some parts of the country since 1846, the section lines are required to close on the corners previously set on the north and west boundaries, the same as on the east, thus doing away with the system of double section corners. ORIGINAL SURVEYS. L'lo The practice in the several surveying districts in the United States does not seem to have been uniform at any time previous to 1860, and perhaps not always since that date. For instance, in the Instructions of the Commis- sioner of the General Land Office to surveyors-general, dated Feb. 22, 1855, which is stated to be a revision of the manual of surveying instructions prepared for Or- egon in 1851, it is expressly ordered that "double corners are to be nowhere except on the base and standard lines;" while in the instructions to deputy surveyors of the United States for the district of Illinois and Missouri, published in 1856, P. 9, the deputy surveyors were directed to plant their closing corners at the intersec- tion of their lines with the north and west boundary and return their direction and distance from the corners of the corresponding sections on the north and west of these boundaries," the surveyor-general of that district thus giving different instructions from those of the Commissioner of the General Land Office. 9. Fractional Areas.. It has been a puzzle to many surveyors to know how the area of the fractional quarter-sections adjoining the north and west boundaries of the township were calculated. It has been just as much of a puzzle to the surveyors-general and Commis- sioners of the General Land Office. Edward Tiffin, surveyor -general of the Northwest Territory, in 1815 issued instructions how to do it, which instructions were made applicable to the surveys in Ohio, Michigan, Arkansas and Missouri. Under these instruc- tions, the calculations of the areas of these fractions were to be made on the assumption that the quarter- posts on the township and range lines were common to the sections on both sides of these lines, thus making the lengths of the fractions more or less unequal where there w r ere double section corners. This plan does not seem to have been in force long, or to have been very generally followed. Another plan quite extensively adopted was to make the calculations on the theory that all the north and south quarter-lines of these fractional sections were to be parallel with the east line of the sec- 214 A MANUAL OF LAND SURVEYING. tions, and all east and west quarter-lines parallel with the south line of the sections. Neither plan was in harmony with the law of 1805, which required "the corners of half and quarter sections not marked on the surveys to be placed as nearly as possible equidistant from those two corners which stand on the same line." The plan under which most if not all the fractional areas of Michigan were calculated was on the theory that the quarter-posts on the township and range lines were to be placed midway between their respective section corners. Previous to 1828, the deputy surveyors were required to return with their field notes plats of all the townships which they surveyed, and to calculate the area of the fractions. These plats were rudely constructed, and in many cases the areas put down on them were erroneous. If this was found out before the land was sold, the areas were re calculated in the surveyor-general's office. In making the calculations of the areas of the fractions along the township and range lines, some of the deputies considered the quarter-section corners along those lines as common to the sections on both sides, some adopted the second method described above, while the areas of many of the fractions appear to have been put down without any calculation whatever. In the U. S. Surveying Instructions of Jan. 1, 1902, the following rules are given : In the north tier of Sections the fractional lots along the boundary are numbered 1 to 4 from east to west. In the west tier they are numbered from north to south. In Section 6 they are numbered from 1 to 7 from the N. E. corner of the Section along the boundary to the S. W. corner. 1. In regular townships, the tracts of land in each sec- tion adjoining the north and west boundaries of such townships, in excess of the regularly subdivided 480 acres (except in section 6), will, in general, be in the form of trapezoids, 80.00 chains in length by about 20 chains in width. On the plats of such townships, each of said tracts will be divided into four lots, by drawing broken lines ORIGINAL SURVEYS. 215 at intervals of 20.00 chains, parallel to the ends of the tracts, which will be regarded as parallel to each other. With the exception of section 6. the south boundaries of sections of the north tier, when within prescribed limits, will be called 80.00 chains. When the above-named conditions obtain, the areas of the lots in any one tract (except in section 6) may be determined, as follows: Divide the difference between the widths of the ends of the tract by 4: if 3 remains, increase the hundredth figure of the quotient by a unit; in all other cases disre- gard the fraction: call the quotient thus obtained, "d :*' then, taking the end widths of the tract in chains and decimals of a chain, the areas of the lots, in acres, will be: Of the smallest lot: twice the width of the lesser end, pfar**d; M Of the largest lot: twice the width of the greater end, minus "d; " Of the smaller middle lot: sum of the widths of the ends, minus "d: " Of the larger middle lot: sum of the widths of the ends, pfc^"d." A check on the computation may be had by multiply- ing the sum of the widths of the ends of the tract by 4; the product should agree exactly with the total area of the four lots. The proper application of the above rules will always give areas correct to the nearest hundredth of an acre; and, as the use of fractions is entirely avoided, the method is recommended for its simplicity and accuracy. Example 1. The i difference of latitudinal boundaries is 0.031 chains; consequently, "d " is .04 chains; then, 18.35X 2 +.04= 36.74 acres, the area of lot 1 ; 18.50X 2 .04= 36.96 acres, the area of lot 4; 18.50+18.35 .04= 36.81 acres, the area of lot 2: 18.50+18.35 +.04= 36.89 acres, the area of lot 3 : Check: [18.35+18 .50] x 4=147.40 acres, tlie area of the four lots. The arithmetical operations are here written in de- tail, for the purpose of illustration; but the practical computer will perform all the work mentally. 216 A MANUAL OF LAND SURVEYING. 2. Section 6. The areas of lots 5, 6, and 7 may be ob- tained by the foregoing rules in all cases, except when the township closes on a base line or standard par- allel; also, the area of lot 4, provided both meridional boundaries are 80.00 chains in length; when the last condition obtains, the areas of lots 1, 2, and 3 will be equal, and each will contain 40.00 acres. In any case where the west boundary of sec. 6, is 80.00 chains, and the east boundary either greater or less than 80.00 chains, the areas of lots 1, 2, 3, and 4 will be com- puted as follows: Determine the difference, "q," between the east boundaries of lots 1 and 4 by the following propor- tion: N. bdy. sec. 6.: diff. of meridional bdrs. sec. 6. ::60chs. : q; then will E. bdy. lot 4 E. bdy. lot l=bq; in which, "q " will be added when the east boundary of sec. 6 is less than 80.00 chains; but subtracted when said east bound- ary is greater than 80.00 chains. Now take one third of "q," and add it to the shorter east boundary of lots 1 or 4, as conditions may require, and thereby determine the length of one of the meridi- onal boundaries of lot 2; to which again add "one third of q," and thus obtain the length of the opposite side of lot 2. The areas of lots 1, 2, and 3, in acres, will be found by taking the sum of their respective meridi- onal boundaries, expressed in chains and decimals of a chain. The area of lot 4 may be had by multiplying its mean width by its mean length. Finally, to test the entire work, multiply the sum Of the latitudinal boundaries by 4, and to the product add the area of the small triangle C A B, if the east boun- dary is greater than 80.00 chains; but subtract the area of said small triangle if the east boundary is less than 80.00 chains. These operations, correctly performed, will give the true area of the section, which should agree exactly with the total area of its legal subdivisions, obtained as directed in the preceding paragraphs. Example 2. Compute areas of lots 5, 6, and 7 of sec. 6, as directed ORIGINAL PURVEYS. 217 in paragraph 1, and illustrated by the example: then write: chs. chs. chs. chs. chs. 77.75 : 0.05 :: 60.00 : 0.0386=q; ^ q=0.0129 chs. chs. chs. 20.0500 0.0386=20.01. the E. bdy. of lot 4; 20.01 14-f-O.Ol 29 =20.02, the E. bdy. of lot 3: 20.0243+0.0129=20.04, the E. bdy. of lot 2'. Then, for the areas of lots 1, 2, 3, and 4, we have: chs. chs. acres. 20.05-1-20.04 = 40.09, the area of lot 1 ; 20.U4-j-20.02 = 40.06, the area of lot 2 : 20.02-j-20.01 = 40.03, the area of lot 3 ; = 35.54, the area of lot 4. 2 2 Also [17.78+17.S7] X3 = 106.95.'the area of lots 5, 6, and 7. Area of regular subdivisions =360. 00 Total =622.67. the area of Sec. 6. chs. chs. Check: [77.87+77.75] X4=622.48 77.75X 0.025 = 0.19, the area of triangle CAB. ' Total =622.67, which agrees with the area of section 6, before determined. 3. The area in acres of a tract 40.00 chains long, ad- joining north or west township boundaries (except in "N". W. i sec. 6), is equal to the sum of its paralkl bound~ aries (expressed in chains and decimals thereof) multi- plied by 2 ; (e. g.} the area of lots 6 and 7, is [17.87-j-17.81] X2=71.36 acres. The area in acres of a tract 60.00 chains long, situated as above described (excluding lot 4, of sec. 6), may be found by multiplying the sum of its parallel "boundaries (expressed in chains and decimals of a chain) by 3; (e. g.) Fig. 6 ; south boundary lot 4=17.78 chs.; area of lots 5, 6, and 7 is [1 7. 78+17. 87] X 3=106. 95 acres. (See example 2.) The area in acres of quarter sections adjoining north and west township boundaries (excluding N. W. i sec. 6), may be obtained by multiplying the sum of their parallel boundaries (taken in chains and decimals of a chain), by 2; (e. g.} the area of S. W. i sec. 6 (Fig. 6), is [37.87+37.81] x 2=151.36 acres. The area in acres of any section along the north and west boundaries of regular townships (except sec. 6) may 218 A MANUAL OF LAND SURVEYING. be had by multiplying the sum of its parallel boundaries (expressed in chains and decimals of a chain) by 4; (e. g.) the area of sec. 1 (Plate IV) is [80.00-|-79.77]x4=:639.08 acres. The area in acres of a theoretical township may be ob- tained by multiplying the sum of its latitudinal bound- aries (expressed in chains and decimals of a chain) by 24 (e. g.) the area of a township is [480. 00-f 479. 34} x 24=23, 024.16 acres. 10. Instructions of i9o2. The U. S. Manual of Surveying Instructions for 1902, is a large volume of 203 pages, and contains minute instructions in regard to all the operations of the survey of the public lands and private land claims. It is furnished to Deputy U. S. Surveyors and may be had by others who apply for it to the Commissioner of the General Land Of- fice at Washington. The following extracts are made from it : SYSTEM OF RECTANGULAR SURVEYING. 1. Existing law requires that in general the public lands of the United States "shall be divided by north and south lines run according to the true meridian, and by others crossing them at right angles so as to form townships six miles square," and that the corners of the townships thus surveyed "must be marked with pro- gressive numbers from the beginning." Also, that the townships shall be subdivided into thirty-six sections, each of which shall contain six hun- dred and forty acres, as nearly as may be, by a system of two sets of parallel lines, one governed by true meridi- ans and the other by parallels of latitude, the latter in- tersecting the former at right angles, at intervals of a mile. 2. In the execution of the public surveys under exist- ing law, it is apparent that the requirements that the lines of survey shall conform to true meridians, and that the townships shall be 6 miles square, taken together, involve a mathematical impossibility due to the con- vergency of the meridians. ORIGINAL SURVEYS. 219 Therefore, to conform the meridianal township lines to the true meridians produces townships of a trape- zoidal form which do not contain the precise area of 23.040 acres required by law, and which discrepancy in- creases with the increase in the convergency of the meridians, as the surveys attain the higher latitudes. In view of these facts, and under the provisions of section 2 of the act of May 18, 1796, that sections of a mile square shall contain 640 acres, as nearly as may 6e, and also under those of section 3 of the act of May 10, 1800, that "in all cases where the exterior lines of the townships, thus to be subdivided into sections and 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 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 shall be sold as containing only the quantity expressed in the returns and plats, respectively, and all others as containing the complete legal quantity." the public lands of the United States shall be surveyed un- der the methods of the system of rectangular surveying, which harmonizes the incompatibilities of the require- ments of law and practice, as follows: First. The establishment of a principal meridian con- forming to the true meridian, and, at right angles to it, a base line conforming to a parallel of latitude. Second. The establishment of standard parallels con- forming to parallels of latitude, initiated from the principal meridian at intervals of 24 miles and extended east and west of the same. Third. The establishment of guide meridians con- forming to true meridians, initiated upon the base line and successive standard parallels at intervals of 24 miles, resulting in tracts of land 24 miles square, as nearly as may he, which shall be subsequently divided into tracts of land 6 miles square by two sets of lines, one conform- ing to true meridians, crossed by others conforming to 220 A MANUAL OF LAND SURVEYING. parallels of latitude at intervals of 6 miles, containing 23,040 acres, as nearly as may be, and designated townships. Such townships shall be subdivided into thirty-six tracts, called sections, each of which shall contain 640 acres, as nearly as may be, by two sets of parallel lines, one set parallel to a true meridian and the other conforming to parallels of latitude, mutually intersecting at intervals of 1 mile and at right angles, as nearly as may be. Any series of contiguous townships situated north and south of each other constitutes a range, while such a series situated in an east and west direction consti- tutes a tier. By the terms of the original law, and by general practice, section lines were surveyed from south to north and from east to west, in order to uniformly place excess or deficiency of measurement on the north and west sides of the townships. But under modern conditions many cases arise in which a departure from this method is necessary. Where the west or the north boundary is sufficiently correct as to course, to serve as a basis for rectangular subdivision, and the opposite line is defective, the section lines should be run by a reversed method. For convenience the well-surveyed lines on which subdivisions are to be based, will be called govern- ing boundaries of the township. 3. The tiers of townships will be numbered, to the north or south, commencing with No. 1, at the base line; and the ranges of the townships, to the east or west, beginning with No. 1, at the principal meridian of the system. 4. The thirty-six sections into which a township is subdivided are numbered, commencing with number one at the northeast angle of the township, and proceed- ing west to number six, and thence proceeding east to number twelve, and so on, alternately, to number thir- ty-six in the southeast angle. In all cases of surveys of fractional townships, the sections will bear the same numbers they would have if the township was full, and where doubt arises as to which section numbers should ORIGINAL SURVEYS. 221 be omitted, the proper section numbers will be used on the side or sides which are governing boundaries, leaving any deficiency to fall on the opposite sides. 5. Standard parallels, formerly called correction lines, shall be established at intervals of every 24 miles, north and south of the base line, and guide meridians at intervals of every 24 miles, east and west of the principal meridian; thus confining the er- rors resulting from convergence of meridians and inac- curacies in measurement within comparatively small areas. Instruments. 6. The surveys of the public lands of the United States, embracing the establishment of base lines, principal meridians, standard parallels, rnean- iler lines, and the subdivisions of townships, will be made with instruments provided with the accessories necessary to determine a direction with reference to the true meridian, independently of the magnetic needle. Burt's improved solar compass, or a transit of ap- proved construction, with or without solar attachment, will be used in all cases. When a transit without solar attachment is employed, Polaris observations and the re- tracements necessary to execute the work in accordance with existing law and the requirements of these instruc- tions will be insisted upon. Observations every clear night will be necessary to secure accuracy in the di- rection of transit reference lines, when solar appa- ratus is not used. The method of connecting surveys with the stellar meridian should distinctly appear in the field notes, as evidence that the courses were not derived from the magnetic needle. 7. Deputies using instruments with solar apparatus will be required to make observations on the star Polaris at the beginning of every survey, and, whenever necessary, to test the accuracy of the solar apparatus. The observations required to test the adjustments of the solar apparatus will be made at the corner where the survey begins, or at the camp of the deputy surveyor nearest said corner ; and in all cases the deputy wiU 222 A MANUAL OF LAND SUBVEYING. fully state in the field notes the exact locatforr of the observing station. Deputy surveyors will examine the adjustments of their instruments, and take the latitude daily, weather per- mitting, while running all lines of the public surveys. They will make complete records in their field notes, under proper dates, of the making of all observations in com- pliance with these instructions, showing the character and condition of the instrument in use, and the precis- ion attained in the survey, by comparing the direction of the line run with the meridian determined by obser- vation. On every survey executed with solar instruments, the deputy will, at least once on each working day, record in his field notes the proper reading of the latitude arc; the declination of the sun, corrected for refraction, set off on the declination arc; and note the correct local mean time of his observation, which, for the record, will be taken at least two hours from apparent noon. In field inspection of contract surveys, the exam- iners are required to obtain the meridian, both by solar and stellar observations, testing their instru- ments fully before reporting on the courses of the deputy's lines. Hence no deputy should incur risk by omitting any of the safeguards here required as essential to accurate work. 8. The construction and adjustments of all surveying instruments used in surveying the public lands of the United States will be tested at least once a year, and oftener, if necessary, on the true meridian, established under the direction of the surveyor general of the dis- trict; and if found defective, the instruments shall un- dergo such repairs or modifications as may be found necessary to secure the closest possible approximation to accuracy and uniformity in all field work controlled by such instruments. ORIGINAL SURVEYS. 223 9. Chaining. The instruments for measuring lines are the chain and pins. Each deputy will be pro- vided with a standard steel chain or steel tape of ap- proved style, precisely ad justed to the standard meas- ures kept by the surveyor general. The deputy's standard measure will not be used on the field work, but be carefully preserved in camp and used for pur- poses of frequent comparison with his field chains or steel tapes, in order that changes due to constant use may be discovered at the beginning of each day's work. All his returns of distance will be made in miles, chains, and links, a chain of 100 links being equal to 66 feet. Engineers' chains reading by feet only are not to be used in public land surveys. Dis- tances of height or depth may be given in feet or inches. In these details the specimen field notes are to be observed. The simple conditions imperatively demanded for all accurate measurements are specified in the chain- man's oath, promising that he will level the chain upon even and uneven ground, will plumb the pins, either by sticking or. dropping them, and will re- port the true distances. These brief rules, faith- fully observed, will render chaining sufficiently exact to stand the test of inspection by strict examiners. Before chainmen are entrusted with their actual duties, they should be exercised for practice and thoroughly instructed, under the eye of their em- ployer, by chaining two or three times over one or more trial lines of hilly or mountainous surface, to ascertain the accuracy and uniformity of the results. The methods used by competent surveyors to obtain true horizontal distance over steep slopes, are too important to be disregarded, yet too elementary to be given here. When using only a portion of the chain, on steep hill-sides, especially in a strong wind, ac- curacy requires a plumb-line or some equivalent means, to mark the vertical. The dropping of flagged pins not loaded, too often in such cases leads to re- peated and serious error, which may be avoided by 224 A MANUAL OF LAND SURVEYING. dropping a more suitable object, such as a piece of metal carried in the pocket. If any other methods of obtaining measurements up or down hills or across ravines be resorted to, ex- cept that here authorized, the facts will be stated in the returns, and the distances must well sustain the tests of the field examiner. 10. Marking Line. The marking of trees and brush along lines was required by law as positively as the erection of monuments, by the act of 1796, which is still in force. The old rules therefor are unchanged. All lines on which are to be established the legal corner boundaries will be marked after this method, viz : Those trees which may be intersected by the line will have two chops or notches cut 011 the sides facing the line, without any other marks whatever. These are called sight trees, or line trees. A sufficient num- ber of other trees standing within-50 links of the line, on either side of it, will be blazed on two sides di- agonally or quartering toward the line, in order to render the line conspicuous, and readily to be traced in either direction, 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 toward the line, the farther the line passes from the blazed trees. In early surveys, an opposite practice prevailed Due care will ever be taken to have the lines so well marked as to be readily followed, and to cut the blazes deep enough to leave recognizable scars as long as the trees stand. This can be attained only by blazing through the bark to the wood. Trees marked less thoroughly will not be considered sufficiently blazed. Where trees two inches or more in diame- ter occur along a line, the required blazes will not be omitted. Lines are also to be marked by cutting away enough of the undergrowth of bushes or other vege- tation to facilitate correct sighting of instruments. Where lines cross deep wooded valleys, by sighting ORIGINAL SURVEYS. 225 over the tops, the usual blazing of trees in the low ground when accessible will be performed, that set- tlers may find their proper limits of land and timber without resurvey. The practice of blazing a random line to a point some distance away from an objective corner, and leaving through timber a marked line which is not the true boundary, is unlawful, and no such surveys are acceptable. The decisions of some State courts make the marked trees valid evidence of the place of the legal boundary, even if such line is crooked, and has the quarter-section corner far off the blazed line. On trial or random lines, therefore, the trees will not be blazed, unless occasionally, from indispen- sable necessity, and then it will 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 bo lopped, and stakes set on the trial or random line, at every ten chain*, to enable the surveyor on his return to follow and cor- ro( t the trial line and establish therefrom the true line. To prevent confusion, the temporary stakes set on the trial or random line will be removed when the surveyor returns to establish the true line. The terms of each act making appropriation for compensation of surveys, allow increased pay for lines passing through lands " covered with dense un- dergrowth." The evident purpose of the increase is to compensate the surveyor for the additional labor and delay of cutting away brush and trees which ob- struct the proper survey of the line, and also of blazing the line as required by law. By dense undergrowth is meant thick bushes, boughs, or other vegetable growth of such height as to obstruct the use of the transit and require cutting away to obtain sights along line ; also bushes, brush, or vines, that are of such character as to seriously impede the work of traversing and chaining the line. Increased rates for heavy timber or dense under- growth will not be allowed for lines on which no 220 A MANUAL OF LAND SURVEYING. cutting away of brush is done or is necessary, or where blazing of timber is generally neglected, if these conditions shall be shown by field inspection. Insuperable Objects on Line Witness Points. 1. Under circumstances where the survey of a line is obstructed by an impassable obstacle, such as a pond, swamp, or marsh (not meanderable), the line will be prolonged across such obstruction by making the necessary right-angle offsets; or,, if such proceeding is impracticable,. a traverse line will be run, or some proper trigonometical operation will be employed to locate the line on the opposite side of the obstruction; and in case the line, either meridional or latitudinal, thus regained, is recovered beyond the intervening obstacle, said line will be surveyed back to the margin of the obstruction and all the particulars, in relation to the field operations, will be fully stated in the field notes. 2. As a guide in alinement and measurement, at each point where the line intersects the margin of an obstacle, a witness point will be established, except when such point is less than 20 chains distant from the true point for a legal corner which falls in the obstruction, in which case a witness corner will be established at the intersec- tion. 3. In a case where all the points of intersection with the obstacle to measurement fall more than 20 chains from the proper place for a legal corner in the obstruction, and a witness corner can be placed on the offset line within 20 chains of the inaccessible corner point, such " witness corner " will be established. Establishing Corners. 1. To procure the faith- ful execution of this part of a surveyor's duty, is a mat- ter of the utmost importance. After true coursing and most exact measurements, the establishment of corners is the consummation of the field work. Therefore, if the corners be not perpetuated in a permanent and workmanlike manner, the principal object of surveying operations will not have been attained. 2. The points at which corners will be established are ORIGINAL SURVEYS. 227 fully stated in the several articles: "Base Lines," "Principal Meridians." "Standard Parallels," etc., following the title " Initial Points." All marking of letters and figures should be done neatly, distinctly, and durably, using the tools best adapted to the purpose, and keeping them in good order. These tools are the chisel and hammer for marking stones, and the scribing-tool or gouge for surfaces of wood. Since the greatest permanency requires, stone corner monuments, and the perishable nature of wood prohibits its use where stones can be found or brought, the deputy should be provided with good chisels, to enable him to mark neatly and expeditiously, using arabic figures for all numbers. Surveying Monuments 1. These consist of what is called the corner, and its accessories. The corner itself should be durable and firmly imbedded. It may consist of an iron monument, rod, or pipe, a cross cut on a ledge, or a marked stone ; or in case these can not be obtained, then a post of durable timber. Where a stone corner has to be set upon a ledge of surface rock, it should be of large size and supported in a well-built stone mound, with its marks well shown ; in addition to which, the usual witness mound will be separately built. Descriptions of Corners. 1. The form and lan- guage used in the following articles, in describing, for each one of the thirteen classes of corners, eight specific constructions and markings, with the stated modifica- tions in certain cases, will be carefully followed by deputy surveyors in their field notes; and their field work will strictly comply with the requirements of the descriptions. 2. When pits and mounds of earth are made accesso- ries to corners, the pits will always have a rectangular plan; while the mounds will have a conical form, with circular base; and in all cases both pits and mounds will have dimensions at least as great as those specified in the descriptions. Deputy surveyors will strictly adhere to 228 A MANUAL OF LAND SURVEYING. these provisions, and no departure from the stated re- quirements will be permitted, either in instructions or practice in tJie field. 3. Kef erring to the numbered paragraphs, the corners described in "3 " will be preferred to those described in either "1" or "2, J1 when corners are established in loose, sandy soil, and good bearing trees are available: under similar conditions, the corners described in "5" and "8" will be preferred to those described in "4" and " 7," respectively. 4. The selection of the particular construction to be adopted in any case will be left, as a matter of course, to the judgment and discretion of the deputy, who will as- sign the greatest weight to the durability of the corner materials and pei-manency of the finished corners. Abbreviations Allowed in Returns, Dimen- sions of stones, posts, and pits should for brevity be expressed in a regular manner, in consecutive order of length, breadth, and thickness, as shown in specimens ; for instance, " a stone 23 x 10 x 8 ins." To describe a mound the material, the altitude, and diameter of base will be given, as "mound of earth 4 ft. base, 2 ft. high." The following contractions are authorized to be used in the preparation of field notes, transcripts, inspection reports, and similar records, and no others should be introduced. The arrangement of lines, blanks, spaces, numbers, and the general form of the specimen notes should be observed. ORIGINAL SURVEYS. 229 A. for acres. M. C. for meander cor- a. m. " forenoon. ner. A. M. C. " aux. meander mer. 1 meridian. corner. mkd. 1 marked. asc. " ascend. N. 1 north. astron. ' astronomical. NE. ' northeast. bdy. bdrs. ' boundary. " boundaries. NW. obs. 1 northwest. " observe. bet. " between. obsn. " observation. B. 0. " bearing ob- ject. p. m. Pol. " afternoon. " Polaris. B. T. " bearing tree. Pr. Mer. " principal me- C. C. " closing cor- ridian. ner. Pt. of Tr. 14 point of trian- chs. . " chains. gulation. cor., cors . " corner, cor- Xsec. " quarter sec- ners. tion. corr. " correction. R., Rs. " range, ranges. decl. " declination red. " reduce, reduc- dep. " departure. tion. desc. " descend. S. ;< south. dia. " diameter. S. C. " standard cor- difT. ' difference. ner. ( 1st. ;< distance. SE. " southeast. D. S. " deputy sur- sec., sees., " section, sec- veyor. tions. E. ;< east. S. M. C. " special mean- elong. frac. " elongation. " fractional. sq. der corner. 1 square. ft. " foot. feet. St. Par. " standard par- G. M. 11 guide merid- allel. ian. SW. " southwest. h., hrs., " hour, hours. T., or Tp. 1 township. ins. ' inches. Ts.,orTps ' townships. lat. " latitude. temp. ' temporary. L. C. " lower culmi- U. C. " upper culmi- nation. nation. Iks. " links. var. ' variation. 1. m. t. " local mean W. ' west. time. W. C. ' witness cor- long. 1 longitude. ner. m. " minutes. w. corr. " watch correc- mag. " magnetic. W. P. tion. ' witness point. w. t. " watch time. 230 A MANUAL OF LAND SURVEYING. * AUTHORIZED FORMS AND DESCRIPTIONS OF CORNERS. The forms given below will guide the surveyor in the choice and erection of monuments and acces- sories, and the same forms will be followed in pre- paring field notes. In case a deputy is compelled to choose another style of corner, he should state in his notes the reasons that made it necessary to depart from the rules, and should erect a monument of equal or greater permanence than the one prescribed. The punctuation marks heretofore shown in former editions, to be used with letters and figures on stones, posts, and trees, are now omitted, for the reason that they are neither made, nor desired to be made, in the actual field work, and hence should not be inserted in the official returns. The stated dimensions of posts are minimum ; if posts are longer than 3 feet, the extra length will be placed in the ground ; the posts will in no case pro- ject more than 12 ins. above the natural surface of the earth. STANDARD TOWNSHIP CORNERS. METHOD OF MARKING. When more than one half of all the standard town- ship and section corners on any 6 miles of a base line or standard parallel are stone corners, the descriptions in paragraphs 1 and 2, if the corners therein described are established, will be modified as follows: Strike out "S. C., on N." After "marked," insert the words: "S. C., 13 N. onN., 22 E. on E., and 21 E. on W. faces; " When under the conditions above specified, the corner described in paragraph 1 is established, a stake may be driven in the east pit and marked instead of the stone, and described as exemplified in the last clause of para- graph 6. ORIGINAL SURVEYS. 231 1. Stone, with P#.s and Mound of Earth. Set a stone, x X ins., ins. in the ground, for standard cor. of (e. g.) Tps. 13 TS., Rs. 21 and 22 E., marked S. C. on N.; with 6 grooves on N., E., and W. faces; dug pits, 30x24x12 ins., crosswise on each line, E. and W., 4 ft., and X. of stone, 8 ft. dist.; and raised a mound of earth, 5 ft. base, 2i ft. high, N. of cor. The direction of the mound, from the corner, will be stated wherever a mound is built. 2. Stone, with Mound of Stone. Set a stone, X X ins., ins. in the ground, for standard cor. of (e. g) Tps. 13 N., Rs. 21 and 22 E., marked S. C., on X.; with 6 grooves on X., E., and W. faces; and raised a mound of stone, 2 ft. base, li ft. high, X. of cor. Pits impracticable. Mound of stone will consist of not less than four stones, and will be at kast H ft. high,, with 2 ft. base. 3. Stone, with Bearing Trees. Seta stone, X X ins., ins. in the ground, for standard cor. of (e. g.) Tps. 13 N., Rs. 21 and 22 E., marked S. C.. on N.; with 6 grooves on N., E., and W. faces; from which A , ins. diam., bears N. E., Iks. dist., marked T. 13 X.. R. 22 E.. S. 31, B. T. A , ins. diam., bears X. W., Iks. dist., marked T. 13 X., R, 21 E., S. 36, B. T. All bearing trees, except those referring to quarter section corners, will be marked with the townshi}), range, md section in which they stand. 4. Post, with Pits and Mound of Earth. Set a post, 3 ft. long, 4 ins. sq., with marked stone (charred stake or quart of charcoal), 24 ins. in the ground, for standard cor. of (e. g.) Tps. 13 X., Rs. 22 and 23 E., marked S. C., T. 13 N. on N. R. 23 E., S. 31 on E., and R. 22 E., S. 36 on W. faces; with 5 grooves on K, E., and W. faces; dug pits, 30x24x12 ins., crosswise on each 232 A MANUAL OF LAND SURVEYING. line, E. and W., 4 ft., and N. of post, 8 ft. dist.; and raised a mound of earth, 5 ft. base, 2i ft. high, N. of cor. 5. Post, with Bearing Trees. Set a post, 3 ft. long, 4 ins. sq., 24 ins. in the ground, for standard cor. of (e. g.) Tps. 18 N., Rs. 22 and 23 JE., marked S. C., T. 13 N. on N., R. 23 E. S. 31 on E., and R. 22 E., S. 36 on W. faces: with 6 grooves on N.,-E M and W. faces, from which A . ins. diam., bears N. - E., Iks. dist., marked T. 13 N., R. 23 E., S. 31, B. T. A , ins. diam., bears N. W., Iks. dist., marked T. 13 N., R. 22 E., S. 36, B. T. 6. Mound of Earth, with Deposit, and Stake in Pit. Deposited a marked stone (charred stake or quart of charcoal) 12 ins. in the ground, for standard cor. of (e. g.} Tps. 13 N., Rs. 22 and 23 E.; dug pits, 30X24X12 ins., crosswise on each line, N., E., and W. of cor., 5 ft. dist.; and raised a mound of earth, 5 ft. base, 2i ft. high, over deposit. In E. pit drove a stake, 2 ft. long, 2 ins. sq., 12 ins. in the ground marked S. C.,T.13N. onN., R. 23 E., S. 31 onE., and R. 22 E., S. 36 on W. faces; with 6 grooves on N., E., and W. faces. 7 Tree Corner, with Pits and Mound of Earth. A , ins. diam., for standard cor. of (e. g.) Tps. 13 ST., Rs. 22 and 23 E., I marked S. C., T. 13 N. on N., R. 23 E., S. 31 on E., and R. 22 E., S. 36 on W. sides; with 6 notches on "N., E., and W. sides; dug pits, 24x18x12 ins., crosswise on each line, N., E., and W. of cor., 5 ft. dist.; and raised a mound of earth around tree. 8. Tree Corner, with Bearing Trees. A , ins. diam., for standard cor. of (e. g.) Tps. 13 ORIGINAL SUKVKYS. ' 233 N., Rs. 22 and 23 E., I marked S. C., T. 13 N. on N., R. 23E., S. 31 on E., and R, 22 E., S. 36 on W. sides; witli 6 notches on N., E., and W. sides; from which .v , ins. diam., bears N. E., Iks. dist., marked T. 13 N., R. 23 E., S. 31, B. T. A , ins. diam., bears N: W., Iks. dist., marked T. 13 X., R. 22 E., S. 36, B. T. Witness Corners. 1. When the true point for any corner described in these instructions falls where prevailing conditions would insure its destruction by natural causes, a witness corner will be established in a secure position, on a surveyed line if possible, and within twenty chains of the corner point thus witnessed. 2. Markings on Witness Corners. A witness corner will bear the same marks that would be placed upon the corner for which it is a witness, and in addition, will have the letters " W. C." (for witness corner), conspicuously displayed above the regular mark- ings; such witness corners will be established, in all other respects, like a regular corner, marking bearing trees with the proper numbers for the sections in which they stand. When bearing trees are described as accessories to a witness corner, the prescribed markings on each tree will be preceded by the letters "W. C.," distinctly cut into the wood. The true bearing and distance of witness corners, from the true point for the corner, will always be clearly stated i n the field notes. 4. Witness Comers to Corner Points Falling in Roads, etc. The point for a corner falling on a railroad, street, or wagon road, will be perpetuated by a marked stone charred stake or quart of charcoal, deposited 24 inches in the ground, and witnessed by two witness corners, one of which will be established on each limiting line of the highway. 234 A MANUAL OF LAND SURVEYING. In case the point for any regular corner falls at the intersection of two or more streets or roads, it will be perpetuated by a marked stone (charred stake or quart of charcoal), deposited 24 inches in the ground, and ivitnessed by two witness corners established on opposite sides of the corner point, and at the mutual intersections of the lines limiting the roads or streets, as the case may be. Witness Points will be perpetuated by corners similar to those described for quarter section corners, with the marking ' W. P." (for witness point), in place of "i,"or "is. ", 'as the case may be. If bearing trees are available as accessories to witness points, each tree will be marked W. P. B. T. (See "In- superable objects on line Witness Points." Miscellaneous. 1. Corners on Rock in Place, or on Boulders. When a corner falls on rock in place, or on a boulder, a cross (X), will be made at the exact corner point, and witnessed by the proper number of bearing trees, if they are available; in the absence of suitable trees, a mound of stones will be raised, or of earth if stones are not found and pits are available. Owing to the diffi- culty of identifying the corner coming upon a flat rock in place, when only a cross is cut thereon, it is imperative that some adequate witness be used and marked. 2. Location of Mounds. When mounds of earth or other material are raised as accessories to corners, they will be placed as specified in the foregoing Description of Corners, and in every case the direction of tlie mound from the corner will be carefully stated. The use of the indefinite description "alongside" will not be approved. In case the character of the land is such that the mound cannot be placed as hereinbefore described, the deputy will state in his notes, by bearing and distance, exactly where the mound is located with reference to the corner, and will give his reasons for placing it as described. 3. Mounds of Stone, Covered with Earth. ORIGINAL SURVEYS. 235 In a case where pit8 are practicable and the deputy prefers raising a mound of stone, or a mound of stone covered with earth, he will use the form given for " Stone with mound of stone," when the corner thus described is established; but when the corner "Stone, with mound of stone covered with earth" is con- structed, the description will be modified as follows: Strike out the words "Pits impracticable;" in place of " Mound of Stone, 2 ft. base, l ft. high," write "Mound of stone covered with earth, ft. base, ft. high," inserting in the blank spaces the dimensions of the mound given in paragraph 1, following the -designation of each class of corners. Mounds of stone, or of stone covered with earth must never be built AROUND the corner stone, but separate. When stones are necessary to hold the corner stone upright and firm, they should be in ad- dition to the witness mound, and not a part of it. 4. Bearing Trees. Bearing trees marked as accessories to standard cor- ners, either township, section, or quarter section, will be selected on the north side of base lines or standard parallels, and bearing trees referring to the closing cor- ners on said lines, will be located on the south side; in general, the bearing trees referring to any particular closing corner, together with one pit and the mound be- longing to such corner, will be located on the same side of tlie line dosed upon, and on the side from which the surveys have been closed. When the requisite number of trees can be found within 300 links of the corner point, two (2) bearing trees will be marked and described for every standard or closing township or section corner, or corner common to two townships or sections, only; four (4) for every corner common to four townships or four sections; one (1) for a corner referring to one township or one section, only; two (2) for every quarter section corner or meander corner, and four (4) for each mile or half mile corner, or corner monument on a reservation or other boundary, not conforming to the system of rectangular surveying. 236 A MANUAL OF LAND SURVEYING. The limit of 300 links will not be held to prohibit the use of bearing trees or rocks beyond that dis- tance. Where such objects are few but accessible, they are too useful as evidences of corners to be dis- regarded by a faithful deputy, even when several chains distant. In the surveys of 50 or 60 years ago, corners were often witnessed by trees 8 or 10 chains distant, with great advantage to subsequent retracements. In case the prescribed number of trees cannot be found within practicable distance, the deputy will state in his field notes, after describing those marked, "No other trees within limits," and add "Dig pits X X ins.," etc., or " Raise a mound of stone, ft. base, ft. high, of cor.," as prevailing conditions may require. Bearing trees, being important accessories to the corners, will have their exact bearings from the true meridian taken with the instrument used in run- ning the lines of survey ; and the distance from the middle of each bearing tree to the middle point of the corner will be carefully measured, and recorded in the field notes. 7. As to the height or position of marks placed on bearing trees, practice differs in various localities. The custom of placing these important evidences high enough to insure their destruction when some woodman, ignorant or careless of the penalty of the law, cuts down the tree, is a direct violation of rules. A tree will be so marked that if inadvert- ently cut down its stump will retain evidence of its importance. Many surveyors have adopted the plan of placing all the marks at the height of 4 or 5 feet, except the letters B T, which are made on another blaze about one foot above the ground. The intent is commendable ; but as a better rule, applicable to trees of every size, the following is now adopted : Place all figures and letters on that part of the tree which would probably remain as the stump ; and make one plain blaze high on the same ORIGINAL SURVEYS. 237 side, to attract notice in case of snow or dense undergrowth. No tree less than 4 inches in diameter should be chosen for a witness, if larger ones are convenient: and if none over 3 inches are found, pits will be dug to witness the corner. 6. Stones for Corners. Stones 18 ins. long, or less, will be set with two thirds of their length in the ground, and those more than 18 ins. long will have three fourths of their length in the ground. ~No stones measuring less than 504 cubic inches, or less than 12 ins. in length, will be used for corners. 7. Lines Discontinued at Legal Corners. No mountainous lands, or lands not classed as survey- able, will be meandered, and all lines approaching such lands will be discontinued at the section or quarter-sec- tion corner nearest the unsurveyed land. 8. Marks to be cut. All letters and figures on posts, trees, or stones, etc., will be cut into the object upon which they are placed. Arabic figures and plain letters will be used for all markings. 9. Orientation of Corners. Corners referring to one, two, or four townships or sections, not identical with standard or closing corners, will be set with their faces directed XE. and SW., and N"W. and SE., while all other corners will be set with their sides facing the cardinal points; except corners on boundaries of reservations and private land claims, which will be set squarely on line. 10. Size of Posts, Hounds, etc. The sizes of wooden posts, mounds, and pits, noted in the foregoing descriptions, will be regarded as minimum, and their dimensions will be increased whenever prac- ticable. 11. Corner Materials. In establishing corners the first preference will be given durable stones when obtainable ; then, posts ; and lastly, mounds, with stake in pit. 238 A MANUAL OF LAND SURVEYING. Wood of a perishable nature will not be used for posts or stakes. 12. Instructions to be studied. Deputy surveyors will carefully read, study, and fa- miliarize themselves with all instructions contained in this volume, and will instruct their assistants as to their duties before commencing work. An extra copy of this Manual may be furnished each deputy, for the use of his assistants. Initial Points. Initial points from which the lines of the public surveys are to be extended will be estab- lished whenever necessary, under such special instruc- tions as may be prescribed in each case by the Commis- sioner of the General Land Office. The locus of such initial points will be selected with great care and due consideration for their prominence and easy identifica- tion, and must be established astronomically. An initial point should have a conspicuous loca- tion, visible from distant points on lines ; it should be perpetuated by an indestructible monument pref- erably a copper bolt firmly set in a rock ledge ; and it should be witnessed by rock bearings, without relying on anything perishable like wood. 115. The initial point having been established the lines of public-land surveys will be extended therefrom. They are classified as follows: Class 1. Base lines and standard parallels. Class 2. Principal and guide meridians. Class 3. Township exteriors (or meridional and latitudinal township boundaries). Class 4. Subdivision and meander lines. Only the base line and principal meridian can pass through the initial point. Base Line. 1. From the initial point the base line will be extended east and west on a parallel of latitude, by the use of transit or solar instruments, as may be ORIGINAL SURVEYS. 269 directed by the surveyor general in his written special instructions. The transit will be used for the aline- ment of all important lines. 2. The direction of base lines will conform to parallels of latitude and will be controlled by true meridians; consequently the correct determination of true meridians by observations on Polaris at Elonga- tion is a matter of prime importance. 3. Certain reference lines, called tangents and *e- cants, having a known position and relation to the required parallel of latitude, will be prolonged as straight lines. Two back and two fore sights are taken at each setting of the instrument, the hori zontal limb being revolved 180 in azimuth between the observations, in one method, taking the mean of observations. Another method, called double back and fore sights, is still more exact and therefore pref- erable. In this process the vertical cross-wire is fixed upon two transit points at some distance apart, in the rear, and then reversed to set one or two new points in advance. This not only insures a straight line, if the transit is leveled, but also de- tects the least error of collimation. 4. Where solar apparatus is used in connection with a transit, the deputy will test the instrument, whenever practicable, by comparing its indications with a meridian determined by Polaris observations; and in all cases where error is discovered, he will make the necessary corrections of his line before pro- ceeding with the survey. All operations will be fully described in the field notes. 5. The proper township, section, and quarter section corners will be established at lawful intervals, and me- ander corners at the intersection of the line with all meanderable streams, lakes, or bayous. 6. In order to detect errors and insure accuracy in measurement, two sets of chainmen will be employed; one to note distances to intermediate points and to lo 240 A MANUAL OF LAND SURVEYING. cate topographical features, the other to act as a check. Each will measure forty chains and in case the difference is inconsiderable, the proper corner will be placed midway between the ending points of the two measurements; but if the discrepancy exceed 8 links on even ground., or 25 links on mountainous surface, the true distance will be found by careful re-chaining by one party or both. The deputy will be present when each corner is thup established, and will record in the body of his field notes the distances to the same, according to the measure- ment by each set of chainmen. To obviate collusion between the sets of chainmen, the second set should commence at a point in advance of the beginning corner of the first set, the initial dif- ference in measurement thus obtained being known only to the deputy Principal Meridian. 1. This line shall conform to a true meridian and will be extended from the initial point, either north or south, or in both directions, as the conditions may require, by the use of transit or solar instruments, as may be directed by the surveyor general in his special written instructions. 2. The methods used for determination of directions, and the precautions to-be observed to secure accuracy in measurement, are fully stated above under the title "Base Line," and will be complied with in every partic- ular. 3. 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 commissioner of the General Land Office, and the survey of such principal meridian shall not be commenced until written author- ity, together with such special instructions as he may deem necessary, shall have been received from the com- missioner. Standard Parallels. 1. Standard parallels, which are also called correction lines, shall be extended- east ORIGINAL SURVEYS. 241 and west from the principal meridian, at intervals of 24 miles north and south of the base line, in the manner prescri bed for running said line, and all require- ments under the title "Base Line" will be carefully observed. 2. Where standard parallels have been placed at inter- vals of 30 or 36 miles, regardless of existing instructions, and where gross irregularities require additional stand- ard* lines, from which to initiate new, or upon which to close old surveys, an intermediate correction line should be established to which a local name may bi given, e. #., "Cedar Creek Correction Line;" and the . same will be run, in all respects, like the regular stan- dard parallels. Guide Meridians- 1. Guide meridians shall be extended north from the base line, or standard parallels, at intervals of 24 miles east and west from the principal meridian, in the manner prescribed for running the principal meridian, and all the provisions for securing accuracy of alinement and measurement found, or referred to under the titles "Base Line," and "Principal Meridian," will apply to the survey of said guide meridians, 2. When existing conditions require that such guide meridians shall be run south from the base or correction lines, they will be initiated at properly established closing corners on such lines marked as closing corners. 3. Where guide meridians have been improperly placed at intervals greatly exceeding the authorized distance of 24 miles, and standard lines are required to limit errors of old, or govern new surveys, a new guide meridian may be run from a standard, or properly estab- lished closing corner, and a local name may be assigned to the same, e. 1.16 S. 8959'.7 1.44 S. ""*,?&,"! S'00*.2 0.69 tos. 31 89 Iff. i 2.01 N. 8958'.6 0.91 N. 8*553 89 59'.2 0.70 S. 8969'.5 1.20 8. 8969'.7 1.60 S. W(E -,? W ! 3' 07".4 0.72 Ins. 32 89Mf.4 3.09 N. 958'.6 0.94 N. .*.. 89 59' .2 0.73 S. 8959'.6 1.25 S. 89 6y.7 1.66 S. W>(E. orW.) 1.67 Si 3' 15".0 O.Vilns. 33 r,fi? 8958'.6 0.97 N. 8958'.8 0.00 89 69M 0.76 S. 8959'.4 1.30 S. 89 59'.7 1.62 S. 90 (E. orW.) 1.738. 3'22". 0.78 Ins. 34 8958'.2 2.25 ?l. 8fl58'.5 1.01 S. 8958'.8 0.00 89 59M 0.79 S. 89 69'.4 1.36 S. 89 59'.7 1.69 S. 90 (E. or W.) 1*80 S* 3'30.4 0.81 Ins. 'So 89 58'.2 2*33 N. 89<>58'.5 1.05 N. -89 t '58'.8 0.00 8959M 0.82 S. 8959'.4 1.40 S. 89 W.l 1.75vS. 9(,( E .orW.) 8' 38".4 0.84 Ins. 36 8968'.l 2.42 N. 8958'.4 1.09 N. W K 89 59'.0 0.86 S. 89 69'.4 1.46 S. 89 59'.7 1.82 S. "Wi! 3' 46".4 0.87 las. 3,7 89 68'.0 3.51 X. 89 BO 1.13 N. 89 68'.6 0.00 89 68'.9 0.88 S. 89 69'.3 1.61 S. WW.l 1.89 S. 90 (E. orW.) 2.01 s! 3' 55".0 0.90 Ins. 38 8958'.0 2.61 N. 89 58'.3 1.17 N. 8968'.6 0.00 89 68'.9 0.91 S. 89 69'.3 1.56 8. ea ^y. 7 1.95 S. 9V>(E 2?08 W 8! 4' 03".6 0.03 Ins. 30 89 57'.9 2.70 N. 89 68'.2 1.21 N. 89 58'.6 0.00 8958'.9 0.04 S. 89 591.3 1.62 S: 89 69'.7 2.02 8. ^^Wn! 4' 12".6 0.97 Ins. 40 8957'.8 2.79 N. 89 58M 1.25 N. 89 58'.5 0.00 89 68'. 9 0.98 8. 89 59'.3 1.68 S. 89 59'. 7 2.10 S. ^(E.orW s ) 4' 2l".6 1.00 Ins. 41 89 57'.7 2.89 N. 8968'.0 1.30 N. 89 58'.4 o.oo 8958'.8 1.02 S. 89 59'.2 1.74 8. 89 59'. 6 2.17 8. ""^WK 4' 31".2 1.04 in*. 42 89 57'. 7 3.00 N. 89 58'.0 1.3d N. "To* 89 58'.8 1.06 S. 89 59'.2 1.80 S. 89 59'.6 2.26S. "^Jtt 4' 40":8 1.1D8 In*. 43 89 57'.6 3.11 N. 89 58'.0 1.40 N. TcS 8 58'.8 1.08 S. 895y.2 1.86 S. 8959'.6 2.33 S. 90 (E. orW.) 2.48 S. 4' 50".8 1.12ln. 44 89 57'.5 3.22 N. 8957'.9 1.46 N. S^SS'.S 0.00 89 58'.7 1.12 S. 8969'.2* 1.93 S. 89 59'.6 2.41 S. 90 (E. or W.) 2.67 S. 6' 01" 1.161ns. 45 89 67'.4 3.33 N. 89 57'.8 1.60 X. 89 58',3 0.00 8968'.7 1.16 S. 89 59M 2.00 S. 89= 5y.5 2.49 S. 90 (E. or W ) 2.66 S. 5'11".8 1.201ns. 46 89 57'.3 3.44 N. 89 57'. 7 1.65 N. 89 68'.2 0.00 8958'.6 1.21 S. 8959M 2.07 S. 89 59'.5 2.59 S. 90 (E. or WO 2.768. 6' 22" 8 1.24 Ins. 47 89 57' .2 3.57 N. 89 57'.6 1.61 N. 8958'.l 0.00 89 58'.6 1.25 S. 89 59M 2.14 S. 89 69' .5 2.07< S. 90 (E orW.) 2.868. 6' 34".2 1.28 Ins. 48 89 57'. 1 3.70 N. 89 57'.5 1.66 N. 89 68'. 0.00 89 58'.5 1.30 S. 8959'.0 2.22 S. 89 ;> 69' 5 2.78 S. ""? 5' 46".2 1.331ns. 49 8957'.0 3.82 N. 89 67'.6 1.72 N. 8958'0 0.00 8968'.5 1.34 S. 89 59' 2.30 S. 89 59'. 5 2.87 8. ""Wi! 5' 58".6 1.38 Ins. 50 89 55'.9 3.96 N. 89 57'.4 1.78 V. 89 57'.9 0.00 89 58'.4 1.39 S. 89 59'.0 2.3S S. 89 69'.5 2.97 S. 90 (E. or W.) 3.i7 s; 6'11".4 1.431ns. Lati- tude. 6 miles. 5i miles. 5 miles. 41 miles. 4 miles. 31 miles. 3 miles. Deflec- tion Angl* and nat. tan, to Bad. 66 ft. Azimuths ^nd offsets at '*;>HJ . J *Q TM*M .t J ORIGINAL SURVEYS. 259 2. The direction of the first secant will be deter- mined at its initial point by observations on Polaris at elongation, and similar observations will be made at intervals not exceeding 18 miles; while observa- tions by the method given on page 96 et seq , or on Polaris at elongation (as the deputy may prefer), will be taken every night when practicable, to guard against mistakes, detect errors, and check the direction of the line. The principal advantage of this method, over that by offsets from a tangent, results directly from the proximity of the secant and the parallel of lati- tude, and the consequent reduced length of the maximum offsets, thereby limiting the cutting, which will contain both secant and parallel, to a single opening less than four feet in width ; avoiding the necessity for clearing out roads for, and instrument- ally laying off, the long offsets inseparable from the tangent method ; and permitting the noting of topo- graphical features on the lines actually run, a con- venience not always attainable by the tangent method. 3. In any given case, the secant lines will bear such relation to the latitude curve, that points on said secants at one and five miles from either end of any secant, will be coincident with two points on the latitude curve four miles apart ; between which points the latitude curve will lie south of the se- cants ; while the curve will lie north of the secant lines on the first and sixth miles ; therefore each secant will run south of sees. 31 and 36, in every range, and through all other sections on the north side of the base line or standard parallel, as the case may be. Each secant, the azimuth and offsets thereof, and the corresponding part of the parallel, will be sym- metrically divided by the middle meridian of each range, i. e., the bearings and offsets at equal dis- tances on opposite sides of the central meridian will 260 A MANUAL OF LAND SURVEYING. be equal; the bearings, which continually change, will always be north of east (or west), on the first three miles, and south of east (or west) , on the last three miles of each secant. The changes of bearing should not be understood to imply a change of direction of any secant with respect to its initial direction ; the change is due to the varying inclina- tion of the meridians to the straight secant, i. e. , the effect of convergency of meridians. 4. Employing the data provided by Table II, the practical application of the method herein outlined will be conducted in the field as follows : Set up the carefully adjusted transit south of the township corner at which the survey will begin, and at a distance therefrom to be interpolated for the given latitude, from the column headed "0 miles." in Table VIII. By observations on Polaris at elongation, determine and mark a true meridian. Lay off the azimuth, found in the table under " miles, " toward the east (or west), as the case may be, and re-measure the angle a sufficient number of times to secure an accurate result. Produce the direction of the secant thus deter- mined, a distance of six miles in a straight line, taking double back and fore sights at each setting of the instrument. At each half-mile and mile point, establish on the standard parallel the proper quarter section and section corners, by offsets of correct length, north or south, as indicated in the table by the initial letters N. or S. The offsets being very short, their direction (per- pendicular to the secant, without sensible error), may be determined by the eye ; the length of offsets should be carefully measured. At 6 miles on the secant, turn off to the north the proper deflection angle, given in the right hand column of the table, thereby defining the direction of a new secant, from which points will be estab- OBIGIXAL SURVEYS. 261 lished on the parallel, as directed in preceding para- graph. 5. Applications of Table VIII. The true bear- ing of the secant at each mile and half-mile point will be expressed by the tabular azimuth preceded by the initial meridional letter X., when the distance argument is found at the top of the table ; but when said argument is found at the bottom of the table, the meridional letter S. will be placed before the azimuth; while the departure letter, E. or W., will be made to agree with the direction of the survey, east or west, as the case may require. The bearings will be taken from the table, to the nearest whole minute only, and entered at the beginning of each mile recorded in the field notes. The direction of the offsets or distances from the secant north or south to the base line or standard parallel, as the case may be, are indicated by the initial letters, X. or S. following the offsets. Example 1. Standard parallel run west, lat. 48 N. ; dist. from initial point of secant, 2 miles ; the bearing is X. 89 59' W., the offset, 2.22 ft., S. ; at 5^ miles the bearing is S.89 57' W.,-the offset 1.66 ft. X. In all latitudes the bearing of the secant at 3 miles will be east or west, agreeing with the direction of the parallel. The offsets may be interpolated for minutes of latitude, by simple proportion, as follows: Multiply thf difference between the offsets corresponding to the whole degrees of latitude, immediately preceding and following the given latitude, by the minutes, ex- pressed in decimals of a degree, and add the prod- uct to the offset corresponding to the lesser lati- tude ; the sum will be the offset required. Example 2 , Lat. 45 34' ,5 ; dist., miles or 6 miles ; the diff. between offsets in latitudes 45 and 46 is 0.11 ft. ; 34 '.5=0. 575 ; 0.11X0.575 0.06 ft. ; and, 3.33+0.06=3.39 ft. the offset required. A ; 262 A MANUAL OF LAND SURVEYING. similar method of interpolation may be applied to the data in the right-hand column. Example, S. Latitude 45 34 '.5; diff. of angles isO' 11"; 11 X 0.575=6". 3 ; and 5' 11". 8 + 6". 3= 5' 18", nearly; also 0'. 04X0.575=0 02 ins.'; and 1.20+0.02=1.22 ins. 6. The deputy should clearly understand from the foregoing rules and directions that the correct establishment of a^ standard parallel on a true latitude curve, by offsets from secant lines, will depend in the order of sequence upon careful attention to the following points : 1. Accurate observations on Polaris at elonga- tion, to determine a true meridian. 2.' Close measurement of the azimuth angle, to define the initial direction of the secant. 3. Careful prolongation of the secant in a straight line. . 4. Correct measurement of the deflection angle 7. With ordinary field instruments, usually read- ing to single minutes only, fractional parts of the "least count" are generally estimated by the eye Greater accuracy may be obtained by making use of a linear measure to lay off deflection angles. Table VIII supplies the requisite data ; "the natural tan- gent of the angle of deflection to a radius of one chain," inserted in the right-hand column, may be employed as follows : Having taken a back sight at the 6-mile point on the secant, at exactly one chain in advance of the center of the instrument, place upon the ground in a horizontal position, and precisely at right angles to the line, a rule or scale divided into decimal parts of an inch, move the scale north or south until one of its principal lines appears coincident with the ver- tical wire ; then with the tangent screw of the ver- nier plate, carry the wire over the scale toward the north, the required distance, i. e. , the length of tan- gent in the right-hand column. The readings of the ORIGINAL SURVEYS. 263 vernier will check the measurement and guard against mistakes. A piece of white paper with two fine parallel lines drawn across it, exactly the proper distance apart, pasted on a thin slip of wood (such as a piece of cigar box 3 inches long by 1 inch wide), will make an accurate and very convenient and portable sub- stitute for a rule or scale. Several copies may be prepared in advance to replace the original in case of loss. 8. To mark the direction of the new secant thus determined, set a flag on line, and as far in advance of the instrument as practicable. The direction will be verified by another similar observation, to be made after revolving the azimuth circle 180. Theoretically, it is immaterial whether the scale tye placed above or below the level of the tele- scope provided the horizontal distance from the center of the instrument is accurately one chain (66 ft.); practically, the most satisfactory result will be had on level ground, suitable for correct meas- urement of the distance. 9. The secant method adapted to transit in- struments exclusively, is recommended for its sim- plicity and accuracy, and the facility with which the line may be extended over rough mountainous land or through dense undergrowth ; in deep valleys or canyons where the sun cannot be observed in favorable positions ; or anywhere during the contin- uance of adverse weather conditions and under cir- cumstances when the use of solar apparatus would be, if not impossible, at least inconvenient and un - reliable. 10. The true bearing of a line joining any two points on a standard parallel will be obtained from Table IX, by taking it from the column headed with one-half of the distance between said points. Ex- 204 A MANUAL OF LAND SURVEYING. ample. Required the bearing from corner of sees. 32 and 33, R. 22 E., to corner of sees. 32 and 33 E., R. 21 E. The latitude is 45 '34' .5, the dis- tance 6 miles. Consequently, the azimuth from the column marked "3 miles" for the given latitude, is N. 8 1 J 57' 20" .9 W., the required true bearing. Tangent Method. This method consists in lay- ing off from a true meridian, established by obser- vations on Polaris at elongation, an angle of 90 pro- ducing the directions thus determined, a distance of 6 miles in a straight line, and measuring north therefrom, at half-mile intervals, distances of correct length, taken from Table X (interpolated if neces- sary), for the given latitude, to attain other points on the latitude curve passing through the tangential or initial point. The azimuth or bearing of the tangent at succes- sive mile points will be taken from Table IX to the nearest whole minute only, and will be inserted in the field notes, no interpolation being required, ex- cept when test sights are taken. The true bearing between two points on a standard parallel will be derived from Table IX by taking it in the column headed with one-half the' distance between said points. The offsets at intervals of one mile are in- serted in Table X ; to obtain the length of offsets at the half-mile points, take one fourth the offset cor- responding to twice the distance of the half-mile, point from the tangential point. This method is suitable for running standard par- allels and latitudinal township lines in a level open country, where no intersections with topographical features will be required ; bu| in all cases the secant method will be found most convenient. ORIGINAL SURVEYS. 265 TABLE IX. Azimutlis oftJie Tangent to the Parotid. fThe azimuth IB the smaller angle the tangent make's with the true meridian and always measured from the north and towards the tangential points.] Lati- tude. 1 mile. 2 miles. 3 miles. 4 miles. 6 nlles. 6 miles. 80 II 88 11 89 59 30.0 89 59 28.8 89 69 27.5 89 59 26.2 89 59 24.9 89 59 23.6 89 58 69.9 89 58 57.5 89 58 55.0 89 58 62.5 89 58 49.9 89 68 47.2 89 53 29.9 89 5S 2C.3 89 68 22.5 89 68 18\7~ 89 5tf 14.8 89 56 10.8 89 67 69.9 89 57 66.0 89 57 60.0 89 67 44.9 89 57 39.7 89 57 34.4 89 67 29.9 89 57 23.8 89 67 17.6 89 67 U.2 89 57 04.6. 89 66 68.0. 89 56 59.8 89 66 52.6 89 66 4&.0 89 66 37.4 89 56 29.S 89 66 21.6 1 89 59 22.2 89 09 20.8 89 59 19.4 89 68 44.4 89 58 41.6 89 68 38.8 89 58 06.8 89 68 02.5 89 57 58.2 89 57 28.9 89 57 23.3 89 57 17.5 89 56 61.1 S9 56 44.1 89 66 36.9 89 66 13.4 89 56 05.Q 89 55 56.3 80 40 41 89 59 17.9 89 59 16.4 89 59 14.8 89 58 35.8 89 58 32:8 89 58 29.6 89 57 63.7 89 57 49.2 89 57 44.4 89 67 11.6 89 6T C5.5 89 56 59.3 89 56 29.6 89 56 21.9 89 56 14.1 89 55 47. & 89 55 38.8 89 55 28. 48 48 M 89 59 13.2 89 59 11.5 89 59 09.8 9 58 26.4 89 58 23.1 89 58 19.6 89 57 39.6 89 57 34.6 89 67 29.5 89 56 52.8 89 56 46.2 89 56 39.3 89 56 06.0 89 55 67.7 89 55 49.1 89 55 19.2 89 55 09.2 89 54 68.9 45 46 47 89 59 08.0 89 59 06.2 89 59 04.3 89 58 16.1 89 58 12.4 89 58 08.6 89 67 24.1 89 57 18.6 89 57 1.9 89 56 32.1 89 56 24.8 89 56 17.1 89 55 40.2 89 55 31.0 89 55 21.4 89 64 48.2 89 54 37.2 89 54 25.7 M 49 50 89 59 02.3 89 69 00.2 8? 68 58.1. 89 58 04.6 89 58 00.5 89 57 56.2 89 57 06.9 89 57 00.7 89 56 54.3 89 56 09.2 89 K 00.9 89 56 W. 89 55 11.5 89 55 01.2 89 54 50.5 89 54 13.8 89 M 01.4 89 63 48.5 Ltl- tmde. 7 miles. 8 miles. 9 miles. Id miles. 11 miles. 12 miles. H 89 56 29.8 89 66 21.3 89 56 12.5 89 M 59.8 89 u5 60.0 89 65 40.0 89 55 29.3 89 55 18.8 89 66 07.6 89 54 59.7 89 54 47.6 89 54 35.1 89 &t 29.7 89 54 16.3 89 54 02.6 89 53 59.7 89 53 45.1 89 63 30.1 & 89 55 03.6 89 55 64.5 89 66 45.2 89 55 29.9 89 55 19.4 89 66 08.8 89 64 56.1 89 54 44.4 89 64 32.3 89 M 2-2.3 89 54 09.3 89 63 65.9 89 58 48.5 89 53 34.2 89- 63 19.5 89 G3 14.8 89 52 63.1 89 62 43.1 80 11 89 65 35.6 89 66 25.8 89 69 15.7 89 54 67.8 89 54 4&6 89 64 36.1 89 64 20.0 89 64 07.4 89 63 64.6 89 63 42.3 89 63 2&2 89 63 13.9 89 53- 04.6 89 52 49.1 89 62 33.2 89 52 26.7 89 52'09.t 89 51 52.6 % 41 iv-.l 42 43 44 89 66 06.4 89 64 64.7 89 64 43.7 89 54 32.4 89 64 20.8 89 54 08.7 89 54 23.3 89 64 11.1 89 63 68.5 89 53 45.6 89 63 32.3 89 63 18.5 89 63 41.2 89 63 27.5 89 63 13.4 89 62 68.8 89 52 43.8 89 62 28.4 89 62 59.1 89 62 43.8 89 62 28.2 89 52 12.0 89 61 'b6.4 89 61 38.2 89 62 17.0 89 62 00.2 89 61 43.0 89 51 26.2 89 61 06.9 89 50 48.0 89 51 34. 1 89 61 16. < 88 50-S7. J 89 50 38. 4 89 60"'18.6 89 49 57.8 46 46 47 89 53 66.3 89 53 43.4 89 53 30.0 89 53 04.3 89 52 49.5 89 52 34.3 89 52 12.3 89 61 65.7 89 51 38.6 89 51 20.4 89 51 01.9 89 50 42.9 89 60 28.4 89 50 08.1 89 49 47.2 89 49 36.4 89 49 14. S 89 48 51.4 43 IS 89 63 16.1 89 63 01.7 89 62 46.6 89 52 18.4 89 52 01.9 89 61 44.7 89 61 20.7 89 51 02.1 89 50 42.8 89 60-23.0 89 60 02.4 89 49 40 9 89 49 25.3 89 49 02.6 89 48 39.0 89 48 27.C 89 48 02.8 89 47 S7.1 283 A MANUAL OF LAND SURVEYING. TABLE X. Offsets, in Chains, from Tangent to Parallel. Utl- tudft. 1 mile. 2 mile*. 3 miles. 4 miles. 6 miles. miles. sr Si Chain*. 0.006 0.006 0.006 Chains. 0.023 0.024 0.025 Chains. 0.053 0.066 0.057 CAain*. 0.09 0.10 0.10 Chains. 0.14 0.15 0.16 Chains. 0.21 0.22 0.23 H 0.007 0.007 0.007 0.026 0.027 0.028 0.059 tss o.lo 0.11 0.11 0.16 0.17 0.18 0.24 0.26 0.26 1 0.007 0.008 0.008 0.029 0.031 0.032 0.066 0.068 0.071 0.12 0.12 0.13 0.18 0.19 0.20 0.26 S:S 8 41 0.008 X 0.033 $ 0.074 0.076 0.079 0.13 0.13 0.14 0.20 0.21 0.22 0.29 o:H 42 S &8S 0.010 0.036 0.038 0.039 0.082 S 0.14 0.15 0.16 0.23 0.24 0.24 i2 0.36 46 t? 0.010 0.010 0.011 0.040 0.042 07044 0.091 0.094 0.097 0.16 0.17 0.17 0.25 0.26 0.27 0.36 0.37 0.39 48 49 60 0.011 0.012 0.012 0.046 0.046 0.048 0.101 0.104 0.108 0.18 0.19 0.19 0.28 0.29 0.30 0.40 0.42 0.43 Latl. tude. 7 miles. 8 mile*. 9 miles. 10 miles. 11 miles. 12 mllt. 80 81 32 Chains. 0.29 0.30 0.31 Chains. 0.37 0.39 0.40 Chains. 0.47 0.49 0.61 Chains. 0.56 0.60 0.63 Chains. 0.71 0.74 0.76 Chnin*. 0.84 0.88 0.91 g 0.32 0.33 0.36 0.42 0.43 0.46 ts 0.67 0.66 0.68 0.70 0.79 0.82 0.86 0.95 ts 8 88 0.36 0.37 0.38 0.17 0.48 0.60 0.69 0.61 0.64 0.73 0.76 0.78 0.89 0.91 0.96 1.06 1.10 1.14 89 8:J? 0.43 0.62 0.64 0.66 0.66 0.68 070 0.81 0.84 0.87 0.99 1.02 1.06 1.18 1.22 1.26 8 0.44 0.46 0.48 0.68 S: 0.73 0.76 0.79 0.90 si 1.09 1.14 1.18 1.31 1.36 1.40 ts 47 0.49 0.51 0.63 0.64 8 0.81 0.84 0.87 1.00 !:S 1.22 1.26 1.31 1.46 1.60 1.66 48 X 0.66 IS |3 0.77 0.91 53 1.12 1.16 1.20 1.36 11 1.61 1.67 1.73 SUBDIVISION OF SECTIONS. 267 CHAPTEE IX. SUBDIVISION OF SECTIONS. 1. Subdivisions of sections are original surveys to be made in the following manner: 1. Section and quarter-section corners set by the gov- ernment surveyors, and the boundaries actually run by them, as well as the length of all lines as returned in their field notes, are to be taken as correct. (See Sec. 2396 R. S., First and Second. P.200, Sec. 100.) 2. The corners of half and quarter sections which were not marked on the government surveys, must be placed as nearly as possible equidistant from those two comers which stand on the same line. (Sec. 2396, First. P.200, Sec. 100.) This applies to the quarter-posts on the north and west lines of the township which were surveyed previous to 1846; also to those townships which, under the act of 1796, were surveyed into blocks of two miles square (P.200. Sec. 99, Third), and to those surveyed under the act of 1800,* where no quarter-section corners were planted on the lines running from south to north. *No. 21. An Act to amend the act entitled "An act providing for the sale of the lands of the United States, in the territory northwest of the Ohio, and above the mouth of the Kentucky River." SEC. 3. And be it further enacted. That the surveyor-general shall cause the townships ^west of the Muskingum, which by the above- mentioned act are directed to be sold in quarter townships, to be sub- divided into half sections of three hundred and twenty acres each, as nearly as may be, by running parallel lines through the same from east to west, and from south to north, at the distance of one mile from each '268 A MANUAL OF LAND SURVEYING. 3. The boundary lines of sections, (see Page 198, Sec. 99, Third), and of half and quarter sections, which were not actually run and marked, are to be ascertained by run- ning straight lines from the established corners to the opposite corresponding corners. Where no such opposite corners have been or can be fixed, the line should be run from the established corner due north and south or east and west, as the case may be, to the water-course or other external boundary. (P.200, Sec. 100, Second.) These due lines are to be found by trial of the boundary lines of the section, as actually run by the government sur- veyor, and the subdivision line, run on a course interme- diate between the courses of the section lines which lie parallel with it. The following figure illustrates the manner of sub- dividing sections. It shows sections 5, 6, 7, and 8, repre- other, and marking corners, at the distance of each half mile on the lines running from east to west, and at the distance of each mile on those running from south to north, and making the marks, notes, and descriptions prescribed to surveyors by the above-mentioned act : And the interior lines of townships intersected by the Muskingum, and of all the townships lying east of that river, which have not been hereto- fore actually subdivided into sections, shall also be run and marked in the manner prescribed by the said act for running and marking the interior lines of townships directed to be sold in sections of six hun- dred and forty acres each. 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 six miles, the excess or deficiency shall be specially noted, arid added to or deducted from the western and northern ranges of sections or half-sections in such township, ac- cording as the error may be in running the lines from east to west, 01 from south to north; the sections and half-sections bounded on the northern and western lines of such townships shall be sold as contain- ing omy the quantity expressed in the returns and plats, respectively, and all others as containing the complete legal quantity. And the President of the United States shall fix the compensation of the dep- uty surveyors, chain-carriers, and axemen: Provided, The whole ex- pense of surveying and marking the lines shall not exceed three dollars for every mile that shall be actually run, surveyed, and marked. SUBDIVISION" OF SECTIONS. 