PRACTICAL SURVEYING 
 
 FOR 
 
 SURVEYORS' ASSISTANTS, 
 VOCATIONAL, AND HIGH SCHOOLS 
 
 BY 
 
 ERNEST McCULLOUGH, C.E. Ph.D. 
 
 Consulting Engineer; Member of the American Society of Civil Engineers. 
 
 Sometime Lieut.-Colonel, United States Army, A. E. F.; Author of 
 
 "Engineering as a Vocation 1 '; "Practical Structural Design," etc. 
 
 229 ILLUSTRATIONS 
 
 SECOND EDITION, REVISED 
 
 NEW YORK 
 D. VAN NOSTRAND COMPANY 
 
 EIGHT WARREN STREET 
 1921 
 
' . 
 
 * v : 
 
 . 
 
 i\A 
 
 Copyright, 1915, 
 By D. VAN NOSTRAND CO. 
 
 Copyright, 1921, 
 By D. VAN NOSTRAND CO. 
 
To 
 LEWIS INSTITUTE (CHICAGO) 
 
 IN RECOGNITION OF THE WORK 
 IT IS DOING 
 
 469981 
 
PREFACE TO THE SECOND EDITION 
 
 During the recent world war, into which the United 
 States entered in 1917, thousands of men had to be in- 
 structed in the principles of surveying. Not only did the 
 engineers need men of non-commissioned grade to serve as 
 instrument-men and makers of topograpnical maps, but the 
 artillery also required them. In the conventionalized 
 courses given in the conventionalized technical schools of 
 the country it had been assumed that surveying could be 
 taught only to men who had completed, in high school and 
 college, algebra, plane and solid geometry, trigonometry and 
 analytical geometry. During the war thousands of intelli- 
 gent men whose mathematical preparation did not extend 
 beyond the arithmetic given in grade schools, were trained 
 in a few weeks in the use of surveying instruments and 
 received adequate instruction in the handling of formulas 
 used by surveyors. They were not finished surveyors, for 
 their training was of a special sort, but with the elements of 
 surveying given them many were able after the close of 
 hostilities to continue their studies. The author benefitted, 
 for this book, the first edition of which appeared in 1915, 
 was the only available modern work on surveying written 
 for men who had not studied enough mathematics, higher 
 than arithmetic, to enable them to read modern texts on 
 surveying written by college and university professors. 
 
 The author in 1915 was entering upon his twenty-eighth 
 year of practice as a civil engineer, of which sixteen years 
 had been spent in the far western states in the Rocky 
 Mountains and along the Pacific Coast. He had served as a 
 town and city engineer in small and medium size munici- 
 
VI PREFACE TO SECOND EDITION 
 
 palities, as a deputy county surveyor and as a United States 
 Deputy Mineral Surveyor. He had served also as a teacher 
 of surveying in evening classes of schools operated not for 
 profit, and numbers of boys and young men appreciated the 
 instruction given them, many of whom later met with 
 considerable success in surveying and civil engineering work. 
 In that year he revised the lessons prepared for his pupils 
 and put them into book form for the benefit of others who 
 might wish to study the subject in a plainly written book. 
 That it has sold well proves his ideas to have been right. 
 The work of the war schools in surveying was additional 
 confirmation that the principles of surveying can be 
 taught to men whose knowledge of mathematics is meager. 
 
 The following specifications must be adhered to in pre- 
 paring a text book; (a) it must be accurate and clear, (b) 
 it must be well balanced, especially when intended for men 
 who will obtain little, or no, help from teachers, (c) it must 
 be readable. The author in attempting to comply with these 
 specifications proved that a book could be prepared which 
 did not lack in vigor because of the assumption of little 
 previous mathematical preparation on the part of the 
 student. It contains more than is commonly given in 
 schools and colleges where some instruction is given in 
 surveying as a part of the course in mathematics, often- 
 times under instructors who have had little or no practical 
 surveying experience. It contains more than is required as 
 a basic surveying course for non-commissioned officers of 
 artillery and the corps of engineers. 
 
 The subject is here presented in a logical manner. As- 
 suming that the student is familiar with common school 
 arithmetic all necessary instruction in mathematics is given 
 step by step as the need is felt for it, and not before. The 
 chapter on trigonometry contains what should be properly 
 considered the minimum a surveyor should possess, although 
 
PREFACE TO SECOND EDITION Vll 
 
 many surveyors do earn a living who know only a few of the 
 formulas there presented. The essentials of algebra were 
 placed in an appendix to serve merely as a useful introduc- 
 tion to the study of algebra. This appendix has been highly 
 spoken of and the author is gratified that it met with the 
 approval of men who are competent judges. 
 
 The author believes he succeeded in showing that there 
 is more to surveying than the ability to solve triangles and 
 read a vernier, important as these items are. The insistence 
 from the first chapter on the large part played by unescap- 
 able errors and the action of courts and juries in passing 
 upon the work of surveyors, gives a view of the subject that 
 is seldom obtained until after the class room is a memory 
 and actual work begins. Land surveying is emphasized, for 
 land survey methods underlie all work done by instru- 
 mentmen. Enough is given of engineering surveying to 
 help local surveyors over many hard places. The work is 
 complete in everything with which the average surveyor 
 will have to deal: titles and prices being given of books 
 which deal fully with special branches of survey work. 
 The prices given are those of before the war and may not 
 be in all cases the prices of to-day. Some however have not 
 changed and changes may go one way as easily as another, 
 so it was thought best to retain the original prices in this 
 second edition. 
 
 The first class surveyor the author has found is a respected 
 person in his community, and rightly so. Every reader of 
 this book is counselled to buy and read "Boundaries and 
 Landmarks," by A. C. Mulford, to obtain the point of view 
 of the successful surveyor who is proud of his calling, the 
 most ancient of the learned arts and sciences. To read that 
 book gives something all young men need and should 
 appreciate, the nearest possible equivalent to association 
 with an elderly, well read, kindly gentleman of broad 
 
Vlll PREFACE TO SECOND EDITION 
 
 experience in his calling. The reproach under which sur- 
 veyors suffer is that many have picked up the business in a 
 haphazard way for lack of comprehendible texts, the 
 majority of graduates in engineering preferring to engage 
 in engineering work and not liking the work of the surveyor. 
 This leaves the field then to self-tutored men working with- 
 out proper instruction, or at least instruction not as com- 
 plete as it should be. That men in this class are willing to 
 study when simply written books are placed in their hands 
 is evidenced by the reception given to the first edition of 
 this work. 
 
 Opportunity we are told is half of life. An old engineer 
 gave the writer in 1888 the following recipe for success: 
 
 Opportunity % part 
 
 Common sense J/ part 
 
 Special training J^j part 
 
 which implies, that, given opportunity, success is due to a 
 mixture of two-thirds natural ability and one-third special 
 training. This book is intended to give the special training, 
 so far as it may be given in books, to self-tutored young 
 men of natural ability who like surveying and to whom the 
 opportunity comes to obtain work in the office or field with 
 practicing surveyors. 
 
 Thousands of men have contributed to the advancement 
 of the science and art of surveying, and it is therefore not 
 possible to give credit to past writers, except to say that in 
 preparing this book the literature of the subject has been 
 well searched. Credit has been given where but one author 
 is involved, matters discussed by several writers, in this 
 subject which has filled hundreds of volumes, being assumed 
 to be common knowledge. The author believes he contrib- 
 uted nothing original, for even things which he learned 
 from experience may not have been original even though 
 
PREFACE TO SECOND EDITION IX 
 
 heretofore not mentioned In text books, but he is gratified 
 that many purchasers took the trouble to write that they 
 liked his manner of presentation and found the book of 
 considerable value. One of the most delightful incidents of 
 this nature was a meeting on the British front in the winter 
 of 1917-8, with an English officer who was studying the 
 book preparatory to following the profession of surveying in 
 Africa, when the war should be over. He went there in 
 1919, fortunately none the worse for wounds, and the 
 acquaintance thus begun promises to endure as a warm 
 friendship. 
 
 The second edition differs from the first merely in the 
 correction of all errors discovered. Readers are invited to 
 call attention to any errors which may still exist, in case 
 they discover any. Hints and suggestions for improvements 
 are invited. It is believed that six years of sales which show 
 increases from year to year, in spite of the fact that three 
 were years of war, may be considered as proof that the 
 book may be now classed as standard. 
 
 THE AUTHOR. 
 
 New York City, N. Y., September, 1921. 
 
TABLE OF CONTENTS 
 
 CHAPTER PAGE 
 
 I. INTRODUCTORY i 
 
 II. CHAIN SURVEYING 14 
 
 III. LEVELING 73 
 
 IV. COMPASS SURVEYING 100 
 
 V. TRIGONOMETRY 148 
 
 VI. TRANSIT SURVEYING 210 
 
 VII. SURVEYING LAW AND PRACTICE 290 
 
 VIII. ENGINEERING SURVEYING 33 
 
 APPENDIX A. THE ESSENTIALS OF ALGEBRA 3 6 3 
 
PRACTICAL SURVEYING 
 
 CHAPTER I 
 INTRODUCTORY 
 
 Gravitation is a natural force acting on all material 
 bodies with the effect of attracting them to each other. 
 
 Terrestrial gravitation (gravity) is the operation of the 
 law of gravitation so that all bodies within the range of its 
 influence tend to be drawn to the center of the earth. 
 That is, a straight line drawn from any point on the surface 
 of the earth in the direction of its center is said to lie in the 
 direction of gravity. In many cases gravity means simply 
 weight, which is a measure of the force of gravity acting 
 on a body. 
 
 A cord having a plumb-bob attached to the lower end 
 points in the direction of gravity and is said to define a 
 vertical line. 
 
 Differences in elevation are measured on vertical lines. 
 
 A line forming a right angle with a vertical line is defined 
 in plane surveying as being level, or horizontal. 
 
 A point has position without length, breadth or thick- 
 ness. A line has position and length without breadth or 
 thickness. A line is terminated by points at the ends; 
 it may be said to be composed of a number of points touch- 
 ing each other; it may be said to be generated by a point 
 changing position in space, the first position being the begin- 
 ning of the line, the final position of the point being the 
 other end, the line between the two points marking the 
 path traveled. This is illustrated every time a line is 
 drawn with a pencil or pen. 
 
 A curved line changes direction at each point and a 
 straight line is the shortest distance between two points. 
 
; PRACTICAL SURVEYING 
 
 A broken line consists of a number of short straight lines 
 joined end to end and changing direction at the junction 
 points. The changes in direction are angular, an angle 
 being the amount of divergence between two lines that 
 join or cross. 
 
 A surface has position, length and breadth without 
 thickness. A plane surface, or plane, is perfectly flat so 
 
 FIG. i. Types of plumb-bobs used by surveyors. 
 
 that a straight line may be drawn to connect any two 
 points in the plane and each point in the line will touch the 
 plane. 
 
 In plane surveying the portion of the earth's surface 
 measured is small in comparison with the circumference of 
 the earth, so the curvature of the earth is safely neglected, 
 an assumption which simplifies operations. Consequently 
 a horizontal plane is dealt with. 
 
 In geodetic surveying the curvature of the earth is taken 
 into account, for the operations are so extensive that 
 boundaries between nations are determined. The use of 
 geodetic methods is confined almost entirely to the highest 
 grade of Government work. Large cities are surveyed by 
 a combination of geodetic and plane surveying methods. 
 
INTRODUCTORY 3 
 
 The methods of plane surveying are used on work of the 
 following character: 
 
 Land surveys to determine boundaries of fields or lots 
 and areas of same. 
 
 Canal, road and railway surveys to determine routes to 
 be followed and the quantities of materials to be moved in 
 forming the excavations and embankments. 
 
 Construction surveys for the purpose of setting stakes 
 for the location of buildings, bridges and other structures; 
 computations of quantities, etc. 
 
 Mining surveys to guide miners in driving tunnels, shafts 
 and other workings. These are a combination of land and 
 construction survey work. 
 
 Topographical surveys made to obtain data for maps on 
 which are shown all natural physical characteristics, such 
 as the shape and heights of hills, extent of lowlands, routes 
 of rivers, streams, roads, etc., so that improvements may 
 be planned. On extensive topographical surveys a frame- 
 work may be used of lines fixed by geodetic methods, the 
 "filling in" being done by plane surveying methods. 
 
 LAND SURVEYS 
 
 The river Nile in Egypt annually overflows its banks and 
 deposits upon the adjacent bottom lands many tons of silt, 
 which, being washed down from rich land on the mountains 
 of the interior of Africa, makes the valley wonderfully 
 fertile. This silt covers the stones and 'posts set for land- 
 marks and swirling water often removes them, so that 
 from the most ancient times the boundaries of land in 
 Egypt have been surveyed annually. The science of geom- 
 etry arose in consequence according to Diodorus and others, 
 although some authorities claim the science of geometry 
 was of indigenous growth, appearing in each country as 
 the people reached a proper development. That survey- 
 ing preceded geometry is indicated by the Latin name, 
 geometria, from the Greek yeunerpia (the measurement 
 of land). The first geometers were land surveyors. 
 
 When a portion of a tract of land is sold the seller and 
 buyer proceed to " establish" the corners where the angular 
 boundary lines join and a surveyor is employed to determine 
 
4 PRACTICAL SURVEYING 
 
 the lengths and directions of the lines between corners. A 
 description of the corners, together with the directions and 
 lengths of all lines, is written in the deed conveying title 
 to the land and sometimes a map is made. Such a survey 
 is termed "original," for by it the boundaries and marks 
 are first described. 
 
 In an original survey the corners are established. This 
 can never again be done. If by any chance the marks are 
 lost surveyors are employed to "re-locate" if possible the 
 position of the corners, but no man can "re-establish" 
 anything once established. A surveyor, making a re- 
 survey, can only say that he has gone over the lines to the 
 best of his ability from information given to him and 
 believes he has set marks .as nearly as possible where the 
 original marks were placed. 
 
 A surveyor, not being an interested party, cannot pre- 
 sume to put in a new corner having the force and effect of 
 an original corner. Some time after he is gone good evi- 
 dence may be found showing that the true corner lies at a 
 considerable distance from the corner set by him. The 
 owners being the only parties having any vital interest in 
 the matter may destroy, if they wish, all evidences of the 
 re-survey and abide by the boundaries originally fixed. 
 The surveyor however should be notified so he may alter 
 his notes. If this is not done trouble may arise in the 
 following generation. 
 
 Sometimes, after several unsuccessful attempts to re- 
 locate a missing corner, the owners affected agree to accept 
 a certain point in lieu of the corner originally set. This, 
 however, can be done only by mutual agreement duly 
 recorded and it does not constitute a "re-establishing" of 
 a corner, but is the "establishing" of a new corner. If the 
 agreement is not a mutual one between all the parties at 
 interest and no proper record is made of it, a discovery at a 
 future time of the original corner may cause trouble. In 
 some states corners mutually held to become fixed after 
 the lapse of a certain number of years, but this does not 
 act always as a final settlement. In the absence of records 
 it is easy for some person to deny that the agreement was 
 mutual. There may be unearthed records to show that 
 the agreement was based on fraud, so it is better to have a 
 
INTRODUCTORY 5 
 
 carefully drawn up agreement placed on record than to 
 depend on the statute of limitations. The fact a student 
 must never forget is that no surveyor, by virtue of any 
 official position he may hold, can "fix," "re-establish," or 
 "establish," any old corner. 
 
 A surveyor on an original survey is employed to describe 
 the intentions of a grantor in selling and of a grantee in buy- 
 ing. A surveyor on a re-survey tries to retrace the lines 
 run by the surveyor on the original survey. If the first 
 man did his work well no troubles should develop even 
 though nearly all the original corner marks have been 
 destroyed. If the first surveyor made mistakes trouble 
 cannot be avoided unless the second surveyor possesses a 
 good knowledge of surveying, a good knowledge of court 
 decisions affecting surveys, and plenty of common sense. 
 
 It is a physical impossibility to make surveys positively 
 free from error, but by proceeding carefully and taking all 
 the time that is necessary errors may be reduced to a small 
 amount which will be satisfactory to all concerned. Good 
 work takes time and surveyors are generally paid by the 
 day. It follows that the accuracy of surveyors' work must 
 be governed by the value of the land surveyed. Various 
 authors give limits of accuracy based on experience, which 
 the author a number of years ago reduced to the following 
 rule : 
 
 The limit of error in land surveys may be expressed by a 
 fraction having a numerator of I and a denominator equal to 
 one-tenth the value of the land per acre expressed in cents, the 
 maximum limit being -gkv, corresponding to a value of $50 
 per acre. 
 
 When a compass is used to obtain directions and a chain 
 in the hands of unskilled men is used to measure distances 
 the error may be as large as I in 500. A very little expe- 
 rience will enable men to do better work, so this limit of 
 error should never be considered satisfactory. A limit of I 
 in 750 should be attempted. 
 
 The limit of error for land worth $100 per acre will be 
 i in looo by the above rule and for land worth $1000 per 
 acre the limit of error will be I in 10,000. An error of I in 
 20,000 is permissible in the business district of any large 
 city, except possibly in the hearts of cities having a popula- 
 
6 PRACTICAL SURVEYING 
 
 tion of more than 1,000,000 where a limit of I in 50,000 may 
 be worth attempting. 
 
 An old legal maxim reads, " Monuments govern courses 
 and courses govern distances." This arose from the fact 
 that monuments are marks set by the grantor as visible 
 evidences of intention. The original survey was an oper- 
 ation performed to obtain data for a record of intention. 
 It was not considered that chaining was a particularly 
 skilled avocation and untrained men usually did such 
 work. It was not uncommon to have the owner and one 
 of his sons act as chainmen in order to impress upon the 
 memory the location of the corners. In Great Britain 
 "whipping the bounds" has not entirely gone out of style, 
 and this was as common in the early history of America as 
 it was in older countries. On a certain day in each year, 
 usually just after the spring planting, boys were taken 
 around every piece of property and whipped soundly at 
 each corner after the names of the owners whose lands the 
 fences and corner determined, were read to them. The 
 evidence painfully so secured was often invaluable in law- 
 suits years later, sometimes when the whipped boys were 
 almost in their second childhood. In the early days chain- 
 ing, therefore, was considered to be on a par with the read- 
 ing of bearings for surveyors were not always well educated 
 and many knew little, or nothing, about the variation 
 (declination) of the needle. The effect of local attraction 
 was generally ignored ; compasses were not always well 
 made and readings were seldom taken closer than the 
 nearest half degree of bearing. 
 
 Nevertheless, with all the shortcomings of the angle 
 work the bearings were given first consideration for they 
 at least gave a close idea of direction, without which the 
 most careful measurement is worthless. 
 
 The compass is not used today for surveys where land 
 is valuable for readings cannot be taken closer than one- 
 quarter of a degree with the best made compass. When 
 skilled chainmen are employed the errors in chaining are 
 apt to be much smaller than the errors in angle. Modern 
 instruments are well made and angles can be read to half 
 minutes on even the lower-priced transits, while many 
 men have instruments graduated to read to ten seconds, 
 
INTRODUCTORY 7 
 
 twenty seconds being the usual degree of accuracy for en- 
 gineering surveys. Modern surveyors are better trained 
 than the surveyors of preceding generations, so the in- 
 strument work leaves little to be desired in point of ac- 
 curacy. Errors today are most apt to occur in the chaining, 
 for only by experience can men be trained to measure 
 with a chain or tape. Young surveyors should impress 
 this fact upon employers who want to save expenses, and 
 object to paying a good price to an experienced chainman. 
 The picking up of some unemployed laborer to do the 
 measuring is a fruitful cause of lawsuits. 
 
 In reading angles the plate of the instrument must be 
 level, and the chain or tape must be level and be drawn 
 
 ' . '\ H&& /,T4W f ' , .' S' t* 
 
 FIG. 2. Measuring on sloping ground. 
 
 taut. The chainmen are carefully "lined in" by the in- 
 strument man in order that the measured line may be a 
 straight, not a broken, line. Each man has a plumb-bob 
 on a cord by means of which he fixes each tape length, for 
 the tape must be held high enough to clear all bushes, 
 stones, etc. On level ground the tape is usually held 
 about the height of the waist. On sloping ground one 
 man holds his end as close as possible to the ground, the 
 other man raising his end until the tape is level, letting 
 the plumb-bob line slide between his fingers slowly until the 
 point of the plumb-bob touches the ground. When the 
 
8 
 
 PRACTICAL SURVEYING 
 
 slope is steep short measurements are taken. In meas- 
 uring up or down hill in short sections like steps the addi- 
 tive process is used to avoid errors. For 
 example assume it is necessary to meas- 
 ure in lengths of approximately 20 ft. 
 The head chainman first pulls the tape 
 (or chain) ahead the full length, then 
 returns to the 20 ft. mark and sets a pin 
 while the rear chainman holds his end on 
 the starting point. Then the rear chain- 
 man drops his end and goes forward to 
 the head chainman, gives him a pin in 
 place of the one just set, holds at the 
 20 ft. mark while the head chainman 
 goes to the 40 ft. mark, when the proc- 
 ess is repeated. In this manner the full 
 length is measured without any adding 
 being done mentally. When the full tape 
 length is set off the number of pins held 
 
 ~_ ' Q by the rear chainman represents the 
 
 FIG. 3. Survey pins. J t f r ,, , i < i 
 
 number of full lengths, for the pins used 
 
 temporarily for the short lengths have been exchanged 
 each time. 
 
 In commencing to measure a line a steel or iron pin is 
 stuck in the ground, or, if 
 the mark is plain, the pin 
 is held by the rear chain- 
 man. When a tape length 
 is measured the head chain- 
 man sticks a pin in the 
 ground at the point marked 
 by the plumb-bob. An- 
 other length is then meas- 
 ured, the rear chainman 
 pulling up the pin at his 
 end after the head chain- ^ 
 man has set his pin. The 
 number of pins held by the 
 rear chainman always indicates the number of tape, or 
 chain, lengths measured, the pin in the ground not being 
 counted for it merely fixes the point from which to measure. 
 
 FIG. 4. 
 
INTRODUCTORY 9 
 
 The head chainman starts with ten pins, the total number 
 being eleven. When ten lengths have been measured the 
 rear chainman goes forward and hands ten pins to the head 
 chainman and they check by counting the pins, making a 
 record of the "tally," so that the correct number of tape 
 lengths can be told when the line is fully measured. The 
 head chainman holds the zero end of the tape and thus the 
 number of feet, or links, past the last full tape length may 
 be read directly. Pins usually have a length of about one 
 foot and are stuck in the ground slanting to one side so the 
 plumb-bob may be held over the point where the pin enters 
 the ground. In order to find the pins readily it is customary 
 to tie white, or brightly colored, strips of cloth in the ring 
 at the upper end. 
 
 Correct measuring cannot be done when weighted pins 
 are used instead of plumb-bobs, although weighted pins 
 were formerly used. A pin cannot be accurately dropped, 
 no matter how well weighted. 
 
 ROUTE SURVEYS 
 
 Route surveys for roads, ditches, railways, etc., are 
 marked by stakes driven in the ground at each tape length, 
 the distances between stakes 100 ft. apart being termed 
 "stations," the numbering of the stations proceeding from 
 zero. Each stake being numbered, the 
 number on any stake multiplied by 100 
 gives the distance in feet from Station o, 
 the starting point. 
 
 In land surveys only the distance 
 between corners is wanted so pins are 
 used merely as counters of tape lengths. 
 In route surveys the elevation of the 
 ground at each station point is required _ 
 for the purpose of fixing grade lines and FlG ' 5 nes ^ t b ak a e nd wlt ' 
 computing quantities of earthwork, there- 
 fore the stakes are numbered and left in the ground for 
 the use of the leveler and slopeman. These stakes are set 
 slanting for the same reason that pins are thus set. At 
 stations where the instrument is placed for the purpose of 
 reading angles a square stake, called a hub, is driven flush 
 
10 PRACTICAL SURVEYING 
 
 with the ground and a tack driven in the top to mark the 
 point exactly. A stake is driven to one side as a witness 
 and on the witness stake the necessary identification marks 
 are placed. 
 
 After the stations have been set, by the transit (or com- 
 pass) party, elevations of the ground are taken at each stake 
 and recorded in a book. These elevations when plotted on 
 ruled paper give a "profile" of the route and on the profile 
 is fixed the grade line. The slope of the ground to the 
 right and left of each station is also measured and from the 
 slope-notes are plotted cross-sections from which earth- 
 work quantities are computed after a grade is adopted. 
 
 MEASURING ON SLOPES 
 
 If all ground sloped the same degree all measurements 
 might be made on the surface and no errors would arise. 
 Since some ground is level and some is very steep, with 
 many degrees of slope between, it is necessary to adopt a 
 standard, so 100 ft. will be 100 ft., no matter what the 
 nature of the ground. This is accomplished by holding 
 the tape level and taut. A level line is parallel to the sur- 
 face of still water and as the earth is practically a sphere 
 a level line is really the circumference of a great circle. The 
 diameter of the circle is so great and the distances, com- 
 
 FIG. 6. McCullough tape level. 
 
 paratively, so short that horizontal lines are assumed for 
 all practical purposes to be level, a horizontal line being 
 perpendicular to a vertical line. 
 
 Unskilled men experience considerable difficulty in hold- 
 ing a tape level. The error is cumulative and the sloping 
 tape measures " short" each time, so the total length is re- 
 corded as greater than it is actually marked on the ground. 
 A number of devices are used to assist inexperienced chain- 
 men, the author, in 1892, patenting a small level for the 
 
INTRODUCTORY 
 
 II 
 
 purpose. The chief merit of this device is that it may be 
 carried in the vest pocket and be quickly put on or taken 
 off any tape. The patent having expired a number of 
 instrument dealers now advertise similar levels, a few 
 crediting the inventor by name. The level is placed on the 
 tape about one foot from the end and the tape is pulled tight. 
 By first experimenting with the tape and level on a level 
 surface, as a floor, the positions of the bubble for different 
 tape lengths and different amounts of sag are noted. With- 
 out some such device a tape may be held horizontally as 
 follows: While the rear chainman holds his end stationary 
 the head chainman raises and lowers his end until he ascer- 
 tains the longest distance which can be obtained, each man 
 using a plumb-bob. On route surveys it is usual to give 
 each man a hand level, the proper use of which will be ex- 
 plained later. 
 
 The numbering of stations has been described. Stakes 
 set between are marked with the number of feet from the 
 
 7 
 
 
 Needle 
 
 
 
 +50 
 
 
 
 
 
 +30 
 
 
 
 
 ""*" Stream 
 
 6 
 
 
 
 
 BlUft 
 
 +59 
 
 
 
 Uj 
 
 <o. 
 
 +50 
 
 
 
 ^ 
 
 352 
 
 5 
 
 
 
 5: 
 
 Jyv 
 
 +67.3 
 
 L ZTIl'R 
 
 N.4Z52'E. 
 
 
 2 
 
 + 50 
 
 
 
 
 
 4 
 
 
 
 - U' 
 
 
 +50 
 
 
 
 \ & 
 
 
 3 
 
 
 
 \ * 
 
 75"&&x 
 
 +50 
 
 
 
 \ 5s 
 
 ^j?& 
 
 2 
 
 
 
 j 30' 
 
 Sg 
 
 +50 
 
 
 
 
 & 
 
 1 
 
 
 
 
 "SZ. 
 
 +50 
 
 
 
 q, , 
 
 -T 
 
 
 
 
 N.I54I'E. 
 
 ^S 
 
 
 \^ 
 
 
 
 
 _y 
 
 FIG. 7. Proper methods for keeping notes of line survey. 
 
 preceding station, preceded by a + sign. Thus 4 + 50 
 indicates a point on the line 450 ft. from the starting point. 
 It is read "Station 4 plus 50." The transitman enters his 
 
12 PRACTICAL SURVEYING 
 
 notes in a book starting on the bottom line on the left- 
 hand page, the notes reading from the bottom up. On the 
 right-hand page the center line represents the line of survey 
 and sketches may be made with objects shown in the proper 
 relative positions. A leveler makes no sketches so stations 
 are numbered from the top down in his field book in order 
 to add and subtract readily. 
 
 Fig. 7 shows some notes of a transit survey. At Sta. 
 o a hub is set and the A mark placed on line indicates an 
 instrument point. The correct bearing is set down in the 
 third column and the reading given by the needle is placed 
 on the sketch line of the survey as a check on the "right" 
 (R) and "left" (L) readings. The needle is always read 
 to check angle readings taken on the horizontal plate. At 
 Sta. 4 + 67.3 a change in direction occurred and this was 
 shown on the sketch line in order to give the draftsman a 
 check, but the sketch line is always kept straight. Bearings 
 are obtained from preceding angles by addition or subtrac- 
 tion. 
 
 When very accurate work is required all measurements 
 are made parallel to the surface of the ground, between 
 marks in the tops of hubs driven at each tape length. 
 With a level and rod the elevations of the tops of the hubs 
 are taken. Then 
 
 H= 
 in which expression 
 
 H = horizontal length, 
 L = length on slope, 
 
 D = difference in elevation between ends of tape on 
 slope. 
 
 (Note. In all problems involving squares or square 
 roots of numbers use the table of squares in Chapter II. 
 Information on the solving of formulas is given in Ap- 
 pendix A.) 
 
 Problem. A line was measured on the surface of the 
 ground with a tape 50 ft. long, hubs being set at each tape 
 length. Elevations were taken on each hub and the differ- 
 ences in elevation thus obtained. What is the horizontal 
 length of the line? 
 
INTRODUCTORY 
 
 Stations. 
 
 Elevations 
 above base. 
 
 Difference 
 in elevations. 
 
 Stations. 
 
 Elevations 
 above base. 
 
 Difference 
 in elevations. 
 
 O 
 
 93-2 
 
 0.0 
 
 + 50 
 
 109. 2 
 
 0.2 
 
 +50 
 
 95-6 
 
 2-4 
 
 4 
 
 II4-5 
 
 5-3 
 
 i 
 
 / 98.2 
 
 2.6 
 
 + 50 
 
 115.2 
 
 -7 
 
 + 50 
 
 101 .4 
 
 3-2 
 
 5 
 
 115.8 
 
 0.6 
 
 2 
 
 105. 1 
 
 3-7 
 
 +50 
 
 117.0 
 
 I. 2 
 
 +50 
 
 108.1 
 
 3-o 
 
 6 
 
 119.4 
 
 2-4 
 
 3 
 
 109.0 
 
 0.9 
 
 +50 
 
 124-3 
 
 4-9 
 
 ANGULAR MEASURING ON SLOPES 
 
 In rough country where extreme accuracy is not re- 
 quired and speed is essential it is a common practice to 
 measure on the slope. Sometimes the tapes used for this 
 work are several hundred feet long. They are held at a 
 height to go over small brush, etc., and plumb bobs at 
 each end transfer the length to the ground. With a clinom- 
 eter, or transit, the vertical angle is measured from the 
 horizontal. The secant of an angle is the length of the 
 hypothenuse of a triangle having a horizontal base = I. 
 Take from a table of secants (pp. 194 200) the secant of 
 the angle; then 
 
 in which 
 
 H = horizontal distance. 
 L = length on the slope. 
 A angle of slope. 
 
 The following table shows deductions to make in feet for 
 each one hundred feet measured on a slope provided the 
 surveyor has an instrument for obtaining the angle of 
 slope. Column A shows the angle and column B the 
 number of feet to be deducted. 
 
 A 
 
 B 
 
 A 
 
 B 
 
 A 
 
 B 
 
 A 
 
 B 
 
 1 
 
 0.015 
 
 6 
 
 0.548 
 
 11 
 
 1.837 
 
 16 
 
 3.874 
 
 2 
 
 0.061 
 
 7 
 
 0-745 
 
 12 
 
 2.185 
 
 17 
 
 4.37 
 
 3 
 
 0.137 
 
 8 
 
 0-973 
 
 13 
 
 2.563 
 
 18 
 
 4.894 
 
 4 
 
 0.244 
 
 9 
 
 1.231 
 
 14 
 
 2.970 
 
 19 
 
 5.448 
 
 5 
 
 0.381 
 
 10 
 
 i-5 J 9 
 
 15 
 
 3-407 
 
 20 
 
 6.031 
 
CHAPTER II 
 CHAIN SURVEYING 
 
 Steel tapes have practically supplanted chains for meas- 
 uring, but the word chain will be used for years to come 
 and the men who make linear measurements will no doubt 
 always be called chainmen. The term "chain surveying" 
 applies only to surveying done without the use of instru- 
 ments to measure angles or read bearings. 
 
 The surveyors' chain was invented by Edward Gunter in 
 1620 and is often referred to as Gunter's chain. It is 
 66 ft. long and is divided into 100 links. One acre contains 
 
 1 60 square rods and the in- 
 vention of the chain by 
 Gunter simplified area calcu- 
 lations. Ten square chains 
 equal one square acre, so the 
 division of the chain into 100 
 equal parts was a practical 
 demonstration of the value 
 and simplicity of the decimal 
 system of notation. In com- 
 mon units one link is equal 
 to 0.66 ft. or 7.92 inches. 
 The surveyors' chain is four rods long, the rod being an 
 old unit for land measurement in Great Britain. An old 
 legend describes the origin of the rod for this purpose as 
 due to the fact that in olden times farmers carried ox- 
 goads, or rods, which were conveniently used for measur- 
 ing land and a royal decree fixed a standard length. In 
 common units of today the length of the rod is sixteen and 
 one-half feet, this being the rod used by Gunter. One 
 mile contains eight furlongs and a furlong has a length of 
 forty rods. Gunter divided the furlong into ten chains, 
 which thus made the chain four rods long. 
 
 14 
 
 FIG. 8. Surveyors' chain. 
 
CHAIN SURVEYING 15 
 
 As long as the acre and mile remain in our system of 
 measurements the Gunter divisions will be retained, but 
 as time passes more men will use the loo-ft. tape with 100 
 links each one foot long. Every piece of land in the United 
 States was originally surveyed with a Gunter chain, so to 
 reduce the descriptions to modern terms involves consider- 
 able labor and already has given rise to mistakes. In this 
 work the word "chain" will refer to the surveyors', or 
 Gunter, standard with a link of 7.92 ins. as the unit. The 
 word "tape" will refer to the engineers' standard, a tape 
 100 ft. long with the decimally divided foot as a unit. 
 
 The surveyors' standard for land measurement made in 
 the form of a chain is heavy and the sag in consequence is 
 so great that each length is less than it should be, con- 
 sidering a horizontal line to be a universal standard. When 
 a standard is short it is applied oftener than necessary 
 when measuring a line, thus the recorded length is too great. 
 Some manufacturers tried to overcome the effect of sag 
 by making each chain a trifle long, or by having one link 
 at each end attached to a threaded bar by means of which 
 the chain could be lengthened to overcome the effect of 
 sag: or to shorten the chain when the rings and ends of 
 links became worn. Some chains of very light steel have 
 links connected end to end but this arrangement lacks flex- 
 ibility. It is usual to make the links less than 7 ins. 
 long and connect them by using two rings between, which 
 provides 600 wearing points. When a chain has three rings 
 between the links there are 800 wearing points. A slight 
 pull has a tendency to open the joints and in the better 
 grade of chains all joints are closed by brazing. 
 
 When a standard is long it is not applied as often as neces- 
 sary when measuring a line, so the recorded length is 
 short. The weight of a chain tending to make it short and 
 the wearing of the joints tending to make it long there 
 never was a definite standard of length in the days when 
 the surveyors' unit of measure was made in the form of a 
 chain. Adjusting screws at the ends only affected the 
 whole length and fractional chain lengths were never 
 correct. Links frequently became bent and the small rings 
 were often snarled, thus shortening the chain. The numer- 
 ous joints made it difficult to pull a chain through brush 
 
l6 PRACTICAL SURVEYING 
 
 and places filled with small obstructions. Fractional parts 
 of links had to be estimated, so for close measurements 
 surveyors carried rods ten links long with the links divided 
 decimally, to measure the ends of lines, the poles being 
 convenient for sighting purposes. In cities where a chain 
 
 FIG. 9. Contact rod. 
 
 would be out of place it was customary up to quite recent 
 times to measure with contact rods. These were made 
 from properly seasoned wood and were shod with brass at 
 each end. The length was a matter of individual prefer- 
 ence, usually ten feet. In the top was inserted a spirit 
 level, in some rods a level being placed at each end. Two 
 or more rods were used, being placed carefully end to end. 
 A later type had marks near the ends and plumb-bobs were 
 used with one rod, instead of using several rods placed end 
 to end. 
 
 Tapes were used for many years in Europe before they 
 were adopted in the United States. Since 1880 tapes have 
 
 (6) (c) 
 
 FIG. 10. Steel tapes: (a) Graduated in links, (b) Graduated in feet and 
 tenths, (c) Graduated in feet and hundredths. Open reels protect 
 against rusting of tapes. 
 
 so grown in favor that the old-style chain is regarded as a 
 curiosity. Tapes are much lighter than chains so the sag 
 is less. The length is standard at a temperature of practi- 
 cally 60 F. with the tape lying on a level surface with a pull 
 of 10 Ibs. applied at the ends. The coefficient of expan- 
 sion for steel is 0.0000067 the length for one degree F. so 
 
CHAIN SURVEYING 17 
 
 that a tape 100 ft. long at a temperature of 60 F. will 
 be 99.981 ft. long at a temperature of 32 F. and will be 
 100.0134 ft. long at 80 F., the formula being 
 
 L t =Z,(id=Cr), 
 in which 
 
 L t = length of tape at assumed temperature, 
 L = length of tape at standard temperature, 
 C = coefficient = 0.0000067, 
 
 T = difference in degrees between standard and as- 
 sumed temperature. 
 
 When a tape is sold the maker should state the temper- 
 ature and pull at which the length is standard. The 
 United States Bureau of Standards, Washington, D. C., 
 will test and certify tapes for a small fee. When work of a 
 very important character is to be performed by an engineer 
 or surveyor the temperature should be read and the proper 
 correction applied for each measurement. The effect of 
 temperature is to shorten a tape when cold and lengthen it 
 when warm. Errors continually alter in amount and direc- 
 tion during the day but they practically balance in ordi- 
 nary work. The most accurate measurements are made 
 during cool foggy weather or at night, the latter being the 
 time selected by Government surveyors for the highest- 
 grade work. The "Invar" tape is made of an alloy of 
 nickel and steel possessing a very small coefficient of ex- 
 pansion. It is used on work of the highest grade. 
 
 The effect of sag is to shorten measurements and being 
 constant is important. When a new tape is purchased it 
 should be laid on a level floor or walk at a temperature as 
 near the standard as possible. A spring balance should be 
 fastened to one end and the tape pulled until the standard 
 pull is indicated, the ends being carefully marked on the 
 level surface. Two men should now hold the tape at a 
 definite height above the level surface and, holding a plumb- 
 bob at each end, pull the tape until the points of the bobs 
 strike the marks, thus eliminating sag. The pull then 
 registered on the spring balance will be from 50 to 100 per 
 cent greater than the standard pull and should be the pull 
 used with this tape thereafter. Only on exceedingly im- 
 portant work is it necessary to use a spring balance, a chain- 
 
l8 PRACTICAL SURVEYING 
 
 man after a few tests becoming expert in applying a pull 
 close enough for ordinary work. Having obtained the 
 proper pull as described, this becomes the standard pull 
 for this particular tape in order to eliminate the effect of 
 sag. There will be some sag with any less pull and the tape 
 will measure short of the proper length. By using ordinary 
 care no correction is required to overcome the effect of sag 
 due to lack of tension but a wind blowing a tape to one side 
 makes it curve, or deflect, and this is a case calling for a 
 correction on important work. The deflection must be 
 obtained and the correction added to the measured length. 
 
 Let C = correction for deflection or sag in feet, 
 d = amount of deflection or sag in feet, 
 L = length of tape in feet, 
 
 r 
 
 then C = = 
 
 3^ 
 
 When a pull greater than the standard pull is used the 
 tape will stretch. The correction must be subtracted from 
 the measured length, on important work. The standard 
 pull here referred to is the new standard obtained by meas- 
 uring with the suspended tape above a level surface. 
 
 LetP = difference in pounds between the standard and 
 
 actual pull, 
 
 A = area of cross-section of tape in square inches, 
 E = modulus of elasticity of steel = 30,000,000, 
 L = length of tape in feet, 
 S = stretch of tape in feet, 
 
 then 
 
 ERRORS IN MEASUREMENT 
 
 Errors in measurement are as follows: 
 
 i. The tape may not be held in a horizontal position. 
 This shortens the length and the error is constant and 
 "cumulative." A cumulative error is one that grows by 
 repetition so errors of this sort cannot be too carefully 
 guarded against. It requires long practice before men 
 learn to hold a tape with the ends at the same elevation, 
 
CHAIN SURVEYING 19 
 
 so work done with inexperienced chainmen is always faulty. 
 A level on the tape, or a hand level, is desirable until expe- 
 rience is gained. 
 
 2. The tape is not held with a constant, or with a proper, 
 tension. This causes an undue amount of sag, the error 
 being cumulative. 
 
 3. Chainmen are not carefully kept to line ("lined in"). 
 This gives a measured broken line instead of a straight line, 
 the error being cumulative. 
 
 4. The plumb-bobs may be too light. The proper 
 weight for a plumb-bob is one of the points on which it 
 is easy to raise discussion between practical surveyors. 
 Wind delays work when light plumb-bobs are used and the 
 error is more likely to be cumulative than compensating 
 (that is, balancing). A light plumb-bob weighs less than 
 one pound. The writer prefers plumb-bobs weighing not 
 less than one and one-half pounds with a decided prefer- 
 ence for a two-pound bob. 
 
 5. Pins may be placed ahead, or back, of the point. If 
 ordinary care is used this error will be compensating, for 
 with 'a series of repetitions the number of pins placed ahead 
 will equal the number placed back of a point, the law of 
 averages applying in this case. The error is only important 
 on short lines. 
 
 6. A mistake may be made in counting pins, thus omit- 
 ting a chain or tape length. This often happens and can 
 only be avoided by careful attention to the work, each 
 man checking the count of the other at the ends of " tallies. " 
 
 7. The chain or tape may not be of standard length. 
 The length should be tested on a level surface side by side 
 with a standard in order to determine a proper correction. 
 The measure may be short or long. 
 
 Let M = length of line as measured with faulty tape or 
 
 chain, 
 T = true length provided the line were measured 
 
 with the standard, 
 / = assumed length of tape or chain (standard 
 
 length), 
 a = actual length, 
 
 then r 
 
20 PRACTICAL SURVEYI-NG 
 
 Example. A line was measured and the length recorded 
 as 1122.5 ft. The tape was later discovered to be 99.8 ft. 
 long instead of 100 ft. What is the true length of the line? 
 
 CORRECTING ERRONEOUS AREAS 
 
 A tract of land is sometimes surveyed with a chain or 
 tape not of standard length and the area is computed before 
 the error is discovered. It is not necessary to re-compute 
 the area by first correcting each line. The correct area may 
 be obtained by making use of the geometrical proposition 
 that similar polygons are to each other as the squares of 
 the like sides. 
 
 Let A = true area of field, 
 
 C = computed area of field, 
 / = standard length of chain or tape, 
 a = actual length of chain or tape, 
 
 then 
 
 PROBLEMS 
 
 1. A line was recorded as having a length of 43.80 chains 
 but the chain was discovered to be 97 links long. What 
 was the true length of the line? 
 
 2. A tape having 0.15 ft. broken off one end was used 
 to measure a line which was reported as a result of the 
 measurement to be 1345.2 ft. long. Assuming that the 
 broken tape was used as if it were 50 ft. long what was 
 the actual length of the line? 
 
 3. A tape standard at 70 F. with a pull of 18 Ibs. when 
 freely suspended was used to measure a line 2910 ft. long. 
 What length was returned by the surveyor who used the 
 same tape with a pull of 26 Ibs. at a temperature of 96 F. 
 and made no corrections? 
 
 4. A tape 200 ft. long was used to measure a line 7800 ft. 
 long on a windy day with a deflection of 0.73 ft. for each 
 tape length. What was the length of the line as returned 
 
CHAIN SURVEYING 
 
 21 
 
 by the surveyor who made no allowance for sag caused 
 by wind? 
 
 5. A line was measured with a ipo-ft. tape and the 
 length returned as 6700 ft., a stake being set at each tape 
 length. The lining-in was done by the chainmen. After- 
 wards an instrument was used to project a perfectly straight 
 line, when half the stakes were found to be 6 ins. on one 
 side and half were found to be 7 ins. on the opposite side 
 of the line. What was the actual length? 
 
 6. The area of a field was returned as 48.9 acres. The 
 tape was later found to be half a link too long. What was 
 the true area? 
 
 7. A chain 98.5 links long was used to measure a field, 
 the area of which was known to be 
 
 93.2 acres. What was the area as 
 found with the defective chain? 
 
 8. A field known to contain 83.4 
 acres was measured with a defec- 
 tive chain and the area given as 
 being 80.7 acres. What was the 
 actual length of the chain? 
 
 RANGING LINES 
 
 Surveyors use poles of wood sev- 
 eral feet long for ranging lines and 
 for sights. These poles are shod 
 with a metal point on one end and 
 are painted with one foot spaces 
 alternately red and white. The top 
 diameter is usually one inch and 
 the diameter at the bottom is one a 
 and one-half inches. For use with FIG. n. 
 instruments having telescopes the 
 sighting poles are of steel five- 
 eighths of an inch in diameter. Steel poles are called line 
 rods. For very accurate work line rods have a relatively 
 short point at one end for setting hubs, the other end having 
 a long point used in setting tack points on hubs. 
 
 To range a line with poles. Three or more points are 
 required. One man can do the work but time is saved 
 
 Sighting poles and 
 line rods. 
 
22 PRACTICAL SURVEYING 
 
 when two or more men are used. The general direction of 
 the line being chosen a pole is set on line at any distance 
 from the starting point and a second pole is set ahead in line 
 with the first pole and the starting point. The first pole 
 is then carried ahead and set in line with the second pole 
 and the starting point, this operation being repeated until 
 the starting point cannot be plainly seen. A third pole is 
 then used and the line carried forward by moving one pole 
 ahead and setting it in line with the two standing poles. 
 A small stake is driven in each hole as the poles are lifted. 
 It is better to measure the line and set the stakes at regular 
 intervals. In setting the poles a plumb line should be used 
 to insure verticality. 
 
 Assume that a line has been ranged with poles with the 
 intention that it shall strike a certain point and it fails to 
 do so. The offset distance is measured from the point to 
 the line and each stake must be moved proportionately. 
 In Fig. 12 the random line from A to C was ranged with 
 
 FIG. 12. Illustrating random line. 
 
 poles, the intention being to strike the point B which was 
 missed by 60 links, the total length being 5 chains and 75 
 links. At the end of each chain a stake was set and these 
 stakes must be moved over to the true line. What correc- 
 tion is to be made at each stake? 
 
 Solution. - = 10.435 links per chain. The stakes 
 
 will be moved 10.43 links at I ; 20.87 nnk s at 2; 31.31 links 
 at 3; 41.74 links at 4 and 52.175 links at 5, stake C being 
 discarded. 
 
 To pass obstacles. Measure to one side of the first pole 
 a distance sufficient to enable a line to pass the obstacle. 
 Measure an equal distance from the third pole and set a 
 middle pole between these two by sighting. There will be 
 three poles set in a straight line parallel to the line first 
 
CHAIN SURVEYING 23 
 
 started. This second line is ranged forward until the ob- 
 stacle is passed when the original line is resumed by meas- 
 uring back to the line from two poles set in line on the 
 offset line beyond the obstacle. 
 
 To range a line over a hill. Let A and J3, Fig. 13, be 
 on the line. One man stands at C, where he can see both 
 A and B. A second man is then lined in to d, between c 
 
 *':L 
 
 FIG. 13. Ranging a line over a hill. 
 
 and A. Standing at d and facing B, the second man lines 
 the first man in to e, on line between d and B. From e 
 the first man then lines the second man to/, on line between 
 e and A. In this manner the poles are successively brought 
 closer to line at each operation until finally one is set at j 
 and the other at k on the line from A to B. 
 
 Further use of poles and tapes calls for the exercise of 
 considerable ingenuity and the problems should be first 
 worked out on a drawing board. When the principles are 
 mastered the work may be repeated on a larger scale in 
 the field. 
 
 FIG. 14. T square. 
 PRACTICAL GEOMETRY 
 
 The student should have the following drafting tools : 
 
 1. A drawing board of soft pine, f in. thick, preferably 
 1 8 ins. by 24 ins. in size. All edges should be straight and 
 true, the lower left-hand corner being a perfect right angle. 
 
 2. AT square with a blade 24 ins. long. The best have 
 transparent celluloid edges and the heads are fixed. A T 
 
'PRACTICAL SURVEYING 
 
 square has a head to slide along the edge of a drafting 
 board so lines ruled along the edge of the blade as the head 
 is moved will be parallel. The head should form a true 
 right angle with the blade and if the edges of the board are 
 true and two of the edges form a perfect right angle, lines 
 drawn along the edge of the T square will form right angles 
 when the head is placed first against one side and then 
 against a side perpendicular to the first. 
 
 For architectural and mechanical 
 drafting and for map drafting where 
 all angles are right angles, or are 
 readily drawn from a horizontal or 
 vertical line as a base, a T square 
 is useful and well-nigh indispensa- 
 ble. For the surveyor a straight- 
 edge is better than a T square. All 
 
 lines do not so run that right angles 
 15. urattsmens' tri- are f ormed and a h eavy steel 
 
 straight-edge may be laid anywhere 
 on a drafting board in any direction and it will not be dis- 
 turbed when triangles are slid along the edge, provided 
 proper care is exercised. 
 
 3. A 3O-deg. triangle, 8 ins. long. 
 
 4. A 45-deg. triangle, 8 ins. long. 
 
 The three interior angles of a triangle equal two right 
 angles. A 3O-deg. triangle contains one right angle, one 
 
 30'x 60*90' 45x45x9i 
 
 FIG . i ^ . D raf tsmens ' 
 angles. 
 
 FIG. 16. Engineers' triangular scale. 
 
 angle of 30 deg. and one angle of 60 deg. A 45-deg. trian- 
 gle contains one right angle (90 deg.) and two 45-deg. 
 angles. Transparent celluloid triangles are better than 
 triangles of rubber or of wood. 
 
 5. An engineers' triangular scale. The six edges are 
 divided into one-inch spaces with each edge divided deci- 
 mally for convenience in plotting lines. The usual gradua- 
 
CHAIN SURVEYING 
 
 of an inch, enabling the 
 scales of 10, 100, 1000, 
 
 tions are T V, ?<i ^ ?V> -fa, eV 
 draftsman to make maps with 
 20, 200, 2000, 30, 300, 3000, etc., feet per 
 inch, inches, feet or yards per mile, etc. 
 
 6. A 3-H lead pencil, although it is well 
 to have a 4-H pencil also. The 4-H pencil 
 will be used to mark points and draw lines 
 lightly, the 3-H pencil being used to draw 
 more prominent lines. 
 
 7. Pencil eraser. 
 
 8. Drawing pen, 5 ins. long. 
 
 9. Compass with pen and pencil points, 
 for drawing circles. The length should be 
 6 ins. or 8 ins. 
 
 Other tools will be required when com- 
 pass surveying is reached but for present 
 use the foregoing will be "sufficient. Use a 
 good quality of drawing paper, the most FlG - I 7- () Rut 
 restful color to the eyes being a light brown 
 or buff. For holding the paper to the draw- 
 ing board use thumb tacks of punched 
 steel f-in. diameter. 
 
 To draw parallel lines. Place a triangle on the paper 
 and hold it by pressing with the tips of the fingers of the 
 
 (a) 
 
 pass with pen 
 point and length- 
 ening bar. 
 
 FIG. i 8. 
 
 FIG. 19. 
 
 left hand. With the right hand slide another triangle 
 along one side of the first. 
 
 In Figs. 1 8 and 19 the line a ... b having been drawn, 
 the triangles are held so the edge c . . . d touches the 
 line a ... b. Holding the larger triangle firmly the 
 
26 
 
 PRACTICAL SURVEYING 
 
 smaller one is held closely in contact with it and moved to 
 the point c, when a pencil drawn along the edge will describe 
 the line c . . . d parallel with a . . . b. Instead of two 
 triangles a straight-edge or T square may be used with one 
 triangle. By using two triangles in combination with a 
 straight-edge or T square a number of angles may be 
 formed as shown in Fig. 20. 
 
 FIG. 20. 
 
 The surveyor seldom uses triangles for setting off angles 
 in the manner shown in Fig. 20. The principal use to 
 which he puts these useful tools is to transfer lines from one 
 part of a map to another. 
 
 PROBLEMS 
 
 I . To erect a perpendicular to a given line passing through 
 a given point outside the line. Let B-C be the line and A 
 be the point. 
 
 1st method. Draw the line BC. Set the needle point 
 of the compasses at A and describe the arc intercepting 
 B-C at D and E. Bisect D-E and from the midway point 
 F draw the line F-A , which will be the required perpendic- 
 ular. Fig. 21. 
 
 2nd method. Describe arc D-E as before. With needle 
 point at E set pencil point at A and describe arc A-E. 
 
CHAIN SURVEYING 27 
 
 With needle point at D and pencil point at A describe arc 
 A-F intersecting at F the first arc drawn from E as the 
 center. The line A-F will be the required perpendicular. 
 Fig. 22. 
 
 $rd method. Let F be the point. From a point A 
 describe an arc D-E-F and a line connecting E and F will 
 
 / 
 
 V 
 
 -' 
 
 FIG. 21. 
 
 FIG. 22. 
 
 FIG. 23. 
 
 be the required perpendicular provided the three points Z>, 
 A and F are in a straight line. To accomplish this draw 
 a line from F to the line jB-C and mark D. Set off the 
 middle point A, so that DA = AF. Fig. 23. 
 
 From the foregoing problems a perpendicular is seen to 
 be a line intersecting another line in such a manner that 
 the angle formed on one side of the line is equal to the angle 
 on the other side. By the second method four equal angles 
 are formed, each being a right angle. 
 A perpendicular line is also called a 
 normal line. 
 
 With the needle point set at the 
 intersection of two lines normal to each 
 other describe with the pencil point a 
 complete circle. Fig. 24. Then 
 <AOB= <BOC= <COD= <DOA, 
 
 the sign/ standing for the word "angle." 
 
 Circles are divided into 360 equal divisions called degrees 
 and one-fourth of 360 = 90, so each of the equal angles 
 formed by the intersection of perpendicular lines, or lines 
 normal to each other, contains 90 deg. Such an angle 
 is called a right angle, and contains a "quadrant," or 
 quarter of a circle. 
 
 An angle is the amount of divergence of two intersecting 
 
28 PRACTICAL SURVEYING 
 
 lines. The amount of divergence, that is, the difference in 
 direction between the line A-0 and the line B-O, is 90 deg. 
 (90). Between A-0 and C-O, proceeding to the right in 
 a clockwise direction, the amount of divergence is 180 deg., 
 the angle being known as a straight angle. Between A-0 
 and D-0 the amount of divergence in a clockwise direc- 
 tion = 270 deg. and in an anticlockwise direction = 90 
 deg. The three figures show the line A-D to be equal to 
 the line A-E. This is proof that the angles formed by the 
 meeting of the perpendicular lines are equal, for lines op- 
 posite angles are proportional to the angles. 
 
 When the problem has been worked on paper the 
 operations should be repeated on the ground. The lengths 
 of the lines are unimportant except as they govern the 
 accuracy of the work; longer lines producing more accu- 
 rate results. The first method is generally used when 
 the line B-C is along a wall arid all the work must be 
 performed on one side. Drive a stake or pin at A and hold 
 the end of the tape there. Take any distance A-D to the 
 wall and make a mark at D. On the other side measure 
 A-E = A-D and mark the point E. Halfway between D 
 and E make the mark F, which will 
 be the point where a perpendicular 
 through A will meet the wall. 
 
 Sometimes it may be necessary to 
 erect a perpendicular from a point. 
 Let the point be F (Fig. 25). From 
 
 FIG. 25. F measure F-D and F-E, equal to 
 
 each other. With any radius describe 
 
 an arc with D as a center and with the same radius 
 describe an arc with E as a center. The arcs will intersect 
 at A. 
 
 The second method may be used when no wall or other 
 obstruction prevents a measurement on both sides of line 
 B-C. For the reason that long radii may be used and the 
 required point is midway between two accurately deter- 
 mined points, this method is apt to be more accurate than 
 the first method. 
 
 The third method is not so good as either of the two others, 
 but is often used when a perpendicular is to be set off from 
 a point on a line and not through a point off the line. 
 
CHAIN SURVEYING 
 
 Let E (Fig. 26) be a point on the line B-C, usually the 
 end of the line, in which case the line is B-E. Select some 
 point A as a center and with a radius = A-E describe the 
 arc D-E-F intersecting the line at D and having F, as 
 nearly as the eye can judge, on the line D-A prolonged. 
 With as a center with radius = E-D describe an arc 
 intersecting the first arc at F. A line through E-F will be 
 the required perpendicular through E. 
 
 FIG. 26. 
 
 FIG. 27. Marking inter- 
 section of two lines. 
 
 2. To mark the intersection of two lines in the field. In 
 Fig. 27 let B-C be one line. It is assumed that the second 
 line D-E will cross B-C near the point 0. A pole may be 
 set at C and the observer standing on line at B sights 
 towards C over pegs driven on line at F and G, with small 
 nails or tacks in the top to mark the exact line. Standing 
 at D with a pole at E, similar pegs are set at H and 7. Tie 
 strings to the projecting nails, connecting F to G and H 
 to /, drawing the strings taut. Under the intersection 
 of the strings at drive a peg and drive a small nail or 
 tack in the top of the peg to mark the exact intersection. 
 
 If the line D-E is an arc, after setting the points F and 
 G, connect them with a string and describe the arc, setting 
 peg with tack at the intersection. In this case pegs H 
 and / are not necessary. This latter method may also be 
 used when line D-E is straight, the peg and tack at being 
 set by sighting an intersecting line after the string is tied 
 to F and G. The best method to use depends upon 
 circumstances, governed by the judgment of the surveyor. 
 
 3. To erect from a point on a given straight line a perpen- 
 dicular to the line. 
 
3 
 
 PRACTICAL SURVEYING 
 
 The square on the hypothenuse of a right-angled triangle 
 is equal to the sum of the squares upon the other two sides ; 
 . that is 
 
 Hyp 2 = base 2 -f- altitude 2 , 
 or Hyp. = \/base 2 -f altitude 2 . 
 
 Designating the angles by capital 
 letters and the sides opposite by small 
 letters 
 
 base 
 
 FIG. 28. 
 
 c 2 = a 2 + b 2 . 
 a = 
 
 - b 2 and b = Vc 2 - a 2 . 
 
 Assume values for some triangle as follows: 
 
 I 
 then < 
 
 3; b = 4; c = 5; 
 
 - 4 2 =A/25 - 16 =A/ 9 = 3, 
 
 which proves that when the sides of a triangle are to each 
 other as 3, 4 and 5, the angle at C will be a right angle. 
 
 Using a multiple of 4, set off the base A-C. Using the 
 same multiple of 5, describe an arc from A as a center, pass- 
 ing through B. Using the same multiple of 3, describe an 
 arc from C as a center, intersecting the first arc at B. A 
 line connecting B and C will be normal (perpendicular) to 
 A-C. 
 
 4. To set out on the ground a line parallel to another line. 
 In Fig. 29 the line C-D is to be laid off parallel to the line 
 A-B. Three methods are in common 
 use. 
 
 ist method. With radius = A-C de- 
 scribe an arc from A and also from B. ~ 
 Sight across the two arcs on the line C-D . 
 and set pegs. The second method is pref- / 
 erable for it is difficult to obtain a good A 
 intersection on flat curves. 
 
 2nd method. From A with a radius 
 A-A' describe an arc intended to pass through C. With 
 the same radius from A' describe an arc intersecting the 
 first arc at C and drive a peg. Repeat these operations at 
 B and B f , setting a peg at D. 
 
 fa 
 
 6 ' B 
 
 FIG. 29. 
 
CHAIN SURVEYING 31 
 
 Whether to use the first or second method depends upon 
 circumstances and the judgment of the surveyor, the 
 accuracy desired being a governing factor. The distance 
 A-B should be not less than ten times the distance A-C, 
 the degree of accuracy depending upon the ratio adopted 
 and the care with which the work is done. 
 
 yd method. This method is based on the geometrical 
 truth that when a straight line crosses two parallel lines 
 the alternate angles ABC and DCS are equal. At B and C, 
 Fig. 30, set pegs and stretch a string to define the line B-C. 
 From B as a center describe the arc e-f and measure the 
 
 FIG. 30. 
 
 FIG. 31. Equilateral triangle. 
 
 chord (the straight line e-f). From C with the same radius 
 describe the arc g-h and set off the chord g-h = chord 
 e-f. From C a straight line through g will fix the location 
 of CD. This problem is most conveniently 
 worked when an equilateral triangle is formed 
 by the lines A-B, A-C and B-C. 
 
 5. To describe an equilateral triangle. 
 In Fig. 31 from C with a radius = A-C 
 describe an arc through B. From A with 
 the same radius describe an arc intersecting 
 the first arc at B. Draw the lines A-B 
 and A-C, forming the triangle, which has 
 three equal sides and three equal angles. 
 
 (Note. An isosceles triangle has two FlG Isosce i es 
 equal sides and two equal angles, Fig. triangle 
 32.) 
 
 6. To pass an obstacle on line. When an obstacle is 
 encountered in ranging a line it may be passed by off- 
 setting as already described. A common method is shown 
 
3 2 
 
 PRACTICAL SURVEYING 
 
 Line D 
 
 {-Backsight 
 
 'oresight 
 
 in Fig. 33. From A and B perpendiculars of equal length 
 are measured and points A' and B' set. Sighting from A' 
 
 past B' stakes are set at 
 C' and D'. Perpendiculars 
 from the offset line are 
 measured from C' to C and 
 from D' to D, equal in 
 FIG. 33. length to A A' and BB', 
 
 then A , B, C, D are on the 
 
 same line. Another method not often used is shown in 
 Fig. 34. On the line A-B set the points I and 2. With a 
 base equal to the distance between 
 these points erect the equilateral 
 triangle 1-2-3. Produce the side 
 1-3 to a point 5 opposite the ob- 
 stacle and construct the equilateral 
 triangle 4-5-6 with sides equal to 
 1-2-3. Produce the line 5-8 with 
 5-6 = 4-5 and 6-7 = 3-4. Make 
 7-8 = 5-6 and construct the equi- 
 lateral triangle 7-8-9. The base 
 8-9 should be on the line A-B. 
 
 FIG. 34. 
 
 The method just described is clumsy and the base 8, 9 
 is very short so a slight error will throw the line off con- 
 siderably. A parallel line set off by means of carefully 
 measured perpendiculars is recommended as the best means 
 for passing obstacles. The triangle method is used only 
 when the survey is small and it is impossible to obtain long 
 sights because of numerous obstacles. 
 
 7. To divide a line into any number of equal 
 parts. 
 
 1st method. Measure the line carefully 
 and divide the length by the number of spaces 
 into which it will be divided. This is the field 
 method. 
 
 2nd method. This is an office method and 
 is shown in Fig. 35. It is useful in making 
 scales for drafting. Assume that on a map 
 a line A-B is known to be a certain length but the paper 
 has shrunk through age and the scale is not known. From 
 one end lay off a line A-C of any length and divide it 
 
 FIG. 35. 
 
CHAIN SURVEYING 33 
 
 into the number of parts, by scaling, into which it is de- 
 sired to divide A-B. Draw the line C-B and by using a 
 straight-edge and triangle draw parallel lines through A-B 
 from the points marked on A C. 
 
 PROBLEM 
 
 A map is to be made to a scale of one-half mile to an inch 
 and no scale is available for the purpose. One-half mile 
 contains 2640 ft. so on a piece of paper draw a line A-B 
 one inch long and the line AC 2.64 ins. long, using the 
 ipo per inch graduation of the triangular scale. On the 
 line A-C a division of T V in. = 100 ft., so through each 
 T Vin. graduation on A-C draw 
 a line through A B parallel with 
 C-B. Each space on A-B equals 
 100 ft., except the last space which 
 equals 40 ft. 
 
 8. To divide a straight line in 
 any proportion, as 3, 5, 7, etc. 
 Let the line be A-B in Fig. 36. 
 
 From A draw the line A-C, making A-E = 3 and E-C = 5: 
 From B draw the line B-D parallel to A-C, making B-F = 
 7 and F-D = 5. Join C to Fand E to D, thus making A-H 
 = 3, H-K = 5 and K-B = 7. 
 
 A number of problems might be given similar to those 
 already treated but it is assumed the student has a scale 
 with which to work in the office and a tape with which to 
 work in the field, so some problems may be more readily 
 solved by arithmetical methods than by strictly geometrical 
 methods. By measurement the last problem would be 
 solved as follows: Add 3 + 5 + 7 = 15. Measure the 
 line and divide by 15. Then A-H = A of A-B; H-K = 
 T 5 s of A-B; K-B = T V of A-B. Annoying differences may 
 creep into arithmetical and measuring work whereas the 
 geometrical method is exact if proper intersections can be 
 obtained. The geometrical method may be used in either 
 the field or the office when no tapes, scales or other measur- 
 ing instruments are available. 
 
 9. To bisect a given angle ABC. From B in Fig. 37 
 with any radius cut the sides in A and C. From A with any 
 
34 
 
 PRACTICAL SURVEYING 
 
 radius A-D, and from C with the same radius, describe arcs 
 
 intersecting at D. Join B-D and then /.ABD = /.CBD. 
 
 10. To divide a right angle into three equal parts. From 
 
 A in Fig. 38 with radius A-B describe the arc B-E-D-C. 
 
 FIG. 37. 
 
 FIG. 38. 
 
 From B with the same radius cut this arc in D and from C 
 with the same radius cut the arc in E. Then Z CAD = 
 ZZX4E = ZE.45 = 30 deg. 
 
 (Note. - No direct method is known for closer division 
 except continual bisection, or dividing by trial. With a 
 protractor, an instrument to be later explained, any re- 
 quired angle may be set off.) 
 
 ii. To construct a triangle when the three sides are known. 
 
 - In Fig. 39 lay off the line A-C to scale. From C with 
 
 a radius = C-B, and from A with a radius = A-B, describe 
 
 FIG. 39. 
 
 FIG. 40. 
 
 arcs intersecting at B. Then A-B-C is the required trian- 
 gle. 
 
 12. To measure an inaccessible distance, for example a 
 line crossing a stream or swamp. 
 
 1st method. The principle of equal triangles illus- 
 trated in Fig. 40 is used. Set out a triangle A-B-C 
 
CHAIN SURVEYING 
 
 35 
 
 with right angle at B. Make C-D = B-C and set a pole 
 vertically at C. Opposite D set pole at E on a line normal 
 to the line B-D. One observer stands at A and sighting 
 past C directs an assistant with a pole into position at F. 
 Another observer stands at D and sighting past E also 
 directs the assistant at F. The lines A-C and D-E inter- 
 sect at F and D-F = A-B. By measuring the length of 
 A-B the distance between D and F is found, the two trian- 
 gles being equal. 
 
 2nd method. By similar triangles. The line B-C may 
 
 be made equal to - - when D-F will be twice the 
 
 length of A-B. If B-C = lme Z) ~ C t hen line D-F will be 
 
 o 
 
 three times the length of AB. The divisor may be any 
 number instead of 2 or 3. Let B-C divided by D-C be any 
 ratio, then 
 
 A-B X D-C 
 
 D-F 
 
 B-C 
 
 In Fig. 41 is illustrated another form in which triangles 
 may appear. The line B-D is produced across the river 
 and at B a perpendicular line, B-C, is set 
 off. At C set off a line perpendicular to 
 
 C-D, intersecting B-D at A . 
 then 
 
 B-D 
 
 Measure A-B, 
 
 A-B 
 
 FIG. 41. 
 
 The length A-B is proportional to B-D 
 
 and any error in measurement will be in 
 
 the same proportion. Line B-C should 
 
 therefore be not less than one-half or two-thirds the length 
 
 of B-D. 
 
 13. To erect a perpendicular to a line from an inaccessible 
 point. In Fig. 42 point C is a fence post from which a line 
 is to cross the river to an intersection with the fence AB, 
 forming an angle of 90 deg. with the fence. At any con- 
 venient point, H, on the line A-B erect perpendicular lines 
 on both sides. By intersection fix F on the line A-C and 
 
PRACTICAL SURVEYING 
 
 make H-F = G-H. Sight from G to C and by intersec- 
 tion fix / on the line A-B. Sighting from F past / and 
 from A past G, fix E by intersection. From E sight to C 
 
 7 
 
 FIG. 43. Surveyors' cross. 
 
 and fix D on line A-B by intersection. Then by similar 
 triangles C-D = D-E and the angle ADC is a right angle. 
 By sighting from E to C the line connecting C and D may 
 be staked out on both sides of the river. 
 
 14. To erect a perpendicular with a surveyors' cross. - 
 The surveyors' cross was once a common instrument, 
 made and sold by all instrument manufacturers. Several 
 patterns were used. Today few dealers sell this instrument 
 but it is very useful in chain surveys 
 when no compass or transit is avail- 
 able. In Fig. 44 a homemade cross 
 is illustrated. A piece of wood 2 ins. 
 thick and 6 ins. square has two cuts in 
 it normal to each other to a depth of 
 about I in. It is fastened to a Jacob 
 staff (a pole 5 ft. 3 ins. long, about \\ 
 ins. diameter, pointed and shod at the 
 lower end with metal) so the top sur- 
 face makes a right angle with the staff. A small pocket 
 level is used with the cross, generally circular in form. If 
 a circular level cannot be obtained then two small levels are 
 used, one parallel with each cut. 
 
 The staff is set in the ground at the point on the line from 
 which the perpendicular is to be set off. By means of the 
 level the staff is brought to a truly vertical position and by 
 
 I 
 
 FIG. 
 
 44. Surveyors' 
 cross. 
 
CHAIN SURVEYING 
 
 37 
 
 sighting forward and back through one slit this slit is placed 
 on the line. Sighting through the other slit stakes may be 
 set on a line normal to that on which the cross is set. It 
 is practically impossible, except with costly tools, to make 
 the two cuts form a perfect right angle, so if the perpen- 
 dicular line is to be set off with more than ordinary care 
 the principle known as "double-centering" must be used. 
 After sighting through the slit across the line and setting 
 a stake turn the staff 180 deg. and get the first slit again on 
 line, the ends being reversed. When the slit 
 is on the line and the staff vertical sight again 
 at the stake set on the perpendicular line. 
 The line of sight will fall to one side if the 
 two slits do not make a perfect right angle. 
 The correct position for the center of the 
 stake is between the first and second points 
 fixed by the two sightings. 
 
 With such a primitive instrument a difference of three 
 or four inches on a sight of more than 100 ft. will not be 
 extraordinary. By "double centering" the sights and 
 bisecting the points the line will be exactly located if proper 
 care is used. By "double centering" the error is doubled 
 and occurs on one side of the line for the first sight and on 
 the opposite side for the second sight. 
 The reversal of observations and 
 doubling of errors, to obtain a mean 
 (average) is the underlying principle 
 '' of all adjustments of instruments. 
 This is illustrated in Fig. 46. 
 
 Let AB be a line with which line 
 C-D is to intersect and form a right 
 angle. The cross is set at with 
 a-b on the line AB and sighting 
 
 FiG. 4 6. Principleofdouble- thr U S h C ^ the P intS ' a " d ^' 
 
 centering. 
 
 are set. The line c d may not be 
 truly perpendicular to line a-b so the 
 cross is turned 180 deg. and by sighting through c-d the 
 points C" and D" are set. No further explanation is 
 required to show that the true position of C is midway 
 between C' and C", and the true position of D is midway 
 between D' and D". 
 
38 PRACTICAL SURVEYING 
 
 Assuming the board in which the slits are cut to be per- 
 pendicular to the staff the board will be level when the staff 
 is truly vertical. If the board is not level all angles set off 
 by using it will be too small. With such an instrument 
 (one which may be turned 180 deg. only by turning the 
 staff) no correction is possible for errors caused by lack of 
 horizontality of the board. With transits or compasses, 
 having a spindle on which the instrument is turned, lack of 
 verticality in the spindle is corrected by " double centering. " 
 
 A surveyors' cross made by a scientific instrument maker 
 sometimes has graduations by means of which angles of 
 30 deg., 45 deg., 60 deg. and 90 deg. may be set off. A 
 compass or transit, however, should be used for angles 
 other than right angles, although a true right angle is the 
 most difficult angle to turn. 
 
 CHAIN SURVEYS 
 
 i . To survey a triangular field. On the left-hand page 
 of the field book set down the lengths of the lines and on 
 the right-hand page draw a diagram. 
 This is shown in Fig. 47. 
 
 Chains. 
 A-B = 20 
 
 B-C = 24 
 C-A = 18 
 
 FIG. 47. 
 
 2. To survey a field with any number of sides. Measure 
 each side and then measure tie lines to divide the field into 
 triangles. This is illustrated in Fig. 48. 
 
 Sides. Chains. Tie lines. Chains. 
 
 A-B 30.60 A-C 45.0 
 
 B-C 20.40 A-D 35.0 
 
 C-D 22 . 40 A-E 24. 20 
 
 D-E 16.20 
 
 E-F 13.50 
 
 F-A 28 . 00 PIG. 48. Tie line survey. 
 
CHAIN SURVEYING 
 
 39 
 
 3. To survey a field with crooked sides. On the inside of 
 the field close to the boundaries lay off straight lines as 
 long as possible. From these lines measure perpendiculars 
 to intersect the boundaries at each angle. Run tie lines 
 on the inside to divide the interior survey into triangles. 
 It is not necessary to measure the boundaries, the tie lines 
 and perpendiculars sufficiently fixing the positions of the 
 corners. This is illustrated in Fig. 49 and the following 
 notes. 
 
 Lengths. 
 
 
 
 Offsets. 
 
 Chains. 
 
 
 
 Chains. 
 
 A-B 
 
 11.20 
 
 I 
 
 0.56 
 
 
 at 5.40 
 
 2 
 
 .... 1 . 4O 
 
 
 8.26 
 
 3 
 
 0.36 
 
 
 11.20 
 
 4 
 
 0.36 
 
 B-C 
 
 7.96 
 
 i 
 
 . . . .O.2O 
 
 
 at 2.36 
 
 2 
 
 0.36 
 
 
 4.28 
 
 3 
 
 . . . .0.96 
 
 
 7.96 
 
 4 
 
 0.30 
 
 C-D 
 
 4. 68 
 
 
 
 
 T" V -" J 
 
 
 0.30 
 
 at 4.34 
 
 D-E 
 
 4.20 
 
 at 
 
 end 0.30 
 
 E-F 
 
 8.20 
 
 i 
 
 . . . .0.40 
 
 
 at i . 04 
 
 2 
 
 ....0.86 
 
 
 2.96 
 
 3 
 
 0.33 
 
 
 5-88 
 
 4 
 
 . . . . i.oo 
 
 
 8.20 
 
 5 
 
 . . . .0. 12 
 
 F-G 
 
 7.96 
 
 i 
 
 1 . 20 
 
 
 at 2.0 
 
 2 
 
 . . . .0.24 
 
 
 7.96 
 
 3 
 
 . . . .0. 16 
 
 G-A 
 
 6 48 
 
 
 
 
 \J 4J.W 
 
 i 
 
 0.80 
 
 at 3.80 
 
 
 6.48 
 
 2 
 
 . . . .0.40 
 
 Tie lines. 
 
 Chains. 
 
 B-G 
 
 
 
 71 
 
 C-G 
 
 
 
 / 
 
 Q-I 
 
 D-G 
 
 
 
 7v5 
 
 . 4-^ 
 
 D-F 
 
 
 8 
 
 T"O 
 
 .68 
 
 FIG. 49. Tie line and offset 
 survey. 
 
40 
 
 PRACTICAL SURVEYING 
 
 FIG. 50. Tie line and offset 
 survey. 
 
 If the adjoining owners do not object, time and labor 
 may sometimes be saved by running lines across boundaries 
 and measuring offsets to the right and left. Care must be 
 
 taken in recording the notes. 
 This method applied to the last 
 field is illustrated in Fig. 50. 
 
 LOCATING OBJECTS 
 
 To locate an object on line. 
 The line is A-B, Fig. 51 , and the 
 object, assumed here to be a 
 house, is to be shown on the 
 map. 
 
 The line is carried past the 
 house and marks placed at / 
 and g where it meets the walls. 
 From some point a on line meas- 
 ure the distances a-c, f-c and a-f. On the other side of 
 the house from the point b measure b-d, and b-e. When 
 the map is drawn points a and b, f and g are marked on the 
 line AB. From a with radius a-c describe an arc and 
 from / with arc f-c describe an arc intersecting the first at c. 
 Join c to / and prolong the line through e. 
 
 From b with radius b e describe an arc cutting the line 
 c-f-e at e. From b with radius b-d describe an arc through 
 d. From e draw a line through g to d. Three corners cde 
 of the building are now located and the remaining corner 
 is at the intersection of lines parallel with c-e and d-e, 
 passing through c and d. 
 
 (Note. When a survey is made for the purpose of draw- 
 ing a plat or map and objects are to be located the method 
 of route surveys should be followed. The first station 
 should be marked o, and pegs set at the end of each tape 
 length. If the surveyors' unit of measure is used the sta- 
 tions will be 66 ft. long and if the engineers' unit of meas- 
 ure is used the stations will be 100 ft. long. Each station 
 contains 100 units so a portion will be designated by a 
 decimal, as 4.37 chains, or by a + sign, as 4 + 37.) 
 
 When the line is marked in stations all objects are 
 located by reference to the nearest stations or plus station. 
 
CHAIN SURVEYING 41 
 
 A sketch is drawn on the right-hand page of the book 
 showing the shape of the object and all the tie lines. Houses 
 are usually rectangular, which simplifies sketching. 
 
 Fences crossing a line. Note the station where a fence 
 crosses, and from points on line 25 ft. each side of the fence 
 measure to points on the fence 25 ft. from the line, as shown 
 
 \5gQ 
 
 y 
 
 FIG. 51. FIG. 52. 
 
 in Fig. 52. When platting this sketch A-B is the line. 
 From Sta. 5 + 00 describe an arc of 28 ft. on the right of 
 the line and from Sta. 5 + 50 describe an arc of 26 ft. on 
 the left. From Sta. 5 + 26 describe arcs with radius of 
 25 ft. intersecting these two. A line drawn through these 
 two points of intersection and through Sta. 5 + 26 repre- 
 sents the fence in position and direction. 
 
 Objects not on line. In Fig. 53 a house is shown at one 
 side of the line. At a a stake is set by intersection, to show 
 where the line of the side of the house produced strikes the 
 survey line A B. At b and c perpendiculars are erected 
 and the distances a-b, b-d and ce measured. The points 
 are easily platted, following the instructions previously given. 
 
 2 53 3 
 FIG. 53. 
 
 In Fig. 54 a more simple method is shown. Let a, b, c 
 and d be four stakes on the line from which measurements 
 are made to e and /. 
 
42 PRACTICAL SURVEYING 
 
 By following the methods given, and such variations as 
 will present themselves with experience, a complete sur- 
 vey of any farm or tract of land may be executed with a 
 satisfactory degree of accuracy. Maps can be made and 
 areas found without the use of compass or transit. 
 
 MAKING MAPS (OR PLATS) OF CHAIN SURVEYS 
 
 To plat the survey of the triangular field, first draw to 
 any scale the line A-B, 20 chains long. From B with a 
 radius of 24 chains describe an arc passing through C. From 
 A with a radius of 18 chains describe an arc intersecting 
 the first arc at C. Draw lines connecting the first arc at C. 
 Draw lines connecting A and C and B and C. 
 
 To plat the second field draw the line A-B. From A with 
 radius A-C describe an arc intercepting an arc described 
 from B with radius B-C. From A with radius A-D de- 
 scribe an arc intercepting an arc described from C with 
 radius C-D. From A with radius A-E describe an arc 
 intercepting an arc described from D with radius D-E. 
 From E with radius E-F and from A with radius A-F 
 describe arcs intercepting at F. 
 
 To plat the third field draw the line A-B and fix the 
 points C, G, D, E and F by intersection. Measure on each 
 line the distances marked and set off the perpendicular 
 offset lines. Connect the ends of the lines. At the corners 
 the outside lines must be produced to intersect. 
 
 The tie lines are measured to divide fields into triangles 
 in order to plat them as described. If the maps are drawn 
 to a large scale they may be divided into triangles or quad- 
 rilaterals, using fine-pointed hard pencils. The lengths of 
 all the lines are then measured with a scale and the area of 
 each interior figure found. The sum of the areas will be 
 the area of the whole field. 
 
 PLANE MENSURATION 
 
 A rectangle is a four-sided figure with the opposite sides 
 parallel and each angle a right angle, Fig. 55. 
 
 A rectangle with all sides equal is called a square. 
 Area of a rectangle = length X breadth, or A = Ib. 
 Area of a right-angled triangle. 
 
CHAIN SURVEYING 
 
 43 
 
 In Fig. 56 two right-angled triangles equal in size are 
 placed as shown, thus forming a rectangle. Since the area 
 of a rectangle = Ib then the area of a right-angled triangle = 
 
 FIG. 55- 
 
 | Ib, or, using the letters commonly denoting the base and 
 altitude of a triangle, A = J ab = 
 
 Area of any triangle, Fig. 57. 
 
 (a) Area ABC = AreaABD + AreaBDC 
 
 AD Xh , DCXh 
 
 2 
 
 (AD 
 
 DQh ACXh 
 
 O D 
 
 fa) 
 
 FIG. 57. 
 
 (b) Area ABD = Area ABC - Area BCD 
 = AC X h CD Xh 
 
 2 2 
 
 = (AC - CD)h = ADXh 
 
 2 2 
 
 ri f ii r * i base X height . 1-1, 
 Therefore the area of any triangle = - , the height 
 
 being measured on a line normal to the base. 
 
44 PRACTICAL SURVEYING 
 
 When the perpendicular height cannot be readily ob- 
 tained the area may be found by means of the expression 
 
 A = Vs(S-a) (S -b)(S -c), 
 where 5 is half the sum of the three sides = a ' C . 
 
 2 
 
 Example. In a triangular field the three sides are as 
 follows : 
 
 A-B = 20 chains, 
 
 B-C = 24 chains, 
 
 C-A = 1 8 chains. 
 Find the area 
 
 24 + 20+18 _ 
 o 31. 
 
 5 - a = 31 - 24 = 7. 
 
 5 b = 31 20 = ii. 
 
 S - c = 31 - 18 = 13. 
 
 S(S-a)(S-b)(S-c) =31 X7X 11X13 = 31,031. 
 
 A/3 1, 03 1 = 176.16 sq. chains = 17.616 acres. 
 
 PROBLEMS 
 
 Compute the area of the field in Fig. 48. Find the area 
 of each triangle into which the field is divided by tie lines, 
 add these areas and the sum will be the area of the field. 
 
 Area of a parallelogram. A par- 
 
 \~~i j ~7 allelogram is a four-sided figure with 
 
 | / j / opposite sides parallel but none of the 
 
 j V angles are right angles. 
 
 FIG g Let Aj B, C, D be a parallelogram 
 
 and from B erect a perpendicular touch- 
 ing C-D at E. From A erect a perpendicular meeting 
 C-D produced to F. 
 
 Then ABEF form a rectangle and the Area = AF X AB. 
 
 Since AD and BC are parallel to each other FD = EC. 
 Since AB and CD are parallel to each other AF = BE. 
 The AAFD = ABEC, therefore the area of the rectangle 
 ABEF = area of the parallelogram A BCD. 
 
 The area of a parallelogram = base X perpendicular height. 
 
CHAIN SURVEYING 
 
 45 
 
 Area of a trapezium. A trapezium is a quadrilateral 
 (four-sided figure) with no parallel sides. 
 
 Divide the trapezium into triangles and find the sum of the 
 areas of the triangles. 
 
 FIG. 59. 
 
 FIG. 60. 
 
 Area of a trapezoid. A trapezoid is a quadrilateral with 
 two sides parallel. It may, or may not, contain two right 
 angles' at the base. 
 
 IST CASE. No included right angles. Divide the 
 trapezium ABCD into two triangles ABC and ADC, then 
 
 Area ABCD = Area (ABC) + Area (ADC) 
 
 _AD Xh , BC X h 
 
 ~~T ~~T 
 
 (AD + BC) h 
 
 2ND CASE. The two parallel sides are perpendicular to 
 the base. D 
 
 H 
 
 Area = AB X AD + BC = AB X EF. 
 
 
 
 FIG. 61. 
 
 The area of a trapezoid = half the sum 
 of the parallel sides X the perpendicular 
 distance between them. 
 
 The area of an irregular field bounded 
 by straight lines may be obtained by 
 reducing the platted figure to a single equivalent tri- 
 angle. 
 
 Let the field be represented by the figure ABCD, Fig. 62. 
 
 Connect BD with a light pencil line (using a hard pencil 
 with chisel-edge point). Draw CF parallel to BD, produc- 
 ing AD to F. 
 
PRACTICAL SURVEYING 
 
 Draw the line BF. ABDC = ABDF for they have a 
 common base BD, and a common altitude, between the 
 parallel lines BD and CF. 
 
 ABGD is common to ABDC and ABDF and when sub- 
 
 FIG. 63. 
 
 % 
 
 tracted leaves ADFG which is added to the figure to replace 
 ABCG of equal area which is subtracted from it. 
 
 AABF = Trapezium AB CD. 
 Draw the perpendicular BE. 
 
 AFX BE 
 Area= - 
 
 ,Fig. 63 shows the principle extended to obtain the area 
 of a five-sided field. 
 
 A FGXCH 
 Area= 
 
 This method is general (applicable to fields having any 
 number of sides). 
 
 Problem. Plat the field shown 
 in Fig. 49. Compute the areas 
 of the triangles formed by the 
 tie lines. The irregular pieces 
 around the edges are to be com- 
 puted as triangles or as trape- 
 zoids, the sum of the areas of 
 all the small divisions being the area of the field. 
 
 Another method is to carefully plat the survey of the 
 field and draw averaging lines through the boundaries, thus 
 reducing the field to a trapezoidal shape (Fig. 64). 
 
 FIG. 64. 
 
CHAIN SURVEYING 
 
 47 
 
 The small pieces on either side of the averaging lines must 
 balance in area. 
 
 Area of a field bounded by a curved line. 
 
 Trapezoidal rule. From a base line erect perpendiculars 
 dividing the field into an even number of strips, the inter- 
 vals between the perpendiculars (or- ^ 
 dinates) being equal. The accuracy 
 of the computation of area is fixed A [._|. 
 by the number of ordinates. The 
 general rule is to have them so close 
 together that each section of the 
 intercepted curve may be considered 
 to be practically straight. 
 
 The sum of the lengths of the ordi- 
 nates, divided by the number of ordi- 
 nates = mean ordinate. 
 
 Area = mean ordinate X length. 
 
 The operation is shortened by 
 adding to half the sum of the lengths 
 of the two end ordinates the sum of the lengths of the in- 
 terior ordinates and multiplying by the width of one strip. 
 
 This rule may be expressed as follows: 
 
 FIG. 65. 
 
 Let 
 
 then 
 A = 
 
 = area, 
 
 ,5 = width of strip, 
 h = ordinate (height), 
 
 . . . h n = each ordinate, the subscript denoting the 
 particular ordinate, the subscript n 
 indicating the last ordinate, 
 
 n I = number of spaces when n = number of 
 ordinates, 
 
 kn-l] 
 
 (Note. When the curve touches the base at one end or 
 at both ends the length of the end ordinate = o.) 
 
 Assume hi = o and h n = 6 then 
 Assume h\ 
 
 hi -f h n + 6 
 
 = 3- 
 
 , o -f- o 
 and h n = o then - - = o. 
 
48 PRACTICAL SURVEYING 
 
 To pass a circle through three points not lying in a straight 
 line. 
 
 Let A, B, C be the points. Join A to B and B to C. 
 Erect a perpendicular from the middle of AB and also from 
 the middle of BC, the perpendiculars intersecting at some 
 point O. From O as a center with a radius 
 = OA describe a circle which will pass 
 through A , B and C. 
 
 Thomas Simpson, Professor of Mathe- 
 matics, Royal Military Academy, Wool- 
 wich, England; born Aug. 10, 1710; died 
 May 14, 1761; was the author of a rule 
 FIG. 66. for finding the area of an irregular figure, 
 
 used in ship-building calculations. The 
 mathematical proof can only be followed by students who 
 have completed a college course in mathematics, but it is 
 not necessary to study the demonstration in order to use 
 the rule. It is based on the assumption that through the 
 ends of three equidistant ordinates we can draw arcs of 
 parabolas, approximating to the curve between these or- 
 dinates, and a series of arcs can be thus drawn to fit the 
 boundary of any irregular figure with a curved outline. 
 
 Simpson 1 s Rule. The figure is divided by an odd number 
 of perpendiculars into an even number of strips of equal width . 
 To the sum of the lengths of the first and last ordinate 
 add twice the sum of the lengths of the remaining odd ordi- 
 nates and four times the sum of the even ordinates. Mul- 
 tiply by one-third the width of one strip. 
 
 Using the notation given for the trapezoidal rule, the 
 area by Simpson's rule may be expressed as follows: 
 
 A = - 
 
 Examples. Find the areas of the following irregular 
 surfaces by the trapezoidal rule and by Simpson's rule. 
 Simpson's rule is the more accurate. 
 
 (Note. If there are nine ordinates, the first, third, 
 fifth, seventh and ninth ordinates are the odd, and the 
 
CHAIN SURVEYING 49 
 
 second, fourth, sixth and eighth are the even, ordinates 
 referred to in the rule.) 
 
 1. Length of base 20 feet. Ordinates = 10, 16, 14, n, 
 1 6 feet. 
 
 2. Length of base 325 links. Ordinates o, 25, 38, 51, 64, 
 TO, 83, 96 links. Here we have a curve cutting the base at 
 one end. 
 
 3. Length of base 252 links. Ordinates = o, 24, 36, 42, 
 54, 67, 76, 58, 49, 33, 19, o. Here the curve cuts the base 
 at both ends. 
 
 4. Length of base 1260 links. Ordinates = 364, 396, 
 418, 453, 512, 554, 578 links. 
 
 5. Length of base 2364 links. Ordinates = o, 335, 417 
 432, 524.587, 642, 758 links. 
 
 Area by weighing. Plat the field accurately on a thick 
 piece of paper and cut it out carefully with a sharp knife. 
 On the same sheet draw a square or rectangle of predeter- 
 mined area to the same scale and cut out. The more nearly 
 the area of the field and regular figure 
 agree the more accurate the method. a | b 
 
 On a light rod hang the cut-out field rf\ ~~~ 
 
 (A) at one end and the regular figure ^^ 
 
 (B) at the other end. Suspend the rod p IG 6? 
 on a thin wire or over a pin and move 
 
 to right or left until it remains in a horizontal position, 
 showing that A balances B. 
 
 , . A A rea of B X length be 
 
 The area of A = - - 
 
 length ac 
 
 Area by planimeters. A planimeter is an instrument 
 used for obtaining the areas of irregular figures. There 
 are several forms and full instructions for use are given 
 with each one sold. Pictures with descriptions are given 
 in the catalogues of instrument dealers. 
 
 A form of planimeter that the student can make is 
 known as the "hatchet" planimeter, because one end 
 slightly resembles a hatchet blade. Some writers call it 
 the "jack-knife" planimeter because it resembles a knife 
 with a projecting blade at each end. For demonstrating 
 purposes a pocket knife is a fair substitute for a properly 
 made instrument of this type. 
 
PRACTICAL SURVEYING 
 
 A piece of wire is bent at A and B. On one leg is fastened 
 a weight F and the lower end is flattened to a shape which 
 gives the instrument the name of "hatchet" planimeter. 
 At E on the hatchet blade end make a mark. The other 
 
 I 
 
 FIG. 68. Hatchet, or jack-knife, planimeter. 
 
 leg has a sleeve c, in which the wire may move freely but 
 not too loosely. The lower end of the leg AD is brought 
 to a point. 
 
 The distance between D and E may be any length but 
 is usually some even number of inches, preferably a decimal 
 division such as 5, 10, etc. Some planimeters of this type 
 have an adjustable arm AB so the length may be altered, 
 enabling the instrument to be used for comparatively large, 
 as well as for small, drawings. 
 
 Estimate as closely as possible the position of the center 
 of the area of the drawing, Fig. 69, and mark it 0, drawing 
 through it the lines AB and CD normal to each other. 
 
 The accuracy of the work de- 
 O f pends upon the accuracy with 
 
 which the point o fixes the 
 center of area (center of grav- 
 ity). Place the point D of 
 the planimeter at O and the 
 mark E on the line CD, press- 
 ing the hatchet edge into the 
 paper slightly to fix the posi- 
 tion of the mark. Holding 
 the sleeve C move the point 
 D along the line AB to A. 
 Then go to the right carefully tracing the perimeter with 
 the point until A is reached, and the point again follows 
 the line AB from A to O. The hatchet edge will be close 
 to the starting point on the line CD and should be pressed 
 into the paper to mark the new position of the point E. 
 Measure the distance between the original and final posi- 
 tions of E normal to line CD and call this a\. 
 
 a 
 
CHAIN SURVEYING 51 
 
 2nd. Place D at O and E on the line CD near D, which is 
 equivalent to changing the position of drawing 180 deg. 
 Move from O to A and move the point around the perim- 
 eter to the left. Call the distance between the first and 
 final positions of R on this second operation 0%. 
 
 Area = length of DE X ai + a *. 
 
 3rd. Closer results may be obtained by next placing 
 the planimeter on the line AB with E near A. Move D 
 along OD towards D and trace the perimeter to the right. 
 Call the distance between the first and final positions of 
 E on this third operation a 3 . 
 
 4th. Place D at O with E on the line AB near B. Move 
 towards D on line OD to the perimeter and trace the perim- 
 eter to the left. Call the distance between the first and 
 final position of E, on this fourth operation a*. 
 
 Area = length of DE X a i + a * + a * + a *. 
 
 To become expert in the use of a planimeter the be- 
 ginner should check plats of regular figures, the areas of 
 which may be readily found by common methods. 
 
 THE USE OF SQUARED PAPER 
 
 Figures may be platted on squared paper and the in- 
 cluded squares counted. The sum multiplied by Jhe area 
 of one square = area of figure. 
 
 CIRCLES 
 
 The perimeter (boundary) of a square f 
 with side d = 4 d. 
 
 The perimeter (circumference) of a ! 
 circle with diameter d = *?- d = 3} d = 7 
 3.1416^. 
 
 The area of a square with side d 
 
 FlG - 
 
PRACTICAL SURVEYING 
 
 The area of a circle with diameter d = - = 0.7854 d 2 . 
 
 4 
 
 A rea of a ring. 
 Let D outer diameter, 
 d = inner diameter, 
 A = 0.7854 > 2 - 0.7854^ 
 = 0.7854 (D*-d 2 ). 
 
 An excellent text on mensuration is 
 " Practical Mathematics " by Knott & 
 Mackay ($2.00). 
 
 3456 78 
 
 STAKING OUT WORK 
 
 To stake out a lot for grading. Grade stakes on lots are 
 generally set in lO-ft. or 25-1 1. squares. At the corner of 
 each square is set a stake about one foot long projecting 
 4 ins. or 6 ins. above the surface, and on this stake is marked 
 the cut or fill. 
 
 In Fig. 72 assume that the square A BCD is to be laid 
 out in squares or rectangles. From A in any convenient 
 way lay off the line AD, perpendicular 
 to AB, setting the stakes and marking 
 them Oa, Ob, Oc, etc. From B set off 
 the line B C, perpendicular to BA, set- 
 ting .the stakes and marking them 8 a, 
 8 b, 8 c, etc. 
 
 Measure from A toB, setting the stakes 
 and marking them I a, 2 a, 3 a, etc. ...._, 
 
 Measure from D to C, setting the 
 stakes and marking them o i, I i, 2 i, etc. IG ' 72> 
 
 A 4-ft. lath, with a cloth tied near the top, is set on the 
 line BC at each stake and also on the line DC at each 
 stake. An observer stands at I a and sights towards I d. 
 Another observer stands at o b and sights towards 8 b. 
 
 A helper with a bag of stakes and a line pole goes along 
 the line b and is lined in by the two observers by the in- 
 tersection process, the observer on the b line remaining in 
 place, the observer on the a line moving towards B after 
 each stake is driven. 
 
CHAIN SURVEYING 
 
 53 
 
 When stake 7 a is set, the observer at o b moves to o c 
 and the helper goes to the c line and sets 7 c. He and the 
 observer on the a line work back to the I line. The other 
 observer then drops to the d line, and the 
 work proceeds thus until all the stakes are 
 set. In Fig. 73 a stake is shown, every stake 
 being marked to indicate the intersection of 
 the two lines on which it stands. The lower 
 figures +7.3 are put on after the elevations 
 are taken. FIG. 73. 
 
 The -f- sign indicates a cut, the surface of 
 the ground at this stake being 7.3 feet above grade. A 
 sign indicates a fill. 
 
 To stake out a building. The 3, 4, 5 method for erect- 
 ing perpendiculars is generally used by contractors when 
 
 234 56 
 
 10 II 
 
 \ 
 
 i _ 
 
 
 
 
 ! 
 i 
 
 L L.. 
 
 
 L 
 
 /?* 
 
 h- 
 
 i 
 
 T 
 
 --{-1 
 
 ! 
 
 ri 
 
 (' 
 
 V A 
 
 X' 
 
 > 1 H 
 
 E 
 
 K 
 
 <? 
 
 IM 
 
 i 
 i 
 
 
 J 
 
 K 
 
 
 
 P 0\ 
 
 
 14 
 
 IKI 
 
 
 i 
 i 
 
 i 
 i 
 
 H\ 
 
 . 
 
 
 
 \ r 
 
 > ! 
 
 
 
 
 - 
 
 .._ 
 
 f 
 ) 
 sp 
 
 1 J 
 
 
 
 __ 
 
 n 
 
 .- 
 _. 
 
 
 
 
 
 . 
 