269 sentiiig the four different cases which occur in a township surveyed previous to 1846. In the later surveys, the de- TVj^n ._s_h jj; J i n B tails would dif- fer a little, ow- ing to the fact that the section and quarter- section corners on the town- ship and range lines are com- mon to the townships o n each side of and ad joining those lines. The prin- ciple of subdi- vision is, how- 3 ' * t~'^'~l *_. j_J^S-2. 1 _ . JLet; 3992, ; F * o V 1 1 " f+0. r>0 \20flO [J./.33 * UJ7 * 5 C . A J ! \ I 11 26 . ^o oo 39.R9, c 39. as. FIG. 69. CASE 1. Section 8. All the quarter posts are at equi- distant points from the section corners which are on the same line. CASE 2. Section 5. Quarter posts on the north and the south are at equidistant points. Those on the east and the west are 40 chains from the south line of the sec- tion. The fraction is on the north half of the section. CASE 3. Section 7. Quarter posts on the north and the south are placed at 40 chains from the east line of the section. Those on the east and the west are at equidistant points. The west half of the section is fractional. CASE 4. Section 6. The quarter posts on the north and the south are placed at 40 chains from the east line of the section. Those on the east and the west are 40 chains from the south line of the section. Fractional both on the north and west. NOTE. In 1856, Thomas A. Hendricks, then Commissioner of the General Land Office, gave the following rule for locating the center of a section: "Run a true line from the quarter-section corner on the east boundary, to that in the west boundary, and at the equidistance between them establish the corner for the center of the section." 270 A MANUAL OF LAND SURVEYING. This was in harmony with an opinion previously given by the Sur- veyor General of Missouri and Illinois, and was very generally fol- lowed by the surveyors in those States. This rule has not been sus- tained by the courts, nor by any other ruling of the Land Office, so far as we can learn. It was expressly overruled by the Secretary of the Interior in 1868. Quarter- sections are to be subdivided into half -quar- ters by lines running north and south. The corners which were not marked are to be placed as nearly as possible equidistant between the two corners of the quarter-section which stand on the same line. Then run straight lines from the established corners to the opposite corresponding corners, (Page 202, Sec. 101.) Half-quarter sections are to be subdivided into quar- ter-quarters in a similar manner, by east and west lines, (P. 202, Sec. 101.) It may be well to remark here, that the instructions from the Gen eral Land Office have not been uniform in regard to the proper manner of subdividing quarter-sections, and, as might be expected, the prac- tice is not uniform among good surveyors Commissioners Wilson and Edmunds held that half-quarter and quarter-quarter lines should be " straight lines running through the section " to points on the sec- tion line. (See Hawes's Manual, p. 142, and Dunn's Land Laws, p, 19.) The foregoing rules are those of the statute, and are endorsed by Commissioners Drummond, Williamson, and McFarland. Commissioner Drummond's instructions are as follows: " In the subdivision of quarter-sections, the quarter-quarter posts are to be placed at points equidistant, and on straight lines between the section and quarter-section corners, and between the quarter-cor- ners and the common center of the section," etc. The difference in the two methods occurs when, as very often happens, the quarter-posts are not in line between the section corners. 2, Fractional sections are to be subdivided ac- cording to the Fifth paragraph of Sec. 2395 of the Kevised Statutes, under such rules and regulations as may be pre- scribed by the Secretary of the Interior. (Sec. 99, Ex. Land Laws, and TJ. S. Instructions, 1881, p. 39.) Under these regulations.* the fractional quarter-sections lying next to the north line of the township are divided * NOTE." Circular to Surveyors-General, Nov. 9, law. SIR: By the first section of the act of April 24, 1820, all the public lands of the Uni- ted States shall be offered at public sale in half-quarter sections; and SUBDIVISION OF SECTIONS. 27 1 into half -quarters by lines running east and west, parallel with and twenty chains distant from the quarter-section line. (See Keasling v. Truitt, 30 Ind. 506.) The quarter- sections lying next to the west line of the township are divided into half-quarters by lines running north and south, parallel with and twenty chains distant from the quarter-section line. 3 . Section 6 adjoins both the north and the west lines of the township, and is subject to both rules. The north half is divided into half -quarters by an east and west line, and the south half by north and south lines. The quarter-post on the north side of section six should be placed on the township line at a point 40 chains of original measure west from the northeast corner of the section. The quarter-post on the west line of section six should be placed at a point on the range line 40 chains of orig- inal measure north from the southwest corner of the section. By anginal measure is meant such measure as was actually laid down on the ground by the deputy sur- veyors who made the original survey. fractional sections containing one hundred and sixty acres and up- ward shall, as nearly as practicable, be divided into half-quarter sec- tions, under such rules and regulations as m&y be prescribed by the Secretary of the Treasury; but fractional sections containing less than one hundred and sixty acres shall not be divided, etc. By the act of May 10, 1800, section 3, the excess or deficiency of regular sections or quarter-sections in any township is to be thrown on the north and west sides of the township, making fractional sections more or less than one hundred and sixty acres. In subdividing such fractional sections to form a half-quarter section, viz., 80 acres, the Secretary of tha Treasury directs that the subdividing line for such fractions as lie on the north side of a township sha.ll be an east and west line, forming the half-quarter section on the south side of the fraction; and for such fractions as lie on the west side, the subdividing line shall be a merid- ian, forming the half-quarter section on the east side of the fraction. This mode of subdivision will preserve the compactness of the tracts with the general divisions, and will not interfere with the rule adopted relative to fractions formed by a stream, a river, etc." 272 A MANUAL OF LAND SURVEYING. Iii further subdividing the northwest quarter of Section 6 into quarter-quarters, it is done by a line parallel with and 20 chains west of the north and south quarter -sectipn line. The foregoing is the general plan adopted for the sub- division of sections of the United States Survey. There have, however, been many exceptions in the earlier official plats, in accordance with which the land was sold. To meet all such cases the rule has been adopted to subdivide in such a way as to suit the calculation of the areas on the official plat. This is sometimes difficult, the areas in some cases seeming to have been put down without any calculation. fractional by FIG. 70. waters, reser- Sections made vations, etc., should be sub- divided in such a manner as to produce the same result as would have been produced had the section been full. This may sometimes be done by ex- tending and by measuring the lines on the ice, or over the res- ervation. figure illustrating the Subdivision of a Section fractional on waters. Commissioner Drummond says (see Copp's Land Laws, p. 761): "In the subdivision of fractional sections, where no opposite corners have been or can be fixed, the subdivision lines should be ascertained by running lines from the es- tablished corners due north, south, east or west, as the case SUBDIVISION OF SECTIONS. 273 may be, to the water-course, Indian boundary line, or other external boundary line of such fractional section. The law presupposes the section lines surveyed and marked in the field by the United States deputy survey- ors 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 the running of subdi visional lines through fractional sections to adopt mean courses where the lines are not due lines, or to run the subdivisional line paral- lel with the section line when there is no opposite section line.' 1 4. Irregular Subdivisions of Fractional Sec- tions. In making irregular subdivisions of fractions bounded on streams or lakes, the following rule has been laid down by the authorities. It has been decided by the Supreme Court of the United States that "the meander lines run in surveying frac- tional portions of the public lands bordering upon navi- gable rivers are run not as boundaries of the tract but for the purpose of defining the sinuosities of the stream and as the means of ascertaining the quantity of land in the fraction, and which is to be paid for by the pur- chaser." R. R. Co. v, Schurmier, 7th Wallace (U.S.) 272. It has been, held that the same lines are to be used in ascertaining the quantity of land in any portion of the fraction. Thus, as often happens, if a deed calls for so many acres off the end of the fraction, the surveyor in making his computations to determine at what point to locate the dividing line, should in the absence of any- thing showing to* the contrary, use the meander line for the purpose of estimating the area of the tract, and lay down the dividing line accordingly. Otherwise there could be no common basis of calculation and as many different results would be arrived at as there were differ- ent surveyors to run the line, or different times of survey. 274 A MANUAL OF LAND SURVEYING. This is especially true of fractions bordering- on lakes whose shore lines are subject to great change from natu- ral causes or artificial drainage. The common law rule for calculating the quantity of land bordering on a non-navigable stream is that no ref- erence is had to what lies between low water mark and the centre of the stream. On navigable waters, high water mark is the line. Lamb v. Eickett, 11 Ohio 311. 5. Exceptional Oases. In the United States sur- veys made previous to 1815, there was much irregularity in the practice of the surveyors in carrying on the sur- veys. The fractional sections were frequently thrown upon the south or east tiers of sections in the township; the surveys being carried on from the north to the south and from the west to the east. Where the township was made fractional by large rivers or lakes, they were fre- quently so laid off as to throw all the fractions into the sections bordering on the water. There was even greater irregularity in the manner of subdividing the fractional sections into the lesser tracts. Many of them had no quarter section corners. In some, the government plats show no subdivision; some are sub- divided in one way and some in another. In making resurveys and subdivisions of these and all other exceptional cases, the surveyor must always make his resurvey conform to the plan as shown by the field- notes and plats of the original survey. 6. Field Notes of the Survey and Subdivi- sion of a Fractional Section. The following notes of an actual survey are intended as an illus- tration of the manner of subdividing a section under the ordinary conditions as they are met with in the field, including the manner of restoring, certain lost corners and to a certain extent the principles governing re-surveys as laid down in SUBDIVISION OF SECTIONS. 275 subsequent chapters. All the corners of the United States survey have equal weight or authority, hence in subdividing a section or restoring lost corners of that survey it makes no difference at what corner the surveyor begins his work or in what order it is done, except so far as his own convenience and that of his assistants are concerned. It will be rarely, if ever, that the work will be done in precisely the same order in any two sections in a settled country. The student should trace the notes of each operation carefully through and verify the results. The first- thing required by the surveyor is a complete and correct copy of all the field notes of record which refer to the section to be subdivided and of adjacent sections when necessary for restoring lost corners. In order that the student may be able to trace through the several operations properly and under - standingly so much of the field notes as were re- quired in the survey are given herewith. Field Notes of U. S. Survey of Section 2 T4 S, E 9 W. North Boundary. Var. 5 10 ' E. West 71.78 80.00 West 22.50 39.00 40.00 59.59 66.44 69.22 80.00 On S. boundary of sec. 36, T 3 S, R 9 W. Black Ash 16 in. Set post cor. to sees. 35 and 36. Beech 10 in., N. 30 W. 10 Iks. Tam'k 6 in., N. 46^ E. 78 Iks. On S. boundary of sec. 35, T 3 S, R 9 W. Lynn 6 in. Left Swamp. Set post X se c. cor. sec. 35. Beech 9 in., N 39 W 39 Iks. Beech 6 in., N 74 E 48 Iks. Sugar 14 in. Stream 20 Iks. wide, course south. Stream 10 Iks. wide, course south. Set post cor. sees. 34 and 35. Beech 6 in., N. 58^ W. 33 Iks. Beech 6 in., N. 52 E. 31 Iks. John Mullett D. S., Nov. 8, 1825. 2*76 A MANUAL OF LAND SURVEYING. West 39.95^ Subdivisions. Var. 5 35' E. Between sections 11 and 12. Stream 25 Iks. wide, course S. E. Set post X se c- cor - Sees. 11 and 12. Water Beech 6 in.N. 55 E. 11 Iks. Sycamore 30 in. N. 77 W. 28^ Iks. Set Post corner to Sees. 1, 2, 11, and 12. Tam'k 6 in. S. 18 W. 37 Iks. Tam'k 8 in. N. 88 E. 21 Iks. Corrected line between Sees. 1 and 12. Set X Sec. post. Beech 16 in. N. 23 E. 31 Iks. Beech 8 in. S. 41 W. 17>^ Iks. Sec. Cor. Between Sees. 1 and 2. Elm HO in. Set post ^ sec. cor. Sees. 1 and 2. Beech 13 in. S. 74 E. 26 Iks. Beech 6 in. N. 30^ W. 11 Iks. Entered swamp. Intersected N. boundary 20 Iks. E. of post. Set post at intersection. Cor. to Sees. I and 2. Black ash 16 in. S. 65 W. 33 Y 2 Iks. Black. ash 15 in. S. 47 E. 39 Iks. Between Sees. 10 and 11. Set post # sec. cor Sees. 10 and 11. Beech 8 in, S. 78 E. 34 Iks. Beech 10 in. S. 79 W. 13^ Iks Set post cor. to Sees. 2, 3, 10, and 11. Beech 7 in. S. 62 E. 25^ Iks. Beech 14 in. N. 63 W. 3 Iks. On random between Sees. 2 and 11. Enter tam'k and birch swamp. Stream 25 Iks. wide, course S. E. Intersected E. boundary 25 Iks. S. of post. "Corrected line between Sees. 2 and 11. Set quarter Sec. post. Beech 12 in. S 50 E. 3 Iks. Beech 8 in. N. 21 W. 18 Iks. Sec. Cor. Between Sees, 2 and 3. Entered swamp. Brook 10 Iks. wide, course S. W. Left Swamp. 40.00 61.48 78-32 SUBDIVISION OF SECTIONS. 277 Set post ^ sec. cor. Sees. 2 and 3. Whitewood 20 in. N. 53 W. 27 Iks. Beech 8 in. N. 89 E. 20 Iks. Beech 12 in. Intersected N. boundary 45 Iks. E. of post. Set post at intersection. Cor. to Sees. 2 and 3. Beech 8 in. S. 74 \V. 33 Iks. Beech 14 in. S. 16 E. 47 Iks. Robert Clark, Jr., D. S. May 6, 1826. Notes of Later Survey. On May 12, 1854, Randolph Nutting found the east and the west quarter section corners of this section, with bearing trees standing and ran the east and west quarter line between them, setting temporary stakes on the true line, dividing it into four equal parts. He also renewed the posts at these quarter section corners. At the corner of Sections 2, 3, 10, and 11 he found the decayed post and bear- ing tree/ Beech 14 N. 63 W. 3 Iks., standing, the other bearing tree destroyed. He planted a granite boulder 12x16x24 in. one foot below the surface and at the corner drilled a hole l in. diameter and 4 in. deep. Thence he ran east on random 40.12 ch. to the quarter section corner and correct- ing back at 20.06 ch. on true line set granite boulder 10x12x18 in. 1 foot beneath the surface, marked it with a drill hole and planted side stones 50 links each north and south from the corner. Re- Survey and subdivision of Section 2, T 4 S, E 9, W., April 4, 5, 6, 1882. (Note : For convenience of reference the corners of the several subdivisions are numbered or lettered as shown in the figure, on the system shown at the close of Chapter VI. ) 278 A MANUAL OF LAND SURVEYING. 5 14 7 PLAT OF SEC 2. r Began at 4, where I found the "rock bound" planted by Nutting, 18 in. below surface at intersec- tion of roads. All parties agree that this corner stone has not been moved from its original position. All traces of bearing trees are destroyed or removed. I accept the corner as correct and mark Sugar 12 in. N. 67 E. 68 Iks. Beech 9 in. N. 37 W. 76 Iks. ,40.15 Ran thence north on random, setting stakes every 10 chains. Intersected the quarter line 7 Iks. east of 8. Cor- ner post dug out in the road. The stump of white- wood bearing tree is standing and I ran thence S. 53 E. 27 Iks. and plant for X Sec. Corner an earth- enware post 3 in. diam. 30 in. long with stones around it and mark Beech 8 in. N. 26 E. 76 Iks. Sugar 10 in. S. 47 E. 104 Iks. SUBDIVISION OF SECTIONS. 279 South corrected line. Set granite boulder 8 x 12 x 24 in. marked -f for corner (15) with side stones 50 Iks. each east and west. Put a back sight on this corner and return to 8. Set up transit over the post, back sight to 15 and prolong line north, setting temporary stakes at 20.00 chs. and 38.32 chs. and search for sec. corner. Bearing trees are both destroyed and obliterated in the jog in the road. Chapm says a new stake has been driven in the old stake hole, and points out the location, but on digging I find nothing of it there. I then look for the corner of Sees. 34 and 35. I find remains of a stump, which from its position, may have belonged to one of the bearing trees. I set transit up over it and run S. 58 > E. 33 Iks. and dig the earth carefully away. I find the decayed remains of a stake from which point I run east 45 links and dig and find the new stake driven in the old stake hole, as described by Chapin. Random line from the south intersects the township boundary 2 Iks. east of the corner of Sees. 2 and 3 at 38.46 chains from the quarter post. I plant new stake in old stake hole, cor. of Sees. 34 and 35 and for cor. of Sees. 2 and 3 plant iron landside of plow packed about with brickbats and broken crockery and mark Sugar 10 in. S. 43*4 W. 76 Iks. Sugar 14 in. S. 32 E. 1.24 Iks. Thence south corrected line. Set stake with broken brick and glass around for corner (16) 148 Iks. south of stump of Beech line tree; Cherry 18 in. N. 72 E. 64 Iks. Elm 24 in. S. 61^ E. 96 Iks. I then return to 8 and offset north 20 itis. and set up transit in the section line. Backsight to (15) and run thence east at right .angles with section line, setting stakes every 10 chs. on random line. To bank of Mill-pond. Set flag in line across pond and then run a line south at right angles with random 5.00 eh. to a point where I set up transit and measure the angle between lines to back-sight and to flag over the pond=54 26 ' whence nat. tan, 1.3985X5.00=6.9925 makes the distance across pond 6.99^ and distance on random over pond= 43.00-j-6.99#=49.99i4:. Continue the line. 280 A MANUAL OF LAND SURVEYING. 80.00 118.00 80.00 38.12 Set temporary stake and look for #. sec. cor. Sees. 1 and 2. There are no bearing' ^tfrees stand- ing and no one knows the location of the corner. I set stakes to mark the random line and go next to 3, where I find no traces or evidence ot the cor- ner. I then go to the # sec. cor. of Sees. 11 and 12, where I find both bearing trees and corner post standing in place. From thence I run north on random, setting temp, stakes at 40.00, 60.00 and 80.00 chains." Set temp, stake and look for corner. The bearing trees of both corners are missing, but there are numerous stumps near by rotted to the ground. I go east along township boundary and find blk ash station tree standing. I take its distance fromTp. cor. at 71.78 chains and continue the measure west on random. Set temp, stake and search again, and finding what appears likely to be the remains of stump of bearing tree, run thence N. 47 W. 39 Iks. stick a pin and run from it S. 65 W. 33^ Iks. to roots of black ash tree lying on the ground decayed and moss covered. Digging at the point where I stuck the pin, I find 18 in. below the surface in wet soil, the sound bottom of the original post, cor. of Sees. 1 and 2. As a check I measure west 20 Iks. and by digging find the post cor. of Sees. 35 and 36 and plant a new post in its place. At the cor. of Sees. 1 and 2 I put in a piece of iron plow beam 24 inches long, pack brick and stone about it and set side stones 50 Iks. each east and south from corner. Random line from the south at 118.39 ch. inter- sects the township boundary 12 Iks. west of corner of Sees. 1 and 2. South between Sees. 1 and 2. Find by digging the remains of decayed post in position 9 Iks. east of random line. ; I also find roots of bearing trees at corresponding points called for by field notes. Random line from the west at 80.04 ch. inter- sects E. bound 45 Iks. north from qr. sec. cor. I plant a stone 5 x 8 x 16 marked 4- with brick- bats and broken glass around for 1 4 mark Beech 12 in. E. 63 Iks., ' Beech 14 in. S. 29 W. 92 Iks. sec. cor. and SUBDIVISION" OF SECTIONS. 281 Set flag in line at 6 and continue thence south. Set temp, stake 6 Iks. east of random line. Search the ground carefully and find no traces of corner stake or of bearing trees. I next return to 4, set up flag on the corner and measure thence east along the highway. Corner 14 "Rock bounds" set by Surveyor Nutting. Set up transit at corner 14, backsight to 4 and prolong the line east. Original corner and bearing trees entirely dug out and destroyed in the highway. Punch a hole with on iron bar 4 ft. deep and three in. diameter and fill it with Portland cement mortar for quarter section cor. and mark maple 10 N. 5 W. 62 Iks. Set temp, stake and continue measure east with- out running the line, the qr. post of Sees. 1 and 12 being known. Find corner post and bearing trees standing at quarter section Cor. See's 1 and 12. I then return to 3 and locate section corner at distances from nearest corners N. S. E. and W. as follows: From quarter sec. cor. Sees. 1 and 2 40.13 chs. From quarter sec. cor. Sees. 11 and 12 40.13 chs. From quarter sec. cor. Sees. 1 and 12 40.05^ chs. From quarter sec. cor. Sees. 2 and 11 40.14 chs. I drive a black walnut stake deep in the ground at the corner and over it put a stone 8 x 12 x 30 in. marked -f- Black Ash 8 in. S. 43 W. 82 Iks., Black Ash 10 in. N. 51 W. 126 Iks. Thence north on true line (Var. of needle 2 34' E.) Set stake with brick around for corner 12 and mark Elm 36 in. N. 87 W. 54 Iks. Cherry 24 in. S. 5 W. 68 Iks. I then return to 7, set up transit over the corner, sight to flag at 14 and turn angle to right of 89 40' and run thence at that angle north on random setting temporary stakes every 10 eh. Intersected E. and W. corrected quarter sec. line. To point on bank of mill pond. Set two flags in Hue over pond to range by, then angle to the left 60 and run 5.00 to 2nd triangulating point 282 A MANUAL OF LAND SURVEYING. 60.00 78.42 18.21^ 20.01 40.13 20.06^ from which I turn off angle of 90 to right and set flag in the random line over the pond. Distance between the triangulating points 1 and 3=5.00 X secant 60 (2. 0000) =10. 00 ch. Point in random over pond. Intersect N. boundary 36 Iks. east of quarter sec- tion cor. Sec. 35. Found both bearing trees stand- ing for quarter section cor. Sec. 35 and decayed stake in right place for corner. Planted a new stake with stones around and ran thence east 32j^ Iks. and planted a piece of l l / 2 iron pipe 3 ft. long for quarter section corner Sec. 2 and put brick around it and marked Birch 10 in. S. 48 E. 116 Iks. Beech 12 in. S. 16 E. 32 Iks. South corrected quarter section line. Set stake with glass and broken crockery around for corner and marked Swamp oak 12 in. S. 2 E. 186 Iks. At intersection of quarter section lines set stone 9 x 13 x 24 with stake underneath and cross on top at corner. Wh. ash 8 in. N. 12 W. 10 Iks. Hickory 12 in. N. 84 E. 63 Iks. Intersection is 40.02 ch. from 8. From C west corrected line. Set stake with brick and glass around for cor- ner d. No trees near. From d south on random at angle .of 90 with random E. and W. ^ line. Intersected Sec. line 4 Iks. west of corner 14. North corrected line. Set. stone 6 x 8 x 16 marked -j- for cor. h. No tree near. Returned to c and on corrected line 20.07 from C and 7 set post with broken glass around. Cherry 10 in. S. 67^ E. 93 Iks. Whitewood 16 in. S. 21 W. 114 Iks. OTHER ORIGINAL SURVEYS. 283 This survey was made with transit and steel tape. On all lines run with the transit temporary stakes were set every ten chains on the random line and afterward corrected to the true line. Let the stu- dent find the amount and direction of the correction applied to each stake to place it in the true line. The measurements on this section were more than usually uniform on different lines, the ground being comparatively level. Usually the quarter section corners set in the old compass surveys are more or less out of line between the section corners, causing a discrepancy in the interior measurements. Differ- ence in the character of the surface of the land along different lines also has a tendency to cause discrepancy of measures. It will be also noticed that in this survey the direction of the lines is con- trolled by the monuments of the original survey and their absolute direction from the true meridian was considered of so little importance that no atten- tion was paid to it. In retracing the more careful surveys now being made by the United States sur- veyors, the direction from the true meridian is a more important factor in the resurvey. 7. Other Original Surveys. In a considerable portion of the United States, the general government never had any ownership of the land. The surveys were there made by the proprietors upon such system or plan as suited themselves. The further subdivision of these tracts is original sur- veying. It is sufficient to say of this work that it should be done with great care, and that ihe marks upon the ground which indicate the boundary lines should be y or to a stream includes flats at least to low-water mark, and in many cases to the middle thread of the river. Thomas v. Hatch, 3 Sumner (U. S.) 170. 5. A boundary on the bank of a river referring to fixed monuments on the bank, limits the grant to the. bank and excludes the flats. (Ibid.) See also Hopkins v. Kent, 9 Ohio, 13. 6. The words "along the bank" are strong and definite enough to exclude the idea that any part of the river or its bed was granted in the navigable or innavigable parts of the river. Howard v. Ingersoll, 13 How. (U. S.) 341, 416 7. A deed describing the land by a boundary run- ning to a stream, and thence along its bank, and re- serving the right to use the river front a specified time, LANDS BORDERING ON WATERS. 309 conveys the land to the water's edge and covers the riparian rights to the middle of the stream. Cole v. Wells, 49 Mich. 450. 8. Congress, in making a distinction between streams navigable and those not so, in the acts relating to the sale of the public lands bordering thereon, in- tended to provide that the common law rules of ripa- rian ownership should apply to the lands bordering on the latter, but that the title to lands bordering on the former should stop at the stream. R. R. Co. v. Schurmeir, 7 Wall. (U. S.) 272. 9. In streams which are not navigable, adjacent proprietors own to the center of the stream measured from low-water mark. Clark v. Caupau, 19 Mich. 325. Moore v. Sanborn, 2 Mich. 519. Lorman v. Benson, 8 Mich. 18. Bay City Gas Light Co. v. Ind. Wks. 8 Mich. 182. Lamb v. Ricketts, 11 Ohio 311. 10. The same principle is applied to Lake Muske- gon, in Michigan (Rice v. Ruddeman, 10 Mich. 125), but not applied to a similar lake in Wisconsin, where the court says (Deidrich v. K W. U. Ry. Co., 42 Wis. 271) : " Riparian owners upon a natural lake or pond take only to the shore." 11. In the case of the State of Indiana v. Milk, Circuit Court of the United States, April term, 1882, llth Bissell, page 197, the court rejects the theory of riparian ownership in the lake, and after presenting its reasons at some length, concludes with the follow- ing : " That while a general grant of land on a rivr or stream non-navigable extends the line of the grantee to the middle or thread of the current, a grant on a natural pond or lake extends only to the water's edge." 12. Islands in rivers fall under the same rule as to ownership as the soil under water does. If not 310 A MANUAL OF LAND SURVEYING. otherwise lawfully appropriated, they belong to . the proprietors on either side of the stream, according to the original dividing line or filum aquce as it would run if the islands were under water. The filum aquce is midway between the lines of ordinary low-water mark, without regard to the channel or depth oi water. When the island is appropriated, the boundary is then midway between it and the mainland. McCullough v. Wall, 4 Rich. (S. C.) 68. Kimball v. Schaff, 40 N. H. 190. 13. The grant includes any land between the mean- der line and the water, in an unnavigable stream. The same principle applies to unnavigable lakes Forsyth v. Smale, 7 Biss. (U. S.) 201. 14. The owners of land bordering on the shore of a meandered non-navigable or dried-up lake, own the bed of the lake in severalty, and their title extends to the center ; the boundary lines of each abutting tract being fixed by extending, from the meander line on each side of the tract, lines converging to a point in the center of the lake. Shell v. Matteson (Minn. 1900) 83 N. W. 491. 15. Where an island springs up in the midst of a stream, it is an accretion to the soil in the bed of the river, and not to the land of the riparian owner. East Omaha Land Co. v. Hansen, 90 N. W. 705 (Iowa, 1902). 16. Where, after submergence, the water disap- pears from the land, either by gradual retirement or elevation of the land by natural or artificial means, and its identity can be established by reasonable marks, or by situation or boundary lines, the pro- prietorship returns to the original owner. Hughes v. Birney's Heirs, 32 So. 30 (La. 1902). 17. High-water mark in the Mississippi River is to be determined from the river bed, and that only is LANDS BORDERING OX WATERS. 311 river bed which the river occupies long enough to wrest it from vegetation. Houghton v. Railway Co., 47 Iowa 370. 18. A bank is the continuous margin where vege- tation ceases. The shore is the sandy space between it and low- water mark. McCullugh v. Wainwright, 14 Penn. St. 59. 19. Where a levee was shown to have been judi- ciously located by a competent engineer and agents of the State acting under authority conferred by the State Legislature, it was held that such levee became the boundary line of high water, and that no private ownership ceuld be acquired to land lying between that and the bed of the stream. Musser v. Hershey, 42 Iowa 356. 20. Grant of a city lot bounded on a river, takes to the center of the stream. Watson v. Peters, 26 Mich. 508. 21. Riparian rights, unless expressly limited, ex- tend to the middle of the navigable channel, and cover any shallows or middle ground not shown in the gov- ernment surveys, but lying between such shallows and the shore, and it makes no difference that the deed conveying the premises to which the rights attach describes them according to a city plat instead of the government entry. Fletcher v. Thunder Bay Boom Co., 51 Mich. 277. 22. But if the plat plainly indicates the proprie- tor's intent to reserve the space between the shore and the thread or main channel, the case would be dif- ferent. Watson v. Peters, 26 Mich. 508. 23. Riparian rights extend laterally into the stream. Rocks and shoals along the margin of navigable riv- ers above tide-water belong to the riparian owner. Moore v. Willamette T. & L. Co., 7 Oregon R. 355. 312 A MANUAL OF LAND SURVEYING. 24. When a navigable stream is meandered in mak- ing the public surveys, and the United States has" granted to the meander line, the grantee takes to the river. The stream, and not the meander line, is the true boundary of the riparian owner. Minto v. Delaney, id., 337. 25. Lands patented by the United States on a tide- water stream extend to the meandered line of the v stream, which is the line of ordinary high water. Parker v. Taylor, id., 435. 26. A boundary by the shore of a mill pond takes to low water mark. Stevens v. King, 76 Maine 197. 27. The fact that a deed described the property conveyed as commencing at a known monument on the shore of a pond, and running thence " along said pond," does not show an intention to convey only to the shore. A deed of land bordering on a small non-navigable lake or pond is presumed to convey title to the center of the lake or pond, unless the contrary appears. Gouverneur v. National Ice Co. (N. Y. App.) 31 N. E. 865. 28. N. conveyed a lot according to a certain piat. The plat represented the lot as bounded north by a street; south by a stream; on the east and west by lines running from the street to the stream, with fig- ures purporting to give the length of these -lines. In fact, the distance to the stream was greater than in- dicated by these figures. Held, that the conveyance of the lot according to the plat included all the land between the street and the stream. Nicolin v. Schneiderham, Minn. 33, N. W. Rep. 33. 29. In Turner v. Holland, the Supreme Court of Michigan gives riparian rights to owners of lots LANDS BORDERING ON WATERS. 313 bounded by a bayou of Saginaw river, described by plat similar to the above. 33 N. W. Rep. 283. 30. In a navigable stream, as the DesMoines river in Iowa, high water mark is the boundary line. When, by action of the water, the river bed changes, high water mark changes and ownership of adjoining land changes with it. The location of meander lines does not affect the question. Meander lines are not boun- dary lines. 'Steele v. Sanchez, 33 X. W. Rep. 367. Krant v. Crawford, 10 Iowa 549. Lockwood v. R. R. Co., 37 Conn. 387. 31. A boundary stated in a deed as a line forty feet above the border of a river at high water mark, is not ambiguous, and if disputed is to be fixed like any other facts, by testimony and an examination of the ground. Bresler v. Pitts, 59 Mich. 348. 32. A patent for a fractional quarter section, which is bounded by a meandered stream, passes title to all land within the lines of said quarter section between the meandered line and the water's edge. Sphung v. Moore, (Ind.) 22 N. E. 319. 33. The owner of land on the margin of a naviga- ble stream, holding under a grant from the United States, does not take to the middle of the stream, but to high water mark, which is determined by the change in the vegetation and the character of the soil, and the beds of all navigable streams, though the tide does not ebb and flow in them, belong to the state. St. Louis, I. M. & S. Ry. Co. v. Ramsey (Ark.) 13 S. W. 931. 34. The owner of land on a bay conveyed an acre at the end of the tract nearest the bay, described as follows : " Beginning * * * by the beach, run- 314 A MANUAL OF LAND SURVEYING. ning * * * along the beach to," etc. In the general description of the tract it was bounded " east- erly by the said beaph." The grantee was given the privilege of a road from the middle of the front of the lot to the bay, and also half the drift coming on shore in front of the lot, and all the other privileges of the beach were reserved by the grantor, who bound himself not to build any house in front of the lot. The courses and distances would not carry the boun- dary to high-water mark. Held, that the beach did not pass by the deed. Benson v. Townsend, 7 N. Y. S. 162. 35. Part of a quarter section of land conveyed was covered by a lake. The deed described the part con- veyed as 140 acres in the east part of said quarter sec- tion. Held, that the deed was not void for uncer- tainty, since the land could be laid off in a strip of equal width off the east side of the quarter section, though such strip included part of the lake. Mendota Club v. Anderson (Wis. 1899) 78 N. W. 185. 36. Where two deeds in plaintiff's chain of title respectively define the boundary, of the land " by the edge of the mill-pond " and as " the bank of said mill-pond," and defendant is entitled to pond as much land as the pond flowed at the time of his purchase, defendant may enter on land orginally covered by the pond, but which has subsequently become dry land by the receding of the water, though plaintiff's deed on its face shows his line to be the center of the pond. Holden v. Chandler (Vt.) 18 A. 310. 37. Where the patentee of " the north half of the southeast quarter, and that part of the northeast fractional quarter, of Section 36," etc., " which lies north of the Kankakee river, containing in all 122.70 acres," conveys " the northeast quarter of Section 36," LANDS BORDERING ON WATERS. 315 etc., " containing 122.70 acres," the deed passes title to all of the land in said northeast fractional quarter lying south of said river. Sphung v. Moore (Ind.) 22 N. E. 319. 38. Where one who owns a tract of land that sur- rounds and underlies a non-navigable lake, the length of which is distinguishably greater than its breadth, conveys a parcel thereof that borders on the lake, by a description which makes the lake one of its boun- daries, the presumption is that the parties do not in- tend that the grantor should retain the title to the land between the edge of the water and the center of the lake, and the title of the purchaser, therefore, will extend to the center thereof. Lembeck v". Nye (Ohio) 24 N. E. 686. 39. A patent from the United States of a surveyed fractional government subdivision, bounded on a me- andered lake, conveys the land to the lake, although the meander line of the survey be found to be not coincident with the shore line. Everson v. City of Waseca (Minn.) 46 N. W. 405. 40. When the United States has disposed of the lands bordering on a meandered lake, by patent, with- out reservation or restriction, it has nothing left to convey, and any patent thereafter issued for land form- ing the bed, or former bed of the lake, is void and in- operative. Lamphrey v. Metcalf (Minn.) 53 N. W. 1139. 41. Where the United States has made grants with- out reservation or restriction of public lands bounded on streams or other waters, the question whether the lands forming the beds of the waters belong to the state, or to the owners of the riparian lands, is to be determined entirely by the law of the state in which the lands lie. Lamprey v. Metcalf (Minn.) 53 N. W. 1139. 316 A MANUAL OF LAND SURVEYING. 42. Where a section is divided by a water course, and is subdivided in lots instead of regular subdivi- sions, and a lot bounds on the water course, the water course itself, and not the meander lines thereof, is the proper boundary; and, if the grantee does not find the water course as called for by his patent, he may go as far as the next " eighth line " to locate his boundary. Lally v. Rossman( Wis.) 15 N. W. 1132. 43. Where the description is by metes and bounds, no reference being made therein to the lake, then only the land included within the lines as fixed by the terms used by the parties to the deed will pass to the grantee, Lembeck v. Nye (Ohio) 24 N. E. 686. 44. If, however, the call in the description be to and thence along the margin of the lake, no such pre- sumption arises, and the title of the purchaser will extend to low water mark only. Lembeck v. Nye (Ohio) 24 N. E. 686. 45. Where a deed conveys land " bounded and des- cribed according to " a certain survey, does not call for a river, but calls for a line run between certain points, designated by the surveyor as on the bank of & navigable river, and it appears that the lines of such survey exclude flats between high and low water marks, evidence aliande is admissible that the bank referred to was an artificial dike ; that the grantee had notice that the grantors reserved the flats; that the grantors refused to execute a deed expressly conveying the flats ; and that the sale was expressly subject to the survey, as tending to show that the flats were excluded, whatever may be the presumption from the deed. Palmer v. Farrell (Pa.) 18 A. 761. GENERAL RULES. 317 Second. 4. In locating the corners and boundary lines on the ground, we will consider: 1. General rules which apply to all resurveys : 2. Special application of these rules to the rect- angular system of United Stat.es surveys. GENERAL RULES. KULE 1. In locating a deed on the ground, we are to rely (1) On the actual lines originally surveyed; (2) On lines run from acknowledged calls and corners. (3) On lines run according to the course and dis- tance in the deed. Avery v. Baum, Wright's Ohio, 576. 1 Rich. (S. C.) 491. 2. When the boundaries of lands are fixed, known and unquestionable monuments, though neither courses, distances, nor computed contents correspond, the monuments must govern. Pernam v. Wead, 6 Mass. 131. Nelson v. Hall, 1 McLean (U. S.) 518. 3. Though known and fixed monuments control where they conflict with the courses and distances, yet where there are two conflicting monuments, only one of which corresponds with the courses and dis- tances, that one should be taken, and the other re- jected as surplusage. Zeibold v. Foster (Mo. Sup.) 24 S. W. 155. 4. While natural objects usually control courses and distances in boundaries to land, the rule will not be applied where the natural object is shown to be variable in its position. Smith v. Hutchinson (Tenn. 1900) 58 S. W. 226. 318 A MANUAL OF LAND SURVEYING. 5. A boundary line described by measurement and without monuments will govern, although the dis- tance be described as so many feet, more or less. Adkins v. Quest (Mo. App. 1899) 79 Mo. App. 36, 2 Mo. App. Rep. 348. 6. Marked lines and corners control courses and distances. Surplus lands do not vitiate a survey nor does a deficiency of acres called for in a survey operate against it. Wherever the boundaries can be estab- lished, they must prevail. Robinson v. Moore, 4 McLean (U. S. C. C.) 279. Morrow v. Whitney, 5 Otto (U. S.) 551. 7. A deed called for posts as corners. The survey was made and the posts set prior to the execution oi' the deed. It was afterward found that there was a shortage of several acres. Held that proof that posts were set up as corners between adjoining owners con- trols the call for course and distance. Alseire v. Hulse, 5 Ohio, 534. 8. The rule that courses, distances and quantities must yield to monuments, is not inflexible, especially when the distances are very short, and the monuments artificial ones, as here, a mill-race, etc. Higinbotham v. Stoddard, 72 N. Y. 94. Ga. R. R. Co. v. Hamilton, 59 Ga. 171. 9. In a case where no mistake could be reasonably supposed in the courses and distances, the reasons of the rule were held to fail, and the rule was not ap- plied. Davis v. Rainsford, 17 Mass. 207. 10. The rule that natural or artificial boundaries will control distances or courses, authorizes no other departure from the course or distance than such as is necessary to effectuate the apparent intent of the grantor. GENERAL RULES. 319 Distances may be increased and courses departed from in order to preserve the boundary, but the rule authorizes no other departure from the course and dis- tance than such as is necessary to preserve the bound- ary. Johnson v. McMillan, 1 Strobh. (S. C.) 143. 11. If the courses and distances cannot be other- wise reconciled with the monuments in a description, a line in a survey which has evidently been omitted will be supplied to prevent the obvious intent of the grantor from being frustrated. Serrano v. Rawson. 47 Cal. 52. See also Schultz v. Young, 3 Iredell, N. C., 385, where two lines must be run instead of the one called for, to best conform with the whole de- scription in the deed. 12. A survey must be closed in some way or other. If this can only be done by following the course the proper distance, then it would seem that distance should prevail; but when the distance falls short of closing, and the course will do it, the reason for ob- serving distance fails. Doe v. King, 3 How. Miss. 125. 13. Where land conveyed forms a triangle, and two sides and the acreage are given, a straight line from point to point will be adopted as the third side, when the boundary thus formed will enclose the number of acres called for. Hostetter v. Los Angeles Terminal Ry. Co. (Cal.) 41 P. 330. 14. Where three sides and the number of acres are known, and it is disputed whether the fourth side is a straight or meandering line, the straight line will be adopted, when the tract thus enclosed contains the number of acres called for, and when the acreage 320 A MANUAL OF LAND SURVEYING. \vould be largely increased if the meandering line were adopted. Hostetter v. Los Angeles Terminal Ry. Co. (Cal.) 41 P. 330. 15. It is a universal rule that course and distance yield to natural and ascertained objects. But where these objects are wanting, and the course and distance can not be 'reconciled, there is no universal rule that obliges us to prefer the one to the other. Cases may exist in which either one may be preferred, according to the circumstances. Preston's Heirs v. Bowman, 6 Wall. (U. S.) 580. 16. If no principle of location be violated by clos- ing from either of two points, that may be closed from which will be more against the grantor and include the greater quantity of land. Johnson v. McMillan, 1 Strobh. (S. C.) 143. 17. The boundary line is to be ascertained by run- ning direct lines from one monument to the other. Melcher v. Merryman, 4 Me. 601. 18. A line actually marked must be adhered to, though not a right line from corner to corner. Where a line has been marked only part of the way, the re- mainder of the line must run direct to the corner called for. Cowan v. Fauntleroy, 2 Bibb (Ky.) 261. 19. A marked line of another tract, when called for in a conveyance, must be run disregarding dis- tance; but where such line can not be established, the distance run must govern. Cause v. Perkins, 2 Jones Law Rep. (N. Y.) 222. 20. Where a line is described as running a certain distance to a particular monument, and that monu- ment has disappeared and its place cannot be ascer- tained, the course and distance, in the absence of other controlling words, must govern. Budd v. Brooke, 3 Gill (S. C.) 198. See also Bruckner v. Lawrence, I Douglass (Mich.) 19. GENERAL RULES. 321 21. Course and distance yield to known, visible and definite objects; but they do not yield unless to calls more material and equally certain. Shipp et. al. v. Miller's Heirs, 2 Wheat. (U. S.) 316. 22. Courses and distances in the deed are not to be controlled by monuments or objects variant therefrom and not called for in the description, but they must yield to such objects and monuments as are refer- red to. Bruckner's Lessee v. Lawrence, 1 Doug., Mich. 29. Moore v. People, 2 Doug., Mich. 424. Bower v. Earle, 18 Mich. 165. 23. Wherever it can be proved that the line was actually run, was marked, and the corners made, the party claiming under the deed will hold accordingly, although there is a mistake in the description in the deed. Cherry v. Slade, 3 Murph, (N. C.) 82. 24. A sold to B lot 7, informing B, at the time of the sale, that it was four rods wide, and marking it out upon the ground. He subsequently sold to C lot 8 and a vacated alley one rod in width between lots 7 and 8, informing C, at the time, that lot 8 was four rods wide, and the alley one rod wide, making five rods in all, and pointing out to C the marks previously made by him for the boundary of lot 7, sold to B, as being also the boundary of the alley sold to C. The premises were occupied by B and C in accordance therewith, without dispute. It was subsequently found, by reference to the plat, that lot 7 was five rods wide, and that there was no alley between the lots; whereupon B claimed the additional rod. Held, that to allow B to hold the rod in width of land which she did not purchase or pay for, and to deprive C of 21 322 A MANUAL OF LAND SURVEYING. land x which he did purchase and pay for, would be both bad law and bad morals. Bolton v. Eggleston, Iowa. N. W. Rep., Vol. 16, P. 62. 25. Boundary may be proved by any evidence which is admissible to establish any_ other fact. Smith v. Prewitt, 2 A. K. Marsh. (Ky.) 158. 26. Where no bounds were established, the dividing line must be run by aid of the measurements in the deeds, the oldest title receiving its full measure first. Talbott v. Copeland, 38 Me. 333. 27. A long established fence is better evidence of actual boundaries, settled by practical location, than any survey made after the monuments of the original survey have disappeared. A resurvey made after the monuments of the original survey have disappeared, is for the purpose of determining where they were, and not where they ought to have been. Diehl v. Zauger, 39 Mich. 601. Hunt's Lessee v. McHenry and Williams, Wright's (Ohio) 599. 28. Where between the plan and the original survey there is a difference in the location of the lines and monuments, the lines and monuments originally marked as such are to govern, however much they may differ from those represented on the plan. Ripley v. Barry, 5 Greenl. (Me.) 24. See also 2 Greenl. (Me.) 214, and 3 Gr. (Me.) 126. 29. But no such rule has obtained where the survey was subsequent to the plan. Thomas v. Patten, 1 Shep. (Me.) 329. 30. Purchasers of town lots have a right to locate them according to the stakes which they find planted and recognized, and no subsequent survey can be allowed to unsettle them. The question afterwards is not where they should have been,, in order to make GENERAL RULES. 323 them correspond with the lot lines as they should be if the platting were done with mathematical accuracy, but it is whether they were planted by authority, and the lots were purchased and taken possession of in reliance on them. If such was the case, they must govern, notwithstanding any errors in locating them. Flynn v. Glenny, 51 Mich. 580. 31. In ascertaining the true line of a city street, fences built by adjoining lot owners on the line of the street, according to stakes set by the surveyor soon after the original survey was made, and maintained for 45 years, are better evidence of the location of such line than a new survey, made 40 years after the original survey, which changes such line. City of Racine v. Emerson (Wis.) 55 N. W. 177. 32. Of two overlapping surveys, the one first made has priority, particularly where the second is bounded with express reference to the first. Van Amburgh v. Hitt (Mo. Sup.) 22 S. W. 636. 33. Any calls of the second survey conflicting with monuments and calls of the first must yield thereto. Van Amburgh v. Hitt (Mo. Sup.) 22 S. W. 636. 34. Where two surveys call for each other, there can be no vacancy unless the lines marked on the ground contradict the call; and in such case the marked lines must govern. McGinnis v. Porter, 20 Penn. 80. 35. Where two surveys made twenty-three years apart are found to disagree, the probabilities favor the earlier survey when the original corners and wit- nesses are gone at the time of the last survey, espe- cially if the line of the first survey has remained un- questioned for many years. Case v. Trapp, 49 Mich. 61. 324 A MANUAL OF LAND SURVEYING. 36. When the same grantor conveyed to two per- sons, to each one a lot of land, limiting each to a certain number of rods from opposite known bounds running in a direction to 'meet if extended far enough, and by measure the lots do not join when it appears from the same deeds that it was the intention that they should join, a rule should be applied which will divide the surplus between the grantees in proportion to the length of the respective lines as stated in their deeds. Lincoln v. Edgecomb, 28 Maine, 275. 37. Where original surveys have been made, and returned as a block into the land office, the location of each tract therein may be proved by proving the loca- tion of the block. In ascertaining the location of a tract, the inquiry is not where it should or might have been located, but where it actually was located. 38. Every mark on the ground tending to show the location of any tract in the block, is some evidence of the location of the whole block, and therefore of each tract therein. Coal Co. v. Clement, 95 Pa. St. 126.. 39. The beginning corner of a survey, as given in the field notes, is of no more dignity than any other corner found on the ground. Cox v. Finks (Tex. Civ. App.) 41 S. W. 95. 40. Where lots are conveyed by number according to a plat which is made from an actual survey, the corners and lines fixed by that survey are to be re- spected. Pyke v. Dyke, 2 Greenl. Me. 214. 41. Streets which are well defined, and designated by some natural or artificial monument, must govern course and distance in fixing boundaries of lands; but streets which are not thus defined, and themselves re- GENERAL RULES. 325 quire to be located, would furnish very uncertain guides in arriving at the boundaries of other lands. Saltenstall v. Riley, 28 Ala. 164. 42. When streets have been opened and long ac- quiesced in, in supposed conformity to the plat, they should be accepted as fixed monuments in locating lots or blocks contiguous thereto or fronting thereon. Van den Brooks v. Correon, 48 Mich. 283. 43. Lands have been laid off into lots and blocks, and platted, before being cleared, when, by reason of inequalities of the surface, logs, and other obstruc- tions, strictly accurate surveys were not and could not be made. Where the blocks and streets were staked out at the time, such monuments would be fixed and permanent, leaving the excess or shortage to be dealt with by itself. So where the streets, although not so designated, have by the parties interested or by the public authorities been opened, used, and acqui- esced in, they thereby become permanent boundaries and form new starting points in subsequent surveys of the premises. Twogood v. Hoyt, 42 Mich, 609. 44. -Ancient reputation and possession in regard to streets in a town are entitled to more respect in de- ciding on the boundaries of lots than any experi- mental survey that may be afterwards made. Ralston v. Miller, 3 Rand. (Va.) 44. 45. Where lots are sold by numbers and a plat, any variance in the distance between known and fixed points as found by actual measure on the ground, and the distance between the same points as laid down on the plat, is to be divided between the lots in pro- portion to the respective lengths as laid down on the plat. Marsh v. Stephenson, 7 Ohio, N. 3, 264. Quinnin v. Reimers, 46 Mich, 605. 326 A MANUAL OF LAND PURVEYING. 46. Surplus or shortage 'in a block is to be divided pro rat a between the lots. Newcomb v. Lewis, 31 Iowa 488. O'Brien v. McGraw, 27 Wis. 446. 47. Where the accuracy of the starting points taken for test surveys is merely matter of speculation, they cannot be used to fix a disputed boundary between two lots when the dispute arises from a discrepancy which affects all the lots in a block, and must there- fore be apportioned among them. Reimers v. Quinnin, 49 Mich. 449. 48. A resurvey is inadmissible in evidence to show that a private boundary is incorrect, if its starting point is outside of and does not belong to the immedi- ate plan or local system by which the original survey was controlled. Burns v. Martin, 45 Mich. 22. 49. If in running the lines of the grant, one line be found which is admitted or proved to be a line of the grant, which will run with a variation from the calls of the grant, if no other marked lines be found, the other calls should be run with the same variation as that found on the marked line. Sevier v. Wilson, Peck. 146. 50. Where a deed convey 3 lots in a town, and refers to a plat to identify them, and, in describing their lines, calls the points of compass as designated on the plat by its lines and angles, a correct survey cannot be based on any other system; and although the lines there delineated are not comformable to the true meridian, the plat and not the compass should govern. Bower v. Earl, 18 Mich. 367. 51. An instruction that, in arriving at a boundary line as originally run, natural objects are controlling calls; artificial objects, second in importance; course, GENERAL RULES. 327 third, and distance, fourth; and that, where there is still uncertainty, that rule should be adopted most consistent with the intent of the grant, is correct. Luckett v. Scruggs (Tex.) 11 S. W. 529. 52. An instruction that the beginning corner of a survey is of no higher dignity or importance than any other corner, and that, " if there are well-known and undisputed original corners established upon the ground around the survey, they would control the other calls of the survey, which are conflicting and contradictory, if there are any such," is correct. Luckett v. Scruggs (Tex.) 11 S. W. 529. 53. Where the beginning corner of a survey is the southwest, but the southeast corner is equally well identified, a charge limiting the jury to finding the unidentified northeast corner by the first and second lines from the southwest corner, is erroneous, as the southeast corner is of equal importance, unless the line from the former corner was actually run and measured, and that from the latter not. Scott v. Pettigrew (Tex.) 12 S. W. 161. Lancaster v. Ayres, Id. 163. 54 An instruction making the importance of an established northeast corner, in locating the north and west lines of a survey, dependent upon the jury's belief that such western line was not run, is erroneous, as such corner has the same weight for the purpose in question, whether the western line was run or not. Scott v. Pettigrew (Tex.) 12 S. W. 161. 55. In the description of lands, as to questions of boundaries the rule is settled in Virginia and West Virginia that natural land-marks, marked lines and reputed boundaries will control mere courses and dis- tances, or mistaken descriptions in surveys and con- veyances. Gwynn v. Schwartz (W. Va.) 9 S. E. 880. 328 A MANUAL OF LAND SURVEYING. 56. The course of the eastern line of the H. tract, as given in the original survey made in 1745, was 14 deg. east. The course of the western line of the B. tract, lying immediately east of the H. tract, as given in the original survey made in 1813, was 17 deg. and 15 min. east. The western line of the B. tract was made of exactly the same length as the eastern line of the H. tract, and the beginning point of the two lines was the same. The difference in the course of the two lines could be satisfactorily explained by the change in the position of the magnetic needle which had taken place in the time intervening between 1745 and 1813. Held, that the two lines must be considered as coincident. Scott v. Yard (K. J.) 18 A. 359. 57. Where neither the corners of plaintiffs' nor defendants' land are satisfactorily established, and there is a well-established and identified corner of another survey, from which, by following course and distance, defendants' survey can be constructed, such course should be followed though the boundaries thus established include land within the boundaries of plaintiffs' junior survey. Griffith v. Rife (Tex.) 12 S. W. 168. 58. A county surveyor, employed to restore the lines and corners of adjoining tracts of land accord- ing to the original government survey, found township corners only, then (the other quarter and section cor- ners being missing) ran a straight line from one town- ship corner to the other, and on this line placed the quarter and section corners, but did not take any testi- mony to ascertain the lines or corners of the original survey, did not attempt to prove his lines or corners by re-establishing the missing corners from all the nearest known original corners, in all directions, did GENERAL RULES. 32S not sufficiently regard the field notes, and did not, where the original monuments had disappeared, regard the boundary lines long recognized and acquiesced in. Held, that such a survey is incomplete, and cannot be approved as the true and correct determination of the boundaries and corners as originally established by the government. ; ^ , Reinert v. Brunt (Kan.) 21 P. 807. 59. Upon an issue as to the location of a line of the government survey, evidence of the location of monu- ments is not overcome by field-notes of the original survey, taken at the time of the erection of said monu- ments or subsequent thereto. Hubbard v. Dusy (Cal.) 22 P. 214. 60. As between complicated descriptions of a line dividing two sections or quarter sections, that one is to be adopted which is most in conformity with the monument established by the government survey. Hubbard v. Dusy (Cal.) 22 P. 214. 61. As between different monuments, those best identified should prevail, independent of anything in the field-notes of the original or any subsequent sur- vey. Hubbard v. Dusy (Cal.) 22 P. 214. 62. Where it is x doubtful which of two lines of monuments is the true government line, other things being equal, that one is to be so considered which most nearly conforms to the field-notes. Hubbard v. Dusy (Cal.) 22 P. 214. 63. Where, in ejectment, the location of the bound- ary line between two lots is in question, and the lots were staked when platted, such monuments are con- clusive of the question; but if they were not staked, other monuments, establishing any given points as platted, furnish starting points to aid in arriving at 330 A MANUAL OF LAND SURVEYING, the true boundary, and, in the absence of either, old monuments indicating user may .be resorted to. Brudin v. Inglis (Mich. 1899) 80 N. W. 115. 64. Where there is a discrepancy, in a government survey, between the monuments and the distances given in the field notes, the monuments will control, event though the result be that some of the quarter sections will contain less than their proper number of acres. Ogilvie v. Copeland (111. Sup.) 33 N. E. 1085. 05 In the rule that monuments control courses and distances, and that when monuments and measure- ments vary, the monuments always control, the refer- ence is to monuments and measurements made by the original survey. Woodbury v. Venia (Mich. 1897) 72 N. W. 189. 66. On a question as to the true location of a land patent, boundaries fixed by reversing the courses and distances must govern when found to coincide with the natural calls of the patent. Ellinwood v. Stancliff, 42 F. 316. 67. When the points fixed by reversing the courses and distances do not coincide with the natural calls of the patent, or the natural calls cannot be identified, then the regular courses and distances must govern. Ellinwood v. Stancliff, 42 F. 316.' 68. When a survey calls for the " Dougherty " survey as one of its adjoiners, an instruction that if the jury find that the " King " is the survey intended by the call for " Dougherty," the former being located, the call would furnish " some evidence " of the loca- tion of the survey in question, is insufficient, as such a finding would locate the survey in the absence of marks upon the ground. Tyrone Min. & Manuf'g Co. v. Cross (Pa.) 18 A. 519. GENERAL RULES. 331 69. Where no marks are found on the boundaries of a survey, and it cannot be located on the ground, evidence of the location of junior surveys which call for the lines of the elder as adjoiners is admissible, as showing where the surveyors upon the ground located such lines. Tyrone Min. & Manuf'g Co. v. Cross (Pa.) 18 A. 519. 70. Where the distances of a survey have been actually measured upon the ground, the courses and distances may be reversed when by so doing they more nearly harmonize with the natural calls of the patent, and the " beginning " corner does not control more than any other corner which is definitely ascertained. Ayers v. Watson, 11 S. Ct. 201. 71. Where the court, in an action of ejectment, in- structs the jury that, " after a survey of blocks had been returned and had remained in the land-office 21 years, it was conclusively presumed that it was run upon the ground, whether marks were found upon the ground or not," but in other portions of his charge repeatedly states the law to be that marks made by the surveyor on the ground are the first and highest evidence of the true survey, the instruction cannot, on the whole, be said to be misleading, as he will be rea- sonably understood to have charged that the presump- tion in favor of returns of surveys on file for 21 years is only applicable to such surveys where no monu- ments or marks on the ground are found to contradict them. Grier v. Pennsylvania Coal Co. (Pa.) 18 A. 480. 72. The exterior of two adjoining interior surveys were undisputed. The boundary line between them had never been surveyed, but its southern end was marked by an oak. North of these surveys were two others. These four surveys were originally returned 332 A MANUAL OF LAND SURVEYING. as being of equal size, and having one common corner. The northern end of the line between these two latter surveys was marked by a sugar-maple; which was not directly opposite the oak, and it was proved that the northern line of these surveys was shorter than the southern line of the others. Held, that the boundary line between the two southern surveys should run from the oak parallel to the end lines, and not diagonally from the oak to the maple. Bloom v. Ferguson (Pa.) 18 A. 488. 73. Where a dividing line is established between tracts of land owned by a county, before purchases are made of land on each side of it, and the deeds under which parties claim have been made, and are known by the parties to have been made with refer- ence to that line, they, and all the persons claiming through them, are bound by it. Briscoe v. Puckett (Tex.) 12 S. W. 978. 74. The northwest corner of a survey was plainly marked, and part of the west line was also marked. The rest of the survey had apparently not been run on the ground, but the southeast corner was ascertainable from the field-notes, being located on an established line of another survey and at a given distance from an established point. The lines of survey as called for in the field-notes were correct as to courses but were too short to reach from one of said corners to the other. Held, that the survey included all the land be- tween the corners bound by the lines as extended so as to reach from one corner to the other. Randall v. Gill (Tex.) 14 S. W. 134. 75. Where a deed . describes a lot conveyed as of a certain width, and a party-wall stands on the south line, the north line may be found by measuring the given distance north from the middle of such wall. Warfel v. Knott (Pa.) 18 A. 390. GENERAL RULES. 333 76. The statement of the quantity of land supposed to be conveyed, and inserted in deeds by way of de- scription, must not only yield to natural land-marks and marked lines, but also to descriptions in deeds by courses and distances. Gwynn v. Schwartz (W. Va.) 9 S. E. 880. 77. A call for a lot by the name or number which it bears on a plat of the land will prevail over courses and distances, and ordinarily over calls for monu- ments. O'Herrin v. Brooks (Miss.) 6 So. 844. 78. Where the descriptions in a deed refer to a sur- vey and a map based thereon, making both a part of the deed, and there is a discrepancy between the map and the survey, the latter will prevail. Whiting v. Gardner (Cal.) 32 P. 71. 79. The owner of a lot in the city of Rochester, of the area of about one-half acre, rectangular in form, fronting 274 feet on a street, and abutting on the rear for the same distance on a canal, the location of both, as well as the other lines, being undisputed, conveyed a portion, by description, of " 137 feet front and rear, measuring from G. H.'s north line on G. street, and also 137 feet from G. H.'s south line on the canal; being the piece of land occupied as a garden by the grantor." The lot was divided by a 'fence, one side being used as a garden ; the fence starting on G. street midway, but striking the back line at the canal at a point 19^ feet from the middle of the lot. That fence was not mentioned in the deed. Held, that the reference to the garden was too indefinite to control the calls for exact distances from known bounds, and the divisional point on the canal should be located 137 feet from G. H 's line. Harris v. Oakley, 7 N. Y. S. 232. *i . 334 A MANUAL OF LAND SURVEYING. 80. Plaintiff owned a village lot, No. 124, and a tract of land lying adjacent thereto on the south and east sides. Eiver street, which lay along a river's edge, was the westerly front of both the lot and the tract. He conveyed the tract to defendant, reserving a part thereof, beginning at the S. W. corner of the lot; thence southeasterly, along River street, 32 feet; thence northeasterly, " on a line with the southeast corner of lot No. 124," 10 rods and 23 links; thence N. to M. street; thence W. to the N. E. corner of the lot; thence southwesterly, to the S. E. corner; thence to the beginning. Locating the beginning point at the S. W. corner of the lot as appeared by the village plat on the easterly side of the street, the line passed directly through the S. E. corner of lot 124, taking no part of the lot, and thus making the reservation wholly- within the tract conveyed; but by beginning at the river's edge, on the westerly side of the street, on the theory that plaintiff's property extended to the river, subject only to the easement of the street, the line would pass through and take part of lot 124. Held, that the former location of the corner w$s cor- rect. Anderson v. Scott (Mich.) 42 N. W. 991. 81. In an action to recover a tract of land lying between a slough and a river, plaintiff claimed title by virtue of a grant which bounded the land granted by the river, and the defendant introduced evidence that the surveyor who surveyed the grant meandered the slough instead of the river. Held, that, in determin- ing the true boundaries of the grant, the sole ques- tion was to ascertain exactly where the surveyor ran his lines, and, if the, jury found that he ran the line along the slough, they should find for the defendant. Allen v. Koepsel (Tex.) 14 S. W. 151. GENERAL RULES. 335 82. Where, in ejectment, a surveyor testified that he ran the boundary line in dispute about 1S68; that he found the original stake of the government survey at the section corner, and used it as a starting point; and it appeared that about the same time defendant built a fence upon this line, which he has ever since maintained this line must prevail over one surveyed 20 years later, when the corner mark was gone, by one who testified that he located the section corner by measurements from various lines and points, and then by digging found a stump which he took to be the original witness, and based his survey upon it. Carpenter v. Monks (Mich.) 45 N. W. 477. 83. The monuments or marks of the surveyor on the ground determine the true survey as against calls for adjoinders or courses and distances as returned; but, each block of surveys being separate and com- plete of itself, the call of a tract in one block for an adjoinder in another does not make the monument of the adjoinder the monument of the later block. Grier v. Pennsylvania Coal Co. (Pa.) 18 A. 480. 84. Where a boundary line is assented to by the owner of a tract of land at a time when there is ni dispute concerning such line, and on the supposition that it is the true boundary, he is not estopped, on discovering that such is not the case, from claiming title to the real boundary. Schraeder Min. & Manuf'g Co. v. Packer, 9 S. Ct. 385. 85. Continuous and uninterrupted possession, under claim of ownership, to the line of a division fence, will not bar title, where it appears that such occupation was under a belief that flie fence was on a true line, and without intention of claiming beyond the true line, as described in the deeds. Skinker v. Haagsma (Mo.) 12 S. W. 659. 336 A MANUAL OF LAND SURVEYING. 86. Lands are not surveyed lands by the United States until a certified copy of the official plat of survey has been filed in the local land office. United States v. Curtner, 38 F. I. 87. One who receives deeds of lots, and conveys to others, according to an unacknowledged plat of a town, is thereby estopped from denying the sufficiency of the dedication for want of the acknowledgment. Giffen v. City of Olathe (Kan.) 24 P. 470. 88. Testimony of declarations of a grantor, before the execution of a deed, tending to establish a bound- ary other than that made by the deed as construed by the court on appeal, is inadmissible, as its effect would be to convey land by parole in contravention of the statute of frauds. Harris v. Oakley, 7 N. Y. S. 232. 89. Where a town site was surveyed and laid out in lots, blocks, streets and alleys, and a plat thereof made and lithographed, and distributed among thy occupants of the town site, and one of the lithographed copies was afterwards recorded in- the office of the register of deeds, but the same was not acknowledged, and the town site was pre-empted by the president of the town site company, and a patent was obtained by him for the benefit of the occupants, under the town^ site act (5 U. S. St. 657), there was a sufficient dedica- tion of the streets and alleys of said town, despite the want of acknowledgment of the recorded plat. Giffen v. City of Olathe (Kan.) 24 P. 470. 90. A deed conveying land in a town, but " reserv- ing streets and alleys according to recorded plat of the town," passes the fee in such streets when such fee was at the time held by the grantor subject to the ease- ment of the public therein. Gould v. Howe (111.) 23 N. E. 602. ALLUVIUM. 337 91. Where surveys of 1837 and 1856 do not agree the former holds. Palmer v. Montgomery-, 26 X. Y. Rep. 536. 92. The boundary lines of water lots fronting on a river extend into the river at right angles with the thread of the stream, without reference to the shape of the shore. Clark v. Campau, 19 Mich. 328. Bay City Gas Light Co. v. Ind. Works, 28 Mich. 182. Twogood v. Hoyt, 42 Mich. 609. Noms v. Hill, 1 Mich. 202. 93. Where a certain distance is called for from a given point on a navigable stream to another point on the stream to be ascertained by measurement, such measurement must be made by its meanders, and not in a straight line. The same rule prevails when dis- tance is called for along a traveled highway. A dif- ferent rule is sometimes adopted when the stream is not navigable. When a tract of land is bounded upon a navigable stream, the distance upon the stream will be ascertained, in the absence of other controlling facts, by measuring in a straight line from the oppo- site boundaries. People v. Henderson, ,40 C'al. 29. 94. In computing the number of acres in a survey, "from," "to," and "with" the bank of a stream mean to low-water mark. Lamb v. Ricketts, 11 Ohio 311. 1. Alluvium means an addition to riparian land gradually and imperceptibly made through causes either natural or artificial by the water to which the land is contiguous. It matters not whether the addi- tion be on streams which overflow their banks, or on those which do not. In each case it is alluvium. County of St. Clair v. Livingston, 23 Wall. (U. S.) 46. 338 A MANUAL OF LAND SURVEYING. 2. Land formed by alluvium in a river is in gen- eral to be divided among the several riparian owners entitled to it, according to the following rule: Meas- ure the whole extent of their ancient line on the river, and ascertain how many feet each proprietor owned on this line. Divide the newly formed river line into an equal number of parts, and appropriate to each owner as many of these parts as he owned feet on the old line; and then draw lines from the points at which the proprietors respectively bounded on the old, to the points thus determined as points of division on the newly formed shore. This rule is to be modified under particular circumstances; for instance, if the ancient margin has deep indentations or sharp pro- jections, the general available line of the river ought to be taken, and not the actual length of the margin as thus changed by the indentations or projections. Deerfield v. Arms, 17 Pick. Mass. 41. Jones, et. a!, v. Johnston, 18 How. (U. S.) 100. 3. Alluvium deposited against an island in a lake and a neighboring lot, so as to connect them, must be equally divided between the owners of both. Bigelow v. Hoover (Iowa) 52 N. W. 124.. 4. Flats situate in a tidal river at a point in its course above the line of low tide, are to be divided among the adjoining properties, by drawing lines from the terminal of the latter on the banks at the ordinary stage of water to and at right angles with the centre line of the river. Tappan v. Boston Water Power Co. (Mass.) 31 N. E. 703; Browne v. Same Id. 5. Under Kev. Stat. U. S. 2396. Held, that in surveying a lot bordering on a river the water-course becomes the boundary, and continues so, no matter ALLUVIUM. 339 how much it shifts by accretion, and conveyances of the lot pass all, including such accretion to that line. East Omaha Land Co. v. Jeffries, 40 F. 386. 6. The facts that rapid changes in the banks' of the Missouri River are constantly going on, and that 40 acres have been added to adjoining land, do not overthrow an averment of a bill to quiet title to such addition, on the ground of accretion, that it was by an imperceptible increase, where it was nearly 20 years in forming. East Omaha Land Co. v. Jeffries, 40 F. 386. 7. The rule that owners of land bounded by streams are entitled to additions to their land formed by accretion is applicable to the Missouri river, notwith- standing the peculiar character of that stream, and of the soil through which it flows, whereby changes in its banks are great and rapid. Jeffries v. East Omaha Land Co., 10 S. Ct. 518. 8. Where the official plat of the survey of govern- ment lands shows a river as one boundary of a certain lot, in accordance with Rev. St. TT. S. 2395, .et seq., a subsequent patent for the lot, describing it by num- ber, and referring to the plat, on which it is marked as containing a certain amount, and deeds, describing the lot by number, pass all accretion 10 the lot up to their respective dates. Jeffries v. East Omaha Land Co., 10 S. Ct. 518. 5. Rules Applicable to the United States Surveys. " All the corners marked in the surveys returned by the surveyor-general shall ~be established as the proper corners of the sections or subdivisions of sections which they were intended to designate." " The boundary lines actually run and marked in the surveys returned by the surveyor-general shall be established as the proper boundary lines of the sections 340 A MANUAL OF LAND SURVEYING. or subdivisions for which they were intended ; and the length of such lines as returned shall 'be held and considered as the true length thereof." The preceding quotation from section 2396 of the Revised Statutes of the United States, settles all questions in regard to any change in the corners, lines or measures of the government survey. They are thereby made unchangeable, the statute thus empha- sizing the common law, which holds the same doctrine to be true of all original surveys after the land has been conveyed in accordance with them. Hence, in making resurveys, the surveyor must find, if possible, the original corners, and make his courses and dis- tances agree with those of the United States survey. The following points have been decided by the courts with reference to these surveys: RULE 1. The original surveys by which the govern- ment sold its land and conveyed it to the purchaser establish the rights of the parties as to the bound- aries. No line which will vary the rights thus ac- quired can afterwards be established without the con- sent of all parties. May v. Baskins, 12 S. and M. (Miss.) 428. 2. AH disputes as to the boundaries of land are to be governed by the United States surveys, unless there is some statute to the contrary. Taylor v. Fomby (Ala. 1897) 22 So. 910. 3. Government corners, fixed by a United States surveyor, will control the field notes of the survey taken at the time the corners were erected, and also the field notes of any subsequent survey. In the absence of a government corner, or of satisfactory proof of its location, the field notes of a government survey will govern, and are prima facie evidence of the true location of the true line of the survey. Knoll v. Randolph, 92 N. W. 195 (Neb. 1902). SPECIAL RULES. 341 4. Land sold under the United States surveys pass according to the description of the legal subdivisions, whether those subdivisions contain the legal quantity or not, more or less. Fulton v. Doe, 6 Miss. 751. . - ':-+ 5. Each section or a subdivision of a section is independent of any other section in the township and must be governed by its marked and established boundaries. Should they be obliterated, a last recourse must be had to the best evidence that can be obtained showing their former situation and place. Lewen v. Smith, 7 Port (Ala.) 428. 6. Field notes must yield to actual monuments erected by the original surveyor. They are only to be relied on as evidence to assist in finding the exact situation of the monuments. McClintock v Rogers, 11 111. 279. 7. The rule that monuments control courses and distances applies to discrepancies in government sur- veys between the courses and distances and the witness trees called for in the field notes. England v. Vandermark (111. Sup.) 35 N. E.. 8. Monuments found at the two extremes of a township line are entitled to no more controlling in- fluence in determining the actual location of an inter- mediate line than the section corners established along the line. All original monuments established in con- nection with the field notes and plats must be referred to in order to define the locality of the line. McClintock v. Rogers, 11 111. 279. 9. The corners established by the original sur- veyors of public lands by authority of the United States are conclusive as to the boundaries of sections and divisions thereof; and no error in placing them 342 A MANUAL OF LAND SURVEYING. can be corrected by any survey made by individuals or a state surveyor. Arnier v. Wallace, 28 Miss. 556. 10. In ascertaining the lost corner of a section, recourse must be had to the unobliterated marks of the original survey, the field notes and plats and subse- quent surveys made under their guidance. If only a portion of one of the boundary lines leading to the lost corner on a township line has been obliterated, the remaining portion must be considered established as marked, and the corner must be presumed, in the absence of evidence to the contrary, to be at the point where the marked line if continued would intersect the township line. But if the lost corner is proved to have been at another point, the lost portion of the boundary must be ascertained by running a straight line from the point where the marks disappear to that corner. Billingley v. Bates, 30 Ala. 378. 11. In determining the line between the quarters of a section, the quarter post established by the gov- ernment surveyors must govern in all cases where its Iocati6n can be ascertained. Vroman v. Dewey, 23 Wis. 530. Britton v. Ferry, 14 Mich. 53. 12. In re-establishing a lost quarter post on a sec- tion line, any difference in the length of such line by actual measure as compared with that indicated by the government survey should be divided between the parts in proportion to their respective lengths as shown by that survey. Jones v. Kimble, 19 Wis. 429. 13. Where a government corner is lost or oblit- erated, so that resort must be had to the government field notes for the purpose of determining its location, SPECIAL RULES. 343 but these field notes are inconsistent, and can not be reconciled, there is no universal rule that certain ones shall be preferred to the others, but, as in a case where living witnesses contradict each other, those should be accepted as correct which, under all the circumstances, are most entitled to credit, and most likely to be in accordance with the actual facts. A witness or bearing tree is not an established corner, but merely a designated object from which in connection with the field notes, the location of the corner may be ascertained. Stadin v^ Helin (Minn. 1899) 79 N. W. 587. 14. The unvarying rule to be followed in estab- lishing a lost corner, is' to start at the nearest known point on one side of the lost corner, on the line on which it was originally established; to then measure to the nearest known corner on the other side, on the same line; then, if the length of the line is in excess of that called for by the original survey, to divide it between the tracts connecting such two known points, in proportion to the length of the boundaries of such tracts on such line, as given in such survey. Lewis v. Prien (Wis. 1897) 73 N. W. 654. 15. Where the original survey and field-notes of a township show all the sections full, but, after all the natural monuments in the two northern tiers of sec- tions have been lost, it appears that there is a short- age somewhere within those two tiers, such shortage will be apportioned between the two tiers, and not im- posed wholly on. the northern tier, though the survey was made by beginning t the southeast corner of the township, and working north. James v. Drew (Miss.) 9 So. 293. 16. If the distance between recognized government corners as originally established . overruns or under- runs that given in the field notes, it should be divided 344 A MANUAL OF LAND SURVEYING. pro rata between the intervening sections. The origi- nal field notes should be the main guide. Section lines being frequently deflected, the true corners must be tested by east and west distances from the recog- nized government corners yet standing in the same township as well as by north and south distances. Martz v. Williams, 67 111. 306. 17. Unknown corners must be found by the cor- roborative testimony of all known corners with as little departure as may be from the system adopted on the original survey, without giving preponderance to the testimony of any one monument above ^another. In re-establishing lost corners between remote cor- ners of the same survey, when the whole length of the line is found to vary from the length called for; we are not permitted to presume that the variance arose from the defective survey of any part, but must con- clude in the absence of circumstances showing the contrary that it arose from the imperfect measurement of the whole line, and distribute such variance be- tween the several subdivisions of the whole line in pro- portion to their respective lengths. Moreland v. Page, 2 Clarkes, Iowa, 139. 18. Quarter posts of the government survey are to be as much respected as the corners of townships or sections however distant from the center line. Campbell v. Clark, 8 Mo. 558. 19. There was a mistake in the government survey of a section by which the quarter section line and the meander line of a river were shown on the official plat to be one and the same line, being the boundary line of the fractional lots. As a matter of fact they were a considerable distance apart. There was no question as to the location of the quarter section corners. In a suit to determine the ownership of the land between SPECIAL RULES." 845 the quarter section line and the river, it was held that 'the quarter section line should be adhered to as the more certain call, and that where the lines of a survey can be run from well ascertained and estab- lished monuments, they are to control and govern a description delineated on a plat, although the quantity in the fraction fell short of the amount laid down in the plat about as much as there, was land contained between the quarter line and the river. Martin v. Carlin, 19 Wis. 454. 20. When a deed designates the land conveyed as one of the subdivisions known in the United States survey, as, for instance, a quarter, half -quarter or quarter-quarter section, the presumption is that the parties intend that the tract shall be ascertained in the same manner as is done in the government sur- veys. Not so, where the deed conveys a tract of land not known in that system of surveys, as, for instance, the east half of a lot, or of a quarter-quarter section. Cogan v. Cook, 22 Minn. 142. - 21. The line between the northeast and northwest quarters of a quarter section is to be extended south from a point midway between the northeast and north- west corners, rathe'r than from a point on such line 1,320 feet from one of the corners. Packscher v. Fuller (Wash.) 33 P. 875. 22. The defendant sold the north half of a lot which is bounded on the west side by the Au Gres river. But the river is not straight at this point, and the north line of the lot is longer than the south line. The bill demands the north half of the lot, and the north half must mean the north half in quantity divided from the remainder by an east and west line. Au Gres Boom Co. v. Whitney, 26 Mich. 44. 15. It is a question of fact to be determined by all the surrounding circumstances whether the land between the 346 A MANUAL OF LAND SURVEYING. meander line and the shore of the lake or water course is included in the survey. Shoemaker v. Hatch, 13 Nev. 267. 23. The lines run to divide sections into halves and quarters, if erroneous, may be corrected, for they are subdivided by law; and if the officer in running the sub- division line makes a mistake, it can be corrected by run- ning the line according to law. Nolin v. Palmer, 21 Ala. 66. 24. An original township was divided into sections " by running through the same, each way, parallel lines at the end of every two miles, and making a corner at the end of every mile," arid afterward a supplemental survey was made under a subsequent statute, which directed that these two mile blocks should be subdivided by running straight lines from the corners thus marked to the oppo- site corresponding corners. Held, that where the original mile corners in a certain block can be clearly identified, the courses of lines of subdivision within the block can- not be determined by proof of monuments, blazes, or other witness marks found in other blocks in the town- ship. Ginn v. Brandon, 29 Ohio St. 656. 25. When a navigable stream intervenes in running the lines of a section, the surveyor stops at that point, and does not continue across the river. The fraction thus made is complete, and its contents can be ascertained. Therefore, when there is a discrepancy between the cor- ners of the section as established by the United States, and the lines as run and marked, the latter do not yield to the former. Lewen v. Smith, 7 Port. (Ala.) 428. 26. In government surveys, the line actually run by the government surveyors is the true line. Goodman v. Myrick, 5 Oregon, 65. 27. In a case where the township lines had been run and marked by the United States survey, but the field SPECIAL KULES. 347 notes of the subdivision lines were fraudulent and re- jected by the surveyor-general, because incorrect, no proper survey of them having been made, it was held that the line between sections one and two must be ascer- tained by running a straight line from the corner of the sections established on the exterior line of the township to the corresponding corner on the opposite side of the township. Hamil v. Carr, 21 Ohio St. 258. 28. Where the initial point in the description of prem- ises in a deed is the southeast corner of the north half of the southeast quarter, fractional, of a section, and the quarter-section is made fractional by a meandered lake so situated as to cover the eastern and central portions thereof; and the parcel described was carved out of the north half within a year after the same was patented, the southeast corner in question is construed to be the point which constituted the southeast corner of the land as it was surveyed out and platted by the government, which located it on the meandered line of the lake. The fact that the waters of the lake have since receded can- not change the boundaries as previously located. Verplanck v. Hall, 27 Mich, 79. 29. Extending fractional lots beyond quarter lines: Etheridge and Stone were the original settlers, pre-empt- ors, and purchasers of fractional section 22. Etheridge's patent called for "the S. W. ^ o f Sec. 22 containing 92.67 acres." Stone's patent called for " S. E. subdiv. Qr. Sec. 22, containing 110.50 acres." These two descriptions were in controversy in Brown's lessees v. Clements, 3d How. 650. In the figure (page 34 8) the full lines show the frac- tional section as it was returned on the official plat. The dotted lines show the quarter lines as they would have been if the section had been f ulL 348 A MANUAL OF LAND SURVEYING. FIG. 71 On the part of the grantees of Etheridge two claims were set up. One was that under the pre-emption laws Etheridge was entitled to a full quarter section of land. The other was that, as his deed called for the S. W. % and the fractional section was of such size and shape that a regular southwest quarter could be laid out from it, he was entitled to it, and that the action of the Surveyor General in returning irregular subdivisions of the section, when he could have made one regular quarter section out of it, was contrary to law, and therefore void. The Supreme Court by a bare majority upheld these claims and decided the case on those grounds. The case of Brown's lessees v. Clements was decided in 1845, several of the judges strongly dissenting from the decision. In 1858 the same tract of land came in question again. Gazzam v. Phillips' lessee and others, 20th Howard 372. Speaking of the sales to Stone and Etheridge, the Court says: " The sales in each case were made in conformity with the plat of the survey then on file in his office," etc. " We deny altogether the right of the court in this ac- tion to go beyond these terms thus explicit and specific and under a supposed equity in favor of Etheridge, arising out of the pre-emption laws, to the whole of the southwest quarter enlarge the description in the grant, or more accurately speaking, determine the tract and quantity of the land granted by this supposed equity instead of by the description of the patent. " We are not satisfied that there was any want of power in the surveyor general in making subdivisions of this SPECIAL RULES. 349 section according to the plat and in conformity with which the sales of the lands in dispute were made. "The Act of 1820 provides that fractional sections containing 160 acres and upwards shall in like manner, as nearly as practicable, be subdivided into half quarter sections under such rules and regulations as may be prescribed by the secretary of the treasury. "The secretary of the treasury, on the 10th of June following the passage of the act, issued regulations through the commissioner of the land office, directing fractional sections containing more than 160 acres to be divided by north and south or east and west lines, so as to preserve the most compact and convenient form. This section was divided by a north and south line according to these instructions. The question came before the secretary of the treasury and before us in 1837, and the construction first given and the practice of the surveyor general under it confirmed. Attorney General Butler in a well considered opinion observed: 'If congress had intended that fractional sections should at all events be divided into half quarter sections when their shape per- mitted the formation of such a subdivision, I think they would have said so in explicit terms, and that the discre- tionary power entrusted to the secretary would have been plainly confined to the residuary parts of the section. And further that the clause in the first section of the act of 1820, concerning fractional sections containing less than 160 acres (which are not to be divided at all) is decisive to show that congress * * did not deem it indispensable that regular half quarter sections should in all practicable cases be formed by the surveyors. On the contrary, it shows that they preferred a single tract though containing more than 80 acres to small incon- venient fractions.'" The court adds: "We entirely concur in this construc- tion of the act," and further goes on to say: "The only difficulty we have had in this case arises from the cir- cumstance that a different opinion was expressed oy a 350 A MANUAL OF LAND SURVEYING. majority of this court in the case of Brown's lessees v. Clement, 3 How. 650. " It is possible some rights may be disturbed by refusing to follow the opinion expressed in that case, but we are satisfied that far less inconvenience will result from this dissent than by adhering to a principle which we think unsound and which in its practical operation will unsettle the surveys and subdivisions of fractional sections of the public land running through a period of some 38 years. We cannot adopt that decision or apply its principles in rendering the judgment in this case." 30. Quarter posts on section lines where there are double sets of section corners : " Quarter section corners are not required to be established on the west boundary of the western tier of sections in a township, nor on the- north boundary of the north tier of sections in a township south of and bordering on a standard parallel. The resurvey of township, standard, or base lines, by the deputy sur- veyor for the purpose of establishing such quarter-posts, is unnecessary and will not be paid for." Instructions to surveyors-general by Commissioner Edmunds, p. 9. 31. "Range lines are run north or south from the base line, and corners for sections and quarter sections are established thereon at every mile and half mile for the sections and quarter sections on the west side of the line, but not for those on the east side" On township lines "the corners of sections and quarter sections are estab- lished at every 80 and 40 chains for the sections and quarter sections on the north side of the line, but not for those on the south side" Instructions to Deputy Surveyors of the United States for the district of Illinois and Missouri, 1856, p. 50. 6. Decisions of the General Land Office With reference to Mineral Surveys. Plats and field notes: Of surveys of mining claims, required to disclose all conflicts with prior surveys, giving areas of all conflicts. DESCRIPTIONS IX DEEDS. 351 In future, surveyor-general will use no coloring on plats. Com'r. (N.) Nov. 16, 1882. Circular. Location (of mine) : Must be marked on the ground so that its boundaries can be readily traced. K. Noonday M'g Co. v. Orient M'g Co., G Saw., C. C., 299 ; Myers et al. v. Spooner et a?., 55 Cal. R. 257; Gleason v. N. White M'g Co., 13 Nev. R., 443; Southern Cross G. and S. M'g Co. v. Europa M'g Co., 15 id., 383. Surface line : Agreement by adjoining claimants, fixing surface boundary line between them, must be construed as extending such line downward, through the dips of the vein or lode, to the earth's centre. Richmond M'g Co. v. Eureka M'g Co., 103 S. C., 389. Bearings and distances must be given in a survey, from the respective survey corners to the location corners, and the same must be shown on the plat. Survey: Of a mining claim should show location of all improvements of a municipal nature, as blocks, alleys, etc. Sec'y Dec. 18, 1880, and Feb. 3, 1881. Little Nettie Lode. 7. Descriptions in Deeds. Surveyors are fre- quently required to make surveys for the purpose of fur- nishing a description of the land to be conveyed. Every surveyor of experience is familiar with the many diffi- culties encountered in correctly locating boundary lines, caused by defective, false or impossible descriptions in the deeds. The description is the controlling guide to the surveyor in locating a man's possessions on the ground, hence it is important that it should be clear, distinct and harmonious in its terms. Where land is conveyed in the regular subdivisions of the United States survey, little difficulty will be met in writing a correct description. The main caution to be observed is to avoid the common clerical error of using the wrong letter or word, such as north instead of south, or east instead of west, thereby locating the deed in a different place from which it was intended. Scrutinize 352 A MANUAL OF LAND SURVEYING. the description closely to see that no such error is made, and write plainly, so that no one need make a mistake in reading or copying the description. A great many of these mistakes are caused by bad penmanship. Similar remarks apply to the description of land by plat, where only clerical errors are likely to be made. It is in the description " by metes and bounds " and by courses and distances, that greatest care should be taken. Do not use two descriptions if one will clearly describe the land. Avoid surplusage and conflicting descriptions. If after writing a description it is found necessary to explain it, lay it aside and if possible write a description that does not need explanation. Let the starting point be well denned and permanent, so that there need be no difficulty in locating it at any time in the future. A striking example of a disregard of this principle was brought to the attention of the writer when he was called to locate the boundary lines of several lots in a village. The descriptions all referred back to a small cherry tree as a starting point. The lines had never been marked on the ground even by fences, and the cherry tree had been gone so long that no one could be found who could remember that there ever was such a tree. Not only the starting point but as many of the angles in the boundary as possible should be described by some- thing permanent and definite on the ground. This is of prime importance. Let it be the plainest and most permanent that the nature of the case permits. If the courses are given by compass bearings, state whether they refer to the magnetic or some other merid- ian. This is put in the form of a statement of the decli- nation of the needle, written for example, Var. 4 2(K E. By this it is understood that the magnetic meridian makes an angle of 4 20' to the east of the meridian of the survey. It was formerly a custom to refer all lines to the magnetic meridian. Since the adoption of the system of the United States Land Surveys it has become a DESCRIPTIONS IX DEEDS. 353 custom, especially in that part of the country surveyed under that system, to refer all surveys to the true merid- ian, or what was supposed to be so. As time has passed and old. descriptions have been retained in the deeds conveying the land from owner to owner, it has become impossible in thousands of cases to tell what meridian controls the description. Hence we see the prime importance of permanent monuments describing the boundaries, and of describing the meridian of the survey. If we must needs figure out courses from the change in direction of the needle, let us have something definite to start from. Do not describe a boundary solely by reference to the boundary of the adjoining tract, if it can be avoided without error. Such a description requires the finding of the description of the adjoining tract whenever a survey is made, and may cause great delay and trouble before the correct definite description can be found. The writer knows of a case where the only description of the bound- ary line between two village lots in either deed is by a reference to the other: A.'s land is bounded on the east by B.'s land, and B.'s land is bounded on the west by A.'s land nothing more. If a boundary line is not intended to be a straight line, but to follow a fence, a wall, a hedge or a stream, say so in the description. Hake everything clear, definite, concise and consistent throughout, so that a surveyor having the description in the deed can locate the boun- daries on the ground, without having to hunt up descrip- tions from other deeds. 8. Illustrations. 1. " The east half of the northeast quarter of Section 16, Township 2 south, Range 10 west? The United States land department in selling land in regular subdivisions of non-fractional sections does not state the quantity in the patent. It is quite customary in later conveyances to add something like the following: "containing 80 acres, more or less, according to the United States survey." ^Nothing is gained by the addi- 23 354 A MANUAL OF LAND SURVEYING. tion. There is a good deal of useless verbiage and repe- tition in deeds, the only effect of which is to add to the expense of making out and recording them. 2. " The north fractional half of the northeast frac- tional quarter of Section 3, Township 3 south, Range 9 west, containing 98.72 acres, according to the official plat of the United States Survey." The area of fractional lots is stated in the United States patents. The word fractional is used and the area given to show that the land is conveyed according to the system of the United States survey. Without them the description would convey the aliquot part of the entire area of the section in the same manner as Description No. 1. 3. "The south fraction of the southeast quarter of Section 28, Township ft north, Range 3 west, containing in. 85 acres" Sections are made fractional by streams, lakes and reservations, making fractional lots of all manner of sizes and shapes. The land department attaches small outlying fractions to the adjacent larger ones, and sells the whole under one description, which takes its name from the larger lot. The above description might con- tain land attached from the southwest quarter. Such descriptions do sometimes contain land attached from other sections, and even from other townships. The official plat of the section shows precisely what land is included in the description. 4. "A piece of land twenty feet wide off from the ewtt side of Lot 99 of the lithographed plat of the milage of Kalamazoo" A description like the above sometimes leads to contro- versy. Suppose the original survey by which the lots were laid out, was made with a long chain, as it was in Kalamazoo, and that there was a surplus in the lot. The purchaser might claim that he was entitled under the common law to his proportional share of the surplus, while the seller, if he owned the balance of the lot, might claim it all as his own. Such questions do fre- DESCRIPTIONS IN DEEDS. 355 quently arise, and it is better to settle them at the outset, by putting it definitely in the description what is meant. In the above case suppose the recorded width of the lot to be sixty feet; then a description calling for the "east one-third of Lot 99 " would show clearly that any surplus or shortage in the lot was to be divided, while a descrip- tion reading " 20 feet off the east side of Lot 99, etc., as surveyed by F. H., May 22nd, 1883," would show that the later surveyor's measure was to govern. The care and accuracy of measurement of land in cities keeps pace with its increase in value, and as a careful, accurate measure cannot be expected to agree with a careless, inaccurate one, it is best to settle such questions in advance, as far as possible. 5. "Commencing at a stone with a hole drilled in it, set in the east and west quarter line of Section 18, Township 4 south. Range 10 west, 22 chains east of the range line, from which stone a White oak 16 inches diameter, bears S. 28 W. t 62 links distant, and running thence (Far. 2 40* E., at 10 A. M., June 12th, 1880\ north 22 east 12.00 chains to a stone marked with a ci'oss, set in an angle of a hedge; Thence east along the hedge 8.00 chains to an iron stake of 1% inch gas pipe, driven on west bank of a ditch; Thence south along the bank of the ditch 5.00 chains to an iron stake of gas pipe driven in the bank where the ditch turns east; Thence south 22 west 6.61 chains to a stake set in the quarter line, from which a Burr Oak 12 in. di. bears N. 16 E., 26 Iks. distant, Burr Oak 18 in. di. bears 8. 46 E., 51 Iks. distant; Thence west along the quarter line 1031 chains to the place of beginning." This is given as a sample of a description by metes and bounds such as a surveyor may furnish under the ordinary circumstances when called on to make a survey for that purpose, and such as he or any other surveyor would have no trouble in locating on the ground at any future time so long as any of the monuments or bearing trees could be found. 356 A MANUAL, OF LAND SURVEYING. CHAPTEE XL BE-LOCATION OF LOST CORNERS. The general principles to be observed in re-locat- ing lost corners are laid down in the Supreme Court deci- sions which have already been quoted. A corner is not lost so long as its position can be deter- mined by evidence of any kind without resorting to sur- veys from distant corners of the same or other surveys. Often after making a survey from a distant corner, the surveyor will come upon some traces or evidence which will enable him to determine the true position of the corner he is seeking. It is an uncertain way at the best to locate corners by running lines and measuring from distant corners, and should only be resorted to in absence of better proof of the original location of the corner sought. It will sometimes happen that the exact spot where a lost corner stood cannot be found or shown by evidence, but it can be proved that it stood within certain limits. In these cases, which are not rare, there is no question but that the corner should be placed at that point within the known limits which best agrees with all the evidence in the case. "Failing of better evidence by which to determine the location of a lost corner, we may next resort to the fol- lowing methods: . GENERAL RULE. Retrace the known lines of the de- scription and find how the lengths and directions of these lines by your survey agree with those of the same lines as laid down in the original description. Then run the RELOCATION OF LOST CORNERS. 357 unknown lines and place the lost corners so that they will bear the same relation to the known lines and cor- ners as they are required to do by the description of the original survey. Example. The four lines of a description are as fol- lows: 1. North 7 east 12.00 chains. 2. South 83 east 6.00 " . 3. South 7 west 12.00 " 4. North 83 west 6.00 " The first line and its termini are known. We retrace that line and find by our survey that it runs north 7 30' east and 12.24 chains. We would then run the remaining lines, making them as follows: 2. South 82 30' east 6.12 chains. 3. South 7 30' west 12.24 i * ir 4. North 82 30' west 6.12 ; * < Or the compass may be set on the known line and the vernier so adjusted that the reading of the needle shall be the same as that given in the original description and the remaining lines run accordingly. 2. Be-looation of Lost Corners of the United States Survey. RULE 1. On base lines, correction parallels, township and range lines. Restore the lost corner in line between the nearest known corners on the same line and at dis- tances from them proportional to those laid down in the field notes of the government survey. This rule supposes the original line to have been a straight line. As a matter of fact this is frequently not the case. If there is reason to suspect the line to have angles in its course, measures from known corners to the right and left of the line will aid in determining its true position. RULE 2. Lost closing section corners upon a town- ship or range line, where the closing distance from the 358 A MANUAL OF LAKD SURVEYING. adjacent corners is not given in the field notes should be restored by prolonging the known portion of the line to its intersection with the township. or range line. RULE 3. Lost interior section corners should be restored at distances from the nearest known corners, north, south, east and west, proportional to those laid down in the field notes of the original survey. This rule supposes that^the measurements of the origi- nal survey were uniform on the several adjacent sections. This is frequently not the case, and it will be well for the surveyor to compare his chaining on each section with the original measure between known corners of the same sections, choosing by preference those lines which on the government survey were measured next previous to the portion of the line closing on the lost corner. RULE 4. Lost township corners, when common to four townships, are to be restored in a similar manner to interior section corners, Rule 3. When common to only two townships, they are to be restored according to Rule 1. RULE 5. Lost quarter section corners are to be re- stored in line between the section corners which stand on the same line and at distances between them proportional to those returned in the field notes of the government survey. RULE 6. Lost meander corners are to be restored by running the line from the nearest known corner the di- rection and distance called for by the notes of the orig- inal survey. When a portion of the line leading to the meander corner is known, it should be prolonged in the same direction. When no portion of the line is known' the surveyor will have to use his own judgment as to what method under the circumstances of the case will most nearly retrace the original line to the corner. There is no rule which will rigidly and inflexibly apply to all cases for restoring lost corners and boundary lines except this that the aim of the surveyor should always RELOCATION OF LOST CORNERS. 359 be to find the exact spot where the original corner or line was located. The thing to find out is not where the cor- ner or line ought to have been, but where it actually was. There are many cases in which other methods for re- storing any of the corners mentioned will prove more satisfactory than the rules heretofore given. For instance, a half-quarter post properly planted at a time when both the section and quarter-section corners adjacent were known, may be used in restoring either of these corners when lost, by prolonging the line over the known corners and doubling the distance. Any other intermediate corner whose location is definitely known may be used in a similar manner. On a similar principle, the Supreme Court of Illinois decided in the case of Noble 0, Chrisman (88 111. 186) that the northwest corner of sec- tion 19 could, in that instance, be better determined by tracing the section lines from known corners east and west of the range line to their intersection with that line, and measuring the jog between the corners, than it could by prorating six miles of the range line. Most of the difficulties which the surveyor has to con- tend with in restoring lost corners arise from errors made in the original survey, or in the field notes thereof. He should bear in mind that errors in the original survey cannot be corrected by him. In any case of a lost corner, find as many of the adjacent corners of the original sur- vey as possible, according to the .best evidence that can be had to prove their exact location. Having done this, the others may be found according to the rules already laid down. But do not give up a corner as lost while any means of finding its exact location are left untried. There is great virtue in a pick and shovel intelligently applied to the finding of corner posts and monuments. This is very important, as it is very difficult, if not impossible, in many cases, to re-locate a lost corner in the exact position it originally occupied, by surveys from distant corners. The following extracts from a paper read by the author 360 A MANUAL OF LAND SURVEYING. before the Michigan Association of Surveyors and Engi- neers, treat more fully of the application of the foregoing principles to finding corners of the United States survey in those regions where wooden posts were planted for corner monuments : " It often happens that one surveyor will fail utterly in finding the marks of an origina* corner, while another, more apt in discovering the evidences, will strike upon it readily. These evidences are of -vari- ous kinds, some of which it is the principal aim of this paper to dis- cuss. I take it that the best possible evidence of the location of an orig- inal corner is the monument fixed at that corner when the survey was made . ( Vide McClintock v. Rogers, 11 111. 279 ; also Gratz v. Hoover, 16 Penn. State Rep. 232 ; 16 Ga. 141.) After this come witness trees, fences, distant corners of the same survey, and the testimony of persons. All these latter kinds of evidence only go to corroborate the first, and may take the place of the first only so far as they may any of them seem to have weight in any particular case. Many of the corners of the United States survey were marked by planting a post or stake in the ground. These stakes had notches cut in them, were squared at the top, and set in certain regular positions fn the ground. These marks tended to distinguish them from other stakes that might chance to be driven in the ground for any purpose. When trees stood conveniently near, two of them were marked, and their directions and distances from the corner were given in the field notes. When no trees were near, a mound was sometimes raised about the post. Some of the posts have been entirely destroyed, but the bottoms of a great many of them still remain, much decayed, but plainly visible when the surface earth is removed from about them. To find them, careful manipulation is required. The surveyor first determines as nearly as he can, from extrinsic evidence, the point where the corner post should be looked for. He then, with a shovel, spade or hoe, carefully removes the surface earth, a little at a time, being particular not to strike deep at first into the earth at the level as it was when the stake was set. The best and sometimes the sole evi- dence of a corner has often been destroyed by an ignorant person striking deep into the ground, expecting to find a sound stake, and casting away the decayed wood and filling up the hole of a rotten one without observing it. If the surveyor is looking in the right place, and the earth has not been previously removed, he will soon come upon the object of his search ; but he must be careful lest he mistake it. If the soil is a stiff clay, packed hard, as in a road, or covered with a sward, he will presently find a hole of the size and shape of the stake which RELOCATION OF LOST CORNERS. 361 made it. This hole will'contain the decayed wood of the stake, and a marking pin may be readily thrust to the bottom. By carefully scrap- ing or cutting away the earth from the top, or cutting down at one side of the hole, its size, shape and direction may be readily discovered. Thus it often happens that the position of a corner is as well and sat- isfactorily marked by the decayed stake as it was by the sound one. It sometimes happens that new stakes have been driven beside the orig- inal stake, so that several different ones will be found by the surveyor. He will seldom have any difficulty in deciding which is the true corner by its appearance, for the first stake will be more completely decayed and of a darker color. As a rule, it will be driven deeper and straighter down than the newer stakes. Then, too, the original stakes were generally round, being cut from whole timber, while the later ones were often cut from rails or other split timber, the sharp corners of which can be readily seen in the holes made by them. There is thus in the appearance of the stakes of the United States survey such peculiarities and such likeness to each other, even when far gone in decay, that the experienced surveyor will be impressed with the appearance of truthfulness pervading them, and will seldom be deceived. This appearance of truthfulness about a stake, which to a surveyor is one of the most valuable parts of the testimony of these silent witnesses, is something that courts and juries can seldom take cognizance of, because, first, they speak in a language that courts and juries do not understand, and secondly, the evidence is itself de- stroyed by the surveyor in the taking, and does not come before court or jury in all its freshness, truth and purity. These decayed stakes may be best observed in the light-colored subsoil after the black sur- face mould has been removed. In sandy soil, the cavity made by the stake is gradually filled by the falling sand as the wood decays, but rotten wood discolors the sand so that where it has not been disturbed the position, size and shape of the stake may be readily traced. In the black muck of our marshes and river bottoms it is more difficult to distinguish the stake near the surface, but as the ground is soft and wet the stakes were driven deep, and we may sometimes find in the wet, peaty subsoil the bottom of the stake so perfectly preserved that even the scratches made in the wood by nicks in the axe are plainly to be seen. When the stakes are constantly wet, they do not decay. Next we consider the bearing or witness trees. These are marked and their directions and distances noted, in order to assist in finding the corner posts set on the survey. These bearing trees are marked with a blaze and a notch near the ground on the side facing the corner. The measures were taken from this notch. At this time most of the living witness trees have grown to such an extent that only a scar re- mains in sight, to indicate the point where the notch was cut. In order 362 A MANUAL OF LAND SURVEYING. to get at the notch, the superincumbent wood, which is in some cases a foot in thickness, will have to be cut away. It will not often be necessary to do this, as we can come sufficiently near the correct point to find the stake without it. But if the stake has been destroyed, or there are several stakes near, we shall need to be exact, and measure from the notch. If the tree has been cut down, and a sound stump remains, the marks will be easily exposed. Sometimes the mark is gone, but a part of the stump is left. At others the stump is gone, but a dish-like cavity remains in the earth to show where the tree once stood. We can almost always find under and around these cavities places where the large roots have penetrated the subsoil, and thus be able to locate within a foot or so the position of the bole of the tree when standing. In looking for a corner post, we may frequently as- sume for the time being that a certain stump or a cavity where a tree had stood was the stump of or the place occupied by a bearing tree. If we then measure the required direction and distance, and find a stake, we may reasonably conclude that our assumption was correct. Such assumptions are frequently of great assistance in finding corners. There may be, and I know there are cases, where the original corner stakes have been destroyed, and can be more nearly restored to their original position by measurements from old stump bottoms or holes in the ground than in any other way. But bearing trees, however good their condition, are by no means infalible witnesses as to the location of a corner. Mistakes in laying down their direction or distance, or both, are not rare. (See McClintock v. Rogers, 11 Ills, 279.) A direc- tion may be given as north instead of south, east instead of west, or vice versa. The limb may have been wrongly read 64 for 56'. The figures denoting the bearing may have been transposed in setting down, as 53 for 35. So, too, the chain may have been wrongly read, as 48 for 52, the links having been counted from the wrong end. Or they may have counted from the wrong tag, as 48 for 38. Mistakes of the nature of these mentioned are common, so that in working from a bearing tree to find a corner, and not finding the stake at the place indicated in the notes, it will be well to test all these sources of error before giving up the search, for as I have said before, the post planted at the timerof the original survey is the best evidence of the corner it was intended to indicate. I next consider fences in their relations to corners. (Potts v Ever- hart, 26 Penn. St. Rep., 493.) Whether any particular fence may be depended on to indicate the true line will depend on the particular cir- cumstances attending that case. In a general and rough way, a fence will indicate to the surveyor where to begin looking for his corner. But the practice has been, and still is common, for the first settlers on a section to clear and fence beyond the line in order to have a clear place on which to set their permanent fence when they get ready to RELOCATION OF LOST CORNERS. 363 build it. Afterward they forget where the line is and set the new fence where the old one stood. Many fences, too, were set without any sur- vey or any accurate knowledge where the line was and left there to await a convenient time to have the line established. So, too, where the land has been long settled and occupied, it is a common custom for adjoining land owners by consent to set the fence on one side of the true line, there to remain until they are ready to rebuild, the one party to have the use of the land for that time in consideration of clear- ing out and subduing the old fence row. The original parties fre- quently sell out or die, and the new owners have no knowledge of the agreement and suppose the fence to be on the true line. For these reasons, fences should be looked on with suspicion, unless corroborated by other evidence, and the surveyor should enquire pretty closely into the history of a fence before placing any great reliance on it to deter- mine the position of a corner. It may be the best of evidence, or it may be utterly worthless. It not unfrequently happens that there are no trustworthy marks near a corner to direct the surveyor in his search for the post or from which to replace it if it be destroyed. In these cases, he must visit the nearest corners he can find in each direction (varying with the circum- stances whether it be section corner or quarter post he wishes to find or restore), go through the process of identification with each of them, and then make his point so that it will bear the same relation to these corners as did the original corner post. Many very intelligent gentle- men suppose that if the surveyor can but find one of the corners of the original United States survey he can readily determine the position of all the rest from it. They were never more mistaken in their lives. The continual change in the direction of the magnetic needle, the un- certainty as to what its direction was when any particular line was run, the difference in the lengths of chains, and the difference in the men who use them, introduce so many elements of uncertainty into the operation as to render it one of little value, and not to be resorted to except in the absence of trustworthy evidence nearer at hand. If it be a section corner you desire to find or replace, and have ad- jacent quarter posts in each direction to work from, you will not be likely on the one hand to fall more than a rod or two out of the way, and on the other hand will not be likely to come within a foot or two of the right place. This method will assist you in seaching for the original stake, and if that be destroyed, and no better evidence pre- sents itself, may be used to determine the point where the corner stake shall be placed. The chief difficulty in applying this method to determine corners arises from the f&ct that the measurements made on the original surveys were not uniform in length on different sec- tions, and frequently not on different parts of the same section. I have measured sections 22 and 23 on a level prairie, along the line of high- 364 A MANUAL OF LAND SURVEYING. ways, where no obstacles of any kind interfered to prevent accurate work. I took the greatest possible care in the chaining to have it as accurate as chain work can be done. On the north Une of section 22 my chaining tallied exactly with that of the United States survey, viz., 79.60. On the north line of section 23, my measure was 80.96, that of the United States survey, 80.40 a difference of 56 links. Fortunately, all the corners of the original survey on this two miles of line were well preserved, and the distance between quarter post and section cor- ners was uniform on the same section in both sections. But suppose that a part of them had been lost, and it was required to restore the middle section corner (n. e. of 22) from the remaining ones. Omit all consideration of corners, north or south, and there remain four differ- ent solutions of the problem, depending on which corners were lost and which preserved. Of these different solutions, one would place the corner 9^ links, one 14 links, one 18M links, and one 28 links, all east of the true corner. This is not by any means an extreme instance, as I have observed discrepancies twice as great. It is given simply to show how unreliable is the evidence drawn from distant corners of the United States survey. Lastly, I shall consider the evidence of living persons. [Weaver v. Eobinett, 17 Mo., 459; Chapman v, Twitchell, 37 Maine, 59; Dagget v. Wiley, 6 Florida, 482: Lewen v. Smith, 7 Port. (Ala.), 428; McCoy v. Gal- loway, 3 Han. (Ohio), 283; and Stover v. Freeman, 6 Mass., 441.] Con- ceding all men to be equally honest in their evidence, there is a vast deal of difference among them with regard to their habits of observa- tion and their ability to determine localities. Some have an exceed- ingly acute sense of locality, if we may so call it, and can determine very accurately the position of any object which they have been accus- tomed to see; while others seem to have little or no capacity of that sort. I have found many men who would describe accurately the sort of monument used to perpetuate a corner, and who would tell you that they could put their foot on the very spot to look for it; but when the trial came I have found but few of them who could locate the point within several feet, unless they had some object near at hand to assist the memory, and even then they would frequently fail. It may happen where a corner post has been destroyed, that its loca- tion can be more nearly determined by the testimony of persons who were familiar with it when standing and can testify to its relations to other objects in its vicinity, than in any other way. But the surveyor in receiving this testimony should ascertain as far as possible what are the habits of accurate observation and the memory of localities pos- sessed by th person testifying, in order to know how much weight to give his testimony." MISCELLANEOUS. 365 CHAPTEE XII. MISCELLANEOUS. 1. Questions of Practice. Answers to most if not all questions which arise in the surveyor's practice will be found in the Supreme Court decisions which have been quoted. The following questions which have been raised in several surveyors' associations, are given with the answers adopted in each case, or a reference to the law decision or principle which governs it. 1. An interior section has its quarter posts out of line and not at equidistant points between the section corners. How shall the centre be determined ? Ans. At the intersection of straight lines from each quarter section corner to its opposite corresponding cor- ner. See page 200, Sec. 100, Second. 2. How shall the quarter posts on the north and west lines of the township which were not established by the U. S, survey be located ? Ans. The corners of half and quarter sections, not marked on the surveys, shall be placed as nearly as possi- ble equidistant from those two corners which stand on the same line. See page 200, Sec. 100, First. Section 6 is an exception to this rule. See page 271. 3. Posts for lines closing on the north and west boun- daries of townships are often off the boundary line to one side or the other. Shall the boundary line be deflected to pass through these posts ? 366 A MANUAL OF LAND SURVEYING. Ans. No. The posts serve to show the position of the section line, but the line itself stops at the township boundary.* Mich. Surv. Rep., 1881. 4. Are the station or line trees marked on the govern- ment surveys and returned in the field notes, monu- ments of the lines? Ans. Yes. See page 200, Sec. 100, Second. Billingsiey v. Bates, 30 Ala. 378. 5. How shall the east and west quarter line of section 30 be located, there having been no quarter post set on the east side of the section by the U. S. survey, because of a lake? Ans. Locate the west quarter post as directed in the answer to question 2. Then run the quarter line east on a course which is intermediate between the courses of the north and the south lines of the section. See page 268. 6. A closing corner on the north or west boundary of the township is lost. The field notes do not give the dis- tance between the closing corner and the adjacent corner on the boundary. How shall it be restored ? Ans. Prolong the known portion of the line to its inter- section with the boundary and there pat the corner. See Billingsiey v. Bates, 30 Ala. 378; see p. 200, * E. F. Best, acting commisioner of the General Land Office, in an opinion given April 16, 1896, says: " These cases seem to be ex- ceptions to the stringent law that the original corners 'must stand as the true corners which they were intended to represent, whether the corners be in place or not.' The improper placing of such a corner results in a change of the true line of the standard parallel into a broken line, if the erroneous corner must be held valid. It is the opinion of this office that the true intent of the law is sub- served by holding that such corners must show the true locus of the line separating the sections but cannot alter the position of the township line; hence, that all corners of small tracts adjacent thereto should be set on the true line originally run." 21 MISCELLANEOUS: 367 7. Should section lines running north and south be run in a straight line between known corners to locate lost corners on interior sections ? Ans. ^N"ot unless the original lines were actually straight lines between the known points, which they seldom are. See Moreland v. Page, 2 Clarkes, Iowa, 139. Martz v. Williams, 67 111., 306. 8. How shall the half-quarter corner on the quarter line be located on those quarter sections which- adjoin the north and west lines of the township ? Ans. Measure the distance from the centre of the sec- tion to the quarter post on the township line. Then place the corner on the quarter line at a distance of twenty chains proportionate measurement from the centre of the section. In order to prorate the distance, your own measure should be compared with a distance which is a mean between the distances given in the field notes as the length of the corresponding lines of the sec- tion on either side. Por example, on section 3 the dis- tance by U. S. survey from the east post to township line is 42.18; from the west J post to township line is 43.20; which gives a mean distance of 42.69. Commissioner McFarland gives the following reply to a similar question* DEPARTMENT OF THE INTERIOR, ) GENERAL LAND OFFICE, Washington, D. C., February 11, 1882. } Isaac Teller, Esq., Webbervffle, Ingham County, Michigan: SIR I am in receipt of your letter of the 5th instant requesting in- formation in regard to the proper method of locating the quarter-quar- ter corners north of the legal centres of the northern tier of sections in a township when the present measurement of the east and west boundaries of the section differs from the original measurement, In reply, I have to state that the length of the quarter line from the south quarter corner to the township line is to be considered as the mean of the east and west boundaries of the section as given in the field notes, and where the present measurement of the section lines differs from the original measurement, the rule of proportionate measurement applies to the quarter line as well as to the section lines in the establishment of quarter-quarter corners on the half mile closing 368 A MANUAi, OF LAND SURVEYING. on the township boundary. See enclosed circular dated November 1, 1879. The mean width of the north half of the section in the case stated by you is 40.18 chains, while by your chaining it is 42.42 chains (calling the distance to the east and west quarter line 40.00 chains), therefore the proportion will be as 40.18 : 42.42 : : 20.00 : 21.11 chains, the distance north of the centre of the section at which by your chaining the quar- cer-quarter corner should be located. Very respectfully, N. C. McFAELAND, Commissioner. 9^ In surveying sections fractional on the township line to restore lost quarter section corners, should the lines be divided pro rata according to the U. S. field notes, or should the south or east quarters be made full and the entire excess or deficiency be thrown into the fraction ? Ans. Any difference between your measure and the government measure must be distributed proportionally between the different parts of the section. See p. 200, Sec. 100, Second. Moreland v. Page, 2 Clarkes, Iowa 139. Jones v. Kimble, 19 Wis. 429. Martz v. Williams, 67 111. 306, In Missouri, the Supreme Court holds (Knight v. Elliott, 67 Mo. 317) a different view, viz., that the difference in measure is all to be thrown into the fraction. It is difficult to see upon what grounds this decision can be upheld in view of the fact that all rights to the land were acquired and held under the law of Congress, which expressly states that the length of such lines as returned by the surveyor-general shall be held and con- sidered as the true length thereof. Northwest Quarter, Sec. 18. FIG. 72. 10. The accompanying figure is a copy of the plat of the U. S. survey of this quarter section. A owns the whole quarter. He sells to B the W. 1 of the N. W. \ of section 18, containing 91^% acres. At about the same time he sells to C the E. of the N. W. i of section 18, con- taining 91 T Vb- acres. MISCELLANEOUS.- 369 Where shall the surveyor run the dividing line between BandC? Ans. The language of the deed clearly shows the inten- tion of A to sell and of B and C to purchase each the half of the area of the quarter section. The surveyor should so locate the line as to carry out the evident intent of the parties. See rule 2, p. 244 and rule 14, p. 284 . The fact that the quarter is differently subdivided on the govern- ment plat has no bearing on this case. 11. Certain early surveyed townships had three sets of corners on the range lines. (1) Those set when the range lines were run; (2) Those set as closing corners running east; (3) Those set as closing corners running west. What use is made of each set of corners ? Ans. The first corners set determine the location of the range line. The second and third sets of corners deter- mine the location of their respective section lines which close on and terminate at the range line. FIG. 73. 12. This figure shows a fractional township on the Ohio River. The figures show the dimensions of section 1, as shown by the field notes of the United States survey. By a subsequent measure, AB 82.25 chains, and AD = 79.50 chains. 370 A MANUAL OF LAND SURVEYING. How shall the northeast quarter of section 1 be laid off, no quarter-posts having been planted ? Ans. Place the quarter-section corners on the north and east sides of the section in line with and midway between their respective section corners. Make the east and west quarter-line parallel with the south line of the section, placing the west quarter-post at the point where the quarter-line thus run intersects the section HDC. From the north quarter-post run the quarter-line south on a course which is a mean between the courses of the east and the west lines of the section, placing the south quar- ter post at the intersection of the section and quarter- section lines. The exceptional features of this case are that no quar- ter-posts were set on the United States survey, and that the east line of the section is just 80 chains in length, having been run from the north to the south. FIG. 74. 13. The description in the deed runs: " Beginning at a stone (A\ at the N.W. corner of lot 401; thence east 112 ft. to a stone (By, thence S. 36i W. 100 ft.; thence west par- allel with AB to the west line of said lot 401 ; thence north on west line of said lot, 66 ft., to the place of beginning." The points A and B, and angle ABC, are fixed. C, by MISCELLANEOUS. 3 r 1 construction and in fact, is 80^ ft. distant, at right an- gles from the line AS. 1. Shall I locate CD parallel with AB, or locate D 66 ft. from A ? 2. Have I any right to consider any apparent intention to locate 66 ft. or 80^ ft. from A ? 3. Have I, if I know it, any authority to consider the actual intention of the grantor to locate CD V 4. If the distance AS should actually measure 114 ft. am I to use it, or shall I make J5 112 ft. from A ? Ans. 1. The answer to this question will depend upon the state of facts brought out in answer to questions 2 and 3. If there be evidence showing what the intention and understanding of the parties to the conveyance was as to which of the two lines should be taken, that evi- dence would settle the question. If not, that construc- tion may be given to the deed which will operate most strongly against the grantor and give the grantee the greater amount of land. So far as anything is shown in the question, the deeds to the adjacent land might fur- nish the necessary evidence. 2 and 3. Yes. Judge Cooley says, (see " Judicial Func- tions of Surveyors"): "The surveyor must inquire into all the facts, giving due prominence to the acts of parlies concerned, and always keeping in mind * * * * that courts and juries may be required to follow after the surveyor over the same ground, and that it is exceedingly desirable that he govern his action by the same lights and the same rules that will govern theirs." 4. The monument controls the distance. 14. A piece of land is sold, and described as commencing at the north quarter-post of section 15, and running thence east 100 rods; thence south 160 rods; thence west 100 rods; thence north 160 rods, to the place of beginning; containing 100 acres, according to the United States sur- vey. 372 A MANUAL OF LAND SURVEYING. Ques. How shall it be set off ? Ans. The deed clearly indicates the understanding of the parties to the conveyance to be that the land should pass according to the rules that govern the United States survey. One of these rules is, that "the length of the boundary lines as returned by the surveyor-general shall be held and considered as the true length thereof." Hence in this case, measure east from the quarter-post along the section line 25 chains of just such measure as the United States surveyors gave ; or in other words, of pro rata measurement. Suppose the distance by the field notes to be 40.32 chains from quarter-post to section corner. Then 25.00 lay off of that distance. Proceed in a similar man- 40.32 ner, running east on the quarter-line from the center of the section, and the two points thus located will be the corners of the 100 acres. To get the length of the south line of the N. E. J of the section by the United States survey, take the half sum of the measure given on the north and the south lines of the section. Supposing it to be 40.32 on the north, and 40.18 on the south, then the distance on the quarter-line would be equal to 40.32 + 40.18 = 40.25, 2 25.00 and you should measure off of this distance for the 40.25 corner. 15. A man buys a tract of land described as the north 40 acres of the northeast quarter of Section 3. This section overruns the government measure when measured with a standard chain. Should this land be measured as 40 acres standard measure or should the division be made so as to include the proper proportion of the overplus? MISCELLANEOUS. 373 Ans. In the absence of evidence to prove a different intention on the part of the parties to the conveyance, it should be measured according to the U. S. Govern- ment measure thereof, as explained in the answer to the previous question. 16. " Section 3 is fractional on Grand Traverse Bay, the center being a few rods out in the bay. How shall the quarter lines be run?" See Figure 75. Sec Line FIG. 75. Ans. As Section 3 is fractional in the north half, run the east and west quarter-line parallel with the south line of the section. Run the north and south quarter-line parallel with the west line of the section. The field notes were not given with the question. A full knowledge of what they contain might show reason for modifying the answer. 17. A sells land to B, described as 30 acres off the west side of Lot 1 of the section. Figure 76 repre- sents Lot 1 as shown on the official plat. A's intention was to sell 30 acres, and have a strip left. Does the surveyor take the government returns for it, or does he have to run it out following the meander lines, and then part off all but the 30 acres, be it more or less? 374 A MANUAL OF LAND SURVEYING. If the land actually measured only 30 acres, would B be entitled to the whole of it or only his proportion of 33.60 acres? FIG. 76. Ans. A's intention cuts no figure in the case, unless it was understood and shared in by B. If there was a mutual understanding and agreement between the parties, it would guide the surveyor in locating the line. In the absence of evidence as to the mutual intent of the parties, B is entitled to his proportion of the lot, as shown by the official plat. It has been decided by the supreme court of Michigan that in such cases the government meander line is to be used in making the computation. 18. " How much variation shall I allow for a period .of, say 24 years, in the notes of the survey of an angling road of many courses, which are as fol- lows," etc.? Ans. I cannot tell. The annual change of declina- tion approximates 4' in Michigan. If you can definite- ly locate any two points of the original survey of the road, you can run a random line between them, and find the exact variation to allow. For methods of doing this where more than one course has to be run, see page 79, and the paper entitled, " That Problem in Land Surveying," in the Michigan Engineers' Annual for 1891, page 36. If there are not two known points of the original survey of the road, you had better not MISCELLANEOUS. 375 try to relocate the original line by surveys. It cannot be done with any certainty, because you have no means of comparing your compass and chain, or transit and tape, with that used on the original survey. Bet- ter make a relocation of the road in accordance with actual occupation, or such other location as may be asked for by the parties interested. 19. Since the original survey of Section 9, the Mis- sissippi River has changed its course, adding by accre- tion nearly an entire section in Section 16 and large amounts in other sections. How shall I survey the accretion in Section 16? The field notes show that the line between Sections 9 and 16 was run, and meander posts and quarter post established. FIG. 77. 876 A MANUAL OF LAND SURVEYING. Ans. The official plat does not show any Section 16, but it does show that the outlying fractions, which would otherwise have constituted a fractional Section 16, were attached to the lots in Section 9. Any accre- tions which have formed against those lots belong to the owners of the lots. To locate the boundaries of the lots on the accretion, relocate the old river line as far as the accretion ex- tends, and mark the points where it is intersected by the lot lines as shown on the official plat. Measure the amount of the old shore line which each lot has. Then measure the new shore line and assign to each lot the same proportion of the new shore line that it has of the old, and connect the corresponding points on the two shore lines by straight lines. A strict carrying out of the principles involved would call for curved lines to correspond with the changing contours as the river receded, but this is hardly practicable, and is not required by the courts. 20. How shall I run the east and west quarter line of Section 6, where the east quarter post has not been and can not be fixed, but the west quarter post and the exterior lines of the section are known? Ans. The rules and regulations prescribed by the Secretary of the Interior, as provided by Section 2397 of the Revised Statutes of the United States, provide that all the east and west subdivision lines of these frac- tional sections adjoining the north boundary of the township shall be made parallel with the south line of the section, and that in those fractional sections adjoin- ing the west boundary of the township, the north and south subdivision lines shall be made parallel with the east line of the section. Hence in this case run the quar- ter line east from the west quarter post on a line parallel with the south line of the section. The rule for running lines on mean courses does not apply in these cases. MISCELLANEOUS. o i i 21. What is the penalty for destroying or removing a " government corner ?" Ans. An act of Congress approved June 10, 1896, contains the following: Provided further, That hereafter it shall be unlaw- ful for any person to destroy, deface, change, or remove to another place, any section corner, quarter-section corner, or meander post, on any government line of survey, or to cut down any witness tree or any tree blazed to mark the line of the government survey, or to deface or remove any monument or bench mark of any government survey. That any person who shall offend against any of the provisions of this paragraph ehall be deemed guilty of a misdemeanor, and upon conviction thereof in any 'court shall be fined not ex- ceeding two hundred and fifty dollars, or be imprisoned not more than one hundred days. All the fines accru- ing under this paragraph shall be paid into the treas- ury, and the informer, in each case of conviction, shall be paid the sum of twenty-five dollars. (29 Stat. L., 343.) 2. The Bights, Duties and Responsibilities of Surveyors. Surveyors, by the consent and acquiescence of the parties concerned, are usually the arbiters of dis- puted boundaries, and their decisions, when thus acqui- esced in by the parties, become in time as binding, and as much respected by the authorities, as the decisions of juries and courts of law. It is probable that at least ninety-nine per cent, of all questions of disputed bound- aries are thus settled by the interested parties themselves, in accordance with the decision of the surveyor. Surveyors, from constantly exercising this seeming authority, come at last in many cases to believe it to be absolute and final, something which must be respected, overlooking the fact that the only force their decisions have comes from the consent of the parties. When that 378 A MANUAL OP LAND SURVEYING. consent is withheld, the case goes to the courts for settle- ment; and thus the courts have in some cases felt called upon to define the surveyor's standing before the law. They say: 1. " Surveyors have no more authority than other men to determine boundaries, of their own motion. All bounds and starting points are questions of fact to be determined by testimony. Surveyors may or may not have in certain cases means of judgment not possessed by others, but the law can not and does. not make them arbiters of private rights. Cronin v. Gore, 38 Mich. 381. 2. The law recognizes surveyors as useful assistants in doing the mechanical work of measurement, and calcu- lation, and also allows such credit to their judgment as belongs to any experience which may give it value in cases where better means of information do not exist. But the determination of facts belongs exclusively to courts and juries. Where a section line or other starting point actually exists, is always a question of fact, and cannot be left to the opinion of an expert for final decis- ion. And where, as is generally the case in an old com- munity, boundaries have been fixed by long use and acquiescence, it would be contrary to all reason to have them interfered with on any abstract notion of science. Stewart v. Carleton, 31 Mich. 273. Gregory v. Knight, 50 Mich. 61. 3. New surveys disturbing old boundaries are not to be encouraged. Toby v. Secor, Wisconsin. N. W. Reporter, Vol. 19, p. 79. 4. Lines long unquestioned ought not to be disturbed upon a mere disagreement among surveyors, especially when the last survey is made under the unfavorable cir- cumstances of corner posts and witness trees being gone, which it is probable to suppose were in existence at the time of the first survey. Case v. Trapp, 49 Mich. 59. MISCELLANEOUS. 379 5. County surveyors' certificate are not admissible in evidence unless they contain all the particulars required by the statute to be entered in the surveyor's record. Smith v. Rich, 37 Mich. 549. The statute of Michigan required the length of all lines run, the area of lands surveyed, and other particulars, to be entered in the county surveyor's record. In the above case the survey was solely to find the location of a corner post. As the surveyor's certificate did not show any area of land surveyed, it was not admitted in evidence. 