 
 _' _- 
 
 _! ^ 
 ! ! 
 i j 
 
 ___ 
 
 
 
 10 
 
 ir 
 
 
 
 /)' 
 
 
 i 
 
 c 
 
 
 
 ! 1 r 
 
 
 
 
 ?fl' 
 
 
 
 ! ' 
 
 ._J 
 
 
 
 ! 
 
 
 <_y 
 ?/' 
 
 ""/' 2' 3' 4' 5' 6' TV 9' 10' //" 
 
 FIG. 74. Staking out a building. 
 
 staking out buildings without an instrument for turning 
 angles. 
 
 In Fig. 74 the property line AB is assumed to be staked 
 
54 
 
 PRACTICAL SURVEYING 
 
 and the location of the building settled. The heavy lines 
 show the outline of the building. 
 
 At I and n stakes are set, the distance between being 
 10 ft. more than the width of the building, each stake 
 being 5 ft. from the building line. A chalk line is stretched 
 tightly from nails driven to mark the exact points in the 
 tops of the stakes and the points 2 to 10 inclusive set and 
 nails are driven to mark them. 
 
 At I and n perpendicular lines (i ... 21 and n to 
 21 are set out, the stakes numbered 21 being 5 ft. from 
 
 the building line. At 2 1 check 
 measurements are made to 
 make it certain that perfect 
 right angles have been turned. 
 Lines are stretched from I 
 to 21 and from n to 21', the 
 intermediate points 12 to 20 
 inclusive measured off and 
 set with nails marking the 
 exact locations. 
 When a stake is used the top is usually I or 2 ft. above 
 the ground and the nail projects an inch or so above the 
 top so a cord may be tied to it. If a number of points fall 
 within a space of 10 or 12 ft. a couple of posts are driven 
 to which a board is spiked on the line and nails are driven 
 in the top edge of the board. 
 
 FIG. 75. Batter boards. 
 
 Batter boards at corner. 
 
 The dotted lines represent cords stretched between nails 
 and the points of intersection fix the building lines. The 
 stakes being at least 5 ft. away from the building are 
 safe from disturbance during the excavating period. It is 
 
CHAIN SURVEYING 55 
 
 never necessary to have all the lines in place at one time. 
 Two men can stretch line 2 . . . 2' and then hold a line 
 on 1 8 . . . 1 8' while a third man drives a stake at F. 
 Moving to 17 ... 17' a stake is driven at G. Line 
 2 ... 2' is moved to 3 ... 3', the cross line held on 
 16 . . . 16' while // is driven and on 19 ... 19' while E 
 is driven. 
 
 The corner line guides are often arranged as shown in 
 Fig. 76, and the cords are tied around the boards, as the 
 pull on a long line will bend a small nail, for all lines must 
 be taut. 
 
 THE USE OF A TABLE OF SQUARES 
 
 Surveyors should become accustomed to the use of 
 labor-saving tables and diagrams. The table here given is 
 very old and of great value. 
 
 Find the square of 138. 
 
 The square of 138 = 138 X 138 = I38 2 . Looking in 
 the column headed No. find 138. In the column headed 
 Square find 19,044 opposite 138. 
 
 Then I38 2 = 19,044. 
 
 Find the cube of 138. 
 
 The cube of 138 = 138 X 138 X 138 = I38 3 . Looking 
 in the column headed No. find 138. In the column headed 
 Cube find 2,628,072 opposite 138. Then I38 3 = 2,628,072. 
 
 Proceeding in a similar manner the square root of 
 138 = V^S = 11.747344 and the cube root of 138 = 
 ^138 = 5.167649. 
 
 The square root of any number is one of two equal factors 
 which multiplied together will produce that number. 
 Thus 5 is the square root of 25, for 25 = 5 X 5- 
 
 The cube root of any number is one of three equal factors 
 which multiplied together produce that number. Thus 5 is 
 the cube root of 125, for 125 = 5 X 5 X 5. 
 
 The fourth root of any number is one of four equal 
 factors which multiplied together produce that number. 
 Thus 5 is the fourth root of 625, for 625 = 5X5X5X5. 
 
 The sixth root of any number is one of six equal factors 
 which multiplied together produce that number. Thus 5 
 is thesixth root of 1,953, 125, for 1,953,125 = 5 X5 X5 X5 X 
 5X5- 
 
56 PRACTICAL SURVEYING 
 
 Let a = any number, then 
 
 a 2 = a X a, 
 
 and a* = aXaXaXa. 
 .'. a 4 = a 2 X a 2 = a 2 + 2 . 
 
 a 3 = aXaXa, 
 
 and a* = aXaXaXaXaXa. 
 .'. a 6 = a 3 X a 3 = a 3 + 3 . 
 
 The exponent = the sum of the times the number is 
 used. It follows that: 
 
 a 5 = a X a X a X a X a, 
 
 = a 2 X a 3 = a 2 + 3 , 
 and a 7 = a 2 X a 2 X a s = a 4 X a 3 = a 4+3 . 
 
 A knowledge of the law of exponents permits the use of 
 the table to find the fourth and higher powers. 
 
 Thus to find the fourth power of any number, first find 
 the square. Using the square as a number find its square. 
 Find the fourth power of 30. 
 
 30 2 = 900, 
 900 2 = 810,000 = 30 4 = 30 2 X 3 2 = 3<> X 30 X 30 X 30- 
 
 The sixth power is found by using the cube. Find the 
 sixth power of 7. 
 
 7* =343, 
 
 343 3 = 40,353,607 = 7 6 = 7 3 X7 3 = 7X7X7X7X7X7. 
 
 To find the fifth power of a number find the square and 
 the cube of the number and multiply. 
 Find the fifth power of 2. 
 
 2 2 = 2 X 2 = 4, 
 
 2 3 = 2X2X2 = 8, 
 
 2 5 = 2 2 X2 3 = 4X8 = 32 =2X2X2X2X2. 
 
 To find the seventh power of a number multiply the 
 third power by the fourth power. 
 Find the seventh power of 2. 
 
 2 3 = 2X2X2 = 8, 
 
 2 4 = 2 X 2 X 2 X 2 = 16, 
 
 2 7 = 2 3 X2 4 = 8Xi6 = 128 = 2X2X2X2X2X2X2. 
 
CHAIN SURVEYING 
 
 57 
 
 In Fig. 77 Aa = and ABCD form a square. 
 
 Aa' = 
 
 a'e = ed = Ad. The square ABCD has an area four times 
 that of the square Aa'ed, having sides one-half the length 
 
 FIG. 78. 
 
 of the larger square. Similarly in Fig. 78 the square ABCD 
 has an area nine times as large as the area of the square 
 Aabc, having sides one-third the length of the larger square. 
 
 The table contains no numbers larger than 1000, there- 
 fore the principle just illustrated must be used when the 
 square of a large number is wanted. 
 
 Rule. Divide by a number giving a quotient contain- 
 ing less than four figures. Multiply the square of the 
 quotient by the square of the divisor. 
 
 Example. Find the square of 1220. 
 
 1220 
 10 
 
 122. 
 
 I22 2 = 14,884. 
 I220 2 = I22 2 X I0 2 = 14,884 X 100 = 1,488,400. 
 
 Find 90 4 . 
 
 90 2 = 8100. 
 810. 
 
 8100 
 
 10 
 
 8io 2 = 656,100. 
 90 4 = 8io 2 X io 2 = 656,100 X 100 = 65,610,000. 
 
 Find 95 4 . 
 
 95 2 = 9025. 
 9025 
 
 V = 902-5. 
 
58 PRACTICAL SURVEYING 
 
 25 250 
 
 36.I 2 = 1303-21. 
 250 2 = 62,500. 
 95 4 = 36.I 2 X 250 2 = 1303.21 X 62,500 = 81,450,625. 
 
 In the last example the method of handling squares of 
 decimal numbers is plainly illustrated. 
 
 The square of 361 = 130,321. 
 The square of 36.1 = 1303.21. 
 The square of 3.61 = 13.0321. 
 
 To extract the square root of any number. The square 
 root of each number less than 1000 is found in the column 
 headed Square Root. If the number is greater than 1000 
 find it in the column headed Square and the square root 
 will be found in the column headed No. 
 
 To extract the cube root of any number. The cube root 
 of each number less than 1000 is found in the column 
 headed Cube Root. If the number is greater than 1000 
 find it in the column headed Cube and the cube root will 
 be found in the column headed No. 
 
 When the cube is wanted of a number larger than 1000 
 it will be best to use logarithms. 
 
 Barlow's Table Book contains squares, square roots, 
 cubes and cube roots for all numbers from I to 10,000, and 
 is sold by all dealers in scientific books. No books are in 
 general circulation containing squares of higher numbers 
 because of the small demand, although tables of squares 
 of all numbers up to 100,000 were once printed. 
 
 8 a = altitude, 
 b = base, 
 c = hypothenuse, 
 
 C v CL i * * 
 
 c b = V(c + a) X (c - a) = V c 2 - a 2 , 
 FIG. 79. a = V(c + b) X (c - b) = Vc 2 - b 2 . 
 
 The above formulas should be memorized for they are 
 very useful to surveyors. 
 
CHAIN SURVEYING 59 
 
 The formulas for rinding the base and altitude of a right- 
 angled triangle illustrate the algebraic theorem 
 
 The product of the sum and difference of two quantities = 
 the difference of their squares. 
 
 Examples. Use the table of squares. 
 
 (i) -> 
 
 6 = 4. Find c. 
 
 a 2 = 3 2 = 9 
 
 1 6 . 
 
 & 2 = 4 2 = ~ c=V 25 = 5 
 
 In the column headed Squares find 25, and in column 
 headed No. find 5. 
 
 Another method is to find 25 in the column headed No. 
 and in the column headed Square Root find 5. 
 
 (2) c = 35. 
 
 b = 28. 
 c 2 = 35 2 = 1225 
 
 441 
 
 When either number contains more than three figures 
 both numbers must be divided by a number that will re- 
 duce them to less than four figure values. 
 
 Algebraically and geometrically we can prove 
 The value of a ratio is not altered when both terms are 
 multiplied or divided by the same quantity. 
 
 3-2 y 1 "2 y C 1 y *7 
 
 _6^6_6^o_6^i e C 
 
 4 4X3 4X5 4X7' 
 
 Using the new values proceed as before. When the 
 square root is found multiply by the number used as a 
 divisor. The result will be the same as though the original 
 values had been used. 
 
 We can prove by geometry 
 
 Triangles which have two angles equal each to each have 
 their sides proportional. 
 
60 PRACTICAL SURVEYING 
 
 Example. 
 
 a = 117.28 ft. 
 
 b = 92.20 ft. Find c. 
 
 Divide by a common divisor. 
 
 4|ii7.28 92.20 
 
 4129.32 23.04 Divisor = 4 X 4 = 16 
 
 7-33 576 + 
 
 7-33 2 = 537289 
 
 5-76 2 = 33.1776 
 
 86.9065 
 
 From table 86.8624 = 9.32 
 
 0.0441 1 6 
 
 5592 
 
 932 
 149.12 
 add 0.05 
 
 149.17 = c. 
 
 Squaring the lengths of the two sides and extracting 
 the square root of the sum by arithmetic the length of 
 c = 149.182. The difference is closer than ordinary work 
 in the field. 
 
 No rational reason can be given for adding the remainder 
 0.0442 (0.05) but we know the root lies between 9.32 and 
 9.33, and experience has shown that when the second 
 significant figure in the remainder is increased by I and 
 the remainder then added, the final error is very small. 
 With a table of squares of numbers up to 10,000 more 
 exact results can be obtained. 
 
 SLIDE RULE 
 
 The slide rule has become an indispensable tool for 
 engineers, and should be used by every surveyor. The 
 principle of the slide rule is simple although some very 
 complicated forms are manufactured. Full instructions 
 for use accompany each instrument. Purchase only from 
 
CHAIN SURVEYING 6l 
 
 a firm of high reputation and test the graduations care- 
 fully to see that the A scale coincides with the B scale, and 
 the C scale with the D scale. 
 
 The best form for the use of a surveyor has the ordinary 
 Mannheim graduations with a reciprocal scale so three 
 numbers can be multiplied at one setting. For office use 
 a i6-in. slide rule is best. For ordinary use either an 
 8-in. or lo-in. rule will be found satisfactory. The man who 
 is a slave to the slide rule carries one 5 ins. long in his 
 pocket. 
 
 MULTIPLYING TABLES 
 
 Crelle's Multiplying and Dividing Tables ($5.00) should 
 be in the office of every man who has to do much figuring. 
 On a large survey the time saved will pay for the book, 
 and mistakes can occur only through grave carelessness. 
 
62 PRACTICAL SURVEYING 
 
 SQUARES, CUBES, SQUARE ROOTS AND CUBE ROOTS 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 i 
 
 t 
 
 i 
 
 I. OOO 
 
 .000 
 
 51 
 
 26 oi 
 
 132 651 
 
 7.141 
 
 3.708 
 
 2 
 
 4 
 
 8 
 
 I.4I4 
 
 .260 
 
 52 
 
 2704 
 
 140 608 
 
 7. 211 
 
 3-733 
 
 3 
 
 
 27 
 
 1-732 
 
 442 
 
 53 
 
 2809 
 
 148 877 
 
 7.280 
 
 3-756 
 
 4 
 
 16 
 
 64 
 
 2. OOO 
 
 587 
 
 54 
 
 29 16 
 
 157 464 
 
 7.349 
 
 3.78o 
 
 5 
 
 25 
 
 125 
 
 2.236 
 
 .710 
 
 55 
 
 3025 
 
 166 375 
 
 7.416 
 
 3-803 
 
 6 
 
 36 
 
 216 
 
 2.449 
 
 .817 
 
 56 
 
 3136 
 
 175616 
 
 7.483 
 
 3-826 
 
 7 
 8 
 
 S 
 
 343 
 
 512 
 
 2.646 
 2.828 
 
 .913 
 
 2.000 
 
 57 
 58 
 
 3249 
 3364 
 
 185 193 
 195 112 
 
 7.550 
 7.616 
 
 3.849 
 3.871 
 
 9 
 
 81 
 
 729 
 
 3.000 
 
 2.080 
 
 59 
 
 348i 
 
 205 379 
 
 7.681 
 
 3-893 
 
 10 
 
 I 00 
 
 I 000 
 
 3.162 
 
 2.154 
 
 60 
 
 3600 
 
 216 ooo 
 
 7.746 
 
 3.915 
 
 ii 
 
 I 21 
 
 I 331 
 
 3.317 
 
 2.224 
 
 61 
 
 3721 
 
 226981 
 
 7.810 
 
 3-937 
 
 12 
 
 144 
 
 I 728 
 
 3-464 
 
 2.289 
 
 62 
 
 3844 
 
 238 328 
 
 7.874 
 
 3-958 
 
 13 
 
 169 
 
 2 197 
 
 3-606 
 
 2-351 
 
 63 
 
 3969 
 
 250 047 
 
 7.937 
 
 3-979 
 
 14 
 
 196 
 
 2744 
 
 3-742 
 
 2.410 
 
 64 
 
 4096 
 
 262 144 
 
 8.000 
 
 4.000 
 
 15 
 
 225 
 
 3375 
 
 3.873 
 
 2.466 
 
 65 
 
 4225 
 
 274 625 
 
 8.062 
 
 4.021 
 
 16 
 
 256 
 
 4096 
 
 4.000 
 
 2.520 
 
 66 
 
 4356 
 
 287496 
 
 8.124 
 
 4.04T 
 
 17 
 
 289 
 
 4913 
 
 4-123 
 
 2.571 
 
 67 
 
 4489 
 
 300763 
 
 8.185 
 
 4.062 
 
 18 
 
 324 
 
 5832 
 
 4-243 
 
 2.621 
 
 68 
 
 4624 
 
 314 432 
 
 8.246 
 
 4.082 
 
 19 
 
 36i 
 
 6859 
 
 4-359 
 
 2.668 
 
 69 
 
 476i 
 
 328509 
 
 8.307 
 
 4.102 
 
 20 
 
 400 
 
 8 ooo 
 
 4.472 
 
 2.714 
 
 70 
 
 4900 
 
 343000 
 
 8.367 
 
 4.121 
 
 21 
 
 441 
 
 9 261 
 
 4.583 
 
 2.759 
 
 71 
 
 5041 
 
 357 9" 
 
 8.426 
 
 4.141 
 
 22 
 
 484 
 
 10648 
 
 4.690 
 
 2.802 
 
 72 
 
 5184 
 
 373 248 
 
 8.485 
 
 4.160 
 
 23 
 
 529 
 
 12 167 
 
 4-796 
 
 2.844 
 
 73 
 
 5329 
 
 389 017 
 
 8.544 
 
 4-179 
 
 24 
 
 576 
 
 13824 
 
 4.899 
 
 2.885 
 
 74 
 
 5476 
 
 405 224 
 
 8.602 
 
 4.198 
 
 25 
 
 625 
 
 15625 
 
 5.000 
 
 2.924 
 
 75 
 
 5625 
 
 421 875 
 
 8.660 
 
 4.217 
 
 26 
 
 676 
 
 17576 
 
 5-099 
 
 2.963 
 
 76 
 
 5776 
 
 438 976 
 
 8.718 
 
 4-236 
 
 27 
 
 729 
 
 19683 
 
 5.196 
 
 3.000 
 
 77 
 
 5929 
 
 456533 
 
 8.775 
 
 4-254 
 
 28 
 
 784 
 
 21952 
 
 5.292 
 
 3-037 
 
 78 
 
 6084 
 
 474 552 
 
 8.832 
 
 4-273 
 
 29 
 
 841 
 
 24389 
 
 5-385 
 
 3.072 
 
 79 
 
 62 41 
 
 493 039 
 
 8.888 
 
 4.291 
 
 30 
 
 9 oo 
 
 27 ooo 
 
 5-477 
 
 3-107 
 
 80 
 
 6400 
 
 512000 
 
 8-944 
 
 4.309 
 
 31 
 
 961 
 
 29791 
 
 5-568 
 
 3.I4I 
 
 81 
 
 6561 
 
 531 441 
 
 9.000 
 
 4-327 
 
 32 
 
 1024 
 
 32768 
 
 5.657 
 
 3-175 
 
 82 
 
 6724 
 
 551 368 
 
 9-055 
 
 4-345 
 
 33 
 
 1089 
 
 35 937 
 
 5-745 
 
 3.208 
 
 83 
 
 6889 
 
 571 787 
 
 9.110 
 
 4.362 
 
 34 
 
 ii 56 
 
 39304 
 
 5.831 
 
 3.240 
 
 84 
 
 7056 
 
 592 704 
 
 9-165 
 
 4.38o 
 
 35 
 
 1225 
 
 42875 
 
 5.916 
 
 3-271 
 
 85 
 
 7225 
 
 614 125 
 
 9.220 
 
 4-397 
 
 36 
 
 1296 
 
 46656 
 
 6.000 
 
 3-302 
 
 86 
 
 7396 
 
 636 056 
 
 9-274 
 
 4.414 
 
 37 
 38 
 
 1369 
 
 1444 
 
 50653 
 54872 
 
 6.083 
 6.164 
 
 3-332 
 3-362 
 
 87 
 88 
 
 7569 
 7744 
 
 658 503 
 
 63i 472 
 
 9-327 
 9-38i 
 
 4-431 
 4-448 
 
 39 
 
 1521 
 
 59319 
 
 6.245 
 
 3-391 
 
 89 
 
 7921 
 
 704 969 
 
 9-434 
 
 4.465 
 
 40 
 
 1600 
 
 64 ooo 
 
 6.325 
 
 3.420 
 
 90 
 
 81 oo 
 
 729 ooo 
 
 9-487 
 
 4.481 
 
 41 
 
 1681 
 
 68 921 
 
 6.403 
 
 3.448 
 
 91 
 
 8281 
 
 753 571 
 
 9-539 
 
 4.498 
 
 42 
 43 
 
 1764 
 1849 
 
 74088 
 79507 
 
 6.481 
 6.557 
 
 3.476 
 3.503 
 
 92 
 93 
 
 8464 
 8649 
 
 778 688 
 804 357 
 
 9-592 
 9.6l4 
 
 4.514 
 4-531 
 
 44 
 
 1936 
 
 85184 
 
 6.633 
 
 3-530 
 
 94 
 
 8836 
 
 830 584 
 
 9.695 
 
 4-547 
 
 45 
 
 20 25 
 
 91 125 
 
 6.708 
 
 3-557 
 
 95 
 
 9025 
 
 857 375 
 
 9-747 
 
 4.563 
 
 46 
 
 21 16 
 
 97 336 
 
 6.782 
 
 3.583 
 
 96 
 
 92 16 
 
 884 736 
 
 9-798 
 
 4-579 
 
 47 
 
 22 09 
 
 103 823 
 
 6.856 
 
 3.609 
 
 97 
 
 9409 
 
 912 673 
 
 9.849 
 
 4-595 
 
 48 
 
 2304 
 
 no 592 
 
 6.928 
 
 3.634 
 
 98 
 
 9604 
 
 941 192 
 
 9.900 
 
 4.610 
 
 49 
 
 24 oi 
 
 117 649 
 
 7.000 
 
 3.659 
 
 99 
 
 98 oi 
 
 970 299 
 
 995Q 
 
 4.626 
 
 50 
 
 25 oo 
 
 125 ooo 
 
 7.071 
 
 3-63 4 
 
 100 
 
 I 00 OO 
 
 I OOO 000 
 
 10.000 
 
 4-642 
 
CHAIN SURVEYING 63 
 
 SQUARES, CUBES, SQUARE ROOTS AND CUBE ROOTS (Continued) 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 101 
 
 I O2 OI 
 
 i 030 301 
 
 10.0499 
 
 4-6570 
 
 151 
 
 2 28 01 
 
 3 442 95i 
 
 12.2882 
 
 5.3251 
 
 IO2 
 
 10404 
 
 i 061 208 
 
 10.0995 
 
 4-6723 
 
 152 
 
 23104 
 
 3 5H 808 
 
 12.3288 
 
 5-3368 
 
 103 
 
 i 06 09 
 
 i 092 727 
 
 10.1489 
 
 4.6875 
 
 153 
 
 23409 
 
 3 58l 577 
 
 12.3693 
 
 5.3485 
 
 104 
 
 I 08 16 
 
 i 124 864 
 
 10.1980 
 
 4.7027 
 
 154 
 
 23716 
 
 3 652 264 
 
 12.4097 
 
 5.3601 
 
 105 
 
 I 10 25 
 
 I 157 625 
 
 10.2470 
 
 4-7177 
 
 155 
 
 24025 
 
 3 723 875 
 
 12.4499 
 
 5-3717 
 
 106 
 
 I 1236 
 
 i 191 016 
 
 10.2956 
 
 4.7326 
 
 156 
 
 24336 
 
 3 796 416 
 
 12.4900 
 
 5-3832 
 
 107 
 
 I 14 49 
 
 i 225 043 
 
 10.3441 
 
 4-7475 
 
 157 
 
 24649 
 
 3 869 893 
 
 12.5300: 5-3947 
 
 108 
 
 i 16 64 
 
 i 259 712 
 
 10.3923 
 
 4.7622 
 
 158 
 
 24964 
 
 3 944 312 
 
 12.5698 5.4061 
 
 109 
 
 I 18 81 
 
 i 295 029 
 
 10.4403 
 
 4.7769 
 
 159 
 
 252 81 
 
 4 019 679 
 
 12.6095 
 
 5.4175 
 
 no 
 
 I 21 OO 
 
 i 331 ooo 
 
 10.4881 
 
 4-7914 
 
 160 
 
 25600 
 
 4 096000 
 
 12.6491 
 
 5.4288 
 
 III 
 
 12321 
 
 I 367 631 
 
 10.5357 
 
 4-8059 
 
 161 
 
 2 59 21 
 
 4 173 281 
 
 12.6886 
 
 5-4401 
 
 112 
 
 I 25 44 
 
 I 404 928 
 
 10.5830 
 
 4-8203 
 
 162 
 
 262 44 
 
 4 251 528 
 
 12.7279 5-4514 
 
 H3 
 
 i 27 69 
 
 I 442 897 
 
 10.6301 
 
 4.8346 
 
 163 
 
 2 65 69 
 
 4 330 747 
 
 12.7671 5.4626 
 
 114 
 
 I 29 96 
 
 I 481 544 
 
 10.6771 
 
 4.8488 
 
 164 
 
 26896 
 
 4 410 944 
 
 12.8062 5-4737 
 
 US 
 
 13225 
 
 I 520 875 
 
 10.7238 
 
 4.8629 
 
 165 
 
 27225 
 
 4 492 125 
 
 12.8452 
 
 5.4848 
 
 116 
 
 I 3456 
 
 I 560 896 
 
 10.7703 
 
 4.8770 
 
 166 
 
 27556 
 
 4 574 296 
 
 12.8841 
 
 5-4959 
 
 117 
 
 13689 
 
 I 601 613 
 
 10.8167 
 
 4.8910 
 
 167 
 
 27889 
 
 4 657 463 
 
 12.9228 
 
 5-5069 
 
 118 
 
 i 39 24 
 
 I 643 032 
 
 10.8628 
 
 4.9049 
 
 168 
 
 2 82 24 
 
 4 741 632 
 
 12.9615 
 
 5.5178 
 
 119 
 
 i 41 61 
 
 i 685 159 
 
 10.9087 
 
 4.9187 
 
 169 
 
 2 85 61 
 
 4 826 809 
 
 13.0000 
 
 5.5288 
 
 120 
 
 14400 
 
 i 728 ooo 
 
 10.9545 
 
 4.9324 
 
 170 
 
 28900 
 
 4 913 ooo 
 
 13-0384 
 
 5-5397 
 
 121 
 
 I 46 41 
 
 i 771 561 
 
 II.OOOO 
 
 4.9461 
 
 171 
 
 29241 
 
 5 ooo 211 
 
 13-0767 
 
 5.5505 
 
 122 
 
 14884 
 
 i 815 848 
 
 11.0454 
 
 4-9597 
 
 172 
 
 29584 
 
 5088448 
 
 13-1149 
 
 5.5613 
 
 123 
 
 151 29 
 
 i 860 867 
 
 11.0905 
 
 4-9732 
 
 173 
 
 29929 
 
 5 177 717 
 
 13-1529 
 
 5.5721 
 
 124 
 
 i 53 76 
 
 i 906624 
 
 II -1355 
 
 4.9866 
 
 174 
 
 30276 
 
 5 268 024 
 
 13-1909 
 
 5.5828 
 
 125 
 
 15625 
 
 i 953 125 
 
 11.1803 
 
 5.0000 
 
 175 
 
 30625 
 
 5 359 375 
 
 13.2288 
 
 5-5934 
 
 126 
 
 15876 
 
 2 000376 
 
 11.2250 
 
 5-0133 
 
 176 
 
 30976 
 
 5 451 776 
 
 13-2665 
 
 5.6041 
 
 127 
 
 i 61 29 
 
 2 048 383 
 
 11.2694 
 
 5-0265 
 
 177 
 
 31329 
 
 5 545 233 
 
 13-3041 
 
 5.6147 
 
 128 
 
 16384 
 
 2 097 152 
 
 11-3137 
 
 5-0397 
 
 178 
 
 3 16 84 
 
 5 639 752 
 
 13.3417 
 
 5.6252 
 
 129 
 
 i 66 41 
 
 2 146 689 
 
 11-3578 
 
 5-0528 
 
 179 
 
 32041 
 
 5735339 
 
 13-3791 
 
 5.6357 
 
 130 
 
 i 6900 
 
 2 197000 
 
 11.4018 
 
 5-0658 
 
 180 
 
 32400 
 
 5 832 ooo 
 
 13.4164 
 
 5.6462 
 
 131 
 
 i 71 60 
 
 2 248 091 
 
 n.4455 
 
 5.0788 
 
 181 
 
 3 27 61 
 
 5 929 741 
 
 I3.4536 
 
 5-6567 
 
 132 
 
 i 7424 
 
 2 299 968 
 
 11.4891 
 
 5.0916 
 
 182 
 
 33124 
 
 6 028 568 
 
 13.4907 
 
 5-6671 
 
 133 
 
 
 2 352 637 
 
 H.5326 
 
 5-1045 
 
 183 
 
 33489 
 
 6 128 487 
 
 13.5277 
 
 5.6774 
 
 134 
 
 i 79 56 
 
 2 406 1O4 
 
 11-5758 
 
 5-II72 
 
 184 
 
 33856 
 
 6 229 504 
 
 13-5647 
 
 5.6877 
 
 135 
 
 i 82 25 
 
 2 460 375 
 
 11.6190 
 
 S.I299 
 
 185 
 
 34225 
 
 6 331 625 
 
 13-6015 
 
 5.6980 
 
 136 
 
 18496 
 
 2 515 456 
 
 11.6619 
 
 5.1426 
 
 186 
 
 34596 
 
 6 434 856 
 
 13-6382 
 
 5.7083 
 
 137 
 
 18769 
 
 2 571 353 
 
 11.7047 
 
 5.I55I 
 
 187 
 
 34969 
 
 6 539 203 
 
 13-6748 
 
 5.7185 
 
 138 
 
 i 9044 
 
 2 628 O72 
 
 11-7473 
 
 5-1676 
 
 188 
 
 35344 
 
 6 644 672 
 
 I3.7H3 
 
 5-7287 
 
 139 
 
 I932I 
 
 2 685 6l9 
 
 11.7898 
 
 S-lSoi 
 
 189 
 
 35721 
 
 6 751 269 
 
 13-7477 
 
 5-7388 
 
 140 
 
 i 96 oo 
 
 2 744000 
 
 11.8322 
 
 5.1925 
 
 190 
 
 3 61 oo 
 
 6859 ooo 
 
 13.7840 
 
 5.7489 
 
 141 
 
 19881 
 
 2 803 221 
 
 II-8743 
 
 5-2048 
 
 191 
 
 36481 
 
 6 967 871 
 
 13-8203 
 
 5.7590 
 
 142 
 
 2 OI 64 
 
 2 863 288 
 
 11.9164 
 
 5.2171 
 
 192 
 
 36864 
 
 7 077 888 
 
 13-8564 
 
 5.7690 
 
 143 
 
 20449 
 
 2 924 207 
 
 H.9583 
 
 5.2293 
 
 193 
 
 37249 
 
 7 189 057 
 
 13.8924 
 
 5.7790 
 
 144 
 145 
 
 2 0736 
 2 1025 
 
 2 985 984 
 
 3 048 625 
 
 I2.OOOO 
 I2.04I6 
 
 5.2415 
 5 2536 
 
 194 
 195 
 
 37636 
 38025 
 
 7 301 384 
 7 4M 875 
 
 13.9284 
 13.9642 
 
 5.7890 
 5.7989 
 
 146 
 
 2 13 16 
 
 3 112 136 
 
 12.0830 
 
 5.2656 
 
 196 
 
 38416 
 
 7 529 536 
 
 14.0000 
 
 5.8088 
 
 147 
 
 2 1609 
 
 3 i?6 523 
 
 12.1244 
 
 5.2776 
 
 197 
 
 38809 
 
 7 645 373 
 
 14.0357 
 
 5.8186 
 
 148 
 
 2 1904 
 
 3 241 792 
 
 12.1655 
 
 5-2896 
 
 198 
 
 39204 
 
 7 762 392 
 
 14.0712 
 
 5.8285 
 
 149 
 
 2 22 01 
 
 3 307 949 
 
 12.2066 
 
 5-3015 
 
 199 
 
 396oi 
 
 7 880 599 
 
 14.1067 
 
 5.8383 
 
 150 
 
 2 25 00 
 
 3375000 
 
 12.2474 
 
 5.3133 
 
 200 
 
 4 oo oo 
 
 8 ooo ooo 
 
 14.1421 
 
 5-8480 
 
4 PRACTICAL SURVEYING 
 
 SQUARES, CUBES, SQUARE ROOTS AND CUBE ROOTS (Continued) 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 root. 
 
 Cube 
 
 root. 
 
 201 
 
 40401 
 
 8 120 601 
 
 14.1774 
 
 5.8578 
 
 251 
 
 6 30 01 
 
 15 813 251 
 
 15 8430 
 
 6.3080 
 
 2O2 
 
 40804 
 
 8 242 408 
 
 14.2127 
 
 5-8675 
 
 252 
 
 6 .35 04 
 
 16 003 008 
 
 15.8745 6.3164 
 
 203 
 
 41209 
 
 8 365 427 
 
 14.2478 
 
 5.8771 
 
 253 
 
 64009 
 
 16 194 277 
 
 15.9060! 6.3247 
 
 204 
 
 4 16 16 
 
 8 489 664 
 
 14.2829 
 
 5.8868 
 
 254 
 
 6 45 16 
 
 16 387 064 
 
 15-9374 6.3330 
 
 205 
 
 42025 
 
 8 615 125 
 
 14.3178 
 
 5-8964 
 
 255 
 
 65025 
 
 16 581 375 
 
 15.9687 6.3413 
 
 206 
 
 42436 
 
 8 741 816 
 
 14.3527 
 
 5.9059 
 
 256 
 
 65536 
 
 16 777 216 
 
 i6.oooo| 6.3496 
 
 207 
 
 42849 
 
 8 869 743 
 
 14-3875 
 
 5-9155 
 
 257 
 
 6 6049 
 
 16 974 593 
 
 16.0312 6.3579 
 
 208 
 
 43264 
 
 8 998 912 
 
 14.4222 
 
 5.9250 
 
 258 
 
 66564 
 
 17 173 512 
 
 16.06241 6.3661 
 
 209 
 
 4368i 
 
 9 129 329 
 
 14.4568 
 
 5-9345 
 
 259 
 
 6 7081 
 
 17 373 979 
 
 16.0935 
 
 6.3743 
 
 210 
 
 4 41 oo 
 
 9 261 ooo 
 
 14.4914 
 
 5-9439 
 
 260 
 
 6 76 oo 
 
 17 576 ooo 
 
 16.1245 
 
 6.3825 
 
 211 
 
 4 45 21 
 
 9 393 93i 
 
 14.5258 
 
 5-9533 
 
 261 
 
 68121 
 
 17 779 58i 
 
 16.1555 
 
 6.3907 
 
 212 
 
 44944 
 
 9 528 128 
 
 14-5602 
 
 5.9627 
 
 262 
 
 686 44 
 
 17 984 728 
 
 16.1864 
 
 9.3988 
 
 213 
 
 45369 
 
 9 663 597 
 
 14-5945 
 
 5-9721 
 
 263 
 
 691 69 
 
 18 191 447 
 
 16.2173 
 
 6.4070 
 
 214 
 
 45796 
 
 9800344 
 
 14.6287 
 
 5.9814 
 
 264 
 
 69696 
 
 18 399 744 
 
 16.2481 
 
 6.4I5I 
 
 215 
 
 4 6225 
 
 9 938 375 
 
 14.6629 
 
 5.9907 
 
 265 
 
 70225 
 
 18 609 625 
 
 16.2788 
 
 6.4232 
 
 216 
 
 46656 
 
 10 077 696 
 
 14-6969 
 
 6.0000 
 
 266 
 
 70756 
 
 18 821 096 
 
 16.3095 
 
 6.4312 
 
 21? 
 
 47089 
 
 10 218 313 
 
 14.7309 
 
 6.0092 
 
 267 
 
 7 1289 
 
 19 034 163 
 
 16.3401 
 
 6.4393 
 
 218 
 
 47524 
 
 10 360 232 
 
 14.7648 
 
 6.0185 
 
 268 
 
 71824 
 
 19 248 832 
 
 i 6 3707 
 
 6.4473 
 
 219 
 
 47961 
 
 10 503 459 
 
 14.7986 
 
 6.0277 
 
 269 
 
 72361 
 
 19 465 109 
 
 16.4012 
 
 6.4553 
 
 220 
 
 48400 
 
 10 648 ooo 
 
 14.8324 
 
 6.0368 
 
 270 
 
 72900 
 
 19683 ooo 
 
 16.4317 
 
 6.4633 
 
 221 
 
 48841 
 
 10 793 861 
 
 14.8661 
 
 6.0459 
 
 271 
 
 73441 
 
 19 902 511 
 
 16.4621 
 
 6.4713 
 
 222 
 
 49284 
 
 10 941 048 
 
 14.8997 
 
 6.0550 
 
 272 
 
 73984 
 
 20 123 648 
 
 16.4924 
 
 6.4792 
 
 223 
 
 49729 
 
 II 089 567 
 
 14.9332 
 
 6.0641 
 
 273 
 
 7 45 29 
 
 20 346 417 
 
 16.5227 
 
 6 . 4872 
 
 224 
 
 501 76 
 
 ii 239 424 
 
 14.9666 
 
 6.0732 
 
 274 
 
 75076 
 
 20 570 824 
 
 16.5529 
 
 6.4951 
 
 225 
 
 50625 
 
 n 390 625 
 
 15.0000 
 
 6.0822 
 
 275 
 
 75625 
 
 20 796 875 
 
 16.5831 
 
 6 5030 
 
 226 
 
 51076 
 
 ii 543 176 
 
 15.0333 
 
 6.0912 
 
 276 
 
 76176 
 
 21 024 576 
 
 16.6132 
 
 6.5108 
 
 227 
 
 51529 
 
 ii 697 083 
 
 15.0665 
 
 6.IOO2 
 
 277 
 
 76729 
 
 21 253 933 
 
 16.6433 
 
 6.5187 
 
 228 
 
 51984 
 
 II 852 352 
 
 15.0997 
 
 6.1091 
 
 278 
 
 77284 
 
 21 484 952 
 
 16.6733 
 
 6.5265 
 
 229 
 
 52441 
 
 12 008 989 
 
 15.1327 
 
 6.1180 
 
 279 
 
 77841 
 
 21 717 639 
 
 16.7033 
 
 6.5343 
 
 230 
 
 52900 
 
 12 167 OOO 
 
 15.1658 
 
 6.1269 
 
 280 
 
 78400 
 
 21 952 OOO 
 
 16-7332 
 
 6.5421 
 
 231 
 
 53361 
 
 12 326 391 
 
 15.1987 
 
 6.1358 
 
 281 
 
 78961 
 
 22 188 041 
 
 16 . 7631 
 
 6.5499 
 
 232 
 
 53824 
 
 12 487 168 
 
 15.2315 
 
 6.1446 
 
 282 
 
 7 95 24 22 425 768 
 
 16.7929 
 
 6.5577 
 
 233 
 
 54289 
 
 12 649 337 
 
 15.2643 
 
 6.1534 
 
 283 
 
 8 oo 89 
 
 22 665 IS? 
 
 16.8226 
 
 6.5654 
 
 234 
 
 54756 
 
 12 8l2 904 
 
 15.2971 
 
 6.1622 
 
 284 
 
 80656 
 
 22 906 304 
 
 16.8523 
 
 6.5731 
 
 235 
 
 55225 
 
 12 977 875 
 
 15.3297 
 
 6.1710 
 
 285 
 
 8 12 25 
 
 23 149 125 
 
 16.8819 
 
 6.5808 
 
 236 
 
 55696 
 
 13 144 256 
 
 15.3623 
 
 6.1797 
 
 286 
 
 8 17 96 
 
 23 393 656 
 
 16.9115 
 
 6.5885 
 
 237 
 
 56169 
 
 13 312 053 
 
 15.3948 
 
 6.1885 
 
 287 
 
 8 23 69 
 
 23 639 903 
 
 16.9411 
 
 6.5962 
 
 238 
 
 56644 
 
 13 481 272 
 
 15.4272 
 
 6.1972 
 
 288 
 
 82944 
 
 23 887 872 
 
 16.9706 
 
 6.6039 
 
 239 
 
 5 71 21 
 
 13 651 919 
 
 15.4596 
 
 6.2058 
 
 289 
 
 83521 
 
 24 137 569 
 
 17.0000 
 
 6.6115 
 
 240 
 
 576oo 
 
 13 824 ooo 
 
 15.4919 
 
 6.2145 
 
 290 
 
 841 oo 
 
 24 389 ooo 
 
 17-0294 
 
 6.6191 
 
 241 
 
 58081 
 
 13 997 521 
 
 15.5242 
 
 6.2231 
 
 291 
 
 84681 
 
 24 642 171 
 
 17.0587 
 
 6.6267 
 
 242 
 
 58564 
 
 14 172 488 
 
 15.5563 
 
 6.2317 
 
 292 
 
 85264 
 
 24 897 088 
 
 17.0880 
 
 6.6343 
 
 243 
 
 5 9049 
 
 14 348 907 
 
 15.5885 
 
 6.2403 
 
 293 
 
 85849 
 
 25 153 757 
 
 17.1172 
 
 6.6419 
 
 244 
 
 59536 
 
 14 526 784 
 
 15.6205 
 
 6.2488 
 
 294 
 
 86436 
 
 25 412 184 
 
 17.1464 
 
 6.6494 
 
 245 
 
 6 oo 25 
 
 14 706 125 
 
 15.6525 
 
 6.2573 
 
 295 
 
 87025 
 
 25 672 375 
 
 17.1756 
 
 6.6569 
 
 246 
 
 605 16 
 
 14 886 936 
 
 15-6844 
 
 6.2658 
 
 296 
 
 87616 
 
 25 934 336 
 
 17.2047 
 
 6.6644 
 
 247 
 
 6 10 09 
 
 15 069 223 
 
 15.7162 
 
 6.2743 
 
 297 
 
 882 09 
 
 26 198 073 
 
 17.2337 
 
 6.6719 
 
 248 
 
 6 15 04 
 
 IS 252 992 
 
 15.7480 
 
 6.2828 
 
 298 
 
 88804 
 
 26 463 592 
 
 17.2627 
 
 6.6794 
 
 249 
 
 6 2001 
 
 15 438 249 
 
 15-7797 
 
 6.2912 
 
 299 
 
 89401 
 
 26 730 899 
 
 17.2916 
 
 6.6869 
 
 250 
 
 6 25 oo 
 
 15 625 ooo 
 
 15.8114 
 
 6.2996 
 
 300 
 
 90000 
 
 27000000 
 
 17-3205 
 
 6.6943 
 
CHAIN SURVEYING 65 
 
 SQUARES, CUBES, SQUARE ROOTS AND CUBE ROOTS (Continued) 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 301 
 
 9 (36 01 
 
 27 270 901 
 
 17.3494 
 
 6.7018 
 
 351 
 
 12 32 oi 
 
 43 243 551 
 
 18-7350 
 
 7.0540 
 
 302 
 
 91204 
 
 27 543 608 
 
 I7-378I 
 
 6.7092 
 
 352 
 
 12 39 04 
 
 43 614 208 
 
 18.7617 
 
 7.0607 
 
 303 
 
 91809 
 
 27 818 127 
 
 17.4069 
 
 6.7166 
 
 353 
 
 124609 
 
 43 986 977 
 
 18.7883 
 
 7.0674 
 
 304 
 
 9 24 16 
 
 28 094 464 
 
 17.4356 
 
 6.7240 
 
 354 
 
 12 53 16 
 
 44 361 864 
 
 18.8149 
 
 7.0740 
 
 305 
 
 93025 
 
 28 372 625 
 
 17.4642 
 
 6.7313 
 
 355 
 
 12 6025 
 
 44 738 875 
 
 18.8414 
 
 7.0807 
 
 306 
 
 93636 
 
 28 652 616 
 
 17.4929 
 
 6.7387 
 
 356 
 
 12 67 36 
 
 45 118 016 
 
 18.8680 
 
 7.0873 
 
 307 
 
 94249 
 
 28 934 443 
 
 17.5214 
 
 6.7460 
 
 357 
 
 12 74 49 
 
 45 499 293 
 
 18.8944 
 
 7.0940 
 
 308 
 
 94864 
 
 29 2l8 112 
 
 17 5499 
 
 6.7533 
 
 358 
 
 12 81 64 
 
 45 882 712 
 
 18.9209 
 
 7.1006 
 
 309 
 
 9548i 
 
 29 503 629 
 
 17.5784 
 
 6.7606 
 
 359 
 
 12 88 81 
 
 46 268 279 
 
 18.9473 
 
 7.1072 
 
 310 
 
 9 61 oo 
 
 29 791 ooo 
 
 17.6068 
 
 6.7679 
 
 360 
 
 12 96 OO 
 
 46 656 ooo 
 
 18.9737 
 
 7.II38 
 
 3ii 
 
 96721 
 
 30 080 231 
 
 17 6352 
 
 6.7752 
 
 36i 
 
 13 03 21 
 
 47 045 881 
 
 19.0000 
 
 7 . 1204 
 
 312 
 
 97344 
 
 30 371 328 
 
 17 6635 
 
 6.7824 
 
 362 
 
 I3I044 
 
 47 437 928 
 
 19.0263 
 
 7.1269 
 
 3U 
 
 97969 
 
 30 664 297 
 
 17.6918 
 
 6.7897 
 
 363 
 
 I3I769 
 
 47 832 147 
 
 19 0526 
 
 7-1335 
 
 3i4 
 
 985 96 
 
 30 959 144 
 
 17.7200 
 
 6.7969 
 
 364 
 
 132496 
 
 48 228 544 
 
 19.0788 
 
 7.1400 
 
 315 
 
 99225 
 
 31 255 875 
 
 17.7482 
 
 6.8041 
 
 365 
 
 133225 
 
 48 627 125 
 
 19 1050 
 
 7.1466 
 
 3i6 
 
 99856 
 
 31 554 496 
 
 17.7764 
 
 6.8113 
 
 366 
 
 13 39 56 
 
 49 027 896 
 
 19.1311 
 
 7 1531 
 
 3i7 
 
 10 04 89 
 
 31 855 013 
 
 17.8045 
 
 6.8185 
 
 367 
 
 134689 
 
 49 430 863 
 
 19.1572 
 
 7.1596 
 
 318 
 
 10 II 24 
 
 32 157 432 
 
 17 8326 
 
 6.8256 
 
 368 
 
 13 54 24 
 
 49 836 032 
 
 19 183,3 
 
 7.1661 
 
 3i9 
 
 10 17 61 
 
 32 461 759 
 
 17.8606 
 
 6.8328 
 
 369 
 
 13 61 61 
 
 50 243 409 
 
 19.2094 
 
 7.1726 
 
 320 
 
 1024 oo 
 
 32 768000 
 
 17.8885 
 
 6.8399 
 
 370 
 
 136900 
 
 50 653 ooo 
 
 19-2354 
 
 7-I79I 
 
 321 
 
 10 30 41 
 
 33 076 161 
 
 17.9165 
 
 6.8470 
 
 371 
 
 13 76 41 
 
 51 064 811 
 
 19.2614 
 
 7.1855 
 
 322 
 
 10 36 84 
 
 33 .386 248 
 
 17-9444 
 
 6.8541 
 
 372 
 
 13 83 84 
 
 Si 478 848 
 
 19.2873 
 
 7.1920 
 
 323 
 
 10 43 29 
 
 33 698 267 
 
 17.9722 
 
 6.8612 
 
 373 
 
 13 9i 29 
 
 51 895 117 
 
 19.3132 
 
 7.1984 
 
 324 
 
 10 49 76 
 
 34 012 224 
 
 18.0000 
 
 6.8683 
 
 374 
 
 139876 
 
 52 313 624 
 
 19 3391 
 
 7.2048 
 
 325 
 
 10 56 25 
 
 34 328 125 
 
 18.0278 
 
 6.8753 
 
 375 
 
 140625 
 
 52 734 375 
 
 19 3649 
 
 7.2112 
 
 326 
 
 10 62 76 
 
 34 645 976 
 
 18.0555 
 
 6.8824 
 
 376 
 
 14 13 76 
 
 53 157 376 
 
 19.3907 
 
 7-2177 
 
 327 
 
 10 69 29 
 
 34 965 783 
 
 18.0831 
 
 6.8894 
 
 377 
 
 14 21 29 
 
 53 582 633 
 
 19.4165 
 
 7.2240 
 
 328 
 
 10 75 84 
 
 35 287 552 
 
 18.1108 
 
 6.8964 
 
 378 
 
 14 28 84 
 
 54 oio 152 
 
 19.4422 
 
 7-2304 
 
 329 
 
 10 82 41 
 
 35 611 289 
 
 18.1384 
 
 6.9034 
 
 379 
 
 14 36 41 
 
 54 439 939 
 
 19.4679 
 
 7-2368 
 
 330 
 
 108900 
 
 35 937 ooo 
 
 18.1659 
 
 6.9104 
 
 38o 
 
 144400 
 
 54 872 ooo 
 
 19.4936 
 
 7.2432 
 
 331 
 
 10 95 61 
 
 36 264 601 
 
 18.1934 
 
 6.9174 
 
 38i 
 
 14 Si 61 
 
 55 306 341 
 
 19.5192 
 
 7.2495 
 
 332 
 
 II 02 24 
 
 36 594 368 
 
 I8.220Q 
 
 6.9244 
 
 382 
 
 14 59 24 
 
 55 742 968 
 
 19-5448 
 
 7 2558 
 
 333 
 
 II 0889 
 
 36 926 037 
 
 18.2483 
 
 6.9313 
 
 383 
 
 
 56 181 887 
 
 19-5704 
 
 7 2622 
 
 334 
 
 ii 15 56 
 
 37 259 704 
 
 18.2757 
 
 6.9382 
 
 384 
 
 14 74 56 
 
 56 623 104 
 
 19.5959 
 
 7^2685 
 
 335 
 
 II 22 25 
 
 37 595 375 
 
 18.3030 
 
 6-9451 
 
 385 
 
 14 82 25 
 
 57 066 625 
 
 19.6214 
 
 7.2748 
 
 336 
 
 II 28 96 
 
 37 933 056 
 
 18.3303 
 
 6.9521 
 
 386 
 
 148996 
 
 57 512 456 
 
 19.6469 
 
 7.2811 
 
 337 
 
 ii 35 69 
 
 38 272 753 
 
 18.3576 
 
 
 387 
 
 149769 
 
 57960603 
 
 19.6723 
 
 7.2874 
 
 338 
 
 II 42 44 
 
 38 614 472 
 
 18.3848 
 
 6.9658 
 
 388 
 
 150544 
 
 58 411 072 
 
 19.6977 
 
 7.2936 
 
 339 
 
 II 49 21 
 
 38 958 219 
 
 18.4120 
 
 6.9727 
 
 389 
 
 15 13 21 
 
 58 863 869 
 
 19-7231 
 
 7.2999 
 
 340 
 
 ii 56 oo 
 
 39 304 ooo 
 
 18.4391 
 
 6.9795 
 
 390 
 
 IS 21 OO 
 
 59 319 ooo 
 
 19-7484 
 
 7.3061 
 
 341 
 
 ii 62 81 
 
 39 651 821 
 
 18.4662 
 
 6.9864 
 
 391 
 
 15 28 81 
 
 59 776 471 
 
 19-7737 
 
 7.3124 
 
 342 
 
 ii 69 64 
 
 40 ooi 688 
 
 18.4932 
 
 6.9932 
 
 392 
 
 15 36 64 
 
 60 236 288 
 
 19.7990 
 
 7.3186 
 
 343 
 
 ii 76 49 
 
 40 353 607 
 
 18.5203 
 
 7.0000 
 
 393 15 44 49 
 
 60 698 457 
 
 19.8242 
 
 7.3248 
 
 344 
 
 ii 83 36 
 
 40 707 584 
 
 18.5472 
 
 7.0068 
 
 394 15 52 36 
 
 61 162 984 
 
 19.8494 
 
 7-3310 
 
 345 
 
 119025 
 
 41 063 625 
 
 18.5742 
 
 7.0136 
 
 395 15 60 25 
 
 61 629 875 
 
 19.8746 
 
 7-3372 
 
 346 
 
 ii 97 16 
 
 41 421 736 
 
 18.6011 
 
 7-0203 
 
 396 
 
 15 68 16 
 
 62 099 136 
 
 19.8997 
 
 7-3434 
 
 347 
 
 12 04 09 
 
 41 781 923 
 
 18.6279 
 
 7.0271 
 
 397 i 15 76 09 
 
 62 570 773 
 
 18.9249 
 
 7.3496 
 
 348 
 
 12 ii 04 
 
 42 144 192 
 
 18.6548 
 
 7.0338 
 
 398 15 84 04 
 
 63 044 792 
 
 19.9499 
 
 7.3558 
 
 349 
 
 12 18 01 
 
 42 508 549 
 
 18.6815 
 
 7.0406 
 
 399 IS 92 oi 
 
 63 521 199 
 
 19.9750 
 
 7.3619 
 
 350 
 
 12 25 OO 
 
 42 875 ooo 
 
 18.7083 
 
 7-0473 
 
 400 
 
 16 oo oo 
 
 64 ooo ooo 
 
 20.0000 
 
 7.3681 
 
66 PRACTICAL SURVEYING 
 
 SQUARES, CUBES, SQUARE ROOTS AND CUBE ROOTS (Continued) 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 401 
 
 160801 
 
 64 481 2OI 
 
 20.0250 
 
 7-3742 
 
 45i 
 
 20 34 01 
 
 91 733 851 
 
 21.2368; 7.6688 
 
 402 
 
 16 16 04 
 
 64 964 808 
 
 20.0499 
 
 7.3803 
 
 452 
 
 20 43 04 
 
 92 345 408 
 
 21.2603 7-6744 
 
 403 
 
 162409 
 
 65 450 827 
 
 20.0749 
 
 7.3864 
 
 453 
 
 20 52 09 
 
 92 959 677 
 
 21.2838 7.6801 
 
 404 
 
 16 32 16 
 
 65 939 264 
 
 20.0998 
 
 7-3925 
 
 454 
 
 20 61 16 
 
 93 576 664 
 
 21.3073! 7-6857 
 
 405 
 
 16 40 25 
 
 66 430 125 
 
 20.1246 
 
 7.3986 
 
 455 
 
 20 70 25 
 
 94 196 375 
 
 21.3307 
 
 7.6914 
 
 406 
 
 16 48 36 
 
 66 923 416 
 
 20.1494 
 
 7.4047 
 
 456 
 
 20 79 36 
 
 94 818 816 
 
 21.3542 
 
 7.6970' 
 
 407 
 
 16 56 49 
 
 67 419 143 
 
 20.1742 
 
 7.4108 
 
 457 
 
 20 88 49 
 
 95 443 993 
 
 21.3776; 7.7026 
 
 408 
 
 16 64 64 
 
 67 917 312 
 
 20.1990 
 
 7-4169 
 
 458 
 
 20 97 64 
 
 96 071 912 
 
 21 . 4009 
 
 7.7082 
 
 409 
 
 16 72 81 
 
 68 417 929 
 
 20.2237 
 
 7-4229 
 
 459 
 
 21 0681 
 
 96 702 579 
 
 21.4243 
 
 7.7138 
 
 410 
 
 1681 oo 
 
 68 921 ooo 
 
 20.2485 
 
 7.4290 
 
 460 
 
 21 l6 OO 
 
 97 336 ooo 
 
 21.4476 
 
 7.7194 
 
 411 
 
 16 89 21 
 
 69 426 531 
 
 20.2731 
 
 7-4350 
 
 461 
 
 21 25 21 
 
 97 972 181 
 
 21.4709 
 
 7.7250 
 
 412 
 
 16 97 44 
 
 69 934 528 
 
 20.2978 
 
 7.4410 
 
 462 
 
 213444 
 
 98 611 128 
 
 21.4942 
 
 7.7306 
 
 4i3 
 
 17 05 69 
 
 70 444 997 
 
 20.3224 
 
 7.4470 
 
 463 
 
 21 43 69 
 
 99 252 847 
 
 21.5174 7.7362 
 
 414 
 
 I7I396 
 
 70 957 944 
 
 20.3470 
 
 7-4530 
 
 464 
 
 21 5296 
 
 99 897 344 
 
 21-5407 
 
 7-7418 
 
 4iS 
 
 17 22 25 
 
 71 473 375 
 
 20.3715 
 
 7-4590 
 
 465 
 
 21 62 25 
 
 loo 544 625 
 
 21.5639 
 
 7-7473 
 
 416 
 
 17 30 56 
 
 71 991 296 
 
 20.3961 
 
 7-4650 
 
 466 
 
 21 71 56 
 
 101 194 696 
 
 21.5870 
 
 7-7529 
 
 417 
 
 17 38 89 
 
 72 511 713 
 
 20.4206 
 
 7-4710 
 
 467 
 
 21 8O 89 
 
 1 01 847 563 
 
 2I.6I02 
 
 7.7584 
 
 418 
 
 17 47 24 
 
 73 034 632 
 
 20.4450 
 
 7.4770 
 
 468 
 
 21 90 24 
 
 102 503 232 
 
 21.6333 
 
 7.7639 
 
 419 
 
 17 55 61 
 
 73 56o 059 
 
 20.4695 
 
 7.4829 
 
 469 
 
 21 99 61 
 
 103 161 709 
 
 21 . 6564 
 
 7.7695 
 
 420 
 
 17 64 oo 
 
 74 088 ooo 
 
 20.4939 
 
 7.4889 
 
 470 
 
 22*6900 
 
 103 823 ooo 
 
 21.6795 
 
 7-7750 
 
 421 
 
 17 72 41 
 
 74 618 461 
 
 20.5183 
 
 7-4948 
 
 47i 
 
 22 l8 41 
 
 104 487 in 
 
 21.7025 
 
 7-705 
 
 422 
 
 17 80 84 
 
 75 151 448 
 
 20.5426 
 
 7.5007 
 
 472 
 
 22 27 84 
 
 105 154 048 
 
 21 . 7256 
 
 7.7860 
 
 423 
 
 17 89 29 
 
 75 686 967 
 
 20.5670 
 
 7.5067 
 
 473 
 
 22 37 29 
 
 105 823 817 
 
 21.7486 
 
 7.7915 
 
 424 
 
 17 97 76 
 
 76 225 024 
 
 20.5913 
 
 7-5126 
 
 474 
 
 22 46 76 
 
 106 490 424 
 
 21.7715 
 
 7-7Q70 
 
 425 
 
 18 06 25 
 
 76 765 625 
 
 20.6155 
 
 7-5185 
 
 475 
 
 22 56 25 
 
 107 171 875 
 
 21-7945 
 
 7.8025 
 
 426 
 
 18 14 76 
 
 77 308 776 
 
 20.6398 
 
 7-5244 
 
 476 
 
 22 65 76 
 
 107 850 176 
 
 21.8174 
 
 7.8079 
 
 427 
 
 18 23 29 
 
 77 854 483 
 
 20 . 6640 
 
 7-5302 
 
 477 
 
 22 75 29 
 
 1 08 531 333 
 
 21 . 8403 
 
 7.8134 
 
 428 
 
 18 31 84 
 
 78 402 752 
 
 20.6882 
 
 7.536i 
 
 478 
 
 22 84 84 
 
 109 215 352 
 
 21.8632 
 
 7.8188 
 
 429 
 
 18 40 41 
 
 78 953 589 
 
 20.7123 
 
 7-5420 
 
 479 
 
 22 94 41 
 
 109 902 239 
 
 2I.886I 
 
 7.8243 
 
 430 
 
 18 4900 
 
 79 507 ooo 
 
 20.7364 
 
 7-5478 
 
 480 
 
 23 04 oo 
 
 no 592 ooo 
 
 21.9089 
 
 7-8297 
 
 43i 
 
 18 57 61 
 
 80 062 991 
 
 20.7605 
 
 7-5537 
 
 481 
 
 23 13 61 
 
 III 284 641 
 
 21.9317 
 
 7.8352 
 
 432 
 
 18 66 24 
 
 80 621 568 
 
 20.7846 
 
 7-5595 
 
 482 
 
 23 23 24 
 
 111980168 21.9545 
 
 7.8406 
 
 433 
 
 18 74 89 
 
 81 182 737 
 
 20.8087 
 
 7.5654 
 
 483 
 
 23 32 89 
 
 112678587 21.9773 
 
 7.8460 
 
 434 
 
 18 83 56 
 
 81 746 504 
 
 20.8327 
 
 7-5712 
 
 484 
 
 23 42 56 
 
 H3 379 904' 22.0000 
 
 7.8514 
 
 435 
 
 18 92 25 
 
 82 312 875 
 
 20.8567 
 
 7-5770 
 
 485 
 
 23 52 25 
 
 114 084 125 
 
 22.0227 
 
 7.8568 
 
 436 
 
 19 oo 96 
 
 82 881 856 
 
 20.8806 
 
 7-5828 
 
 486 
 
 23 61 96 
 
 114 791 256 
 
 22.0454 
 
 7.8622 
 
 437 
 
 19 09 69 
 
 83 453 453 
 
 20.9045 
 
 7.5886 
 
 487 
 
 23 7i 69 
 
 115 501 303 
 
 22.0681 
 
 7.8676 
 
 438 
 
 19 18 44 
 
 84 027 672 
 
 20.9284 
 
 7-5944 
 
 488 
 
 23 81 44 
 
 116 214 272 
 
 22.0907 
 
 7.8730 
 
 439 
 
 19 27 21 
 
 84 604 519 
 
 20.9523 
 
 7.6001 
 
 489 
 
 23 91 21 
 
 116 930 169 
 
 22.1133 
 
 7.8784 
 
 440 
 
 193600 
 
 85 184 ooo 
 
 20.9762 
 
 7.6059 
 
 490 
 
 24 01 oo 
 
 117 649 ooo 
 
 22.1359 
 
 7-8837 
 
 441 
 
 19 44 8l 
 
 85 766 121 
 
 2I.OOOO 
 
 7.6117 
 
 491 
 
 24 10 81 
 
 118 370 771 
 
 22.1585 
 
 7.8891 
 
 442 
 
 19 53 64 
 
 86 350 888 
 
 .21.0238 
 
 7.6i74 
 
 492 
 
 24 20 64 
 
 119 095 488 
 
 22.I8II 
 
 7.8944 
 
 443 
 
 19 62 49 
 
 86 938 307 
 
 21.0476 
 
 7-6232 
 
 493 
 
 24 30 49 
 
 119 823 157 
 
 22.2036 
 
 7.8998 
 
 444 
 
 19 71 36 
 
 87 528 384 
 
 21.0713 
 
 7.6289 
 
 494 
 
 24 40 36 
 
 120 553 784 
 
 22.2261 
 
 7.9051 
 
 445 
 
 19 80 25 
 
 88 121 125 
 
 21.0950 
 
 7.6346 
 
 495 
 
 24 50 25 
 
 121 287 375 
 
 22 . 2486 
 
 7.9105 
 
 446 
 
 19 89 16 
 
 88 716 536 
 
 2I.II87 
 
 7.6403 
 
 496 
 
 2460 16 
 
 122 023 936 
 
 22.2711 
 
 7.9158 
 
 447 
 448 
 
 199809 
 20 07 04 
 
 89 314 623 
 89 915 392 
 
 21.1424 
 21 . 1660 
 
 7.6460 
 7-6517 
 
 497 
 498 
 
 24 70 09 
 24 80 04 
 
 122 763 473 
 123 505 992 
 
 22.2935 
 22.3159 
 
 7-92II 
 7.9264 
 
 449 
 
 20 16 01 
 
 90 518 849 
 
 21 . 1896 
 
 7-6574 
 
 499 
 
 24 90 01 
 
 124 251 499! 22.3383 
 
 7 9317 
 
 450 
 
 20 25 oo 
 
 91 125 ooo 
 
 21.2132 
 
 7-6631 
 
 500 
 
 25 oo oo 
 
 125 ooo ooo 22.3607 
 
 1 
 
 7-9370 
 
CHAIN SURVEYING 67 
 
 SQUARES, CUBES, SQUARE ROOTS AND CUBE ROOTS (Continued) 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 SGI 
 
 25 10 01 
 
 125 751 501 
 
 22.3830 
 
 7.9423 
 
 551 
 
 30 36 oi 
 
 167 284 151 
 
 23.4734 
 
 8.1982 
 
 502 
 
 25 20 04 
 
 126 506 008 
 
 22.4054 
 
 7.9476 
 
 552 
 
 30 47 04 168 196 608 
 
 23-4947 
 
 8.2031 
 
 503 
 
 25 30 09 
 
 127 263 527 
 
 22.4277 
 
 7-9528 
 
 553 
 
 30 58 03 
 
 169 112 377 
 
 23.5160 
 
 8.2081 
 
 504 
 
 25 40 16 
 
 128 024 064 
 
 22.4499 
 
 7.958i 
 
 554 
 
 30 69 16 
 
 170 031 464 
 
 23.5372 
 
 8.2130 
 
 505 
 
 25 So 25 
 
 128 787 625 
 
 22.4722 
 
 7.9634 
 
 555 
 
 30 80 25 
 
 170 953 875 
 
 23.5584 
 
 8.2180 
 
 506 
 
 25 60 36 
 
 129 554 216 
 
 22.4944 
 
 7.9686 
 
 556 
 
 30 91 36 
 
 171 879 616 
 
 23.5797 
 
 8.2229 
 
 So? 
 
 25 70 49 
 
 130 323 843 
 
 22.5167 
 
 7-9739 
 
 557 
 
 31 02 49 
 
 172 808 693 
 
 23.6008 
 
 8.2278 
 
 508 
 
 25 80 64 
 
 131 096 512 
 
 22.5389 
 
 7-9791 
 
 558 
 
 3i 13 64 
 
 173 741 H2 
 
 23 . 6220 
 
 8.2327 
 
 509 
 
 25 90 81 
 
 131 872 229 
 
 22.5610 
 
 7-9843 
 
 559 
 
 3i 24 81 
 
 174 676 879 
 
 23.6432 
 
 8.2377 
 
 5io 
 
 26 01 oo 
 
 132 651 ooo 
 
 22.5832 
 
 7.9896 
 
 5 6o 
 
 313600 
 
 175 616 ooo 
 
 23.6643 
 
 8.2426 
 
 Sii 
 
 26 II 21 
 
 133 432 831 
 
 22.6053 
 
 7.9948 
 
 56l 
 
 31 47 21 
 
 176 558 481 
 
 23.6854 
 
 8.2475 
 
 512 
 
 26 21 44 
 
 134 217 728 
 
 22.6274 
 
 8.0000 
 
 562 
 
 31 58 44 
 
 177 504 328 
 
 23.7065 
 
 8.2524 
 
 513 
 
 26 31 69 
 
 135 005 697 
 
 22.6495 
 
 8.0052 
 
 563 
 
 31 69 69 
 
 178 453 547 
 
 23.7276 
 
 8.2573 
 
 514 
 
 26 41 96 
 
 135 796 744 
 
 22.6716 
 
 8.0104 
 
 564 
 
 318096 
 
 179 406 144 
 
 23-7487 
 
 8.2621 
 
 515 
 
 26 52 25 
 
 136 590 875 
 
 22.6936 
 
 8.0156 
 
 565 
 
 31 92 25 
 
 180 362 125 
 
 23.7697 
 
 8.2670 
 
 Si6 
 
 26 62 56 
 
 137 388 096 
 
 22.7156 
 
 8.0208 
 
 566 
 
 32 03 56 
 
 181 321 496 
 
 23.7908 
 
 8.2719 
 
 Si? 
 
 26 72 89 
 
 138 188 413 
 
 22.7376 
 
 8.0260 
 
 567 
 
 32 14 89 
 
 182 284 263 
 
 23.8118 
 
 8.2768 
 
 Si8 
 
 26 83 24 
 
 138 991 832 
 
 22.7596 
 
 8.0311 
 
 568 
 
 32 26 24 
 
 183 250 432 
 
 23.8328 
 
 8.2816 
 
 519 
 
 26 93 61 
 
 139 798 359 
 
 22.7816 
 
 8.0363 
 
 569 
 
 32 37 61 
 
 184 220 009 
 
 23.8537 
 
 8.2865 
 
 520 
 
 27 04 oo 
 
 140 608 ooo 
 
 22.8035 
 
 8.0415 
 
 570 
 
 324900 
 
 185 193 ooo 
 
 23-8747 
 
 8.2913 
 
 521 
 
 27 14 41 
 
 141 420 761 
 
 22.8254 
 
 8.0466 
 
 571 
 
 326041 
 
 186 169 411 
 
 23.8956 
 
 8.2962 
 
 522 
 
 27 24 84 
 
 142 236 648 
 
 22.8473 
 
 8.0517 
 
 572 
 
 32 71 84 
 
 187 149 248 
 
 23.9165 
 
 8.3010 
 
 523 
 
 27 35 29 
 
 143 055 667 
 
 22.8692 
 
 8.0569 
 
 573 
 
 32 83 29 
 
 188 132 517 
 
 23-9374 
 
 8.3059 
 
 524 
 
 27 45 76 
 
 143 877 824 
 
 22.8910 
 
 8.0620 
 
 574 
 
 32 94 76 
 
 189 119 224 
 
 23.9583 
 
 8.3107 
 
 525 
 
 27 56 25 
 
 144 703 125 
 
 22.9129 
 
 8.0671 
 
 575 
 
 330625 
 
 190 109 375 
 
 23.9792 
 
 8.3155 
 
 526 
 
 27 66 76 
 
 145 531 576 
 
 22.9347 
 
 8.0723 
 
 576 
 
 33*776 
 
 191 102 976 
 
 24.0000 
 
 8.3203 
 
 527 
 
 27 77 29 
 
 146 363 183 
 
 22.9565 
 
 8.0774 
 
 577 
 
 33 29 29 
 
 192 100 033 
 
 24.0208 
 
 8-3251 
 
 528 
 
 27 8? 84 
 
 147 197 952 
 
 22.9783 
 
 8.0825 
 
 578 
 
 334084 
 
 193 100 552 
 
 24.0416 
 
 8.3300 
 
 529 
 
 27 98 41 
 
 148 035 889 
 
 23.0000 
 
 8.0876 
 
 579 
 
 33 52 41 
 
 194 104 539 
 
 24 . 0624 
 
 8-3348 
 
 530 
 
 280900 
 
 148 877 ooo 
 
 23.0217 
 
 8.0927 
 
 58o 
 
 33 64 oo 
 
 195 l\2 OOO 
 
 24.0832 
 
 8.3396 
 
 531 
 
 28 19 61 
 
 149 721 291 
 
 23 0434 
 
 8.0978 
 
 58l 
 
 33 75 61 
 
 196 122 941 
 
 24.1039 
 
 8.3443 
 
 532 
 
 28 30 24 
 
 150 568 768 
 
 23.0651 
 
 8.1028 
 
 582 
 
 338724 
 
 197 137 368 
 
 24.1247 
 
 8-3491 
 
 533 
 
 28 40 89 
 
 151 419 437 
 
 23.0868 
 
 8.1079 
 
 583 
 
 33 98 89 
 
 198 155 287 
 
 24.1454 
 
 8.3539 
 
 534 
 
 28 51 56 
 
 152 273 304 
 
 23.1084 
 
 8.1130 
 
 584 
 
 34 10 56 
 
 199 176 704 
 
 24.1661 
 
 8.3587 
 
 535 
 
 28 62 25 
 
 153 130 375 
 
 23.1301 
 
 8.1180 
 
 585 
 
 34 22 25 
 
 2OO 2OI 625 
 
 24.1868 
 
 8.3634 
 
 536 
 
 287296 
 
 153 990 656 
 
 23.1517 
 
 8.1231 
 
 586 
 
 343396 
 
 201 230 056 
 
 24.2074 
 
 8.3682 
 
 537 
 
 288369 
 
 154 854 153 
 
 23-1733 
 
 8.1281 
 
 587 
 
 34 45 69 
 
 202 262 003 
 
 24.2281 
 
 8.3730 
 
 538 
 
 28 94 44 
 
 155 720 872 
 
 23.1948 
 
 8.1332 
 
 588 
 
 34 57 44 
 
 203 297 472 
 
 24.2487 
 
 8.3777 
 
 539 
 
 29 05 21 
 
 156 5QO SlQ 
 
 23.2164 
 
 8.1382 
 
 589 
 
 34 69 21 
 
 204 336 469 
 
 24.2693 
 
 8.3825 
 
 540 
 
 29 1600 
 
 157 464 ooo 
 
 23.2379 
 
 8.1433 
 
 590 
 
 348100 
 
 205 379 ooo 
 
 24.2899 
 
 8.3872 
 
 541 
 
 29 26 81 
 
 158 340 421 
 
 23.2594 
 
 8.1483 
 
 591 
 
 34 92 8l 
 
 206 425 071 
 
 24.3105 
 
 8.3919 
 
 542 
 
 29 37 64 
 
 159 220 088 
 
 23.2809 
 
 8.1533 
 
 592 
 
 35 04 64 
 
 207 474 688 
 
 24-3311 
 
 8.3967 
 
 543 
 
 29 48 49 
 
 160 103 007 
 
 23.3024 
 
 8.1583 
 
 593 
 
 35 16 49 
 
 208 527 857 
 
 24.3516 
 
 8.4014 
 
 544 
 
 29 59 36 
 
 160 989 184 
 
 23-3238 
 
 8.1633 
 
 594 
 
 35 28 36 
 
 209 584 584 
 
 24.3721 
 
 8.4061 
 
 545 
 
 29 70 25 
 
 161 878 625 
 
 23.3452 
 
 8.1683 
 
 595 
 
 35 40 25 
 
 210 644 8-5 
 
 24-. 3926 
 
 8.4108 
 
 546 
 
 29 81 16 
 
 162 771 336 
 
 23.3666 
 
 8.1733 
 
 596 
 
 35 52 16 
 
 211 708 736 
 
 24.4131 
 
 8.4155" 
 
 547 
 
 2992 09 
 
 163 667 323 
 
 23.3880 
 
 8.1783 
 
 597 
 
 35 6409 
 
 212 776 173 
 
 24.4336 
 
 8.4202^ 
 
 548 
 
 30 03 04 
 
 164 566 592 
 
 23.4094 
 
 8.1833 
 
 598 
 
 35 76 04 
 
 213 847 192 
 
 24.4540 
 
 8.4249 
 
 549 
 
 30 14 oi 
 
 165 469 I4Q 
 
 23.4307 
 
 8.1882 
 
 599 
 
 35 88 oi 
 
 214 921 799 
 
 24.4745 
 
 8.4296 
 
 550 
 
 302500 
 
 166 375 ooo 
 
 23 4521 
 
 8.1932 
 
 600 
 
 36 oo oo 
 
 2l5 OOO OOO 
 
 24.4949 
 
 8.4343 
 
68 PRACTICAL SURVEYING 
 
 SQUARES, CUBES, SQUARE ROOTS AND CUBE ROOTS (Continued) 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 root. 
 
 Cube 
 
 root. 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 601 
 
 36 12 01 
 
 217 081 801 
 
 24.5153 
 
 8.4390 
 
 651 
 
 42 38 01 
 
 275 894 451 
 
 25.5147 
 
 8.6668 
 
 602 
 
 36 24 04 
 
 218 167 208 
 
 24-5357 
 
 8.4437 
 
 652 
 
 42 51 04 
 
 277 167 808 
 
 25.5343 
 
 8.6713 
 
 603 
 
 36 36 09 
 
 219 256 227 
 
 24.5561 
 
 8.4484 
 
 653 
 
 42 64 09 
 
 278 445 077 
 
 25 5539 
 
 8.6757 
 
 604 
 
 36 48 16 
 
 220 348 864 
 
 24-5764 
 
 8.4530 
 
 654 
 
 42 77 16 
 
 279 726 264 
 
 25.5734 
 
 8.6801 
 
 605 
 
 36 60 25 
 
 221 445 125 
 
 24.5967 
 
 8.4577 
 
 655 
 
 429025 
 
 281 oil 375 
 
 25 5930 
 
 8.6845 
 
 606 
 
 367236 
 
 222 545 016 
 
 24.6171 
 
 8.4623 
 
 656 
 
 43 03 36 
 
 282 300 416 
 
 25.6125 
 
 8.6890 
 
 607 
 
 36 84 49 
 
 223 648 543 
 
 24-6374 
 
 8.4670 
 
 657 
 
 43 16 49 
 
 283 593 393 
 
 25.6320 
 
 8.6934 
 
 608 
 
 369664 
 
 224 755 712 
 
 24.6577 
 
 8.4716 
 
 658 
 
 43 29 64 
 
 284 890 312 
 
 25 6515 8.6978 
 
 609 
 
 37 08 81 
 
 225 866 529 
 
 24.6779 
 
 8.4763 
 
 659 
 
 43 42 81 
 
 286 191 179 
 
 25.6710! 8.7022 
 
 610 
 
 372100 
 
 226 981 ooo 
 
 24-6982 
 
 8.4809 
 
 660 
 
 435600 
 
 287 496 ooo 
 
 25.6905 
 
 8.7066 
 
 611 
 
 37 33 21 
 
 228 099 131 
 
 24.7184 
 
 8.4856 
 
 661 
 
 43 69 21 
 
 288 804 781 
 
 25.7099 
 
 8.7110 
 
 612 
 
 37 45 44 
 
 229 220 928 
 
 24.7386 
 
 8.4902 
 
 662 
 
 43 82 44 
 
 290 117 528 
 
 25.7294 
 
 8.7154 
 
 613 
 
 37 57 69 
 
 230 346 397 
 
 24.7588 
 
 8.4948 
 
 663 
 
 43 95 69 
 
 291 434 247 
 
 25.7488 
 
 8.7198 
 
 614 
 
 376996 
 
 231 475 544 
 
 24.7790 
 
 8.4994 
 
 664 
 
 440896 
 
 292 754 944 
 
 25.7682 
 
 8.7241 
 
 615 
 
 37 82 25 
 
 232 608 375 
 
 24.7992 
 
 8.5040 
 
 665 
 
 44 22 25 
 
 294 079 625 
 
 25.7876 
 
 8.7285 
 
 616 
 
 37 94 56 
 
 333 744 896 
 
 24.8193 
 
 8.5086 
 
 666 
 
 44 35 56 
 
 295 408 296 
 
 25 8070 
 
 8-7329 
 
 617 
 
 380689 
 
 234885 113 
 
 24.8395 
 
 8.5132 
 
 667 
 
 444889 
 
 296 740 963 
 
 25 . 8263 
 
 8.7373 
 
 618 
 
 38 19 24 
 
 236 029 032 
 
 24.8596 
 
 8.5178 
 
 668 
 
 44 62 24 
 
 298 077 632 
 
 25 8457 
 
 8.7416 
 
 619 
 
 38 31 61 
 
 237 176 659 
 
 24.8797 
 
 8.5224 
 
 669 
 
 44 75 6l 
 
 299 418 309 
 
 25 . 8650 
 
 8.7460 
 
 620 
 
 38 44 oo 
 
 238 328 ooo 
 
 24.8998 
 
 8.5270 
 
 670 
 
 448900 
 
 300 763 ooo 
 
 25.8844 
 
 8.7503 
 
 621 
 
 38 56 41 
 
 239 483 061 
 
 24.9199 
 
 8.5316 
 
 671 
 
 45 02 41 
 
 302 III 711 
 
 25.9037 
 
 8.7547 
 
 622 
 
 386884 
 
 240 641 848 
 
 24-9399 
 
 8.5362 
 
 672 
 
 45 15 84 
 
 303 464 448 
 
 25.9230 
 
 8-7590 
 
 623 
 
 38 81 29 
 
 241 804 367 
 
 24.9600 
 
 8.5408 
 
 673 
 
 45 29 29 
 
 304 821 217 
 
 25.9422 
 
 8.7634 
 
 624 
 
 38 93 76 
 
 242 970 624 
 
 24.9800 
 
 8.5453 
 
 674 
 
 45 42 76 
 
 306 182 024 
 
 25.9615 
 
 8.7677 
 
 625 
 
 390625 
 
 244 140 625 
 
 25.0000 
 
 8.5499 
 
 675 
 
 45 56 25 
 
 307 546 875 
 
 25.9808 
 
 8.7721 
 
 626 
 
 39 18 76 
 
 245 314 376 
 
 25 . 02OO 
 
 8.5544 
 
 676 
 
 456976 
 
 308 915 776 
 
 26.0000 
 
 8.7764 
 
 627 
 
 39 3i 29 
 
 246 491 883 
 
 25.0400 
 
 8.5590 
 
 677 
 
 45 83 29 
 
 310 288 733 
 
 26.0192 
 
 8.7807 
 
 628 
 
 39 43 84 
 
 247 673 152 
 
 25-0599 
 
 8 5635 
 
 678 
 
 459684 
 
 311 665 752 
 
 26.0384 
 
 8.7850 
 
 629 
 
 39 56 41 
 
 248 858 189 
 
 25.0799 
 
 8.5681 
 
 679 
 
 46 10 41 
 
 313 046 839 
 
 26.0576 
 
 8.7893 
 
 630 
 
 396900 
 
 250 047 ooo 
 
 25.0998 
 
 8.5726 
 
 680 
 
 462400 
 
 314 432 ooo 
 
 26.0768 
 
 8-7937 
 
 631 
 
 39 81 61 
 
 251 239 59i 
 
 25-1197 
 
 8.5772 
 
 681 
 
 46 37 6l 
 
 315 821 241 
 
 26.0960 
 
 8.7980 
 
 632 
 
 39 94 24 
 
 252 435 968 
 
 25.1396 
 
 8.5817 
 
 682 
 
 46 51 24 
 
 317 214 568 
 
 26.1151 
 
 8.8023 
 
 633 
 
 400689 
 
 253 636 137 
 
 25.1595 
 
 8.5862 
 
 683 
 
 46 64 89 
 
 318 611 987 
 
 26.1343 
 
 8.8066 
 
 634 
 
 40 19 56 
 
 254 840 104 
 
 25.1794 
 
 8.5907 
 
 684 
 
 46 78 56 
 
 320 013 504 
 
 26.1534 
 
 8.8109 
 
 635 
 
 40 32 25 
 
 256 047 875 
 
 25.1992 
 
 8.5952 
 
 685 
 
 46 92 25 
 
 321 419 125 
 
 26.1725 
 
 8.8152 
 
 636 
 
 404496 
 
 257 259 456 
 
 25.2190 
 
 8.5997 
 
 686 
 
 47 05 96 
 
 322 828 856 
 
 26.1916 
 
 8.8194 
 
 637 
 
 40 57 69 
 
 258 474 853 
 
 25.2389 
 
 8.6043 
 
 687 
 
 47 19 69 
 
 324 242 703 
 
 26.2107 
 
 8.8237 
 
 638 
 
 40 70 44 
 
 259 694 07- 
 
 25.2587 
 
 8.6088 
 
 688 
 
 47 33 44 
 
 325 660 672 
 
 26 . 2298 
 
 8.8280 
 
 639 
 
 40 83 21 
 
 260 917 119 
 
 25.2784 
 
 8.6132 
 
 689 
 
 47 47 21 
 
 327 082 769 
 
 26.2488 
 
 8.8323 
 
 640 
 
 40 96 oo 
 
 262 144 ooo 
 
 25.2982 
 
 8.6177 
 
 690 
 
 47 61 oo 
 
 328 509 ooo 
 
 26.2679 
 
 8.8366 
 
 641 
 
 41 08 81 
 
 263 374 721 
 
 25.3180 
 
 8.6222 
 
 691 
 
 47 74 81 
 
 329 939 371 
 
 26.2869 
 
 8.8408 
 
 642 
 
 41 21 64 
 
 264 609 288 
 
 25 3377 
 
 8.6267 
 
 692 
 
 47 88 64 
 
 331 373 888 
 
 26.3059 
 
 8.8451 
 
 643 
 
 41 34 49 
 
 265 847 707 
 
 25-3574 
 
 8.6312 
 
 693 
 
 48 02 49 
 
 332 812 557 
 
 26.3249 
 
 8.8493 
 
 644 
 
 41 47 36 
 
 267 089 984 
 
 25-3772 
 
 8.6357 
 
 694 
 
 48 16 36 
 
 334 255 384 
 
 26.3439 
 
 8.8536 
 
 645 
 
 416025 
 
 268 336 125 
 
 25.3969 
 
 8.6401 
 
 695 
 
 48 30 25 
 
 335 702 375 
 
 26 . 3629 
 
 8.8578 
 
 646 
 
 41 73 16 
 
 269 586 136 
 
 25.4165 
 
 8.6446 
 
 696 
 
 48 44 16 
 
 337 153 536 
 
 26.3818 
 
 8.8621 
 
 647 
 
 41 86 09 
 
 270 840 023 
 
 25.4362 
 
 8.6490 
 
 697 
 
 485809 
 
 338 608 873 
 
 26.4008 
 
 8.8663 
 
 648 
 
 41 99 04 
 
 272 097 792 
 
 25.4558 
 
 8.6535 
 
 698 
 
 48 72 04 
 
 340 068 392 
 
 26.4197 
 
 8.8706 
 
 649 
 
 42 12 OI 
 
 273 359 449 
 
 25.4755 
 
 8.6579 
 
 699 
 
 48 86 01 
 
 341 532 099 
 
 26.4386 
 
 8.8748 
 
 650 
 
 4225 oc 
 
 274 625 ooo 
 
 25.4951 
 
 8.6624 
 
 700 
 
 49 oo oo 
 
 343000000 
 
 26.4575 
 
 8.8790 
 
CHAIN SURVEYING 69 
 
 SQUARES, CUBES, SQUARE ROOTS AND CUBE ROOTS (Continued) 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 701 
 
 49 14 01 
 
 344 472 101 
 
 26.4764 
 
 8.8833 
 
 751 
 
 56 40 01 
 
 423 564 75i 
 
 27.4044 
 
 9.0896 
 
 702 
 
 49 28 04 
 
 345 948 408 
 
 26.4053 
 
 8.8875 
 
 752 
 
 56 55 04 
 
 425 259 008 
 
 27 . 4226 
 
 9-0937 
 
 703 
 
 49 42 09 
 
 347 428 927 
 
 26.5141 
 
 8.8917 
 
 753 
 
 56 7009 
 
 426 957 777 
 
 27.4408 
 
 9-0977 
 
 704 
 
 49 56 16 
 
 348 913 664 
 
 26.5330 
 
 8.8959 
 
 754 
 
 56 85 16 
 
 428 66l 064 
 
 27.4591 
 
 9.1017 
 
 70S 
 
 49 70 25 
 
 350 402 625 
 
 26.5518 
 
 8.9001 
 
 755 
 
 570025 
 
 430 368 875 
 
 27-4773 
 
 9 1057 
 
 706 
 
 49 84 36 
 
 351 895 816 
 
 26.5707 
 
 8.9043 
 
 756 
 
 57 15 36 
 
 432 081 216 
 
 27-4955 
 
 9.1098 
 
 707 
 
 49 98 49 
 
 353 393 243 
 
 26 . 5895 
 
 8.9085 
 
 757 
 
 57 30 49 
 
 433 798 093 
 
 27.5136 
 
 9 H38 
 
 708 
 
 50 12 64 
 
 354 894 912 
 
 26 . 6083 
 
 8.9127 
 
 758 
 
 57 45 64 
 
 435 519 512 
 
 27.5318 
 
 9.1178 
 
 709 
 
 50 26 81 
 
 356 400 829 
 
 26.6271 
 
 8.9169 
 
 759 
 
 57 6081 
 
 437 245 479 
 
 27.5500 
 
 9.1218 
 
 710 
 
 50 41 oo 
 
 3579HOOO 
 
 26.6458 
 
 8.9211 
 
 760 
 
 57 76oo 
 
 438 976 ooo 
 
 27.5681 
 
 9 1258 
 
 711 
 
 50 55 21 
 
 359 425 43i 
 
 26.6646 
 
 8.9253 
 
 ?6l 
 
 57 91 21 
 
 440711 081 
 
 27.5862 
 
 9.1298 
 
 712 
 
 50 69 44 
 
 360 944 128 
 
 26.6833 
 
 8.9295 
 
 762 
 
 580644 
 
 442 450 728 
 
 27.6043 
 
 9 1338 
 
 713 
 
 50 83 69 
 
 362 467 097 
 
 26.7021 
 
 8.9337 
 
 763 
 
 58 21 69 
 
 444 194 947 
 
 27.6225 
 
 9-1378 
 
 7U 
 
 50 97 96 
 
 363 994 344 
 
 26.7208 
 
 8.9378 
 
 764 
 
 58 36 96 
 
 445 943 744 
 
 27 . 6405 
 
 9.1418 
 
 715 
 
 51 12 25 
 
 365 525 875 
 
 26.7395 
 
 8.9420 
 
 765 
 
 58 52 25 
 
 447 697 125 
 
 27.6586 
 
 9 1458 
 
 716 
 
 51 26 56 
 
 367 061 696 
 
 26.7582 
 
 8.9462 
 
 766 
 
 58 67 56 
 
 449 455 096 
 
 27.6767 
 
 9.1498 
 
 717 
 
 51 40 89 
 
 368 601 813 
 
 26.7769 
 
 8.9503 
 
 767 
 
 58 82 89 
 
 451 217 663 
 
 27.6948 
 
 9-1537 
 
 718 
 
 Si 55 24 
 
 370 146 232 
 
 26.7955 
 
 8.9545 
 
 768 
 
 58 98 24 
 
 452 984 832 
 
 27.7128 
 
 9-1577 
 
 719 
 
 51 69 61 
 
 371 694 959 
 
 26.8142 
 
 8.9587 
 
 769 
 
 59 13 61 
 
 454 756 609 
 
 27.7308 
 
 9.1617 
 
 720 
 
 518400 
 
 373 248 ooo 
 
 26.8328 
 
 8.9628 
 
 770 
 
 592900 
 
 456 533 ooo 
 
 27.7489 
 
 9.1657 
 
 721 
 
 51 98 41 
 
 374 805 361 
 
 26.8514 
 
 8.9670 
 
 771 
 
 59 44 4i 
 
 458 314 on 
 
 27.7669 
 
 9.1696 
 
 722 
 
 52 12 84 
 
 376 367 048 
 
 26.8701 
 
 8.9711 
 
 772 
 
 59 59 84 
 
 460 099 648 
 
 27.7849 
 
 9.I736 
 
 723 
 
 52 27 29 
 
 377 933 067 
 
 26.8887 
 
 8.9752 
 
 773 
 
 59 75 29 
 
 461 889 917 
 
 27.8029 
 
 9-1775 
 
 724 
 
 52 41 76 
 
 379 503 424 
 
 26.9072 
 
 8.9794 
 
 774 
 
 599076 
 
 463 684 824 
 
 27.8209 
 
 9.I8I5 
 
 '25 
 
 52 56 25 
 
 381 078 125 
 
 26.9258 
 
 8.9835 
 
 775 
 
 600625 
 
 465 484 375 
 
 27.8388 
 
 9.1855 
 
 726 
 
 52 70 76 
 
 382 657 176 
 
 26.9444 
 
 -8.9876 
 
 776 
 
 6021 76 
 
 467 288 576 
 
 27.8568 
 
 9-1894 
 
 727 
 
 52 85 29 
 
 384 240 583 
 
 26.9629 
 
 8.9918 
 
 777 
 
 6037 29 
 
 469 097 433 
 
 27.8747 
 
 9 1933 
 
 728 
 
 529984 
 
 385 828 352 
 
 26.9815 
 
 8.9959 
 
 778 
 
 605284 
 
 470 910 952 
 
 27.8927 
 
 9-1973 
 
 729 
 
 53 14 4i 
 
 387 420 489 
 
 27.0000 
 
 9.0000 
 
 779 
 
 6068 41 
 
 472 729 139 
 
 27.9106 
 
 9.2012 
 
 730 
 
 532900 
 
 389 017 ooo 
 
 27.0185 
 
 9.0041 
 
 780 
 
 608400 
 
 474 552 ooo 
 
 27.9285 
 
 9.2052 
 
 731 
 
 53 43 61 
 
 390 617 891 
 
 27.0370 
 
 9.0082 
 
 78l 
 
 609961 
 
 476 379 541 
 
 27.9464 
 
 9.2091 
 
 732 
 
 53 58 24 
 
 392 223 168 
 
 27.0555 
 
 9 0123 
 
 782 
 
 61 15 24 
 
 478 211 768 
 
 27.9643 
 
 9 2130 
 
 733 
 
 53 72 89 
 
 393 832 837 
 
 27.0740 
 
 9.0164 
 
 783 
 
 61 30 89 
 
 480 048 687 
 
 27.9821 
 
 9.2170 
 
 734 
 
 53 87 56 
 
 395 446 904 
 
 27.0924 
 
 9.0205 
 
 784 
 
 61 46 56 
 
 481 890 304 
 
 28.0000 
 
 9.2209 
 
 735 
 
 54 02 25 
 
 397 065 375 
 
 27.1109 
 
 9.0246 
 
 785 
 
 61 62 25 
 
 483 736 625 
 
 28.0179 
 
 9.2248 
 
 736 
 
 541696 
 
 398 688 256 
 
 27.1293 
 
 9.0287 
 
 786 
 
 61 77 96 
 
 485 587 656 
 
 28.0357 
 
 9.2287 
 
 737 
 
 54 31 69 
 
 400 315 553 
 
 27-1477 
 
 9.0328 
 
 787 
 
 61 93 69 
 
 487 443 403 
 
 28.0535 
 
 9.2326 
 
 738 
 
 54 46 44 
 
 401 947 272 
 
 27.1662 
 
 9-0369 
 
 788 
 
 620944 
 
 489 303 872 
 
 28.0713 
 
 9-2365 
 
 739 
 
 54 61 21 
 
 403 583 419 
 
 27.1846 
 
 9.0410 
 
 789 
 
 62 25 21 
 
 491 169 069 
 
 28.0891 
 
 9.2404 
 
 740 
 
 5476oo 
 
 405 224 ooo 
 
 27.2029 
 
 9.0450 
 
 790 
 
 62 41 oo 
 
 493 039 ooo 
 
 28.1069 
 
 9-2443 
 
 741 
 
 54 90 8l 
 
 406 869 021 
 
 27.2213 
 
 9.0491 
 
 791 
 
 62 56 81 
 
 494 913 671 
 
 28.1247 
 
 9.2482 
 
 742 
 
 55 05 64 
 
 408 518 488 
 
 27.2397 
 
 9.0532 
 
 792 
 
 62 72 64 
 
 496 793 088 
 
 28.1425 
 
 9.2521 
 
 743 
 
 55 20 49 
 
 410 172 407 
 
 27 . 2580 
 
 9.0572 
 
 793 
 
 6288 49 
 
 498 677 257 
 
 28.1603 
 
 9-2560 
 
 744 
 
 55 35 36 
 
 411 830 784 
 
 27.2764 
 
 9-0613 
 
 794 
 
 63 04 36 
 
 500 566 184 
 
 28.1780 
 
 9-2599 
 
 745 
 
 55 50 25 
 
 413 493 625 
 
 27.2947 
 
 9-0654 
 
 795 
 
 63 20 25 
 
 502 459 875 
 
 28.1957 
 
 9-2638 
 
 746 
 
 55 65 16 
 
 415 160 936 
 
 27.3130 
 
 9.0694 
 
 796 
 
 63 36 16 
 
 504 358 336 
 
 28.2135 
 
 9.2677 
 
 747 
 
 558-009 
 
 416 832 723 
 
 27.3313 
 
 9-0735 
 
 797 
 
 63 52 09 
 
 506 261 573 
 
 28.2312 
 
 9.2716 
 
 748 
 
 55 95 04 
 
 418 508 992 
 
 27.3496 
 
 9-0775 
 
 798 
 
 63 68 04 
 
 508 169 592 
 
 28.2489 
 
 9-2754 
 
 749 
 
 56 10 01 
 
 420 189 749 
 
 27.3679 
 
 9.0816 
 
 799 
 
 63 84 01 
 
 510 082 399 
 
 28.2666 
 
 9-2793 
 
 750 
 
 562500 
 
 421 875 ooo 
 
 27.3861 
 
 9.0856 
 
 800 
 
 64 oo oo 
 
 512 ooo ooo 
 
 28.2843 
 
 9.2832 
 
70 PRACTICAL SURVEYING 
 
 SQUARES, CUBES, SQUARE ROOTS AND CUBE ROOTS (Continued) 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 801 
 
 64 16 01 
 
 513 922 401 
 
 28.3019 
 
 9 . 2870 
 
 851 
 
 72 42 01 
 
 616 295 051 
 
 29.1719 
 
 9.4764 
 
 802 
 
 643204 
 
 515 849 608 
 
 28.3196 
 
 9-2909 
 
 852 
 
 72 59 04 
 
 618 470 208 
 
 29.1890 
 
 9.4801 
 
 803 
 
 64 48 09 
 
 517 781 627 
 
 28.3373 
 
 9.2948 
 
 853 
 
 72 76 09 
 
 620 650 477 
 
 29 . 2062 
 
 9.4838 
 
 804 
 
 64 64 16 
 
 519 7i8 464 
 
 28.3549 
 
 9.2986 
 
 854 
 
 72 93 16 
 
 622 835 864 
 
 29.2233 
 
 9.4875 
 
 805 
 
 64 80 25 
 
 521 660 125 
 
 28.3725 
 
 9 3025 
 
 855 
 
 73 10 25 
 
 625 026 375 
 
 29.2404 
 
 9.4912 
 
 806 
 
 649636 
 
 523 606 616 
 
 28.3901 
 
 9.3063 
 
 856 
 
 73 27 36 
 
 627 222 Ol6 
 
 29-2575 
 
 9.4949 
 
 807 
 
 65 12 49 
 
 525 557 943 
 
 28.4077 
 
 9-3102 
 
 857 
 
 73 44 49 
 
 629 422 793 
 
 29.2746 
 
 9-4986 
 
 808 
 
 65 28 64 
 
 527514112 
 
 28.4253 
 
 9 3140 
 
 858 
 
 73 61 64 
 
 631 628 712 
 
 29.2916 
 
 9.5023 
 
 809 
 
 65 44 81 
 
 529 475 129 
 
 28.4429 
 
 9.3179 
 
 859 
 
 73 78 81 
 
 633 839 779 
 
 29.3087 
 
 9.5060 
 
 810 
 
 65 61 oo 
 
 531 441 ooo 
 
 28.4605 
 
 9.3217 
 
 860 
 
 7396oo 
 
 636 056 ooo 
 
 29.3258 
 
 9-5097 
 
 811 
 
 65 77 21 
 
 533 4ii 731 
 
 28.4781 
 
 9 3255 
 
 861 
 
 74 13 21 
 
 638 277 38i 
 
 29.3428 
 
 9-5134 
 
 812 
 
 65 93 44 
 
 535 387 328 
 
 28.4956 
 
 9.3294 
 
 862 
 
 74 30 44 
 
 640 503 928 
 
 29.3598 
 
 9.5I7I 
 
 813 
 
 66 09 69 
 
 537 367 797 
 
 28.5132 
 
 9-3332 
 
 863 
 
 74 47 69 
 
 642 735 647 
 
 29.3769 
 
 9.5207 
 
 814 
 
 662596 
 
 539 353 144 
 
 28.5307 
 
 9-3370 
 
 864 
 
 74 64 96 
 
 644 972 544 
 
 29-3939 
 
 9-5244 
 
 815 
 
 66 42 25 
 
 541 343 375 
 
 28.5482 
 
 9.3408 
 
 865 
 
 74 82 25 
 
 647 214 625 
 
 29.4109 
 
 9-5281 
 
 816 
 
 66 58 56 
 
 543 338 496 
 
 28.5657 
 
 9-3447 
 
 866 
 
 74 99 56 
 
 649 461 896 
 
 29.4279 
 
 9-5317 
 
 817 
 
 66 74 89 
 
 545 338 513 
 
 28.5832 
 
 9.3485 
 
 867 
 
 75 16 89 
 
 651 714 363 
 
 29.4449 
 
 9-5354 
 
 818 
 
 66 91 24 
 
 547 343 432 
 
 28.6007 
 
 9-3523 
 
 868 
 
 75 34 24 
 
 653 972 032 
 
 29.4618 
 
 9.5391 
 
 819 
 
 67 07 61 
 
 549 353 259 
 
 28.6182 
 
 9.356I 
 
 869 
 
 75 51 61 
 
 656 234 909 
 
 29.4788 
 
 9-5427 
 
 820 
 
 67 24 oo 
 
 551 368 ooo 
 
 28.6356 
 
 9-3599 
 
 870 
 
 75 6900 
 
 658503000 29.4958 
 
 9.5464 
 
 821 
 
 67 40 41 
 
 553 387 601 
 
 28.6531 
 
 9.3637 
 
 871 
 
 75 86 41 
 
 660 776 311 
 
 29.5127 
 
 9.5501 
 
 822 
 
 67 56 84 
 
 555 412 248 
 
 28 . 6705 
 
 9.3675 
 
 872 
 
 76 03 84 
 
 663 054 848 
 
 29.5296 
 
 9-5537 
 
 823 
 
 67 73 29 
 
 557 441 767 
 
 28.6880 
 
 9.3713 
 
 873 
 
 76 21 29 
 
 665 338 617 
 
 29.5466 
 
 9-5574 
 
 824 
 
 67 89 76 
 
 559 476 224 
 
 28.7054 
 
 9-3751 
 
 874 
 
 76 38 76 
 
 667 627 624 
 
 29.5635 
 
 9.5610 
 
 825 
 
 68 06 25 
 
 561 515 625 
 
 28.7228 
 
 9.3789 
 
 875 
 
 76 56 25 
 
 669 921 875 
 
 29.5804 
 
 9.5647 
 
 826 
 
 68 22 76 
 
 563 559 976 
 
 28.7402 
 
 9-3827 
 
 876 
 
 76 73 ?6 
 
 672 221 376 
 
 29.5973 
 
 9-568.3 
 
 827 
 
 68 39 29 
 
 565 609 283 
 
 28.7576 
 
 9.3865 
 
 877 
 
 76 91 29 
 
 674 526 133 
 
 29.6142 
 
 9.5719 
 
 828 
 
 68 55 84 
 
 567 663 552 
 
 28.7750 
 
 9.3902 
 
 878 
 
 77 08 84 
 
 676 836 152 
 
 29-6311 
 
 9.5756 
 
 829 
 
 68 72 41 
 
 569 722 789 
 
 28.79^4 
 
 9-3940 
 
 879 
 
 77 26 41 
 
 679 151 439 
 
 29.6479 
 
 9-5792 
 
 830 
 
 688900 
 
 571 787 ooo 
 
 28.8097 
 
 9.3978 
 
 880 
 
 77 4400 
 
 68 1 472 ooo 
 
 29.66*8 
 
 9.5828 
 
 831 
 
 69 05 61 
 
 573 856 191 
 
 28.8271 
 
 9.4016 
 
 881 
 
 77 61 61 
 
 683 797 841 
 
 29.6816 
 
 9.5865 
 
 1832 
 
 69 22 24 
 
 575 930 368 
 
 28.8444! 9.4053 
 
 882 
 
 77 79 24 
 
 686 128 968 
 
 29-6985 
 
 9-5901 
 
 i 833 
 
 69 38 89 
 
 578 009 537 
 
 28.8617 
 
 9.4091 
 
 883 
 
 77 96 89 
 
 688 465 387 
 
 29-7153 
 
 9-5937 
 
 834 
 
 69 55 56 
 
 580 093 704 
 
 28.8791 
 
 9.4129 
 
 884 
 
 78 14 56 
 
 690 807 104 
 
 29.7321 
 
 9-5973 
 
 ,835 
 
 69 72 25 
 
 582 182 875 
 
 28.8964 
 
 9.4166 
 
 885 
 
 78 32 25 
 
 693 154 125 
 
 29.7489 
 
 9.6010 
 
 836 
 
 69 88 96 
 
 584 277 056 
 
 28.9137 
 
 9.4204 
 
 886 
 
 784996 
 
 695 506 456 
 
 29.7658 
 
 9.6046 
 
 837 
 
 70 05 69 
 
 586 376 253 
 
 28.9310 
 
 9.4241 
 
 887 
 
 78 67 69 
 
 697 864 103 
 
 29.7825 
 
 9.6082 
 
 838 
 839 
 
 70 22 44 
 70 39 21 
 
 588 480 472 
 590 589 719 
 
 28.9482 
 28.9655 
 
 9-4279 
 9.4316 
 
 888 
 889 
 
 78 85 44 
 79 03 21 
 
 700 227 072 
 702 595 369 
 
 29-7993 
 29.8161 
 
 9.6118 
 9-6154 
 
 840 
 
 70 56 oo 
 
 592 704 ooo 
 
 28.9828 
 
 9-4354 
 
 890 
 
 792100 
 
 704 969 ooo 
 
 29.8329 
 
 9.6190 
 
 841 
 
 70 72 81 
 
 594 823 321 
 
 29.0000 
 
 9.4391 
 
 891 
 
 79 38 81 
 
 707 347 97i 
 
 29 . 8496 
 
 9.6226 
 
 842 
 
 70 89 64 
 
 596 947 688 
 
 29.0172 
 
 9.4429 
 
 892 
 
 79 56 64 
 
 709 732 288 
 
 29.8664 
 
 9.6262 
 
 843 
 
 71 06 49 
 
 599 077 107 
 
 29.0345 
 
 9.4466 
 
 893 
 
 79 74 49 
 
 712 121 957 
 
 29.8831 
 
 9.6298 
 
 844 
 
 71 23 36 
 
 601 211 584 
 
 29.0517 
 
 9 - 4503 
 
 894 
 
 79 92 36 
 
 714 5i6 984 
 
 29.8998 
 
 9-6334 
 
 845 
 
 71 40 25 
 
 603 351 125 
 
 29.0689 
 
 9-4541 
 
 895 
 
 80 10 25 
 
 716 917 375 
 
 29.9166 
 
 9.6370 
 
 846 
 
 71 57 16 
 
 605 495 736 
 
 29.0861 
 
 9.4578 
 
 896 
 
 80 2g 16 
 
 719 323 136 
 
 29-9333 
 
 9.6406 
 
 847 
 
 71 7409 
 
 607 645 423 
 
 29.1033 
 
 9.4615 
 
 897 
 
 80 46 09 
 
 721 734 273 
 
 29.9500 
 
 9.6442 
 
 848 
 
 71 91 04 
 
 609 800 192 
 
 29.1204 
 
 9.4652 
 
 898 
 
 80 64 04 
 
 724 150 792 
 
 29.9666 
 
 9-6477 
 
 849 
 
 72 08 01 
 
 6n 960 049 
 
 29.1376 
 
 9.4690 
 
 899 
 
 80 82 or 
 
 726 572 699 
 
 29.9833 
 
 9.6513 
 
 850 
 
 722500 
 
 614 125 ooo 
 
 29.1548 
 
 9-4727 
 
 900 
 
 81 oo oo 
 
 729 ooo ooo 
 
 30.0000 
 
 9 6549 
 
CHAIN SURVEYING 71 
 
 SQUARES, CUBES, SQUARE ROOTS AND CUBE ROOTS (Continued) 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 Nos. 
 