6. A surveyor was called on to survey the line of a highway. He performed the work so unskillf ully as to render a new survey necessary. A large amount of road constructed at great expense, on the line designated by the surveyor before the mistake was discovered, had to be abandoned. Action was bfrought to recover damages. Held, that whether the defendant was a professional or official surveyor, or represented himself as such, his under- taking was that he should bring to the work the neces- sary knowledge and skill to perform the same properly and correctly; and if he failed to do so, and the plaintiff suffered damage in consequence of such failure, the plain- tiff will be entitled to recover. Commissioner of Highways v. Beebe, Mich. Sup. Court. N. W. Rep., Vol. 20, No. 16. The following paper, by Chief Justice Cooley, of the Supreme Court of Michigan, discusses more fully the surveyor's functions : 3 The Judicial Functions of Surveyors. When a man has had a training in one of the exact sciences, where every problem within its purview is sup- posed to be susceptible of accurate solution, he is likely to be not a little impatient when he is told that, under some circumstances, he must recognize inaccuracies, and govern his action by facts which lead him away from the 380 A MANUAL OF LAND SURVEYING. results which theoretically he ought to reach. Observa- tion warrants us in saying that this remark may fre- quently be made of surveyors. In the State of Michigan, all our lands are supposed to have been surveyed once or more, and permanent monu- ments fixed to determine the boundaries of those who should become proprietors. The United States, as orig- inal owner, caused them all to be surveyed once by sworn officers, and as the plan of subdivision 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 missing. The truth unfortunately is, that the lines were very carelessly run, the monuments inaccurately placed; and, as there- corded witnesses to these were many times wanting iu permanency, 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 the witnesses it refers to have disappeared. It is, perhaps, generally supposed that our town plats were 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 pur- poses. Many of them, however, were laid out in the woods; some of them by .proprietors 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 determine 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 surveys in what is now the State of Michigan MISCELLANEOUS. 381 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 be- gan. A generation has passed away since they were con- verted 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 be relied upon after the stake was gone. Too often the first settlers were careless in fixing their lines with accuracy while monu- ments remained, and an irregular brush fence, or some- thing 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 determined the boundary line be- tween the proprietors. However erroneous may have been the original survey, the monuments that were set must nevertheless govern, even though the effect be to make one half -quarter section ninety acres and the one adjoining seventy ; 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 remark- able how many there are who mistake altogether the duty 382 A MANUAL OF LAND SURVEYING. that now devolves upon the surveyor. It is by no means uncommon 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 alluded to, is supposed to have found expression in our legislation; though it is possible that the real intent of the act to which we shall refer is not what is com- monly supposed. An act passed in 1869, (Compiled Laws, 593), amending the laws respecting the duties and powers of county sur- veyors, after providing for the case of corners which can be identified by the original field notes or other unques- tionable testimony, directs as follows: " /Second. Extinct interior section corners must be re-established 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 fractional lines, must be re-established equidistant and in a right line between the section corners ; in all other cases at its proportionate distance between the nearest original corners on the same line." The corners thus determined, the surveyors are required to perpetuate by noting bearing trees when timber is near. To estimate properly this legislation, we must start with the admitted and unquestionable fact that each purchaser from government 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 never- theless determined the extent of his possessions, and he MISCELLANEOUS. 383 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 establish 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 any surveyor will be able to determine with almost abso- lute 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 dis- appearance ; 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 author- ity 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 govern- ment surveyor where the field notes indicated it to be. 2. But this is only a presumption, and may be over- come 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 par- ties concerned. Nor is this a question merely of conflict between State and'federal law; it is. a question of prop- 384 A MANUAL OF LAND SURVEYING. erty 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 legis- lation, 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 deter- mination 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 adjudged. 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. Occupation, especially if long con- tinued, often affords very satisfactory 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 w.ere, and whether a claim of title has always accompanied the possession, and give all the facts due force as evidence. Unfortun- ately, 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 neigh- borhoods into quarrels and litigation 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 parties concerned have acquiesced in lines which were traced by the guidance of some other corner or landmark, which may or 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 subject the surveyor MISCELLANEOUS. 335 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 lino should be than any survey made after the original monuments have dis- appeared. (Stewart v. Carleton, 31 Mich. Reports, 270; Diehl v. Zanger, 39 Mich. Reports, 601.) And county sur- veyors, no more than any others, can conclude parties by their surveys. The mischiefs of overlooking the facts of possession most often appear in cities and villages. In towns the block and lot stakes soon disappear; there are no witness trees, and no monuments to govern except such as have been put in their places, or where their places were sup- posed to be. The streets are likely to be soon marked oft by fences, and the lots in a block will be measured off from these, without looking farther. Now it may per- haps be known in a particular case that a certain monu- ment 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 depart- ure 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 distance according to the plat, measuring and estimating from his point of departure. This procedure might unsettle every line and every mon- ument existing by acquiescence in the town; it would be very likely to change the lines of streets, and raise con- troversies 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 proprietors in a certain block have laid off their lots in reference to this practical location. Two lot own- ers quarrel, and one of them calls in a surveyor, that he 386 A MANUAL OF LAND SURVEYING. may make sure his neighbor shall not get an inch of land from him. This surveyor undertakes to make his survey accurate, whether the original was so 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 gener- ally will allow the new survey to unsettle their posses- sions, but there is always a probability of finding some one disposed to do so. We shall then have a lawsuit; 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 bound- ary, and all concerned have cultivated and claimed up to it Public policy requires that such- lines be not lightly disturbed, or disturbed at all after the lapse of any con- siderable 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 surveyor 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 original monument was at one place, when at the same time he is satisfied that acquies- cence has fixed the rights of parties as if it were at an- other. 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 MISCELLANEOUS. 387 go, as an officer of the law, is to express his opinion where the monument should be, at the same time that he im- parts the information to those who employ him, and who mignt 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 conclusive evidence of such settlement. The peace of the commu- nity absolutely requires this rule. 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 survey under wnich 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 surveyor 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 them 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 expres- sion, 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 be- comes 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, accord- ing to some inflexible rule. The surveyor, on the other hand, must inquire into all the facts: giving due promi- nence to the acts of parties concerned, and always keep- ing in mind, first, that neither his opinion nor his survey 388 A MANUAL OF LAND SURVEYING. 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 exceedingly desirable that he govern his action by the same lights and the same rules that will govern theirs. It is always possible, when corners are extinct, that the surveyor may usefully act as a mediator between parties, and assist in preventing legal controversies by settling doubtful lines. Unless he is made for this purpose an arbitrator by legal submission, the parties, of course, even if they consent to follow his judgment, cannot on the basis of mere consent, be compelled 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 desirable that all such agreements be reduced to writing; but this is not absolutely indispensable if they are carried into effect without. 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 ia 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 impression still lingers in some minds that the mean- der lines are boundary lines, and that 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 that this was the rule on all fresh-water streams, large MISCELLANEOUS. 389 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 banks all riparian rights. The practical difference is not very im- portant. In this State, the rule that the center line is the boundary line, is applied to all our great rivers, including 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 them- selves when the boundary line between two contiguous estates is to be continued from the meander line to the center 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 be- cause 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 diminu- tion of the flow of water. Now the dividing line be- tween two government subdivisions might strike the meander line at right angles, or obliquely; and, in some cases, if it were continued in the same direction to the center line of the river, might cut off from the 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 390 A MANUAL OF LAND SURVEYING. 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 divi- sion between the two parcels from the meander line to the center line of the river, as nearly as possible 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 nctes 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 is not limited to government surveys, but applies as well to water lots which appear as such on town plats. '(Bay City Gas Light Co. v. The In- dustrial Works, 28 Mich. Reports, 182.) It often happens, therefore, that the lines of city lots bounded on naviga- ble 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 deter- mine 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 circum- stances will admit, the general 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 circumstances, 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 MISCELLANEOUS. 391 the legal boundary, namely, the center 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 apportion- ment unless the islands were surveyed out as government subdivisions in the original admeasurement. Wherever that was the case, the purchaser 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 afterward, 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 separately surveyed and sold, will pass with the shore in the government sale: and having this right, anything which their purchase would include under it cannot afterward be taken from them. It is believed, however that the federal courts would not recognize 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 ex- pansions near their mouths of the rivers passing through them such as the Muskegon Pere Marquette and Manis- tee 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. Euddiman, 10 Mich., 125.) If such a lake were circular, the lines would converge to the center; if oblong or irregular, there might be a line in the middle on which they would terminate, whose course would bear 392 A MANUAL OF LAND SURVEYING. 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 sometimes important, how- ever, to run the lines out for considerable 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. (Lor- man v. Benson, 8 Mich., 18; People's Ice Co. o. Steamer Excelsior, recently decided.) What is said above will show how unfounded is the notion, which is sometimes advanced, that a riparian 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 sur- veyed out and sold by the United States, can this be done now? Others wii. ^e suggested by the answers give~ to these. It seems obvious that the rules of private ownership which are applied to rivers cannot be applied to the great MISCELLANEOUS. 393 lakes. Perhaps it should be held that the boundary is at low water mark, but improvements beyond this would only becoifte unlawful when they became nuisances Islands in the great lakes would belong to the United States until sold, and might be surveyed and measured for sale 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 to the United States; aod so, it is believed, does the bed of a lake also. (Pollord v. Hagan, 3 Howard's U. S. Keports.) But such rights are not generally consid- ered proper subjects of sale, but like the right to make use of the public highways, they are held by the state ID 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 ex- ample, as Houghton, Higgins, Cheboygan, Burt's, MoBet. Whitmore, and many others. But there are many little lakes or ponds which are gradually disappearing, and the shore proprietorship advances part passu as the waters recede. If these are of any considerable size say, even a mile across there may be questions of conflicting rights which no adjudication hitherto made could settle Let any surveyor, lor example, take the case of a pond ot irregular form, occupying a mile square or more of terri- tory, and undertake to determine the rights of the shore proprietors to its bed when it shall totally disappear, anrt he will find he is in the midst of problems such as proba- bly 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 settle- ment. Where a pond is so small as to be included within the lines of a private purchase from the government, it is not 394 A MANUAL OF LAND SURVEYING. believed the public have have any rights in it whatever. Where it is not so included, it is believed they have rights cf fishery, rights to take ice and water, and rights of nav- igation 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 legislation. 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 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 condition 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, unless their rights were appropriated and paid for. Upon many questions that might arise between the state as owner of the bed of a little lake and the shore owners, it would be presumptuous to express an opinion now, and fortunately the occasion does not require it. I have thus indicated a few of the questions with which surveyors may now and then have occasion to deal, and to which they should bring good sense and sound judg- ment. Surveyors are not and cannot be judicial officers, but in a great many cases they act in a quasi judicial capacity with the acquiescence 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 en- lightenment, but I trust will not be wholly without value. LEVELING. 395 CHAPTER XIII. BEVELING AND DRAINAGE SURVEYING. 1. Leveling is the operation of measuring the differ- ence in height of two or more points. The surface of water at rest is an example of a level surface. If the earth was a perfect sphere, a line of true level would be an arc or a circle having its centre at the centre of gravity of the earth. So far as common level- ing is concerned it may be so considered, as the error arising therefrom is so small as to be of no practical consequence. The line of apparent level is a straight horizontal line passing through the point of observation, tangent to the line of true level. In precise leveling the difference between true and apparent level is measured, the instruments used are of the best, and all the operations are performed so as to reduce the error to the smallest possible amount. In common leveling for streets, railroads, drains, water powers and the like operations, a lower degree of ac- curacy is required and the refinements of precise level- ing are dispensed with." No attention is paid to the dif- ference between true and apparent level, it being too small to affect the practical result. 2. The deviation of the true from the apparent level between two points is equal to the square of the distance between the points t divided by the diameter of the earth. Also, The deviations for different distances are pro- portional to the squares of the distances. Calling the diameter of the earth 7920 miles and ta- king points one mile apart, we find the deviation = 0.000126 miles == 0.665 ft. = 7.98 inches. For m miles, deviation = 7.98 m 2 inches. 396 A MANUAL OF LAND SURVEYING. The effect of the refraction of light is to apparently increase the difference between true and apparent level. For considerable distances the correction for curvature as above found is sometimes diminished by about one- sixth of itself. If the instrument is placed midway between the points whose difference in height is required, the errors are bal- anced and eliminate each other, giving a correct result. 3. In leveling, two instruments are required, one to find a horizontal line, and the other to measure vertical distances. These instruments are called a Level and a Leveling rod. Level lines, for many common purposes, on a limited area, when no instruments are at hand, can be obtained by the following method : Suspend from some fixed point of support P by stout cords as indicated, a pole of any shape A B, having the longer end sharpened to a fine point. From this pole hang a heavy weight R s- as shown. Set two stakes S& so that | A db JlL the point of the pole when swung around will just touch them. Smooth a place on each stake to receive marks. After taking the twist out of the supporting cord, care- fully swing the pole around and mark the exact place where the point of the pole touches each stake. Repeat this, and take the most satisfactory points. They will determine a level line of sight. A cheap instrument which almost any one can make, having a more extended range, is made as follows : Take two pieces of glass tubing three or four "^ inches long and connect them with a rubber tube two or three feet long, so as to make a continuous water tight tube, with glass ends. Pass the ends of the tube through holes in a cross bar FIG. 76. made of a piece of board of suitable LEVELING. 397 size, as shown in the cut, and fasten them with the tops projecting an inch or more above the bar. The cross bar may be fastened with a bolt and nut to a staff so that it may be set up and adjusted to a level line. Colored fluid is poured into the tube. The surface of the fluid in the glass tubes determines the level line. Sights of horse hair or fine wire may be attached close to the glass tubes and the cross bar adjusted to bring them into a level line. An instrument can thus be made at the expense of a few cents in money and a few minutes' labor that will do very satisfactory work. ..-?' 4. If a tube be nearly filled with any liquid, as water, alcohol or ether, and closed, the liquid will seek the lowest part, and the vacant space or bubble, as it is called, will be found at the highest part of the tube. If the tube is of glass, and very truly ground on the inside to a segment of a circle, it furnishes the best known means for determin- ing a level line. Such tubes are made and nearly filled with ether or alcohol, leaving a small space or bubble. When such a tube is placed convex side uppermost, the bubble seeks the highest point. Then a vertical line passing through the center of the bubble will coincide with the radius of the arc to which the tube is ground. A perpendicular to this vertical line is a line of apparent level. Such a tube is the most essential part of the level. It is encased in a brass tube, having an opening so that the bubble and as much of the glass tube as necessary can be seen. A graduated scale is attached to it, or marked on the tube, by means of which the bubble is measured and its position with relation to other parts of the instrument is determined. The tube thus prepared is attached to a telescope in such a manner that it can be adjusted so as to bring the radius of the ground glass perpendicular to the line of sight in the telescope.' The telescope is mounted in such a manner as to permit it to revolve freely in a horizontal plane and to be readily adjusted to the line of apparent level. A MANUAL OF LAND SURVEYING. FIG. 77. The plan of mounting the telescope most in favor in the United States is by a horizontal bar with forked arms called wyes. The telescope rests upon the wyes and is held in place by clips which may be loosened, per- mitting the telescope to be rolled over in the wyes. The bar is connected by a spindle to the tripod socket and leveling head similar to that used upon the transit. By permission of Messrs. Buff & Berger, of Boston, the following quotation is taken from their catalogue: 5. " The Adjustments. In a theoretically perfect level the following points are established: 1. The object and eye-glasses are perpendicular to the optical axis at all distances apart. 2. The optical axis coincides with the axis of rotation in the wyes. 3. The axis of collimation coincides with the optical axis. 4. The axis of collimation is parallel to the telescope level. 5. The collars resting in the wyes are circles of the same diameter and concentric with the line of collima- tion of the telescope. 6. The wyes are exactly similar, and similarly placed with reference to the line of collimation of the telescope. 7. The level bubble moves over equal spaces for equal displacements of the telescope in altitude. LEVELING. 399 8. The level bubble expands or contracts equally from the centre in both directions, during changes of tempera- ture. 9. The vertical axis of revolution is perpendicular to the line of collimation of the telescope. Of the above, the maker establishes points numbered 1, 2, 5, 6, 7 and 8. The remaining points, 3, 4 and 9, are established when the instrument leaves the- shop, but being liable to derangement from rough usage, they are made adjustable in the field. Adjusting. After the engineer has set up the instru- ment and adjusted the eye-piece for parallax, the hor- izontal cross-line had better be made to lie in the plane of the azimuthal rotation of the instrument. This may be accomplished by rotating the reticule, after loosening the capstan-headed screws, until a point remains bisected throughout the length of the line when the telescope is moved in azimuth. In making this adjustment, the level tube is to be kept directly beneath the telescope tube. When made, the small set screw attached to one of the wyes may be set so that by simply bringing the project- ing pin from the telescope against it, the cross-lines will be respectively parallel and perpendicular to the motion of the telescope in azimuth. The first collimating of the instrument may be made, using an edge of some building, or any profile which is vertical. Make the vertical cross-line tangent to any such profile, and then turn the telescope half-way round in its wyes. If the vertical cross-line is still tangent to the edge selected, the vertical cross-line is collimated. Select some horizontal line, and cause the horizontal cross-line to be brought tangent to it. Again rotate the telescope half way round in its wyes, and if the hori- zontal cross-line is still tangent to the edge selected, the horizontal cross-line is collimated. Having adjusted the two wires separately in this man- ner, select some well denned point which the cross-lines are made to bi-sect. Now rotate the telescope half way 400 A MANUAL OF LAND SURVEYING. round in its wyes. If the point is still bi-sected, the tel- escope is collimated. A very excellent mark to use is the intersection of the cross-lines of a transit instrument. Centre the eye-piece by the four capstan-headed screws nearest the eye end. This is done by moving the opposite screws in the same direction until a distant object under observation is without the appearance of a raise or fall throughout an entire rotation of the telescope in its wyes. The telescope is now adjusted. Next, bring the level bar over two of the leveling screws, focus the telescope upon some object about 300 :feet distant, and put on the sun-shade. These precau- tions are necessary to a nice adjustment of the level tube. Throw open the two arms which hold the telescope down in its wyes, and carefully level the instrument over the two level screws parallel to the telescope. Lift the tele- scope out of its wyes, turn it end for end and carefully replace it. If the level tube is adjusted, the level will indicate the same reading as before. If it does not, cor- rect half the deviation by the two leveling screws and the remainder by moving the level tube vertically by means of the two cylinder nuts which secure the level tube to the telescope tube at its eye-piece end. Loosen the upper nut with an adjusting pin, and then raise or lower the lower nut as the case requires, and finally clamp that end of the level tube by bringing home the upper nut. This adjustment may require several repetitions before it is perfect. The level is now to be adjusted so that its axis may be parallel to the axis of the telescope. Rotate the telescope about 20 in its wyes, and note whether the level bubble has the same reading as when the bubble was under the telescope. If it has, this adjustment is made. If it has not the same reading, move the end of the level tube nearest the object-glass in a horizontal direction, when the telescope is in its proper position, by means of the two small capstan-headed screws which secure that end of the level to the telescope tube. If the level bubble goes LEVELING. 401 to the object-glass end when that end is to the engineer's right hand, upon rotating the telescope level toward him, then these screws are to be turned in the direction of a left-handed screw, as the engineer sees them, and vice versa. Having completed this adjustment, the level bar itself must now be made parallel to the axis of the level. To do this, level the instrument carefully over two of its leveling screws, the other two being set as nearly level as may be; turn the instrument 180 in azimuth, and if the level indicates the same inclination, the level bar is adjusted. If the level bubble indicates a change of incli- nation of the telescope in turning 180, correct half the amount of the change by the two level screws, and the remainder by the two capstan-headed nuts at the end of the level bar, which is to the engineer's left hand when he can read the firm's name. Turn both nuts in the same direction, an equal part of a revolution, starting that nut first which is in the direction of the desired movement of the level bar. Many engineers consider this adjustment of little importance, preferring to bring the level bubble in the middle of its tube at each sight by means of the leveling screws alone, rather than to give any considera- tion to this adjustment, should it require to be made." 6. Leveling rods are made in a variety of styles and are of two principal classes, viz. : target rods and speak- ing or self reading rods. Target rods are made, of hard wood in two or more parts, which are grooved and tongued to slide upon each other, by which means they are lengthened out to 12 or more feet. They are graduated to feet, tenths and hun- dredths, the decimal notation being more convenient for computation than the division into inches and fractions of an inch. The target is a disc of brass made to slide up and down on the rod and to be clamped fast to the rod at any desired place. It is divided into quadrants painted alternately white and red. When used in level- ing the target is moved up and down on the rod until the horizontal line between these divisions is brought to coin- cide with the line of sight in the level. The target has a 402 A MANUAL OF LAND SURVEYING. vernier attached by which the distance on the rod is read to the nearest Tc ^ 7 part of a foot. In common leveling it is a useless refinement to carry the reading to thousandths of a foot, as it is out of harmony with the other conditions of the rod and the work to be done. The target on the rod, as a rule, is not capable of being set as closely and accurately to the level line as the ver- nier will read, nor will the rod be held so truly plumb as to justify so close a reading. Generally the line be- tween the quadrants of the target is not perpendicular to the rod and does not coincide with the zero of the vernier within several thousandths. Speaking rods are plain, straight rods, having the gradua- tions marked on them so boldly and distinctly that they can be read from the instrument. No targets are used with them, although some rods, like the Philadelphia rod, are made so as to be used either as target or speak- ing rods. There are many devices for marking the speaking rod, all of which are intended to facilitate ac- curate reading by the observer. A simple form of graduation and let- tering which gives excellent results in actual service is shown on a re- duced scale in the cut. The gradu- ations are to tenths and half tenths of a foot. Distances less than half a tenth are estimated by the eye. This is facilitated by having the figures for tenths made either .04 or .06 feet in length and accurately spaced on the rod. The student having a level and a rod for use in practice may now solve the following problems in the FIG. 78. fie i d: I LEVELING. 403 7. Prob. I. To find the difference of level of two points. CASE l.When the difference of level may be found by one setting of the instrument. FIG. 7t). Suppose A and B to be the points. Set up the level at a point about equidistant from A and B, though not necessarily in a line between them. Plant it firmly on the ground, placing the legs so as to bring the instrument nearly level, leaving as little as possible to be done with the leveling screws. If set up on yielding ground con- stant care will be required to be sure that the instrument is level at the instant the observation is taken. When the level is set up on ice or frozen ground, the legs will settle into the frost. It is well to set the instrument in the shade whenever convenient, as the rays of the sun, a passing cloud or a sudden breeze will throw the instru- ment out of level by causing unequal expansion and con- traction of the metal. In .precise leveling the instrument must be shaded. Having the instrument firmly planted, bring the telescope in line with one pair of the leveling screws and turn them in or out till the bubble is brought to the middle. Then bring the level in line with the other pair and again level it. Repeat until the bubble will remain in the middle of the tube through an entire revo- lution of the telescope around the spindle. The rod-man holds the rod at A, and its reading, Aa is taken. This is called a Back Sight. All observa- tions on other points taken at the same setting of the instrument are termed Fore Sights. The distance Aa shows how much the line of collimation of the level is above the point A and is called the height of instru- ment. The rod-man now holds the rod on the point B and its reading is taken. The difference between the 404 A MANUAL OF LAND SURVEYING. back sight and the fore sight is the difference in height of the points A and B. If the back sight is 9.20 and the fore sight 6.40, then B is 2.80 higher than A. If the fore sight were 11.45 instead of 6.40, then B is 2.25 lower than A. The rod-man should stand square behind his rod and hold it plumb. Sometimes small levels are attached to the rod to plumb it by. If they are not used the leveler when necessary directs the rod-man to move the top of the rod to the right or left to plumb it that way, and the rod-man also moves it gently back and forth towards the level, until the smallest reading of the rod is obtained. It is manifest that as many points may be taken as can be reached from the instrument and that their relative heights will be shown by the distances they are below the horizontal plane of the instrument, which is told by the readings on the rod. CASE 2. When the difference of level can not be found by one setting of the instrument. Suppose A and E to be the points, and that it is necessary to set the instrument four times to find the difference between them. We find by the first setting the difference between the points A and B, as already described. We then go forward and find successively the differences between the points B and C, C and D, and D-and E. The algebraic sum of these differences is the difference in height of the points A and E. A convenient form of field notes in cases like the above consists of three columns as shown in the following Example. Required the difference of level between the points A and E from the accompanying notes: Sta. Back Sights. Fore Sights. A 3.28 B 2.14 7.15 C 3.25 8.50 D 4.70 3.45 E 2.75 Which point is the higher, and how much? LEVELING. 405 A Bench Mark or Bench is-a fixed point used for reference in finding the heights of other points. It is indicated in the notes by the letters B. M. It is customary to establish bench marks at convenient distances along a line of levels by which the work may be reviewed, or at which it may be resumed after temporary cessation. The most convenient permanent objects are selected for the location of these bench marks, such as foundation stones in buildings, rocks or large boulders, or shoulders cut in the roots of large tree's, so situated that the rod can be set up on them and the level readily taken. Where a line of levels is run taking a number of points it is customary to refer the heights to an assumed level plane called a datum. This is generally assumed to be far enough below the first or principal bench mark so that it shall be below the lowest station likely to be found in any part of the survey for which it is used. Negative heights are thus avoided. A line of levels is usually marked by stakes set at uni- form distances apart, marked and numbered consecu- tively from zero upwards. 100 feet is the distance most usually adopted between stations, although in levels for country drains it is sometimes found more convenient to space the stations by chains to correspond with the measures of the land surveys. Intermediate stakes are usually referred to as plus stations, and are so marked on the stakes and in the notes. For instance, a stake set between stakes No. 6 and 7 at 40 feet from No. 6, is marked 6 -f 40 or simply -f 40. 8. Prob. 2. To find the heights above a datum plane, of several stations on a given line. SUGGESTIONS. Let AB (Fig. 80, page 406) be the given line and DP the datum plane assumed at any con- venient distance, say 10 ft., below a bench near A. Set up the level at some convenient point, for example between stations 2 and 3. Take the reading of the rod upon the bench and add it 406 A MANUAL OF LAND SURVEYING. FIG. 80. to the assumed height of the bench above the datum. The sum is the height of the instrument. Take the readings upon stations 0, 1, 2, 3, 4 and 5 in succession, and subtract each from the height of the instrument. The remainders are the heights, respect- ively, of those stations above the datum. Carry the instrument forward to another position, as between stations 6 and 7. Take the reading of the rod a second time on station 5, and add it to the height of station 5 as before found. The sum is the new height of instrument, with which proceed as before. A point used as station 5, as above indicated, is called a Turning Point. In practice, a bench is often adopted as a turning point. The reading of the rod upon a turning point or bench- mark is usually taken with somewhat greater precision than upon other points. A reading upon a bench or turning point -is added to the height of the point above the datum in finding height of instrument; and a reading upon any point is sub- tracted from the height of instrument in finding the height of the point. Accordingly, an observation for the former is called a Plus Sight, denoted by -j-S, and for the latter, a Minus Sight, denoted by S. LEVELING. 407 The height of instrument is denoted by H. In., and the height of any point above the datum, by H. or elev. The following is an example of the notes made in solv- ing the above problem : Sta. + s. H. In. S. H. Remarks. B.M. 3.426 13.426 10.000 A stone 20 ft. S. E.of 0. 5.45 7.976 I 7.30 6.126 2 f 5.35 8.08 3 5.40 8.03 4 6,23 7.20 5 8.274 3.76 9.666 6 17.040 5.25 1 2.69 7 5.10 12.84 8 5.00 3 2.1)4 9. Prob. 3. To find the cut or fill, to grade, at points between two given points. SUGGESTIONS. Let A and B (Fig. 80) denote the given points. Beginning at A, for example, measure the distance AE, at the same time marking it off into con- venient divisions of equal length, as 33 ft., 50 ft., 66 ft., or 100 ft., for example, by driving pegs down to the sur- face of the ground. The last division will ^usually be frac- tional. Number the divisions, 0, 1, 2, 3, etc., beginning at A. Find now (Prob. 2) the heights of the points, 0, 1, 2, 3, etc., above some convenient datum. For illustration, suppose the heights to be as given in the above Table (Prob. 2). Also suppose the height of the grade line at A to be 5 ft., and at B, 9 ft. TJbe distance from A to B consisting of 8 equal parts, say of 50 ft., we should then have (9 ft. - 5 ft.) -j- 8 = 0.5 ft. =-rise per station. 408 A MANUAL OF LAND SURVEYING. Beginning at A or station 0, we have 7.98 _ 5. = 2.98 = cut at 6.13 5.50 = 0.63 = cut at 1 8.08 6.00 = 2.08 = cut at 2 8.03 6.50 = 1.53 = cut at 3 6.20 7.00 = 0.80 = fill at 4 9.67 7.50 = 2.17 = cut at 5 etc. etc. etc. Observe that we take the difference in height between the grade* line and the station at each station ; and since we have here proceeded from lower to higher points of the grade, we have added the rise of the grade per station to the height of the grade at the last preceding station. Let the student find the cut at each station, beginning at B, all other things being as above. Again, supposing the heights of the stations to be as above, let the student find the depths of cut and fill under the supposition that the height of the grade at A is 6 ft., and at B, 8.4 ft. 10. Drawing Profile. Fig. 80 represents a section formed by a vertical plane passing through the points A and B, and meeting the datum plane in the line DP. The irregular line AB represents the intersection of the vertical plane with the surface of the ground, and is called the Profile. The manner of drawing the profile is as follows : Draw a horizontal line to represent the datum line, on which lay off to a convenient scale the distance between the stations. At the points of division of the datum line, erect per- pendiculars, on which lay off the surface heights of the several stations, in their order, but to a scale usually ten times greater than that used for the horizontal distances. A line drawn through the points thus located forms the profile. The use of a larger scale in drawing the vertical dis- tances serves to render the irregularities of the surface DRAINAGE SURVEYING. 409 more apparent to the eye than they would be if drawn to the same scale with the horizontal measurements. The grade line is drawn through any two points at the proper distances from the datum line. The position and inclination of the grade line depend upon certain condi- tions required to be fulfilled by the work, such as the flowage of water, ease of travel, economy of construc- tion, etc. In road work the grade is often adopted with reference to an equalization of " cut " and " fill," so that the mate- rial furnished by excavations shall make the embank- ments. The required position of the grade line, in order to fulfill this condition most advantageously, is conven- iently got by stretching a thread across the profile, vary- ing the position of the thread until the areas intercepted by it and the profile on opposite sides appear to be equal. EXERCISES. ii. 1. Find depths of cut or fill, and draw profile and grade line from the following notes: Sta. +s. H. In. S. H. H. Gr. Cut. Fill. 1 2 3 3 4 5 S 25 4.26 4.12 14.26 15.15 .30 8.45 3.23 8.20 4.63 5.53 5.75 10.00 8.00 0.575 Distance between stations, 100 ft. 25. Examples made by the student in the " Field." II. DRAINAGE SURVEYING. 12. Of the many applications of leveling, the most common, perhaps, in the province of the ordinary sur- veyor, is that relating to drainage. Almost every neigh- borhood offers occasions for work of this kind. 13. Drains are of two forms: the Open Drain or Ditch, and the Under Driin. The former is adapted to the case of water lying upon 410 A MANUAL OF LAND SURVEYING. the surface of the ground, and the latter to water under- lying the surface. Under drains 'are usually discharged into open drains, which are thus rendered an essential auxiliary to thorough drainage. 14. Making the Survey. This will be, in the first place, a careful reconnoissance of the locality respecting the general " lay of the land," natural water courses, etc. In this will be determined the proper commencement, route and terminus of the drain. The term commence- ment will be here understood to mean the upper end of the drain, and terminus the outlet. The word commence- ment in connection with open drains will also be taken as significant of the proper place to begin the survey. Preliminaries having been settled, a stake marked is driven at the point of commencement, and the survey, proper, begins by setting the transit over the stake and taking the bearings and distances of two convenient ob- jects near by as witnesses of the point of commencement. The location of the commencement should be described also by distances and direction from some neighboring monument or line of original survey. Thus, 10 ch. E. and 7.15 ch. N. of V 4 post bet. Sees. 11 and 14, T. 2 N. R. 5E. These items are to be entered in the column of remarks in the Transit book, opposite the station 0. The instrument is then turned upon the first angle in the line of the drain and its bearing entered in the col- umn of bearings opposite station 0. Ax-men are required in clearing away bushes, making and driving stakes, etc. Two chain-men, the forward one carrying a transit-rod, now begin to measure at in the direction of the first angle, and stakes marked 1, 2, 3, etc., are driven at uniform distances from each other. A 100-ft. tape is a convenient measure, and locates the stations at ordinarily suitable distances. A stake should be set also at each angle of the drain, and its distance from the last preceding station entered in the notes. The points of meeting of any land-lines, roads, etc., should be noted by distances in a similar manner. DRAINAGE SURVEYING. 411 The number of acres in farms whose lines are met may, very properly, be made a matter of memorandum. The following is a specimen of the form of notes which are taken, in accordance with the above suggestions : TRANSIT NOTES. Sta. Bearing Distance, of Course Remarks. 1 2 S. 70 E. 0. A point 10 ch. E. and 7.15 ch. N. of VA post on line bet. Sees. 11 and 14, T 2 N., K 5 E. W. Oak 15, N. 23^, E. t 57 ft.; Hickory 12, S. 40 E., 34 ft. 3 4 M Land owned by John Doe, 80 A. ; about 6 A. wet. 5 " 5" S. 28%E. 528 ft. 5. 1st Angle. G i 7 " 8 8 <> 8 40 . Line bet. Sees. 13 and 24. 9 10 11 K B. Onk 10, S. 35Vi W., 10 ft. ; W. Oak 18, N. 63 W., 28 ft. Richard Rowe, 160 A. on south, 30 A. swamp. II 80 East. 652 ft. 11". 2d Angle. 12 " 13 " . H v' ; il 23 u. 23 M 1163 ft. 23". Terminus in drain by road side on Township line. Marked Boulder, N. 20 E., 15 ft. Ash 14, S. 27 W., 10 ft. 412 A MANUAL OF LAND SURVEYING. 15. Taking the Levels. The line of the drain hav- ing been established, the next thing is to take the levels. This is done in the manner previously described. Beside the engineer or principal surveyor, two men are required a rod-man, and an ax-man to make and drive pegs. The pegs should be driven down even with the surface of the ground and at such a distance from the stakes marking the stations that they may be used without dis- turbance in excavating. Some practice driving them, say six inches, in front of the stakes ; other set them opposite and at such a uniform distance from the record stakes as not to be disturbed by the digging. Bench marks should be made at convenient distances, for example, at every tenth station, and far enough from the line not to be disturbed. 1 6. Platting. The field work having been completed, the next thing is to make a plat of the line and also of the sections or tracts of land which will be affected by the drain, writing the owner's name and number of acres on each. On some convenient part of the plat, the courses and their corresponding distances should be noted, also the number of linear feet of drain on each separate tract. Next comes the drawing of the profile. This is most conveniently done by use of paper, called Profile paper, prepared specially for the purpose. Taking a piece of the proper width and of sufficient length to contain also the title and necessary explanatory notes, at the left hand, we begin on the edge next to us and write the num- bers of all the stations in their order toward the right, upon' the vertical lines. We then mark with the point of a sharp pencil the point of elevation of each station as taken from the column of elevations in the level notes. Connecting the points thus marked,, by an ink line, we have the profile of the surface of the ground on the line DRAINAGE SURVEYING. 413 of the drain. We then take a black thread and stretch it on the profile between the points assumed as grade, at the first and the last station. From this inspection, it will be seen whether it is necessary or desirable to introduce one or more changes of grade between the extreme points in order to avoid objectionable cuts. Having determined the situation of the grade lines, we then draw them in their places, preferably with red ink. Under the grade lines and upon the vertical lines of the several stations should be written in red ink the ele- vations of the grade, and below that, in black ink, the elevations of the surface. In a similar manner, above the profile may be written first, in red ink, the depths of the cuts, and, second, the widths of the ditch at bottom and top. The names of the land owners through whose land the ditch passes, with the number of linear feet on each, may be conveniently written upon the datum line. 17. The writer has saved himself and assistants a great many miles of tramping and wading through swamps and morasses in drainage surveys by running the transit and level lines for the drains both at one operation. It was found by repeated tests on long lines that the level on the transit gave very nearly if not quite as accurate results in leveling as the wye level. Hence the wye level was left at home and the transit line and levels were both run at the same time with the transit. A condensed form of keeping the notes was used. All the rod readings are kept in one column. The back or plus sights, to be added to the elevation for height of instrument, are marked " B. S." The others are all to be' subtracted from " Ht. Inst." for elevation of stations. The following is a sample extract : Commencing at a point in the Section line 4.53 chains east of the quarter post between Sections 11 and 14, and running thence S. 16 W. Stations 2.00 chains apart. 414 A MANUAL OF LAND SURVEYING. Sta. Obs. Ht. Inst. Elev. Grade Ht. Cut Remarks. B.S.on B.M. 4.96 104.96 100.00 On Elm 40' to rt. of Sta. 1. 1 5.21 5.30 99.75 99.66 96.00 95.90 3.75 3.76 Elm and Black Ash. 2 3 5.28 5.46 99.68 99.50 95.80 95.70 3.88 3.80 +50, enter thick Willows. 4 5.72 99.24 95.60 3.64 5 5.83 99.13 95.50 3.63 -f GO Angle rt. 12 24'= S. 28 24' W. Cross line fence be- tween Smith and Jones. C 5.84 99.12 B.S. 2.91 102.03 6 2.95 99.08 95.40 3.68 Open marsh. Saw grass. 7 3.06 98.97 95.30 3.67 1 8. Depth and Width. The depth of a drain obvi- ously depends upon the situation of the grade line with respect to the surface. In adjusting the grade line it is more important to guard against the drain being too shallow rather than too deep; most open drains are too shallow. Again, it should be taken into account, if the drain is to run through soft marshes and hard ridges, that the soft ground, on the withdrawal of the water, will settle; ard so the drain may need to be dug deeper in some places than would otherwise be necessary. The necessary width of a drain of given depth and grade depends upon the quantity of water it is required to discharge in a given time. The width at the top is determined from the width at the bottom and the slope or inclination given the sides, DRAINAGE SURVEYING. 415 which is usually from one to one and one-half feet on the horizontal to each foot in depth. 19. Quantity of Discharge. The amount of water which a drain may discharge in a given time obviously depends upon the area of the water-way or cross-section of the drain and the velocity of the stream. Thus, denoting by Q the quantity of discharge, by a the area of the water-way, and by v the mean velocity of discharge, we should have Q=av (1) As an approximate formula for computing the mean velocity of water flowing in an open canal of uniform cross-section and fall, Trautwine gives the formula i a/* X 8975 ) % - I -.1089 (2) in which V= mean velocity in feet per second, a = area of water-way in square feet, f=fall in feet per foot, and p wet perimeter or the water border of the channel. REMARK. In applying the above formula, It is customary to use 9000 for 8975 and .11 for .1089. Example. Required the velocity and the capacity of a drain 5 ft. wide at the bottom, the sides having a slope of 1 to 1, depth of water 3 ft., and the fall 2 ft. to 1,000 ft. Solution.^- Width at top = 5 ft. + 2 X 3 ft. = 11 ft. Area of water-way = 1% (11 ft. + 5 f t. ) = 24 sq. ft. Wet perimeter = 5 ft. -f 6\/2 ft. = 13.5 ft. Fall per foot = 0.002 ft. ( 24X0.002X9000 \ % Substituting in (2),7=<( V 0.11 ( 13.5 ) =5.55. Substituting in (1), # = 24X5.55 = 133.2 cu. ft per second, or 11,508,480 cu. ft. per day. 416 A MANUAL OF LAND SURVEYING. Trautwine gives also the following formula, with the remark that it is applicable also to sewers : in which a and p are as above described, and F is the fall in feet per mile. REMARK. In connection with the above formulas, as well as with others of similar import, Trautwine re'peats again and again the caution that they are to be regarded only as approx- imately true. Table XII shows approximately the number of acres served by drains having bottom widths of 1 to 10 ft., with side slopes of 1 to 1, and various rates of fall per station, on the supposition of 1 inch rainfall in 24 hours, one- half of which reaches the drain. 20. Amount of Rainfall. All calculations of requisite capacity of drains must be based upon the probable amount or number of inches of rainfall in a given time. The soil, however, acts as a reservoir up to the point of saturation, depending upon its texture, keeping from the drains altogether a portion of the rainfall, which passes off by evaporation or is absorbed by plants. The average annual rainfall in Michigan, Indiana, Illinois and Missouri is about 35 inches. In Ohio, for a period of ten years, it was reported to be 37.86 inches. In the matter of rainfall in Michigan, we are indebted to Prof. Carpenter for the following data : " By a consultation of the meteorological records of the Agri- cultural College we learn that, although large showers in which the rainfall exceeds one inch occur comparatively seldom (on the average only four times a year), yet they bring with them twen- ty-eight per cent of our total rainfall during that period, and consequently they must be fully provided for in any works for thorough drainage. The following table is compiled from the meteorological records kept at the college, and shows the com- parative depth and number of showers from the months of March to December for five years. The last column shows the total per- centage of rainfall in all the showers of a given depth. The last DRAINAGE SURVEYING. 417 column but one shows the total percentage of the number of showers compared with the whole number. Although this table is not extended sufficiently far back to give very accurate results, it is thought (since one year's rainfall does not differ greatly from that of another year) to be sufficiently reliable to produce data for any ordinary case of farm drainage in this part of the United States. TABLE OF SHOWERS FROM MARCH TO DECEMBER. Depth of Rain- fall in Inches. Number of Showers. Percentage of Total. 1872 1873 1874 1875 187C Total No. of Showers. Am't of Rainfall. .00 to .25 . .25 to .50 .50 to .75 .75 to 1.00 1.00 to 1.25 1.25 to 1.50 1.50 to 1.75 1.75 to 2.00 2.00 to 2.25 2.25 to 2 50 19 20 C 2 40 14 8 G 28 13 6 5 33 9 10 2 43 11 5 3 2 3 105 67 35 18 2 8 3 2 o~ 54.2 22.0 11.5 OC.O 00.7 02.C 01.0 00.7 OOJ3 00.3 ~~66.Y~ 17 21 21 13 2 9 4 3 2 2 .... 3 1 2 -- 1 2.50 to 2.75 2.75 to 3.00 3.00 to 3.25 "-- -- 1 Totals 304 100.00 100 "The amount of discharge of drains as compared with the rainfall is usually estimated at about 50 per cent. So that in order to produce thorough drainage it is necessary to assume that the capacity of the drains shall be sufficient to carry off during twenty-four hours one-half the water that fell the pre- vious twenty-four hours. The probability of the rainfall in any day exceeding one inch is so slight that we shall be safe in assuming as the necessary carrying capacity of drains one-half of 3,630 cu. ft., or 1,815 cu. ft. of water for each acre drained." 21. Under Drains are formed in various ways; sometimes of brush, rails or loose stone trenched in, sometimes of tubes made of logs or of iron, sometimes of plank or of brick or stone laid in cement, and again of earthen tubes, of which there are various forms, called Tiles. The prevailing method of under-drainage for agricul- tural purposes consists in the use of cylindrical tiles, which are made of different sizes and usually about a foot in length. 418 A MANUAL OF LAND SURVEYING. It is of this form of under drain, only, that we propose to write briefly. 22. Surveying for Under Drains. Very much of what has been said upon surveying for the ditch or open drain applies also to the tile drain. The same pre- liminary inspection is required to determine the best location of the outlet and the proper directions of trunk and branch lines. Indications as to source of water, whether from springs on the premises or on lands sit- uated above, whether from rainfall, merely, upon the particular tract or also as flowing off from neighboring areas; the directions of slopes, whether of surface or of underlying strata; the character of the soil, etc., all have to be carefully observed and their bearing duly con- sidered. 23. Location of Drains. As above intimated, any well-conducted survey for under drains contemplates the execution of a system of drains working together and depending upon each other. This will include usually a principal drain, called a Main, and lateral drains, called Minors, which discharge into the main. In ah extended system, auxiliary mains called Sub-Mains are also introduced. Since it is the direct office of the minors to remove the surplus water from the ground, it is of the first impor- tance that they be so located as successfully to perform their functions. To do this requires the exercise of care- ful judgment on the part of the engineer, respecting the proper directions of the minors and also their distances from each other. Equal care is requisite also in regard to the location of the main, so as properly to receive the water from the minors and discharge it at the principal outlet. As a rule, the main should be located at the foot of the regular slopes, or along the valleys of the field; and, in DRAINAGE SURVEYING. 419 general, the minors should run directly down the slopes, discharging themselves obliquely into the main. Cases, however, will sometimes occur that require de- parture from the above rules, but these are to be regarded as " exceptions which prove the rule." The distances of the minors from each other will be governed largely by the character of the soil as to per- meability, and to some extent by the depth of the drains. In a porous soil, as a general rule, the deeper the drain the further it will draw. Circumstances are infinitely varied. Every situation is a new one and must be treated on its own merits. None but the most general instruction on this point can be given In any treatise. About as practical a suggestion as may be afforded the student is, Go into the field and there mix plenty of brains with your work. 24. Running the Lines. Having settled the ques- tion of the proper system of drains to be adopted, the next thing to be done is to lay out and measure the lines. This is perhaps most conveniently done in the case of under drains, by beginning at the outlet, measuring and staking out, first, the main lines of the system and then the branches. A distance of 50 ft. between stations is a convenient one in tile draining. In some instances, as where the fall is very slight, a less distance may be desirable ; in others a -greater one may give equally good results. In addition to the stakes driven at the uniform distances of the sta- tions, a stake should mark the entrance of each minor, and the distance to it should be entered in the notes, in the usual manner. Such stakes mark the points of beginning in running out the minors. To facilitate examinations for " faults," the points of entrance of the branches in the main drain should be established by witnesses. 420 A MANUAL OF LAND SURVEYING. 25. Taking the Levels. This is done in the same manner as in the case of open drains, but, perhaps, with a somewhat greater degree of care and precision. The point assumed for the outlet must, of course, be suf- ficiently low to receive all the water of the field; and at the same time the outlet ought to be high enough to be at all times above the back water of the stream into which the drain empties. A drain is of little more use under a violation of the latter condition than under a disregard of the former. In assuming the grade, due consideration must be had for proper depth consistently with required fall. The depth of an under drain should be, at the least, two feet; all the better if three or four feet in most soils. Henry F. French, author of " Farm Drainage," says : " We cannot, however, against the overwhelming weight of authority, and against the reasons for deeper drain- age, which to us seem so satisfactory, conclude that even three feet is, in general, deep enough for under drains. Three-foot drains will produce striking results on almost any wet lands, but four-foot drains will be more secure and durable, will give wider feeding-ground to the roots, better filter percolating water, warm and dry the land earlier in Spring, furnish a larger reservoir for heavy rains, and, indeed, more effectually perform every office of drains." Accordingly, the rule should be to approximate as closely as possible to what are thus regarded as desirable depths, admitting depths very much below the standard only when we must, in order to have any drains at all. Upon the question of necessary amount of fall, with which the surveyor is so often confronted in connection with the requirement of desirable depths, it is to be ob served in the first place that large, deep streams require less fall than small ones; and, again, the form and the DRAINAGE SURVEYING. 421 condition of the channel have much to do with the move- ment of water. " It has been found in practice that a water-course thirty feet wide and six feet deep will flow at the rate of one mile per hour, with a fall of no more than six inches per mile." Examples are cited of successful operation of drains with three inches or even two and one-half inches fall to one hundred feet, or even on a dead level. The contour of the ground will determine the grade of the drain, which will be given all the fall there is. A level drain will work successfully provided it is laid to a true line and kept free from obstructions. The writer successfully lowered two lakes, covering about 120 acres, by an open ditch two feet wide on the bottom, a mile long, and laid perfectly level. The less fall there is, the larger should be the tile used and the greater should be the care taken in laying them to a true grade line and keeping out leaves or other obstructions. Changes of grade, though undesirable, are admissible when not easily avoided. If possible, the heaviest grades should be in the direction of the outlet. When this can not be, it may be desirable to introduce silt wells at points of any considerable change of grade. The heights of the outlets of minor drains into the main are usually the heights of grade in the main drain for the same points. 26. Constructing the -Drain. The principal point is the method of opening the trench and laying the tiles on the grade line. To do this systematically requires a measuring rod six or eight feet in length divided into feet, tenths, and hun- dredths of feet, the larger divisions being numbered up- ward, as in the ordinary leveling rod. A cord or wire. 422 A MANUAL OF LAND SURVEYING. also is needed, which is to be stretched above the line of the drain and adjusted to a position parallel, to the grade line. This is done by inverting the measuring rod on the grade peg and bringing the cord or wire to the division of the rod indicating the cut at that point. The cord is thus placed at the full length of the measuring rod from the grade line or intended bottom of the trench. The cord may be held each fifty or one hundred feet by two slats, each about seven feet long, and movable about a bolt passing through a little distance from the upper end. These are called Shears. The cord or wire is pre- vented from slipping by a couple of turns, and is tied to a stake eight or ten feet from the shears. Another device consists in the use of stakes or posts driven on opposite sides of the ditch, and connected with a cross-bar arranged so that either end may be raised or lowered to a level, and fastened to the posts by a clamp and thumb-screw. The cross-bars being adjusted to the proper height, as above described, the cord or wire is drawn tightly across them, directly over the center line of the drain. Again, single stakes or posts, driven on one side of the ditch, each having attached at right angles an arm which may be raised or lowered, and secured in place by a clamp and screw, are sometimes employed. By such means as the above, the ditch is readily dug to just the proper depth, and the tile laid to grade with ex- ceeding accuracy and with great rapidity. The proper distance from the top of the tile to the cord may be indi- cated by an arm attached to the measuring rod. 27. Size of Tile. The size of tile required in a given case will depend upon the quantity of water to be re- moved and the fall available to remove it. Formulas are given in works upon hydraulics, to express the veloci- ty and discharge of water flowing in pipes, but the condi- DRAINAGE SURVEYING. " 423 tions are so different in case of tiles that such formulas, at best, give only the most roughly approximate results. Thus, for example, the following, which is Poncelet's formula : F=48 L + 54D in which, F= approximate velocity in feet per second, D = diameter of pipe in feet, H = total head in feet, and L = total length of pipe in feet. Having found the velocity, we have Discharge in cu. ft. = vel. X cross-section of pipe. Tables XII and XIII are used for the above purpose, the latter quite extensively by drainage engineers and has been found to give good results. As regards size of tile for main and sub-main drains a good authority says, " that can be regulated only by the person in charge of the drainage at any particular place, after seeing the land opened up and the minor drains dis- charging. As a rule, a circular pipe of three inches inter- nal diameter will discharge the ordinary drainage of six statute acres, and give sufficient space for the circulation of the air." This estimate is based upon an amount of annual rain- fall of from twenty-six to thirty inches, which differs but slightly from -that of Michigan and adjoining States. In addition to the above, it may be remarked that if the fall in the main is slight, a larger size of tile would be required than if the fall was considerable. And, again in order to provide suitably for the accu- mulation of water which occurs toward the outlet, a larger size may be there required than that used in the upper part of the main. 424 A MANUAL OF LAND SURVEYING. 28. Protection at Outlets. The outlets of under- drains should be protected by some construction to pre- vent the earth from falling down in front of the drain. A retaining wall of masonry laid in hydraulic cement is the best provision for the purpose. The outlets should be protected also by a coarse grating of some sort in front of the tile to prevent muskrats and other creatures from getting in. A common practice is to introduce at the outlet a box made of plank a few feet in length, into which the tile is made to discharge. 29. Silt Well. This is a well sunk below the level of the tile for catching the silt gathered by the drains above it. It serves also the purpose of affording a means of inspecting the working of the drains. Silt wells may be constructed with a view, chiefly, to facilitating the movement of the water at an abrupt bend in the dram. And again, they may be constructed somewhat with ref- erence to convenience of obtaining a pail of water for any purpose, in the field. SURVEYORS' TABLES. SUGGESTIONS ON USE OF TABLES. TABLES. SUGGESTIONS TO YOUNG SURVEYORS ON THE USES OF THE TABLES. Traverse Table. The table calculated to quarter degrees is adapted to the simplest work of compass surveying, where great accuracy is neither required nor expected. When the transit is used, and the angles are taken to minutes or less, the author prefers the tables of logarithms and logarithmic sines and cosines to any traverse table yet made. They are capable of any re- quired degree of accuracy, and require the use of no more figures than the ordinary traverse table. In transit work, where latitudes and departures are to be calculated, it is well to refer the angles of all lines to a common base, just as in compass surveying all lines are referred to the meridian as a base. Then, in any course, Latitude = co-sine of angle X length of the course. Departure = sine of angle X length of the course. Using the logarithmic tables, this is a short and simple computation. Example 1. Angle, 36 22'. Distance, 47.63. Eequired the latitude and departure. Log. of 47.63 = 1.677881 to which add log. sine, 36 22' = 9.773018 11.450899 the log of 28.244- = departure. Log. of 47.63 = 1.677881 to which add log. cos., 36 22' = 9.905925 11.583806 the log. of 38.35+ = latitude. 2. Course N. 57 21' 20" E. 34.36^ chains. Required the latitude and departure. A MANUAL OF LAND SURVEYING. 1. The Table of Tangents is convenient in estimating courses of lines to be run. Example 1. From the quarter post on the east side of Section 2 I wish to run a line for a road straight to a point 80 rods north of the southwest corner of Section 30. What course shall I run ? Solution. Distance west, 5 miles; distance south, 4.25 miles, which divided by 5 equals the natural tangent of the angle which the course makes with an east and west line, = .850. Find this number in the table of natural tangents and take out the corresponding angle, = 40 22', which is the same as S. 49 38' W. 2. What is the course from the village of Climax, at the east quarter post of Section 3, Township 3 south, Range 9 west, to the village of Richland, at the southwest corner of Section 14, Township 1 south, Range 10 west? To the village of Schoolcraft, at the southeast corner of Section 19, T. 4 S., R. 11 W., from Climax? What to Schoolcraft from Richland? 2. The Table of Secants is convenient for finding the hypothenuse of a triangle, thus simplifying many computations in the field. Secants not given in the table may be found by interpolation or by the formula: 1 Secant = . cosine The following example indicates one of the practical applications in the field: Example. Lots in a city are laid out with their lines perpendicu- lar to N Street and running through to M Street. Required the width (x) of the lots on M Street. Call the width of the N.5t lots on N Street r. Measure the angle A. FIG. 8i. SUGGESTIONS ON USE OF TABLES. Ill Then x = r, sec. A. If r = 100, as is common, x may be taken directly from the table. If r = 100, A *= 21 4(X, then re = 107.6. In laying, out such lots it is generally easier and quicker to measure this distance on the street line than it is to set up the transit for each lot line and run it in. 3. Table of Departures. This table has many convenient uses, of which a few examples ar,e given. Examples. 1. I wish to stake out a line along an oid hedge row from quarter-post to section corner. On one side is a clear field. I go to the section corner, and make an offset of 25 links and set up a flag. 1 then go to the quarter-post, and, making an equal offset, find that I cannot see the flag; so I offset until I can see it say 37 links more. I sight to the flag, find from the table of departures the angle corresponding to 37 links at a dis- tance of 40 chains = 32'. turn off the angle on the transit, and run the line back parallel with the section line, setting stakes on the true line, by 62 link offsets, as often as required. 2. To run a true half-quarter-line when one end is inaccessible. Fig. 82 repre- sents the whole section, and ab the line to be run. Bisect eg, setting stake at a. Meas- ure the angle acd, which we will call 89 24'. By the field notes the north line of the section measures 80.22, hence . C= 180 (A + B). - sin A tan i (AB) = T tan Area = K= \ ab sin C. f< S) then sin i^=J^- 6 )^-). be tan /(s 6) (s c \ * (.v d) sin J. =r- v x s (s a) (s 6) (s c). 2 (. b} (s c). versin A = Area = be Area = (s a) (sb) (s c). a 2 sin B sin C TABLE I. LOGARITHMS OF NUMBERS. TABLES. LOGAEITHMS OF NTJMBEKS FROM 1 TO 10000. N. Log. N. Log. N. Log. N. Log. 1 0000000 26 414973 51 1 707570 76 1880814 2 301030 27 431364 52 1716003 77 1 886491 3 477121 28 447158 53 1 724276 78 1892095 4 06020t>0 29 462398 54 1 732394 79 1 897627 5 0698970 30 477121 55 1740363 80 1 903090 6 778151 31 491362 56 1 748188 81 1908485 7 0845098 32 505150 57 1755875 82 1 913814 8 0903090 33 518514 58 1763428 83 1 919078 9 09M243 34 531479 59 1770852 84 1 924279 10 1 000000 35 544068 60 1 778151 85 1929419 11 041393 36 556303 61 1 785330 86 1 934498 12 079181 37 568202 62 1792392 87 1 939519 13 113943 38 579784 63 1 799341 88 1 944483 14 146128 39 591065 64 1 806180 89 1949390 15 176091 40 602060 65 1 812913 90 1954243 16 20*120 41 612784 66 1 819544 91 1959041 17 230449 42 623249 67 1 826075 92 1 963788 18 255273 43 633468 68 1 832509 93 1 968483 19 278754 44 643453 69 1 838849 94 1 973128 20 301030 45 653213 70 1 845098 95 1 977724 21 322219 46 662758 71 1 851258 96 1982271 22 342423 47 672098 72 1 857332 97 1 986778 23 361728 48 681241 73 1 863323 98 1 991238 24 380211 49 690196 74 1 869232 99 1 995686 25 1 397940 50 698970 75 1 87o061 100 2000000 TABLE I. LOGARITHMS OF NUMBERS. No. o I 2 8 4 | 5 6 7 s 9 Diff. 100 000000 000434 000868 001301 001734 002166 002598 003 ft 29 003461 003891 432 1 4321 4751 5181 5609 6038 6466 1 6894 1 7321 7748 8174 428 2 8600 9026 9451 9876 010300 010724 011147011570 011993 012415 424 3 012837 013259 013680 014100 4521 4940 5360 5779 6197 6616 '419 4 7033 7451 7868 8284 8700 9116 9532 9947 020361 020775 416 5 021189 021603 022016 022428 022841 023252 023664 1024075 4486 4896 412 6 5306 5715 6125 6533 6942 7350 77571 8164 8571 8978 408 7 9384 9789 030195 030600 031004 031408 031812 032216 032619 033021 404 8 033424 033826 4227 4628 5029 5430 5830 6230 6629 7028 400 9 7426 7825 8223 8620 9017 9414 9811 04C207 040602 040998 396 110 041393 041787 042182 042576 042969 043362 043755 044148,044540 044932 393 1 5323 5714 6105 6495 6885 7275 7664 8053 8442 8830 389 2 9218 9606 9993 050380 050766 051153 051538 051924 052309 052694 386 3 053078 053463 053846 4230 4613 4996 5378 5760 6142 6524 382 4 6905 7286 7666 8046 8426 8805 9185 9563 9942 060320 379 5 060698 061075 061452 061829 062206 062582 062958 063333 063709 4083 376 6 4458 48321 52061 5580 5953 6326 6699 7071 7443 7815 373 7 8186 8557] 8928 9298 9668 070038 070407 070776 071145 071514 369 8 071882 072250 072617 072985 073352 3718 4085 4451 4816 5182 366 9 5547 5912 6276, 6640 7004 7368 7731 8094 8457 8819 363 120 079181 079543 079904 080266 080626 080987 081347 081707 082067 082426 360 1 082785 083144 083503 3861 4219 4576 4934 5291 5647 6004 357 2 6360 6716 7071 7426 7781 8136 8490 8845 9198 9552 355 3 9905 090258 090611 090963 091315 091667 092018 092370 092721 093071 351 4 093422 3772 4122 4471 4820 5169 5518 5866 6215 6562 349 5 6910 7257 7604 7951 8298 8644 8990 9335 9681 100026 346 6 100371 100715 101059 101403 101747 102091 102434 102777 103119 3462 343 7 3804 4146 4487 4828 5169 5510 5851 6191 6531 6871 341 8 7210 7549 7888 8227 8565 8903 9241 9579 9916 110253 338 9 110590 110926 111263 111599 111934 112270 112605 112940 113275 3609 335 130 113943 114277 114611 114944 115278 115611 115943 116276 116608 116940 333 1 7271 7603 7934 8265 8595 8926 9256 9586 9915 120245 330 2 120574 120903 121231 121560 121888 122216 122544 122871 123198 3525 328 3 3852 4178 4504 4830 5156 5481 5806 b!31 6456 6781 325 4 7105 7429 7753 8076 8399 8722 9045 9368 9690 130012 323 5130334 130655 130977 131298 131619 131939 132260 132580 132900 3219 321 6 3539 3858 4177 4496 4814 5133 5451 5769 6086 6403 318 7 6721 7037 7354 7671 7987 8303 8618 8934 9249 9564 315 8 9879 140194 140508 140822 141136 141450 141763 142076 142389 142702 314 9143015 3327 3639 3951 4263 4574 4885 5196 5507 5818 311 140 146128 146438 146748 147058 147367 147676 147985 148294 148603 148911 S09 1 9219 9527 9835 150142 150449 150756 151063 151370 151676 151982 307 2 152288 152594 152900 3205 3510 3815 4120 4424 4728 5032 305 3 5336 5640 5943 6246 6549 6852 7154 7457 7759 8061 303 4 8362 8664 8965 9266 9567 9868 160168 160469 160769 161068 301 5 161368 161667 161967 162266 162564 162863 3161 3460 3758 4055 299 6 4353 4650 4947 5244 5541 5838 6134 6430 6726 7022 297 7 7317 7613 7908 8203 8497 8792 9086 9380 9674 9968 295 8 170262 170555 170848 171141 171434 171726 172019 172311 172603 172895 293 9 3186 3478 3769 4060 4351 4641 4932 5222 5512 5802 291 150 176091 176381 176670 176959 177248 177536 177825 178113 178401 178689 289 1 8977 9264 9552 9839 180126 180413 180699 180986 181272 181558 287 2 181844 182129 182415 182700 2985 3270 3555 3839 4123 4407 285 3 4691 4975 5259 5542 5825 6108 6391 6674 6956 7239 283 4 7521 7803 8084 8366 8647 8928 9209 9490 9771 190051 281 5 190332 190612 190892 191171 191451 191730 192010 192289 192567 2846 279 6 3125 3403 3681 3959 4237 4514 4792 5069 5346 5623 278 7 5900 6176 6453 6729 7005 7281 7556 7832 8107 8382 276 8 8657 8932 9206 9481 9755 200029 200303 200577 200850 201124 274 9 201397 201670 201943 202216 202488 2761 3033 3305 3577 3848 272 No. O 1 '2 I 4 5 6 7 8 9 Diff. TABLE I. LOGARITHMS OF NUMBERS. No 1 2 3 4 5 G 7 8 9 Diff. 16020412C 204391 2046G32049& 205204 205475 205746 206016 206286 20655G 271 1 6826 7096 j 7365! 7634 7904 8173 8441 8710! 8979 9247 269 2 9515 9783.210051 210319 210586 210853 211121 211388-211654 211921 267 3212188 212454 2720! 298G 3252 3518! 3783 4049! 4314 4579 266 4 4844 5109 5373 5638 5902 G1GG 6430 66941 6957 7221 264 I 7484 7747 8010 8273 8536 8798 9060 9323J 9585i 9846 262 6220108 220370220631220892 221153 221414 221675 221936|222196;222456 261 71 2716 2976 3236 3496 OK 40151 4274 45331 4792| 5051 259 8 6309 5568| 5S26I 6084 6342 6600! 6858 7115 7372! 7630 258 9 7887 8144 8400 8657 8913 9170 9426 9682 9938230193 256 170230449 230704 230960 ! 231215 231470 231724'231979 2322341232488 232742 264 1 2996 3250 3504| 3757 4011 4264 4517 4770 5023 5276 253 2 5528 5781 6033! 6285 6537 6789 | 7041 7292 7544 7795 252 3 8046 8297 85481 8799 9049 9299 9550 9800 240050 240300 250 4240549 240799^241048 241297 241546 241795242044 242293! 2541 | 2790 249 6 CC38 3286 3534 3782 4030 4277 4525 4772 5019! 6266 248 G 5513 6759 6006 6252 64991 6745 6991 7237 7482| 7728 246 7 7973 8219 8464 8709 8954 9198 9443 9687 9932250176 245 1 250420250664:250908 251151 251395 251638 251881 252125 252368i 2610 243 9 2853 3096, 3338 3580 3822 4064 4306 4548 4790 5031 242 180'255273 255514 2H755 255996 256237 256477 '256718! 256958]257198 257439 241 l| 7679 7918 8158 8398 8637 1 8377 9116; 9355 9594 9833 239 2,260071 260310 260548 260787 2G1025 261263 261501 261739 261976 262214 238 3 2451 2688 2925 3162 3399 3636 3873 4109 4346 4582 237 4 4818 6054 5290 5525 5761 5996 6232 6467 6702; 6937J 236 5 7172 7406 7641 7875^ 811w 8344 8578 88121 9046! 9279 234 6 9513 9746 9980 270213 27014G 270679 270912 271144271377271609 233 7 271842 272074 272306 25381 2770 3001 3233 34641 3696; 39271 232 8 4158 4389 4620 4850J 6081 5311 5542 6772 6002 6232| 230 9 6462 66S2 6921 7151 7380 7609 7838 8067 8296 8525j 229 190 278754'278982 279211 279439 2796G7 279895 280123 280351280578 280806 228 1 281033281261,281488 281715i281942 282169 2396 2622 j 2849 3075 227 2 3301 3527 3753 3979 4205 4431 4656 4882 5107 5332, 226 3 6557 6782 6007 6232 6456 6S81 6906 7130 7354 7578 225 4 7802 8026 8249 8473 8696 8920 9143 9366 9589 9812 223 6290035 290257 290480 290702290925 291147 291369 291591 291813 292034. 222 61 2256 2478 2699 2920 3141 3363 3584 3804! 4025 4240J 221 7 4466 4687 4907 6127 5347 6567 5787 6007J 6226 6440 220 8 6665 6884 7104 7323 7542 7761 7979 8198 8416 8635 21 n 9 8853 9071 9289 9507 9725 9943300161 300378300595 3008131 218 2CO 301030 301247 301464 301681 301898 302114 302331 302547302764 302980 217 1 3196 3412 3628 3844 4059 4275 4491 4706 4921 6136 216 2 5351 6566 5781 5996 6211 6425 6639 6854 7068 7282 214 3 7496 7710 7924 8137 8351 8564 8778 8S91 9204 9417 213 4 B 9630 311754 9843 310056 311966 2177 310268 2389 310481 2600 310693 2812 310906 3023 311118 3234 311330 3445 311542| 212 3656 211 1 3867 4078 4289 4499 4710 4920 5130 6340 5551 5760 210 7 5970 6180 6390 6599 6809 7018 7227 7436 7646 7854 209 8 8063 8272 8481 8689 8898 9106 9314 9522 9730 9938: 208 I 320146 320354J320562 320769 320977 321184 321391 321598 321805 322012 207 216 322219 322426322633322839 323046 323252 323458 323665 323871 324077 206 1 4282 4488 4694 4899 5105 6310 5516 5721 5926 6131 205 2 6336 6541 6745 6950 7155 7359 7563 7767 7972 8176 204 3 8380 8583 8787 8991 9194 9398 9601 9805 330008 330211 203 4 330414 330617 330819 331022 331225 331427 331630 331832 2034 2236 202 5 2438 2640 2842 3044 3246 3447 3649 3850 4051 4263 202 I 4454 4655 4856 5057 5257 6458 6658 5859 6059 6260 201 1 6460 6660 6860 7060 7260 7459 7659 7858 8058 8257 200 8 8456 8G56 8855 9054 9253 9451 9650 9849 340047 340246 199 9 340444 340642 340841 341039 341237 341435 341632 341830 2028 2225 198 No- 1 | 2 3 4 5 6 7 8 9 Dlff. TABLE L IX)GAEITHMS OF NUMBERS. 0. o 1 2 a. 4 5 6 7 ff. 220 342423 342620 (42317 343014 (43212 (43409 (43606 3802 (43999 344196 97 1 4392 4589 4785 4981 5178 5374 5570 5766 5962 6157 96 2 6353 6549 6744 6939 7135 7330 7525 7720 7915 8110 95 3 8305 8500 8694 8880 9083 9278 9472 9666 9860 350054 194 4 350248 350442 {50636 350829 351023 (51216 351410 (51603 (51796 1989 193 6 2183 2375 2568 2761 2954 3147 3339 3532 3724 3916 193 6 4108 4301 4493 4685 4876 5068 5260 5452 5643 6834 192 7 6026 6217 6408 6599 6790 6981 7172 7363 7554 7744 191 8 9 7935 9835 8125 360025 8316 360215 8506 360404 8696 360593 8886 360783 9076 360972 9266 361161 9456 361350 9646 361539 190 189 230 361728 361917 362105 362294 362482 362671 362859 363048 363236 363424 188 1 3612 3800 3988 4176 4363 4551 4739 4926 5113 5301 188 2 5488 6675 6862 6049 6236 6423 6610 6796 6983 7169 187 3 7356 7542 7729 7915 8101 8287 8473 8659 8845 9030 186 4 9216 9401 9587 9772 9958 70143 70328 370513 70698 70883 185 5 371068 71253 371437371622 371806 1991 2175 2360 2544 2728 184 6 2912 3096 3280 3464 3647 3831 4015 4198 4382 4565 184 4748 4932 6115 6298 5481 5664 5846 6029 6212 6394 183 8 6577 6759 6942 7124 7306 7488 7670 7852 8034 8216 182 9 8398 8580 8761 8943 9124 9306 9487 9668 9849 380030 181 240 1 380211 2017 380392 2197 380573 2377 380754 2557 380934 2737 381115 2917 381296 3097 381476 3277 381656 3456 381837 3636 181 180 c 3815 3995 4174 4353 4533 4712 4891 5070 5240 6428 179 * 5606 5785 5964 6142 6321 6499 6677 6856 7034 7212 178 i 5 7390 9166 7568 9343 7746 9520 7923 9698 8101 9875 8279 90051 8456 390228 8634 90405 8811 390582 8989 390759 178 177 ( 390935 391112 391288 391464 391641 1817 1993 2169 2345 2521 176 2697 2873 3048 3224 3400 3575 3751 392 4101 4277 176 j 4452 4627 4802 4977 6152 6326 6501 567 5850 6025 175 9 6199 6374 6548 6722 6896 707 7245 741 7592 7766 174 25( 397940 9674 98114 9847 398287 400020 398461 400192 398634 40036 39880 40053 398981 40071 399154 40088 399328 401056 39950 40122 173 17J i 401401 401573 1745 1917 208 226 2433 260 2777 294 172 3121 3292 3464 3635 380 397 414 432 4492 466 171 i 483- 6005 5176 6346 651 668 685 602 6199 637 171 J 6540 6710 6881 705 722 739 756 773 790 807 170 8240 8410 8579 874 891 9087 925 942 959 9764 169 1 9933 411620 410102 1788 410271 1956 410440 2124 410609 229 41077 246 41094 2629 41111 2796 411283 2964 41145 313 169 168 j 3300 3467 3635 380 397 413 4305 447 463 4806 167 260 414973 415140 415307 41547 41564 415808 41597 41614 41630 41647 167 664 6807 6973 713 7306 747 763 7804 797 8135 166 2 3 830 995 8467 42012 8633 420286 879 42045 8964 42061 912S 42078 929 42094 9460 42111 962c 42127 979 42143 165 165 4 421604 176 1933 209 226 242 2590 2754 291 308 164 3246 3410 3574 373 390 406 4228 4392 455 471 164 I 488 5045 5208 537 5534 669 6860 602, 618 634 163 651 667 6836 6999 716 732 7486 764 781 797 162 813 829 MS 862 8783 8944 9106 926? 942 959 162 9 975 991 430075 430236 43Q39 43055 430720 43088 43104. 43120 161 270 ] 431364 296 43152 3130 43168E 329C 43184 345C 432007 361 43216 377 432328 3930 43248 409C 43264 424 32808 440 161 160 j 456 4729 4888 604 620 636 652 668E 584 600 159 j 61& 6322 648 664 679 695 711 727* 743. 759 159 ^ 775 790 806 822 838 854 870 8851 > 901 917 158 i 933 949 964 9806 9964 44012 44027 44043 r 44059^ 44075 158 ( 44090S 441066 44122 44138 44153 169 185 2001 ) 216 232 157 248 263 279 295C 310( 326, 341 357( J 373 388 157 i 404 420 435 451 466 482 498 613' r 529 64*! 156 i 5604 676C 591 607 622 638 653 6692 1 6841 700. 155 Mo O 1 2 3 4 5 6 7 8 9 Dlff. TABLE I. LOGARITHMS OF NUMBERS. No, o 1 2 3 Dlff. 280 447158,447313 447468 447623 447778 447933 448088 448242 448397 448552 155 1 8706J 8861 9015 9170 9324 9478 9633 9787) 9941 450095 154 2 450249;450403 450557 450711 450865,451018 451172 451 326! 451479 1633 154 3 1786 1940 2093 2247 2400 2553 2706 2859 3012 3165 153 4 3318 3471 3624 3777 3930 4082 4235 4387 4540 4692 153 5 48451 4997 5150 5302 5454 5606 5758 5910 6062 6214 152 6 6366; 6518 6670 6821 6973 7125 7276 7428 7579 7731 152 7 78821 8033 8184 8336 8487 8638 8789 8940 9091 9242 151 8 9392 9543 9694 9845 999 460146 460296 460447 460597 460748 151 9 460898 ; 461048 461198 461348 461499 1649 1799 1948 2098 2248 150 290 462398 462548 462697 462847 462997 463146 463296 463445463594 463744 150 1 3893 4042 4191 4340 4490 4639 4788 4936 5085 5234 149 2 5383 5532 5680 5829 5977 6126 6274 6423 6571 6719 149 3 6868 7016 7164 7312 7460 7608 7756 7904 8052 8200 148 4 8347 8495 8643 8790 8938 9085 9233 9380 9527 9675 148 5 9822 9969 470116 470263 470410 470557 470704 470851 470998 471145 147 6 471292 471438 1585 1732 1878 2025 2171 2318 2464 610 147 7 2756 2903 3049 3195 3341 3487 3633 3779 3925 40/1 146 8 4216 4362 4508 4653 4799 4944 5090 5235 5381 6526 146 9 5671 5816 5962 6107 6252 6397 6542 6687 6832 6976 145 300 1 477121 8566 477266 8711 477411 8855 477555 8999 477700 9143 477844 9287 477989 9431 478135 9575 478278 9719 4784221 145 986C 1 144 2 480007 480151 480294 480438 480682 480725 480869 481012 481156 481290 144 3 1443 1586 1729 1872 2016 2159 2302 2445 2588 2731 143 4 2874 3016 3159 3302 3445 3587 3730 3872 4015 4157 143 5 4300 4442 4585 4727 4869 5011 5153 5295 5437 5579 142 6 5721 5863 6005 6147 6289 6430 6572 6714 6855 6997! 142 7 7138 7280 7421 7563 7704 7845 7986 8127 8269 8410! 141 8 8551 8692 8833 8974 9114 9255 9396 9537 9677 9818 141 9 9958 490099 490239 490380 490520 490661 490801 490941 491081 491222 140 310 491362 491502 491642 491782 491922 492062492201 492341 492481 492621 140 1 2760 2900 3040 3179 3319 3458 3597 3737 3876 4015 139 2 4155 4294 4433 4572 4711 4850 4989 5128 5267 5406 139 3 5544 5683 5822 5960 6099 6238 6376 6515 6653 6791 139 4 6930 7068 7206 7344 7483 7621 7759 7897 8035 8173 138 6 8311 8448 8586 8724 8862 8999 9137 9275 9412 9550 138 6 9687 9824 9962 500099 500236 500374 500511 500648 500785 500922 137 7 501059;501196 501333 1470 1607 1744 1880 2017 2154 2291 137 8 2427 2564 2700 2837 2973 3109 3246 3382 3518 3655 136 9 3791 3927 4063 4199 4335 4471 4607 4743 4878 5014 136 320 505150 505286 505421 505557 505693 505828 505964 506099 506234 506370 136 1 6505 6640 6776 6911 7046 7181 7316 7451 7586 7721 135 2 7856 7991 8126 8260 8395 8530 8664 8799 8934 9068 135 3 9203 9337 9471 9606 9740 9874 510009 510143 510277 510411 134 4 510545 510679 510813 510947 511081 511215 1349 1482 1616 1750 134 5 1883 2017 2151 2284 2418 2551 2684 2818 2951 3084 133 6 3218 3351 3484 3617 3750 3883 4016 4149 4282 4415 133 7 4548 4681 4813 4946 5079 5211 5344 5476 5609 5741 133 8 5874 6006 6139 6271 6403 6535 6668 6800 6932 7064 132 9 7196 7328 7460 7592 7724 7855 7987 8119 8251 8382 132 330 518514 518646 518777 518909 519040 519171 519303 519434 519566 519697 131 1 9828 9959 520090 520221 520353 520484 520615 520745 520876 521007 131 2 521138 521269 1400 1530 1661 1792 1922 2053 2183 2314 131 3 2444 2575 2705 2835 2966 3096 3226 3356 3486 3616 130 4 3746 3876 4006 4136 4266 4396 4526 4656 4785 4915 130 5 5045 5174 5304 5434 5563 5693 5822 5951 6081 6210 129 6 6339 6469 6598 6727 6856 6985 7114 7243 7372 7501 129 7 7630 7759 7888 8016 8145 8274 8402 8531 8660 8788 129 8 8917 9045 9174 9302 9430 9559 9687 9815 9943 530072 128 9 530200 530328 530456 530584 530712 530840 530968 531096 531223 1351 128 No. O 1 a 3 4 6 7 8 9 Dlff. TABLE I. LOGARITHMS OF NUMBERS. No. O 1 2 o 4 5 6 7 8 9 Dlff. 340 531479 531607 531734531862 531990 532117 532245 532372532500 532627 128 1 2754 2882 30091 3136 3264 3391 3518 3645 3772 3899 127 2 4026 4153 4280 4407 4534 4661 4787 4914 6041 5167 127 3 5294 5421 5547 5674 5800 5927 6053 6180 6306 6432 126 4 6558 6685 6811 6937 7063 7189 7315 7441 7567 7693 126 5 7819 7945 8071 8197 8322 8448 8574 8699 8825 8951 126 6 9076 9202 9327 9452 9578 9703 9829 9954 540079 540204 125 7 540329 540455 540580 540705 540830 540955 541080 541205 1330 1454 125 8 1579 1704 1829 1953 2078 2203 2327 2452 2576 2701 125 9 2825 2950 3074 3199 3323 3447 3571 3696 3820 3944 124 350 544068 544192 544316 544440 544564 544688 544812 544936 545060 545183 124 1 5307 5431 5555 5678 5802 5925 6049 6172 6296 6419 124 2 6543 6666 6789 6913 7036 7159 7282 7405 7529 7652 123 3 7775 7898 8021 8144 8267 8389 8512 8635 8758 8881 123 4 9003 9126 9249 9371 9494 9616 9739 9861 9984 550106 123 5 6 550228 1450 550351 1572 550473 1694 550595 1816 550717 1938 550840 2060 550962 2181 551084 2303 551206 2425 1328 2547 122 122 7 2668 2790 2911 3033 3155 3276 3398 3519 3640 3762 121 8 3883 4004 4126 4247 4368 4489 4610 4731 4852 4973 121 9 5094 5215 5336 5457 5578 5699 5820 5940 6061 6182 121 3GO 556303 556423 556544 556664 556785 556905 557026 557146 557267 557387 120 1 7507 7627 7748 7868 7988 8108 8228 8349 8469 8589 120. 2 8709 8829 8948 9068 9188 9308 9428 9548 9667 9787 120 3 9907 560026 560146 560265 560385 560504 560624 560743 560863 560982 120 4 561101 1221 1340 1459 1578 1698 1817 '1936 2055 2174 119 5 2293 2412 2531 2650 2769 2887 3006 3125 3244 3362 119 6 3481 3600 3718 3837 3955 4074 4192 4311 4429 4548 119 7 4666 4784 4903 5021 5139 5257 5376 5494 5612 5730 118 8 5848 5966 6084 6202 6320 6437 6555 6673 6791 6909 118 9 7026 7144 7262 7379 7497 7614 7732 7849 7967 8084 118 370 1 568202 9374 568319 9491 568436 568554 9608 9725 568671 9842 568788 9959 568905 569023 570076 570193 569140 570309 569257 570426 117 117 2 570543 570660 570776 570893 571010 571126 1243 1359 1476 1592 117 3 1709 1825 1942 2058 2174 2291 2407 2523 2639 2755 116 4 2872 2988 3104 3220 3336 3452 3568 3684 3800 3915 116 5 4031 4147 4263 4379 4494 4610 4726 4841 4957 5072 116 6 5188 5303 5419 5534 5650 5765 5880 5996 6111 6226 115 7 6341 6457 6572 6687 6802 6917 7032 7147 7262 7377 115 8 7492 7607 7722 7836 7951 8066 8181 8295 8410 8525 115 9 8639 8754 8868 8983 9097 9212 9326 9441 9555 9669 114 380 579784 579898 580012 580126 580241 580355 580469 580583 580697 580811 114 1 580925 581039 1153 1267 1381 1495 1608 1722 1836 1950 114 2 2063 2177 2291 2404 2518 2631 2745 2858 2972 3085 114 3 3199 3312 3426 3539 3652 3765 3879 3992 4105 4218 113 4 4331 4444 4557 4670 4783 4896 5009 5122 5235 5348 113 5 5461 5574 1 5686 5799 5912 6024 6137 6250 6362 6475 113 6 6587 6700 6812 6925 7037 7149 7262 7374 7486 7599 112 7 7711 7823 7935 8047 8160 8272 8384 8496 8608 8720 112 8 8832 8944 9056 9167 9279 9391 9503 . 9615 9726 9838 112 9 9950 590061 590173 590284 590396 590507 590619 590730 590842 590953 112 390 591065 591176 591287 591399 591510 591621 591732 591843 591955 592066 111 1 2177 2288 2399 2510 2621 2732 2843 2954 3064 3175 111 2 3286 3397 3508 3618 3729 3840 3950 4061 4171 4282 111 4393 4503 4614 4724 4834 4945 5055 5165 5276 5386 110 j 5496 5606 5,717 5827 5937 6047 6157 6267 6377 6487 110 5 6597 6707 6817 6927 7037 7146 7256 7366 7476 7586 110 6 7695 7805 7914 8024 8134 8243 8353 8462 8572 8681 110 8791 8900 9009 9119 9228 9337 9446 9556 9665 9774 109 8 9883 9992 600101 600210 600319 600428 600537 600646 600755 600864 109 9 600973 601082 1191 1299 1408 1517 1625 1734 1843 1951 109 No. O 1 2 3 4 5 6 7 8 9 Diff. TABLE I. LOGARITHMS OF NUMBERS. No. O 1 2 3 4 5 6 7 8 O Diff. 400 602060 602169 602277 602386 602494 602603 602711 602819 602928 603036 108 1 3144 3253 3361 3469 3577 3686 3794 3902 4010 4118 108 2 4226 4334 4442 4550 4658 4766 4874 4982 5089 5197 108 3 5305 5413 6521 5628 5736 5844 5951 6059 6166 6274 108 4 6381 6489 6596 6704 6811 6919 7026 7133 7241 7348 107 5 7455 7562 7669 7777 7884 7991 8098 82051 8312 8419 107 6 8526 8633 8740 8847 8954 9061 9167 9274! 9381 9488 107 7 9594 9701 9808 9914610021610128 610234 610341 610447 610554 107 8 610660 610767 610873 610979 1086! 1192 1298 1405 1511 1617 106 9 1723 1829 1936 2042 2148 2254 2360 2466 2572 2678 106 410 612784 612890 612996 613102 613207 613313 613419 613525 613630 613736 106 1 3842 3947 4053 4159 426> 4370 4475 4581 4686 4792 106 2 4897 5003 5108 5213 5319 5424 5529 5634 5740 5845 105 3 5950 6055 6160 6265 6370 6476 6581 6686 6790 6895 105 4 7000 7105 7210 7315 74201 7525 7629 7734 7839 7943 105 K o 8048 8153 8257 8362 8466 8571 8676 8780 8884 8989 105 6 9093 9198 93021 9K)6 9511; 9615 9719 9824 9928 620032 104 7 620136620240 620344 620448 620552 620656 620760 620864 620968 1072 104 8 1176! 1280 1384 1438 1592 1695 1799 1903 2007 2110 104 9 2214 2318 2421 2525 2628 2732 2835 2939 3042 3146 104 420 623249623353 623456 623559 623663 623766 623869 623973 624076 624179 103 1 4282 4385 4488j 4591 | 4695 4798 4901 5004 5107 5210 103 2 5312 5415 6518 6621 5724 5827 5929 6032 6135 6238 103 3 6340 6443 6546 6648 6751 1 6853 6956 7058 7161 7263 103 4 7366 7468 7571 76731 7775! 7878 7980 8082 8185 8287 102 5 8389 8491 8593 8695 8797 8900 9002 9104 9206 9308 102 6 9410 9512 9613! 9715 9817 9919 080081 630123630224 63<>:; 102 7630428 630530 630631 630733 630835 63093(3 1038 11391 1241 1342 102 8 1444 1545 1647i 1748 1849 1951 2052 2153 2255 2356 101 9 2457 2559 2660 2761 2862J 2963 3064 3165 3266 3367 101 430 633468 633559 633670 633771 633872633973 634074 634175634276 634376 101 1 4477 4578 4679 4779 4880; 4981 5081 6182 6283 5383 101 2 5484 6584 6685 5785 5886 5986 6087 6187 6287 6388 100 3 6488 6588 6688 6789 6889 6989 7089 7189 7290 7390 100 4 7490 7590 7690 7790 7890 7990 8090 8190 8290 8389 100 5 8489 8589 86S9 8789 8888 8988 9088 9188 9287 9387 99 6 9486 9586 9686 9785 9885 9984640084 640183640283 640382 99 7640481 640581 640680 640779 640879 640978 1077 1177 1276 1375 99 8 1474 1573, 1672 1771 1871 1970 2069 2168 2267 2366 99 9 2465 2563 2662 2761 2860 2959 3058 3156 3255 3354 99 440 643453 643551 643650 643749 643847 643946 644044 644143 644242 644340 98 1 4439 4537 4636 4734 4832 4931 6029 5127 5226 5324 98 2 5422 5521 5619 5717 5815 5913 6011 6110 6208 6306 98 3 6404 6502 6600 6698 6796 G894 6992 7089 7187 7285 98 4 7383 7481 7579 7676 7774 7872! 7969 8067 8165! 8262 98 5 8360 8458 8555 8653 8750 8848 8945 9043 9140 9237 97 6 9335 9432 9530 9627 9724 9821 9919650016 650113650210 97 7650308 650405 650.302 650399 650696,650793 650890 0987 1084 1181 97 8 1278 1375 Ii72 1569 1666 1762 1859 1956 2053: 2150 97 9 2246 2343 2140 2536 2633 2730 2826 2923 3019 3116 97 450653213 653309 653405 653502 653598 653695 653791 653888 653984 654080 96 1 4177 4273 4369 4465 4562 4658 4754 4850 4946 6042 96 2 5138 5235 5331 5427 5523 5619 5715 5810 5906 6002 96 3 6098 6194 6290 6386 6482 6577 6673 6769 6864 6960 96 4 7056 7152 7247 7343 7438 7534 7629 7725 7820 7916 96 5 8011 8107 8202 8298 8393 8488 8584 8679 8774 8870 95 6 8965 9060 9155 9250 9346 9441 9536 9631 9726 9821 95 7 9916 660011 660106 660201 660296 660391 660486 660581 660676 660771 95 8 660865 0960 1055 1150 1245 1339 1434 1529 1623 1718 95 9 1813) 1907 2002 2096 2191 2286 2380 2475 2569 2663 95 No. 123 4 5 G 7 8 9 Diff. TABLE I. LOGARITHMS OF NUMBERS. No. O 1 3 3 4 5 6 7 8 9 iff. 460 662758 1 3701 2 4642 3 5581 4 6518 5 7453 6 8386 7 9317 8 670246 9 1173 62852 3795 4736 5675 6612 7546 8479 9410 70339 1265 2947 3889 4830 5769 6705 7640 8572 9503 0431 1358 63041 3983 4924 5862 6799 7733 8665 9596 70524 1451 63135 4078 5018 5956 6892 7826 8759 9689 0617 1543 63230 4172 5112 6050 6986 7920 8852 9782 0710 1636 63324 4266 5206 6143 7079 8013 8945 9875 0802 1728 63418 4360 5299 6237 7173 8106 9038 9967 0895 1821 4454 5393 6331 7266 8199 9131 0060 0988 1913 63607 4548 5487 6434 7360 8293 9224 70153 1080 2005 94 94 94 94 94 93 93 93 93 93 470 672098 1 3021 2 3942 3 4861 4 5778 5 6694 6 7607 7 8518 8 9428 9680336 72190 3113 4034 4953 5870 6785 7698 8609 9519 80426 2283 3205 4126 5045 5962 6876 7789 8700 9610 580517 72375 3297 4218 5137 6053 6968 7881 8791 9700 680607 72467 3390 4310 5228 6145 7059 7972 8882 9791 80698 72560 3482 4402 5320 6236 7151 8063 8973 9882 80789 72652 3574 4494 5412 6328 7242 8154 9064 9973 80879 2744 3666 4586 5503 6419 7333 8245 9155 80063 0970 2836 3758 4677 5595 6511 7424 8336 9246 80154 1060 572929 3850 4769 5687 6602 7516 8427 9337 680245 1151 92 92 92 92 92 91 91 91 91 91 480 81241 1 2145 2 3047 3 3947 4 4845 6 5742 6 6636 7 7529 8 8420 9 9309 81332 2235 3137 4037 4935 5831 6726 7618 8509 9398 81422 2326 3227 4127 5025 5921 6815 7707 8598 9486 681513 2416 3317 4217 5114 6010 6904 7796 8687 9575 j81603 2506 3407 4307 5204 6100 6994 7886 8776 9664 581693 2596 3497 4396 5294 6189 7083 7975 8865 9753 E581784 2686 3587 4486 5383 6279 7172 8064 8953 9841 81874 2777 3677 4576 5473 6368 7261 8153 9042 9930 81964 2867 3767 4666 5563 6458 7351 8242 9131 90019 682055 2957 3857 4756 5652 6547 7440 8331 9220 690107 90 90 90 90 90 89 89 89 89 89 490 690196 1 1081 2 1965 3 284 4 372 5 460 6 548 7 635 8 722 9 810 690285 1170 2053 2935 381 469 556 6444 731 818 90373 125 214 302 390 478 565 653 7404 827 69046 134 223 311 399 486 5744 66l 749 836 90550 1435 2318 319 407 495 583 6706 757 844 90639 1524 2406 3287 4166 5044 591 679 7665 853 90728 1612 2494 3375 4254 513 600 688 775 862 90816 1700 2583 3463 434 521 6094 696 783 8709 9090 178 2671 3551 4430 5307 6182 7055 7926 -8796 690993 1877 275 363 451 539 626 714 801 888 89 88 88 88 88 88 87 87 87 87 500 69897 1 9838 69905 992 69914 70001 69923 70009 69931 700184 699404 70027 69949 70035 69957 700444 699664 700531 69975 70061 87 87 270070 3 156 4 2431 700790 1654 251 087 174 2602 096 1827 268 1050 191 277 113 1999 286 1222 208 294 1309 217 303 1395 2258 31 1 148 2344 320 86 86 86 5 3291 6 4151 7 5008 8 5864 337 4236 5094 594 346J 4325 617< 603f 354 440 526v > 6126 3635 449 535C 6206 372 457 6436 629 380 4665 5522 637 389 475 560 646 397 483'J 569C 654' 406^ 492 577 66 86 86 85 9 671* ( 680 688* 5 697 705 714 722 731 746X 74 85 510 707576 1 8421 2 9276 3 71011' ) 70765 8506 ) 935 r 71020* j 104 707746 859 944< 71028' 113' ) 70782 L 867 ) 952 r7103 I 12 70791 876 96CK 71045< 130 707996 884 9694 71054 138 70808 893 977 71062 147 708166 901 986k 71071 1554 708251 916X 9945 71079^ 163< 708336 91 7100 08 17 85 85 85 85 84 6 180' 6 26M r 189 \ ty-r'. 197 281 5 206 5 290 21 29 222 30 23 31 239 32 248 332 256< 34 84 84 7 349 8 433 9 516 L 35 ) 44 r 52 365 449 533 ) 37 r 45 5 54 38 466 550 1 39 47 55 39 48 56 40 49 7 416 56X> 583 42 50 59 84 84 Mo 1 2 3 4 5 6 7 8 9 Dif. TABLE I. LOGARITHMS OF NUMBERS. No. 1 2 3 4 5 6 7 8 9 Dlff. 520 716003 716087 716170 716254 716337 716421 716504 716588 716671 716754 83 1 6838 6921 7004 7088 7171 7254 7338 7421 7504 7587 83 2 7671 7754 7837 7920 8003 8086 8169 8253 8336 8419 83 3 8502 8585 8668 8751 8834 8917 9000 9083 9165 9248 83 4 9331 9414 9497 9580 9663 9745 9828 9911 9994 720077 83 5 720159 720242 720325 720407 720490 720573 720655 720738 720821 0903 83 6 0986 1068 1151 1233 1316 1398 1481 1563 1646 1728 82 7 1811 1893 1975 2058 2140 2222 2305 2387 2469 2552 82 g 2634 2716 2798 2881 2963 3045 3127 3209 3291 3374 82 9 3456 3538 3620 3702 3784 3866 3948 4030 4112 4194 82 530 1 724276 5095 724358 5176 724440 5258 724522 5340 724604 5422 724685 6503 724767 5585 724849 5667 724931 5748 725013 6830 82 82 2 5912 5993 6075 6156 6238 6320 6401 6483 6564 6646 82 3 6727 6809 6890 6972 7053 7134 7216 7297 7379 7460 81 4 7541 7623 7704 7785 7866 7948 8029 8110 8191 8273 81 5 8354 8435 8516 8597 8678 8759 8841 8922 9003 9084 81 6 7 9165 9974 9246 730055 9327 730136 9408 730217 9489 730298 9570 730378 9651 730459 9732 730540 9813 730621 9893 730702 81 81 8 730782 0863 0944 1024 1105 1186 1266 1347 1428 1508 81 9 1589 1669 1750 1830 1911 1991 2072 2152 2233 2313 81 540 1 732394 3197 732474 3278 732555 3358 732635 3438 732715 3518 732796 3598 732876 3679 732956 3759 733037 3839 733117 3919 80 80 2 3999 4079 4160 4240 4320 4400 "4480 4560 * 4640 4720 80 3 4800 4880 4960 5040 5120 5200 5279 6359 6439 6519 80 4 6599 5679 5759 5838 5918 5998 6078 6157 6237 6317 80 5 6397 6476 6556 6635 6715 6795 6874 6954 7034 7113 80 6 7193 7272 7352 7431 7511 7590 7670 7749 7829 7908 79 7 7987 80G7 8146 8225 8305 8384 8463 8543 8622 8701 79 g 8781 8860 8939 9018 9097 9177 9256 9335 9414 9493 79 9 9572 9651 9731 9810 9889 9968 740047 740126 740205 740284 79 550 740363 740442 740521 740600 740678 740757 740836 740915 740994 741073 79 1 1152 1230 1309 1388 1467 1546 1624 1703 1782 1860 79 2 1939 2018 2096 2175 2254 2332 2411 2489 2568 2647 79 3 2725 2804 2882 2961 3039 3118 3196 3275 3353 3431 78 4 3510 3588 3667 3745 3823 3902 3980 4058 4136 -4215 78 5 4293 4371 4449 4528 4606 4684 4762 4840 4919 4997 78 6 5075 5153 5231 6309 6387 5465 6543 6621 6699 5777 78 7 5855 5933 6011 6089 6167 6245 6323 6401 6479 6666 78 g 6634 6712 6790 6868 6945 7023 7101 7179 7256 7334 78 9 7412 7489 7567 7645 7722 7800 7878 7955 8033 8110 78 560 748188 748266 748343 748421 748498 748576 748653 748731 748808 748885 77 8963 9040 9118 9195 9272 93501 9427 9504 95821 9659 77 2 9736 9814 9891 9968 750045;750123 750200 750277 760354 75O431 77 3 750508 750586 750663 750740 0817 0894 0971 1048 1125 1202 77 '4 1279 1356 1483 1510 1587 1664 1741 1818 1895 1972 77 5 2048 2125 2202 2279 2356 2433 2509 2686 2663 2740 77 6 2816 2893 2970 3047 3123 3200 3277 3353 3430 3506 77 7 3583 3660 3736 3813 3889 3966 4042 4119 4195 4272 77 g 4348 4425 4501 4578 4654 4730 4807 4883 4960 6036 76 9 6112 5189 5265 6341 6417 6494 6570 5646 5722 6799 76 570 1 755875 6636 755951 6712 756027 6788 756103 6864 756180 756256 6940 7016 756332 7092 756408 7168 756484 7244 756560 7320 76 76 2 7396 7472 7548 7624 7700 7775 7851 7927 8003 8079 76 3 8155 8230 8306 8382 8458 8533 8609 8685 8761 8836 76 4 8912 8988 9063 9139 9214 9290 9366 9441 9517 9692 76 5 9668 9743 9819 9894 9970760045 760121 760196 760272 760347 76 6 760422 760498 760573760649 760724 0799 0875 0950| 1025 1101 76 7 1176 1251 1326 1402 1477 1552 1627 1702 1778 1853 76 g 1928 2003 2078 2153 2228 2303 2378 2453 2529 2604 75 9 2679 2754 2829 2904 2978 3053 3128 3203 3278 3353 75 No. 1 2 ! 3 4 5 6 7 8 9 Dlff. 10 TABLE I. LOGARITHMS OF NUMBERS. Ho.]- 1 2 3 4 5 6. 7 8 9 Diff. 580 763428 763503 763578 763653 763727 763802 763877 763952 764027 764101 75 1 4176 4251 4326 4400 4475 4550 4G24 4699 4774 4848 75 2 4923 4998 5072 5147 5221 5296 5370 5445 5520 5594 75 3 5669 5743 5818 5892 5P66 6041 6115 6190 6264 6338 74 4 6413 6487 6562 6636 6710 6785 6859 6933 7007 7082 74 5 7156 7230 7304 7379 7453 7527 7601 7675 7749 . 7823 . 74 6 7898 7972 8046 8120 8194 8268 8342 8416 8490 8564 74 7 8638 8712 8786 8860 8934 9008 9082 9156 9230 9303 74 8 9377 9451 9525 9599 9673 9746 9820 9894 9968 770042 74 9 770115 770189 770263 770336 770410 770484 770557 770631 770705 0778 74 590 770852 770926 770999 771073 771146 771220 771293 771367 771440 771514 : 74 1 1587 1661 1734 1808 1881 1955 2028 2102 2175 2248 73 2 2322 2395 2468 2542 2615 2688 2762 2835 2908 2981 73 3 3055 3128 3201 3274 8348 3421 3494 3567 3640 3713 73 4 3786 3860 3933 4006 4079 4152 4225 4298 4371 4444 73 5 4517 4590 4663 4736 4809 4882 4955 5028 5100 5173 73 6 5246 5319 5392 5465 5538 5610 5683 5756 5829 5902 73 7 5974 6047 6120 6193 6265 6338 6411 6483 6556 6629 73 8 6701 6774 6846 6919 6992 7064 7137 7209 7282 7354 73 9 7427 7499 7572 7644 7717 7789 7862 7934 8006 8079 72 600 778151 778224 778296 778368 778441 778513 778585 778658 778730 778802 72 1 8874 8947 9019 9091 9163 9236 9308 9380 9452 9524 72 2 9596 9669 9741 9813 9885 9957 780029 780101 780173 780245 72 3 780317 780389 780461 780533 780605 780677 0749 0821 0893 0965 72 4 1037 1109 1181 1253 1324 1396 1468 1540 1612 1684 72 5 1755 1827 1899 1971 2042 2114 2186 2258 2329 2401 72 6 2473 2544 2616 2688 2759 2831 2902 2974 3046 3117 72 7 3189 3260 3332 3403 3475 3546 3618 3689 3761 3832 71 8 3904 3975 4046 4118 4189 4261 4332 4403 4475 4546 71 9 4617 4689 4760 4831 4902 4974 5045 5116 5187 5259 71 610 785330 785401 785472 785543 785615 785686 785757 785828 785899 785970 71 1 6041 6112 6183 6254 6325 6396 6467 6538 6609 6680 71 2 6751 6822 6893 6964 7035 7106 7177 7248 7319 7390 71 3 7460 7531 7602 7673 774* 7815 7885 7956 8027 8098 71 4 8168 8239 8310 8381 8451 8522 8593 8663 8734 8804 71 5 8875 8946 9016 -9087 9157 9228 9299 9369 .9440 9510- 71 6 9581 9651 9722 9792 9863 9933 790004 790074 790144 790215 70 7 790285 790356 790426 790496 790567 790637 0707 0778 0848 0918 70 8 0988 1059 1129 1199 1269 1340 1410 1480 1550 1620 70 9 1691 1761 1831 1901 1971 2041 2111 2181 2252 2322 70 620 792392 792462 792532 792602 792672 792742 792812 792882 792952 793022 70* -1 3092 3162 3231 3301 3371 3441 3511 3581 3651 3721 70 2 3790 3860 3930 4000 4070 4139 4209 4279 4349 4418 70 3 4488 4558 4627 4697 4767 4836 4906 4976 5045 5115 70 4 5185 5254 5324 5393 5463 5532 5602 5672 5741 5811 70 5 5880 5949 6019 6088 6158 6227 6297 6366 6436 6505 69 6 6574 6644 6713 6782 6852 6921 6990 7060 7129 7198 69 7 7268 7337 7406 7475 7545 7614 7683 7752 7821 7890, 69 8 7960 8029 8098 8167 8236 8305 8374 8443 8513 8582 69 9 8651 8720 8789 8858 8927 8996 9065 9134, 9203 9272 69 630 799341' T99409 799478 799547 799616 799685 799754 799823 799892 799961 69 1 800029 800098 800167 800236 800305 800373 800442 800511 800580 800648 69 2 0717 0786 0854 0923 0992 1061 1129 1198 1266 1335 69 3 1404 1472 1541 1609 1678 1747 1815 1884 1952 2021 69 4 2089 2158 2226 2295 2363 2432 2500 2568 2637 2705 69 5 2774 2842 2910 2979 3047 3116 3184 3252 3321 3389 68 6 3457 3525 3594 3662 3730 3798 3867 3935 4003 4071 68 7 4139 4208 4276 4344 4412 4480 4548 4616 4685 4753 68 8 4821 4889 4957 5025 5093 5161 5229 5297 5365 5433 68 9 5501 5569 5637 5705 5773 5841 5908 5976 6044 6112 68 No. 1 2 3 4 5 6 7 8 9 ~m. TABLE I. LOGARITHMS OF NUMBERS. 11 So. 1 2 3 4 5 6 7 8 9 Dif. 640 806180,806248 6858 6926 806316 - 6994 806384J 806451 7061 7129 806519 7197 806587 7264 806655 7332 806723 7400 806790 7467 68 68 2 7535 7603 7670 7738 7806 7873 7941 8008 8076 8143 68 3 8211 8279 8346 8414 8481 8549 8616 8684 8751 8818 67 4 8886 8953 9021 9088 9156 9223 9290 9358 9425 9492 67 5 9560 9627 9694 9762 9829 9896 9964 810031 810098 810165 67 G 810233 810300 810367 810434 810501 810569 810636 0703 0770 0837 67 0901 0971 1039 1106 1173 1240 1307 1374 1441 1508 67 8 1575 1642 1709 1776 1843 1910 1977 2044 2111 2178 67 9 2245 2312 2379 2445 2512 2579 2646 2713 2780 2847 67 6.50 812913 812980 813047 813114 813181 813247 813314 813381 813448 813514 67 1 3581 3648 3714 3781 3848 3914 3981 4048 4114 4181 67 2 4248 4314 4381 4447 4514 4581 4647 4714 4780 4847 67 3 4913 4980 5046 5113 5179 5246 5312 5378 6445 5511 66 4 5578 5644 5711 5777 5843 5910 5976 6042 6109 3175 66 5 6241 6308 6374 6440 6506 6573 6639 6705 677f 6838 66 6 6904 6970 7036 7102 7169 7235 7301 7367 7433 7499 66 7 7565 7631 7698 7764 7830 7896 7962 8028 8094 8160 66 8 8226 8292 8358 8424 8490 8556 8622 8688 8754 8820 66 9 8885 8951 9017 9083 9149 9215 9281 9346 9412 9478 66 660 819544 819610 819676 819741 819807 819873 819939 820004 820070 820136 66 1 820201 820267 820333 820399 820464 820530820595 0661 0727 0792 66 2 0858 0924 0989 1055 1120 1186 1251 1317 1382 1448 66 3 1514 1579 1645 1710 1775 1841 1906 1972 2037 2103 66 4 2168 2233 2299 2364 2430 2495 2560 -2626 2691 2756 65 5 2822 2887 2952 3018 3083 3148 3213 3279 3344 3409 65 6 3474 3539 3605 3670 3735 3800 3865 3930 3996 4061 66 7 4126 4191 4256 4321 4386 4451 4516 4581 4646 4711 65 8 4776 4841 4906 4971 5036 5101 5166 5231 6296 5361 65 9 5426 5491 5556 5621 5686 575r 6815 5880 6945 6010 65 670 826075 826140 826204 826269.826334 826399 826464 826528 826593:826658 65 1 6723 6787 6852 6917 6981 7046 7111 7175 7240 7305 65 2 7369 7434 7499 7563 7628 7692 7757 7821 78861 7951 66 3 8015 8080 8144 82Q9 8273 8338 8402 8467 8531 8595 64 4 8660 8724 8789 8853 8918 8982 9046 9111 9175 9239 64 5 9304 9368 9432 9497 9561 9625 9690 9754 9818 9882 64 6 9947 830011 830075 830139 S3<>_M4 sso-f;8 830332 830596,830460830525 64 7 830589 0653 0717 078JL 0845 09^)9 0973 1037 1102 1166 64 8 1230 1294 . 1358 1422 1486 1550 1614 1678 1742 1806 64 9 1870 1934 1998 2062 2126 2189 2253 2317 2381 2445 64 680 832509 832573 832637 832700 832764 832828 832892 832956 833020 833083 64 1 3147 3211 3275 3338 3402 3466 3530 3593 3657 3721 64 2 3784 3848 3912, 3975 4039 4103 4166 4230 4294 4357 64 3 4421 4484 4548 4611 4675 4739 4802 4866 4929 4993 64 4 5056 5120 5183 5247 5310 5373 5437 5500 5564 5627 63 5 5691 5754 5817 5881 5944 6007 6071 6134 6197 6261 63 6 6324 6387 6451 6514 6577 6641 6704 6767 6830 6894 63 7 6957 7020 7083 7146 7210 7273 7336 7399 7462 7525 63 8 7588 7652 7715 7778 7841 7904 7967 8030 8093 8156 63 9 8219 8282 8345 8408 8471 8534 8597 8660 872 8786 63 690 838849 838912 838975 839038 839101 839164 839227 839289 839352 839415 63 1 9478 9541 9604 96671 9729 9792 98551 9918! 9981,840043 63 2 840106 840169 840232 840294 840357 840420 840482 840545840608 0671 63 3 0733 0796 0859 0921 0984 1046 1109 1172 1234 1297 63 4 1359 1422 1485 1547 1610 1672 1735 1797 1860 1922 63 5 1985 2047 2110 2172 2235 2297 2360 2422 2484 2547 62 6 2609 2672 2734 2796 2859 2921 2983 3046 3108 3170 62 7 3233 3295 3357 3420 3482 3544 3606 3669 3731 3793 62 8 3855 3918 3980 4042 4104 4166 4229 4291 4353 4415 62 9 4477 4539 4601 4664 4726 4788 4850 4912 4974 5036 62 No. 1 2 4 5 6 7 8 9 ~m 12 TABLE I. LOGARITHMS OF NUMBERS. 0. 1 2 3 4 5 6 7 8 9 Biff. 700 845098 845160 845222 845284 845346 845408 845470 845532 845594 845656 62 5718 5780 5842 5904 5966 6028 6090 6151 6213 6275 62 2 6337 6399 6461 6523 6585 6646 6708 6770 6832 6894 62 3 6955 7017 7079 7141 7202 7264 7326 7388 7449 7511 62 4 7573 7634 7696 7758 7819 7881 7943 8004 8066 8128 62 5 8189 8251 8312 8374 8435 8497 8559 8620 8682 8743 62 6 8805 8866 8928' 8989 9051 9112 9174 9235 9297 9358 61 7 9419 9481 9542 9604 9665 9726 9788 9849 9911 9972 61 8 850033 850095 850156 850217 850279 850340 850401 850462 850524 850585 61 9 0646 0707 0769 0830 0891 0952 1014 1075 1136 1197 61 710 851258 851320 851381 851442 851503 851564 851625 851686 851747 851809 61 1 1870 1931 1992 2053 2114 2175 2236 2297 2358 2419 61 2 2480 2541 2602 2663 2724 2785 2846 2907 2968 3029 61 3 3090 3150 3211 3272 3333 3394 3455 3516 3577 3637 61 4 3698 3759 3820 3881 3941 4002 4063 4124 4185 4245 61 5 4306 4367 4428 4488 4549 4610 4670 4731 4792 4852 61 6 4913 4974 5034 5095 5156 5216 5277 5337 5398 5459 61 7 5519 5580 5640 5701 5761 5822 5882 5943 6003 6064 61 8 6124 6185 6245 6306 6366 6427 6487 6548 6608 6668 60 9 6729 6789 6850 6910 6970 7031 7091 7152 7212 7272 60 720 857332 857393 857453 857513 857574 857634 857694 857755 857815 857875 60 1 7935 7995 8056 8116 8176 8236 8297 8357 8417 8477 60 2 8537 8597 8657 8718 8778 8838 8898 8958 9018 9078 60 3 9138 9198 9258 9318 9379 9439 9499 9559 9619 9679 60 4 9739 9799 9859 9918 9978 860038 860098 860158 860218 860278 60 5 860338 860398 860458 860518 860578 0637 0697 0757 0817 0877 60 6 0937 0996 1056 1116 1176 1236 1295 1355 1415 1475 60 7 1534 1594 1654 1714 1773 1833 1893 1952 2012 2072 60 8 2131 2191 2251 2310 2370 2430 2489 2549 2608 2668 60 9 2728 2787 2847 2906 2966 3025 3085 3144 3204 3263 60 730 863323 863382 863442 863501 863561 863620 863680 863739 863799 863858 59 3917 3977 4036 4096 4155 4214 4274 4333 4392 4452 59 2 4511 4570 4630 4689 4748 4808 4867 4926 4985 5045 59 3 5104 5163 5222 5282 5341 5400 5459 5519 5578 5637 59 4 5696 5755 5814 5874 5933 5992 6051 6110 6169 6228 59 5 6287 6346 6405 6465 6524 6583 6642 6701 6760 6819 59 6 6878 6937 6996 7055 7114 7173 7232 7291 7350 7409 59 7 7467 7526 7585 7644 7703 7762 7821 7880 7939 7998 59 8 8056 8115 8174 233 8292 8350 8409 8468 8527 8586 59 9 8644 8703 8762 8821 8879 8938 8997 9056 *9114 9173 59 740 869232 869290 869349 869408 869466 869525 869584 869642 869701 869760 59 1 9818 9877 9935 9994 870053 870111 870170 870228 870287 870345 59 2 870404 870462 870521 870579 0638 0696 0755 0813 0872 0930 58 3 0989 1047 1106 1164 1223 1281 1339 1398 1456 1515 58 4 1573 1631 1690 1748 1806 1865 1923 1981 2040 2098 53 5 2156 2215 2273 2331 2389 2448 2506 2564 2622 2681 58 C 2739 2797 2855 2913 . 2972 3030 3088 3146 3204 3262 58 3321 3379 3437 3495 3553 3611 3669 3727 3785 3844 58 8 3902 3960 4018 4076 4134 4192 4250 4308 4366 4424 58 9 4482 4540 4598 4656 4714 4772 4830 4888 4945 5003 58 750 875061 875119 875177 875235 875293 875351 875409 875466 875524 875582 58 1 5640 5698 5756 5813 5871 5929 5987 6045 6102 6160 58 6218 6276 6333 6391 6449 6507 6564 6622 6680 6737 58 i 6795 6853 6910 6968 7026 7083 7141 7199 7256 7314 58 4 7371 7429 7487 7544 7602 7659 7717 7774 7832 7889 58 5 7947 8004 8062 8119 8177 8234 8292 -S349 8407 8464 57 ( 8522 8579 8637 8694 8752 8809 8866 8924 8981 9039 57 9096 9153 9211 9268 9325 9383 9440 9497 9555 9612 57 8 9669 9726 9784 9841 9898 9956 880013 880070 880127 880185 57 9 880242 880299 880356 880413 880471 880528 0585 0642 0699 0756 57 Ho. O 1 2 3 4 5 6 7 8 9 Diff. TABLE I. LOGARITHMS OF NUMBERS. 13 Ho. 1 2 3 4 5 6 7 8 9 Diff. 760 880814 880871 880928 880985 881042 881099 881156 881213 881271 881328 67 1 1385 1442 1499 1556 1613 1670 1727 1784 1841 1898 57 2 1955 2012 2069 2126 2183 2240 2297 2354 2411 2468 67 3 2525 2381 2638 2695 2752 2809 2866 2923 2980 3037 57 4 3093 3150 3207 3264 3321 3377 3434 3491 3548 3605 57 5 3661 3718 3775 3832 3888 3945 4002 4059 4115 4172 57 6 4229 4285 4342 4399 4455 4512 4569 4625 4682 4739 57 7 4795 4852 4909 4965 5022 5078 5135 5192 5248 5305 67 8 6361 6418 5474 5531 5587 5644 5700 5757 5813 5870 57 9 5926 5983 6039 6096 6152 6209 6265 6321 6378 6434 66 770 886491 886547 886604 886660 886716 886773 886829 886885 886942 886998 66 1 7054 7111 7167 7223 7280 7336 7392 7449 7505 7561 56 2 7617 7674 7730 7786 7842 7898 7955 8011 8067 8123 66 3 8179 8236 8292 8348 8404 8460 8516 8573 8629 8685 66 4 8741 8797 8853 8909 8965 9021 9077 9134 9190 9246 66 5 9302 9358 9414 9470 9526 9582 9638 9694 9750 9806 66 6 9862 9918 9974 890030 890086 890141 890197 890253 890309 890365 66 7 890421:890477890533 0589 0645 0700 0756 0812 0868 0924 66 8 0980 1035 1091 1147 1203 1259 1314 1370 1426 1482 66 9 1537 1593 1649 1705 1760 1816 1872 1928 1983 2039 66 780 892095 892150 892206 892262 892317 892373 892429 892484 892540 892595 66 1 2651 2707 2762 2818 2873 2929 2985 3040 3096 3151 66 2 3207 3262 3318 3373 3429 3484 3540 3595 3651 3706 66 3 3762 3817 3873 3928 3984 4039 4094 4150 4205 4261 66 4 4316 4371 4427 4482 4538 4593 4648 4704 4759 4814 65 1 4870 4925 4980 5036 5091 5146 5201 5257 5312 5367 66 G 5423 5478 5533 5588 5644 5699 5754 5809 5864 5920 66 7 5975 6030 6085 6140 6195 6251 6306 6361 6416 6471 66 8 6526 6581 6636 6692 6747 6802 6857 6912 6967 7022 55 9 7077 7132 7187 7242 7297 7352 7407 7462 7517 7572 55 790 897627 897682 897737 897792 897847 897902 897957 898012 898067 898122 55 1 8176 8231 8286 8341 8396 8451 8506 8561 8615 8670 65 2 8725 8780 8835 8890 8944 8999 9054 9109 9164 9218 55 3 9273 9328 9383 9437 9492 9547 9602 9656 9711 9766 55 4 9821 9875 9930 9985 900039 900094 900149 900203 900258 900312 55 5 900367 900422 900476 900531 0586 0640 0695 0749 0804 0859 65 6 0913 0968 1022 1077 1131 1186 1240 1295 1349 1404 55 7 1458 1513 1567 1622 1676 1731 1785 1840 1894 1948 54 8 2003 2057 2112 2166 2221 2275 2329 2384 2438 2492 54 9 2547 2601 2655 2710 2764 2818 2873 2927 2981 3036 54 800 903090 903144 903199 903253 903307 903361 903416 903470 903524 903578 54 1 3633 3687 3741 3795 3849 3904 3958 4012 4066 4120 54 2 4174 4229 4283 4337 4391 4445 4499 4553 4607 4661 54 3 4716 4770 4824 4878 4932 4986 5040 5094 5148 5202 64 4 5256 5310 5364 5418 5472 5526 5580 5634 5688 5742 64 5 5796 5850 5904 5958 6012 6066 6119 6173 6227 6281 64 6 6335 6389 6443 6497 6551 6604 6658 6712 6766 6820 64 7 6874 6927 6981 7035 7089 7143 7196 7250 7304 7358 64 8 7411 7465 7519 7573 7626 7680 7734 7787 7841 7895 54 9 7949 8002 8056 8110 8163 8217 8270 8324 8378 8431 64 810 908485 908539 908592 908646 908699 908753 908807 908860 908914 908967 64 1 9021 9074 9128 9181 9235 9342 93961 94491 9503 64 2 9556 9610 9663 9716 9770 $B23 9877 9930 : 9984,910037 63, 3 910091 910144 910197 910251 910304 910358 910411 910464,910518 0571 53 4 0624 0678 0731 0784 0838 0891 0944 0998 1051 1104 63 5 1158 1211 1264 1317 1371 1424 1477 1530 1584 1637 63 6 1690 1743 1797 1850 1903 1956 2009 2063 2116 2169 63 7 2222 2275 2328 2381 2435 2488 2541 2594 2647 2700 63 8 2753 2806 2859 2913 2966 3019 3072 3125 3178 3231 63 9 3284 3337 3390 3443 3496 3549 3602 3655 3708 3761 63 No. O 1 2 3 4 5 6 7 8 9 Diff. TABLE I. LOGARITHMS OF NUMBERS. Ho. 1 2 3 4 5 6 7 8 9 riff. 820 913814 913867 913920 913973 914026 914079 914132 914184 914237 914290 53 1 4343 4396 4449 4502 4555 4608 4660 4713 4766 4819 53 2 4872 4925 4977 5030 5083 5136 5189 5241 5294 5347 53 3 5400 5453 5505 5558 5611 5664 5716 5769 5822 5875 53 4 5927 5980 6033 6085 6138 6191 6243 6296 6349 6401 53 5 6454 6507 6559 6612 6664 6717 6770 6822 6875 6927 53 6 6980 7033 7085 7138 7190 7243 7295 7348 7400 7453 53 7 7506 7558 7611 7663 7716 7768 7820 7873 7925 7978 52 8 8030 8083 8135 8188 8240 8293 8345 8397 8450 8502 52 9 8555 8607 8659 8712 8764 8816 8869 8921 8973 9026 52 830 919078 919130 919183 919235 919287 919340 919392 919444 919496 919549 - 52 1 9601 9653 9706 9758 9810 9862 9914 9967 920019 920071 52 2 920123 920176 920228 920280 920332 920384 920436 920489 0541 0593 52 3 0645 0697 0749 0801 0853 0906 0958 1010 1062 1114 52 4 1166 1218 1270 1322 1374 1426 1478 1530 1582 1634 52 5 1686 1738 1790 1842 1894 1946 1998 2050 2102 2154 52 6 2206 2258 2310 2362 2414 2466 2518 2570 2622 2674 52 7 2725 2777 2829 2881 2933 2985 3037 3089 3140 3192 52 8 3244 3296 3348 3399 3451 3503 3555 3607 3658 3710 52 9 3762 3814 3865 3917 3969 4021 4072 4124 4176 4228 52 840 924279 924331 924383 924434 924486 924538 924589 924641 924693 924744 52 1 4796 4848 4899 4951 5003 5054 5106 5157 5209 5261 52 2 5312 5364 5415 5467 5518 5570 5621 5673 5725 5776 52 3 5828 5879 5931 5982 6034 6085 6137 6188 6240 6291 51 4 6342 6394 6445 6497 6548 6600 6651 6702 6754 6805 51 5 6857 6908 6959 7011 7062 7114 7165 7216 7268 7319 51 6 7370 7422 7473 7524 7576 7627 7678 7730 7781 7832 51 7 7883 7935 7986 8037 8088 8140 8191 8242 8293 8345 51 8 8396 8447 8498 8549 8601 8652 8703 8754 8805 8857 51 9 8908 8959 9010 9061 9112 9163 9215 9266 9317 9368 51 850 929419 929470 929521 929572 929623 929674 929725 929776 929827 929879 51 1 9930 9981 930032 930083 930134 930185 930236930287 930338 930389 51 2 930440 930491 0542 0592 0643 0694 0745 0796 0847 0898 51 3 0949 1000 1051 1102 1153 1204 1254 1305 1356 1407 51 4 1458 1509 1560 1610 1661 1712 1763 1814 1865 1915 51 5 1966 2017 2068 2118 2169 2220 2271 2322 2372 2423 51 6 2474 2524 2575 2626 2677 2727 2778 2829 2879 2930 51 7 2981 3031 3082 3133 3183 3234 3285 3335 3386 3437 51 8 3487 3538 3589 3639 3690 3740 3791 3841 3892 3943 51 9 3993 4044 4094 4145 4195 4246 4296 4347 439Y 4448 51 860 934498 934549 934599 934650 934700 934751 934801 934852 934902 934953 50 1 5003 5054 5104 5154 5205 5255 5306 5356 5406 5457 50 2 5507 5558 5608 5658 5709 5759 5809 5860 5910 5960 50 3 6011 6061 6111 6162 6212 6262 6313 6363 6413 6463 50 4 6514 6564 6614 6665 6715 6765 6815 6865 6916 6966 50 5 7016 7066 7117 7167 7217 7267 7317 7367 7418 7468 50 6 7518 7568 7618 7668 7718 7769 7819 7869 7919 7969 50 7 8019 8069 8119 8169 8219 8269 8320 8370 8420 8470 50 8 8520 8570 8620 8670 8720 8770 8820 8870 8920 8970 50 9 9020 9070 9120 9170 9220 9270 9320 9369 9419 9469 50 870 939519 939569 939619 939669 939719 939769 939819 939869 939918 939968 50 1 940018 940068 940118 940168 940218 940267 940317 940367 940417 940467 50 2 0516 0566 0616 0666 0716 0765 0815 0865 0915 0964 50 3 1014 1064 1114 1163 1213 1263 1313 1362 1412 1462 50 4 1511 1561 1611 1660 1710 1760 1809 1859 1909 1958 50 6 2008 2058 2107 2157 2207 2256 2306 2355 2405 2455 50 6 2504 2554 2603 2653 2702 2752 2801 2851 2901 2950 50 7 3000 3049 3099 3148 3198 3247 3297 3346 3396 3445 49 8 3495 3544 3593 3643 3692 3742 3791 3841 3890 3939 49 9 3989 4038 4088 4137 - 4186 4236 4285 4335 4384 4433 49 No! O 1 2 3 4 5 6 7 8 9 Diff. TABLE I. LOGARITHMS OF NUMBERS. No. O 1 2 3 4 5 6 7 - 8 9 " Diff. 880^944483 944532 944581 944631 944680 944729 944779 944828 944877 944927 49, 1 4976 5025 5074 5124 5173 5222 5272 5321 5370 5419 49 I 5469 5518 5567 5616 5665 5715 5764 5813 5862 5912 49 3 5961 6010 6059 6108 6157 6207 6256 6305 6354 6403 49 4 6452 6501 6551 6600 6649 6698 6747 6796 6845 6894 49 5 6943 6992 7041 7090 7140 7189 7238 7287 7336 7385 49 6 7434 7483 7532 7581 7630 7679 "7728 7777 7826 7875 49 7 7924 7973 8022 8070 8119 8168 8217 8266 8315 8364 49 8 8413 8462 8511 8560 8609 8657 8706 8755 8804 8853 49 9 8902 8951 8999 9048 9097 9146 9195 9244 9292 9341 49 890949390 1| 9878 2950365 949439949488 9926 9975 950414 950462 949536949585 950024950073 0511 0560 949634 950121 0608 949683 949731 9501701950219 0657 0706 949780 950267 0754 949829 950316 0803 49 49 49 3! 0851 0900 0949 0997 1046 1095 1143 1192 1240 1289 49 4 1338 1386 1435 1483 1532 1580 1629 1677 1726 1775 49 5 1823 1872 1920 1969 2017 2066 2114 2163 2211 2260 48 6 2308 2356 2405 2453 2502 2550 2599 2647 2696 2744 48 7 2792 2841 2889 2938 2986 3034 3083 3131 3180 3228 48 8 3276 3325 3373 3421 3470 3518 3566 3615 3663 3711 48 9 3760 3808 3856 3905 3953 4001 4049 4098 4146 4194 48 900 954243 954291 954339 954387 954435 954484 954532 954580 954628 C54677 48 1 4725 4773 4821 4869 4918 4966 5014 5062 5110 5158 48 2 5207 5255 5303 5351 5399 5447 5495 5543 5592 5640 48 3 5688 5736 5784 5832 5880 5928 5976 6024 6072 6120 48 4 6168 6216 6265 6313 6361 6409 6457 6505 6553 6601 48 5 6649 6697 6745 6793 6840 6888 6936 6984 7032 7060 48 6 7128 7176 7224 7272 7320 7368 7416 7464 7512 7559 48 7 7607 7655 7703 7751 7799 7847 7894 7942 7990 8038 48 8 8086 8134 8181 8229 8277 8325 8373 8421 8468 8516 48 9 8564 8612 8659 8707 8755 8803 8850 8898 8946 8994 48 910 959041 959089 959137 959185 959232 959280 959328 959375 I 959423;959471 48 1 9518 9566 9614 9661 9709 9757 9804 9852 9900| 9947 48 2 9995 960042 960090 960138 960185 960233 960280 : 960328 960376 t;i4-.'3 48 3 960471 0518 0566 0613 0661 0709 0756 08041 0851 0899 48 4 0946 0994 1041 1089 1136 1184 1231 1279 1326 1374 47 5 1421 1469 1516 1563 1611 1658 1706 1753 1801 1848 47 6 1895 1990 2038 2085 2132 2180 2227 2275 2322 47 7 2369 2417 2464 2511 2559 2606 2653 2701 2748 2795 47 8 2843 2890 2937 2985 3032 3079 3126 3174 3221 3268 47 9 3316 3363 3410 3457 3504 3552 3599 3646 3693 3741 47 920'963788 963835 963882 963929 963977 964024 964071 964118 964165 964212 47 1 4260 4307 4354 4401 4448 4495 4542 4590 4637 4684 41 2 4731 4778 4825 4812 4919 4966 5013 5061 5108 6155 47 3 5202 5249 5296 5343 5390 5437 5484 5531 5578 5625 47 4 5672 5719 5766 5813 5860 5907 5954 6001 6048 6095 47 5 6142 6189 6236 6283 6329 6376 6423 6470 6517i 6564 47 6 6611 6658 6705 6752 6799 6845 6892 6939 6986< 7033 47 7 7080 7127 7173 7220 7267 7314 7361 7408 7454 i 7501 47 8 7548 7595 7642 7688 7735 7782 7829 . 7875 7922 7969 47 9 8016 8062 8109 8156 8203 8249 8296 8343 8390 8436 47 930 968483 968530 968576 968623 968670 968716 968763 968810968856968903 47 1 8950 8996 90431 9090 9136 9183 9229 9276 9323 9369 47 2 9416 3 9882 9463 9928 9509 9556 9975! 970021 9602 970068 9649 970114 9695 970161 9742 970207 9789 970254 9835 970300 47 47 41970347 970393 970440 0486 0533 0579 0626 0672 0719 0765 46 5 0812 0858 0904 0951 0997 1044 1090 1137 1183 1229 46 6 1276 1322 1369 1415 1461 1508 1554 1601 1647 1693 46 7 1740 1786 1832 1879 1925 1971 2018 2064 2110 2157 46 8 2203 2249 2295 2342 2388 2434 2481 2527 2573 2619 46 9 2666 2712 2758 2804 2851 2897 2943 2989 3035 3082 46 No. O 1 2 3 4 5 6 7 8 9 Diff. 16 TABLE I. LOGAKITHMS OF NUMBERS. No. O 1 2 3 4 5 6 7 8 9 iff. 940 973128 973174 73220 73266 73313 73359 73405 )73451 J73497 973543 46 1 3590 3636 3682 3728 3774 3820 3866 3913 3959 4005 46 2 4051 4097 4143 4189 4235 4281 4327 4374 4420 4466 46 3 4512 4558 4604 4650 4696 4742 4788 4834 4880 4926 46 4 4972 5018 5064 5110 5156 5202 5248 5294 5340 5386 46 6 5432 5478 5524 5570 5616 5662 5707 5753 5799 5845 46 6 5891 5937 5983 6029 6075 6121 6167 6212 6258 6304 46 7 6350 6396 6442 6488 6533 6579 6625 6671 6717 6763 46 g 6808 6854 6900 6946 6992 7037 7083 7129 7175 7220 46 9 7266 7312 7358 7403 7449 7495 7541 7586 7632 7678 46 950 977724 977769 77815 77861 77906 77952 77998 978043 978089 978135 46 1 8181 8226 8272 8317 8363 8409 8454 8500 8546 8591 46 2 8637 8683 8728 8774 8819 8865 8911 8956 9002 9047 46 3 9093 9138 9184 9230 9275 9321 9366 9412 9457 9503 46 4 9548 9594 9639 9685 9730 9776 9821 9867 9912 9958 46 5 80003 980049 980094 80140 80185 80231 80276 980322 980367 980412 45 6 0458 0503 0549 0594 0640 0685 0730 0776 0821 0867 45 7 0912 0957 1003 1048 1093 1139 1184 1229 1275 1320 45 8 1366 1411 1456 1501 1547 1592 1637 1683 1728 1773 45 9 1819 1864 1909 1954 2000 2045 2090 2135 2181 2226 45 960 82271 982316 982362 982407 982452 982497 82543 982588 982633 982678 45 1 2723 2769 2814 2859 2904 2949 2994 3040 3085 3130 45 2 3175 3220 3265 3310 3356 3401 3446 3491 3536 3581 45 3 3626 3671 3716 3762 3807 3852 3897 3942 3987 4032 45 4 4077 4122 4167 4212 4257 4302 4347 4392 4437 4482 45 6 4527 4572 4617 4662 4707 4752 4797 4842 4887 4932 45 6 4977 5022 5067 5112 5157 5202 5247 5292 5337 5382 45 7 5426 5471 5516 5561 5606 5651 5696 6741 5786 5830 45 g 5875 5920 5965 6010 6055 6100 6144 6189 6234 6279 45 9 6324 6369 6413 6458 6503 6548 6593 6637 6682 6727 45 /970 986772 986817 986861 986906 986951 986996 987040 987085 987130 987175 45 1 7219 7264 7309 7353 7398 7443 7488 7532 7577 7622 45 2 7666 7711 7756 7800 7845 7890 7934 7979 8024 8068 45 3 8113 8157 8202 8247 8291 8336 8381 8425 8470 8514 45 4 8559 8604 8648 8693 8737 8782 8826 8871 8916 8960 45 5 9005 9049 9094 9138 9183 9227 9272 9316 9361 9405 45 ( 9450 9494 9539 9583 9628 9672 9717 9761 9806 9850 44 *3 9895 9939 9983 990028 990072 990117 990161 990206 990250 990294 44 8 990339 990383 990428 0472 0516 0561 0605 0650 0694 0738 44 9 0783 0827 0871 0916 0960 1004 1049 1093 1137 1182 44 980 991226 991270 991315 991359 991403 991448 991492 991536 991580 991625 44 1 1669 1713 1758 1802 1846 1890 1935 1979 2023 2067 44 2111 2156 2200 2244 2288 2333 2377 2421 2465 2509 44 2554 2598 2642 2686 2730 2774 2819 2863 2907 2951 44 j 2995 3039 3083 3127 3172 3216 3260 3304 3348 3392 44 5 3436 3480 3524 3568 3613 3657 3701 3745 3789 3833 44 ( 3877 3921 3965 4009 4053 4097 4141 4185 4229 4273 44 4317 4361 4405 4449 4493 4537 4581 4625 4669 4713 44 j 4757 4801 4845 4889 4933 4977 5021 5065 6108 5152 44 9 5196 5240 5284 5328 5372 5416 5460 5504 6547 6591 44 990 995635 995679 995723 995767 995811 995854 995898 995942 995986 996030 44 6074 6117 6161 6205 6249 6293 6337 6380 6424 6468 44 i 6512 6555 6599 6643 6687 6731 6774 6818 6862 6906 44 ! 6948 6993 7037 7080 7124 7168 7212 7255 7299 7343 44 ] 7386 7430 7474 7517 756 7605 7648 7692 7736 777 44 j 7823 7867 7910 7954 799 804 8085 8129 8172 821 44 I 825S 8303 8347 8390 8434 847 8521 8564 8608 865 44 869E 8739 8782 8826 886 891 8956 9000 904S 908 44 1 9131 9174 9218 926 930 934 9392 9435 947 952 44 956 9609 9652 9696 973 978. 9826 987C 991J 995 43 No O 1 2 3 4 5 6 7 8 9 riff. TABLE II. NATURAL SINES AND COSINES. 18 TABLE II. NATURAL SINES AND COSINES. ine.l Cos. 0000 0029 0058 0087 0116 0145 0175 0204 0233 0262 K)291 X)320 W349 0378 )0407 K)436 X)i65 )0495 )0524 )0553 W582 K)611 )0640 )0669 W698 X)727 )0756 30785 00814 00844 One. One. One. One. One. One". One. One. One. One. One. 99999 .99998 .99997 .99996 .99996 009021.99996 00931 .99996 00960 .99995 01018 .99995 01047 .99995 01076 .9999; 01105 .9999 01134 .9999 01164 .99991 01193 01222 01251 01280 01309 01338 01367 01396 01425 01454 01483 01513 01542 ,01571 ,01600 .01629 .01658 .01687 .01716 .01745 9999, 9999 9999 9999 .9998 .9998 .999? Cos. i Sin 89~ ne. Cos. 1745 .99985 17741.99984 18031.99984 1832 .99983 1862 .99983 1891 .99982 1920'. 99982 1349. 00981 Sine. Cos. 1978 2007 2036 2065 .09980 .99979 6\JVO . J7i7Mi7 209 1 1 . 99978 2123 1 . 99977 2152'.99977 21811.99976 2211 .99976 2240 .99975 2269,. 99974 2298 .99974 2327!. 99973 2356 .99972 02385 ! .99972 32414 .99971 32443 .99970 02472 ! . 99969 025011.99969 32530 .99968 32560 1 . 99967 32589;. 99966 02618 .99966 92647 . 02676 .99964 02705 .99963 02734 .99963 02763 .99962 02792 .9996: 02821 . 99961 02850 .9995! 02879 .99951 02908 .9995! 02938 .9995 02967 .9995 02996 .9995 ,03490 .03519 .03548 .03577 03606 03635 03664 03693 3723 3752 3781 3810 3839 3868 3897 392G 3955 03025 03054 03083 03112 03141 03170 03199 03228 03257 .9995 .9995 .9995 .9995 .9995 .9995 .9994 .9994 03286 .9994 033161.9994 ,03345 ! . 9994 .03374^.9994 .03403 .9994 .034321.9994 . 03461 j. 999^ Cos. I Sin 88" Sine.' Cos. .05234 .99863 .99936 .99935 05263 .05292 .05321 .05350 .05379 .99933 ,99931 34013 34042 4071 34100 04129 34159 34188 34217 04246 3427E 04304 04333 04362 04391 0442( 0444C 0447* 04507 04536 04565 04594, 04623 04653 04682 04711 04740 ,04769 ,04798 ,04827 ,04856 .04885 .04914 .0494J .0497', .05001 .0503C .05058 .05088 .0511 .05146 .05171 .05205 .05234 99929 99927 99926 99925 99924 99923 99922 99921 ,9991 ,99918 .99861 .99858 .99857 .99855 4 Sine, i Cos. 99756 99754 99752 .05408 .99854' .05437:. 99852 .05466 .99851 .05495 .99849 . 05524 j. 99847 .05553 .99846 . 05582 1. 99844 ,05611 .99842 .05640 .99841 ,99916 ,99915 .99913 .99912 .999111 .99910 .99909 .99907 .99906 '.05727|.99836 .05756;. 99834 05785 !. 99833 .05814 .99831 .05844 .99829 .05873:. 99827 .05902 .99826 .05931 .99824 .05960 .99822 05989 .99821 06018 .9981S 99904 99902 99901 99900 99898 99891 99896 99894 ,9987, .9987 .9987 .9987 .9986' .9986 .9986 .99864 Cos. Sine 87' 06047 06076 06105 06134 .99817 .99811 .9981; .9981! .06970 .07005 .07034 .07063 .07092 .07121 07150 07179 07208 07237 07266 37295 07324 07353 07382 07411 0744C 07469 07498 07527 07556 0758 0761 0764. 07672 07701 07730 07759 06163 06192 06221 06250 . 06279 .9980 06308 0633 06366 06424 06453 .06482 ,06511 .06540 06569 .06627 .06656 _____ .9979 .9979 9979 .9979 .9979( .9978 .9978 .9978 .9978 .99781 .9977 06714 .06743 .06773 .06802 .06831 .06860 .06889 .06918 ,06947 .06976 .99746 .99744 .9974C .99738 .99736 .99734 .9973 .99729 .9972 9972! .99719 .99716 .99714 .99712 .99710 .99708 .99705 99703 99701 M. 60 59 58 57 56 55 54 53 52 61 GO 49 48 47 46 45 44 43 12 41 40 07817 07846 ,07875 07004 07 in 99377 99374 99370 99367 '.09498 .09527 .09556 .09585 .09614 .09642 .09671 .09700 1 .09729 .09758 .09787 .09816 .09845 .09874 .09903 .09932 .09961 99156 .14695 99152 .14723 99344 99341 99337 99334 99331 99327 99324 99320 99317 99314 99310 99523 99520 99517 99514 995111 99508 99506 99503 .16706 .16734 .16763 .16792 .16820 .16849 .16878 .16906 .16935 9SS.-4 98849 96846 96841 98836 .10019 .10048 .10077 .10106 .10135! .10164 .10192! .10221 .10250 .10279! .10308! .10337 13514 13543 13572 13600 .15241 .15270 .15299 .15827 .15356 .15385 15414 .15442 .15471 .15500 .15529 .15557 .15586 .15615 .15643 .17021 .17050 .17078 .17107 .17136 .17164 .17193 .17222 .17250 .17279 .17308 ,17336 ,17365 .99482 ,9947 .9947 . 99473 ; .99470 .99467 .99464 .99461J i 99458 .99455 .99452 11927 11956 i ,11985! ,12014! ,12043 .12071! .121001 .12129 .12158 .12187 98516 98511 98506 98501 98496 99055 99051 99047 90043 98796 98791! 98787 98782 98778 98773 13744 13773 13802 13831 20 TABLE ii. NATURAL SINES AND COSINES. 1O 11 12 13 14 M. Sine. Cos. Sine. Cos. Sine. Cos. Sine. Cos. Sine. Cos. M. 17365 .98481 .19081 .98163 .20791 .97815 .22495 97437 24192 97030 60 1 17393 .98476 .1U109 .98157 .20820 .97809 .22523 97430 24220 97023 59 2 17422 .98471 .19138 98152 .20848 .97803 .22552 97424 24249 97015 58 3 17451 .98466 .19107 98146 .20877 .97797 .22580 97417 24277 97008 57 4 17479 .984G1 19195 98140 .20905 .97791 .22608 97411 24305 .97001 56 5 17508 .98455 19224 98135 .20933 .97784 .22637 97404 24333 .96994 55 6 17537 .98450 .19252 98129 .20962 .97778 .22665 97398 24362 .96987 54 7 17565 .98445 .19281 93124 .20990 .97772 .22693 97391 24390 .96980 53 8 17594 .98440 .19309 93118 .21019 .97766 .22722 97384 24418 .96973 52 9 17623 .98435 19338 98112 .21047 .97760 .22750 97378 24446 .96966 51 10 17651 .98430 .19366 98107 .21076 .97754 .22778 97371 24474 .96959 50 11 17680 .98425 19395 98101 .21104 .97748 .22807 97365 24503 .96952 49 12 17708 .98420 .19423 98096 .21132 .97742 .22835 97358 24581 .96945 48 13 17737 .98414 .19452 98090 .211C1 .9773.5 .22863 97351 .24559 .96937 47 14 17766 .98409 .19481 98084 .21139 .97729 .22892 97345 .24587 .96930 46 15 17794 .98404 .19509 .98079 .21218 .97723 .22920 97338 .24615 .96923 45 16 17823 .98399 .19538 98073 .21246 .97717 .22948 97331 .24644 .96916 44 17 17852 .98394 .19566 98067 .21275 .97711 .22977 97325 .24672 .96909 43 18 17880 .98389 .19595 98061 .21303 .97705 .23005 .97318 .24700 .96902 42 19 17909 .98383 .19623 98056 .21331 .97698 .23033 .97311 .24728 .96894 41 20 17937 .98378 .19652 98050 .21360 .97692 .23062 .97304 .24756 .96887 40 21 17966 .98373 .19680 98044 .21388 .97686 .23090 .97298 .24784 .96880 39 22 17995 .98368 .19709 98039 .21417 .97680 .23118 .97291 .24813 .96873 38 23 18023 .98362 .19737 98033 .21445 .97673 .23146 .97284 .24841 .96866 37 24 18052 .98357 .19766 98027 .21474 .97667 .23175 .97278 .24869 .96858 36 25 18081 .98352 19794 98021 .21502 .97661 .23203 .97271 .24897 .96851 35 26 18109 .98347 .19823 98016 .21530 .97655 .23231 .97264 .24925 .96844 34 27 18138 .98341 19851 98010 .21559 .97648 .23260 .97257 .24953 .96837 33 28 18166 .98336 19880 98004 .21587 .97642 .23288 .97251 .24982 .96829 32 29 18195 .98331 19908 97998 .21616 .97636 .23316 .97244 .25010 .96822 31 30 18224 .98325 .19937 97992 .21644 .97630 .23345 .97237 .25038 .96815 30 31 18252 .98320 .19965 97987 .21672 .97623 .23373 .97230 .25066 .96807 29 32 18281 .98315 .19994 97981 .21701 .97617 .23401 .97223 .25094 .96800 28 33 18309 .98310 .20022 97975 .21729 .97611 .23429 .97217 .25122 .96793 27 34 18338 .98304 .20051 97969 .21758 .97604 .23458 .97210 .25151 .96786 26 35 18367 .98299 .20079 97963 .21786 .97598 .23486 .97203 .25179 .96778 25 36 18395 .98294 .20108 97958 .21814 .97592 .23514 .97196 .25207 .96771 24 37 .18424 .98288 .20136 97952 .21843 .97585 .23542 .97189 .25235 .96764 23 38 .18452 .98283 .20165 97946 .21871 .97579 .23571 .97182 .25263 .96756 22 39 .18481 .98277 .20193 97940 .21899 .97573 .23599 .97176 .25291 .96749 21 40 .18509 .98272 .20222 .97934 .21928 .97566 .23627 .97169 .25320 .96742 20 41 .18538 .98267 .20250 .97928 .21956 .97560 .23656 .97162 .25348 .96734 19 42 .18567 .98261 .20279 .97922 .21985 .97553 .23684 .97155 .25376 .9672" 18 43 .18595 .98256 .20307 .97916 .22013 .97547 .23712 .97148 .25404 .96719 17 44 .18624 .98250 .20336 .97910 .22041 .97541 .23740 .97141 .25432 .96712 16, 45 .18652 .98245 .20364 .97905 .22070 .97534 .23769 .97134 .25460 .96705 15 46 .18681 .98240 .20393 .97899 .22098 .97528 .23797 .97127 .25488 .96697 14 47 .18710 .98234 .20421 .97893 .22126 .97521 .23825 .97120 .25516 .96690 13 48 .18738 .98229 .20450 .97887 .22155 .97515 .23853 .97113 .25545 .96682 12 49 .18767- .98223 .20478 .97881 .22183 .97508 .23882 .97106 .25573 .96675 11 50 .18795 .98218 .20507 .97875 .22212 .97502 .23910 .97100 .25601 .96667 10 51 .18824 .98212 .20535 .97869 .22240 .97496 .23938 .97093 .25629 .96660 9 52 .18852 .98207 .20563 .97863 .22268 .97489 .23966 .97086 .25657 .96653 8 53 .18881 .98201 .20592 .97857 .22297 .97483 .23995 .97079 .25685 .96645 7 54 .18910 .98196 .20620 .97851 .22325 .97476 .24023 .97072 .25713 .96638 6 55 .18938 .98190 .20649 .97845 .22353 .974701.24051 .97065 .25741 .96630 5 56 .18967 .98185 .20677 .97839 .22382 .97463 .24079 .97058 .25769 .96623 4 57 .18995 .98179 .20706 .97833 .22410 .97457 .24108 .97051 .25798 .96615 3 58 .19024 .98174 .20734 .97827 .22438 .97450 .24136 .97044 .25826 .96608 2 59 .19052 .98168 .20763 .97821 .22467 .97444 .24164 .97037 .25854 .96600 1 60 .19081 .98163 .20791 .97815 .22495 .97437 .24192 .97030 .25882 .96593 M. COS. Sine. Cos. Sine. Cos. Sine. COS. Sine; Cos. Sine. M. 79" 78 77 76 75 TABLE II. NATURAL SINES AND COSINES. 21 13" 16 17 18 19* M. Sine. Cos. Sine.: Cos. Sine. Cos. Sine. Cos. Sine. | Cos. M. .25882 .96593 .27564 1 . 96126!. 29237 .95630 .30902 .95106 .32557 .94552 ^50 1 .25910 .96585 .27592 .961 18 1.29265 .95622 .30929 .95097 .32584 .94542 59 2 .25938 .96578 .27620 .961 10 .29293 .95613 .30957 .95088 .32612 .94533 58 3 .25966 .96570 .27648 1.96102!. 29321 .95605 .30985 .95079 .32639 .9452b 57 4 .25994 .96562 .27676 .96094 .29348 .95596 .31012 .95070 .32667! .945141 56 5 i. 26022 .96555 .27704 .96086 .29376 .95588 .31040 .95061 .32694! .94504; 55 6 .26050 .96547 .27731 .96078 .29404 .95579 .31068 .95052 .327221. 944951 54 7 .2607$ .96540 .27759 .96070 .29432 .95571 .31095 .95043 .32749 .94485 53 8 2610" .96532 .27787 .96062 .29460 .95562 .31123 .95033 .32777 .94476 52 9 .26135 .96524 .27815 .96054 .29487 .95554 .31151 .95024 .32804 .94466 51 10 . 26163 .96517 .27843 .96046 .29515 .95545 .31178 '. 95015 .32832 .94457] 50 11 i 26191 .96509 .27871 .96037 .29543 .95536 .31206 .95006 .32859 .94447 49 12 '.26219 .96502 .27899 .96029 .29571 .95528 .31233 .94997 .32887 .94438 48 13 1.26247 .96494 .27927 .96021 .29599 .95519 .31261 .94988 .32914 .94428 47 14 .26275 .96486, .27955 .96013 1 . 29626 .95511 1. 312891.94979 .32942 .94418 46 15 .26393.96479 .27983 .96005 .29654,^86502 -.31316, .94970 .32969 .94409 45 16 .26331!. 96471 .28011 .95997 .29682 1 . 95493 .31344 .94961 .32997 .94398 44 17 1.26359 .06463 .28039 .95989 .29710 .95485 .31372 .94952 .33024 .94390 43 18 .26387 .96456 .28067 .95981 .29737 .95476 .3139S .94943 .33051 .94380 42 19 .26415 .96448 .28095 .95972 .29765 .95467 .31427 .94933 .33079 .94370 41 20 .26443 .96440 .28123 .95964 .29793 .95459 .31454 .94924 .33106 .94861 40 21 .26471 .96433 .28150 .95956 .29821 .95450 .31482 .94915 .33134 .94351' 39 22 .26500 .96425 .28178 .95948 .29849 .95441 1.315101.94906 .33161 .94342 38 23 .26528 .96417 .28206 .95940 .29876 ,954.x .31537 .94897 .33189 .94332 37 24 .26556 .96410 .28234 .95931 .2M<-4 .95424 I.3156E .P4s>> .33216 .94322 36 25 .26584 .96402 .28262 .95923 .29932 .95415!. 315931 .94878 .33244 .94313 35 26 .26612 .96394 .28290 .95915 .29960 .95407 .3162C .94869 .33271 .94303 34 27 .26640 .96386 .28318 .95907 .29987 .95398 .31G4 .94860 .33298 .94293 33 28 .26668 .96379 .28346 .95898 .30015 .95389 .31675 . 94851 1.3332C .94284 32 29 .26696 .96371 .28374 .95890 .30043 .95380 .3170: .94842 .33353 .94274 31 30 .26724 .96363 .28402 .96882 .30071 .95372 j .31730 .94832 .33381 .94264 30 31 .26752 .96355 .28429 .95874 .30098! .95363 1 ,. 31758! .94823 .33408 .94254 29 32 L 26780 .96347 .28457 .95865 .30126 .95354 .3178 .94814 .33436 .94245 28 33 1.26808 .96340 .284851.95857 .30154 .95345 .3181J .94805 ;. 33463 .94236 27 34 1.26836 .96332 .28513 .95849 .30182!. 95337J. 31841 .94795 j.334% .94225 26 35 1.26864 .96324 .28541! .95841 j .30209' .9532? .31868 , .94786 .33518 .94215 25 36 i. 26892 .96316 .285691.95832 .302371.95318 .3189( .94777 1.33545 .94206 24 37 i. 26920 .96308 .28597 . 95824 1.30265 1 . 95310 .319231.94768 .33573 .94196 23 38 .26948 .96301 .28625 .95816 .30292J .95301 .31951 .94758 .33600 .94186; 22 39 .26976 .96293 .28652 .95807 .30320| .95293 .31979|. 94749 .33627 .94176: 21 40 .27004 .96285 .28680 .95799 .30348| .95284 .32001 ,.94740 . 33655 .94167, 20 41 42 .27032 .27060 .96277 .96269 .28708 .95791 .30376 .95275 .32034! .9473C .28736 .95782 .30403 .95266 .320611.94721 .33682 .94157' 19 .337101.94147 18 43 .27088 .96261 .28764 .95774 .30431 .95251 .3208J ,.94712 .33737 .94137 17 44 .27116 .96253 .28792 .95766 .30459 .95248 .32116] . 94702 .33764 .94127 Iff 45 .27144 .96246 .28820 .95757 . 304861. 9524C .32144 ,.94693 .33792 .94118 15 46 .27172 ! . 96238 .28847 .95749 .30514 .95231 .32171 1. 94684 .33819 .94108 14 47 .27200 .96230 .28875 .95740 .30542 .95222 .321& .94674 .33846 .94098 13 48 .27228'. 96222 .28903 .95732 .30570 .95213 .322271. 94665). 33874 .94088 12 49 .27256 .96214 .28931 .95724 .30597 .95204 .32254 i. 94656 1.33901 .94078 11 50 .27284 .96206 .28959 .95715 .30625 .9519S .32281 .94646 1.33929 .94068 10 51 .273121.96198 .28987 .95707 .30653 .95186 .323091.94637 1.33956 .94068 9 52 .27340 .96190 .29015 .95698 .30680.95177 .3233" .94627 .33983 .94049 8 53 .27368 1 . 96182 .29042 .95690 .30708 .95168 .32364|.94618 .34011 .94039 7 54 .27396 .96174 .29070 .95681 . 307361. 9515 .32391 .94609 j. 34038 .94029 6 55 .274241.96166 .29098 .95673 . 30763 . 95150 . 32419 ;. 94599 .34C65 .94019 5 56 .27452 .96158 .29126 .95664 .30791 .95142 .3244" .94590 .34093 .94009 4 57 .27480;. 96150 .29154 .95656 .308191.95133 .32474:. 94580 .34120 .93999 3 53 .27508J.96142 .29182 .95647 .30846 .95124 '.32502 .94571 .34147 .93989 2 59 .27536 .96134 .29209 .95639 .308741.95115 .3252S .94561 .34175 .93979 1 60 .27564 '96126 .29237 .956301.30902 .95106 .32557 !. 94552 j. 34202 .93969 M. Cos. Sine. COS. Sine. Cos. | Sine. Cos. 1 Sine. Cos. Sine. M. 74' 73' 72 71' 70' TABLE II. NATURAL SINES AND COSINES. 2 21 22 22 24 M. Sine. Cos. Sine. Cos. Sine. Cos. Sine.] Cos. Sine. Cos. VI. 34202 93969 35837 93358 37461 92718 39073 92050 . 40874 91355 60 1 34229 93959 35864 93348 37488 92707 39100 92039 40700 91343 59 2 34257 93949 35891 93337 37515 92697 39127 92028 40727 91331 58 3 34284 93939 35918 93327 37542 92686 39153 92016 40753 91319 57 4 34311 93929 35945 93316 37569 92675 39180 92005 40780 91307 56 5 34339 93919 35973 93306 37595 92664 39207 91994 40806 91295 55 6 34366 93909 36000 93295 37622 92653 39234 91982 40833 91283 54 7 34393 93899 36027 93285 37649 92642 39260 91971 40860 91272 53 8 34421 93889 36054 93274 37676 92631 39287 91959 40886 91260 52 9 34448 93879 36081 93264 37703 92620 39314 91948 40913 91248 51 10 34475 93869 36108 93253 .37730 92609 39341 91936 40939 91236 -50 11 34503 93859 36135 93243 .37757 92598 39367 91925 .40966 91224 49 12 34530 93849 .36162 93232 .37784 92587 .39394 91914 .40992 91212 48 13 34557 93839 36190 93222 .37811 92576 .39421 91902 .41019 91200 47" 14 34584 93829 36217 93211 .37838 92565 .39448 91891 .41045 91188 46 15 34612 93819 36244 93201 .37805 92554 .39474 91879 .41072 91176 45 16 34639 93809 36271 93190 .37892 92543 .39501 91868 .41098 91164 44 17 34666 93799 36298 93180 .37919 92532 .39528 .91856 .41125 .91152 43 18 34694 93789 36325 93169 .37946 92521 .39555 ,91845 .41151 .91140 42 19 34721 93779 36352 93159 .37973 92510 .39581 .91833 .41178 .91128 41 20 34748 93769 36379 93148 .37999 92499 .39608 .91822 .41204 .91116 40 21 34775 .93759 36406 93137 .38026 92488 .39635 .91810 .41231 .91104 39 22 34803 .93748 36434 93127 .38053 92477 .39661 .91799 .41257 .91092 38 23 34830 .93738 36461 93116 .38080 92466 .39688 .91787 .41284 .91080 37 24 34857 .93728 36488 93108 .38107 92455 .39715 .91775 .41310 .91068 36 25 34884 .93718 36515 93095 .38134 92444 .39741 .91764 .41337 .91056 35 26 34912 .93708 .36542 93084 .38161 92432 .39768 .91752 .41363 .91044 34 27 34939 .93698 .36569 93074 .38188 .92421 .39795 .91741 .41390 .91032 33 28 34966 .93688 .36596 93063 .38215 .92410 .39822 .91729 .41416 .91020 32 29 34993 .93677 .36623 93052 .38241 .92399 .39848 .917181.41443 .91008 31 30 35021 .93667 .36650 93042 .38268 .92388 .39875 .91706 .41469 .90996 30 31 35048 .93657 .36677 93031 .38295 .9237 .39902 .91694 .41496 .90984 29 32 .35075 .93647 .36704 93020 .38322 .9236 .39928 .91683 .4152 .90972 28 33 .35102 .93637 .36731 93010 .38349 .9235 .39955 .91671 .4154 .90960 27 34 .35130 .93626 .36758 92999 .38376 .9234 .39982 .91660 .4157 .90948 26 35 .35157 .93616 .36785 92988 .38403 .9233 .4000 .91648 .4160 .90936 25 36 .35184 .93606 .36812 92978 .38430 .9232 .40035 .91636 .4162 .90924 24 37 .35211 .93596 .36839 92967 .38456 .9231 .4006 .91625 .4165 .909111 23 38 .35239 .93585 .36867 .92956 .38483 .92299 .4008 .91613 .41681 .90899 22 39 .35266 .93575 .3689 .92945 .38510 .92287 .4011 .91601 .41707 .90887 21 40 .3529 .93565 .3692 .92935 .38537 .92276 .40141 .91590 .41734 .90875 20 4t .3532 .93555 .36913 .92924 .3856 .92265 .40168 .91578 .41760 .90863 19 42 .3534 .93544 .3697o .92913 .3859 .92254 .40195 .91566 .41787 .90851 18 43 44 .3537 .3540 .93534 .93524 .37002 .37029 .92902 .92892 .3861 .38644 .92243 .92231 .40221 .40248 .91555 .91543 .41813 .90839 17 .41840.90826 16 45 .3542 .93514 .37056 .92881 .3867 .9222C .40275 .91531 .41866 1.90814 15 46 .35456 .93503 .3708 .92870 .3869 .9220 .40301 .91519 .41892 .90802 14 47 .35484 .93493 .3711 .92859 .3872 .92198 .40328 .91508 .4191S .90790; 13 48 .35511 .93483 .3713 .92849 .3875 .9218( .40355 .91496 .41945 .90778 1 12 49 .35538 .93472 .3716 .92838 .3877 .92171 .40381 .91484 .41972 .9076C 11 50 .35565 .93462 .3719 .92827 .3880 .92164 .40408 .91472 .41998 .90752 10 51 .35592 .93452 .3721 .9281G .3883 .9215$ .40434 .91461 .42024 .90741 c 52 .3561S .93441 .3724 .92805 .3885 .92141 .40461 .91443 .42051 .9072 I 53 .35647 .93431 .3727 .92794 .3888 .9213( ) .40488 .91437 .42077 .90717 54 .35674 .9342C .3729 .92784 .3891 .9211< .40514 .91425 .42104 .90704 6 55 .35701 .9341C .3732 .92772 .3893 .92103 ' .40541 .91414 .4213C .9069$ 5 56 .3572* .9340C .3735 .92765 .3896 .9209f > .40567 .91402 .4215C .9068C 4 57 .3575 .9338< .3738 .92751 .3899 .9208J > .40594 .9139C .42182 .90668 c 68 .3578$ > .9337< > .3740 .9274C .3902 .9207; i .40621 .91378 .4220 .9065f > 2 59 .358K ) .9336? * .37434 .9272f .3904 .9206$ > .40647 .91366 .4223 .9064: S 1 60 .3583' r .9335! ? .3746 .9271* ( .3907 .9205( ) .40674 .91355 .4226$ .90631 M Cos. Sine COS Sine COS Sine COS. Sine Cos. Sine M, 68 s 68* 67 66 65 TABL.& iL NATURAL SINES AND COSINES. 23 25 26 27 28 2 M. Sine. Cos. Sine. Cos. Sine. Cos. Sine. Cos. Sine._ Cos. M. .42262 .90631 .43837 .898791.45399 .89101' .46947 .88295 .48481 '.87462 60 1 .42288 .90618j. 43863 .898671. 45425;. 89087 .46973 .88281 .48506i.87448 59 2 .42315 .906061.43889 .89854 1.45451!. 89074 .46999 .88867 .48532 .87434 58 3 .42341 .905941.43916 .89841 .45477 '.89061 .47024 .88254 .48557 .87420 57 4 .42367 .90582 .43942 .89828'. 45503 .89048 .470.oO:.88240 .485831.87406 56 5 .42394 .905691.43968 .89816 .45529 .89035 .470761.88226 .486081.87391 55 6 .42420 .90557 .43994 .89803 .45554 .89021 .47101 .88213 .48634 .87377 54 7 .4244G .905451.44020 .89790 : .45580!. 89008 .47127 .88199 .48800 .87363 53 8 .42473 .90532 .44046 .89777 .45606 .88995 .47153 .88185 .486841.87349 52 9 .42499 .90520 .44072 .897645.45632 .88981 .47178 .88172 .48710 .87335 51 10 .42525 ,90507 .44098 . 89752 >. 45658 .88968 .47204 .88158 .48735 .87321 50 1] .42^2 .90495 .44124 . 89739 '. 45684 .88955 .47229 .88144 .48761 .87306 49 12 .42578 .904831. 44151 .897261.45710 .88942 .47255 .88130 .48786 .87292 48 13 .42604 .90470 :.44177 .89713 .45736 .88928 .47281 .88117 .48811 .87278 47 14 .42631 .90458 .44203 .89700 .45762 .88915 .47306 .88103 .48837 .87264 46 15 .42657 .90446 .44229 .89687 ; 45787 .88902 .47332 .88089 .48862 .87250 45 16 .42683 .90433 .44255 .89674 .45813 .88888 .47358 .88075 .48S88 .87235 44 17 .42709 .90421 .44281 .89662 .45839 .88875 .47383 .88062 .48913 .87221 43 18 .42736 .90408 .44307 .89649 .45865 .88862 .47409 .88048 .48938 .87207 42 19 .42762 .90396 .44333 .89636 .45891 .88848 .47434 .88034 .48964 .87193 41 20 .42788 .90383 .44359 .89623 .45917 .88835 .47460 .88020 .48989 .87178 40 21 .42815 .90371 .44385 .89610 .45942 .S>xJ .47486 .88006 .49014 .87164 39 22 .42841 .90358 .44411 .89597 .45968 .88808 .47511 .87993 .49040 .87150 38 23 .42867 .90346 .44437 .89584 .459941.88795,47537 .87979 .49065 .87136 37 24 .42894 .90334 .44464 .89571 .46020 .88782 .47562 .879651.49090 .87121 36 25 .42920 .90321 .44490 .89558 .46046 .88768 .47588 .87951 1.49116 .87107 35 26 .42946 .90309 .44516 .89545 .46072 .88755 .47614 .87937 .49141 .87093 34 27 .42972 .90296 .44542 .89532 .46097 .88741 .47639 .87923 .49166 .87079 33 23 .42999 .90284 .44568 .89519 .46123 .88728 .47665 .87909 .49192 .87064 32 29 .43025 .90271 .44594 .89506 .46149 .88715 .47690 .87896 .49217 .87050 31 30 .43051 .90259 .44620 .89493 .46175 .88701 .47716 .87882 .49242 .8703C 30 31 .43077 .90246 .44646 .89480 .46201 .88688 .47741 .87868 .49268 .87021 29 32 .43104 .90233 .44672 .89467 '.46226 .88674 .47767 .87854 .49293 .87007 28 33 .43130 .90221 .44098 .89454 .46252 .88661 .47793 .87840 .49318 .86993 27 34 .43156 .90208 .44724 .89441 .46278 ! 88647 .47818 .87826 .49344 .86978 26 35 .43182 .90196 .44750 .89428^.46304 .88634 .47844 .87812 .49369 .86964 25 36 .43309 .90183 .44776 .89415 .46330 .88620 .47869 .87798 .49394 .86949 24 37 .43235 .90171 .44802 .894021.46355 .88607J. 47895 .87784 .49419 .86935 23 38 .43261 .90158 .44828 .89389 .46381 .88593 .47920 .87770 .49445 .86921 22 39 .43287 .90146 .44854 .89376 .46407 .88580 .47946 .877561.49470 .8690C 21 40 .43313 .90133 .44880 .89363 .46433 .88566 .47971 .87743 .49495 .86892 20 41 .43340 .90120 .44906 . 89350 ! . 46458 .88553 .47997 .87729 .49521 .86878 19 42 1.43366 .90108 .44932 .89337!. 46484 .88539 .48022 .87715 .4954G .8686C 18 43 .43392 .90095 .44958 .89324 .46510 .88526 .48048 .877011.49571 .8684C 17 44 .43418 .90082 .44984 .893111.46536 .88512 .48073 .87687 .4959C .86834 16 45 .43445 .90070 .45010 .89298 .46561 .88499 .48099 .87673 .48622 .86820 15 46 .43471 .90057 .45036 .89285 .46587 .88485 .48124 .87659 .49647 .86805 14 47 .43497 .900451.45062 . 89272 ; . 46613 1.88472 .48150 .87615 .49672 .867C1 13 48 .43523 .90032 . 45088 . 89259 . 46639i . 88458 . 48175;. 876C1 1.49697 .86777 12 49 .43549 .90019 .45114 .89245 .46664!. 88445 .4820H.87G17 .4972C .867C2 11 50 .43575 .90007 .45140 .89232 .46690! .88431 .48226 .8760,? .49748 .86748 10 51 .43602 .89994 .45166 .89219 .46716 .88417 .48252 .87589 .49773 .86733 9 52 .43628 .89981 .45192'. 89206. 46742!. 88404 .482771.87575 .49798 .86719 8 53 .43654 .89968 .45218 .89193'. 467671. 88390 .483031.87561 .49824 .86704 7 54 .43680 . 89956 . 45243 . 891 80 . 46793 . 88377 .483281.87546 .49849 .86690 6 |55 .43706 .89943 .45269, .89167 .46819: .88363 .48."54 .87532 .49874 .86675 5 56 .43733 .89930;.45295 .89153 .46844 .88349 .48379:. 87518 .49899 .86661 4 57 .43759 1.89918!. 45321'. 89140'. 46870 .88336 .48405 .87504 .40924 .86646 3 58 .43785 .899051.45347 .89127 ; .46896 .88322 .484301.87490 .4C9EO .86632 2 69 .43811 .89892 .45373 .89114 .46921 .88308 .48456 .87476 .49975 .86617 1 60 .43837 . 89879 . 45399 . 89101 ; . 46947 . 88295 .484811.87462 50000 .86603 M. Cos. 1 Sine. Cos. 1 Sine. Cos. i Sine. Cos. i Sine. Cos. Sine. M. 64 C 63 62' 61 60 24: TABLE II. NATURAL SINES AND COUINES. SO 3f 32 33 34- M. Sine. Cos. Sine. Cos. Sine. Cos. Sine. Cos. Sine. Cos. M. .50000 .86603 .51504 .85717 .52992 .84805 .54464 .83867 .55919 .82904 60 1 .50025 .86588 .51529 .85702 .53017 .84789 .54488 .83851 .55943 .82887 59 2 ..50050 .86573 .51554 .85687 .53041 .84774 .54513 .83835 .55968 .82871 58 3 .50076 .86559 .51579 .85672 .53066 .84759 .54537 .83819 .55992 .82855 57 4 .50101 .86544 .51604 .85657 .53091 .84743 .54561 .83804 .56016 .82839 56 5 .50126 .86530 .51628 .85642 .53115 .84728 .54586 .83788 .56040 .82822 55 6 .50151 .86515 .51653 .85627 .53140 .84712 .54610 .83772 .56064 .82806 54 7 .50176 .86501 .51678 .85612 .53164 .84697 .54635 .83756 .56088 .82790 53 8 .50201 .86486 .51703 .85597 .53189 .84681 .54659 .83740 .56112 .82773 52 9 .50227 .86471 .51728 .85582 .53214 .84666 .54683 .83724 .56136 .82757 ,51 10 .50252 .86457 .51753 .85567 .53238 .84650 .54708 .83708 .56160 .82741 50 11 .50277 .86442 .51778 .85551 .53263 .84635 .54732 .83692 .56184 .82724 49 12 .50302 .86427 .51803 .85536 .53288 .84619 .54756 .83676 .56208 .82708 48 13 .50327 .86413 .51828 .85521 .53312 .84604 .54781 .836(30 .56232 .82692 47 14 .50352 .86398 .51852 .85506 .53337 .84588 .54805 .83645 .56256 .82675 46 15 .50377 .86384 .51877 .85491 .53361 .84573 .54829 .83629 .56280 .82659 45 16 .50403 .86369 .51902 .85476 .5338G .84557 .548541.83613 .563051.82643 44 17 .50428 .86354 .51927 .85461 .53411 .84542 .54878 .83597 .56329 .82626 43 18 .50453 .86340 .51952 .85446 .53435 .84526 .54902 .83581 .56353 .82610 42 19 .50178 .86325 .51977 .85431 .53460 .84511 .54927 .83565 .50377 .82593 41 20 .50503 .86310 .52002 .85416 .53484 .84495 .54951 .83549 .56401 .82577 40 21 .50528 .86295 .52026 .85401 .53509 .84480 .54975 .83533 .56425 .82561 39 22 .50553 .86281 .52051 .85385 .53534 .84464 .54999 .83517 .56449 .82544 38 23 .50578 >86266 .52076 .85370 .53558 .84448 .55024 .83501 .56473 .82528 37 2t .50603 .86251 .52101 * 85355 .53583 .84433 .55048 .83485 .56497 .82511 36 23 .50628 .86237 .52126 .85340 .53607 .84417 .55072 .83469 .56521 .82495 35 26 .506.54 86222 .52151 .85325 .53632 .84402 .55097 .83453 .565451.82478 34 27 .50679 .86207 .52175 .85310 .53656 .84386 .55121 .83437 .56569 .82462 33 28 .50704 .86192 .52200 .85294 .53681 .84370 .55145 .83421 .56593 .82446 32 29 .50729 .86178 .52225 .85279 .53705 .84355 .55169 .83405 .56617 .82429 31 30 .50754 .86163 .52250 .85264 .53730 .84339 .55194 .83389 .56641 .82413 30 31 .50779 .86148 .52275 .85249 .53754 .84324 .55218 .83373 .56665 .82396 29 32 .50804 .86133 .52299 .85234 .53779 .84308 J 55242 .833C6 .56689 .82380 28 33 .50829 .86119 .52324 .85218 .53804 .84292 .55266 .83340 .56713 .82363 27 34 .50854 .86104 .52349 .85203 .53828 .84277 .55291 .83324 .56736 .82347 26 35 .50879 .86089 .52374 .85188 .53853 .84261 .55315 .83308 .56760 .82330 25 36 .50904 .86074 .52399 .85173 .53877 .84245 .55339 .83292 .56784 .82314 24 37 .50929 .86059 .52423 .85157 .53902 .84230 .55363 .83276 .56808 .82297 23 38 .50954 .86045 .52448 .85142 .53926 .84214 .55388 .83260 .56832 .82281 22 39 .50979 .86030 .52473 |.85127 .53951 .84198 .55412 .83244 .56856 .82264 21 40 .51004 .86015 .52498'. 85112 .53975 .84182 .55436 .83228 .56880 .82248 20 41 .51029 .86000 .52522 .85096 .54000 .84167 .55460 .83212 .56904 .82231 19 42 .51054 .85985 .52547 .85081 .54024 .84151 .55484 .83195 .56928 .82214 18 43 .51079 .85970 .52572 .85066 .54049 .84135 .55509 .83179 .56952 .82198 17 44 .51104 .85956 .52597 .85051 .54073 .84120 .55533 .83163 .56976 .82181 16 45 .51129 .85941 .52621 .85035 .54097 .84104 .65557 .83147 .57000 .82165 15 46 .51154 .85926 .52646 .85020 .54122 .84088 .55581 .83131 .57024 .82148 14 47 .51179 .85911 .52671 .85005 .54146 .84072 .55605 .83115 .57047 .82132 13 48 .51204 .85896 .52696 .84989 .54171 .84057 .55630 .83098 .57071 .82115 12 49 .51229 .85881 .52720 .84974 .54195 .84041 .55654 .83082 .57095 .82098 11 50 .51254 .85866 .52745 .84959 .54220 .84025 .55678 .83066 .57119 .82082 10 61 .51279 .85851 .52770 . 84943 ! . 54244 .84009 .55702 .83050 .57143 .82065 9 52 .51304 .85836 .52794 .84928 .54269 .83994 5572(5 .83034 .57167 .82048 8 53 .51329 .85821 .52819 .84913 .54293 83978 55750 .83017 .57191 .82032 7 54 .51354 .85806 .52844 .84897 .54317 83962 55775 .83001 .57215 .82015 6 55 .51379 .85792 .52869 . 84882 j . 54342 83946 .55799 82985 .57238 .81999 5 56 .51404 .85777 .52893 .84866 .54366 83930 .55823 82969 .57262 .81982 4 57 .51429 .85762 .52918 .84851 .54391 .83915 .55847 82953 .57286 .81965 3 58 .51454 .85747 .52943 .84836 .54415 .83899 55871 82936 57310 .81949 2 59 .51479 .85732 .52967 .84820 .54440 .83883 55895 82920 57334 .81932 1 60 .51504 .85717 .52992 .84805^54464 .83867 55919 .82904 573581 .81915 5T COS. Sine. Cos. Sine. Cos. Sine. COS. Sine. Cos. Sine. 57 59 58 57 56' 55' TABLE IL NATURAL SINES AND COSINES. 