 Squares. 
 
 Cubes. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 901 
 
 81 18 01 
 
 73i 432 701 
 
 30.0167 
 
 9.6585 
 
 95: 
 
 904401 
 
 860 085 351 
 
 30.8383 
 
 9 8339 
 
 902 
 
 81 36 04 
 
 733 870 808 
 
 30.0333 9.6620 
 
 952 
 
 906304 
 
 862 801 408 
 
 30.8545 
 
 9.8374 
 
 903 
 
 81 5409 
 
 736 314 327 
 
 30.0500! 9.6656 
 
 953 
 
 9082 09 
 
 865 523 :77 
 
 30.8707 
 
 9.8408 
 
 904 
 
 81 72 16 
 
 738 763 264 
 
 30.0666 
 
 9.6692 
 
 954 
 
 91 01 16 
 
 868 250 664 
 
 30.8869 
 
 9.8443 
 
 905 
 
 81 9025 
 
 741 217 625 
 
 30.0832 
 
 9.6727 
 
 955 
 
 91 20 25 
 
 870 983 875 
 
 30.903: 
 
 9.8477 
 
 906 
 
 820836 
 
 743 677 4i6 
 
 30.0998 
 
 9-6763 
 
 956 
 
 9: 39 36 
 
 873 722 816 
 
 30.9:92 
 
 9-85II 
 
 90? 
 
 82 26 49 
 
 746 142 643 
 
 30.1164 9.6799 
 
 957 
 
 91 58 49 
 
 876 467 493 
 
 30.9354 
 
 9.8546 
 
 908 
 
 82 44 64 
 
 748 613 312 
 
 30.1330 9-6834 
 
 958 
 
 91 77 64 
 
 879 217 9:2 
 
 30.95:6 
 
 9.8580 
 
 909 
 
 82 62 81 
 
 751 089 429 
 
 30.1496 
 
 9.6870 
 
 959 
 
 91 9681 
 
 88 1 974 079 
 
 30.9677 
 
 9.8614 
 
 910 
 
 82 81 oo 
 
 753 571 ooo 
 
 30.1662 
 
 9-690S 
 
 960 
 
 92 16 oo 
 
 884 736 ooo 
 
 30.9839 
 
 9.8648 
 
 911 
 
 82 99 21 
 
 756 058 031 
 
 30.1828 
 
 9.694: 
 
 96! 
 
 92 35 21 
 
 887 503 681 
 
 3:. oooo 
 
 9.8683 
 
 912 
 
 83 17 44 
 
 758 550 528 
 
 30.1993 
 
 9.6976 
 
 962 
 
 925444 
 
 890 277 128 
 
 31.0161 
 
 9-87:7 
 
 
 83 35 69 
 
 761 048 497 
 
 30.2159 
 
 9.7012 
 
 963 
 
 92 73 69 
 
 893 056 347 
 
 3: 0322 
 
 9.875: 
 
 914 
 
 835396 
 
 763 551 944 
 
 30.2324 
 
 9-7047 
 
 964 
 
 929296 
 
 895 841 344 
 
 3:. 0483 
 
 9.8785 
 
 
 83 72 25 
 
 766 060 875 
 
 30.2490 
 
 9.7082 
 
 965 
 
 93 :2 25 
 
 898 632 125 
 
 3:. 0644 
 
 9.8819 
 
 916 
 
 839056 
 
 768 575 296 
 
 30.2655 
 
 9.7118 
 
 966 
 
 93 31 56 
 
 901 428 696 
 
 3:. 0805 
 
 9-8854 
 
 917 
 
 840889 
 
 771 095 213 
 
 30.2820 
 
 9.7:53 
 
 967 
 
 93 50 89 
 
 904 231 063 
 
 3:. 0966 
 
 9.8888 
 
 918 
 
 84 27 24 
 
 773 620 632 
 
 30.2985 
 
 9-7:88 
 
 968 
 
 93 70 24 
 
 907 039 232 
 
 3:. "27 
 
 9.8922 
 
 919 
 
 84 45 6l 
 
 776 151 559 
 
 30.3150 
 
 9.7224 
 
 969 
 
 93 89 61 
 
 909 853 209 
 
 31.1288 
 
 8.8956 
 
 920 
 
 8464 oo 
 
 778 688 ooo 
 
 30.3315 
 
 9-7259 
 
 970 
 
 940900 
 
 912 673 ooo 
 
 3:. 1448 
 
 9.8990 
 
 921 
 
 84 82 41 
 
 781 229 961 
 
 30.3480 
 
 9.7294 
 
 971 
 
 94 28 4: 
 
 915498611 
 
 3:.:6o9 
 
 9.9024 
 
 922 
 
 850084 
 
 783 777 448 
 
 30.3645 
 
 9-7329 
 
 972 
 
 94 47 84 
 
 918 330 048 
 
 3:.: 769 
 
 9.9058 
 
 923 
 
 85 19 29 
 
 786 330 467 
 
 30.3809 
 
 9.7364 
 
 973 
 
 94 67 29 
 
 921 167 317 
 
 31.1929 
 
 9.9092 
 
 924 
 
 85 37 76 
 
 788 889 024 
 
 30.3974 
 
 0.7400 
 
 974 
 
 94 86 76 
 
 924 oio 424 
 
 31.2090 
 
 9.9126 
 
 925 
 
 85 S6 25 
 
 791 453 125 
 
 30.4138 
 
 9-7435 
 
 975 
 
 950625 
 
 926 859 375 
 
 3:- 2250 
 
 9.9160 
 
 926 
 
 85 74 76 
 
 794 022 776 
 
 30.4302 
 
 9.7470 
 
 976 
 
 95 25 76 
 
 929 7:4 :?6 
 
 3:- 24:0 
 
 9.9:94 
 
 927 
 
 85 93 29 
 
 796 597 983 
 
 30.4467 
 
 9.7505 
 
 977 
 
 95 45 29 
 
 932 574 833 
 
 3:. 2570 
 
 9.9227 
 
 928 
 
 86 II 84 
 
 799 178 752 
 
 30.4631 
 
 9-7540 
 
 978 
 
 95 64 84 
 
 935 44: 352 
 
 3:- 2730 
 
 9.9261 
 
 929 
 
 86 30 41 
 
 801 765 089 
 
 30.4795 
 
 9-7575 
 
 979 
 
 95 84 41 
 
 938 3:3 739 
 
 31.2890 
 
 9.9295 
 
 930 
 
 864900 
 
 804 357 ooo 
 
 30.4959 
 
 9.7610 
 
 980 
 
 96 04 oo 
 
 941 192 ooo 
 
 31.3050 
 
 9.9329 
 
 931 
 
 86 67 61 
 
 806 954 491 
 
 30.5:23 
 
 9.7645 
 
 98l 
 
 962361 
 
 944 076 141 
 
 31.3209 
 
 9.9363 
 
 932 
 
 86 86 24 
 
 809 557 568 
 
 30.5287 
 
 9.7680 
 
 982 
 
 9643 24 
 
 946 966 168 
 
 
 9-9396 
 
 933 
 
 87 04 89 
 
 812 166 237 
 
 30.5450 
 
 9.77:5 
 
 983 
 
 96 62 89 
 
 949 862 087 
 
 31.3528 
 
 9-9430 
 
 934 
 
 87 23 56 
 
 814 780 504 
 
 30.5614 
 
 9-7750 
 
 ?s- 
 
 968256 
 
 952 763 904 
 
 31 .3688 
 
 9.9464 
 
 935 
 
 87 42 25 
 
 817 400 375 
 
 30.5778 
 
 9.7785 
 
 
 97 02 25 
 
 955 671 625 
 
 3: '3847 
 
 9-9497 
 
 936 
 
 876096 
 
 820 025 856 
 
 30.5941 
 
 9.7819 
 
 986 
 
 972196 
 
 958 585 256 
 
 31.4006 
 
 9-9531 
 
 937 
 
 87 79 69 
 
 822 656 953 
 
 30.6105 
 
 9.7854 
 
 987 
 
 974169 
 
 961 504 803 
 
 3:. 4:66 
 
 9.9565 
 
 938 
 
 8? 98 44 
 
 825 293 672 
 
 30.6268 
 
 9.7889 
 
 988 
 
 97 61 44 964 430 272 
 
 3: 4325 
 
 9 9598 
 
 939 
 
 88 17 21 
 
 827 936 019 
 
 30.6431 
 
 9.7924 
 
 989 
 
 97 81 21 
 
 967 361 669 
 
 3:. 4484 
 
 
 940 
 
 88.3600 
 
 830 584 ooo 
 
 30.6594 
 
 9-7959 
 
 990 
 
 9801 oo 
 
 970 299 ooo 
 
 3:- 4643 
 
 9.9666 
 
 941 
 
 885481 
 
 833 237 621 
 
 30.6757 
 
 9-7993 
 
 991 
 
 98 2081 
 
 973 242 271 
 
 31.4802 
 
 9.9699 
 
 942 
 
 88 73 64 
 
 835 896 888 
 
 30.6920 
 
 9.8028 
 
 992 
 
 984064 
 
 976 191 488 
 
 3:. 496o 
 
 9-9733 
 
 943 
 
 889249 
 
 838 561 807 
 
 30.7083 
 
 9.8063 
 
 993 
 
 986049 
 
 979 146 657 
 
 3:.5"9 
 
 9.9766 
 
 944 
 
 89 II 36 
 
 841 232 384 
 
 30.7246 
 
 9.8097 
 
 994 
 
 988036 
 
 982 107 784 
 
 3:- 5278 
 
 9.9800 
 
 945 
 
 89 30 25 
 
 843 908 625 
 
 30.7409 
 
 9-8132 
 
 995 
 
 990025 
 
 985 074 875 
 
 3:- 5436 
 
 9.9833 
 
 946 
 
 89 49 16 
 
 846 590 536 
 
 30.757: 
 
 9.8167 
 
 996 
 
 99 20 16 
 
 988 047 936 
 
 3:- 5595 
 
 9-9866 
 
 947 
 
 8968 09 
 
 849 278 123 
 
 30.7734 
 
 9.8201 
 
 997 
 
 994009 
 
 991 026 973 
 
 31-5753 
 
 9-9900 
 
 948 
 
 89 87 04 
 
 851 97i 392 
 
 30.7896 
 
 9.8236 
 
 998 
 
 996004 
 
 994 oil 992 
 
 3:- S9" 
 
 
 949 
 
 900601 
 
 854 670 349 
 
 30.8058 
 
 9.8270 
 
 993 
 
 99 80 01 
 
 997 002 999 
 
 31 . 6070 
 
 9.9967 
 
 950 
 
 90 25 oo 
 
 857 375 ooo 
 
 30.8221 
 
 9-8305 
 
 IOOO 
 
 I OO OO OO 
 
 I OOO OOO OOO 
 
 31.6228 
 
 10. OOOO 
 
72 PRACTICAL SURVEYING 
 
 MEASURES OF LENGTH AND AREA. 
 
 12 inches I foot = 0.3047973 meter. 
 
 3 feet I yard = 36 ins. = 0.9143919 meter. 
 
 5^ yards. ...'..... I rod, pole or perch = i6j ft. = 198 ins. 
 
 40 rods I furlong = J mile = 220 yds. = 660 ft. 
 
 8 furlongs I statute mile = 320 rods = 1760 yds. 
 
 = 5280 ft. 
 
 3 miles I league = 24 furlongs = 960 rods 
 
 = 5280 yds. = 15,840 ft. 
 
 Gunter's chain for surveyors = 66 ft. = 4 rods = 100 
 links. 
 
 I link = 0.66 ft. = 7.92 ins. 
 I section of land = I sq. mile = 640 acres. 
 I acre contains 43,560 sq. ft. and measures 208.71 X 
 208.71 ft. = 10 sq. Gunter's chains. 
 
 The vara is an old Spanish measure of length. It is 
 used in Mexico and in some of the western states. The 
 legal vara in California = 33.372 ins. In San Francisco, 
 Cal., the vara = 33 ins. The vara of Castile = 32.8748 ins. 
 
 The metric system is decimal and is based on the meter 
 (39.370428 ins.). The decimeter = T V m., centimeter = 
 T tf m., millimeter = y^V^ m. The dekameter = 10 m., 
 hectometer = 100 m., kilometer = 1000 m., myriameter = 
 10,000 m. The metric square measures have the word 
 "square" prefixed to the measures of length except the 
 square hectometer which is known as the hectare = 2.4711 
 acres. 
 
CHAPTER III 
 LEVELING 
 
 The object of leveling is to determine the difference in 
 elevation between two or more points. To do this work 
 requires no mathematical knowledge beyond the ability to 
 add and subtract. 
 
 A vertical line points to the center of the earth and a 
 line perpendicular to a vertical line is a horizontal line. 
 
 A level line is parallel with the surface of still water 
 and each point marks an equal distance from the center 
 of the earth. In plane surveying the distances between 
 points are so short that for all practical purposes a hori- 
 zontal line is considered to be a level line. 
 
 A level line in plane surveying is one having the same 
 elevation throughout the length and a horizontal line is 
 actually only a line of apparent level. 
 
 In Fig. 80 let BE represent an arc of the A 
 earth's surface. AD represents an arc about B 
 the height of the eye of an observer par- ^^Ss. '" IA 
 
 allel to BE. When C is seen from A the p IG> o 
 points A and D are on the same true level 
 and the points A and C are on the same apparent level. 
 In a distance of one mile the difference CD is practically 
 8 ins. 
 
 Let D = distance in miles, 
 
 h difference in feet between true and apparent 
 level, 
 
 . 2D* 
 
 then h = 
 
 3 
 When D = distance in feet, h = 0.000,000,024 D 2 . 
 
 Refraction causes objects near the horizon to appear 
 higher than they are actually. For very long sights, 
 
 73 
 
74 PRACTICAL SURVEYING 
 
 especially when taken early or late in the day, a correction 
 must be made for refraction. The formulas then become, 
 
 when D = distance in miles, 
 
 9 
 
 and, when D = distance in feet r 
 
 h = 0.000,000,021 > 2 . 
 
 With instruments ordinarily used and with usual lengths 
 of sights curvature of the earth and refraction cannot 
 affect the work. If a backsight, however, is very short 
 and a foresight is very long both the aforementioned 
 factors may have a slight effect, and if the instrument is 
 not in good adjustment errors will be multiplied in the 
 proportion the length of backsight bears to the length of 
 foresight. The instrument should be set as nearly as 
 possible equidistant between points on which the rod is 
 held. This is important. 
 
 Very simple leveling instruments were used by surveyors 
 and engineers in early times, and are just as useful today 
 when well-made modern instruments are not available. 
 
 The miner's triangle. The early 
 miners in California set grade pegs on 
 hundreds of miles of ditches and roads 
 with this primitive instrument. When 
 the grade pegs were set at intervals 
 of one rod (i6J ft.) the distance from 
 D to E = i rod. When this made 
 the triangle inconveniently large the 
 FiG.Si. Miner's triangle. d i stance was 8 J f t>> or IO ft 
 
 To adjust the triangle two pegs were driven so the ends 
 of the triangle could rest on them, the tops of the pegs 
 being, as nearly as the eye could judge, at the same ele- 
 vation. From the apex of the triangle a plumb-bob was 
 suspended by a thread or fine cord and the feet of the tri- 
 angle placed on the pegs. 
 
 The point where the plumb line crossed the brace was 
 then marked (say at A). The ends of the triangle were 
 then reversed, and the place where the line crossed on this 
 trial was marked (B). Halfway between A and B a 
 
LEVELING 75 
 
 mark C was cut on the brace. Whenever the triangle was 
 held so the vertical plumb line crossed the brace at C, the 
 ends D and E were at the same elevation. 
 
 To run a grade line with a miner's triangle. Suppose 
 the grade is to be | in. in one rod. At E drive a nail 
 with the head projecting \ in. When the first grade peg 
 is driven rest the leg D on it and swing the triangle until 
 E rests on the ground and the plumb line crosses the mark 
 at C. Swing E far enough to one side to permit a peg to 
 be driven where the end E had touched the ground. Now 
 place E so the nail head rests on the peg and drive the peg 
 until the plumb line crosses C. The top of the peg under 
 E will be J in. lower than the top of the peg under D. 
 
 To run a grade with a carpen- 
 ter s level and straight-edge. 
 A straight-edge is more con- 
 venient to use than a miner's 
 triangle, and is used in the FlG . 82> straight-edge and level, 
 same manner on smooth land 
 
 and in ditches. When the ground is covered with rocks 
 or vegetation the triangle is better although clumsy. A 
 straight-edge is usually made from a 2-in. plank. In 
 the middle for a length of 2 or 3 ft. the depth is about 
 7 ins. and at the ends about 3 ins. The bottom is first 
 made perfectly straight after which the top is made parallel 
 with it. A carpenter's level is used on top, this being- 
 more convenient than a plumb-bob and line. The grade 
 nail is driven in the bottom at one end when the edges 
 have been made truly parallel. 
 
 To adjust a straight-edge so the top and bottom will be 
 parallel. First test the level and be certain the bubble is 
 in correct adjustment. There being no grade nail in the 
 bottom of the straight-edge, drive a peg at each end and 
 with the carpenter's level held on top drive the pegs until 
 the bubble indicates the tops of the pegs to be at the same 
 elevation. 
 
 Mark the glass at the end of the bubble and holding the 
 level in position reverse the ends of the straight-edge, 
 resting them on the pegs. The bubble will move and the 
 new position of the end is to be marked. Midway be- 
 tween the two marks make a third mark. Holding the 
 
76 PRACTICAL SURVEYING 
 
 level in place and the straight-edge on the pegs drive down 
 the high peg gently until the end of the bubble moves to 
 the middle mark. 
 
 The tops of the pegs are now at exactly the same ele- 
 vation. Holding the straight-edge in place reverse the 
 level and plane down the high end of the top of the straight- 
 edge until the end of the bubble touches the middle mark. 
 The bubble in the carpenter's level should remain in the 
 middle no matter how the level is placed on top of the 
 straight-edge. 
 
 Road percenter. This instrument was in common use 
 for several centuries and the author once used one in lay- 
 
 B m & out a roa< ^ ^ or a mmm g com- 
 
 1 pany when no better instrument 
 
 1 was available. The same principle 
 
 is seen today in the straight-edges 
 used by brick masons for plumbing 
 walls. 
 
 A piece of wood 2 ins. by 4 ins. is 
 made perfectly straight on top, with 
 pieces of tin at A and B for sights. 
 
 Through each sight at points the 
 FIG. 83. Road percenter. same h?ight aboye the tQp of the 
 
 wood a hole ^ in. in diameter is made. At D a i-in. hole 
 is bored through for the upper part of a Jacob staff. A 
 f-in. by 3-in. piece 15 ins. long is attached to the side so 
 that a line scratched down the middle is exactly perpen- 
 dicular to the top. The space C is cut out and a 12-oz. 
 plumb-bob hung by a silk thread one foot long attached 
 near the top swings in this space. 
 
 Theoretically when the instrument is placed on the 
 Jacob staff and the silk plumb line covers the vertical 
 scratch the line of sight through the sighting holes is hori- 
 zontal. A level rod is used with this instrument. It is 
 termed a percenter because a graduated arc is sometimes 
 placed above the opening in which the plumb-bob swings. 
 When the eye end is lowered by tipping the Jacob staff 
 until the plumb line crosses the 2 per cent mark the line 
 of sight instead of being horizontal is inclined 2 per cent 
 to the horizontal. Road grades are expressed in rise or 
 fall per 100 ft., that is in per cent of rise per foot and the 
 
LEVELING 
 
 77 
 
 angle corresponding to any per cent of rise being known 
 the line of sight may be set at the proper angle and the 
 grade stakes set. The percenter in theory is good but 
 practically is of little service except on straight lines. Used 
 as a level fairly good results may be obtained with careful 
 work. 
 
 Water-tube level. This level consists of a metal tube 
 two or three feet long bent up at 
 
 the ends with glass vials proj ecting (\ fss= _ i j) 
 
 above the ends and open at the top. 
 In the middle a socket for the head 
 of a Jacob staff or a tripod is fas- 
 tened. The tube is rilled with col- 
 ored liquid and the line of sight 
 across the top of the liquid is level. 
 
 Water tube level. 
 
 Leveling with rubber hose. For leveling line shafts in 
 mills where it is often difficult to use levels and rods the 
 use of small rubber tubes is common. One end of the tube 
 is fastened to a tank containing water with the surface at 
 nearly the elevation of the shafting. In the other end is 
 a graduated glass tube corked to prevent the loss of water 
 while the tube is being carried. Holding the glass tube 
 against a post or beam on which a mark is to be placed 
 and raising it until the top of the water is a few inches be- 
 low the end, the cork is removed. The surface of the water 
 when the cork is removed, so the compression of enclosed 
 air will have no effect, will be level with the surface of the 
 
 water in the tank and the 
 height can be marked by a 
 cut or by driving a nail. A 
 number of points are placed , 
 from which the millwright 
 can measure to level the 
 shafting. 
 
 A modern level consists 
 of a telescope mounted in 
 a rigid frame together with 
 FIG. 85. Dumpy level ordinary type. a j Qn g j evel tube conta ining 
 
 a sensitive bubble. By means of right-and-left-threaded 
 screws the bubble is brought to the middle of the tube. 
 When the instrument is in adjustment the bubble remains 
 
PRACTICAL SURVEYING 
 
 stationary while the frame is revolved on the vertical 
 axis so the observer may look in any direction through the 
 telescope. 
 
 A ring inside the telescope tube carries cross-hairs, or 
 fine wires, to define the line of sight. The wires are fo- 
 cussed, or brought into the field of view, by moving the 
 
 FIG. 86. Dumpy level Bausch and Lomb type. 
 
 eyepiece. The line of sight passes through the center of 
 the eyepiece and the intersection of the wires. When the 
 instrument is in adjustment the line of sight is horizontal 
 when the level bubble is in the middle of the glass tube, 
 which is graduated so the position of the bubble can be 
 located. 
 
 In Fig. 87 the usual arrangement of wires is shown at 
 (a). The vertical wire is of some assistance in enabling 
 the leveler to tell when the rod is vertical one way while 
 
 waving the rod insures vertical- 
 ity the other way. To assist in 
 obtaining equal backsights and 
 foresights two additional wires are 
 
 A B c sometimes used as shown at (6) 
 
 p IG g 7 and (c). The horizontal line is 
 
 caught with the middle wire (b] 
 
 and the upper and lower wires are spaced so they win 
 intercept one foot on the rod at a distance of 100 ft., two 
 feet at a distance of 200 ft., etc. Three horizontal wires 
 are used by some men to prevent mistakes in reading the 
 rod, by taking readings on the three wires at each turning 
 point or bench mark. When a level has three horizontal 
 
LEVELING 79 
 
 wires and one only is read mistakes often occur, so the 
 style shown at (c) is sometimes used. With this arrange- 
 ment the rodman holds his rod horizontally for the distance 
 to be read, after which he sets the turning point and holds 
 the rod vertically on it. 
 
 Three kinds of levels are in common use for the best work 
 and they are known as the wye (Y), dumpy and precision 
 level. The latter is used only for the highest grade of 
 government work and is made as a form of wye level by 
 some makers and as a form of dumpy level by others. The 
 latest style of precision level is of the dumpy type and is 
 known as the United States Coast and Geodetic Survey level. 
 
 The wye level for some unexplained reason obtained a 
 strong footing in the United States nearly a century ago. 
 It is comparatively easy to adjust, but the adjustments are 
 those of the shop rather than the field, so the instrument 
 requires frequent adjustment. European engineers gen- 
 erally express surprise when they first learn that the wye 
 is so extensively used in this country. The author sold 
 his wye nearly twenty years ago and purchased a dumpy, 
 since which time he has been a strong advocate of the 
 latter type. A description of the wye level and its adjust- 
 ments is given in every instrument maker's catalogue. A 
 wye level may be adjusted in the same manner as a dumpy. 
 
 The dumpy was so named because the earlier ones had 
 inverting telescopes (that is all objects were seen upside 
 down) and the omission of one set of lenses called for short 
 (dumpy) tubes. Inverting telescopes gather more light 
 than erecting telescopes and are used for government work 
 requiring the highest precision. For most of the work 
 done by engineers and surveyors nothing is gained by 
 using an inverting telescope, and the amount of training 
 necessary to become accustomed to viewing objects in an 
 unnatural position is trying. 
 
 The dumpy weighs less than a wye of the same power; 
 costs less; retains adjustments longer and stands rough 
 usage better. 
 
 The bubble should be sensitive. The sensitiveness of a 
 bubble is a test of workmanship. No maker will put a 
 sensitive bubble on a poorly made instrument and a slug- 
 gish bubble is a warning to the prospective purchaser. 
 
8o PRACTICAL SURVEYING 
 
 TO ADJUST A DUMPY LEVEL 
 
 1st adjustment. The level must be perpendicular to the 
 vertical axis. 
 
 Set the level up by spreading the tripod legs so the tele- 
 scope will be about five feet above the ground. Push each 
 leg into the ground firmly. The two plates should first be 
 made parallel by means of the leveling screws and when 
 the instrument is set up the plates should be as nearly 
 horizontal as possible. 
 
 Turn the telescope so it is over two screws. Then turn 
 the screws together and bring the bubble to the middle of 
 the tube. The thumbs move towards or from each other 
 but never move in the same direction. The bubble travels 
 in the direction in which the right thumb moves. The 
 screws should move easily without binding but never 
 loosely, for a firm seating of the ends is necessary. If 
 screwed tight the instrument becomes strained and the 
 slightest touch disturbs the bubble. Making the screws 
 tight also injures the threads. After the bubble is brought 
 to the middle over one pair of screws revolve the telescope 
 ninety degrees and repeat the leveling process over the 
 other pair of screws. 
 
 The foregoing instructions are general and apply to the use 
 of the level as well as when adjustments are being tested. 
 
 The level having been brought to the middle of the tube 
 over each pair of screws turn the telescope end for end 
 over one pair and if the bubble remains in the middle the 
 level is perpendicular to the vertical axis. If it moves 
 away from the middle bring it halfway back by means of 
 one pair of screws and the rest of the way by turning the 
 capstan head screws, on one end of the level tube, with an 
 adjusting pin. Then level it over the other pair and test. 
 
 2nd adjustment. The horizontal wire must be perpendic- 
 ular to the vertical axis. 
 
 The instrument maker fixes the wires in the reticule per- 
 pendicular to each other. Hang a heavy plumb-bob in a 
 sheltered place so the plumb line will not move. After 
 adjusting the bubble sight on the plumb line. If the 
 vertical wire covers it the horizontal wire is perpendicular 
 to the vertical axis. 
 
LEVELING 
 
 8l 
 
 If the wire and plumb line form a small angle loosen 
 the capstan head screws on the sides of the telescope, hold- 
 ing the reticule, and tap lightly with the finger until the 
 reticule is shifted enough to bring the wire into a vertical 
 position. Then tighten the screws. Fig. 88. 
 
 $rd adjustment. The line of sight must be perpendicular 
 to the vertical axis. 
 
 After the instrument has been tried for the first and 
 second adjustments, and the adjustments made, drive two 
 
 FIG. 89. 
 
 FIG. 90. 
 
 stakes from 200 to 300 ft. apart and set the level exactly 
 midway. 
 
 Level it carefully and read the rod held on A. The rod 
 is next held on B and a reading taken with the instrument 
 level, that is with the bubble in the middle of the tube. 
 The difference of the readings is the difference in elevation 
 between A and B, no matter how badly the instrument may 
 be out of adjustment. 
 
 The instrument is then set as close as possible to a rod 
 held vertically on one stake with the eyepiece next to the 
 rod. The bubble is brought to the middle of the tube and 
 the observer looking through the object glass focuses the 
 telescope until he can read the rod, which appears to be at 
 a great distance. The cross-hairs cannot be seen so the 
 elevation is taken in the center of the field of view, a lead 
 pencil or pointed stick marking the point. 
 
 Assuming, for example, that the instrument is placed in 
 front of stake A , and that B is lower than A , add the differ- 
 
82 PRACTICAL SURVEYING 
 
 ence to the rod reading. If B is higher than A subtract the 
 difference in elevation from the rod reading. Set the target 
 at the height thus obtained and have the rodman hold the 
 rod on B. 
 
 The effect of the curvature of the earth and refraction, 
 assuming the distance of B from A to be 250 ft., will 
 amount to 0.000,000,021 X 25O 2 = o.ooi ft., by which 
 amount the target should be lowered if the correction is 
 applied. Errors in observation will probably offset this 
 small correction which may therefore be disregarded. 
 * The rod being held vertically on B the telescope is 
 pointed towards the target and the leveling screws turned 
 until the bubble is exactly in the middle of the tube. By 
 means of the capstan screws on the top and bottom of the 
 telescope the reticule is raised or lowered until the hori- 
 zontal wire intercepts the center of the target. This com- 
 pletes the adjustment. This adjustment on a wye level 
 makes the wye adjustment unnecessary. 
 
 THE DATUM 
 
 It has been shown that a level is merely an instrument 
 by means of which a horizontal line may be determined. 
 
 The horizontal line passing through the center of the eye- 
 piece and the intersection of the cross wires is a base, but 
 for convenience in platting and computing, this base is 
 assumed to be at some definite height above a parallel 
 base termed a "datum," or "datum plane." An "arbi- 
 trary datum" is one arbitrarily chosen for a particular 
 piece of work and is often assumed as being 100 ft. below 
 the line of sight at the starting point. 
 
 In city work the datum is usually assumed to be 100 ft. 
 below the lowest point on the streets of the city. When- 
 ever any Government bench marks are close by the sur- 
 veyor uses the elevation marked on the nearest one, the 
 Government datum (o) being the mean of low tides. In 
 some seacoast cities the mean^of lower-low tides is used as 
 datum. If it is believed that excavations will be made below 
 the Government datum it is best to use an elevation of 
 100 ft. for the datum instead of o to avoid mistakes likely 
 to arise when minus elevations are used. This in effect 
 
LEVELING 83 
 
 fixes an arbitrary datum one hundred feet below the mean 
 of low tides. 
 
 If a hole is dug ten feet below Government datum then 
 the elevation, referred to datum, is 10. A wall lo-ft. 
 high with the bottom at datum will have an elevation on 
 top of +10. The use of the -f- and signs has been a 
 fruitful source of error and the simple expedient of assum- 
 ing the zero datum below the lowest point an excavation 
 may reach stops all trouble. 
 
 In the example just cited assume that an arbitrary datum 
 has been selected 100 ft. below the Government datum. 
 The bottom of the hole will have an elevation of looft. 
 above datum and the top of the wall will have an elevation 
 of no ft. above datum. 
 
 LEVEL RODS 
 
 There are several types of level rods known as Boston, 
 New York, Philadelphia, etc. A description 
 of each rod is not necessary for this informa- 
 tion is contained in the catalogues of instru- 
 ment makers. 
 
 Rods graduated with thin black lines on 
 varnished wood on which a target is neces- 
 sary are not much used today. The favorite 
 type is some form of Philadelphia rod. This 
 is known as "self-reading" because a target 
 is not necessary. The face of the rod is 
 painted white and originally was graduated 
 in feet and tenths of a foot. When closer 
 readings were wanted a scale on the sliding 
 target was used by means of which half-hun- 
 dred ths could be read. 
 
 The rod was later made with graduations 
 of one-hundredth of a foot stenciled on the 
 face, the marks being alternately black and 
 white. A target with scale was attached to 
 the rod so that readings could be taken to FIG. 91. Phila- 
 the nearest half-hundredth. An improve- delphia self- 
 ment was made when the graduations were reading rod. 
 cut into the white face of the rod with the alternate gradua- 
 tions painted black. Another form has the graduations cut 
 
84 PRACTICAL SURVEYING 
 
 without the alternate black and white spaces. The author, 
 and probably the majority of engineers and surveyors, 
 prefers the broad marks, as the reading of the rod does not 
 tax the eyes and it is "self-reading" at long distances. A 
 target with a vernier reading to a thousandth of a foot is 
 used on all rods except those with a stenciled face. 
 
 The vernier is a scale attached to the target. It is 
 divided into ten equal divisions with the zero on the center 
 line of the target. The ten divisions on the vernier cover 
 nine divisions (nine one-hundred ths) on the rod, each 
 division on the vernier being thus one-tenth of one-hun- 
 dredth, or one- thousandth, of a foot less than the smallest 
 graduation on the rod. 
 
 To read less than one-hundredth of a foot, read upward on 
 the rod until the hundredth below the zero 
 of the vernier is reached. Then read up- 
 wards from zero on the vernier until a line 
 is reached that coincides with some line on 
 the rod. The vernier in Fig. 92 reads 6 feet, 
 I tenth, 2 hundredths and 4 thousandths 
 (6.124) and is read "six point one, two, 
 four." 
 FIG. 92. Vernier A level rod is merely a rod graduated in 
 
 tarret^ 1 ^ feet ' t enths and hundred ths of a foot; or in 
 feet, inches and eighths of an inch, the divi- 
 sion in inches being used only by architects and building 
 mechanics. The decimally divided foot is used by en- 
 gineers and surveyors. 
 
 The graduations proceed upward, the foot of the rod being 
 zero. It is customary to mark the feet in red, the figures 
 being "one- tenth" high and the tenths in black, the figures 
 being a half-tenth high. Each five-hundredth mark pro- 
 jects slightly beyond the marks denoting hundredths. 
 
 When a level is set up it may be turned in any direction 
 and a rod is used to determine the height of the horizontal 
 line above the ground. The rod must be held perfectly 
 vertical, or plumb, and wherever the horizontal line of 
 sight strikes it a "reading" is obtained. 
 
 A rodman must always stand directly back of the rod 
 and face the leveler, so the latter will see the face of the 
 rod fully and not at an angle. The bottom of the rod must 
 
LEVELING 
 
 rest firmly on the point on which it is held. The hands 
 should be about the height of the chin to support the rod 
 properly and the fingers should grasp the sides, never en- 
 circling the rod, for they will cover some of the graduations 
 and may interfere with a sight if placed across the face. 
 
 FIG. 93. Rod level. 
 
 FIG. 94. Folding rod level. 
 
 For holding the rod in a vertical position several forms 
 of rod levels are on the market. Some men use a light 
 plumb-bob, generally a nuisance in a wind. A time-honored 
 method is " waving the rod." 
 
 In Fig. 95 A-B is the horizontal line and C-D the rod. 
 When the rod is held at C and allowed to lean towards D' 
 or fall back towards D" ', the horizon- 
 tal line gives a greater reading than A 
 when C D is perpendicular to AB. 
 When the rodman has no level or 
 other means for insuring a vertical 
 position of the rod and holds the 
 rod on a point on which* a close read- 
 ing is desired, the rod is waved to- 
 wards and from the leveler until he 
 obtains the shortest reading possible. 
 
 FIG. 95. 
 
 The "waving" must be done very slowly. Some instru- 
 ment makers furnish targets made in the shape of an angle 
 and the horizontal division line on the target cannot be 
 completely covered by the cross wire unless the rod is ver- 
 tical. This necessitates signaling by the leveler and the 
 author believes waving a rod on particular points cannot 
 be improved upon. If a rodman is not experienced he 
 should have a rod level for intermediate sights. 
 
 A rod is read with target to the nearest "thousandth" 
 on bench marks; to the nearest "half-hundredth" on 
 
86 
 
 PRACTICAL SURVEYING 
 
 turning points, and the nearest "tenth" for intermediate 
 points. The use of the target on turning points is optional 
 but is customary so the reading by the leveler can be 
 .checked by the rodman. The target is never used on 
 intermediate sights or for "stations" except when grade 
 pegs are set. In setting grades it is very convenient and 
 easy on the eyes to use a target. 
 
 FIG. 96. Angle target. 
 
 Checking a reading is done as follows : The leveler reads 
 the rod and records the reading. He then motions to the 
 rodman to set the target and directs him in moving it up 
 or down until it is practically right. Then the rod is 
 waved and the target moved until it is set, when the rod- 
 man clamps it. He then reads the rod and compares his 
 reading with that of the leveler as the latter goes past to 
 a new set up. When the backsight is taken he gives his 
 reading to the leveler as a check before going ahead on 
 line. 
 
 When the leveler wishes to have the target used he raises 
 his right arm vertically as high as possible and describes 
 a small horizontal circle. This means "turning point." 
 The rodman selects a good point, holds the rod on it, slides 
 the rod to where he thinks the line of sight will intersect 
 it and awaits orders. The target is moved up when the 
 leveler puts his right hand out horizontally to the side. 
 It is moved down when he holds out his left hand. He 
 thrusts out both arms when it is right. When the rod is 
 to be waved he holds up his right arm and waves it forward 
 and back. 
 
 For readings on line at stations the rod is held on the 
 ground. Turning points must be hard and so firm they 
 
LEVELING 87 
 
 will not move when the rod is held on top. In stony coun- 
 try the tops of stones make excellent turning points and in 
 timbered land exposed roots and notches in trees are used. 
 In some sections where the soil is light or sandy the rod- 
 man carries pegs to drive in the ground for use 
 as turning points. 
 
 A convenient turning point consists of an 
 iron pin attached by a light chain to the 
 rodman's belt. The pin is pushed into the 
 ground and the rod held on top of the ring. 
 The chain is used to keep the rodman from 
 going off without the pin after holding the rod 
 on it. 
 
 To find the difference in elevation between two 
 points not far apart. 
 
 Assume the points to be so close together FtG 97 
 that they may both be seen from an inter- 
 mediate point. Let A and B be the points on the surface of 
 the earth and C and D be points vertically above them on a 
 true level with F, on the line of sight. 
 
 If F (position of level) is midway between C and D the 
 difference between true level and apparent level will be the 
 same. That is 
 
 CA ^EA 
 BG 
 
 FIG. 98. If the level is not midway between 
 
 A and B it will be necessary to meas- 
 
 ure the distances AH and BH and use the formulas al- 
 ready given to determine the difference between the true 
 level on the arc CFD and the apparent level on the hori- 
 zontal line EFG. The difference in elevation is obtained 
 by setting the instrument up at some point H between A 
 and B and leveling it so the line of sight EFG is hori- 
 zontal. A graduated rod is held vertically at A and the 
 reading taken where the line of sight intersects the face of 
 the rod. The rod is then held vertically at B and a read- 
 ing taken. The difference between these readings gives the 
 difference in elevation between the two points, provided 
 they differ in elevation* 
 
88 
 
 PRACTICAL SURVEYING 
 
 PROBLEMS 
 
 1. Let AH = 240 ft. and BH = 300 ft. AE = 10 ft., 
 BG = 6 ft. What is the true difference in elevation be- 
 tween A and B? If AH = BH what will be the true differ- 
 ence in elevation? 
 
 2. Let AH = 1040 ft. and BH = 1820 ft. AE = 5 ft., 
 BG = 6 ft. Find true and apparent difference in elevations. 
 
 3. Let AH = 792 ft. and BH = ii88ft. AE =3.17 ft., 
 BG = 5.67 ft. Find true and apparent difference in 
 elevations. 
 
 DIFFERENTIAL LEVELING 
 
 To find the difference in elevation between two points far apart. 
 The line ABCD is a profile of the ground. The differ- 
 ence in elevation between A and D is wanted and the points 
 
 are assumed to be about 2000 
 ft. apart. Since sights should 
 not be taken more than 400 
 ft. and the level should be as 
 nearly as possible equidistant 
 from turning points the level 
 must be set up three times. 
 The leveler starts from A and walks in the direction of 
 D, counting his paces until he is about 100 paces from A. 
 Going to one side of the line until he believes a horizontal 
 line will be above the ground at A, he sets up the level and 
 levels the telescope. The elevation of A is assumed to be 
 some definite distance above datum if the actual eleva- 
 tion is not known and the rodman holds his rod on A . 
 
 The leveler reads the rod and puts 
 the rod reading (a) in his field book in 
 a column headed by a -f- sign. The 
 rodman now paces from A to E and an 
 equal number of paces beyond E to B 
 where he selects a good turning point 
 and holds his rod. The leveler reads 
 the rod and puts the rod reading (b) in 
 the column headed by a sign. He 
 
 FIG. 99. 
 
 Sta. 
 
 + 
 
 - 
 
 A 
 
 3 
 
 o 
 
 B 
 
 6 
 
 9 
 
 C 
 
 4 
 
 2 
 
 D 
 
 o 
 
 3 
 
 
 13 
 
 14 
 
 
 
 13 
 
 
 
 I 
 
 now goes to Fand levels the instrument, - 
 
 the rodman holding the rod on the turning point (or turn, 
 
 as it is commonly called) but turning it so the leveler can 
 
LEVELING 89 
 
 see the face. The leveler takes the reading (b l ) and records 
 it in the + column while the rodman paces to Fand an equal 
 distance beyond and holds the rod on a turn at C. The 
 leveler now reads the rod and puts the reading (c) in the 
 column, then sets the level at G, judging as closely as pos- 
 sible with the eye, for no great accuracy is required, that the 
 distance from C to G = G to D. He then reads the rod 
 and puts the reading (c r ) in the -f column. The rodman 
 then holds the rod on D and the leveler places the reading 
 (d) in the column. 
 
 The -f- readings and readings are separately added 
 and the difference between them is the difference in eleva- 
 tion between A and D. The forward point is lower than 
 the starting point if the + readings are less than the 
 readings. It is of course higher if the sum of the read- 
 ings exceeds the sum of the -f- readings. 
 
 The rod readings given in the example are marked in the 
 figure for identification. The profile is not drawn in the 
 
 FIG. 100. Bench mark on tree. 
 
 field book, the record of readings on the ruled page being 
 sufficient. A "backsight" is a + reading and a "fore- 
 sight" is a reading. The intermediate points are not 
 marked in differential leveling, the leveler entering on his 
 book, instead of B and C, the letters "T. P. " (turning point) 
 or "Peg," the latter word being a survival from the days 
 when every turning point was actually a peg left in the 
 ground. 
 
 BENCH MARKS AND TURNING POINTS 
 
 A bench mark is a permanent point with a recorded eleva- 
 tion. In the country a stone is used when available or a 
 tree is cut and marked. A cut is made in the side as if for 
 a broad blaze. The lower part, however, is cut in like a 
 
90 PRACTICAL SURVEYING 
 
 shelf and sloped each way from a ridge in the middle so 
 there will be a point on which to hold the rod. At the edge 
 a nail or spike is driven flush. If the head projects it may 
 be hammered down and the elevation thus disturbed. 
 The object in driving the spike is to furnish a hard point 
 on which to hold the rod. The bench mark may be used 
 for years and if the wood rots the elevation will not be 
 preserved unless a metal point is used. 
 
 On some smooth place as close as possible to the bench 
 mark the elevation should be marked. When trees are 
 blazed the figures are cut into the wood with a surveyor's 
 scribe, which is a useful tool sold by all instrument dealers. 
 An accurate description of each bench mark is written in 
 the level book and recorded in the office. 
 
 On route surveys bench marks are placed about one-third 
 of a mile apart as close as possible to the line but far enough 
 to the side to be preserved during the construction period. 
 Turning points are merely temporary benches and no record 
 is usually made of them. For the most accurate work the 
 distance between turning points should be not less than 
 200 ft. (100 ft. from the instrument) nor more than 600 
 ft. For ordinary work the distance between turning 
 points should not exceed 1200 ft. in the middle of a com- 
 fortable day. Sights should be short in the early morning 
 and late afternoon; also on very bright, hot days. The 
 instrument must be level, in perfect adjustment and 
 midway between turning points. The rod must be 
 vertical. 
 
 In cities or on construction work there cannot be too 
 many bench marks. Corners of iron or masonry steps; 
 curb stones; tops of hydrants, etc., are used and recorded. 
 Always have two benches close together on such work and 
 read on both. The records give the difference in elevation 
 of the benches and if this is not verified then one must 
 have been disturbed and each must be compared with 
 some near-by bench until two are found to agree with the 
 records. These are used and the faulty ones removed 
 from the records. 
 
 Errors in adjustment, effect of the curvature of the earth 
 and effect of refraction are taken out by equality of sights 
 to turning points. A difference of one-tenth the length of 
 
LEVELING QI 
 
 the average sight, provided the differences are about 
 equally plus and minus, will not affect the work noticeably. 
 
 The rod may be held approximately vertical on inter- 
 mediate sights for ground reading but lack of verticality 
 on bench marks and turning points causes errors of a cumu- 
 lative nature. 
 
 The sun heats the end of the level toward it and the 
 metal expanding raises that end. The bubble always 
 seeks the high end and this error, which is cumulative, can 
 only be guarded against by bringing the bubble to the center 
 after setting the target, and then obtaining a new reading. 
 On very careful work the instrument should be shaded. 
 
 Errors in reading, of one foot, one-tenth, etc., are often 
 made, but are more common with target rods than with 
 self-reading rods. Requiring the rodman to carry a book 
 for "peg readings" so he can check the leveler is a practice 
 that should never be omitted. The only real check is to 
 duplicate the work in an opposite direction, so on all route 
 work, or "profile leveling," the leveler "checks benches" 
 every ten miles and sometimes for shorter distances. 
 
 All persons using instruments carry their personality 
 into their work so a recognized difference in results obtained 
 by different individuals has been termed by scientific men 
 the "personal equation." With increase in experience 
 errors due to personality become small and tend to balance, 
 thus eliminating error due to this cause. The personal 
 equation shows up when a leveler checks on a bench set 
 by another man before the personal errors of either have 
 balanced. 
 
 Careful leveling is a continual balancing of small errors 
 but the residual errors are cumulative. To duplicate a 
 line of levels in an opposite direction does not change the 
 sign (+ or ) of the residual error. The error increases 
 with distance so to run a line of levels ten miles and re-run 
 it in the opposite direction is the same as running one line 
 twenty miles. All work should be thus checked when 
 possible and one-half the error found at the starting point 
 should be used as a correction to the bench elevation at the 
 other end and proportionately to intermediate benches. 
 
 Failures to make a close check when one route survey 
 crosses another and a leveler reads a bench set by another 
 
PRACTICAL SURVEYING 
 
 man often happens because rods of different makers were 
 used without comparing the graduations. When a num- 
 ber of rods are used they should be purchased under good 
 specifications from one manufacturer and before sending 
 to the field should be carefully tested for accuracy of the 
 graduations with a standard steel tape. 
 
 On long lines of levels the error increases as the square 
 root of the distance, while on short lines the error is 
 much less. In city work benches should be established if 
 possible on the corner of each block, in pairs, and in the 
 middle of long blocks. The limit of error in feet should 
 never exceed the following: 
 
 e a Vm, 
 
 in which e = error in feet, 
 a = error factor, 
 m = distance in miles between the benches. 
 
 For city work and all accurate leveling a = 0.017. 
 
 For ordinary route leveling a = 0.034. 
 
 When leveling down hill time is saved if the horizontal 
 line is brought close to the foot of the rod for a backsight. 
 When going up hill the horizontal line should be brought 
 close to the top of the rod. It often happens that an at- 
 tempt to read near one end. of the rod results in the hori- 
 zontal line striking below or above the rod, so the level 
 must again be set up. To avoid this the leveler should 
 have a hand level by means of which he can obtain a hori- 
 zontal line at the height of his eyes. The level is then set 
 up with the telescope at this height and leveled. 
 
 ROUTE, OR PROFILE, LEVELING 
 
 Differential leveling has been illustrated and such work 
 is done when the only elevations wanted are those at the 
 ends of the line, every intermediate reading being on a 
 turning point. "Check leveling," to check elevations of 
 bench marks, is differential leveling. 
 
 Profile leveling is for the purpose of making a profile 
 and obtain a record of heights at all changes of elevation 
 on the line. 
 
 The route is laid off in stations and the leveler obtains 
 the elevation of the ground at each station, and also at 
 
LEVELING 
 
 93 
 
 intermediate points where decided changes occur, as 
 gullies, etc. Between adjacent T. P.'s there may be sev- 
 eral readings and on steep ground there may be several 
 turns in one station. 
 
 The level is set up and a reading taken on a convenient 
 bench mark. This reading added to the elevation of the 
 bench gives the elevation of the horizontal wire and is 
 recorded in the column headed H. I. (Height of Instrument). 
 Readings are then taken on the ground at each station and 
 -f- station and subtracted from the H. I., the difference being 
 the ground elevation. A reading taken to a bench mark 
 or turning point to get the H. I. is known as a backsight 
 and placed in the B. S. column. A reading taken on the 
 ground or on a turning point or bench mark to obtain the 
 elevation is known as a foresight and is placed in the F. S. 
 column. The notes are placed on the left-hand page of the 
 field book, as follows: 
 
 Station. 
 
 B. S. 
 
 H.I. 
 
 P. S. 
 
 Elevation. 
 
 Remarks. 
 
 B. M. 
 
 o 
 
 1.234 
 
 ioi . 234 
 
 2 7 
 
 100.00 
 
 98 5 
 
 B. M. on oak tree 
 20 ins. diam. 
 ii ft. to left of Sta. i 
 
 i 
 
 
 
 e 4 
 
 y x 
 qe 8 
 
 
 + 50 
 
 
 
 7.8 
 
 07 .4 
 
 
 4-85 
 
 
 
 3 - 
 
 98. 2 
 
 
 T. P. 2 A 
 
 2 
 
 2.176 
 
 102.723 
 
 0.687 
 
 I .7C 
 
 100.547 
 IOI .OO 
 
 T. P. on hub at Sta. 2 
 
 P 4 eg 
 
 
 
 2. 2 
 8.431 
 
 100.5 
 94.292 
 
 
 In Fig. ioi is illustrated a common feature in profile 
 leveling. A gully runs across the surveyed line and the 
 chainmen when setting stakes meas- 
 ured the -f- distances to the edges 
 and bottom. When the levels are 
 taken the rodman paces the distance 
 from the station to the bank and a 
 reading is obtained. 
 
 Leaving his level the leveler by 
 
 means of his hand level obtains the elevation at the bot- 
 tom, the process being differential leveling. The notes 
 
94 
 
 PRACTICAL SURVEYING 
 
 are placed on the right-hand page of the level book. He 
 returns to his level and the rodman holds on the opposite 
 bank. After a reading is taken he 
 paces the distance to the next station 
 and calls it to the leveler who then 
 writes down the + stationing. The 
 three + distances are known to be 
 only closely approximate but they 
 check the distances previously meas- 
 ured with the tape and so identify the place. In rough 
 country the hand level saves much time. 
 
 The notes are platted on profile paper, there being two 
 rulings in common use, plate A with 20 and plate B with 30 
 horizontal lines to an inch. Each plate has four vertical 
 
 FIG. 102. Two types of 
 hand levels. 
 
 FIG. 103. Plate A profile paper. 
 
 lines to an inch. When plotting the profile each horizontal 
 space represents a station and each vertical space repre- 
 sents one foot, the distorted scale showing plainly all minor 
 irregularities in the surface. Cuts and fills and differences 
 in elevation are easily read to the nearest half foot and es- 
 timated to the nearest quarter of a foot. For estimating 
 earthwork quantities this is convenient but the principal 
 value of the exaggerated vertical scale is the ease with which 
 grade lines are selected. 
 
 The purpose in making a profile survey is usually to 
 select the grade for a road, railway or ditch. When the 
 profile is made a thread is stretched between the hands and 
 held on the profile so the projections above the line are 
 equal to those below. In common language the "cuts and 
 fills" balance. If this gives a steeper grade than the 
 
LEVELING 95 
 
 maximum allowed, the position of the thread must be 
 altered. Two considerations therefore enter in the prob- 
 lem of fixing a grade line, one being economy in construction, 
 the other economy in hauling loads up the grade. In irri- 
 
 FIG. 104. Plate B profile paper. 
 
 gation ditches the grade must be light so water can be 
 delivered to the highest point of the land yet steep enough 
 to secure a velocity that will prevent the deposit of silt and 
 the growth of vegetation. 
 
 In drainage ditches the grade should be as steep as 
 possible without producing a velocity of flow great enough 
 to scour the bottom and make a gully of the ditch. 
 
 TO SET GRADE STAKES 
 
 Contour road and ditch surveys are generally made with 
 a target rod. The first stake is driven so that the top is 
 on grade. The fall per station is given to the rodman. 
 A reading is taken with the rod held on the grade peg, the 
 target set and the reading recorded. A chainman holds 
 one end of the tape or chain at the grade peg and the rod- 
 man holding the other end draws the tape taut and holds 
 the rod vertically for a new peg. At each station the 
 target is moved the amount of fall per station. If the line 
 is going uphill the target is moved down and it is moved 
 up if the line is going downhill. The leveler calls the cor- 
 rect reading to the rodman, after the latter has clamped 
 the target, as a check each time. 
 
 In setting grade stakes for sewers the tops are placed at 
 some definite height above the bottom of the sewer when 
 
g6 PRACTICAL SURVEYING 
 
 the street is already on grade. The line being straight a 
 peg is driven to grade at each end of the block. Over one 
 is set the level and the height from the top of the grade 
 peg to the center of the telescope is carefully measured. 
 
 ^ ^_L A _A 
 
 y "N- t ' 
 
 A 
 
 ^^^/P^ 
 
 FIG. 105. Plunging a grade. 
 
 A light rod is driven in the ground at the other peg and 
 a white card tacked to it so the middle of the card is at a 
 height above the top of the peg equal to the H. I. above 
 the other peg. The exact height is measured and marked 
 on the card. By manipulating the leveling screws the 
 telescope is pointed to the mark on the card. The line of 
 sight is then parallel to the grade line instead of being 
 horizontal. The target is set on the rod to a height equal 
 to the height of the cross-hairs above the grade line. The 
 rod being held vertically the bottom is on line with the 
 tops of the grade pegs when the horizontal wire cuts the 
 center of the target and stakes are driven to grade wherever 
 desired. This method is termed "plunging a grade." 
 
 FIG. 106. Sewer grade line on street. 
 
 When the ground is irregular so that some stakes would 
 be driven with tops below the surface and others would 
 stand high if driven to grade, it is best to drive a peg with 
 the top flush with the ground at each station. Beside each 
 peg is driven a stake marked with figures indicating the dis- 
 tance to grade, a + sign indicating a cut and a sign a fill. 
 
LEVELING 97 
 
 To lay pipes to grade, all the witness stakes will be 
 marked +, for they will all be above grade. Beside each 
 grade peg drive a post and on the opposite side of the 
 trench drive another. Measure 
 
 up from the grade peg some dis- ^4T ! 'Rod level 
 
 tance to obtain a line a certain 
 number of feet above the flow 
 line and at this height nail or 
 clamp a plank to the two posts, 
 the plank to be perfectly level. 
 This is done at each station and 
 a cord stretched from plank to 
 plank will show the grade and 
 be out of the way of the work- 
 men in the trench. A piece of 
 wood about one inch square FlG I0? Transferring surface 
 with a small bracket fastened grade to pipe in trench, 
 at the lower end is used to ob- 
 tain the grade at the bottom, the projecting arm resting in 
 the bottom of the pipe when an upper mark touches the cord. 
 The rod must be vertical and a rod level should be used. 
 
 MANUAL SIGNALS FOR SURVEYORS 
 
 It is not always possible to shout instructions to assist- 
 ants in the field so signals are necessary. The author 
 mentioned his signals for telling a rodman when to move a 
 target, and the direction in which to move it. For con- 
 veying information into which numbers enter a very old 
 custom is to write the number with the right hand on an 
 imaginary vertical surface. When this is done by the rod- 
 man the leveler reads the number in a reversed direction 
 and the rodman reads it in a reversed direction when the 
 writing is done by the leveler. Some men in making such 
 signals stand with their back to the recipient and so manage 
 to overcome the difficulty of reading numbers reversed. 
 
 In Engineering News, May 22, 1913, Mr. F. T. Darrow 
 illustrated a set of signals in use on the Burlington Lines, 
 these signals being shown in Fig. 108. The reader will 
 notice that one hand is used, the other presumably holding 
 the rod or instrument. 
 
PRACTICAL SURVEYING 
 
 6789 
 
 FIG. 108. Signals for surveyors' assistants. 
 
 Rod up or Move to Right Plumb the Set Hub Set onThis 
 Give Line (or Left) RodCRiqht orqiveTP 
 
 or Left) 
 
 O.K. Come Here Go Back Cant Get You Move Target 
 
 Up Cor Down") 
 FIG. 109. Signals for surveyors' assistants. 
 
LEVELING 99 
 
 In Engineering News, Aug. 21, 1913, Mr. Robert S. 
 Beard illustrated a set of signals requiring the use of both 
 hands. The author has omitted from Fig. 109 the signals 
 for numbers as he prefers signals for which one hand only 
 is used. It is necessary, however, to convey other ideas 
 than numbers so those signals in the article by Mr. Beard 
 which relate to common instructions are shown in Fig. 109. 
 The signals may be used by instrument men, rodmen, 
 chainmen or other helpers and require no explanation as the 
 information is placed under each figure. These signals are 
 those in common use by experienced men the world over. 
 
 I 
 
CHAPTER IV 
 
 COMPASS SURVEYING 
 
 Nearly all land surveys before the middle of the last 
 century were made with the compass. It is still used to- 
 day where land is not high in price. 
 
 About 70 or 80 years ago a fairly certain method was 
 developed for eliminating the effects of local attraction, but 
 it was not in common use. Until about 100 years ago 
 little attention was paid to the declination of the needle, 
 or variation as it was commonly called. The declination 
 is the angle between the true meridian and the meridian 
 indicated by the compass needle. The variation is the 
 change in declination. In 1836 William A. Burt patented 
 the solar compass of which he was the inventor jointly with 
 John Mullett, both Government surveyors in Michigan 
 where local ore bodies attracted the needle and rendered it 
 valueless for running lines. With the solar compass a true 
 north and south line could be run from observations on the 
 sun. It is now obsolete so will net be described in this 
 book. The engineers' transit with solar attachment super- 
 seded the solar compass and today few men use a solar 
 attachment, direct observations on the sun tending to 
 greater accuracy and simplicity. 
 
 After 1836 all Government land surveys had to be made 
 with a solar compass. About 50 years later a transit with 
 solar attachment was required and now direct solar obser- 
 vations are permitted. The General Land Office at present 
 prohibits employees and contracting surveyors from de- 
 pending to any extent on courses derived from the needle. 
 
 The compass possesses the following merits: 
 
 1. It is light. 
 
 2. It is readily set up and therefore rapid work can be 
 done with it. 
 
 3. When there is no local attraction an error made in 
 
 100 
 
COMPASS SURVEY I&C-. 
 
 reading the needle at any station is not cumulative, but is 
 confined to the one course. If a "back reading" is taken 
 the error is caught and may be corrected. 
 
 4. Two points are not required from which to start, as 
 with a transit. The needle pointing always in one direc- 
 tion (when there is no local attraction) any station may be 
 used as a starting point. 
 
 5. Fewer helpers are needed than with the transit. 
 
 6. On preliminary lines and on random lines considerable 
 time may be saved in cutting brush, for the compass can 
 be set to one side and a parallel offset line run past an ob- 
 struction by merely setting the sight on the' proper course 
 indicated by the needle. 
 
 7. In re- tracing old surveys it is better than a more 
 exact instrument for the work is duplicated more nearly 
 when the methods adopted closely copy those originally 
 used. 
 
 8. When land is very low in value or when the informa- 
 tion wanted is only closely approximate, the compass is 
 the best instrument to use, because men of ordinary ability 
 can do the work and do it quickly. 
 
 Before mentioning the defects of the compass it is neces- 
 sary to say that all land survey methods and terms now 
 used were developed during the era preceding the intro- 
 duction of well-made accurate instruments. The only 
 difference is increased accuracy so that an understanding 
 of compass work is necessary for a full knowledge of land 
 surveying. 
 
 When the compass was invented no one knows and the 
 name of the inventor has not been preserved. The in- 
 ventor is supposed to have been a Chinese and he lived at 
 least 1900 years ago. The first mention in print of the 
 compass is found in a Chinese dictionary printed in 121 
 A.D., in which book the lodestone is defined as "a stone 
 with which an attraction can be given to the needle." 
 The first mention of the compass by any European writerwas 
 in 1190 but it was in common use by seamen in 1250. Its 
 use in surveying seems to be first mentioned about 1300 and 
 on his voyage to America in 1492 Christopher Columbus 
 verified the fact long suspected, that the needle did not 
 point constantly to the North Star. In 1600 Gilbert 
 
102 PRACTICAL SURVEYING 
 
 showed that the earth is a great magnet with lines of force 
 flowing from pole to pole in which the needle settled and 
 that angels and demons and the stars in the heavens 'had 
 no influence upon the needle. 
 
 The needle does not point to the North Pole but to 
 a shifting magnetic pole located in Canadian territory. 
 When a compass is set so that a line drawn through the 
 north pole, the magnetic pole and the instrument will be 
 straight (that is in the plane of a great circle) we have a 
 line of no variation, an agonic line. 
 
 At all points east of the agonic line the needle points west 
 of true north and has a west declination. At all points 
 west of the agonic line the needle points east of the true 
 north and has an east declination. Since all meridians 
 pass through the earth's north pole a true north and south 
 line is spoken of as "the meridian." 
 
 This edition contains no Isogonic Chart as "one was 
 issued by the United States Coast and Geodetic Survey for 
 January, 1915, in "Special Publication on Terrestrial 
 Magnetism," which should be owned by every land sur- 
 veyor. A chart for 1920 will appear about 1922 or 1923 and 
 it is the intention to issue such charts at five year intervals. 
 
 The red lines show the actual declinations for January, 
 1915, as determined in 1914 by 5,830 observations in North 
 America. The blue lines show the annual change; the 
 north end of the compass needle moving to the westward at 
 all places west of the line of no annual change and to the 
 eastward at all places west of that line. 
 
 To use the Isogonic Chart. Find the declination for 191 5 at 
 the point desired, as shown by the figures in red. Add to this 
 the amount of annual change at that point multiplied by the 
 number of years from i9i5todate. The publication referred to 
 enables one to trace changesindeclination from the year 1750. 
 
 Many old time surveyors knew little about the decli- 
 nation of the compass needle and cared less, so their lines 
 do not agree with the bearings set down in the records. 
 
 The annual change (variation) of the declination was 
 unknown to the majority of surveyors 100 years ago. 
 
 That minor variations in the pointing of the needle 
 occur during the day is not known to all surveyors of the 
 present time. 
 
COMPASS SURVEYING 103 
 
 Many surveyors paid small attention to the probable 
 effects of local attraction and some neglected correcting 
 readings which were incorrect because of local attraction. 
 
 Badly made instruments, some containing small quanti- 
 ties of iron; bent pivots; bent needles; neglected adjust- 
 ments; weakly magnetized needles; mistakes in reading 
 bearings; bearings read only to half degrees or quarter 
 degrees; all these things during the 400 years during which 
 the compass was the principal instrument used by sur- 
 veyors finally resulted in its abandonment as an instru- 
 ment of precision. 
 
 It is not wise to purchase a second-hand compass except 
 from a reputable maker, and then only with his certificate 
 that it has been carefully tested and found to contain no 
 metal liable to deflect the needle. Cheap surveying in- 
 struments are a poor investment and no instruments should 
 be purchased from stores or agents. The needle of a sur- 
 veying compass should be at least \\ ins. long and deli- 
 cately balanced so it appears to be always quivering while 
 the ends remain in one plane. A swinging of the ends 
 indicates local attraction which may be caused by iron or 
 steel in the compass box, bodies of iron ore under the sur- 
 face of the ground, wire fences, wire in the hat brim of 
 the instrument man, or steel or iron on his person. When 
 a compass is set up chains, tapes, hatchets and all tools 
 containing metal which might attract the needle should be 
 removed to a considerable distance. 
 
 Diurnal variation. In addition to the secular variation 
 of the declination the needle swings backward and for- 
 ward each day through an arc sometimes as great as 20 
 minutes in size. This variation cannot be predicted for it 
 alters with the time of day, seasons of the year, temperature 
 and amount of sunshine. It is different in localities not 
 far apart. In practical work it is ignored, except in hot 
 weather when the sun is bright. On such days it is best 
 to work only before 9 A.M. and after 3 P.M. 
 
 It is impossible to make all compass needles alike. All 
 makers testify that a number of needles can be made alike 
 in shape and of equal weight from one piece of steel, yet 
 when magnetized and placed on pivots the readings will 
 differ as much as 10 min. 
 
104 
 
 PRACTICAL SURVEYING 
 
 Several men reading the bearing shown by a needle will 
 obtain different results. This personal error taken with 
 the differences in needles shows that any attempt to allow 
 for the diurnal variation is generally a needless refinement. 
 
 Notwithstanding all the drawbacks mentioned, a well- 
 trained, conscientious surveyor using a well-made modern 
 compass can do very good work. 
 
 READING THE COMPASS 
 
 Let the line N ... 5 represent the true meridian in 
 which the needle is supposed to lie and the direction (bear- 
 ing) of the line A ... B is wanted. The 
 compass is set at some point O on the line 
 A ... B and the needle allowed to swing 
 freely after the instrument is leveled. When 
 the needle comes to rest the observer looks 
 through the sights towards point B. The 
 compass circle is graduated from north to 
 east and north to west, the north point 
 being marked O and the east and west 
 points 90. It is graduated similarly from 
 FIG no south to east and west, this method be- 
 ing termed "quadrantal graduation," be- 
 cause the circle is divided into four quadrants, or 
 quarters. 
 
 FIG. in. Surveyor's compass with open sights. 
 
 When the sights are set on the line A . . . B the needle 
 still points to N. The zero being on the line A ... B 
 
COMPASS SURVEYING 105 
 
 (in the line of sight) the end of the needle points to some 
 number on the circle, which number indicates the angle 
 N, 0, B, between the lines ... TV and . . . B, this 
 angle being called the bearing of the line O-B. 
 
 Fig. in illustrates a good type of compass with variation 
 plate by means of which the declination of the needle is set 
 off. Underneath the compass is the ball-and-socket joint 
 by means of which the compass is leveled. Leveling screws, 
 although a great improvement, are seldom used on com- 
 passes. Two levels at right angles show when the plate 
 is level before clamping the joint. A is the sight at the 
 south end and B the sight at the north end of the compass. 
 Each sight has a very fine slit, and at the ends of the slits 
 round openings are made so objects may be readily found. 
 
 Sometimes each sight has two slits as shown. In the 
 upper slit in one sight and the lower slit in the other sight 
 a fine wire or horse hair is stretched to enable closer sights 
 to be taken. The surveyor looks through the plain slit 
 and bisects the object by means of the wire in the opposite 
 slit. 
 
 The right edge of B is graduated to half degrees for 
 angles of elevation to be read through the lower peephole 
 on sight A. The left edge is graduated for angles of de- 
 pression to be read through the upper peephole on sight A . 
 To read an angle of elevation or depression, look through 
 the proper peephole at a point the same height above the 
 ground as the compass plate. Slide a card up sight B 
 until the top edge strikes the line of sight. The card is 
 held against the sight until the angle is read. 
 
 At C is shown a variation plate. By means of a milled 
 screw the compass circle can be revolved about the vertical 
 axis about 30 degrees each side of the line of sight, thus 
 enabling the declination to be set off on the plate at C. 
 The declination may be set off in one of three ways. If 
 a well-defined line can be found of which the bearing is 
 known, set the instrument on it and sight to a stake on the 
 line. By means of the screw attached to the variation 
 (declination) plate move the compass circle until the needle 
 points to the course of the line given in the record. The 
 declination can then be read on the plate, which should be 
 clamped and not again touched. Another method is to 
 
io6 
 
 PRACTICAL SURVEYING 
 
 lay off a line in the true meridian, set the compass on it and 
 turn the compass ring until the needle points to zero, when 
 the declination may be read on the plate at C. For closely 
 approximate work the declination may be obtained from 
 the Isogonic Chart and set off on the plate. 
 
 The dial at D is used to keep tally in chaining. Under 
 the compass plate is a screw by means of which the needle 
 
 is raised and held against the 
 glass cover when not in use, 
 so it will not dull or bend the 
 pivot when the instrument is 
 being carried. 
 
 On the compass plate the 
 letter E is on the left and the 
 letter W is on the right. The 
 needle points to the north and 
 if the line of sight is pointed 
 in an easterly direction the 
 needle will show the angle, 
 as already explained, between 
 the meridian and the line of 
 sight. The north end of the 
 needle will be to the left of 
 the line of sight, so by trans- 
 posing the E and W the 
 direction and amount of di- 
 vergence from the meridian 
 are both indicated by the 
 needle. The student must 
 never forget that the needle 
 indicates the line of sight but 
 
 FIG. 112. 
 
 Surveyor's compass with 
 
 not jj e j n 
 
 Fig. 112 shows a compass 
 with a telescope instead of open sights, a vertical circle for 
 reading angles of elevation and depression and a level adapter 
 instead of ball and socket/ 
 
 COMPASS ADJUSTMENTS 
 
 There are five adjustments of the compass: 
 I . The compass circle must be perpendicular to the vertical 
 axis. The manufacturer alone can make this adjustrrent. 
 
COMPASS SURVEYING 107 
 
 2. The levels must be perpendicular to the line of sight. 
 This adjustment cannot be made if the compass circle is 
 not perpendicular to the vertical axis, this fact being a 
 check on the first adjustment. Level the compass by 
 bringing the bubbles to the middle of their respective tubes. 
 Turn the instrument 180 deg. and if the bubbles have moved 
 correct half the difference by means of the capstan head 
 screws holding the level tubes. Check and repeat until 
 the bubbles remain stationary during a complete revolution 
 of the compass on the vertical axis. 
 
 3. The needle must be perpendicular to the vertical axis. 
 This means the needle must be horizontal and also straight. 
 The north end of the needle tends to dip towards the earth 
 and a coil of fine wire is placed near the south end to counter- 
 act this tendency. 
 
 Before making any adjustments involving the needle it 
 should be re-magnetized. Holding the needle in one hand 
 hold the magnetic pole of a permanent magnet firmly with 
 the other hand against the needle near the middle and pass 
 the magnet to the north end of the needle. Before each 
 pass describe a circle about one foot in diameter with the 
 magnet in a plane with the needle. This operation should 
 be repeated for the south end of the needle but care should 
 be taken that the north and the south ends are applied to 
 the opposite poles of the magnet or the work will be wasted. 
 About twenty-five passes will generally be sufficient. This 
 operation should be performed three or four times each 
 year. 
 
 A weak needle is affected by the friction on the pivot so 
 it is necessary to keep it charged in order to do good work. 
 The method above described is not always satisfactory but 
 in these days of electrical power plants the surveyor can 
 readily charge his needle. Place the needle in the magnetic 
 field of a dynamo and then test to see if the magnetism is 
 reversed. If the needle points south instead of north put 
 it again in the magnetic field of the dynamo in a reverse 
 position from that used first. 
 
 When satisfied that the needle is properly charged level 
 the compass, bringing the north end of the needle to zero 
 at the north end. Read both ends and reverse the instru- 
 ment so the north end of the needle will read the same as 
 
108 PRACTICAL SURVEYING 
 
 the south end of the needle on the first trial. If now the 
 south end of the needle does not read zero correct half 
 the difference by bending the needle. 
 
 4. The point of the pivot must lie in the vertical axis. 
 Having performed the third adjustment, or made a test that 
 showed the needle to be straight, read the north end of the 
 needle at N, E, S and W (that is at points 90 deg. apart). 
 Note the reading of the south end as each quadrant is read. 
 If in any quadrant the south end passes over less than 90 deg. 
 while the north end passes 90 deg., bend the pivot away from 
 that quadrant. Test each quadrant in this manner until 
 the needle swings freely, the two ends reading the same in 
 each quadrant on points 180 deg. apart. 
 
 5. The line of sight must lie in the vertical axis. This 
 adjustment can be made only by an instrument maker, but 
 the surveyor must be certain that the slits in the sighting 
 vanes are vertical. Sight at a long plumb line and if it 
 does not coincide with an edge of the slit, put paper under 
 the low side or file the bottom of the high side under the 
 foot of the sight and screw tight. This is to be done with 
 both sights so the edges of the slits will be truly vertical 
 when the instrument is in adjustment. 
 
 Caution. Good work, even with an instrument in per- 
 fect adjustment, cannot be done if any unnecessary walking 
 around the instrument is allowed. When the compass is 
 set up and leveled it must not be disturbed. 
 
 CARE OF THE COMPASS 
 
 Treat the instrument with care and it will give good ser- 
 vice for several generations. Avoid all shocks and jars, 
 and handle it so there will be no danger of bending any of 
 the parts. Carry it in the hollow of the arm and not as 
 one carries a pail or basket. Do not use the sights as 
 handles. 
 
 Be careful of the pivot that it will not be dulled or bent. 
 Lower the needle gently. When the needle is let down on 
 the pivot check the vibrations by lifting it off the point 
 at each swing until it settles. VVhen moving to another 
 station lift the needle before taking up the compass and on 
 arriving at the next station level the compass carefully, 
 
COMPASS SURVEYING 
 
 IOQ 
 
 sight backward to the station left and let the needle down 
 gently. It will then be parallel to the position last occupied 
 and will rest on the pivot without swinging. 
 
 The needle should always be held off the pivot when not 
 in use. When returned to the case to remain until again 
 required hold the plate level and let the needle down gently 
 on the pivot until it swings in the magnetic meridian. 
 Then raise it off the pivot. 
 
 Avoid riding in electric cars with a compass. If such 
 cars must be used it is well to hold the compass as nearly 
 level as possible and let the needle swing. Rubbing the 
 glass with a cloth often causes trouble because of f fictional 
 electricity, so the surface should be slightly moistened by 
 breathing on it after cleaning. The compass needle should 
 not be played with by drawing the end from side to side 
 with magnets or pieces of 
 metal. 
 
 THE USE OF THE COMPASS 
 
 \j 
 
 The field shown in Fig. 113 p 
 was surveyed and the first 
 station occupied was at 0, 
 the work proceeding with the 
 field on the right hand of the 
 surveyor, " surveying with the 
 sun, " to use an old expression. 
 There is no reason for this 
 other than convenience. Some 
 surveyors begin a survey at 
 the most easterly or the 
 most westerly corner to keep 
 the signs either all -f or 
 all when computing areas. 
 When a station other than 
 the most easterly or westerly 
 is taken as a starting point 
 the signs will be mixed and 
 unless the computer is careful he may meet with difficulties 
 in making his work close. It makes no difference which 
 station is chosen from which to start and the lines may be 
 
 FIG. 113. 
 
HO PRACTICAL SURVEYING 
 
 run to the right or to the left. In computing areas it is 
 not necessary to begin the computations at the first station 
 of the field work. 
 
 At each station a hub is set and the instrument is set 
 over it carefully with a plumb-bob marking the center, if 
 a tripod is used. If the compass is mounted on a Jacob 
 staff the staff is thrust into the ground close beside the hub. 
 The compass is carefully leveled and the sight at the N 
 end placed ahead. Looking back through the sights to 
 the station last set the needle is lowered. When it stops 
 swinging the reading is taken, thus checking the forward 
 reading. At station O no backsight is taken, the instru- 
 ment being placed so the needle, as nearly as one can judge, 
 points to the north before lowering it. The more nearly 
 the needle lies in the meridian before being released the more 
 quickly it comes to rest with least wear on the pivot. 
 
 A back reading (backsight) should always be taken to 
 guard against effects of local attraction and as a check 
 on readings, for it is easy to make a mistake of 10 degrees 
 and sometimes letters are transposed in recording. A back- 
 sight should give the same reading as a foresight, that is, 
 the bearing should be the same at all points on a line. 
 
 A reading is recorded by writing the letters and the in- 
 cluded angle as shown in Fig. 113. With the instrument 
 at the forward reading was ^49 15' E to i. At I the 
 bearing of the line should be 5 49 15' W to O. Beginners 
 are often told to read the south end of the needle on back- 
 sights. Practical surveyors seldom do it for it may lead 
 to mistakes and is often confusing. The common method 
 is to set the instrument and while the needle is still clamped 
 sight backward at the last station. Then lower the needle 
 gently and it will be in the line last read. When the needle 
 rests in this bearing sight carefully through the slits, 
 standing at the N end of the compass. Then go to the S 1 
 end and read the needle, which should give the same bear- 
 ing as at the last station. Then the next bearing may be 
 read. This procedure gives a check reading without mental 
 computation or any changing of letters. A common ejror 
 when the south end of the needle is read is to take 86 deg. to 
 be 84 deg., 83 deg. to be 87 deg., or vice versa when the 
 reading is almost due east or west; and, when the reading 
 
COMPASS SURVEYING 
 
 III 
 
 is so close to 90 E is often read for Wor W for E. It being 
 customary to complete a survey before correcting the bear- 
 ings, such errors may be serious. 
 
 The field is gone around in the manner stated until 
 station is occupied a second time, to obtain a backsight 
 on Sta. 5, if a backsight was not taken to this station at 
 the beginning of the work. The notes are placed on the 
 left-hand page, the right-hand page being used for sketches 
 and memoranda. 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 Station. 
 
 Corrected 
 bearing. 
 
 Distance 
 (chains) . 
 
 F. S. 
 
 B. S. 
 
 Corr. 
 
 
 
 N 4 9 i5' E 
 
 7.00 
 
 N 5 i 15' E 
 
 
 -2 00' 
 
 
 
 
 
 N 4 7 oo'W 
 
 
 I 
 
 S 46 15' E 
 
 8.00 
 
 S 48 30' E 
 
 
 -2 I 5 ' 
 
 
 
 
 
 S 45 45' E 
 
 
 2 
 
 S 29 30' W 
 
 9.00 
 
 S 30 oo' W 
 
 
 -o 30' 
 
 
 
 
 
 S 29 30' W 
 
 
 3 
 
 S 61 45'W 
 
 5-07 
 
 S 61 45'W 
 
 
 
 
 
 
 
 
 S 61 45'W 
 
 
 4 
 
 N 22 30' E 
 
 6-43 
 
 N22 3 0' E 
 
 
 
 
 
 
 
 
 N 25 15' E 
 
 
 5 
 
 N 4 i 30' W 
 
 7.00 
 
 N 3 8 45' W 
 
 
 + 2 45' 
 
 
 
 
 
 N 39 30' W 
 
 
 In the field the first, third, fourth and fifth columns are 
 used. When the survey is completed the corrections are 
 entered in the sixth column and the corrected courses placed 
 in the second column. The plus and minus signs before 
 the corrections refer to the difference between the corrected 
 course and the backsight. 
 
 TO CORRECT LOCAL ATTRACTION 
 
 When sufficient care is exercised it may be assumed that 
 all differences discovered between bearings from the two 
 ends of a line are due to local attraction. The reference 
 to sufficient care must not be overlooked, for surveyors often 
 make mistakes of several degrees in reading angles. 
 
 The following method for correcting errors due to local 
 attraction has been taken from Flint's " Survey," the first 
 
112 PRACTICAL SURVEYING 
 
 edition of which was published in 1805. Sometime between 
 1832 and 1851, L. W. Meech, A.M., was employed by the 
 publishers to revise the work, which had gone through six 
 editions before 1835, and he is credited with being the 
 author of the method since used by a number of men, but 
 mentioned in few texts. 
 
 Examining the field notes it is found that the forward 
 reading from Sta. 3 agrees with the backsight from Sta. 4. 
 Evidently there was no local attraction at either station so 
 the forward and back readings are assumed to be correct. 
 Therefore the bearing of the line from 2 to 3 is 5 29 30' W, 
 from 3 to 4 is S 61 45' W and from 4 to 5 is N 22 30' E. 
 
 Start at Sta. 4 to make corrections : 
 
 4. Correct bearing .......................... #22 30' E 
 
 Backsight ............................... ^25 15' E 
 
 + 2 45' 
 
 The backsight is greater so the sign of the difference is 
 plus. 
 
 The sign of the difference is minus when the backsight is 
 less than the forward reading. 
 
 The difference is placed under the course following. If 
 this course and that preceding are in the same or opposite 
 quadrant a plus sign becomes minus and a minus sign be- 
 comes plus. The signs are not altered when the courses 
 are not in the same or an opposite quadrant. When the 
 proper sign is determined the correction is applied. 
 
 5. Forward reading ......................... ^38 45', W 
 
 Correction .............................. + 2 45' 
 
 30' W 
 
 The correction is added here, for courses 4 and 5 are in 
 adjacent quadrants and the backsight was greater than the 
 forward bearing. 
 
 5. Corrected bearing ........................ N 41 3' w 
 
 Backsight ............................... ^39 30' W 
 
 2 00' 
 
 o. Forward reading .......................... N 51 15' 
 
 Correction ............................... - 2 oo' 
 
 N 49 15' E 
 
COMPASS SURVEYING 113 
 
 The correction is subtracted as the backsight was less 
 than the forward bearing and the two bearings are in ad- 
 jacent quadrants. 
 
 o. 
 
 i. 
 
 Corrected bearing ^49 15' E 
 
 Backsight ^47 oo' E 
 
 - 2 15' 
 
 Forward reading S 48 30' E 
 
 Correction 
 
 - 2 15' 
 
 5 46 15' E 
 
 I. Corrected bearing. 5 46 15' E 
 
 Backsight 5 45 45' E 
 
 - o 30' 
 
 Forward reading 5 30 oo' W 
 
 Correction - o 30' 
 
 2. 
 
 2. 
 
 5 29 30' W 
 
 Corrected bearing S 29 30' W 
 
 Backsight 5 29 30' W 
 
 If the last corrected bearing checks with its backsight 
 the courses may be assumed to be correct, but such a close 
 check is not always obtained. 
 
 This method also takes care of the diurnal declination of 
 the needle and all errors are greatly reduced even when not 
 entirely eliminated. 
 
 In the following example the first course is assumed to 
 be correct. The difference between the forward and back 
 readings gives a correction of +3 deg. but courses I and 2 
 being in the same quadrant the sign is changed. 
 
 Station. 
 
 Bearing. 
 
 Distance, 
 chains. 
 
 P. S. 
 
 B.S. 
 
 Corr. 
 
 
 
 N8 5 W 
 
 120.00 
 
 N85W 
 
 
 
 
 
 
 
 
 88 
 
 
 I 
 
 Ni4W 
 
 60.00 
 
 Ni7W 
 
 10 
 
 -3 
 
 2 
 
 N 7 5*E 
 
 90.00 
 
 N 7 4E 
 
 I5a 
 
 + i* 
 
 
 
 
 
 73* 
 
 
 3 
 
 S 27^ E . 
 
 102.60 
 
 S 29*E 
 
 
 -2 
 
 
 
 
 
 27* 
 
 
114 PRACTICAL SURVEYING 
 
 Correct the following field notes: 
 
 Station. 
 
 F. S. 
 
 B. S. 
 
 Station. 
 
 F. S. 
 
 B. S. 
 
 I 
 
 Ni 9 E 
 
 18 
 
 I 
 
 S 25 W 
 
 25 
 
 2 
 
 S 78 E 
 
 79 
 
 2 
 
 S 10 W 
 
 w|' 
 
 3 
 
 4 
 
 S 29 E 
 S 52|W 
 
 2 6r 
 
 53 
 
 3 
 4 
 
 S 7SiW 
 Nn E 
 
 76 
 
 5 
 
 S i 4 iE 
 
 is* 
 
 5 
 
 N 2 E 
 
 si 
 
 
 
 
 6 
 
 N8siE 
 
 8 S i 
 
 In correcting bearings by the foregoing method a check 
 will not always be obtained because needle readings are 
 taken to the nearest quarter degree with the best com- 
 passes. Closer readings are only estimates. The correc- 
 tion for local attraction will be found generally to be more 
 accurate than the reading of bearings. 
 
 All compass surveys should be corrected for the effects 
 of local attraction and thus the greatest source of error is 
 guarded against. When several lines seem to show an 
 agreement, as in the worked example, it is taken for granted 
 the true bearings were obtained, although the local attrac- 
 tion may have been the same in amount and direction at 
 certain stations, in which case only the average probable 
 bearings were obtained. If local attraction is proved to be 
 present at all stations then that course where it seems to 
 be least may be taken as the standard. 
 
 When the object of the survey is to obtain the area of a piece 
 of land the corrected bearings will give accurate results no 
 matter how far wrong the selected initial bearing may be. 
 The effect of local attraction being eliminated the correct angles 
 between the lines are obtained. 
 
 When the survey is to be recorded and future surveyors 
 are to be guided by the record it will be necessary to have 
 the true bearings. 
 
 Set two stakes on the true meridian, at some place not 
 far from the field, where it is known that no local attrac- 
 tion exists. The compass is set over the stake at the south 
 end of the line and sighted to the stake at the north end. 
 By means of the declination plate screw the compass box 
 is revolved on the vertical axis until the needle reads zero 
 and the declination is set off on the plate. A sight is then 
 
COMPASS SURVEYING 
 
 taken to some corner on the survey and the bearing is read. 
 The line is then run perfectly straight by backsights and 
 foresights without using the needle, until the corner is 
 reached. Setting up on the corner a backsight is taken 
 along the line just run (the starting point of which is within 
 an area having no local attraction) and the compass box is 
 turned until the needle points to the correct bearing. This 
 of course alters the declination, which makes no difference 
 as it is not necessary to record the declination when a survey 
 is stated to have been run from a true meridian. Having 
 set the needle the first forward reading will be correct and 
 all other bearings may be corrected by using it as a standard. 
 
 THE TRUE MERIDIAN 
 
 The declination of the needle may be set off on the com- 
 pass plate after using the Isogonic Chart as already de- 
 scribed. If this method is not available the true meridian 
 may be obtained with an accuracy equal to that of good 
 compass work, from a record of equal altitudes of the sun. 
 
 On a level area set a straight pole 
 vertically. About three hours before 
 noon drive a small stake in the ground 
 at the end of the shadow of the pole. 
 With a chain or tape attached to a 
 ring around the pole strike a semi- 
 circle with a radius equal to the 
 length of the shadow. If a cord is 
 used a strong pull may stretch it and a 
 wire will do instead of a chain or tape. 
 
 In the afternoon drive a small stake on the end of the 
 shadow when it touches the circumference of the circle. 
 Midway between the two stakes make a mark. A line 
 through this mark and the center of the pole will define 
 the true meridian, which may be laid off to any length by 
 using a chalk line stretched from the pole over the meridian 
 
 point. 
 
 MAKING A COMPASS SURVEY 
 
 Field notes should be clear and the surveyor must never 
 forget that other men may have to use his notes after he is 
 dead. Nothing essential should be omitted, nothing non- 
 essential should be put down. 
 
 vertical Polo 
 
 FIG. 114. True meridian 
 from equal altitudes of 
 sun. 
 
Il6 PRACTICAL SURVEYING 
 
 Common custom indicates that for compass work as 
 well as transit work it is most convenient to use the left- 
 hand page of the book for notes and the right-hand page for 
 sketches. The stations should begin with and at the 
 bottom of the page. One line should be used for each 
 chain or on long lines every tenth chain, even when no 
 stakes are set between instrument points, or stations. In 
 this way fences, roads, buildings, etc., may be located by 
 measurement and be shown by sketches in their relative 
 positions. See Fig. 7. 
 
 From each instrument point bearings to objects are 
 often taken and measurements made and the surveyor 
 should carry a thin flexible ruler in his field book for sketch- 
 ing purposes, drawing lines as nearly as possible in the 
 right direction. 
 
 A good workman is judged by the evidences of his work, 
 and neat full notes go far towards fixing the reputation of 
 a surveyor, even when the notes may fall into the hands of 
 persons ignorant of surveying. The criterion by which the 
 quality of field notes may be judged is that they may be 
 sent to a draftsman so he can map the survey and no further 
 
 PIG. 115. 
 
 information is required by him. When an instrument 
 man must plat his own notes or stay near a draftsman 
 who is using them, he is incompetent. 
 
 It is seldom possible to place an instrument on a corner, 
 for fences or hedges usually cover the lines, so surveys are 
 made on offset lines. 
 
 Fig. 115 illustrates a method used for offsetting when 
 using a compass. The point D is selected by eye to bisect 
 the included angle between AB and BC. The instrument 
 is set over D but the offset line parallel to AB is measured 
 
COMPASS SURVEYING 117 
 
 to d' ', this point being, as nearly as may be estimated, per- 
 pendicular to AB opposite B. Similarly the offset line 
 parallel to B C is measured from d. 
 
 When boundaries are crooked one line may be run. At 
 each angle the perpendicular distance is measured from 
 the offset line and recorded. Sometimes the surveyed 
 line is run on one side entirely, so all measurements will be 
 
 FIG. 116. 
 
 to the right or to the left, instrument men occasionally 
 forgetting to set down the right letter. To prevent errors 
 sketches should be made to supplement the written 
 notes. 
 
 When buildings, fences or other objects are to be located 
 for the purpose of showing them on a map, bearings are read 
 to them and the distances measured. 
 
 Sometimes the surveyor surveys an interior polygon, 
 from the corners of which bearings are taken to the corners 
 of the field and the distances measured as shown in Fig. 117, 
 where the instrument was set only at 
 the corners A, B, C, D and E with bear- 
 ings and distances taken to the corners 
 o, i, ... 8 of the field. This saves 
 time in the field, when the surveyor and 
 his assistants are under pay, and in- 
 creases the time in the office when the 
 draftsman alone is working. The possibility of mistakes 
 is increased because of the greater number of compu- 
 tations and lack of checks. A good surveyor never neg- 
 lects to check every operation when possible. 
 
 The office work in the case of a field surveyed by radiating 
 courses from an interior polygon begins by making correc- 
 tions for local attraction at each station and then correct- 
 ing the radiating bearings. This method of surveying by 
 radiation is very old. 
 
Il8 PRACTICAL SURVEYING 
 
 ANGLES FROM BEARINGS 
 
 When much local attraction exists it is a good plan to 
 mark on the map the angles at the corners after correcting 
 the bearings. A note should call attention to the fact that 
 these angles are to be used in preference to the bearings on 
 re-surveys, for the cause of the local attraction may be 
 removed after the original survey. 
 
 The angles may be deflections or included angles. 
 
 An angle is the amount of divergence between two inter- 
 secting lines in a plane. 
 
 A deflection or exterior angle is the difference in direc- 
 tion between two courses, shown by d in Fig. 118. 
 
 FIG. 118. Interior and deflection angles. FIG. 119. 
 
 An included, or interior, angle is the supplement of the 
 exterior angle, that is 180 d. In Fig. 118 the letter i 
 is used to indicate the included angles. 
 
 RULES FOR OBTAINING DEFLECTION ANGLES ' 
 
 I. First letters alike and last letters alike. Fig. 119. 
 
 Rule. The deflection is equal to the difference between 
 the courses. 
 
 The deflection is to the right in the TV E and 5 W quad- 
 rants when the following course is the greater. 
 