25 35 36 37* 38* 39* M. Sine. Cos. Sine. Cos. Sine. Cos. Sine. Cos. Sine. CfB. M. .57358 .81915 .58779 .80902 1 .60182 79864 .61566 .78801 .62932 .77715 60 1 .57381 .81899 .58802 .80885' .60203 .79846 G1689 .78783 .62955 .77696 59 2 .57405 .81882 .58826 .80867J .60228 .79829 .61612 .78765 .62977 .77678 58 3 .57429 .81865 .58849 .80850 .60251.. 79811 ;.61635 .78747 .63000 .77660 57 4 .57453 .81848 .58873 .80833 . 60274 1. 79793 .61658 .78729 .63022 .77641 56 5 57477 .81832 .588961. 808 16j .602981.79776 .61681 .78711 .63045 .77623 55 6 57501 .81815 . 58920!. 80799j .G032H.79758I.61704 .78694 .630681.77605 54 7 57524 .81798 . 58943 |. 80782 . 60344 ! . 79741 .61726 .78676 .G30901. 77586 53 8 57548 .81782 .58967 .80765' .60367 .79723 .61749 .78658 .631131.77568 52 9 57572 .81765 .58990 .80748' .60390;. 79706 .61772 .78640 .63135 .77550 51 10 575961.81748 .59014 .80730! .604141.79688 .61795 .78622 .63158 .77531 50 11 57619 .81731 . 59037 j. 80713! . 60437 1. 79671 .61818 .78604 .63180J.77513 49 12 57643 .81714 . 59061 !.80696 : .60460 .79653 .61841 .78586 .63203 .77494 48 13 57667 .81698 . 59084 J.80679J .60483! .79635! .61864 .78568 .63225 .77476 47 14 57691 .81681 .59108;. 80662; .60506 .79618 .618871.78550 .63248^.77458 46 15 57715 .81664 .59131 .80644 .60629 .79600 .61909 .78532 .63271 .77439 45 16 57738 .81647 .59154 .80627 .60553 1 . 79583 .61932 .78514 .632931.77421 44 17 57762 .81631 .59178 .80610 .60576'. 79565 .61955 .73496 .63316] .77402 43 18 57786 '81614 .59201 .80593! .60599 .79547 .61978 .78478 .63338 .77384 42 19 57810 .81597 .59225 .80576' .606221.79530 .62001 .78460 .63361 .77366 41 20 57833 .81580 .59248 .80558 .60645J. 79512 .62024 .78442 .63383 .77347 40 21 57857 .81563 .59272 .80541! .60668 .79494 .62046 .78424 .63406 .77329 39 22 57881 .81546 .59295 .80524; .60691 .79477 .62069 .78405 63428 .77310 38 23 57904 .81530 .59318 . 80507 ! .60714!. 79459 .62092 .783871.63451 .77292 37 24 57928 .81513 .59342 '.80489' .60738 .79441 .62115 .78369 .63473 .77273 36 25 57952 .81496 .59365 .80472 .607611.79424 .62138 .78351:. 63496 .77255 35 26 57976 .81479 .593891.80455 .60784 .79406 .62160 .78333! .63518 .77236 34 27 57999 .81462 .59412 .80438 .608071.79388 .62183 .78315 .63540 .77218 33 28 58023 .81445 .59436 .80420 .60830:. 79371 .62206 .78297 .63563 .77199 32 29 58047 .81428 .59459 .80403 .60853 .79353 .62229 .78279! .63585 .77181 31 30 58070 .81412 .59482 .80386 .60876 .79335 .62251 .78261 .63608 .77162 30 31 58094 .81395 .59506 .80368 .60899 .79318 .62274 .78243 .63630 .77144 29 32 58118 .81378 .59529 .80351 .60922 .79300 .62297 .782251.63653 .77125 28 33 58141 .81361 .59552 .80334 .60945 .79282 .62320 .782061.63675 .77107 27 34 .58165 .81344 .59576 .80316 .60968 .79264 .62342 .78188 .63698 .77088 26 35 .58189 .81327 .59599 .80299 .60991 .79247 .62365 .78170 .63720 .77070 25 36 .58212 .81310 .59622 .80282 .61015 .79229 .62388 .78152 .63742 .77051 24 37 .58236 .81293 .59646 .80264 .61038 .79211 .62411 .78134 .63765 .77033 23 38 .58260 .81276 .59669 .80247 .61061 .79193 .62433 .78116|.63787 .77014 22 39 .58283 .81259 .59693 .80230 .61084 .791761.62456 .78098 .63810 .76996 21 40 .58307 .81242 .59716 .S0212 .61107 .79158-62479 .78079 .63832 .76977 20 41 .58330 .81225 .59739 .80195 .61130 . 79140; . 62502 1 . 78061 1 . 63854 .76959 19 42 .58354 .81208 .59763 .80178 .61153 .79122 .62524 .78043 .63877 .76940 18 43 .58378 .81191 .59786 .80160 .61176 .79105 .625471.78025 .63899 .76921 17 44 .58401 .81174 .59809 .80143 .61199 .79087 .62570 .78007 .63922 .76903 16 45 .58425 .81157 .59832 .80125 .61222 .79069 .62592 .77988 .63944 .76884 15 46 .58449 .81140 .59856 .80108 .61245 .790511 .62615, .7797o! .63966 .76866 14 47 .58472 .81123 .59879 .80091 .61268!. 790331. 62638 .77952 .63989 .76847 13 48 .58496 .81106 .59902 .80073 .61291.790151.62660 .77934 .64011 .76828 12 49 58519 .81089 .59926 .80056 .61314.789981.62683 .77916 .64033 -76810 11 50 .58543 .81072 .59949 .80038 .61337 .78980 .62706 .77897!. 64056,. 76791 10 51 .58567 .81055 .59972 .80021 .61360 .78962 .62728 .77879 .64078 .76772 9 52 .58590 .81038 .59995 .80003 .61383 .78944' .62751 .77861 .641001.76754 8 53 .58614 .81021 ,60019 .79986 .61406 .78926 .62774 .77843 .64123 .76735 7 54 .58637 .81004 .60042 .79968 .61429 .78908 .62796 .77824 .64145'. 76717 6 55 .586611.80987 .60065 .79951 .61451 . 78891 ; . 62819 .77806 .64167 .76698 5 56 .58684 .80970 .60089 .79934 .61474 .78873J . 62842 .77788 .64190 .76679 4 57 .58708 .80953 .60112 .79916 .61497 .78855;. 62864 .77769 64212 .76661 3 58 .58731 .80936 .60135 .79899 .61520 .78837 .62887 .77751 .64234 .76642 2 59 .58755 .80919 .601581.79881 .61543 .78819 .62909 .77733 .64256'. 76623 1 60 .58779 .80902 .60182 '.79864 .61566 .78801 .62932 .77715 .64279'. 76604 M Cos. 1 Sine. Cos. ' Sine. Cos. Sine. Cos. Sine. Cos. Isine. M. 54 53* 52" 51 50' 26 TABLE II. NATURAL SINES AND COSINES. 4 .74412 .68093 .73234 .69361 .7203, .7060? .7081 j 56 .6551? .75547 .6682' .74392 .69382 .7201 .7062? .7079 < 57 .6554C .7552? .66848 .74373 '.68136 .7319E , .6940C .7199 .7064* .7077 '< 58 .65562 .7550 .66870 .74353 .6815 r .7317f > .69424 .7197 .7067( .7075 ! 59 ,65584 .7549( .6689 L .74334 .68171 .7315E > .6944E .7195^ .7069( .7073 60 .6560* .7547] .6691, J .74314 .68201 .7313, i .6946< .7193^ .70711 .7071 M 7 Cos. Sine Cos. Sine Cos. Sine . Cos. Sine Cos. Sin M. 49* 48 47 4G 45 TABLE III. NATURAL TANGENTS AND COTANGENTS. 28 TABLE III. NATUBAL TANGENTS, ETC. O 1 2 3' M' Tang. Cotang Tang. Cotang. Tang. Cotang. Tang. Cotang M- .00000 Infinite .01746 57.2900 .03492 28.6363 .05241 19.0811 .60 1 .00029 3437.75 .01775 56.3506 .03521 28.3994 .05270 18.9755 69 2 .00058 1718.87 .01804 55.4415 .03550 28.1664 .05299 18.3711 68 3 .00087 1145.92 .01833 54.5613 .03579 27.9372 .05328 18.7678 67 4 .00116 859.436 .01862 53.7086 .03609 27.7117 .05357 18.6656 66 6 .00145 687.549 .01891 52.8821 .03638 27.4899 .05387 18.5645 65 6 .00175 572.957 .01920 52.0807 .03667 27.2715 .05416 18.4645 64 7 .00204 491.106 .01949 61.3032! .03696 27.0566 .05445 18.3655 63 8 .00233 429.718 .01978 60.5485 .03725 26.8450 .05474 18.2677 62 9 .00262 381.971 .02007 49.8157 03754 26.6367 .05503 18.1708 61 10 .00291 343.774 .02036 49.1039 .03783 26.4316 .05533 18.0750 ,60 11 .00320 312.521 .02066 48.4121 .03812 26.2296 .05562 17.9802 49 12 .00349 286.478 .02095 47 7395 .03842 26.0307 .05591 17.8863 48 13 .00378 264.441 .02124 47.0853 .03871 25.8348 .05620 17.7934 47 14 .00407 245.552 .02163 46 4-189 .03900 25.6418 .05649 17.7015 46 15 .00436 229.182 .02182 45 8294 .03929 25.4517 .05678 17.6106 45 16 .00465 214.858 .02211 45.2861 .03958 25.2644 .05708 17.5205 44 17 .00495 202.219 .02240 44.6386 .03987 25.0798 .05737 17.4314 43 18 .00524 190.984 .02269 44 0661 .04016 24.8978 .05766 17.3432 42 19 .00553 180 932 .02298 43 5081 .04046 24.7185 .05795 17.2558 41 20 .00582 171.885 .02328 42 96411 .04075 24.5418 .05824 17.1693 40 21 .00611 163.700 .02357! 42.4335 .04104 24.3675 .05854 17.10837 39 22 .00640 156.259 .02386 41.9158 .04133 24.1957 .05883 16.9990 38 23 .00669 149.465 .02415 41.4106 .04162! 24.0263 .05912 16.9150 37 24 .00698 143.237 .02444 40.9174 .04191 23.8593 .05941 16.8319 36 25 .00727 137.507 .02473 40.4358 .04220 23.6945 .05970 16.7496 35 26 .00756 132.219 .02502 39.9655 .04250 23.5321 .05999 16.6681 34 27 .00785 127.321 .02531 39.5059 .04279 23.3718 .06029 16.5874 33 28 .00815 122.774 .02560 39.0568 .04308 23.2137 .06058 16.5075 32 29 .00844 118.540 .02589 38.6177 .04337 M.0577 .06087 16.4283 31 30 .00873 114.589 .02619 38 1885 .04366 .9038 .06116 16.3499 30 31 .00902 110 892 .02648 37.7686 .04395 22.7519 .06145 16.2722 29 32 .00931 107.426 .02677 37.3579 .04424 22.6020 .06175 16-. 1952 28 33 .00960 104.171 .02706 36.9560 .04454 22.4541 .06204 16.1190 27 34 .00989 101.107 .02735 36.5627 .04483 22.3081 .06233 16.0435 26 35 .01018 98.2179 .02764 36.1776 .04512 22.1640 .06262 15.9687 25 36 .01047 95.4895 .02793 35.8006 .04541 22.0217 .06291 15.8945 24 37 .01076 92.9085 .02822 35.4313 .04570 21.8813 .06321 15.8211 23 38 .01105 90.4633 .02861 35.0695 .04599 21.7426 .06350 15.7483 22 39 .01135 88.1436 .02881 34.7151 .04628 21.I5056 .06379 15.6762 21 40 .01164 85.9398 .02910 34.3678 .04658 21.4704 .06408 15.6048 20 41 .01193 83.8435 .02939 34.0273 .04687 21.3369 .06437 15.5340 19 42 .01222 81.8470 .02968 33.6935 .04716 21.2049 .06467 15.4638 18 43 .01251 79.9434 .02997 33.3662 .04745 21.0747 ,06496 15.3943 17 44 .01280 78.1263 .03026 33.0452 .04774 20.9460 .06525 15.3254 16 45 .01309 76.3900 .03055 32.7303 .04803 20.8188 .06554 15.2571 15 46 .01338 74.7292 .03084 32.4213 .04833 20.6932 .06584 15.1893 14 47 .01367 73.1390 .03114 32.1181 .04862 20.5691 .06613 15.1222 13 48 .01396 71.6151 .03143 31 S205 .04891 20.4465 .06642 15.0557 12 49 .01425 70.1533 .03172 31.5284 .04920 20.3253 .06671 14.9898 11 50 .01455 68.7501 .03201 31 2416 .04949 20.2056 .06700 14.9244 10 51 .01484 67.4019 .03230 30.9599 .04978 20.0872 .06730 14.8596 9 52 .01513 66.1055 .03259 30.6833 .05007 19.9702 .06759 14.7954 8 53 .01542 64.8580 .03288 30.4116 .05037 19.8546 .06788 14.7317 7 54 .01571 63.6567 .03317 30.1446 .05066 19.7403 .06817 14.6685 6 55 .01600 62.4992 .03346 29.8823 .05095 19.6273 .06847 14.6059 5 66 .01629 61.3829 .03376 29.6245 !. 05124 19.5156 .06876 14.5438 4 67 .01658 60.3058 .03405 29.3711 .05153 19.4051 .06905 14.4823 3 58 .01687 59.2659 .03434 29.1220 .05182 19.2959 .06934 14.4212 2 69 .01716 58.2612 .03463 28.8771 .05212 19.1879 .06963 14.3607 1 60 .01746 57.2900 .03492 28.6363 .05241 19.0811 .06993 14.3007 M. Cotang. I Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. M. 89 88* 87 86 TABLE III. NATURAL TANGENTS, ETC. 4- 5 l 6- 7 M Tang. Cotang Tang. Cotang Tang. Cotang Tang. Cotang M. .06993 14.3007 .08749 11.4301 .10510 9.51436 .12278 8.14435 60 1 .07022 14.2411 .08778 11.3919 .10540 9.48781 .12308 8.12481 59 2 .07051 14.1821 .08807 11.3540 .10569 9.46141 .12338 8.10536 58 3 .07080 14.1235 .08837 11.3163 .10599 9.43515 12367 8.08600 57 4 .07110 14.0655 .08866 11.2789 .10628 9.40904 .12397 8.06674,56 5 .07139 14.0079 .08895 11.2417 .10657 9.38307 .12426 8.0475* 55 6 .07168 13.9507 .08925 11.2048 .10687 9.35724 .12456 8.0284*154 7 .07197 13.8940 .08954 11.1681 .10716 9.33155 .12485 8.00948|53 8 .07227 13.8378 .08983 11.1316 .10746 9.30599 .12515 7.9905852 9 .07256 13.7821 .09013 11.0954 .10775 9.28058 .12544 7.97176 ! 51 10 .07285 13.7267 .09042 11.0594 .10805 9.25530 .12574 7.95302J50 11 .07314 13.6719 .09071 11.0237 .10834 9.23016 .12603 7.93438 49 12 .07344 13.6174 .09101 10 9882 .10863 9.20516 .12633 7.9158248 13 .07373 13.5634 .09130 10.9529 .10893 9.18028 .12662 7.89734 47 14 .07402 13.5098 .09159 10.9178 .10922 9.15554 .12692 7.87895 46 15 .07431 13.4566 .09189 10.8829 ..10952 9.13093 .12722 7.8606445 16 .07461 13.4039 .09218 10.8483 .10981 9.10646 .12751 7.84242 44 17 .07490 13.3515 .09247 10.8139 .11011 9.08211 .12781 7.82428 43 18 .07519 13.2996 .09277 10.7797 .11040 9.05789 .12810 7.80622 42 19 .07548 13.2480 .09306 10.7457 .11070 9.03379 .12840 7.78825 41 2<> .07578 13.1969 .09335 10.7119 .11099 9.00983 .12869 7.77035 40 21 .07607 13.1461 .09365 10.6783 .11128 8.98598 .12899 7.75254 39 22 .07636 13.0958 .09394 10.6450 .11158 8.96227 .12929 7.73480 38 23 .07665 13.0458 .09423 10.6118 .11187 8.93867 .12958 7.71715 37 24 .07695 12.9962 .09453 10.5789 .11217 8.91520 .12988 7.69957 36 2.-, .07724 12.9469 .09482 10.5462 .11246 8.89185 .13017 7.68208 35 26 .07753 12.8981 .09511 10.5136 .11276 8.86862 .13047 7.66466 34 27 28 .07782 .07812 12.8496 12.8014 .09541 .09570 10.4813 10.4491 .11305 .11335 8 84551 8.82252 .13076 .13106 7.64732J33 7.63005^32 29 .07841 12.7536 .09600 10.4172 .11364 8.79964 .13136 7.61287 31 30 .07870 12.7062 .09629 10.3854 .11394 8.77689 .13165 7.5957530 31 .07899 12.6591 .09658 10.3538 .11423 8.75425 .13195 7.5787229 32 .07929 12.6124 .09688 10.3224 .11452 8.73172 .13224 7.5617628 33 .07958 12.5660 .09717 10.2913 .11482 8.70931 .13254 7.5448727 34 .07987 12.5199 .09746 10.2602 .11511 8.68701 13284 7.5280626 3.-, .08017 12.4742 .09776 10.2294 .11541 8.66482 .13313 7.51132 25 30 .08046 12.4288 .09805 10.1988 .11570 8.64275 .13343 7.4946524 37 .08075 12.3838 .09834 10.1683 .11600 8.62078 .13372 7.47806 23 38 .08104 12.3390 .09864 10.1381 .11629 8.59893 .13402 7.46154 22 39 .08134 12.2946 .09898 10.1080 .11659 8.57718 .13432 7.4450921 40 .08163 12.2505 .09923 10.0780 .11688 8.55555 .13461 7.4287120 41 .08192 12.2067 .09952 10.0483 .11718 8.53402 .13491 7.41240 19 42 .08221 12.1632 .09981 10.0187 .11747 8.51259 .13521 7.39616 18 43 .08251 12.1201 .10011 9.98931 .11777 8.49128 .13550 7.37999 17 44 .08280 12.0772 .10040 9.96007 .11806 8.47007 .13580 7.36389 16 45 .08309 12.0346 .10069 9.93101 .11836 8.44896 .13609 7.34786 15 46 .08339 11.9923 .10099 9.90211 .11865 8.42795 .13639 7.33190 14 47 .08368 11.9504 .10128 9.87338 .11895 8.40705 .13669 7.31600 13 48 .08397 11.9087 .10158 9.84482 .11924 8.38625 .13698 7.30018 12 49 .08427 11.8673 .10187 9.81641 .11954 8.36555 .13728 7.28442 It 50 .08456 11.8262 .10216 9 78817 .11983 8.34496 .13758 7 26873 10 51 .08485 11.7853 .10246 9.76009 .12013 8.32446 .13787 7.25310J 9 52 .08514 11.7448 .10275 9.73217 .12042 8.30406 .13817 7.23754J 8 53 .08544 11.7045 .10305 9 70441 .12072 8.28376 .13846 7.22204J 7 .54 .08573 11.6645 .10334 9.67680 .12101 8.26355 .13876 7.20661! 6 55 .08602 11.6248 .10363 9.64935 .12131 8.24345 .13906 7.19125 5 56 ,08632 11.5853 .10393 9.62205 .12160 8.22344 .13935 7.17594 4 57 .08661 11.5461 .10422 9.59490 .12190 8.20352 .13965 7.16071 3 .58 .08690 11.5072 .10452 9.56791 .12219 8.18370 .13995 7.14553 2 59 .08720 11.4685 .10481 9.54106 .12249 8.16398 .14024 7.13042 1 60 .08749 11.4301 .10510 9.51436 .12278 8.14435 .14054 7.11537 M. Cotang. Tang. Cotang, Tang. Cotang. Tang. Cotang. Tang. M. 85' 84 83' 82 30 TABLE III. NATURAL TANGENTS, ETC. 8 9 1O 11* M- Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. 1 M. .14054 7 11537 .15838 6.31375 .17633 5.67128 .19438 6.14455 60 .14084 7.10038 .15868 6.30189 .17663 5.66165 .19468 5.13658 59 2 .14113 7.08546 .15898 6.29007 .17693 5.65205 .19498 5.12862 68 8 .14143 7.07059 .15928 6.27829 .17723 5.64248 .19529 5.12069 67 4 .14173 7.05579 .15958 6.26655 .17753 5.63295 .19559 6.11279 56 5 .14202 7.04105 116888 6.25486 .17783 5.62344 .19589 5.10490 B5 6 .14232 7.02637 .16017 6.24321 .17813 6.61397 .19619 5.09704 54 7 .14262 7.01174 .16047 6.23160 .17843 6.60452 .19649 5.08921 53 8 .14291 6.99718 .16077 6.22003 .17873 6.59511 .19680 B. 08139 52 9 .14321 6.98268 .16107 6.20851 .17903 6.58573 .19710 B. 07360 51 10 .14351 6.96823 .16137 6.19703 .17933 5.57638 .19740 B. 06584 50 11 .14381 6.95385 .16167 6.18559 .17963 5.56706 .19770 5.05809 49 12 .14410 6.93952 .16196 6.17419 .17993 5.55777 .19801 5.05037 48 13 .14440 6.92525 .16226 6.16283 .18023 5.54851 .19831 5.04267 47 14 .14470 6.91104 .16256 6.15151 .18053 5.53927 .19861 5.03499 46 15 .14499 6.89688 .16286 6.14023 .18083 5.53007 .19891 5.02734 45 16 .14529 6.88278 .16316 6.12899 .18113 6.52090 .19921 5.01971 44 17 .14559 6.86874 .16346 6.11779 .18143 5.51176 .19952 5.01210 43 18 .14588 6.85475 .16376 6.10664 .18173 5.50264 .19982 5.00451 42 19 .14618 6.84082 .16405 6.09552 .18203 5.49356 .20012 4.99695 41 20 .14648 6.82694 .16435 6.08444 .18233 5.48451 .20042 4.98940 40 21 .14678 6.81312 .16465 6.07340 .18263 5.47548 .20073 4.98188 39 22 .14707 6.79936 .16495 6.06240 .18293 5.46648 .20103 4.97438 38 23 .14737 6.78564 .16525 6.05143 .18323 5.45751 .20133 4.96690 37 24 25 .14767 6.77199 .14796 6.75838 .16555 .16585 6.04051 6.02962 .18353 .18384 5.44857 5.43966 .20164 .20194 4.95945 4.95201 36 35 26 .14826 6.74483 .16615 6.01878 .18414 5.43077 .20224 4.94460 34 27 .14856 6.73133 .16645 6.00797 .18444 5.42192 .20254 4.93721 33 28 .14886 6.71789 .16674 6.99720 .18474 5.41309 .20285 4.92984 32 29 .14915 6.70450 .16704 5.98646 .18504 5.40429 .20315 4.92249 31 30 .14945 6.69116 .16734 5.97576 .18534 5.39552 .20345 4.91516 30 31 .14975 6.67787 .16764 5.96510 .18564 5.38677 .20376 4.90785 29 32 .15005 6.66463 .16794 5.95448 .18594 5.37805 .20406 4.90056 28 33 .15034 6.65144 .16824 5.94390 .18624 5.36936 .20436 4.89330 27 34 .15064 6.63831 .16854 5.93335 .18654 5.36070 .20466 4.88605 26 36 .15094 6.62523 .16884 5.92283 .18684 5.35206 .20497 4.87882 26 36 .15124 6.61219 .16914 5.91236 .18714 5.34345 .20527 4.87162 24 37 .15153 6.59921 .16944 5.90191 .18745 5.33487 .20557 4.86444 23 38 .15183 6.58627 .16974 5.89151 .18775 5.32631 .20588 4.85727 22 39 .15^13 6.57339 .17004 5.88114 .18805 5.31778 .20618 4.85013 21 40 .15243 6.56055 .17033 5.87080 .18835 5.30928 .20648 4.84300 20 41 .15272 6.54777 .17063 5.86051 .18865 5.30080 .20679 4 . 83590 19 42 .15302 6.53503 .17093 5.85024 .18895 5.29235 .20709 4 . 82882 18 43 .15332 6.52234 .17123 5.84001 .18925 5.28393 .20739 4.82175 17 44 .15362 6.50970 .17153 5.82982 .18955 5.27553 .20770 4 81471 16 45 .15391 6.49710 .17183 5 81966 .18986 5.26715 .20800 4.80769 15 46 .15421 6.48456 .17213 5.80953 .19016 5.25880 .20830 4.80068 14 47 .15451 6.47206 .17243 5.79944 .19046 5.25048 .20861 4.79370 13 48 .15481 6.45961 .17273 5.78938 .19076 5.24218 .20891 4.78673 12 49 .15511 6.44720 .17303 5.77936 .19106 5.23391 .20921 4.77978 11 50 .15540 6.43484 .17333 5.75937 .19136 5.22566 .20952 4.77286 10 51 .15570 6.42253 .17363 5.75941 .19166 5.21744 .20982 4.76595 9 52 .15600 6.41026 .17393 5.74949 .19197 5.20925 .21013 4.75906 8 53 .15630 6.39804 .17423 5.73960 .19227 5.20107 .21043 4.752191 7 54 .15660 6.38587 .174531 6.72974 .19257 5.19293 .21073 4.74534 6 55 .15689 6.37374 .17483 5.71992 .19287 5.18480 .21104 4.73851 5 56 .15719 6 36165 .17513 5.71013 .19317 5.17671 .21134 4.73170 4 57 .15749 6.34961 .17543 5.70037 .19347 5.16863 .21164 4.72490 3 58 .15779 6.33761 .17573 5.69064 .19378 5.16058 .21195 4.71813 2 59 .15808 6.32566 .17603 5.68094 .19408 5.15256 .21225 4.71137! 1 60 .15838 6.31375 .17633 5.67128 .19438 5.14455 .21256 4.70463 u M Cotang.i Tang. Cotang. Tang. Cotang.i Tang. Cotang Tang. M. 81" 80 79' 78 TABLE III. NATURAL TANGENTS, ETC. 31 12 1* 14 15 M. Tang. Cotang. Tang. | Cotang Tang. Cotang Tang. rotang. M. .21256 4.70463 .230871 4.33148 .24933) 4.01078 .26795 3.73205 60 ] .21286 4.69791 .23117 4.32573 .24964 4.00582 .26826 3.72771 59 2 .21316 4.69121 .23148 4.32001 .24995 4.0008G .26857 3.72338 58 3 .21347 4.68452 .23179 4.31430 .25026 3.99592 .26888 3.71907 57 4 .21377 4.67786 .23209 4.30860 .25056 3.9909? .26920 3.71476 56 5 .21408 4.67121 .23240 4.30291 .25087 3.98607 26951 3.71046 55 C .21438 4.66458 .23271 4.29724 .25118 3.98117 .26982 3.70616 54 7 .21469 4.65797 .23301 4.29159 .25149 3.97627 .27013 3.70188 53 8 .21499 4.65138 .23332 4.28595 .25180 3.97139 .27044 3.69761 52 9 .21529 4.64480 .23363 4.28032 .25211 3.96651 .27076 3.69335 51 10 .215601 4.63825 .23393 4.27471 .25242 3.96165 .27107 3.68909 50 11 .21590 4.63171 .23424 4.26911 .25273 3.95680 .27138 3.68485 49 12 .21621 4.62518 .23455 4.26352 .25304 3.951% .27169 3.68061148 13 .21651 4.61868 .23485 4.25795 .25335 3.94713 .27201 3.67638] 47 14 .21682 4.61219 .23516 4.25239 .25366 3.94232 .27232 3.67217 46 15 .21712 4.60572 .23547 4.24685 .25397 3.93751 .27263 3.66796 45 16 .21743 4.59927 .23578 4.24132 .25428 3.93271 .27294 3.66376 44 17 .21773 4.592*3 .23608 4.23580 .25459 3 92793 .27326 3.65957 43 18 .21804 4.58641 .23639 4.23030 .25490 3.92316 .27357 3.65538! 42 1'J .21834 4.5NH1 .23670 4.22481 .25521 3.91839 .27388 3.65121 41 20 .21864 4.57363 .23700 4.21933 .25552 3.91364 .27419 3.64705 40 21 .21895 4.56726 .23731 4.21387 .25583 3.90890 .27451 3.64289 39 22 .21925 4.56091 .23762 4.20S42 .25614 3.90417 .27482 3.63874 38 23 .21956 4.55458 .23793 4.20298 .25645 b. 89945 .27513 3.63461 37 24 .21986; 4.54826 .23823 4.19756 .25676 3.89474 .27545 3.63048 36 25 .22017! 4.54196 .23854 4.19215 .25707 3.89004 .27576 3.62636 35 28 .22047; 4.53568 .23885 4.18675 .25738 3.88536 .27607 3.62224 34 27 .22078 4.52941 .23916 4.18137 .25769 3>M-,x .27638 3.61814 33 26 .22108: 4.52316 .23946 4.17600 .25800 a. 87601 .27670 3.61405 32 29 .22139 4.51693 .23977 4.17064 .25831 S. 871 36 .27701 3.60996 31 30 .22169 4.51071 .24008 4.16530 .25862 3.86671 .27732 3.60588' 30 31 .22200 4.50451 .24039 4.15997 .25893 3.86208 .27764 3.60181 29 32 .22231 4.49832 .24069 4.15465 .25924 3.85745 .27795 3.59775 28 33 .22261 4.49215 .24100 4.14934 .25955 3.85284 .27826 3.59370 27 34 .22292 4.48600 .24131 4.14405 .25986 3.84824 .27858 3.58966 26 35 .22322 4.47986 .24162 4.13877 .26017 3.84364 .27889 3.58562 25 33 .22353 4.47374 .24193 4.13350 .26048 3.83906 .27921 3.58160 24 37 .22383 4.46764 .24223 4.12825 .26079 3.83449 .27952 3.57758 23 n .224141 4.46155 .24254 4.12301 .26110 3.82992 .27983 3.57357 22 99 .22444 4.45548 .24285 4.11778 .26141 3.82537 .28015 3.56957 21 40 .22475 4.44942 .24316 4.11256 .26172 3.82083 .28046 3.56557 20 41 .22505 4.44338 .24347 4.10736 .26203 3.81630 .28077 3.56159 19 42 .22536' 4.43735 .24377 4.10216 .26235 3.81177 .28109 3.55761 18 43 .22567 4.43134 .24408 4.09699 .26266 3.80726 .28140 3.55364 17 44 .22597 4.42534 .24439 4.09182 .26297 3.80276 .28172 3.54968 16 45 .22628 4.41936 .24470 4.08666 .26328 3.79827 .2820! 3.54573 15 46 .22658 4.41340 .24501 4.08152 .26351 3.79378 .28234 3.54179 14 47 .22689 4.40745 .24532 4.07639 .26390 3.78931 .28266 3.53785 13 48 .22719; 4.40152 .24562 4.0712" .26421 3.78485 .28297 3.53393 12 49 .22750 4. 39,560 .24593 4.06616 .26452 3.78040 .28328 3.53001 11 i .22781 4.38969 .2462-t 4.06107 .26483 3.77595 .28360 3.52609 10 51 .22811 4.38381 .24655 4.05599 .26515 3.77152 .2839 3.52219 9 52 .22842 4.37793 .24686 4.05092 .26546 3.76709 .28423 3.5182S 8 53| .22872 4.37207 .24717 4.(H. r )S6 .26577 3.76268 .28454 3-51441 7 54 .22903 4.36623 .24747! 4.04081 .26608 3.75828 .28486 3.51063 6 55 .22934 4.360401 .247781 4.a3578 -.26639 3.75388 .28517 3.50666 5 56 .22964 4.a>t59 .24809 4.03076 .26670 3.74950 .28549 3.50278 4 57 .22995 4.34879 .24840 4.02574 .26701 3.74512 .2858C 3.49894 3 58 .23026 4.34300 .24871 4.02074 .26733 3.74075 .28612 3.49508 2 59 .23056 4.33723 .24902 4.01576 .26764 3.73640 .28643 3.4912E 1 60 .23087 4.33148 .24933 4.01078 .26795 3.73205 .28675 3.48741 M Cotang Tang. Cotang Tang, Cotang Tang. Cotang Tang. M. 77 76 75 74 C TABLE HI. NATURAL TANGENTS, ETC. 16 17 C 18" 19 M Tang. Cotang Tang. Cotang Tang. Cotang. j Tang. Cotang M. ( > .2867 3.4874 .3057 3.2708E .32492 3.0776? .344 2.90421 60 .2870 3.4835 .3060 3.26745 .32524 3.07464 .3446 2. aoui 59 .28738 3. 4797 .3063 3.26406 .32556! 3.0716C .34498! 2.8987 68 .2876 3.4759 .3066 3.26067 .32588' 3.06857 .3453C 2.8960C 57 .2880C 3,4721 .30700 3.25729 .32621 3.06554 .3456 2.8932 56 .28832 3.4683 .30732 3.25392 .32653 3.06252 .34596 2.8905 65 .28864 3.4645* .30764 3.25055 .32685 3.059501 .3462* 2.8878 5< .28895 3.4608 .30796 3.24719 .32717 3.05649L .34661 2.8851 63 .28927 3.4570 .30828 3.24383 .32749 3.05349! .34692 2.8824 52 .28958 3.4532 .30860 3.24049 .32782 3.05049 .34726 2.8797 51 1 .28990 3.4495 .3089 3.23714 .32814 3.04749 .34758 2.87700 60 1 .29021 3.4457 .30923 3.23381 .32846 3.04450 .34791 2.87430 49 1 .29053 3.4420 .30955 3.23048 .32878 3.04152 .34824 2.8716 48 13 .29084 3.4382 .3098-- 3.22715 .32911 3.03854 .34856 2.86892 47 14 .29116 3.4345 .31019 3.22384 .32943 3.03556 .34889 2.8662 46 lo .2914 3 .43084 .31051 3.22053 =32975 3.03260 .34922 2.86356 45 1C .29179 3.42713 .31083 3.21722 .33007 3.02963 .34954 2.8608 M 17 .29210 3.42343 .31115 3.21392 .33040 3.02667 .34987 2.8582 43 18 .29242 3.41973 .31147 3.21063 .33072 3.02372 .35020 2.85555 42 19 .29274 3.41604 .31178 3.20734 .33104 3,02077 .35052 2.85289 41 20 .29305 3.41236 .31210 3.20406 .33136 3.01783 .35085 2.85023 40 21 .29337 3.40869 .31242 3.20079 .33169 3.01489 .35118 2.84758 39 22 .29368 3.40502 .31274 3.19752 .33201 3.01196 .35150 2.84494 38 23 .29400 3.40136 .31306 3.19426 .33233 3.00903 .35183 2.84229 24 .29432 3.39771 .31338 3.19100 .33266 3.00611 .35216 2.83965 36 25 .29463 3.39406 .31370 3.18775 .33298 3.00319 .35248 2.83702 35 26 .29495 3.39042 .31402 3.18451 .33330 3.00028 .35281 2.83439 34 21 .29526 3.38679 .31434 3.18127 .33363 -2.99738 .35314 2.83176 33 28 .29558 3.38317 .31466 3.17804 .33395 2.99447 .35346 2.8291' 32 29 .29590 3.37955 .31498 3.17481 .33427 2.99158 .35379 2.82653 31 30 .29621 3.37594 .31530 3.17159 .33460 2.98868 .35412 2.82391 30 31 .29653 3.37234 .31562 3.16838 .33492 2.98580 .35445 2.82130 29 32 .29685 3.36875 .31594 S. 16517 .33524 2.98292 .35477 2.81870 28 33 .29716 3.36516 .31626 3.16197 .33557 2.98004 .35510 2.816K 27 34 .29748 3.36158 .31658 3.15877 .33589 2.97717 .35543 2.81350 26' 35 .29780 3.35800 .31690 3.15558 .33621 2.97430 .35576 2.81091 25 36 .29811 3.35443 .31722 3.15240 .33654 2.97144 .35608 2.80833 24 37 .29843 3.35087 .31754 3.14922 .33686 2.96858 .35641 2.80574 23 38 .29875 3.34732 .31786 3.14605 .33718 2.96573 .35674 2.80316 22 39 .29906 3.34377 .31818 3.14288 .33751 2.96288 .35707 2.80059 21 40 .29938 3.34023 .31850 3.13972 .33783 2.96004 .35740 2.79802 20 41 .29970 3.33670 .31882 3.13656 .33816 2.95721 .35772 2.79545 19 42 .30001 3.33317 .31914 3.13341 .33848 2.95437 .35805 2 . 79289 18 43 .30033 3.32965 .31946 3.13027 .33881 2.95155 .35838 2.79033 17 44 .30065 3.32614 .31978 3.12713 .33913 2.94872 .35871 2.78778 16 45 .30097 3.32264 .32010 3.12400 .33945 2.94591 .35904 2.78523 15 46 .30128 3.31914 .32042 3.12087 .33978 2.94309 .35937 2.78269 14 47 .30160 3.31565 .32074 3.11775 .34010 2.94028 .35969 2.78014 13 48 .30192 3.31216 .32106 3.11464 .34043 2.93748 .36002 2.77761 12 49 .30224 3.30868 .32139 3.11153 .34075 2.93468 ,36035 2.77507 [1 50 .30255 3.30521 .32171 3.10842 .34108 2.93189 .36068 2.77254 LO 51 .30287 3.30174 .32203 3.10532 .34140 2.92910 .36101 2.77002 9 52 .30319 3.29829 .32235 3.10223 .34173 2.92632 .36134 2,76750 g 53 .30351 3.29483 .32267 3.09914 .34205 2.92354 .36167 2.76498 7 54 .30382 3.29139 .32299 3.09606 .34238 2.92076 .36199 2.76247 6 55 .30414 3.28795 .32331 3.09298 .34270 2.91799 .36232 2.75996 5 56 .30446 3.28452 .32363 3.08991 .34303 2.91523 .36265 2.75746 4 57 .30478 3.28109 .32396 3.08685 .34335 2.91246 .36298 2.75496 3 58 .30509 3.27767 .32428 3.08379 .34368 2.90971 .36331 2.75246 2 59 .30541 3.27426 .32460 8.08073 .34400 2.90696 .36364 2.74997 1 60 .30573 3.27085 .32492 3.07768 .34433 2.90421 .36397 2.74748 M. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. M. 73* 72 71' 70' i TABLE III. NATURAL TANGENTS, ETC. 33 2O 21' 22 23 M. Tang. Cotang Tang. Cotang. Tang. Cotang. Tang. Cotang. M. .36397 2.74748 .38386 2.60509 .40403 2.47509 .42447 2.35585 60 1 .36430 2.74499 .38420 2.60283 .40436 2.47302 .42482 2 35395 59 2 .36463 2.74251 .38453 2.60057 .40470 2.47095 .42516 2.35205 58 3 .36496 2.74004 .38487 2.59831 .40504 2.46888 .42551 2.35015 57 4 56529 2.73756 .38520 2.59606 .40538 2.46682 .42585 2.34825 56 5 .36562 2.73509 .38553 2.59381 .40572 2.46476 .42619 2.3463655 6 .36595 2.73263 .38587 2.59156 .40606 2.46270 .42654 2.3444754 7 .36628 2.73017 .38620 2.58932 .40640 2.46065 .42688 2.3425853 8 ,36661 2.72771 .38654 2.58708 .40674 2.45860 .42722 2.3406952 9 .36694 2.72526 .38687 2.58484 .40707 2.45655 .42757 2.3388151 10 .36727 2.72281 .38721 2.58261 .40741 2.45451 .42791 2.3369350 11 .36760 2.72036 .38754 2.58038 .40775 2.45246 .42826 2.3350549 12 .36793 2.71792 .38787 2.57815 .40809 2.45043 .42860 2.33317;48 13 .36826 2.71548 .38821 2.57593 .40843 2.44839 .42894 2.3313047 14 .36859 2.71305 .38854 2.57371 .40877 2.44636 .42929 2.3294346 15 .36892 2.71062 .38888 2.57150 .40911 2.44433 .42963 2.3275645 16 .36925 2.70819 .38921 2.56928 .40945 2.44230 .42998 2.3257044 17 .36958 2.70577 .38955 2.56707 .40979 2! 44027i .43032 2.3238343 1?* .36991 2.70335 .38988 2.56486 .41013 2.43825 .43067 2.32197 42 19 .37024 2.70094 .39022 2.56266 .41047 2.43623 .43101 2.32012 41 20 .37057 2.69853 .39055 2.56046 .41081 2.43422 .43136 2.31826 40 21 .37090 2.69612 .39089 2.55827 .41115 2.43220 .43170 2.31641 39 22 .37123 2.69371 .39122 2.55608 .41149 2.43019 .43205 2.3145638 28 .37157 2.69131 .39156 2.55389 .41183 2.42819 .43239 2.31271 37 24 .37190 2.68892 .39190 2.55170 .41217 2.42618 .43274 2.3108636 25 .37223 2.68653 .39223 2.54952 .41251 2.42418 .43308 2.3090235 20 .37256 2.68414 .39257 2.54734 .41285 2.42218 .43343 2.3071834 27 .37289 2.68175 .39290 2.54516 .41319 2.42019 .43378 2.30534 33 H .37322 2.67937 .39324 2.54299 .41353 2.41819 .43412 2.30351 32 29 .37355 2.67700 .39357 2.54082 .41387 2.41620 .43447 2.30167 31 30 .37388 2.67462 ,39391 2.53865 .41421 2.41421 .43481 2.29984 30 31 .37422 2.67225 .39425 2.53648 .41455 2.41223 .43516 2.29801 29 32 .37455 2.66989 .39458 2.53432 .41490 2.41025 .43550 2.29619 28 33 .37488 2.66752 .39492 2.53217 .41524 2.40827 .43585 2.29437 27 34 .37521 2.66516 .39526 2.53001 .41558 2.40629 .43620 2.29254 26 a^ .37554 2.66281 .39559 2.52786 .41592 2.40432 .43654 2.29073 25 36 .37588 2.66046 .39593 2.52571 .41626 2.40235 .43689 2.SMH 24 37 .37621 2.65811 .39626 2.52357 .41660 2.40038 43724 2.28710 23 38 .37654 2.65576 .39660 2.52142 .41694 2.39841 .43758 2.28528 22 39 .37687 2.65342 39694 2.51929 .41728 2.39645 .43793 2.28348 21 40 .37720 2.65109 .39727 2.51715 .41763 2.39449 .43828 2.28167 20 41 .37754 2.64875 .39761 2.51502 .41797 2.39253 .43862 2.27987 19 42 .37787 2.64642 .39795 2.51288 .41831 2.39058 .43897 2.27806 18 43| .37820 2.64410 .39829 2.51076 .41865 2.38863 .43932 2.27626 17 44 .37853 2.64177 .39862 2.50864 .41899 2.38688 .43966 2.27447 16 45 .37887 2.63945 .39896 2.50652 .41933 2.38473 .44001 2.27267 15 46 .37920 2.63714 .39930 2.50440 .41968 2.38279 .44036 2.27088 14 47 .37953 2.63483 .39963 2.50229 .42002 2 38084 .44071 2.26909 13 48 .37986 2.63252 .39997 2.50018 .42036 2.37891 .44105 2.26730 12 49 .38020 2.63021 .40031 2.49807 .42070 2.37697 .44140 2.26552 11 50 .38053 2.62791 .40065 2.49597 .42105 2.37504 .44175 2.26374 10 51 .38086 2.62561 .40098 2.49386 .42139 2.37311 .44210 2.26196 9 52 .38120 2.62332 .40132 2.49177 .42173 2.37118 .44244 2.26018 8 53 .38153 2.62103 .40166 2.48967 .42207 2.36925 .44279 2.25840 7 54 .38186 2.61874 .40200 2.48758 .42242 2.36733 .44314 2.25663 6 55 .38220 2.61646 .40234 2 48549 .42276 2.36541 .44349 2.25486 5 56 .38253 2.61418 .40267 2.48340 .42310 2.36349 .44384 2.25309 4 57 .38286 2.61190 .40301 2.48132 .42345 2.36158 .44418 2.25132 3 58 .38320 2.60963 .40335 2.47924 .42379 2.35967 .44453 2.24556 2 59 .38353 2.60736 .40369 2.47716 .42413 2.35776 .44488 2.24780 1 60 .38386 2.60509 .40403 2.47509 .42447 2.35585 .44523 2.24604 j M. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. M. 69* 68* 67 66* TABLE IH. NATUBAL TANGENTS; ETC. 24 5 U 26 27 M Tang. Cotarig Tang. Cotang Tang. Cotang Tang. Cotang M. .44523 2.24604 .46631 2.1445 .4877 2.0503 .5095 1.9626 60 .44558 2.24428 .46666 2.1428 .4880 2.0487 .5098 1.96120 59 .44593 2.24252 .46705 2.1412 .4884 2.0472 .5102 1.9597 58 , .44627 2.24077 .46737 2.1396J .4888 2.0457 .5106 1.9583 57 - .44662 2.23902 .46772 2.1380 .4891 2.0442 .5109 1.9569 50 .44697 2.23727 .46808 2.13639 .48953 2.04276 .51136 1.9555 55 i .44732 2.23553 .46843 2.1347' .48989 2.04125 .51173 1.9541 54 ' .44767 2.23378 .4687S 2.13316 .49026 2.03975 .51209 1.9527 53 1 .44802 2.23204 .46914 2.13154 .49062 2.03825 .51246 1.9513 52 c .44837 2.23030 .46950 2.12993 .49098 2.03675 .51283 1.94997 51 10 .44872 2.22857 .46985 2.12832 .49134 2.03526 .51319 1.94858 50 11 .44907 2.22683 .47021 2.12671 .49170 2.03376 .51356 1.94718 49 12 .44942 2.22510 .47056 2.12511 .49206 2.03227 .51393 1.94579 48 13 .44977 2.22337 .47092 2.12350 .49242 2.03078 .51430 1.94440 47 14 .45012 2.22164 .47128 2.12190 .49278 2.02929 .51467 1.94301 46 15 .450417 2.21992 .47163 2.12030 .49315 2.02780 .51503 1.94162 45 16 .45082 2.21819 .47199 2.11871 .49351 2.02631 .51540 1.94022 44 17 .45117 2.21647 .47234 2.11711 .49387 2.02483 .51577 1.93885 43 18 .45152 2.21475 .47270 2.11552 .49423 2.02335 .51614 1.93746 42 19 .45187 2.21304 .47305 2.11392 .49459 2.02187 .51651 1.93608 41 20 .45222 2.21132 .47341 2.11233 .49495 2.02039 .51688 1.93470 40 21 .45257 2.20961 .47377 2.11075 .49532 2.01891 .51724 1.93332 39 22 .45292 2.20790 .47412 2.10916 .49568 2.01743 .51761 1.93195 38. 23 .45327 2.20619 .47448 2.10758 .49604 2.01596 .51798 1.93057 37 24 .45362 2.20449 .47483 2.10600 .49640 2.01449 .51835 1.92920 36 25 .45397 2.20278 .47519 2.10442 .49677 2.01302 .51872 1.927C2 35 26 .45432 2.20108 .47555 2.10284 .49713 2.01155 .51909 1.92645 34 27 .45467 2.19938 .47590 2.10126 .49749 2.01008 .51946 1.92508 33 28 .45502 2.19769 .47626 2.09969 .49786 2.00862 .51983 1.92371 32 2 .45538 2.19599 .47662 2.09811 .49822 2.00715 .52020 1.92235 31 30 .45573 2.19430 .47698 2.09654 .49858 2.00569 .52057 1.92098 30 31 .45608 2.19261 .47733 2.09498 .49894 2.00423 .52094 1.91962 29 32 .45643 2.19092 .47769 2.09341 .49931 2.00277 .52131 1.91826 28 33 .45678 2.18923 .47805 2.09184 .49967 2.00131 .52168 1.91690 27 34 .45713 2.18755 .47840 2.09028 .50004 1.99986 .52205 1.91554 26 35 .45748 2.18587 .47876 2.08872 .50040 1.99841 .52242 1.91418 25 36 .45784 2.18419 .47912 2.08716 .50076 1.99695 .52279 1.91282 24 37 .45819 2.18251 .47948 2.08560 .50113 1.99550 .52316 1.91147 23 38 .45854 2.18084 .47984 2.08405 .50149 1.99406 .52353 1.91012 22 39 .45889 2.17916 .48019 2.08250 .50185 1.99261 .52390 1.90876 21 40 .45924 2.17749 .48055 2.08094 .50222 1.99116 .52427 1.90741 20 41 .45960 2.17582 .48091 2.07939 .50258 1.98972 .52464 1.90607 19 42 .45995 2.17416 .48127 2.07785 .50295 1.98828 .52501 1.90472 18 43 .46030 2.17249 .48163 2.07630 .50331 1.98684 .52538 1.90337 17 44 .46065 2.17083 .48198 2.07476 .50368 1.98540 .52575 1.90203 16 45 .46101 2.16917 .48234 2.07321 .50404 1.98396 .52613 1.90069 15 46 .46136 2.16751 .48270 2.07167 .50441 1.98253 .52650 1.89935 14 47 .46171 2.16585 .48306 2.07014 .50477 1.98110 .52687 1.89801 13 48 .46206 2.16420 .48342 2.06860 .50514 1.97966 .52724 1.89667 12 49 .46242 2.16255 .48378 2.06706 .50550 1.97823 .52761 1.89533 11 60 .46277 2.16090 .48414 2.06553 .50587 1.97681 .52798 1.89400 10 51 .46312 2.15925 .48450 2.06400 .50623 1.97538 .52836 1.89266 9 52 .46348 2.15760 .48486 2.06247 .50660 1.97395 .52873 1.89133 8 53 .46383 2.15596 .48521 2.06094 .50696 1.97253 .52910 1.89000 7 54 .46418 2.15432 .48557 2.05942 .50733 1.97111 .52947 1.88867 6 65 .46454 2.15268 .48593 2.05790 .50769 1.96969 .52985 1.88734 6 66 .46489 2.15104 .48629 2.05637 .50806 1.96827 .53022 1.88602 4 67 .46525 2.14940 .48665 2.05485 .50843 1.96685 .53059 1.88469 3 68 .46560 2.14777 .48701 2.05333 .50879 1.96544 .53096 1.88337 2 59 .46595 2.14614 .48737 2.05182 .50916 1.96402 .53134 1.88205 1 60 .46631 2.14451 .48773 2.05030 .50953 1.96261 .53171 1.88073 M. Cotang. Tang. Cotang Tang. Cotang. Tang. Cotang Tang. M. 65 64 63' 62 TABLE III. NATURAL TANGENTS, ETC. 35 28" 29 30 81- M. Tang. Cotang. Tang. | Cotang. Tang. Cotang. Tang. Cotang. M. .53171 .88073 .55431 1.80405 .57735 73205 .60086 .66428 60 1 .53208 .87941 .55469 1.80281 .57774 73089 .60126 .66318 59 2 .53246 .87809 .55507 1.80158 .57813 .72973 .60165 .66209 58 3 .53283 .87677 .55545 1.80034 .57851 .72857 .60205 .66099 57 4 .53320 .87546 .55583 1.79911 .57890 .72741 .60245 .65990 56 5 .53358 .87415 .55621 1.79788 .57929 .72625 .60284 .65881 55 6 .53396 .87283 .55659 1.79665 .57968 .72509 .60324 .65772 54 7 .53432 .87152 .55697 1.79542 .58007 .72393 .60364 .65663 53 8 .53470 .87021 .55736 1.79419 .58046 .72278 .60403 .65554 52 9 .53507 .86891 .55774 1.79296 .58085 .72163 .60443 .65445 51 .53545 .86760 .55812 1.79174 .58124 .72047 .60483 .65337 50 .53582 .86630 .55850 1.79051 .58162 .71932 .60522 .65228 49 .53620 .86499 .55888 1.78929 .58201 .71817 .60562 .65120 48 .53657 .86369 .55926 1.78807 .58240 .71702 .60602 .65011 47 .53694 .86239 .55964 1.78685 .58279 .71588 .60642 .64903 46 .53732 .86109 .56003 1 78563 .58318 .71473 .60681 .64795 45 .53769 .85979 .56041 1.78441 .58357 .71358 .60721 .64687 44 .53807 .85850 .56079 1.78319 .58396 .71244 .60761 .64579 43 .53844 .85720 .56117 1.78198 .58435 .71129 .60801 .64471 42 .53882 .85591 .56156 1.78077 .58474 .71015 .60841 .64363 41 .53920 .85462 .56194 1.77955 .58513 .70901 .60881 .64256 40 .53957 .85333 .56232 1.77834 .58552 .70787 .60921 .64148 39 .53995 .85204 .56270 1.77713 .58591 .70673 .60960 .64041 38 .54032 .85075 .56309 1.77592 .58631 .70560 .61000 .63934 37 .54070 .84946 .56347 1.77471 .58670 .70446 .61040 .63826 36 .54107 .84818 .56385 1.77351 .58709 .70332 .61080 .63719 85 .54145 .84689 .56424 1.77230 .58748 .70219 .61120 .63612 84 .54183 .84561 .56462 1.77110 .58787 .70106 .61160 .63505 83 .54220 .84433 .56501 1.76990 .58826 .69992 .61200 .63398 3^ .54258 .84305 .56539 1.76869 .58865 .69879 .61240 .63292 31 .54296 .84177 .56577 1.76749 .58905 .69766 .61280 .63185 80 .54333 .84049 .56616 1.76629 .58944 .69653 .61320 .63079 29 .54371 .83922 .56654 1.76510 .58983 .69541 .61360 .62972 28 .54409 .83794 .56693 1.76390 .59022 .69428 .61400 .62866 27 3* .54446 .83667 .56731 1.76271 .59061 .69316 .61440 .62760 26 35 .54484 .83540 .56769 1.76151 .59101 .69203 .61480 .62654 25 36 .54522 .83413 .56808 1.76032 .59140 .69091 .61520 .62548 24 37 .54560 .83286 .56846 1.75913 .59179 .68979 .61561 .62442 23 38 .54597 .83159 .56885 1.75794 .59218 .68866 .61601 .62336 2-2 39 .54635 .83033 .56923 1.75675 .59258 .68754 .61641 .62230 21 40 .54673 .82906 .56962 1.75556 .59297 .68643 .61681 .62125 20 41 .54711 .82780 .57000 1.75437 .59336 .68531 .61721 .62019 19 42 .54748 .82654 .57039 1.75319 .59376 .68419 .61761 .61914 18 43 .54786 .82528 .57078 1.75200 .59415 .68308 .61801 .61808 17 44 .54824 .82402 .57116 1.75082 .59454 .68196 .61842 .61703 16 45 .54862 .82276 .5.7155 1.74964 .59494 .68085 .61882 .61598 15 46 .54900 .82150 .57193 1.74846 .59533 .67974 .61922 .61493 14 47 .54938 .82025 .57232 1.74728 .59573 .67863 .61962 .61388 13 48 .54975 .81899 .57271 1.74610 .59612 .67752 .62003 .61283 12 49 .55013 .81774 .57309 1.74492 .59651 .67641 .62043 .61179 11 50 .55051 .81649 .57348 1.74375 .59691 .67530 .62083 .61074 10 51 .55089 .81524 .57386 1.74257 .59730 .67419 .62124 .60970 g 52 .55127 .81399 .57425 1.74140 .59770 .67309 .62164 .60865 8 53 .55165 .81274 .57464 1.74022 .59809 .67198 .62204 .60761 7 5t .55203 .81150 .57503 1.73905 .59849 .67088 .62245 .60657 6 55 .55241 .81025 .57541 1.73788 .59888 .66978 .62285 .60553 6 56 .55279 1.80901 .57580 1.73671 .59928 .66867 .62325! 1.60449 4 57 .55317 1.80777 .57619 1.73555 .59967 .66757 .623661 1.60345 8 58 .55355 1.80653 .57657 1.73438 .6000< .66647 .62406 1.60241 2 59 .55393 1.80529 .57696 1.73321 .60046 .66538 .62446 1.60137 1 60 .55431 1.80405 .57735 1.73205 .60086 .66428 .62487 1.60033 M 1 Cotang. Tang. Cotang. Tang. Cotang. Tang. rotang. Tang. 5T 61 60' 59 58 TABLE III. NATURAL TANGENTS, ETC. 3 3 2 3 3 4 1 * 5 M Tang. Cotang Tang. Cotang Tang. Cotang Tang. Cotang M- i .62487 1.6003*. .6494 1.5398 .6745 1.48256 .7002 1.4281 60 .62527 1.5993C .6498 1.53888 >6749 1,4816 .70064 1.4272 59 i .62568 1.59826 .6502 1.5379 .6753 1.4807 .7010 1.4263 58 j .62608 1.59723 .6506 1.5369 .6757 1.4797 .7015 1.4255 57 i .6264S 1.5962C .65106 1.5359 .6762 1.4788 .7019 1.4246 66 i .62689 1.59517 .6514 1.5349 .6766 1.4779 .7023 1.4237 65 i .62730 1.59414 .6518 1.5340C .6770 1.4769 .7028 1.4228 54 1 .62770 1.5931 .6523 1.5330 .6774 1.4760 .7032 1.4219 53 .62811 1.59208 .6527 1.5320 .67790 1.4751 .7036 1.4211 62 ; .62852 1.5910S .6531 1.5310 .6783 1.4742 .7041 1.4202 61 1 .62892 1.59002 .6535 1.5301 .6787 1.4733( .7045 1.41934 60 .62933 1.58900 .6539 1.5291 .6791 1.4723 .7049 1.4184 49 .62973 1.58797 .6543 1.5281 .6796 1.4714 .7054 1.4175 48 .63014 1.58695 .6548 1.5271 .6800 1.4705 .7058 1.4167 47 i .63055 1.58593 .6552 1.5262 .6804 1.4696 .7062 1.4158 46 15 .63095 1.5849C .6556 1.5252 .6808 1.4687 .7067 1.4149' 45 16 .63136 1.58388 .6560 1.5242 .6813 1.4677 .7071 1.4140 44 17 .63177 1.58286 .6564 1.5233 .6817 1.4668 .7076 1.4132 43 18 .63217 1.58184 .6568 1.52235 .6821 1.4659 .70804 1.4123, 42 19 .63258 1.58083 .65729 1.5213 .6825 1.4650 .7084 1.4114 41 20 .63299 1.57981 .6577 1.5204 .6830 1.4641 .7089 1.4106 40 21 .63340 1.57879 .65813 1.5194 .6834 1.4632 .7093 1.4097 89 22 .63380 1.57778 .65854 1.51850 .6838 1.4622 .7097 1.4088 38 23 .63421 1.57676 .65896 1.51754 .6842 1.4613 .7102 1.4080C 37 24 .63462 1.57575 .65938 1.51658 .6847 1.4604 .7106 1.4071 36 25 .63503 1.57474 .65980 1.51562 .6851 1.4595 .7111 1.4062 35 26 .63544 1.57372 .66021 1.51466 .6855 1.4586 .71154 1.4054 34 27 .63584 1.57271 .66063 1.51370 .68600 1.4577 .71198 1.4045 33 28 .63625 1.57170 .66105 1.51275 .68642 1.4568 .71242 1.4036 32 29 .63666 1.57069 .66147 1.51179 .68685 1.4559 .71285 1.4028 31 30 .63707 1.56969 .66189 1.51084 .68728 1.4550 .71329 1.4019 30 31 .63748 1.56868 .66230 1.50988 .68771 1.4541 .71373 1.4010 29 32 .63789 1.56767 .66272 1.50893 .68814 1.4532 .71417 1.4002 28 33 .63830 1.56667 .66314 1.50797 .68857 1.4522 .71461 1.399E 27 3* .63871 1.56566 .66356 1.50702 .68900 1.45139 .71505 1.3985 26 35 .63912 1.56466 .66398 1.50607 .68942 1.45049 .71549 1.3976 25 36 .63953 1.56366 .66440 1.50512 .68985 1.44958 .71593 1.3967 24 37 .63994 1.56265 .66482 1.50417 .69028 1.44868 .71637 1.3959 23 38 .64035 1.56165 .66524 1.50322 .69071 1.44778 .71681 1.3950 22 39 .64076 1.56065 .66566 1.50228 .69114 1.44688 .71725 1.3942 21 40 .64117 1.55966 .66608 1.50133 .69157 1.44598 .71769 1.39336 20 41 .64158 1.55866 .66650 1.50038 .69200 1.44508 .71813 1.39250 19 42 .64199 1.55766 .66692 1.49944 .69243 1.44418 .71857 1.39165 18 43 .64240 1.55666 .66734 1.49849 .69286 1.44329 .71901 1.39079 17 44 .64281 1.55567 .66776 1 . 49755 .69329 1.44239 .71946 1.38994 16 45 .64322 1.55467 .66818 1.49661 .69372 1.44149 .71990 1.38909 15 46 .64363 1.55368 .66860 1.49566 .69416 1.44060 .72034 1.38824 14 47 .64404 1.55269 .66902 1.49472 .69459 1.43970 .72078 1.38738 13 48 .64446 1.55170 .66944 1.49378 .69502 1.43881 .72122 1.38653 12 49 .64487 1.55071 .66986 1.49284 .69545 1.43792 .72167 1.38568 11 50 .64528 1.54972 .67028 1.49190 .69588 1.43703 .72211 1.38484 10 51 .64569 1.54873 .67071 1.49097 .69631 1.43614 .72255 1.38399 9 62 .64610 1.54774 .67113 1.49003 .69675 1.43525 .72299 1.38314 8 53 .64652 1.54675 .67155 1.48909 .69718 1.43436 .72344 1.38229 7 54 .64693 1.54576 .67197 1.48816 .69761 1.43347 .72388 1.38145 6 55 .64734 1.54478 .67239 1.48722 .69804 1.43258 .72432 1.38060 5 56 .64775 1.54379 .67282 1.48629 .69847 1.43169 .72477 1.37976 4 57 .64817 1.54281 .67324 1.48536 .69891 1.43080 .72521 1.37891 3 58 .64858 1.54183 .67366 1.48442 .69934 1.42992 .72565 1.37807 2 59 .64899 1.54085 .67409 1.48349 .69977 1.42903 .72610 1.37722 1 60 .64941 1.53986 .67451 1.48256 .70021 1.42815 .72654 1.37638 M^ Cotang. Tang. Cotang. Tang. Cotang. Tang. Uotang. Tang. M. 5 7* 5 8 a 5' 54 f TABLE III. NATURAL TANGENTS, ETC. 37 36 37 38 39* M. Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. M. .72654 1.37638 .75355 .32704 .78129 .27994 .80978 1.23490 60 1 .72699 1.37554 .75401 .32624 .78175 .27917 .81027 1.23416 59 2 .72743 1.37470 .75447 .32544 .78222 .27841 .81075 1.23343 58 3 .72788 1.37386 .75492 .32464 .78269 .27764 .81123 1.23270 57 4 .72832 1.37302 .75538 .32384 .78316 .27688 .81171 1.23196 56 5 .72877 1.37218 .75584 .32304 .78363 .27611 .81220 1.23123 55 6 .72921 1.37134 .75629 .32224 .78410 .27535 .81268 1.23050 54 7 .72966 1.37050 .T5675 .32144 .78457 .27458 .81316 1.22977 53 8 .73010 1.36967 .75721 .32064 .78504 .27382 .81364 1.22904 52 9 .73055 1.36883 .75767 .31984 .78551 .27306 .81413 1.22831 51 10 .73100 1.36800 .75812 .31904 .78598 .27230 .81461 1.22758 50 11 .73144 1.36716 .75858 .31825 .78645 .27153 .81510 1.22685 49 12 .73189 1.36633 .75904 .31745 .78692 .27077 .81558 1.22612 48 13 .73234 1.36549 .75950 .31666 .78739 .27001 .81606 1.22539 47 14 .73278 1.36466 .75996 .31586 .78786 .26925 .81655 1.22467 46 15 .73323 1.36383 .76042 .31507 .78834 .26849 .81703 1.22394 45 16 .73368 1.36300 .76088 .31427 .78881 .26774 .81752 1.22321 44 17 .73413 1.36217 .76134 .31348 .78928 .26698 .81800 1.22249 43 18 .73457 1.36134 .76180 .31269 .78975 .26622 .81849 1.22176 42 19 .73502 1.36051 .76226 .31190 .79022 .26546 .81898 1.22104 41 20 .7a547 1.35968 .76272 .31110 .79070 .26471 .81946 1.22031 40 21 .73592 1.35885 .76318 .31031 .79117 .26395 .81995 1.21959139 22 .73637 1.35802 .76364 .30952 .79164 .26319 .82044 1.21886 38 83 .73681 1.35719 .76410 .30873 .79212 .26244 .82092 1.21814 37 24 .73726 1.35637 .76456 .30795 .79259 .26169 .82141 1.21742 36 25 .73771 1.35554 .76502 .30716 .79306 .26093 .82190 1.21670 35 2.: .73816 1.35472 .76&8 .30637 .79354 .26018 .82238 1.21598 34 27 .73861 1.35389 .76594 .30558 .79401 .25943 .82287 1.21526 33 28 .73906 1.35307 .76640 .30480 .79449 .25867 .82336 1.21454 32 29 .73951 1.35224 .76686 .30401 .79496 .25792 .82385 1.21382 31 30 .73996 1.35142 .76733 .30323 .79544 .25717 .82434 1.21310 30 31 .74041 1.35060 .76779 .30244 .79591 .25642 .82483 1.21238 29 82 .74086 1.34978 .76825 .30166 .79639 .25567 .82531 1.21166 28 83 .74131 1.34896 .76871 .30087 .79686 .25492 .82580 1.21094 27 84 .74176 1.34814 .76918 .30009 .79734 .25417 .82629 1.21023 26 85 .74221 1.34732 .76964 .29931 .79781 .25343 .82678 1.20951 25 86 .74267 1.34650 .77010 .29853 .79829 .25268 .82727 1.20879 24 37 .74312 1.34568 .77057 .29775 .79877 .25193 .82776 1.20808 23 3s .74357 1.34487 .77103 .29696 .79924 .25118 .82825 1.20736 22 3i .74402 1.34405 .77149 .29618 .79972 .25044 .82874 1.20665 21 40 .74447 1.34323 .77196 .29541 .80020 .24969 .82923 1.20593 20 41 .74492 1.34242 .77242 .29463 .80067 .24895 .82972 1.20522 19 42 .74538 1.34160 .77289 .29385 .80115 .24820 .83022 1.20451 18 43 .74583 1.34079 .77335 .29307 .80163 .24746 .83071 1.20379 17 44 .74628 1.33998 .77382 .29229 .80211 .24672 .83120 1.20308 16 45 .74674 1.33916 .77428 .29152 .80258 .24597 .83169 1.20237 15 46 .74719 1.33835 .77475 .29074 .80306 .24523 .83218 1.20166 14 47 .74764 1.33754 .77521 .28997 .80354 .24449 .83268 1.20095 13 48 .74810 1.33673 .77568 .28919 .80402 .24375 .83317 1.20024 12 49 .74855 1.33592 .77615 .28842 .80450 .24301 .83366 1.19953 11 50 .74900 1.33511 .77661 .28764 .80498 .24227 .83415 1.19882 10 51 .74946 1.33430 .77708 .28687 .80546 .24153 .83465 1.19811 9 52 .74991 1.33349 .77754 .28610 .80594 .24079 .83514 1.19740 8 53 .75037 1.33268 .77801 - .28533 .80642 .24005 .83564 1.19669 7 54 .75082 1.33187 .77848 .28456 .80690 .23931 .83633 1.19599 6 55 .75128 1.33107 .77895 .28379 .80738 .23858 .83662 1.19528 5 56 .75173 1.33026 .77941 .28302 .80786 .23784 .83712 1.19457 4 57 .75219 1.32946 .77988 .28225 .80834 .23710 .83761 1.19387 3 58 .75264 1.32865 .78035 .28148 .80882 .23637 .83811 1.19316 2 59 .75310 1.32785 .78082 .28071 .80930 .23563 .83860 1.19246 1 60 .75355 1.32704 .78129 .27994 .80978 .23490 .83910 1.19175 M. Cotang. Tang. Cotang. Tang, Cotang. Tang. Cotang. Tang. M. I 52 51 5O TABLE III. NATUKAL TANGENTS, ETC. 40 41 42 43 1 M. Tang. Cotang. Tang. Cotang. Tang. Cotang. Tang. Cotang. M .83910 1.19175 .86929 1 . 15037 .90040 .11061 .93252 1.07237 60 1 .83960 1.19105 .86980 1 . 14969 .90003 .10996 .93306 1.07174 59 2 .84009 1.19035 .87031 1.U902 .90146 . 10931 .93360 1.07112 58 3 .84059 1 . 18964 .37082 1.11834 .90199 . 1G867 .93415 1.07049 57 4 .84108 1.18894 .87133 1.14767 .90251 . 10802 .93469 1.06987 ec 5 .84158 1.18824 .87184 1.14699 .90304 . 10737 .93524 1.06925 55 6 .84208 1 . 18754 .87236 1.14C32 .90357 .1067* .93578 1.06862 54 7 .84258 1 . 18684 .87287 1.14565 .90410 .10607 .93633 1 .06800 53 8 .84307 1.18614 .87338 1.14498 .90463 1.10543 .93688 1.06738 52 9 .84357 1 . 18544 .87389 1.14430 .90516 1.10478 .93742 1.06676 51 10 .84407 1.18474 .87441 1.14363 .90569 1.10414 .93797 1.06613 50 11 .84457 1 . 18404 .87492 1.14296 .90621 1 . 10349 .93852 1.06551 57.19 56.74 Kf OA .999708 .999704 .999699 .'08 .08 no .564291 .567727 .571137 Dl . v 57.27 56.82 .435709 .432273 .428863 54 53 52 9 .574214 oo. ou 65.87 .999694 Uo .08 .574520 55^95 .425480 61 10 8.577566 55 44 9.999689 AQ 8.577877 KK Kf) 11.422123 CO 11 .580892 KK fi9 .999685 .Uo fift .581208 DO. O^ err -\f\ .418792 .49 12 .584193 OD. \)& .999680 Uo .584514 Do. 1U .415486 48 13 .587469 M1O .999675 *AQ .587795 54.68 KA f)i-f .412205 47 14 15 .590721 .593948 . 1 J 63.79 pro OQ .999670 .999665 .Uo .08 /\Q .591051 .594283 D4 . Zi 53.87 .408949 .405717 46 45 16 17 18 19 .597152 .600332 .603489 .606623 DO. t>y 53.00 52.61 52.23 51.86. .999660 .999655 .999650 .999645 Uo .08 .08 .08 .09 .597492 .600677 .603839 .606978 53.47 63.08 52.70 52.32 51.94 .402508 .399323 .396161 .393022 44 43 41 20 8.609734 51 49 9.999640 8.610094 11.389906 40 21 .612823 61 12 .999635 no .613189 f-t 01 .386811 9 22 23 24 25 26 .615891 .618937 .621962 .624965 .627948 50!76 50.41 50.06 49.72 .999629 .999624 .999619 .'999614 .999608 Uy .03 .09 .03 .03 .616262 .619313 .622343 .625352 .628340 ol .^1 60.85 50.50 60.15 49.81 .383738 .380687 .377657 .374648 .371660 38 37 - 36 35 27 28 .630911 .633854 49!04 48 71 .999603 .999597 !09 f)O .631308 .634256 49^13 AQ Qf\ .368692 .365744 3 32 29 .636776 ^to. i L 48.39 .999592 .\j*7 .09 .637184 4o. oO 48.48 .362816 31 30 31 32 33 34 35 36 37 38 39 8.639680 .642563 .645428 .648274 .651102 .653911 .656702 .659475 .662230 .664968 48.06 47.75 47.43 47.12 46.82 46.52 46.22 45.92 45.63 45.35 9.999586 .999581 .999575 .999570 .999564 .999558 .999553 .999547 .999541 .999535 .09 .09 .03 .03 .09 .10 .10 .10 .10 .10 8.640093 .642982 .645853 .648704 .651537 .654352 .657149 .659928 .6626C9 .6654S3 48.16 47.84 47.53 47.22 46.91 46.61 46.31 46.02 45.73 45.44 11.359907 .357018 .354147 .351296 .348463 .345848 .342851 .340072 .337311 .334567 30 29 28 27 25 23 22 21 40 41 42 43 44 45 46 47 43 49 8.667689 .670393 .673080 .675751 .678405 .681043 .683665 ! 688863 .691438 45.06 44.79 44.51 44.24 43.97 43.70 43.44 43.18 42.92 42.67 9.999529 .999524 .999518 ".999312 .999506 .999500 .999493 .999487 .999481 .999475 .10 .10 .10 .10 .10 .10 .10 .10 .10 .10 8.6681CO .670870 .673503 .6762S9 .678900 .681544 .684172 .68G7F4 .6893C1 .691263 45.16 44.88 44.61 44.34 44.07 43.80 43.54 43.28 43.03 42.77 11.331840 .329120 .326437 .323761 .321100 .3184C6 .315818 .313216 .310619 .308037 20 18 17 16 . 15 14 13 12 11 60 61 8.693998 .696543 42.42 A.O 1 7 9.9991C9 .99D4G3 .10 8.094529 .697081 42.52 11.305471 .302919 10 9 62 63 64 65 66 .699073 .701589 .704090 .706577 .709049 *- . 1 f 41.92 41.68 41.44 41.21 Af\ Q7 .9994C6 .999450 .999443 .999437 .999431 .11 .11 .11 .11 .699617 .7021 9 .704646 707140 42.^8 42. f 3 41.79 41.55 41.32 .300383 .297861 .295354 .292860 .290CS2 8 7 6 5 4 67 68 69 60 .711507 .713952 .716383 .718800 4v.y* 40.74 40.51 40.29 .999424 .999418 .999411 .999404 .11 .11 .11 71208-5 .714534 .716972 .719396 41 .08 40.85 40. 62 40.40 .287917 .285465 .283028 .280604 3 2 1 M. Cosine. D.l . Sine. D.l - Cotailg. D.l". 92 87^ TABLE IV. LOGARITHMIC SINES, ETC. 43 176 Sine. D.l. Cosine. |D. l. Tang. D.I". Cotang. M. 8.718800 .721204 .723595 .725972 .728337 .783027 .735354 .737667 8.742259 .744536 .746802 .751297 .753528 .755747 .757955 .760151 .762337 8.764511 .766675 .770970 .773101 .775223 .777333 .779434 .781524 .783605 8.785675 .787736 .789787 .791828 .793859 .795881 .797894 .801892 .807819 .809777 .811726 .813667 .815599 .817522 8.825130 .827011 .830749 .832607 .S34456 .836297 .838130 .841774 M. I Cosine. 93 40.06 39.41 39.19 38.77 38.57 38.16 37.96 37.76 37.56 37.37 37.17 36.98 36.79 36.61 36.42 36.24 36.06 35.70 35.53 35.35 35.18 35.01 34.84 34.67 34.51 34.31 34.18 34.02 33.70 33.54 33.39 33.23 33.08 32.93 32.78 32.63 32.49 32.34 32.19 32.05 31.91 31.77 31.63 31.49 31.35 31.22 31.08 30.95 30.56 30.43 30.30 30.17 .999371 .999315 9.999265 .999257 .999250 .999212 .999205 .999197 .999181 .999174 .999150 .999142 9.999110 .999102 .999077 .[99069 .9S9061 .999053 0.999027 .999019 .999002 .11 .11 .11 .11 .11 .11 .12 .12 .12 .12 .12 .12 .12 .12 .12 .12 .12 .12 .12 .12 .12 .12 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .13 .14 .14 .14 .14 .14 .14 .11 .14 .14 .14 .14 .14 .14 .14 .14 .15 .15 .15 D 1". Sine. ID.i '. Cotang. .721806 .724204 .726588 .731317 .733663 .735996 .738317 .740626 8.742922 .745207 .747479 .749740 .751989 .754227 .756453 .760872 .763065 8.765246 .767417 .769578 .771727 .773866 .775995 .778114 .780222 .784408 8.786486 .788554 .790613 .792662 .794701 .796731 .798752 .800763 .802765 .804758 8.806742 .808717 .812641 .814589 .816T29 .818461 .820384 .824205 8.82C103 .827992 .829874 .831748 .835471 .837321 .8?9163 .842*25 .844644 40.17 39.95 39.74 39.52 39.30 38.48 38.27 38.07 37 87 37.68 37.49 37.29 37.10 36.92 36.73 36. 5 36.18 36.00 35.83 35.65 35.48 35.31 35.14 34.97 34.80 34.64 34.47 34.31 34.15 33.99 33.83 33.68 33.52 33.37 33.22 33.07 32.92 32.78 32.62 32.48 32.33 32.19 32.05 31.91 31.77 31.63 31.50 31.36 31.23 31.10 30.83 30.70 30.57 bO.45 30.32 D.I". 1.280604 .278194 .275796 .273412 .271041 .264004 .261683 .259374 1.257078 .254793 .252521 .250260 .248011 .245773 .243547 .241332 .239128 .236935 1.234754 .232583 ! 228273 .226134 .224005 .221886 .219778 .217680 .215592 11.213514 .211446 .208387 .201248 ! 197236 .195242 11.193258 .191283 .189317 .187359 .185411 .183471 .181539 .179616 .177702 .175795 11.173897 .172008 .170126 .168252 .166387 .164529 .162679 .160837 .159002 .157175 .155356 Tang. 86 C 44 TABLE IV. LOGARITHMIC SINES, ETC. 175 a M. Sine. D.I". Cosine. D.I". Tang. D.I". Cotang. M. 1 2 3 4 5 6 7 8 9 8.843585 .845387 -847183 .848971 .850751 .852525 .854291 .856049 .857801 .859546 30.05 29.92 29.80 29.67 29.55 29.43 29.31 29.19 29.07 28.96 9.993941 .998932 .993923 .998914 ,99:905 .993896 .993887 .993878 .993869 .993860 .15 .15 .15 .15 .15 .15 .15 .15 .15 .15 8.844644 .846455 .848260 .850057 .851846 - .853628 .855403 .857171 .858932 .860686 30.19 30.07 29.95 29.82 29.70 29.58 29.46 29.35 29.23 29.11 11.155356 .153545 .151740 .149943 .148154 .146372 .144597 .142829 .141068 .139314 60 59 58 57 56 55 54 53 52 51 10 11 12 13 14 8.861283 .863014 .864738 .866455 .868165 28.84 28.73 28.61 28.50 9.998S51 .998841 .998832 .998823 .998813 .15 .15 .15 .16 8.862433 .864173 .865906 .867632 .869351 29.00 28.88 28.77 28.66 11.137567 .135827 .134094 .132363 .130649 50 49 48 47 46 15 16 17 18 19 .869868 .871565 .873255 .874938 .876615 28.28 28.17 28.06 27.95 27.86 .998804 .998795 .998785 .998776 .998766 .16 .16 .16 .16 .16 .871064 .872770 .874469 .876162 .877849 28.43 28.32 28.21 28.11 28.00 .128936 .127230 .125531 .123833 .122151 45 44 43 42 41 20 21 8.878285 .879949 27.73 9.998757 .998747 .16 8.879529 .881202 27.89 11.120471 .118798 40 39 22 23 24 25 26 27 28 29 .881607 .883258 .884903 .886542 .888174 .889801 .891421 .893035 27.52 27.42 27.31 27.21 27.11 27.00 26.90 26.80 .993738 .998728 .998718 .998708 .998690 .998689 .993679 .998669 .16 .16 .16 .16 .16, .16 .16 .17 .882869 .884530 .886185 .887833 .889476 .891112 .892742 .894365 27.68 27.58 27.47 27.37 27.27 27.17 27.07 26.97 .117131 .115470 .113815 .112167 .110524 .108883 .107253 .105634 If 36 35 34 33 32 31 30 31 32 33 34 35 8.894643 .896246 .897842 .899432 .901017 .902596 26.70 26.60 26.51 26.41 26.31 9.998659 .993649 .998639 .998629 .998619 .998609 .17 .17 .17 .17 .17 8.895984 .897596 .890203 .900803 .902398 .903987 26.87 26.77 26.67 26.58 26.48 11.104016 .102404 .100797 .099197 .097602 .096013 30 29 28 27 26 25 36 37 38 .904169 .905736 .907297 .908853 26.12 26.03 25.93 25.84 .993599 .998589 .998578 .998568 .17 .17 .17 .17 .905570 .907147 .908719 .910285 26.29 26.20 26.10 26.01 .094430 .092853 .091281 .089715 24 23 22 21 40 41 42 43 44 45 43 47 43 49 8.910404 .911949 .913488 .915022 .916550 .918073 .919591 .921103 .922610 .924112 25.75 25.66 25.56 25.47 25.38 25.29 25.20 25.12 25.03 24.94 9.998558 .998548 .998537 .998527 .998516 .908506 .998495 .998485 .993474 .998464 .17 .17 .17 .17 .18 .18 .18 .18 .18 .18 8.911846 .913401 .914951 .916495 .918034 .919568 .921096 .922619 .924136 .925649 25.92 25.83 25.74 25.65 26.56 25.47 25.38 25.30 25.21 25.12 11.088154 .086599 .085049 .083505 ' .081966 .080432 .078904 .077381 .075864 .074351 20 19 18 17 16 15 14 13 12 11 59 51 8.925609 .927100 24.86 9.998453 .998442 .18 8.927156 .928658 25.03 11.072844 .071342 10 9 52 53 54 55 56 57 58 59 60 .928587 ,930068 .931544 .933015 .934481 .935942 .937398 .938850 .940296 24.77 24.69 24.60 24.52 24.43 24.35 24.27 24.19 24.11 .998431 .998121 .998410 .998399 .998388 .998377 .998366 .998355 .998344 .18 .18 .18 .18 .18 .18 .18 .18 .18 .930155 .931647 .933134 .934616 .936093 .937565 .939032 .940494 .941952 24.95 24.86 24.78 24.70 24.61 24.53 24.45 24.37 24.30 .069845 .068353 .066866 .065384 .063907 .062435 .060968 .059506 .058048 8 7 6 5 4 3 2 1 M. Co^ino. D.I". Pino, n i". r-otang. D.I". Tang. M. 94 C 85 TABLE IV. LOGARITHMIC SINES, ETC. 46 5" 174 8 M. Sine. D.l". Cosine. D.I V Tang. D.I". Cotang. M. 1 2 3 4 5 6 7 8 9 8.940296 1941738 .943174 .944606 .946034 .947456 .948874 .950287 .951696 .953100 24.03 23.94 23.87 23.79 23.71 23.63 23.55 23.48 23.40 23.32 9.998344 .998333 .998322 .998311 .998300 .998289 .998277 .998266 .998255 .998243 .19 .19 .19 .19 .19 .19 .19 .19 .19 .19 8.941952 .943404 .944852 .946295 .947734 .949168 .950597 .952021 .953441 .954856 24.21 24.13 24.05 23.97 23.90 23.82 23.74 23.67 23.60 23.51 11.058048 .056596 .055148 .053705 .052266 .050832 .049403 .047979 .0-16559 .045144 60 59 58 57 56 55 54 53 52 51 10 11 12 8.954499 .955894 .957284 23.25 23.17 9.998232 .998220 .998209 .19 .19 8.956267 .957674 .959075 23.44 23.37 11.043733 .042326 .040925 50 49 48 13 14 15 16 17 18 19 .958670 .960052 .961429 .S62801 .964170 .C65534 .166893 23.10 23.02 22.95 22.88 22.80 22.73 22.66 22.59 .998197 .998186 .998174 .998163 .998151 .998139 .998128 .19 .19 .19 .19 .19 .19 .20 .20 .960473 .961866 .963255 .964639 .966019 .967394 .968766 23.29 23.22 23.14 23.07 23.00 22.93 22.86 22.79 .039537 .038134 .036745 .035361 .033981 .032606 .031234 47 46 45 44 43 42 41 20 8.968249 9.998116 8.970133 11.029867 40 21 .969600 22.52 .998104 .20 .971496 22.72 .028501 39 22 .970947 .972289 22.45 22.38 .998092 .998080 .20 .20 OA .972855 .974209 22.65 22.57 077145 38 37 24 26 26 27 28 .973628 .974962 .976293 .977619 .978941 22.31 22.24 22.17 22.10 22.03 .998068 .998056 .998044 .998032 .998020 .M .20 .20 .20 .20 .975560 .976906 .978248 .979586 .980921 22.51 22.44 22.37 22.30 22.23 .0:)4440 02309* .021753 .020414 .019079 36 35 34 33 32 29 .'80259 21.97 21.90 .998008 .20 .20 .982251 22.17 22.10 .017749 31 30 31 8.981573 .982883 21.83 9.9979S6 .997984 .20 8.983577 .984899 22.04 11.016423 .015101 30 29 32 33 34 35 36 37 38 39 .984189 .985491 .986789 .988083 .989374 .990660 .891943 .993222 21 .77 21.70 21.63 21.57 21.50 21.44 21.38 21.31 21.25 .997972 .997959 .997947 .997935 .997922 .997910 .997897 .997885 .20 .20 .20 .20 .21 .21 .21 .21 .21 .986217 .987532 .988842 .990149 .991451 .992750 .994045 .995337 21.97 21.91 21.84 21.78 21.71 21.65 21.58 21.52 21.46 .013783 .012468 .011158 .009851 .C08549 .007250 .005955 .004663 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 8.994497 .995768 .997036 ,998299 .999560 9.000816 .C02069 21.19 21.12 21.06 21.00 20.94 20.88 9.997872 .997860 .997847 .997835 .997822 .997809 .S97797 .21 .21 .21 .21 .21 .21 8.996624 .997908 .999188 9.C00465 .C01738 .C03007 .C04272 21.40 21.34 21.27 21.21 21.15 21.09 11.003376 .002092 .OC0812 10 999535 I9S263 9nti093 .91)5728 20 19 18 17 16 15 14 47 48 49 .003318 .004563 .005805 20^76 20.70 20.64 .997784 .997771 .997758 .21 .21 .21 .21 .CC5534 .C06792 .008047 21.03 20.97 20.91 20.86 .994466 .993208 .991953 13 12 11 50 51 52 53 54 55 66 57 58 59 60 9.007044 .008278 .009510 .010737 .011962 .013182 .014400 .015613 .016824 .018031 .019235 20.58 20.52 20.46 20.40 20.34 20.29 20.23 20.17 20.12 20.46 9.997745 .997732 .997719 .997706 .997693 .997680 .997667 .997654 .997641 .997628 .997614 .21 .21 .21 .21 .22 .22 .22 .22 .22 .22 9.009298 .010546 .011790 .013031 .014268 .015502 .016732 .017959 .019183 .020403 .021320 20.80 20.74 20.68 20.62 20.56 20.51 20.45 20.40 20.33 20.28 10.990702 .989154 .988210 .986969 .985732 .984498 .983268 .982041 .980817 .979C97 .978380 10 9 8 7 6 5 4 3 1 M. Cosine, D.r'. Sine. Drl". Cotang. D.I". Tang. liT 95' 84 46 TABLE IV. LOGARITHMIC SINES, ETC. 6" 173 M. Sine. D.l". Cosine. D.l". Tang. D.l". Cotang. M. 1 2 3 4 5 6 9.019235 .020435 .021632 .022825 .024016 .025203 .026386 20.00 19.95 19.89 19.84 19.78 19.73 1Q &7 9.997614 .997601 .997588 .997574 .997561 .997547 .997534 .22 .22 .22 .22 .22 .22 9.021620 .022834 .024044 .025251 .026455 .027655 .028852 20.23 20.17 20.12 20.06 20.01 19.95 .0.978380 .977166 .975956 .974749 .973545 .972345 .971148 61 5J 53 57 53 53 51 7 .027567 ly.Di .997520 ,23 .030046 19.90 .969954 53 8 .028744 19.62 .997507 .23 .031237 19.85 .968763 52 9 .029918 19.57 19.52 .997493 .23 .23 .032425 19.79 19.74 .967575 51 10 9.031089 9.997480 9,033609 10.966391 50 11 12 13 14 15 16 17 18 19 .032257 .033421 .034582 .035741 .036896 .038048 .039197 .040342 .041485 19!41 19.36 19.30 19.25 19.20 19.15 19.10 19.05 19.00 .997466 .997452 .997439 .997425 .997411 .997397 .997383 .997369 .997355 23 .23 .23 .23 .23 .23 .23 .23 .23 .034791 o 035969 .037144 . 03831 6 .039485 .040651 .0418"13 .042973 .044130 19. '64 19.58 19.53 19.48 19.43 19.38 19.33 19.28 19.23 .965209 .964031 .962856 .961684 .960515 .959349 .958187 .957027 .955870 49 '48 47 46 45 44 43 42 41 20 21 22 9.042625 .043762 .044895 18.95 18.90 1ft QK 9.997341 .997327 .997313 .23 .24 ()A 9.045284 .046434 .047582 19.18 19.13 10.954716 .953566 .952418 40 39 38 23 .046026 -lu.oO 1Q QA .997299 .zA .048727 19.08 .951273 37 24 25 .047154 .048279 Jo.oU 18.75 .997285 .997271 .24 .24 .049869 .051008. 19.03 18.98 .950131 .948992 36 35 26 .049400 1 Q K .997257 .24 .052144 18.93 .947856 34 27 28 29 .050519 .051635 .052749 lo.oo 18.60 18.55 18.50 .997242 .997228 .997214 .24 .24 .24 .24 .053277 .054407 ,055535 18.89 18.84 18.79 18.74 .946723 .945593 .944465 33 32 31 30 31 32 33 34 35 9.053859 .054966 .056071 .057172 .058271 .059367 18.46 18.41 18.36 18.31 18.27 1 Q OO 9.997199 .997185 .997170 .997156 .997141 .997127 .24 .24 .24 .24 .24 9.056659 .057781 .058900 .060016 .061130 .062240 18.70 18.65 18.60 18.56 18.51 10.943341 .942219 .941100 .939984 .938870 .937760 30 29 28 27 26 25 36 37 38 39 .060460 .061551 .062639 .063724 lo. ZiL 18.17 18.13 18.08 18.04 .997112 ,997098 .997083 o 997068 ^24 .24 .25 .25 .063348 .064453 .065556 .066655 18.46 18.42 18.37 18.33 18.28 .936652 .935547 .934444 .933345 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.064806 .065885 .066962 .068036 .069107 .070176 .071242 .072306 .073366 .074424 17.99 17.95 17.90 17.86 17.81 17.77 17.72 17.68 17.64 17.59 9.997053 .997039 .997024 .997009 .996994 .996979 o 996964 .996949 .996934 .996919 .25 .25 .25 .25 .25 .25 .25 ,25 ,25 .25 9.067752 .068846 .069938 .071027 .072113 .073197 .074278 .075356 .076432 .077505 18.24 18.19 18.15 18.10 18.06 18.02 17.97 17.93 17.89 17.84 10.932248 .931154 .930062 .928973 .927887 .926803 .925722 .924644 .923568 .922495 20 19 18 17 16 13 14 13 1 2 11 50 51 52 53 54 55 56 57 58 9.075480 .076533 .077583 .078631 .079676 .080719 .081759 .082797 .083832 17.55 17.51 17.46 17.42 17.38 17.34 17.29 17.25 17 21 9.996904 .996889 .996874 .996858 .996843 .996828 .996812 .996797 .996782 .25 .25 .25 .25 .25 .25 .26 .26 9.078576 . 079644 .080710 .081773 .082833 .083891 .084947 .086000 .087050 17.80 17.76 17.72 17.67 17.63 17.59 17.55 17.51 nftf 10.921424 .920356 .919290 .918227 .917167 .916109 .915053 ,914000 .912950 10 9 8 7 6 5 4 3 2 59 60 .084864 .085894 17 '.17 .996766 .996751 !26 .088098 .089144 .47 17.43 .911902 .910856 1 1*7 Cosine. D.l". Sine. D.l". Cotang. D.l". Tang. M. 36 TABLE IV. LOGARITHMIC SINES, ETC. 47 172 M. Sine.' D.I". Cosine. |D. 1 Tang. D.I". Cotang. M. 1 2 3 4 9.085894 .086922 .087947 .088970 .089990 17.13 17.09 17.05 17.00 9.996751 .996735 .996720 .996704 .26 .26 .26 .26 9.089144 .090187 .091228 .092266 .093302 17.39 17.35 17.31 17.27 10.910856 .909813 .908772 .907734 .906698 60 59 58 57 FO 5 6 .091008 .092024 16.96 16.92 .996673 .996657 .26 .26 .094336 .095367 17.19 .905664 .904633 55 54 7 8 9 .093037 .094047 .095056 16.84 16.80 16.76 .996641 .996625 .996610 .26 .26 .26 .096395 .097422 .098446 17.11 17.07 17.03 .903605 .902578 .901554 53 52 51 10 11 12 13 14 15 16 17 13 19 9.096062 .097065 .098066 .099065 .100062 .101056 .102048 .103037 .104025 .105010 16.73 16-68 16.65 16.61 16.57 16.53 16.49 16.46 16.43 16.38 9.996594 .996578 .996562 .996546 .996530 .996514 .996498 .996482 .996465 .996449 .26 .27 .27 .27 .27 .27 27 .27 .27 .27 9.099468 .100487 .101504 .102519 .103532 .104542 .105550 .106556 .107559 .108560 16.99 16.95 16.91 16.88 16.84 16.80 16.76 16.72 16.69 16.65 10.900532 .899513 .898496 .897481 .896468 .895458 .894450 .893444 .892441 .891440 50 49 48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 28 29 9.105992 .106973 .107951 .108927 .109901 .110873 .111842 .112809 .113774 .114737 16.34 16.30 16.27 16.23 16.19 16.16 16.12 16.08 16.05 16.01 9.996433 .996417 .996400 .996384 .996368 .996351 .996335 .996318 .996302 .996285 .27 .27 .27 .27 .27 .27 .27 .27 .28 .28 9.109559 .110556 .111551 .112543 .113533 .114521 .115507 .116491 .117472 .118452 16.61 16.58 16.54 16.50 16.47 16.43 16.39 16.36 16.32 16.29 10.890441 .889444 .888449 .887457 ! 886467 .885479 .884493 .883509 .882528 .881548 40 39 33 37 30 35 S4 33 32 31 30 31 32 33 34 35 36 37 38 9.115698 .116656 .117613 .118567 .119519 .120469 .121417 .122362 .123306 15.98 15.94 15.90 15.87 15.83 15.80 15.76 15.73 9.996269 .996252 .996235 .996219 .996202 .996185 .996168 .996151 .996134 .28 .28 .28 .28 .28 .28 .28 .28 9.119429 .120404 .121377 .122348 .123317 .124284 .125249 .126211 .127172 16.25 16.22 16.18 16.15 16.11 16.08 16.04 16.01 10.880571 .879596 .878623 .877652 .876683 .875716 .874751 .873789 .872828 30 29 28 2G 25 24 23 22 39 .124248 15.66 .996117 .28 .128130 15.94 .871870 21 40 41 42 43 44 9.125187 .126125 .127060 .127993 .128925 15.62 15.59 15.56 15.52 9.996100 .996083 .996066 .996049 .996032 .29 ..29 .29 .29 9Q 9.129087 .130041 .130994 .131944 .132893 15.91 15.87 15.84 15.81 I* *7 10.870913 .869959 .869006 .868056 .867107 20 ID 18 17 16 45 46 47 48 49 .129854 .130781 .131706 .132630 .133551 15.45 15.42 15.39 15.35 15.32 .996015 .995998 .995980 .995963 .995946 .29 .29 .29 .29 .29 .133839 .134784 .135726 .136667 .137605 15.74 15.71 15.68 15.64 15.61 .866161 .865216 .864274 .863333 .862395 15 14 13 12 11 50 9.134470 15.29 9.995928 .29 9.138542 15.58 10.861458 10 51 52 53 54 55 56 57 58 59 60 .136303 .137216 .138128 .139037 .139944 .140850 .141754 .142655 .143555 15.26 15.22 15.19 15.16 15.13 15.09 15.06 15.03 15.00 .995894 .995876 .995859 .995841 .995823 .995806 .995788 .995771 .995753 .29 .29 .29 .29 .29 .29 .29 .29 .29 .140409 .141340 .142269 .143196 .144121 .145044 .145966 .146885 .147803 15.55 15.51 15.48 15.45 15.42 15.39 15.36 15.32 15.29 .859591 .858660 .857731 .856804 .855879 .854956 .854034 .853115 .852197 8 7 6 5 4 3 2 1 M. Cosine. D.I". Sine. D.I . rotang D. 1". Tang. M. 97 48 TABLE IY. LOGARITHMIC SINES, ETC. 8 171' M. Sine. D.I". Cosine. D.I". Tang. D. 1". Cotang. M. 9.143555 -f A AT 9.995753 QA 9.147803 1*> 9R 10.852197 60 1 .144453 14.y i .995735 .oU QA .148718 IO. Z\y IK OO .851282 59 2 3 .145349 .146243 14 .'90 UQ7 .995717 .995699 oU .30 .149632 .150544 10. o 15.20 IK 17 .850368 .849456 58 57 4 5 6 .147136 .148026 .148915 . O* 14.84 14.81 1/1 7Q .995681 .995664 .995646 !30 .30 Q/\ .151454 .152363 .153269 10. 1 1 15.14 15.11 -IK AQ .848546 .847637 .846731 56 55 54 7 8 9 .149802 .150686 ,151569 14. *O 14.75 14.72 14.69 .995628 .995610 .995591 oU .30 .30 .30 .154174 .155077 .155978 10. Uo 15.05 15.02 14.99 .845826 . 844923 .844022 53 52 51 io 9.152451 1,4 fid 9.995573 QA 9.156877 14 Qfi 10.843123 50 11 12 13 14 15 16 17 18 19 .153330 .154208 .155083 .155957 .156830 .157700 .158569 .159435 .160301 14 OU 14.63 14.60 14.57 14.54 14.51 14.48 14.45 14.42 14.39 .995555 .995537 .995519 .995501 .995482 .995464 .995446 .995427 .995409 .oU .30 .30 .30 .31 .31 .31 .31 .31 .31 .157775 .158671 .159565 .160457 .161347 .162236 .163123 .164008 .164892 14! 93 14.90 14.87 14.84 14.81 14.78 14.75 14.73 14.70 .842225 .841329 .840435 .839543 .838653 .837764 .836877 .835992 .835108 49 48 47 46 45 44 43 42 41 20 9.1611G4 9.995390 91 9.165774 10.834226 40 21 22 23 .162025 .162885 .163743 14.36 14.33 14.30 1/4 O7 .993372 .993353 .995331 ol .31 .31 91 .166654 .167532 .168409 14.67 14.64 14.61 1/4 FCC .833346 .832468 .831591 33 33 37 24 25 26 27 28 .164600 .165454 .166307 .167159 .168008 14 . Zi 14.24 14.22 14.19 14.16 .995316 .995297 .995278 .995260 .995241 ol .31 .31 .31 .31 .169284 .170157 .171029 .171899 .172767 14. 5o 14.56 14.53 14.50 14.47 .830716 .829843 .828971 .828101 .827233 36 35 :u 33 32 29 .168856 14.13 14.10 .995222 .32 .32 .173634 14.44 14.42 .826366 31 30 31 9.169702 .170547 14.07 -1 4 APi 9.995203 .995184 .32 OO 9.174499 .175362 14.39 1/1 Q 10.825501 .824638 30 29 32 .171389 14. Uo .995165 cOufi .176224 14.OD .823776 28 33 34 .172230 .173070 14.02 13.99 .995146 .995127 '.32 .177084 .177942 14.33 14.31 .822916 .822058 27 26 35 36 37 .173908 .174744 .175578 13.96 13.94 13.91 .995108 .995089 .995070 '.32 .32 OO .178799 .179655 .180508 14.28 14.25 14.23 .821201 .820345 .819492 25 24 23 38 39 .176411 .177242 13.88 13.85 13.83 .995051 .995032 .0* .32 .32 .181360 .182211 14.20 14.17 14.15 .818640 .817789 22 21 40 41 42 43 44 9.178072 .178900 .179726 .180551 .181374 13.80 13.77 13.75 13.72 9.995013 .994993 .994974 .994955 .994935 .32 .32 .32 .32 9.183059 .183907 .184752 .185597 .186439 14.12 14.09 14.07 14.04 10.816941 .816093 .815248 .814403 .813561 20 19 18 17 16 45 46 47 48 49 .182190 .183016 .183834 .184651 .185466 13.69 13.67 13.64 13.61 13.59 13.56 .994916 .994896 .994877 .994857 .994838 .32 .33 .33 .33 .33 .33 .187280 .188120 .188958 .189794 .190629 14.02 13.99 13.97 13.94 13.91 13.89 .812720 .811880 .811042 .810206 .809371 15 14 13 12 11 50 51 52 53 9.186280 .187092 .187903 .188712 13.54 13.51 13.48 9.994818 .994798 .994779 .994759 .33 .33 .33 9.191462 .192294 .193124 .193953 13.86 13.84 13.81 10.808538 .807706 .806876 .806047 10 9 8 7 54 55 56 57 .189519 .190325 .191130 .191933 13.46 13.43 13.41 13.38 f O nf* .994739 .994719 .994700 .994680 .33 .33 .33 .33 OQ .194780 .195606 .196430 .197253 13.79 13.76 13.74 13.71 .805220 .804394 .803570 .802747 6 5 4 3 68 59 60 .192734 .193534 .194332 16. OO 13.33 13.31 .994660 .994640 .994620 .DO .33 .33 .198074 .198894 .199713 13! 66 13.64 .801926 .801106 .800287 2 1 M. Cosine. D.I". Sine. D.I". Cotang. D.I". Tang. M. 81* TABLE IV. LOGARITHMIC SINES, ETC. 49 170= M. Sine. D. 1". Cosine. D.I . Tang. D. 1". Cotang. M- 1 2 3 4 5 9.194332 .195129 .195925 .196719 .197511 .198302 1 <\ .767174 .766414 .765655 18 17 16 45 46 47 48 49 .228784 .229518 .230252 .230984 .231715 Mj6mlBv 12.24 12.22 12.20 12.18 12.16 .993681 .993660 .993638 .993616 .993594 .00 .36 .36 .36 .36 .36 .235103 .235859 .236614 .237368 .238120 IZ.Oo 12.60 12.58 12.56 12.54 12.52 .764897 .764141 .763386 .762632 .761880 15 14 13 12 11 50 51 52 53 54 55 56 67 9.232444 .233172 .233899 .234625 .235349 .236073 .236795 .237515 12.14 12.12 12.10 12.07 12.05 J2.03 12.01 1 1 OQ 9.993572 .993550 .993528 .993506 .993484 .993462 .993440 .993418 .36 .37 .37 .37 .37 .37 .37 9.238872 .239622 .240371 .241118 .241865 .242610 .243354 .244097 12.50 12.48 12.46 12.44 12.42 12.40 12.38 1O Q 10.761128 ,760378 .759629 .758882 .758135 .757390 .756646 .755903 10 9 8 7 -6 6 4 3 58 59 60 .238235 .238953 .239670 11 . yy 11.97 11.95 .993396 .993374 .993351 .37 .37 .37 .244839 .245579 .246319 U.dO 12.34 12.32 .755161 .754421 .753681 2 1 M. Cosine. D.r. Sine. D.I" Cotang. D. 1". Tang. M 99 80 50 TABLE IV. LOGARITHMIC SINES, ETC. 1O' 169 M. Sine. D.I". Cosine. D.l". Tang. D.l". Cotang. M. 1 2 3 4 5 6 9.239670 .240386 .241101 .241814 .242526 .243237 .243947 11.93 11.91 11.89 11 .37 11.85 11.83 noi 9.993351 .993329 .993307 .993284 .993262 .993240 .993217 .37 .37 .37 .37 .37 .37 OQ 9.246319 .247057 .247794 .248530 .249264 .249998 .250730 12.30 12.28 12.26 12.24 12.22 12.20 10.753681 .752943 .752206 .751470 .750736 .750002 .749270 60 59 58 57 56 55 54 7 8 .244656 .245363 .OJL 11.79 nil .993195 .993172 .OO .38 OQ .251461 .252191 12.18 12.17 .748539 .747809 53 52 9 .246069 ..< 11.75 .993149 .OO .38 .252920 12.15 12.13 .747080 61 10 11 9.246775 .247478 11.73 n71 9.993127 .993104 .38 00 9.253648 .254374 12.11 10.746352 .745626 50 49 12 .248181 k \. 11 69 .993081 .00 OQ .255100 12.09 .744900 48 13 14 15 16 17 18 19 .248883 .249583 .250282 .250980 .251677 .252373 .253067 ll!67 11.65 11.63 11.61 11.59 11.58 11.56 .993059 .993036 .993013 .992990 .992967 .992944 .992921 OO .38 .38 .38 .38 .38 .38 .38 .255824 .256547 .257269 .257990 .258710 .259429 .260146 12l 05 12.03 12.01 12.00 11.98 11.96 11.94 .744176 .743453 .742731 .742010 .741290 .740571 .739854 47 46 45 44 43 42 41 20 21 22 9.253761 .254453 .255144 11.54 11.52 11 50 9.992898 .992875 .992852 .38 .38 OQ 9.260863 .261578 .262292 11.92 11.90 nQO 10.739137 .738422 .737708 40 39 38 23 24 .255834 .256523 ll!48 nAf* .992829 .992806 OO .39 .263005 .263717 .89 11.87 .736995 .736283 37 36 25 26 27 28 .257211 .257898 .258583 .259268 VO 11.44 11.42 11.41 11 3Q .992783 .992759 .992736 .992713 .39 .39 .39 .39 'Ml .264428 .265138 .265847 .266555 11.85 11.83 11.81 11.79 .735572 .734862 .734153 .733445 35 34 33 32 29 .259951 11 .'37 .992690 .o9 .39 .267261 11.78 11.76 .732739 31 30 81 32 33 9.260633 .261314 .261994 .262673 11.35 11.33 11.31 11 30 9.992666 .992643 .992619 .992596 .39 .39 .39 9.267967 .268671 .269375 .270077 11.74 11.72 11.70 10.732033 .731329 .730625 .729923 30 29 28 27 34 35 36 37 38 39 .263351 .264027 .264703 .265377 .266051 .266723 1L28 11.26 11.24 11.22 11.20 11.19 .992572 .992549 .992525 .992501 .992478 .992454 '.39 .39 .39 .39 .40 .40 .270779 .271479 .272178 .272876 .273573 .274269 11 .69 11.67 11.65 11.64 11.62 11.60 11.58 .729221 .728521 .727822 .727124 .726427 .725731 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.267395 .268065 .268734 .269402 .270069 .270735 .271400 .272064 .272726 .273388 11.17 11.15 11.13 11.12 11.11 11.08 11.06 11.05 11.03 11.01 9.992430 .992406 .992382 .992359 .992335 .992311 .992287 .992263 .992239 .992214 .40 .40 .40 .40 .40 .40 .40 .40 .40 .40 9.274964 .275658 .276351 .277043 .277734 .278424 .279113 .279801 .280488 .281174 11.57 11.55 11.53 11.51 11.50 11.48 11.47 11.45 11.43 11.41 10.725036 .724342 .723649 .722957 .722266 .721576 .720887 .720199 .719512 .718826 20 19 18 17 16 15 14 13 12 11 60 61 62 63 64 65 66 67 68 69 60 9.274049 .274708 .275367 .276025 .276681 .277337 .277991 .278644 .279297 .279948 .280599 10.99 10.98 10.96 10.94 10.92 10.91 10.89 10. 8T 10.86 10.84 9.992190 .992166 .992142 .992118 .992093 .992069 .992044 .992020 .991996 .991971 .991947 .40 .40 .40 .41 .41 .41 .41 .41 .41 .41 9.281858 .282542 .283225 .283907 .284588 .285268 .285947 .286624 .287301 .287977 .288652 11.40 11.38 11.36 11.35 11.33 11.31 11.30 11.28 11.26 11.25 10.718142 .717458 .716775 .716093 .715412 .714732 .714053 .713376 .712699 .712023 .711348 10 9 8 7 6 5 4 3 2 1 M. Cosine. D. 1". Sine. D.I '. Cotang. D. 1 . Tang. M. 100 79 TABLE IT. LOGAKITHMIC SINES, ETC. 51 11 168 M. Sine. D.I". Cosine. D.I". Tang. D.I". Cotang. M. 1 2 3 4 5 6 7 8 9.280599 .281248 .281897 .282544 .283190 .283836 .284480 .285124 .285766 10.82 10.81 10.79 10.77 10.76 10.74 10.72 10.71 9.991947 .991922 .991897 .991873 .991848 .991823 .991799 .991774 .991749 .41 .41 .41 .41 .41 .41 .41 .42 49 9.288652 .289326 .289999 .290671 .291342 .292013 .292682 .293350 .294017 11.23 11.22 11.20 11.18 11.17 11.15 11.14 11.12 10.711348 .710674 .710001 .709329 .708658 .707987 .707318 .706650 .705983 60 59 68 67 66 65 64 53 62 9 .286408 10.67 .991724 .42 .294684 11.09 .705316 61 10 11 12 13 14 9.287048 .287688 .288326 .288964 .289600 10.66 10.64 10.63 10.61 9.991699 .991674 .991649 .991624 .991599 .42 .42 .42 .42 9.295349 .296013 .296677 .297339 .298001 11.07 11.06 11.04 11.03 10.704651 .703987 .703323 .702661 .701999 60 49 48 47 46 15 16 17 18 19 .290236 .290870 .291504 .292137 .292768 10.58 10.56 10.55 10.53 10.51 .991574 .991549 .991524 .991498 .991473 .42 .42 .42 .42 .42 .298662 .299322 .299980 .300638 .301295 11.00 10.98 10.97 10.95 10.93 .701338 .700678 .700020 .699362 .698706 45 44 43 42 41 20 21 22 23 24 26 26 27 28 29 9.293399 .294029 .294658 .295286 .295913 .296539 .297164 .297788 .298412 .299034 10.50 10.48 10.47 10.45 10.43 10.42 10.40 10.39 10.37 10.36 9.991448 .991422 .991397 .991372 .991346 .991321 .991295 .991270 .991244 .991218 .42 .42 .42 .43 .43 .43 .43 .43 .43 .43 9.301951 .302607 .303261 .303914 .304567 ' .305218 .305869 .306519 .307168 .307816 10.92 10.90 10.89 10.87 10.86 10.84 10.83 10.81 10.80 10.78 10.698049 .697393 .696739 .696086 .695433 .694782 .694131 .693481 .692832 .692184 40 39 38 37 36 35 34 33 32 31 30 31 32 35 36 37 38 39 9.299655 .300276 .300895 .301514 .302132 .302748 .303364 .303979 .304593 .305207 10.34 10.33 10.31 10.30 10.28 10.26 10.25 10.23 10.22 10.20 9.991193 .991167 .991141 .991115 .991090 .991064 .991038 .991012 .990986 .990960 .43 .43 .43 .43 .43 .43 .43 .43 .43 .43 9.308463 .309109 .309754 .310399 .311042 .311685 .312327 .312968 .313608 .314247 10.77 10.76 10.74 10.73 10.71 10.70 10.68 10.67 10.65 10.64 10.691537 .690891 .690246 .689601 .688958 .688315 .687673 .687032 .686392 .685753 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.305819 .306430 .307041 .307650 .308259 .308867 .309474 .310080 .310685 .311289 10.19 10.17 10.16 10.14 10.13 10.12 10.10 10.09 10.07 10.06 9.990934 .990908 .990882 .990855 .990829 .990803 .990777 .990750 .990724 .990697 .44 .44 .44 .44 .44 .44 .44 .44 .44 .44 9.314885 .315523 .316159 .316795 .317430 .318064 .318697 .319329 .319961 .320592 10.62 10.61 10.60 10.58 10.57 10.55 10.54 10.53 10.51 10.50 10.685115 .684477 .683841 .683205 .682570 .681936 .681303 .680671 .680039 .679408 20 19 18 17 16 15 14 13 12 11 60 51 62 63 64 65 66 67 68 69 60 9.311893 .312495 .313097 .313698 .314297 .314897 .315495 .316092 .316689 .317284 .317879 10.04 10.03 10.01 10.00 9.98 9.97 9.96 9.94 9.93 9.91 9.990671 .990645 .990618 .990591 .990565 .990538 .990511 .990485 .990458 .990431 .990404 .44 .44 .44 .44 .44 .44 .45 .45 .45 .45 9.321222 .321851 .322479 .323106 .323733 .324358 .324983 .325607 .326231 .326853 .327475 10.48 10.47 10.46 10.44 10.43 10.41 10.40 10.39 10.37 10.36 10.678778 .678149 .677521 .676894 .676267 .675642 .675017 .674393 .673769 .673147 .672525 10 9 8 7 6 6 4 3 2 1 M. Cosine. D. 1". Sine. D.I'. Cotang. D.I". Tang. M. 101 78* 52 TABLE IT. LOGABITHMIC SINES, ETC. M Sine. D.I" Cosine D.r Tang. D.I" Cotang M. { 9.317879 .318473 .319066 .319658 .320249 .320840 9.90 9.88 9.87 9.86 9.84 9.990404 .990378 .990351 .990324 .990297 .990270 .4 .4 .45 .45 .45 9-. 327475 .32809 .328715 .329334 .329953 .330570 10.35 10.3 10.3 10.3 10.29 10.67252 .67190 .67128 .67066 .67004 66943 60 69 68 57 56 65 ( .321430 9 82 .990243 .45 .331187 10.28 .66881 54 9 .322019 .322607 .323194 9.80 9.79 9.77 .990215 .990188 .990161 Ao .45 .45 .331803 .332418 .333033 10.27 10.25 10.24 10 23 .66819 .66758 .66696 63 62 61 10 1 16 17 18 19 9.323780 .324366 .324950 .325534 .326117 .326700 .327281 .327862 .328442 .329021 9.76 9.75 9.73 9.72 9.70 9.69 9.68 9.66 9.65 9.64 9.990134 .990107 .990079 .990052 .990025 .989997 .989970 .989942 .989915 .989887 .45 .46 .46 .46 .46 .46 .46 .46 .46 .46 9.333646 .334259 .334871 .335482 .336093 .336702 .337311 .337919 .338527 .339133 10.21 10.20 10.19 10.17 10.16 10.15 10.14 10.12 10.11 10 10 10.66635 .66574 .66512 .664518 .663907 .663298 .662689 .662081 .661473 .660867 60 49 48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 28 29 9.329599 .330176 .330753 .331329 .331903 .332478 .333051 .333624 .334195 .334767 9.62 9.61 9.60 9.58 9.57 9.56 9.54 9.53 9.52 9.50 9.989860 .989832 .989804 .989777 .989749 .989721 .989693 .989665 .989637 .989610 .46 .46 .46 .46 .47 .47 .47 .47 .47 .47 9.339739 .340344 .340948 .341552 .342155 .342757 .343358 .343958 .344558 .345157 10.08 10.07 10.06 10.05 10.03 10.02 10.01 10.00 9.98 9 97 10.660261 .659656 .659052 .658448 .657845 .657243 .656642 .656042 .655442 .654843 40 39 38 37 36 35 34 33 32 31 30 31 32 33 34 35 36 9.335337 .335900 .336475 .337043 .337610 .338176 .338742 9.49 9.48 9.46 9.45 9.44 9.43 9.41 9.989582 .989553 .989525 .989497 .989469 .989441 .989413 .47 .47 .47 .47 .47 .47 47 9.345755 .346353 .346949 .347545 .348141 .348735 .349329 9.96 9.95 9.93 9.92 9.91 9.90 10.654245 .653647 .653051 .652455 .651859 .651265 .650671 30 29 28 27 26 25 24 38 39 .339871 .340434 9.40 9.39 9.37 .989385 .989356 .989328 .47 .47 .47 .349922 .350514 .351106 9.87 9.86 9oe .650078 .649486 .648894 23 22 21 40 41 42 43 44 45 46 47 48 49 9.340996 .341558 .342119 .342679 .343239 .343797 .344355 .344912 .345469 .346024 9.36 9.35 9.34 9.32 9.31 9.30 9.29 9.27 9.26 9.25 9.989300 .989271 .989243 .989214 .989186 .989157 .989128 .989100 .989071 .989042 .47 .47 .47 .47 .47 .47 .48 .48 .48 48 9.351697 .352287 .352876 .353465 .354053 .354640 .355227 .355813 .356398 .356982 9.84 9.82 9.81 9.80 9 79 9.78 9.76 9.75 9.74 0.648303 .647713 .647124 .646535 .645947 .645360 .644773 .644187 .643602 .643018 20 19 18 17 16 15 14 13 2 11 50 51 52 53 54 55 56 57 58 59 60 9.346579 .347134 .347687 .348240 .348792 .349343 .349893 .350443 .350992 .351540 .352088 9.24 9.22 9.21 9.20 9.19 9.17 9.16 9.15 9.14 9.13 9.989014 .988985 .988956 .988927 .988898 .988869 .988840 .988811 .988782 .988753 .988724 .48 .48 .48 .48 .48 .48 .48 .49 .49 .49 9.357566 .358149 .358731 .359313 .359893 .360474 .361053 .361632 .362210 .362787 .363364 9.72 9.70 9.69 9.68 9.67 9.66 9.65 9.63 9.62 9.61 0.642434 .641851 .641269 .640687 .640107 .639520 .638947 .638368 .637790 .637213 .636636 9 8 7 6 5 4 3 2 1 M. Cosine, j D.I". Sine. .1'. Cotang. D.I . Tang. TABLE IV. LOGARITHMIC SINES, ETC. 18 166* M. Sine. D.l . Cosine. D.r Tang. D.I". Cotang. M. 9.352088 .352635 .353181 .353726 .354271 .354815 .355358 .355901 .356443 9.357524 .358064 .358603 .359678 .360215 .360752 .361287 .361822 .363422 .363954 .364485 .365016 .365546 .367131 .367659 9.368185 .368711 .369761 .370285 .370808 .371330 .371852 .372373 9.373414 .373933 .374452 .374970 .375487 .376003 .376519 .377035 .377549 .378063 9.378577 .379089 .379601 .380113 .381134 .381643 .382152 .383168 .383675 9.11 9.10 9.07 9.05 9.04 9.03 9.02 9.01 8.99 8.98 8.97 8.96 8.95 8.94 8.92 8.91 8.90 8.87 8.86 8.84 8.83 8.82 8.81 8.80 8.79 8.78 8.76 8.75 8.74 8.73 8.72 8.71 8.70 8.65 8.64 8.63 8.62 8.61 8.58 8.57 8.56 8.55 8.53 8.52 8.51 8.50 8.49 8.48 8.47 8.46 8.45 9.988724 .988578 .988548 .988519 .988371 .988193 .988163 9.988133 .988073 .988043 .988013 .987953 .987922 .987892 .987862 9.987832 .987801 .987771 .987740 .987710 .987679 .987649 .987618 .987588 .987557 9.987526 .987465 .987434 .987403 .987372 .987341 .987310 .987279 .987248 .987217 .987186 .987155 .987124 .987092 .987061 .987030 .49 .49 .49 .49 .49 .49 .49 .49 .49 .49 .49 .49 .49 .50 .50 .50 .50 .50 .50 .50 .50 .50 .50 .50 .50 .50 .50 .50 .50 .51 .51 .51 .51 .51 .51 .51 .51 .51 .51 .61 .51 .51 .51 .52 .52 .52 .52 .52 .52 .52 .52 .52 .52 .52 .52 .364515 .365090 .986237 .367382 .367953 9.369094 .370232 .370799 .371367 .371933 .372499 .373064 .373629 .374193 9.374756 .375319 .376442 .377003 .377563 .378122 .379797 .380910 .381466 .382020 .382575 .383129 .3*4234 .384786 .389178 .390270 .390815 .391903 .392447 .394073 .394614 .395154 .396771 9.60 9.59 9.58 9.57 9.55 9.54 9.53 9.52 9.51 9.50 9.49 9.48 9.47 9.45 9.44 9.43 9.42 9.41 9.40 9.39 9.38 9.37 9.36 9.35 9.33 9.32 9.31 9.30 9.29 9.28 9.27 9.26 9.25 9.24 9.23 9.22 9.21 9.20 9.19 9.18 9.17 9.16 9.15 9.14 9.12 9.11 9.10 9.07 9.06 9.05 9.04 9.03 9.02 9.01 9.00 8.97 10.636636 .636060 .635485 .634910 .634336 .633763 .633190 .632618 .632047 .631476 10.630906 .630337 .629768 .629201 .628633 .628067 .627501 .626936 .626371 .625807 10.625244 .624681 .624119 .623558 .622997 .622437 .621878 .621319 .620761 .620203 10.619646 .619090 .618534 .617980 .617425 .616871 .616318 .615766 .615214 .614663 10.614112 .613562 .613013 .612464 .611916 .611369 .610822 .610276 .609730 .607553 .607011 .604846 .604306 .603767 M. 1 Cosine. D.l". Sine. D.l". Cotang. D.l". Tang. M. 29 W 54: TABLE IV. LOGARITHMIC SINES, ETC. 165" M. Sine. D.l". Cosine. D.I" Tang. D.l". Cotang. M. 1 2 3 4 6 6 7 8 9.383675 .384182 .384687 .385192 .385697 .386201 .386704 .387207 .387709 8.44 8.43 8.42 8.41 8.40 8.39 8.38 8.37 9.986904 .986873 .986841 .986809 .986778 .986746 .986714 .986683 .986651 .53 .53 .53 .53 .53 .53 .53 .53 9.396771 .397309 .397846 .398383 .398919 .399455 .399990 .400524 .401058 8.96 8.96 8.95 8.94 8.93 8.92 8.91 8.90 10.603229 .602691 .602154 .601617 .601081 .600545 .600010 .599476 598942 60 59 58 57 56 55 54 53 52 9 .388210 8.35 .986619 .53 .401591 8.89 8 88 .598409 51 10 11 12 13 14 15 16 17 18 19 9.388711 .389211 .389711 .390210 .390708 .391206 .391703 .392199 .392695 .393191 8.34 8.33 8.32 8.31 8.30 8.28 8.27 8.26 8.25 8.24 9.986587 .986555 .986523 .986491 .986459 .986427 .986395 .986363 .986331 .986299 .53 .53 .53 .53 .53 .53 .53 .54 .54 .64 9.402124 .402656 .403187 .403718 .404249 .404778 .40530 .405836 .406364 .406892 8.87 8.86 8.85 8.84 8.83 8.82 8.81 8.80 8.79 8.78 10.597876 .597344 .596813 .596282 .595751 .595222 .594692 .594164 .593636 .593108 50 49 48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 9.393685 .394179 .394673 .395166 .395658 .396150 .396641' .397132 8.23 8.22 8.21 8.20 8.19 8.18 8.17 9.986266 .986234 .986202 .986169 .986137 .986101 .986072 .986039 .54 .54 .54 .54 .54 .54 .54 9.407419 .407945 .408471 .408997 .409521 .410045 .410569 .411092 8.77 8.76 8.75 8.74 8.74 8.73 8.72 10.592581 .592055 .591529 .591003 .590479 .589955 .589431 588908 40 39 38 37 36 35 34 33 28 29 .397621 .398111 8.16 8.15 .986007 .985974 .54 .54 .54 .411615 .412137 8.71 8.70 8 69 .588385 .587863 32 31 30 9.398600 8.14 9.985942 54 9.412658 8 68 10.587342 30 81 82 83 84 85 86 37 38 89 .399088 .399575 .400062 .400549 .401035 .401520 .402005 .402489 ,402972 8.13 8.12 8.11 8.10 8.09 8.08 8.07 8.06 8.05 .985909 .985876 .985843 .985811 .985778 .985745 .985712 .985679 .985646 .55 .55 .55 .55 .55 .55 .55 .55 .55 .413179 .413699 .414219 .414738 .415257 .415775 .416293 .416810 .417326 8.67 8.66 8.65 8.65 8.64 8.63 8.62 8.61 8.60 .586821 .586301 .585781 .585262 .584743 .584225 .583707 .583190 .582674 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.403455 .403938 .404420 .404901 .405382 .405862 .406341 .406820 .407299 .407777 8.04 8.03 8.02 8.01 8.00 7.99 7.98 7.97 7.96 7.95 9.985613 .985580 .985547 .985514 .985480 .985447 .985414 .985380 .985347 .985314 .55 .55 .55 .55 .55 .55 .56 .56 .56 .56 9.417842 .418358 .418873 .419387 .419901 .420415 .420927 .421440 .421952 .422463 8.59 8.58 8.57 8.56 8.56 8.55 8.54 8.53 8.52 8 51 10.582158 .581642 .581127 .580613 .580099 .579585 .579073 .578560 .578048 .577537 20 19 18 17 16 15 14 13 12 11 60 61 62 63 64 65 66 67 68 69 60 9.408254 .408731 .409207 .409682 .410157 .410632 .411106 .411579 .412052 .412524 .412996 7.94 7.94 7.93 7.92 7.91 7.90 7.89 7.88 7.87 7.86 9.985280 .985247 .985213 .985180 .985146 .985113 .985079 .985045 .985011 .984978 .984944 .56 .56 .56 .56 .56 .56 .56 .56 .56 .56 9.422974 .423484 .423993 .424503 .425011 .425519 .426027 .426534 .427041 .427547 .428052 8.50 8.49 8.49 8.48 8.47 8.46 8.45 8.44 8.43 8.43 10.577026 .576516 .576007 .575497 .574989 .574481 .573973 .573466 .572959 .572453 .571948 10 9 8 7 6 5 4 3 2 1 M. Cosine, D.I". Sine. D.I". Cotang. D.l ". Tang. M 104' 7K . TABLE IV. LOGARITHMIC SINES, ETC. 55 164 M. Sine. D.l". Cosine. D.l". Tang. D.l . Cotang. M. 1 9.412996 .413467 7.85 7 R4 9.984944 .984910 .57 9.428052 .428558 8.42 10.571948 .571442 60 59 2 3 .413938 .414408 7i84 .984876 .984842 '.57 KTT .429062 .429566 S.40 .570938 .570434 58 '57 4 .414878 7.83 700 .984808 .57 CT .430070 2 .569930 56 6 6 .415347 .415815 . O-rfJ 7.81 7OA .984774 .984740 Oi .57 KT .430573 .431075 8i38 807 .569427 .568925 55 54 7 .416283 .oU .984706 O KfJ .431577 Of 8o/ .568423 53 8 .416751 7* 7ft .984672 Ol *7 .432079 .OO 8QK .567921 62 9 .417217 4 . 4O 7.77 .984637 Ol .57 .432580 . oO 8.34 .567420 51 10 11 12 9.417684 .418150 .418615 7.76 7.75 77K 9.984603 .984569 .984535 .57 .57 eff 9.433080 .433580 .434080 8.33 8.33 10.566920 .566420 .565920 50 49 48 13 14 15 .419079 .419544 .420007 . 45 7.74 7.73 .984500 .984466 .984432 .Ol .67 .57 CO .434579 .435078 .435576 sisi 8.30 ,565421 .564922 .564424 47 46 45 16 .420470 7.72 771 .984397 .5o KQ .436073 98 .563927 44 17 .420933 . ll .984363 Do ro .436570 a oa .563430 43 18 .421395 7.70 7 fiQ .984328 .5o CO .437067 8*07 .562933 42 19 .421857 .oy 7.68 .984294 .Do .58 .437563 .i 8.23 .562137 41 20 9.422318 9.984259 KQ 9.438059 10.561941 40 21 22 .422778 .423238 7.67 7.67 Tea .984224 .984190 .Do .58 .438554 .439048 8. '24 8O/4 .561446 .560952 39 38 23 .423697 DO Tfift .984155 CO .439543 & 00 .560457 37 24 25 .424156 .424615 DO 7.64 7/jO .984120 .984085 .Do .58 KQ .440036 .440529 o. _> 8.22 801 .559964 .559471 36 35 26 .425073 .DO 7s o .984050 .on .441022 .l 8Ort .558978 34 27 .425530 .62 7(M .984015 .58 CO .441514 .20 8OA .558486 33 28 .425987 bl 7*1 .983981 .Do .442006 . A) 8-m .557994 32 29 .426443 61 7.60 .983946 .58 .58 .442497 .19 8.18 .557503 31 30 9.426899 7KCL 9.983911 9.442988 817 10.557012 30 31 .427354 .5y 7CQ .983875 .58 CO .443479 .17 8-1 jt .556521 29 32 .427809 ,5o 7KT .983840 .Do .443968 . lo 81 ^ .556032 28 33 34 .428263 .428717 .57 7.56 7Cf .983805 .983770 .59 .59 KO .444458 .444947 .lo 8.15 81 j .555542 .555053 27 26 35 36 37 .429170 .429623 .430075 .55 7.54 7.53 .983735 .983700 .983664 .59 .59 .59 .445435 .445923 .446411 .14 8.13 8.13 C 1 > .554565 .554077 .553589 25 24 23 38 .430527 7.52 7 CO .983629 .59 .446898 .553102 22 39 .430978 .52 7.51 .983594 .59 .59 .447384 8.11 8.10 .552616 21 40 9.431429 7 Kfl 9.983558 9.447870 800 10.552130 20 41 .431879 4 .DU .983523 .59 .448356 .uy 8AO .551644 19 42 .432329 7.49 .983487 .59 .448841 .uy O AQ. .551159 18 43 .432778 7.49 7 JO .983452 -.59 449326 o.uo o /yr .550674 17 44 45 .433226 .433675 ,4o 7.47 7 -ir .983416 .983381 !59 .449810 .450294 O.U4 8.06 .550190 .549706 16 15 46 47 .434122 .434569 7^45 7 MM .983345 .983309 .59 .59 .450777 .451260 8.06 8.05 .549223 .548740 14 13 48 49 .435016 .435462 .44 7.44 7.43 .983273 .983238 .69 .60 .60 .451743 .452225 8.04 8.03 8.03 .548257 .547775 12 11 60 9.435908 7 An 9.983202 9,452706 10.547294 10 61 .436353 .42 741 .983166 .60 .453187 8.02- 8A1 .546813 9 62 63 .436798 .437242 .41 7.40 7>IA .983130 .983094 .60 .60 .453668 .454148 .01 8.00 .546332 .545852 8 7 64 .437686 .40 700 .983058 .60 .454628 8.00 .545372 6 65 .438129 .0*7 TOO .983022 .60 .455107 7.99 .544893 *5 56 .438572 .00 .982986 .60 .455586 7.98 .544414 4 57 .439014 7.37 70 .982950 .60 .456064 7.97 .543936 3 68 .439456 OO .982914 .60 .456542 7.97 .543458 2 59 .439897 7.36 7 OK .982878 .60 .457019 7nK .542981 1 60 .440338 .OO .982842 .60 .457496 .95 .542504 M. Cosine. D.l". Sine. D.l . Cotang. D.l". Tang. M. 1O5 740 56 TABLE IV. LOGARITHMIC SINES, ETC. 163 M. Sine. D.I". | Cosine. D 1" Tang. D.1". Cotang. M. 9.440338 9.982842 9.457496 7 94 10.542504 60 1 2 3 4 5 6 7 .440778 .441218 .441658 .442096 .442535 .442973 .443410 7^33 7.32 7.31 7.31 7.30 7.29 7OQ .982805 .982769 .982733 .982696 .982660 .982624 .982587 !co .61 .61 .61 .61 .61 d .457973 .458449 .458925 .459400 .459875 .460349 .460823 7. '94 7.93 7.92 7.91 7.91 7.90 .542027 .541551 .541075 .540600 .540125 .539651 .539177 59 58 57 56 55 54 53 8 .443847 .Jo .982551 Ol .461297 7QQ .538703 52 9 .444284 7.27 7.27 .982514 .61 .61 .461770 .00 7.88 .538230 51 10 11 9.444720 .445155 7.26 9.982477 .982441 .61 9.462242 .462714 7.87 7QA 10.537758 .537286 eo 49 12 .445590 7.25 .982404 M .463186 . OO 7QA .536814 48 13 14 15 16 .446025 .446459 .446893 .447326 7. 24 7.24 7.23 7.22 7 21 .982367 .982331 .982294 .982257 Ol .61 .61 .61 A1 .463658 .464129 .464599 .465069 .OO 7.85 7.84 7.83 7QO .536342 .535871 .535401 .534931 47 46 45 44 17 18 19 .447759 .448191 .448623 7l20 7.20 7.19 .982220 .982183 .982146 .01 .62 .62 .62 .465539 .466008 .466476 .OO 7.82 7.81 7.81 .534461 .533992 .533524 43 42 41 20 9.449054 71 Q 9.982109 9.466945 10.533055 40 21 .449485 .lo 717 .982072 fi9 .467413 77Q .532587 39 22 23 24 .449915 .450345 .450775 . 14 7.17 7.16 71 . .982035 .981998 .981961 '.62 .62 .467880 .468347 .468814 . i J 7.78 7.78 777 .532120 .531653 .531186 38 37 36 25 26 .451204 .451632 . 10 7.14 71 Q .981924 .981886 ^62 AO .469280 .469746 . 4 i 7.76 77A .530720 .530254 35 34 27 28 .452060 .452488 . lo 7.13 7 19 .981849 .981812 ftvQI .62 AO .470211 .470676 . (0 7.75 7 74 .529789 .529324 33 32 29 .452915 4 . 1 w 7.11 .981774 yQI .62 .471141 4 . I 'i 7.74 .528859 31 30 31 9.453342 .453768 7.10 71A 9.981737 .981700 .62 9.471605 .472068 7.73 10.528395 .527932 20 29 32 33 .454194 .454619 . l\) 7.09 .981662 .981625 !63 .472532 .472995 Tin .527468 .527005 28 27 34 .455044 7.08 .981587 .63 AQ .473457 7.71 77A .526543 26 35 .455469 7*07 .981549 .OO .473919 . lU 7AQ .526081 25 36 .455893 1 .1/4 7A/ .981512 .474381 . oy 7AQ .525619 24 37 .456316 .Ut> 7f\K .981474 /.Q .474842 .oy 7/Q .525158 23 38 .456739 .(JO .981436 .DO .475303 .Oo 7A7 .524697 22 39 .457162 7!04 .981399 '.63 .475763 . Ol 7.67 .524237 21 40 41 42 43 9.457584 .458006 .458427 _ .458848 7.03 7.02 7.01 9.981361 .981323 .981285 .981247 .63 .63 .63 AQ 9.476223 .476683 .477142 .477601 7.66 7.65 7.65 7AA 10.523777 .523317 .522858 .522399 20 19 18 17 41 .459268 7AA .981209 Oo .478059 .O'l 7AO. .521941 10 4> 46 .459688 .460108 .UU 6.99 .981171 .981133 '.63 /o .478517 .478975 .Oo 7.63 .521483 .521025 15 14 47 .460527 6 no .981095 Oo .479432 7/21 .520568 13 43 .460946 .y<5 6G7 .981057 AA .479889 .01 7A1 .520111 12 49 .461364 .y i 6.96 .981019 .O'l .64 .480345 . 01 7.60 .519655 11 EO 9.461782 9.980981 AA 9.480801 7 10.519199 10 51 .462199 K. .980942 wB .481257 7.59 .518743 9 52 .462616 C'od .980904 AA. .481712 7EQ .518288 8 53 .463032 .U4 .980866 .0% .482167 .Oo 7K7 .517833 7 5i 55 .463448 .463864 e!93 .980827 .980789 '.64 Al .482621 .483075 .Ol 7.57 7KA .517379 .516925 6 5 56 57 .464279 .464694 6. '91 6QA .980750 .980712 .01 .64 AA .483529 .483982 .00 7.55 7KK .516471 .516018 4 3 58 59 .465108 .465522 . yu 6.90 .980073 .980635 .0* .64 fid .484435 .484887 .00 7.54 7 CO .515565 .515113 2 1 60 .465935 ' .980596 .04 .485339 .Oo . 51466 I M. Cosine. D.I . Sine. D.I". Cotang. D.I '. Tang. M. 106 73 I7 C TABLE IV. LOGARITHMIC SINES, ETC. 67 162 M. Sine. D.l". Cosine. |D.l". Tang. D.I". Cotang. M. 1 2 3 9.465935 .466348 .466761 .467173 6.88 6.88 6.87 6 on 9.980596 .980558 .980519 .980480 .64 .64 .65 OK 9.485339 .48.3791 .480242 .486693 7.53 7.52 7.51 T Pil 0.514661 .514209 .513758 .513307 60 59 68 57 4 5 6 7 8 9 .467585 .467996 .468407 .468817 .469227 .469637 . oO 6.85 6.85 6.84 6.83 6.83 6.82 .980442 .980403 .980364 .980325 .980286 .980247 OD .65 .65 .65 .65 .65 .65 .487143 .487593 .488043 .488492 .488941 .489390 t .01 7.50 7.50 7.49 7.48 7.48 7.47 .512857 .612407 .511957 .511508 .511059 .510610 56 55 54 53 52 51 10 11 12 13 9,470046 .470455 .470863 .471271 6.81 6.81 6.80 9.980208 .980169 .980130 .980091 .65 .65 .65 9.489838 .490286 .490733 .491180 7.46 7.46 7.45 0.510162 .509714 .509267 .508820 50 49 48 47 14 15 16 .471679 .472086 .472492 6.79 6.78 6.78 .980052 .980012 .979973 .65 .65 .65 .491627 .492073 .492519 7.44 7.44 7.43 .508373 .507927 .507481 46 45 44 17 .472898 6.77 6 'ret .979934 .65 act .492965 7.43 7j) .507035 43 18 .473304 .7o .979895 .DO .493410 .d .506590 42 19 .473710 6.76 6.75 .979855 .66 .66 .493854 7.41 7.41 506146 41 20 9.474115 6TJ 9.979816 9.494299 74 A 0.505701 40 21 22 23 .474519 .474923 .475327 .7* 6.74 6.73 .979776 .979737 .979697 .66 .66 .66 .494743 .495186 .495630 .4(1 7.39 7.39 .505257 .504814 .504370 39 38 37 24 25 .475730 .476133 6 72 6.72 .979658 .979618 .66 .66 .496073 .496515 7.38 7.38 .503927 36 35 26 .476536 6.71 .979579 .66 .496957 7.37 ! 50*H3 34 27 .476938 6.70 .979539 .66 .497399 7.36 7o/- .502601 33 28 .477340 6.69 .979499 .66 .497841 .00 .502159 32 29 .477741 6.69 6.68 .979459 .66 .66 .498282 7.35 7.34 .501718 31 30 9.478142 9.979420 9^498722 10.501278 30 31 .478542 6.67 .979380 .66 .499163 7.34 .500837 29 32 .478942 6.67 .979340 .66 .499603 7.33 .500397 28 33 .479342 6.66 .979300 .67 .500042 7.33 .499958 27 34 35 36 37 .479741 .480140 .480539 .480937 6.65 6.65 6.64 6.63 .979260 .979220 .979180 .979140 .67 .67 .67 .67 r*7 .500481 .500920 .501359 .501797 7.32 7.31 7.31 7.30 .499519 .499080 .498641 .498203 26 25 24 23 38 .481334 6.63 .979100 .o7 .502235 7.30 .497765 22 39 .481731 6.62 6.61 .979059 .67 .67 .502672 7.29 7.28 .497328 21 40 41 42 43 44 45 46 47 9.482128 .482525 .482921 .483316 .483712 .484107 .484501 .484895 6.61 6.60 6.59 6.59 6.58 6.57 6.57 9.979019 .978979 .978939 .978898 .978sr,8 .978817 .978777 .978736 .67 .67 .67 .67 .67 .67 .67 9.503109 .503546 .503982 .504418 .504854 .505289 .505724 .506159 7.28 7.27 7.27 7.26 7.25 7.25 7.24 7O4 10.496891 .496454 .496018 .49558-2 .495146 .494711 .494276 .493841 20 19 18 17 16 15 14 13 48 .485289 6.56 .978696 .68 .506593 .24 .493407 12 49 .485682 6.55 6.55 .978655 .68 .68 .507027 7.23 7.2S .492973 11 50 9.486075 6" 1 9.978615 f\Q 9.507460 TOO 10.492540 10 51 52 .486467 .486860 -O4 6.54 .978574 .978533 Do .68 .r> .507893 .508326 XI 7.21 70-f .492107 .491674 9 8 53 .487251 6.53 .978493 .00 .508759 .zl .491241 7 54 55 .487643 .488034 6.52 6.52 .978452 .978411 .68 .68 .509191 .509622 7.20 7.20 .490809 .490378 6 5 56 57 58 .488424 .488814 .489204 6.51 3.50 6.50 .978370 .978329 .978288 .68 .68 .68 .510054 .510485 .510916 7. 19 7.18 7.18 .489946 .489515 .489084 4 3 2 59 .489593 6.49 3 AO .978247 .68 (0 .511346 7.17 717 .488654 1 60 .489982 .978206 .00 .511776 .14 .488224 M. Cosine rD.i". Sine. r.v fotanp. D1 '. Tang. M. 107 C 72 C 58 TABLE IV. LOGARITHMIC SINES, ETC. 18 D 161 M. Sine. D.l . Cosine. D.I". Tang. D.I ". Cotang, M. 1 9 .489982 .490371 6.48 fi 47 9.978206 .978165 .68 9.511776 .512206 7.16 7%m 10.488224 .487794 60 59 2 ,490759 .978124 ro .512635 . lb .487365 58 3 .491147 6*4fi .978083 f\Q .513064 7.15 71 1 .486936 67 4 .491535 40 .978042 U.7 .513493 .14 .486507 56 6 .491922 6*45 .978001 ViQ .613921 7.14 71 o .486079 55 6 .492308 6*44. .977959 . uy ftQ .514349 . lo 71 o. .485651 54 7 .492695 4- .977918 HP .514777 . lo .485223 53 8 .493081 6 43 .977877 ro .515204 7. 12 .484796 52 9 .493466 6.42 .977835 .'69 .515631 7^11 .484369 51 10 11 12 9.493851 .494236 .494621 6.41 6.41 9.977794 .977752 .977711 .69 .69 9.516057 .516484 .516910 7.10 7.10 10.483943 .483516 .483090 50 49 48 13 .495005 5*15 .977669 .69 .517335 7.09 .482665 47 14 .495388 ?* 2 .977628 CO .517761 7.09 .482239 46 15 .495772 6*38 .977586 *2 .518185 7.08 .481815 45 16 .496154 6OQ .977544 *jj .518610 7.08 .481390 44 17 18 .496537 .496919 .OO 6.37 60.fi .977503 .977461 !70 .519034 .519458 7.07 7.07 .480966 .480542 43 42 19 .497301 .00 6.36 .977419 !70 .519882 7.06 7.05 .480118 41 20 9.497682 6 OK 9.977377 9.520305 10.479695 40 21 22 23 24 25 26 .498064 .498444 .498825 .499204 .499584 .499963 .oD 6.34 6.34 6.33 6.33 6.32 6Q1 .977335 .977293 .977251 .977209 .977167 .977125 .70 .70 .70 .70 .70 .520728 .521151 .521573 .521995 .522417 .522838 7.05 7.01 7.04 7.03 7.03 7.02 .479272 .478849 .478427 .478005 .477583 .477162 39 38 37 36 35 34 27 28 29 .500342 .500721 .501099 ol 6.31 6.30 6.30 .977083 .977041 .976999 .70 .70 .70 .70 .523259 .523680 .524100 7.02 7.01 7.01 7.00 .476741 .476320 .475900 33 32 31 30 31 9.501476 .501854 6.29 9.976957 .976914 .70 "71 9.524520 .524939 6.99 10.475480 .475061 30 29 32 .502231 oa .976872 4 1 .525359 6.99 .474641 28 33 .502607 6 '07 .976830 71 .525778 6.98 .474222 27 34 35 36 .502984 .503360 .503735 .XI 6.27 6.26 .976787 .976745 .976702 71 .71 .71 *71 .526197 .526615 .527033 6.98 6.97 6.97 .473803 .473385 .472967 26 25 24 37 38 39 .504110 .504485 .504860 6^25 6.24 6.24 .976660 .976617 .976574 ll .71 .71 .71 .527451 .527868 .528285 6.96 6.96 6.95 6.95 .472549 .472132 .471715 23 22 21 40 41 42 9.505234 .505608 .505981 6.23 6.22 9.976532 .976489 .976446 .71 .71 9.528702 .529119 .529535 6.94 6.94 10.471298 .470881 .470465 20 19 18 43 44 45 46 47 .506354 .506727 .507099 .507471 .507843 e!2i 6.21 6.20 6.19 .976404 .976361 .976318 .976275 .976232 '.71 .71 .72 .72 .529950 .530366 .530781 .531196 .531611 6.93 6.93 6.92 6.91 6.91 .470050 .469634 .469219 .468804 .468389 17 16 15 14 13 48 49 .508214 .508585 6. 19 6.18 6.18 .976189 .976146 .72 .72 .72 .532025 .532439 8.90 6.90 6.89 .467975 .467561 12 11 50 9.508956 617 9.976103 9.532853 10.467147 10 51 52 .509326 .509696 li 6.16 .976060 .976017 .72 .72 .533266 .533679 6.89 6.88 .466734 .466321 9 8 53 54 55 .510065 .510434 .510803 6.16 6.15 6.15 6 14 .975974 .975930 .975887 .72 .72 .72 ITO .534092 .534504 .534916 6.88 6.87 6.87 ,465908 .465496 .465084 7 6 5 56 57 58 59 60 .511172 .511540 .511907 .512275 .512642 e!i4 6.13 6.12 6.12 .975844 .975800 .975757 .975714 .975670 i .72 .72 .72 .72 .535328 .535739 .536150 .536561 .536972 eise 6.85 6.85 6.84 .464672 .464261 .463850 .463439 .463028 4 3 2 1 M. Cosine. D.I". Sine. D.I". Cotang. D.I". Tang. M. 108 71' TABLE IV. LOGARITHMIC SINES, ETC. 59 160* M. Sine. D. 1 '. Cosine. D.I . Tang. D. l". Cotang. M. 9.512642 6-f 9.975670 7O 9.536972 10.463028 60 1 .513009 .11 6-1 1 .975627 .7o *TO .537382 6QO .462618 59 2 .513375 . 11 61 A .975583 . Io 170 .537791 .OO 6Qr .462209 58 3 .513741 . 1U 6AO .975539 . IO 79 .538202 .00 6QO .461798 57 4 .514107 .uy 6AQ .975496 7o 7Q .538611 .82 .461389 56 5 .514472 .uy 6AQ .975452 IO 170 .539020 6Q1 .460980 55 6 .514837 .Uo 6 Aft .975408 IO 70 .539429 .ol 601 .460571 54 7 .515202 .Uo 6A7 .975365 . IO fTft .539837 .ol 6QA .460163 53 8 .515566 .UY 6A7 .975321 Io 75 .540245 . oU 6QA .459755 52 9 .515930 .U7 6.06 .975277 Io .73 .540653 .oU 6.79 .459347 51 10 9.516294 6AK 9.975233 M 9.541061 10.458939 50 11 .516657 .UO 6 AC .975189 "70 .541468 6.79 670 .458532 49 12 .517020 .UO .975145 . IO .541875 . IO 6rJQ .458125 48 13 14 .517382 .517745 6.' 04 6AQ .975101 .975057 .'73 70 .542281 .542688 . Io 6.77 f* 17*7 .457719 .457312 47 46 15 16 17 18 19 .518107 .518468 .518829 .519190 .519551 .Uo 6.03 6.02 6.02 6.01 6.00 .975013 .974969 .974925 .974880 .974836 IO .74 .74 .74 .74 .74 .543094 .543499 .543905 .544310 .544715 . 1 i 6.76 6.76 6.75 6.75 6.74 .456906 .456501 .456095 .455690 .455285 45 44 43 42 41 20 21 9.519911 .520271 6.00 9.974792 .974748 .74 9.545119 .545524 6.74 6 TO 10.454881 .454476 40 39 22 23 .520631 .520990 5 99 5.99 5 no .974703 .974659 !?4 .545928 .546331 . 7o 6.73 6 TO .454072 .453669 38 37 24 .521349 .yo .974614 .546735 .14 .453265 36 25 .521707 5.98 .974570 .547138 6.72 6T1 .452862 35 26 .522066 rvr .974525 : .547540 . ll 6*71 .452460 34 27 28 .522424 .522781 5.97 5.96 5QK .974481 .974436 ; .74 .547943 .548345 .71 6.70 67A .452057 .451655 33 32 29 .523138 .yo 5.95 .974391 1 .75 .548747 . lU 6.69 .451253 31 30 9.523495 5/VJ 9.974347 ITK 9.549149 6/Q 10.450851 30 31 .523852 .y* .974302 . lO .549550 .D .450450 29 32 .524208 5.94 .974257 *7* .549951 6.68 6O .450049 28 33 .524564 Q0 .974212 . Io *TK .550352 .DO 6 AT .449648 27 34 35 36 .524920 .525275 .525630 5^92 5.92 K ft] .974167 .974122 .974077 . *O .75 .75 .550752 .551152 .551552 .Ol 6.67 6.67 .449248 .448843 .448448 26 25 24 37 .525984 o.yi - . u i .974032 *7* .551952 ft'fifi .448048 23 38 39 .526339 .526693 o.yu 5.90 5.89 .973987 .973942 Io .75 .75 .552351 .552750 6^65 6.65 .447649 .447250 22 21 40 9.527046 9.973897 7K 9.553149 fi ft4 10.446851 20 41 .527400 500 .973852 4U .553548 6mA .446452 19 42 .527753 .00 5QO .973807 *7fc .553946 .54 6/50 .446054 18 43 44 .528105 .528458 .00 5.87 507 .973761 .973716 !?5 7fi .554344 .554741 .Do 6.63 .445656 .445259 17 16 45 .528810 .01 6 Off .973671 ID .555139 6.62 .444861 15 46 .529161 .00 5OJ! .973625 Tfi .555536 6.62 444464 14 47 .529513 .OD 5oe .973580 .76 7o .354484 .354143 .353801 9 8 7 54 55 56 57 58 59 60 .C07607 .607892 .608177 .608461 .608745 .609029 .609313 . (O 4.75 4.74 4.74 4.74 4.73 4.73 .961067 .961011 .960955 .960899 .960843 .960786 .960730 .93 .93 .93 .94 .94 .94 .646540 .646881 .647222 .647562 .647903 .648243 .648583 o.oy 5.68 5.68 5.68 5.67 5.67 5.67 .353460 .353119 .352778 .352438 .352097 .351757 .351417 6 5 4 3 2 1 M. Cosine. D.i". Sine. D.I" Cotang. D.I". Tang. M. 113 66 64: TABLE IV. LOGARITHMIC! SINES, ETC 24* M Sine. D.I". Cosine. D.r Tang. D.I". Cotang M. 1 2 3 4 5 6 7 8 9 9.609313 .609597 .609880 .610164 .610447 .610729 .611012 .611294 .611576 .611858 4.73 4.72 ' 4.72 4.72 4.71 4.71 4.71 4.70 4.70 4.69 9.960730 .960674 .960618 .960561 .960505 .960448 .960392 .960335 .960279 .960222 .94 .94 .94 .94 .94 .94 .94 .94 .94 .94 9.648583 .648923 .649263 .649602 .649942 .650281 .650620 .650959 .651297 .651636 5.67 5.G6 5.G6 5.66 5.65 5.65 5. 65 5.64 5.64 5 64 10.351417 .351077 .350737 .350398 .350058 .349719 .349380 .349041 .348703 .348364 60 59 58 57 56 55 54 53 52 51 10 11 12 13 14 15 16 17 18 19 9.612140 .612421 .612702 .612983 .613264 .613545 .613825 .614105 .614385 .614665 4.69 4.69 4.68 4.68 4.68 4.67 4.67 4.67 4.66 4.66 9.960165 .960109 .960052 .959995 .959938 .959882 .959825 .959768 .959711 .959654 .95 .95 .95 .95 .95 .95 .95 .95 .95 .95 9.651974 .652312 .652650 .652988 .653326 .653663 .654000 .654337 .654674 .655011 5.64 5.63 5.63 5.63 5.62 5.62 5.62 5.62 5.61 5 61 10.348026 .347688 .347350 .347012 .346674 .346337 .346000 .345663 .345326 .344989 50 49 "48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 28 29 9.614944 .615223 .615502 .615781 .616060 .616338 .616616 .616894 .617172 .617450 4.65 4.65 4.65 4.64 4.64 4.64 4.63 4.63 4.63 4.62 9.959596 .959539 .959482 .959425 .959368 .959310 .959253 .959195 .959138 .959081 .95 .95 .95 .95 .96 .96 .96 .96 .96 .96 9.655348 .655684 .656020 .656356 .656692 .657028 .657364 .657699 .658034 .658369 5.61 5.61 5.60 5.60 5.60 5.59 5.59 5.59 5.58 5 58 0.344652 .344316 .343980 .343644 .343308 .342972 .342636 .342301 .341966 .341631 40 39 38 37 36 35 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.617727 .618004 .618281 .618558 .618834 .619110 .619386 .619662 .619938 .620213 4.62 4.61 4.61 4.61 4.60 4.60 4.60 4.59 4.59 4.59 9.959023 .958965 .958908 .958850 .958792 .958734 .958677 .958619 .958561 .958503 .96 .96 .96 .96 .96 .96 .96 .97 .97 .97 9.658704 .659039 .659373 .659708 .660042 .660376 .660710 .661043 .661377 .661710 5.58 5.58 5.57 5.57 5.57 5.56 5.56 5.56 5.56 5 55 0.341296 .340961 .340627 .340292 .339958 .339624 .339290 .338957 .338623 .338290 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.620488 .620763 .621038 .621313 .621587 .621861 .622135 .622409 .622682 .622956 4.58 4.58 4.58 4.57 4.57 4.57 4.56 4.56 4.56 4.55 9.958445 .958387 .958329 .958271 .958213 .958154 .958096 .958038 .957979 .957921 .97 .97 .97 .97 .97 .97 .97 .97 .97 .97 9.662043 .662376 .662709 .663042 .663375 .663707 .664039 .664371 .664703 .665035 5.55 5.55 5.54 5.54 5.54 5.54 5.53 5.53 5.53 5 53 0.337957 .337624 .337291 .336958 .336625 .336293 .335961 .335629 .335297 .334965 20 19 18 17 16 15 14 13 12 11 60 51 52 53 54 9.623229 .623502 .623774 .624047 .624319 4.55 4.54 4.54 4.54 9.957863 .957804 .957746 .957687 .957628 .97 .98 .98 .98 9.665366 .665697 .666029 .666360 .666691 5.52 5.52 5.52 5.51 0.334634 .334303 .333971 .333640 : 333309 10 9 8 6 55 .624591 4KO .957570 .667021 5.51 .332979 5 56 57 58 59 60 .624863 .625135 .625406 .625677 .625948 4.53 4.52 4.52 4.52 .957511 .957452 .957393 .957335 .957276 .98 .98 .98 .98 .667352 .667682 .668013 .668343 .668672 5.51 5.51 5.50 5.50 5.50 .332648 .332318 .331987 .331657 .331328 4 3 2 1 M. Cosine. D.I". Sine. D.I". Cotang. D.I". Tang. M. 114 65 TABLE IV. LOGARITHMIC SINES, ETC. 65 25 M. Sine. D. 1". Cosine. D.I". Tang. D. 1". Cotang. M. 9.625948 4*1 9.957276 QQ 9.668673 K KA 10. 331327 60 1 2 .626219 .626490 .01 4.51 4K| .957217 .957158 .yo .98 .669002 .669332 O.Ov 5.49 .330998 .330668 59 58 3 4 .626760 .627030 .Ol 4.50 4trt .957099 .957040 ^98 QQ .669661 .669991 5\49 K A(\ .330339 .330009 67 66 5 6 7 .627300 .627570 .627840 .DU 4.50 4.49 4AQ .956981 .956921 .956862 .yy .99 .99 QQ .670320 .670649 .670977 5i48 5.48 5AQ. .329680 .329351 .329023 65 64 53 8 .628109 *y 4 49 .956803 .yy QQ .671306 4o 547 .328694 62 9 .628378 4i48 .956744 yy .99 .671634 4i 5.47 .328366 61 10 11 9.628647 .628916 4.48 4JQ 9.956684 .956625 .99 QQ 9.671963 .672291 5.47 547 10.328037 .327709 50 49 12 .629185 ,*O 4A7 .956566 . yy QQ .672619 *' t 54fS .327381 48 13 .629453 ffl 4 AT ,956506 .yy .672947 4O 5 Aft .327053 47 14 .629721 .41 447 .956447 .99 QQ .673274 .TO 5AR .326726 46 15 .629989 4f 4 46 .956387 . yy QQ .673602 4O 54f* .326398 45 16 17 .630257 .630524 4'.46 .956327 .956268 .yy .99 QQ .673929 .674257 4O 5.45 5 45 .326071 .325743 44 43 18 19 .630792 i 631059 4*.45 4.45 .956208 .956148 .yy .00 .00 .674584 .674910 5*45 5.45 .325416 .325090 42 41 20 21 22 23 24 25 26 27 28 29 9.631326 .631593 .631859 .632125 .632392 .632658 .632923 .633189 .633454 .633719 4.45 4.44 4.44 4.44 4.43 4.43 4.43 4.42 4.42 4.42 9.956089 .956029 .955969 .955909 .955849 .955789 .955729 .955669 955609 .955548 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 9.675237 .675564 .675890 .676216 .676543 .676869 .677194 .677520 .677846 .678171 5.44 5.44 5.44 5.44 5.43 5.43 5.43 5.42 5.42 5.42 10.324763 .32443G .324110 .323784 .323457 .323131 .322806 .322480 .322164 .321829 40 39 38 37 36 35 34 33 32 31 30 31 32 33 9.633984 .634249 .634514 .634778 4.41 4.41 4.41 4 40 9.955488 .955428 .955368 .955307 .00 .01 .01 Ol 9.678496 .678821 .679146 .679471 5.42 5 41 5.41 K Al 10.321504 .321179 .320854 .320529 30 29 28 27 34 35 36 37 .635042 .635306 .635570 .635834 4^40 4.40 4.39 40Q .955247 .955186 .955126 .955065 .Ul .01 .01 .01 A1 .679795 .680120 .680444 .680768 O.41 5.41 6.40 5.40 5Af\ .320205 .319880 .319556 .319232 26 25 24 23 38 .636097 .>y 40Q .955005 .Ul 01 .681092 .4U 5Af\ .318908 22 39 .636360 .O7 4.38 .954944 .Ul .01 .681416 4U 5.39 .318584 21 40 9.636623 40Q 9.954883 A1 9.681740 10.318260 20 41 .636886 .OO .954823 .01 A1 .682063 R*M .317937 19 42 .637148 4 Q7 .954762 .01 A1 .682387 50Q .317613 18 43 .637411 .Of 407 .954701 .Ul Al .682710 .oy 500 .317290 17 44 .637673 .04 4Q7 .954640 .Ul AO .683033 . OO 500 .316967 16 45 .637935 Ol .954579 ,U 9.949881 .949816 .949752 .949688 1.07 1.07 1.07 IAQ 9.707166 .707478 .707790 .708102 5.20 5.20 5.20 5 on 10.292834 .292522 .292210 .291898 60 69 68 57 4 5 6 .658037 .658284 .658531 . 1*5 .12 .12 .949623 .949558 .949494 .Uo 1.08 1.08 .708414 .708726 .709037 .Aj 5.20 6.19 5 1O .291586 .291274 290963 66 65 64 . 7 8 .658778 .659025 .11 .949429 .949364 lios .709349 .709660 . 1*7 5.19 .290651 .290340 63 52 9 .659271 '. .11 .10 .949300 1.08 1.08 .709971 5.19 5.18 .290029 51 10 11 9.659517 .659763 .10 9.949235 .949170 1.08 IAQ 9.710282 - .710593 5.18 51Q. 10.289718 .289407 60 49 12 .660009 .949105 . Uo IAQ .710904 lo 51Q .289096 48 13 .660255 AQ .949040 .Uo IAQ .711215 .lo 51ft .288785 47 14 .660501 .uy .948975 . UO IAQ .711525 .lo 51T .288475 46 15 .660746 .09 .948910 .Uo IAQ .711836 ! 5-ffT .288164 45 16 .660991 .09 , AQ .948845 .Uo IAQ .712146 .17 61T .287854 44 17 .661236 .Uo .948780 .uy .712456 .14 .287544 -43 18 .661481 .08 > no .948715 1.09 1 OQ .712766 5.17 517 .287234 42 19 .661726 .Uo .08 .948650 i .uy 1.09 .713076 * 1 i 5.16 .286924 41 20 9.661970 9.948584 1/Vl 9.713386 5flM 10.286614 40 21 22 .662214 .662459 .07 .948519 .948454 .uy 1.09 Irwi .713696 .714005 .lo 5.16 51/5 .286304 .285995 39 38 23 24 .662703 .662946 !oe .948388 .948323 .uy 1.09 .714314 .714624 . ID 5.15 51 K. .285686 .285376 37 36 25 26 .663190 .663433 4i<)6 .948257 .948192 l!( .714933 .715242 . ID 6.15 .285067 .284758 35 34 27 .663677 4.05 .948126 1.09 .715551 5.15 .284449 33 28 .663920 4.05 A A*\ .948060 1.09 IAQ .715860 5.15 511 .284140 32 29 .664163 4.UD 4.05 .947995 .uy 1.10 .716168 . 1% 5.14 .283832 31 30 31 32 33 34 35 36 37 38 39 9.664406 .664648 .664891 .665133 .665375 .665617 .665859 .666100 .666342 .666583 4.04 4.04 4.04 4.03 4.03 4.03 4.03 4.02 4.02 4.02 9.947929 .947863 .947797 .947731 .947665 .947600 .947533 .947467 .947401 .947335 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 9.716477 .716785 .717093 .717401 .717709 .718017 .718325 .718633 .718940 .719248 5.14 6.14 5.14 6.13 5.13 6.13 6.13 5.13 6.12 6.12 10.283523 .283215 .282907 .282599 .282291 .281983 .281675 .281367 .281060 .280752 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 9.666824 .667065 .667305 .667546 .667786 .668027 .668267 .668506 4.01 4.01 4.01 4.01 4.00 4.00 4.00 o (\f) 9.947269 .947203 .947136 .947070 .947004 .946937 .946871 .946804 1.10 1.11 1.11 1.11 1.11 1.11 1.11 1-1-1 9.719555 .719862 .720169 .720476 .720783 .721089 .721396 .721702 6.12 6.12 5.11 5.11 6.11 6.11 6.11 61 f\ 10.280445 .280138 .279831 .279524 .279217 .278911 .278604 .278298 20 19 18 17 16 15 14 13 48 49 .668746 .668986 >.yy 3.99 3.99 .946738 .946671 .11 1.11 1.11 .722009 .722315 .10 5.10 5.10 .277991 .277685 12 11 50 51' 9.669225 .669464 3.99 3 no 9.946604 .946538 1.11 11 1 9.722621 .722927 5.10 51 A 10.277379 .277073 10 9 52 .669703 .yo .946471 . 11 111 .723232 . 1U K Afl .276768 8 53 54 55 56 57 58 59 60 .669942 .670181 .670419 .670658 .670896 .671134 .671372 .671609 3^98 3.98 3.97 3.97 3.97 3.96 3.96 .946404 .946337 .946270 .946203 .946136 .946069 .946002 .945935 .11 1.11 1.12 1.12 1.12 1.12 1.12 1.12 .723538 .723844 .724149 .724454 .724760 .725065 .725370 .725674 o.uy 5.09 5.09 5.09 5.09 6.08 5.08 5.08 .276462 .276156 .275851 .275546 .275240 .274935 .274630 .274326 7 6 5 4 3 2 1 M. Cosine. D.I". Sine. Dl". Cotane. D.I". Tang. M. 117 C 62 TABLE IV. LOGAKITHMIO SINES, ETC. 151' M. Sine. D.I". Cosine. D.I". Tang. D.I". Cotang. M. 9.671609 9.945935 11 O 9.725674 5AQ 10.274326 60 1 .671847 3.96 .945868 - \ .725979 .Uo .274021 69 2 3 4 5 6 7 8 9 .672084 .672321 .672558 .672795 .673032 .673268 .673505 .673741 3.96 3.95 3.95 3.95 3.94 3.94 3.94 3.94 3.93 .945800 .945733 .945666 .945598 .945531 .945464 .945396 .945328 l'.12 1.12 1.12 1.12 1.12 1.13 1.13 1.13 .726284 .726588 .726892 .727197 .727501 .727805 .728109 .728412 5.08 5.07 5.07 6.07 5.07 5.07 6.06 5.06 5.06 .273716 .273412 .273108 .272803 .272499 .272195 .271891 .271588 58 57 56 65 64 53 52 51 10 11 12 13 14 15 16 9.673977 .674213 .674448 .674684 .674919 .675155 .675390 3.93 3.93 3.93 3.92 3.92 3.92 3 O1 9.945261 .945193 .945125 .945058 .944990 .944922 .944854 1.13 1.13 1.13 1.13 1.13 1.13 11 Q 9.728716 .729020 .729323 .729626 .729929 .730233 .730535 6.06 5.06 6.05 5.05 5.05 6.05 10.271284 .270980 .270677 .270374 .270071 .269767 .269465 50 49 48 47 46 45 44 17 18 19 .675624 .675859 .676094 .yi S.91 3.91 3.91 .944786 .944718 .944650 . lo 1.13 1.13 1.13 .730838 .731141 .731444 6.05 5.05 5.04 6.04 .269162 .268859 .268556 43 42 41 20 21 22 23 24 9.676328 .676562 .676796 .677030 .677264 3.90 3.90 3.90 3.90 3QQ 9.944582 .944514 .944446 .944377 .944309 1.14 1.14 1.14 1.14 1 14 9.731746 .732048 .732351 .732653 .732955 6.04 6.04 5.04 5.04 K A'-i 10.268254 .267952 .267649 .267347 .267045 40 39 38 37 36 25 26 27 28 .677498 .677731 .677964 .678197 .Oi7 3.89 3.89 3.88 3QQ .944241 .944172 .944104 .944036 l!l4 1.14 1.14 11 1 .733257 .733558 .733860 .734162 O.Uo 6.03 5.03 5.03 .266743 .266442 .266140 .265838 35 34 33 32 A .678430 .OO 3.88 .943967 . 14 1.14 .734463 5.02 6.02 .265537 31 30 9.678663 3QO 9.943899 111 9.734764 10.265236 30 31 32 33 34 35 36 37 .678895 .679128 .679360 .679592 .679824 .680056 .680288 .OO 3.87 3.87 3.87 3.87 3.86 3.86 3Q/> .943830 .943761 .943693 .943624 .943555 .943486 .943417 .14 1.14 1.15 1.15 1.15 1.15 1.15 11 K .735066 .735367 .735668 .735969 .736269 .736570 .736871 5.02 5.02 5.02 5.01 5.01 5.01 5.01 .264934 .264633 .264332 .26403^ .263731 .263430 .263129 29 28 27 26 25 24 23 38 39 .680519 .680750 .OD 3.86 3.85 .943348 .943279 .ID 1.15 1.15 .737171 .737471 5.01 6.01 5.00 .262829 .262529 22 21 40 41 42 43 44 45 9.680982 .681213 .681443 .681674 .681905 .682135 3.85 3.85 3.84 3.84 3.84 3QJ. 9.943210 .943141 .943072 .943003 .942934 .942864 1.15 1.15 1.15 1.15 1.15 11 A 9.737771 .738071 .738371 . 738671 .738971 .739271 5.00 5.00 5.00 5.00 4.99 4 Art 10.262229 .261929 .261629 .261329 .261029 .260729 20 19 18 17 16 15 46 47 48 .682365 .682595 .682825 .tr* 3.83 3.83 3QO .942795 .942726 .942656 . lu 1.16 1.16 .739570 .739870 .740169 .yy 4.99 4.99 4 no .260430 .260130 .259831 14 13 12 49 .683055 .OO 3.83 .942587 1.16 1.16 .740468 .yy 4.98 .259532 11 50 9.683284 9.942517 Iic 9.740767 4OQ 10.259233 10 51 52 53 54 .683514 .683743 .683972 .684201 3.82 3.82 3.82 3Q1 .942448 .942378 .942308 .942239 .ID 1.16 1.16 1.16 Itfi. .741066 .741365 .741664 .741962 .98 4.98 4.98 4.98 .258934 .258635 .258336 .258038 9 8 7 6 55 56 57 .684430 .684658 .684887 . ol 3.81 3.81 Sort .942169 .942099 .942029 . lo 1.16 1.16 117 .742261 .742559 .742858 4L97 4.97 .257739 .257441 .257142 5 4 3 68 59 .685115 .685343 .Ov/ 3.80 O QA .941959 .941889 ll 1.17 .743156 .743454 4.97 4.97 .256844 .256546 2 1 60 .685571 o.oU .941819 1.17 .743752 4.97 .256248 M. Cosine. D.I". Sine. D.I". Cotang. D.I". Tang. M. 118 29' TABLE IV. LOGARITHMIC SINES, ETC. 69 150 M. 37 Sine. 9.685571 .685799 .686264 .686709 .687163 .687389 .687616 9.687843 .690772 .690996 .691220 .691444 .692115 .692785 .693231 .693453 .694120 9.694564 .695007 .695229 .695671 .696334 .696554 9.696775 .697215 .697435 .697874 3.80 3.79 3.79 3.79 3.79 3.78 3.78 3.78 3.78 3.77 3.77 3.77 3.77 3.76 3.76 3.76 3.76 3.75 3.75 3.75 3.75 3.74 3.74 3.74 3.74 3.73 3.73 3.73 3.73 3.72 3.72 3.72 3.72 3.71 3.71 3.71 3.71 3.70 3.70 3.70 3.70 3.69 3.69 3,69 3.69 3.68 3.68 3.68 3.68 3.67 3.67 3.67 3.67 3.66 3.66 3.66 3.66 3.65 3.65 3.65 M. Cosine. D. 1". 119 30 Cosine. D.I'. Tang. D.I". Cotang. M. 9.941819 .941749 .941679 .941539 .941398 .941328 .941258 .941187 9.941117 .941046 .940975 .940905 .940834 .940763 .940622 .940.V,1 9.940409 .940338 .940267 .940125 .940054 .939768 .939554 .939482 .939410 .939267 .939195 .939123 .938703 .938047 .938475 .938402 .938258 .938185 .938113 .937967 .937749 .937604 .937531 Sine. 1.17 1.17 .17 .17 .17 .17 .17 .17 .17 .17 .18 .18 .18 .18 .18 .18 .18 .18 .18 .13 .18 .18 .19 .19 .19 .19 .19 .19 .19 .19 .19 .19 .19 .19 .19 .20 .20 .20 .20 .20 .20 .90 .90 .20 .90 .90 .20 .21 .21 .21 .21 .21 .21 .21 .21 .21 .21 .21 .21 1.22 D.I' 9.743752 .744050 .744348 .744645 .744943 .745240 .745538 .745835 .746132 .746429 9.746726 .747023 .747319 .747616 .747913 .748209 .748505 .748801 .749393 .749985 .750281 .750576 .750372 .751167 .751462 .751757 .752052 .752347 9.752642 .752937 .753231 .753526 .753820 .754115 .754409 .754703 .754997 .755291 9.755585 .755878 .756172 .756465 .756759 .757052 .757345 .757638 .757931 .758224 9.758517 .758810 .759102 .759687 .759979 .760272 .760564 .760856 .761148 .761439 Cotang. 4.96 4.96 4.96 4.96 4.96 4.96 4.95 4.95 4.95 4.95 4.95 4.95 4.94 4.94 4.94 4.94 4.94 4.93 4.93 4.93 4.93 4.93 4.93 4.92 4.92 4.92 4.92 4.92 4.92 4.91 4.91 .91 .91 .91 .91 .90 .90 .90 .90 .90 D.r 10.256248 .255950 .255652 .255355 .255057 .254760 .254462 .254165 .253868 .253571 10.253274 .252977 .252681 .251791 .251495 .251199 .250903 10.250311 .250015 .249719 .249424 .249128 .248833 .248538 .248243 .247948 .247653 10.247358 .247063 .246769 .246474 .246180 .245885 .245591 .245297 .245003 .244709 10.244415 .244122 .243828 .243535 .243241 .242948 .242655 .242362 .242069 .241776 10.241483 .241190 .240898 .240605 .240313 .240021 .239728 .239436 .239144 .238852 .238561 Tang. 60 70 TABLE IV. LOGARITHMIC SINES, ETC. 3O 149' M. Sine. D.I". Cosine. D.I". Tang. D. l". Cotang. M. 9.698970 3/*e 9.937531 9.761439 40ft 10.238561 60 1 2 3 .699189 .699407 .699626 .DO 3.64 3.64 .937458 .937385 .937312 l'.22 1.22 .761731 .762023 .762314 .OU 4.86 4.86 .238269 .237977 .237686 59 58 57 4 5 6 7 .699844 .700062 .700280 .700498 3.64: 3.63 3.63 .937238 .937165 .937092 .937019 l!22 1.22 1.22 .762606 .762897 .763188 .763479 4^86 4.85 4.85 4QK .237394 .237103 .236812 .236521 56 55 54 53 8 .700716 3.63 3/50 .936946 1 .22 Ion .763770 .OO 4 OK .236230 52 9 .700933 .DO 3.62 .936872 .352 1.22 .764061 .oO 4.85 .235939 51 10 11 12 13 14 15 9.701151 .701368 .701585 .701802 .702019 .702236 3.62 3.62 3.62 3.61 3.61 3/M 9.936799 .936725 .936652 .936578 .936505 .936431 1.22 1.23 1.23 1.23 1.23 1OQ 9:764352 .764643 .764933 .765224 , .765514 .765805 4.85 4.84 4.84 4.84 4.84 484 10.235648 .235357 .235067 .234776 .234486 .234195 50 49 48 47 46 45 16 17 .702452 .702669 . Ol 3.61 .936357 .936284 .Zo 1.23 .766095 .766385 . .231876 .231587 37 36 25 26 27 28 .704395 .704610 .704825 .705040 -Oi/ 3.59 3.58 3.58 .935692 .935618 .935543 .935469 1^24 1.24 1.24 .768700 .768992 .769281 .769571 . oZ 4.82 4.82 4.82 4 CO .231297 .231008 .230719 .230429 35 34 33 32 29 .705254 3.58 3.58 .935395 l!24 .769860 . OJ& 4.82 .230140 31 80 81 82 83 9.705469 .705683 .705898 .706112 3.57 3.57 3.57 3*7 9.935320 .935246 .935171 .935097 1.24 1.24 1.24 i .24 9.770148 .770437 .770726 .771015 4.81 4.81 4.81 A Q1 10.229852 .229563 .229274 .228985 30 29 28 27 84 .706326 .Ol .935022 .771303 4. ol 4Qi .228697 26 85 86 87 38 39 .706539 .706753 .706967 .707180 .707393 3^56 3.56 3.56 3.55 3.55 .934948 .934873 .934798 .934723 .934649 l!24 1.25 1.25 1,25 1.25 .771592 .771880 .772168 .772457 .772745 .ol 4.81 4.80 4.80 4.80 4.80 .228408 .228120 .227832 .227543 .227255 25 24 23 22 21 40 41 42 43 44 45 9.707606 .707819 .708032 .708245 .708458 .708670 3.55 3.55 3.54 3.54 3.54 3KA 9.934574 .934499 .934424 .934349 .934274 .934199 1.25 1.25 1.25 1.25 1.25 1OK 9.773033 .773321 .773608 .773896 .774184 .774471 4.80 4.80 4.80 4.79 4.79 4 79 10.226967 .226679 .226392 .226104 .225816 .225529 20 19 18 17 16 15 46 .708882 .D*x Q KA .934123 .ZD 1OK .774759 .225241 14 47 .709094 O.O4 .934048 2Sv .775046 47O .224954 13 48 49 .709306 .709518 3.53 3.53 3.53 .933973 .933898 1 .25 1.26 1.26 .775333 .775621 . li) 4.79 4.78 .224667 .224379 12 11 50 61 52 63 54 55 56 9.709730 .709941 .710153 .710364 .710575 .710786 .710997 3.53 3.52 3.52 3.52 3.52 3.51 3K1 9.933822 .933747 .933671 .933596 .933520 .933445 .933369 1.26 1.26 1.26 1.26 1.26 1.26 10ft 9.775908 .776195 .776482 .776769 .777055 .777342 .777628 4.78 4.78 4.78 4.78 4.78 4.78 477 10.224092 .223805 .223518 .223231 .222945 .222658 .222372 10 9 8 7 6 5 4 57 58 59 60 .711208 .711419 .711629 .711839 .Ol 3.51 3.51 3.51 .933293 .933217 .933141 .933066 . \) 1.26 1.26 1.26 .777915 .778201 .778488 .778774 44 4.77 '4.77 4.77 .222085 .221799 .221512 .221226 3 2 1 M. Cosine. D.I". Bine. D.I". Cotang. D. 1". Tang. 1S7 120 TABLE IV. LOGARITHMIC SINES, ETC 71 3r 148 M. Sine. D.l . Cosine. D.l" Tang. D.l". Cotang. M. 1 2 3 4 5 9.711839 .712050 .712260 .712469 .712679 .712889 3.50 3.50 3.50 3.50 3.49 9.933066 .932990 .932914 .932838 .932762 .932685 1.27 1.27 1.27 1.27 1.27 9.778774 .779060 .779346 .779632 .779918 .780203 4.77 4.77 4.77 4.76 4 76 10.221226 .220940 .220654 .220368 .220082 .219797 60 59 68 57 56 65 6 7 9 .713098 .713308 .713517 .713726 3.49 3.49 3.49 3.48 3.48 932609 .932533 .932457 .932380 1 .27 1.27 1.27 1.27 1.27 .780489 .780775 .781060 .781346 4.76 4 76 4.76 4.76 4.76 .219511 .219225 .218940 .218654 64 63 62 51 10 11 12 9.713935 .714144 .714352 3.48 3.48 9.932304 .932228 .932151 1.27 1.27 9.781631 .781916 .782201 4.75 4.75 4TK 10.218369 .2*6084 .217799 60 49 48 13 14 15 .714561 .714769 .714978 3.48 3.47 3.47 3 AT .932075 .931998 .931921 1.28 1.28 1.28 1QQ .782486 .782771 .783056 .75 4.75 4.75 4fK .217514 .217229 .216944 47 46 45 16 .715186 .4if SAT .931845 Zo 100 .783341 iD 4TK. .216659 44 17 .715394 .47 .931768 .6 .783626 . IO 4*TA .216374 43 18 19 .715602 .715809 3.46 3.46 3.46 .931691 .931614 1.28 1.28 1.28 .783910 .784195 .74 4.74 4.74 .216090 .215805 42 41 20 9.716017 9.931537 1 28 9.784479 474. 10.215521 40 21 22 .716224 .716432 3.'46 .9314GO .931383 l!28 1f)0 .784764 .785048 1 4 4.74 .215236 .214952 39 38 23 24 25 .716639 .716846 .717053 3.45 3.45 3.45 .931306 .931229 .931152 .ZO 1.28 1.29 1 ' ' ' .785332 .785616 .785900 4.74 4.74 4.73 .214668 .214384 .214100 37 36 35 26 27 28 29 .717259 .717466 .717673 .717879 3.45 3.44 3.44 3.44 3.44 .931075 .930998 .930921 .930843 1-29 1.29 1.-29 1.29 .786184 .786468 .786752 .787036 4-73 4.73 4.73 4.73 4.73 .213816 .213532 .213248 .212964 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.718085 .718291 .718497 .718703 .718909 .719114 .719320 .719525 .719730 .719935 3.43 3.43 3.43 3.43 3.43 3.42 3.42 3.42 3.42 3.41 9.930766 .930688 .930611 .930533 .930456 .930378 .930300 .930223 .930145 .930067 1.29 1.29 l!29 1.29 1.29 1.30 1.30 1.30 1.30 9.787319 .787603 .787886 .788170 .788453 .788736 .789019 .789302 .789585 .789868 4.73 4.72 4.72 4.72 4.72 4.72 4.72 4.72 4.71 4.71 0.212681 .212397 .212114 .211830 .211547 .211264 .210981 .210698 .210415 .210132 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 9.720140 720345 .720549 .720754 .720958 .721162 3.41 3.41 3.41 3.41 3.40 9.929989 .929911 .929833 .929755 .929677 .929599 1.30 1.30 1.30 1.30 1.30 19A 9.790151 .790433 .790716 .790999 .791281 .791563 4.71 4.71 4.71 4.71 4.71 4TA 0.209849 .209567 .209284 .209001 .208719 .208437 20 19 18 17 16 15 46 47 48 .721366 .721570 .721774 3^40 3.40 QO .929521 .929442 .929364 . oU 1.30 1.31 191 .791846 .792128 .792410 .70 4.70 4.70 .208154 .207872 .207590 14 13 12 49 .721978 o.oy 3.39 .929286 .ol 1.31 .792692 4.70 4.70 .207308 11 50 9.722181 QO 9.929207 1Vf 9.792974 4TA 0.207026 10 51 52 53 .722385 .722588 .722791 o.ov 3.39 3.39 OQ .929129 .929060 .928972 .ol 1.31 1.31 Iof .793256 .793538 .793819 :.7O 4.70 4.70 .206744 .206462 .206181 9 8 7 54 55 .722994 .723197 o.oo 3.38 00 .928893 .928815 .ol 1.31 101 .794101 .794383 4.69 4.69 .205899 .205617 6 5 56 67 58 59 60 .723400 .723603 .723805 .724007 .724210 O.Oo 3.38 3.37 3.37 3.37 .928736 .928657 .928578 .928499 .928420 .ol 1.31 1.31 1.31 1.32 .794664 .794945 .795227 .795508 .795789 4.69 4.69 4.69 4.69 4.69 .205336 .205055 .204773 .204492 .204211 4 3 2 1 M. Cosine. D.l" ' Sine. D.l". Cotang. D.l". Tang. M. 121 68 72 TABLE IV. LOGARITHMIC SINES, ETC. 32 147 o M. Sine. D.l". Cosine. D.I" Tang. D.I . Cotang. M. 1 2 3 4 5 6 7 8 9 9.724210 .724412 .724614 .724816 .725017 .725219 .725420 .725622 .725823 .726024 3.37 3.37 3.36 3.36 3.36 3.36 3.36 3.35 3.35 3.35 9.928420 .928342 .928263 .928183 .928104 .928025 .927946 .927867 .927787 .927708 1.32 1.32 1.32 1.32 1.32 1.32 1.32 1.32 1.32 1.32 9.795789 .796070 .796351 .796632 .796913 .797194 .797475 .797755 .798036 .798316 4.68 4.68 4.68 4.68. 4.68 4.68 4.68 4.68 4.67 4.67 10.204211 .203930 .203649 .203368 .203087 .202806 .202525 .202245 .201964 .201684 60 69 58 57 66 65 64 63 62 61 10 11 12 13 14 15 16 17 18 19 9.726225 .72W26 .726626 .726827 .727027 .727228 .727428 .727628 .727828 .728027 3.35 3.34 3.34 3.34 3.34 3.34 3.33 3.33 3.33 3.33 9.927629 .927549 .927470 .927390 .927310 .927231 .927151 .927071 .926991 .926911 1.32 1.33 1.33 1.33 1.33 1.33 1.33 1.33 1.33 1.33 9.798596 .798877 .799157 .799437 .799717 .799997 .800277 .800557 .800836 .801116 4.67 4.67 4.67 4.67 4.67 4.66 4.66 4.66 4.66 4.66 10.201404 .201123 .200843 .200563 .200283 .200003 .199723 .199443 .199164 .198884 50 49 -48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 28 29 9.728227 .728427 .728626 .728825 .729024 .729223 .729422 .729621 .729820 .730018 3.33 3.32 3.32 3.32 3.32 3.31 3.31 3.31 3.31 3.31 9.926831 .926751 .926671 .926591 .926511 .926431 .926351 .926270 .926190 .926110 1.33 1.33 1.33 1.34 1.34 1.34 1.34 1.34 1.34 1.34 9.801396 .801675 .801955 .802234 .802513 .802792 .803072 .803351 .803630 .803908 4.66 4.66 4.66 4.65 4.65 4.65 4.65 4.65 4.65 4 65 10.198604 .198325 .198045 .197766 .197-187 .197208 .196928 .196640 .196370 .196092 40 39 38 37 36 35 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.730216 .730415 .730613 .730811 .731009 .731206 .731404 .731602 .731799 .731996 3.30 3.30 3.30 3.30 3.30 3.29 3.29 3.29 3.29 3.28 9.926029 .925949 .925868 .925788 .925707 .925626 .925545 .925465 .925384 .925303 1.34 1.34 1.34 1.34 1.35 1.35 1.35 1.35 1.35 1.35 9.804187 .804466 .804745 .805023 .805302 .805580 .805859 .806137 .806415 .806693 4.65 4.64 4.64 4.64 4.64 4.64 4.64 4.64 4.64 4 63 10 195813 . 195534 . 195255 .194977 .194698 .194420 .194141 .193863 .193585 .193307 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 43 49 9.732193 .732390 .732587 .732784 .732980 .733177 .733373 .733569 .733765 .733961 3.28 3.28 3.28 3.28 3.27 3.27 3.27 3.27 3.27 3.26 9.925222 .925141 .925060 .924979 .924897 .924816 .924735 .924654 .924572 .924491 1.35 1.35 1.35 1.35 1.35 1.35 1.36 1.36 1.36 1.36 9.806971 .807249 .807527 .807805 .808083 .808361 .808638 808916 .809193 .809471 4.63 4.63 4.63 4.63 4.63 4.63 4.63 4.62 4.62 4.62 10.193029 .192751 .192473 .192195 .191917 .191639 .191362 .191084 .190807 .190529 20 19 18 17 16 15 14 13 12 11 50 61 62 53 54 55 56 67 58 9.734157 .734353 .734549 .734744 .734939 .735135 .735330 .735525 .735719 3,26 3.26 3.26 3.26 3.25 3.25 3.25 3.25 9.924409 .924328 .924246 .924164 .924083 .924001 .923919 .923837 .923755 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.37 9.809748 .810025 .810302 .810580 .810857 .811134 .811410 .811687 .811964 4.62 4.62 4.62 4.62 4.62 4.61 4.61 4.61 10.190252 .189975 .189698 .189420 .189143 .188866 .188590 .188313 188036 10 9 8 7 6 5 4 3 2 69 60 .735914 .736109 3.24 .923673 .923591 1.37 1.37 .812241 .812517 4.61 4.61 .187759 .187483 1 M Cosine. D.I". Sine. D.I". Cotang. D.I". Tang. M 122 57 C TABLE IV. LOGARITHMIC SINES, ETC. 73 33 o 146 M. Sine. D.I". Cosine. JD.l" Tang D.I". ! Cotang. M. 9.736109 .736303 3.24 o 04 9.923591 .923509 1.37 1 37 9.812517 .812794 4.C1 4.61 10.187483 .187206 60 59 2 4 5 6 7 8 9 .736498 .736692 .736886 .737080 .737274 .737467 .737661 .737855 O./-X 3.24 3.23 3.23 3.23 3.23 3.23 3.22 3.22 .923427 .923345 .923263 .923181 .923098 .923016 .922933 .922851 1^37 1.37 1.37 1.37 1.37 1.37 1.37 1.38 .813070 .813347 .813623 .813899 .814176 .814452 .814728 .815004 4'.61 4.61 4.60 4.60 4.60 4.60 4.60 4.60 .186930 .186653 .186377 .186101 .185824 .185548 .185272 .184996 58 57 56 55 54 53 52 51 10 9.738048 3 ^ 9.922768 1OQ 9.815280 10.184720 50 11 12 13 14 15 16 17 .738241 .738434 .738627 .73882" .739013 .739206 .739398 3.22 3.22 i 3.21 3.21 3.21 3.21 .922686 .922003 .922520 ! 9223^5 .922272 .922189 .OO 1.38 1.38 1.38 1.38 1.38 1.38 .815555 .815831 .816107 .816382 .816658 .816933 .817209 4'.60 4.59 4.59 4.59 4.59 4.59 .184445 .184169 .183893 .183618 .183342 .183067 .182791 49 48 47 46 45 44 43 18 19 .739590 .739783 3.21 3.20 3.20 .922106 .922023 1.38 1.38 1.38 .817484 .817759 4.59 4.59 4.59 .182516 .182241 42 41 20 21 22 23 21 25 26 9.739975 .740167 .740359 .740550 .740742 .740934 .741125 3.20 3.20 3.20 3.19 3.19 3.19 9.921940 .921857 .921774 .921691 .921607 .921524 .921441 1.39 1.39 1.39 1.39 1.39 1.39 9.818035 .818310 .818585 .818860 .819135 .819410 .819684 4.59 4.58 4.58 4.58 4.58 4.58 4KQ 10.181965 .181690 .181415 .181140 .180865 .180590 .180316 40 39 38 37 36 35 34 '27 .741316 3.19 3m .921357 1.39 .819959 .00 4KQ .180041 33 28 .741508 . 1U .921274 Ion .820234 .Oo 4 to .179766 32 29 .741699 3. 18 3.18 .921190 . oy 1.39 .820508 .Do 4.58 .179492 31 30 31 32 33 34 35 36 37 9.741889 .742080 .742271 .742462 .742652 .742842 .743033 .743223 3.18 3.13 3.18 3.17 3.17 3.17 3.17 9.921107 .921023 .920909 .920856 .920772 JB6B8 .920604 .920520 1.39 1.39 1.40 1.40 1.40 1.40 1.40 9.820783 .821057 .821332 .821606 .821880 .822154 .822420 .82270:; 4.57 4.57 4.57 4.57 4.57 4.57 4.57 4KT 10.179217 .178943 .178668 .178394 .178120 .177846 .177571 .177297 30 29 28 27 26 25 24 23 38 39 .743413 .743602 3J6 3.16 .986488 .920352 l!40 1.40 .822977 .823250 .Dl 4.57 4.56 .177023 .176750 22 21 40 41 9.743792 .743982 3.16 9.920268 ..920184 1.40 1JA 9.823524 .823798 4.56 10.176476 .176202 20 19 42 43 44 45 46 .744171 .744361 .744550 .744739 .744928 3.16 3.16 3.15 3.15 3.15 3 IK. .920099 .920015 .919931 .919846 .919762 .40 1.40 1.41 1.41 1.41 1 41 .824072 .824345 .824619 .824893 .825166 4.56 4.56 4.56 4.56 , 4.56 .175928 .175655 .175381 .175107 .174834 18 17' 16 15 14 47 48 49 .745117 .745306 .745494 . ID 3.15 3.14 3.14 .919677 .919593 .919508 l!4l 1.41 1.41 .825439 .825713 .825986 4^56 4.55 4.55 .174561 .174287 .174014 13 12 11 60 51 9.745683 .745871 3.14 9.919424 .919339 1.41 9.826259 .826532 4.55 10.173741 .173468 10 9 62 63 64 65 66 67 58 69 60 .746060 .746248 .746436 .746624 .746812 .746999 .747187 .747374 .747562 3. 14 3.14 3.13 3.13 3.13 3.13 3.13 3.12 3.12 .919254 .919169 .919085 .919000 .918915 .918830 .918745 .918659 .918574 1 .41 1.41 1.41 1.42 1.42 1.42 1.42 1.42 1.42 .826805 .827078 .827351 .827624 .827897 .828170 .828442 .828715 .828987 4.55 4.55 4.55 4.55 4.55 4.55 4.54 4.54 4.54 .173195 .172922 .172649 .172376 .172103 .171830 .171558 .171285 .171013 8 7 6 5 4 3 2 1 M. Cosine. D.I". Sine. D.I Cotang. D.I". Tang. M. 74 TABLE IV. LOGAEITHMIO SINES, ETC. 84' 145" M. Sine. 0.1". Cosine. D.l . Tang. D.I "... Cotang. M. 9.747562 3 1O 9.918574 1 49 9.828987 4KA 10.171013 60 1 2 3 4 5 .747749 .747936 .748123 .748310 .748497 . 14 3.12 3.12 3.11 3.11 3-t 1 .918489 .918404 .918318 .918233 .918147 J . 4.4 1 42 1.42 1.42 1.42 1.4O .829260 .829532 .829805 .830077 .830349 .0'* 4.54 4.54 4.54 4.54 4CJ .170740 .170468 .170195 .169923 .169651 59 58 57 56 55 6 7 8 9 .748683 .748870 .749050 .749243 . 11 3.11 3.11 3.10 3.10 .918062 .917976 .917891 .917805 . 4t> 1.43 1.43 1.43 1.43 .830621 .830893 .831165 .831437 . D^ 4.53 4.53 4.53 4.53 .169379 . 169107 .168835 .168563 54 53 52 51 10 11 12 13 14 9.749429 .749615 .749801 .749987 .750172 3.10 3.10 3.10 3.10 3 no 9.917719 .917634 .917548 .917462 .917376 1.43 1.43 1.43 1.43 14 Q 9.831709 .831981 x .832253 .832525 .832796 4.53 4.53 4.53 4.53 4KQ 10.168291 .168019 .167747 .167475 .167204 50 49 48 47 46 15 .750358 .uy 3 An .917290 .43 .830068 .OO .166932 45 16 .750543 .uy 3AQ .917204 1 .43 1/lQ .833339 4.53 4rO .166661 44 17 18 19 .750729 .750914 .751099 .Uif 3.09 3.09 3.08 .917118 .917032 .916946 .4o 1.44 1.41 1.44 .833611 .833882 .834154 Di 4.52 4.52 4.52 .166389 .166118 .165846 43 42 41 20 9.751284 3 08 9.916859 1 A/1 9.834425 4Kf\ 10.165575 40 21 22 .751469 .751654 3^08 3f\Q .916773 .916687 J . 44 1.44 .834696 .834967 . &A 4.52 .165304 .165033 39 38 23 .751839 . Oo 3f\Q .916600 1 .44 .835238 4.52 .164762 37 24 25 .752023 .752208 .(JO 3.07 3 07 .916514 .916427 1.44 1.44 1A4 .835509 .835780 4.52 4.52 4KO .164491 .164220 6 35 26 27 .752392 .752576 3^07 3A7 .916341 .916254 . 44 1 44 .836051 .836322 .& 4.51 .163949 .163678 34 33 28 29 .752760 .752944 .Ui 3.07 3.06 .916167 .916081 1.44 1.45 1.45 .836593 .836864 4.51 4.51 4.51 .163407 .163136 32 31 30 31 32 33 34 35 9.753128 .753312 .753495 .753679 .753862 .754046 3.06 3.06 3.06 3.06 3.05 3 Arc 9.915994 .915907 .915820 .915733 .915646 .9155.59 1.45 1.45 1.45 1.45 1.45 9.837134 .837405 .837675 .837946 .838216 .838487 4.51 4.51 4.51 4.51 4.51 10.162866 .162595 .162325 .162054 .161784 .161513 30 29 28 27 26 25 36 37 38 39 .754229 .754412 .754595 .754778 .UD 3.05 3.05 3.05 3.05 .915472 .915385 .915297 .915210 1 .45 1.45 1.45 1.45 1.46 .838757 .839027 .839297 .839568 4.51 4.50 4.50 4.50 4.50 .161243 .160973 . 160703 .160432 24 23 22 21 40 41 42 43 9.754960 .755143 .755326 .755508 3.04 3.04 3.04 3 A/I 9.915123 .915035 .914948 .914860 1.46 1.46 1.46 9.839838 .840108 .840378 .840648 4.50 4.50 4.50 10.160162 .159892 .159622 .159352 20 19 18 17 44 45 46 47 48 49 .755690 .755872 .756054 .756236 .756418 .756600 .U4 3.04 3.03 3.03 3.03 3.03 3.03 .914773 .914685 .914598 .914510 .914422 .914334 1 .46 1.46 1.46 1.46 1.46 1.46 1.46 .840917 .841187 .841457 .841727 .841996 .842260 4.50 4.50 4.49 4.49 4.49 4.49 4.49 .159083 .158813 .158543 .158273 .158004 .157734 16 15 14 13 12 11 50 51 52 63 64 65 66 67 68 59 60 9.756782 .756963 .757144 .757326 .757507 .757688 .757869 .758050 .758230 .758411 .758591 3.02 3.02 3.02 3.02 3.02 3.02 3.01 3.01 3.01 3.01 9.914246 .914158 .914070 .913982 .913894 .913806 .913718 .913630 .913541 .913453 .913365 1.47 1.47 1.47 1-47 1.47 1.47 1.47 1.47 1.47 1.47 9.842535 .842805 .843074 .843343 .843612 .843882 .844151 .844420 .844689 .844958 .845227 4.49 4.49 4.49 4.49 4.49 4.49 4.48 4.48 4.48 4.48 10.157465 .157195 .156926 .156657 .156388 .156118 .155849 .155580 .155311 .155042 .154773 10 9 8 7 6 6 4 3 2 1 M. Cosine. D.I . Sine. D.I" Cotang. D. 1 . Tang. M. 124 C 55 C TABLE IV. LOGARITHMIC SINES, ETC. 75 35 144 M. Sine. D.I . Cosine. D.I Tang. D.I . Cotang. M. 1 2 3 4 5 6 9.758591 .758772 .758952 .759132 .759312 .759492 .759672 3.01 3.00 3.00 3.00 3.00 3.00 2 Oft 9.913365 .913276 .913187 .913099 .913010 .912922 .912833 1.47 1.48 1.48 1.48 1.48 1.48 1AQ 9.845227 .845496 .845764 .846033 .846302 .846570 .846839 4.48 4.48 4.48 4.48 4.48 4.48 4AQ 10.154773 .154504 .154236 .153967 .153698 .153430 .153161 60 69 58 67 66 65 54 7 8 .759852 .760031 .yy 2.99 2 GO .912744 .912655 .4o 1.48 1AQ .847108 .847376 .4o 4.47 447 .152892 .152624 63 62 9 .760211 .yy 2.99 .912566 .4o 1.48 .847644 .41 4.47 .152356 51 10 9.760390 9.912477 140 9.847913 4*1 10.152087 60 11 12 13 .760569 .760748 .760927 2!99 2.98 .912388 .912299 .912210 .4o 1.48 1.49 11' 1 .848181 .848449 .848717 JBI 4.47 4.47 44.7 .151819 .151551 .151283 49 48 47 14 15 16 .761106 .761285 .761464 2!98 2.98 2 Oft .912121 .912031 .911942 .4y 1.49 1.49 It.. .848986 .849254 .849522 .44 4.47 4.47 4 AT .151014 .150746 .150478 46 45 44 17 .761642 . yo rt Q-T .911853 . .4y .849790 4i 4. 4fi .150210 43 18 19 .761821 .761999 2i97 2.97 .911763 .911674 1^49 1.49 .850057 .850325 4^46 4.46 .149943 .149675 42 41 20 21 22 9.762177 .762356 .762534 2.97 2.97 2 97 9.911584 .911495 .911405 1.49 1.49 9.850593 .850861 .851129 4.46 4.46 4 46 10.149407 .149139 .148871 40 39 38 23 .762712 2'ofi .911315 1Kf\ .851396 4 46 .148604 37 24 25 .762889 .763067 .yo 2.96 2 96 .911226 .911136 .ou 1.50 1CA .851664 .851931 4^46 4 46 .148336 .148069 36 35 26 27 28 .763245 .763422 .763600 96 2.96 2QK .911046 .910956 .910866 .OU 1.50 1.50 1CA .852199 .852466 .852733 4^46 4.46 44fi .147801 .147534 .147267 34 33 32 29 .763777 .yo 2.95 .910776 .OU 1.50 .853001 .40 4.45 .146999 31 30 31 32 33 34 35 36 9.763954 .764131 .764308 .764485 .764662 .764838 .765015 2.95 2.95 2.95 2.95 2.94 2.94 2 94 9.910686 .910596 .910506 .910415 .910325 .910235 .910144 1.50 1.50 1.50 1.51 1.51 1.51 Ir-i 9.853268 .853535 .853802 .854069 .854336 .854603 .854870 4.45 4.45 4.45 4.45 4.45 4.45 4AK 10.146732 .146465 .146198 .145931 .145664 .145397 .145180 30 29 28 27 26 25 24 37 38 39 .765191 .765367 .765544 2i94 2.94 2.93 .910054 .909963 .909873 .01 1.51 1.51 1.51 .855137 .855404 .855671 4t> 4.45 4.45 4.44 .144863 .144596 .144329 23 22 21 40 9.765720 200 9.909782 1 11 9.855938 4 44 10.144062 20 41 .765896 .yo .909691 1 .01 Ie.1 .856204 4AA .143796 19 42 43 .766072 .766247 2i93 .909601 .909510 Ol 1.51 1K1 .856471 .856737 .44 4.44 4 A A .143529 .143263 18 17 44 45 46 47 48 49 .766423 .766598 .766774 .76C949 .767124 .767300 2i93 2.92 2.92 2.92 2.92 2.92 .909419 .909328 .909237 .909146 .909055 .908964 01 1.52 1.52 1.52 1.52 1.52 1.52 .857004 .857270 ,857537 .857803 .858069 .858336 .44 4.44 4.44 4.44 4.44 4.44 4.44 .142996 .142730 .142463 .142197 .141931 .141664 16 15 14 13 12 11 60 51 62 63 9.767475 .767649 .767824 .767999 2.91 2.91 2.91 9.908873 .908781 .908690 .908599 1.52 1.52 1.52 Ieo 9.858602 .858868 .859134 .R59400 4.44 4.43 4.43 10.141398 .141132 .140866 .140600 10 9 8 7 54 .768173 2.91 201 .908507 .OJ Iff) .859666 4.43 4 JO .140334 6 65 66 f>7 .768348 .768522 7/>o/Q7 .yi 2.91 2.90 .908416 .908324 OAQOQQ flfli 1.53 1.53 .859932 .8H0198 ftttTUtfl iff 4.43 4.43 .140068 .139802 5 4 Of 68 . *uooy < .768871 2.90 .yuo^oo .908141 1.53 . OOU4U4 .800730 4.43 .139536 .139270 3 2 59 60 .769045 .769219 2!90 .908049 .907958 1 .53 1.53 .860995 .861261 4.43 4.43 .139005 .138739 1 "57 Cosine. D.I". Sine. D.I". Cotang. D.I". Tang. "UTi 125' 76 TABLE IV. LOGARITHMIC SINES, ETC. 36 143 M. Sine. D.l". Cosine. D.l". Tang. D.l". Cotang. M. 1 2 3 9.769219 .769393 .769566 .769740 2.90 2.90 2.89 9.907958 .907866 .907774 .907682 1.53 1.53 1.53 9.861261 .861527 .861792 .862058 4.43 4.43 4.43 10.138739 .138473 .138208 .137942 60 59 68 57 4 5 6 7 8 .769913 .770087 .770260 .770433 .770606 2.89 '2.89 2.89 2.89 2.88 200 .907590 .907498 .907406 .907314 .907222 1.53 1.53 1.53 1.54 1.54 .862323 .862589 .862854 .863119 .863385 4^42 4.42 4.42 4.42 4jif\ .137677 .137411 .137146 .136881 .136615 66 55 54 63 62 9 .770779 .00 2.88 .907129 1 .54 1.54 .863650 .4J 4.42 .J36350 61 10 9.770952 200 9.907037 9.863915 4 Act 10. 136085 60 11 .771125 .00 2QQ .906945 1.54 1KX .864180 .4J 4Acy .135820- 49 12 .771298 .OO 200 .906852 .DC: .864445 '..T.A .135555 48 13 .771470 .00 207 .906760 1 .54: 1 F\*. .864710 4 JO .135290 47 14 15 16 17 18 .771643 .771815 .771987 .772159 .772331 .01 2.87 2.87 2.87 2.87 2O7 .906667 .906575 .906482 .906389 .906290 l!54 1.54 1.55 1.55 .864975 .865240 .865505 .865770 .866035 [J9 4.42 4.41 4.41 4.41 .135025 .134760 .134495 .134230 .133965 46 45 44 43 42 19 .772503 .04 2.86 .906204 1 .55 1.55 .866300 4.41 4.41 .133700 41 20 9.772G75 ft 9.906111 IKK 9.866564 4A1 10.133436 40 21 .772847 20/5 .906018 . OD .866829 :.41 A 11 .133171 39 22 23 .773018 .773190 oO 2.86 2 Oft .905925 .905832 1.55 IKK .867094 .867358 4.41 4.41 4A1 .132906 .132642 38 37 24 25 26 27 28 .773361 .773533 .773704 .773875 .774046 .oO 2.85 2.85 2.86 2.85 2 OK .905739 .905645 .905552 .905459 .905366 . DD 1.55 1.55 1.55 1.56 .867623 .867887 .868152 .868416 .868680 .41 4.41 4.41 4.41 4.41 4 Art .132377 .132113 .131848 .131584 .131320 36 35 34 33 32 29 .774217 .Oc) 2.85 .905272 l!56 .868945 :.4U 4.40 .131055 31 30 31 9.774388 .774558 2.84 204 9.905179 .905085 1.56 IKf* 9.869209 .869473 4.40 4AC\ 10.130791 .130527 30 29 32 33 .774729 .774899 .cr* 2.84 .904992 .904898 . OO 1.56 .869737 .870001 4U 4.40 4 Art .130263 .129999 28 27 34 35 30 37 .775070 .775240 .775410 .775580 2^84 2.84 2.83 200 .904804 .904711 .904617 .904523 l!56 1.56 1.56 1K7 .870265 .870529 .870793 .871057 .41) 4.40 4.40 4.40 .129735 .129471 .129207 .128943 26 25 24 23. 38 39 .775750 .775920 .00 2.83 2.83 .904429 .904335 . .04 1.57 1.57 .871321 .871585 4^40 4.40 .128679 .128415 22 21 40 41 42 43 44 9.776090 .776259 .776429 .776598 .776768 2.83 2.83 2.82 2.82 2QO 9.904241 .904147 .904053 .903959 .903864 1.57 1.57 1.57 1.57 It- FT 9.871849 .872112 .872376 .872640 .872903 4.40 4.39 4.39 4.39 10.128151 .127888 .127624 .127360 .127097 20 19 18 17 16 45 46 47 .776937 .777106 .777275 .04 2.82 2.82 2Qi> .903770 .903676 .903581 .07 1.57 1.57 1 57 .873167 .873430 .873694 4^39 4.39 .126833 .126570 .126306 15 14 13 48 .777444 Odd .903487 .873957 4.0*7 .126043 12 49 1 .777613 2.81 2.81 .903392 1 .58 1.58 .874220 4.39 4.39 .125780 11 50 61 52 9.777781 .777950 .778119 2.81 2.81 Q1 9.903298 .903203 .903108 1.58 1.58 1KQ 9.874484 .874747 .875010 4.39 4.39 10.125516 .125253 .124990 10 9 8 63 54 55 66 .778287 .778455 .778624 .778792 Z.ol 2.81 2.80 2.80 2Qrt .903014 .902919 .902824 .902729 .Oo 1.58 1.58 1.58 1KQ .875273 .875537 .875800 .876063 4! 39 4.38 4.38 40Q .124727 .124463 .124200 .123937 7 6 5 4 57 .778960 .oU 20/1 .902634 .OO 1KQ .876326 .OO 40Q .123674 3 68 .779128 .OU 2Qrt .902539 .Oo 1 . 9 .876589 .OO 40Q .123411 2 59 .779295 .oU .902444 .876852 OO 40Q .123148 1 60 .779463 2.79 .902349 1 . 59 .877114 OO .122886 M. Cosine. D.l". Sine. D.l". Cotaiig. D.l". Tang. M 37' TABLE IV. LOGABITHMIC SINES, ETC. 77 142 M. Sine. D.l" Cosine. D.l". Tang. D.l". Cotang. M. 1 9.779463 .779631 2.79 9.902349 .902253 1.59 9.877114 .877377 4.38 10.122886 .122623 60 69 2 .779798 .902158 .877640 .122360 r>8 3 4 5 6 7 8 9 .779966 .780133 .780300 .780467 .780634 .780801 .780968 2.79 2.79 2.78 2.78 2.78 2.78 2.78 .902063 .901967 .901872 .901776 .901681 .901585 .901490 1.59 1.59 1.59 1.59 1.59 1.59 1.60 .877903 .878165 .878428 .878691 .878953 .879216 .879478 4.38 4.38 4.38 4.38 4.38 4.37 4.37 .122097 .121835 .121572 .121309 .121047 .120784 .120522 67 C6 65 54 63 62 51 10 11 12 13 14 15 16 17 18 19 9.781134 .781301 .781468 .781634 .781800 .781966 .782132 .782298 .782464 .782630 2.78 2.77 2.77 2.77 2.77 2.77 2.77 2.76 2.76 2.76 9.901394 .901298 .901202 .901106 .901010 .900914 .900818 .900722 .900626 .900529 1.60 1.60 1.60 1.60 1.60 1.60 1.60 1.60 1.60 1.61 9.879741 .880003 .880265 .880528 .880790 .881052 .881314 .881577 .881839 .882101 4.37 4.37 4.37 4.37 4.37 4.37 4.37 4.37 4.37 4.37 10.120259 .119997 .119735 .119472 .119210 .118948 .118686 .118423 .118161 .117899 60 49 48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 28 29 9.782796 .782961 .783127 .783292 .783458 .783623 .783788 .783953 .784118 .784282 2.76 2.76 2.76 2.75 2.75 2.75 2.75 2.75 2.75 2.74 9.900433 .900337 .900240 .900144 .900047 .899951 .899854 .899757 .899660 .899564 1.61 1.61 1.61 1.61 1.61 1.61 1.61 1.61 1.61 1.62 9.882363 .882625 .882887 .883148 .883410 .883672 .883934 .884196 .884457 .884719 4.37 4.37 4.36 4.36 4.36 4.36 4.36 4.36 4.36 4.36 10.117637 .117375 .117113 .116852 .116590 .116328 .116066 .115804 .115543 .115281 40 39 38 37 36 35 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.784447 .784612 .784776 .784941 .785105 .785269 .785433 .785597 .785761 .785925 2.74 2.74 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.73 9.899467 .899370 .899273 .899176 .899078 .898981 .898884 .898787 .898689 .898592 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 9.884980 .885242 .885504 .885765 .886026 .886288 .886549 .886811 .887072 .887333 4.36 4.36 4.36 4.36 4.36 4.36 4.36 4.35 4.35 4.35 10.115020 .114758 .114496 .114235 .113974 .113712 .113451 .113189 .112928 .112667 30 29 28 27 26 25 24 23 22 21 40 41 42 43 44 45 46 47 48 49 9.786089 .786252 .786416 .786579 .786742 .786906 .787069 .787232 .787395 .787557 2.73 2.73 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 9.898494 .898397 .898299 .898202 .898104 .898006 .897908 .897810 .897712 .897614 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.C3 9.887594 .887855 .888116 .888378 .888639 .888900 .889161 .889421 .889682 .889943 4.35 4.35 4.35 4.35 4.35 4.35 4.35 4.35 4.35 4.35 10.112406 .112145 .111884 .111622 .111361 .111100 .110839 .110579 .110318 .110057 20 19 18 17 16 15 14 13 12 11 60 61 62 63 64 55 66 67 68 69 60 9.787720 .787883 .788045 .788208 .788370 .788532 .788694 .788856 .789018 .789180 .789342 2.71 2.71 2.71 2.71 2.70 2.70 2.70 2.70 2.70 2.70 9.897516 .897418 .897320 .897222 .8*7123 .897025 .896926 .896828 .896729 .896631 .896532 1.64 1.64 1.64 1.64 1.64 1.64 1.64 1.64 1.64 1.64 9.890204 .890465 .890725 .890986 .891247 .891507 .891768 .892028 .892289 .892549 .892810 4.35 4.35 4.34 4.34 4.34 4.34 4.34 4.34 4.34 4.34 10.109796 .109535 .109275 .109014 .108753 .108493 .108232 .107972 .107711 .107451 .107190 10 9 8 7 6 5 4 3 2 1 M. Cosine. D.l". Sine. D.l". Cotang. D.l". Tang. M. 127 78 TABLE IV. LOGARITHMIC SINES, ETC. 141* Sine. D.l . osine. D.I . Tang. D.l". otang. M. 1 .789342 .789504 2.69 2.69 .896532 .896433 1.65 1.C5 .892810 .893070 4.3i 4.34 .107190 .106930 106669 60 59 KQ 2 3 4 5 6 7 8 9 .789665 .789827 .789988 .790149 .790310 .790471 .790632 .790793 2.69 2.69 2.69 2.69 2 68 2.68 2.68 2 68 .896236 .896137 .896038 .895939 .895840 .895741 .895641 1.65 1.65 1.65 1.65 1.65 1.65 1.65 1.65 .893591 .893851 .894111 .894372 .894632 .894892 .895152 4.34 4.34 4.34 4.34 4.34 4.34 4.33 4.33 .106409 .106149 .105889 .105628 .105368 .105108 .104848 57 56 55 54 53 52 51 10 11 32 .790954 .791115 .791275 2.68 2.68 9.895542 .895443 .895343 1.66 1.66 9.895412 .895672 .895932 4.33 4.33 4 33 .104588 .104329 .104068 50 49 48 13 14 15 16 17 18 19 .791436 .791596 .791757 .791917 .792077 .792237 .792397 2.67 2.67 2.67 2.67 2.67 2.67 2.66 .895244 .895145 .895045 .894945 .894846 .894746 .894646 .bo 1.6$ 1.66 1.66 1.66 1.66 1.66 1.66 .896192 .896452 .896712 .896971 .897231 .897491 .897751 4.33 4.33 4.33 4.33 4.33 4.33 4.33 .103808 .103548 .103288 .103029 .102769 .102509 .102249 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 28 29 9.792557 .792716 .792876 .793035 .793195 .793354 .793514 .793673 .793832 .793991 2.66 2.66 2.66 2.66 2.66 2.65 2. 65 2.65 2.65 2.65 9.894546 .894446 .894346 .894246 .894146 .894046 .893946 .893846 .893745 .893645 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 9.898010 .898270 .898530 .898789 .899049 .899308 .899568 .899827 .900086 .900346 4.33 4.33 4.33 4.33 4.33 4.32 4.32 4.32 4.32 4.32 0.101990 .101730 .101470 .101211 .100951 .100692 .100432 .100173 .099914 .099654 40 39 38 36 35 34 33 32 31 30 31 32 33 34 35 36 37 38 39 9.794150 .794308 .794467 .794626 .794784 .794942 .795101 .795259 .795417 .795575 2.65 2.64 2.64 2.64 2.64 2.64 2.64 2.64 2.63 2.63 9.893544 .893444 .893343 .893243 .893142 .893041 .892940 .892839 .892739 .892638 1.68 1.68 1.68 1.68 1.68 1.68 1.68 1.68 1.68 1.68 9.900605 .900864 .901124 .901383 .901642 .901901 .902160 .902420 .902679 .902938 4.32 4.32 4.32 4.32 4.32 4.32 4.32 4.32 4.32 4.32 0.099395 .099136 .098876 .098617 .098358 .098099 .097840 .097580 .097321 .097062 30 29 28 27 26 25 24 23 22 21 40 41 42 4: 9.795733 .795891 .796049 .796206 .796364 .796521 2.63 2.63 2.63 2.63 2.62 9 892536 .892435 .892334 .892233 .892132 .892030 1.69 1.69 1.69 1.69 1.69 9.903197 .903456 .903714 .903973 .904232 .904491 4.32 4.32 4.31 4.31 4.31 10.096803 .096544 .096286 .096027 .095768 .095509 20 19 18 17 16 15 ( .796679 .796836 .796993 .797150 2.62 2.62 2.62 2.62 2.61 .891929 .891827 .891726 .891624 1.69 1.69 1.69 1.6 .904750 .905008 .905267 .905526 4.31 4.31 4.3 4.3 .095250 .094992 .094733 .094474 14 13 12 11 50 9.797307 9.891523 9.90578 LO. 094215 10 5 .797464 .891421 .90604 .093957 9 52 5 54 5 56 5 5 5 .797621 .797777 .797934 .798091 .798247 .798403 .798560 .798716 2.6 2.6 2.6 2.6 2.6 2.6 2.60 .891319 .891217 .891115 .891013 .890911 .890809 .890707 .890605 1.70 1.70 1.70 1.70 1.70 1.70 1.70 .90630 .90656 .90681 .90707 .90733 .90759 .90785 .90811 4.3 4.3 4.3 4.3 4.3 4.3 4.3 .093698 .093440 .093181 .092923 .092664 .092406 .092148 .091889 8 7 6 5 .4 3 2 1 60 .798872 2.60 .890503 1.70 .90836 .091631 M Cosine. D.r Sine. D.r . Cotang D.l". | Tang. M. TABLE IY. LOGABITHMIO SINES, ETa 79 140 M. Sine. D.I". Cosine. D.I'. Tang. D.I". Cotang. >. 798872 .799184 .799651 .800117 .800272 9.800427 .800582 .800737 .801047 .801201 .801356 .801511 .801665 .801819 9.801973 .802128 .802282 .802436 .802743 .803050 .803204 >.803511 .803664 .803817 .804123 .804276 .804428 .804581 .804734 .805191 .805343 .805495 .805647 .805799 .805951 .806103 .806254 9.806557 .807011 ,807163 .807314 .807465 .807615 .807766 .807917 .808067 2.60 2.60 2.60 2.59 2.59 2.59 2.59 2.59 2.59 2.59 2.58 2.58 2.58 2.58 2.58 2.58 2.57 2.57 2.57 2.57 2.57 2.57 2.57 2.56 2.56 2.56 2.56 2.56 2.56 2.55 2.55 2.55 2.55 2.55 2.55 2.55 2.54 2.51 2.54 2.54 2.54 2,54 2.54 2.53 2.53 2.53 2.53 2.53 2.53 2.52 2.52 2.52 2.52 2.52 2.52 2.52 2.51 2.51 2.51 2.51 9.890503 .890400 .890298 .890195 9.889477 .889374 .889271 .888755 .888651 .888548 9.888444 .887926 .887822 .887718 .887614 .887510 K887406 .887302 .887198 .887093 .886571 .8864G6 .885837 .885627 .885522 9.885311 .885205 .885100 .884783 .884677 .884572 1.71 1.71 1.71 1.71 1.71 1.71 1.71 1.71 1.71 1.71 1.72 1.72 1 72 1.72 1.72 1.72 1.72 1.72 1.72 1.72 1.73 1.73 1.73 1.73 1.73 1.73 1.73 1.73 1.73 1.74 1.74 1.74 1.74 1.74 1.74 1.74 1.74 1.74 1 74 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.76 1.76 1.76 1.76 1.76 1.76 1.76 1.76 1.76 1.77 1.71 .909144 .909402 .910177 .910435 9.910951 .911209 .911467 .911724 .911982 .912240 .912756 .913014 .913271 .913787 .914044 .914302 .914560 .914817 .915075 .915332 .915590 .915847 .916104 .916362 .916619 .916877 .917131 .917391 .91764S .917900 .918420 9.918677 .918934 .919191 .919443 .919705 .919962 .920219 .920476 .920733 9.921247 .921503 .921760 .922017 .922274 .922530 .922787 4 30 4 30 4.30 4.iJO 4 30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.30 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.29 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 4.28 0091631 .091372 .091114 .090856 .090598 .090340 .090082 .089823 .089565 .089307 0.089049 .088791 .088533 .087760 .087502 .087244 .086986 .086729 0086471 086213 89 37 .OS54-JO .085183 .084925 .084608 .084410 .084153 0.083896 .083638 .083381 .083123 .081838 .081580 10.081323 .081066 .080809 .080552 .079781 .079524 .079267 .079010 10.078753 .078497 .078240 .077983 .077726 .077470 .077213 .076956 .076700 .076443 .076187 5O Cosine. D.I". Sine. D.I" Cotang. D. 1". Tang. 129 80 40 TABLE IV. LOGARITHMIC SINES, ETC. M. Sine. D.I". Cosine. D.I". Tang. D.I". Cotang. M. 1 2 3 4 5 9.808067 .808218 .808368 .808519 .808669 .808819 2.51 2.51 L.51 2.50 2.50 9 50 9.884254 .884148 .884042 .883936 .883829 .883723 1 77 1.77 1.77 1.77 1.77 1 77 9.923813 .924070 .924327 .924583 .924840 .925096 4.28 4.28 4,27 4.27 4.27 4 07 10.076187 .075930 .075673 .075417 .075160 .074904 60 59 58 57 56 55 6 7 8 9 .808969 809119 .809269 .809419 z.ou 2.50 2.50 2.50 2.50 .883617 .883510 .883404 .883297 1 . i 4 1.77 1.77 1.78 1.78 .925352 .925609 .925865 .926122 :.Z< 4.27 4.27 4.27 4.27 .074648 .074391 .074135 .073878 54 53 52 51 10 11 12 13 14 15 16 17 18 19 9.809569 .809718 .809868 .810017 .810167 .810316 .810465 .810614 .810763 .810912 2.49 2.49 2.49 2.49 2.49 2.49 2.48 2.48 2.48 2.48 9.883191 .883084 .882977 .882871 .882764 .882657 .882550 .882443 .882336 .882229 1.78 1.78 1.78 1.78 1.78 1.78 1.78 1.79 1.79 1.79 9.926378 .926634 .926890 .927147 .927403 .927659 .927915 .928171 .928427 .928683 4.27 4.27 4.27 4.27 4.27 4.27 4.27 4.27 4.27 4.27 10.073622 .073366. .073110 .072853 .072597 .072341 .072085 .071829 .071573 .071317 50 49 48 47 46 45 44 43 42 41 20 21 22 23 24 25 26 27 9.811061 .811210 .811358 .811507 .811655 .811804 .811952 .812100 2.48 2.48 2.48 2,47 2.47 2.47 2.47 2 AT 9.882121 .882014 .881907 .881799 .881692 .881584 .881477 .881369 1.79 1.79 1.79 1.79 1.79 1.79 1.79 1Q{\ 9.928940 .929196 .929452 .929708 .929964 .930220 .930475 .930731 4.27 4.27 4.27 4.27 4.27 4.27 4.26 10.071060 .070804 .070548 .070292 .070036 .069780 .069525 .069269 40 39 38 37 36 35 34 33 28 .812248 .4< 2AJ7 .881261 .oU .930987 4na .069013 32 29 .812396 .4< 2.47 .881153 1^80 .931243 . Zo 4.26 .068757 31 30 9.812544 2A.fi 9.881046 9.931499 10.068501 30 31 32 .812692 .812840 ,4O 2.46 O AjR .880938 .880830 i.'so .931755 .932010 4 '.26 .068245 .067990 29 28 33 .812988 2A(* .880722 1 'flA .932266 4 Of? .067734 27 34 .813135 ,4o .880613 1.80 .932522 ,OQ .067478 26 35 .813283 2.46 .880505 1QA .932778 4.26 .067222 25 36 37 38 39 .813430 .813578 .813725 .813872 2^46 2.45 2.45 2.45 .880397 .880289 .880180 .880072 . oU 1.81 1.81 1.81 1.81 .933033 .933289 .933545 .933800 4^26 4.26 4 26 4.26 .066967 .066711 .066455 .066200 24 23 22 21 40 9.814019 Q JK 9.879963 1Q1 9.934056 4f>fi 10.065944 20 41 42 43 44 .814166 .814313 .814460 .814607 2!45 2.45 2.45 2 A^ .879855 .879746 .879637 .879529 . Ol 1.81 1.81 1.81 81 .934311 .934567 .934823 .935078 .Zo 4.26 4.26 4.26 .065689 .065433 .065177 .064922 19 18 17 16 45 .814753 2AA .879420 .ol Q1 .935333 4 Of* .064667 15 46 .814900 .44 O A .879311 .ol 89 .935589 ,Zo .064411 14 47 48 .815046 .815193 2.4A .879202 .879093 . o J .82 .935844 .936100 4! 26 .064156 .063900 13 12 49 .815339 2!44 .878984 '.82 .936355 4.26 4.26 .063645 11 50 51 9.815485 .815631 2.44 9.878875 .878766 .82 9.936611 .936866 4 26 10.063389 .063134 10 9 52 .815778 2 A ft .878656 QO .937121 4.Zo .062879 8 53 .815924 ,4o 2 A*> .878547 .oZ 00 .937376 4 OK .062624 7 54 .816069 4o .878438 . oZ .937632 . Zt) 4 OK < .062368 6 55 .816215 2AO .878328 QO .937887 4 OK .062113 5 56 57 58 .816361 .816507 .816652 .4o 2 A3 2.43 2 An .878219 .878109 .877999 . OO .83 .83 QO .938142 .938398 .938653 . ZD 4.25 4.25 4 OK .061858 .061602 .061347 4 3 2 59 60 .816798 .816943 me 2.42 .877890 .877780 . oO .83 .938908 .939163 . ZD 4.25 .061092 .060837 1 IT Cosine. D.I". Sine. D.I". Cotang. D.I". Tang. 1*7. TABLE IV. LOGARITHMIC SINES. ETC. 81 41 138 M. Sine. D.i . Cosine. | D.I". Tang. | D.I . j Cotang. M. 1 2 3 9.816943 .817088 .817233 .817379 2.42 2.42 2.42 9.8777SO .877670 .8775CO .877450 83 .83 .83 9.50163 .939418 .939673 .939928 4.25 4.25 4.25 4Ot 10.060837 .060582 .060327 .060072 60 59 T7 4 .817524 '% .877340 Ql .940183 .