 ^70^: 
 
 N20 E 
 
 50 deflection. If course N 70 E follows N 20 E 
 the deflection is 50 R. If N 20 E follows N 70 E the 
 deflection is 50 L. 
 
COMPASS SURVEYING 
 
 IIQ 
 
 The deflection is to the left in the S E and N W quadrants 
 when the following course is the greater. 
 
 N 70 W 
 
 N20W 
 
 50 deflection. If course N 70 W follows N 20 W 
 the deflection is 50 L. If N 20 W follows N 70 W the 
 deflection is 50 R. 
 
 (Note. N 70 E and 5 70 W describe the same line 
 viewed from different ends.) 
 
 FIG. 120. 
 
 FIG. 121. 
 
 2. /<Yr.y/ letters alike and last letters unlike. Fig. 120. ** 
 
 Rule. Add the courses. 
 
 Bearing changing from W to E, going north the deflec- 
 tion is to the right; going south the deflection is to the left. 
 
 Bearing changing from E to W, going north the deflection 
 is to the left; going south the deflection is to the right. 
 
 -V 
 
 B 
 
 tb) 
 
 FIG. 122. 
 
 3. First letters unlike and last letters alike. Fig. 12 1. 
 
 Rule. Subtract the sum of the courses from 1 80. "" 
 
 Going east the deflection is to the right when N changes 
 to 5, and to the left when S changes to N. 
 
 Going west the deflection is to the right when S changes 
 to N and to the left when ^V changes to S. 
 
120 PRACTICAL SURVEYING 
 
 4. First letters unlike and last letters unlike. Fig. 122. 
 Rule. Subtract the difference of the courses from 180 deg. 
 Going north if the bearing changes toward the west, or 
 
 going south the bearing changes toward the east the de- 
 flection is left. 
 
 Going north if the bearing changes toward the east, or 
 going south the bearing changes toward the west the de- 
 flection is right. 
 
 RULES FOR OBTAINING INCLUDED ANGLES 
 
 5. If the first letters are alike and the last are alike sub- 
 tract the greater course from 180 deg. and add the smaller 
 course. Fig. 123. 
 
 6. If the first letters are alike and the last are unlike 
 subtract the sum of the courses from 1 80 deg. Fig. 124. 
 
 V 
 
 I 
 
 I 
 I 
 
 S 
 
 FIG. 125. FIG. 126. 
 
 7. If the first letters are unlike and the last are alike, 
 add the courses. Fig. 125. 
 
 8. If the first letters are unlike and the last are unlike 
 the included angle is equal to the difference of the courses. 
 Fig. 126. 
 
 It will be seen that the required angle in each case is ^4, B, 
 C in the last four rules. 
 
 The student should have considerable exercise in obtain- 
 ing angles from bearings. This can be done very well on 
 a table by using a circular protractor. Procure a paper 
 protractor 8 ins. in diameter and trim it to circle. Number 
 the graduations each way from a meridian line from o deg. to 
 90 deg. , lettering the quadrants like a compass, with E and W 
 reversed. Tack down a sheet of drawing paper and on it 
 draw a straight line to represent the true meridian. Lay the 
 protractor on this line so the zero points touch it. Draw a 
 
COMPASS SURVEYING 
 
 121 
 
 line normal to the meridian and set the center of the pro- 
 tractor at the intersection of the two lines. 
 
 FIG. 127. Obtaining angles from bearings. 
 
 To plot a course turn the protractor until the required 
 angle touches the meridian line. On the drawing paper 
 mark a dot at the N zero point and opposite it write the 
 
122 PRACTICAL SURVEYING 
 
 course. Set off the second course. Then with the N zero 
 at one course the included angle, or the deflection angle, 
 can be read off as a check on the calculations. 
 To obtain the true bearing from a random line. 
 
 In Fig. 128 two cases are shown 
 of random lines. In (a) the dotted 
 line represents a boundary and the 
 surveyor started from A to re-trace 
 the line but when the required dis- 
 tance was measured found himself 
 at BI instead of B. The difference 
 d was then measured. 
 In (b) the case is similar but an offset line A\B\ was run 
 so that instead of the offset at each end being 0, it was O 
 at Ai and O 1 at B\. 
 
 The difference d = 0' - 0. 
 
 Then angle a = *** = * * d . 
 
 AtXBi AXBi 
 
 Assume the length of the line to be 20 chains and d = 20 
 links = 0.20 chain. 
 
 0-573 X 60 = 34.38 min. 
 
 The line AB\ (or A\B\) was run on a bearing of N 45 
 oo' E, but as it ran to the right of the true line the bearing 
 must be corrected to the left and becomes N 44 25' E. 
 
 Suppose the original field notes to have called for a bear- 
 ing of N 45 R for line AB and the re-tracement as above 
 showed a difference of 35 min. between the actual bear- 
 ing and that given in the old field notes ; this difference can 
 be set off on the variation plate. The whole survey can 
 then be re-traced according to the recorded bearings, pro- 
 vided the original surveyor made no serious errors in read- 
 ing the needle. 
 
 MAKING THE MAP 
 
 In the center of a sheet of paper rule two long straight 
 lines perpendicular to each other. The line from the 
 bottom to the top is the true meridian and the line from left 
 to right is an east and west line. 
 
COMPASS SURVEYING 
 
 123 
 
 Graduate a paper protractor in quadrants but do not 
 transpose the E and W as on a compass plate. On the 
 protractor, which should be 14 ins. in diameter, draw a 
 fine ink line joining the N and S zeros and another joining 
 the E and W, 90, points. At the center where the lines 
 
 FIG. 129. Paper protractor for plotting angles. 
 
 intersect and at a couple of places on each line cut square 
 or round holes about \ in. across, with clean smooth edges. 
 Trim the protractor along the edge into circular form, for 
 it is printed on a square sheet. 
 
 Lay the protractor on the drawing paper over the two 
 ruled lines so they may be seen through the holes and their 
 intersection will indicate the exact center of the protractor. 
 
124 
 
 PRACTICAL SURVEYING 
 
 To keep it in position two thumb tacks may be used, or, 
 if the holes will be considered objectionable in the map 
 use paper weights. Very good paper weights are made 
 of chamois skin bags containing two Ibs. of fine shot, an 
 outer bag of cloth being used as a cover. 
 
 The protractor being carefully set, with a sharp-pointed 
 pencil mark all the angles on the drawing paper, numbering 
 each dot to correspond with the number of the course, or 
 write the course after each dot. When all the angles are 
 marked remove the protractor and connect by fine lines 
 each dot with the center. The result will look like Fig. 130. 
 
 FIG. 130. Plotting bearings. FIG. 131. Plat made by bearings. 
 
 Select a starting point and with triangles, or with trian- 
 gle and straight-edge, transfer the first course to that point 
 and draw a line parallel to the course. With the scale set 
 off the length on the line and mark the end for Sta. I. 
 From this station set off the second course and proceed in 
 this manner until the boundary is all platted. The com- 
 pleted result is shown in Fig. 131. 
 
 On long lines of survey the true meridian is carefully 
 transferred by straight-edge and triangle to a convenient 
 place so it will not be necessary to transfer the courses 
 farther than is convenient with a triangle, thus avoiding 
 danger of slipping and consequent error. 
 
 PLOTTING BY DEFLECTIONS 
 
 Some inexperienced, or poorly instructed, men will use 
 small protractors and plot deflections. Each course is 
 laid off to a considerable* length and the distance marked. 
 
COMPASS SURVEYING 
 
 125 
 
 The protractor is placed with center on the station and the 
 deflection angle set off. A line is then drawn through the 
 two points, the length of the next course marked and the 
 
 FIG. 132. Platting deflections. 
 
 operation repeated until all the courses are plotted. The 
 method is troublesome and very inaccurate for the errors 
 are cumulative. 
 
 Another method is to use a heavy 
 steel straight-edge and large triangle 
 and carry the meridian through the 
 end of each course. A large pro- 
 tractor is placed on each station and 
 the following course laid off. This 
 is an improvement on the method of plotting deflections, 
 but slow and not so accurate as the method first described. 
 
 Other methods of plotting, requiring some knowledge of 
 Trigonometry, are described in Chapter VI. 
 
 DISTRIBUTING ERRORS 
 
 Errors in surveys show up when making the map, for 
 the last distance will carry the end past the last station 
 (the starting point), or perhaps fall short. It will also be 
 
 FIG. 133. Platting from 
 meridians. 
 
126 
 
 PRACTICAL SURVEYING 
 
 to one side. A plat of a survey is apt to look like Fig. 
 
 134. 
 
 Measure the distance from 0' to o and call it a. Call 
 
 the sum of all the courses b. The closing error = , 
 
 Vtf/b 
 
 If this exceeds 5 <y, or any limit 
 of error selected, then each course 
 should be mentally examined to 
 discover if possible whether a mis- 
 take was made. If no evidence is 
 found it will be best to re-run the 
 lines and bring the error within 
 proper limits. 
 
 An error within proper limits may 
 be distributed over the courses in 
 proportion to the actual or weighted lengths. 
 
 To weight errors the survey is mentally reviewed and 
 those courses on which no difficulties were encountered are 
 given a weight of I, those on marshy land, 1.5, those on 
 steep land, 2, these values being here chosen merely for 
 illustrative purposes. The values indicate the relative 
 probability of errors occurring. 
 
 v. Multiply each course by the weight assigned to it and 
 add the new lengths progressively. This new total length 
 
 FIG. 134. Plat in which 
 error is not distributed. 
 
 14 '8 21 18 7-5 6 6. 6.5 
 
 FIG. 135. Graphical method for distributing errors. 
 
 is measured off on a straight line and at one end the total 
 error is measured off on a perpendicular line and a triangle 
 formed. From each station point a perpendicular is 
 erected to intersect the hypothenuse and the length of the 
 perpendicular at any station represents the amount of 
 
 error. 
 
 A line is drawn from O' to O on the map and parallel 
 lines are transferred to each station as shown in Fig. 136. 
 The ends are connected and the survey is closed. 
 
COMPASS SURVEYING 
 
 127 
 
 Multiplying a distance by the weight assigned lengthens 
 it and thus gives it that much greater proportion of error. 
 This is evident from the law of pro- 
 portionality of triangles. Instead of 
 making a triangle the error may be 
 distributed as follows: 
 
 Divide the error by the total length 
 of all the courses, after weighting same. 
 Multiply the quotient by each course 
 to obtain the error for each course. 
 
 Example. - - Closing error 0.19 
 chain. 
 
 FIG. 136. Plat of field 
 with errors distributed. 
 
 No. of course. 
 
 Distance, 
 chains. 
 
 'Weight. 
 
 Weighted 
 distance. 
 
 Error. 
 
 O 
 
 4-75 
 
 I 
 
 4-75 
 
 O.OlS 
 
 I 
 
 3-lS 
 
 I 
 
 3-15 
 
 0.012 
 
 2 
 
 6.22 
 
 I 
 
 6.22 
 
 0.024 
 
 3 
 
 7.10 
 
 i-S 
 
 10.65 
 
 0.041 
 
 4 
 
 9-05 
 
 2 
 
 I8.IO 
 
 0.069 
 
 5 
 
 2.76 
 
 2-5 
 
 6.90 
 
 O.O26 
 
 
 
 
 49-77 
 
 O.igo 
 
 0.19 
 4977 
 
 = 0.0038 + per chain. 
 
 When the lines are adjusted and the survey is closed the 
 map may be completed. From the notes and sketches in 
 the field book plat the fences, etc. Draw meridians through 
 
 FIG. 137. 
 
 stations from which bearings were taken to objects; place 
 a protractor on the meridians and lay off the bearings. 
 For this work a transparent protractor with scale on edge 
 is very convenient. It is also a useful tool to carry in the 
 
128 PRACTICAL SURVEYING 
 
 field book to assist in making sketches. The center is 
 placed on the station and the meridian passes through the 
 angle. The edge then indicates the bearing to the object, 
 the distance to which is measured by means of the gradu- 
 ations. 
 
 Protractors are made of horn, paper, metal, celluloid, 
 ivory and rubber. The prices range from fifteen cents to 
 eighty-five dollars. For ninety per cent of the work done 
 by surveyors and engineers, I4~in. paper protractors grad- 
 uated to J degrees are better than the more expensive 
 kinds. 
 
 To find the area of a surveyed field, first plot it as de- 
 scribed and when the lines are closed divide it into quad- 
 rilaterals, triangles, etc.; obtain the area of each, and the 
 sum of these areas is the area of the field. 
 
 OBTAINING AREAS BY COMPUTATION WITHOUT MAKING 
 
 A MAP 
 
 All compass bearings are angles measured to the right or 
 left of the true meridian. 
 
 A course may therefore be drawn, forming the hypothe- 
 nuse of a right-angled triangle the base of 
 which is on the meridian. The base is called 
 the latitude, for latitude is measured north 
 and south from the equator. The alti- 
 tude is called the departure, for it measures 
 the amount by which the course "departs" 
 from a true north and south line. 
 
 By some writers within recent years, the 
 "departure" has been termed "longitude 
 difference." The reasoning is that through 
 each point a true meridian may be drawn 
 and longitude being measured on circles 
 normal to meridians the distance between 
 F g meridians is the difference in longitude. A 
 Traverse Table (page 142), used for compass 
 surveys, contains the latitude and departure for all bearings 
 from o to 90, varying by quarters of a degree. 
 
 Example I. Find the latitude and departure for 
 N 5i W 23.77 chains. 
 
COMPASS SURVEYING 
 
 129 
 
 Looking in the column headed Course, trace down to the 
 
 horizontal line 
 follows : 
 
 opposite 5 15'. Arrange in columns as 
 
 
 Latitude. 
 
 Depart tire. 
 
 Dist. 2X10 = 
 Dist. 3X1 = 
 Dist. 7X0.1 = 
 Dist. 7X.oi = 
 
 19.916 
 2.9874 
 0.6970 
 0.0697 
 
 1-83 
 0-2745 
 o . 0645 
 0.0064 
 
 23.6701 
 
 2.I7S4 
 
 Therefore in going AT 5 15' W 23.77 chains the line 
 runs north 23.67 chains and west 2.18 chains. 
 
 For an angle of 45 the latitude and departure are equal. 
 
 For an angle of o the latitude is I and the departure = o. 
 
 For an angle of 90 the latitude is zero and the departure 
 = i. 
 
 Therefore the latitude of an angle less than 45 is equal 
 to the departure of an angle of the same size between 45 
 and 90. For example the latitude for 13 is the depar- 
 ture for 77 and vice versa. This fact saves much labor 
 in the computation of tables and economizes space in the 
 printing of tables. For all angles under 45 read the courses 
 on the left-hand edge of the page going down, using the 
 Lat. and Dep. columns at the top. For 
 all angles over 45 read the courses on 
 the right-hand edge of the page going 
 up, using the Lat. and Dep. columns at 
 the bottom. 
 
 North latitude is positive ( + ) and 
 south latitude is negative ( ) . Similarly 
 east departure is positive ( + )and west 
 departure negative ( ). This is illus- 
 trated in Fig. 139. All distances due 
 north and due east are said to be meas- 
 
 FIG. 139. 
 
 ured in a positive sense and all distances due south and due 
 west, the opposite of north and east respectively, are said 
 to be measured in a negative sense. If a man starts from 
 a certain point and measures north +10 chains, then meas- 
 ures back on the line south 4 chains, he ends at a point 
 + io 4 = +6 chains north of the starting point. 
 
130 PRACTICAL SURVEYING 
 
 Example. A line is surveyed as follows : 
 
 N 50 E, 1 1 chains, 
 N 7 W, 12 chains, 
 ,S 14 E, 22 chains. 
 
 How far apart are the two ends of the line, measured by 
 the latitude and departure of a line connecting the points? 
 
 Bearing. 
 
 Distance. 
 
 N+ 
 
 S- 
 
 E+ 
 
 W- 
 
 N sE 
 
 II 
 
 10.958 
 
 
 O.CKQ 
 
 
 N 7W 
 
 12 
 
 II 911 
 
 
 
 I 4.62 
 
 S i4E 
 
 22 
 
 
 21-347 
 
 0-532 
 
 
 
 
 22.869 
 
 21-347 
 
 I-49I 
 
 1 .462 
 
 After ruling a sheet of paper as above and filling in the 
 bearings and distances use the traverse table and place the 
 results in the proper columns. 
 
 NSE. 
 
 N7W. 
 
 S 14 E. 
 
 Dist. 
 
 Lat. 
 
 Dep. 
 
 Dist. 
 
 Lat. 
 
 Dep. 
 
 Dist. 
 
 Lat. 
 
 Dep. 
 
 IO 
 
 I 
 
 9-96I9 
 0.9962 
 
 0.8716 
 0.0872 
 
 IO 
 
 2 
 
 9-9255 
 1-9851 
 
 1.2187 
 0.2437 
 
 20 
 2 
 
 19.406 
 I-94I 
 
 0.4838 
 . 0484 
 
 10.9581 
 
 0.8588 
 
 II .9106 
 
 1.4624 
 
 21-347 
 
 -53 2 3 
 
 North ( + ) latitudes 
 South (-) latitudes 
 
 East (+) departures 
 West ( ) departures 
 
 22 . 869 
 21.347 
 1.522 Diff. (+) 
 
 i . 462 
 
 0.029 Diff. (+) 
 
 The last point set is 1.522 chains north and 0.029 chain 
 east from the starting point. 
 
 In computing areas of compass surveys the process 
 commonly followed is essentially that of reducing the figure 
 to a triangle of equivalent area and then finding the area 
 of the triangle. This is the method of Double Meridian 
 
COMPASS SURVEYING 131 
 
 Distances. In some old American textbooks it was known 
 as the Pennsylvania Method but the original discoverer 
 was Thomas Burgh of Ireland in 1778, to whom it is said 
 the Irish Parliament awarded a pension equal to the inter- 
 est on twenty thousand pounds as a recognition of the great 
 value of his work. 
 
 In the April, 1879, number of Van Nostrand's Engineering 
 Magazine a new rule was proposed by Professor J. Wood- 
 bridge Davis, which is less laborious than the old rule. 
 The author used it instead of Double Meridian Distances 
 during those years in which much of his work was surveying 
 and cannot understand why the old rule is not abandoned. 
 The newer one is called the Total Latitude rule and will 
 alone be given in this book. It is based on trapezoids 
 instead of triangles. 
 
 Rule. Multiply the total latitude of each station by the 
 sum of the departures of the two adjacent courses. The 
 algebraic half sum of these products is the area. 
 
 Algebraic sum must first be understood and the rules 
 memorized. 
 
 To add positive (+) and negative ( ) quantities, subtract 
 the lesser from the greater and prefix to the result the sign of 
 the greater. This has been illustrated in the use of the 
 traverse table. 
 
 To subtract positive and negative quantities change the sign 
 of the subtrahend and then proceed as in addition. 
 
 To multiply positive and negative quantities the rules 
 are: 
 
 (a) Like signs produce plus. 
 
 (+5) X (+6) = +30. 
 (-5) X (-6) = +30. 
 
 (b) Unlike signs produce minus. 
 
 (-6) =-30. 
 
 In practical work the positive (+) sign is generally under- 
 stood so is seldom written. 
 
 The following example is taken from the article of 
 Professor Davis. 
 
132 
 
 PRACTICAL SURVEYING 
 
 Sta. 
 
 Bearing. 
 
 Dist. 
 
 Lat. 
 
 Dep. 
 
 Total 
 lat. 
 
 Adj. 
 dep. 
 
 Double 
 areas. 
 
 N+ 
 
 2.21 
 O.IS 
 
 1-7* 
 
 4.14 
 
 S- 
 
 E+ 
 
 W- 
 
 o 
 
 i 
 
 2 
 
 3 
 4 
 
 N 3 5 E 
 N8 3 iE 
 S 57 E 
 S 34iW 
 
 Ns6*w 
 
 2.70 
 1.29 
 
 2.22 
 
 3-55 
 
 3-23 
 
 I.2I 
 
 2-93 
 
 i-55 
 
 1.28 
 1.86 
 
 4~69~ 
 
 
 
 
 6.2543 
 7.4104 
 o. 1610 
 8.3482 
 
 2.OO 
 2.69 
 4.6 9 
 
 Sq 
 
 2.21 
 
 2-36 
 
 i. S 
 
 -1.78 
 
 iiare cl: 
 
 2.83 
 3-14 
 0.14 
 -4.69 
 
 .ains 
 
 4.14 
 
 2)21 .8519 
 10.9259 
 
 Area 1.0926 acres. 
 
 When using the Double Meridian Distance rule certain 
 considerable advantage is secured by starting the work of 
 computation at the most easterly or the most westerly 
 station. When using the Total Latitude rule any station 
 may be used as a starting point but if that station is chosen 
 for which the latitude is most nearly the average of all, the 
 fewest possible figures will be used in the double area 
 factors. 
 
 In the example the work was performed as follows: 
 
 Total latitude: \ 2 2l 
 
 +Q.I5- 
 +2.36 
 
 1.21 
 
 -2.93 
 
 - 1 . 78 Check 
 
 Check. The total latitude for the last station must be 
 equal to the latitude of that station, with opposite sign. 
 The total latitude of the first station is zero. 
 
 The adjacent departure for the first station is zero. 
 2nd station. 
 
 Dep. o 
 Dep. i 
 Dep. i 
 Dep. 2 
 Dep. 2 
 Dep. 3 
 Dep. 3 
 Dep. 4 
 
 = +1-55 
 = +1.28 
 
 + 2.83 
 + 3-14 
 
 o. 14 
 
 -4.69 
 
 = +1.28 
 = +1.86 
 = +1.86 
 
 = 2.OO 
 
 = 2.OO 
 = 2.69 
 
COMPASS SURVEYING 133 
 
 The only checks on the adjacent departures addition are 
 care and re-computing. 
 
 Following the algebraic law for dealing with positive and 
 negative signs the column of double areas is rilled. For 
 Sta. i and 2 the sign is -f for both the total latitude 
 and the adjacent departure, therefore the double area is 
 positive. For Sta. 3 the first is + and the second is , 
 therefore the double area is negative. For Sta. 4 both 
 signs are and the double area is positive. 
 
 DISTRIBUTING ERRORS 
 
 When measuring with a chain, using a compass for the 
 angles, lengths are usually taken to the nearest link. 
 Some surveyors record the nearest half link but when this 
 'is done they attempt to read the bearing more closely than 
 a quarter of a degree by estimating the position of the north 
 end of the needle between graduations. 
 
 The maximum error in angle when the nearest quarter 
 of a degree is recorded is seven minutes, corresponding to a 
 departure of 0.2 link per chain or I in 500. The maximum 
 allowable error in chaining for ordinary farming land should 
 not exceed I in 500, so that the work of chainmen having 
 some experience is about equal in accuracy to that of 
 ordinary compass work. 
 
 The errors here referred to are the accidental errors of 
 the work. Errors due to a chain or tape being longer or 
 shorter than standard do not show until a re-survey is 
 made with a standard. 
 
 Minor errors of closure are distributed proportionately 
 to lengths in compass surveys, the effect being to change 
 bearings and distances. The transit being an instrument 
 by means of which angles can be read very closely, all 
 closing errors are distributed over the lengths only, it being 
 easily possible to check the angles and obtain exact angular 
 closures. 
 
 The method here given for distributing errors is applicable 
 alike to compass work or transit work since it is bad 
 practice to alter bearings and distances when the error of 
 closure is found to be within reasonable limits. Forcing a 
 closure is called "fudging" and notes of a compass survey 
 
134 
 
 PRACTICAL SURVEYING 
 
 are looked upon with suspicion when they close without 
 error. 
 
 Example. A field was surveyed and the following notes 
 recorded : 
 
 
 
 N c 
 
 ' .. 
 
 Chains 
 2. 71 
 
 
 TV 8v 
 
 t E 
 
 I "*O 
 
 1 
 
 S S7' 
 
 1 E" 
 
 2 21 
 
 7 
 
 5 34^ 
 
 \ W 
 
 I C/1 
 
 4. 
 
 TV& 
 
 \ c w.., 
 
 .1.22 
 
 When the latitudes and departures were computed the 
 field did not close. Before the area of a field can be com- 
 puted the northings must equal the southings and the 
 eastings must equal the westings. In other words, in 
 measuring the boundaries of a field the surveyor goes 
 as far north as he goes south and as far east as he" 1 goes 
 west. 
 
 Sta. 
 
 Bearing. 
 
 Dist. 
 
 Lat. 
 
 Dep. 
 
 Balanced 
 
 Lat. 
 
 Dep. 
 
 N+ 
 
 S- 
 
 E+ 
 
 W- 
 
 N+ 
 
 S- 
 
 E+ 
 
 W- 
 
 o 
 i 
 
 2 
 
 3 
 
 4 
 
 N 3 S E 
 N 83* E 
 S 57 E 
 S 3 4lW 
 N S 6i W 
 
 2.73 
 1.30 
 
 2.21 
 
 3.54 
 
 3-22 
 
 2.214 
 0.147 
 
 i 209 
 
 i -SSo 
 1.292 
 i 862 
 
 
 2.2132 
 0.1469 
 
 1.2094 
 2.9271 
 
 I.S456 
 1.2883 
 1.8566 
 
 1.9978 
 2.6927 
 
 1.777 
 
 2.926 
 
 
 1.992 
 2.685 
 
 1.7764 
 
 13.00 4.138 4.135 4.704 4.677 4.1365 4.1365 4.6905 4.6905 
 4.135 4.677 
 
 Error in lat.= 0.003 
 
 o . 027 = Error in dep. 
 
 /O -2 i o 7 2 
 
 Error of closure = V -^ - = 0.00208, 
 V 1300 
 
 - = i in 480. 
 0.00208 
 
 The error, being within reasonable limits, is distributed 
 as follows: 
 
COMPASS SURVEYING 
 
 Latitudes 
 
 '- = 4.1365 correct sum. 
 
 4-1380 
 4-I365 
 0.0015 to be subtracted from the northings. 
 
 4-1365 
 4-I350 
 0.0015 to be added to the southings. 
 
 . 0.0015 
 
 I H -- = 0.00036 to increase southings. 
 4- I 3 e> 5 
 
 0.0015 
 
 i --- - = i 0.00036 = 0.99964 to decrease northings. 
 4- I 3 t> 5 
 
 
 Northings. 
 
 Southings. 
 
 2. 214 X O. 99964 
 
 2. 2132 
 
 
 o. 147 X o. 99964 
 
 o. 1469 
 
 
 i . 209 X i . 00036 
 
 
 I 2019 
 
 2. 926 X I. 00036 
 
 
 2. 9271 
 
 
 4- 1365 
 
 4- 1365 
 
 The results are placed in the "Balanced" columns. 
 Departures 
 
 -- = 4.6905 correct sum. 
 
 4.7040 
 4-69Q5 
 0.0135 to be subtracted from eastings. 
 
 4-6905 
 
 4-677 
 
 0.0135 to be added to westings. 
 
 i + ' ^ = 1.00287 to increase westings. 
 1 ~ 4^6905 = a " 713 to decrease eastings. 
 
136 
 
 PRACTICAL SURVEYING 
 
 
 Eastings. 
 
 Westings. 
 
 i .55 X 0.99713 
 
 I . 5456 
 
 
 1.292 X 0.99713 
 
 I . 2883 
 
 
 i 862 X o 99713 
 
 1.8566 
 
 
 i .992 X i .00287 
 2.685 X 1-00287 
 
 
 1-9978 
 2.6927 
 
 
 4-6905 
 
 4-6905 
 
 In actual practice few surveyors do all the figuring here 
 shown, the corrections being made mentally and distrib- 
 uted in approximately proportionate amounts. The practi- 
 cal surveyor also pays great attention to the weighting of 
 the various courses when his chainmen are inexperienced 
 and their work is therefore not wholly reliable. 
 
 OMISSIONS 
 
 It is not possible always to measure each side of a field 
 or take every bearing and the missing parts must be sup- 
 plied by platting or by computation, this latter method 
 requiring a knowledge of trigonometry. If a map is not 
 accurately made to a very large scale omissions cannot 
 be obtained from it within the limits of even reasonable 
 accuracy. For the purpose, however, of computing areas 
 by mensuration missing lines may be supplied by plotting. 
 The methods to be now described should be carefully 
 studied and worked out in full by the student, for he will 
 then fully understand the methods of computation in the 
 chapter following. 
 
 There are six cases in omissions 
 and one field will be used to illus- 
 trate them. 
 
 Chains 
 
 0. N i6%E 22.00 
 
 1. N82 E T9.6o 
 
 2. S 17 E 24.00 
 
 3- S 37 W 22.00 
 
 4. #49 W 25.20 
 
COMPASS SURVEYING 
 
 CASE I. Bearing and distance omitted. Fig. 141. 
 
 Assume the course from 2 to 3, 5 17 E, 24 chains to be 
 omitted. 
 
 Plat the two courses o and I, beginning at Sta. o. Re- 
 verse the bearings for courses 3 and 4 so they may be 
 platted beginning at Sta. o. The line from 2 to 3 may 
 then be drawn. A close check on the bearing may be ob- 
 tained by drawing a meridian line through one end of the 
 line and applying a protractor. 
 
 FIG. 141. 
 
 FIG. 142. 
 
 CASE II. Bearing read but length not measured. Fig. 142. 
 
 The omitted length is for course 2. Begin at Sta. o and 
 lay off the bearings and distances to Sta. 2 and from Sta. 2 
 lay off by the protractor a long line 
 having the bearing S 17 E. Re- 
 versing the bearings of courses 3 and 
 4 as in the first example, plot them; 
 Sta. 3 should be on the long line 
 drawn from Sta. 2 provided the field 
 work and platting are correct. 
 
 CASE III. Length measured, 
 bearing not read. Fig. 143. 
 
 Assume same course to be 
 affected. Beginning at Sta. o plot 
 courses o and I in a forward direc- 
 tion and courses 3 and 4 reversed. 
 From Sta. 2 as a center with a 
 radius equal to 24 chains, describe an arc. If the work 
 was properly done the arc should pass through Sta. 3. 
 
 FIG. 143. 
 
138 
 
 PRACTICAL SURVEYING 
 
 CASE IV. Two bearings read but lengths not measured. 
 
 (a) Adjacent courses. Fig. 144. 
 
 The omitted lengths are assumed to be for courses 2 and 
 3. Begin at Sta. o and plot in a forward direction to Sta. 2, 
 from which point draw a long line on the given bearing. 
 Then plot the reversed courses; the intersection of the 
 long line from Sta. 2 with that from Sta. 4 fixes the loca- 
 tion of Sta. 3. 
 
 (b) Courses not adjacent. Fig. 145. 
 
 Assume the omitted lengths to be in courses I and 4. 
 Begin at o and lay off the line from o to I . The length of 
 
 the next line is not known so this course is disregarded 
 and from Sta. I lay off the bearing and distance of course 
 2. 5" 17 E 24 chains and follow that with the next course. 
 Then from o lay off the bearing 5 82 W (the reverse of 
 N 82 E). From 4' lay off the bearing N 49 W. These 
 lines will intersect at i'. By using triangles transfer the 
 line I '-2' to 1-2 and thus locate 2. From 2 draw 2-3 equal 
 and parallel to 1-3' and from 3 draw line 3-4 equal and 
 parallel to 3'~4'. Then the line 4-0 will be equal and 
 parallel to Iine4'-i / . 
 
 CASE V. Two lengths measured but bearings not 
 read. 
 
 (a) Adjacent courses. Fig. 146. 
 
 Assuming the bearings of courses 2 and 3 to have been 
 omitted plot the other courses from Sta. o as before. From 
 
COMPASS SURVEYING 
 
 139 
 
 Sta. 2 describe an arc with a radius equal to the length of 
 
 course 2 and from Sta. 4 describe an arc with a radius 
 
 equal to the length of course 3. 
 
 These arcs will intersect in 3 and in 
 
 3'. It will be necessary to view the 
 
 land in order to know which point 
 
 is correct. 
 
 (b) Courses not adjacent. Fig. 147. 
 
 The bearings of courses 2 and 4 
 were not read. The remaining three 
 courses are plotted from o to I, I 
 to 2, 2 to 3'. From o an arc with 
 radius equal to the length of course 
 4 is described and from 3' an arc with 
 radius equal to the length of course 
 2 is described. Fig. 147 (a) shows 
 the work, (b) the figure resulting when courses 1-2 and 04 
 are omitted and (c) the figure resulting when o-i and 4-3 
 are omitted. In this case a knowledge of the shape of the 
 field is necessary in order to know which solution is right. 
 
 FIG. 146. 
 
 co; 
 
 CASE VI. Bearing of one course read but length not 
 measured and length measured but bearing omitted on 
 another course. 
 
 (a) Adjacent courses. Fig. 148. 
 
 Plat as before. The bearing from Sta. 2 having been 
 read lay off a dotted line. The length of course 3 having 
 been measured describe an arc with this radius from Sta. 4. 
 This gives two possible solutions so that actual knowledge 
 of the shape of the field is necessary. 
 
140 
 
 PRACTICAL SURVEYING 
 
 (b) Courses not adjacent. Fig. 149. 
 
 It is assumed that the bearing from Sta. I was omitted 
 and the length of course 4 was omitted. 
 
 Plot the course o-i. From Sta. I describe an arc with 
 radius equal to the length of course I and from o lay off 
 the line 0-4 with the bearing of course 4. On a piece of 
 
 FIG. 148. 
 
 tracing paper plot the line 3-4 and from 3 the line 3-2, 
 extending the line considerably beyond the mark at 2. 
 From 4 plot a line having the bearing of course 4. Place 
 the line 4-0 on the tracing paper over the same line on the 
 drawing paper and move it along this line until the line 
 23 is tangent to the arc and the point 2 is on the arc. A 
 needle put through at 2, 3 and 4 will mark the stations and 
 lines may then be drawn to complete the map. 
 
 A careful examination of the problems given shows that 
 in reality there are only four "cases, in which the lost, or 
 omitted, parts may be: 
 
 CASE I. Bearing and length of one course. 
 
 CASE II. Lengths of two courses. 
 
 CASE III. Bearing of two courses. 
 
 CASE IV. Length of one course and bearing of another. 
 
COMPASS SURVEYING 
 PROBLEMS 
 
 141 
 
 Find by computation the areas of the following fields. 
 Plat same carefully and check the computations by men- 
 suration. 
 
 Chains 
 
 1. o. N i6 E 22.00 
 
 1. NS2 E 19-60 
 
 2. 5 17 E 24.00 
 
 3-537 W 22.00 
 
 4. N 49 W 25.20 
 
 Chains 
 
 2. o. #15 E 80 
 
 1. #37! E* 40 
 
 2. East 30 
 
 3. S 11 E 50 
 
 4. South 54 
 
 5. West 40 
 
 6. S 3 6i W 40 
 
 7- N&VW 34 
 
 Chains 
 
 3. o. #75 E 13.70 
 
 1. N 20\ E 10.30 
 
 2. East 16.20 
 
 3- S 33 J" W 35-30 
 
 4.576 W 16.00 
 
 5. North 9.00 
 
 6.584 W ii. 60 
 
 7- Ns&W ii. 60 
 
 8. N 36! E 19. 20 
 
 9. N 22\ E 14.00 
 
 10. 5 76! E 12.00 
 
 11. 5 15 W 10.85 
 
 12. 5 i6f W 10.12 
 
142 
 
 PRACTICAL SURVEYING 
 
 TRAVERSE TABLE 
 
 Course. 
 
 Dist. i. 
 
 Dist. 2. 
 
 Dist. 3. 
 
 Dist. 4. 
 
 Dist. 5. 
 
 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 
 o IS 
 
 I.OOOO 
 
 0.0044 
 
 2.OOOO 
 
 0.0087 
 
 3.0000 
 
 0.0131 
 
 4.0000 
 
 0.0175 
 
 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 
 
 I 
 
 9998 
 
 0175 
 
 9997 
 
 0349 
 
 9995 
 
 0524 
 
 9994 
 
 0698 
 
 9992 
 
 0873 
 
 89 o 
 
 IS 
 
 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 o 
 
 IS 
 
 9992 
 
 0393 
 
 9985 
 
 0785 
 
 9977 
 
 1178 
 
 9969 
 
 1570 
 
 996i 
 
 1963 
 
 45 
 
 30 
 
 9990 
 
 0436 
 
 9981 
 
 0872 
 
 9971 
 
 1309 
 
 9962 
 
 1745 
 
 9952 
 
 2181 
 
 3O 
 
 45 
 
 0.9988 
 
 0.0480 
 
 1.9977 
 
 0.0960 
 
 2.9965 
 
 0.1439 
 
 3-9954 
 
 0.1919 
 
 4-9942 
 
 0.2399 
 
 15 
 
 3 o 
 
 9986 
 
 0523 
 
 9973 
 
 1047 
 
 9959 
 
 1570 
 
 9945 
 
 2093 
 
 9931 
 
 2617 
 
 87 o 
 
 15 
 
 9984 
 
 0567 
 
 9968 
 
 H34 
 
 9952 
 
 1701 
 
 9936 
 
 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 o 
 
 9976 
 
 0698 
 
 9951 
 
 1395 
 
 9927 
 
 2093 
 
 9903 
 
 2790 
 
 9878 
 
 3488 
 
 86 o 
 
 IS 
 
 
 0741 
 
 9945 
 
 1482 
 
 99i8 
 
 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 o 
 
 9962 
 
 0872 
 
 9924 
 
 1743 
 
 9886 
 
 2615 
 
 9848 
 
 3486 
 
 9810 
 
 4358 
 
 85 o 
 
 IS 
 
 0.9958 
 
 0.0915 
 
 1.9916 
 
 0.1830 
 
 2.9874 
 
 0.2745 
 
 3.9832 
 
 0.3660 
 
 4.9790 
 
 0.4575 
 
 45 
 
 30 
 
 9954 
 
 0958 
 
 
 1917 
 
 9862 
 
 2875 
 
 9816 
 
 3834 
 
 9770 
 
 4792 
 
 30 
 
 45 
 
 9950 
 
 1 002 
 
 9899 
 
 2004 
 
 9849 
 
 3006 
 
 9799 
 
 4008 
 
 9748 
 
 
 15 
 
 6 o 
 
 9945 
 
 1045 
 
 9890 
 
 2091 
 
 9836 
 
 3136 
 
 978i 
 
 4181 
 
 9726 
 
 5226 
 
 84 o 
 
 IS 
 
 9941 
 
 1089 
 
 9881 
 
 2177 
 
 9822 
 
 3266 
 
 9762 
 
 4355 
 
 9703 
 
 5443 
 
 45 
 
 30 
 
 9936 
 
 1132 
 
 9871 
 
 2264 
 
 9807 
 
 3390 
 
 9743 
 
 4528 
 
 9679 
 
 5660 
 
 3o 
 
 45 
 
 9931 
 
 1175 
 
 9861 
 
 2351 
 
 9792 
 
 
 9723 
 
 4701 
 
 9653 
 
 5877 
 
 15 
 
 7 o 
 
 9925 
 
 1219 
 
 9851 
 
 2437 
 
 9776 
 
 3656 
 
 9702 
 
 4875 
 
 9627 
 
 6093 
 
 83 o 
 
 IS 
 
 9920 
 
 1262 
 
 9840 
 
 2524 
 
 9760 
 
 3786 
 
 9680 
 
 5048 
 
 9600 
 
 6310 
 
 45 
 
 30 
 
 9914 
 
 1305 
 
 9829 
 
 2611 
 
 9743 
 
 39i6 
 
 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 
 
 IS 
 
 8 o 
 
 9903 
 
 1392 
 
 9805 
 
 2783 
 
 9708 
 
 4175 
 
 9611 
 
 5567 
 
 9513 
 
 6959 
 
 82 o 
 
 IS 
 
 9897 
 
 1435 
 
 9793 
 
 2870 
 
 9690 
 
 4305 
 
 9586 
 
 5740 
 
 9483 
 
 7175 
 
 45 
 
 30 
 
 
 1478 
 
 978o 
 
 2956 
 
 9670 
 
 4434 
 
 956i 
 
 5912 
 
 9451 
 
 7390 
 
 30 
 
 45 
 
 9884 
 
 1521 
 
 9767 
 
 3042 
 
 9651 
 
 4564 
 
 9534 
 
 6085 
 
 9418 
 
 7606 
 
 IS 
 
 9 o 
 
 9877 
 
 1564 
 
 9754 
 
 3129 
 
 9631 
 
 4693 
 
 9508 
 
 6257 
 
 9384 
 
 7822 
 
 81 o 
 
 IS 
 
 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 
 
 971 1 
 
 3387 
 
 9567 
 
 5080 
 
 9422 
 
 6774 
 
 9278 
 
 8467 
 
 15 
 
 10 o 
 
 9848 
 
 1736 
 
 9696 
 
 3473 
 
 9544 
 
 5209 
 
 9392 
 
 6946 
 
 9240 
 
 8682 
 
 80 o 
 
 IS 
 
 0.9840 
 
 0.1779 
 
 1.9681 
 
 0.3559 
 
 2.9521 
 
 o . 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 
 
 IS 
 
 II 
 
 9816 
 
 1908 
 
 9633 
 
 3816 
 
 9449 
 
 5724 
 
 9265 
 
 7632 
 
 9081 
 
 9540 
 
 79 o 
 
 IS 
 
 9808 
 
 1951 
 
 9616 
 
 3902 
 
 9424 
 
 5853 
 
 9231 
 
 7804 
 
 939 
 
 9755 
 
 45 
 
 30 
 45 
 
 9799 
 9790 
 
 1994 
 2036 
 
 9598 
 958i 
 
 3987 
 
 4073 
 
 9398 
 9371 
 
 598i 
 6109 
 
 9197 
 9162 
 
 7975 
 8146 
 
 8952 
 
 9968 
 1.0182 
 
 30 
 IS 
 
 12 O 
 
 978i 
 
 2079 
 
 9563 
 
 4158 
 
 9344 
 
 6237 
 
 9126 
 
 8316 
 
 8907 
 
 0396 
 
 78 o 
 
 is 
 
 9772 
 
 2122 
 
 9545 
 
 4244 
 
 9317 
 
 6365 
 
 9089 
 
 848? 
 
 8862 
 
 0609 
 
 45 
 
 30 
 
 9763 
 
 2l64 
 
 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 
 
 I 1035 
 
 IS 
 
 13 
 
 9744 
 
 2250 
 
 9487 
 
 4499 
 
 9231 
 
 6749 
 
 8975 
 
 8998 
 
 8719 
 
 1248 
 
 77 o 
 
 IS 
 
 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 o 
 
 15 
 
 
 2462 
 
 9385 
 
 4923 
 
 9077 
 
 7385 
 
 8769 
 
 9846 
 
 8462 
 
 2308 
 
 45 
 
 30 
 
 9681 
 
 2504 
 
 9363 
 
 5008 
 
 9044 
 
 75" 
 
 8726 
 
 1.0015 
 
 8407 
 
 2519 
 
 30 
 
 45 
 
 9670 
 
 2546 
 
 9341 
 
 5092 
 
 9011 
 
 7638 
 
 8682 
 
 0184 
 
 8352 
 
 2730 
 
 IS 
 
 15 o 
 
 9659 
 
 2588 
 
 9319 
 
 5176 
 
 8978 
 
 7765 
 
 8637 
 
 0353 
 
 8296 
 
 2941 
 
 75 o 
 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. Lat. 
 
 Dep. 
 
 Lat. 
 
 Course. 
 
 
 Dist. I. 
 
 Dist. 2. 
 
 Dist. 3. 
 
 Dist. 4. 
 
 Dist. 5. 
 
 
COMPASS SURVEYING 
 
 TRAVERSE TABLE (Continued) 
 
 
 Dist. 6. 
 
 Dist. 7. 
 
 Dist. 8. 
 
 Dist. 9- 
 
 Dist. 10. 
 
 
 Course 
 
 
 
 
 
 
 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 
 o IS 
 
 5-9999 
 
 0.0262 
 
 6.9999 
 
 0.0305 
 
 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 
 
 9995 
 
 0785 
 
 9994 
 
 0916 
 
 9993 
 
 1047 
 
 9992 
 
 1178 
 
 9991 
 
 1309 
 
 IS 
 
 I 
 
 9991 
 
 1047! 
 
 9989 
 
 1222 
 
 9988 
 
 1396 
 
 9986 
 
 IS7I 
 
 9985 
 
 1745 
 
 89 o 
 
 IS 
 30 
 
 9986 
 9979 
 
 1309 
 1571 1 
 
 
 1527 
 1832 
 
 998i 
 9973 
 
 1745 
 
 2094 
 
 9979 
 9969 
 
 1963 
 2356 
 
 9966 
 
 2181 
 2618 
 
 45 
 30 
 
 45 
 
 9972 
 
 1832 
 
 9967 
 
 2138 
 
 9963 
 
 2443 
 
 9958 
 
 2748 
 
 9953 
 
 3054 
 
 IS 
 
 2 
 
 9963 
 
 2094 
 
 9957 
 
 2443 
 
 9951 
 
 2792 
 
 9945 
 
 3i4i 
 
 9939 
 
 3490 
 
 88 o 
 
 IS 
 
 9954 
 
 2356 
 
 9946 
 
 2748 
 
 9938 
 
 3141 
 
 9931 
 
 3533 
 
 9923 
 
 3926 
 
 45 
 
 30 
 
 9943 
 
 2617! 
 
 9933 
 
 3053 
 
 9924 
 
 3490 
 
 9914 
 
 3926| 
 
 9905 
 
 4362 
 
 30 
 
 45 
 
 5-9931 
 
 0.2879 
 
 6.9919 
 
 0.3358 
 
 7.9908 
 
 0.3838 
 
 8.9896 
 
 0.4318 
 
 9 9885 
 
 0.4798 
 
 15 
 
 3 
 
 9918 
 
 3140 
 
 9904 
 
 3664 
 
 9890 
 
 4187 
 
 9877 
 
 4710 
 
 '9863 
 
 5234 
 
 87 o 
 
 IS 
 
 9904 
 
 3402 
 
 9887 
 
 3968 
 
 9871 
 
 4535 
 
 9855 
 
 5102 
 
 9839 
 
 5669 
 
 45 
 
 30 
 
 9888 
 
 3663 
 
 9869 
 
 4273 
 
 9851 
 
 4884 
 
 9832 
 
 5494 
 
 98i3 
 
 6105 
 
 30 
 
 45 
 
 9872 
 
 3924 
 
 9850 
 
 4578 
 
 9829 
 
 5232 
 
 9807 
 
 5836 
 
 9786 
 
 6540 
 
 IS 
 
 4 o 
 
 9854 
 
 4185 
 
 9829 
 
 4883 
 
 9805 
 
 5S8i 
 
 978i 
 
 6278 
 
 9756 
 
 6976 
 
 86 o 
 
 IS 
 
 9835 
 
 4447 
 
 9808 
 
 5188 
 
 978o 
 
 5929 
 
 9753 
 
 6670 
 
 9725 
 
 7411 
 
 45 
 
 30 
 
 98i5 
 
 4708 
 
 9784 
 
 5492 
 
 9753 
 
 6277 
 
 9723 
 
 7061 
 
 9692 
 
 7846 
 
 30 
 
 45 
 
 9794 
 
 4968 
 
 9760 
 
 5797 
 
 9725 
 
 6625 
 
 9691 
 
 7453 
 
 9657 
 
 8281 
 
 IS 
 
 5 
 
 9772 
 
 5229' 
 
 9734 
 
 6101 
 
 9696 
 
 6972 
 
 9658 
 
 7844 
 
 96i9 
 
 8716 
 
 85 o 
 
 IS 
 
 5.9748 
 
 0.5490' 
 
 6.9706 
 
 0.6405 
 
 7.9664 
 
 0.7320 
 
 8.9622 
 
 0.8235 
 
 9.958o 
 
 0.9150 
 
 45 
 
 30 
 
 9724 
 
 5751 
 
 9678 
 
 6709 
 
 9632 
 
 7668 
 
 9586 
 
 8626 
 
 9540 
 
 9585 
 
 30 
 
 45 
 
 9698 
 
 6011 
 
 9648 
 
 7013 
 
 9597 
 
 8015 
 
 9547 
 
 9017 
 
 9497 
 
 1.0019 
 
 IS 
 
 6 o 
 
 9671 
 
 6272 
 
 96i7 
 
 7317 
 
 9562 
 
 8362 
 
 9507 
 
 9408 
 
 9452 
 
 0453 
 
 84 o 
 
 IS 
 
 9643 
 
 6532 
 
 9584 
 
 7621 
 
 9525 
 
 8709 
 
 9465 
 
 9798 
 
 9406 
 
 '0887 
 
 45 
 
 30 
 
 9614 
 
 6792 
 
 9550 
 
 7924 
 
 9486 
 
 9056 
 
 9421 
 
 1.0188 
 
 9357 
 
 1320 
 
 30 
 
 45 
 
 9584 
 
 7052 
 
 9515 
 
 8228 
 
 9445 
 
 9403 
 
 9376 
 
 0578 
 
 9307 
 
 1754 
 
 IS 
 
 7 o 
 
 9553 
 
 7312 
 
 9478 
 
 8531 
 
 9404 
 
 9750 
 
 9329 
 
 0968 
 
 9255 
 
 2187 
 
 83 o 
 
 IS 
 
 9520 
 
 7572 
 
 9440 
 
 8834 
 
 936o 
 
 1.0096 
 
 9280 
 
 1358 
 
 9200 
 
 2620 
 
 45 
 
 30 
 
 9487 
 
 7832 
 
 9401 
 
 9137 
 
 9316 
 
 0442 
 
 9230 
 
 1747 
 
 9144 
 
 3053 
 
 30 
 
 45 
 
 5-9452 
 
 0.8091 
 
 6.9361 
 
 0.9440 
 
 7.9269 
 
 1.0788 
 
 8.9178 
 
 I. 2137 
 
 9-9087 
 
 1.3485 
 
 15 
 
 8 o 
 
 9416 
 
 8350 
 
 9319 
 
 9742 
 
 9221 
 
 "34 
 
 9124 
 
 2526 
 
 9027 
 
 3917 
 
 82 o 
 
 IS 
 
 9379 
 
 8610 
 
 9276 
 
 1.0044 
 
 9172 
 
 1479 
 
 9069 
 
 2914 
 
 8965 
 
 4349 
 
 45 
 
 3 
 
 9341 
 
 8869 
 
 9231 
 
 0347 
 
 9121 
 
 1825 
 
 9011 
 
 3303 
 
 8902 
 
 478i 
 
 30 
 
 45 
 
 9302 
 
 9127 
 
 9185 
 
 0649 
 
 9069 
 
 2170 
 
 8953 
 
 3691 
 
 8836 
 
 5212 
 
 IS 
 
 9 o 
 
 9261 
 
 9386 
 
 9138 
 
 0950 
 
 9015 
 
 2515 
 
 8892 
 
 4079 
 
 8769 
 
 5643 
 
 8l o 
 
 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 
 
 8989 
 
 1854 
 
 8844 
 
 3548 
 
 8700 
 
 5241 
 
 8556 
 
 6935 
 
 IS 
 
 10 o 
 
 9088 
 
 0419 
 
 8937 
 
 2155 
 
 8785 
 
 3892 
 
 8633 
 
 5628 
 
 8481 
 
 7365 
 
 80 
 
 is 
 
 5.9042 
 
 1.0677 
 
 6.8883 
 
 1.2456 
 
 7-8723 
 
 1.4235 
 
 8.8564 
 
 1.6015 
 
 9.8404 
 
 1-7794 
 
 45 
 
 30 
 
 8995 
 
 0943 
 
 8828 
 
 2756 
 
 8660 
 
 4579 
 
 8493 
 
 6401 
 
 8325 
 
 8224 
 
 30 
 
 45 
 
 8947 
 
 1191 
 
 8772 
 
 3057 
 
 8596 
 
 4922 
 
 8421 
 
 6787 
 
 8245 
 
 8652 
 
 IS 
 
 II 
 
 8898 
 
 1449 
 
 8714 
 
 3357 
 
 8530 
 
 5265 
 
 8346 
 
 7173 
 
 8163 
 
 9081 
 
 79 o 
 
 IS 
 
 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 
 
 4255 
 
 8324 
 
 6291 
 
 8114 
 
 8328 
 
 7905 
 
 2.0364 
 
 IS 
 
 12 O 
 
 8689 
 
 2475 
 
 8470 
 
 4554 
 
 8252 
 
 6633 
 
 8033 
 
 8712 
 
 7815 
 
 0791 
 
 78 o 
 
 IS 
 
 8634 
 
 2731 
 
 8406 
 
 4852 
 
 8178 
 
 6974 
 
 7951 
 
 9096 
 
 7723 
 
 1218 
 
 45 
 
 30 
 
 8578 
 
 2986 
 
 8341 
 
 5151 
 
 810^ 
 
 7315 
 
 7867 
 
 9480 
 
 7630 
 
 1644 
 
 30 
 
 45 
 
 5.8521 
 
 1.3242 
 
 6.8274 
 
 1.5449 
 
 7.8027 
 
 1.7656 
 
 8.7781 
 
 1.9863 
 
 9-7534 
 
 2 . 2O70 
 
 IS 
 
 13 o 
 
 8462 
 
 3497 
 
 8206 
 
 5747 
 
 7950 
 
 7996 
 
 7693 
 
 2.0246 
 
 7437 
 
 2495 
 
 77 o 
 
 IS 
 
 8403 
 
 3752 
 
 8i37 
 
 6044 
 
 7870 
 
 8336 
 
 7604 
 
 0628 
 
 7338 
 
 2920 
 
 45 
 
 30 
 
 8342 
 
 4007 
 
 8066 
 
 6341 
 
 7790 
 
 8676 
 
 7513 
 
 IOIO 
 
 7237 
 
 3345 
 
 30 
 
 45 
 
 8281 
 
 4261 
 
 7994 
 
 6638 
 
 7707 
 
 9015 
 
 7421 
 
 1392 
 
 7134 
 
 3769 
 
 IS 
 
 14 o 
 
 8218 
 
 4515 
 
 7921 
 
 6935 
 
 7624 
 
 9354 
 
 7327 
 
 1773 
 
 7030 
 
 4192 
 
 76 o 
 
 IS 
 
 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 
 
 IS o 
 
 7956! 5529 
 
 7615 
 
 8117 
 
 7274 
 
 0706 
 
 6933 
 
 3294 
 
 6593 
 
 5882 
 
 75 o 
 
 
 Dep. Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 
 
 Dist. 6. 
 
 Dist. 7. 
 
 Dist. 8. 
 
 Dist. 9. 
 
 Dist. ic. 
 
 Course. 
 
144 
 
 PRACTICAL SURVEYING 
 
 TRAVERSE TABLE (Continued) 
 
 Course. 
 
 Dist. I. 
 
 Dist. 2. 
 
 Dist. 3. 
 
 Dist. 4. 
 
 Dist. 5. 
 
 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 
 IS IS 
 
 0.9648 
 
 0.2630 
 
 1.9296 
 
 0.5261 
 
 2.8944 
 
 0.7891 
 
 3.8591 
 
 1.0521 
 
 4-8239 
 
 1.3152 
 
 74 45 
 
 30 
 
 9636 
 
 2672 
 
 9273 
 
 5345 
 
 8909 
 
 8017 
 
 8545 
 
 0690 
 
 8182 
 
 3362 
 
 30 
 
 45 
 
 9625 
 
 2714 
 
 9249 
 
 5429 
 
 8874 
 
 8i43 
 
 8498 
 
 0858 
 
 8123 
 
 3572 
 
 15 
 
 16 o 
 
 96i3 
 
 2756 
 
 9225 
 
 5513 
 
 8838 
 
 8269 
 
 8450 
 
 1025 
 
 8063 
 
 3782 
 
 74 o 
 
 IS 
 
 9600 
 
 2798 
 
 9201 
 
 5597 
 
 8801 
 
 8395 
 
 8402 
 
 H93 
 
 8002 
 
 3991 
 
 45 
 
 30 
 
 9588 
 
 2840 
 
 9176 
 
 5680 
 
 8765 
 
 8520 
 
 8353 
 
 1361 
 
 794i 
 
 4201 
 
 30 
 
 45 
 
 9576 
 
 2882 
 
 9i5i 
 
 5764 
 
 8727 
 
 8646 
 
 8303 
 
 1528 
 
 7879 
 
 4410 
 
 15 
 
 17 o 
 
 9563 
 
 2924 
 
 9126 
 
 5847 
 
 8689 
 
 877i 
 
 8252 
 
 1695 
 
 7815 
 
 4619 
 
 73 o 
 
 IS 
 
 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 
 
 1.9048 
 
 0.6097 
 
 2.8572 
 
 0.9146 
 
 3.8096 
 
 1.2195 
 
 4.7620 
 
 I.S243 
 
 IS 
 
 18 o 
 
 9Si i 
 
 3090 
 
 9021 
 
 6180 
 
 8532 
 
 9271 
 
 8042 
 
 2361 
 
 7553 
 
 5451 
 
 72 
 
 IS 
 
 9497 
 
 3132 
 
 8994 
 
 6263 
 
 8491 
 
 9395 
 
 7988 
 
 2527 
 
 7485 
 
 5658 
 
 45 
 
 30 
 
 9483 
 
 3173 
 
 8966 
 
 6346 
 
 8450 
 
 9519 
 
 7933 
 
 2692 
 
 74i6 
 
 586s 
 
 30 
 
 45 
 
 9469 
 
 3214 
 
 8939 
 
 6429 
 
 8408 
 
 9643 
 
 7877 
 
 2858 
 
 7347 
 
 6072 
 
 15 
 
 19 o 
 
 9455 
 
 3256 
 
 8910 
 
 6511 
 
 8366 
 
 9767 
 
 7821 
 
 3023 
 
 7276 
 
 6278 
 
 71 o 
 
 IS 
 
 9441 
 
 3297 
 
 8882 
 
 6594 
 
 8323 
 
 9891 
 
 7764 
 
 3188 
 
 7204 
 
 6485 
 
 45 
 
 30 
 
 9426 
 
 3338 
 
 8853 
 
 6676 
 
 8279 
 
 1.0014 
 
 7706 
 
 3352 
 
 7132 
 
 6690 
 
 30 
 
 45 
 
 94" 
 
 3379 
 
 8824 
 
 6758 
 
 8235 
 
 0138 
 
 7647 
 
 3517 
 
 7059 
 
 6896 
 
 IS 
 
 20 
 IS 
 
 9397 
 0.9382 
 
 3420 
 0.3461 
 
 8794 
 1.8764 
 
 6840 
 0.6922 
 
 8191 
 2.8146 
 
 0261 
 
 7588 
 3.7528 
 
 3681 
 1.3845 
 
 6985 
 4.6910 
 
 7101 
 1.7306 
 
 70 o 
 45 
 
 30 
 
 9367 
 
 3502 
 
 8733 
 
 7004 
 
 8100 
 
 0506 
 
 7467 
 
 4008 
 
 6834 
 
 75io 
 
 30 
 
 45 
 
 9351 
 
 3543 
 
 8703 
 
 7086 
 
 8054 
 
 0629 
 
 7405 
 
 4172 
 
 6757 
 
 7715 
 
 IS 
 
 21 
 
 9336 
 
 3584 
 
 8672 
 
 7167 
 
 8007 
 
 0751 
 
 7343 
 
 4335 
 
 6679 
 
 7918 
 
 69 o 
 
 15 
 
 9320 
 
 3624 
 
 8640 
 
 7249 
 
 796o 
 
 0873 
 
 7280 
 
 4498 
 
 6600 
 
 8122 
 
 45 
 
 30 
 
 9304 
 
 3665 
 
 8608 
 
 7330 
 
 7913 
 
 0995 
 
 7217 
 
 4660 
 
 6521 
 
 8325 
 
 30 
 
 45 
 
 9288 
 
 3706 
 
 8576 
 
 74" 
 
 7864 
 
 1117 
 
 7152 
 
 4822 
 
 6440 
 
 8528 
 
 IS 
 
 22 
 
 9272 
 
 3746 
 
 8544 
 
 7492 
 
 7816 
 
 1238 
 
 7087 
 
 4984 
 
 6359 
 
 8730 
 
 68 o 
 
 IS 
 
 9255 
 
 3786 
 
 8511 
 
 7573 
 
 7766 
 
 1359 
 
 7022 
 
 5146 
 
 6277 
 
 8932 
 
 45 
 
 30 
 45 
 
 9239 
 0.9222 
 
 3827 
 0.3867 
 
 8478 
 1.8444 
 
 7654 
 0.7734 
 
 77i6 
 2.7666 
 
 1481 
 1.1601 
 
 
 5307 
 1.5468 
 
 6194 
 4.6110 
 
 9134 
 1.9336 
 
 30 
 15 
 
 23 o 
 
 9205 
 
 3907 
 
 8410 
 
 7815 
 
 7615 
 
 1722 
 
 6820 
 
 5629 
 
 6025 
 
 9537 
 
 67 o 
 
 IS 
 
 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 
 
 5766 
 
 2.0137 
 
 IS 
 
 24 o 
 
 9135 
 
 4067 
 
 8271 
 
 8i35 
 
 7406 
 
 22O2 
 
 6542 
 
 6269 
 
 5677 
 
 0337 
 
 66 o 
 
 IS 
 
 9118 
 
 4107 
 
 8235 
 
 8214 
 
 7353 
 
 2322 
 
 6470 
 
 6429 
 
 5588 
 
 0536 
 
 45 
 
 30 
 
 9100 
 
 4147 
 
 8i99 
 
 8294 
 
 7299 
 
 2441 
 
 6398 
 
 6588 
 
 5498 
 
 0735 
 
 30 
 
 45 
 
 9081 
 
 4187 
 
 8163 
 
 8373 
 
 7244 
 
 2560 
 
 6326 
 
 6746 
 
 5407 
 
 0933 
 
 IS 
 
 25 
 
 9063 
 
 4226 
 
 8126 
 
 8452 
 
 7i89 
 
 2679 
 
 6252 
 
 6905 
 
 5315 
 
 1131 
 
 65 o 
 
 IS 
 
 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 
 
 IS 
 
 26 o 
 
 8988 
 
 4384 
 
 7976 
 
 8767 
 
 6964 
 
 3151 
 
 5952 
 
 7535 
 
 4940 
 
 1919 
 
 64 o 
 
 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 
 
 35O3 
 
 5719 
 
 8004 
 
 4649 
 
 2505 
 
 IS 
 
 27 o 
 
 8910 
 
 4540 
 
 7820 
 
 9080 
 
 6730 
 
 3620 
 
 5640 
 
 8160 
 
 4550 
 
 2700 
 
 63 o 
 
 15 
 
 8890 
 
 4579 
 
 7780 
 
 9157 
 
 6671 
 
 3736 
 
 556i 
 
 8315 
 
 4451 
 
 2894 
 
 45 
 
 30 
 
 8870 
 
 4617 
 
 7740 
 
 9235 
 
 6610 
 
 3852 
 
 548o 
 
 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 
 
 IS 
 
 28 
 
 8829 
 
 4695 
 
 7659 
 
 9389 
 
 6488 
 
 4084 
 
 53i8 
 
 8779 
 
 4147 
 
 3474 
 
 62 o 
 
 IS 
 
 8809 
 
 4733 
 
 7618 
 
 9466 
 
 6427 
 
 4200 
 
 5236 
 
 8933 
 
 4045 
 
 3666 
 
 45 
 
 30 
 45 
 
 8788 
 8767 
 
 4772 
 4810 
 
 7576 
 7535 
 
 9543 
 9620 
 
 6365 
 6302 
 
 4315 
 4430 
 
 5069 
 
 9086 
 9240 
 
 3941 
 3836 
 
 3858 
 4049 
 
 30 
 IS 
 
 29 o 
 
 8746 
 
 4848 
 
 7492 
 
 9696 
 
 6239 
 
 4544 
 
 4985 
 
 9392 
 
 3731 
 
 4240 
 
 61 o 
 
 IS 
 
 8725 
 
 4886 
 
 7450 
 
 9772 
 
 6i75 
 
 4659 
 
 4900 
 
 9545 
 
 3625 
 
 4431 
 
 45 
 
 30 
 
 8704 
 
 4924 
 
 7407 
 
 9848 
 
 6m 
 
 4773 
 
 4814 
 
 9697 
 
 35i8 
 
 4621 
 
 30 
 
 45 
 
 8682 
 
 4962 
 
 7364 
 
 9924 
 
 6046 
 
 4886 
 
 4728 
 
 9849 
 
 34io 
 
 4811 
 
 IS 
 
 30 o 
 
 8660 
 
 5000 
 
 7321 
 
 I.OOOO 
 
 598T 
 
 5000 
 
 4641 
 
 2.OOOO 
 
 33QI 
 
 SOOQ 
 
 60 o 
 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Bourse" 
 
 
 Dist. i. 
 
 Dist. 2. 
 
 Dist. 3. Dist. 4. 
 
 Dist. 5. 
 
 
COMPASS SURVEYING 
 
 145 
 
 TRAVERSE TABLE (Continued) 
 
 Course. 
 
 Dist. 6. 
 
 Dist. 7. 
 
 Dist. 8. 
 
 Dist. 9. 
 
 Dist. 10. 
 
 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 o / 
 
 IS 15 
 
 S.7887 
 
 1.5782 
 
 6.7535 
 
 i .8412 
 
 7.7183 
 
 2 . IO42 
 
 8.6831 
 
 2.3673 
 
 9.6479 
 
 2.6303 
 
 74 45 
 
 30 
 
 7818 
 
 6034 
 
 7454 
 
 8707 
 
 7090 
 
 1379 
 
 6727 
 
 4051 
 
 6363 
 
 6724 
 
 30 
 
 45 
 
 7747 
 
 6286 
 
 7372 
 
 9001 
 
 6996 
 
 1715 
 
 6621 
 
 4430 
 
 6246 
 
 7144 
 
 IS 
 
 16 o 
 
 7676 
 
 6538 
 
 7288 
 
 9295 
 
 6901 
 
 2051 
 
 6514 
 
 4807 
 
 6126 
 
 7564 
 
 74 o 
 
 15 
 
 7603 
 
 6790 
 
 7203 
 
 9588 
 
 6804 
 
 2386 
 
 6404 
 
 5185 
 
 6005 
 
 7983 
 
 45 
 
 30 
 
 7529 
 
 7041 
 
 7117 
 
 9881 
 
 6706 
 
 2721 
 
 6294 
 
 556i 
 
 5882 
 
 8402 
 
 30 
 
 45 
 
 7454 
 
 7292 
 
 7030 
 
 2.0174 
 
 6606 
 
 3056 
 
 6181 
 
 5938 
 
 5757 
 
 8820 
 
 15 
 
 17 o 
 
 7378 
 
 7542 
 
 6941 
 
 0466 
 
 6504 
 
 3390 
 
 6067 
 
 6313 
 
 5630 
 
 9237 
 
 73 o 
 
 IS 
 
 7301 
 
 7792 
 
 6851 
 
 0758 
 
 6402 
 
 3723 
 
 5952 
 
 6689 
 
 5502 
 
 9654 
 
 45 
 
 30 
 
 7223 
 
 8042 
 
 6760 
 
 1049 
 
 6297 
 
 4056 
 
 5835 
 
 7064 
 
 5372 
 
 3.0071 
 
 30 
 
 45 
 
 5-7144 
 
 1.8292 
 
 6.6668 
 
 2.1341 
 
 7.6192 
 
 2.4389 
 
 8.5716 
 
 2.7438 
 
 9.5240 
 
 3.0486 
 
 IS 
 
 18 o 
 
 7063 
 
 8541 
 
 6574 
 
 1631 
 
 6085 
 
 4721 
 
 5S9S 
 
 7812 
 
 5106 
 
 0902 
 
 72 
 
 IS 
 
 
 8790 
 
 6479 
 
 1921 
 
 5976 
 
 5053 
 
 5473 
 
 8185 
 
 4970 
 
 1316 
 
 45 
 
 30 
 
 6899 
 
 9038 
 
 6383 
 
 221 1 
 
 5866 
 
 5384 
 
 5349 
 
 8557 
 
 4832 
 
 1730 
 
 30 
 
 45 
 
 6816 
 
 9286 
 
 6285 
 
 25OI 
 
 5754 
 
 5715 
 
 5224 
 
 8930 
 
 4693 
 
 2144 
 
 IS 
 
 19 o 
 
 6731 
 
 9534 
 
 6186 
 
 2790 
 
 5641 
 
 6045 
 
 5097 
 
 9301 
 
 4552 
 
 2557 
 
 71 
 
 IS 
 
 6645 
 
 978i 
 
 6086 
 
 3078 
 
 5527 
 
 6375 
 
 4968 
 
 9672 
 
 4409 
 
 2969 
 
 45 
 
 30 
 
 6558 
 
 2.0028 
 
 5985 
 
 
 5411 
 
 6705 
 
 4838 
 
 3 0043 
 
 4264 
 
 338i 
 
 30 
 
 45 
 
 6471 
 
 0275 
 
 5882 
 
 3654 
 
 5294 
 
 7033 
 
 4706 
 
 0413 
 
 4118 
 
 3792 
 
 15 
 
 20 
 
 6382 
 
 0521 
 
 5778 
 
 3941 
 
 5175 
 
 7362 
 
 4572 
 
 0782 
 
 3969 
 
 4202 
 
 70 
 
 IS 
 
 5.6291 
 
 2.0767 
 
 6.5673 
 
 2.4228 
 
 7-5055 
 
 2.7689 
 
 8.4437 
 
 3-iiSi 
 
 9.3819 
 
 3.4612 
 
 45 
 
 '30 
 
 6200 
 
 1012 
 
 5567 
 
 4515 
 
 
 -8017 
 
 4300 
 
 1519 
 
 3667 
 
 5021 
 
 30 
 
 45 
 
 6108 
 
 1257 
 
 5459 
 
 4800 
 
 4811 
 
 8343 
 
 4162 
 
 1886 
 
 3514 
 
 5429 
 
 IS 
 
 21 
 
 6015 
 
 1502 
 
 5351 
 
 5086 
 
 4686 
 
 8669 
 
 4022 
 
 2253 
 
 3358 
 
 5837 
 
 69 o 
 
 15 
 
 5920 
 
 1746 
 
 5241 
 
 5371 
 
 456i 
 
 8995 
 
 3881 
 
 2619 
 
 3201 
 
 6244 
 
 45 
 
 30 
 
 5825 
 
 1990 
 
 5129 
 
 5655 
 
 4433 
 
 9320 
 
 3738 
 
 2985 
 
 3042 
 
 6650 
 
 30 
 
 45 
 
 5729 
 
 2233 
 
 5017 
 
 
 4305 
 
 9645 
 
 3593 
 
 33SO 
 
 2881 
 
 7056 
 
 IS 
 
 22 
 
 5631 
 
 2476 
 
 4903 
 
 6222 
 
 4175 
 
 9969 
 
 3447 
 
 3715 
 
 2718 
 
 746i 
 
 68 o 
 
 IS 
 
 5532 
 
 2719 
 
 4788 
 
 6505 
 
 4043 
 
 3.0292 
 
 3299 
 
 4078 
 
 2554 
 
 7865 
 
 45 
 
 30 
 
 5433 
 
 2961 
 
 4672 
 
 6788 
 
 3910 
 
 0615 
 
 3149 
 
 4442 
 
 2388 
 
 8268 
 
 30 
 
 45 
 
 5-5332 
 
 2.3203 
 
 6.4554 
 
 2.7070 
 
 7.3776 
 
 3-0937 
 
 
 3-4804 
 
 9 . 2220 
 
 3.8671 
 
 IS 
 
 23 o 
 IS 
 
 5230 
 SI27 
 
 3444 
 3685 
 
 4435 
 4315 
 
 39 
 
 3640 
 3503 
 
 1258 
 1580 
 
 2691 
 
 Si66 
 5527 
 
 2O50 
 1879 
 
 9073 
 9474 
 
 67 o 
 45 
 
 30 
 
 5024 
 
 3925 
 
 4194 
 
 79" 
 
 3365 
 
 1900 
 
 2535 
 
 S88 7 
 
 1706 
 
 9875 
 
 30 
 
 45 
 
 4919 
 
 4165 
 
 4072 
 
 8192 
 
 3225 
 
 222O 
 
 2378 
 
 6247 
 
 1531 
 
 4.0275 
 
 IS 
 
 24 o 
 
 4813 
 
 4404 
 
 3948 
 
 8472 
 
 3084 
 
 2539 
 
 2219 
 
 6606 
 
 1355 
 
 0674 
 
 66 o 
 
 IS 
 
 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 
 
 93o6 
 
 2651 
 
 3493 
 
 1733 
 
 7679 
 
 0814 
 
 1866 
 
 IS 
 
 25 o 
 
 4378 
 
 5357 
 
 3442 
 
 9583 
 
 2505 
 
 3809 
 
 1568 
 
 8036 
 
 0631 
 
 2202 
 
 65 o 
 
 IS 
 
 5 . 4267 
 
 2.5594 
 
 6.3312 
 
 2.9860 
 
 7.2356 
 
 3.4125 
 
 8.1401 
 
 3.8391 
 
 9.0446 
 
 4-2657 
 
 45 
 
 30 
 
 4IS5 
 
 5831 
 
 3181 
 
 3-0136 
 
 
 4441 
 
 1233 
 
 8746 
 
 0259 
 
 3051 
 
 30 
 
 ' 45 
 
 4042 
 
 6067 
 
 3049 
 
 0411 
 
 2056 
 
 4756 
 
 1063 
 
 9100 
 
 0070 
 
 3445 
 
 IS 
 
 26 o 
 
 3928 
 
 6302 
 
 2916 
 
 0686 
 
 1904 
 
 5070 
 
 0891 
 
 9453 
 
 8.9879 
 
 3837 
 
 64 o 
 
 IS 
 
 3812 
 
 6537 
 
 2781 
 
 0960 
 
 1750 
 
 5383 
 
 0719 
 
 9806 
 
 9687 
 
 4229 
 
 45 
 
 30 
 
 3696 
 
 6772 
 
 2645 
 
 1234 
 
 1595 
 
 5696 
 
 0544 
 
 4-0158 
 
 9493 
 
 4620 
 
 30 
 
 45 
 
 3579 
 
 7006 
 
 2509 
 
 1507 
 
 1438 
 
 6008 
 
 0368 
 
 0509 
 
 9298 
 
 5010 
 
 IS 
 
 27 o 
 
 3460 
 
 7239 
 
 2370 
 
 1779 
 
 1281 
 
 6319 
 
 0191 
 
 o8sQ 
 
 9101 
 
 5399 
 
 63 o 
 
 IS 
 
 3341 
 
 7472 
 
 2231 
 
 2051 
 
 II2I 
 
 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 
 
 7.9649 
 
 4.1905 
 
 8.8499 
 
 4.6561 
 
 IS 
 
 28 o 
 
 2977 
 
 8168 
 
 1806 
 
 
 0636 
 
 7558 
 
 9465 
 
 2252 
 
 8295 
 
 6947 
 
 62 o 
 
 IS 
 
 2853 
 
 8399 
 
 1662 
 
 3132 
 
 0471 
 
 7866 
 
 9280 
 
 2599 
 
 8089 
 
 7332 
 
 45 
 
 30 
 
 2729 
 
 8630 
 
 1517 
 
 3401 
 
 0305 
 
 8173 
 
 9094 
 
 2944 
 
 7882 
 
 7716 
 
 30 
 
 45 
 
 2604 
 
 8859 
 
 1371 
 
 3669 
 
 0138 
 
 8479 
 
 8905 
 
 3289 
 
 7673 
 
 8099 
 
 IS 
 
 29 o 
 
 2477 
 
 9089 
 
 1223 
 
 3937 
 
 6.9970 
 
 8785 
 
 8716 
 
 3633 
 
 7462 
 
 
 61 o 
 
 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 
 
 9282 
 
 4.0000 
 
 7942 
 
 5000 
 
 6603 
 
 5.0000 
 
 60 o 
 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 
 
 Dist. 6. 
 
 Dist. 7. 
 
 Dist. 8. 
 
 Dist. 9. 
 
 Dist. 10. 
 
 L-oursc* 
 
I 4 6 
 
 PRACTICAL SURVEYING 
 
 TRAVERSE TABLE (Continued) 
 
 Course. 
 
 Dist. I. 
 
 Dist. 2. 
 
 Dist. 3. 
 
 Dist. 4. 
 
 Dist. 5. 
 
 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 
 30 is 
 
 0.8638 
 
 0.5038 
 
 1.7277 
 
 1.0075 
 
 2-5915 
 
 i.5ii3 
 
 3- 4553 
 
 2.0151 
 
 4.3192 
 
 2.5189 
 
 59 45 
 
 30 
 
 8616 
 
 5075 
 
 7233 
 
 0151 
 
 5849 
 
 5226 
 
 4465 
 
 0302 
 
 3081 
 
 5377 
 
 30 
 
 45 
 
 8594 
 
 5ii3 
 
 7188 
 
 0226 
 
 5782 
 
 5339 
 
 4376 
 
 0452 
 
 2970 
 
 5565 
 
 IS 
 
 31 o 
 
 8572 
 
 5150 
 
 7142 
 
 0301 
 
 5715 
 
 5451 
 
 4287 
 
 0602 
 
 2858 
 
 5752 
 
 59 o 
 
 IS 
 
 8549 
 
 5188 
 
 7098 
 
 0375 
 
 5647 
 
 5563 
 
 4196 
 
 0751 
 
 2746 
 
 5939 
 
 45 
 
 30 
 
 8526 
 
 5225 
 
 7053 
 
 0450 
 
 5579 
 
 5675 
 
 4106 
 
 0900 
 
 2632 
 
 6125 
 
 30 
 
 45 
 
 8504 
 
 5262 
 
 7007 
 
 0524 
 
 SSii 
 
 5786 
 
 4014 
 
 1049 
 
 2518 
 
 6311 
 
 15 
 
 32 
 
 8480 
 
 5299 
 
 6961 
 
 0598 
 
 5441 
 
 5898 
 
 3922 
 
 H97 
 
 2402 
 
 6496 
 
 58 o 
 
 IS 
 
 8457 
 
 5336 
 
 6915 
 
 0672 
 
 5372 
 
 6008 
 
 3829 
 
 1345 
 
 2286 
 
 6681 
 
 45 
 
 30 
 
 8434 
 
 5373 
 
 6868 
 
 0746 
 
 5302 
 
 6119 
 
 3736 
 
 1492 
 
 2170 
 
 6865 
 
 30 
 
 45 
 33 o 
 
 0.8410 
 8387 
 
 0.5410 
 5446 
 
 1.6821 
 6773 
 
 1.0819 
 0893 
 
 2.5231 
 5160 
 
 1.6229 
 6339 
 
 3.3642 
 3547 
 
 2.1639 
 1786 
 
 4.2052 
 1934 
 
 2.7049 
 7232 
 
 IS 
 57 o 
 
 IS 
 
 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 
 
 mi 
 
 4944 
 
 6667 
 
 3259 
 
 2223 
 
 1573 
 
 7779 
 
 15 
 
 34 o 
 
 8290 
 
 5592 
 
 6581 
 
 1184 
 
 4871 
 
 6776 
 
 3162 
 
 2368 
 
 1452 
 
 7960 
 
 56 o 
 
 IS 
 
 8266 
 
 5628 
 
 6532 
 
 1256 
 
 4798 
 
 6884 
 
 3064 
 
 2512 
 
 1329 
 
 8140 
 
 45 
 
 30 
 
 8241 
 
 566 4 
 
 6483 
 
 1328 
 
 4724 
 
 6992 
 
 2965 
 
 2656 
 
 1206 
 
 8320 
 
 30 
 
 45 
 
 8216 
 
 5700 
 
 6433 
 
 1400 
 
 4649 
 
 7100 
 
 2866 
 
 2800 
 
 1082 
 
 8500 
 
 IS 
 
 35 o 
 
 8192 
 
 5736 
 
 6383 
 
 1472 
 
 4575 
 
 7207 
 
 2766 
 
 2943 
 
 0958 
 
 8679 
 
 55 o 
 
 15 
 30 
 
 0.8166 
 8141 
 
 0-5771 
 5807 
 
 
 I . 1543 
 1614 
 
 2.4499 
 4423 
 
 I.73I4 
 7421 
 
 3.2666 
 2565 
 
 2.3086 
 . 3228 
 
 4.0832 
 0706 
 
 2.8857 
 9035 
 
 45 
 30 
 
 45 
 
 8116 
 
 5842 
 
 6231 
 
 1685 
 
 4347 
 
 7527 
 
 2463 
 
 3370 
 
 0579 
 
 9212 
 
 15 
 
 36 o 
 
 8090 
 
 5878 
 
 6180 
 
 1756 
 
 4271 
 
 7634 
 
 2361 
 
 35" 
 
 0451 
 
 9389 
 
 54 o 
 
 15 
 
 8064 
 
 5913 
 
 6129 
 
 1826 
 
 4193 
 
 7739 
 
 2258 
 
 3652 
 
 0322 
 
 9565 
 
 45 
 
 30 
 
 8039 
 
 5948 
 
 6077 
 
 1896 
 
 4116 
 
 7845 
 
 2154 
 
 3793 
 
 oi93 
 
 9741 
 
 30 
 
 45 
 
 8013 
 
 5983 
 
 6025 
 
 1966 
 
 4038 
 
 7950 
 
 2050 
 
 3933 
 
 0063 
 
 9916 
 
 IS 
 
 37 o 
 
 7986 
 
 6018 
 
 5973 
 
 2036 
 
 
 8054 
 
 1945 
 
 4073 
 
 3-9932 
 
 3.0091 
 
 S3 o 
 
 15 
 
 7960 
 
 6053 
 
 5920 
 
 2106 
 
 3880 
 
 8i59 
 
 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 
 
 1.8367 
 
 3.1628 
 
 2.4489 
 
 3-9534 
 
 3.o6n 
 
 IS 
 
 38 o 
 
 7880 
 
 6i57 
 
 576o 
 
 2313 
 
 3640 
 
 8470 
 
 1520 
 
 4626 
 
 9400 
 
 0783 
 
 52 
 
 IS 
 
 7853 
 
 6191 
 
 57o6 
 
 2382 
 
 356o 
 
 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 
 
 "95 
 
 5037 
 
 8994 
 
 1296 
 
 IS 
 
 39 o 
 
 7771 
 
 6293 
 
 5543 
 
 2586 
 
 3314 
 
 8880 
 
 1086 
 
 5173 
 
 8857 
 
 1466 
 
 Si o 
 
 IS 
 
 7744 
 
 6327 
 
 5488 
 
 2654 
 
 3232 
 
 8981 
 
 0976 
 
 5308 
 
 8720 
 
 1635 
 
 45 
 
 30 
 
 77i6 
 
 6361 
 
 5432 
 
 2722 
 
 3149 
 
 9082 
 
 0865 
 
 5443 
 
 8581 
 
 1804 
 
 30 
 
 45 
 
 7688 
 
 6394 
 
 5377 
 
 2789 
 
 3065 
 
 9i83 
 
 0754 
 
 5578 
 
 8442 
 
 1972 
 
 IS 
 
 40 o 
 
 7660 
 
 642$ 
 
 5321 
 
 2856 
 
 2981 
 
 9284 
 
 0642 
 
 5712 
 
 8302 
 
 2139 
 
 SO 
 
 15 
 
 0.7632 
 
 0.6461 
 
 1.5265 
 
 1.2922 
 
 2.2897 
 
 1.9384 
 
 3-0529 
 
 2.5845 
 
 3.8162 
 
 3.2306 
 
 45 
 
 30 
 45 
 
 7604 
 7576 
 
 6494 
 6528 
 
 5208 
 SISI 
 
 2989 
 3055 
 
 2812 
 2727 
 
 9483 
 9583 
 
 0416 
 0303 
 
 5978 
 6110 
 
 8020 
 7878 
 
 2638 
 
 30 
 IS 
 
 41 o 
 
 7547 
 
 6561 
 
 5094 
 
 3121 
 
 2641 
 
 9682 
 
 0188 
 
 6242 
 
 7735 
 
 2803 
 
 49 o 
 
 IS 
 
 75i8 
 
 6593 
 
 5037 
 
 3187 
 
 2555 
 
 978o 
 
 0074 
 
 6374 
 
 7592 
 
 2967 
 
 45 
 
 30 
 
 7490 
 
 6626 
 
 4979 
 
 3252 
 
 2469 
 
 9879 
 
 2.9958 
 
 6505 
 
 7448 
 
 3131 
 
 30 
 
 45 
 
 746i 
 
 6659 
 
 4921 
 
 3318 
 
 2382 
 
 9976 
 
 9842 
 
 6635 
 
 7303 
 
 3294 
 
 15 
 
 42 o 
 
 7431 
 
 6691 
 
 4863 
 
 3383 
 
 2294 
 
 2.0074 
 
 9726 
 
 6765 
 
 7157 
 
 3457 
 
 48 o 
 
 15 
 
 7402 
 
 6724 
 
 4804 
 
 3447 
 
 2207 
 
 0171 
 
 9609 
 
 6895 
 
 7011 
 
 3618 
 
 45 
 
 30 
 
 7373 
 
 6756 
 
 4746 
 
 3512 
 
 2118 
 
 0268 
 
 9491 
 
 7024 
 
 6864 
 
 378o 
 
 30 
 
 45 
 
 0.7343 
 
 0.6788 
 
 1.4686 
 
 1.3576 
 
 2.2030 
 
 2.0364 
 
 2.9373 
 
 2.7152 
 
 3.6716 
 
 3-3940 
 
 IS 
 
 43 o 
 
 7314 
 
 6820 
 
 4627 
 
 3640 
 
 1941 
 
 0460 
 
 9254 
 
 7280 
 
 6568 
 
 4100 
 
 47 o 
 
 IS 
 
 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 
 
 IS 
 
 44 o 
 
 7193 
 
 6947 
 
 4387 
 
 3893 
 
 1580 
 
 0840 
 
 8774 
 
 7786 
 
 5967 
 
 4733 
 
 46 o 
 
 IS 
 
 7163 
 
 6978 
 
 4326 
 
 3956 
 
 1489 
 
 0934 
 
 8652 
 
 7912 
 
 58i5 
 
 4890 
 
 45 
 
 30 
 
 7133 
 
 7009 
 
 4265 
 
 4018 
 
 1398 
 
 1027 
 
 8530 
 
 8036 
 
 566 3 
 
 5045 
 
 30 
 
 45 
 
 7102 
 
 7040 
 
 4204 
 
 4080 
 
 1306 
 
 1 120 
 
 8407 
 
 8161 
 
 5509 
 
 5201 
 
 IS 
 
 45 o 
 
 7071 
 
 7071 
 
 4142 
 
 4142 
 
 1213 
 
 1213 
 
 8284 
 
 8284 
 
 5355 
 
 5355 
 
 45 
 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 C-/OtU"3C. 
 
 
 Dist. I. 
 
 Dist. 2. 
 
 Dist. 3. 
 
 Dist. 4. 
 
 Dist. 5. 
 
 
COMPASS SURVEYING 
 
 147 
 
 TRAVERSE TABLE (Continued) 
 
 < r V*iir 
 
 Dist. 6. 
 
 Dist. 7. 
 
 Dist. 8. 
 
 Dist. 9. 
 
 Dist. 10. 
 
 
 V^OuTSc. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 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 
 
 59 45 
 
 30 
 
 
 0452 
 
 0314 
 
 5528 
 
 8930 
 
 0603 
 
 7547 
 
 5678 
 
 6163 
 
 0754 
 
 30 
 
 45 
 
 1564 
 
 0678 
 
 0158 
 
 5791 
 
 8753 
 
 0903 
 
 7347 
 
 6016 
 
 5941 
 
 1129 
 
 IS 
 
 31 o 
 
 1430 
 
 0902 
 
 0002 
 
 6053 
 
 8573 
 
 1203 
 
 7145 
 
 6353 
 
 5717 
 
 1504 
 
 59 o 
 
 15 
 
 1295 
 
 1126 
 
 5.9844 
 
 6314 
 
 8393 
 
 1502 
 
 6942 
 
 6690 
 
 5491 
 
 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 
 
 IS 
 
 32 o 
 
 0883 
 
 1795 
 
 9363 
 
 7094 
 
 7844 
 
 2394 
 
 6324 
 
 7693 
 
 4805 
 
 2992 
 
 58 o 
 
 15 
 
 0744 
 
 2017 
 
 9201 
 
 7353 
 
 7658 
 
 2689 
 
 6116 
 
 8025 
 
 4573 
 
 336i 
 
 45 
 
 30 
 
 0603 
 
 2238 
 
 9037 
 
 7611 
 
 7471 
 
 2984 
 
 5905 
 
 8357 
 
 4339 
 
 3730 
 
 30 
 
 45 
 
 5.0462 
 
 3.2458 
 
 5.8S73 
 
 3.7868 
 
 6.7283 
 
 4.3278 
 
 7.5694 
 
 4.8688 
 
 8.4104 
 
 5.4097 
 
 IS 
 
 33 o 
 
 0320 
 
 2678 
 
 8707 
 
 8125 
 
 7094 
 
 3571 
 
 548o 
 
 9018 
 
 3867 
 
 4464 
 
 57 
 
 15 
 
 0177 
 
 2898 
 
 i 8540 
 
 8381 
 
 6903 
 
 386 3 
 
 5266 
 
 9346 
 
 3629 
 
 4829 
 
 45 
 
 30 
 
 0033 
 
 3116 
 
 ! 8372 
 
 8636 
 
 6711 
 
 4155 
 
 5050 
 
 9674 
 
 3389 
 
 5194 
 
 3 
 
 45 
 
 4.9888 
 
 3334 
 
 8203 
 
 8890 
 
 6518 
 
 4446 
 
 4832 
 
 S.oooi 
 
 3147 
 
 5557 
 
 IS 
 
 34 o 
 
 9742 
 
 3552 
 
 8033 
 
 9144 
 
 6323 
 
 4735 
 
 4613 
 
 0327 
 
 2904 
 
 5919 
 
 56 o 
 
 15 
 
 9595 
 
 3768 
 
 ! 786i 
 
 
 6127 
 
 5024 
 
 4393 
 
 0652 
 
 2659 
 
 6280 
 
 45 
 
 30 
 
 9448 
 
 3984 
 
 l 7689 
 
 9648 
 
 5930 
 
 5312 
 
 4171 
 
 0977 
 
 2413 
 
 6641 
 
 30 
 
 45 
 
 9299 
 
 4200 
 
 I 7515 
 
 9900 
 
 5732 
 
 5600 
 
 3948 
 
 1300 
 
 2165 
 
 7000 
 
 IS 
 
 35 o 
 
 9149 
 
 4415 
 
 734i 
 
 4.0150 
 
 5532 
 
 5886 
 
 3724 
 
 1622 
 
 1915 
 
 7358 
 
 55 o 
 
 IS 
 
 4.8998 
 
 3.4629 
 
 5.7165 
 
 4.0400 
 
 6.5331 
 
 4.6172 
 
 7.3498 
 
 S.I943 
 
 8.1664 
 
 5-77IS 
 
 45 
 
 30 
 
 8847 
 
 4842 
 
 6988 
 
 0649 
 
 5129 
 
 6456 
 
 3270 
 
 2263 
 
 1412 
 
 8070 
 
 30 
 
 45 
 
 8694 
 
 5055 
 
 6810 
 
 0897 
 
 4926 
 
 6740 
 
 3042 
 
 2582 
 
 1157 
 
 8425 
 
 IS 
 
 36 o 
 
 8541 
 
 5267 
 
 6631 
 
 1145 
 
 4721 
 
 7023 
 
 2812 
 
 2901 
 
 0902 
 
 8779 
 
 54 o 
 
 IS 
 
 8387 
 
 5479 
 
 6451 
 
 
 45i6 
 
 
 2580 
 
 3218 
 
 0644 
 
 9UI 
 
 45 
 
 3 
 
 8231 
 
 5689 
 
 6270 
 
 1638 
 
 4309 
 
 7586 
 
 2347 
 
 3534 
 
 0386 
 
 9482 
 
 3 
 
 45 
 
 8075 
 
 5899 
 
 6088 
 
 1883 
 
 4100 
 
 7866 
 
 2113 
 
 3849 
 
 0125 
 
 9832 
 
 IS 
 
 37 o 
 
 7918 
 
 6109 
 
 5904 
 
 2127 
 
 3891 
 
 8I4S 
 
 1877 
 
 4163 
 
 7.9864 
 
 6.0182 
 
 53 o 
 
 IS 
 
 7760 
 
 6318 
 
 5720 
 
 2371 
 
 3680 
 
 8424 
 
 1640 
 
 4476 
 
 9600 
 
 0529 
 
 45 
 
 30 
 
 7601 
 
 6526 
 
 5535 
 
 2613 
 
 3468 
 
 8701 
 
 1402 
 
 4789 
 
 9335 
 
 0876 
 
 30 
 
 45 
 
 4-7441 
 
 3.6733 
 
 5.5348 
 
 4.2855 
 
 6-3255 
 
 4.8977 
 
 7.1162 
 
 5.5100 
 
 7.9069 
 
 6.1222 
 
 IS 
 
 38 o 
 
 7281 
 
 6940 
 
 5161 
 
 3096 
 
 3041 
 
 9253 
 
 0921 
 
 5410 
 
 8801 
 
 1566 
 
 52 o 
 
 IS 
 
 7119 
 
 7146 
 
 4972 
 
 3337 
 
 2825 
 
 9528 
 
 0679 
 
 5718 
 
 8532 
 
 1909 
 
 45 
 
 30 
 
 6956 
 
 7351 
 
 4783 
 
 3576 
 
 2609 
 
 9801 
 
 0435 
 
 6026 
 
 8261 
 
 22SI 
 
 30 
 
 45 
 
 6793 
 
 7555 
 
 4592 
 
 3815 
 
 2391 
 
 5-0074 
 
 0190 
 
 6333 
 
 7988 
 
 2592 
 
 IS 
 
 39 o 
 
 6629 
 
 7759 
 
 4400 
 
 4052 
 
 2172 
 
 0346 
 
 6.9943 
 
 6639 
 
 7715 
 
 2932 
 
 51 o 
 
 IS 
 
 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 
 
 476l 
 
 ISO? 
 
 "55 
 
 9196 
 
 7550 
 
 6884 
 
 3944 
 
 IS 
 
 40 o 
 
 5963 
 
 8567 
 
 3623 
 
 4995 
 
 1284 
 
 1423 
 
 8944 
 
 7851 
 
 6604 
 
 4279 
 
 50 o 
 
 IS 
 
 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 
 
 546i 
 
 0832 
 
 1956 
 
 8437 
 
 8450 
 
 6041 
 
 4945 
 
 30 
 
 45 
 
 5454 
 
 9166 
 
 3030 
 
 5693 
 
 0605 
 
 2221 
 
 8181 
 
 8748 
 
 5756 
 
 5276 
 
 15 
 
 41 o 
 
 5283 
 
 9364 
 
 2830 
 
 5924 
 
 0377 
 
 2485 
 
 7924 
 
 9045 
 
 5471 
 
 5606 
 
 49 
 
 IS 
 
 5110 
 
 956i 
 
 2629 
 
 6i54 
 
 0147 
 
 2748 
 
 7666 
 
 9341 
 
 SI84 
 
 5935 
 
 45 
 
 30 
 
 4937 
 
 9757 
 
 2427 
 
 6383 
 
 5.9916 
 
 3OIO 
 
 7406 
 
 9636 
 
 4896 
 
 6262 
 
 30 
 
 45 
 
 4763 
 
 9953 
 
 2224 
 
 6612 
 
 9685 
 
 3271 
 
 7145 
 
 9929 
 
 4606 
 
 6588 
 
 IS 
 
 42 o 
 
 4589 
 
 4-0148 
 
 2O2O 
 
 6839 
 
 9452 
 
 3530 
 
 6883 
 
 6.O222 
 
 4314 
 
 6913 
 
 48 o 
 
 IS 
 
 4413 
 
 0342 
 
 I8l5 
 
 7066 
 
 9217 
 
 3789 
 
 6620 
 
 0513 
 
 4022 
 
 7237 
 
 45 
 
 30 
 
 4237 
 
 0535 
 
 1609 
 
 7291 
 
 8982 
 
 4047 
 
 6355 
 
 0803 
 
 3728 
 
 7559 
 
 3 
 
 45 
 
 4.4059 
 
 4.0728 
 
 5 1403 
 
 4.7516 
 
 5.8746 
 
 5-4304 
 
 6.6089 
 
 6.IO92 
 
 7.3432 
 
 6.7880 
 
 IS 
 
 43 o 
 
 3881 
 
 0920 
 
 1195 
 
 7740 
 
 8508 
 
 4560 
 
 5822 
 
 1380 
 
 3135 
 
 8200 
 
 47 
 
 IS 
 
 3702 
 
 IIII 
 
 0986 
 
 7963 
 
 8270 
 
 4815 
 
 
 1666 
 
 2837 
 
 8518 
 
 45 
 
 30 
 
 3522 
 
 1301 
 
 0776 
 
 8185 
 
 8030 
 
 5068 
 
 5284 
 
 1952 
 
 2537 
 
 8835 
 
 3 
 
 45 
 
 3342 
 
 1491 
 
 0565 
 
 8406 
 
 7789 
 
 S32I 
 
 5013 
 
 22 3 6 
 
 2236 
 
 9151 
 
 IS 
 
 44 o 
 
 3160 
 
 1680 
 
 0354 
 
 8626 
 
 7547 
 
 5573 
 
 4741 
 
 2519 
 
 1934 
 
 9466 
 
 46 o 
 
 15 
 
 2978 
 
 1867 
 
 0141 
 
 8845 
 
 7304 
 
 5823 
 
 4467 
 
 2801 
 
 1630 
 
 9779 
 
 45 
 
 30 
 
 2795 
 
 2055 
 
 4.9928 
 
 9064 
 
 7060 
 
 6073 
 
 4193 
 
 3O82 
 
 1325 
 
 7.0091 
 
 30 
 
 45 
 
 2611 
 
 2241 
 
 9713 
 
 9281 
 
 6815 
 
 6321 
 
 3917 
 
 3361 
 
 1019 
 
 0401 
 
 IS 
 
 45 o 
 
 2426 
 
 2426 
 
 9497 
 
 9497 
 
 6569 
 
 6569 
 
 3640 
 
 3640 
 
 0711 
 
 0711 
 
 45 o 
 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 
 
 Dist. 6. 
 