^D 4 5 .059817 C6 5 .817668 o'TT . cl C 1 .940438 4 OK .059562 C5 6 .817813 ;} .877120 .0* .940694 . - D 4 25 .059306 54 7 .817958 I 7, ,1 .877010 C 1 .940949 4 25 .059051 53 8 9 .818103 .818247 A.ti. 2.41 2.41 .876899 .876789 . 89 .84 .84 .941204 .941458 4^25 4.25 .058796 .058542 ro 51 10 11 9:818392 .818536 2.41 2 1 9.876678 .876568 1.81 9.941713 .941968 4.25 4 25 10.058287 .058032 EO 49 12 13 14 13 13 .818681 .818825 .818969 .819113 .819257 . i L 2.40 2.41 2.40 2.40 .876457 .876347 .876236 .876125 .76014 i!si 1.84 1.85 .85 op? .942223 .942478 .942733 .942986 .943243 4!25 4.25 4.25 4.25 4o*t .057777 .057522 .057267 .057012 .056707 48 47 48 45 44 17 .819401 o tn .875904 .CO OK .943498 - 4-D 4 OK .056502 43- 13 19 .819545 .819689 2^40 2.39 .875793 .875682 .CO .85 .85 .943752 .944007 . -. y 4.25 4.25 .056248 .OGC993 42 41 20 21 22 23 9.819832 .819976 .820120 .820263 2.39 2.39 2.39 9.875571 .875459 .875348 .875237 .S5 85 .85 9.944262 .944517 .944771 .945026 4.25 4.25 4 24 10.055738 .0.35483 .055229 .054974 40 39 38 37 24 23 .820406 .820550 2!.39 .875126 .875014 .88 .945281 .945535 4^24 4O4 .054719 .054465 36 35 23 .820693 29ft .874903 .945790 :. --t .054210 34 27 .820836 oO 2r>0 .874791 .86 '.946045 J .053955 33 23 .820979 .00 f> OQ .874680 .86 Of* .946299 4.24 4 24 .053701 32 29 .821122 2.38 .874568 .00 .86 .946554 4'.24 .053446 31 30 9.821265 20Q 9.874456 Qf* 9.946808 4 Of 10.053192 30 31 32 .821407 .821550 .Jo 2.38 200 .874344 .874232 .eo .86 .947063 .947318 . Jl 4.24 4OA .052937 .052682 29 28 33 31 .821693 .821835 .*X> 2 37 .874121 .874009 '.SI .947572 .947826 . Z4 4.24 .052428 .052174 27 26 33 O "* cr C3 39 .821977 .822120 .822262 .822404 .822546 2.37 2.37 2.37 2.37 2.37 2.37 .873896 .873784 .87:3672 .873560 .873448 .87 87 .87 .87 .87 .87 .948081 948336 .948590 .948844 .949099 4.24 4.24 4.24 4.24 424 4.24 .051919 .051664 .051410 .051156 .050901 25 24 23 22 21 40 41 42 9.822688 .822830 .822972 2.37 2.36 9.873335 .873223- .873110 .87 .88 9.949353 .949608 .949862 4.24 4.24 10.050647 .050392 .050133 20 19 18 43 .823114 2.36 .872998 .88 .950116 }?} ! .049884 17 44 .823253 2.36 .88 oo .950371 m.M .049629 16 45 .823397 2w 's72772 .00 .950625 4.24 j Of .049375 15 46 47 .823539 .823680 .00 2.36 .872659 .872547 .88 .88 950879 .951133 4!24 4O4 .049121 .048867 14 13 48 49 .823821 .823963 2. 35 2.35 .872434 .872321 !88 1.88 .951388 .951642 2M 4.24 4.24 .048612 .048358 12 11 50 51 9.824104 .824243 2.35 9.872208 .872095 .89 9.951896 .932150 4.24 10.048104 .047850 10 9 52 53 54 55 .824386 .824527 .824668 .824808 2.35 2.35 2.35 2.35 204 .871981 .871868 .871755 .871641 .89 1.89 1.89 1.89 .952405 .952659 .952913 .953167 4.24 4.24 4.24 4.24 4rtJ .047595 .047341 .047087 .046833 8 7 6 5 56 57 58 J) .824949 .825090 .825230 .825371 .825511 . vr* 2.34 2.34 2.34 2.34 .871528 .871414 .871301 .871187 .871073 1.89 1.89 1.89 1.89 1.90 .953421 .953675 .953929 .954183 .954437 .Ji 4.24 4.23 4.23 4.23 .046579 .046325 .046071 .045817 .045563 4 3 2 M. Cosine. D,l . i Sine. Dl . Cotaner. 1 D.I . Tang. M. "53 48* TABLE IV. LOGARITHMIC SINES, ETC. 42 137^ M. Sine. D.I . Cosmo. D.I . Tang. D.I". Cotung. M. 1 2 3 4 5 9.825511 .825651 .825791 .825931 .826071 .826211 2.34 2.34 2.33 2.33 2.33 9.871073 .870960 .870846 .870732 .870618 .870504 1.90 .90 .90 .90 .90 Ofi 9.954437 .954691 .954945 .955200 .955454 .955707 4.23 4.23 4.23 4.23 4.23 10.045563 .045309 .045055 .044800 .044546 .044293 60 59 58 57 56 55 6 7 8 .826351 .826491 .826631 2 '.33 2.33 .870390 .870276 .870161 . yu .90 .90 Ol .955961 .956215 .956469 4i23 4.23 .044039 .043785 .043531 54 53 52 9 .826770 2 '.33 .870047 .yi .91 .956723 4.23 4.23 043277 51 10 9.826910 200 9.869933 Q1 9.956977 400 10.043023 50 11 12 .827049 .827189 . oZ 2.32 2 32 .869818 .869704 .yi .91 Ol .957231 .957485 Zo 4.23 400 .042769^ .042515 49 48 13 14 15 16 17 18 19 .827328 .827467 .827606 .827745 .827884 .828023 .828162 2^32 2.32 2.32 2.32 2.31 2.31 2.31 .869589 .869474 .869360 .869245 .869130 .869015 .868900 . yi .91 .91 .91 .91 .92 .92 .92 .957739 .957993 .958246 .958500 .958754 .959008 .959262 . Zo 4.23 4.23 4.23 4.23 4.23 4.23 4.23 .042261 .042007 .041754 .041500 .041246 .040992 .040738 47 46 45 44 43 42 41 20 21 22 9,828301 .828439 .828578 2.31 2.31 2O1 9.868785 .868670 .868555 .92 .92 OO 9.959516 .959769 .960023 4.23 4.23 400 10.040484 .040231 .039977 40 39 38 23 .828716 .ol 20.1 .868440 . . ' ' -_ .960277 . Zo .039723 37 24 23 .828855 .828993 .ol 2.31 2 on .868324 .868209 !92 oo .960530 .960784 4.23 4.23 4" OQ .039470 .039216 36 35 26 27 28 .829131 .829269 .829407 .oU 2.30 2.30 OA .868093 .867978 .867862 . yz .93 .93 .961038 .961292 .961545 .2.6 4.23 4.23 4" oo .038962 .038708 .038455 34 33 32 29 .829545 Z*o(J 2.30 .867747 '93 .961799 .2,6 4.23 .038201 31 30 9.829683 2 on 9.867631 9.962052 400 10.037948 30 31 32 .829821 .829959 .oU 2.30 2 on .867515 .867399 !93 .962306 .962560 . Zo 4.23 400 .037694 .037440 29 28 33 34 35 .830097 .830234 .830372 .ZJ 2.29 2.29 200 .867283 .867167 .867051 !93 .93 .962813 .963067 .963320 Zo 4.23 4.23 400 .037187 .036933 .036680 27 26 25 36 37 .830509 .830646 . ZJ 2.29 .866935 .866819 !94 .963574 .963828 t Zo 4.23 .036426 .036172 24 23 38 39 .830784 .830921 2.29 2.29 2.29 .866703 .866586 .94 .94 .94 .964081 .964335 4.23 4.23 4.23 .035919 .035665 22 21 40 9.831058 20Q 9.866470 9.964588 4 OO 10.035412 20 41 42 .831195 .831332 Zo 2.28 .866353 .866237 !94 .964842 .965095 . zz 4.22 .035158 .034905 19 18 43 44 .831469 .831606 2.28 2.28 .866120 .866004 .94 .94 .965349 .965602 4.22 4.22 .034651 .034398 17 16 45 .831742 2.28 .865887 .95 .965855 4.22 .034145 15 46 47 48 .831879 .832015 .832152 2. 28 2.28 2.27 .865770 .865653 .865536 .95 .95 .95 .966109 .966362 .966616 4.22 4.22 4.22 .033891 .033638 .033384 14 13 12 49 .832288 2.27 2.27 .865419 .95 .95 .966869 4.22 4.22 .033131 11 50 9.832425 9.865302 9.967123 10.032877 10 51 52 53 .832561 .832697 .832833 2.27 2.27 2.27 .865185 .865068 .864950 .95 .95 .95 .967376 .967629 .967883 4.22 4.22 4.22 .032624 .032371 .032117 9 8 7 54 55 56 .832969 .833105 .833241 2.27 2.27 2.26 .864833 .864716 .864598 .96 .96 .96 .968136 .968389 .968643 4.22 4.22 4.22 .031864 .031611 .031357 6 5 4 57 58 59 .833377 .833512 .833648 2.26 2.26 2.26 .864481 .864363 .864245 .96 .96 .96 .968896 .969149 .969403 4.22 4.22 4.22 .031104 .030851 .030597 3 2 1 60 .833783 2.26 .864127 .96 .969656 4.22 .030344 M. Cosine. Dr. Sine. D.r . Cotang. D.r . Tang. M. 132 4T 43 TABLE IV. LOGAKITHMIG SINES, ETC. 83 136* M. Sine. D.I". Cosine. D.I". Tang. D. 1". Ootang. M. 9.833783 9.864127 9.969656 4 22 10.030344 60 1 2 3 4 5 6 .833919 .834054 .834189 .834325 .834400 .834595 2^26 2.25 2.25 2.25 2.25 2 OK .864010 .863892 .863774 .863656 .863538 .863419 1^97 .97 .97 .97 .97 .969909 .970162 .970416 .970669 .970922 .9711-75 4^22 4.22 4.22 4.22 4.22 A .,. .030091 .029838 .029584 .029331 .029078 .028825 59 58 57 66 55 54 7 8 9 .834730 8348G5 .834999 . JO 2.25 2.25 2.25 .863301 .863183 .863064 .97 .97 .97 .971429 .971682 .971935 4i22 4.22 4.22 .028571 .028318 .028065 53 52 51 ib 11 3.835134 .835269 2.24 9.862946 .862827 .98 9.972188 .972441 4.22 10.027812 .027559 60 49 12 .835403 2.24 .862709 .98 .972694 4.22 .027306 48 13 .835538 2.24 .862590 .98 flQ .972948 4.22 400 .027052 47 14 .835672 2.24 .862471 .98 .973201 lot .026799 46 15 .835807 2.24 .862353 .98 .973454 4.22 .026546 45 16 .835941 2.24 .862234 .98 .973707 4.22 400 .026293 44 17 .836075 2.24 .862115 .98 .973960 .at .02C040 43 18 .830209 2.23 .861996 .98 .974213 4.22 .025787 42 19 .836343 2.23 2.23 .861877 .98 .99 974466 4.22 4.22 .025534 41 20 9.836477 9.861758 9.974720 10.025280 40 21 .836611 2.23 .861638 .99 .974973 4.22 .025027 39 22 .836745 2.23 .861519 .99 .975226 4.22 .024774 S3 23 .836878 2.23 .861400 .99 .975479 4.22 .024521 37 24 .837012 2.23 .861280 .99 .975732 4.22 .02-1268 S6 25 .837146 2.23 .861161 .99 .975985 4.22 .024015 35 26 .837279 2.22 .861041 .99 .976238 4.22 .02S762 34 27 28 29 .837412 .837546 .837679 2.22 2.22 2.22 2.22 .860922 .860802 .860682 .99 2.00 2.00 2.00 .976491 .976744 .976997 4.22 4.22 4.22 4.22 .022509 .02C256 .02C003 33 C2 SI 30 31 32 33 34 35 36 37 38 39 9.837812 .837945 .838078 .838211 .838344 .838477 .838610 .838742 .838875 .839007 2.22 2.22 2.22 2.21 2.21 2.21 2.21 ' 2.21 2.21 2.21 9.860562 .860442 .860322 .860202 .860082 .859962 .859842 .859721 .859601 .869480 2.00 2.00 2.00 2.00 2.00 2.00 2.01 2.01 2.01 2.01 "> "77250 .^7503 .977756 .978009 .978262 .978515 .978768 .979021 .979274 .979527 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 4.22 10.022750 .022497 .022244 .021991 .021738 .021485 .021232 .020979 .020726 .020473 30 29 28 27 26 25 24 23 22 21 40 9.839140 A *%1 9.859360 9.979780 10.020220 20 41 42 43 44 45 46 47 43 49 .839272 .839404 .839536 .839668 .839800 .839932 .840064 .846196 .840328 2.21 2.20 2.20 2.20 2.20 2.20 2.20 2.20 2.19 2.19 .859239 .859119 .858998 .858877 .858756 .858635 .858514 .858393 .858272 2.01 2.01 2.01 2.01 2.02 2.02 2.02 2.02 2.02 2.02 .980033 .980286 .980538 .980791 .981044 .981297 .981550 .981803 .982056 4.22 4.22 4.22 4.22 4.22 4.21 4.21 4.21 4 21 4.21 .019967 .019714 .019462 .019209 .018956 .018703 .018450 .018197 .017944 19 18 17 16 15 14 13 12 11 50 51 62 63 54 55 56 67 58 59 CO 9.840459 .840591 .840722 .840854 .840985 .841116 .841247 .841378 .841509 .841640 .841771 2.19 2.19 2.19 2.19 2.19 2.19 2.18 2.18 2.18 2.18 9.858151 .858029 .857908 .857786 .857665 .857543 .857422 .857300 .857178 .857056 .856934 2.02 2.02 2.02 2.03 2.03 2.03 2.03 2.03 2.03 2.03 9.982309 .982562 .982814 .983067 .983320 .983573 .983826 984079 .984331 .984584 .984837 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 10.017691 .017438 .017186 .016933 .016680 .016427 .016174 .015921 .015669 .015416 .015163 10 9 7 6 5 4 3 2 1 M. Cosine. D.I . Sine. D.I" rotanc. D. 1 . Tane. M. 46 84 TABLE IV. LOGARITHMIC SINES, ETC. 44 135 M. Sine. D.I". Cosine. D.I". Tang. D.I". Cotang. M. 1 2 3 4 9.841771 .841902 .842033 .842163 .842294 2.18 2.18 2.18 2.18 21 7 9.856934 .856812 .856690 .856568 .856446 2.03 2.04 2.04 2.04 2ni 9.984837 .985090 .985343 .985596 .985848 4.21 4.21 4.21 4.21 10.015163 .014910 .014657 .014404 .014152 60 59 58 57 56 5 .842424 . 14 217 .856323 ,\r .986101 4O1 .013899 55 6 .842555 11 .856201 O i .986354 . Zl. 4 O1 .013646 54 7 .842685 2. 17 21 7 .856078 2f\A .986607 ,zi 401 .013393 53 8 9 .842815 .842946 . 14 2.17 2.17 .855956 .855833 . U-i 2.04 2.04 .986860 .987112 . Zl 4.21 4.21 .013140 .012888 52 51 10 11 9.843076 .843206 2.17 O 17 9.855711 .855588 2.05 9.987365 .987618 4.21 4 O1 10.012635 .012382 50 49 12 13 14 15 16 17 .843336 .843466 .843595 .843725 .843855 .843984 Z. 14 2.16 2.16 2.16 2.16 2.16 21 S .855465 .855342 .855219 .855096 .854973 .854850 2^05 2.05 2.05 2.05 2.05 .987871 .988123 .988376 .988629 .988882 .989134 . ZL 4.21 4.21 4.21 4.21 4.21 4 O1 .012129 .011877 .011624 .011371 .011118 .010866 48 47 46 45 44 43 18 19 .844114 .844243 lb 2.16 2.16 .854727 .854603 2.05 2.06 2.06 .989387 .989640 Zi 4.21 4.21 .010613 .010360 42 41 20 21 22 9.844372 .844502 .844631 2-15 2.15 9.854480 .854356 .854233 2.06 2.06 9.989893 .990145 .990398 4.21 4.21 4O1 10.010107 .009855 .009602 40 39 38 23 24 25 26 27 28 29 .844760 .844889 .845018 .845147 .845276 .845405 .845533 2. 15 2.15 2.15 2.15 2.15 - 2.15 2.14 2.14 .854109 .853986 .853862 .853738 .853614 .853490 .853366 2.06 2.06 2.06 2.06 2.06 2.07 2.07 2.07 .990651 .990903 .991156 .991409 .991662 .991914 .992167 .ZL. 4.21 4.21 4.21 4.21 4.21 4.21 4.21 .009349 .009097 .008844 .008591 .008338 .008086 .007(333 37 35 34 33 32 31 30 31 32 33 34 35 36 9.845662 .845790 .845919 .846047 .846175 .846304 .846432 2.14 2.14 2.14 2.14 2.14 2.14 210 9.853242 .853118 .852994 .852869 .852745 .852620 .852496 2.07 2.07 2.07 2.07 2.07 2.08 2 no 9.992420 .992672 .992925 .993178 .993430 .993683 .993936 4.21 4.21 4.21 4.21 4.21 4.21 4 O1 10.007580 .007328 .007075 .006822 .006570 .006317 .006064 30 29 28 27 26 25 24 37 38 39 .846560 .846688 .846816 .lo 2.13 2.13 2.13 .852371 .852247 .852122 .Uo 2.08 2.08 2.08 .994189 .994441 .994694 ./I 4.21 421 4.21 .005811 .005559 .005306 23 22 21 40 9.846944 210 9.851997 2f\Q 9.994947 4 O1 10.005053 20 41 .847071 . lo 2 10 .851872 . Uo 2f\Q .995199 /I 4 O1 .004801 19 42 43 44 45 46 47 48 49 .847199 .847327 .847454 .847582 .847709 .847836 .847964 .848091 .lo 2.13 2.13 2.12 2.12 2.12 2.12 2.12 2.12 .851747 .851622 .851497 .851372 .851246 .851121 .850996 .850870 .Us 2.08 2.09 2.09 2.09 2.09 2.09 2.09 2.09 .995452 .995705 .995957 .996210 .996463 .996715 .996968 .997221 .Zl 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 .004548 .004295 .004043 .003790 .003537 .003285 .003032 .002779 18 17 16 15 14 13 12 11 50 51 62 63 54 55 56 57 58 59 9.848218 .848345 .848472 .848599 .848726 .848852 .848979 .849106 .849232 .849359 2.12 2.12 2.11 2.11 2.11 2.11 2.11 2.11 2.11 9.850745 .850619 .850493 .850368 .850242 .850116 .849990 .849864 .849738 .849611 2.09 2.10 2.10 2.10 2.10 2.10 2.10 2.10 2.10 9.997473 .997726 .997979 .998231 .998484 .998737 .998989 .999242 .999495 .999748 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 4.21 10.002527 .002274 .002021 .001769 .001516 .001263 .001011 .000758 .000505 .000252 10 9 8 7 6 5 4 3 2 1 60 .849485 2.11 .849485 2.11 10.000000 4.21 .000000 M. Cosine. D.I". Sine. D.I". Cotang. D.I". Tang. M. 45 TABLE V. LATITUDES AND DEPARTURES, OR TRAVERSE TABLE. 86 TABLE V. TBAVERSE TABLE* B'ng Di*t. 1. Dist. 2. Dist. 3. Dist. 4. Dist. 5. B'ng , Lat. Dep. Lat. Dep. Lat. Dep. Lat | Dep. Lat. Dep. / 15 1.0000 0.0044 2.0000 0.0087 3.0000 0.01314.00000.0175 5.0000 0.0218 89 45 30 0000 0087 1.9999 0175 2.9999 0262 3.9998 0349-4.9998 0436 30 45 0.9999 0131 9998 0262 9997 0393 9997 0524 9996 0654 15 1 9998 0175 9997 0349 9995 0524 9994 0698 9992 0873 89 15 9998 0218 9995 0436 9993 0654 9990 0873 9988 1091 45 30 9997 0262 9993 0524 9990 0785 9986 1047 9983 1309 30 45 9995 0305 9991 0611 9986 0916 9981 1222 9977 1527 15 2 9994 0349 9988 0698 9982 1047 9976 1396 9970 1745 88 15 9992 0393 9985 0785 9977 1178 9969 1570 9961 1963 45 30 9990 0436 9981 0872 9971 1309 9962 1745 9952 2181 30 45 0.9988 0.0480 1.9977 0.0960 2.9965 0.1439 3.9954 0.1919 4.9942 0.2399 15 3 9986 0523 9973 1047 9959 1570 9945 2093 9931 2617 87 15 9984 0567 9968 1134 9952 1701 993f 2268 9920 2835 45 30 9981 0610 9963 1221 9944 1831 9925 2442 9907 3052 30 45 9979 0654 9957 1308 9936 1962 9914 2616 9893 3270 15 4 9976 0698 9951 1395 9927 2093 9903 2790 9878 3488 86 15 9973 0741 9945 1482 9918 2223 9890 2964 9863 3705 45 30 9969 0785 9938 1569 9908 2354 9877 3138 9846 3923 30 45 9966 0828 9931 1656 9897 2484 9863 3312 9828 4140 15 5 9962 0872 9924 1743 9886 2615 9848 3486 9810 4358 85 15 0.9958 0.0915 1.9916 0.1830 2.9874 0.2745 3.9832 0.3660 4.9790 0.4575 45 30 9954 0958 9908 IP 1 .? 9862 2875 9816 3834 9770 4792 30 45 9950 1002 9899 20 '* 9849 3006 9799 4008 9748 5009 15 6 9945 1045 9890 2091 9836 3136 9781 4181 9726 5226 84 15 9941 1089 9881 2177 9822 3266 9762 4355 9703 5443 45 30 9936 1132 9871 2264 9807 3396 9743 4528 9679 5660 30 45 9931 1175 9861 2351 9792 3526 9723 4701 9653 5877 15 7 9925 1219 9851 2437 9776 3656 9702 4875 9627 6093 83 15 9920 1262 9840 2524 9760 3786 9680 5048 9600 6310 45 30 9914 1305 9829 2611 9743 3916 9658 5221 9572 6526 30 45 0.9909 0.1349 1.9817 0.2697 2.9726 0.4046 3.9635 0.5394 4.9543 0.6743 15 8 9903 1392 9805 2783 9708 4175 9611 5567 9513 6959 82 15 9897 1435 9793 2870 9690 4305 9586 5740 9483 7175 45 30 9890 1478 9780 2956 9670 4434 9561 5912 9451 7390 30 45 9884 1521 9767 3042 9651 4564 9534 6085 9418 7606 15 9 9877 1564 9754 3129 9631 4693 9508 6257 9384 7822 81 15 9870 1607 9740 3215 9610 4822 9480 6430 9350 8037 45 30 9863 1650 9726 3301 9589 4951 9451 6602 9314 8252 30 45 9856 1693 9711 3387 9567 5080 9422 6774 9278 8467 15 10 9848 1736 9C96 3473 9544 5209 9392 6946 9240 8682 80 15 0.9840 0.1779 1.9681 0.3559 2.9521 0.5338 3.9362 0.7118 4.9202 0.8897 45 30 9833 1822 9665 3645 9498 5467 9330 7289 9163 9112 30 45 9825 1865 9649 3730 9474 5596 9298 7461 9123 9326 15 11 9816 1908 9633 3816 9449 5724 9265 7632 9081 9540 79 15 9808 1951 9616 3902 9424 5853 9231 7804 9039 9755 45 30 9799 1994 9598 3987 9398 5981 9197 7975 8996 9968 30 45 9790 2036 9581 4073 9371 6109 9162 8146 8952 1.0182 15 12 9781 2079 9563 4158 9344 6237 9126 8316 8907 0396 78 15 9772 2122 9545 4244 9317 6365 9089 8487 8862 0609 45 30 9763 2164 9526 4329 9289 6493 9052 8658 8815 0822 30 45 0.9753 0.2207 1.9507 0.4414 2.9260 0.6621 3.9014 0.8828 4.8767 .1035 15 13 9744 2250 9487 4499 9231 6749 8975 8998 8719 1248 77 15 9734 2292 9468 4584 9201 6876 8935 9168 8669 1460 45 30 9724 2334 9447 4669 9171 7003 8895 9338 8618 1672 30 45 9713 2377 9427 4754 9140 7131 8854 9507 8567 1884 15 14 9703 2419 9406 4838 9109 7258 8812 9677 8515 2096 76 15 9692 2462 9385 4923 9077 7385 8769 9846 8462 2308 45 30 9681 2504 9363 5008 9044 7511 8726 1.0015 8407 2519 30 45 9670 2546 9341 5092 9011 7638 8682 0184 8352 2730 15 15 9659 2588 9319 5176 8978 7765 8637 0353 8296 2941 75 ' Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. ' B'ng Dlst. 1. Dist. 2. Dist. 3. Dlst. 4. Dlst. 5. B'ng TABLE V. TKAVERSE TABLE. Liijt. 6. i>ist. 7. iMst. 8. : Dist. 9. Dist. 1O. B'ag , . Lat. Dep. Lat. Dep, Lat. Dep. Lat. J Dep. Lat. Dep. , , 15 5. 9999 0. 0262 6. 99990. 0305*7. 9999 0.0349 8. 9999 0.0393 ; 9. 9999 0.0436 89 45 30 9998 ' 0524 9997 0611 9997) 0698! 9997 0785 9996 0873 30 45 9903 0785 9994 0916 9993 1047 9992 1178 9991 1309 15 1 9991 1047 9989 1222 9988 ! 1396 9986 1571 9985 174589 15 9986 1309 9983 1527 9981 1745 9979 1963 9976 2181 j 45 30 9979 1571 9976 1832 9973 2094 9969 2356 9966 26181 30 45l 9972 1832 9967 '2138 9963 2443 9958 2748 9953 3054! 15 2 9963 2094 9957 2443 9951 2792 99451 3141 9939 3490 88 15 9l54 2.r,; 9946| 2748) 9938' 3141 1 99311 3533 9923 3926 45 30 9943 2617 9933 3053 9924 3490) 9914 3926 9905 4362 30 455.99310.28796.9919 0.3358 7.9908 0.3838 8.9896 0.43189.9885.0.4798 15 3 9918 3140 9904 3664 9890 4187 9877 4710J 9863 5234 87 15 9904 3402 9887 3968 9871 4535 9855 5102 98391 5669 45 30 9888 3663 9869 4273 9851 4884 9832 5494 9813 6105 30 45 9872 3924 9850 4578 9829 5232 9807 5886 9786 6540 15 4 o! 9854 4185 9829 4883 9805 5581 9781 6278 9756 697686 15 9835 4447 9808 5188 9780 5929 9753 6670 9725 7411 45 30; 9815 4708 9784 5492 9753 6277 9723 7061 9692 7846 30 45 9794' 4968 9760 5797 9725 6625 9691 7463 9657 8281 15 5 9772 5229 9734 6101 96S6 6972 9658 7844 9619 8716 85 15 5.97480.5490 6.97060.6405 7.9664 0.73208.96220 8235 9.9580;0.9150 45 30 9724 5751 907* 6709 9632 7668 9586' 8626 9540 9585 30 45 969* 6011 9648; 70131 9597 8015 9547 9017 9497:1.0019 15 6 9671 9617 7317 9562 8362 9507 9408 9452 0453 84 15 9643 6532 9584i 7621 9525 8709 9465 9798 9406 0887 45 30 9614 6792 9550: 79241 9486 9056 9421 1.0188 9357 1320 30 45 9584 7052 ! 9515! 8228 9445 9403 9376 0578 j 9307 17.-4 15 7 95,'3 7312! 9478| 8531 9404 9750 93291 0968 9255 218783 15 9520 7572 9440! 8834 93601.0096 92801 1358 9200 2620 45 30 9487 7832 9401 9137 9316! 0442 9230 1747 9144 3053 30 455.9452 0.8091 6.93(31 0.9440 7.92691.0788 8.91781.21379.90871.3485 15 8 941H 83501 9319! 9742 9221 1134' 9124 2526! 9027 391782 15 9379 8610 92761.0044 9172 1479 90C9 2914 8965 4349 45 30 9341 88691 9231 0347 9121 1825 9011 3303 89021 4781 30 45 9302 9127 9185 0649 9069; 2170 8953 3691 8836 5212 15 9 9261 9386 9138 0950 9015! 2515 8892 4079 8769 564381 15 9220 9645 9090 1252 8960 2859 8830 4467 8700 6074 45 30 9177 9903 9040 1553 8903 3204 8766 4854 8629 6505 30 45 9133 1.0161 1 8989 1854 8844 3548 8700 5241 8556 6935 15 10 9088 0419J 8937 2155 8785 3892 8633 5628 8481 736580 155.9042 1. 0677 i 6. 8883 1.2456 7.8723 1.42358.8564 1.60159.8404 1.7794! 45 30 8995 0934 8828 2756 8660 4579 8493 6401 8325 8224 30 45 8947 1191 8772 3057 .8596 4922 8421 6787 8245 8652 15 11 8898 1449 8714 3357 85301 5265 8346 7173 8163 9081 79 15 8847 1705 8655 3656 8463! 5607 8271 7558 8079 9509 45 30 8795 1962 8595 3956 8394; 5949 8193 7943 7992 9937 30 45 8743 2219 8533 42-55 8324 6291 8114 8328 7905 2.0364 15 12 8689 2475 8470 4554 6633 8033 8712 7815' 0791 78 15 8634 2731 8406 4852 81781 6974, 7951 9096 7723 1218 45 30, 8578 2986 8341 5151 8104 7315 78G7 9480 7630 1644 30 455.8521 1.32426.8274 1 5449 7.8027 1.76568.7781 1.98639.7534 2.2070 15 13 8462 i 3497; 8206 5747 7950 7996 7693 2.0246 7437 249577 15 8403; 3752! 8137 6044 7870 8336 7604 0628 7338 2920 45 30 8342 4007 8066 6341 7790 8676 7513 1010 7237 3345 30 45 8281 4261 7994 6638 7707 9015 7421 1392 7134 3769 15 14 8218 4515 79211 6935 7624 93.34 7327 1773 7030 4192 76 15 8154 4769 7846 7231 7538 9692 7231 2154 6923 4615 45 30 8089 5023 7770 7527 7452 2.0030 7133 2534 6815 5038 30 45 8023 5276 7693 7822 7364 0368 7034 2914 6705 5460 15 15 7956 5529 7615 8117 7274 0706 6933 3294 6593 5882 75 ' ' Dep. Lat. Dep. Lat. Dep. Lat. Dep. j Lat. Dep. Lat. . . B'n gi Dist. 6. Dist. 7. Dist. 8. Dist. 9. Dist. 1O. ;B'ng 88 TABLE V. TRAVERSE TABLE. B'ng I>ist. 1. Dist. 3. Dist. 3. Iis Lat. t.4. IMst. 5. B'ng . . Lat. Dep. Lat. Dep. Lat. Dep. Dep. Lat. Dep. . 15 15 0.9648 0.2630 1.9296 0.5261 2.8944 0.7891 3.8591 1 .0521 4.8239 1.3152 7445 30 9636 2672 9273 5345 8909 8017 8545 0690 8182 3362 30 45 9625 2714 9249 5429 8874 8143 8498 0858 8123 3572 15 16 9613 2756 9225 5513 8838 8269 8450 1025 8063 3782 74 15 9600 2798 9201 5597 8801 8395 8402 1193 8002 3991 45 30 9588 2840 9176 5680 8765 8520 8353 1361 7941 4201 30 45 9576 2882 9151 5764 8727 8646 8303 1528 7879 4410 15 17 9563 2924 9126 5847 8689 8771 8252 1695 7815 4619 73 15 9550 2965 9100 5931 8651 8896 8201 1862 7751 4827 45 30 9537 3007 9074 6014 8612 9021 8149 2028 7686 5035 30 45 0.9524 0.3049 .9018 0.6097 2.8572 0.9146 3.8096 1.2195 4.7620 .5243 15 18 9511 3090 9021 6180 8532 9271 8042 2361 7553 5451 72 15 9497 3132 8994 6263 8491 9395 7988 2527 7485 5658 45 30 9483 3173 8966 6346 8450 9519 7933 2692 7416 5865 30 45 9469 3214 8939 6429 8408 9643 7877 2858 7347 6072 15 19 9455 3256 8910 6511 8366 9767 7821 3023 7276 6278 71 15 9441 3297 8882 6594 8323 9891 7764 3188 7204 6485 45 30 9426 3338 8,853 6676 8279 .0014 7706 3352 7132 6690 30 45 9412 3379 8824 6758 8235 0138 7647 3517 7059 6896 15 20 9397 3420 8794 6840 8191 0261 7588 3681 6985 7101 70 15 0.9382 0.3461 .8764 0.6922 2.8146 .0384 3.7528 .3845 4.6910 1.7306 45 30 9367 3502 8733 7004 8100 0506 7467 4008 6834 7510 30 45 9351 3543 8703 7086 8054 0629 7405 4172 6757 7715 15 21 9336 3584 8672 7167 8007 0751 7343 4335 6679 7918 69 15 9320 3624 8640 7249 7960 0873 7280 4498 6600 8122 45 30 9304 3665 8608 7330 7913 0995 7217 4660 6521 8325 30 45 9288 3706 8576 7411 7864 1117 7152 4822 6440 8528 15 22 9272 3746 8544 7492 7816 1238 7087 4984 6359 8730 68 15 9255 3786 8511 7573 7766 1359 7022 5146 6277 8932 45 30 9239 3827 8478 7654 7716 1481 6955 5307 6194 9134 30 45 0.9222 0.3867 .8444 0.7734 2.7666 .1601 3.6888 1.5468 4.6110 1.9336 15 23 9205 3907 8410 7815 7615 1722 6820 5629 6025 9537 67 15 9188 3947 8376 7895 7564 1842 6752 5790 5940 9737 45 30 9171 3987 8341 7975 7512 1962 6682 5950 5853 9937 30 45 9153 4027 8306 8055 7459 2082 6612 6110 5760 ^.0137 15 24 9135 4067 8271 8135 7406 2202 6542 6269 5677 0337 66 15 9118 4107 8235 8214 7353 2322 6470 6429 5588 0536 45 30 9100 4147 8199 8294 7299 2441 6398 6588 5498 0735 30 45 9081 4187 8163 8373 7244 2560 6326 6746 5407 0933 15 25 9063 4226 8126 8452 7189 2679 6252 6905 5315 1131 65 15 0.9045 0.4266 1.8089 0.8531 2.7134 1.2797 3.6178 1.7063 4.5223 2.1328 45 30 9026 4305 8052 8610 7078 2915 6103 7220 5129 1526 30 45 9007 4344 8014 8689 7021 3033 6028 7378 5035 1722 15 26 8988 4384 7976 8767 6964 3151 5952 7535 4940 1919 64 15 8969 4423 7937 8846 6906 3269 5875 7692 4844 2114 45 30 8949 4462 7899 8924 6848 3386 5797 7848 4747 2310 ' 30 45 8930 4501 7860 9002 6789 3503 5719 8004 4649 2505 15 27 8910 4540 7820 9080 6730 3620 5640 8160 4550 2700 63 15 8890 4579 77SO 9157 6671 3736 5561 8315 4451 2894 45 30 8870 4617 7740 9235 6610 3852 5480 8470 4351 3087 30 45 0.8850 0.4656 1.7700 0.9312 2.6550 1.3968 3.5400 1.8625 4.4249 2.3281 15 28 8829 4695 7659 9389 6488 4084 5318 8779 4147 3474 62 15 8809 4733 7618 9466 6427 4200 5236 8933 4045 3666 45 30 8788 4772 7576 9543 6365 4315 5153 9086 3941 3858 30 45 8767 4810 7535 9620 6302 4430 5069 9240 3836 4049 15 29 8746 4848 7492 9696 6239 4544 4985 9392 3731 4240 61 15 8725 4886 7450 9772 6175 4659 4900 9545 3625 4431 45 30 8704 4924 7407 9848 6111 4773 4814 9697 3518 4621 30 45 8682 4962 7364 9924 6046 4886 4728 9849 3410 4811 15 30 8660 5000 7321 1.0000 5981 5000 4641 2.0000 3301 5000 60 , Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. . B'ng IMst. 1. JMst. 2. Dist. 3. 1 IHst. 4. IMst. 5. B'ng TABLE V. TRAVERSE TABLE. 89 B"'ng Dist. 6. lust. 7. l>ist. 8. uit . 9. DiSt. 10. B'ng . Lafe | Dep. Lat. Dep. Lat. JDep. Lat. Dep, Lat. Dep. . . 15 15 5.78871.57826.75351.84127.71832.10428.68312.3673 9.6479 2.6303 7445 30 7818 6034 7454 ! 8707 7090 1379 6727 4051 6363 6724 30 45 7747 62861 7372, 9001i 6996i 1715 6621 4430 6246 7144 15 16 7676 6538 7288 9295 6901, 2051 6514 4807 6126 i 7564 74 15 7603 6790 7203 1 9588; 6804 2386 6404 5185 6005 7983 45 30 7529 7041 7117 9881 1 6706 2721 6294 5561 5882 8402 30 45 7454 7292 7030 2.0174 6606 3056 6181 5938 5757 8820 15 17 7378 7542 6941; 0466 6504! 3390 6067 63131 5630 9237 73 15 7301 7792 6851: 0758; 6402 | 3723 5952 6689 5502J 9654 45 30 7223 8042 6760 1049 6297 4056 5835 7064 5372'3.0071 30 45 5.7144'l.8292 6.66682.1341 7.61922.43898.5716 2.7438 9. 5240 '3.0486 15 18 7063 8541 6574 1631 60851 4721 5595 7812 5106 0902 72 15 6982 8790 6479 1921 5976 5053 5473 8185 4970J 1316 45 30 6899 9038 6383 2211 5866 5384 5349 8557 4832 1730 30 45 6816 9286 6285 2501 5754 5715 ! 5224 8930 4693 2144 15 19 6731 9534 6186 2790 5641 i 6045J 5097 9301 4552 2557 71 15 6645 9781 6086 3078 5527 6375' 4968 9672 4409 2969 45 30 6558 2.0028 59851 3366 5411, 1 6705 4838 3.0043 4264 3381 30 45 6471 0275 "5882 3654 5294 7033 4706 0413 4118 3792 15 20 6382 0521 5778 3941 5175 7362i 4572 0782 3969 4202 70 15 5.6291 2.0767 6.5673 2.4228 7.5055 2.7689 8.4437 3.1151 9.3819 3.4612 45 30 6200 1012 55671 4515 ! 4934 8017 4300 1519 3667 5021 30 45 6108 1257 5459J 4800 4811 8343 4162 1886 3514 5429 15 21 6015 1502 5351! 5086 4686 8669 4022 22531 3358 5837 69 15 5920 1746 5241 5371! 4561 8995 3881 2619 3201 6244 45 30 5825 1990 5129 5655 4433 9320 3738 2985 3042 6650 30 45 5729 2233 5017 5939; 4305! 9645 3593 3350 2881 7056 15 22 5631 2476 4903 6222 4175 9969 3447 3715 2718 7461 68 15 5532 2719 4788 - 6505 4043 3.0292 3299 4078 2554 7865 45 30 5433 2961 46721 6788; 3910j 0615 3149 4442 2388 8268 30 45 5.5332 2.3203 6.4554 2.7070 7.3776 3.0937 8.29983.4804 9.2220 3.8671 15 23 5230 3444 4435 7351 3640 1258 2845 5166 2050 9073 67 15 5127 3685 4315 76321 3503 1580 2691 5527 1879 9474 46 30 5024 3925 4194 7912 3365 1900 2535J 5887 1706 9875 30 45 4919 4165 4072 8192 3225 2220 23781 6247 1531 4.0275 15 24 4813 4404 3948 8472 3084 2539 2219 6606 1355 0674 66 15 4706 4643 3823 8750 2941 2858 2059 6965 1176 1072 45 30 4598 4882 3697 9029 2797 3175 1897 7322 0996 1469 30 45 4489 5120 3570 9306 2651 3493 1733 7679 0814 1866 15 25 4378 5357 3442 9583 2505 3809 1568 8036 0631 2262 65 155.4267 2.5594 6.3312 2.9860 7.23563.41258.1401 3.8391 9.0446 4.2S57 45 30 4155 5831 3181 3.0136 2207 4441 1233 8746 0259 3051 30 45 4042 6067 3049 0411 2056 4756! 1063 9100 0070 3445 15 26 3928 6302 2916 0686 1904 5070 0891 9453 8.9879 3837 64 15 30 3812 3696 6537 6772 2781 2645 0960 1234 17501 5383 1595 5696 0719 9806 05444.0158 9687 9493 4229 4620 45 30 45 3579 7006 2509 15071 1438 6008 0368 0509 9298 5010 15 27 3460 7239 2370 17791 1281 6319 0191 0859 9101 5399 63 15 3341 7472 2231 2051 1121 6630 0012 1209 8902 5787 45 30 3221 7705 2091 2322 0961 6940 7.9831 1557 8701 6175 30 45 5.3099 2.7937 6.1949 3.2593 7.0799 3. 7249 17.9649 4.1905 8.8499 4.6561 15 28 2977 8168 18061 2863 0636 7558 9465 2252 8295 6947 62 15 2853 8399 1662! 3132 0471! 786C 9280; 2599 8089 7332 45 30 2729 8630 1517 3401 0305 8173 9094 2944 7882 7716 30 45 2604 8859 1371 3669 01381 8479 8905 3289 7673 8099 15 29 2477 9089 1223 39376.9970 8785 8716 3633 7462 8481 61 15 2350 9317 1075 4203 9800 9090 8525 3976 7250 8862 45 30 2221 9545 0925 4470 9628; 9394 8332 4318 7036 9242 30 45 2092 9773 0774 4735 9456: 9697 8138 4659 6820 9622 15 30 1962 3.0000 0622 5000 92824.0000 7942 5000 6603 5.0000 60 ' . Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat i B'ng Dist. 6. Blfct. 7. Dial. 8. i lHt.9. J>iMt.lO. B'ng G 90 TABLE V. TEA VERSE TABLE. B'ng J>ist. 1. | J*ist. 2. l>ist. 3. J>i*i. 4. I>ist. 5. B'ng . , Lat. Dep Lat. Dep Lat Dep. Lat. Dep Lat. Dep. , 30 15 0.8638 0.503* 1.727 1.007 2.591 1.511 3.455 2.015 4.3192 2.518S 5945 30 8616 507 7233 015 584 522 446 0302 308 5377 30 45 8594 5112 7188 022 578 533 437 045 2970 5565 15 31 8572 51 5C 7142 030 571 545 428 0602 2858 5752 59 15 8549 5188 7098 037 564 5563 419 075 2746 5939 45 30 8526 5225 7053 0450 557 5675 4106 0900 2632 6125 30 45 8504 5262 7007 0524 551 5786 4014 1049 2518 6311 15 32 8480 5299 6961 0598 544 5898 3922 1197 2402 6496 58 15 8457 5336 6915 0672 5372 6003 3829 1345 2286 668 45 30 8434 5373 6868 0746 5302 6119 3736 1492 2170 6865 30 45 0.8410 0.5410 1.6821 1.0819 2.5231 1.6229 3.3642 2.1639 4.2052 2.7049 15 33 8387 5446 6773 0893 5160 6339 3547 1786 1934 7232 57 15 8363 5483 6726 0966 5089 6449 3451 1932 1814 7415 45 30 8339 5519 6678 1039 5017 6558 3355 2077 1694 7597 30 45 8315 5556 6629 1111 4944 6667 3259 2223 1573 7779 15 34 8290 5592 6581 1184 4871 6776 3162 2368 1452 7960 56 15 8266 5628 6532 1256 4798 6884 3064 2512 1329 8140 45 30 8241 5664 6483 1328 4724 6992 2965 2656 1206 8320 30 45 8216 5700 6433 1400 4649 7100 2866 2800 1082 8500 15 35 8192 5736 6383 1472 4575 7207 2766 2943 0958 8679 55 15 0.8166 0.5771 1.6333 1.1543 2.4499 .7314 3.2666 2.3086 4.0832 2.8857 45 30 8141 5807 6282 1614 4423 7421 2565 3228 0706 9035 30 45 8116 5842 6231 1685 4347 7527 2463 3370 0579 9212 15 36 8090 5878 6180 1756 4271 7634 2361 3511 0451 9389 54 15 8064 5913 6129 1826 4193 7739 2258 3652 0322 9565 45 30 8039 5948 6077 1896 4116 7845 2154 3793 0193 9741 30 45 8013 5983 6025 1966 4038 7950 2050 3933 0063 9916 15 37 7986 6018 5973 2036 3959 8054 1945 4073 3.9932 3.0091 53 15 7960 6053 5920 2106 3880 8159 1840 4212 9800 0265 45 30 7934 6088 5867 2175 3801 8263 1734 4350 9668 0438 30 45 0.7907 0.6122 1.5814 1.2244 2.3721 .8367 3.1628 2.4489 3.9534 3.0611 15 38 7880 6157 5760 2313 3640 8470 1520 4626 9400 0783 52 15 7853 6191 5706 2382 3560 8573 1413 4764 9266 0955 45 30 7826 6225 5652 2450 3478 8675 1304 4901 9130 1126 30 45 7799 6259 5598 2518 3397 8778 1195 5037 8994 1296 15 39 7771 6293 5548 2586 3314 8880 1086 5173 8857 1466 51 15 7744 6327 5488 2654 3232 8981 0976 5308 8720 1635 45 30 7716 6361 5432 2722 3149 9082 0865 5443 8581 1804 30 45 7688 6394 5377 2789 3065 9183 0754 5578 8442 1972 15 40 7660 6428 5321 2856 2981 9284 0642 5712 8302 2139 50 15 0.7632 0.6461 1.5265 1.2922 2.2897 .9384 .0529 2.5845 3.8162 3.2306 45 30 7604 6494 5208 2989 2812 9483 0416 5978 8020 2472 30 45 7576 6528 5151 3055 2727 9583 0303 6110 7878 2638 15 41 7547 6561 5094 3121 2641 9682 0188 6242 7735 2803 49 15 7518 6593 6037 3187 2555 9780 0074 6374 7592 2967 45 30 7490 6626 4979 3252 2469 9879 .9958 6505 7448 3131 30 45 7461 6659 4921 3318 2382 9976 9842 6635 7303 3294 15 42 7431 6691 4863 3383 2294 .0074 9726 6765 7157 3457 48 15 7402 6724 4804 3447 2207 0171 9609 6895 7011 3618 45 30 7373 6756 4746 3512 2118 0268 9491 7024 6864 3780 30 45 0.7343 0.6788 1.4686 .3576 .2030 .0364 .9373 2.7152 .6716 3.3940 15 43 7314 6820 4627 3640 1941 0460 9254 7280 6568 4100 47 15 7284 6852 4567 3704 1851 0555 9135 7407 6419 4259 45 30 7254 6884 4507 3767 1761 0651 9015 7534 6269 4418 30 45 7224 6915 4447 3830 1671 0745 8895 7661 6118 4576 16 44 7193 6947 4387 3893 1580 0840 8774 7786 5967 4733 46 15 7163 6978 4326 3956 1489 0934 8652 7912 5815 4890 45 30 7133 7009 4265 4018 1398 1027 8530 8036 5663 5045 30 45 7102 7040 4204 4080 1306 1120 8407 8161 5509 5201 15 45 7071 7071 4142 4142 1213 1213 8284 8284 5355 5355 45 e < Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. * B'ng JMst. 1. IMs t . 2. Dist. 3. Dist. 4. Dist. 5. Bn TABLE V. TRAVERSE TABLE. 91 ,'B'ng Dist. 6. Jisi 7. Dist . 8. Dist. 9. Dist. 1O. B'ng . . Lat Dep. Lat. Pep. Lat. Dep. Lat Dep. Lat". Dep. . . 30 15 5.1830 3.0226 6.0468 3.5264 6.9107 4.0302 7.7745 4.5340 8.6384 5.0377 5945 30 1698 0452 0314 5528 8930 0603 7547 5678 6163 0754 30 45 1564 0678 0158 5791 8753 0903 7347 6016 5941 1129 15 31 1430 0902 0002 6053 b573 1203 7145 6353 5717 1504 59 15 1295 1126 5.9844 6314 8393 1502 6942 6690 6491 1877 45 30 1158 1350 9685 6575 8211 1800 6738 7025 5264 2250 30 45 1021 1573 9525 6835 8028 2097 6532 7359 5035 2621 15 32 0883 1795 9363 7094 7844 2394 6324 7693 4805 2992 58 15 0744 2017 9201 7353 7658 2689 6116 8025 4573 3361 45 30 0603 2238 9037 7611 7471 2984 5905 8357 4339 3730 30 45 5.0462 3.2458 5.8873 3.7868 6.7283 4.3278 7.5694 4.8688 8.4104 5.4097 15 33 0320 2678 8707 8125 7094 3571 5480 9018 3867 4464 57 15 0177 2898 8540 8381 6903 3863 5266 9346 3629 4829 45 30 0033 3116 8372 8636 6711 4155 6050 9674 3389 5194 30 45 4.9888 3334 8203 8890 6518 4446 4832 5.0001 3147 5557 15 34 9742 3552 8033 9144 6323 4735 4613 0327 2904 5919 56 15 9595 3768 7861 9396 6127 5024 4393 0652 2659 6280 45 30 9448 3984 7689 9648 5930 5312 4171 0977 2413 6641 30 45 9299 4200 7515 9900 5732 5600 3948 1300 2165 7000 15 35 9149 4415 7341 4.0150 5532 5886 3724 1622 1915 7358 55 15 4.8998 3.4629 5.7165 4.0400 6.5331 4.6172 7.3498 5.1943 8-1664 5.7715 45 30 8847 4842 6988 0649 5129 6456 3270 2263 1412 8070 30 45 8694 5055 6810 0897 4926 6740 3042 2582 1157 8425 15 36 8541 5267 6631 1145 4721 7023 2812 2901 0902 8779 54 15 8387 5479 6451 1392 4516 7305 2580 3218 0644 9131 45 30 8231 5689 6270 1638 4309 7586 2347 3534 0386 9482 30 45 8075 5899 6088 1883 4100 7866 2113 3849 0125 9832 15 37 7918 6109 5904 2127 3891 8145 1877 4163 7.9864 6.0182 53 15 7760 6318 5720 2371 3680 8424 1640 4476 9600 0529 45 30 7601 6526 5535 2613 3468 8701 1402 4789 9336 0876 30 45 4.7441 3.6733 5.5348 4.2855 6.3255 4.8977 7.1162 5.5100 7.9069 6-1222 15 38 7281 6940 5161 3096 3041 9253 0921 6410 8801 1566 52 15 7119 7146 4972 3337 2825 9528 0679 5718 8532 1909 45 30 6956 7351 4783 3576 2609 9801 0435 6026 8261 2251 30 45 6793 7555 4592 3815 2391 5.0074 0190 6333 7988 2592 15 39 6629 7759 4400 4052 2172 0346 6.9943 6639 7715 2932 51 15 6464 7962 4207 4289 1951 0616 9695 6943 7439 3271 45 30 6297 8165 4014 4525 1730 0886 9446 7247 7162 3608 30 45 6131 8366 3819 4761 1507 1155 9196 7550 6884 3944 15 40 5963 8567 3623 4995 1284 1423 8944 7851 6604 4279 50 15 4.5794 3.8767 5.3426 4.5229 6.1059 5.1690 6.8691 5.8151 7.6323 6.4612 45 30 5624 8967 3228 5461 0832 1956 8437 8450 6041 4945 30 45 5454 9166 3030 5693 0605 2221 8181 8748 5756 5276 15 41 5283 9364 2830 5924 0377 2485 7924 9045 5471 5606 49 15 5110 9561 2629 6154 0147 2748 7666 9341 5184 6935 46 30 4937 9757 2427 6383 5.9916 3010 7406 9636 4896 6262 30 45 4763 9953 2224 6612 9685 3271 7145 9929 4606 6588 15 42 4589 4.0148 2020 6839 9452 3530 6883 6.0222 4314 6913 48 15 4413 0342 1815 7066 9217 3789 6620 0513 4022 7237 46 30 4237 0535 1609 7291 8982 4047 6355 0803 3728 7559 30 45 4.4059 4.0728 5.1403 4.7516 5.8746 5.4304 6.6089 6.1092 7.3432 6.7880 15 43 3881 0920 1195 7740 8508 4560 5822 1380 3135 8200 47 15 3702 1111 0986 7963 8270 4815 5553 1666 2837 8518 45 30 3522 1301 0776 8185 8030 5068 5284 1952 2537 8835 30 45 3342 1491 0565 8406 7789 5321 5013 22361 2236 9151 16 44 3160 1680 0354 8626 7547 5573 4741 2519 1934 9466 46 15 2978 1867 0141 8845 7304 5823 4467 2801 1630 9779 46 30 2795 2055 4.9928 9064 7060 6073 4193 3082 1325 7.0091 30 45 2611 2241 9713 9281 6815 6321 3917 3361 1019 0401 15 45 2426 2426 9497 9497 6569 6569 3640 3640 0711 0711 45 ' Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. Dep. Lat. ' iB'ng DIst. 6. DIst. 7. Dist. 8. j Dist. 9. Dist 10. B^S 92 TABLE VL DEPABTUBE8, For Correction of Courses on Random LAnes. Minutes. 10 Chains. 20 Chains. 40 Chains. 80 Chains. Minutes. 1 .003 .006 .012 .023 l 2 006 012 023 046 2 3 009 017 035 070 3 4 012 023 046 093 4 5 014 029 058 116 5 6 017 035 070 140 6 7 020 041 081 163 8 023 046 093 186 - 8 9 026 052 105 209 9 10 029 058 116 233 10 11 032 064 128 256 11 12 035 070 140 279 12 13 038 076 151 302 13 14 041 081 163 326 14 15 044 087 174 349 15 16 046 093 186 372 16 17 049 099 198 396 17 18 052 105 209 419 18 19 055 110 221 442 19 20 058 116 233 466 20 21 061 122 244 488 21 22 064 128 256 512 22 23 067 134 268 535 23 24 070 140 279 558 24 25 073 145 291 581 25 26 - 076 151 302 605 26 27 078 157 314 628 27 28 081 163 326 651 28 29 084 169 337 674 29 30 087 174 349 698 30 31 090 180 361 722 31 32 093 186 372 744 32 33 096 192 384 767 33 34 099 198 395 790 34 35 102 204 407 814 35 36 105 209 419 837 36 37 108 215 430 860 37 38 110 221 442 883 38 39 113 227 454 906 39 40 116 233 465 929 40 41 119 238 477 953 41 42 122 244 488 976 42 43 125 250 500 999 43 44 128 256 512 1.022 44 45 131 262 523 .045 45 46 134 268 535 .068 46 47 137 273 546 .092 47 48 140 279 558 .115 48 49 142 285 570 .138 49 60 145 291 581 .161 50 51 148 297 593 1.184 51 52 151 302 605 1.207 52 53 154 308 616 1.230 53 54 157 314 628 1.253 54 55 160 320 639 1.276 55 56 163 326 651 1.299 56 57 166 331 663 1.323 67 58 169. 337 674 1.346 58 59 172 343 686 1.369 69 60 174 349 698 1.392 60 TABLE VIL NATURAL SECANTS. 1 - - - 11 !!-- 21 C 21*-- 81- 31'-- 41- Angle. Secant. Angle. Secant. Angle. Secant Angle. Secant. 1 1.00015 11 1.01872 21 1.07115 31 1.16663 10 1.00021 10 1.01930 10 1.07235 10 1.16868 20 1.00027 20 1.01989 20 .07356 20 1.17075 30 1.00034 30 1.02049 30 .07479 30 1.17283 40 1.00042 40 1.02110 40 .07602 40 1.17493 50 1.00051 50 1.02171 60 .07727 60 1.17704 2 1.00061 12 1.02234 22 .07853 32 1.17918 10 1.00072 10 1.02298 10 .07981 10 1.18133 20 1.00083 20 1.02362 20 .08109 20 1.18350 30 1.00095 30 1.02428 30 .08239 30 1.18569 40 1.00108 40 1.02494 40 .08370 40 1.18790 60 1.00122 60 1.02562 60 .08503 60 1.19012 3 1.00137 13 1.02630 23 .08636 33 1.19236 10 1.00153 10 1.02700 10 .08771 10 1.19463 20 1.00169 20 1.02770 20 .08907 20 1.19691 30 1.00187 30 1.02842 30 .09044 30 1.19920 40 1.00205 40 1.02914 40 .09183 40 1.20152 60 1.00224 60 1.02987 60 .09323 60 1.20386 4 1.00244 14 1.03061 24 .09464 34 1.20622 10 1.00265 10 1.03137 10 .09606 10 1.20859 20 1.00287 20 1.03213 20 .09750 20 1.21099 30 1.00309 30 1.03290 30 .09895 30 1.21341 40 1.00333 40 1.03368 40 .10041 40 1.21584 60 1.00357 60 1.03447 50 .10189 60 1.21830 5 1.00382 15 1.03528 25 .10338 36 1.22070 10 1.00408 10 1.03609 10 .10488 10 1.22327 20 1.00435 20 1.03691 20 .10640 20 1.22579 30 1.00463 30 1.03774 30 .10793 30 1.22833 40 1.00491 40 1.03858 40 .10947 40 1.23089 60 1.00521 60 1.03944 60 .11103 60 1.23347 6 1.00551 16 1.04030 26 .11260 36 1.23607 10 1.00582 10 1.04117 10 .11419 10 1.23869 20 1.00614 20 1.04206 20 .11579 20 1.24134 30 1.00647 30 1.04295 30 .11740 30 1.24400 40 1.00681 40 1.04385 40 .11903 40 1.24669 60 1.00715 60 1.04477 50 .12067 60 1.24940 7 1.00751 17 1.04569 27 .12233 37 1.26214 10 1.00787 10 1.04663 10 .12400 10 1.25489 20 1.00825 20 1.04757 20 .12568 20 1.25767 30 1.00863 30 1.04853 30 .12738 30 1 26047 40 1.00902 40 1.04950 40 .12910 40 1 26330 50 1.00942 60 1.05047 60 .13083 60 1.26615 8 1.00983 18 1.05146 28 .13257 38 1.26902 10 1.01024 10 1.05246 10 .13433 10 1.27191 20 1.01067 20 1.05347 20 .13610 20 1.27483 30 1.01111 30 1.05449 30 .13789 30 1.27778 40 1.01155 40 1.05552 40 .13970 40 1.28075 , 60 1.01200 60 1.05657 60 .14152 60 1.28374 9 1.01247 19 1.05762 99 .14335 39 1.28676 10 1.01294 10 1.05869 10 .14521 10 1.28980 20 1.01342 20 1.05976 20 .14707 20 1.29287 30 1.01391 30 1.06085 30 .14896 30 1.29597 40 1.01440 40 1.06195 40 .15085 40 1.29909 60 1.01491 50 1.06306 60 .15277 60 1.30223 10 1.01543 20 1.06418 30 .15470 40 1.30541 10 1.01595 10 1.06531 10 .15665 10 1.30831 20 1.01649 20 1.06645 20 .15861 20 1.31183 30 1.01703 30 1.06761 30 .16059 30 1.31509 40 1.01768 40 1.06878 40 1.16259 40 1.31837 60 1.01815 50 1.06995 60 1.16460 60 1.32168 94 TABLE VII. NATUEAL SECANTS. 41 46 _jg . 51 51 56 56 151 1 Angle. Secant. Angle. Secant. Angle. Secant. Angle. Secant. 41 1.32501 46 1.43956 51 1.58902 56 1.78829 10 1.32838 10 1.44391 10 1.59475 10 1.79604 20 1.33177 20 1.44831 20 1.60054 20 1.80388 30 1.33519 30 1.45274 30 1.60639 30 1.81180 40 1.33864 40 1.45721 40 .61229 40 1.81981 50 1.34212 50 1.46173 50 .61825 50 1.82790 42 1.34563 47 1.46628 52 .62427 57 1.83608 10 1.34917 10 1.47087 10 .63035 10 1.84435 20 1.35274 20 1.47551 20 .63648 20 1.85271 30 1.35634 30 1.48019 30 .64268 30 1.86116 40 1.35997 40 1.48491 40 .64894 40 1.86990 50 1.36363 50 1.48967 50 .65526 50 1.87834 43 1.36733 48 1.49448 53 .66164 58 1.88708 10 1.37105 10 1.49933 10 .66809 10 1.89591 20 .37481 20 1.50422, 20 .67460 20 1.90485 30 .37860 30 1.50916 30 .68117 30 1.91388 40 .38242 40 1.51415 40 .68782 40 1.92302 50 .38628 50 1.51918 50 .69452 50 1.93226 44 .39016 49 1.52425 54 .70130 59 .94160 10 .39409 10 1.52938 10 .70815 10 .95106 20 .39804 20 1.53455 20 .71506 20 .96062 30 .40203 30 1.53977 30 .72205 30 .97029 40 1.40606 40 1.54504 40 .72911 40 .98008 50 1.41012 50 1.55036 50 .73624 50 .98998 45 1.41421 50 1.55572 55 .74345 60 2.00000 10 1.41835 10 1.56114 10 .75073 ,10 2.01014 20 1.42251 20 1.56661 20 .75808 20 2.02039 30 1.42670 30 1.57213 30 .76552 30 2.03077 40 1.43096 40 1.57771 ' 40 .77303 40 2.04128 50 1.43524 50 1.58333 50 .78062 50 2.05191 JAN. 1. TABLE VIII. JAN. 1. AZIMUTHS OP POLARIS AT EXTREME ELONGATIONS. 3 1906 1907 1908 1909 J9IO a J 1906 1907 1908 i 909 I9IO 25 I 19 -I i 18. i 18.4 i 18.1 I 17-7 50 i 51-5 I 5I.O I 5O.6 i 50.1 I 49-6 26 19.8 19. 19.1 18.7 18. 51 54-0 53-5 53-0 52.5 52.0 27 20.5 20. 19.8 19.4 19- 52 56.4 55-9 55-4 54-9 54.4 28 21.3 2O. 20.5 20.1 19- 53 59-1 58.6 58.1 57 6 57.1 29 22.1 21. 21.3 20.g 20. 54 2 O2.0 2 OI.5 2 OO.g 2 OO.4 59;9 30 22.8 22. 22.1 21.7 21. 55 O5.0 04-4 03-9 03-4 2 02.8 31 23.6 23. 22.9 22.5 22. 56 08.2 07-7 07-1 06.6 06.0 32 24-5 24. 23-8 23-4 23- 57 ii. 7 II. I 10.5 IO.O 09.4 33 25-5 25. 24-7 24-3 24. 58 15-3 14-7 14.2 13-6 13.0 34 26.5 26. 25-7 25-3 25- 59 19.2 18.6 18.0 17-4 16.8 35 27-5 27- 26.8 26.4 26. 60 23-4 22.8 22.1 21.5 20. Q 36 28.6 28. 27-9 27-5 27. 61 27-9 27.1 26.6 25-9 25.3 37 29.7 29- 29-0 28.6 28. 62 32-7 32.1 31-4 30.8 30.1 38 31-0 30.6 30.2 29.8 29- 63 38.0 37-3 36.6 35-9 35.2 39 32-3 31.8 31.4 31.0 3o. 64 43-6 42.9 42.2 41-5 40.8 40 33-6 33.2 32.8 32.4 32. 65 49-7 49-0 48.3 47-5 46.8 41 35-0 34.6 34-2 33-8 _ 33- 66 56-3 55-6 54.8 54-1 53-3 42 36.5 36.0 35-6 35-2 34-8 67 3 03.6 3 02.8 3 02.0 3 OI.2 3 00.4 43 38.1 37.6 37-2 36.8 36.3 68 ii. 5 10.7 09.8 09.0 08.2 44 39-7 39.2 38.8 38.4 37-9 69 20. i 19-3 18.4 17.6 16.7 45 41-4 40.9 40-5 40.1 39-6 70 -29-7 28.8 27-9 27.0 26.1 46 43-2 42.7 42.3 41-9 41.4 7i 40.3 39-4 38.4 37-5 36.5 47 45-1 4 4 .6 44-2 43-7 43-3 72 52.1 51- 1 50.1 49-1 48.1 48 47-2 46.7 46.3 45-8 45-3 49 I 49-3 i 48.8 I 48.4 I 47-9 I 47-4 TABLE IX. MULTIPLIERS OF R, 95 For one revolutionof Gradienter Screw, used in finding d' and d. Page 117. Elevation. Multipliers of r. i Elevation. Multipliers of r. Elevat'n. Multipliers of r. Inc. Hor. ' Inc. Hor. Inc. Hor. e. Dist. Dist. e. Dist. Dist. e. Dist. Dist. 1 00 99.97 99.95 o / 14 96.79 93.91 t 22 30 92.01 85.01 2 99.90 99.84 14 30 96.56 93.49 23 91.66 84.37 3 99.81 99.67 15 96.33 93.05 23 30 91.31 83.73 4 99.69 99.44 15 30 96.09 92.59 24 90.95 83.08 5 99.53 99.15 16 95.85 92.13 24 30 90.58 82.42 6 99.35 98.80 16 30 95.60 91.66 25 90.21 81.75 7 99.13 98.39 17 95 34 91.17 25 30 89.83 81.08 8 98.89 97.92 17 30 95.07 90.67 26 89.44 80.39 9 98.61 97.39 18 94.80 90.15 20 30 89.05 79.69 10 98.31 96.81 18 30 94.52 89.63 27 88.65 78.99 10 30 98.14 96.50 19 94.23 89.09 27 30 88.24 78.27 11 97.97 96.17 19 30 93.93 88.54 28 87.83 77.55 11 30 97.79 95.83 20 93.63 87.97 28 30 87.40 76.81 12 97.61 95.47 20 30 93.32 87.41 29 86.98 76.07 12 30 97.41 95.10 21 93.00 86.82 29 30 86.54 75.32 13 97.21 94.72 21 30 92.68 86.23 30 86.10 74.67 13 30 97.00 94.22 22 92.34 85.61 30 30 85.66 73.81 TABLE X. ANGLES OF ELEVATION, Corresponding to numbers of Revolution of the Gradienter Screw, Screw. Angle. Screw. Angle. Screw. Angle. Rev. Div. . Rev. Div. ' /' Rev. Div. 1 00 21 10 3 26 1 00 34 23 2 41 20 6 53 2 1 08 45 3 1 02 30 10 19 3 1 43 06 1 23 40 13 45 4 2 17 26 1 43 50 17 11 5 2 51 45 2 04 20 38 6 3 26 01 2 24 70 24 04 7 4 00 15 2 45 80 27 30 8 4 34 26 9 3 06 90 30 56 9 5 08 34 10 3 26 1 00 34 23 10 00 5 42 38 TABLE XI. MEAN REFRACTIONS, In Declination, for use with Solar Compass. i -< I Declinations. For Latitude 30. +2O + 15 + 1O +5 5" 1O 15 ao Oh. 2 3 4" 5 10" 14 20 32 I'OO 15" 19 26 39 1 10 21" 25 32 46 1'24 27" 31 39 52 1'52 33" 38 47 1'06 207 40" 46 55 110 244 48" 54 1'06 1 35 346 57" 1'05 119 1 57 543 ros" 1 18 136 229 1306 For Latitude 32 30'. Oh. 2 3 4 5 13" 17 23 35 1'03 18" 22 29 43 1*15 24" 28 35 51 1'31 30" 35 43 I'Ol 1 53 36" 42 51 1*13 220 44" 50 I'Ol 127 305 52" I'OO 1 13 1 46 425 1'02" 1 11 128 213 736 114" 1 26 147 254 For Latitude 35. Oh. 2 3 4 5 15" 20 26 39 1'07 21" 25 33 47 1'20 27" 32 39 56 1'38 33" 38 47 1'07 200 40" 46 56 1'20 234 48" 55 1'07 1 36 329 57" 1'05 121 159 514 1'08" 1 18 138 232 1016 1'21" I 35 200 325 For Latitude 37 30'. Oh. 2 3 4 5 18" 22 29 43 I'll 24" 28 36 51 1'26 30" 35 43 I'Ol 154 36" 42 52 1'13 210 44" 50 1'02 127 249 52" I'OO 114 149 355 1'02" 112 129 214 615 114" 126 149 254 1458 1'29" 145 216 405 For Latitude 40. Oh. 2 3 4 5 21" 25 33 47 115 27" 32 40 55 1'31 33" 39 48 1'06 151 40" 46 57 1'19 220 48" 52 ros 136 305 57" 106 121 158 425 1'08" 119 138 230 734 1'21" 1 35 202 321 2518 1'39" 157 236 459 For Latitude 42 30'. ill. 2 3 4 5 24" 28 36 50 1'16 30" 35 43 I'OO 136 36" 39 52 I'll 1 58 44" 50 1'02 126 230 52" I'OO 113 144 322 1'02" 1 12 129 210 500 114" 126 149 249 924 1'29 ' 145 217 355 1'49" 211 259 616 For Latitude 45. Oh. 2 3 4 5 27" 32 40 54 1'23 33" 39 47 104 141 40" 46 56 1*16 205 48" 52 1'07 133 241 57" 106 1 21 154 340 1'08" 1 19 1 38 224 540 1'21" 135 200 311 1202 1'39" 157 234 438 2'02" 229 329 81ft For Latitude 47 30" oh. 2 3 4 6 30" 35 43 56 1'27 36" 42 51 1'09 146 44" 50 roi 123 212 52" I'OO 113 140 252 1'02" 1 12 128 205 401 114" 126 147 240 630 1'29" 145 215 339 1619 1'49" 201 256 537 218" 251 408 1118 TABLE XH. ACREAGE OF OPEN DRAINS. 97 Showing Number of Acres served try drains having bottom urfdth* fron* 1 ft to 10 ft, with side slopes of l to 1, on the supposition of 1 Ineh rain fatt in 2t hours, one-half of which reaches the drain. Computed by B. F. WELLES, C. K, Marshall, Mich. Fall in feet per Bottom Widths. 1ft. 2ft. 3ft. imi. 100ft. 3rd. 2ft. deep. 3ft deep. 2ft. deep. 3ft. deep. 2ft, deep. 3ft. deep. 1.6 2.0 2.4 2.8 3.2 36 4.0 4.8 5.6 6.4 72 8.0 .030 .038 .045 .053 .060 .070 .076 .091 .110 .120 .136 .150 .04 .05 .06 .07 .08 .09 .10 .12 .14 .16 .18 .20 407 462 508 553 592 631 666 733 794 852 905 956 981 1105 1218 1319 1416 1505 1590 1748 1895 2030 2154 2273 594 665 732 797 853 939 959 1057 1143 1225 1300 1373 1311 1473 1622 1762 1889 2009 2115 2333 2523 2700 2869 3031 780 879 968 1053 1128 1198 1264 1391 1499 1612 1715 1809 1649 1861 2047 2217- 2377 2529 2665 2935 3172 . 3401 3612 3815 Fall in Feet per Bottom Widths. 4ft. 5ft. 6ft. imi. 100ft. 8rd. 2ft. deep. 3ft. deep. 2ft. deep. 3ft. deep. 2ft. deep. 3ft. deep. 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.8 5.6 6.4 7.2 8.0 .030 .038 .045 .053 .060 .070 .076 .091 .110 .120 .136 .150 .04 .05 .06 .07 .08 .09 .10 .12 .14 .16 .18 .20 976 1094 1206 1308 1404 1494 1579 1731 1878 2013 2137 2256 2003 2249 2477 2684 2872 3049 3227 3553 3849 4115 4372 4609 1171 1316 1448 1572 1684 1790 1894 2089 2257 2415 2566 2705 2357 2650 2910 3158 3384 3598 3800 4173 4512 4833 5141 5412 1368 1541 1699 1835 1970 2097 2211 2436 2632 2820 3001 3165 2716 3046 3362 3642 3908 4150 4322 4810 5203 5571 5927 6257 Fall in Feet per Bottom Widths. j 7ft. 8ft. g 10ft. imi. 100ft. 8rd. 2ft. deep. 3ft. deep. 2ft. 89 58 59.9 89 58 29.9 89 67 69.9 31 59 28.8 58 57.5 58 26.3 57 55.0 32 69 27.5 58 55.0 58 22.5 67 60.0 33 59 26.2 68 52.5 58 18.7 67 44.9 34 59 24.9 68 49.9 58 14.8 67 39.7 35 69 23.6 58 47.2 58 10.8 67 34.4 36 59 22.2 68 44.4 58 06.8 67 28.9 37 59 20.8 58 41.6 58 02.5 67 23.3 38 69 19.4 58 38.8 67 68.2 67 17.5 39 69 17.9 68 35.8 67 63.7 67 11.6 40 69 16.4 58 32.8 57 49.2 67 05.5 41 69 14.8 58 29.6 57 44.4 66 69.3 42 69 13.2 58 26. 67 39.6 66 52.8 43 69 11.5 58 23. 57 34.6 66 46.2 44 59 09.8 58 19. 67 29.5 66 39.3 45 59 08.0 58 16. 57 24.1 66 32.1 46 69 06.2 58 12. 57 18.6 66 24.8 47 89 59 04.3 89 58 08.6 89 57 12.9 89 56 17.1 Lati- tude. 5 miles. 6 miles. 7 miles. 8 miles. 30 89 57 29.9 89 56 59.8 89 56 29.8 89 55 69.8 31 57 23.8 66 52.5 66 21.3 65 60.0 32 57 17.5 56 45.0 56 12.5 65 40.0 33 57 11.2 56 37.4 66 03.6 65 29.9 34 67 04.6 66 29.6 55 54.5 65 19.4 35 66 68.0 66 21.6 55 45.2 65 08.8 36 66 51.1 66 13.4 65 35.6 64 57.8 37 66 44.1 66 06.0 65 25.8 54 46.6 38 66 36.9 65 56.3 55 15.7 64 36.1 39 66 29.6 65 47.5 55 05.4 64 23.3 40 66 21.9 55 38.3 54 64.7 64 11.1 41 66 14.1 55 28.9 64 43.7 63 68.6 42 56 06.0 55 19.2 64 32.4 63 46.6 43 65 57.7 55 09.2 64 20.8 53 32.3 44 65 49.1 64 68.9 64 08.7 63 18.5 45 65 40.2 54 48.2 53 56.3 53 04.3 46 65 31.0 54 37.2 53 43.4 52 49.5 47 89 55 21.4 89 54 25.7 89 53 30.0 89 52 34.3 lati- tude. Smiles. 10 miles. 11 miles. 12 miles. o 30 89 65 29.8 89 64 59.7 89 54 29.7 89 63 69.7 31 55 18.8 64 47.6 64 16.3 63 46.1 32 65 07.6 54 35.1 54 02.6 63 30.1 33 64 56.1 64 22.3 53 48.5 63 14.8 34 54 44.4 64 09.3 53 34.2 52 69.1 35 54 32.3 63 55.9 53 19.5 62 43.1 36 64 20.0 63 42.3 53 04.5 62 26.7 37 54 07.4 63 28.2 52 49.1 52 09.9 38 63 64.5 63 13.9 62 33.2 51 62.6 39 63 41.2 , 52 59.1 52 17.0 51 34.9 40 63 27.5 52 43.8 52 00.2 61 16.6 41 63 13.4 62 28.2 51 43.0 60 57.8 42 52 58.8 62 12.0 51 25.2 50 38.4 43 52 43.8 51 55.4 51 06.9 50 18.5 44 52 28.4 61 38.2 50 48.0 49 67.8 45 52 12.3 51 20.4 50 28.4 49 36.4 46 61 55.7 51 01.9 50 08.1 49 14.3 47 89 51 38.6 89 50 42.9 89 49 47.2 89 48 51.4 TABLE XVI. OFFSETS FROM TANGENT. Lati- tude. imile. 2 miles. Smiles. 4 miles. Feet. Feet. Feet. Feet. 30 0.39 .54 3.47 6.17 31 0.40 .60 3.61 6.42 32 0.42 .67 3.76 6.67 33 0.43 .73 3.90 6.93 34 0.45 .80 4.05 7.20 36 0.47 .87 4.20 7.47 36 0.48 .94 4.36 7.75 37 0.50 2.01 4.52 8.04 38 0.52 2.08 4.69 8.33 89 0.54 2.16 4.86 8.63 40 0.56 2.24 5.03 8.95 41 0.58 2.32 6.21 9.27 42 0.60 2.40 6.40 9.69 43 0.62 2.48 6.59 9.93 44 0.64 2.57 6.79 10.29 45 0.67 2.66 5.99 10.65 46 0.69 2.76 6.20 11.02 47 0.71 2.85 6.42 11.41 Lati- tude. 5 miles. 6 miles. 7 miles. 8 miles. Feet. Feet. Feet. Feet. 30 9.64 13.88 18.89 24.67 31 10.03 14.44 19.66 25.68 32 10.42 15.02 20.44 26.69 33 10.82 15.60 21.23 27.74 34 11.25 16.20 22.06 28.80 35 11.68 16.81 22.89 29.89 36 12 11 17.41 23.74 31.01 37 12.57 18.09 24.62 32.16 38 13.02 18.75 25.52 33.33 39 13.49 19.43 26.44 34.54 40 13.98 20.11 27.40 35.78 41 14.48 20.85 28.37 37.06 42 14.99 21.59 29.38 38.38 43 15.52 22.35 30.42 39.74 44 16.07 23.14 31.50 41.14 45 16.64 23.96 32.61 42.59 46 17.21 24.80 33. 7S 44.10 47 17.83 25.68 34.95 45.65 Lati- tude. 9 miles. 10 miles. 11 miles. 12 miles. . Feet Feet. Feet. Feet. 30 31.23 38.55 46.65 55.52 31 32.49 40.12 48.54 67.77 32 33.78 41.71 60.47 60.06 33 35.10 43.34 62.44 62.41 34 36.45 45.00 64.45 64.80 35 37.83 46.71 66.62 67.26 36 39.25 48.45 68.63 69.77 37 40.70 60.24 60.79 72.35 38 42.19 62.08 63.02 75.00 39 43.71 63.97 65.30 77.71 40 45.29 65.91 67.65 80.51 41 46.90 67.91 70.07 83.39 42 48.57 69.97 72.56 86.35 43 60.29 62.09 75.13 89.41 44 62.07 64.28 77.78 92.57 45 63.91 66.55 80.53 95.84 48 65.81 68.90 83.37 99.22 47 57.78 71.34 86.32 102.72 TABLE XVIL Minutes In Decimals jf a Be^rde.- 1' .0167 11' .1833 21' .3500 31' 5167 41 ' .6833 61' .8500 2 .0333 12 .2000 22 .3667 32 5333 .7000 62 .8667 3 .0500 18 .2167 23 .3833 33 5-500 4J .7167 63 8833 4 .0667 14 .2333 24 .4000 34 5667 44 .7333 64 .'9000 6 .0833 16 .2500 25 .4167 35 5833 45 .7500 66 .9167 ft .1000 16 .2667 26 .4333 36 6000 4fc .7667 66 .9333 7 .1167 17 .2833 27 .4500 37 6167 47 .7833 67 .9500 8 .1333 18 .3000 28 .4667 38 6333 4S .8000 68 .9667 9 .1500 19 .3167 29 .4833 39 .6500 4< > .8167 69 .9833 10 .1667 20 .3333 30 .5 ,a333 60 1.0000 TABLE xvm. Inches in Decimals of a Toot. 1-16 3-32 > 3-16 M 5-16 % U % % % .0052 .0078 .0104 .0156 .0208 .0260 .0313 .0 417 .0521 .0625 .0729 I 2 3 4567 8 9 10 11 .0833 .1667 .2500 .3333 .4167 .5000 .5833 .6 98.4 691.4 50 1742.6 259.1 50 2305.2 446.4 50 2908.9 696.1 34 1751.7 261.8 44 2314.9 450.0 54 2919.4 700.9 10 1760.8 264.5 10 2324.6 453.6 10 2929.9 705.7 20 1770.0 267.2 20 2334.3 457.3 20 2940.4 710.5 30 1779 1 2699 30 2344.1 461.0 30 2951.0 715.3 40 1788.2 272,6 40 2353.8 464.6 40 2961.5 720.1 50 1797.4 275.3 50 2363.5 468.4 50 2972.1 725.0 35 1806.6 278.1 45 2373.3 472.1 55 2982.7 729.9 10 1815.7 280.8 10 2383,1 475.8 10 2993.3 734.8 20 1824.9 283.6 20 2392.8 479.6 20 3003.9 739.7 30 1834.1 286.4 30 2402.6 483.8 30 3014.5 744.6 40 1843.3 289.2 40 2412.4 487.2 40 30i5.2 749.6 50 1852.5 292.0 50 2422.3 491.0 50 3035.8 754.6 86 1861.7 2949 46 2432.1 494.8 56 3046.5 7596 10 1870.9 297.7 10 2441.9 498.7 10 3057.2 7646 20 1880.1 300.6 20 2451.8 502.5 20 3067.9 769.7 30 1889.4 303.5 30 2461.7 506.4 30 3078.7 774.7 40 1898.6 306.4 40 2471.5 510.3 40 3089.4 779.8 50 1907.9 309.3 50 2481.4 514.3 60 3100.2 784.9 37 1917.1 312.2 47 2491.3 518.2 57 3110.9 790.1 10 1926.4 315.2 10 2501.2 622.2 10 3121.7 795.2 20 19&5.7 318.1 20 2511.2 526.1 20 3132.6 800.4 30 1945.0 321.1 30 2521.1 530.1 30 3143.4 805.6 40 1954.3 324.1 40 2531.1 534.2 40 3154.2 810.9 50 1963.6 327.1 50 2541.0 538.2 50 3165.1 816.1 88 1972.9 330.2 48 2651.0 542.2 58 3176.0 821.4 10 1982.2 333.2 10 25610 546.3 10 3186.9 826.7 20 1991.5 336.3 20 2571.0 550.4 20 3197.8 832.0 30 2000.9 339.3 30 2581.0 554.5 30 3208.8 837.3 40 2010.2 342.4 40 2591.0 558.6 40 3219.7 842.7 50 2019.6 345.5 50 2601.1 -662.8 50 3230.7 848J 89 2029.0 348.6 49 2611.2 566.9 69 3241.7 853.5 10 2038.4 351.8 10 2621.2 571.1 10 3252.7 858.9 20 2047.8 3549 20 2631.3 575.3 20 3263.7 864.3 30 2057.2 358.1 30 2641.4 579.5 30 3274.8 869.8 40 2066.6 361.3 40 2651.5 583.8 40 3285.8 875.3 60 2076.0 364.5 50 2661.6 688.0 60 3296.9 880.8 40 2085.4 367.7 50 2671.8 592.3 60 3308.0 886.4 10 2094.9 371.0 10 2681.9 596.6 10 3319.1 892.0 20 2104.3 3742 20 2692.1 600.9 20 3330.3 897.5 30 2113.8 377.5 30 2702.3 605.3 30 3341.4 903.a 40 2123.3 380.8 40 2712.5 609.6 40 3352.6 908.8 50 2132.7 384.1 50 2722.7 614.0 50 3363.8 914.5 104 TABLE xx Tangents and Externals to a 1 Curve. Ingle Tangent Externa Angle Tangent Externa Angle. Tangent External 61 10 20 30 40 50 3375.0 3386.3 3397.5 3408.8 3420.1 3431.4 920.2 925.9 931.6 937.3 943.1 948.9 71 10 20 30 40 50 4086.9 4099.5 4112.1 4124.8 4137.4 4150.1 1308.2 1315.6 1322.9 1330.3 1337.7 1345.1 81 10' 20 30 40 50 4893.6 4908.0 4922.5 4937.0 4951.5 4966.1 1805.3 1814.7 1824.1 1833.6 1843.1 1852.6 62 10 20 30 40 50 3442.7 3454.1 3465.4 3476.8 34883 3499.7 954.8 960.6 966.5 972.4 978.3 984.3 72 10 20 30 40 50 -4162.8 4175.6 4188.5 4201.2 4214.0 4226.8 1352.6 1360.1 1367.6 1375.2 1382.8 1390.4 82 10 20 30 40 50 4980.7 4995.4 5010.0 5024.8 5039.5 5054.3 1862.2 1871.8 1881.5 1891.2 190C9 1910,7 63 10 3511.1 3522.6 990.2 996.2 78 10 4239.7 4252.6 1398.0 1405.7 83 10 5069.2 50840 1920.5 1930.4 20 30 3534.1 3545.6 10023 100S.3 20 30 4265.6 4278.5 1413.5 1421.2 20 30 5099.0 5113.9 1940!3 1950 3 40 50 3557.2 3568.7 1014.4 1020.5 40 50 4291.5 4304.6 1429.0 1436.8 40 50 5128.9 5143.9 1980.2 1970.3 64 3580.3 1026.6 74 4317.6 1444 6 84 5159 1980.4 10 3591.9 1032.8 10 4330.7 1452.5 10 5174 1 20 3603.5 1039 20 4343.8 1460.4 20 61893 2000.'6 3J 3615.1 1045.2 30 4356.9 1468.4 30 5204.4 20108 40 50 3626.8 3638.5 1051.4 1057.7 40 50 4370.1 4383.3 1476.4 1484.4 40 50 5219.7 5234.9 202U 2031.4 65 10 20 30 40 50 3650.2 3661.9 3673,7 3685.4 3697.2 3709.0 1063,9 1070.2 1076.6 1082.9 1089.3 1095.7 75 10 20 30 40 50 4396.5 4409.8 4423.1 4436.4 4449.7 4463.1 1492.4 1500.6 1508.6 1516.7 1524.9 1533.1 85 10 20 30 40 50 5250.3 5265.6 5281.0 5296.4 5311.9 6327.4 2041.7 2052.1 2062.5 2073.0 2083.5 2094.1 66 3720.9 1102.2 76 4476.5 1541.4 86 5343.0 2104.7 10 20 30 40 50 3732.7 3744.6 3756.5 3768.5 3780.4 1108.6 1115.1 1121.7 1128.2 1134.8 10 20 30 40 50 4489.9 4503.4 4516.9 4530.4 4544.0 1549.7 1558.0 1566.3 1574.7 1583.1 10 20 30 40 50 5358.6 5374.2 5389.9 5405.6 5121.4 2115.3 21260 2136.7 2147.5 2158.4 67 10 20 3792.4 3804.4 38164 1141.4 1148.0 1154.7 77 10 20 4557.6 4571.2 4584.3 1591.6 1600.1 16086 87 10 20 5437.2 5453.1 54690 2169.2 2180.2 2191 1 30 40 60 3828.4 3840.5 3852.6 1161.3 1168.1 1174.8 30 40 60 4598.5 4612.2 4626.0 1617.1 1625.7 1634.4 30 40 50 5484.9 5500.9 5517.0 2202,2 2213.2 2224.3 68 10 3864.7 3876,8 4181.6 1188.4 78 10 4639.8 4653.6 1643.0 1651.7 88 10 5533.1 5549.2 2235.5 2246.7 20 3889.0 1195.2 20 4667.4 16605 20 5565.4 2258 30 3901.2 1202.0 30 4681.3 1669.2 30 5581 6 22693 40 50 3913.4 3925.6 1208.9 1215.8 40 50 4695.2 4709.2 1678.1 1686.9 40 50 5597^8 5614.2 2280^6 2292.0 69 3937.9 1222.7 79 4723.2 1695.8 89 56305 23035 10 20 3950.2 3962.5 1229.7 1236.7 10 20 4737.2 4751.2 1704.7 17137 10 20 5646.9 .'j663.4 2315.0 23266 30 40 3974.8 3987.2 1243.7 , 1250.8 30 40 4765.3 4779.4 1722.7 1731.7 30 40 5679.9 5696.4 2388.2 2349 8 50 3999.5 1257.9 50 4793.6 1740.8 60 5713.C 236L5 70 4011.9 1265.0 80 4807.7 1749.9 90 5729.7 23733 10 20 4024.4 4036.8 1272.1 1279.3 10 20 4822.0 4836.2 1759.0 1768.2 10 20 5746.3 5763.1 2385.1 2397 130 4049.3 1286.5 30 4850.5 1777.4 30 57799 2408.9 40 4061.8 1293.6 40 4864.8 17*6.7 40 6796.7 24209 60 4074.4 1300.9 50 4879.2 1796.0 50 5813.6 2432.9 TABLE xx Tangents and Externals to a 1 Curve. 105 Ingle. Tangent External Angle. Tangent External ' Angle. Tangent External 91 583U.5 2444.9 101 6950.6 3278.1 111 8336.7 4386.1 Iff 5847.5 2457.1 10' 6971.3 3294.1 HX 8362.7 4407.6 20 5864.6 2469.3 20 6992.0 3310.1 20 83SS.9 4429.2 30 5881.7 2481.5 30 7012.7 326. 1 30 8415.1 44509 40 5898.8 2493.8 40 7033.6 3342.3 40 8441.5 4472.7 60 5916.0 2506.1 50 7054.5 3358.5 60 846j.O 4494.6 92 5933.2 2518.5 102 7075.5 3374.9 112 8494.6 4516.6 10 59-50.5 2531.0 10 7096.6 3391.2 10 6521.3 4538.8 20 5967.9 2543.5 20 7117.8 3407.7 20 8548.1 4561.1 30 5985.3 2556.0 30 7139.0 3124.3 30 8575.0 45^3.4 40 6' ii)2.7 2-568.6 40 7160.3 3440.9 40 8602.1 4606.0 50 6020.2 2581.3 5'J 7181.7 3157.6 5J 8629.3 4628.6 93 6037.8 2594.0 103 7203.2 3474.4 113 8656.6 4651.3 10 6055.4 26)68 10 7224.7 3491.3 10 8684.0 4674.2 20 6073.1 2619.7 20 7246.3 3508.2 20 8711.5 4697.2 30 6090.8 2632.6 30 7268.0 3525.2 30 8739.2 4720.3 40 6108.6 2645.5 40 72898 3542.4 40 8767.0 4743.6 50 6126.3 2658.5 50 7311.7 3559.6 60 8794.9 4766-9 94 6144.3 2671.6 104 73336 3576.8 114 8822.9 4790.4 10 6162.6 26S4.7 10 7355.6 594.2 10 SS51.0 4814.1 20 6180.2 2697.9 20 7377.8 3611.7 20 8879.3 4887.8 30 6198.3 2711.2 30 7399.9 3629.2 30 8907.7 4861.7 40 6216.4 2724.5 40 7422.2 3646.8 40 8936.3 4885.7 50 6234.6 2737.9 50 7444.6 3664.5 60 8965.0 4909.9 95 6252.8 2751.3 105 7467.0 3682.3 115 8993.8 4934.1 10 6271.1 2764,8 10 74896 3700.2 10 9022.7 4958.6 20 6289.4 2778.3 20 7.512.2 3718.2 20 9051.7 4983.1 30 6307.9 2792.0 30 7534 9 3736.2 30 9080.9 5007.8 40 6326.3 2805.6 40 75-57.7 3754.4 40 9110.3 .5032.6 50 6344.8 2819.4 50 75o0.5 3772.6 50 9139.8 6057.6 96 6363.4 2833.2 106 7603.5 3791.0 116 9169.4 5082.7 10 6382. 1 2847.0 10 7626.6 3809.4 10 9199.1 5107.9 20 6400.8 2861.0 20 7649.7 3827.9 20 9229.0 5133.3 30 6419.5 2875.0 30 7672,9 3846.5 30 9259.0 51-58.8 40 6438.4 2889!o 40 7696.3 3*65.2 40 9289.2 5184.5 50 6457.3 2903.1 50 7719.7 38*4.0 50 9319.5 6210.3 97 6476.2 2917.3 107 7743,2 3902.9 117 9349.9 5236.2 10 6495.2 2931.6 10 77668 8921.9 10 9380.5 5262.3 20 6514.3 2945.9 20 7790.5 3940.9 20 9411.3 5288.6 30 6.533.4 2960.3 30 7814.3 3960.1 30 9442.2 5315.0 40 6-552.6 2974.7 40 7838-1 3979.4 40 9473.2 5341.5 50 6571.9 2989.2 60 7862.1 3998.7 50 9504.4 5368.2 98 6591.2 3003.8 108 7886.2 4018.2 118 9535.7 6395.1 10 6610.6 3018.4 10 791U.4 4037.8 10 9567.2 5422.1 20 6630.1 3033.1 20 7934.6 4057.4 20 9598.9 5449.2 30 6649.6 3047.9 30 79.50.0 4077. '_' 30 9630.7 5476.5 40 6669.2 3062.8 40 7983.5 4097.1 40 9662.6 5504.0 60 6688.8 3077.7 50 8008.0 4117.0 50 9694.7 5531.7 99 6708.6 3092.7 109 8032.7 4137.1 119 9727.0 5559.4 10 6728.4 3107.7 10 8057.4 4157.3 10 9759.4 5587.4 20 6748.2 3122.9 20 &H2.3 4177.5 20 9792.0 5615.5 30 6768.1 3138.1 30 8107.3 4197.9 30 9824.8 5613.8 40 67HS.1 3153.3 40 81323 4218.4 40 9857.7 5672.3 50 6808.2 3168.7 60 8157.5 4239.0 60 9890.8 5700.9 100 6828.3 3184.1 110 8182.8 4259.7 120 9924.0 5729.7 10 6848.5 3199.6 10 8208.2 4280.5 10 9957.5 5758.6 20 8B8&8 3215.1 20 8233.7 4301.4 20 9991.0 5787.7 30 6889.2 3230.8 30 8259.3 4322.4 30 10025.0 5817.0 40 6909.S 3246.5 40 8285.0 4343.6 40 lix 1.59.0 5846.5 50 6930.1 3262,3 60 8310.8 4364.8 60 10093.0 6876.1 106 TABLE XXI. STADIA REDUCTIONS BY ARTHUR WINSLOW STADIA REDUCTIONS FOR READING 100 1 2 3 Minutes. Hor. Diff. Hor. Diff. Hor. Diff. Hor. Diff. Dist. Elev. Dist. Elev. Dist. Elev. Dist. Elev. 0' 100.00 .00 99.97 1.74 99.88 3.49 99.73 5.23 2 .06 .80 99.87 3.55 99. 73 5.28 4 .12 " .86 3.60 99.71 5.34 6 .17 99.96 .92 3.66 44 5.40 8 .23 .98 99.86 3.72 99.70 5.46 10 .29 " 2.04 3.78 99.69 5,52 12 .35 2.09 99". 85 3.84 " 5.57 14 .41 99.95 2.15 " 3.90 99.68 5.63 16 ' .47 " 2.21 99.84 3.95 5.69 18 " .52 " 2.27 .01 99.67 5.76 20 .58 '* 2.33 99.83 .07 99.66 5.80 22 " .64 99.94 2.38 .13 14 5.6 24 " .70 44 2.44 99.82 .18 99.65 5.92 26 99.99 .76 " 2.50 " .24 99.64 5.98 28 .81 99.93 2,56 99.81 .30 99.63 6.04 30 .87 44 ' 2.62 .36 44 6.09 32 .93 " 2.67- 99.80 4.42 99^62 6.15 34 .99 2.73 44 4.48 6.21 36 " 1.05 99.92 2.79 99.79 4.53 99.61 6.27 38 li 1.11 " 2.85 "-" 4.59 99.60 6.38 40 " 1.16 2.91 99.78 4.65 99.59 6.38 42 " .22 99.91 2.97 4.71 a " 6.44 44 99.98 .28 3.0-2 99.77 4.76 99.58 6.50 46 .34 99.90 3.08 4.8.' 99.57 6.56 48 .40 3.14 99.76 4.88 99.56 6.61 50 " .45 3.20 4.94 6.67 52 " .51 99.89 3.26 99.75 4.99 09.55 6.73 54 .57 3.31 99.74 5.05 99.54 6.78 56 99.97 .63 " 3.37 5.11 99.53 6.84 58 .09 99.88 3.43 99.73 5.17 99.52 6.90 60 .74 3.49 5.23 99.51 6.96 c+/= .75 .75 .01 .75 .02 .75 .03 .75 .05 c-f/= 1.00 1.00 .01 1.00 -.03 1.00 .04 1.00 .06 c+/= 1.25 1.25 .0> 1.25 -03 1.25 .05 1.25 .08 TABLE XXI. STADIA REDUCTIONS 107 TABLE XXI. STADIA REDUCTIONS FOR READING 100 Minutes 4 5 6 7 Her. Diff. Dist. Elev. Hor. Diff. Dist. Elev. Hor. Diff. Dist. Elev. Hor. Diff. Dist. Elev. 0' 2 4 6 8 10 99.51 6.96 7.02 99.50 7.0? 99.49 7.13 99..4S .19 99.47 .25 99.24 8.68 99.23 8 74 99.22 8.80 99.21 8.85 99.20 8.91 99.19 8.97 98.91 10.40 98.90 10.45 68.88 10.51 98.87 10.57 9S.86 10 62 98.85 10.63 98.5! 12.10 93.50 12.15 98.48 12 21 S8.47 12.20 98.46 12.3:.' 98.44 12.38 ! 12 14 16 18 20 99.46 .-30 .36 99.45 .42 99.44 .48 99.43 .53 99.18 9.03 99.17 9.08 99.16 9.14 99.15 9.20 99.14 9.25 98.33 10.74 98.82 10.79 98.81 10.85 98.80 10.91 98.78 10.96 98.43 12.43 93.41 12.49 98.40 12.55 98.39 12.60 98.3? 12.66 oo 24 2f> 28 30 99.42 .59 90.41 .65 99.40 .71 99.39 .76 99.38 .82 99.13 9.31 99.11 9.37 ,99.10 9.43 ;99.09 9.48 99.08 9.54 98.77 11.02 98.76 11.08 98.74 11.13 98.73 11.19 98.72 l;.25 98.36 12.72 98.34 12.7? 93.33 12.83 98.31 12.88 93.89 12.94 32 34 36 38 40 99.38 .88 99.3? .94 99.36 7.99 99.35 8.05 99.34 8.11 99.07 9.60 99.06 9.G5 99.05 9.71 99.04 9.7? 99.03 9.83 98.71 11.30 9869 11. 30 98.68 11. -52 98.67 11.47 98.65 11.53 98.28 13.00 98.27 13.05 98.25 13.11 98.24 13.1? 98.22 13.22 42 44 46 48 50 99.33 8.17 99.32 8.22 99.31 S 28 99.30 8.34 99.29 8.40 99.01 -9.88 99.00 9.94 9S.99 10.00 98.98 10.05 98.9? 10.11 98.64 11.59 98.63 11.64 98.61 11.70 98.60 11.76 98.58 11.81 98.20 13t28 98.19 13.33 93.17 13 39 98.16 13.45 98.14 13.50 52 54 56 58 60 99.28 8.45 99.27 8.51 99.26 8.57 99.25 8.63 9924 8.68 98.90 10.17 98.94 10.22 98.93 10. 2S 98.92 10.34 98.91 10.40 98.57 11.87 93 .56 11.93 98:54 11.98 98.53 12.04 98.51 12.10 98.13 13.56 !>8.11 13.61 1^.10 13.6? 9S.08 13.73 Sfc'.OG 13.78 c+/= -T5 c+/= 1.00 c + /= 1.25 .75 .06 I. 00 .OS 1.25 .10 .75 .07 .99 .09 1.24 .11 .75 .08 .99 .11 1.24 .14 .74 .10 .99 .13 I.:54 .16 108 TABLE XXI. STADIA REDUCTIONS TABLE XXI. STADIA KFDUCTIONS FOR READING 100 Minutes. 8 9 10 11 Hor. Diff. Dist. Elev. Hor. Diff. Dist. Elev. Hor. Diff. Dist. Elev. Hor. Diff. Dist. Elev. 0' 2 4 6 8 10 98.06 13.78 98.05 13.84 98.03 13,89 98.01 13.95 98.00 14.01 97.98 14.06 97.55 15.45 97.53 15.51 97.52 15.56 97.50 15.62 97.48. 15.67 97.46 15.73 96.98 17.10 96.96 17.16 96.94 17.21 96.92 17.26 96.90 17.32 96.88 17.37 96.36 18.73 9634 18.78 96.32 18.84 96.29 18.89 96.27 18.95 96.25 .19.00 12 14" 16 18 20 97.97 14.12 97.95 14.17 97.93 14.23 97.92 14.28 97.90 14.34 97.44 15.78 97.43 15.84 97.41 15.89 97.39 15.95 97.37 16.00 96.88 17.43 96.84 17.48 96.82 17.54 96.80 .17.59 96.78 17.65 96,23 19.05 96.21 J9.ll 96.18 19.16 90.16 19.21 96.14 19.27 22 24 26 28 30 97.88 14.40 97.87 14.45 97.85 14.51 97. a3 14.56 97.82 14.62 97.35 16.06 97.33 16.11 97.31 16.17 97.29 16.22 97.28 16.28 96.76 17.70 96.74 17.76 96.72 17.81 96.70 17.86 96.68. 17.92 96.12 19.82 96.09 19.38 96.07 19.43 90.05 19.48 96.03 19.54 32 34 36 38 40 97.80 14/67 97.78 14.73 97.76 14.79 97.75 14.84 97.73 14.90 97.26 16.33 97.24 16.39 97.22 16.44 97.20 16.50 97.18 16.55 96.66 17.97 96.64 18.03 96.62 18.08 96.60 18.14 96.57 18.19 96.00 19.59 95.98 19.64 95.96 1&.70 95.93 19.75 95.91 19.80 42 44 46 48 50 97.71 14.95 97.69 15.01 97.68 15.06 97.66 15.12 97.64 15.17 97.16 16.61 97.14 16.66 97.12 16.72 97.10- 16.77 97.08 16.83 96.55 18.24 96.53 18.30 96.51 18.35 96.49 18.41 96.47 18.46 95.89 19.86 95.86 19.9! 95.84 19.96 95.82 20.02 95.79 20.07 52 i 54 56 58 60 97.62 15.23 97.61 15.28 97.59 15.34 97.57 15.40 97.55 15.45 97.06 16.88 97.04 36.94 97.02 16.99 97.00 17.05 96.98 17.10 96.45 18.51 96.42 18.57 96.40 18.62 96.38 18.68 96.36 18.73 95.77 20.12 95.75 20.18 95.72 20.23 95.70 20.28 95.68 20.34 (.+/= .75 t; 4-/ = 'i.OO c + /= 1.25 .74 .11 .09 .15 i.as .13 .74 .12 .99 .16 1.23 .21 .74 .14 .98 .18 1.23 .23 .73 .15 .93 .20 1.22 .25 M'jfc, TABLE XXI. STADIA REDUCTIONS 109 TABLE XXI. STADIA 11EDUCTIONS FUIi HEADING 100 1 Minutes. 12 13 : 14 15 Hor. Diff. Dist. Elev. Hor. Diff. Dist. Elev. Hor. Diff. Dist. Elev. Hor. Diff. Dist. Elev. 0' 2 4 6 8 10 95.68 20.34 95.65 20.39 95.63 20.44 95.61 20.50 95-58 20.55 95.56 20.60 94.94 21.92 94.91 21.97 94.89 22.02 94.86 22.08 94.84 22.13 M.81 22.18 94.15 23.47 94.12 23.52 94.09 23.58 94.07 23.63 94.04 23.68 94.01 23.73 93.30 25.00 93.27 25.05 93.24 25.10 93.21 25.15 93.18 25.20 93.16 25.25 12 14' 16 18 20 95.53 20.66 C5.51 20.71 95. 49 20.76 95.46 20.81 95.44 20.87 94.79 22.23 94.76 2-2.28 94.73 22.34 94.71 22.39 94.68 22.44 93 93 23.78 93. 55 23 .S3 93.93 23.88 93.90 23.93 93.87 23.99 93.13 25.30 93.10 25.35 93.07 25.40 93.04 25.45 93.01 25.50 22 24 26. . 2? 30 95.41 20.92 95.39 20.97 95.36 21.03 95.34 21.08 95.32 21.13 94.66 22.49 94.63 22.54 94.60 22.60 34.58 22.65 94.55 22.70 93.84 24.04 93.81 24.09 93.79 24.14 93.76 24.19 93.73 24.24 92.98- 25.55 92.95 25.60 92.9-2 25.65 92.89 25.70 92.86 25.75 32 34 36 38 40 95.29 21.18 95.27 21.24 95.24 21.29 95.22 21 31 95.19 21.39 94.52 22.75 94.50 22.80 94.47 22.85 94.44 22.91 94.42 22.96 53.70 24.29 93.67 24.34 93.65 24.39 93.62 24.44 93.59 24.49 92.83 25.80 92.80 25 85 92.77 25.90 92.74 25.95 92.71 26.00 42 44 46 48 50 95.17 21.45 95.14 21.50 95.12 21.55 95.09 21.60 95.07 21.66 94.39 2301 94.36 23.06 94.34 23.11 94.31 23.16 94.28 23.22 93.56 24.55 93.53 24.60 93.50 24.65 93.47 24.70 93.45 24'. 75 92 68 26.05 92.65 26.10 92.62 26.15 92.59 26.20 92.56 26.25 52 54 56 58 60 95.04 21.71 95,02 2L76 94.99 21.81 94.97 21.87 94.94 21.92 '94.26 23.27 94.23 23.32 94.20 23 37 94.17 23.42 94.15 23.47 93.42 24.80 93.39 24.85 93.36 24.90 93.33 24.95 93.30 25.00 92.53 26.30 92.49 26.35 92.46 26.40 92.43 26.45 92.40 26.50 c+f= .75 c-t-/= 1.00 c-(-/=1.25 .73 .16 .98 .22 1.22 .27 .73 .17 .97 .23 1.21 .29 .73 .19 .97 .25 1.21 .31 .72 .20 .96 .27 1.20 .34 no TABLE XXI. STADIA REDUCTIONS TABLE XXI. STADIA REDUCTIONS FOR READING 100 Minutes. 13 17 ' 18 19 Hor. Diff. Dist. Elev. Hor. Diff. Dist. Elev. Hor. Diff. Dist. Elev. H-.r. Diff. Dist. Elev. 0' 2 4 6 8 10 92.40 26.50 92.37 26.55 92.34 26.59 P-"'.31 26.64 9^.28 26.69 92.25 26.74 91.45 27.96 91.42 28.01 91.39 28.06 91.35 28.10 91.32 28.15 91.29 28.20 90.45 29.39 90.42 29.44 90.38 29.48 90.35 29.53 90.31 29.58 90.28 29.62 89.40 30.78 89 36 30.83 89.33 30.87 89.29 30.92 89.26 30.97 89.22 3K01 12 14 16 18 20 92.22 26.79 92.19 2684 92.15 26.89 92.12 26.94 92.09 26.99 91.26 28.25 91.22 28.30 91.19 28.34 91.16 28.39 91.12 28.44 90.24 29.67 90.21 29.72 90.18 29.76 90.14 29.81 90.11 29.86 89.18 31.06 89.15 31.10 89.11 31.15 89.08 31.19 89.04 31.24 22 24 26 28 30 92.06 27.04 92.03 27.09 92.00 27.13 91.97 27.18 91.93 27.23 91.09 28.49 91.06 28.54 91. OJ 28.58 90.99 28.63 90.96 28.68 90.07 .29.90 90.04 29.95 90.00 30.00 89'. 97 30.04 89.93 30.09 89.00 31.28 88.96 31.33 88.93 31.38 88.89 31.42 88.86 31.47 32 34 36 38 40 91.90 27.28 91.87 27.33 91.84 27. 3S 91.81 27.43 91.77 27.48 90.92 28.73 90.89 28.77 90.56 28.82 90.82 28.87 90.79 28.92 89.90 80.14 89.86 30.19 ,S9 83 30.23 89.79 30.28 89.76 3tf.32 88.82 31.51 88.78 31.56 &S.75 31.60 88.71 31.65 8^67 31.69 42 44 46 48 50 91.74 27.52 91.71 27.57 91.68 27.02 91.65 27.67 91.61 27.72 90.76 28.96 90.72 29.01 90.69 29.06 90.66 29.11 90.62 29.15 89.72 30.37 89.69 30.41 89.65 30.46 89.61 30.51 89.58 30.55 88.64 31.74 88.60 31.78 88.56 31.83 88.53 81.87 88.49 31.92 52 54 56 58 60 91.58 27.77 91.55 27.81 91.52 27.86 91.48 27.91 91.45 27.96 90.59 29.20 90. M 29.25 90.52 29.30 90.48 29.34 90.45 29.39 89.54 30.60 89.51 30.65 69.47 30.69 89.44 30.74 89.40 30.78 88.45 31.96 88.41 32.01 88.38 32.05 88.34 32.09 88.30 32.14 c-f/= -75 c-f /= I -00 c+/ = 1 25 .72 .21 .J6 .28 1.20 .30 .72 .23 .95 .30 1.19 .38 .71 .24 .95 .32 1.1'J .40 .71 .25 .94 .33 1.18 .42 TABLE XXI. STADIA REDUCTIONS 111 TABLE XXI. STADIA REDUCTIONS FOR READING 100 Minutes. 20 21 22 23 Hor. Diff. Dist. Elev. Hor. Diff. Dist. Eler. Hor. Diff. Dist. Elev. Hor. Diff. Dist. Elev. 0' 2 4 6 8 10 88.30 32.14 88.26 32.18 88.23 32.;>3 88.19 32.27 88.15 32.32 88.11 32. 3'ti 87.16 3:5.46 87.12 33.50 87.08 33.54 87.04 3:5.59 87.00 33.63 86.96 33.67 85.97 34.73 85.93 31.77 85.89 34.82 85.85 34.86 85.80 34 90 85.76 34.94 84.73 35.97 84.69 36.01 84.65 36.05 84 01 36.09 84.57 36.13 84.52 36.17 12 14 16 18 20 88.08 32.41 88.04 32.45 S8.00 32.4!) 87.56 32.54 87.93 32.58 86.92 33 72 86.88 33 76 86.84 3.< 80 86.80 33.84 86.77 33.89 85.72 34.98 85.68 35.02 85.64 35.07 85.60 35.11 85.56 35.15 84.48 3.?2 87.77 32.76 87.74 32.80 86.73 33 93 86.69 33.97 86.65 34.01 86.61 34.06 86.57 34.10 85.52 35.19 85.48 3.-). 23 85.44 35.27 85.40 35.31 85.36 35 36 84.27 36.41 f>4.23 36.45 84.18 36.49 84.14 3b.53 84.10 36.57 32 34 36 38 40 87.70 32.85 87.66 32.89 87.62 32.93 87.58 35.98 87.54 33.02 86.53 34.14 86:49 34.18 8645 34.23 86.41 31.27 86.37 34.31 85.31 35.40 85.27 35.44 85.23 35.48 85.19 35.52 85.15 35.56 84.06 36.C.1 84.01 36 U5 83.97 3t> 69 83.93 36.73 83.89 36 77 42 . 44 46 48 50 87.51 33.07 87.47 33.11 87.43 33.15 87.39 33.20 87.35 33.24 86.33 34.35 86.29 34.40 86.25 34.44 86 21 34 48 80.17 34. J2 85.11 35.60 85.07 35.64 85.0-.' 35.6* 84.98 35.72 84.94 35.76 83.84 36.80 83.80 36.84 83.76 36. S8 83.72 36.92 83.67 36.96 52 54 56 58 60 87.31 33.28 87.27 33.33 87.24 33.37 87.20 33.41 87.16 33.46 86.13 34 57 86.09 34.01 86.05 34.6.-) 80.01 34. C9 85.97 34.73 84.90 35.80 84.86 35.85 84.82 35.89 84.77 35.93 84.73 35.97 83. G3 37.00 83.59 37 01 83.51 37 08 83.50 37.1.' 83.46 37.16 c+/= .75 c-f/=1.00 c+/=l-25 .70 .26 .94 .35 1.17 .44 .70 .27 .93 .37 1.16 .46 .69 .29 .92 .38 1.15 .48 .69. .30 .92 .40 1.13 .50 TABLE XXI. STADIA REDUCTIONS TA^BLE XXI. STADIA SEDUCTIONS FOTC HEADING 100 Minutes. 24 25 2CT 27 Hor. Diff. Dist. Elev. Hor. Diff. Dist, Elev. Hor. Diff. Dist. Elev. Hor. ' Diff. Dist. Elev. 0' 2 4 6 8 10 83.46 37.16 83.41 37.20 83.37 37.23 83.33 37.27 83.528 37.31 83.24 37.35 82.14 38.30 82.09 38.34 82.05 38.38 82.01 38.41 81.96 38.45 81.92 38.49 80.78 39.40 80.74 39.44 80.69 39.47 80.65 39.51 80.60 39.54 80.55 39.58 79.39 40. ' 79.34 40 49 79.30 40.52 79.25 40.55 79.20 40.59 79.15 40.62 . 12 14 16 18 20 83.20 37.39 83.15 37.43 83.11 37.47 83.07 37.51 83.02 37.54 81.87 38.53 81.83 38.56 81.78 38 60 81.74 38.64 81.69 38.67 80.51 39.61 80.46 39.65 80.41 39.69 80.37 39.72 80.32 39.76 79.1.1 40.66 79.06 40. 69 79.01 40=72 78.96 40.76 78.92 40.79 22 24 26 28 30 82.98 37.58 82.93 37.62 82.89 37.66 82.85 37.70 82.80 37.74 81.65 38.71 81.60 38.75 81.56 38.78 81.51- 38.82 81.47 38.86 80.28 39.79 80.23 39.83 80.18 39.86 80.14 39.90 80,09 39.93 78.87 40.82 78.82 40.86 78.77 40 89 78 73 40.92 78'.68 40.96 32 34 36 38 40 82.76 37.77 82.72 37.81 82.67 37.85 82.63 37.89 82.58 37 ..93 81,42 38.89 81.38 38.93 81.33 38.97 81.28 39.00 81.24 39.04 .80.04 39.97 80.00 40.00 79.95 40.04 79.90 40.07 79.86 40.11 78.63 40.99 78.58 41.02 78.54 41.06 78.49 41.09 78.44 41.12 42 44 46 48 50 82.54 37.96 82.49 38.00 82.45 38.04 82.41 38.08 82.36 38.11 81.19 39.08 81.15 39.11 81.10 39.15 81.06 39.18 81.01 39.22 79.81 40.14 79.76 40.18 79.72 40.21 79.67 40.24 79.6-2 40.28 78.39 41.16 78.34 41.19 78.30 41.22 78.25 41.26 78.'20 41.29 52 54 56 58 60 82.32 38.15 82.27 38.19 82.23 38.23 82.18 38.26 82.14 38.30 80.97 39.26 80.92 39.29 80.87 39.33 80.83 39.36 80.78 39.40 79.58 40.31 79.53 40.35 79.48 40.38 79.44 40.4^ 79.39 40/45 78.15 41.3-2 78.10 41.35 78.06 41.39 78.01 41.42 77.96 41.45 .c-f/= -75 c+/ = 1.00 c+/=1.25 .68 .31 .91 .41 1.14 .52 .68 .32 .90 .43 1.13 .54 .67 .33 .89 .45 1,13 - Kfl .66 .35 .89 .46 1.11 .03 A MANUAL OF LAND SURVEYING. T = R tan. H T _ 50 tan. ^ Sin. D Sin. D = K Sin D Cl JRVE FORMUL, R = T cot. M I R E. Chord def.- cbord9a R No. chords - D Tan. def.= & chord def. Sin. D jt:x>q E = R ex. sec. H I E T tan. H I T The square of any distance, divided by twice the ra- dius, will equal the distance from Tangent to Curve, very nearly. Table XX contains Tangents and Externals to a 1 curve. Tan. and Ext. to any other radius may be found, nearly enough, by dividing the Tan. or Ext. opposite the given Central Angle by the given degree of curve. To find Deg. of Curve, having the central Angle and Tangent : Divide Tan. opposite the given Central An- gle by the given Tangent. To find Deg. of Curve, having the Central Angle and External : Divide Ext. opposite the given Central An- gle by the given External. To find Nat. Tan. and Nat. Ex. Sec. for any angle by Table XX : Tan., or Ext. of twice the given angle di- vided by the radius of a 1 curve will be the Nat. Tan. or Ex. Sec. To find angle for a given distance and deflection. Rule 1. Multiply given distance by .01745 (def. for 1 for 1 ft.), and divide given deflection by the product. Rule 2. Multiply given deflection by 57.3, and divide the product by the given distance. To find deflection for a given angle and distance: Multiply the angle by .01745. and the product by the distance. MANUAL OF LAND SURVEYING BY F. HODGMAN, M. S., C. E., Practical Surveyor and Civil Engineer. Over 500 pages, printed on strong, light paper, and bend in leather with flap. The Land Surveyor's Best Pocket Companion. PRICE, - $2.6O STAR EDITION, $3.OO The star edition differs only in the binding, which is of the best Morocco and workmanship to be had to stand hard use. SURVEYOR'S TABLES. Being the tables from the " MANUAL OF LAND SURVEYING,' bound separately. The handiest little pocket table book for stu- dents, surveyors, and mining engineers. Bound in leather, with round corners. The Star Edition is of the very best paper, bind- ing and workmanship to be had in the market, and has 33 pages of blank cross-section paper for memoranda. A premium given to the first person who discovers and reports any error in the tables. PRICE .... $1.00 5TAR EDITION 1.50 HODGMAN'S BOOK FOR SURVEYORS. FOR TAKING DOWN NOTES IN THE FIELD. Ruled In small cross-sections, and having tables of Natural Sines, Tangents, Secants, Departures, Azimuths of Polaris, Radi, and Deflections, Tangents and External Secants of a 1 Curve. Curve Formulas, Traverse Table, 176 blank pages and Index. Strong linen paper bound in red Russia, with flap and pencil holder. Single Copies by mail - - - $ -75 Per Dozen by Express - - - 7.00 The above books published and for sale by THE F. HODGMAN CO., Climax, Michigan. THE T. F. RANDOLPH co. MANUFACTURERS, IMPORTERS, AND DEALERS IN SURVEYORS' AND ENGINEERS' INSTRUMENTS AND SUPPLIES OF ALL KINDS. ROE, CHE8- TERMAN AND PAINE'S TAPES. SOLE MANUFACTURERS OF Randolph's Patent Telescope Compasses, Patent Telescope Attachment for Common Compasses, Patent (Jem Transit, Patent Quick Leveling Tripod, Patent Transit Level, Patent Daisy Level, Patent Sole Leather Boxes. 232 E, FIFTH ST., CINCINNATI, O, ESTABLISHED 1853. Send for CATALOGUED 11*3 JOGV3A5! . " . " " "' ' . ' -833 ,308 .2fHH3( JJA 10 231^18 OHA .M1AT B'MI OMA MMfl37 ' rt UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped beloi ENGINEERING JUN 8 183 21-100m-9,'48 (B399sl6) 476 YA 03069 793955 Ubrary UNIVERSITY OF CALIFORNIA LIBRARY