 Dist. 7. 
 
 Dist. 8. 
 
 Dist. 9. 
 
 Dist. 10. 
 
 Course. 
 
CHAPTER V 
 TRIGONOMETRY 
 
 When the surveyor reads the bearing of a compass 
 needle he reads an angle and during his working life he is 
 dealing with angular measurements. It is therefore neces- 
 sary that he be skilled in that special branch of mathematics 
 termed trigonometry. 
 
 Trigonometry was formerly said to deal with the prop- 
 erties of triangles and was divided into Plane, Analytical 
 and Spherical Trigonometry. It is now defined as a branch 
 of mathematics dealing with the functions of angles and 
 their application in the solution of triangles. Plane and 
 Analytical Trigonometry have been merged, so we now have 
 only Plane and Spherical Trigonometry, the latter merely a 
 particular application to cases where the sides of triangles 
 are arcs of circles. It is proposed in this chapter to give 
 the minimum amount of trigonometry every surveyor 
 requires. 
 
 Every triangle consists of six parts, three sides and three 
 angles. When three parts are known, one of which must 
 be a side, the other three can be found. To the foregoing 
 statement the exception must be made that when two 90 
 angles are given with a side opposite one of them the solu- 
 tion is indeterminate. However the surveyor will not 
 meet with this particular case. 
 
 In arithmetic the three sides of a right-angled triangle 
 are known as the base, the altitude and the hypothenuse. 
 
 Carpenters and practical men usually call the three sides 
 the run, the rise and the slope. 
 
 For convenience it is usual in mathematical work to 
 letter the sides and angles. The acute angle at the base 
 is known as ''Angle A," the side opposite being marked 
 with a small letter and called "side a." The word " alti- 
 tude" begins with "a," so the upright line designating the 
 
 148 
 
TRIGONOMETRY 
 
 149 
 
 altitude may be easily remembered as "line a." Similarly 
 the base is "line b" and the acute angle at the top is 
 "Angle B." The right angle at the base is "Angle C," 
 the hypothenuse opposite the angle being "line c. " 
 
 Base 
 
 FIG. 150. 
 
 Run 
 
 FIG. 151. 
 
 FIG. 152. 
 
 The right-angled triangle was used for illustration but 
 the same conventions are used for oblique-angled triangles; 
 i.e., capital letters indicate angles and small letters indicate 
 sides. 
 
 THE PYTHAGOREAN THEOREM 
 
 Pythagoras, a noted mathematician of ancient times, is 
 said to have been the first to demonstrate that 
 
 In a right-angled triangle the square on the hypothenuse is 
 equal to the sum of the squares on the other two sides, so the 
 statement is known as the Pythagorean Theorem and is 
 the basis for a number of trigonometrical expressions. 
 
 Stated in a modern way (algebraically) it appears as 
 follows : 
 
 or 
 
 then 
 
 and 
 
 a = V(c + b)(c- b) = Vc 2 - b\ 
 
 b = V(c + a)(c- a) = Vc* - a 2 
 
 GRAPHICAL SOLUTION OF A RIGHT TRIANGLE 
 
 Fig. 153 is reproduced, with some of the finer gradua- 
 tions omitted, from a diagram copyrighted in 1912 by 
 Constantine K. Smoley. 
 
PRACTICAL SURVEYING 
 
 The numbers may be taken to represent inches, feet, 
 yards or meters. Given two sides of any triangle the third 
 side may be found. 
 
 Given the base and altitude to find the hypothenuse. Find 
 the base on A-B and the altitude on B-C. Follow the 
 
 & 
 
 /* 
 
 $ 
 
 $ 
 
 Y 
 
 4 
 
 V 
 
 2\ v& v ^ v a 
 
 >c 
 
 I 
 
 ''/ 
 
 s 
 
 
 
 \ 
 
 \ 
 
 
 
 s 
 
 
 \ 
 
 \ 
 
 N 
 
 
 
 
 \ 
 
 
 
 x 
 
 ^ 
 
 
 
 
 \ 
 
 ^ 
 
 
 ^ 
 
 I 
 
 
 
 a 1 
 
 | 
 
 6 
 
 4 
 3 
 2 
 
 i 
 
 L^ 
 
 \ 
 
 
 
 ^ 
 
 \ 
 
 
 
 
 
 \ 
 
 2E 
 
 
 
 \ 
 
 
 
 
 ^ 
 
 \ 
 
 \ 
 
 1 
 
 
 k \ 
 
 \ 
 
 
 
 \ 
 
 \ 
 
 s 
 
 
 
 
 
 \ 
 
 V 
 
 
 
 L 
 
 r 
 
 
 
 
 
 \ 
 
 X 
 
 \ 
 
 
 
 0^ 
 
 i) 
 
 /, 
 
 
 
 \ 
 
 5 
 
 
 
 \ , 
 
 \ 
 
 \ 
 
 
 
 
 I 
 
 ^ 
 
 \ 
 
 ^ 
 
 \ 
 
 
 
 ^ 
 
 \ 
 
 \ 
 
 
 
 
 j 
 
 \ 
 
 \ 
 
 \ 
 
 ^ 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 \ 
 
 ^ 
 
 \ 
 
 \ 
 
 
 
 
 
 
 .-OSS 
 
 
 
 S 
 
 
 ^ 
 
 \ 
 
 
 
 ^ 
 
 \ 
 
 \ 
 
 
 
 
 
 
 \ 
 
 \ 
 
 \ 
 
 J 
 
 ^ 
 
 ^ 
 
 
 
 ^ 
 
 N 
 
 \ 
 
 
 
 
 
 v 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 \ 
 
 ^ 
 
 
 
 
 \ 
 
 \ 
 
 
 \ 
 
 \ \ 
 
 
 
 
 
 
 
 
 \ 
 
 4 
 
 \ 
 >/ 
 
 s 
 t* 
 / 
 
 
 V 
 
 x 
 
 / 
 
 s 
 
 c 
 
 
 
 
 \ 
 
 \ 
 
 \ 
 
 
 
 
 
 
 ^ 
 
 \ 
 
 \ 
 
 \ 
 
 [ 
 
 \ 
 
 \ 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 \ 
 
 i 
 
 \ 
 
 5 
 
 \ 
 
 j 
 
 
 1 
 
 
 
 
 
 ^ 
 
 \ 
 
 
 
 
 
 \ 
 
 J 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 
 \ \ 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 \ 
 
 ^ 
 
 ^ 
 
 \ 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 \ 
 
 
 1 
 
 
 \ 
 
 \ 
 
 s 
 
 \ 
 
 \ 
 
 i 
 
 
 
 
 
 
 
 
 j 
 
 
 J 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 
 \ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 1 
 
 i 
 
 \ 
 
 
 
 
 
 \ 
 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 
 
 
 
 j 
 
 \ 
 
 \ 
 
 \ 
 
 s 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 I 
 
 I 
 
 \ 
 
 i 
 
 
 
 
 i 
 
 
 
 \ 
 
 
 
 
 V/A" 
 
 5 
 
 \ 
 
 \ 
 
 \ 
 
 j 
 
 I 
 
 \ 
 
 | 
 
 ; , 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \XN> 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 AVQ J 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 J / , J < 
 
 . 
 
 ; 
 
 
 4~56i7?8 
 
 9 i I0 1 1! 
 
 ? /2i 13114-15. 
 
 i 
 
 FIG. 153. Graphical solution of a right triangle. 
 
 lines to an intersection and then trace the arc found at this 
 point to line A D, on which read the length of the hypothe- 
 nuse. 
 
 Given the hypothenuse and another side to find the third 
 side. Place a ruler on the line marking the second side. 
 If this side is the altitude the ruler will be parallel with 
 A-B, starting from B-C. If the second side is the base 
 the ruler will be parallel with B C, starting from A-B. 
 Find the hypothenuse on A-D and follow the arc inter- 
 secting A-D at this point to an intersection with the ruler. 
 Going from this intersection on a line perpendicular to 
 the ruler read the required third side. 
 
 The diagram presented appears in the seventh edition 
 of Smoley's Tables, the standard tables for engineering 
 draftsmen, and is in inches. The author used similar 
 diagrams for many years drawn on cross-section paper 
 
TRIGONOMETRY 
 
 divided decimally instead of in eighths. Such diagrams are 
 in fairly common use and are of considerable value in cer- 
 tain kinds of work. For instructional purposes they are 
 very good. 
 
 TRIGONOMETRIC FUNCTIONS 
 
 In trigonometry all angles are assumed to have angle A 
 at the center of a circle, side b and side c being radial lines 
 intercepting an arc on the circumference. Either b or c 
 may be a radius. 
 
 If an arc is described and a perpendicular let fall from 
 the end of the radius to the base, then 
 
 a = sine, 
 b = cosine, 
 c = radius, 
 c b = versed sine, 
 
 written sin, 
 written cos, 
 written rod, 
 written versin. 
 
 b-co-slne C 
 
 FIG. 154. 
 
 b-radius 
 
 FIG. 155. 
 
 If an arc is described and a perpendicular erected from 
 the end of the radius as the base, then 
 
 a = tangent, written tan or tangt, 
 b = radius, written rod, 
 c = secant, written sec. 
 
 The sine, tangent, etc., are called functions of an angle, 
 a function in mathematics being any algebraic expression 
 or quantity dependent for its value on another one. 
 
 The difference between any angle in a quadrant and 90 
 degrees, the full quadrantal angle, is said to be the comple- 
 ment of the angle. Thus in Fig. 156, angle B is the com- 
 plement of angle A and vice versa. 
 
152 
 
 PRACTICAL SURVEYING 
 
 -Cotangent 
 
 The functions of the complement are said to be co-func- 
 tions of the angle. 
 
 The sine of A cosine of B. 
 
 The sine of B = cosine of A . 
 
 The tangent of A = cotangent of B. 
 
 The cotangent of B = tangent of 
 A, etc., as shown in Fig. 156. 
 
 The difference between any angle 
 and 1 80 degrees is called the supple- 
 ment of the angle. The functions 
 of the supplement are the functions 
 of the angle. 
 
 Cosine- r 1 
 FIG. 156. Functions of 
 angles. 
 
 Side a may be the sine or tangent of angle A and side b 
 may be the cosine, or it may be the radius of the circle in 
 which a triangle is drawn having angle A at the center. 
 
 TO avoid confusion it is considered best to regard the 
 functions as ratios, that is, as pure numbers and not as 
 lines. Side a and side b are lines but sin 
 A , tan A and cos A are not always lines. 
 Tangent A does not become a line un- 
 til it is multiplied by the run of the tri- 
 angle having angle A at the base, when 
 it becomes side a. If sin A is multi- 
 plied by th$ slope it becomes side a. 
 
 Using the "ratio concept" the functions 
 as follows, referring to Fig. 157. 
 
 Sin A =- 
 c 
 
 FIG. 157. 
 
 are described 
 
 Cot A = 
 
 Sec A =T 
 
 Cosec A = -- 
 a 
 
TRIGONOMETRY 153 
 
 The relations (ratios) here shown are true, no matter 
 what may be the lengths of the lines. The ratios must be 
 memorized or the student will find himself as helpless in 
 trigonometrical work as he would be in arithmetic with- 
 out knowing the multiplication tables. 
 
 The following trigonometrical equivalents must also be 
 memorized, an easy task when the connection with the 
 Pythagorean theorem is noted. 
 
 cos 
 
 Sin = - = - = V(i - cos 2 ) = V(i+cos)(i -cos), 
 cosec cot 
 
 Cos = ^ = = sin X cot = V(i - sin 2 ) 
 tan sec 
 
 = V(i -+- sin) (i sin). 
 
 ^ sin I 
 
 Tan = -~ = 
 cos cot 
 
 cos i 
 Cot = --' 
 sin tan 
 
 Sec = t ^ = = Vrad 2 + tan 2 . 
 sin cos 
 
 Cosec = 
 sin 
 
 Rad = tan X cot = Vsin 2 -f cos 2 , 
 c-b 
 
 Versin = rod cos = 
 Coversin = rod sin = 
 
 c 
 c a 
 
 NUMERICAL VALUES FOR THE TRIGONOMETRICAL RATIOS 
 
 Angle o 
 
 Draw a square A, B, C, D and in it draw the diagonal 
 
 
 
 00 
 A, B. Angle x = 45 = - because side a = side b. 
 
 c 2 = a 2 + b 2 , 
 and since a = b = I, 
 
 c 2 = 2b 2 = 2a 2 ; or c = \/2 X b = \/2 X a. 
 
154 
 
 PRACTICAL SURVEYING 
 
 .o b i 
 
 Cos 45' ----i-. 
 
 Tan 45-? -I. 
 
 FIG. 158. Functions of 45. FIG. 159. Functions of 30 and 60. 
 
 Angle of 60. 
 
 In the equilateral triangle (Fig. 159) each of the three 
 angles is equal to 60. Drop the perpendicular BD to b. 
 Then, assuming the length of each side = I, 
 
 that is, 
 
 2 AD = AB = AC = BC = a = b = c = I, 
 = BD 2 + AD 2 = a 2 =b 2 = ( 2 AD) 2 = 4 AD* 
 4 AD 2 = BD 2 + AD 2 or BD 2 = 3 AD 2 ; 
 
 4 X i 2 = BD 2 + i 2 or 
 ... BD = V 3 X i = 
 
 3 X i. 
 
 Since angle ^4 = 6o c 
 
 Cos 60 = 
 
 AD 
 
 I 2 
 
TRIGONOMETRY 
 
 155 
 
 Angle of 30. 
 
 Sin 30 = cos 60 = J. 
 
 V\ 
 Cos 30 = sin 60 = ^ 
 
 Tan 30 = cot 60 = 
 
 tan 60 
 
 Sin 15 = 
 
 Sin i 8 = 
 
 _ 
 
 V2 
 
 To obtain the remaining functions of the foregoing angles 
 refer to the list of Trigonometrical Equivalents. 
 
 SIGNS OF THE TRIGONOMETRICAL RATIOS 
 
 It is convenient (hence the word convention) to assume 
 that all directions up or down are measured from a hori- 
 zontal line (or axis) X . . . X' and 
 that all directions right or left are 
 measured from a vertical line (or axis) 
 Y . . . Y f . The vertical axis is termed 
 an ordinate and the horizontal axis an 
 abscissa. The axes are known as co- 
 ordinates and the point of intersection, 
 0, the origin of co-ordinates. 
 
 Describing a circle with O as a center 
 and O ... X as a radius the axes di- 
 vide it into four quarters, or quadrants. 
 Proceeding around the circle in a direction contrary to 
 that followed by the hands of a clock anti-clockwise 
 the quadrants are numbered as shown in Fig. 160. 
 
 All motion toward X (to the right) and towards Y (up) 
 is considered positive ( + ). Motion to the left and down is 
 considered negative ( ). 
 
 The sine and cosine cannot extend beyond the circum- 
 ference of a circle having a radius = I but the tangent, 
 cotangent, secant and cosecant may extend to infinity (<*>). 
 
156 
 
 PRACTICAL SURVEYING 
 
 Representing the functions by lines, the signs of the func- 
 tions, together with their limiting values, are shown in the 
 following table, x being the angle at the origin. 
 
 If the angle is in 
 quadrant. 
 
 Sin*. 
 
 Cos*. 
 
 Tan*. 
 
 Cot *. 
 
 Sec*. 
 
 Cosec *. 
 
 T (Sign 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 H Value... 
 
 o to i 
 
 I tO 
 
 o to oo 
 
 oo to o 
 
 I tO 00 
 
 00 tO I 
 
 TT jSign 
 1L IValue... 
 
 i to o 
 
 o to i 
 
 00 tO O 
 
 o to oo 
 
 oo to i 
 
 i to oo 
 
 TTT (Sign 
 [IL IValue ... 
 
 o to i 
 
 I tO 
 
 o to oo 
 
 oo to o 
 
 I tO CO 
 
 00 tO I 
 
 IV ( Sign 
 
 
 
 _l_ 
 
 
 
 
 
 + 
 
 
 
 * 1 Value . . . 
 
 i to o 
 
 o to i 
 
 oo to o 
 
 o to oo 
 
 oo to i 
 
 I tO 00 
 
 Summarizing the results of the investigation of the nu- 
 merical values the following table has been prepared: 
 
 ^^^ 
 
 oor 
 360 
 
 30 
 
 45 
 
 60 
 
 90 
 
 180 
 
 270 
 
 Sine . . . 
 
 o 
 
 jj 
 
 I 
 
 V3 
 
 
 o 
 
 I 
 
 
 
 
 V 2 
 
 2 
 
 
 
 
 Cosine 
 
 i 
 
 ^1 
 
 I 
 
 \ 
 
 o 
 
 I 
 
 o 
 
 
 
 2 
 
 V 2 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 Tangent 
 
 o 
 
 / 
 
 I 
 
 \/3 
 
 00 
 
 
 
 00 
 
 
 
 V 3 
 
 
 
 
 
 
 Sin 30 = J = 0.5 = cos 60. 
 
 Sin 45 = 4= = - - = 0.7071 = cos 45 
 
 I 4 I 4 
 
 Sin 60 = 
 Tan 30 = 
 Tan 60 = 
 
 
 = 0.866 = cos 30. 
 = -5774 = cot 60. 
 
 1.732 = 
 
TRIGONOMETRY 157 
 
 TABLE OF NATURAL FUNCTIONS 
 
 The circumference of a circle is divided into 360 equal 
 parts called degrees. 
 
 Each degree is divided into 60 equal parts called minutes. 
 
 Each minute is divided into 60 equal parts called seconds. 
 
 The size of an angle is indicated by 
 the number of parts of a circle contained 
 in the intercepted arc; thus, 25 15' 10" 
 which is read 25 degrees, 15 minutes, 10 
 seconds. 
 
 In Fig. 161 the angle at A = 30, 
 at C - 90 and at B = 60. The line 
 A-B is 75.3 ft. long. What are the lengths of the other 
 sides? 
 
 Referring to the list of Trigonometrical Equivalents 
 
 c tan 
 Sec = -^-i 
 sin 
 
 therefore 
 
 sin X sec = tan. 
 
 In the 30 triangle the sin = 0.5 and as the slope has a 
 length of 75.3 ft., the tangent (line B-C) has a length of 
 0.5 X 75-3 = 37.65 ft. 
 
 Also the secant = > 
 cos 
 
 therefore cos X sec = I = radius. 
 
 The cos of 30 = 0.866 and the length of A-C = 75.3 X 
 0.866 = 65.21 ft. 
 
 Use of tangent. 
 
 (i) A triangle has a base of 65.21 ft. and an altitude of 
 37.65 ft. What is the angle A ? 
 
 sin alt. a 37. 65 
 
 The tangent for an angle of 30 = 0.5774 so the angle 
 A = 30. 
 
 (2) A 30 right-angled triangle has a base of 65.21 ft. 
 What is the altitude? 
 
-158 PRACTICAL SURVEYING 
 
 The tangent of 30 = 0.5774 so the altitude = 0.5774 X 
 65.21 = 37.65 ft. 
 
 By methods given in college textbooks, tables have been 
 computed which contain values of all the functions. So 
 many tables are in existence it is unnecessary for any one 
 save professional mathematicians, as a part of their train- 
 ing, to compute such tables. 
 
 The table of Natural Functions of Angles here presented 
 gives values of the ratios for each ten minutes of arc, this 
 being sufficient for the examples in this book, which are 
 intended to give practice in the solution of triangles and 
 the use of tables. Tables in common use give values for 
 each minute of arc, with numbers whereby interpolations 
 may be made for smaller angles. Books of tables usually 
 contain directions for use. 
 
 Example. Find the sine of 33 20'. 
 
 Turning to the table find 33 in the column headed (Deg.) 
 and find 20 in the column headed (Min.). The sine is in 
 the column headed Sine and = 0.564007. The radius has 
 a value = i.oooooo. 
 
 The values of other functions are found in the same way 
 in the proper column. 
 
 Example. Find the tangent of 54 30'. 
 
 The tangent of 54 30' = 1.4019483. 
 
 Read the note at the bottom of each page of the table. 
 The functions of angles are co-functions of the complements 
 of the angles so it is therefore necessary to compute the 
 values of functions for angles between o and 45 only. 
 For angles between 45 and 90 read up from the bottom 
 of the pages. Notice that the column with 
 
 Sine at the top has Cosine at the bottom, 
 Tangent at the top has Cotangent at the bottom, 
 Secant at the top has Cosecant at the bottom 
 and vice versa. 
 
 Example. Find the sine of 146 40'. 
 Subtract from 179 60' (180) 
 
 146 40' 
 and find sine of 33 20' 
 
 Example. Find the tangent of 125 30'. 
 
TRIGONOMETRY 159 
 
 Subtract from 179 60' (180) 
 
 125 3Q' 
 and find tangent of 54 30' 
 
 These two examples show that the functions of an angle 
 are the functions of the supplement of the angle. 
 
 In adding and subtracting angles it is a good safe habit 
 to write the work on paper instead of attempting to do it 
 mentally until considerable experience is had. It is well 
 to see such things. 
 
 In subtraction lessen the degrees by I and add 60 to the 
 minutes of the larger angle. When seconds are given lessen 
 the number of minutes by I and add 60 to the seconds. 
 
 The tables in this book give values to 10 minutes of arc 
 but for all practical purposes the values for intermediate 
 angles may be obtained by interpolation. 
 
 Example. Find sine of 27 13'. 
 
 Sine 27 20' = 0.459166 
 Sine 27 10' = 0.456580 
 
 0.002586 diff. for 10' 
 3 minutes = 0.3 of 10 minutes, therefore 
 0.002586 diff. for 10' 
 
 0-3 
 
 0.0007758 
 Sine 27 10' 04565800 
 
 04573558 = sin27!3 / . 
 
 In using the tables in this book for arcs smaller than 10 
 minutes, the final results should be considered as being 
 correct for the first five places (figures). If the sixth figure 
 is 5 or more, increase the fifth figure by I and reject the 
 figures following. Thus, the sine of 27 13' correct to five 
 places = 045736. If the sixth figure is less than 5 drop all 
 figures following the fifth. 
 
 GRAPHICAL NATURAL FUNCTIONS 
 
 Paper protractors 8 and 14 ins. in diameter are printed 
 from engine divided plates on rectangular sheets. On 
 such a sheet draw a line through the center passing through 
 the 270 and 90 graduations. Normal to this line draw 
 
160 PRACTICAL SURVEYING 
 
 a line from the center through the o point. Make each 
 line 5 or 10 ins. long and divide in ten equal parts. Draw 
 lines through each division so the paper will be ruled in 
 squares which are to be numbered from o to 10, starting 
 at the center of the protractor. Each division should be 
 divided into ten parts. A fine line drawn from the center 
 through any angle will represent the hypothenuse of a 
 triangle. This hypothenuse is the secant of the angle with 
 a radius = i.o. In making practical use of such a diagram 
 a fine line is drawn on a narrow strip of tracing cloth or 
 transparent celluloid and this line is graduated to corre- 
 spond with the horizontal and vertical lines. A fine needle 
 is put through the zero on this line and the center of the 
 protractor. The transparent strip may then be swung so 
 the line will intersect any angle and the lengths, of the three 
 sides of a triangle having this angle at the base may be 
 read. 
 
 If the angle A alone is given the secant may be extended 
 an infinite distance and an infinite number of triangles be 
 formed in which 
 
 B = 90 -4, 
 C = 90, 
 
 for the sum of the three interior angles of any triangle = 
 2 x 90 = 1 80. 
 
 An angle measures the amount of divergence between two 
 lines starting from a common point. The base is one line 
 and the secant another line defining an angle. When the 
 secant is defined on the diagram as described and the angle 
 A thus marked, the angle B may also be read off, for it is 
 the complement of the angle at the base. To assist in 
 obtaining the complementary angle a second set of gradu- 
 ations may be placed on the circumference of the protractor 
 in an opposite direction to the regular marks. This second 
 set should be in red ink to avoid mistakes. 
 
 The radius being 1 .00 the functions of an angle are ratios 
 expressed in per cent of the radius. 
 
 If the radius is ten inches, the heavy lines drawn at 
 intervals of one inch and the lighter lines at intervals of 
 one- tenth of an inch, values of the functions may be read 
 to three decimal places. 
 
TRIGONOMETRY 
 
 161 
 
 Fig. 162 is an engine-divided diagram used in Lewis 
 Institute, Chicago, 111., and reproduced by permission of 
 Prof. Phillip B. Wood worth. The only difference between 
 this engine-divided protractor and the diagram just de- 
 scribed is in the angular graduations, which are placed on 
 vertical and horizontal lines outside of the quadrille ruling. 
 
 EH 
 
 5' &>' 75" ft 
 
 vvvyv^vv^^ 
 
 V ' 
 
 45y 
 /^ 
 
 3 | 
 
 35* 
 7 <: 
 
 Aftf". 
 
 
 
 
 
 --^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -^ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 x 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 x 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 x 
 
 c 
 a 
 
 CJi 
 
 c 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 . 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 x 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 x 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 x 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 X 1 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 X 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 >^ 
 
 
 
 
 
 
 
 
 
 A^ 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 CO 
 
 
 
 
 \ 
 
 
 
 
 
 . 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 > 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 x 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 , 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 
 
 Cosine. 6 , 
 
 
 
 
 
 
 
 
 FIG. 162. Trigonometer. 
 
 Arranged as shown in the illustration this device is called 
 a Trigonometer and is of considerable value in checking 
 calculations for latitudes and departures. Metal trigo- 
 nometers were formerly sold by instrument dealers but the 
 ease with which a good draftsman may make one probably 
 caused the manufacture to be discontinued, for they seem- 
 ingly are no longer advertised. The writer used one for 
 many years. 
 
 SOLUTIONS OF RIGHT TRIANGLE 
 
 Given A and c to find B, a and b. 
 B = 90 -A, 
 a c sin A , 
 b = c cos A . 
 
 D 
 
 FIG. 163. 
 
1 62 PRACTICAL SURVEYING 
 
 Given A and a, to find B, b and c. 
 B = 90 -A, 
 b = a cot A , 
 
 a 
 
 sin A 
 
 Given A and b, to find B, a and c. 
 5 = 90 -A, 
 a = b tan ^4 , 
 
 cos A 
 Given c and a, to find A , and b. 
 
 Sin ^4 = - 
 c 
 
 5 = 90-^, 
 6 = a cot ^4 . 
 
 Given a and 6, to find A, B and c. 
 
 Tan A = -7 > 
 
 D 
 
 B = 90 -4, 
 
 a 
 
 c = 
 
 sn ^ 
 
 A a b 
 
 Area = 
 
 Either acute angle may be assumed to be the angle at 
 the base in which case the adjacent side becomes the base. 
 
 PROBLEMS 
 
 1. A = 48 if, c = 324 ft. Find B, a, b, area. 
 
 2. A = 51 19', b = 1254 ft. Find B, a, c, area. 
 
 3. A = 43 38', a = 1 86 ft. Find B, b, c, area. 
 
 4. a = 249 ft., c = 415 ft. Find A, B, b, area. 
 
 5. a = 67, b = 53. Find A, B, c, area. 
 
 6. c = 893, b = 586. Find ^4, 5, a, area. 
 
 7. yl = 64 40', & = 326. Find B, a, c, area. 
 
 8. A = 71 24', a = 286. Find B, 6, c, area. 
 
 9. ^4 = 41 48', c = 963. Find 5, b, a, area. 
 
TRIGONOMETRY 163 
 
 In solving problems the student is greatly assisted by 
 drawing the triangle free hand and writing the given values 
 in the proper places. Each part as found should be placed 
 on the sketch. Check. Draw the triangles to scale, using 
 a protractor to measure the angles. 
 
 The expression abc means a X b X c, the multiplication 
 sign being understood when letters are used. Sin A means 
 the sine of the angle A , and b sin C means side b X sin C, 
 the value of the sine being given in the tables. 
 
 Sine J A means the sine of one-half the angle A and does 
 not mean half the sine of A, which is something entirely 
 different. Beginners often get into trouble over this matter. 
 
 ALGEBRAIC THEOREMS 
 
 I. The square of the sum of two quantities is equal to the 
 square of the first, plus twice the product of the first multiplied 
 by the second, plus the square of the second. 
 
 Example. (a -f b) 2 = a 2 + 2 ab -f b 2 . 
 a +b 
 a +b 
 
 + ab + 
 
 a 2 + 2 ab + b 2 
 
 2. The square of the difference of two quantities is equal to 
 the square of the first minus twice the product of the first by 
 the second, plus the square of the second. 
 
 Example. (a - b) 2 = a 2 - 2 ab + b 2 . 
 a -b 
 a -b 
 a 2 ab 
 -ab + b 2 
 
 a 2 - 2 ab + b 2 
 
 3. The product of the sum and difference of two quantities 
 is equal to the difference of their squares. 
 
164 PRACTICAL SURVEYING 
 
 Example. (a + b) (a - b) = a 2 - b 2 . 
 a +b 
 a -b 
 a 2 + ab 
 -ab-b* 
 
 (Note. In the preceding chapter it was explained that 
 in multiplication like signs produce + and unlike signs 
 produce .) 
 
 TRIGONOMETRIC LAWS 
 
 The following laws should be thoroughly understood and 
 in their development appears the Pythagorean Theorem 
 and the algebraic rules and theorems just given. The 
 
 C D A A b C A 
 
 FIG. 164. 
 
 student should work the expressions by assuming the fol- 
 lowing values, a = 9, b = 12, c = 15. 
 Law of sines. 
 
 In any triangle the sides are to one another as the sines 
 of their opposite angles. 
 
 a _ sin A b _ sin B a __ sin A 
 b sin B ' c sin C ' c sin C 
 
 Law of cosines. 
 
 a 2 = b 2 + c 2 2 be cos A . 
 
 tf = a 2 + c 2 - 2accosB. 
 
 c* = a 2 -f b 2 - 2 ab cos c. 
 Law of tangents. 
 
 a - b = tan \ (A - B) 
 
 a + b ~ tan \ (A + B) ' 
 
 a - c = tan \ (A - C) 
 
 a + c ~ tan | (A + C) ' 
 
 b - c _ tan %(B - C) 
 
TRIGONOMETRY 165 
 
 Half the difference of two unequal quantities AB and BC, 
 added to half their sum, gives the greater, and half the differ- 
 ence taken from half the sum, 
 gives the less. * 
 
 Proof.- Draw AB + BC. '^^ 
 
 Make AD = BC. 
 
 Then A C = their sum and 
 BD = their difference. 
 
 Bisect BD in E. 
 
 Then BE = ED = half their difference and 
 AE = EC = half their sum. 
 
 Consequently AE + EB = AB, the greater and 
 EC - EB = BC, the less. 
 
 Also, half the difference BE, added to the less BC, or 
 taken from the greater AB, gives half the sum. 
 
 SOLUTION OF OBLIQUE TRIANGLES 
 FIRST CASE. 
 Given A, B, a, to find C, 6, c. 
 
 C= 180- (A +3). (i) 
 
 FIG. 166. 
 
 
 c = sin C- 7- (3) 
 
 sin A 
 
 SECOND CASE. 
 
 Given A, a, b, to find B, C, c. 
 
 (4) 
 
 Cuse (i). 
 c use (3). 
 
l66 PRACTICAL SURVEYING 
 
 THIRD CASE. 
 
 Given A, b, c, to find a, B, C, area. 
 
 a = v6 2 -f c 2 2 be cos yl . (5) 
 
 B use (4). 
 C use (i). 
 
 a = sin A ^; or sin A ^- (6) 
 
 sin B sin C 
 
 . 
 
 Area = -- - -- (7) 
 
 The second case has an angle at the base with the base 
 and side opposite the angle given. 
 
 The third case has an angle and the two including sides 
 given. 
 
 The second case often presents difficulties to the student. 
 If the given angle is acute and the side opposite is the lesser 
 side, then the angle found may be either obtuse (greater 
 than 90), or acute (less than 90). The conditions of the 
 problem should be known. 
 
 The sign < means. "less than" and the sign > means 
 "greater than." The sign = means "equal to or greater 
 than." 
 
 Let 6, c, B be the parts known. If b < c sin B, no 
 solution is possible; if b = c sin B, then sin C = 90; if 
 b > c sin B and b < c, and B is acute, two solutions are 
 possible, but if b = c only one solution is possible and C is 
 an acute angle. 
 
 Professor Stang in Engineering News, Dec. 25, 1913, 
 presented the following formula for solving a triangle in 
 which is given two sides and the included angle: 
 
 cotB = a _ - cotC. 
 osin C 
 
 FOURTH CASE. 
 
 Given a, b, c, the three sides, to find A, B, C, the three 
 angles. 
 
 Let 
 
 
 SinM =V/^ -^ 
 
TRIGONOMETRY 167 
 
 B use 4. 
 Cuse I. 
 
 Area = Vs (s - a) (s - b) (s - c). (9) 
 
 The following formulas may also be used for this case. 
 
 -v/ 
 
 Co.SB-V^*^ 2 - Tania-V/''"*^'"'' 
 
 Cos C = l. Tan C = 
 
 s (s c) 
 
 The following item appeared in Engineering News, July 
 25, 1912. 
 
 A triangle formula, to obtain the angles when the three 
 sides are known, is given by Prof. C. Frank Allen (Mass. 
 Inst. of Technology). While not new it may be unknown 
 to many of our readers, as the formula 
 
 t l( 
 
 = V 
 
 (s - b) (s - c) 
 
 is usually quoted without reference to the other method of 
 solution. In the above the sides are a, b and c, with the 
 opposite angles A , B and C, and s is half the sum of the sides. 
 The derivation and formula, given by Professor Allen, 
 are as follows : From angle C drop a perpendicular on side 
 c, dividing c into the segments k and g, adjacent respec- 
 tively to sides a and b. Calling the length of the perpen- 
 dicular h, we have: 
 
 w = p - g 2 = b 2 - (c - k) 2 , 
 
 and 
 
 h* = a 2 - k 2 . 
 Then 
 
1 68 PRACTICAL SURVEYING 
 
 Whence 
 
 and 
 
 (2) 
 
 Since g = c k, we get 
 
 When g and & are computed, the angles are readily found, 
 as 
 
 n k 
 
 cos A = | cos B = - 
 
 b a 
 
 For many uses these formulas are more convenient than 
 Eq. (i), being probably less liable to errors of computation, 
 and more quickly derived if forgotten. 
 
 PROBLEMS 
 
 In the following problems give both solutions when two 
 are possible. Find in each problem all the parts not given 
 and the area. Check the work by drawing to scale and 
 read angles with a protractor. 
 
 1. c = 532, b = 358, C = 107 40'. 
 
 2. c = 232, b = 345, C = 37 20'. 
 
 3. b = 560, a = 258, B = 63 28'. 
 
 4. B = 63 48', A = 49 25', b = 275. 
 
 5. A = 49 25', C = 63 48', 6 = 275. 
 
 6. A ship sailing due north observes a cape bearing 
 N 54 12' W and after sailing 27 miles, the cape bore 
 5 70 30' W. Required her distances from it. 
 
 7. c = 133, b = 176, A = 73 16'. 
 
 8. c = 237, a- = 482, B = 77 48'. 
 
 9 . & = 78, a = 168, C = 128 26'. 
 
 10. a = 230, b = 365, c = 426. 
 
 11. a = 1248, b = 728, c = 956. 
 
 12. a = 375, b = 275, c = 196. 
 
TRIGONOMETRY 
 
 169 
 
 FUNCTIONS OF HALF AN ANGLE 
 In Fig. 167 let 0, 0, c be the angle z, then 
 
 b, o,f = c, 0,/ = |z, 
 sin c, 0, / = sin \ z = c . . . g, 
 tan c, 0, / = tan \ z = e . . . /, 
 sin z = a . . . c, 
 tan z = b . . d. 
 
 A study of this figure will show why 
 the student must be careful to note the FIG. 167. Functions 
 difference between \ sin A and sin \ A . of half angles. 
 
 Sinjz = 
 
 Cosiz = 
 
 cosz 
 
 cosz 
 H- cosz 
 + cosz 
 
 -+- cosz 
 cosz 
 
 Work the above expressions with z = 45. 
 The sign means "positive or negative." 
 
 CIRCULAR MEASURE OF AN ANGLE 
 
 In certain kinds of calculations the radian is the unit of 
 measure when the magnitude of an angle is to be measured. 
 In Fig. 1 68 the arc ab is equal in length 
 to the radius oa. 
 
 The ratio 
 
 arc 
 
 is called the circular or 
 
 radius 
 radian measure of an angle. 
 
 The circumference of a circle = ird, in 
 which TT = 3.1416 (pronounced pi) and d = 
 diameter = 2 X radius. 
 The circumference of a circle of radius r is 2 wr, or, if the 
 radius is unity, 2 w. 
 
 FlG radfan 
 
170 
 
 PRACTICAL SURVEYING 
 
 The angle 360 corresponds to an arc with the length 2 TT; 
 the angle 180 to an arc = ?r; the angle of 90 to an arc = 
 JTT; etc. 
 
 The radian has a value when expressed in degrees as 
 follows : 
 
 360 = 1 80 
 
 27T " 7T 
 
 i8o c 
 
 3.I4I6 
 
 = 57 if 44.8" = 57-295. 
 
 The value usually given is 57.3. 
 
 The radian, or 57.3 rule, is convenient when the size of 
 angle between a random line and true line is wanted. If 
 the angle is found to be less than 6 the rule is all right, 
 but for larger angles may introduce a considerable error. 
 This is because the offset is always measured on a straight 
 
 line and in the 57.3 rule it 
 
 is assumed to be an arc. 
 
 In Fig. 169 this is illus- 
 
 trated. 
 
 The line AC is measured 
 
 on the supposition that it is 
 the line AB, which it however misses at B by the distance 
 BC. For a small angle the difference between BC straight 
 and curved is so small that 
 
 Angle BAG 
 
 FIG. 169. Application of 57.3 rule. 
 
 A C 
 
 distance 
 
 In higher mathematics the radian measure of an angle is 
 necessary but the instance just given is the only one in 
 which the surveyor can advantageously use it. 
 
 HEIGHTS AND DISTANCES 
 
 In Fig. 170, DA is parallel to BC and 
 the angle ABC = 90. 
 
 47 25'. 
 
 FIG. 170. 
 
 Find height AB. 
 
 2. J3C=i36ft., angle .4 C5 
 Find height AB. 
 
 3. The angular elevation of a wall, taken from the edge 
 of a ditch 18 ft. wide, was 62 40'. Required the height 
 of the wall and the length of a ladder to reach the top of it. 
 
TRIGONOMETRY 171 
 
 4. Let the sloping side of a hill AC be 268 ft., and the 
 angle of depression at its top DAC be 33 45'. Required 
 the horizontal distance BC and the vertical height AB. 
 To measure an inaccessible height AB on level ground. 
 Measure any distance CD in a straight ^ 
 
 line towards the object and at C and D /fi 
 
 read the angles of elevation ; their differ- /' / \ 
 
 ence is the angle CAD. /' / 
 
 /' / 
 
 sin C X sin D X CD c f B 
 
 sin (C - D) FlG - 171- 
 
 To measure an inaccessible height which has no level 
 ground before it. 
 
 Take two stations C and D, in a vertical plane, and meas- 
 ure CD] at C read the vertical angle GCD and the two 
 
 vertical angles ACF and BCF. 
 At D take the angle ADE. 
 
 Since the angle EDC = DCG 
 .'. ADC = ADE + DCG and 
 DAC = ACF - ADE. Then 
 in the triangle ADE, the two 
 angles ADC and DAC and the 
 side CD are given to find the 
 FlG I?2 side AC. In the triangle ACB 
 
 are given the angles. ACB = 
 
 ACF BCF, and ABC = 90 d= BCF, and the side AC 
 to find the side AB. 
 
 If DE is above A, the angle DAC is the sum of ACF 
 and ADE; otherwise it is their difference. Also in this 
 case ADC is the difference of DCG and ADE; otherwise 
 it is their sum. Also when F is below B, the angle ACB 
 is the difference of ACF and BCF; otherwise it is their 
 sum. 
 
 If the stations C and D cannot be conveniently taken in 
 a vertical plane, they may be taken anywhere, and then 
 the angles ADC and A CD must be measured. The triangle 
 A CD will give the side AC. 
 
 To measure a line across a river or canyon. 
 With the instrument at B, Fig. 173, sight to the opposite 
 bank and set the point C, so the points A , B and C will be 
 
172 
 
 PRACTICAL SURVEYING 
 
 in the same straight line. An assistant standing on C 
 sights back to B and marks out the line CD normal to line 
 ABC. At D, just 10 ft. from C, set a point. The instru- 
 ment man at B reads the angle 
 CBD- then 
 
 Length BC = CD X cot CBD. 
 
 Another method. The line CD 
 may be any length and it is not 
 FIG. 173. necessary that the triangle be right 
 
 angled at C. The instrument is 
 
 set at B, then at C and finally at D. Each angle is read. 
 The surveyor then has a triangle with one side CD, and 
 the three angles known, the distance from B to C being 
 obtained according to the Law of Sines. 
 
 To obtain the depth and front of a lot on a diagonal 
 street. 
 
 The streets shown in Fig. 174 have 
 an angle x at the intersection of the 
 lot lines, all the lines between the 
 lots being normal to B Avenue. 
 
 To obtain the lengths of the lines p IG I7 
 
 between the lots from A Street to B 
 
 Avenue multiply the tangent of angle x by the distance of 
 the line from the point ; for example : 
 
 Length of line between I and 2 = 75 tan x t 
 Length of line between 2 and 3 = 100 tan x, 
 Length of line between 3 and 4 = 125 tan x. 
 
 To obtain the frontage on A Street multiply the secant 
 of angle x by the width of the lot on B Avenue ; for example : 
 
 Front of lot I on A Street = 75 sec x, 
 Front of lot 2 on A Street = 25 sec x, 
 Front of lot 3 on A Street = 25 sec x, etc. 
 
 B Ave. 
 
 TRAVERSES 
 
 In Fig. 175 it is desired to obtain the bearing and length 
 of a straight line from A to B. Houses and shrubbery are 
 in the way. The obvious thing to do is to run lines through 
 
TRIGONOMETRY 
 
 173 
 
 the cleared spaces and calculate the bearing and length of 
 the line AB. 
 
 Starting from Sta. o (A) the line was run down the street 
 
 FIG. 175. Running a traverse line. 
 
 6*7 oo' E 249 ft.; thence N 43 15' E 156 ft.; thence to 
 B, N 11 30' 143 ft. 
 
 To compute the missing line set the work down as fol- 
 lows: 
 
 Station. 
 
 Bearing. 
 
 Distance. 
 
 N+ 
 
 S- 
 
 E+ 
 
 W- 
 
 O 
 
 I 
 2 
 
 S 7 oo'E 
 N 4 3 is'E 
 Nn oo'E 
 
 249 
 156 
 143 
 
 '113-63 
 140.37 
 
 247.16 
 
 30-35 
 106 . 89 
 27.28 
 
 .... 
 
 
 
 
 254.00 
 247.16 
 
 247.16 
 
 164.52 
 
 
 
 
 
 6.84 
 
 
 
 
 The southings lack 6.84 ft. of balancing the northings 
 and the westings lack 164.52 ft. of balancing the eastings. 
 This gives a triangle of the following shape and the required 
 bearing will be southwest. 
 
174 PRACTICAL SURVEYING 
 
 This is to be solved as a right triangle with a and b given, 
 to find A , B and c ; 
 
 C = 90. 
 
 (Yesf /64.K . 
 
 T 3 6.84 
 
 LanA = T = ~r~ 0.0440. 
 b 164.52 
 
 .-. A =2 31' 
 FIG. 176. = 90 _ ^4 = gy 29 / 
 
 The required bearing is S 87 29' W. 
 
 Thelength C = ^ = ( ^= I55 . 77 ft. - , 
 
 If it is necessary to run the line from A to B without 
 wanton cutting of shrubbery the bearing of the line is 
 found in the manner just described and it is then run in 
 from B towards A , or the bearing is made to read N 87 
 29' E and the line is run from A to B. 
 
 If the work is carefully done the bearing and distance 
 should check with the computations. 
 
 A surveyor must never neglect to check his work. An 
 experienced surveyor would make a " closed traverse ' ' by 
 endeavoring to run his line through open spaces from B 
 to 4, for example; then to 5 and back to A. He would 
 compute the latitudes and departures and get them to 
 balance, as described in the chapter on Compass Surveying. 
 Then the bearing and length of line AB should be calculated 
 as already described, using courses o, 1,2, to Sta. B. An 
 independent calculation would then be made using courses 
 3, 4, 5 to Sta. A. The differences will be small and can be 
 averaged, after which the line can be staked out. 
 
 "Running a traverse" is a common operation in survey- 
 ing. Any number of courses may be run and the bearing 
 and length of a straight line joining the ends of the first and 
 last obtained. In fact the bearing and length of a line join- 
 ing any two points on a traverse may be .found by tabulat- 
 ing only the courses involved. 
 
 The captain of a ship finds her position by traversing, 
 the work of surveyors and navigators being in many 
 respects similar. 
 
TRIGONOMETRY 
 
 175 
 
 OMISSIONS 
 
 The supplying of omitted bearings and distances in sur- 
 veys is an extension of the principles of traversing. 
 
 Many reasons may be given to explain why a portion of 
 the field notes are omitted but a principal reason is forget- 
 fulness, which is one manifestation of carelessness. In the 
 example under the head of traversing one reason for the 
 omission to run one course was given. It sometimes hap- 
 pens that two courses may lie in swampy, or otherwise 
 inaccessible, places, in which case the surveyors should run 
 a closing line to make a closed survey of all accessible 
 corners, the inaccessible courses then forming with the 
 closing line a separate survey. All errors are thrown into 
 the omitted parts. 
 
 There are four cases in omissions, each of which will now 
 be illustrated. 
 
 CASE I . Bearing and length of one course omitted. 
 
 Station. 
 
 Bearing. 
 
 Distance. 
 
 Latitudes. 
 
 Departures. 
 
 N+ 
 
 S- 
 
 E+ 
 
 W- 
 
 
 I 
 2 
 
 3 
 4 
 
 Ni6 30' E 
 N8 2 05' E 
 S 16 54' E 
 S 36 58' W 
 Omitted 
 
 22. IO 
 19.62 
 
 23-97 
 22. II 
 
 Omitted 
 
 21.190 
 2.702 
 
 
 6.276 
 
 19-433 
 6.968 
 
 
 
 
 22-935 
 17.666 
 
 13.296 
 
 
 
 
 23.892 
 
 40.601 
 23-892 
 
 32.677 
 13.296 
 
 13.296 
 
 16.709 
 
 I9-38I 
 
 Nat. tan of bearing = 
 
 
 Length of course 
 
 Bearing = TV 49 14' W. 
 dep. 
 
 1.15991 
 
 19.381 
 
 sin bearing 0.75738 
 
 = 25.59 chains. 
 
 The traverse table in the chapter on compass surveying 
 gives values to quarter degrees only, whereas in work done 
 with a transit angles are read to one minute of arc, except 
 in extremely high-grade work when angles are read to one 
 
176 PRACTICAL SURVEYING 
 
 second. Surveyors' transits usually read to minutes and 
 engineers' transits read to one-half or one-third of a minute. 
 When traversing is done by means of an instrument reading 
 closer than a compass it is plain that a compass traverse 
 table cannot be used. 
 
 It is common to use a table of sines and cosines, for 
 
 Cosine X distance latitude. 
 Sine X distance = departure. 
 
 The student must memorize this. 
 
 In all tables of latitudes and departures, for any angle, 
 the value given in column I for the latitude is the cosine of 
 that angle, and the value given in column I for the depar- 
 ture is the sine of that angle. 
 
 The actual labor of multiplying and dividing can be 
 greatly lessened by using Crelle's Tables ($5.00), by using 
 a slide rule, or by logarithms. Crelle's Tables contain the 
 products of all numbers up to 1000 X 1000 and by following 
 the instructions, products of much larger numbers may be 
 obtained by inspection. The tables are as readily used for 
 division of one number by another. The slide rule is not 
 so good for this class of problems as for others and is not 
 recommended for traversing calculations except for check- 
 ing, in which work it is better than a trigonometer. Log- 
 arithms are great labor savers and the degree of accuracy 
 of logarithmic work is fixed by the number of significant 
 figures to which the tables are computed. 
 
 The greatest labor savers in calculating latitudes and 
 departures are traverse tables, which reduce all multiplica- 
 tion to addition. The author has read in several textbooks 
 written by teachers, that traverse tables do not save time 
 as compared with tables of logarithms and logarithmic 
 functions of angles. For a number of years the work of 
 the author was principally surveying and he found that 
 traverse tables are far superior to logarithms in saving 
 time and lessening liability of errors. 
 
 The oldest known traverse tables with Latitude and De- 
 parture computed for each minute of angle are those of 
 Gen. J. T. Boileau ($5.00). In 1900 H. Louis and G. W. 
 Caunt brought out a book somewhat more convenient to 
 
TRIGONOMETRY 
 
 177 
 
 use, with larger and clearer type. The price of this book 
 is two dollars, the page measuring six by nine inches. 
 
 The most complete Traverse Tables are those of R. L. 
 Gurden. The latitudes and departures are computed to 
 four places of decimals and for every minute of angle up to 
 100 of distance. The size is 9 ins. by 14 ins. ; the binding is 
 cloth and half morocco. When the author bought his copy 
 in 1889, the price was $12.50, but is now $7.50. The Louis 
 and Caunt tables are admirable for field use while the Gur- 
 den tables are ideal for the orifice. The difference in the 
 tables lies in the fact that the Boileau and the Louis and 
 Caunt tables are computed only for all distances up to 
 10. The following example will serve as an illustration. 
 
 Find the latitude and departure of S 1 6 54' E 23.97 
 chains. 
 
 3. Gurden. 
 16 54' 
 
 20.00 
 3.00 
 0.90 
 0.07 
 
 i. Boileau. 
 2. Louis & Caunt. 
 16 54' 
 
 Lat. 
 
 Dep. 
 
 i I9-I36 
 
 > 2 . 8704 
 
 5.814 
 0.8721 
 
 i 0.86II3 
 0.06698 
 
 0.26163 
 0.02035 
 
 22.93451 
 
 6.96808 
 
 Lat. Dep. 
 
 23.00 22.0064 6.6861 
 
 0.97 0.92811 0.28198 
 
 22.93451 6.96808 
 
 To obtain the same results by using logarithms it will be 
 necessary to take from one table the logarithm of 23.97 
 and from another table the logarithm of the sine and of 
 the cosine of 16 54'. These are added and the numbers 
 corresponding to the resulting logarithms found. 
 
 By logarithms: 
 
 Dep. 
 
 = 1.379668 
 = 9.463448 
 
 10.843116 
 
 10. 
 
 = 0.843116 
 = 6.968 
 
 CASE II. Lengths of two courses omitted. 
 This might be a case where bearings are taken to a cor- 
 ner, which is visible but inaccessible. 
 
 log. 23.97 
 log. cos 1 6 54' 
 
 subtract 
 log. lat. 
 Latitude 
 
 Lat. 
 - 1.379668 
 = 9.980827 
 11.360495 
 
 10. 
 
 log. 23.97 
 log. sin 16 54' 
 
 subtract 
 log. dep. 
 Departure 
 
 = 1.360495 
 = 22.9345 
 
17 8 
 
 PRACTICAL SURVEYING 
 
 In the assumed problem the two courses are adjacent. 
 When the courses are not adjacent the method is the same, 
 or the method for Case IV may be used. 
 
 If the two sides are parallel the problem is indeter- 
 minate. 
 
 The whole process in Cases II, III and IV is to discard 
 the omitted parts and calculate the bearing and length of 
 a closing line. A triangle is then formed on this closing 
 line as a base and the missing parts computed. 
 
 Station. 
 
 Bearing. 
 
 Distance. 
 
 Latitudes. 
 
 Departure. 
 
 N+ 
 
 s- 
 
 E+ 
 
 W- 
 
 O 
 
 I 
 2 
 
 3 
 
 4 
 
 Ni6 30' E 
 N8 2 05' E 
 S 16 54' E 
 S 36 58' W 
 N49 14' W 
 
 22. IO 
 
 19.62 
 
 23-97 
 
 Omitted 
 Omitted 
 
 21 .190 
 2.7O2 
 
 
 6.276 
 
 19-433 
 6.968 
 
 
 22.935 
 
 
 
 
 23.892 
 22-935 
 
 22.935 
 
 32.677 
 
 
 0-957 
 
 Nat. tan of bearing = 
 
 Length of course 
 
 Bearing = S 88 19' W. 
 dep. 
 
 34-14524- 
 
 32.677 
 
 sin bearing sin 88 19' 
 
 = 3 ' '' = 32.691 chains. 
 0-99957 
 
 B/ 
 
 FIG. 177- 
 
 Draw a triangle (free hand) as shown 
 in Fig. 177 and from the bearings com- 
 pute angles A , B and C. 
 
 S88i 9 'W = S8 7 7 9 'W 
 S 36 58' W 
 
 angle B. 
 
 51 21' 
 
 85 72' = 86 12' = angle C 
 
TRIGONOMETRY 
 
 I 79 
 
 S 88 
 N 4 9 
 
 19' W 
 14' W 
 
 To Check 
 ' 
 
 137 33' 
 6o' 
 
 12 
 
 179 
 
 42 27' 
 
 & 
 
 4 2 
 
 = angle A 179 
 
 sin A sin 5 
 
 32.691 
 
 ,/ 32.691 \ 0.67495X32.691 
 
 .'. a = sin 42 27 f (-^-~ 7 = = 22. 1 1 chains 
 
 ' \sin86 127 0.99780 
 
 and 
 
 .78098X32.691 
 
 099780 
 
 CASE III. Bearings of two courses omitted. 
 
 This problem may have two solutions but the ambiguity 
 is practically unimportant as the surveyor is presumed to 
 have a clear idea of the shape of the field. 
 
 Station. 
 
 Bearing. 
 
 Distance. 
 
 Latitudes. 
 
 Departures. 
 
 N+ 
 
 s- 
 
 E+ 
 
 W- 
 
 O 
 I 
 2 
 3 
 4 
 
 Ni6 30' E 
 N8 2 05' E 
 S 16 54' E 
 Omitted 
 Omitted 
 
 22. TO 
 
 19.62 
 
 23-97 
 22.11 
 
 25-59 
 
 21 .190 
 2.7O2 
 
 
 6.276 
 
 19-433 
 6.968 
 
 
 22-935 
 
 
 
 
 
 
 
 
 23.892 
 22-935 
 
 
 32.677 
 
 
 0-957 
 
 This gives the same bearing and 
 length for the closing line as were 
 found for the problem illustrating 
 Case II. The triangle is shown in 
 Fig. 178.^ 
 
 If the instrument man does not make 
 a sketch in his notes to guide the com- 
 puter in the office, the latter might 
 assume the courses to run as indicated 
 by the dotted lines. The interior angles 
 
 -4V 
 
 FIG. 178. 
 
i8o 
 
 PRACTICAL SURVEYING 
 
 of the closing triangle will not be affected but the shape and 
 area of the field will differ considerably from the truth. 
 
 Lei s = 
 
 2559 + 32.69 
 
 A = 42 28'. 
 
 I4.6i X 7-5i 
 J5-59 X 32.69 
 
 = 40 20 
 
 = 0.36216. 
 
 fiSM 
 
 V 22.11 
 
 09 X 7-5i 
 
 ac r 22.11 X 32.69 
 
 B = 51 22'. 
 
 C = 180 -(A+B) = 179 60' - 93 50' 
 
 = 0.43355. 
 
 86io / . 
 
 S 88 19' W 
 
 51 22' L 
 
 a = S 36 57: 
 
 86 10' L 
 
 836 47' W = N 36 57' E 
 b = N 49 13' W 
 
 CASE IV. Length of one course and bearing of another 
 omitted. 
 
 The field is to be revolved until the course of which only 
 the bearing is known is in the meridian, thus throwing all 
 the unknown departure into the course of which only the 
 length is known. The omitted parts may be on adjacent or 
 non-adjacent courses. 
 
 Station. 
 
 Bearing. 
 
 Revolved to 
 left. 
 
 New bearing. 
 
 Distance. 
 
 
 
 Ni6 30' E 
 
 36 58' 
 
 N 20 28' W 
 
 22. IO 
 
 I 
 
 N8 2 05' E 
 
 36 58' 
 
 N 4 5 07' E 
 
 19.62 
 
 2 
 
 S 16 54' E 
 
 36 58' 
 
 S 53 52' E 
 
 23-97 
 
 3 
 
 4 
 
 S 36 58' W 
 Omitted 
 
 36 5 8' 
 
 South 
 Omitted 
 
 Omitted 
 
 2$ "?Q 
 
 
 
 
 
 
 When the computations are finished a doubt sometimes 
 arises as to whether the latitude of the course whose bear- 
 ing is omitted is a northing or a southing. This produces 
 two sets of values, either of which will satisfy the problem, 
 though one will give a wrong area. The knowledge the 
 surveyor has of the general directions of the courses and 
 the shape of the field must settle all such ambiguities. 
 
TRIGONOMETRY 
 
 181 
 
 The habit of making free-hand sketches to supplement the 
 field notes is one every instrument man should early ac- 
 quire. 
 
 Sta. 
 
 Bearing. 
 
 Distance. 
 
 Latitudes. 
 
 Departures. 
 
 N+ 
 
 S- 
 
 E+ 
 
 W- 
 
 o 
 
 I 
 
 2 
 
 3 
 
 4 
 
 N 20 28' W 
 
 N4So 7 / E 
 S 53 52' E 
 South 
 Omitted 
 
 22.10 
 19.62 
 23-97 
 
 Omitted 
 25-59 
 
 20.705 
 I3-845 
 
 H.I34 
 
 13.902 
 19-360 
 
 7.727 
 
 
 
 
 
 34-550 
 14.134 
 
 14-134 
 
 33.262 
 
 7.727 
 
 7.727 
 
 20.416 
 
 25-535 
 
 All the difference in departure is thrown into course 4, 
 the length of which is 25.59 chains. The following triangle 
 is obtained. 
 
 The sine of A = - 
 
 thus making the bearing of 
 course 4 = 86 14' but some un- FIG. 179. 
 
 certainty exists as to whether it 
 
 is a northing or a southing. From the differences in lati- 
 tudes and departures it is apparently S 86 14' W. 
 To settle the matter apply the angle of revolution. 
 
 86 
 36 
 
 14' W 
 
 123 
 179 
 
 12' 
 60' 
 
 N 56 48' W 
 
 The bearing was known (from the study of Cases I, II 
 and III) to be N 49 14' W so another trial must be made. 
 
 N86 
 36 
 
 14' W 
 
 58' R 
 
 N 49 16' W 
 
182 
 
 PRACTICAL SURVEYING 
 
 The bearing is evidently a northing. The difference of 
 two minutes is due to the computations in some of the 
 examples being carried to a greater degree of refinement 
 than in others. It will make no appreciable difference in 
 the length of course 4, but the difference in angle introduces 
 a closing error of about ij links. No survey is free from 
 error, but all errors being thrown into the courses supplied 
 by computation a false idea of accuracy is obtained by 
 doing work in the office that belongs to the field. "Meas- 
 ure in haste and repent in the office, " is a true proverb. 
 
 The latitude of the course = cosine 86 16' X 25.59 = 
 1 .68 1, and, this amount is to be added to the latitude 
 difference to obtain the southing for course 3, so the north- 
 ings and southings will balance. 
 
 Latitude for course 3 = 20.416 + 1.681 = 22.097. 
 
 The latitudes and departures are now 
 
 Station. 
 
 Lati.uJes. 
 
 Departures. 
 
 + 
 
 - 
 
 + 
 
 - 
 
 
 
 I 
 2 
 
 3 
 4 
 
 20.705 
 I3-84S 
 
 
 
 7.727 
 
 14-134 
 22.097 
 
 13.902 
 19.360 
 
 25-535 
 
 I.68I 
 
 36.231 
 
 36.231 
 
 33.262 
 
 33.262 
 
 and the bearings may be restored by revolving the field 
 to the right through an angle of 36 58'. 
 
 The field is not altered in size or shape by being revolved 
 so the area may be computed by using the values already 
 found for the latitudes and departures. 
 
 LOGARITHMS 
 
 A logarithm is an exponent, and in an earlier chapter it 
 was shown that 
 
 and 
 
 
 a. 
 
TRIGONOMETRY 183 
 
 A series of quantities which increase or decrease by a 
 common difference is called an Arithmetical Progression-; 
 as i, 2, 3, 4, 5, etc., or 75, 72, 69, 66, etc. 
 
 A series of quantities which increase by a constant multi- 
 plier, or decrease by a constant divisor, is called a Geometri- 
 cal Progression; as 2, 8, 32, 128, etc., the multiplier being 
 4; or 567, 189, 63, etc., the divisor being 3. 
 
 We can write down a line of figures in arithmetical pro- 
 gression over a line in geometrical progression, and the 
 upper line will contain the exponents of the second line so 
 that multiplication of two quantities in the lower line may 
 be accomplished by adding their exponents. 
 
 0.5 0.4 0.3 02 o.i 01234 5 6 
 
 O.OOOOI O.OOOI O.OOI O.OI O.I I IO IOO IOOO IO.OOO IOO.OOO I.OOO.OOO 
 
 Add 2 and 3, the sum is 5. Under 5 is found 100,000 
 which is plainly the product of 100 (under 2) and 1000 
 (under 3). 
 
 Subtract 4 from 5, the difference is i. Under 5 is found 
 100,000 and under 4 is found 10,000. Under I is found 
 
 100,000 
 10 = - 
 10,000 
 
 Multiply 2 by 3, the product is 6. Under 6 is found 
 1,000,000 which is the square of 1000, found under 3. 
 
 Divide 6 by 2, the quotient is 3. Under 3 is found 1000 
 which is the square root of 1,000,000 found under 6. 
 
 The student for an exercise may carry these series, as- 
 cending and descending, to 10 or more places each way, 
 and multiply and divide, raise to powers and extract roots, 
 keeping always in an ascending or a descending series. 
 
 Logarithms are a series of artificial numbers used as 
 exponents for numbers placed in arithmetical progression. 
 By their use addition takes the place of multiplication and 
 subtraction takes the place of division. Numbers are 
 raised to any power by multiplying the logarithm of the 
 number by the exponent representing that power, and roots 
 are extracted by dividing the logarithm by the index of 
 the required root. 
 
 Logarithms are useful for many calculations, especially 
 those involving proportion, or a combination of multi- 
 plication and division. 
 
184 PRACTICAL SURVEYING 
 
 An expression containing a positive or negative sign is not 
 in shape for logarithmic computation and must be differ- 
 ently written to eliminate every such sign. For example 
 
 sin (x + y) + sin (x y) 
 
 can be adapted for logarithmic computation by writing it 
 as follows : 
 
 2 sin x X cos y. 
 
 Adding logarithms is equivalent to multiplying their 
 numbers, and subtracting logarithms is equivalent to per- 
 forming the operation of division with their numbers. 
 
 Some surveyors never use logarithms. Other surveyors 
 use them upon every occasion. Whether to use "logs" or 
 "naturals" is something each man must settle for himself 
 and is not worth discussing. For involved calculations 
 logarithms save time and lessen the liability of error. 
 
 TO USE A TABLE OF LOGARITHMS 
 
 A complete logarithm consists of two parts. The man- 
 tissa is the decimal part printed in the table. The charac- 
 teristic is a whole number and depends upon the number 
 of figures in the number. 
 
 The log. of 2500 is 3.3979 of which 0.3979 is the mantissa 
 taken from the table and 3 is the characteristic. 
 
 The log. of 8435 = 3.926085. The log. of 8.435 = 0.926085. 
 The log. of 843.5 = 2.926085. The log. of 0.8435 = 1.926085. 
 The log. of 84.35 = 1.926085. The log. of 0.08435 = - 2.926085. 
 
 The characteristic is seen by the above illustration to be 
 a number indicating I less than the number of integral 
 figures of which the number consists. In decimal num- 
 bers the characteristic is negative and indicates the distance 
 of the first significant figure from the decimal point. 
 
 In some tables of logarithms there is a column headed 
 "Tabular Difference." In the table in this book the 
 tabular difference is not shown. The logarithm of 243 = 
 2.38561 and the logarithm of 244 = 2.38739. The differ- 
 ence is 0.00178 and would be printed as Tab. Diff. 178. 
 The Tabular Difference for each horizontal line is therefore 
 the average difference between the units figures on the line. 
 
TRIGONOMETRY 185 
 
 TO FIND THE LOGARITHM OF A NUMBER 
 
 First fix the characteristic. Then look in the table for 
 the mantissa. The number may have only three figures, 
 in which case the mantissa will be found at once. If it 
 contains four or more figures the mantissa for the three 
 first figures is found and the difference between this and the 
 next higher figure used in the following way : 
 
 Find the log. of 2369. 
 
 The characteristic = 3. The mantissa for 2360 = 
 0.37291. To find the mantissa find 23 in the column 
 headed "No.," and proceed horizontally to the right to 
 the column headed "6." 
 
 The remainder of 9 cannot be ignored. Find the man- 
 tissa for 237 = 0.37475. The difference between 0.37475 
 and 0.37291 = 0.00184 an d this must be multiplied by 0.9 = 
 0.001656, the product being added to the mantissa for 2360. 
 The complete log. is as follows : 
 
 Mantissa of 236 = 0.37291 
 Mantissa of 0.9 = 0.00166 
 
 0-37457 
 Log. of 2369 = 3-37457- 
 
 The reason for designating the 9 remainder as 0.9 may be 
 found by considering the mantissas as being given in the 
 table for 2370 and for 2360, but not for fractional parts, 
 therefore 9 is nine-tenths the difference between 60 and 70. 
 Had the number been 236,924 the process would have been 
 the same but the characteristic would be 5 and the differ- 
 ence 0.00184 would be multiplied by 0.924 = 0.00170. 
 
 Mantissa of 236 = 0.37291 
 Mantissa of 0.924 = 0.00179 
 
 0.37470 
 Log. of 236924 = 5-37470. 
 
 Tables of logarithms having a column of tabular differ- 
 ences show the difference of 0.00184 as a whole number, 
 184, to save space. 
 
 The student must remember that the mantissa is a deci- 
 mal fraction and therefore the tabular difference is a decimal 
 fraction containing the same number of figures, the differ- 
 
1 86 PRACTICAL SURVEYING 
 
 ence being made up by placing ciphers in front of the 
 significant figures. 
 
 To find the number corresponding to a given logarithm. 
 
 The log. is 3-37457- 
 
 In the table of logs, find the mantissa nearest to the given 
 mantissa but below. This we find to be 0.37291, corre- 
 sponding to number 2360, for as we have 3 for a character- 
 istic there must be four figures in the number. 
 
 Given log. = 3-37457 
 
 Log. of 2360 = 3.37291 
 Diff. = 0.00166 
 
 Tabular difference between mantissas of 2370 and 2360 = 
 0.00184. 
 
 0.00166 
 
 which gives a number = 2369.13. The 0.13 is an exceed- 
 ingly small per cent of 2369 but proves that a table of 
 logarithms of numbers from o to 1000 is not accurate for 
 numbers containing more than four figures, when the man- 
 tissa is carried to only five places. Tables of logarithms 
 from o to 10,000 are accurate for numbers containing as 
 many figures as there are decimal places in the mantissa. 
 Tables of logarithms from o to 108,000 are accurate enough 
 for practical use for numbers containing one figure more 
 than the decimal figures in the mantissa. For all the work 
 of surveyors the five-place table here given is accurate 
 enough for all practical purposes for numbers containing 
 not more than five figures. A table for numbers from o 
 to 10,000 is more convenient to use. In the examples 
 following notice the inaccuracy of the work due to the 
 table used. 
 
 To multiply two numbers by using logs. 
 
 I. 54.3 X 6.19 = ? 
 
 Log. 54.3 = 1.73480 
 
 Log. 6.19 = 0.79169 
 
 2.52649 
 
 Log. of 336 = 2.52633 
 0.00016 
 
TRIGONOMETRY 187 
 
 Tab. diff. 336 and 337 = 0.52763 - 0.52633 = 0.00130. 
 
 0.00016 
 
 - = 0.123. 
 0.00130 
 
 By logs. 54.3 X 6.19 = 336.123. 
 By arithmetic =336.117. 
 
 2. 54.3 X 6.19 X 27 = ? 
 
 Log. 54.3 = 1.73480 
 
 Log. 6.19 = 0.79169 
 
 Log. 27 = 1.43136 
 
 3.95785 
 Log. of 9070 = 3.95761 
 
 0.00024 
 
 Tab. diff. 9070 and 9080 = 0.95809 0.95761 = 0.00048. 
 0.00024 _ 
 0.00048 ~ 
 
 By logs. 54.3 X 6.19 X 27 = 9075.000. 
 By arithmetic = 9075.159. 
 
 The rule for multiplying numbers by using logs, is to add 
 the logarithms of the numbers and find from the table the 
 number corresponding to the new logarithm thus obtained. 
 
 To divide a number by another by using logs. 
 
 54-3 = ? 
 
 6.19 
 
 Log. 54.3 = i.7348o 
 Log. 6.19 = 0.79169 
 
 0.943H 
 Log. 8.77 = 0.94300 
 
 Diff. = o.oooi i 
 
 Tab. diff. 878 and 877 = 0.94349 0.94300 = 0.00049. 
 
 o.oooi i 
 
 - = 0.225. 
 0.00049 
 
 By logs. 54.3 + 6.19 = 8.77225. 
 By arithmetic = 8.77205. 
 
 To divide one number by another find the difference 
 between their logarithms. The resulting logarithm is the 
 log. of the quotient of the numbers. 
 
l88 PRACTICAL SURVEYING 
 
 To raise a number to any power. 
 Find the square of 6.19. 
 
 Log. 6.19 = 0.79169 
 
 2 
 
 1.58338 
 
 Log- 38.3 = 1-58320 
 0.00018 
 
 Diff. = 0.00018 _ 
 Tab. diff. " 0.00133 " 
 
 6TQ2 (By logs. = 38.3135- 
 
 9 (By arithmetic = 38.3161. 
 
 The rule, which is general, is to multiply the log. by the 
 exponent of the power. The number corresponding to the 
 log. thus obtained is the power sought. 
 
 To extract the root of any number. 
 
 What is the square root of 54.3? 
 
 Log. 54-3 = '-^ = 0.86740 
 Log. 7.36 = 0.86688 
 
 0.00052 
 
 = 0-QOQ52 = Q 8gl 
 Tab. diff. 0.00059 
 /-(By logs. =7.36881. 
 
 v 543JBy arithmetic = 7.368 + . 
 
 The rule, which is general, is to divide the log. of the 
 number by the index of the root. The number correspond- 
 ing to the log. thus obtained is the root sought. 
 
 ARITHMETICAL COMPLEMENT 
 
 A logarithm may be mentally subtracted from 10, an 
 integer, or the right-hand figure may be subtracted from 
 10, and all the rest from 9. 
 
 By using the arithmetical complement of a logarithm, 
 division instead of being subtraction becomes addition. 
 That is, adding an arithmetical complement is equivalent 
 to subtracting its logarithm. In the product, 10 must be 
 subtracted from the characteristic. 
 
TRIGONOMETRY 189 
 
 NEGATIVE CHARACTERISTIC 
 
 The characteristic is negative when the number is a 
 decimal fraction. 
 
 A negative characteristic must be subtracted when the 
 logarithm is added and must be added when the logarithm 
 is subtracted. 
 
 After multiplying a negative index subtract from the 
 resulting index the amount carried from the mantissa. 
 
 Example. Raise 0.009 to the third power. 
 
 Log. 0.009 = 3-95424 
 
 3 
 
 7.86272 
 
 The 7 was obtained as follows : 2 was carried from the 
 multiplication of the mantissa. The product of 3 X (3) 
 = 9 and +2 9= 7. It is customary to place the 
 negative sign above the characteristic instead of placing 
 it in front. 
 
 Sometimes a positive characteristic is used, in which 
 case 10 times the exponent of the power lessened by I 
 must be taken from the final characteristic, and the result 
 added to 10. 
 
 Example. Raise 0.0437 to the fourth power. 
 
 Neg. char. Pos. char. 
 
 Log. 0.0437 = 2.64048 or 8.64048 
 
 = 4 4 
 
 Log. 0.000003649 = 6.56192 + 4.56192 
 
 IP 
 
 6.56192 
 
 The 6 was obtained when the negative characteristic was 
 used by subtracting the 2 carried from the mantissa from 
 the product of 2 X 4 = 8. 
 
 In the case of the positive characteristic 4 X 8 = 32 and 
 adding the 2 carried from the mantissa the result was 34. 
 From this was subtracted 10 X (4 i) = 30. Then 
 +4 10 = 6. 
 
 When extracting roots if the given number be a decimal, 
 and its characteristic positive, subtract I from the index of 
 
190 PRACTICAL SURVEYING 
 
 the root and add the remainder to the characteristic before 
 dividing. 
 
 If the characteristic be negative, add to it the least num- 
 ber that will make the sum divisible by the index of the 
 root; the quotient is the characteristic of the log. of the 
 root. In dividing the mantissa, only the number added is 
 to be considered as the characteristic. 
 
 Example. ^0.00130321 = ? 
 Log. 0.00130321 = 3 
 Add i 
 
 Divide by 4 = 1.27875 
 
 The method by which the characteristic I was obtained 
 is clearly seen. In dividing the mantissa the character- 
 istic 3 was ignored and the I substituted. 
 
 To WORK PROPORTION BY LOGARITHMS 
 
 Add the logarithms of the second and third terms to- 
 gether. From their sum subtract the logarithm of the first 
 term. The remainder is the logarithm of the fourth term. 
 
 Or; add together the arithmetical complement of the 
 first term and the logarithms of the other two. The sum, 
 with 10 subtracted from the characteristic, is the logarithm 
 of the answer. 
 
 Example. Illustrating the two methods. 
 
 36 : 144 :: 28 : x 
 Log. 144 = 2.15836 2.15836 
 
 Log. 28 = 1.44716 1.44716 
 
 3-60552 
 Log. 36 = 1.55630 a.c. = 8.44370 
 
 2.04922 2.04922 
 
 X = 112. 
 
 (The 10 is subtracted mentally.) 
 
 LOGARITHMIC FUNCTIONS OF ANGLES 
 
 Logarithmic functions of angles are merely logarithms 
 of the natural functions. Characteristics are placed in the 
 tables but as the natural functions are decimal fractions 
 
TRIGONOMETRY 191 
 
 the actual characteristics are negative. The numbers 
 used as characteristics are positive for convenience, being 
 the difference between 10 and the true, negative charac- 
 teristic. Tables of logarithmic functions are for use with 
 tables of logarithms of numbers. 
 
 Example. Refer to example in Case III of Omissions, 
 and compare the work required when using "naturals" 
 and using "logs. " 
 
 \l 
 
 V 
 
 I4>61 x 
 
 25-59 X 32-69 
 Log. 14.61 = 1.16465 
 Log. 7.51 = 0.87564 2.04029 
 
 Log. 25.59 = 140807 
 Log. 32.69 = 1.51442 
 
 2.92249 a.c. = 7.07751 
 
 9.11780 = 1.11780 
 1.11780 
 i 
 
 T.55890 = 9-55890 = sin 21 14' 
 A = 21 14' X 2 = 42 28'. 
 
 The student, for exercise, should work all the examples in 
 this chapter by using logarithms. In this way a compar- 
 ison can be made of the advantages and disadvantages of 
 logarithms and their limitations in some kinds of work, 
 bearing in mind the degree of accuracy possible with the 
 small table of logs, in this book. 
 
 PROPORTIONAL PARTS IN LOGARITHMIC TABLES 
 
 The Five Place Logarithmic-Trigonometric Tables by 
 Constantine Smoley, C.E., are printed on a thin, tough 
 bond paper and are bound in flexible cloth. The price is 
 50 cents. They consist of logarithms of numbers from 
 I to 10,000; Logarithms of the sine and tangent varying by 
 ten seconds from o to 3 and of the cosine and cotangent 
 varying by ten seconds from 87 to 90; logarithms of the 
 sine, cosine, tangent and cotangent, secant and cosecant 
 for each minute of arc; tables of squares, cubes, square and 
 cube roots, etc. 
 
I 9 2 
 
 PRACTICAL SURVEYING 
 
 In the table of logarithms of numbers a star is frequently 
 used as follows: 
 
 N. 
 
 o 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 426 
 
 941 
 
 951 
 
 96l 
 
 972 
 
 982 
 
 992 
 
 *002 
 
 *OI2 
 
 *022 
 
 *033 
 
 427 
 
 63,043 
 
 053 
 
 06 3 
 
 073 
 
 08 3 
 
 094 
 
 104 
 
 114 
 
 124 
 
 134 
 
 The mantissa is carried to five figures but the first two 
 figures are omitted wherever possible in order to save space 
 and make the tables more easily read. The mantissa of 
 4260 is 62,941, of 4265 it is 62,992, etc. The star in front of 
 the mantissa for 4266 indicates that the first two figures 
 are increased, so this mantissa is 63,002; the mantissa for 
 4268 is 63,022, etc. 
 
 On the right hand of each page is a column headed P.P. 
 (Proportional Parts) in which figures appear as follows: 
 
 10 
 
 I.O 
 2.O 
 
 3-0 
 4-0 
 
 .0 
 .0 
 
 7-o 
 8.0 
 9.0 
 
 i:: 
 
 Example. Find the log. of 426,758. 
 The tabular difference between logs, of 4267 and 4268 = 
 10, which number heads the P.P. column. 
 
 Log. of 426,700 = 5.63012 
 P.P. 50 = 0.000050 
 
 P.P. 8 = 0.000008 
 
 5.630178 
 
 (Note. These values of P.P. for differences of 50 and 8 
 apply only to tab. diff. of 10. 
 
 When tab. diff. = 14 then P.P. for 5 = 7.0 and for 8 = 
 1 1. 2, etc.) 
 
TRIGONOMETRY 193 
 
 Example. Find number corresponding to log. 5.63018. 
 
 Log- 5-63018 
 
 No. 426,700 = Log. 5.63012 
 0.00006 
 
 The difference between log. of 426,700 and 426,800 = 10, 
 so in the P.P. column find 10. On the right of the vertical 
 line in the column find 6 and on the left of the line is found 
 6.0 which is placed after the 7 in the number and we thus 
 obtain 
 
 No. 426,760 log. = 5.63018. 
 
 In the tables of logarithms of functions of angles the P.P. 
 column refers to seconds. Few surveyors or engineers 
 read angles closer than the nearest minute and few problems 
 are encountered in which seconds appear in the angles. 
 When the angles do contain seconds the proportional parts 
 are used in the way just described for logarithms of num- 
 bers. For angles smaller than 3 the differences are so 
 great between consecutive minutes that a separate table is 
 given in which the values of the functions differ by 10" 
 and the P.P. column gives the differences for single seconds. 
 With the explanation given, the student can with a little 
 practice use the columns of proportional parts readily. 
 
194 
 
 PRACTICAL SURVEYING 
 NATURAL FUNCTIONS OF ANGLES 
 
 Deg. 
 
 Min. 
 
 Sine. 
 
 Cosec. 
 
 Tang. 
 
 Cotang. 
 
 Sec. 
 
 Cosine. 
 
 Min. 
 
 Deg. 
 
 
 
 
 
 o.oooooo 
 
 Infinite. 
 
 o.oooooo 
 
 Infinite. 
 
 .00000 
 
 I. 000000 
 
 
 
 90 
 
 
 10 
 
 0.002909 
 
 343.77516 
 
 0.002909 
 
 343-77371 
 
 .00000 
 
 0.999996 
 
 So 
 
 
 
 20 
 
 0.005818 
 
 171.88831 
 
 0.005818 
 
 171.88540 
 
 .00002 
 
 0.999983 
 
 40 
 
 
 
 30 
 
 0.008727 
 
 114.59301 
 
 0.008727 
 
 114.58865 
 
 .00004 
 
 0.999962 
 
 30 
 
 
 
 40 
 
 0.011635 
 
 85.945609 
 
 0.011636 
 
 85.939791 
 
 .00007 
 
 0-999932 
 
 20 
 
 
 
 So 
 
 0.014544 
 
 68.75736o 
 
 0.014545 
 
 68.750087 
 
 .OOOII 
 
 0.999894 
 
 IO 
 
 
 1 
 
 o 
 
 0.017452 
 
 57.298688 
 
 0.017455 
 
 57.289962 
 
 .00015 
 
 0.999848 
 
 
 
 89 
 
 
 10 
 
 0.020361 
 
 49.114062 
 
 0.020365 
 
 49.103881 
 
 .00021 
 
 0.999793 
 
 50 
 
 
 
 20 
 
 0.023269 
 
 42.975713 
 
 0.023275 
 
 42.964077 
 
 .00027 
 
 0.999729 
 
 40 
 
 
 
 30 
 
 0.026177 
 
 38.201550 
 
 0.026186 
 
 38.188459 
 
 .00034 
 
 0.999657 
 
 30 
 
 
 
 40 
 
 o. 023085. 
 
 34.382316 
 
 0.029097 
 
 34.367771 
 
 .00042 
 
 0.999577 
 
 20 
 
 
 
 So 
 
 0.031992 
 
 31.257577 
 
 0.032009 
 
 31.241577 
 
 .00051 
 
 0.999488 
 
 IO 
 
 
 2 
 
 o 
 
 0.034899 
 
 28.653708 
 
 0.034921 
 
 28.636253 
 
 .00061 
 
 0.999391 
 
 
 
 88 
 
 
 IO 
 
 0.037806 
 
 26.450510 
 
 0.037834 
 
 26.431600 
 
 .00072 
 
 0.999285 
 
 So 
 
 
 
 20 
 
 0.040713 
 
 24.562123 
 
 0.040747 
 
 24.541758 
 
 .00083 
 
 0.999171 
 
 40 
 
 
 
 30 
 
 0.043619 
 
 22.925586 
 
 0.043661 
 
 22.903766 
 
 .00095 
 
 0.999048 
 
 30 
 
 
 
 40 
 
 0.046525 
 
 21.493676 
 
 0.046576 
 
 21.470401 
 
 .00108 
 
 0.998917 
 
 20 
 
 
 
 So 
 
 0.049431 
 
 20.230284 
 
 0.049491 
 
 20.205553 
 
 .00122 
 
 0.998778 
 
 10 
 
 
 3 
 
 
 
 0.052336 
 
 19.107323 
 
 0.052408 
 
 19.081137 
 
 .00137 
 
 0.998630 
 
 
 
 87 
 
 
 10 
 
 0.055241 
 
 18.102619 
 
 0.055325 
 
 18.074977 
 
 .00153 
 
 0.998473 
 
 So 
 
 
 
 20 
 
 0.058145 
 
 17.198434 
 
 0.058243 
 
 I7.I69337 
 
 .00169 
 
 0.998308 
 
 40 
 
 
 
 30 
 
 0.061049 
 
 16.380408 
 
 0.061163 
 
 16.349855 
 
 .00187 
 
 o.998i3S 
 
 30 
 
 
 
 40 
 
 0.063952 
 
 15-636793 
 
 0.064083 
 
 15.604784 
 
 .00205 
 
 0.997357 
 
 20 
 
 
 
 50 
 
 0.066854 
 
 14.957882 
 
 0.067004 
 
 14.924417 
 
 .00224 
 
 0.997763 
 
 10 
 
 
 4 
 
 o 
 
 0.069756 
 
 14.335587 
 
 0.069927 
 
 14.300666 
 
 .00244 
 
 0.997564 
 
 O 
 
 86 
 
 
 IO 
 
 0.072658 
 
 13.763115 
 
 0.072851 
 
 13.726738 
 
 .00265 
 
 0.997357 
 
 50 
 
 
 
 20 
 
 0.075559 
 
 13.234717 
 
 0.075776 
 
 13.196888 
 
 .00287 
 
 0.997141 
 
 40 
 
 
 
 30 
 
 0.078459 
 
 12.745495 
 
 o . 078702 
 
 12.706205 
 
 .00309 
 
 0.996917 
 
 30 
 
 
 
 40 
 
 0.081359 
 
 12.291252 
 
 0.081629 
 
 12.250505 
 
 .00333 
 
 0.996685 
 
 20 
 
 
 
 So 
 
 0.084258 
 
 11.868370 
 
 0.084558 
 
 11.826167 
 
 .00357 
 
 0.996444 
 
 10 
 
 
 5 
 
 
 
 0.087156 
 
 U.4737I3 
 
 0.087489 
 
 11.430052 
 
 .00382 
 
 o.996i95 
 
 
 
 85 
 
 
 10 
 
 0.090053 
 
 11.104549 
 
 0.090421 
 
 11.059431 
 
 .00408 
 
 0.995937 
 
 50 
 
 
 
 20 
 
 o . 092950 
 
 10.758488 
 
 0.093354 
 
 10.711913 
 
 .00435 
 
 0.995671 
 
 40 
 
 
 
 30 
 
 o . 095846 
 
 10.433431 
 
 0.096289 
 
 10.385397 
 
 .00463 
 
 0.995396 
 
 30 
 
 
 
 40 
 
 0.098741 
 
 10.127522 
 
 o . 099226 
 
 10.078031 
 
 .00491 
 
 0-995II3 
 
 20 
 
 
 
 So 
 
 0.101635 
 
 9.8391227 
 
 0.102164 
 
 9.7881732 
 
 .00521 
 
 0.994822 
 
 IO 
 
 
 6 
 
 o 
 
 0.104528 
 
 9.5667722 
 
 o . 1 051x34 
 
 9.SI43645 
 
 .00551 
 
 0.994522 
 
 
 
 84 
 
 
 10 
 
 o . 107421 
 
 9.3091699 
 
 0.108046 
 
 9.2553035 
 
 .00582 
 
 0.994214 
 
 50 
 
 
 
 20 
 
 0.110313 
 
 9.0651512 
 
 0.110990 
 
 9.0098261 
 
 .00614 
 
 0.993897 
 
 40 
 
 
 
 30 
 
 0.113203 
 
 8.8336715 
 
 0.113936 
 
 8.7768874 
 
 .00647 
 
 0.993572 
 
 30 
 
 
 
 40 
 
 0.116093 
 
 8.6137901 
 
 0.116883 
 
 8.5555468 
 
 .00681 
 
 0.993238 
 
 20 
 
 
 
 So 
 
 0.118982 
 
 8.4045586 
 
 0.119833 
 
 8.3449558 
 
 .00715 
 
 0.992896 
 
 10 
 
 83 
 
 Deg. 
 
 Min. 
 
 Cosine. 
 
 Sec. 
 
 Cotang. 
 
 Tang. 
 
 Cosec. 
 
 Sine. 
 
 Min. 
 
 Deg. 
 
 For functions from 83 10' to 90 read fronvbottom of table upward. 
 
TRIGONOMETRY 
 
 NATURAL FUNCTIONS OF ANGLES (Continued) 
 
 195 
 
 Deg. 
 
 Min. 
 
 Sine. 
 
 Cosec. 
 
 Tang. 
 
 Cotang. 
 
 Sec. 
 
 Cosine. 
 
 Min. 
 
 Deg. 
 
 7 
 
 o 
 
 0.121869 
 
 8.2055090 
 
 0.122785 
 
 8.1443464 
 
 .00751 
 
 0.992546 
 
 o 
 
 83 
 
 
 10 
 
 0.124756 
 
 8.0156450 
 
 0.125738 
 
 7.9530224 
 
 .00787 
 
 0.992187 
 
 So 
 
 
 
 , 2Q 
 
 0.127642 
 
 7.8344335 
 
 0.128694 
 
 7.7703506 
 
 .00825 
 
 0.991820 
 
 40 
 
 
 
 " 30 
 
 0.130526 
 
 7 . 6612976 
 
 0.131653 
 
 7-5957541 
 
 .00863 
 
 0.991445 
 
 30 
 
 
 
 40 
 
 0.133410 
 
 7.4957100 
 
 0.134613 
 
 7.4287064 
 
 .00902 
 
 0.991061 
 
 20 
 
 
 
 50 
 
 0.136292 
 
 7-3371909 
 
 O.I37S76 
 
 7.2687255 
 
 .00942 
 
 0.990669 
 
 IO 
 
 
 8 
 
 
 
 O.I39I73 
 
 7-1852965 
 
 0.140541 
 
 7.H53697 
 
 .00983 
 
 0.990268 
 
 
 
 88 
 
 
 10 
 
 0.142053 
 
 7.0396220 
 
 0.143508 
 
 6.9682335 
 
 .01024 
 
 0.989859 
 
 50 
 
 
 
 20 
 
 0.144932 
 
 6.8997942 
 
 0.146478 
 
 6.8269437 
 
 .01067 
 
 0.989442 
 
 40 
 
 
 
 30 
 
 0.147809 
 
 6.7654691 
 
 O.I4945I 
 
 6.6911562 
 
 .OIIII 
 
 0.989016 
 
 30 
 
 
 
 40 
 
 0.150686 
 
 6.6363293 
 
 0.152426 
 
 6.5605538 
 
 .01155 
 
 0.988582 
 
 20 
 
 
 
 50 
 
 o.i5356i 
 
 6.5120812 
 
 0.155404 
 
 6.4348428 
 
 .01200 
 
 0.988139 
 
 10 
 
 
 9 
 
 
 
 0.156434 
 
 6.3924532 
 
 0.158384 
 
 6.3I375I5 
 
 .01247 
 
 0.987688 
 
 
 
 81 
 
 
 10 
 
 0.159307 
 
 6.2771933 
 
 0.161368 
 
 6.1970279 
 
 .01294 
 
 0.987229 
 
 50 
 
 
 
 20 
 
 0.162178 
 
 6.1660674 
 
 0.164354 
 
 6.0844381 
 
 .01342 
 
 0.986762 
 
 40 
 
 
 
 30 
 
 0.165048 
 
 6.0588980 
 
 0.167343 
 
 5.9757644 
 
 .01391 
 
 0.986286 
 
 30 
 
 
 
 40 
 
 0.167916 
 
 5.9553625 
 
 0.170334 
 
 5.8708042 
 
 .01440 
 
 0.985801 
 
 20 
 
 
 
 50 
 
 0.170783 
 
 S.855392I 
 
 0.173329 
 
 5.7693688 
 
 .01491 
 
 0.985309 
 
 10 
 
 
 10 
 
 o 
 
 0.173648 
 
 5.7587705 
 
 0.176327 
 
 5.6712818 
 
 .01543 
 
 0.984808 
 
 
 
 80 
 
 
 10 
 
 0.176512 
 
 5.6653331 
 
 0.179328 
 
 5.5763786 
 
 .01595 
 
 0.984298 
 
 50 
 
 
 
 20 
 
 0.179375 
 
 5.5749258 
 
 0.182332 
 
 5.4845052 
 
 .01649 
 
 0.983781 
 
 40 
 
 
 
 30 
 
 0.182236 
 
 5.4874043 
 
 0.185339 
 
 5.3955172 
 
 .01703 
 
 0.983255 
 
 30 
 
 
 
 40 
 
 0.185095 
 
 5.4026333 
 
 0.188359 
 
 5.3092793 
 
 .01758 
 
 0.982721 
 
 20 
 
 
 
 So 
 
 0.187953 
 
 5.3204860 
 
 0.191363 
 
 5.2256647 
 
 .01815 
 
 0.982178 
 
 10 
 
 
 11 
 
 
 
 0.190809 
 
 5.2408431 
 
 0.194380 
 
 5.1445540 
 
 .01872 
 
 0.981627 
 
 O 
 
 79 
 
 
 10 
 
 0.193664 
 
 5.1635924 
 
 0.197401 
 
 5.0658352 
 
 .01930 
 
 0.981068 
 
 So 
 
 
 
 20 
 
 0.196517 
 
 5.0886284 
 
 o . 200425 
 
 4.9894027 
 
 .01989 
 
 o . 980500 
 
 40 
 
 
 
 30 
 
 0.199368 
 
 5.0158317 
 
 0.203452 
 
 4.9I5I570 
 
 . 02049 
 
 0.979925 
 
 30 
 
 
 
 40 
 
 0.202218 
 
 4.9451687 
 
 o . 206483 
 
 4.8430045 
 
 .02110 
 
 0-979341 
 
 20 
 
 
 
 So 
 
 0.205065 
 
 4.8764907 
 
 0.209518 
 
 4.7728568 
 
 .02171 
 
 0.978748 
 
 IO 
 
 
 12 
 
 o 
 
 0.207912 
 
 .8097343 
 
 0.212557 
 
 4.7046301 
 
 .02234 
 
 0.978148 
 
 
 
 78 
 
 
 IO 
 
 0.210756 
 
 .7448206 
 
 0.215599 
 
 4.6382457 
 
 ,02298 
 
 0.977539 
 
 50 
 
 
 
 20 
 
 0.213599 
 
 .6816748 
 
 0.218645 
 
 4-5736287 
 
 .02362 
 
 0.976921 
 
 40 
 
 
 
 30 
 
 0.216440 
 
 .6202263 
 
 0.221695 
 
 4.5107085 
 
 .02428 
 
 0.976296 
 
 30 
 
 
 
 40 
 
 0.219279 
 
 .5604080 
 
 0.224748 
 
 4.4494181 
 
 .02494 
 
 0.975662 
 
 20 
 
 
 
 50 
 
 0.222116 
 
 .5021565 
 
 0.227806 
 
 4-3896940 
 
 .02562 
 
 0.975020 
 
 10 
 
 
 13 
 
 
 
 0.224951 
 
 4.4454115 
 
 0.230868 
 
 4.3314759 
 
 . 02630 
 
 0-974370 
 
 O 
 
 77 
 
 
 10 
 
 0.227784 
 
 4.3901158 
 
 0.233934 
 
 4.2747066 
 
 .O27OO 
 
 0.973712 
 
 50 
 
 
 
 20 
 
 0.230616 
 
 4.3362150 
 
 0.237004 
 
 4.2193318 
 
 .02770 
 
 0.973045 
 
 40 
 
 
 
 30 
 
 0.233445 
 
 4.2836576 
 
 0.240079 
 
 4.1652998 
 
 .02842 
 
 0.972370 
 
 30 
 
 
 
 40 
 
 0.236273 
 
 4.2323943 
 
 0.243158 
 
 4.1125614 
 
 .02914 
 
 0.971687 
 
 20 
 
 
 
 So 
 
 0.239098 
 
 4.1823785 
 
 0.246241 
 
 4.0610700 
 
 .02987 
 
 0.970995 
 
 10 
 
 76 
 
 Deg. 
 
 Min. 
 
 Cosine. 
 
 Sec. 
 
 Cotang. 
 
 Tang. 
 
 Cosec. 
 
 Sine. 
 
 Min. 
 
 Deg. 
 
 For functions from 76 10' to 83 oo' read from bottom of table upward. 
 
196 PRACTICAL SURVEYING 
 
 NATURAL FUNCTIONS OF ANGLES (Continued) 
 
 Deg. 
 
 Min. 
 
 Sine. 
 
 Cosec. 
 
 Tang. 
 
 Cotang. 
 
 Sec. 
 
 Cosine. 
 
 Min. 
 
 Deg. 
 
 14 
 
 
 
 0.241922 
 
 4.I33S655 
 
 0.249328 
 
 4 . 0107809 
 
 .03061 
 
 0.970296 
 
 o 
 
 76 
 
 
 10 
 
 0.244743 
 
 4.0859130 
 
 0.252420 
 
 3.9616518 
 
 .03137 
 
 0.969588 
 
 So 
 
 
 
 20 
 
 0.247563 
 
 4-0393804 
 
 0.255517 
 
 3.9136420 
 
 .03213 
 
 0.968872 
 
 40 
 
 
 
 30 
 
 0.250380 
 
 3.9939292 
 
 0.258618 
 
 3.8667131 
 
 .03290 
 
 0.968148 
 
 30 
 
 
 
 40 
 
 0.253195 
 
 3.9495224 
 
 0.261723 
 
 3.8208281 
 
 .03363 
 
 0.967415 
 
 '20 
 
 
 
 50 
 
 0.256008 
 
 3.9061250 
 
 0.264834 
 
 3.7759519 
 
 03447 
 
 0.966675 
 
 10 
 
 
 IS 
 
 o 
 
 0.258819 
 
 3.8637033 
 
 0.267949 
 
 3.7320508 
 
 .03528 
 
 0.965926 
 
 O 
 
 75 
 
 
 10 
 
 0.261628 
 
 3.8222251 
 
 0.271069 
 
 3.6890927 
 
 .03609 
 
 0.965169 
 
 So 
 
 
 
 20 
 
 0.264434 
 
 3.7816596 
 
 0.274195 
 
 3.6470467 
 
 .03691 
 
 0.964404 
 
 40 
 
 
 
 30 
 
 0.267238 
 
 3.7419775 
 
 0.277325 
 
 3.6058835 
 
 .03774 
 
 0.963630 
 
 30 
 
 
 
 40 
 
 0.270040 
 
 3.7031506 
 
 0.280460 
 
 3-5655749 
 
 .03858 
 
 0.962849 
 
 20 
 
 
 
 So 
 
 o . 272840 
 
 3.6651518 
 
 0.283600 
 
 3.5260938 
 
 03944 
 
 0.962059 
 
 10 
 
 
 16 
 
 
 
 0.275637 
 
 3-6279553 
 
 0.286745 
 
 3.4874144 
 
 .04030 
 
 0.961262 
 
 o 
 
 74 
 
 
 10 
 
 0.278432 
 
 3.5915363 
 
 0.289896 
 
 3-4495120 
 
 .04117 
 
 0.960456 
 
 So 
 
 
 
 20 
 
 0.281225 
 
 3.55587io 
 
 0.293052 
 
 3.4123626 
 
 .04206 
 
 0.959642 
 
 40 
 
 
 
 30 
 
 0.284015 
 
 3.5209365 
 
 0.296214 
 
 3-3759434 
 
 .04295 
 
 0.958820 
 
 30 
 
 
 
 40 
 
 0.286803 
 
 3.4867110 
 
 0.299380 
 
 3.3402326 
 
 .04385 
 
 0.957990 
 
 20 
 
 
 
 So 
 
 0.289589 
 
 3-4531735 
 
 0.302553 
 
 3.3052091 
 
 .04477 
 
 0.957I5I 
 
 IO 
 
 
 17 
 
 o 
 
 0.292372 
 
 3.4203036 
 
 0.305731 
 
 3.2708526 
 
 .04569 
 
 0.956305 
 
 
 
 73 
 
 
 IO 
 
 0.295152 
 
 3.3880820 
 
 0.308914 
 
 3-2371438 
 
 .04663 
 
 0.955450 
 
 50 
 
 
 
 20 
 
 0.297930 
 
 3-3564900 
 
 0.312104 
 
 3.2040638 
 
 .04757 
 
 0.954588 
 
 40 
 
 
 
 30 
 
 0.300706 
 
 3-3255095 
 
 0.315299 
 
 3.I7I5948 
 
 .04853 
 
 0.953717 
 
 30 
 
 
 
 40 
 
 0.303479 
 
 3.2951234 
 
 0.318500 
 
 3.I397I94 
 
 .04950 
 
 0.952838 
 
 20 
 
 
 
 So 
 
 0.306249 
 
 3.2653149 
 
 0.321707 
 
 3.1084210 
 
 .05047 
 
 0.9SI95I 
 
 10 
 
 
 18 
 
 o 
 
 0.309017 
 
 3.2360680 
 
 0.324920 
 
 3.0776835 
 
 .05146 
 
 0.951057 
 
 
 
 72 
 
 
 10 
 
 0.311782 
 
 3-2073673 
 
 0.328139 
 
 3.04749IS 
 
 .05246 
 
 0.950154 
 
 So 
 
 
 
 20 
 
 0.314545 
 
 3.1791978 
 
 0.331364 
 
 3.0178301 
 
 .05347 
 
 0.949243 
 
 40 
 
 
 
 30 
 
 0.317305 
 
 3 -I5I5453 
 
 0.334595 
 
 2.9886850 
 
 05449 
 
 0.948324 
 
 30 
 
 
 
 40 
 
 0.320062 
 
 3-1243959 
 
 0.337833 
 
 2 . 9600422 
 
 .05552 
 
 0.947397 
 
 20 
 
 
 
 So 
 
 0.322816 
 
 3.0977363 
 
 0.341077 
 
 2.9318885 
 
 .05657 
 
 0.946462 
 
 10 
 
 
 19 
 
 
 
 0.325568 
 
 3.0715535 
 
 0.344328 
 
 2.9042109 
 
 .05762 
 
 0.945519 
 
 o 
 
 71 
 
 
 10 
 
 0.328317 
 
 3.0458*352 
 
 0.347585 
 
 2.8769970 
 
 .05869 
 
 0.944568 
 
 So 
 
 
 
 20 
 
 0.331063 
 
 3.0205693 
 
 0.350848 
 
 2.8502349 
 
 .05976 
 
 0.943609 
 
 40 
 
 
 
 30 
 
 0.333807 
 
 2.9957443 
 
 0.354II9 
 
 2.8239129 
 
 .06085 
 
 0.942641 
 
 30 
 
 
 
 40 
 
 0.336547 
 
 2.9713490 
 
 0.357396 
 
 2.7980198 
 
 .06195 
 
 0.941666 
 
 20 
 
 
 
 50 
 
 0.339285 
 
 2.9473724 
 
 0.360680 
 
 2.7725448 
 
 .06306 
 
 0.940684 
 
 10 
 
 
 20 
 
 o 
 
 0.342020 
 
 2.9238044 
 
 0.363970 
 
 2.7474774 
 
 .06418 
 
 0.939693 
 
 o 
 
 70 
 
 
 10 
 
 0.344752 
 
 2 . 9006346 
 
 0.367268 
 
 2 . 7228076 
 
 .06531 
 
 0.938694 
 
 So 
 
 
 
 20 
 
 0.347481 
 
 2.8778532 
 
 0.370573 
 
 2.6985254 
 
 .06645 
 
 0.937687 
 
 40 
 
 
 
 30 
 
 0.350207 
 
 2.8554510 
 
 0.373885 
 
 2.6746215 
 
 .06761 
 
 0.936672 
 
 30 
 
 
 
 40 
 
 0.352931 
 
 2.8334185 
 
 0.377204 
 
 2.6510867 
 
 .06878 
 
 0.935650 
 
 20 
 
 
 
 So 
 
 0.355651 
 
 2.8117471 
 
 0.380530 
 
 2.6279I2I 
 
 .06995 
 
 0.934619 
 
 10 
 
 69 
 
 Deg. 
 
 Min. 
 
 Cosine. 
 
 Sec. 
 
 Cotang. 
 
 Tang. 
 
 Cosec. 
 
 Sine. 
 
 Min. 
 
 Deg. 
 
 For functions from 69 10' to 76 oo' read from bottom of table upward. 
 
TRIGONOMETRY 
 NATURAL FUNCTIONS OF ANGLES (Continued) 
 
 197 
 
 Deg. 
 
 Min. 
 
 Sine. 
 
 Cosec. 
 
 Tang. 
 
 Cotang. 
 
 Sec. 
 
 Cosine. 
 
 Min. 
 
 Deg. 
 
 21 
 
 o 
 
 0.358368 
 
 2.7904281 
 
 0.383864 
 
 2.6050891 
 
 .07115 
 
 0.933580 
 
 o 
 
 69 
 
 
 10 
 
 0.361082 
 
 2.7694532 
 
 0.387205 
 
 2.5826094 
 
 .07235 
 
 0.932534 
 
 50 
 
 
 
 20 
 
 0.363793 
 
 2.7488144 
 
 0.390554 
 
 2.5604649 
 
 07356 
 
 0.931480 
 
 40 
 
 
 
 30 
 
 0.366501 
 
 2 . 7285038 
 
 0.3939" 
 
 2.5386479 
 
 .07479 
 
 0.930418 
 
 30 
 
 
 
 40 
 
 o . 369206 
 
 2.7085139 
 
 0.397275 
 
 2.5171507 
 
 .07602 
 
 0.929348 
 
 20 
 
 
 
 So 
 
 0.371908 
 
 2.6888374 
 
 0.400647 
 
 2.4959661 
 
 .07727 
 
 0.928270 
 
 IO 
 
 
 22 
 
 o 
 
 0.374607 
 
 2.6694672 
 
 0.404026 
 
 2.4750869 
 
 .07853 
 
 0.927184 
 
 
 
 68 
 
 
 IO 
 
 0-377302 
 
 2 . 6503962 
 
 0.407414 
 
 2.4545061 
 
 .07981 
 
 0.926090 
 
 50 
 
 
 
 20 
 
 0.379994 
 
 2.6316180 
 
 0.410810 
 
 2.4342172 
 
 .08109 
 
 0.924989 
 
 40 
 
 
 
 30 
 
 0.382683 
 
 2.6I3I259 
 
 0.414214 
 
 2.4142136 
 
 .08239 
 
 0.923880 
 
 30 
 
 
 
 40 
 
 0.385369 
 
 2.5949137 
 
 0.417626 
 
 2.3944889 
 
 .08370 
 
 0.922762 
 
 20 
 
 
 
 50 
 
 0.388052 
 
 2.5769753 
 
 0.421046 
 
 2.3750372 
 
 .08503 
 
 0.921638 
 
 IO 
 
 
 23 
 
 o 
 
 0.390731 
 
 2.5593047 
 
 0.424475 
 
 2.3558524 
 
 .08636 
 
 0.920505 
 
 
 
 67 
 
 
 10 
 
 0.393407 
 
 2.5418961 
 
 0.427912 
 
 2.3369287 
 
 .08771 
 
 0.919364 
 
 50 
 
 
 
 20 
 
 0.396080 
 
 2.5247440 
 
 0.431358 
 
 2.3182606 
 
 .08907 
 
 0.918216 
 
 40 
 
 
 
 30 
 
 0.398749 
 
 2.5078428 
 
 0.434812 
 
 2.2998425 
 
 .09044 
 
 0.917060 
 
 30 
 
 
 
 40 
 
 0.401415 
 
 2.49II874 
 
 0.438276 
 
 2.2816693 
 
 .09183 
 
 0.915896 
 
 20 
 
 
 
 So 
 
 o . 404078 
 
 2.4747726 
 
 0.441748 
 
 2.2637357 
 
 .09323 
 
 0.914725 
 
 10 
 
 
 24 
 
 
 
 0.406737 
 
 2.4585933 
 
 0.445229 
 
 2.2460368 
 
 .09464 
 
 0.913545 
 
 o 
 
 66 
 
 
 10 
 
 0.409392 
 
 2.4426448 
 
 0.448719 
 
 2.2285676 
 
 .09606 
 
 0.912358 
 
 50 
 
 
 
 20 
 
 0.412045 
 
 2.4269222 
 
 0.452218 
 
 2.2113234 
 
 .09750 
 
 0.911164 
 
 40 
 
 
 
 30 
 
 0.414693 
 
 2.4II42IO 
 
 0.455726 
 
 2.1942997 
 
 .09895 
 
 0.909961 
 
 30 
 
 
 ii 
 
 40 
 
 0.4I733S 
 
 2.3961367 
 
 0.459244 
 
 2.1774920 
 
 .10041 
 
 0.908751 
 
 20 
 
 
 
 50 
 
 0.419980 
 
 2.3810650 
 
 0.462771 
 
 2.1608958 
 
 . 10189 
 
 0.907533 
 
 IO 
 
 
 25 
 
 o 
 
 0.422618 
 
 2.3662016 
 
 0.466308 
 
 2.1445069 
 
 -10338 
 
 0.906308 
 
 o 
 
 66 
 
 
 10 
 
 0.425253 
 
 2.3515424 
 
 0.469854 
 
 2.1283213 
 
 .10488 
 
 0.905075 
 
 50 
 
 
 
 20 
 
 0.427884 
 
 2.3370833 
 
 0.473410 
 
 2.1123348 
 
 .10640 
 
 0.903834 
 
 40 
 
 
 
 30 
 
 0.430511 
 
 2 . 3228205 
 
 0.476976 
 
 2.0965436 
 
 . 10793 
 
 0.902585 
 
 30 
 
 
 
 40 
 
 0.433135 
 
 2.3087501 
 
 0.480551 
 
 2.0809438 
 
 .10947 
 
 0.901329 
 
 20 
 
 
 
 50 
 
 0.435755 
 
 2.2948685 
 
 0.484137 
 
 2.0655318 
 
 .11103 
 
 0.900065 
 
 10 
 
 
 26 
 
 o 
 
 0.438371 
 
 2.28II72O 
 
 0.487733 
 
 2 . 0503038 
 
 .11260 
 
 0.898794 
 
 
 
 64 
 
 
 10 
 
 0.440984 
 
 2.2676571 
 
 0.491339 
 
 ft.0352565 
 
 .11419 
 
 0.897515 
 
 50 
 
 
 
 20 
 
 0.443593 
 
 2 . 2543204 
 
 0.494955 
 
 2 . O203862 
 
 .11579 
 
 0.896229 
 
 40 
 
 
 
 30 
 
 0.446198 
 
 2.2411585 
 
 0.498582 
 
 2.0056897 
 
 .11740 
 
 0.894934 
 
 30 
 
 
 
 40 
 
 0.448799 
 
 2.228l68l 
 
 0.502219 
 
 .9911637 
 
 11903 
 
 0.893633 
 
 20 
 
 
 
 50 
 
 0.451397 
 
 2.2153460 
 
 0.505867 
 
 .9768050 
 
 .12067 
 
 0.892323 
 
 10 
 
 
 27 
 
 o 
 
 0.453990 
 
 2 . 2O26893 
 
 0.509525 
 
 .9626105 
 
 .12233 
 
 0.891007 
 
 o 
 
 63 
 
 
 IO 
 
 0.456580 
 
 2.I90I947 
 
 0.5I3I95 
 
 .9485772 
 
 .12400 
 
 0.889682 
 
 50 
 
 
 
 20 
 
 0.459166 
 
 2.1778595 
 
 0.516876 
 
 .9347020 
 
 .12568 
 
 0.888350 
 
 40 
 
 
 
 30 
 
 0.461749 
 
 2.1656806 
 
 0.520567 
 
 .9209821 
 
 .12738 
 
 0.887011 
 
 30 
 
 
 
 40 
 
 0.464327 
 
 2.I536S53 
 
 0.524270 
 
 .9074147 
 
 .12910 
 
 0.885664 
 
 20 
 
 
 
 50 
 
 0.466901 
 
 2.I4I7808 
 
 0.527984 
 
 .8939971 
 
 .13083 
 
 0.884309 
 
 10 
 
 62 
 
 Deg. 
 
 Min. 
 
 Cosine. 
 
 Sec. 
 
 Cotang. 
 
 Tang. 
 
 Cosec. 
 
 Sine. 
 
 Min. 
 
 Deg. 
 
 For functions from 62 10' to 69 oo' read from bottom of table upward. 
 
198 PRACTICAL SURVEYING 
 
 NATURAL FUNCTIONS OF ANGLES (Continued} 
 
 Deg. 
 
 Min. 
 
 Sine. 
 
 Cosec. 
 
 Tang. 
 
 Cotang. 
 
 Sec. 
 
 Cosine. 
 
 Min. 
 
 Deg. 
 
 28 
 
 o 
 
 0.469472 
 
 2.1300545 
 
 0.531709 
 
 .8807265 
 
 .13257 
 
 0.882948 
 
 o 
 
 62 
 
 
 10 
 
 0.472038 
 
 -1184737 
 
 0.535547 
 
 . 8676003 
 
 . 13433 
 
 0.881578 
 
 So 
 
 
 
 20 
 
 0.474600 
 
 . 1070359 
 
 0.539I9S 
 
 .8546159 
 
 .13610 
 
 0.880201 
 
 40 
 
 
 
 30 
 
 0.477159 
 
 .0957385 
 
 0.542956 
 
 .8417409 
 
 13789 
 
 0.878817 
 
 30 
 
 
 
 40 
 
 0.479713 
 
 .0845792 
 
 0.546728 
 
 .8290628 
 
 13970 
 
 0.877425 
 
 20 
 
 
 
 50 
 
 0.482263 
 
 .0735556 
 
 0.550515 
 
 .8164892 
 
 .14152 
 
 0.876026 
 
 10 
 
 
 29 
 
 
 
 0.484810 
 
 .0626653 
 
 0.554309 
 
 .8040478 
 
 14335 
 
 0.874620 
 
 
 
 61 
 
 
 IO 
 
 0.487352 
 
 .0519061 
 
 0.558118 
 
 .7917362 
 
 .14521 
 
 0.873206 
 
 So, 
 
 
 
 20 
 
 0.489890 
 
 .0412757 
 
 0.561939 
 
 7795524 
 
 14707 
 
 0.871784 
 
 40 
 
 
 
 30 
 
 0.492424 
 
 0307720 
 
 0.565773 
 
 .7674940 
 
 .14896 
 
 0.870356 
 
 30 
 
 
 
 40 
 
 0.494953 
 
 . 0203929 
 
 0.569619 
 
 . 7555590 
 
 15085 
 
 0.868920 
 
 20 
 
 
 
 50 
 
 0.497479 
 
 .0101362 
 
 0.573478 
 
 7437453 
 
 15277 
 
 0.867476 
 
 10 
 
 
 30 
 
 o 
 
 0.500000 
 
 .0000000 
 
 0.577350 
 
 .7320508 
 
 15470 
 
 o . 866025 
 
 
 
 60 
 
 
 IO 
 
 0.502517 
 
 .9899822 
 
 0.581235 
 
 .7204736 
 
 .15665 
 
 0.864567 
 
 50 
 
 
 
 20 
 
 0.505030 
 
 .9800810 
 
 0.585134 
 
 .7090116 
 
 . 15861 
 
 0.863102 
 
 40 
 
 
 
 30 
 
 0.507538 
 
 .9702944 
 
 0.589045 
 
 .6976631 
 
 . 16059 
 
 0.861629 
 
 30 
 
 
 
 40 
 
 0.510043 
 
 .9606206 
 
 0.592970 
 
 .6864261 
 
 . 16259 
 
 0.860149 
 
 20 
 
 
 
 50 
 
 0.512543 
 
 .9510577 
 
 0.596908 
 
 .6752988 
 
 . 16460 
 
 0.858662 
 
 IO 
 
 
 31 
 
 
 
 0.515038 
 
 .9416040 
 
 0.600861 
 
 .6642795 
 
 .16663 
 
 0.857167 
 
 
 
 59 
 
 
 IO 
 
 0.517529 
 
 .9322578 
 
 0.604827 
 
 .6533663 
 
 .16868 
 
 0.855665 
 
 50 
 
 
 
 20 
 
 0.520016 
 
 .9230173 
 
 0.608807 
 
 .6425576 
 
 .17075 
 
 0.854156 
 
 40 
 
 
 
 30 
 
 0.522499 
 
 .9138809 
 
 0.612801 
 
 .6318517 
 
 .17283 
 
 0.852640 
 
 30 
 
 
 
 40 
 
 0.524977 
 
 .9048469 
 
 0.616809 
 
 .6212469 
 
 17493 
 
 0.851117 
 
 20 
 
 
 
 50 
 
 0.527450 
 
 .8959138 
 
 0.620832 
 
 .6107417 
 
 . 17704 
 
 0.849586 
 
 IO 
 
 
 32 
 
 o 
 
 0.529919 
 
 .8870799 
 
 0.624869 
 
 .6003345 
 
 .17918 
 
 0.848048 
 
 O 
 
 68 
 
 
 10 
 
 0.532384 
 
 .8783438 
 
 0.628921 
 
 .5900238 
 
 .18133 
 
 o . 846503 
 
 50 
 
 
 
 20 
 
 0.534844 
 
 .8697040 
 
 0.632988 
 
 .5798079 
 
 . 18350 
 
 0.844951 
 
 40 
 
 
 
 30 
 
 0.537300 
 
 .8611590 
 
 0.637079 
 
 .5696856 
 
 .18569 
 
 0.843391 
 
 30 
 
 
 
 40 
 
 0.539751 
 
 .8527073 
 
 0.641167 
 
 .5596552 
 
 .18790 
 
 0.841825 
 
 20 
 
 
 
 50 
 
 0.542197 
 
 .8443476 
 
 0.645280 
 
 5497155 
 
 .19012 
 
 0.840251 
 
 10 
 
 
 33 
 
 O 
 
 0.544639 
 
 .8360785 
 
 0.649408 
 
 .5398650 
 
 . 19236 
 
 0.838671 
 
 o 
 
 67 
 
 
 10 
 
 0.547076 
 
 .8278985 
 
 0.653531 
 
 .5301025 
 
 . 19463 
 
 0.837083 
 
 So 
 
 
 
 20 
 
 0.549509 
 
 .8198065 
 
 o. 657710 
 
 .5204261 
 
 . 19691 
 
 0.835488 
 
 40 
 
 
 
 30 
 
 0.551937 
 
 .8118010 
 
 0.661886 
 
 .5108352 
 
 .19920 
 
 0.833886 
 
 30 
 
 
 
 40 
 
 o.55436o 
 
 .8038809 
 
 0.666077 
 
 .5013282 
 
 .20152 
 
 0.832277 
 
 20 
 
 
 
 So 
 
 0.556779 
 
 .7960449 
 
 0.670285 
 
 .4919039 
 
 .20386 
 
 0.830661 
 
 10 
 
 
 34 
 
 
 
 0.559193 
 
 . 7882916 
 
 0.674509 
 
 .4825610 
 
 .20622 
 
 0.829038 
 
 O 
 
 66 
 
 
 IO 
 
 0.561602 
 
 .7806201 
 
 0.678749 
 
 .4732983 
 
 .20859 
 
 0.827407 
 
 So 
 
 
 
 20 
 
 0.564007 
 
 .7730290 
 
 0.683007 
 
 .4641147 
 
 .21099 
 
 o . 825770 
 
 40 
 
 
 
 30 
 
 o . 566406 
 
 . 7655173 
 
 0.687281 
 
 .4550090 
 
 .21341 
 
 0.824126 
 
 30 
 
 
 
 40 
 
 0.568801 
 
 .7580837 
 
 0.691573 
 
 .4459801 
 
 .21584 
 
 0.822475 
 
 20 
 
 
 
 50 
 
 0.57H9I 
 
 .7507273 
 
 0.695881 
 
 .4370268 
 
 .21830 
 
 0.820817 
 
 10 
 
 55 
 
 Deg. 
 
 Min. 
 
 Cosine. 
 
 Sec. 
 
 Cotang. 
 
 Tang. 
 
 COGCC. 
 
 Sine. 
 
 Min. 
 
 Deg. 
 
 For functions from 55 10' to 62 oo' read from bottom of table upward. 
 
TRIGONOMETRY 
 NATURAL FUNCTIONS OF ANGLES (Continued) 
 
 199 
 
 Deg. 
 
 Min. 
 
 Sine. 
 
 Cosec. 
 
 Tang. 
 
 Cotang. 
 
 Sec. 
 
 Cosine. 
 
 Min. 
 
 Deg. 
 
 35 
 
 o 
 
 0.573576 
 
 .7434468 
 
 o . 700208 
 
 .4281480 
 
 1.22077 
 
 0.819152 
 
 o 
 
 66 
 
 
 10 
 
 o:57S957 
 
 .7362413 
 
 0.704552 
 
 .4193427 
 
 1.22327 
 
 0.817480 
 
 50 
 
 
 
 20 
 
 0.578332 
 
 . 7291096 
 
 0.708913 
 
 .4106098 
 
 1.22579 
 
 0.815801 
 
 40 
 
 
 
 30 
 
 0.580703 
 
 . 7220508 
 
 0.713293 
 
 .4019483 
 
 1.22833 
 
 0.814116 
 
 30 
 
 
 
 40 
 
 0.583069 
 
 .7150639 
 
 0.717691 
 
 -3933571 
 
 1.23089 
 
 0.812423 
 
 20 
 
 
 
 So 
 
 0.585429 
 
 . 7081478 
 
 0.722108 
 
 .3848355 
 
 1.23347 
 
 0.810723 
 
 10 
 
 
 36 
 
 o 
 
 0.587785 
 
 . 7013016 
 
 0.726543 
 
 .3763810 
 
 1.23607 
 
 0.809017 
 
 o 
 
 64 
 
 
 10 
 
 0.590136 
 
 .6945244 
 
 0.730996 
 
 -3679959. 
 
 1.23869 
 
 0.807304 
 
 50 
 
 
 
 20 
 
 0.592482 
 
 .6878151 
 
 0.735469 
 
 .3596764 
 
 1.24134 
 
 0.805584 
 
 40 
 
 
 
 30 
 
 0.594823 
 
 .6811730 
 
 o.73996i 
 
 .3514224 
 
 1.24400 
 
 0.803857 
 
 30 
 
 
 
 40 
 
 0.597IS9 
 
 .6745970 
 
 0.744472 
 
 .3432331 
 
 1.24669 
 
 0.802123 
 
 20 
 
 
 
 50 
 
 0.599489 
 
 .6680864 
 
 0.749003 
 
 -3351075 
 
 1.24940 
 
 0.800383 
 
 IO 
 
 
 37 
 
 o 
 
 0.601815 
 
 .6616401 
 
 0.753554 
 
 .3270448 
 
 1.25214 
 
 0.798636 
 
 
 
 53 
 
 
 IO 
 
 0.604136 
 
 .6552575 
 
 0.758125 
 
 .3190441 
 
 1.25489 
 
 0.796882 
 
 50 
 
 
 
 20 
 
 0.606451 
 
 .6489376 
 
 0.762716 
 
 .3111046 
 
 1-25767 
 
 0.795I2I 
 
 40 
 
 
 
 30 
 
 0.608761 
 
 .6426796 
 
 0.767627 
 
 .3032254 
 
 i . 26047 
 
 0-793353 
 
 30 
 
 
 
 40 
 
 0.611067 
 
 .6364828 
 
 0.771959 
 
 .2954057 
 
 1.26330 
 
 0.791579 
 
 20 
 
 
 
 50 
 
 0.613367 
 
 .6303462 
 
 0.776612 
 
 .2876447 
 
 1.26615 
 
 0.789798 
 
 10 
 
 
 38 
 
 
 
 0.615661 
 
 .6242692 
 
 0.781286 
 
 .2799416 
 
 1.26902 
 
 0.788011 
 
 o 
 
 62 
 
 
 10 
 
 0.617951 
 
 .6182510 
 
 0.785981 
 
 .2722957 
 
 1.27191 
 
 0.786217 
 
 50 
 
 
 
 20 
 
 0.620235 
 
 .6122908 
 
 0.790698 
 
 .2647062 
 
 1.27483 
 
 0.784416 
 
 40 
 
 
 
 30 
 
 0.622515 
 
 .6063879 
 
 0.795436 
 
 2571723 
 
 1.27778 
 
 0.782608 
 
 30 
 
 
 
 40 
 
 0.624789 
 
 .6005416 
 
 0.800196 
 
 .2496933 
 
 1.28075 
 
 0.780794 
 
 20 
 
 
 
 So 
 
 o . 627057 
 
 59475" 
 
 0.804080 
 
 .2422685 
 
 1.28374 
 
 0.778973 
 
 IO 
 
 
 39 
 
 o 
 
 0.629320 
 
 .5890157 
 
 0.809784 
 
 .2348972 
 
 1.28676 
 
 0.777146 
 
 o 
 
 61 
 
 
 IO 
 
 0.631578 
 
 .5833318 
 
 0.814612 
 
 .2275786 
 
 1.28980 
 
 0.775312 
 
 50 
 
 
 
 20 
 
 0.633831 
 
 -5777077 
 
 0.819463 
 
 .2203121 
 
 1.29287 
 
 0.773472 
 
 40 
 
 
 
 30 
 
 0.63*078 
 
 .5721337 
 
 0.824336 
 
 .2130970 
 
 1.29597 
 
 0.771625 
 
 30 
 
 
 
 40 
 
 0.638320 
 
 .5666121 
 
 0.829234 
 
 .2059327 
 
 1.29909 
 
 0.769771 
 
 20 
 
 
 
 So 
 
 0.640557 
 
 .5611424 
 
 0.834155 
 
 .1988184 
 
 1.30223 
 
 0.767911 
 
 IO 
 
 
 40 
 
 o 
 
 0.642788 
 
 .5557238 
 
 0.839100 
 
 .1917536 
 
 1.30541 
 
 0.766044 
 
 o 
 
 60 
 
 
 IO 
 
 o . 645013 
 
 .5503558 
 
 0.844069 
 
 1847376 
 
 1.30861 
 
 0.764171 
 
 50 
 
 
 
 20 
 
 0.647233 
 
 .5450378 
 
 0.849062 
 
 .1777698 
 
 1.31183 
 
 0.762292 
 
 40 
 
 
 
 30 
 
 0.649448 
 
 .5397690 
 
 0.854081 
 
 . 1708496 
 
 1-31509 
 
 o . 760406 
 
 30 
 
 
 
 40 
 
 0.651657 
 
 -5345491 
 
 0.859124 
 
 . 1639763 
 
 1.31837 
 
 0.758514 
 
 20 
 
 
 
 50 
 
 0.653861 
 
 .5293773 
 
 0.864193 
 
 .1571495 
 
 1.32168 
 
 0.756615 
 
 10 
 
 
 41 
 
 o 
 
 0.656059 
 
 .5242531 
 
 0.869287 
 
 .1503684 
 
 1.32501 
 
 0.754710 
 
 
 
 49 
 
 
 IO 
 
 0.658252 
 
 .5191759 
 
 0.874407 
 
 . 1436326 
 
 1.32838 
 
 0.752798 
 
 50 
 
 
 
 20 
 
 0.660439 
 
 .5141452 
 
 0.879553 
 
 . 1369414 
 
 I.33I77 
 
 0.750880 
 
 40 
 
 
 
 30 
 
 0.662620 
 
 .5091605 
 
 0.884725 
 
 . 1302944 
 
 I.335I9 
 
 0.748956 
 
 30 
 
 
 
 40 
 
 0.664796 
 
 .5042211 
 
 0.889924 
 
 .1236909 
 
 1.33864 
 
 0.747025 
 
 20 
 
 
 
 So 
 
 0.666966 
 
 .4993267 
 
 0.895151 
 
 .1171305 
 
 1.34212 
 
 0.745088 
 
 IO 
 
 48 
 
 Deg. 
 
 Min. 
 
 Cosine. 
 
 Sec. 
 
 Cotang. 
 
 Tang. 
 
 Cosec. 
 
 Sine. 
 
 Min. 
 
 Deg. 
 
 For functions from 48 10' to 55 oo' read from bottom of table upward. 
 
200 PRACTICAL SURVEYING 
 
 NATURAL FUNCTIONS OF ANGLES (Continued) 
 
 Deg. 
 
 Min. 
 
 Sine. 
 
 Cosec. 
 
 Tang. 
 
 Cotang. 
 
 Sec. 
 
 Cosine. 
 
 Min. 
 
 Deg. 
 
 42 
 
 
 
 0.669131 
 
 .4944765 
 
 0.900404 
 
 .1106125 
 
 .34563 
 
 0.743145 
 
 o 
 
 48 
 
 
 10 
 
 0.671289 
 
 .4896703 
 
 0.905685 
 
 . 1041365 
 
 .34917 
 
 0.741195 
 
 So 
 
 
 
 20 
 
 0.673443 
 
 .4849073 
 
 0.910994 
 
 . 0977020 
 
 35274 
 
 0.739239 
 
 40 
 
 
 
 30 
 
 0.675590 
 
 .4801872 
 
 0.916331 
 
 .0913085 
 
 .35634 
 
 0.737277 
 
 30 
 
 
 
 40 
 
 0.677732 
 
 .4755095 
 
 0.921697 
 
 .0849554 
 
 35997 
 
 c.735309 
 
 20 
 
 
 
 50 
 
 0.679868 
 
 4708736 
 
 0.927091 
 
 .0786423 
 
 .36363 
 
 0.733335 
 
 10 
 
 
 43 
 
 o 
 
 0.681998 
 
 .4662792 
 
 0.932515 
 
 .0723687 
 
 .36733 
 
 0.731354 
 
 o 
 
 47 
 
 
 IO 
 
 0.684123 
 
 .4617257 
 
 0.937968 
 
 .0661341 
 
 37105 
 
 0.729367 
 
 So 
 
 
 
 
 
 0.686242 
 
 .4572127 
 
 0.943451 
 
 .0599381 
 
 .37481 
 
 0.727374 
 
 40 
 
 
 
 30 
 
 0.688355 
 
 4527397 
 
 0.948965 
 
 .0537801 
 
 .37860 
 
 0.725374 
 
 30 
 
 
 
 40 
 
 0.690462 
 
 .4483063 
 
 o.9545o8 
 
 .0476598 
 
 .38242 
 
 0.723369 
 
 20 
 
 
 
 So 
 
 0.692563 
 
 .4439120 
 
 0.960083 
 
 .0415767 
 
 .38628 
 
 0.721357 
 
 IO 
 
 
 44 
 
 o 
 
 0.694658 
 
 .4395565 
 
 0.965689 
 
 .0355303 
 
 .39016 
 
 0.719340 
 
 o 
 
 46 
 
 
 IO 
 
 0.696748 
 
 .4352393 
 
 0.971326 
 
 .0295203 
 
 .39409 
 
 0.717316 
 
 50 
 
 
 
 20 
 
 0.698832 
 
 .4309602 
 
 0.976996 
 
 .0235461 
 
 .39804 
 
 0.715286 
 
 40 
 
 
 
 30 
 
 0.700909 
 
 .4267182 
 
 0.982697 
 
 .0176074 
 
 .40203 
 
 0.713251 
 
 30 
 
 
 
 40 
 
 0.702981 
 
 .4225134 
 
 0.988432 
 
 .0117088 
 
 .40606 
 
 0.711209 
 
 20 
 
 
 
 50 
 
 0.705047 
 
 .4183454 
 
 0.994199 
 
 .0058348 
 
 .41012 
 
 0.709161 
 
 10 
 
 
 45 
 
 
 
 0.707107 
 
 1.4142136 
 
 I.OOOOOO 
 
 I. 0000000 
 
 1.41421 
 
 0.707107 
 
 
 
 45 
 
 Deg. 
 
 Min. 
 
 Cosine. 
 
 Sec. 
 
 Cotang. 
 
 Tang. 
 
 Cosec. 
 
 Sine. 
 
 Min. 
 
 Deg. 
 
 For functions from 45 o' to 48 oo' read from bottom of table upward. 
 
TRIGONOMETRY 
 LOGARITHMS OF NUMBERS FROM o TO 1000 
 
 201 
 
 No. 
 
 o 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 o 
 
 ooooo 
 
 30103 
 
 47712 
 
 /___/- 
 
 OO2OO 
 
 69897 
 
 77815 
 
 84510 
 
 90309 
 
 95424 
 
 10 
 
 ooooo 
 
 00432 
 
 00860 
 
 01284 
 
 01703 
 
 02119 
 
 02531 
 
 02938 
 
 03342 
 
 03743 
 
 II 
 
 04139 
 
 04532 
 
 04922 
 
 05308 
 
 05690 
 
 06070 
 
 06446 
 
 06819 
 
 07188 
 
 07555 
 
 12 
 
 07918 
 
 08279 
 
 08636 
 
 08991 
 
 09342 
 
 09691 
 
 10037 
 
 10380 
 
 10721 
 
 11059 
 
 13 
 
 1 1394 
 
 11727 
 
 12057 
 
 12385 
 
 12710 
 
 13033 
 
 13354 
 
 13672 
 
 13988 
 
 14301 
 
 14 
 
 14613 
 
 14922 
 
 15229 
 
 15534 
 
 15836 
 
 16137 
 
 16435 
 
 16732 
 
 17026 
 
 I73I9 
 
 IS 
 
 17609 
 
 17898 
 
 18184 
 
 18469 
 
 18752 
 
 19033 
 
 19312 
 
 19590 
 
 19866 
 
 20140 
 
 16 
 
 20412 
 
 20683 
 
 20952 
 
 21219 
 
 21484 
 
 21748 
 
 220II 
 
 22272 
 
 22531 
 
 22789 
 
 17 
 
 23045 
 
 23300 
 
 23553 
 
 23805 
 
 24055 
 
 24304 
 
 24551 
 
 24797 
 
 25042 
 
 25285 
 
 18 
 
 25527 
 
 25768 
 
 26007 
 
 26245 
 
 26482 
 
 26717 
 
 26951 
 
 27184 
 
 27416 
 
 27646 
 
 19 
 
 27875 
 
 28103 
 
 28330 
 
 28556 
 
 28780 
 
 29003 
 
 29226 
 
 29447 
 
 29667 
 
 29885 
 
 20 
 
 30103 
 
 30320 
 
 30535 
 
 30750 
 
 30963 
 
 3H75 
 
 31387 
 
 31597 
 
 31806 
 
 32015 
 
 21 
 
 32222 
 
 32428 
 
 32634 
 
 32838 
 
 33041 
 
 33244 
 
 33445 
 
 33646 
 
 33846 
 
 34044 
 
 22 
 
 34242 
 
 34439 
 
 34635 
 
 34830 
 
 35025 
 
 35218 
 
 354H 
 
 35603 
 
 35793 
 
 35984 
 
 23 
 
 36173 
 
 36361 
 
 36549 
 
 36736 
 
 36922 
 
 37107 
 
 37291 
 
 37475 
 
 37658 
 
 37840 
 
 24 
 
 38021 
 
 38202 
 
 38382 
 
 38561 
 
 38739 
 
 38917 
 
 39094 
 
 39270 
 
 39445 
 
 39620 
 
 25 
 
 39794 
 
 39967 
 
 40140 
 
 40312 
 
 40483 
 
 40654 
 
 40824 
 
 40993 
 
 41162 
 
 41330 
 
 26 
 
 41497 
 
 41664 
 
 41830 
 
 41996 
 
 42160 
 
 42325 
 
 42488 
 
 42651 
 
 42813 
 
 42975 
 
 27 
 
 43136 
 
 43297 
 
 43457 
 
 43616 
 
 43775 
 
 43933 
 
 44091 
 
 44248 
 
 44404 
 
 4456o 
 
 28 
 
 44716 
 
 44871 
 
 45025 
 
 45179 
 
 45332 
 
 45484 
 
 45637 
 
 45788 
 
 45939 
 
 46090 
 
 29 
 
 46240 
 
 46389 
 
 46538 
 
 46687 
 
 46835 
 
 46982 
 
 47129 
 
 47276 
 
 47422 
 
 47567 
 
 30 
 
 47712 
 
 47857 
 
 48001 
 
 48144 
 
 48287 
 
 48430 
 
 48572 
 
 48714 
 
 48855 
 
 48996 
 
 31 
 
 49136 
 
 49276 
 
 49415 
 
 49554 
 
 49693 
 
 49831 
 
 49969 
 
 50106 
 
 50243 
 
 50379 
 
 32 
 
 50515 
 
 50651 
 
 50786 
 
 50920 
 
 51055 
 
 51188 
 
 51322 
 
 51455 
 
 51587 
 
 51720 
 
 33 
 
 5I85I 
 
 SI983 
 
 52H4 
 
 52244 
 
 52375 
 
 52504 
 
 52633 
 
 52763 
 
 52892 
 
 53020 
 
 34 
 
 53148 
 
 53275 
 
 53403 
 
 53529 
 
 53656 
 
 53782 
 
 53908 
 
 54033 
 
 54158 
 
 54283 
 
 35 
 
 54407 
 
 54531 
 
 54654 
 
 54777 
 
 54900 
 
 55023 
 
 55145 
 
 55267 
 
 55388 
 
 55509 
 
 36 
 
 55630 
 
 55751 
 
 55871 
 
 55991 
 
 56110 
 
 56229 
 
 56348 
 
 56467 
 
 56585 
 
 56703 
 
 37 
 
 56820 
 
 56937 
 
 57054 
 
 57I7I 
 
 57287 
 
 57403 
 
 57519 
 
 57634 
 
 57749 
 
 57864 
 
 38 
 
 57978 
 
 58093 
 
 58206 
 
 58320 
 
 58433 
 
 58546 
 
 58659 
 
 58771 
 
 58883 
 
 58995 
 
 39 
 
 59io6 
 
 592i8 
 
 59329 
 
 59439 
 
 59550 
 
 59660 
 
 59770 
 
 59879 
 
 59988 
 
 60097 
 
 40 
 
 60206 
 
 60314 
 
 60423 
 
 60531 
 
 60638 
 
 60746 
 
 6o853 
 
 60959 
 
 61066 
 
 61172 
 
 41 
 
 61278 
 
 61384 
 
 61490 
 
 61595 
 
 61700 
 
 61805 
 
 61909 
 
 62014 
 
 62118 
 
 62221 
 
 42 
 
 62325 
 
 62428 
 
 62531 
 
 62634 
 
 62737 
 
 62839 
 
 62941 
 
 63043 
 
 63144 
 
 63246 
 
 43 
 
 63347 
 
 63448 
 
 63548 
 
 63649 
 
 63749 
 
 63849 
 
 63949 
 
 64048 
 
 64147 
 
 64246 
 
 44 
 
 64345 
 
 64444 
 
 64542 
 
 64640 
 
 64738 
 
 64836 
 
 64933 
 
 65031 
 
 65128 
 
 65225 
 
 45 
 
 65321 
 
 65418' 
 
 65514 
 
 65610 
 
 65706 
 
 65801 
 
 65896 
 
 65992 
 
 66087 
 
 66181 
 
 46 
 
 66276 
 
 66370 
 
 66464 
 
 66558 
 
 66652 
 
 66745 
 
 66839 
 
 66932 
 
 67025 
 
 67117 
 
 47 
 
 67210 
 
 67302 
 
 67394 67486 
 
 67578 
 
 67669 
 
 67761 
 
 67852 
 
 67943 
 
 68034 
 
 48 
 
 68124 
 
 68215 
 
 68305 68395 
 
 68485 
 
 68574 
 
 68664 
 
 68753 
 
 68842 
 
 68931 
 
 49 
 
 69020 
 
 69108 
 
 69197 
 
 69285 
 
 69373 
 
 69461 
 
 69548 
 
 69636 
 
 69723 
 
 69810 
 
 50 
 
 69897 
 
 69984 
 
 70070 
 
 70157 
 
 70243 
 
 70329 
 
 70415 
 
 70501 
 
 70586 
 
 70672 
 
 Si 
 
 70757 
 
 70842 
 
 70927 
 
 71012 
 
 71096 
 
 71181 
 
 71265 
 
 71349 
 
 71433 
 
 7I5I7 
 
 52 
 
 71600 
 
 71684 
 
 71767 
 
 71850 
 
 71933 
 
 72016 
 
 72099 
 
 72181 
 
 72263 
 
 72346 
 
 53 
 
 72428 
 
 72509 
 
 72591 
 
 72673 
 
 72754 
 
 72835 
 
 72916 
 
 72997 
 
 73078 
 
 73159 
 
 54 
 
 73239 
 
 73320 
 
 73400 
 
 7348o 
 
 7356o 
 
 73640 
 
 73719 
 
 73799 
 
 73878 
 
 73957 
 
202 PRACTICAL SURVEYING 
 
 LOGARITHMS OF NUMBERS FROM o TO 1000 (Continued] 
 
 No. 
 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 55 
 
 74036 
 
 74H5 
 
 74194 
 
 74273 
 
 74351 
 
 74429 
 
 74507 
 
 74586 
 
 74663 
 
 74741 
 
 56 
 
 748i9 
 
 74896 
 
 74974 
 
 75051 
 
 75128 
 
 75205 
 
 75282 
 
 75358 
 
 75435 
 
 7551 1 
 
 57 
 
 75587 
 
 7566 4 
 
 75740 
 
 75815 
 
 75891 
 
 75967 
 
 76042 
 
 76118 
 
 76193 ' 76268 
 
 58 
 
 76343 
 
 76418 
 
 76492 
 
 76567 
 
 76641 
 
 76716 
 
 76790 
 
 76864 
 
 76938 i 77012 
 
 59 
 
 77085 
 
 77159 
 
 77232 
 
 77305 
 
 77379 
 
 77452 
 
 77525 
 
 77597 
 
 77670 
 
 77743 
 
 60 
 
 77815 
 
 77887 
 
 77960 
 
 78032 
 
 78104 
 
 78176 
 
 78247 
 
 78319 
 
 78390 
 
 78462 
 
 61 
 
 78533 
 
 78604 
 
 78675 
 
 78746 
 
 78817 
 
 78888 
 
 78958 
 
 79029 
 
 79099 79169 
 
 62 
 
 79239 
 
 79309 
 
 79379 
 
 79449 
 
 79518 
 
 79588 
 
 79657 
 
 79727 
 
 79796 79865 
 
 63 
 
 79934 
 
 80003 
 
 80072 
 
 80140 
 
 80209 
 
 80277 
 
 80346 
 
 80414 
 
 80482 , 80550 
 
 64 
 
 80618 
 
 80686 
 
 80754 
 
 80821 
 
 80889 
 
 80956 
 
 81023 
 
 81090 
 
 81158 81224 
 
 65 
 
 81291 
 
 81358 
 
 81425 
 
 81491 
 
 81558 
 
 81624 
 
 81690 
 
 81756 
 
 81823 81889 
 
 66 81954 
 
 82020 
 
 82086 
 
 82151 
 
 82217 
 
 82282 
 
 82347 
 
 82413 
 
 82478 82543 
 
 67 
 
 82607 
 
 82672 
 
 82737 
 
 82802 
 
 82866 
 
 82930 
 
 82995 
 
 83059 
 
 83123 83187 
 
 68 
 
 83251 
 
 83315 
 
 83378 
 
 83442 
 
 83506 
 
 83569 
 
 83632 
 
 83696 
 
 83759 83822 
 
 69 
 
 83885 
 
 83948 
 
 84011 
 
 84073 
 
 84136, 
 
 84198 
 
 84261 
 
 84323 
 
 84386 
 
 84448 
 
 70 
 
 84510 
 
 84572 
 
 84634 
 
 84696 
 
 84757 
 
 84819 
 
 84880 
 
 84942 
 
 85003 
 
 85065 
 
 7i 
 
 85126 
 
 85187 
 
 85248 
 
 85309 
 
 85370 
 
 85431 
 
 85491 
 
 85552 
 
 85612 85673 
 
 72 
 
 85733 
 
 85794 
 
 85854 
 
 853I4 
 
 85974 
 
 86034 
 
 86094 
 
 86153 
 
 86213 86273 
 
 73 
 
 86332 
 
 86392 
 
 86451 
 
 86510 
 
 86570 
 
 86629 
 
 86638 
 
 86747 
 
 86806 
 
 86864 
 
 74 
 
 86923 
 
 86982 
 
 87040 
 
 87093 
 
 87157 
 
 87216 
 
 87274 
 
 87332 
 
 87390 
 
 87448 
 
 75 
 
 87506 
 
 87564 
 
 87622 
 
 87680 
 
 87737 
 
 87795 
 
 87852 
 
 87910 
 
 87967 
 
 88024 
 
 76 
 
 88081 
 
 88138 
 
 88196 
 
 88252 | 88309 
 
 88366 
 
 88423 
 
 88480 
 
 88536 
 
 88593 
 
 77 
 
 88649 
 
 88705 
 
 88762 
 
 88818 i 88874 
 
 88930 
 
 88986 
 
 89042 
 
 89098 
 
 89154 
 
 78 
 
 89209 
 
 89265 
 
 89321 
 
 89376 
 
 89432 
 
 89487 
 
 89542 
 
 89597 
 
 89653 ! 89708 
 
 79 
 
 89763 
 
 89818 
 
 89873 
 
 89927 
 
 89982 
 
 90037 
 
 90091 
 
 90146 
 
 90200 
 
 90255 
 
 80 
 
 90309 
 
 90363 
 
 90417 
 
 90472 
 
 90526 
 
 90580 
 
 90634 
 
 90687 
 
 90741 
 
 90795 
 
 81 
 
 90849 
 
 90902 
 
 90956 
 
 91009 
 
 91062 91116 
 
 91169 
 
 91222 
 
 91275 
 
 91328 
 
 82 
 
 9I38I 
 
 91434 
 
 91487 
 
 91540 
 
 91593 91645 
 
 91698 
 
 9I75I 
 
 91803 
 
 91855 
 
 83 
 
 91908 
 
 91960 
 
 92012 
 
 92065 
 
 92117 92169 
 
 92221 
 
 92273 
 
 92324 
 
 92376 
 
 84 
 
 92428 
 
 92480 
 
 92531 
 
 92583 
 
 92634 92686 
 
 92737 
 
 92788 
 
 92840 
 
 92891 
 
 85 
 
 92942 
 
 92993 
 
 93044 
 
 93095 
 
 93146 93197 
 
 93247 
 
 93298 
 
 93349 
 
 93399 
 
 86 
 
 93450 
 
 935QO 
 
 93551 
 
 936oi 
 
 93651 93702 
 
 93752 
 
 93802 
 
 93852 
 
 93902 
 
 87 
 
 93952 
 
 94002 
 
 94052 
 
 94IOI 
 
 94151 94201 
 
 94250 
 
 94300 
 
 94349 
 
 94399 
 
 88 
 
 94448 
 
 94498 
 
 94547 
 
 94596 
 
 94645 946Q4 
 
 94743 
 
 94792 
 
 94841 
 
 94890 
 
 89 
 
 94939 
 
 94988 
 
 95036 
 
 95085 
 
 95134 
 
 95182 
 
 95231 
 
 95279 
 
 95328 
 
 95376 
 
 90 
 
 95424 
 
 95472 
 
 95521 
 
 95569 
 
 95617 
 
 95665 
 
 95713 
 
 9576i 
 
 95809 
 
 95856 
 
 9i 
 
 95904 
 
 95952 
 
 95999 
 
 96047 
 
 96095 96142 
 
 96190 
 
 96237 
 
 96284 
 
 96332 
 
 92 
 
 96379 
 
 96426 
 
 96473 
 
 96520 
 
 96567 96614 
 
 96661 
 
 96708 
 
 96755 
 
 96802 
 
 93 
 
 96848 
 
 96895 
 
 96942 
 
 96988 
 
 97035 97081 
 
 97128 
 
 97174 
 
 97220 
 
 97267 
 
 94 
 
 97313 
 
 97359 
 
 97405 
 
 97451 
 
 97497 97543 
 
 97589 
 
 97635 
 
 97681 
 
 97727 
 
 95 
 
 97772 
 
 978i8 
 
 97864 
 
 97909 
 
 97955 , 98000 
 
 98046 
 
 98091 
 
 98137 
 
 98182 
 
 96 
 
 98227 
 
 98272 
 
 98318 
 
 98363 
 
 98408 
 
 98453 
 
 98498 
 
 98543 
 
 98588 
 
 98632 
 
 97 
 
 98677 
 
 98722 
 
 98767 
 
 98811 
 
 98856 
 
 98900 
 
 98945 
 
 98989 
 
 99034 
 
 99078 
 
 98 
 
 99123 
 
 99167 
 
 9921 1 
 
 99255 
 
 99300 
 
 99344 
 
 99388 
 
 994.V 
 
 99476 
 
 99520 
 
 99 
 
 99564 
 
 99607 
 
 99651 
 
 99695 
 
 99739 
 
 99782 
 
 99826 
 
 99870 
 
 99913 
 
 99957 
 
TRIGONOMETRY 
 LOGARITHMIC FUNCTIONS OF ANGLES 
 
 203 
 
 Deg. 
 
 Min. 
 
 Sine. 
 
 Cosec. 
 
 Tang. 
 
 Cotang. 
 
 Sec. 
 
 Cosine. 
 
 Min. 
 
 Deg. 
 
 
 
 
 
 oc 
 
 OC 
 
 OC 
 
 OC 
 
 lO.OOOOO 
 
 IO. OOOOO 
 
 o 
 
 90 
 
 
 10 
 
 7.46373 
 
 12.53627 
 
 7.46373 
 
 12.53627 
 
 IO.OOOOO 
 
 9.99999 
 
 So 
 
 
 
 20 
 
 7 76475 
 
 12.23525 
 
 7.76476 
 
 12.23524 
 
 10.00000 
 
 9-99999 
 
 40 
 
 
 
 30 
 
 7.94084 
 
 12.05916 
 
 7.94086 
 
 12.05914 
 
 IO.OOO02 
 
 9.99998 
 
 30 
 
 
 
 40 
 
 8.06578 
 
 11.93422 
 
 8.06581 
 
 11.93419 
 
 10.00003 
 
 9-99997 
 
 20 
 
 
 
 50 
 
 8.16268 
 
 11.83732 
 
 8.16272 
 
 II..83727 
 
 10.00005 
 
 9-99995 
 
 10 
 
 
 1 
 
 
 
 8.24186 
 
 11.75814 
 
 8.24192 
 
 11.75808 
 
 10.00007 
 
 9-99993 
 
 
 
 89 
 
 
 10 
 
 8.30879 
 
 11.69121 
 
 8.30888 
 
 11.69111 
 
 10.00009 
 
 9-99991 
 
 50 
 
 
 
 20 
 
 8.36678 
 
 11.63322 
 
 8.36689 
 
 11.63311 
 
 IO.OOOI2 
 
 9-99988 
 
 40 
 
 
 
 30 
 
 8.41792 
 
 11.58208 
 
 8.41807 
 
 II.58I93 
 
 10.00015 
 
 9-99985 
 
 30 
 
 
 
 40 
 
 8.46366 
 
 11.53634 
 
 8.46385 
 
 II.536I5 
 
 10.00018 
 
 9.99982 
 
 20 
 
 
 
 So 
 
 8.50504 
 
 11.49496 
 
 8.50527 
 
 11-49473 
 
 IO.OO022 
 
 9-99978 
 
 IO 
 
 
 2 
 
 
 
 8.54282 
 
 II.457I8 
 
 8.54308 
 
 11.45692 
 
 IO.OOO26 
 
 9-99974 
 
 
 
 88 
 
 
 IO 
 
 8.57757 
 
 11.42243 
 
 8.57788 
 
 11.42212 
 
 10.00031 
 
 9.99969 
 
 50 
 
 
 
 20 
 
 8.60973 
 
 11.39027 
 
 8.61009 
 
 11.38990 
 
 10.00036 
 
 9.99964 
 
 40 
 
 
 
 30 
 
 8.63968 
 
 11.36032 
 
 8.64009 
 
 II-3599I 
 
 10.00041 
 
 9 99959 
 
 30 
 
 
 
 40 
 
 8.66769 
 
 II.3323I 
 
 8.66816 
 
 II.33I84 
 
 IO.OOO47 
 
 9-99953 
 
 20 
 
 
 
 50 
 
 8.69400 
 
 11.30600 
 
 8.69453 
 
 H.30547 
 
 10.00053 
 
 9-99947 
 
 IO 
 
 
 a 
 
 
 
 8.71880 
 
 11.28120 
 
 8.71940 
 
 11.28060 
 
 IO.OOO6O 
 
 9-99940 
 
 o 
 
 87 
 
 
 IO 
 
 8.74226 
 
 n.25774 
 
 8.74292 
 
 11.25708 
 
 10.00066 
 
 9-99934 
 
 So 
 
 
 
 20 
 
 8.76451 
 
 H.23549 
 
 8.76525 
 
 11.23475 
 
 IO.OOO74 
 
 9-99926 
 
 40 
 
 
 
 30 
 
 8.78567 
 
 11.21432 
 
 8.78648 
 
 11.21351 
 
 10.00081 
 
 9.99919 
 
 30 
 
 
 
 40 
 
 8.80585 
 
 11.19415 
 
 8.80674 
 
 11.19326 
 
 10.00089 
 
 9-999H 
 
 20 
 
 
 
 50 
 
 8.82513 
 
 11.17487 
 
 8.82610 
 
 11.17390 
 
 10.00097 
 
 9.99903 
 
 IO 
 
 
 4 
 
 
 
 8.84358 
 
 11.15642 
 
 8.84464 
 
 11.15536 
 
 IO.OOI06 
 
 9.99894 
 
 o 
 
 86 
 
 
 10 ' 
 
 8.86128 
 
 11.13872 
 
 8.86243 
 
 II 13757 
 
 IO.OOII5 
 
 9-99885 
 
 50 
 
 
 
 20 
 
 8.87829 
 
 11.12171 
 
 8.87953 
 
 11.12047 
 
 IO.OOI24 
 
 9-99876 
 
 40 
 
 
 
 30 
 
 8.89464 
 
 11.10536 
 
 8.89598 
 
 11.10402 
 
 10.00134 
 
 9.99866 
 
 30 
 
 
 
 40 
 
 8.91040 
 
 11.08960 
 
 8.91185 
 
 11.08815 
 
 IO.OOI44 
 
 9.99856 
 
 20 
 
 
 
 50 
 
 8.92561 
 
 11.07439 
 
 8.92716 
 
 11.07284 
 
 10.00155 
 
 9.99845 
 
 10 
 
 
 5 
 
 
 
 8.04030 
 
 11.05970 
 
 8.94195 
 
 11.05805 
 
 IO.OOI66 
 
 9.99834 
 
 o 
 
 85 
 
 
 10 
 
 8.95450 
 
 11.04550 
 
 8.95627 
 
 n.04373 
 
 IO.OOI77 
 
 9 99823 
 
 50 
 
 
 
 20 
 
 8.96825 
 
 11-03175 
 
 8-97013 
 
 11.02987 
 
 10.00188 
 
 9.99812 
 
 40 
 
 
 
 30 
 
 8.03157 
 
 11.01843 
 
 8.98358 
 
 11.01642 
 
 10 . OO2OO 
 
 9.998oo 
 
 30 
 
 
 
 40 
 
 8.99450 
 
 I I . 00550 
 
 8.99662 
 
 11.00337 
 
 IO.OO2I3 
 
 9.99787 
 
 20 
 
 
 
 50 
 
 9.00704 
 
 10.99296 
 
 9.00930 
 
 10.99070 
 
 IO.OO225 
 
 9-99775 
 
 10 
 
 
 6 
 
 o 
 
 9 01923 
 
 10.98076 
 
 9.02162 
 
 10.97838 
 
 10.00239 
 
 9-9976i 
 
 
 
 84 
 
 
 IO 
 
 9.03109 
 
 10.96891 
 
 9 03361 
 
 10.96639 
 
 10.00252 
 
 9-99748 
 
 50 
 
 
 
 20 
 
 9 . 04262 
 
 10.95738 
 
 9-04528 
 
 10.95472 
 
 10.00266 
 
 9-99734 
 
 40 
 
 
 
 30 
 
 9 05386 
 
 10.94614 
 
 9.05666 
 
 10.94334 
 
 10.00280 
 
 9-99720 
 
 30 
 
 
 
 40 
 
 9.06480 
 
 10.93519 
 
 9-06775 
 
 10.93225 
 
 IO.OO295 
 
 9-99705 
 
 20 
 
 
 
 50 
 
 9.07548 
 
 10.92452 
 
 9.07858 
 
 10.92142 
 
 IO.O03IO 
 
 9.99690 
 
 10 
 
 83 
 
 
 
 Cosine. 
 
 Sec. 
 
 Cotang. 
 
 Tang. Cosec. 
 
 Sine. 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 For functions from 83 10' to 90 oc/ read from bottom of table upward. 
 
204 PRACTICAL SURVEYING 
 
 LOGARITHMIC FUNCTIONS OF ANGLES (Continued) 
 
 Deg. 
 
 Min. 
 
 Sine. 
 
 Cosec. 
 
 Tang. 
 
 Cotang. 
 
 Sec. 
 
 Cosine. 
 
 Min. 
 
 Deg. 
 
 7 
 
 o 
 
 9-08589 
 
 10.91411 
 
 9.08914 
 
 10.91086 
 
 10.00325 
 
 9.99675 
 
 
 
 83 
 
 
 IO 
 
 9.09606 
 
 10.90394 
 
 9.09947 
 
 10.90053 
 
 10.00341 
 
 9.99659 
 
 So 
 
 
 
 20 
 
 9-10599 
 
 10.89401 
 
 9-10956 
 
 10.89044 
 
 10.00357 
 
 9.99643 
 
 40 
 
 
 
 30 
 
 9.11570 
 
 10.88430 
 
 9-II943 
 
 10.88057 
 
 10.00373 
 
 9.99627 
 
 30 
 
 
 
 40 
 
 9.12519 
 
 10.87481 
 
 9-12909 
 
 10.87091 
 
 10.00390 
 
 9.99610 
 
 20 
 
 
 
 So 
 
 9-13447 
 
 10.86553 
 
 9.13854 
 
 10.86146 
 
 10.00407 
 
 9 99593 
 
 10 
 
 
 8 
 
 o 
 
 9.14356 
 
 10.85644 
 
 9.14780 
 
 10.85220 
 
 10.00425 
 
 9 99575 
 
 
 
 82 
 
 
 IO 
 
 9.15245 
 
 10.84755 
 
 9.15688 
 
 10.84312 
 
 10.00443 
 
 9-99557 
 
 50 
 
 
 
 20 
 
 9.16116 
 
 10.83884 
 
 9.16577 
 
 10.83423 
 
 10.00461 
 
 9-99539 
 
 40 
 
 
 
 30 
 
 9.16970 
 
 10.83030 
 
 9-17450 
 
 10.82550 
 
 10.00480 
 
 9-99520 
 
 30 
 
 
 
 40 
 
 9.17807 
 
 10.82193 
 
 9.18306 
 
 10.81694 
 
 10.00499 
 
 9-99501 
 
 20 
 
 
 
 50 
 
 9.18628 
 
 10.81372 
 
 9.19146 
 
 10.80854 
 
 10.00518 
 
 9.99482 
 
 IO 
 
 
 9 
 
 O 
 
 9-19433 
 
 10.80567 
 
 9.19971 
 
 10.80029 
 
 10.00538 
 
 9.99462 
 
 o 
 
 81 
 
 
 IO 
 
 9.20223 
 
 10.79777 
 
 9.20782 
 
 10.79218 
 
 10.00558 
 
 9.99442 
 
 50 
 
 
 
 20 
 
 9.20999 
 
 10.79001 
 
 9-21578 
 
 10.78422 
 
 10.00579 
 
 9.99421 
 
 40 
 
 
 
 30 
 
 9.21761 
 
 10.78239 
 
 9.22361 
 
 10.77639 
 
 10.00600 
 
 9.99400 
 
 30 
 
 
 
 40 
 
 9.22509 
 
 10.77491 
 
 9-23130 
 
 10.76870 
 
 10.00621 
 
 9-99379 
 
 20 
 
 
 
 50 
 
 9-23244 
 
 10.76756 
 
 9-23887 
 
 10.76113 
 
 10.00643 
 
 9-99357 
 
 10 
 
 
 10 
 
 
 
 9-23967 
 
 10.76033 
 
 9-24632 
 
 10.75368 
 
 10.00665 
 
 9-99335 
 
 
 
 80 
 
 
 IO 
 
 9-24677 
 
 10.75323 
 
 9-25365 
 
 10.74635 
 
 10.00687 
 
 9-99313 
 
 50 
 
 
 
 20 
 
 9.25376 
 
 10.74624 
 
 9.26086 
 
 10.73914 
 
 10.00710 
 
 9.99290 
 
 40 
 
 
 
 30 
 
 9.26063 
 
 10.73937 
 
 9.26797 
 
 10.73203 
 
 10.00733 
 
 9.99267 
 
 30 
 
 
 
 40 
 
 9.26739 
 
 10.73261 
 
 9.27496 
 
 10.72503 
 
 10.00757 
 
 9-99243 
 
 20 
 
 
 
 50 
 
 9-27405 
 
 10.72595 
 
 9.28186 
 
 10.71814 
 
 10.00781 
 
 9.99219 
 
 IO 
 
 
 11 
 
 
 
 9.28060 
 
 10.71940 
 
 9.28865 
 
 10,71135 
 
 10.00805 
 
 9.99195 
 
 o 
 
 79 
 
 
 10 
 
 9.28705 
 
 10.71295 
 
 9-29535 
 
 10.70465 
 
 10.00830 
 
 9.99170. 
 
 50 
 
 
 
 20 
 
 9-29340 
 
 10.70660 
 
 9.30195 
 
 10.69805 
 
 10.00855 
 
 9-99M5 
 
 40 
 
 
 
 30 
 
 9-29966 
 
 10.70034 
 
 9-30846 
 
 10.69154 
 
 10.00881 
 
 9.99119 
 
 30 
 
 
 
 40 
 
 9-30582 
 
 10.69418 
 
 9.31489 
 
 10.68511 
 
 10.00901 
 
 9-99093 
 
 20 
 
 
 
 50 
 
 9.3"89 
 
 10.68811 
 
 9.32122 
 
 10.67878 
 
 10.00933 
 
 9.99067 
 
 IO 
 
 
 12 
 
 o 
 
 9.31788 
 
 10.68212 
 
 9.32747 
 
 10.67253 
 
 10.00960 
 
 9.99040 
 
 O 
 
 78 
 
 
 10 
 
 9-32378 
 
 10.67622 
 
 9.33365 
 
 10.66635 
 
 10.00987 
 
 9.99013 
 
 50 
 
 
 
 20 
 
 9.32960 
 
 10.67040 
 
 9-33974 
 
 10.66026 
 
 10.01014 
 
 9.98986 
 
 40 
 
 
 
 30 
 
 9-33534 
 
 10.66466 
 
 9.34576 
 
 10.65424 
 
 10.01042 
 
 9.98958 
 
 30 
 
 
 
 40 
 
 9-34100 
 
 10.65900 
 
 9.35170 
 
 10.64830 
 
 10.01070 
 
 9.98930 
 
 20 
 
 
 
 50 
 
 9.34658 
 
 10.65342 
 
 9-35757 
 
 10.64243 
 
 10.01099 
 
 9.98901 
 
 IO 
 
 
 13 
 
 o 
 
 9.35209 
 
 10.64791 
 
 9.36336 
 
 10.63664 
 
 10.01128 
 
 9.98872 
 
 o 
 
 77 
 
 
 IO 
 
 9-35752 
 
 10.64248 
 
 9-36909 
 
 10.63091 
 
 10.01157 
 
 9.98843 
 
 So 
 
 
 
 20 
 
 9.36289 
 
 10.63711 
 
 9.37476 
 
 10.62524 
 
 10.01187 
 
 9-98813 
 
 40 
 
 
 
 30 
 
 9-36819 
 
 10.63181 
 
 9-38035 
 
 10.61965 
 
 10.01217 
 
 9.98783 
 
 30 
 
 
 
 40 
 
 9-37341 
 
 10.62659 
 
 9-38589 
 
 10.61411 
 
 10.01247 
 
 9.98752 
 
 20 
 
 
 
 50 
 
 9.37858 
 
 10.62142 
 
 9-39r36 
 
 10.60864 
 
 10.01278 
 
 9.98722 
 
 IO 
 
 76 
 
 
 
 Cosine. 
 
 Sec. 
 
 Cotang. 
 
 Tang. 
 
 Cosec. 
 
 Sine. 
 
 
 
 For functions from 76 10' to 83 oo' read from bottom of table upward. 
 
TRIGONOMETRY 
 LOGARITHMIC FUNCTIONS OF ANGLES (Continued) 
 
 205 
 
 Deg. 
 
 Min. 
 
 Sine. 
 
 Cosec. 
 
 Tang. 
 
 Cotang. 
 
 Sec. 
 
 Cosine. 
 
 Min. 
 
 Deg. 
 
 14 
 
 
 
 9-38368 
 
 10.61632 
 
 9-39677 
 
 10.60323 
 
 10.01310 
 
 9.98690 
 
 
 
 76 
 
 
 IO 
 
 9-38871 
 
 10.61129 
 
 9.40212 
 
 10.59788 
 
 10.01341 
 
 9.98659 
 
 So 
 
 
 
 20 
 
 9.39369 
 
 10.60631 
 
 9.40742 
 
 10.59258 
 
 10.01373 
 
 9.98627 
 
 40 
 
 
 
 30 
 
 9.3986o 
 
 10.60140 
 
 9.41266 
 
 10.58734 
 
 10.01406 
 
 9.98594 
 
 30 
 
 
 
 40 
 
 9.40346 
 
 10.59654 
 
 9.41784 
 
 10.58216 
 
 10.01439 
 
 9.98561 
 
 20 
 
 
 
 50 
 
 9.40825 
 
 io.59i75 
 
 9.42297 
 
 10.57703 
 
 10.01472 
 
 9-98528 
 
 IO 
 
 
 15 
 
 
 
 9.41300 
 
 10.587* 
 
 9.42805 
 
 10.57195 
 
 10.01506 
 
 9.98494 
 
 o 
 
 75 
 
 
 10 
 
 9.41768 
 
 10.58232 
 
 9.43308 
 
 10.56692 
 
 10.01540 
 
 9-98460 
 
 50 
 
 
 
 20 
 
 9.42232 
 
 10.57768 
 
 9.438o6 
 
 10.56194 
 
 10.01574 
 
 9.98426 
 
 40 
 
 
 
 30 
 
 9.42690 
 
 io.573io 
 
 9.44299 
 
 10.55701 
 
 10.01609 
 
 9.98391 
 
 30 
 
 
 
 40 
 
 9.43I43 
 
 10.56857 
 
 9.44787 
 
 10.55213 
 
 10.01644 
 
 9.98356 
 
 20 
 
 
 
 50 
 
 9-43590 
 
 10.56409 
 
 9.45271 
 
 10.54729 
 
 10.01680 
 
 9-98320 
 
 IO 
 
 
 16 
 
 
 
 9-44034 
 
 10.55966 
 
 9-45750 
 
 10.54250 
 
 10.01716 
 
 9.98284 
 
 
 
 74 
 
 
 10 
 
 9-44472 
 
 10.55528 
 
 9.46224 
 
 10.53776 
 
 10.01752 
 
 9.98248 
 
 50 
 
 
 
 20 
 
 9.44905 
 
 10.55095 
 
 9-46694 
 
 10.53305 
 
 10.01789 
 
 9.98211 
 
 40 
 
 
 
 30 
 
 9-45334 
 
 10.54666 
 
 9.47160 
 
 10.52840 
 
 10.01826 
 
 9.98174 
 
 30 
 
 
 
 40 
 
 9-45758 
 
 10.54242 
 
 9.47622 
 
 10.52378 
 
 10.01864 
 
 9-98136 
 
 20 
 
 
 
 50 
 
 9.46178 
 
 10.53822 
 
 9.48080 
 
 10.51920 
 
 10.01902 
 
 9.98098 
 
 IO 
 
 
 17 
 
 o 
 
 9.46594 
 
 10.53406 
 
 9.48534 
 
 10.51466 
 
 10.01940 
 
 9.98060 
 
 o 
 
 73 
 
 
 IO 
 
 9-47005 
 
 10.52995 
 
 9.48984 
 
 10.51016 
 
 10.01979 
 
 9.98021 
 
 50 
 
 
 
 20 
 
 9-474" 
 
 10.52589 
 
 9-49430 
 
 10.50570 
 
 10.02018 
 
 9.97982 
 
 40 
 
 
 
 30 
 
 9.47814 
 
 10.52186 
 
 9.49872 
 
 10.50128 
 
 10.02058 
 
 9-97942 
 
 30 
 
 
 
 40 
 
 9-48213 
 
 10.51787 
 
 9.50311 
 
 10.49689 
 
 10.02098 
 
 9.97902 
 
 20 
 
 
 
 50 
 
 9.48607 
 
 I0.5I393 
 
 9.50746 
 
 10.49254 
 
 10.02138 
 
 9.97861 
 
 10 
 
 
 18 
 
 o 
 
 9.48998 
 
 10.51002 
 
 9.5H78 
 
 10.48822 
 
 10.02179 
 
 9.97821 
 
 o 
 
 78 
 
 
 10 
 
 9.49385 
 
 10.50615 
 
 9.51606 
 
 10.48394 
 
 10.02221 
 
 9-97779 
 
 50 
 
 
 
 20 
 
 9-49768 
 
 10.50232 
 
 9-52031 
 
 10.47969 
 
 10.02262 
 
 9-97738 
 
 40 
 
 
 
 30 
 
 9-50I48 
 
 10.49852 
 
 9.52452 
 
 10.47548 
 
 IO.O23O4 
 
 9.97696 
 
 30 
 
 
 
 40 
 
 9.50523 
 
 10.49477 
 
 9.52870 
 
 10.47130 
 
 10.02347 
 
 9.97653 
 
 20 
 
 
 
 50 
 
 9.50896 
 
 10.49104 
 
 9.53285 
 
 10.46715 
 
 IO.O2390 
 
 9.97610 
 
 10 
 
 
 19 
 
 
 
 9-5I264 
 
 10.48736 
 
 9.53697 
 
 10.46303 
 
 10.02433 
 
 9-97567 
 
 
 
 71 
 
 
 IO 
 
 9.51629 
 
 10.48371 
 
 9.54io6 
 
 10.45894 
 
 10.02477 
 
 9.97523 
 
 50 
 
 
 
 20 
 
 9.5I99I 
 
 10.48009 
 
 9-54512 
 
 10.45488 
 
 10.02521 
 
 9-97479 
 
 40 
 
 
 
 30 
 
 9.52350 
 
 10.47650 
 
 9-54915 
 
 10.45085 
 
 10.02565 
 
 9-97435 
 
 30 
 
 
 
 40 
 
 9.52705 
 
 10.47295 
 
 9-55315 
 
 10.44685 
 
 10.02610 
 
 9-97390 
 
 20 
 
 
 
 50 
 
 9-53057 
 
 10.46944 
 
 9-55712 
 
 10.44288 
 
 10 . 02656 
 
 9-97344 
 
 IO 
 
 
 20 
 
 
 
 9-53405 
 
 10.46595 
 
 9.56107 
 
 10.43893 
 
 10.02701 
 
 9.97299 
 
 o 
 
 70 
 
 
 10 
 
 9-53751 
 
 10.46249 
 
 9.56498 
 
 10.43502 
 
 10.02748 
 
 9.97252 
 
 50 
 
 
 
 20 
 
 9-54093 
 
 10.45907 
 
 9.56887 
 
 10.43113 
 
 10.02794 
 
 9.97206 
 
 40 
 
 
 
 30 
 
 9-54433 
 
 10.45567 
 
 9.57274 
 
 10.42726 
 
 10.02841 
 
 9-97159 
 
 30 
 
 
 
 40 
 
 9.54769 
 
 10.45231 
 
 9.57658 
 
 10.42342 
 
 10.02889 
 
 9.97111 
 
 20 
 
 
 
 50 
 
 9 55102 
 
 10.44898 
 
 9-58039 
 
 10.41961 
 
 10.02936 
 
 9-97063 
 
 10 
 
 69 
 
 
 
 Cosine. 
 
 Sec. 
 
 Cotang. 
 
 Tang. 
 
 Cosec. 
 
 Sine. 
 
 
 
 For functions from 69 lo 7 to 76 oo' read from bottom of table upward. 
 
206 PRACTICAL SURVEYING 
 
 LOGARITHMIC FUNCTIONS OF ANGLES (Continued) 
 
 Deg. 
 
 Min. 
 
 Sine. 
 
 Cosec. 
 
 Tang. 
 
 Cotang. 
 
 Sec. 
 
 Cosine. 
 
 Min. 
 
 Deg. 
 
 21 
 
 
 
 9-55433 
 
 10.44567 
 
 9-58418 
 
 10.41582 
 
 10.02985 
 
 9-97015 
 
 
 
 69 
 
 
 10 
 
 9.5576i 
 
 10.44239 
 
 9.58794 
 
 10.41206 
 
 10.03034 
 
 .9-96966 
 
 50 
 
 
 
 20 
 
 9-56085 
 
 io.439i5 
 
 9.59168 
 
 10.40832 
 
 10.03083 
 
 9.96917 
 
 40 
 
 
 
 30 
 
 9.56408 
 
 10.43592 
 
 9-59540 
 
 10.40460 
 
 10.03132 
 
 9.96868 
 
 30 
 
 
 
 40 
 
 9-56727 
 
 10.43273 
 
 9-59909 
 
 10.40091 
 
 10.03182 
 
 9.96818 
 
 20 
 
 
 
 50 
 
 9-57044 
 
 10.42956 
 
 9 . 60276 
 
 10.39724 
 
 10.03233 
 
 9-96767 
 
 10 
 
 
 22 
 
 
 
 9.57358 
 
 10.42642 
 
 9.60641 
 
 10.39359 
 
 10.03283 
 
 9.96717 
 
 o 
 
 68 
 
 
 10 
 
 9-57669 
 
 10.42331 
 
 9.61004 
 
 10.38996 
 
 10.03335 
 
 9-96665 
 
 50 
 
 
 
 20 
 
 9.57978 
 
 10.42022 
 
 9-61364 
 
 10.38636 
 
 10.03386 
 
 9.96614 
 
 40 
 
 
 
 30 
 
 9-58284 
 
 10.41716 
 
 9.61722 
 
 10.38278 
 
 10.03438 
 
 9.95562 
 
 30 
 
 
 
 40 
 
 9-53583 
 
 10.41412 
 
 9.62079 
 
 10.37921 
 
 10.03491 
 
 9-96509 
 
 20 
 
 
 
 50 
 
 9.58889 
 
 10.41111 
 
 9-62433 
 
 10.37567 
 
 10.03544 
 
 9-96456 
 
 10 
 
 
 23 
 
 
 
 9.59188 
 
 10.40812 
 
 9-62785 
 
 10.37215 
 
 10.03597 
 
 9.96403 
 
 
 
 67 
 
 
 10 
 
 9.59484 
 
 10.40516 
 
 9.63135 
 
 10.36365 
 
 10.03651 
 
 9.96349 
 
 50 
 
 
 
 20 
 
 9.59778 
 
 10.40222 
 
 9.63484 
 
 10.36516 
 
 10.03706 
 
 9-96294 
 
 40 
 
 
 
 30 
 
 9 . 60070 
 
 io.3993o 
 
 9-63830 
 
 10.36170 
 
 10.03760 
 
 9.96240 
 
 30 
 
 
 
 40 
 
 9.60359 
 
 10.39641 
 
 9.64175 
 
 10.35825 
 
 10.03815 
 
 9.96185 
 
 20 
 
 
 
 50 
 
 9.60646 
 
 10.39354 
 
 9.64517 
 
 10.35483 
 
 10.03871 
 
 9.96129 
 
 10 
 
 
 24 
 
 
 
 9 60931 
 
 10.39069 
 
 9-64858 
 
 10.35142 
 
 10.03927 
 
 9-96073 
 
 
 
 C6 
 
 
 10 
 
 9.61214 
 
 10.38786 
 
 9.65197 
 
 10.34803 
 
 10.03983 
 
 9.96017 
 
 50 
 
 
 
 20 
 
 9.61494 
 
 10.38506 
 
 9.65535 
 
 10.34465 
 
 10.04040 
 
 9.9596o 
 
 40 
 
 
 
 30 
 
 9.6i773 
 
 10.38227 
 
 9-65870 
 
 10.34130 
 
 10 . 04098 
 
 9-95902 
 
 30 
 
 
 
 40 
 
 9-62049 
 
 10.37951 
 
 9.66204 
 
 10.33795 
 
 10.04156 
 
 9.95845 
 
 20 
 
 
 
 50 
 
 9-62323 
 
 10.37677 
 
 9.66537 
 
 10.33463 
 
 10.04214 
 
 9.95786 
 
 IO 
 
 
 25 
 
 
 
 9-62595 
 
 10.37405 
 
 9.66867 
 
 10 33133 
 
 10.04272 
 
 9 95728 
 
 
 
 66 
 
 
 10 
 
 9.62865 
 
 10.37135 
 
 9.67196 
 
 10.32804 
 
 10.04332 | 9.95668 
 
 50 
 
 
 
 20 
 
 9-63I33 
 
 10.36867 
 
 9-67524 
 
 10.32476 
 
 10.04391 
 
 9.95609 
 
 40 
 
 
 
 30 
 
 9.63398 
 
 10.36602 
 
 9.67850 
 
 10.32150 
 
 10.04451 
 
 9-95549 
 
 30 
 
 
 
 40 
 
 9.63662 
 
 10.36338 
 
 9.68174 
 
 10.31826 
 
 10.04512 
 
 9.95488 
 
 20 
 
 
 
 50 
 
 9.63924 
 
 10.36076 
 
 9-68497 
 
 10.31503 
 
 10.04573 
 
 9.95427 
 
 10 
 
 
 26 
 
 
 
 9.64184 
 
 10.35816 
 
 9.68818 
 
 10.31182 
 
 10.04634 
 
 9.95366 
 
 o 
 
 64 
 
 
 10 
 
 9-64442 
 
 10.35558 
 
 9.69138 
 
 10.30862 
 
 10.04696 
 
 9-95304 
 
 50 
 
 
 
 20 
 
 9.64698 
 
 10.35302 
 
 9-69457 
 
 10.30543 
 
 10.04758 
 
 9.95242 
 
 40 
 
 
 
 30 
 
 9.64953 
 
 10.35047 
 
 9.69774 
 
 10.30226 
 
 10.04821 
 
 9.95179 
 
 30 
 
 
 
 40 
 
 9-65205 
 
 10.34795 
 
 9.70089 
 
 '10.29911 
 
 10.04884 
 
 9-9SII6 
 
 20 
 
 
 
 50 
 
 9.65456 
 
 10.34544 
 
 9.70404 
 
 10.29596 
 
 10.04948 
 
 9-95052 
 
 10 
 
 
 27 
 
 o 
 
 9.65705 
 
 10.34295 
 
 9.70717 
 
 10 . 29283 
 
 10.05012 
 
 9.94988 
 
 o 
 
 63 
 
 
 10 
 
 9.65952 
 
 10.34048 
 
 9.71028 
 
 10.28972 
 
 10.05077 
 
 9-94923 
 
 50 
 
 
 
 20 
 
 9.66197 
 
 10.33803 
 
 9.71338 
 
 10.28661 
 
 10.05142 
 
 9.94858 
 
 40 
 
 
 
 30 
 
 9.66441 
 
 10.33559 
 
 9-71648 
 
 10.28352 
 
 10.05207 
 
 9-94793 
 
 30 
 
 
 
 40 
 
 9.66682 
 
 I0.333I8 
 
 9-71955 
 
 10.28045 
 
 10.05273 
 
 9.94727 
 
 20 
 
 
 
 50 
 
 9-66923 
 
 10.33078 
 
 9.72262 
 
 10.27738 
 
 10.05340 
 
 9.94660 
 
 10 
 
 62 
 
 
 
 Cosine. 
 
 Sec. 
 
 Cotang. 
 
 Tang. 
 
 Cosec. 
 
 Sine. 
 
 
 
 For functions from 62 10' to 69 oo' read from bottom of table upward. 
 
TRIGONOMETRY 
 
 LOGARITHMIC FUNCTIONS OF ANGLES (Continued) 
 
 207 
 
 Deg. 
 
 Min. 
 
 Sine. 
 
 Cosec. 
 
 Tang. 
 
 Cotang. 
 
 Sec. 
 
 Cosine. 
 
 Min. 
 
 Deg. 
 
 28 
 
 o 
 
 9.67161 
 
 10.32839 
 
 9.72567 
 
 10.27433 
 
 10.05406 
 
 9-94593 
 
 o 
 
 62 
 
 
 10 
 
 9.67398 
 
 10.32602 
 
 9.72872 
 
 10.27128 
 
 10.05474 
 
 9-94526 
 
 50 
 
 
 
 20 
 
 9-67633 
 
 10.32367 
 
 9-73175 
 
 10.26825 
 
 10.05542 
 
 9-94458 
 
 40 
 
 
 
 30 
 
 9.67866 
 
 10.32134 
 
 9-73476 
 
 10.26524 
 
 10.05610 
 
 9-94390 
 
 30 
 
 
 
 40 
 
 9.68098 
 
 10.31902 
 
 9-73777 
 
 10.26223 
 
 10.05679 
 
 9-94321 
 
 20 
 
 
 
 SO 
 
 9.68328 
 
 10.31672 
 
 9.74077 
 
 10.25923 
 
 10.05748 
 
 9.94252 
 
 10 
 
 
 29 
 
 
 
 9-68557 
 
 10.31443 
 
 9-74375 
 
 10.25625 
 
 10.05818 
 
 9.94182 
 
 o 
 
 61 
 
 
 IO 
 
 9.68784 
 
 10.31216 
 
 9.74673 
 
 10.25327 
 
 10.05888 
 
 9.94112 
 
 50 
 
 
 
 20 
 
 9.69010 
 
 10.30990 
 
 9.74969 
 
 10.25031 
 
 10.05959 
 
 9 94041 
 
 40 
 
 
 
 30 
 
 9-69234 
 
 10.30766 
 
 9.75264 
 
 10.24736 
 
 10.06030 
 
 9-93970 
 
 30 
 
 
 
 40 
 
 9.69456 
 
 10.30544 
 
 9-75558 
 
 10.24442 
 
 10.06102 
 
 9.93898 
 
 20 
 
 
 
 50 
 
 9-69677 
 
 10.30323 
 
 9.75852 
 
 10.24148 
 
 10.06174 
 
 9-93826 
 
 10 
 
 
 SO 
 
 
 
 9.69897 
 
 10.30103 
 
 9.76144 
 
 10.23856 
 
 10.06247 
 
 9-93753 
 
 
 
 60 
 
 
 IO 
 
 9.70115 
 
 10.29885 
 
 9.76435 
 
 10.23565 
 
 10.06320 
 
 9.9368o 
 
 50 
 
 
 
 20 
 
 9-70332 
 
 10.29668 
 
 9.76726 
 
 10.23275 
 
 10.06394 
 
 9.936o6 
 
 40 
 
 
 
 30 
 
 9.70547 
 
 10.29453 
 
 9-77015 
 
 10.22985 
 
 10.06468 
 
 9-93532 
 
 30 
 
 
 
 40 
 
 9.70761 
 
 10.29239 
 
 9.77303 
 
 10.22697 
 
 10.06543 
 
 9-93457 
 
 20 
 
 
 
 So 
 
 9.70973 
 
 10.29027 
 
 9-77591 
 
 10.22409 
 
 10.06618 
 
 9-93382 
 
 10 
 
 
 31 
 
 
 
 9.71184 
 
 10.28816 
 
 9.77877 
 
 10.22123 
 
 10.06693 
 
 9.93307 
 
 o 
 
 59 
 
 
 10 
 
 9-71393 
 
 10.28607 
 
 9-78163 
 
 10.21837 
 
 10.06770 
 
 9-93230 
 
 50 
 
 
 
 20 
 
 9.71602 
 
 10.28398 
 
 9.78448 
 
 10.21552 
 
 10.06846 
 
 9-93154 
 
 40 
 
 
 
 30 
 
 9.71809 
 
 10.28191 
 
 9.78732 
 
 10.21268 
 
 10.06923 
 
 9.93077 
 
 30 
 
 
 
 40 
 
 9.72014 
 
 10.27986 
 
 9-79015 
 
 10.20985 
 
 10.07001 
 
 9.92999 
 
 20 
 
 
 
 50 
 
 9.72218 
 
 10.27782 
 
 9.79297 
 
 10 . 20703 
 
 10.07079 
 
 9.92921 
 
 10 
 
 
 32 
 
 o 
 
 9.72421 
 
 10.27579 
 
 9-79579 
 
 10.20421 
 
 10.07158 
 
 9.92842 
 
 O 
 
 58 
 
 
 10 
 
 9.72622 
 
 10.27378 
 
 9.79860 
 
 10.20140 
 
 10.07237 
 
 9.92763 
 
 50 
 
 
 
 20 
 
 9.72823 
 
 10.27177 
 
 9.80140 
 
 10.19860 
 
 10.07317 
 
 9.92683 
 
 40 
 
 
 
 30 
 
 9-73022 
 
 10.26978 
 
 9.80419 
 
 10.19581 
 
 10.07397 
 
 9.92603 
 
 30 
 
 
 
 40 
 
 9.73219 
 
 10.26781 
 
 9-80697 
 
 10.19303 
 
 10.07478 
 
 9.92522 
 
 20 
 
 
 
 50 
 
 9.73416 
 
 10.26584 
 
 9.80975 
 
 10.19025 
 
 10.07559 
 
 9.92441 
 
 10 
 
 
 33 
 
 
 
 9.73611 
 
 10.26389 
 
 9.81252 
 
 10.18748 
 
 10.07641 
 
 9 92359 
 
 o 
 
 57 
 
 
 10 
 
 9.73805 
 
 10.26195 
 
 9.81528 
 
 10.18472 
 
 10.07723 
 
 9.92277 
 
 50 
 
 
 
 20 
 
 9-73997 
 
 10.26003 
 
 9.81803 
 
 10.18197 
 
 10.07806 
 
 9.92II94 
 
 40 
 
 
 
 30 
 
 9.74189 
 
 10.25811 
 
 9.82078 
 
 10.17922 
 
 10.07889 
 
 9.92111 
 
 30 
 
 
 
 40 
 
 9-74379 
 
 10.25621 
 
 9-82352 
 
 10.17648 
 
 10.07973 
 
 9.92027 
 
 20 
 
 
 
 50 
 
 9.74568 
 
 10.25432 
 
 9.82626 
 
 10.17374 
 
 10.08058 
 
 9.91942 
 
 IO 
 
 
 34 
 
 o 
 
 9.74756 
 
 10.25244 
 
 9-82899 
 
 10.17101 
 
 10.08143 
 
 9.91857 
 
 
 
 56 
 
 
 IO 
 
 9-74943 
 
 10.25057 
 
 9-83I7I 
 
 10.16829 
 
 10.08228 
 
 9.91772 
 
 50 
 
 
 
 20 
 
 9.75I28 
 
 10.24872 
 
 9.83442 
 
 10.16558 
 
 10.08314 
 
 9.91686 
 
 40 
 
 
 
 30 
 
 9-75313 
 
 10.24687 
 
 9.837I3 
 
 10.16287 
 
 10.08401 
 
 9.91599 
 
 30 
 
 
 
 40 
 
 9.75496 
 
 10.24504 
 
 9.83984 
 
 10.16016 
 
 10.08488 
 
 9.9ISI2 
 
 20 
 
 
 
 50 
 
 9.75678 
 
 10.24322 
 
 9.84254 
 
 10.15746 
 
 10.08575 
 
 9-9I425 
 
 10 
 
 55 
 
 
 
 Cosine. 
 
 Sec. 
 
 Cotang. 
 
 Tang. 
 
 Cosec. 
 
 Sine. 
 
 
 
 For functions from 55 10' to 62 oc/ read from bottom of table upward. 
 
208 PRACTICAL SURVEYING 
 
 LOGARITHMIC FUNCTIONS OF ANGLES (Continued) 
 
 Deg. 
 
 Min. 
 
 Sine. 
 
 Cosec. 
 
 Tang. 
 
 Cotang. 
 
 Sec. 
 
 Cosine. 
 
 Min. 
 
 Deg. 
 
 35 
 
 o 
 
 9.75859 
 
 10.24141 
 
 9-84523 
 
 10.15477 
 
 10.08664 
 
 9.91336 
 
 o 
 
 55 
 
 
 10 
 
 9-76039 
 
 10.23961 
 
 9.84791 
 
 10.15209 
 
 10.08752 
 
 9-91248 
 
 50 
 
 
 
 20 
 
 9.76218 
 
 10.23782 
 
 9-85059 
 
 10.14941 
 
 10.08842 
 
 9.9H58 
 
 40 
 
 
 
 30 
 
 9.76395 
 
 10.23604 
 
 9-85327 
 
 10.14673 
 
 10.08931 
 
 9-91069 
 
 30 
 
 
 
 40 
 
 9.76572 
 
 10.23428 
 
 9.85594 
 
 10.14406 
 
 10.09022 
 
 9-90978 
 
 20 
 
 
 
 50 
 
 9.76747 
 
 10.23253 
 
 9.85860 
 
 10.14140 
 
 10.09113 
 
 9-90887 
 
 IO 
 
 
 36 
 
 o 
 
 9.76922 
 
 10.23078 
 
 9.86126 
 
 10.13874 
 
 10.09204 
 
 9.90796 
 
 
 
 54 
 
 
 10 
 
 9.77095 
 
 10.22905 
 
 9-86392 
 
 10.13608 
 
 10 . 09296 
 
 9.90704 
 
 50 
 
 
 
 20 
 
 9.77268 
 
 10.22732 
 
 9.86656 
 
 10.13344 
 
 10.09389 
 
 9.90611 
 
 40 
 
 
 
 30 
 
 9-77439 
 
 10.22561 
 
 9.86921 
 
 10.13079 
 
 10.09482 
 
 9-90518 
 
 30 
 
 
 
 40 
 
 9.77609 
 
 10.22391 
 
 9-87185 
 
 10.12815 
 
 10.09576 
 
 9.90424 
 
 20 
 
 
 
 50 
 
 9-77778 
 
 I O. 22222 
 
 9.87448 
 
 10.12552 
 
 10.09670 
 
 9.90330 
 
 10 
 
 
 37 
 
 o 
 
 9.77946 
 
 IO.22O54 
 
 9-87711 
 
 10.12289 
 
 10.09765 
 
 9.90235 
 
 
 
 53 
 
 
 10 
 
 9.78H3 
 
 IO.2I887 
 
 9.87974 
 
 10.12026 
 
 10.09861 
 
 9-90139 
 
 50 
 
 
 
 20 
 
 9.78280 
 
 I0.2I72O 
 
 9.88236 
 
 10.11764 
 
 10.09957 
 
 9-90043 
 
 40 
 
 
 
 30 
 
 9.78445 
 
 10.21555 
 
 9.88498 
 
 10.11502 
 
 10.10053 
 
 9.89947 
 
 30 
 
 
 
 40 
 
 9-78609 
 
 10.21391 
 
 9.88759 
 
 10.11241 
 
 10.10151 
 
 9.89849 
 
 20 
 
 
 
 50 
 
 9.78772 
 
 10.21228 
 
 9.89020 
 
 10.10980 
 
 10.10248 
 
 9-89752 
 
 10 
 
 
 38 
 
 o 
 
 9.78934 
 
 IO.2IO66 
 
 9.89281 
 
 10.10719 
 
 10.10347 
 
 9.89653 
 
 o 
 
 52 
 
 
 IO 
 
 9-79095 
 
 10.20905 
 
 9.89541 
 
 10.10459 
 
 10.10446 
 
 9.89554 
 
 So 
 
 
 
 20 
 
 9.79256 
 
 10.20744 
 
 9.89801 
 
 10.10199 
 
 10.10545 
 
 9.89455 
 
 40 
 
 
 
 30 
 
 9-79415 
 
 10.20585 
 
 9.90061 
 
 10.09939 
 
 10.10646 
 
 9.89354 
 
 30 
 
 
 
 40 
 
 9-79573 
 
 10.20427 
 
 9.90320 
 
 10.09680 
 
 10.10746 
 
 9.89254 
 
 20 
 
 
 
 50 
 
 9-79731 
 
 10.20269 
 
 9.90578 
 
 10.09422 
 
 10.10848 
 
 9.89152 
 
 10 
 
 
 39 
 
 o 
 
 9.79887 
 
 IO.2OII3 
 
 9.90837 
 
 10.09163 
 
 10.10950 
 
 9-89050 
 
 o 
 
 51 
 
 
 10 
 
 9-80043 
 
 10.19957 
 
 9-91095 
 
 10.08905 
 
 10.11052 
 
 9.88948 
 
 50 
 
 
 
 20 
 
 9.80197 
 
 10.19803 
 
 9.91353 
 
 10.08647 
 
 10.11156 
 
 9.88844 
 
 40 
 
 
 
 30 
 
 9-80351 
 
 10.19649 
 
 9.91610 
 
 10.08390 
 
 10.11259 
 
 9.88741 
 
 30 
 
 
 
 40 
 
 9-80504 
 
 IO.I9496 
 
 9.91868 
 
 10.08132 
 
 10.11364 
 
 9.88636 
 
 20 
 
 
 
 50 
 
 9.80656 
 
 10.19344 
 
 9.92125 
 
 10.07875 
 
 10.11469 
 
 9-88531 
 
 10 
 
 
 40 
 
 o 
 
 9.80807 
 
 10.19193 
 
 9.92381 
 
 10.07619 
 
 10.11575 
 
 9.88425 
 
 O 
 
 50 
 
 
 TO 
 
 9.80957 
 
 IO.I9O43 
 
 9-92638 
 
 10.07362 
 
 10. 11681 
 
 9.88319 
 
 50 
 
 
 
 2O 
 
 9.81106 
 
 10.18894 
 
 9.92894 
 
 10.07106 
 
 10.11788 
 
 9.88212 
 
 40 
 
 
 
 30 
 
 9-81254 
 
 10.18746 
 
 9.93150 
 
 10.06850 
 
 10.11895 
 
 9.88105 
 
 30 
 
 
 
 40 
 
 9.81402 
 
 10.18598 
 
 9.93406 
 
 10.06594 
 
 10.12004 
 
 9.87996 
 
 20 
 
 
 
 50 
 
 9-81548 
 
 IO.I845I 
 
 9.93661 
 
 10.06339 
 
 10.12113 
 
 9.87887 
 
 IO 
 
 
 41 
 
 
 
 9.81694 
 
 10.18306 
 
 9.939i6 
 
 10.06084 
 
 10.12222 
 
 9.87778 
 
 o 
 
 49 
 
 
 10 
 
 9 81839 
 
 10. 18161 
 
 9.94171 
 
 10.05829 
 
 10.12332 
 
 9.87668 
 
 50 
 
 
 
 20 
 
 9 81983 
 
 10.18017 
 
 9.94426 
 
 10.05574 
 
 10.12443 
 
 9-87557 
 
 40 
 
 
 
 30 
 
 9.82126 
 
 10.17874 
 
 9.94681 
 
 10.05319 
 
 10.12554 
 
 9.87446 
 
 30 
 
 
 
 40 
 
 9.82269 
 
 10.17731 
 
 9-94935 
 
 10.05065 
 
 10.12666 
 
 9-87334 
 
 ?o 
 
 
 
 50 
 
 9.82410 
 
 10.17590 
 
 9.95190 
 
 10.04810 
 
 10.12779 
 
 9.87221 
 
 10 
 
 48 
 
 
 
 Cosine. 
 
 Sec. 
 
 Cotang. 
 
 Tang. 
 
 Cosec. 
 
 Sine. 
 
 
 
 For functions from 48 10' to 55 oo' read from bottom of table upward 
 
TRIGONOMETRY 
 LOGARITHMIC FUNCTIONS OF ANGLES (Continued) 
 
 209 
 
 
 
 
 
 
 1 
 
 
 
 
 Deg. 
 
 Min. 
 
 Sine. 
 
 Cosec. 
 
 Tang. 
 
 Cotang. 
 
 Sec. 
 
 Cosine. 
 
 Min. 
 
 Deg. 
 
 42 
 
 o 
 
 9-82551 
 
 10.17449 
 
 9-95444 
 
 10.04556 
 
 10.12893 
 
 9.87107 
 
 o 
 
 48 
 
 
 10 
 
 9.82691 
 
 10.17309 
 
 9.95698 
 
 10.04302 
 
 10.13007 
 
 9.86993 
 
 So 
 
 
 
 20 
 
 9.82830 
 
 10.17170 
 
 9-95952 
 
 10.04048 
 
 10.13121 
 
 9.86879 
 
 40 
 
 
 
 30 
 
 9.82968 
 
 10.17032 
 
 9.96205 
 
 10.03795 
 
 10.13237 
 
 9.86763 
 
 30 
 
 
 
 40 
 
 9.83106 
 
 10.16894 
 
 9.96459 
 
 10.03541 
 
 10.13353 
 
 9.86647 
 
 20 
 
 
 
 So 
 
 9.83242 
 
 10.16758 
 
 9.96712 
 
 10.03288 
 
 10.13470 
 
 9.86530 
 
 10 
 
 
 43 
 
 
 
 9.83378 
 
 10.16622 
 
 9.96966 
 
 10.03034 
 
 10.13587 
 
 9-86413 
 
 o 
 
 47 
 
 
 10 
 
 9-83513 
 
 10.16487 
 
 9.97219 
 
 10.02781 
 
 10.13705 
 
 9-86295 
 
 So 
 
 
 
 20 
 
 9.83648 
 
 10.16352 
 
 9-97472 
 
 10.02528 
 
 10.13824 
 
 9.86176 
 
 40 
 
 
 
 30 
 
 9.83781 
 
 10.16219 
 
 9-97725 
 
 10.02275 
 
 10.13944 
 
 9.86056 
 
 30 
 
 
 
 40 
 
 9.83914 
 
 10.16086 
 
 9.97978 
 
 10.02022 
 
 10.14064 
 
 9 85936 
 
 20 
 
 
 
 SO 
 
 9.84046 
 
 10.15954 
 
 9.98231 
 
 IO.OI7O9 
 
 10.14185 
 
 9-85815 
 
 IO 
 
 
 44 
 
 o 
 
 9.84177 
 
 10.15823 
 
 9-98484 
 
 IO.OI5I6 
 
 10.14307 
 
 9-85693 
 
 O 
 
 46 
 
 
 10 
 
 9.84308 
 
 10.15692 
 
 9.98737 
 
 10.01263 
 
 10.14429 
 
 9.85571 
 
 So 
 
 
 
 20 
 
 9-84437 
 
 10.15563 
 
 9.98989 
 
 IO.OIOII 
 
 10.14552 
 
 9.85448 
 
 40 
 
 
 
 30 
 
 9.84566 
 
 10.15434 
 
 9.99242 
 
 10.00758 
 
 10.14676 
 
 9-85324 
 
 30 
 
 
 
 40 
 
 9-84694 
 
 10.15306 
 
 9-99495 
 
 IO.OO5O5 
 
 10.14800 
 
 9.85200 
 
 20 
 
 
 
 So 
 
 9.84822 
 
 10.15178 
 
 9-99747 
 
 IO.O0253 
 
 10.14926 
 
 9-85074 
 
 10 
 
 
 46 
 
 
 9.84949 
 
 10.15052 
 
 10.00000 
 
 10.00000 
 
 IO . 15052 
 
 9.84949 
 
 
 46 
 
 
 
 Cosine. 
 
 Sec. 
 
 Cotang. 
 
 Tang. 
 
 Cosec. 
 
 Sine. 
 
 
 
 For functions from 45 09' to 48 oo' read from bottom of table upward. 
 
CHAPTER VI 
 
 TRANSIT SURVEYING 
 
 Surveying was an established calling many centuries be- 
 fore the compass was known, there being a well-developed 
 system of mensuration in Egypt in the time of Joseph. 
 Nothing however is known of the instruments used in the 
 most ancient times. 
 
 In the second century B.C. there lived in Alexandria 
 Heron the Elder, a mathematician and practical surveyor. 
 He has been styled "The first engineer," because of a 
 number of inventions, one being the aeolipile, the first 
 steam engine. A book entitled "Dioptra," the first 
 known treatise on surveying, is supposed to have been 
 written by him although some writers believe it to be the 
 work of a later writer of the same name. The "Dioptra" 
 is a treatise on the use of the diopter, a surveying instru- 
 ment in common use up to the end of the Middle Ages. 
 With this instrument the Romans laid out their cities, 
 roads, aqueducts and all public works. 
 
 Venturi wrote : " Dioptra were instruments resembling the 
 modern theodolites. The instrument consisted of a rod, 
 four yards long, with little plates at the end for aiming. 
 This rested upon a circular disk. The rod could be moved 
 horizontally and also vertically. By turning the rod 
 around until stopped by two suitably located pins on the 
 circular disk, the surveyor could work off a line perpendic- 
 ular to a given direction. The level and plumb line were 
 also used." From an illustration given by Heron of a 
 simple diopter and from an illustration given by Venturi of 
 a later type it is evident that the instrument was merely a 
 large surveyors' cross for setting out perpendiculars. At 
 what date the plate was graduated so other angles could 
 be set off no one seems to know. The compass was not 
 long in use before a needle was placed on the plate of the 
 
 210 
 
TRANSIT SURVEYING 
 
 211 
 
 diopter, later followed by two concentric plates so that angles 
 could be read independently of the needle. Transversals 
 were used to subdivide the graduations to enable fine 
 readings to be taken. 
 
 About the year 1608 a Dutch spectacle maker named 
 Lippershey discovered the principle of the telescope and 
 Galileo, in 1609, made the first telescope. In 1631 the 
 vernier was invented and in 1640 cross-hairs were used to 
 define the optical axis, or line of sight, of a telescope. 
 When the vernier was used to read the graduated circles 
 and the improved telescope 
 took the place of the sighting 
 disks, the diopter became a 
 theodolite. This instrument 
 seems to have been first men- 
 tioned in print about the year 
 1674. The first English tele- 
 scopic theodolite is believed 
 to have been made in 1723. 
 Fig. 1 80, copied from an old 
 edition of Davies' Survey- 
 ing, shows the type in use 
 in 1835. The etymology of 
 the word is doubtful, some 
 writers believing it to come 
 from three Greek words mean- 
 ing "to see a way plainly," 
 while others believe it to have 
 been named as a compliment 
 to M. Theodolus, a French 
 mathematician who wrote a treatise concerning its use, and 
 who may have been the man who was responsible for the 
 final form the diopter assumed before the name was changed. 
 The first may have been called "Dioptra Theodolus." 
 
 The theodolite was not well adapted to the work re- 
 quired of a surveying instrument in America and the 
 compass held sway until the commencement of steam rail- 
 way construction. The graduated plates with verniers 
 made the theodolite a good instrument for laying out rail- 
 way curves, a work for which the compass was plainly riot 
 fit. The telescope with its cross-hairs enabled points to 
 
 FIG. 1 80. Cradle theodolite. 
 
212 PRACTICAL SURVEYING 
 
 be set with an accuracy equal to that of the graduations 
 but rapid work could not be done because the telescope 
 was mounted in wyes (cradles) and could only be reversed 
 for backsights by changing it in the wyes end for end. 
 In 1831 Mr. Young, an instrument maker in Philadelphia, 
 applied the principle of the astronomical transit instru- 
 ment to a mounting for a theodolite telescope and the 
 "Portable American Transit for Engineers" appeared. It 
 is the standard surveying instrument in America today 
 for everything but work of the highest character. On 
 such work theodolites are used, the word in the United 
 States being limited to high-grade surveying instruments 
 without a compass, the omission of the needle permitting 
 of great rigidity in the mounting of the telescope. Few, if 
 any, "cradle" theodolites are made. In Europe the word 
 theodolite is used to describe all surveying instruments 
 having graduated horizontal and vertical circles, with and 
 without needles, and regardless of how the telescopes are 
 mounted. 
 
 The transit is mounted on a vertical compound center. 
 That is, the vertical shaft has a central solid spindle fitting 
 in an outer one which works in a deep socket attached to 
 the leveling device. The outer portion of the vertical 
 center carries a horizontal disk with a circumferential band 
 of silver on top, the inner edge of the band being graduated. 
 The solid spindle carries a horizontal disk extending over 
 the graduated ring. Upon this covering plate is placed the 
 standards carrying the telescope, verniers for reading the 
 graduations closely and a compass. 
 
 The compass needle is as long as possible, for much work 
 is done with the needle in sections of the country .where 
 land is cheap. The compass is always fitted with a screw 
 for lifting the needle and is usually supplied with a varia- 
 tion plate. The compass box occupies much space and it 
 is necessary to place the telescope standards near the edge 
 of the plate. This lessens the stability somewhat, in the 
 opinion of hypercritical persons, but not enough to affect 
 much of the work done by the majority of engineers and 
 surveyors. For high-grade city surveying the standards 
 are U-shape, thus bringing them closer to the center at 
 the base, and a trough compass is used. The trough com- 
 
TRANSIT SURVEYING 
 
 213 
 
 pass consists of a needle mounted in a narrow box so it 
 can swing only a few degrees to the right or left of the 
 meridian. A needle thus mounted serves to set the line of 
 sight in the magnetic meridian, from which base line other 
 courses are run wholly by angles read on the horizontal 
 plate. 
 
 In Fig. 181 is shown a typical American transit used for 
 most of the work done by engineers and surveyors. The 
 transit is attached to a tripod, 
 the upper end of which is shown. 
 
 A. The ring containing the 
 threads for attaching the transit 
 to the tripod. It is often called 
 the "lower screw plate." 
 
 B. The leveling screws. These 
 screws work in dustproof screw 
 caps attached to a plate, made 
 in the shape of a cross for light- 
 ness. This cross, "upper screw 
 plate," contains the socket in 
 which the instrument center 
 spindle works. On the lower 
 end of this socket is attached a 
 ball joint working in a plate 
 underneath the lower screw 
 plate. The hole in this plate is 
 much larger than the socket 
 
 stem so by loosening the screws the entire instrument 
 may be moved from one side to the other, the device being 
 termed a "shifting center." 
 
 Four screws are commonly used, but for many years 
 only three screws have been customary on the highest 
 grade instruments for geodetic work. Within recent years 
 three-screw bases have been made which are almost as 
 compact as four-screw bases, and many engineers now 
 have three instead of four leveling screws on their transits 
 and levels. No man who is accustomed to using three 
 screws goes back willingly to four. The latter are much 
 the slower, are not as stable and require the use of two 
 hands to keep the bubbles centered in the plate levels. 
 Leveling screws are used to keep the horizontal plate 
 
 FIG. 181. 
 
 Modern American 
 transit. 
 
214 PRACTICAL SURVEYING 
 
 level. If the plate is not level all horizontal angles read 
 will be too small. 
 
 C. The lower clamp screw. This screw works in a 
 clamp attached to the spindle socket, the inner end of the 
 screw resting against a block. When turned it presses the 
 block against the outer spindle thus clamping the spindle 
 to the socket. 
 
 D. Lower tangent screw. This is fastened to the upper 
 screw plate and works against the lower clamp screw for the 
 purpose of making a final adjustment. Formerly two op- 
 posing screws were used but now there is but one screw as 
 shown, the cylindrical case on the other side of the clamp 
 screw containing a strong German silver spring. 
 
 E. The upper clamp screw used to clamp the inner and 
 outer spindles together.. 
 
 F. The upper tangent screw with opposing spring for 
 making a final adjustment of the line of sight. 
 
 G. The needle lifter. A similar screw not shown is 
 used to shift the variation plate. 
 
 H. Clamp screw for the vertical arc. This screw is on 
 an arm which it clamps to the axis of the telescope. The 
 lower end of the arm is between the vertical arc tangent 
 screw and opposing spring. 
 
 /. Vertical arc tangent screw. After the clamp screw 
 clamps the telescope axis the tangent screw is used to make 
 the final small vertical movement required for accurate 
 pointing. 
 
 The graduated disk shown ahead of the vertical arc 
 tangent screw is a gradienter. The author had two tran- 
 sits equipped with gradienters and used these transits for 
 more than sixteen years. In all that time he found no 
 occasion when he could use a gradienter to advantage. 
 Furthermore he has yet to meet an engineer who uses a 
 gradienter in preference to stadia wires or the vertical arc. 
 In purchasing a transit five dollars are saved when the 
 gradienter is omitted. If the gradienter is to be of service 
 the threads of the tangent screw must be accurately cut and 
 be in perfect condition. After a few years of use all screw 
 threads are worn and while slight wear does not affect them 
 for use on tangent screws it does make them unfit for mi- 
 crometer work, the gradienter being a form of micrometer. 
 
TRANSIT SURVEYING 215 
 
 Two small levels are used to level the horizontal plate. 
 One is set parallel with the line of sight, usually on the left 
 standards as shown in the cut. The other is set across the 
 line of sight on the edge of the plate under the object end 
 of the telescope, and is the more important of the two. 
 The bubbles are centered by means of the leveling screws B. 
 
 Through a rectangular opening in the upper plate the 
 graduations and vernier are viewed. This opening is 
 covered with glass and a reflector of celluloid, or ground 
 glass, is used to enable the graduations to be seen readily. 
 
 On the best transits double verniers are used, 180 de- 
 grees apart, so readings may be checked and to assist in 
 repeating readings. Some makers place the vernier on the 
 sides between the standards; others directly under the tele- 
 scope, but the best place is 30 degrees to the left of the 
 line of sight and is most common. In this position the 
 vernier may be read without disturbing the telescope and 
 the surveyor does not have to step around the instrument. 
 
 The telescope is mounted on an axis revolving in bear- 
 ings on top of the standards. It is generally of such a 
 length as to permit of a complete revolution, but with a 
 sunshade on the object end the eye end only will clear the 
 compass glass. The small capstan screws on the telescope 
 tube indicate the location of the cross-wires, which are 
 brought into the field of view by moving the eyepiece in 
 or out. This focusing is done in one of several ways: by 
 a straight pull, by a screwing motion or by means of a 
 screw near the eyepiece which moves a rack and pinion 
 within the tube. The object glass is focused by means 
 of a screw on the right-hand side of the telescope and is 
 therefore not shown in the cut. Some makers place this 
 screw on top. 
 
 For taking vertical angles a vernier is attached to the 
 left-hand standards and an arc or a full circle is attached to 
 the telescope axis. For practically ninety-five per cent of 
 work done by surveyors an arc like that shown in the cut 
 is sufficient and the author prefers it to a full circle which 
 is more apt to be injured because of its projection above 
 the top of the standards. A full circle is particularly ex- 
 posed to damage in brushy land. When long lines are 
 run and elevations are taken by means of vertical angles, 
 
21 6 PRACTICAL SURVEYING 
 
 instead of a regular level, as happens frequently in under- 
 ground surveying, a full circle with double and opposite 
 verniers is advisable. It should be enclosed in a protective 
 shield with glazed verniers. 
 
 Under the telescope is shown a long level. When in ad- 
 justment this may be used for leveling, the transit being 
 therefore a leveling instrument as well as an angle measurer. 
 First level the instrument so the bubbles in the plate 
 levels remain stationary during a complete revolution 
 horizontally. Then place the telescope in as nearly a 
 horizontal position as possible and clamp the axis. By 
 means of the vertical tangent screw bring the bubble of 
 the long level under the telescope to the center. If in ad- 
 justment it will remain stationary during a revolution of 
 the instrument on the vertical axis. If left in this position 
 it will not be necessary on succeeding "set-ups" to first 
 level the plate, the telescope being alternately leveled over 
 each pair of screws, precisely like an engineer's level. 
 Accurate leveling may be done with a transit but it is 
 slow compared with a level. 
 
 When the transit is not being used as a level the telescope 
 should be pointed vertically, or in line with the vertical 
 axis when the instrument is carried between stations. It 
 should be clamped as lightly as possible. In this way it 
 offers the least obstruction in brush and a blow will cause 
 it to revolve on the axis. The lower tangent screw should 
 be loose, leaving the instrument free to revolve if acci- 
 dently struck. 
 
 A plain transit has no level under the telescope and no 
 vertical arc or circle. Instrument makers quote on plain 
 transits, and give prices for extras. Plain transits are used 
 on railway surveys. If the surveyor expects to do little 
 compass work a large transit is not necessary if of a good 
 make. When the author was young he bought a transit 
 with a 5-in. needle. The weight was 18 Ibs. and the tripod 
 weighed 10 Ibs. Although similar transits are now made 
 the majority of engineers prefer transits weighing not to 
 exceed 13 Ibs. with 7-lb. tripods. For general use the 
 surveyor will obtain very satisfactory results with a transit 
 having a 3i-in. needle, the combined weight of transit 
 and tripod being under 16 Ibs. The writer for a number 
 
TRANSIT SURVEYING 217 
 
 of years held a commission as a U. S. Deputy Mineral 
 Surveyor in the far west, and for mountain work had a 
 transit made to order. The needle was about 2\ ins. long 
 and the weight of the transit was 4^ Ibs. It was carried 
 in a knapsack held by straps over the shoulders. The 
 folding tripod weighed 3^ Ibs., and was. carried in a small 
 case slung to the bottom of the knapsack. Intended for 
 mountain work only the instrument was gradually intro- 
 duced on other work until finally it was used on all the 
 work done by the author for nearly ten years. Because 
 of its lightness it suffered less damage from falls than 
 heavier transits. For the same reason it exhibited great 
 steadiness in a wind. This instrument of course repre- 
 sented one extreme as the first one purchased by the 
 author represented another extreme. 
 
 A man with thick thumbs and large fingers cannot be 
 made to believe the smallest-sized transit is a reliable in- 
 strument. Lightness and portability are greatly appre- 
 ciated in rough country as also in high altitudes where 
 breathing is a conscious effort. In lower country more 
 weight is not objectionable when it implies more easily 
 read graduations. For the highest grade of engineering 
 surveying, such as re-tracing boundaries of expensive city 
 property, setting out lines for great bridges, important 
 tunnels, subways, etc., the need for absolute accuracy 
 makes the question of weight of relatively small importance. 
 With such instruments, however, a compass is not required 
 and this fact effects a saving in weight by a gain in com- 
 pactness. 
 
 Graduations were formerly made on silver-coated brass 
 plates. Now they are cut in the surface of a silver ring 
 attached to the plate. All of the best makers today cut 
 the graduations in solid silver, for by this means only can 
 proper results be obtained. 
 
 For the horizontal circle five methods are used for num- 
 bering the graduations. 
 
 /. Figured clockwise (in the direction followed by the hands 
 of a clock) from o to 360 with single opposite verniers. 
 
 //. Figured in quadrants with double opposite verniers. 
 These quadrants correspond to the compass quadrants so 
 angles may be readily checked by the needle. 
 
2i8 PRACTICAL SURVEYING 
 
 ///. A combination of I and II with double opposite 
 verniers. 
 
 IV. Figured in one row, clockwise and anti-clockwise, 
 from o to 1 80 each way with double opposite verniers. 
 
 V. Figured in two rows, one clockwise and one anti- 
 clockwise, from o to 360, with double opposite verniers. 
 Some makers incline the figures in the direction in which 
 they are to be read and some make further provision against 
 error by coloring one set red and the other black. 
 
 Method V is the best for all purposes and the one used 
 by instrument makers when the purchaser expresses no 
 choice. Vertical circles and arcs are best graduated 
 quadran tally. 
 
 VERNIERS 
 
 Horizontal and vertical circles are divided into degrees 
 by short marks. Each fifth degree mark projects a little 
 and the tenth degree marks project still more, the gradu- 
 ations being figured at each tenth degree only. On some 
 circles the degree is further divided into quarter-degree, 
 third-degree or half-degree graduations. For closer read- 
 ings a vernier must be employed. 
 
 The vernier is a uniformly divided scale on a circle ad- 
 jacent to the graduated circle, or limb. The end marks 
 , on the vernier coincide with marks on the limb but in the 
 included space there is one more interval on the vernier 
 than on the limb. If we assume a limb graduated to half 
 degrees (30 minutes) and 29 divisions on the limb equal 
 30 divisions on the vernier then each division on the ver- 
 nier is 3*0 smaller than a division on the limb and the 
 vernier reads to minutes. 
 
 Rule to find reading of a vernier. Divide the least reading 
 of the limb by number of spaces on the vernier. 
 
 Rule for reading an angle by- nteans^ of a vernier. Read 
 the angle on the limlLtp the 'graduation nearest tfie"zero of the 
 vernier. Then read along the vernier in the same direction 
 until a line] is found that coincides with some line on the 
 limb. Add the vernier reading to the limb reading. 
 
 The following illustrations and descriptions will serve to 
 make the matter clear. The student is advised to study 
 the readings obtained. 
 
TRANSIT SURVEYING 
 
 219 
 
 Fig. 
 
 Reading of 
 limb. 
 
 Divisions! f Divisions 
 of limb. J ~~ lof vernier 
 
 Reading of 
 vernier. 
 
 Kind, of vernier. 
 
 182 
 
 Degrees 
 
 II 
 
 12 
 
 5 minutes 
 
 Double direct 
 
 183 
 
 30 minutes 
 
 29 
 
 30 
 
 i minute 
 
 Double direct 
 
 184 
 
 20 minutes 
 
 39 
 
 40, 
 
 30 seconds 
 
 Double direct 
 
 iS 
 
 20 minutes 
 
 59 
 
 60 
 
 20 seconds 
 
 Folded 
 
 186 
 
 30 minutes 
 
 29 
 
 30 
 
 i minute 
 
 Folded 
 
 187 
 
 15 minutes 
 
 44 
 
 45 
 
 20 seconds 
 
 Double direct 
 
 188 
 
 15 minutes 
 
 49 
 
 50 
 
 yfa degree 
 
 Double direct 
 
 Fig. 182 shows a double direct vernier, that is, one 
 which reads from the center to either extreme division (60), 
 that part being used in which 
 the direction of the number- 
 ing corresponds to the direc- 
 tion in which the limb is num- 
 bered and read. The limb 
 is graduated to degrees and 
 the vernier (from o to 60) comprises 12 divisions, there- 
 fore the reading of the vernier is 60 minutes -r- 12 = 5 
 minutes. 
 
 The figure reads 3 + 50' = 3 50' from right to left. 
 
 Fig. 183 represents the usual graduations of an en- 
 gineer's transit with its vernier. This is an ordinary 
 
 FIG. 182. 
 
 double-direct vernier reading from the center to either 
 extreme division (30). The limb is graduated to half de- 
 grees and the vernier (from o to 30) comprises 30 divisions, 
 therefore the reading of the vernier is 30 minutes -f- 30 = I 
 minute. 
 
 The figure reads 27 + 25' = 27 25' from left to right, 
 and 152 30' + 05' = 152 35' from right to left. 
 
220 
 
 PRACTICAL SURVEYING 
 
 Fig. 184 represents the graduations and vernier of a 
 transit having finer graduations than that shown in Fig. 
 183, with a double-direct vernier reading from the center 
 to either extreme division (20). The limb is graduated to 
 
 20 minutes and there are 40 divisions in the vernier, con- 
 sequently the reading of the vernier is 20 minutes -f- 40 = 
 J minute = 30 seconds. 
 
 The figure reads 17 40' + 12' 30" = 17 52' 30" from 
 left to right, and 162 + f 30" = 162 f 30" from right 
 to left. 
 
 In Fig. 185 the transit has still finer divisions than those 
 already considered. The vernier is a folded one reading 
 from the center, indicated by the arrow, to either of the 
 
 extreme divisions (10), and then forward in the same 
 direction from the other extreme division (10) to the 
 center division (20), the direction being determined by the 
 numbering and reading of the limb. The limb is graduated 
 to 20 minutes, while the vernier is composed of 60 equal 
 parts, consequently the reading of the vernier is 20 minutes 
 -j- 60 = J minute = 20 seconds. 
 
 The figure reads 49 + 14' 20" = 49 14' 20" from left 
 to right, and 130 40' + 5' 40" = 130 45' 40" from right 
 to left. 
 
TRANSIT SURVEYING 
 
 221 
 
 A transit still more finely graduated is shown in Fig. 
 1 86. This has a double-direct vernier reading from the 
 center to either extreme division (45). The limb is grad- 
 uated to 15 minutes and there are 45 divisions in the 
 vernier, consequently the reading of the vernier is 15 
 minutes -f- 45 = ^ minute = 20 seconds. 
 
 The figure reads 30 + 4' 20" = 30 4' 20" from left to 
 right and 149 45' + 10' 40" = 149 55' 40" from right to 
 left. 
 
 FIG. 187. 
 
 Fig. 187 represents a portion of a vertical circle or arc 
 with folded vernier. The graduations are to half degrees, 
 and the vernier is divided into 30 equal parts, consequently 
 the reading of the vernier is 30 minutes -5- 30 = I minute. 
 
 The figure reads 7 30' + 21' = 7 51' from right to 
 left, an angle of depression. 
 
 In many kinds of work a decimal vernier has some 
 minor advantages to recommend it and it is extremely 
 useful in laying out railway curves. Its use is not common 
 and many surveyors and engineers have never seen one. 
 Fig. 1 88 shows the method of graduating the horizontal 
 limb and vernier to read to decimals of a degree. This 
 vernier is a double-direct 
 vernier reading from the 
 center to either extreme di- 
 vision (25), that part being 
 used on which the direc- 
 tion of the numbering cor- 
 responds to the direction in which the limb is numbered 
 and read. The limb is graduated to J degree (0.25) 
 and the vernier divided into 50 parts, consequently the 
 reading of the vernier is 0.25 -5- 50 = 0.005 which equals 
 sU of a degree. 
 
 The figure reads 45 + 0.055 = 45-55 f rom Wt to 
 right and 314.75 + 0.195 = 3- 14945 f rom r ig nt to 
 
 FIG. 188. 
 
222 PRACTICAL SURVEYING 
 
 In reading a vernier when the limb has two rows of 
 figures care must be used to avoid reading the wrong 
 row. 
 
 Never read the vernier by looking only at one line. 
 The adjacent lines on each side of the coinciding line 
 should be observed to see that they fail to coincide by 
 equal amounts with lines on the limb. 
 
 Sometimes no coinciding lines can be found but two 
 adjacent lines will be found which fail by equal amounts 
 to coincide with lines on the limb. The true reading is 
 between these lines; thus a closer reading may be ob- 
 tained than is indicated by the least count of the vernier. 
 
 Every transit supplied with a vertical circle or arc 
 should be equipped with stadia wires, the use of which will 
 be described in the next chapter. 
 
 TO USE THE TRANSIT 
 
 The instrument should be set up firmly, the tripod legs 
 being pressed into the ground, so as to bring the plates as 
 nearly level as convenient. The plates should then be 
 carefully leveled and properly clamped. 
 
 For precise work, in addition to leveling by the plate 
 levels, it is always advisable, if the transit has such at- 
 tachment, to level the plates by the telescope level, as this 
 is much more sensitive than the levels on the plate. In 
 this operation the position of the level on telescope must 
 be observed over each pair of leveling screws in turn, and 
 one-half the correction made by the axis tangent screw, 
 the other half by the leveling screws. 
 
 Before an observation is made with the telescope, the 
 eyepiece should be focused until the object is seen clear 
 and well-defined, and the wires appear as if fastened to 
 its surface. The intersection of the wires should be 
 brought precisely upon the object to which the telescope is 
 directed. 
 
 The zeros of the verniers and limb should be brought 
 into line by the tangent screw of the leveling head. The 
 angles taken are then read off upon the limb, without sub- 
 tracting from those given by the verniers in any other 
 position. 
 
TRANSIT SURVEYING 223 
 
 TO ADJUST THE TRANSIT 
 
 Every instrument should leave the hands of the maker 
 in complete adjustment, but all adjustments are liable to 
 derangement by accident or careless use so it is necessary 
 to describe particularly those which are most likely to 
 need attention. 
 
 The principal adjustments of the transit are: The 
 Levels, the Line of Collimation, the Standards. 
 
 To adjust the levels. Set the instrument upon its tri- 
 pod as nearly level as may be, and having undamped the 
 plates, bring the two levels above, and on a line with, the 
 two pairs of leveling screws. Clasp the heads of two 
 opposite screws, and, turning both in or out, as may be 
 needed, bring the bubble of the level directly over the 
 screws exactly in the middle of the opening. Without 
 moving the instrument, proceed in the same manner to 
 bring the other bubble to the middle. The level first cor- 
 rected may now be thrown a little out; if so, bring it in 
 again, and when both are in place turn the instrument 
 halfway around. If the bubbles are both in the middle 
 they need no correction; but if not, turn the nuts at the 
 end of the levels with the adjusting pin, until the bubbles 
 are moved over half the error, bring the bubbles again into 
 the middle by the leveling screws, and repeat the oper- 
 ation until the bubbles will remain in the middle during a 
 complete revolution of the instrument. 
 
 To adjust the line of collimation. This adjustment is 
 to bring the cross-wires into such a position that the in- 
 strument, when placed at the middle of a straight line, 
 will, by the transit of the telescope, cut the extremities of 
 the line. Having leveled the instrument, determine if the 
 vertical wire is plumb, by focusing on a defined point and 
 observing if the wire remains on that point when the tele- 
 scope is elevated or depressed. If not, loosen the cross- 
 wire screws and by their heads turn the ring until correct, 
 the openings in the telescope tube being slightly larger 
 than the screws so that when the latter are loosened the 
 ring can be rotated a short distance in either direction. 
 Direct the intersection of the cross-wires on an object two 
 or three hundred feet distant. Set the clamps and transit 
 
224 PRACTICAL SURVEYING 
 
 to an object about the same distance in the opposite di- 
 rection. Unclamp, turn the plates halfway around, and 
 direct again to the first object; then transit to the second 
 object. (Note. To transit is to revolve the telescope on 
 its axis. The telescope is thus reversed to get a sight on 
 an object on the line in rear of the instrument.) If it 
 strikes the same place the adjustment is correct. If not, the 
 space which intervenes between the points bisected in the 
 two observations will be double the deviation from a true 
 straight line, since the error is the result of two observations. 
 
 In Fig. 189 let A represent the center of the instrument, 
 and BC the imaginary straight line, upon the extremities 
 of which the line of collimation is to be 
 adjusted. B represents the object first 
 selected, and D the point which the 
 wires bisected when the telescope was 
 F reversed. 
 
 When the instrument is turned half 
 
 around, and the telescope again directed to B_, and once 
 more reversed, 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. It must be kept 
 in mind that the eyepiece apparently inverts the position 
 of the wires, and therefore, 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. 
 
 The wires being adjusted, their intersection may now 
 be brought into the center of the field of view by moving 
 the screws holding the ring, which are slackened and 
 tightened in pairs, the movement being now direct, until 
 the wires are seen in their proper position. 
 
 The position of the line of collimation depends upon that 
 of the objective solely, so that the eyepiece may, as in the 
 case just described, be moved in any direction, or even 
 removed and a new one substituted, without at all de- 
 ranging the adjustment of the wires. 
 
TRANSIT SURVEYING 225 
 
 In case it becomes necessary to remove the cross-wire 
 ring, the operator should proceed as follows: Take out the 
 eyepiece, together with the ring by which it is centered, 
 remove two opposite cross-wire screws, s and with the others 
 turn the ring until one of the screw holes is brought into 
 view from the open end of the telescope tube. In this 
 screw hole thrust a splinter of wood or a wire to hold the 
 ring when the remaining screws are withdrawn. The ring 
 can then be removed. It may be replaced by returning 
 it to its position in the tube, and after either pair of screws 
 is inserted the splinter or wire is removed, and the ring is 
 turned until the other screws can be replaced, care being 
 taken that the face of the diaphragm is turned toward the 
 eyepiece. The eyepiece is next inserted, and its centering- 
 ring brought into such a position that the screws in it can 
 be replaced, and the ring into which the eyepiece is fixed 
 is then screwed to the end of the telescope. 
 
 To adjust the standards. In order that the point of in- 
 tersection of the wires may trace a vertical line as the 
 telescope is elevated or depressed, it is necessary that 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 properly adjusted, set up 
 the instrument in a position where points of observation, 
 such as the apex and base of a lofty spire, can be selected, 
 giving a long range in a vertical direction. 
 
 Level the instrument, direct the telescope to the top of 
 the object, and clamp to the spindle; then bring the tele- 
 scope down until the wires bisect some well-defined point 
 at the base. Turn the instrument half around, direct the 
 telescope to the lower point, clamp to the spindle, and 
 raise the telescope to the highest point. If the wires 
 bisect it, the vertical adjustment is effected; if they are 
 thrown to either side, this proves that the standard op- 
 posite to that side is the highest, the apparent error being 
 double that actually due to this cause. When a transit 
 does not have any means for adjusting the height of the 
 standard it should be sent to the maker for adjustment. 
 
 Cross-wires are usually made of platinum. Occasionally 
 spider webs are used although few modern instruments for 
 
226 PRACTICAL SURVEYING 
 
 ordinary work are so equipped, as rough usage and long 
 periods of damp weather injure spider hairs. When it is 
 necessary to replace a broken wire or hair, hunt for the 
 small black spider, generally found in trees and under- 
 brush, for all other spider web is too coarse. Allow the 
 spider to run out on a pencil or small stick. When the end 
 is reached it attaches a line and drops, thus stretching the 
 line taut. The diaphragm having been removed from the 
 telescope and the broken wires removed, grooves made by 
 the instrument maker will be found. A drop of shellac is 
 placed in each groove, and the stretched web fitted. The 
 spider will not spin when cold and when sulky must be 
 tossed and rolled gently until anxiety to escape is exhibited. 
 
 CARE OF TRANSITS AND LEVELS 
 
 Remarks on care of the compass apply equally to the 
 compass attached to a transit. 
 
 All instruments should be protected during a rain by 
 covering with a bag. For a transit use only oiled silk, for 
 oiled linen and the sulphur in rubber will blacken the silver 
 on the horizontal and vertical limbs. 
 
 In bright sunshine, on a hot day, it is a sensible idea to 
 shade the instrument, to avoid errors due to expansion. 
 Expansion sometimes injures the finer parts of an instru- 
 ment. The use of a bag during wet weather is common 
 but very few surveyors use a sunshade on hot days, a habit 
 all should acquire. 
 
 Covering an instrument during a fog is wise and cover- 
 ing it in cold weather is to be done cautiously, for it may 
 " sweat" with the frost. A soft linen cloth should be 
 carried and the instrument wiped often to remove dust and 
 moisture. When dry go over it with a large, soft camels- 
 hair brush. 
 
 Never place an instrument in a box when damp. Be sure 
 it is dry and then wipe and brush it before locking in the 
 box. To dry an instrument place it in a dry, warm room. 
 
 Never leave an instrument standing in an open space 
 without a person to guard it. In a room place it in a corner 
 with the points of the tripod legs set in floor cracks. An 
 instrument left on a tripod out in a room is easily knocked 
 
TRANSIT SURVEYING 227 
 
 over. After an instrument is dry and has been cleaned, 
 place it in a box. 
 
 If, after exposure of an instrument in extremely hot or 
 cold weather, it is found that the centers do not revolve 
 as freely as usual, .clean them as soon as possible. 
 
 In cleaning object and eyepiece glasses, use a soft rag 
 or chamois leather. If the glasses should become greasy 
 or very dirty, wash them with alcohol. As the fine polish 
 on the object glass will be destroyed by wiping too often, 
 the instrument man should be careful in this respect. 
 Should telescope, compass or vernier glasses become moist, 
 place the instrument in a room which is dry and moderately 
 warm. Should it be impossible to follow this method, the 
 glasses may be wiped dry, this latter process, however, 
 affording* an opportunity for dirt and dust to get into the 
 instrument while the glasses are removed. The inner 
 surfaces of protected glasses need seldom be wiped. 
 
 Due attention should be given the screws confining tele- 
 scope bearings. They should be tightened sufficiently to 
 make the bearing firm and still permit the telescope to re- 
 volve freely, yet be kept in position by the friction thus 
 obtained. When there is too much friction in the telescope 
 slide, take it out immediately and first scrape the rough 
 place with the blade of a pocket knife, having its edge in- 
 clined a trifle, then use the back of blade for burnishing 
 the spot. If possible, treat inside of tube in a similar 
 manner, or at least wipe it out. The slide should then be 
 slightly greased with watch oil, but all the grease must be 
 wiped off before slide is replaced. No emery or emery 
 paper should be used on any part of an instrument. When 
 fretting begins it should be sent to an instrument maker 
 for repairs. If this cannot be done promptly, no man 
 should be afraid to take an instrument apart for cleaning. 
 It is of course foolish to take an instrument apart when not 
 necessary, but many surveyors, afraid to "examine the 
 insides," keep on using a dirty instrument until it gets 
 into such bad shape that an instrument maker loses money 
 putting it into shape again. 
 
 A skilful man can take apart and put together an in- 
 strument without damage; while an unskilful man will 
 certainly do harm by attempting such work. 
 
228 PRACTICAL SURVEYING 
 
 Vaseline will clean the surface of the silvered ring on 
 the limbs and should be thoroughly wiped off. Use only 
 good watch oil for lubricating centers and cleaning screw 
 threads. Use as little as possible and wipe it off so it will 
 not gather dust and grit. Fretting centers should be 
 treated in the same manner as fretting telescope tubes. 
 
 Overstraining of screws should be guarded against as it 
 either stretches the threads, causing them to wear out in a 
 short time, or gradually loosens some part of the instru- 
 ment. When screws are too tight the instrument is sen- 
 sitive to temperature changes; when too loose it is unsteady 
 and reliable work cannot be done. 
 
 Tripod legs must be firm. If the screws are loose the 
 instrument will be shaky and if too tight warping may 
 cause the telescope to shift off the line of sight. The 
 proper degree of restraint may be determined by raising 
 the legs one at a time to a horizontal position. If the 
 screw holds a leg up, it is too tight. If the leg falls rapidly, 
 the screw is too loose. The downward turning should be 
 slow and uniform. 
 
 TO READ ANGLES WITH A TRANSIT 
 
 The instrument is set at B (Fig. 190) and leveled by 
 means of the leveling screws. All clamp screws being 
 loose, bring the zeros on the horizontal 
 " A limb and vernier together and clamp with 
 the upper clamp screw. Then use the 
 reading glass which hangs by a cord 
 around the neck and with the upper 
 FIG loo " tangent screw bring the zeros exactly 
 
 together. 
 
 Revolve the instrument horizontally on its vertical axis, 
 without touching the upper clamp or tangent screw, until 
 the vertical cross- wire bisects the stake (or tack) at A. 
 Clamp the lower clamp screw and with the lower tangent 
 screw bring the cross-hair to the exact point desired. 
 
 The instrument is now clamped with zeros together and 
 cross-hairs bisecting the object. It cannot be revolved 
 on the vertical axis. Loosen the upper clamp screw and 
 direct the telescope to C. Clamp the screw and by means 
 
TRANSIT SURVEYING 229 
 
 of the upper tangent screw bring the cross-hair to exactly 
 bisect the point at C. Now with the reading glass the 
 angle may be read. If the instrument is in perfect ad- 
 justment, well made, with no errors in graduation, and it 
 is not disturbed during the operation, the angle is obtained 
 correctly. 
 
 To check. Unclamp the lower clamp screw without 
 touching the upper screws. Reverse the telescope. Re- 
 volve the instrument horizontally and bisect A with the 
 vertical wire. Clamp the lower clamp and bring the wire 
 to an exact bisection by means of the lower tangent screw, 
 the plate meanwhile remaining set at the angle read. 
 Unclamp the upper screw and direct the telescope to C. 
 Clamp, and with the upper tangent screw bisect C with 
 the vertical wire. Now read the angle, which should be 
 double the exact angle. The operation just described is 
 called "Double-centering." The mean of the two read- 
 ings is correct because every error in one direction on the 
 first reading is balanced by an equal error in the opposite 
 direction on the second reading. This, however, does not 
 apply to errors in graduation. When running lines every 
 angle "on lii\e" should be "double-centered," but only one 
 reading is necessary for side shots and lines where the 
 closest possible accuracy is not required. 
 
 Repeating angles. In triangulation work the method 
 of repetition is followed. Read the angle by sighting to 
 A and then to C. Keep the telescope erect and repeat the 
 readings from A to C until a total of six are taken. The 
 final reading divided by 6 gives an approximate value for 
 the angle. Both verniers are read at the start and after 
 the sixth repetition. The total reading of each is divided 
 by 6. If there is a difference the mean of the final values 
 is taken. 
 
 Reverse the telescope and repeat the process, but this 
 time reading from right to left. Read both verniers, 
 divide the totals by 6, and take the mean value of the six 
 readings of each vernier. 
 
 We now have a mean value for the direct readings 
 and one for the reversed readings. The mean of the 
 means should be the true angle, plus or minus a probable 
 error. 
 
230 PRACTICAL SURVEYING 
 
 Let S = "the sum of all the quantities in the paren- 
 thesis," 
 v = variation (or deviation) of each angle from the 
 
 mean, provided each angle is read, 
 n = number of readings, 
 
 r = probable error, 
 then 
 
 r = d= 0.8453 ^ / \ ' =. for single observation, 
 and 
 
 r = 0.8453 , for all observations. 
 
 n Vn i 
 
 When the deviation gives a larger angle than the mean 
 it is positive and when smaller it is negative, but in ob- 
 taining the sum in the formulas above given the signs are 
 disregarded. The formulas are known as Peter's Ap- 
 proximations of Bessel's Formula, which is based on the 
 principle of least squares and is laborious to work. 
 
 "The probable error is not 'the most probable error,' 
 nor, 'the most probable value of the actual error.' It de- 
 termines the degree of confidence we may have in using 
 the mean as the best representative value oT a series of 
 observations." (Mellor.) If the mean value of the angle 
 is 17 31' 42" and the probable error is 5.33" then the 
 odds are even that the true value lies somewhere between 
 17 3i' 47-33" and 17 36' 36.67." When the three angles 
 of a triangle have been measured and the sum does not 
 equal 180, a computation of the most probable error for 
 each angle will be a guide to balance the angles to obtain 
 1 80. If the triangle has sides more than a mile in length 
 it may be the error in closure is due to "spherical excess" 
 by reason of the curvature of the earth, the triangle being 
 spherical. Geodetic methods must then be used. 
 
 The computation for probable error is sometimes used in 
 the case of a line which has been measured several times 
 with a tape forward and back by different sets of chain- 
 men. A small probable error is often said to be a measure 
 of the accuracy, but as it refers only to the proportion in 
 which errors of different magnitude occur it cannot be a 
 measure of accuracy. 
 
TRANSIT SURVEYING 231 
 
 Formulas determining the probable error are worthless 
 for a small number of observations. We may select ten 
 as the least number, and the first step is to eliminate all 
 constant errors. If a tape is too long or too short the con- 
 stant error in each tape length is the difference between the 
 length of the tape and a standard. If an error in gradu- 
 ation exists in some part of the circle of a transit, the part 
 in which the error exists must not be used. In chaining, 
 three sets of chainmen should measure the line forward 
 and back at least twice, making a total of twelve measure- 
 ments. By using one set of chainmen the personal error 
 would not be eliminated, and as it is a constant error difficult 
 to find and evaluate, other men must be used to give this 
 error something of the character of a series of accidental 
 errors. The careful measuring above described is done 
 only on the highest grade work. 
 
 Constant errors are cumulative and accidental errors 
 are compensating. The probable error is practically a 
 mean square value of the positive and negative deviations 
 from the arithmetical mean of all the accidental errors. 
 The probable error is diminished by increasing the number 
 of observations. 
 
 In chaining, the accuracy of the work is increased by 
 taking plenty of time, and not working too fast. In re- 
 peating angles with a transit the accuracy of the work is 
 increased when speed is increased. Clamping and un-, 
 clamping disturbs an instrument and may throw it out of 
 level. Standing a long time in one place and the handling 
 tend to make one of the metal shod tripod legs sink into the 
 earth. The temperature effects may be equalized in rapid 
 work, for one side may be unduly heated or cooled if al- 
 lowed to remain too long in one position. 
 
 A transit should always be tested for "index error." 
 This may be due to faulty graduation, in which case the 
 instrument should be returned to the maker, or it may be 
 caused by a fall bending the plate. Even straightening 
 the plate may not restore it, and the surveyor for some 
 good reason may have to use it without re-graduation. 
 
 Set three stakes with tacks in the top so the small acute 
 angle will lie between 10 and 20. Start from o and read 
 the angle successively around the circle without double 
 
232 PRACTICAL SURVEYING 
 
 centering. This must be done rapidly. The angles should 
 be equal. When a place is found where the reading 
 differs from readings preceding or following it, test each 
 degree in the space until the error is exactly located. 
 The amount does not matter but the place must be marked 
 and not used thereafter. Sometimes errors of very small 
 value are found which show up only in a series of readings, 
 and are distributed around the circle. Double centering 
 will take care of them. 
 
 Unnecessary stepping around an instrument must be 
 avoided. A good instrument-man never "straddles" a 
 tripod leg. Transit work being closer than 
 compass work every instrument station must 
 be marked with a tack. The plumb-bob 
 hanging from the center of the vertical axis 
 must be centered over the tack, this being 
 accomplished with a shifting center. The 
 lower end of the axis is enlarged in a spheri- 
 cal shape and fits in a spherical socket on a 
 small plate, thus enabling the leveling screws 
 to bring the axis to a vertical position when 
 the tripod head may not be level. The open- 
 ing in the lower plate, which is screwed to 
 the tripod head, is larger than the ball but 
 smaller than the socket plate. By loosening the screws 
 the center may be shifted to carry the plumb-bob to any 
 side. When finally centered the screws are tightened and 
 the plate leveled. 
 
 Plumb-bob lines must be lengthened and shortened and 
 a number of devices are used to accomplish this without 
 the line slipping in the hook at the upper end. Nearly all 
 catch the wind and are objectionable for this reason. A 
 knot in the string is best, but when loose enough to allow 
 the cord to be slipped through it is a nuisance. By loop- 
 ing the cord twice around the hook the knot may be loose 
 and the cord will not slip, the friction of the loop holding 
 the plumb-bob in place. 
 
 In sighting lines, steel line rods are used and it takes 
 considerable experience to enable a helper to hold the rod 
 vertical. Sometimes a plumb line is held over the point 
 but no man has so steady a hand that confidence can be 
 
TRANSIT SURVEYING 233 
 
 had in such a sight. Sometimes a line rod is stuck in the 
 ground at a slant so it is above the tack. To this the plumb 
 line is tied and moved along the rod until the point of the 
 bob is centered on the tack. For quicker sights the helper 
 places one knee on the ground and rests one hand, holding 
 the plumb line, on the other knee. The plumb-bob is then 
 centered over the tack and the white line is viewed with 
 the leg as a background. 
 
 A target of white celluloid made in such size and manner 
 that it may be carried in the pocket and attached to a 
 plumb line when needed has been developed 
 by Kolesch & Company of New York. It 
 is circular in form with a diamond-shaped 
 cut-out, which offers a strong contrast to the 
 body of the target. It can be moved up and 
 down the plumb line at will. The applica- 
 tion of this device will be recognized by all sur- 
 veyors who have taken sights against a back- p 
 ground of shubbery or buildings, and it will 
 enable the taking of longer "shots." It has the advantage 
 also of being a more accurate means of setting a true point 
 than can be attained with a flag or pole, that may be 
 slightly out of plumb when the sight is taken. 
 
 A piece of wood about the width of a lead pencil, sharp- 
 ened on the lower end and stuck in a crack behind the 
 tack is frequently used for a backsight. 
 
 In Fig. 190 let the transit be set at A with point B set. 
 An angle is to be laid off and point C located. Turn off 
 the angle and set a stake at C. The helper holds the rod 
 on the stake and the transitman motions right or left un- 
 til the rod is in line, when it is pressed slightly to make a 
 mark. The telescope is reverse^, the lower clamp loosened, 
 the vertical hair set on B and the lower clamp tightened. 
 The upper clamp is loosened and the telescope directed to 
 C, clamped, and the upper tangent screw used to obtain 
 exactly double the angle. The rod is now directed again 
 to line and another mark made. The true point lies mid- 
 way between the marks and a tack is driven to fix it. 
 
 In Fig. 193 is shown the general method of conducting 
 a transit survey. Assume A and F to be set. The transit 
 is set up at A with horizontal plate and vernier clamped at 
 
234 PRACTICAL SURVEYING 
 
 zero. The lower clamp is loose. Reverse (transit) the tele- 
 scope and sight F. Clamp the lower screw and with the 
 lower tangent screw bisect the tack at F. Transit the tele- 
 scope so it sights forward on the line FA. 
 Unclamp the plate, turn the angle to 
 ifs. the right and set point B. Move to B 
 and set C from a backsight on A. Pro- 
 ceed around the field in this manner, fi- 
 nally closing on A , which may be occupied 
 a second time to check the traverse. 
 IG ' I93 ' In the example given all the angles were 
 
 to the right. In Fig. 194 the angles are turned to the right 
 and left, but the work is the same, the entries in the field 
 book having the letter R, or L, as the case may be, follow- 
 ing the amount of the angle. 
 
 The needle should be al- 
 lowed to swing to check the 
 angles. This is a rule to be 
 followed in all transit work. P IG IQ4 
 
 When carrying the transit 
 
 lift the needle off the pivot and hold against the glass cover 
 of the compass. The needle reading is not so accurate as 
 angles read on the limb, but it affords a check so a mistake 
 in setting down R for L, or vice versa, is quickly discovered. 
 Such a mistake is common. 
 
 A mistake never gets past the person making it. An 
 error is a mistake discovered by others and reflects upon 
 the ability of the person responsible. All the field and 
 office work of surveyors must be checked and re-checked so 
 no errors can be found. Checking is a most important 
 duty and cannot be neglected. 
 
 Angles may be read on a survey by setting the limb and 
 vernier at zero on each station and double centering each 
 time on important work. The bearing of the first line is 
 set down in the field book and all calculations of bearings 
 start from it as a base. The following rules are used to re- 
 duce angles to bearings: 
 
 Reduce bearings to azimuth angles, as follows: 
 
 All N E bearings are less than 90. 
 
 S E bearings lie between 90 and 180 and are reduced to 
 azimuth by subtracting from 180. 
 
TRANSIT SURVEYING 235 
 
 S W bearings lie between 180 and 270 and are reduced 
 to azimuth by adding to 180. 
 
 N W bearings lie between 270 and 360 and are re- 
 duced to azimuth by subtracting from 360. 
 
 The meridian being selected as an azimuth line and the 
 meridian bearing of the starting course having been re- 
 duced to an azimuth bearing by the above method, con- 
 sider all angles to the right as positive and all angles to 
 the left as negative. 
 
 Sum less than 90. Bearing N E. 
 
 Sum between 90 and 1 80. Subtract from 180 and 
 call bearing S E. 
 
 Sum between 180 and 270. Subtract 180 and call 
 bearing S W. 
 
 Sum between 270 and 360. Subtract from 360 and 
 call bearing N W. 
 
 Example. Starting from the end of a line bearing 
 S 1 8 2 1 ' E a line was run as follows : 
 
 Feet 
 
 0. R 14 13' ...................................... 280 
 
 1. R 74 21' ...................................... 500 
 
 2. L 47 11' .......................... : ........... 320 
 
 3. Lioo5' ...................................... 211 
 
 4. R 27 09' ...................................... 419 
 
 Find the bearings of the lines, referred to the true 
 meridian. 
 
 179 60' Course o. 179 60' 
 
 S 18 21' E 175 52' 
 
 161 39' = Azimuth of reference S 40 08' E 
 
 0. 14 13' R course. 
 
 175 52' Course I. 250 13' 
 
 1. 74 21' R 1 80 oo / 
 249 73' S 70 13' W 
 
 2. 47 n' L 
 
 202 62' Course 2. 203 02' 
 
 o , 
 219 66' 
 
 R Course 3. 192 57' 
 
 S 12 57' W 
 
 4. 220 06' 
 I 80 00 
 
 S 40 06' W 
 
 Course 4. 220 06' 
 i 80 oo 
 
236 PRACTICAL SURVEYING 
 
 From a study of the foregoing example, the following 
 shorter method is found: 
 
 S 18 2i'E 
 
 14 13' R 
 S 4 08' E 
 
 74 21' R 
 S 70 13' W 
 
 47 n'L 
 S 23 02' W 
 
 10 05' L 
 Si2 57' W 
 
 27 09' R 
 S 40 06' W 
 
 The student may reduce angles to bearings by either 
 method as the example given illustrates the principle. 
 
 The reason for recording the needle reading each time is 
 now plain. If the R and L are. always recorded properly 
 the needle readings will check the computed bearings 
 within a few minutes. With bearings nearly due north 
 and south or east and west the check may not be valuable 
 if local attraction is present, but it should never be omitted 
 for it is the only check to be had when all angles start 
 from zero and are read right and left. 
 
 Azimuth, according to the dictionary, is "the angle com- 
 prised between two vertical planes, one passing through 
 the elevated pole, the other through the object." Herschel 
 counted azimuth from o to 360, and in this sense it is 
 used by surveyors; that is, it indicates all angles referred 
 to some meridian and independent of bearings. Bearings 
 are never greater than 90 and are reckoned to the east 
 and west from the north or south pole. When azimuth 
 is reckoned from the north, the true meridian being the 
 azimuth meridian, it is easy to convert azimuth into com- 
 pass bearings, as in the example just given. 
 
 There is good reason for following this custom as com- 
 pass bearings have been used many centuries because of 
 the needle pointing north. When azimuth is reckoned 
 clockwise from the north the needle readily checks all 
 computed bearings. For about a generation a number 
 of writers of surveying textbooks have reckoned azimuth 
 from the south clockwise because the astronomical pole is 
 
TRANSIT SURVEYING 237 
 
 south. This may be scientifically correct for a very few 
 astrophysicists, but it is confusing to the surveyor to 
 assume one pole for angle work and another pole distant 
 1 80 for work involving compass bearings. It really 
 makes no difference to the educated man which pole he 
 uses but the surveyor is dealing with the public and the 
 public always "starts from the north." The surveyor 
 who reckons azimuth from the south must use two standard 
 meridians and some day will make mistakes. The writer 
 one summer employed a young graduate who reckoned 
 azimuth from the south and was continually mixing his 
 notes until he abandoned the practice. When the writer 
 went to school, in the 8o's, azimuth was any angle read 
 clockwise from a chosen meridian to an object, so that an 
 object 20 L would have an azimuth of 340, thus prevent- 
 ing errors due to misplacing the letters R and L; no hint 
 was given that azimuth should be referred to the south 
 pole or the north pole. 
 
 With an instrument in perfect adjustment and handled 
 by a competent man surveys are often made by azimuth 
 bearings and no double centering is done. The instrument 
 is set up at the starting point, the variation plate indicat- 
 ing the correct declination. The needle is let down and 
 allowed to settle so it points to the true north. The limb 
 and vernier are set to zero and the instrument clamped. 
 The needle is then lifted and not again used. 
 
 Unclamping the plates the telescope is directed to the 
 next station and the angle read and recorded after clamp- 
 ing the plates. Carrying the instrument to the next sta- 
 tion, the telescope is reversed and sighted to the station 
 just left; and the instrument clamped below. The tele- 
 scope is transitted and then read forward on the line, 
 the limb being still set at the azimuth. Unclamping the 
 plate the telescope is directed to the next station and the 
 azimuth read and recorded. These operations are re- 
 peated at each station and when the first station is again 
 occupied and a sight taken to the second station the first 
 azimuth should be obtained. Actually there may be. a 
 difference of one or two minutes, which may be distributed. 
 On a closed traverse errors in reading azimuth tend to 
 compensate, for the right and left readings balance; but 
 
238 PRACTICAL SURVEYING 
 
 on other surveys errors are apt to be cumulative. Azimuth 
 notes for the last example will read as follows: 
 
 Reference line 161 39' Needle S 18 21' E 
 
 0. 175 52' 280 ft. 
 
 1. 250 13' 500 ft. 
 
 2. 203 02' 320 ft. 
 
 3. 192 57' 211 ft. 
 
 4. 220 06' 419 ft. 
 
 All angles are azimuth angles on stadia surveys. Not 
 all surveyors double center their angles, a surprisingly large 
 number using the azimuth method exclusively. 
 
 When surveying cheap land or doing work requiring no 
 greater accuracy than may be obtained with a compass, 
 the transit may be used as a compass and all readings 
 taken with the needle. 
 
 A very accurate method for setting off 
 an angle is illustrated in Fig. 195. 
 
 On a given line are set two hubs A and 
 B. A line is to run from B through a point 
 c D and this line must be laid off correctly. 
 Setting on B the telescope is directed to A , 
 FIG. 195. " plates clamped to zero and an angle ABC 
 read, one minute smaller than the angle 
 ABD, and stake C set. A tack is placed in C and the 
 angle ABC is read by repetition to obtain the exact value. 
 The length BC is measured with the greatest possible 
 accuracy. The exact value of angle ABC is subtracted 
 from the angle ABD and the tangent of the difference 
 multiplied by the length BC gives the small distance CD, 
 which may be measured off and a tack set. Or the angle 
 ABD may be read and a broad stake driven at D with 
 a tack in the top. The angle is then read by the repeti- 
 tive method and if a difference is found from the true 
 angle a correction may be measured and the tack set in the 
 proper place. 
 
 A transit may be set on a line between two points, by 
 setting as nearly as possible on line and leveling the in- 
 strument. Then sight on one point, clamp the plates and 
 reverse the telescope for a sight on the other point. If it 
 does not strike it, shift the transit and try again. A skil- 
 ful instrument-man can get on a line in three trials. After 
 
TRANSIT SURVEYING 239 
 
 getting on line reverse the telescope for a sight each way, 
 to eliminate errors in adjustment and disturbance due to 
 handling. 
 
 Fig. 196 illustrates the operation, A and B being the 
 points and C the transit sta- 
 tion, the small circles near C o ^ 
 
 showing trial set-up points. F i 6 
 
 To prolong a line, the ap- 
 proved method is to set on a stake and take a backsight 
 with telescope reversed on a stake on line in the rear. 
 
 Transit the telescope and set a stake ahead. Then double 
 center on the stake and set a tack between the two marks. 
 Move to the new stake and repeat for the next one. 
 
 To set a line of stakes between two stakes set on one, 
 sight to the other and clamp the plates. The line of sight 
 being fixed the telescope is plunged * on the horizontal axis 
 and the cross-hairs set on the line rod used to locate the 
 positions of the intermediate stakes and tacks are driven in 
 them. 
 
 A good compass is graduated to J degree and by using 
 care one-half this angle may be estimated. If this is done 
 the maximum error in angle will be 4 minutes, of which the 
 natural tangent is 0.00116. Few surveyors read a bearing 
 closer than J degree (15') in which case the maximum 
 error will be 8 minutes, of which the natural tangent is 
 0.00233. 
 
 Ordinary transits read by means of verniers to single 
 minutes so the maximum error in angle cannot exceed 31 
 seconds, of which the natural tangent is practically 0.000146. 
 It is not uncommon for transits to be graduated to read 
 angles as small as 30, 20 or 10 seconds, while if angles are 
 read by the repetitive method the exact size of an angle 
 may be ascertained to the nearest 5 seconds with a vernier 
 reading to one minute, or to the nearest second with a 
 vernier reading to 20 seconds. 
 
 With a transit a skilful instrument-man can do all the 
 angular work with as high a degree of accuracy as skilled 
 chainmen can do the measuring with first-class tapes. 
 
 Methods for transit surveying are the same as methods 
 
 * To plunge a telescope is to turn it on its axis to set a line ahead. 
 The telescope is clamped and is not inverted or reversed for a rear sight. 
 
240 PRACTICAL SURVEYING 
 
 for compass work plus the increased accuracy. The 
 angles right and left, or the azimuth bearings, are reduced 
 to compass bearings and the computations for area are 
 the same as those described in the chapter dealing with 
 the compass. Computations for traverse work and for 
 supplying omissions are described in the chapter on 
 trigonometry. 
 
 Surveys are made for one of three purposes, sometimes 
 for all: 
 
 1. To describe boundaries. 
 
 2. To obtain areas. 
 
 3. To make a map. 
 
 Boundary surveys are usually made by running out the 
 boundary lines, offsetting a few feet when the boundary is 
 obstructed with hedges, fences, etc. 
 
 Area surveys not calling for a description may be made 
 by methods already described for chain surveys. The 
 oldest book on surveying of which we have knowledge, 
 Hero's Dioptra, shows that the custom fully twenty centu- 
 ries ago was to lay off in the field a rectangular figure and 
 erect on the sides of it, at regular intervals, perpendiculars 
 to strike the boundary lines. The areas of all the divisions 
 were added. 
 
 When the diopter was improved so angles other than 
 right angles could be turned off, surveyors turned angles 
 from points on the boundary of the interior rectangle to 
 corners of the field and measured to the corners. The 
 proceeding was duplicated in making the map and the 
 areas of all the interior figures were computed by men- 
 suration and added. With the introduction of plane 
 trigonometry and the use of the compass the "radiation" 
 method became standard and areas were computed by 
 trigonometric methods. Sometimes the surveyor merely 
 laid off a carefully measured base line and from the ends 
 and some intermediate points took bearings to corners and 
 measured the distances. This divided the field into tri- 
 angles, the areas of which were computed trigonometri- 
 cally. When the "double meridian " method for computing 
 areas was developed in Ireland by Thomas Burgh, the 
 custom came in of running out the exterior lines. The 
 reason this was considered advisable was to save labor. 
 
TRANSIT SURVEYING 241 
 
 Prior to Burgh's discovery lines were run around the 
 boundary to obtain data for writing a legal description, 
 but surveys to compute areas were still made by mensu- 
 ration and trigonometric methods. When the double 
 meridian distance method for computing areas made it 
 possible to use the boundary survey data, only the one 
 survey was necessary. 
 
 With the passing years the "radiation" method of sur- 
 veying land received less attention in textbooks, although 
 all practical surveyors used the method where it proved to 
 be time saving. The writer had to make several surveys 
 by this method when at school and used it frequently in 
 practical work. 
 
 The three approved methods of surveying a piece of 
 land are: 
 
 1. The traverse method. 
 
 2. The radiation method. 
 
 3. The intersection method. 
 
 In the 1912 Proceedings of the Illinois Society of En- 
 gineers and Surveyors, a paper by Mr. G. W. Pickets, then 
 an instructor in Civil Engineering at the University of 
 Illinois, compared the three methods. 
 
 The results were as follows: 
 
 Speed in chaining. Intersection method, first; radi- 
 ation method, second; traverse method, third. In most 
 instances the radiation method called for 20 per cent less 
 chaining than the traverse method. 
 
 Speed in instrument work. Intersection method, first; 
 radiation method, second; traverse method, third. 
 
 Speed in computing results. Traverse method, first; 
 radiation method, second; intersection method, third. 
 
 The above comparisons prove the traverse method to be 
 the more expensive, for slow work in the field means the 
 employment of a party of men for a longer time. A little 
 extra time in the office usually affects one man only. 
 Accurate work involves careful checking and half the work 
 of engineers and surveyors seems to consist in checking 
 lengths, angles and computations. 
 
 Comparison of accuracy. Assuming that no blunders 
 are made, any one of the three methods is accurate enough 
 for all practical work. With respect to actual accuracy 
 
242 PRACTICAL SURVEYING 
 
 the intersection method is first; radiation method, second; 
 traverse method, third. 
 
 The traverse method consists in running out the bound- 
 aries; or running on offset lines as closely as possible to 
 the boundaries. To guard against error in reading the 
 needle, or setting down the wrong angle, it is a common 
 practice to read an angle or take a bearing to some one 
 object from each station. 
 
 In Fig. 197 bearings were taken to one corner of the 
 chimney of the house in the field. When making the map 
 
 these bearings are all drawn and 
 should intersect at a common 
 point. This method has always 
 been used by careful surveyors 
 on compass surveys. Modern 
 transits are equipped with stadia 
 wires, or with gradienters, and 
 distances should be checked with 
 one or the other. In this one 
 instance the gradienter, if new 
 with sharp clean threads, has an 
 advantage over the stadia. The gradienter can be used 
 if one foot of rod can be seen, while the stadia requires a 
 graduated rod. If a plain rod, however, is used the sur- 
 veyor can direct the horizontal cross-hair to one mark and 
 read the vertical angle. Then drop the wire two, three, or 
 more feet on the rod and read a second angle. One-half 
 the sum of the two angles gives the vertical angle to the 
 mid-point, the rod being held vertically. 
 
 Let A = half the sum of the two angles, 
 
 D = distance on rod intercepted between the wire 
 
 on the two sights in feet, 
 L = length of line in feet, 
 then L = D X Cos 2 A + I. 
 
 The formula is only closely approximate, but errors in 
 measuring within reasonable limits are readily caught by 
 such a check. 
 
 In the radiation method the transit is set up at one or 
 more points, from which angles and lines are measured to 
 the corners of the field. 
 
TRANSIT SURVEYING 243 
 
 Fig. 198 illustrates a field of less than 160 acres in fairly 
 compact shape. Here all the corners could be seen from 
 one set-up. To survey the field by the traverse method 
 would call for the measurement of the five sides and for the 
 occupation of each corner with the instrument. The sum 
 of the lengths of the lines OA + OB + OC + OD + OE 
 is usually less than the perimeter of the field, but all 
 lengths should be checked. Surveyors skilled in stadia 
 work can dispense with the chain or tape. 
 
 FIG. 198. FIG. 199. 
 
 Fig. 199 illustrates a larger field, the corners of which 
 cannot be seen from one point. The instrument is set at 
 O and P, the angles at each point being turned from a 
 backsight on the other. All the radiating lines are meas- 
 ured except PB and PE. 
 
 Fig. 200 illustrates a long narrow field and neither point 
 nor P is visible from the other. Flags are set on the 
 boundaries at E and F and angles read to them, the lengths 
 OE, OF, PE and PF being measured. 
 
 V ^: 
 
 /'N 
 
 p 
 
 FIG. 200. FIG. 201. 
 
 Fig. 20 1 illustrates the method to use with fields of con- 
 siderable size, say exceeding 320 acres, in which the method 
 shown in Fig. 200 may not be applicable. A small traverse 
 is run within the field and angles and distances measured 
 to the corners of the field. 
 
 The radiation method divides a field into triangles, two 
 sides and the included angle of each being measured. 
 
244 PRACTICAL SURVEYING 
 
 The area of each triangle = \ be Sin A and the sum of the 
 areas of the triangles equals the area of the field, Fig. 201 
 being an exception. If a description of the field is wanted 
 each triangle must be solved for the third side and the base. 
 "The intersection method consists in establishing and 
 carefully measuring a base line, from each end of which 
 all the corners of the field are visible, and in measuring all 
 the angles around each end formed by lines radiating to 
 the corners. If the field is a small one and topography 
 permits, one base line is sufficient, but in the case of a 
 large field, two or more base lines will be necessary. The 
 base line is taken inside the field when it is possible, as the 
 lengths of the sights is thereby reduced to a minimum; but 
 it is not essential that it be so taken, and it may be all out- 
 side of the field or partly inside and partly outside. The 
 only imperative condition using one base line is that 
 all corners must be visible from each end of the base line." 
 "In locating the base line care must be taken to avoid 
 small angles, as the sines of small angles change rapidly. 
 (Angles under 15 and over 165 must be obtained to the 
 nearest 10 sees., which can be done by trebling the angle 
 on the limb of the transit.) In the case of a rectangular 
 field the best location for the base line is parallel to the 
 short axis. This arrangement should be followed as closely 
 as the topography will permit. The more irregular the 
 field, the harder it is to avoid small angles, and more time 
 and thought are required in the selection of the base line 
 than in the case of the rectangular field. The length of 
 the base line depends upon local conditions. In general, 
 the longer it is, the more accurate the results; but it should 
 never be less than one-third the length 
 of the average sight. This is based on 
 the assumption that the instrument used 
 reads to 30 sees., and that each angle is 
 doubled on the limb." 
 
 Fig. 202 illustrates a field surveyed by 
 FIG 202 ^ ne intersection method, AB being the 
 
 base line, the only line measured. 
 "The traverse method is applicable to any field regard- 
 less of the crops or topographical conditions. There is 
 always a strip of land along the fence line that is not culti- 
 
TRANSIT SURVEYING 245 
 
 vated, and an open sight may be had along this strip, so 
 the distance between corners may be chained without 
 interfering with the growing crops. The radiation method 
 may be used equally as well, provided that the growing 
 crops do not interfere with the chaining or obscure the 
 lines of sight. Farms are generally surveyed upon chang- 
 ing hands, and the principal thing to be ascertained is the 
 acreage. Possession is taken in the spring before the crops 
 are put in, and hence the surveyor is not limited in his 
 choice of methods. The intersection method can be used 
 only in level, open country, as all the corners of the field 
 can seldom be seen from both ends of a base line in hilly 
 wooded country. Hence with respect to applicability, the 
 traverse method has a slight advantage over the radiation 
 method, and both of these are far ahead of the inter- 
 section method." 
 
 "Which method of surveying should be used in any given 
 case depends entirely upon the conditions peculiar to that 
 particular case. Each of the methods is best in some in- 
 stances. The use of the intersection method is limited, it 
 is true, by topographical features, but it is the most ac- 
 curate of all, and should be used when the nature of the 
 country will permit. The radiation method should be 
 given equal standing with the traverse method, for its 
 saving in time more than compensates for its slightly 
 limited use. Frequently the sides of a field have to be 
 measured in order to re-locate the corners, and in such 
 cases the traverse method is the quickest and should be 
 used. As any one of the methods is accurate enough for 
 all practical purposes, the one that is most applicable to 
 the case in hand should be used." 
 
 Figs. 198 to 202 inclusive, and the quoted paragraphs are 
 from the paper by Mr. Pickels. 
 
 ANGULAR LEVELING 
 
 The long level under the telescope of a transit may be 
 used for leveling, as already described, or vertical angles 
 may be read and the rise or fall computed by the formula 
 
 H = I tan A, 
 in which H = height or difference in elevation in feet, 
 
 / = length of sight in feet (horizontal), 
 A = angle of elevation or depression. 
 
246 PRACTICAL SURVEYING 
 
 The intersection of the cross-hairs should be directed to 
 a point at their height above the ground. A rod is generally 
 used to sight on. The horizontal distance must be carefully 
 measured. The vertical angle should be read at both 
 stations, being an angle of elevation ( + ) at one and an 
 angle of depression ( ) at the other. If there is a small 
 difference use the mean value. A large difference calls for 
 a check and may be due to lack of adjustment. The long 
 bubble should be adjusted so it will remain centered as the 
 instrument revolves on its vertical axis. The cross-hairs 
 should then be adjusted as described for the dumpy level, 
 so the line of sight will be parallel to a horizontal line as 
 indicated by the bubble. The zero of the vernier should 
 be adjusted to coincide with the zero of the vertical arc. 
 
 Leveling in mining surveys is generally angular work 
 underground. On steep ground and in rolling country 
 leveling by vertical angles is rapid and when care is used 
 is very accurate. It is not well adapted for work requir- 
 ing elevations at intervals of about 100 ft. or less. 
 
 STADIA WIRES 
 
 When equipped with stadia wires so distances may be 
 read on a rod, the transit becomes, as it has been adver- 
 tised, "A universal surveying instrument." 
 
 Stadia wires are two horizontal wires, one above and 
 one below the horizontal wire found in all transit tele- 
 scopes. These two wires are fixed in a ring or diaphragm 
 in such a way that the interval usually intercepts a space 
 of one foot on a rod held at a point distant d ft. from 
 the center of the transit, plus a small constant C. 
 
 In Fig. 203 the relations are shown: 
 
 c = distance from telescope axis to center of objective, 
 / = focal length, the distance from the stadia wires to 
 
 the objective, 
 D = d + C and C = 
 
 The rays cross each other so that the vertex of the visual 
 angle is not at the center of the instrument, but at a dis- 
 tance in front of the objective equal to its focal length, 
 measured when the telescope is focused on a distant object. 
 
TRANSIT SURVEYING 
 
 247 
 
 The relation between the size and distance of an object 
 and the size of its image in a telescope is given by the 
 formula 
 
 &_f , _fR 
 
 R~ d = R 1 ' 
 
 in which R = space on rod intercepted between the two 
 
 wires and called the "rod reading," 
 R 1 = space between the wires, 
 d = distance from principal focal point to rod. 
 
 .Stadia wires 
 
 ] ^objective 
 
 FIG. 203. 
 
 When the wires are set with a space = of the focal 
 
 100 
 
 length between them we have the relation = = , so 
 
 the wires will read one foot on a rod held at a distance of 
 100 ft., one yard at a distance of 100 yds., one meter at 
 a distance of 100 m., etc. Substituting this value in the 
 above expression d = 100 R, to which must be added the 
 constant C. 
 
 This constant never varies for any given instrument and 
 is independent of the distance. Focus the telescope on an 
 object several miles away and measure from the axis to 
 the objective for c\ and from the screws holding the stadia 
 diaphragm to the objective for/. The constant C = c +/ 
 is noted by all instrument makers on a card or label affixed 
 to the box of every instrument provided with stadia wires. 
 
 When using the stadia the rod is held vertically. The 
 middle wire is directed to a point on the rod at the same 
 height above the ground as the axis of the telescope. The 
 difference between readings of the upper and lower wires 
 gives the rod reading, if a level rod is used, or a rod having 
 
248 PRACTICAL SURVEYING 
 
 numbers on the face. Regular stadia rods are seldom num- 
 bered, the intercepted space being read directly. 
 
 The rod is held vertically and as the line of sight is 
 seldom horizontal this introduces relations between the 
 resulting angles which are finally resolved into the follow- 
 ing expressions: 
 
 D = Rcos 2 A + CcosA, 
 
 . D sin 2 A . . 
 
 H = R - h C sin A , 
 
 in which D = horizontal distance from center of transit to 
 
 rod, 
 
 R = rod reading, 
 A = vertical angle, 
 C = c+f, 
 H = height or difference in elevation. 
 
 For ordinary work add C instead of C cos A , to obtain 
 horizontal distance; and add 0.015 ^4 instead of C sin A, 
 to obtain difference in elevation. The tables here given 
 save a great deal of time in reducing inclined stadia read- 
 ings to horizontal and vertical equivalents. 
 
 Example. An angle +18 12' was read to a rod with 
 a rod reading of 5.72 ft. Find horizontal distance and 
 difference in elevation. 
 
 Turning to column headed 18 on the line opposite 12' 
 find: Hor. Dist. = 90.24, Diff. Elev. = 29.67. At the 
 bottom find the values for C = 0.95 and 0.32, for we 
 here assume C = i.oo ft. 
 
 Hor. Dist. = (5.72 X 90-24) + 0.95 = 517.12 ft. 
 
 Diff. Elev. = (5.72 X 29.67) + 0.32 = 170.03 ft. 
 
 A number of excellent labor-saving slide rules and dia- 
 grams are sold by instrument dealers for reducing stadia 
 rod readings. 
 
 When running lines with the stadia, readings should be 
 taken forward and back for a check both for angle and rod 
 reading. Angles should be kept below 20 when possible 
 and readings distorted by heat are unreliable. When con- 
 ditions are favorable and the surveyor is careful the limit 
 
 of error can be kept within , a degree of accuracy supe 
 
 1500 
 
 rior to ordinary farm survey chaining. 
 
TRANSIT SURVEYING 249 
 
 The ratio I : 100 is convenient for the reason that ordi- 
 nary level rods may be used and also the decimally divided 
 rods sold by all dealers. If some other ratio is used the 
 rods must be specially divided or all reduced readings be 
 multiplied by a constant. 
 
 Instrument dealers advise the use of fixed wires and so 
 do men in the government service who are constantly 
 employed on stadia work. The author however worked 
 for a number of years in regions where instrument repair 
 shops were many days distant and not all instrument re- 
 pairers were first class. Sudden changes in temperature 
 and long-heated terms disturbed the wire interval so many 
 times that finally adjustable wires were placed in his tran- 
 sits and all trouble ceased. He believes that "all surveyors 
 similarly circumstanced should use adjustable stadia wires 
 and this after an extended experience in stadia work. 
 
 To adjust the wires, select a piece of ground practically 
 level and lay off on it with a steel tape an accurately meas- 
 ured line several hundred feet long plus C. Set the transit 
 at one end and have the rod held vertically at the other end. 
 Level the telescope and sight on the rod. Deduct C from 
 the exact distance and with the adjusting screws set the 
 upper and lower wires to intercept this interval on the 
 rod, taking care to keep them equidistant from the middle 
 wire. Assume that the transit is 701 ft. from the rod and 
 C = i.o ft.; the rod reading will be 7 ft. To catch changes 
 in the wire interval the writer checked with a steel tape the 
 first and last rod reading in the forenoon and the first and 
 last rod reading in the afternoon of each day. Frequently 
 the wires held the proper interval for 6 months and once 
 no change was observed for a year. When the interval 
 did change it was sudden and the ratio became as much as 
 i : 104, generally however being I : 102.5 or l ' IO 3-3 or 
 a similarly annoying ratio. Instrument repairers generally 
 blamed the cement used to hold the wires in place. Some- 
 times temperature effects on the diaphragm or too tight 
 screwing up of adjusting screws may cause these alter- 
 ations in the wire interval, which have been noticed by 
 other writers. Alterations due to such .causes disappear 
 within a short time and the interval is apparently correct. 
 By checking rod readings occasionally with a steel tape 
 
250 PRACTICAL SURVEYING 
 
 and correcting readings by applying a constant thus found 
 the stadia becomes one of the most useful of the tools of 
 the surveyor. (See pp. 1041-2, Vol. LXXVII, Trans. Am. 
 Soc. C.E.) 
 
 Stadia surveys of farms are made with a degree of ac- 
 curacy suitable for the work. Either the radiation or the 
 traverse method may be used, or a combination of these 
 methods. Because the vertical angle must be read to 
 reduce the inclined readings to horizontal distances informa- 
 tion as to slope of surface is obtained while the measur- 
 ing is being done. Distances are readily obtained with the 
 stadia but cannot beset off, as with a tape or chain. 
 
 Stadia is a word with the same root as "stadium" the 
 Latin form of "Stadion," the principal Greek measure of 
 length, a little less than an eighth of a mile. Stadia rods are 
 often called "telemeter" ("afar-off measuring") rods. In 
 Europe a theodolite, designed to be used principally for 
 topographical work and equipped with gradienter, stadia 
 wires, etc., is called a "tachymeter," that is, "a rapid 
 measurer." 
 
TRANSIT SURVEYING 
 STADIA REDUCTION TABLE 
 
 251 
 
 M. 
 
 
 
 
 1 
 
 
 2 
 
 
 r o'.. 
 
 i 2' 
 
 Hor. 
 
 dist. 
 
 IOO.OO 
 IOO OO 
 
 Diff. 
 Elev. 
 
 o.oo 
 o 06 
 
 Hor. 
 dist. 
 
 99-97 
 QQ Q7 
 
 Diff. 
 elev. 
 
 1.74 
 I 80 
 
 HOT. 
 
 dist. 
 
 99.88 
 
 QQ 87 
 
 Diff. 
 elev. 
 
 3-49 
 
 3r ir 
 
 A 
 
 IOO OO 
 
 O 12 
 
 QQ Q7 
 
 I 86 
 
 QQ 87 
 
 3 60 
 
 6 
 
 IOO OO 
 
 O 17 
 
 QQ q6 
 
 I Q2 
 
 QQ 87 
 
 3 66 
 
 8 
 
 IOO OO 
 
 O. 23 
 
 QQ Q6 
 
 I 98 
 
 QQ 86 
 
 2 72 
 
 10 
 
 IOO.OO 
 
 O. 2Q 
 
 QQ.Q6 
 
 2 .04 
 
 99 86 
 
 3 78 
 
 12 
 
 IOO OO 
 
 o 3< 
 
 QQ Q6 
 
 2 OQ 
 
 QQ 81; 
 
 3 84 % 
 
 14 
 
 IOO OO 
 
 o 4.1 
 
 QQ QC 
 
 2 1C 
 
 QQ 8s 
 
 3QO 
 
 16 
 
 IOO OO 
 
 O.47 
 
 QQ QC 
 
 2 21 
 
 QQ 84 
 
 1 Qf 
 
 18 
 
 IOO.OO 
 
 O. <\2 
 
 QQ.QC 
 
 2 . 27 
 
 QQ 84 
 
 4 OI 
 
 20 
 
 IOO OO 
 
 o ^8 
 
 QQ Qf 
 
 2 73 
 
 QQ 8 3 
 
 4O7 
 
 22 
 
 IOO OO 
 
 o 64 
 
 QQ Q4 
 
 2 38 
 
 QQ 8l 
 
 41 3 
 
 24 
 
 IOO OO 
 
 o 70 
 
 QQ Q4 
 
 2 44 
 
 i7^-"O 
 
 QQ 82 
 
 4 l8 
 
 26 
 
 no no 
 
 o. 76 
 
 QQ Q4 
 
 2 . SO 
 
 QQ 82 
 
 4 24 
 
 28 
 
 QQ QQ 
 
 o 81 
 
 QQ Q3 
 
 2 ?6 
 
 QQ 8 1 
 
 47Q 
 
 30 
 32. . 
 
 99-99 
 QQ QQ 
 
 0.87 
 
 O Q3 
 
 99-93 
 
 QQ Q3 
 
 2.62 
 
 2 67 
 
 99-81 
 QQ 80 
 
 4.36 
 
 4 42 
 
 34 
 36 
 38 
 
 4O 
 
 99-99 
 99-99 
 99-99 
 
 QQ QQ 
 
 0.99 
 I-OS 
 I. II 
 
 i 16 
 
 99-93 
 99-92 
 99-92 
 
 QQ Q2 
 
 2.73 
 
 2-79 
 
 2.85 
 
 2 QI 
 
 99-80 
 
 99-79 
 99-79 
 
 QQ 78 
 
 4.48 
 
 4-53 
 4-59 
 46c 
 
 42 
 
 QQ QQ 
 
 22 
 
 QQ QI 
 
 2Q7 
 
 QQ 78 
 
 471 
 
 44 
 46 
 
 48 
 
 99.98 
 99.98 
 
 QQ Q8 
 
 .28 
 
 34 
 4.0 
 
 99.91 
 99.90 
 QQ QO 
 
 3-02 
 
 3.08 
 
 314 
 
 99-77 
 99-77 
 
 QQ 76 
 
 4.76 
 4.82 
 
 4 88 
 
 eo. . 
 
 QQ Q8 
 
 4 1 ? 
 
 QQ QO 
 
 3 2O 
 
 QQ 76 
 
 4Q4 
 
 <?2 
 
 QQ Q8 
 
 (?T 
 
 QQ 8Q 
 
 3 26 
 
 QQ 7t 
 
 4QQ 
 
 "?4 
 
 QQ Q8 
 
 (?7 
 
 QQ 8Q 
 
 3-2T 
 
 QQ 74 
 
 5QC 
 
 *3 
 e6 
 
 QQ Q7 
 
 63 
 
 QQ 8Q 
 
 3-37 
 
 QQ 74 
 
 5 II 
 
 5 8 
 
 QQ Q7 
 
 69 
 
 QQ 88 
 
 3A-1 
 
 QQ 73 
 
 r 17 
 
 60 
 
 QQ Q7 
 
 74 
 
 yy "-> 
 QQ 88 
 
 -} AQ 
 
 QQ 72 
 
 e 23 
 
 
 
 
 
 
 
 
 5=o.75 
 
 0-75 
 
 O.OI 
 
 0-75 
 
 O.O2 
 
 0-75 
 
 0.03 
 
 C=i.oo 
 
 i .00 
 
 O.OI 
 
 I .OO 
 
 0.03 
 
 I .OO 
 
 O.O4 
 
 C=I.2S 
 
 1-25 
 
 O.O2 
 
 1.25 
 
 0.03 
 
 1-25 
 
 0.05 
 
252 
 
 PRACTICAL SURVEYING 
 STADIA REDUCTION TABLE (Continued) 
 
 M. 
 
 3 
 
 
 4 
 
 
 
 5 
 
 o' 
 
 HOT. 
 
 dist. 
 
 00.72 
 
 Diff. 
 elev. 
 
 < . 23 
 
 HOT. 
 dist. 
 
 qq. Si 
 
 Diff. 
 elev. 
 
 6 96 
 
 HOT. 
 dist. 
 
 qq 24 
 
 Diff. 
 elev. 
 
 8 68 
 
 2 
 
 QQ.72 
 
 q.28 
 
 qq. 51 
 
 7 .02 
 
 qq 23 
 
 8 74 
 
 A 
 
 QQ 71 
 
 tf -2A 
 
 QQ <?O 
 
 7 Q7 
 
 QQ 22 
 
 8 80 
 
 6 
 
 QQ 71 
 
 ^ 4O 
 
 QQ 4Q 
 
 7 13 
 
 QQ 21 
 
 8 85 
 
 8 
 
 QQ. 7O 
 
 ^ 46 
 
 qq 48 
 
 7 IQ 
 
 QQ 2O 
 
 8 QI 
 
 10 
 
 qq.6q 
 
 <\ . ^2 
 
 QQ 47 
 
 7 2< 
 
 qq iq 
 
 8 q? 
 
 12 
 
 QQ 6Q 
 
 e r7 
 
 qq 46 
 
 7 3O 
 
 qq 1 8 
 
 902 
 
 14. 
 
 qq 68 
 
 5 62 
 
 qq 46 
 
 7 ^6 
 
 qq 17 
 
 q 08 
 
 16 
 
 qq 68 
 
 5 69 
 
 QQ 4^ 
 
 7 42 
 
 qq 16 
 
 Q 14 
 
 18 
 
 qq 67 
 
 c 7C 
 
 QQ 44 
 
 7 48 
 
 QQ je 
 
 920 
 
 20 
 
 99-66 
 
 5 80 
 
 QQ 43 
 
 7 "?3 
 
 qq 14 
 
 92C 
 
 22 
 
 qq 66 
 
 S 86 
 
 QQ 42 
 
 7 S9 
 
 qq 13 
 
 q 31 
 
 24. 
 
 QQ 6<? 
 
 ^ Q2 
 
 QQ 41 
 
 / oy 
 7 6<; 
 
 qq ii 
 
 q 37 
 
 26 .... 
 
 QQ 64 
 
 e q8 
 
 QQ 4O 
 
 7 71 
 
 qq 10 
 
 943 
 
 28 .... 
 
 qq 63 
 
 6.04 
 
 QQ. 3Q 
 
 7.76 
 
 qq oq 
 
 to 
 
 Q 48 
 
 3O 
 
 qq.63 
 
 6.oq 
 
 qq. ^8 
 
 7.82 
 
 qq 08 
 
 Q ^4 
 
 32 . . 
 
 99.62 
 
 6.15 
 
 99.38 
 
 7.88 
 
 qq.O7 
 
 q.6o 
 
 24. 
 
 qq 62 
 
 6 21 
 
 qq 37 
 
 7 q4 
 
 qq 06 
 
 q 6s 
 
 II::::::::.: 
 3.... 
 
 4O 
 
 99 -61 
 99-6o 
 qq. eq 
 
 6.27 
 
 6-33 
 6.38 
 
 99-36 
 99-35 
 qq.34 
 
 7-99 
 8.05 
 8 ii 
 
 99-05 
 99-04 
 qq 03 
 
 9.71 
 
 9-77 
 q 83 
 
 42 
 44 
 
 99-59 
 99.58 
 
 6.44 
 6.50 
 
 99-33 
 99.32 
 
 8-17 
 
 8.22 
 
 99-oi 
 99.00 
 
 9-88 
 
 q.q4 
 
 46 
 48 
 50.... ...... 
 
 ^2 
 
 99-57 
 99.56 
 99-56 
 
 qq "i 1 ? 
 
 6.56 
 6.61 
 6.67 
 
 6 73 
 
 99-31 
 99-30 
 99-29 
 
 QQ 28 
 
 8.28 
 
 8-34 
 8.40 
 
 8 4<? 
 
 98.99 
 98.98 
 98.97 
 
 q8 q6 
 
 10.00 
 
 10.05 
 
 IO.II 
 IO 1 7 
 
 tA 
 
 qq ">4 
 
 6 78 
 
 QQ 27 
 
 8 51 
 
 q8 q4 
 
 IO 22 
 
 It.:::.:::: 
 
 "?8 
 
 99-53 
 
 qq. 52 
 
 6.84 
 6.qo 
 
 99-26 
 
 qq. 2< 
 
 8-57 
 8.63 
 
 98.93 
 q8 q2 
 
 10.28 
 IO 34 
 
 60 
 
 qq. ci 
 
 6.96 
 
 QQ. 24 
 
 8.68 
 
 q8 qi 
 
 10.40 
 
 
 
 
 
 
 
 
 C=o. 7 5 
 
 0-75 
 
 0.05 
 
 0.75 
 
 0.06 
 
 0.75 
 
 O.O7 
 
 C=i.oo 
 
 I.OO 
 
 0.06 
 
 I.OO 
 
 0.08 
 
 0.99 
 
 0.09 
 
 C=i.25 
 
 1.25 
 
 0.08 
 
 1.25 
 
 O.IO 
 
 1.24 
 
 O. II 
 
TRANSIT SURVEYING 
 
 STADIA REDUCTION TABLE (Continued) 
 
 253 
 
 M. 
 
 6 
 
 
 7 
 
 
 
 3 
 
 o' 
 
 HOT. 
 dist. 
 
 q8 qi 
 
 Diff. 
 elev. 
 
 10 40 
 
 Hor. 
 dist. 
 
 qg ci 
 
 Diff. 
 elev. 
 
 12 IO 
 
 HOT. 
 dist. 
 
 98 06 
 
 Diff. 
 elev. 
 
 13 78 
 
 2 
 
 Q8 QO 
 
 10 4C 
 
 y u - o* 
 q8 co 
 
 12 1C 
 
 qg oc 
 
 13 84 
 
 4 
 
 98.88 
 
 IO C-i 
 
 98 48 
 
 12 21 
 
 qg 03 
 
 13 8q 
 
 6 
 
 98.87 
 
 IO. C7 
 
 98 47 
 
 12 26 
 
 y^-^o 
 q8 01 
 
 13 CK 
 
 8 
 
 98.86 
 
 10.62 
 
 98.46 
 
 12.32 
 
 J v 
 98 oo 
 
 UOI 
 
 10 
 12 
 
 98.85 
 98.83 
 
 10.68 
 10. 74 
 
 98.44 
 98 43 
 
 12.38 
 12 43 
 
 97-98 
 97 97 
 
 14.06 
 
 UI? 
 
 14 
 
 16 
 18 
 20 
 
 22 
 
 98.82 
 98.81 
 98.80 
 98.78 
 
 98.77 
 
 10.79 
 10.85 
 10.91 
 10.96 
 
 ii .02 
 
 98.41 
 98.40 
 98-39 
 98-37 
 
 98 36 
 
 12.49 
 
 12.55 
 I 2. 60 
 
 12.66 
 
 1272 
 
 97-95 
 97-93 
 97-92 
 97-90 
 
 97 88 
 
 14.17 
 14.23 
 14.28 
 14.34 
 
 14 4O 
 
 24 
 
 98.76 
 
 n.o8 
 
 98.34 
 
 12 . 77 
 
 97 87 
 
 U4C 
 
 26 
 
 q8 74 
 
 ii i 3 
 
 qg -}> 
 
 12 83 
 
 q7 8c 
 
 14 CI 
 
 28 
 30 
 
 32 . . 
 
 98.73 
 98.72 
 
 Q8.7I 
 
 11.19 
 
 11-253 
 
 ii ^o 
 
 98.31 
 98.29 
 
 98 28 
 
 12.88 
 
 12.94 
 
 13 OO 
 
 97.83 
 97.82 
 
 Q7 80 
 
 14.56 
 14.62 
 
 U6? 
 
 34 
 
 98.69 
 
 ii . 36 
 
 98.27 
 
 13 .OC 
 
 y/ w 
 
 97 78 
 
 14 73 
 
 36 
 
 98.68 
 
 ii .42 
 
 98.21? 
 
 13.11 
 
 97 ?6 
 
 U79 
 
 ?8 
 
 98 67 
 
 II 47 
 
 qg 24 
 
 13 17 
 
 Q7 7C 
 
 U84 
 
 40 
 42 
 
 98.65 
 08.64 
 
 n-53 
 II . cq 
 
 98.22 
 98 2O 
 
 13.22 
 13 28 
 
 97-73 
 
 Q7 71 
 
 14.90 
 14 QC 
 
 44 
 
 98.63 
 
 II .64 
 
 98 iq 
 
 1333 
 
 97 6q 
 
 1C OI 
 
 46 
 
 q8 61 
 
 1 1 7O 
 
 Q8 17 
 
 1 3 3Q 
 
 q? 68 
 
 \o '-'* 
 1C 06 
 
 48 
 50 
 
 C2. . 
 
 98.60 
 98.58 
 
 98. C7 
 
 11.76 
 11.81 
 
 11.87 
 
 98.16 
 98.14 
 
 Q8 13 
 
 13-45 
 I3-50 
 
 13 ^6 
 
 97-66 
 97-64 
 
 q? 62 
 
 15-12 
 I5.I7 
 
 1C 23 
 
 54 
 
 98.56 
 
 II .Q3 
 
 q8 ii 
 
 13 61 
 
 y/ 
 97 61 
 
 IS 28 
 
 56 
 58 
 
 98.54 
 q8 C3 
 
 11.98 
 
 12 O4 
 
 98.10 
 qg 08 
 
 13.67 
 
 13 73 
 
 97-59 
 
 Q7 C7 
 
 15-34 
 I C 4O 
 
 60 
 
 98 c i 
 
 12 IO 
 
 q8 06 
 
 13 78 
 
 Q7 CC 
 
 1C 4C 
 
 
 
 
 
 
 
 
 C=o. 7S 
 
 0.75 
 
 0.08 
 
 0-74 
 
 O.IO 
 
 0.74 
 
 O.II 
 
 C=i .00 
 
 0-99 
 
 O.II 
 
 o-99 
 
 0.13 
 
 o-99 
 
 0.15 
 
 C=i. 25 
 
 1.24 
 
 0.14 
 
 1.24 
 
 o. 16 
 
 1.23 
 
 0.18 
 
254 
 
 PRACTICAL SURVEYING 
 STADIA REDUCTION TABLE (Continued} 
 
 M. 
 
 9 
 
 
 10 
 
 
 n 
 
 
 ' 
 
 Hor. 
 dist. 
 
 Q7 I C. 
 
 Diff. 
 elev. 
 
 T c A e 
 
 Hor. 
 dist. 
 
 06 08 
 
 Diff. 
 elev. 
 
 Hor. 
 dist. 
 
 06 26 
 
 Diff. 
 
 elev. 
 
 18 72 
 
 2 
 
 Q7 13 
 
 ICCT 
 
 Q6 Q6 
 
 17 16 
 
 yu-o" 
 q6 74. 
 
 10 - 16 
 
 18 78 
 
 4 
 
 Q7 12 
 
 i i 16 
 
 Q6 QA 
 
 17 21 
 
 y^-^4 
 
 Q6 32 
 
 1 8 84 
 
 6 
 8 
 10 
 
 97-50 
 97.48 
 Q7 4.6 
 
 15.62 
 15.67 
 
 jr 77 
 
 9<3.92 
 96.90 
 
 96 88 
 
 17. 26 
 17.32 
 17 37 
 
 96.29 
 96.27 
 Q6 2^ 
 
 18.89 
 i8.95 
 
 IQ OO 
 
 12 
 
 07.44 
 
 ic 78 
 
 96 86 
 
 17 43 
 
 Q6 23 
 
 IQ O< 
 
 14. 
 
 Q7 4.3 
 
 K 84 
 
 06 84. 
 
 17 4.8 
 
 yu.z^ 
 
 q5 21 
 
 19 1 1 
 
 16 
 18 
 20 .... 
 
 97-41 
 
 97-39 
 
 Q7 37 
 
 15-89 
 15-95 
 16 oo 
 
 96.82 
 96.80 
 Q6 78 
 
 17-54 
 17-59 
 17 6c. 
 
 96.18 
 96.16 
 
 Q6 14. 
 
 19.16 
 19.21 
 IQ 27 
 
 22 
 
 07 2C 
 
 16 06 
 
 Q6 76 
 
 17 7O 
 
 96 12 
 
 IQ 32 
 
 24 
 26 
 28 
 3O . 
 
 97-33 
 97-31 
 97-29 
 Q7 28 
 
 16.11 
 16.17 
 16.22 
 16 28 
 
 96.74 
 96.72 
 96.70 
 
 96 68 
 
 17.76 
 17.81 
 17.86 
 17 Q2 
 
 96.09 
 96.07 
 96.05 
 Q6 O3 
 
 19-38 
 19-43 
 19.48 
 IQ 14. 
 
 32 . . 
 
 Q7 26 
 
 16 33 
 
 96 66 
 
 17 Q7 
 
 06 oo 
 
 IQ ^Q 
 
 34- 
 
 Q7 24. 
 
 16. 3Q 
 
 0.6.64 
 
 18 03 
 
 Q 1 ^ Q8 
 
 IQ 64 
 
 36 
 38 
 40 
 
 42 
 
 97-22 
 
 97-20 
 97.18 
 
 Q7 l6 
 
 16.44 
 16.50 
 16.55 
 
 16 61 
 
 96.62 
 96.60 
 96.57 
 
 Q6 << 
 
 18.08 
 18.14 
 18.19 
 
 18 24 
 
 95.96 
 
 95-93 
 95-91 
 
 qc 8q 
 
 19.70 
 
 19-75 
 I9-80 
 
 IQ 86 
 
 44 < . . . . 
 
 Q7 14 
 
 16 66 
 
 Q6 ^ 
 
 18 30 
 
 95 86 
 
 IQ QI 
 
 46 
 48 
 So 
 
 >2. . 
 
 97.12 
 97.10 
 97-08 
 
 Q7 06 
 
 16.72 
 16.77 
 16.83 
 
 16 88 
 
 96.51 
 
 96.49 
 96.47 
 
 q6 41 
 
 18.35 
 18.41 
 18.46 
 
 18 <i 
 
 95.84 
 95.82 
 
 95-79 
 qc 77 
 
 * ~J 
 
 I9-96 
 2O.O2 
 20.07 
 
 20. 12 
 
 C.4. 
 
 Q7 O4. 
 
 1 6 QJ. 
 
 Q6 4.2 
 
 18 ^7 
 
 qe 7 e 
 
 20 18 
 
 \l:::::..:.. 
 
 $8 
 
 97-02 
 
 Q7 OO 
 
 16.99 
 
 17 OC. 
 
 96.40 
 
 06 a8 
 
 18.62 
 
 18 68 
 
 95-72 
 
 CK 70 
 
 20.23 
 20 28 
 
 60 
 
 06 08 
 
 17 IO 
 
 57V'<9" 
 
 06 ^6 
 
 18 73 
 
 qc 68 
 
 2O 34 
 
 
 
 
 
 
 
 
 C=o.75 
 
 0.74 
 
 O.I2 
 
 0.74 
 
 0.14 
 
 0-73 
 
 0.15 
 
 C=i.oo 
 
 0-99 
 
 0.16 
 
 0.98 
 
 0.18 
 
 0.98 
 
 O.2O 
 
 C=1.25 
 
 1.23 
 
 O.2I 
 
 1.23 
 
 0.23 
 
 I . 22 
 
 0.25 
 
TRANSIT SURVEYING 
 STADIA REDUCTION TABLE (Continued) 
 
 255 
 
 M. 
 
 12 
 
 
 13 
 
 
 14 
 
 
 o' 
 
 Hor. 
 dist. 
 
 OC 68 
 
 Diff. 
 elev. 
 
 20 34 
 
 Hor. 
 dist. 
 
 Q4-94 
 
 Diff. 
 elev. 
 
 21 .02 
 
 Hor. 
 
 dist. 
 
 04. ic 
 
 Diff. 
 elev. 
 
 23 47 
 
 2 
 
 CK 6< 
 
 2O 3Q 
 
 94- Qi 
 
 21 .07 
 
 04-12 
 
 23 . "\2 
 
 4 
 
 CK 63 
 
 20.44 
 
 94.89 
 
 22.O2 
 
 04.00 
 
 23.58 
 
 6 
 8 
 
 95.61 
 
 Q5.58 
 
 20.50 
 20.55 
 
 94-86 
 94.84 
 
 22.08 
 22.13 
 
 94-07 
 94.04 
 
 23-63 
 23-68 
 
 10 
 
 yj o 
 
 gc. c6 
 
 20.60 
 
 94.81 
 
 22. IS 
 
 94.01 
 
 23.73 
 
 12 . . 
 
 qe c-2 
 
 20.66 
 
 94- 7Q 
 
 22.23 
 
 93.98 
 
 23.78 
 
 14 
 
 qc . ci 
 
 20. 71 
 
 04.76 
 
 22.28 
 
 93-95 
 
 23-83 
 
 16 
 
 qc .4q 
 
 20.76 
 
 94-73 
 
 22.34 
 
 93.93 
 
 23.88 
 
 18 
 
 015.46 
 
 20.81 
 
 94.71 
 
 22.39 
 
 93.90 
 
 23.93 
 
 20 
 
 qc.44 
 
 20.87 
 
 94.68 
 
 22.44 
 
 93.87 
 
 23.99 
 
 22 
 
 QC 41 
 
 2O. Q 2 
 
 04.66 
 
 22.49 
 
 93.84 
 
 24.04 
 
 24 
 
 qc 2Q 
 
 2O. Q7 
 
 94.63 
 
 22.54 
 
 03.81 
 
 24.00 
 
 26 
 
 qc.a6 
 
 21 .03 
 
 yt o 
 04.6o 
 
 22.6O 
 
 93-79 
 
 24.14 
 
 28 
 
 qc 74. 
 
 21 08 
 
 04 ^8 
 
 22 6< 
 
 03 76 
 
 24 IO 
 
 2Q 
 
 qr 52 
 
 21 13 
 
 QA cr 
 
 22 . 70 
 
 03 73 
 
 24 24 
 
 32 . 
 
 0"? 2Q 
 
 21 .l8 
 
 04. 52 
 
 22.7"? 
 
 03-7O 
 
 24. 2Q 
 
 74 
 
 05 27 
 
 21 . 24 
 
 94. 5O 
 
 22.80 
 
 yo ' 
 93.67 
 
 24. 34 
 
 36 - 
 
 qc.24 
 
 21 . 2q 
 
 94-47 
 
 22.85 
 
 93- 6< 
 
 24.39 
 
 38. . 
 
 QC.22 
 
 21 . 34 
 
 94 44 
 
 22.91 
 
 03.62 
 
 24 .44 
 
 4O 
 
 CK.IQ 
 
 21 .39 
 
 94.42 
 
 22.96 
 
 93-59 
 
 24.49 
 
 42 
 
 44 ... 
 
 95-17 
 
 qc 14 
 
 21.45 
 
 21 . ^O 
 
 94-39 
 04- 36 
 
 23.01 
 23.06 
 
 93-56 
 
 Q2 . C7 
 
 24-55 
 24.60 
 
 46 
 48 
 
 95-12 
 Q">.oq 
 
 21-55 
 21.60 
 
 94-34 
 94- 31 
 
 23.11 
 23.16 
 
 93-So 
 93-47 
 
 24-65 
 24.70 
 
 c o. . 
 
 95-O7 
 
 21.66 
 
 94.28 
 
 23.22 
 
 93-45 
 
 24. 75 
 
 *\2 . . 
 
 (K 04 
 
 21 . 71 
 
 Q4. 26 
 
 23 . 27 
 
 03.42 
 
 24.80 
 
 54. . 
 
 Qf? .02 
 
 21 . 76 
 
 04. 23 
 
 23.32 
 
 93 39 
 
 24.85 
 
 56 
 
 Q4 QQ 
 
 21 8l 
 
 04 2O 
 
 f-2 77 
 
 03 36 
 
 24 QO 
 
 58 
 
 Q4 Q7 
 
 21 8? 
 
 04 17 
 
 23 42 
 
 03 33 
 
 24 QC 
 
 60 
 
 Q4 Q4 
 
 21 Q2 
 
 04 IS 
 
 23 47 
 
 03 . 3O 
 
 2< .OO 
 
 
 
 
 
 
 
 
 C=o. 7 S 
 
 0-73 
 
 0.16 
 
 0.73 
 
 0.17 
 
 0-73 
 
 O.I9 
 
 C-i.oo 
 
 0.98 
 
 O. 22 
 
 0-97 
 
 0.23 
 
 0.97 
 
 0.25 
 
 C=I.2 S 
 
 I .22 
 
 0.27 
 
 I . 21 
 
 O.2O 
 
 I .21 
 
 0.31 
 
256 
 
 PRACTICAL SURVEYING 
 STADIA REDUCTION TABLE (Continued) 
 
 M. 
 
 15 
 
 
 16 
 
 
 17 
 
 
 o' 
 
 2 
 
 Hor. 
 
 dist. 
 
 93-30 
 Q3 . 27 
 
 Diff. 
 elev. 
 
 25.00 
 2^ OS 
 
 Hor. 
 dist. 
 
 92.40 
 Q2 37 
 
 Diff. 
 elev. 
 
 26.50 
 26 ^ ^ 
 
 Hor. 
 
 dist. 
 
 91-45 
 QI 42 
 
 Diff. 
 elev. 
 
 27.96 
 28 01 
 
 4 
 
 Q3 24 
 
 2S IO 
 
 Q2 34 
 
 oo 
 
 26 ^Q 
 
 QI 3Q 
 
 28 06 
 
 6 
 
 Q3 21 
 
 2< . I"> 
 
 Q2 31 
 
 26 64 
 
 QI 3 ^ 
 
 28 10 
 
 8 
 
 93.18 
 
 2 5 2O 
 
 Q2. 28 
 
 26 69 
 
 QI 32 
 
 28 15 
 
 10 
 
 Q2 l6 
 
 2C 2S 
 
 Q2 2 S 
 
 26 74. 
 
 QI 2Q 
 
 
 12 
 
 93- 13 
 
 25 3O 
 
 Q2 22 
 
 26 7Q 
 
 91 26 
 
 28 25 
 
 14 
 
 Q3 10 
 
 2^.31; 
 
 Q2 IQ 
 
 26 84 
 
 QI 22 
 
 28 30 
 
 16 
 
 93.07 
 
 25.40 
 
 Q2 I < 
 
 26.89 
 
 QI IQ 
 
 28 34 
 
 18 
 
 07 O4 
 
 2"> 4<? 
 
 Q2 1 2 
 
 26 Q4. 
 
 QI l6 
 
 '^ 
 28 3Q 
 
 20 
 22 
 
 93-Qi 
 
 Q2.Q8 
 
 25-50 
 2<J. 
 
 92.09 
 92 06 
 
 *;-3f* 
 
 26.99 
 
 27 OA 
 
 91 . 12 
 QI OQ 
 
 28.44 
 
 28 dQ 
 
 24 
 
 92.01; 
 
 25.60 
 
 Q2 .03 
 
 27 OQ 
 
 91 06 
 
 28 s4 
 
 26 
 
 92.92 
 
 25.65 
 
 Q2 .OO 
 
 27 . 1^ 
 
 QI O2 
 
 28 & 
 
 28 
 30 
 
 32 . . 
 
 92.89 
 92.86 
 
 92.8^ 
 
 25.70 
 
 25-75 
 
 25.80 
 
 91-97 
 91-93 
 
 QI QO 
 
 27.18 
 27-23 
 
 27.28 
 
 90.99 
 90.96 
 
 QO Q2 
 
 28.63 
 28.68 
 
 28 73 
 
 34. . 
 
 Q2.8o 
 
 25.85 
 
 91.87 
 
 27 . 33 
 
 QO SQ 
 
 28 77 
 
 2 
 36. . 
 
 92.77 
 
 25.90 
 
 QI .84 
 
 27.38 
 
 QO 86 
 
 28 82 
 
 38 
 40 
 
 42 
 
 44 
 46 
 48 
 TO 
 
 92-74 
 92.71 
 
 92.68 
 92.65 
 92.62 
 
 92.59 
 Q2 . c6 
 
 25-95 
 26.0O 
 
 26.05 
 26.10 
 26.15 
 26. 2O 
 26. 2< 
 
 9I.8I 
 91-77 
 
 91-74 
 
 9 x -7i 
 
 91.68 
 
 9I-65 
 QI 6l 
 
 27-43 
 27.48 
 
 27.52 
 27-57 
 27.62 
 
 27.67 
 
 27 72 
 
 yw.u^ 
 
 90.82 
 90.79 
 
 90.76 
 90.72 
 90.69 
 90.66 
 
 GO 62 
 
 28.87 
 28.92 
 
 28.96 
 29.01 
 29.06 
 29.11 
 2Q I ^ 
 
 52 . . 
 
 92.53 
 
 26.30 
 
 QI. ^8 
 
 27.77 
 
 QO "CQ 
 
 2Q 2O 
 
 "\4 
 
 Q2 4Q 
 
 26 3< 
 
 QI ^s 
 
 27 8l 
 
 QO ^ tj 
 
 2Q 2 ^ 
 
 It::::::::: 
 
 58. 
 
 92.46 
 Q2 4^ 
 
 26.40 
 26.4 1 ? 
 
 91.52 
 
 91 48 
 
 27.86 
 
 27 QI 
 
 90.52 
 QO 48 
 
 29 30 
 2Q 34 
 
 60 
 
 Q2 .40 
 
 26. 50 
 
 QI 4$ 
 
 27 96 
 
 QO 4.< 
 
 2Q 3Q 
 
 
 
 
 
 
 
 
 C=o. 7S 
 
 0.72 
 
 O. 2O 
 
 0.72 
 
 O.2I 
 
 0.72 
 
 0.23 
 
 C=i.oo 
 
 0.96 
 
 0.27 
 
 0.96 
 
 0.28 
 
 0-95 
 
 0.30 
 
 C=1.2 5 
 
 1.20 
 
 0-34 
 
 1 . 2O 
 
 0.36 
 
 I.I9 
 
 0.38 
 
TRANSIT SURVEYING 
 STADIA REDUCTION TABLE (Continued) 
 
 257 
 
 M. 
 
 18 
 
 
 19 
 
 
 J 
 
 
 
 o' 
 
 HOT. 
 dist. 
 
 QO.45 
 
 Diff. 
 elev. 
 
 2Q. 3Q 
 
 HOT. 
 
 dist. 
 
 SQ .40 
 
 Diff. 
 elev. 
 
 30.78 
 
 HOT. 
 
 dist. 
 
 88.30 
 
 Diff. 
 elev. 
 
 32 14 
 
 2 
 
 QO 42 
 
 2Q 44 
 
 SQ 36 
 
 3O 83 
 
 88 26 
 
 32 l8 
 
 
 yw.^ 
 
 QO 38 
 
 *y -H-t 
 2Q 48 
 
 <_>y . jv 
 SQ 33 
 
 o'-'-"*) 
 3O 8? 
 
 88 23 
 
 32 23 
 
 6 
 
 QO 3S 
 
 2Q ^3 
 
 SQ 2Q 
 
 3O Q 2 
 
 88 19 
 
 32 27 
 
 8 
 
 QO. 31 
 
 2Q.58 
 
 89 26 
 
 3O Q7 
 
 88 15 
 
 32 32 
 
 IO .... 
 
 00.28 
 
 2Q .62 
 
 89.22 
 
 31 OI 
 
 88 ii 
 
 32 36 
 
 12 
 14 
 
 90.24 
 
 QO 21 
 
 29.67 
 
 2Q . 72 
 
 89.18 
 SQ 15 
 
 31.06 
 31 IO 
 
 88.08 
 88 04 
 
 32.41 
 72 AC 
 
 16 
 
 90.l8 
 
 2Q . 76 
 
 rSJ'*3 
 
 SQ ii 
 
 31 I"? 
 
 88 oo 
 
 32 4Q 
 
 18 . .. . 
 
 QO. 14 
 
 2Q.8l 
 
 SQ 08 
 
 31 IQ 
 
 87 Q6 
 
 32 CJ4 
 
 20 
 
 22 
 24 .... 
 
 90.11 
 90.07 
 
 QO O4 
 
 29.86 
 
 29.90 
 
 2Q Q5 
 
 89.04 
 
 89.00 
 88 96 
 
 3!- 2 4 
 
 31.28 
 31 33 
 
 87.93 
 
 87.89 
 87 85 
 
 32.58 
 32.63 
 
 32 6? 
 
 26 
 
 QO OO 
 
 3O.OO 
 
 88 Q3 
 
 31 38 
 
 87 81 
 
 32 72 
 
 28 
 
 8Q.Q7 
 
 3O.O4 
 
 88 89 
 
 31 42 
 
 87 77 
 
 32 76 
 
 3O. . 
 
 8Q.Q3 
 
 3O. OQ 
 
 88.86 
 
 31 47 
 
 87.74 
 
 32.80 
 
 72 
 
 SQ QO 
 
 3O 14 
 
 88 82 
 
 31 51 
 
 87 7O 
 
 32 85 
 
 34 
 
 89 86 
 
 3O IQ 
 
 88 78 
 
 A O x 
 
 31 56 
 
 87 66 
 
 32 8Q 
 
 36 
 
 89.83 
 
 3O. 23 
 
 88 75 
 
 31 60 
 
 87 62 
 
 32 Q3 
 
 38.. 
 
 8Q.7Q 
 
 30.28 
 
 88 71 
 
 31 65 
 
 87 58 
 
 32 Q8 
 
 4O 
 
 89.76 
 
 ^O. ^2 
 
 88 67 
 
 31 6Q 
 
 87.54 
 
 33 O2 
 
 42 
 
 SQ 72 
 
 3O 37 
 
 88 64 
 
 31 74 
 
 87 51 
 
 33 O7 
 
 44 
 
 89 69 
 
 3O 41 
 
 88 60 
 
 31 78 
 
 87 47 
 
 33 II 
 
 4 6 
 
 89.65 
 
 3O 46 
 
 88 56 
 
 31 83 
 
 87 43 
 
 3315 
 
 4 8 
 
 89.61 
 
 ^O. ^1 
 
 88 53 
 
 31 87 
 
 87 3Q 
 
 33 2O 
 
 SO. . 
 
 89.58 
 
 30. 5^ 
 
 88 4Q 
 
 31 Q2 
 
 87 35 
 
 33 24 
 
 $2 . . 
 
 SQ $A 
 
 30 60 
 
 88 4<; 
 
 31 Q6 
 
 87 31 
 
 33 28 
 
 ^4- - 
 
 SQ 51 
 
 30 6^ 
 
 88 41 
 
 32 OI 
 
 W -O A 
 87 27 
 
 33 33 
 
 56.. 
 
 8Q.47 
 
 30 60 
 
 88 38 
 
 32 O5 
 
 87 24 
 
 33 37 
 
 f 
 
 60 
 
 89.44 
 89.40 
 
 30.74 
 30.78 
 
 88.34 
 88.30 
 
 32.09 
 32.14 
 
 87.20 
 87.16 
 
 33-41 
 33-46 
 
 =0.75 
 
 0.71 
 
 0.24 
 
 0.71 
 
 0.25 
 
 0.70 
 
 0.26 
 
 C=i .00 
 
 0-95 
 
 0.32 
 
 0-94 
 
 0-33 
 
 0-94 
 
 0-35 
 
 C-I.25 
 
 1.19 
 
 0.40 
 
 1.18 
 
 0.42 
 
 I.I7 
 
 0.44 
 
PRACTICAL SURVEYING 
 
 STADIA REDUCTION TABLE (Continued) 
 
 M. 
 
 21 
 
 
 22 
 
 
 23 
 
 
 o' 
 
 Hor. 
 
 dist. 
 
 87 16 
 
 Diff. 
 elev. 
 
 33 .46 
 
 Hor. 
 dist. 
 
 8<?.Q7 
 
 Diff. 
 elev. 
 
 34- 73 
 
 Hor. 
 
 dist. 
 
 84 73 
 
 Diff. 
 elev. 
 
 2C 07 
 
 2 
 
 87.12 
 
 33. SO 
 
 8<?.Q3 
 
 34- 77 
 
 84.69 
 
 36 .01 
 
 
 87.08 
 
 33.54 
 
 85.89 
 
 34.82 
 
 84.65 
 
 - 
 36.05 
 
 6 . .. . 
 
 87.04 
 
 33-59 
 
 85-85 
 
 34-86 
 
 84.61 
 
 36.09 
 
 8 
 
 87.00 
 
 33.63 
 
 85.80 
 
 34.90 
 
 84.57 
 
 36.13 
 
 10 
 
 86.96 
 
 33.67 
 
 85.76 
 
 34-94 
 
 84.52 
 
 36.17 
 
 12 .... 
 
 86.Q2 
 
 33.72 
 
 85.72 
 
 34.98 
 
 84.48 
 
 36. 21 
 
 14. 
 
 86.88 
 
 33.76 
 
 85.68 
 
 35.02 
 
 84.44 
 
 36.25 
 
 16 
 
 86 84 
 
 33 80 
 
 85 64 
 
 3^ .07 
 
 84 40 
 
 36 2Q 
 
 18 
 
 86 80 
 
 33.84 
 
 85.60 
 
 35.11 
 
 84 3^ 
 
 36. 33 
 
 20 
 
 86.77 
 
 33.8Q 
 
 85.56 
 
 35.15 
 
 84.31 
 
 36. 37 
 
 22 
 
 86.73 
 
 33.93 
 
 85.52 
 
 35.19 
 
 84.27 
 
 36.41 
 
 24. 
 
 86.69 
 
 33.97 
 
 85.48 
 
 35.23 
 
 84.23 
 
 36.45 
 
 26 
 
 86.65 
 
 34.01 
 
 85.44 
 
 35-27 
 
 84.18 
 
 36.49 
 
 28 
 
 86. 61 
 
 34.00 
 
 85.40 
 
 35.31 
 
 84. 14 
 
 r 
 
 30. 53 
 
 2Q 
 
 86.57 
 
 34.10 
 
 85.36 
 
 35.36 
 
 84 . 10 
 
 36.57 
 
 32 
 
 86.53 
 
 34.14 
 
 85.31 
 
 35.40 
 
 84.06 
 
 36.6l 
 
 34 . .... 
 
 86.49 
 
 34-i8 
 
 85.27 
 
 35-44 
 
 84.01 
 
 36.65 
 
 36 
 
 86.45 
 
 34-23 
 
 85-23 
 
 35-48 
 
 83-97 
 
 36.69 
 
 38 
 
 86.41 
 
 34.27 
 
 85.19 
 
 35.52 
 
 83-93 
 
 36.73 
 
 40 
 
 86.37 
 
 34.31 
 
 85.15 
 
 35.56 
 
 83.89 
 
 36.77 
 
 42 
 
 86.33 
 
 34-35 
 
 85.11 
 
 35 -60 
 
 83.84 
 
 36.80 
 
 44 
 
 86.29 
 
 34-40 
 
 85.07 
 
 35 -64 
 
 83-80 
 
 36.84 
 
 46 
 
 86.25 
 
 34-44 
 
 85.02 
 
 35-68 
 
 83.76 
 
 36.88 
 
 48 
 
 86.21 
 
 34-48 
 
 84.98 
 
 35-72 
 
 83.72 
 
 36.92 
 
 "?O 
 
 86.17 
 
 34.52 
 
 84.94 
 
 35-76 
 
 83.67 
 
 36.96 
 
 ^2 
 
 86 13 
 
 34- 57 
 
 84.90 
 
 35.80 
 
 83-63 
 
 37-oo 
 
 ^4 
 
 86. oq 
 
 Ot Jl 
 
 34.61 
 
 84.86 
 
 35-85 
 
 83-59 
 
 37-04 
 
 s6 
 
 86.05 
 
 34.65 
 
 84.82 
 
 35-89 
 
 83-54 
 
 37/o8 
 
 <;8 
 
 86.01 
 
 34.69 
 
 84.77 
 
 35-93 
 
 83.50 
 
 37.12 
 
 60" 
 
 85.97 
 
 34-73 
 
 84.73 
 
 35-97 
 
 83.46 
 
 37.16 
 
 
 
 
 
 
 
 
 C=o. 7 5 
 
 0.70 
 
 0.27 
 
 0.69 
 
 0.29 
 
 0.69 
 
 0.30 
 
 C=i.oo 
 
 0-93 
 
 0-37 
 
 0.92 
 
 0.38 
 
 0.92 
 
 0.40 
 
 C=I.2S 
 
 1.16 
 
 0.46 
 
 1.1$ 
 
 0.48 
 
 I-I5 
 
 0.50 
 
TRANSIT SURVEYING 
 STADIA REDUCTION TABLE (Continued) 
 
 259 
 
 M. 
 
 24 
 
 
 25 
 
 
 
 26 
 
 o' .. 
 
 Hot. 
 dist. 
 
 83 46 
 
 Diff. 
 elev. 
 
 37 . 16 
 
 . HOT. 
 dist. 
 
 82 14 
 
 Diff. 
 elev. 
 
 38 30 
 
 HOT. 
 
 dist. 
 
 80 78 
 
 Diff. 
 elev. 
 
 3Q 4O 
 
 2 . ... 
 
 83.41 
 
 37 . 20 
 
 82 09 
 
 38 34 
 
 80 74 
 
 3Q 44 
 
 4 
 
 8^.17 
 
 37-23 
 
 82. o<; 
 
 38 38 
 
 80.69 
 
 3Q.47 
 
 6 
 
 83.33 
 
 37.27 
 
 82.01 
 
 38.41 
 
 80.65 
 
 30 . ci 
 
 8 
 
 83.28 
 
 37.31 
 
 81.96 
 
 38. 4"? 
 
 80.60 
 
 30. C4 
 
 10 
 
 83.24 
 
 37-35 
 
 81.92 
 
 38.49 
 
 8o.<;<; 
 
 39.";8 
 
 12 
 
 83.20 
 
 37 3Q 
 
 81 87 
 
 38 S3 
 
 80 <?i 
 
 3Q 6l 
 
 14 
 
 16 
 
 83-iS 
 83.11 
 
 37-43 
 37-47 
 
 81.83 
 81.78 
 
 38.56 
 38.60 
 
 80.46 
 80.41 
 
 39-65 
 
 30.60 
 
 18 
 20 
 
 22 
 
 83.07 
 83.02 
 
 82.98 
 
 37-51 
 37-54 
 
 37.58 
 
 81.74 
 81.69 
 
 81.65 
 
 38.64 
 38.67 
 
 38 71 
 
 80.37 
 80.32 
 
 80 28 
 
 39.72 
 39-76 
 
 70.70 
 
 24 
 
 82.03 
 
 37.62 
 
 81.60 
 
 38.7^ 
 
 80.23 
 
 39.83 
 
 26 
 
 82.89 
 
 37-66 
 
 81.56 
 
 38.78 
 
 80. 18 
 
 39-86 
 
 28...' 
 
 7Q 
 
 82.85 
 82 80 
 
 37-70 
 37 74 
 
 81.51 
 
 8l 4.7 
 
 38.82 
 38 86 
 
 80.14 
 
 80 OQ 
 
 39-90 
 3Q Q3 
 
 32. . 
 
 82.76 
 
 37-77 
 
 81.42 
 
 38 89 
 
 80.04 
 
 30.07 
 
 34. . 
 
 82.72 
 
 37.81 
 
 81.38 
 
 38.03 
 
 SO.OO 
 
 40100 
 
 36 
 38 
 40 
 
 42 
 
 82.67 
 82.63 
 82.58 
 
 82.54 
 
 37-85 
 37.89 
 37-93 
 
 37.96 
 
 81.33 
 81.28 
 81.24 
 
 81 19 
 
 38.97 
 39-00 
 39-04 
 
 3Q.o8 
 
 79-95 
 79-90 
 
 79-86 
 7Q.8l 
 
 40.04 
 40.07 
 4O.II 
 
 40. 14 
 
 44 
 
 82.49 
 
 38.00 
 
 81.1? 
 
 ^Q. II 
 
 70.76 
 
 40. 18 
 
 46 
 48 
 
 82.45 
 82 41 
 
 38.04 
 38 08 
 
 81.10 
 81 06 
 
 39-15 
 
 3Q l8 
 
 79.72 
 7Q 6? 
 
 40.21 
 
 4O 24 
 
 eo. . 
 
 82.36 
 
 38.11 
 
 81 01 
 
 30 22 
 
 7Q 62 
 
 4.0 28 
 
 52 
 54 
 56 
 
 82.32 
 82.27 
 82. 23 
 
 38.15 
 38.19 
 
 38 23 
 
 80.97 
 80.92 
 80 87 
 
 39-26 
 
 39-29 
 30 33 
 
 79-58 
 
 79-53 
 
 7Q 48 
 
 40.31 
 40-35 
 40 38 
 
 58. . 
 
 82.18 
 
 38.26 
 
 80 83 
 
 3Q 36 
 
 7Q 44 
 
 4.O 42 
 
 60 -.. 
 
 82.14 
 
 38.30 
 
 80 78 
 
 3Q 4O 
 
 7Q 3Q 
 
 AO 45 
 
 
 
 
 
 
 
 
 =0.75 
 
 0.68 
 
 0.31 
 
 0.68 
 
 0.32 
 
 0.67 
 
 0.33 
 
 C=i.oo 
 
 0.91 
 
 0.41 
 
 0.90 
 
 0-43 
 
 0.89 
 
 0.45 
 
 C=I.2 S 
 
 1.14 
 
 0.52 
 
 1-13 
 
 0-54 
 
 I .12 
 
 0.56 
 
260 
 
 PRACTICAL SURVEYING 
 STADIA REDUCTION TABLE 
 
 M. 
 
 27 
 
 28 
 
 29 
 
 30 
 
 o' 
 
 Hor. Difif. 
 dist. elev. 
 
 79-39 4 -45 
 
 Hor. Diff. 
 dist. elev. 
 
 77.96 41.45 
 
 Hor. Diff. 
 dist. elev. 
 
 76.50 42.40 
 
 Hor. Diff. 
 dist. elev. 
 
 75.00 43.30 
 
 2 
 
 4 
 
 79.34 40.49 
 79.30 40.52 
 
 77.91 41.48 
 77.86 41.52 
 
 76.45 42.43 
 76.40 42.46 
 
 74 - 95 43 33 
 74.90 43.36 
 
 6 
 
 7Q 2 ^ 40 . ^ t; 
 
 77 .81 41 . 55 
 
 76.3? 42.40 
 
 74 8 <? 43 3Q 
 
 8 
 
 7Q .20 4O . ZQ 
 
 77.77 41.58 
 
 76.30 42.53 
 
 74.80 43 42 
 
 10 
 
 12 
 
 79.15 40.62 
 
 79 ii 40 66 
 
 77.72 41.61 
 77.67 41.65 
 
 76.25 42.56 
 76.20 42 59 
 
 74-75 43-45 
 74 70 43 47 
 
 14 
 
 79 . 06 40 69 
 
 77.62 41 .68 
 
 76.15 42.62 
 
 74 6< 43 550 
 
 16 
 
 79.01 40.72 
 
 77.57 41.71 
 
 76.10 42.65 
 
 74.60 43 . 53 
 
 18 
 20 
 
 78.96 40.76 
 78.92 40.79 
 
 77.52 41.74 
 77.48 41.77 
 
 76.05 42.68 
 76.00 42.71 
 
 74-55 43-56 
 74.49 43.59 
 
 22 
 
 24 
 26 
 28 
 3O. . 
 
 78.87 40.82 
 78.82 40.86 
 78.77 40.89 
 78.73 40.92 
 78 .68 40 . 96 
 
 77.42 41.81 
 77.38 41-84 
 77,33 41.87 
 77.28 41-90 
 77.23 41.93 
 
 75-95 42.74 
 75.90 42.77 
 75.85 42.80 
 75.80 42.83 
 75-75 42.86 
 
 74.44 43.62 
 
 74-39 43-65 
 74.34 43.67 
 74 .29 43 . 70 
 74 .24 43 . 73 
 
 32 . 
 
 78.63 4O. QQ 
 
 77.18 41.97 
 
 75.70 42.89 
 
 74.19 43.76 
 
 34 
 36.. . 
 
 78.58 41.02 
 78.54 41.06 
 
 77.13 42.00 
 77.09 42.03 
 
 75.65 42.92 
 75.60 42.95 
 
 74.14 43.79 
 74.09 43.82 
 
 38 
 
 78.49 41.09 
 
 77.04 42.06 
 
 75-55 42.98 
 
 74 .04 43 . 84 
 
 40 
 
 42 
 44 
 46 
 48 
 So 
 
 52. . 
 
 78.44 41.12 
 
 78.39 41.16 
 78.34 41.19 
 78.30 41.22 
 78.25 41.26 
 78.2O 41.29 
 
 78.15 41.32 
 
 76.99 42.09 
 
 76.94 42.12 
 76.89 42.15 
 76.84 42.19 
 76.79 42.22 
 76.74 42.25 
 
 76.69 42.28 
 
 75.50 43.01 
 
 75.45 43-04 
 75.40 43.07 
 75-35 43-iQ 
 75.30 43.13 
 75.25 43.16 
 
 75.20 43.18 
 
 73-99 43-87 
 
 73-93 43-90 
 73.88 43.93 
 73-83 43-95 
 73-78 43.98 
 73.73 44.01 
 
 73 .68 44 . 04 
 
 $: 
 
 78.10 41.35 
 
 78 06 41 39 
 
 76.64 42.31 
 76. <Q 42 . 34 
 
 75.15 43.21 
 75 .10 43 . 24 
 
 73.63 44.07 
 73.58 44.09 
 
 58. 
 
 78.01 41 42 
 
 76.55 42.37 
 
 75.05 43.27 
 
 73.52 44.12 
 
 60 
 
 77.96 41.45 
 
 76.50 42.40 
 
 75.00 43.30 
 
 73.47 44.15 
 
 
 
 
 
 
 C=o. 7 5 
 
 0.66 0.35 
 
 0.66 0.36 
 
 0.65 0.37 
 
 0.65 0.38 
 
 C=i.oo 
 
 o . 89 o . 46 
 
 0.88 0.48 
 
 0.87 0.49 
 
 0.86 0.51 
 
 C=i.25 
 
 I. I I 0.58 
 
 i.io 0.60 
 
 1.09 0.62 
 
 i . 08 o . 64 
 
TRANSIT SURVEYING 261 
 
 The subdividing of land into building lots; the re-survey 
 of lots; keeping records; filing notes and records; setting 
 monuments, etc., are all fully dealt with in the author's 
 book ''Engineering Work in Towns and Cities." ($3.00.) 
 
 In making re-surveys a random line must first be run; 
 sometimes several such lines. If the compass, and chain 
 were employed on the original survey, it is advisable to 
 run the random lines with the needle and employ green 
 hands to measure with an old-fashioned chain. In this way 
 the locations of monuments will be discovered more readily 
 than if modern methods are employed. When the corners 
 are located the true lines should be run with the transit 
 and a steel tape used by skilled helpers. The new notes 
 should then be recorded. 
 
 MAKING THE MAP 
 
 A complete map should contain all the data necessary 
 to enable a competent surveyor to re-trace the lines. Not 
 less than two permanent monuments should be shown on 
 the map, connected to the lines of the survey, with tie 
 lines to reference points. 
 
 The following items should be shown on a map and the 
 list is given as a sort of "specification reminder" for the 
 surveyor and draftsman. 
 
 1. Scale of map. 
 
 2. Meridian line with declination of needle. 
 
 3. A short title with the date of survey. 
 
 4. Monuments: Location; description; tie lines. 
 
 5. Lengths of all lines. 
 
 6. Bearings of all lines. 
 
 7. Angles of intersection of all lines. 
 
 8. Name of owner of record. 
 
 9. Names of adjoining owners of record placed on their 
 land with location of common corners. 
 
 10. Names of all recognized landmarks within the area 
 embraced in the map. 
 
 11. Statement over signature of the surveyor that he 
 has carefully checked the map, and that it truly represents 
 the survey made by him, and that he set the monuments 
 described and drove a stake at each corner. 
 
262 PRACTICAL SURVEYING 
 
 If the map is of an addition to a village, town or city, 
 or is of a subdivision of a tract of land into small parcels 
 or lots, it should contain all the foregoing and in addition 
 the following: 
 
 12. The number of each block and lot. 
 
 13. The width and name of each thoroughfare. 
 
 14. The dedication, acknowledged before a qualified 
 legal officer, of the public thoroughfares to the use of the 
 public. 
 
 Every owner whose property is shown on the map as 
 being included within the areas dedicated to public use 
 must sign the dedication. 
 
 Mr. S. N. Howard of Chicago, in the Proceedings of the 
 Illinois Society of Engineers and Surveyors, gave the fol- 
 lowing list of items to be shown on maps of city lot 
 surveys : 
 
 "Lines, grades, angles, location and elevation of sewer, 
 elevation of walks in front of lot and adjoining the same 
 and elevation of ground; and sometimes in addition is the 
 size and location of water, gas and electric mains; the 
 location of point where water, gas and electric service pipes 
 come through the curb; location of house drains; the 
 location, height and plumb of adjoining buildings; thick- 
 ness of party walls at the several stories, and location and 
 depth of the foundations of same." 
 
 City lot surveys are usually made for building purposes 
 and an architect needs all the above data. 
 
 Maps of transit surveys may be plotted like maps of 
 compass surveys with protractors. (See Chapter on Com- 
 pass Surveying.) 
 
 When for any reason the protractor angles are not ac- 
 curate enough, angles may be set out to the nearest minute 
 by using either a table of chords or a table of natural 
 tangents. If the surveyor has no book containing a table 
 of chords he can use natural sines. Twice the sine of half 
 the angle is the chord of the whole angle. 
 
 First lay off a base ten inches (or units) long. The 
 tables express the values of the functions as decimal frac- 
 tions, so this length of 10 = unity (i). The chord (or 
 natural tangent) is measured with the scale used for the 
 base. 
 
TRANSIT SURVEYING 263 
 
 In Fig. 204 let AC = base = unity and from A as a 
 center describe the arc BC. Set the compass to scale the 
 chord BC of the angle BA C, and from C as a center with 
 radius = chord, intersect arc BC at B. Draw a straight 
 line from A through B, thus forming the angle BA C. 
 
 FIG. 204. FIG. 205. 
 
 In Fig. 205 let AC = base = unity, and at C erect the 
 perpendicular BC. On the perpendicular lay off the tan- 
 gent of the angle A. The line is then drawn from A 
 through B to form the angle BA C. 
 
 The angles may be set off from the ends of lines, thus 
 resembling deflection work, or two lines may be drawn 
 normal to each other in the middle of the sheet and a 
 circle drawn with radius = 10 units of scale, with the in- 
 tersection as the center of the circle. Chords may be set 
 off on the circumference or tangents measured from the 
 end of the radius for all the angles. The lines can then 
 be transferred by using triangles and straight-edge. 
 
 If the error in the survey is large enough to show on the 
 map a closure must be "fudged," no matter how accu- 
 rately the angles are drawn. If the survey is balanced so 
 the errors are distributed it will be necessary to calculate 
 new bearings and distances if angles are to be plotted. 
 This however is not right, besides being very laborious. 
 The "latitude and departure method" enables a drafts- 
 man to make an accurate map on which the lines will close 
 and it is used by all experienced men who are noted for 
 careful work. 
 
 The draftsman must remember that the field work has 
 been done with all possible care. In spite of this there is 
 an error of closure, the amount of which depends on the 
 value of the land. The error is probably distributed 
 throughout all the courses so the bearings and distances 
 recorded by the instrument man must be placed on the 
 map. The error of closure is discovered by the computa- 
 
264 
 
 PRACTICAL SURVEYING 
 
 tions, the field work not disclosing it. The same conditions 
 must appear on the map and inaccuracies in plotting added 
 
 to the field error may prevent 
 the lines closing by a quite ap- 
 preciable amount. To plat the 
 work by using latitudes and 
 departures, the lines must close, 
 and furthermore the computa- 
 tions give the exact amount of 
 & error and the surveyor knows 
 whether it lies within proper 
 limits. 
 
 The computations for the 
 field shown in Fig. 206 are 
 given in table on the following page. 
 
 The latitudes and departures are tabulated, the error 
 found and distributed. The balanced latitudes and de- 
 partures are then algebraically as follows: 
 
 FIG. 206. 
 
 Latitudes 
 
 - 599-51 
 + 839.89 
 
 + 240.38 
 + H28.58 
 + 1668.96 
 1059.22 
 + 609.74 
 + 347.89 
 957.63 
 
 ~ 1743.62 
 
 Departures 
 
 + I273.96 
 -j- 109.28 
 
 +1383.24 
 
 - 1075.26 
 
 + 307.98 
 1283.46 
 
 - 975.48 
 
 - 927.94 
 
 - 1903.42 
 + 927-66 
 
 The starting point has no latitude or departure. In 
 going around a field the travel north equals the travel 
 south and the travel east equals the travel west, so the 
 algebraic latitude and departure for the last station must 
 equal the balanced latitude and departure with the sign 
 reversed. This checks the summation. 
 
 The method of plotting is shown in Fig. 207. Through 
 the point selected for Sta. draw the horizontal line OX, 
 OX' and the vertical line OY, OF'. Divide the sheet into 
 squares 100 ft. by 100 ft. Number them as shown and in 
 each square in which a corner will be located measure the 
 
TRANSIT SURVEYING 
 
 265 
 
 i 
 
 
 VD Tj- 00 00 M VD 
 O <T <N C"> 'd" ^ t~ 
 
 O co co t^ to <O to 
 _|j t^ oo O t^ O f^ 
 
 1 I + I f 1 
 
 
 
 Total 
 latitude. 
 
 
 M OO ^O ^ <^ O*i 
 
 O CJ> O CO O^ *^ 1 O 
 
 fl (T ^" ^D O IO OO 
 
 n 10 cs ^D ^O O^ t^-* 
 
 1 + 4: + + 1 
 
 
 
 H 
 
 i 
 
 r^ oo CN ' ' 
 O <N en 
 
 
 
 s 
 
 CM 
 
 
 . * 
 
 + 
 
 w 
 
 t^. O <N r^ 
 
 M 
 
 VD 
 
 VD 
 
 vo' 
 oo 
 
 
 3 1 
 
 1 
 
 w 
 
 M CN d 
 
 10 d VD 
 
 O% to TJ- 
 to O t*i ' 
 
 M M 
 
 to 
 
 i 
 
 
 3 
 
 
 
 O> OO O^ O^ 
 
 00 10 00 CT> 
 co M ^ CO 
 
 oo T}- -co r>. 
 
 M 
 
 to 
 
 CO 
 
 1 
 
 
 i 
 
 1 
 
 ^- o% 
 
 to co I s - 
 
 r^. oo o 
 
 MM 
 
 VO* 
 00 
 
 CM 
 
 
 i 
 
 w 
 
 O CO t^ 00 
 
 Tf CT> l^- IO 
 t^. O M t->. 
 
 M c~i C7 
 
 00 10 
 
 VD* VD* 
 00 OO 
 
 CO 
 
 
 i 
 
 1 
 
 C/3 
 
 IO CS VD 
 
 o*> to ^ 
 10 O t^ 
 
 O 
 
 CO 
 
 
 +2 
 
 
 O^ vD O^ O 
 
 ^J- co 
 
 M 
 
 JS 
 
 2 
 
 O^ OO f^* VD 
 00 <* -co t~ 
 
 M 
 
 CM CM 
 CO CO 
 
 O 
 
 !i 
 
 
 oo r>- oo rj- M to co 
 O ^f OO vD O~> t^ to 
 
 M MM MM 
 
 
 
 
 
 w w ^ ^ ^ w & 
 
 
 
 ! 
 
 
 00 10 00 00 t^ M O> 
 Tt d 10 CN CM O O 
 
 o o o o o o o 
 ^ t^- VD O CT OO M 
 VD co to vD CM 10 
 
 CO ^ ^ CO ^1 rn Z 
 
 
 
 
 
 C M cs co ^t- to vo 
 
 
 
266 
 
 PRACTICAL SURVEYING 
 
 fractional parts, draw the intersecting latitude and depar- 
 ture lines and locate the corner. Finish the outline of the 
 field by connecting the corners. When the outlines and all 
 
 FIG. 207. Platting by latitudes and departures. 
 
 bearings and lengths are inked in, the co-ordinate lines are 
 erased, for they are drawn with a sharp, hard lead pencil. 
 
 Maps expected to be in fairly constant service (working 
 maps) should be drawn on the best quality of cloth mounted 
 paper and the co-ordinate lines should be very fine red ink 
 lines. These maps never leave the surveyor's office, 
 tracings being made of such portions as may be wanted 
 by clients. In many cities maps are drawn on sheets with 
 co-ordinate lines drawn in ink. The co-ordinates (latitude 
 and departure) are marked in figures at each monument 
 and lot corner. The bearing and distance between any 
 two points may then be obtained readily by trigonometry. 
 It is an admirable way to keep records and working maps. 
 Steel straight-edges should be used in drawing the lines. 
 To project a line too long to be drawn with the straight- 
 edge, drive a fine needle at one end to which fasten a fine 
 silk thread. Stretch the thread exactly over the line and 
 at points which may be included within the length of the 
 straight-edge make fine marks on the paper. Draw the 
 extended line through these marks. 
 
 REPRODUCING MAPS 
 
 For reproducing drawings smaller than 12 ins. by 18 ins. 
 hektographs or clay process pads are good. The hekto- 
 graph is a pad of gelatine in a shallow pan, the clay pad 
 
TRANSIT SURVEYING 267 
 
 being a substitute for gelatine and in appearance re- 
 sembling putty. The drawings are made with specially 
 prepared hektograph inks on a smooth hard paper. This 
 is laid face down on the pad and rubbed to remove wrinkles 
 and air bubbles. It is left in place for three minutes. 
 After removal plain pieces of paper, preferably not so hard 
 as the original, are laid on the pad in rapid succession and 
 rubbed smooth. Each sheet receives a print, the im- 
 pressions becoming gradually more faint. Fifteen to 
 twenty good impressions is the limit. The inks can be had 
 in a number of colors. If only a few copies are wanted, 
 less than six or seven, the original drawing may be made 
 with copying pencils, also to be obtained in several colors. 
 
 For larger drawings tracings are used. Tracing cloth is 
 costly, and medium thick "Parchment," "Vellum," 
 "Colonna" or "Ionic" tracing papers are well suited to 
 the use of the surveyor. When tracings are to receive 
 much handling or are expected to be used for many years 
 they should be on tracing cloth. The drawing is made on 
 paper and traced in ink on the transparent cloth or paper. 
 To save time on unimportant work the drawing is made 
 directly on tracing paper or on the rough side of tracing 
 cloth in lead pencil and then inked in. Tracing cloth and 
 paper are greatly affected by moisture in the atmosphere 
 so it is best to make accurate maps first on paper. Sur- 
 veyors should use a good quality of paper for maps. 
 
 When several copies of a drawing are wanted the tracing 
 is blue printed. Blue-print papers are sold by all dealers 
 in surveying supplies. In a frame holding a sheet of plate 
 glass, the tracing is placed on the glass. The sensitized 
 surface of a sheet of blue-print paper is laid on the tracing. 
 On top is placed a piece of felt or blanket, and the whole 
 is covered with a wooden cover held in place by springs 
 and clamps. All bubbles of air and wrinkles in the tracing, 
 the blue-print paper and the felt must be eliminated. The 
 glass is exposed to the direct rays of the sun for several 
 minutes, depending on the sensitiveness of the prepared 
 paper. The blue-print paper is afterwards washed in clean 
 water until white lines appear on a blue ground, and then 
 hung on a line to dry. 
 
 Men who have occasionally to make blue prints find it 
 
268 PRACTICAL SURVEYING 
 
 convenient to prepare paper for their own use, for this 
 material does not have good keeping properties. Take 
 one-half ounce each of potassium ferrid-cyanide and cit- 
 rate of iron and ammonia; dissolve in from 6 ozs. to 8 ozs. 
 of clean water. Put in a bottle and shake thoroughly un- 
 til the chemicals are dissolved, which process requires about 
 10 minutes. The mixture does not keep well so a fresh lot 
 should be prepared for each job of printing. Only chemi- 
 cally pure (C. P.) materials should be used and they can 
 be bought in crystal form from any druggist. Keep in a 
 light-proof bottle with ground-glass stopper. 
 
 Eight ounces of the mixture will coat about 100 square 
 feet of paper. Lay in a shallow dish a piece of thin cotton 
 or linen cloth and pour in the mixture. Then lift up the 
 cloth to allow the liquid to strain through so all undis- 
 solved lumps will be removed. The work should be done 
 in a room lighted by a red or orange light. Use a broad 
 flat brush for coating the paper, which should be a good 
 bond paper with smooth surface. Hang in the dark until 
 dry and keep in the dark until used. 
 
 White lines on a blue ground are not satisfactory when 
 a map is to be colored, but with special inks or erasing fluids 
 white, red and yellow lines may be drawn on blue prints. 
 Several manufacturers sell paper which is used like blue- 
 print paper, the lines coming out blue, black or brown on 
 a white ground. There is also on the market a thin paper 
 which, used like blue-print paper, shows white lines on a 
 brown ground. Such a print may be used with ordinary 
 blue-print paper, as a photographic negative, the result be- 
 ing a print with blue lines on a white ground. 
 
 PRACTICAL ASTRONOMY 
 
 An ephemeris is a table giving the place of a planet for 
 a number of successive days. The solar ephemeris is of 
 value to surveyors and engineers, for by observations on 
 the sun the local time at a place may be ascertained, the 
 latitude found and the true meridian determined. 
 
 Practically all instrument makers issue annually small 
 books containing the solar ephemeris for the year, together 
 with much other useful information. These vest pocket 
 
TRANSIT SURVEYING 269 
 
 books are sold for a small price, usually ten cents. From 
 the 1912 edition of the twenty-five cent book issued by 
 Wm. Ainsworth & Sons, Denver, Colo., the following con- 
 cise description of practical astronomical work has been 
 taken. 
 
 The Sun is the center of the solar system, remaining con- 
 stantly fixed in its position, although often spoken of as in 
 motion around the earth. 
 
 The Earth makes a complete revolution around the sun 
 in three hundred and sixty-five days, five hours, forty-eight 
 minutes and forty-six seconds. 
 
 It also rotates about an imaginary line passing through 
 its center, and termed its axis, once in twenty-three hours, 
 fifty-six minutes and four seconds, mean time, turning 
 from west to east. 
 
 The Poles are the extremities of the earth's axis. The pole 
 in our own hemisphere, known as the North Pole, if pro- 
 duced indefinitely toward the concave surface of the heavens, 
 would reach the North Pole of the heavens a point situated 
 near the Polar star. 
 
 The Equator is an imaginary line passing around the 
 earth, equidistant from the poles, and in a plane at right 
 angles with the axis. 
 
 If the plane of the equator be produced to the heavens, 
 it forms what is termed the Celestial Equator. 
 
 The Orbit of the earth is the path in which it moves in 
 making its yearly revolution. If the plane of this orbit 
 were produced to the heavens, it would form the Ecliptic, 
 or the sun's apparent path in the heavens. 
 
 The earth's axis is inclined to its orbit at an angle of 
 about 23 27', making an angle of the same amount between 
 the earth's orbit and its equator, or between the Celestial 
 Equator and the Ecliptic. 
 
 The Equinoxes are the two points in which the Ecliptic 
 and the Celestial Equator intersect one another. 
 
 The Horizon of a place is the surface which is defined by 
 a plane supposed to pass through the place at right angles 
 with a vertical line, and to bound our vision at the surface 
 of the earth. The horizon, or a horizontal surface, is de- 
 termined by the surface of any liquid when at rest, or by 
 the spirit-levels of an instrument. 
 
270 PRACTICAL SURVEYING 
 
 The Zenith of any place is the point directly overhead, in 
 a line at right angles with the horizon. 
 
 The Meridian of any place is a great circle passing through 
 the zenith of a place and the poles of the earth. 
 
 The Latitude of a place is its distance north or south of 
 the Equator, measured on a Meridian. At the equator the 
 latitude is o, at the poles 90. 
 
 Refraction. By reason of the atmosphere, the rays of 
 light from the sun are bent out of their course, so as to 
 make its altitude appear greater than is actually the case. 
 
 The amount of refraction varies according to the altitude 
 of the body observed, being zero when it is in the zenith, 
 about one minute when midway from the zenith to the 
 horizon, and almost thirty-four minutes when in the 
 horizon. 
 
 The Longitude of a place is its angular distance east or 
 west of a given place taken as the starting point or first 
 meridian; it is measured on the equator or on any parallel 
 of latitude and usually from Greenwich, England. 
 
 As the earth makes a complete rotation upon its axis 
 once a day, every point on its surface must pass over 360 
 in twenty-four hours, or 15 in one hour, and so on in the 
 same ratio. And as the rotation is from west to east, the 
 sun would come to the meridian of every place 15 west 
 of Greenwich just one hour later than the time given in 
 the Ephemeris for apparent noon at that place. 
 
 To an observer situated at Denver, Colo., the longitude 
 of which is, in time, seven hours, the sun would come to 
 the meridian seven hours later than at Greenwich, and 
 thus when it was 12 M. at that place it would be but 5 A.M. 
 in Denver. 
 
 TIME 
 
 A Solar Day is the interval of time between two suc- 
 cessive upper transits of the sun across the same meridian. 
 Solar days are of unequal length. A mean solar day is 
 the average for a year. 
 
 A Sideral Day is the interval of time between two suc- 
 cessive upper transits of a fixed star across the same 
 meridian; it is invariable and is equal to twenty- three 
 hours, fifty-six minutes, four and nine-hundredths seconds 
 
TRANSIT SURVEYING 271 
 
 of mean solar time. The earth makes one complete 
 rotation on its axis in a sideral day. 
 
 Mean Noon. A clock keeps mean solar time when it 
 divides a mean solar day into twenty-four equal parts or 
 hours. Noon as shown by such a clock is mean noon. 
 
 Apparent Noon for any place is the time of the upper 
 transit of the sun across the meridian of that place; it 
 may occur several minutes earlier or later than mean noon. 
 
 Equation of Time. The column headed "Equation of 
 Time" in the Ephemeris shows the quantity to add to 
 or subtract from mean time to obtain the corresponding 
 apparent time. 
 
 Standard Time. Since November, 1883, in the United 
 States, the mean solar time of the meridians 60, 75, 90, 
 105, and 1 20 west of Greenwich is standard time. The time 
 spaces are known respectively as Colonial, Eastern, Central, 
 Mountain and Pacific time. Each differs from the next in 
 time by one hour. Instead of employing the local mean 
 solar time, the time used 1s the mean solar time at the 
 nearest of the standard meridians. 
 
 Hour Angle. The number of hours from the meridian. 
 
 To Set a Watch to Mean Local Time by the Sun. Set up 
 the transit and adjust the telescope to the true meridian, 
 then note the exact time that the center of the sun's image 
 crosses the vertical cross-hair. This is apparent noon, and 
 at this instant, set -the minute hand to as many minutes 
 before 12 as the equation of time for the given day shows 
 is to be added to mean time, or to as many minutes after 
 12 as it shows is to be subtracted. The correction may be 
 noted in case it is not desired to set the watch. 
 
 DECLINATION 
 
 The Declination of the sun is its angular distance north 
 or south of the celestial equator; when the sun is at the 
 equinoxes, that is, about the 2ist of March and the 2ist 
 of September of each year, its declination is o, or it is said 
 to be on the equator; from these points its declination in- 
 creases from day to day and from hour to hour, until on 
 the 2ist of June and the 2ist of December it is 23 27' 
 distant from the equator. 
 
272 PRACTICAL SURVEYING 
 
 It is the declination which causes the sun to appear so 
 much higher in summer than in winter, its altitude in the 
 heavens being about 46 54' more on the 2ist of June than 
 it is on the 2ist of December. 
 
 The Ephemeris gives the sun's declination for mean noon 
 at Greenwich for each day in the year. The declination of 
 the sun at any place for any hour of the day is determined 
 from the Ephemeris as follows: 
 
 1. Divide the longitude of the place (reckoned from 
 Greenwich) by 15 to obtain the corresponding difference of 
 time in hours. 
 
 2. Find the corresponding Greenwich time by adding 
 the difference of time to the mean time at the given place, 
 when west from Greenwich, and subtracting when east. 
 
 3. Multiply the difference for one hour, as found in the 
 table opposite the given day of the year, by the number of 
 hours from noon by Greenwich time. 
 
 4. This product is the change in declination to be ap- 
 plied as indicated in the following expression: 
 
 Declination for 
 given time 
 and place 
 
 {apparent dec. 
 for given day 
 of year 
 
 change in 
 declina- 
 tion. 
 
 The sign of the last term is + for time after noon, Green- 
 wich, when declination is increasing, and for time before 
 noon when declination is decreasing. 
 
 The sign is to be used for time after noon, Greenwich, 
 when declination is decreasing and for time before noon 
 when declination is increasing. 
 
 An inspection of the Ephemeris will show whether the 
 declination is increasing or decreasing from day to day. 
 
 N in the column of apparent declination indicates north, 
 and S indicates south declination. 
 
 Example. Required the declination at 10 A.M., Aug. 
 10, 1911, at Denver, Colo., U. S. A., latitude 39 46' 31" 
 north, longitude 105 west. 
 
 1. Diff. of time = = 7 hrs. 
 
 2. As Denver is west from Greenwich, add the diff. of 
 time, obtaining 10 A.M., Denver time = 10 -f- 7 = 17, or 5 
 P.M., Greenwich time. 
 
TRANSIT SURVEYING 273 
 
 3. Change in dec. for I hr. = 43". 32 
 Change for 5 hrs. = 43". 32 X 5 = o 03' 36".6o 
 
 4. Sun's apparent dec. at 
 
 Greenwich, mean noon = N 15 49' 53". 60 
 
 Dec. at 10 A.M., Denver = N 15 46' 17" 
 
 The change is subtracted as the time is afternoon and 
 dec. is decreasing, since the dec. the next day is less. 
 
 LATITUDE 
 
 Determination of Latitude by Direct Observation of the Sun. 
 Carefully level the transit a few minutes before apparent 
 noon, and if it is not provided with solar hairs, bring the 
 horizontal cross-hair tangent to the upper limb of the sun 
 and keep it tangent by the slow motion screw until the 
 sun ceases to rise, then read the vertical angle, and from 
 this angle subtract the semi-diameter of the sun, as given 
 in the Ephemeris for the proper month of the year, also 
 the refraction corresponding to the observed angle. The 
 resulting angle will be the true altitude of the sun's center. 
 Calculate the sun's declination for noon, apparent time, for 
 place of observation as described above. If the declina- 
 tion is N, subtract, but if it is S, add it to the true altitude 
 of the sun. The result is the co-latitude of the place of 
 observation and the lat. = 90 co.-lat. 
 
 Note. In direct solar observations it is necessary to 
 protect the eye by a darkened glass, which should be used 
 at the eye end of the telescope unless both its surfaces are 
 true and parallel. When the altitude is high, a diagonal 
 eyepiece will be found convenient, or an image of the sun 
 and cross-hairs may be formed on a screen, a card or a 
 blank page of a notebook held a few inches from the eye- 
 piece. If this method is used no darkener is required. An 
 average of several observations is preferable to a single 
 observation. When more than one observation is made, 
 the alternate ones should be taken with the telescope re- 
 versed in order to eliminate instrumental errors. When 
 great accuracy is not essential, standard time may be used 
 in computing the declination. As the difference between 
 standard and local time is seldom more than 30 min. and 
 the greatest hourly change in declination is about one 
 
274 
 
 PRACTICAL SURVEYING 
 
 minute of an arc, the maximum error in declination due to 
 using standard time would not be greater than 30 seconds. 
 Example. On April 17, 1910, an observation was made 
 of the sun to determine the latitude of Denver. The hori- 
 zontal cross-hair was kept tangent to the lower limb of the 
 sun until it ceased to rise, then the vertical circle reading 
 
 Apparent alt. lower limb 
 Refraction for 60 20' oo' 
 True alt. lower limb 
 Add sun's semi-diameter 
 True alt. sun's center 
 Subtract declination 
 
 Co-lat. 
 
 Latitude 
 
 = 60 20' oo" 
 = 00 oo' 34" 
 = 60 19' 26" 
 = 00 15' 58" 
 
 = 60 35' 24" 
 = 10 21' 55" 
 = 50 13' 29" 
 = 39 46' 3i" 
 
 Latitude may be determined by reference to an accurate 
 map, from which the latitude of a neighboring point may 
 be found and then due allowance made for the distance 
 north or south to the station. 
 
 The table on page 275 gives the length, in feet, of one 
 minute of arc and the number of minutes of arc in a mile 
 for latitude and longitude, from o to 60 latitude by one- 
 degree intervals. 
 
 MEAN REFRACTION 
 
 (To be Subtracted from Observed Altitude, in Direct Solar Obser- 
 vations.) Barometer 30 Inches; Thermometer 50 F. 
 
 Altitude. 
 
 Refraction. 
 
 Altitude. 
 
 Refraction. 
 
 10 
 
 5' i9' 
 
 20 
 
 2 39' 
 
 11 
 
 4' 5i' 
 
 25 
 
 2 04' 
 
 12 
 
 4' 27' 
 
 30 
 
 I 41' 
 
 13 
 
 4' 07' 
 
 35 
 
 I 2 3 ' 
 
 14 
 
 3' 49' 
 
 40 
 
 I 9 ' 
 
 15 
 
 3' 34' 
 
 45 
 
 58' 
 
 16 
 17 
 
 3' 20' 
 3' 08' 
 
 
 
 49; 
 
 34 
 
 18 
 
 2' 57' 
 
 70 
 
 ax' 
 
 19 
 
 2' 48' 
 
 80 
 
 tot 
 
TRANSIT SURVEYING 
 
 275 
 
 TABLE SHOWING FEET PER MINUTE (ARC) AND MINUTES (ARC) 
 PER MILE OF LATITUDE AND LONGITUDE 
 
 
 Length of I 1 
 
 Vtin. in Feet. 
 
 No. of Min 
 
 . in i Mile. 
 
 Latitude. 
 
 Latitude. 
 
 Longitude. 
 
 Latitude. 
 
 Longitude. 
 
 
 
 6045 
 
 6087 
 
 0-8734 
 
 0.8674 
 
 i 
 
 6045 
 
 6085 
 
 0-8734 
 
 0.8677 
 
 2 
 
 6045 
 
 6083 
 
 0.8734 
 
 0.8679 
 
 3 
 
 6045 
 
 6078 
 
 0.8734 
 
 0.8687 
 
 4 
 
 6045 
 
 6071 
 
 0-8734 
 
 0.8697 
 
 5 
 
 6045 
 
 6063 
 
 0.8734 
 
 o . 8708 
 
 6 
 
 6045 
 
 6053 
 
 0.8734 
 
 0.8722 
 
 7 
 
 6046 
 
 6041 
 
 0-8733 
 
 0.8740 
 
 8 
 
 6046 
 
 6027 
 
 0.8733 
 
 0.8760 
 
 9 
 
 6046 
 
 6oi2 
 
 0-8733 
 
 0.8782 
 
 10 
 
 6047 
 
 5994 
 
 0.8731 
 
 0.8808 
 
 11 
 
 6047 
 
 5975 
 
 0.8731 
 
 0.8836 
 
 12 
 
 6048 
 
 5954 
 
 0.8730 
 
 0.8867 
 
 13 
 
 6048 
 
 593i 
 
 0.8730 
 
 o . 8902 
 
 14 
 
 6049 
 
 5907 
 
 0.8728 
 
 0.8938 
 
 15 
 
 6049 
 
 5880 
 
 0.8728 
 
 0.8979 
 
 16 
 
 6050 
 
 5852 
 
 0.8727 
 
 0.9022 
 
 17 
 
 6050 
 
 5822 
 
 0.8727 
 
 o . 9069 
 
 18 
 
 6051 
 
 5790 
 
 0.8725 
 
 0.9119 
 
 19 
 
 6052 
 
 5757 
 
 0.8724 
 
 0.9171 
 
 20 
 
 6052 
 
 572i 
 
 0.8724 
 
 0.9229 
 
 21 
 
 6053 
 
 5684 
 
 0.8722 
 
 0.9280 
 
 22 
 
 6054 
 
 5646 
 
 0.8721 
 
 0.9351 
 
 23 
 
 6054 
 
 5605 
 
 0.8721 
 
 0.9420 
 
 24 
 
 6055 
 
 5563 
 
 0.8720 
 
 0.9491 
 
 25 
 
 6056 
 
 55i9 
 
 0.8718 
 
 0.9566 
 
 26 
 
 6057 
 
 5474 
 
 0.8717 
 
 0.9645 
 
 27 
 
 6058 
 
 5427 
 
 0.8715 
 
 0.9729 . 
 
 28 
 
 6059 
 
 5378 
 
 0.8714 
 
 0.9817 
 
 29 
 
 6060 
 
 5327 
 
 0.8712 
 
 0.9911 
 
 30 
 
 6061 
 
 5275 
 
 0.8711 
 
 .0009 
 
 3i 
 
 6061 
 
 5222 
 
 0.8711 
 
 .OIII 
 
 32 
 
 6062 
 
 5i66 
 
 0.8709 
 
 .0220 
 
 33 
 
 6063 
 
 5109 
 
 0.8708 
 
 0334 
 
 34 
 
 6064 
 
 5051 
 
 0.8707 
 
 0453 
 
 35 
 
 6065 
 
 499i 
 
 0.8705 
 
 0579 
 
 36 
 
 6066 
 
 4930 
 
 0.8704 
 
 .0709 
 
 37 
 
 6067 
 
 4867 
 
 0.8702 
 
 .0848 
 
 38 
 
 6068 
 
 4802 
 
 0.8701 
 
 0995 
 
 39 
 
 6070 
 
 4736 
 
 o . 8698 
 
 .1148 
 
 40 
 
 6071 
 
 4669 
 
 0.8697 
 
 . 1308 
 
 4i 
 
 6072 
 
 4600 
 
 0.8695 
 
 .1478 
 
 42 
 
 6073 
 
 4530 
 
 0.8694 
 
 .1655 
 
 43 
 
 6074 
 
 4458 
 
 0.8692 
 
 .1846 
 
276 
 
 PRACTICAL SURVEYING 
 
 TABLE SHOWING FEET PER MINUTE (ARC) AND MINUTES (ARC) 
 PER MILE OF LATITUDE AND LONGITUDE (Continued) 
 
 Latitude. 
 
 Length of i Min. in Feet. 
 
 No. of Min. in i Mile. 
 
 Latitude. 
 
 Longitude. 
 
 Latitude. 
 
 Longitude. 
 
 44 
 
 6075 
 
 4385 
 
 0.8691 
 
 . 2040 
 
 45 
 
 6076 
 
 4311 
 
 0.8689 
 
 .2247 
 
 46 
 
 6077 
 
 4235 
 
 0.8688 
 
 .2467 
 
 47 
 
 6078 
 
 4158 
 
 0.8687 
 
 .2698 
 
 48 
 
 6079 
 
 4080 
 
 0.8685 
 
 .2941 
 
 49 
 
 6080 
 
 4OOI 
 
 o . 8684 
 
 .3196 
 
 50 
 
 6081 
 
 3920 
 
 0.8682 
 
 3469 
 
 
 6082 
 
 3838 
 
 0.8681 
 
 3757 
 
 52 
 
 6084 
 
 3755 
 
 0.8678 
 
 .4061 
 
 53 
 
 6085 
 
 3671 
 
 0.8677 
 
 .4383 
 
 54 
 
 6086 
 
 3586 
 
 0.8675 
 
 4723 
 
 55 
 
 6087 
 
 3499 
 
 0.8674 
 
 .5090 
 
 56 
 
 6088 
 
 3413 
 
 0.8672 
 
 5470 
 
 57 
 
 6089 
 
 3323 
 
 0.8671 
 
 .5888 
 
 58 
 
 6090 
 
 3233 
 
 0.8669 
 
 6331 
 
 
 6091 
 
 3142 
 
 0.8668 
 
 .6804 
 
 6 
 
 6092 
 
 3051 
 
 0.8667 
 
 7305 
 
 ERRORS IN AZIMUTH FOR i MINUTE ERROR IN DECLINATION OR 
 LATITUDE 
 
 
 For i Min. Error in Declination. 
 
 For i Min. Error in Latitude. 
 
 
 Lat. 30. 
 
 Lat. 40. 
 
 Lat. 50. 
 
 Lat. 30. 
 
 Lat. 40. 
 
 Lat. 50. 
 
 H M 
 
 Min. 
 
 Min. 
 
 Min. 
 
 Min. 
 
 Min. 
 
 Min. 
 
 o 30 
 
 I OO 
 
 8.85 
 4.46 
 
 IO.OO 
 
 5.05 
 
 12.90 
 6.01 
 
 8-77 
 4-33 
 
 9-9 2 
 4.87 
 
 II.80 
 5-80 
 
 2 00 
 
 2.31 
 
 2.61 
 
 3-H 
 
 2.00 
 
 2.26 
 
 2.70 
 
 3 oo 
 
 I.6 3 
 
 1.85 
 
 2. 2O 
 
 I-I5 
 
 1.30 
 
 1.56 
 
 4 oo 
 
 i-34 
 
 1-51 
 
 I. 80 
 
 0.67 
 
 0-75 
 
 0.90 
 
 5 oo 
 
 i .20 
 
 1.35 
 
 1.61 
 
 0.31 
 
 0-35 
 
 0-37 
 
 6 oo 
 
 i. IS 
 
 1.30 
 
 1.56 
 
 o.oo 
 
 0.00 
 
 0.00 
 
 By the use of the above table the amount of the azimuth 
 error, resulting from the use of erroneous declination or 
 latitude at the different hours of the day, may be de- 
 termined. 
 
TRANSIT SURVEYING 277 
 
 If the South Polar Distance used be too great, the ob- 
 served meridian falls to the right of the true south point in 
 the forenoon and to the left in the afternoon and vice versa 
 if too small. 
 
 If the latitude used be too great, the observed meridian 
 falls to the left of the true south point in the forenoon and to 
 the right in the afternoon and vice versa if too small. 
 
 DETERMINATION OF THE MERIDIAN 
 
 By Direct Solar Observation. The best time of day for a 
 solar observation is from 8 to 10 A.M., and from 2 to 4 P.M. 
 Observations greater than five or less than one hour from 
 noon should not be relied upon. Use colored glass over 
 eye end. 
 
 The transit should be accurately adjusted and carefully 
 leveled. Set the limb at zero when the telescope is di- 
 rected to some convenient mark, then with the lower 
 motion clamped, point the telescope to the sun and bring 
 the vertical and horizontal cross-hairs tangent to its image, 
 say in the lower left-hand quarter; read the vertical circle 
 and the limb. Note the time, quickly reverse the tele- 
 scope in altitude and azimuth and again bring the" cross- 
 hairs tangent to the image, but in the opposite quarter; 
 read the vertical circle and the limb.* The mean of the 
 vertical circle readings will give the apparent altitude of 
 the center of the sun. The mean of the readings on the 
 limb will give the angle between the sun and the selected 
 mark. 
 
 Formula A . Let Z = the angle between the sun and 
 the meridian, then its value may be obtained from the 
 equation 
 
 cos * 
 
 sin co-alt. X sin co-lat. 
 where 5 = co-dec. + co-alt. + co-lat. 
 
 * When the transit has a vertical arc instead of a full circle the in- 
 tersection of the cross-hairs may be directed to the center of the sun's 
 image. In fact this method is common even with full circles for the 
 error thus introduced, if any, is small. M C C. 
 
2 7 8 
 
 PRACTICAL SURVEYING 
 
 Example. The following notes are of an observation 
 made at Denver at 10 A.M., April 17, 1910. Latitude 
 39 46' 31" 
 
 Telescope. 
 
 Horizontal angle. 
 
 Vertical angle. 
 
 Position of 
 sun's image. 
 
 Direct 
 
 89 17' R 
 
 + 50 14' 
 
 ,J. 
 
 Reversed 
 
 89 07' R 
 
 + 51 07' 
 
 -P 
 
 
 
 
 
 Reduction of notes. 10 A.M., Denver time 
 17 or 5 P.M., Greenwich time. 
 Change in dec. since noon 
 
 10 
 
 Greenwich 
 
 Dec. at noon Greenwich 
 Dec. at 10 A.M. Denver 
 co-dec. 
 
 5 X 53 -02 
 
 = +00 04' 25". i 
 = N 10 1 5' 44" 4 
 = N 10 20' 09 
 = 79 39' 50 
 
 50 40' 30" 
 00 oo' 49" 
 
 50 39' 4i" 
 39 20' 19" 
 39 46' 3i" 
 50 13' 29" 
 
 = \ [(79 39' 5<>".5) + (39 20' 19") + (50 13' 29")] 
 
 8 4 36'49".2 
 
 Subtract refraction correction = 
 True-alt. 
 Co-alt. 
 Lat. 
 Co-lat. = 
 
 Substituting in equation for cos J Z 
 
 cos i z = t /(sin 84 36' 49".2) X (sin 
 V si 
 
 58" .7) 
 
 (sin 39 20' 19") X (sin 50 13' 29") 
 
 +9.9980780" 
 
 + 8.9359106 
 
 9.8020222 
 .-9.8856776J 
 65io'.t8", Z = 130 20' 36". 
 
 = 9-6231444 
 
 Interpretation of result. Fig. 208 shows the relative posi- 
 tions of the sun, the meridian, and the point of reference. 
 
TRANSIT SURVEYING 
 
 279 
 
 Before noon Z is to the left of the line to the sun and to 
 the right after noon. 
 
 The true bearing of the reference point is N 41 08' 36" E. 
 
 Formula B. Identical results will be obtained by the 
 use of the following equation and it may be preferred on 
 account of its containing the direct angles instead of the 
 co-angles, but it is necessary to pay careful attention to 
 the algebraic signs when it is used, 
 
 cos O = 
 
 cos lat. X cos alt. 
 
 tan lat. X tan alt. 
 
 The sign of the first term of the right-hand side of the 
 equation is negative when declination is S ; the second term 
 is positive where the latitude is S. If the algebraic sign of 
 
 the result is positive, Q is the angle between the sun and 
 the north point, but if it is negative, it is the angle between 
 the sun and the south point. 
 
 Example. The solution of the above example by this 
 equation is as follows: 
 
280 PRACTICAL SURVEYING 
 
 COS Q = 
 
 sin 10 20' 09" .5 
 
 (cos 39 46' 31") X (cos 50 39' 41 
 
 - (tan 39 46' 31") X (tan 50 39' 41 
 
 log sin 10 20' 09" .5 = + 9.2538706 
 log cos 39 46' 31" = + 9.8856775 
 log cos 50 39' 41 " = 9.8020222 
 
 9.5661709 = log 0.368274 
 log tan 39 46' 31" = + 9.9203518 
 log tan 50 39' 41 " = + 0.0863893 
 
 0.0067411 = log 1.015640 
 cos Q = - 0.647366 
 ' Q = 49 39' 24" 
 
 Interpretation of result. Q is the angle between the sun 
 and the south point, since the declination is N, the lati- 
 tude is N and the algebraic sign of the right-hand side of 
 the equation is negative. The true bearing of the reference 
 point is therefore N 41 08' 36" E, the same as obtained by 
 the first equation; see Fig. 208. 
 
 Time may also be determined from the foregoing data 
 by the equation (for a spherical triangle) 
 
 . sin Z sin co-alt. 
 
 sin 1 = : > 
 
 sin co-dec. 
 
 in which T is the hour angle of the sun at the time of ob- 
 servation. This is reduced to time by dividing by 15 and 
 corrected by the equation of time to obtain mean local 
 time and still further corrected by the difference between 
 mean local time and standard time if the latter is desired. 
 
 Example. Using the data of the example already given 
 to find the time. 
 
 . T _ (sin 130 20' 36") (sin 39 20' 19") 
 
 Sill JL / f\ ff * 
 
 (sin 79 39' 50.5") 
 
 Log sin 130 20' 36" = +9.8820570 
 Log sin 39 20' 19" = +9.8020222 
 
 9.6840792 
 
 Log sin 79 39' 50.5" = -9.9928945 
 Log sin T = 9.6911847 
 
 T = 29 24' 50" 
 
TRANSIT SURVEYING 281 
 
 .*. Time before noon = -^ - = I h. 57 m. 39 s. 
 
 Noon 12 h. oo m. oo s. 
 
 Deduction I h. 57 m. 39 s. 
 
 Sun time 10 h. 02 m. 21 s. A.M. 
 
 Equation of time to nearest sec. , 1 6 s. 
 
 Mean local time 10 h. 02 m. 05 s. A.M. 
 
 As mean local time and standard time are the same at 
 Denver no further correction is necessary. 
 
 According to the above result, the time used was 02 m. 
 05 s. slow, making an error of about 02" in the declination 
 which, from the table on page 276, makes an error in the 
 azimuth of about 05", being well within the allowable 
 limits of error for ordinary field work. 
 
 Note. The signs preceding the logarithms in the three 
 examples above are used to indicate whether the logarithm 
 is to be added or subtracted. 
 
 THE MERTOIOGRAPH 
 
 Mr. Louis Ross, a civil engineer in San Francisco, Cali- 
 fornia, placed on the market, in 1913, a device by means of 
 which the true north may be found at any time of the day, 
 without any computations or preparation of tables, the 
 only instrumental operation being an observation on the 
 sun with the transit. Between 10 A.M. and 2 P.M. the re- 
 sults are unreliable, for the best time is when the altitude 
 of the sun is from 15 to 25. Results obtained when the 
 altitude is between 25 and 35 are only fair, while an alti- 
 tude of more than 45 should not be used. The fore- 
 going remarks apply as well to direct observations on the 
 sun when the solution of a spherical triangle is made as 
 previously described. 
 
 The surveyor needs a table of the sun's ephemeris, but 
 instead of going to the labor of making a table for the 
 declination for each hour he merely takes from the table 
 the declination for Greenwich time. A diagram ac- 
 companying the meridiograph is then consulted and the 
 declination for the time and place is at once obtained. 
 Another diagram gives the refraction to be subtracted. 
 Declination is D. 
 
282 PRACTICAL SURVEYING 
 
 The latitude of the place may be taken from a good map. 
 If not available measure at noon the sun's altitude H; then, 
 if declination is north, Latitude = 90 H -f- D; if declina- 
 tion is south, Latitude = 90 H D. 
 
 "Apparent local time" can be found with the meridio- 
 graph, without any added work, by a single setting of the 
 proper scales nearly as fast as by referring to a watch and 
 to an accuracy, so the inventor claims, of one-half minute 
 or less. 
 
 From a circular issued by Mr. Ross, the following de- 
 scription is taken, which should be clear to the reader by 
 referring to the illustration. 
 
 The meridiograph is a protractor which solves graphi- 
 cally the problem of finding a true meridian by the sun. 
 It consists of two circular scaled disks, 7 in. max. diam., 
 with a reading arm. The names of all scales are on the 
 arm, exactly over the graduations; -- this obviates search- 
 ing for any desired scales. Nearly all graduations are 5 
 or 10 minutes spaces; angles may therefore be read to an 
 accuracy of I minute. 
 
 The scales, beginning with the outermost, are: 
 
 NUMBER o.io to i.oo, complete circle, for numbers A, B, (A + B). 
 BEARING 20 to 88 40' approx. if loops, for true bearing of sun. 
 
 b 10 to 60 approx. \ loop, for either alt. or lat. 
 
 a 10 to 60 approx. I loop, for either alt. or lat. 
 
 a 10 to 60 approx. I loop, for either alt. or lat. 
 
 b 10 to 60 approx. \ loop, for either alt. or lat. 
 DECL. i to 23 30' approx. \\ loops, for declination. 
 
 The transparent celluloid cover prevents the disks from 
 moving accidentally after they have been set and also pro- 
 tects the scales from wear and dirt. The inner disk is ro- 
 tated through the finger slots above and below. 
 
 To find true north, measure sun's altitude, take its 
 declination from the Ephemeris, and take latitude of 
 station from a map. Set these data on the meridiograph 
 by means of the reading arm, thus: 
 
 On scales a set alt. against lat., opposite index read number A. 
 
 On scales b set alt. against lat., opposite given declination read number B. 
 
 Opposite (A -+- B) read true bearing of sun. 
 
 On the two a scales the alt. is set against the lat. ; it is 
 
TRANSIT SURVEYING 283 
 
 immaterial which is set on which as the scales are identi- 
 cal, but set arm on outer disk first, then turn inner disk 
 until proper reading is under reading line. The same applies 
 to the two b scales. Note that the two a scales produce 
 number A, while the Decl. with the two b scales produce 
 number B. 
 
 Check solution of example given on face of meridio- 
 graph. 
 
 FIG. 209. The Meridiograph. 
 
 The solar compass was superseded by the solar attach- 
 ment to the transit but the method of direct observation on 
 the sun is preferred by the majority of surveyors. The 
 computations however are tedious and the amount of 
 figuring required makes many men slight the work some- 
 what. With the most careful work the true north is found 
 
284 PRACTICAL SURVEYING 
 
 only within the nearest I or 2 minutes of angle. The 
 meridiograph is claimed to give results within this degree 
 of accuracy. 
 
 BY OBSERVATIONS OF POLARIS 
 
 At Elongation. A few minutes before elongation, set 
 up the transit and center the plumb-bob over a tack driven 
 into a stake. Level up very carefully and keep the verti- 
 cal cross-hair on Polaris, using the tangent screw of the 
 vernier plate, until elongation is reached. This is easily 
 recognized since the azimuth then remains practically con- 
 stant for several minutes. 
 
 When elongation has been reached, depress the telescope 
 and carefully fix a stake on line, reverse the telescope on 
 its axis and rotate the instrument 180 on its vertical axis, 
 fix the vertical hair on Polaris, depress the telescope and 
 fix another stake on line, if the vertical hair does not bisect 
 the first one. These two observations must be made be- 
 fore Polaris has appreciably commenced its return motion 
 in azimuth. 
 
 When it is necessary to set two stakes,* a third stake 
 midway between them will be in a vertical plane through 
 the plumb line of the transit and Polaris at elongation. 
 By daylight, lay off from this plane the proper azimuth. 
 North is to the right, if Polaris was at western, and to the 
 left, if at eastern, elongation. 
 
 In making this and the following observation it is neces- 
 sary to illuminate the stakes and the cross-hairs. The 
 latter may be accomplished by a suitable lamp held at one 
 side of the transit, so that sufficient light is reflected into 
 the telescope. 
 
 At Culmination. On account of the great difficulties at- 
 tending this method, it should be used only when the 
 method of elongation is impracticable. 
 
 This method is based on the fact that Polaris is very 
 nearly on the meridian when it is in the same vertical plane 
 with the star Delta, in the constellation Cassiopeia, or 
 
 * When the transit is in correct adjustment it will be necessary to 
 set but one stake, whose position will correspond with that of the third 
 one, as given above. 
 
TRANSIT SURVEYING 
 
 285 
 
 mOt ' ^80-LOQ inoav 
 
 JLV3i/f) 
 
 JO 
 
 POLARIS. 
 
 X] 
 
 Zeta of the Great Dipper, the star at the bend in the handle. 
 It consists in watching either Delta or Zeta until it comes 
 into the same vertical plane with Polaris and then waiting 
 a known interval of time,* until Polaris is on the meridian. 
 The vertical cross-hair must be placed on Polaris precisely 
 at the end of this interval as the motion, in azimuth, is 
 most rapid at culmination. The telescope is now in the 
 meridian, which may be marked in any suitable manner. 
 
 Limitations. On account of the 
 haziness of the atmosphere near 
 the horizon, the lower culminations 
 of Zeta and Delta cannot be used 
 below about 38 north latitude; 
 neither can their upper culmina- 
 tions be used north of about 25 
 and 30 respectively, on account of 
 their being too near the zenith. 
 
 Selecting the Star. The dia- 
 gram, Fig. 209, shows Delta Cassi- 
 opeia on the meridian below the 
 pole at midnight about April 10. 
 It may therefore be used in the 
 above method for two to three 
 months before and after that date. 
 Likewise Zeta, of the Great Dipper, 
 may be used for two to three 
 months before and after October 10. 
 
 Time of Elongation and Culmina- 
 tion. Fig. 210 shows Polaris near 
 eastern elongation at midnight about July 10, at western 
 at midnight about January 10, at upper culmination at 
 midnight about October 10, and at lower at midnight about 
 April 10. The approximate time of elongation or culmi- 
 nation for other dates may be determined by noting the 
 position of the line adjoining Zeta of the Great Dipper 
 and Delta Cassiopeia. When this line is vertical, Polaris 
 is near culmination and when horizontal it is near elonga- 
 tion. Polaris is on the opposite side of the Pole from Zeta 
 of the Great Dipper, thus furnishing a convenient means 
 
 * The waiting time for 1912 is 6 min. 48 sec. for Zeta of the Great 
 Dipper and 7 min. 21 sec. for Delta Cassiopeia. 
 
 Directior 
 
 notion 
 
 of apparent 
 
 
 Vr,' 
 
 tr 
 
 IOPEIA 
 
 C* S S 
 
 MIDNIGHT ABOUT APRIL JO TH 
 FIG. 210. 
 
286 PRACTICAL SURVEYING 
 
 of distinguishing eastern from western elongation and upper 
 from lower culmination. When Zeta is west, Polaris is east 
 of the pole, and when Zeta is above, Polaris is below the 
 pole. 
 
 MEAN POLAR DISTANCE 
 
 The azimuth of Polaris at elongation for any year is 
 found from the following table, which gives the mean 
 polar distance. The maximum error in azimuth for any 
 month cannot exceed one minute from the mean polar 
 distance for the year, this being within the limits of per- 
 missible error. 
 
 The sine of the true azimuth in any latitude is found by 
 the formula. 
 
 ~ . _ Sine of Polar Distance 
 Cosine Latitude 
 
 1915 
 1916 
 1917, 
 1918, 
 1919, 
 1920, 
 
 09 53-51 
 
 08' 34.97" 
 
 08' 16.45" 
 
 07' 57-94" 
 
 07' 39-45" 
 
 07 20.98 
 
 In Engineering News, April 22, 1915, appeared the fol- 
 lowing article : 
 
 AZIMUTH OBSERVATIONS ON POLARIS BY DAYLIGHT 
 
 By ROBERT V. R. REYNOLDS* 
 
 Polaris may always be found in clear weather as soon 
 as the sun has set, and very frequently for five or ten min- 
 utes before sunset or after sunrise. It is stated on good 
 authority that under very favorable conditions an observa- 
 tion has been successful as late as 10 A.M. In the northern 
 parts of the United States the cross-wires may remain 
 visible for a long time after sunset. 
 
 For the novice it is often difficult at the first few attempts 
 to see Polaris while the sky is still bright, but having once 
 
 * Forest Examiner, U. S. Forest Service, Washington, D. C. 
 
TRANSIT SURVEYING 
 
 287 
 
 found the star, which appears as a small white dot in the 
 field, he will never thereafter feel in doubt. Tabulated 
 
 TABLE TO FIND POLARIS BY DAYLIGHT 
 
 Hour Angle of Polaris 
 Approximate* (Use Suit- 
 able Interpolations) 
 
 Azimuth Setting 
 N E or N W 
 Depending upon Position 
 of Polaris E or W of 
 Meridian 
 
 Altitude Setting 
 Latitude Plus or Minus 
 the Tabulated 
 Quantities 
 
 o . o or 1 2 . o 
 
 oo' 
 
 08' 
 
 0.5 or 11.5 
 
 12' 
 
 -1 07' i 
 
 i . o or 1 1 . o 
 
 24' 
 
 9 06' * 
 
 1.5 or 10.5 
 
 36' 
 
 * *' -5. 
 
 2.0 or 10.0 
 
 47' 
 
 J3 OO 13^ 
 
 2.5 or 9.5 
 
 57' 
 
 o 55' Jjf 
 
 3 . o or 9.0 
 
 1 06' 
 
 j 5o' .3 "8 
 
 3-5 or 8.5 
 
 J 5' 
 
 / ,- u 
 
 4 . o or 8.0 
 
 22' 
 
 ~ C 35' ^ OT 
 
 4-5 or 7.5 
 
 27' 
 
 "o"2 2 7' |-ii 
 
 5 . o or 7.0 
 
 3' 
 
 *" i I ^' ** rt 
 
 5-5 or 6.5 
 6.0 hours 
 
 33' 
 
 35' 
 
 3" S' ^ 
 
 * The hour angle used as the argument in this table needs only to be approximate. If 
 it is correct within 5 min. of time, sufficiently accurate settings will be indicated provided 
 interpolation is made. Hence, there is no need of correcting for longitude until the sur- 
 veyor has made the observation and is preparing to enter the table of Azimuths of Polaris. 
 
 The table is computed for a mean latitude of 42, but purposely modified slightly to 
 make it more useful along the 49th parallel, where much of the Forest Service work is 
 being done. It is accurate enough to bring Polaris within the field of an ordinary transit 
 between latitudes of 10 and 58 N. It is not for use to determine the true azimuth after 
 the observations. 
 
 settings (such as accompany this article) sufficiently accu- 
 rate to bring the star into the field are required, and there 
 remain several factors which must be given consideration 
 before success can be assured: 
 
 1. A slight haziness, which may hardly be obvious to the 
 eye, is sufficient to conceal the star until darkness comes 
 on. 
 
 2. The telescope must be in exact focus for celestial ob- 
 jects. This may be accomplished either by focusing at 
 night upon the moon and making a slight scratch upon the 
 objective slide to show the point to which it should be ex- 
 tended, or the surveyor may focus at the time of observa- 
 
288 PRACTICAL SURVEYING 
 
 tion upon a well-defined object 3 or 4 miles distant, which 
 focus will usually be found sufficiently close. Accurate 
 focusing is one of the most important factors in finding 
 the star. 
 
 3. For the purpose of cutting off objectionable light, the 
 sunshade should always be attached. Certainty of finding 
 the star is assured by throwing a coat or other dark cloth 
 over the head when searching through the telescope, as a 
 photographer uses a focusing cloth. 
 
 4. An approximate meridian must be had, from which 
 the azimuth settings are turned off. Commonly, the sur- 
 veyor will already have such a meridian from his backsight. 
 Otherwise, a meridian determination from a solar attach- 
 ment in reasonable adjustment will suffice. Sometimes, 
 when the magnetic declination is closely known, it w T ill even 
 be possible to turn upon the star from the needle. A refer- 
 ence meridian which is true within 5' or 10' will be precise 
 enough to locate the star when the table of approximate 
 settings is used. 
 
 Polaris having been found, the angle from the reference 
 mark to the star should be measured twice, the second 
 time with the telescope inverted. The mean time of obser- 
 vation and the mean angle are then used to find the azimuth 
 of the mark by the simplified hour-angle method. There is 
 practically no chance that any other star will be seen and 
 mistaken for Polaris. (Finis.) 
 
 BY ANY STAR AT EQUAL ALTITUDES 
 
 In high latitudes neither the sun nor Polaris give re- 
 liable results. The sun is low and the refraction is un- 
 certain, while, on account of the height of Polaris, the 
 observation is difficult to make and instrumental errors are 
 magnified. In southern latitudes Polaris is not visible at 
 all. Although this method may be used in any latitude, it 
 is particularly applicable under the above conditions. 
 
 The method consists in observing a star, when at equal 
 altitudes, east and west of the meridian. The meridian 
 will then be halfway between these two positions of the 
 star. The star selected should be 30 or more from the 
 zenith when on the meridian and, at least, the same dis- 
 
TRANSIT SURVEYING 289 
 
 tance from the pole. The observations should be made 
 when the star is three to four hours from the meridian. 
 
 Making the observation. To make the first observation 
 level the transit carefully, direct the telescope to the star, 
 clamp all motions and fix the intersection of the cross-hairs 
 on the star by the slow motion screws, read the star's 
 altitude, unclamp the telescope axis, depress the telescope 
 and fix a point on line. 
 
 To make the second observation re-level the transit care- 
 fully, set the telescope at the altitude determined by the 
 first observation, clamp the limb and lower motion when 
 the star comes near to the horizontal hair, keeping the ver- 
 tical hair on the star by means of the slow-motion screw of 
 the vernier plate until it reaches the intersection of the 
 cross-hairs, then unclamp the telescope axis, depress the 
 telescope and fix a point on line. A third point set half- 
 way between these two will mark the meridian through the 
 transit. 
 
 If the transit has a vertical circle, the error due to the 
 adjustment of the height of standards may be eliminated 
 by making the first observation with the telescope direct 
 and the second with it reversed in altitude and azimuth. 
 In this method artificial illumination must be used for the 
 cross-hairs and for setting the points on line. 
 
 GREEK LETTERS 
 
 of Alpha e Epsilon 
 
 ft Beta 17 Eta 
 
 5 A Delta f Zeta 
 
 7 Gamma *c Kappa 
 
CHAPTER VII 
 
 SURVEYING LAW AND PRACTICE 
 
 The business of the surveyor is firmly bound up with that 
 of the lawyer, but much of the litigation over land lines 
 would be eliminated if the lawyer could be prevented from 
 interfering with the surveyor in the doing of his work. 
 When it comes to the courts that is another matter, for 
 the decisions of judges based upon precedent, and latterly 
 upon changes in customs and advances in civilization, are 
 generally right. Even if occasionally wrong a decision 
 must stand until a better informed judge has a similar case 
 presented to him. The decisions of courts are based upon 
 a very few common sense principles of law, but it should 
 not be necessary to have cases go into court in order to 
 settle lines and the location of monuments. 
 
 The surveyor is presumed to have enough skill to measure 
 angles and lines and make a record of same so competent 
 surveyors can re-trace the work and re-locate boundaries. 
 The average attorney-at-law does not recognize "THE 
 ERROR," a thing that looms up large before every surveyor 
 and which the courts must recognize. 
 
 Mr. J. Francis Le Baron, Member of the American So- 
 ciety of Civil Engineers, on page 425, in the Transactions 
 of the American Society of Civil Engineers, Vol. LXXV 
 (1912), presents the following as a sample of instructions 
 for re-surveys, given him by lawyers, instructions that are 
 positively insulting to men of extended experience in such 
 work. They resemble similar instructions which lawyers 
 at times attempted to give the author in the years when 
 his work was closely associated with land surveying. 
 
 "I want you to start from the beginning corner, as given 
 in the deed, run the exact course and distance and set a 
 stake there. It will not take you long. I suppose you 
 make an allowance for the variation of the needle. The 
 
 290 
 
SURVEYING LAW AND PRACTICE 2 91 
 
 needle, you know, does not point exactly north, and you 
 must make an allowance. This deed reads 'due north' 
 so many chains. Now does that mean true north or the 
 way the needle points ? I suppose when you chain down 
 hill you make an allowance, don't you, because I think 
 the distance wouldn't come right if you didn't? I don't 
 know how much you allow, but I suppose you have some 
 custom about it. You see this distance reads so many 
 chains and links, so you must measure it with a chain and 
 links, and not any other kind of measure, or I am inclined 
 to think that the Court would reject your survey." 
 
 The surveyor is also employed by real estate agents to 
 make surveys for clients and fully ninety-five per cent of 
 these real estate agents demand a commission of no small 
 amount. This is even customary with some attorneys and 
 managers of estates. The surveyor who is compelled to 
 obtain work under such conditions cannot afford to do the 
 work properly and furthermore few competent surveyors 
 will accept work on such terms. The foregoing remarks 
 do not indict all lawyers, real estate agents and managers 
 of estates, for many high-minded men occupy positions of 
 trust and jealously safeguard the interests of their clients 
 to a degree that they would not in work for themselves. 
 To work for an intelligent and honest man occupying such 
 a position is an experience so filled with satisfaction that 
 the surveyor often feels a trifle shamefaced in presenting 
 his bill. If he could afford it he would do the work for 
 the mere satisfaction it affords, yet work done for such 
 men is generally the most highly paid. They are high- 
 priced themselves and prefer to have work done by men who 
 set a proper value upon their services. There is no heart- 
 breaking competition in such work, yet it is seldom offered 
 to a man until after he has served an apprenticeship of 
 many years in competitive work and has established a 
 reputation for accuracy and honesty. 
 
 To properly re-survey a piece of ground requires consider- 
 able preliminary work in obtaining starting points, for in a 
 community every lot is tied to 'adjoining lots. Some- 
 times to re-locate one line, means the running of many lines, 
 some at a considerable distance from the one wanted. 
 Ma\iy men do not want to pay for such work. Once the 
 
2Q2 PRACTICAL SURVEYING 
 
 author had to work for ten days before he could definitely 
 locate the line he had been employed to survey, this work 
 consuming only three hours' time. When his bill was pre- 
 sented the lawyer offered him pay for the three hours and 
 told him to go to the other owners, whose land he had first 
 run out, for the remainder of the bill. It took a lawsuit to 
 recover the amount due and some evidence obtained in 
 the suit resulted in the disbarment of the attorney. This 
 leads naturally to the statement that not every attorney 
 is a lawyer and with real lawyers there is seldom any diffi- 
 culty. The average attorney, therefore, is the man who tries 
 to hold the surveyor down to a definite procedure and who 
 is ignorant of THE ERROR. It is difficult however to dis- 
 tinguish between the real lawyer and the pseudo-lawyer, 
 who is really only a licensed attorney-at-law, until some 
 experience is had with the man. 
 
 Every student who contemplates following surveying as 
 a vocation should procure a copy of Paper No. 1242, Trans- 
 actions of the American Society of Civil Engineers, entitled 
 "Retracement Surveys Court Decisions and Field Pro- 
 cedure," by N. N. Sweitzer, M. Am. Soc. C. E., together 
 with the complete discussion by a number of experienced 
 men. He should also read "Boundaries and Landmarks," 
 by A. C. Mulford ($1.00) and "Descriptions of Lands," by 
 R. W. Cautley ($1.00) in order to obtain the point of view 
 of competent surveyors and hasten the acquirement of 
 knowledge he can obtain otherwise only by years of expe- 
 rience. The surveyor must not forget that his work is not 
 to merely re- trace old lines but to FIND THE LAND. To re- 
 run old field notes exactly as recorded is a simple matter, 
 but to determine the location of the original monuments 
 and lines calls for skill gained only by practical experience. 
 The surveyor's work is as much legal as technical and many 
 times the legal side is the most important. The importance 
 of the legal nature of a surveyor's work is so great that a 
 number of years ago F. Hodgman, then secretary of the 
 Michigan Engineering Society, wrote a book giving the gist 
 of a number of court decisions as a guide for surveyors. 
 This work should be in the possession of every land sur- 
 veyor. It is entitled U A Manual of Land Surveying" 
 ($2.50). John Cassan Wait also wrote a book entitled 
 
SURVEYING LAW AND PRACTICE 293 
 
 "Law of Operations Preliminary to Construction in En- 
 gineering and Architecture," which covers very completely 
 the subject of the law of boundaries. 
 
 A number of years ago Chief Justice Cooley, of the Mich- 
 igan Supreme Court, delivered an address at a meeting of 
 the Michigan Engineering Society on "The Judicial Func- 
 tions of Surveyors," and no modern textbook on surveying 
 is presumed to be complete unless this address is made a 
 part of the contents. 
 
 THE JUDICIAL FUNCTIONS OF SURVEYORS 
 
 By CHIEF JUSTICE COOLEY 
 
 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 results 
 which theoretically he ought to reach. Observation war- 
 rants us in saying that this remark may frequently be made 
 of surveyors. 
 
 In the State of Michigan all our lands are supposed to 
 have been surveyed once or more, and permanent monu- 
 ments fixed to determine the boundaries of those who should 
 become proprietors. The United States, as original 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 monu- 
 ments 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 the recorded witnesses to these 
 were many times wanting in permanency, it is often the 
 case that when a monument was not correctly placed, it is 
 impossible to determine by the record, by the aid of any- 
 thing 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 
 
294 PRACTICAL SURVEYING 
 
 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 purposes. 
 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 sur- 
 veys 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 were 
 made under the authority of the United States. Of the 
 lands south of Lansing, it is now forty years since the major 
 part were sold, and the work of improvement began. A 
 generation has passed away since they were converted into 
 cultivated farms, and few if any of the original corner and 
 quarter stakes now remain. 
 
 The corner and quarter stakes were often nothing but 
 green sticks driven into the ground. Stones might be put 
 around or over these if they were handy, but often they 
 were not, and the witness trees must be relied upon after 
 the stake was gone. Too often the first settlers were care- 
 less in fixing their lines with accuracy while monuments re- 
 mained, and an irregular brush fence, or something equally 
 untrustworthy, may have been relied upon to keep in mind 
 where the blazed line once was. A fire running through 
 this might sweep it away, and if nothing was substituted in 
 its place, the adjoining proprietors might in a few years be 
 found disputing over their lines, and perhaps rushing into 
 litigation, as soon as they had occasion to cultivate the 
 land along the boundary. 
 
 If now the disputing parties call in a surveyor, it is not 
 likely that any one summoned would doubt or question that 
 his duty was to find, if possible, the place of the original 
 stakes which determined the boundary line between 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 
 
SURVEYING LAW AND PRACTICE 295 
 
 parties buy, or are supposed to buy, in reference to these 
 monuments, and are entitled to what is within their lines, 
 and no more, be it more or less. While the witness trees 
 remain, there can generally be no difficulty in determining 
 the locality of the stakes. 
 
 When the witness trees are gone, so that there is no 
 longer record evidence of the monuments, it is remarkable 
 how many there are who mistake altogether the duty that 
 now devolves upon the surveyor. It is by no means un- 
 common 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 mis- 
 take. 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 commonly 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 monu- 
 
296 PRACTICAL SURVEYING 
 
 ment 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 line east 
 and west and north and south of it," it nevertheless 
 determined the extent of his possessions, and he gained 
 or lost, according as the mistake did or did not favor 
 him. 
 
 It will probably be admitted that no man loses title to 
 his land or any part thereof merely because the evidences 
 become lost or uncertain. It may become more difficult 
 for him to establish it as against an adverse claimant, but 
 theoretically the right remains; and it remains as a poten- 
 tial fact so long as he can present better evidence than any 
 other person. And it may often happen that notwithstand- 
 ing 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 absolute certainty where the 
 original boundary was between two government sub- 
 divisions. 
 
 There are two senses in which the word extinct may be 
 used in this connection: One, the sense of physical disap- 
 pearance ; the other, the sense of loss of all reliable evidence. 
 If the statute speaks of extinct corners in the former sense, 
 it is plain that a serious mistake was made in supposing that 
 surveyors could be clothed with authority to establish new 
 corners by an arbitrary rule in such cases. As well might 
 the statute declare that if a man loses his deed, he shall lose 
 his land altogether. 
 
 But if by extinct corner is meant one in respect to the 
 actual location of which all reliable evidence is lost, then 
 the following remarks are pertinent: 
 
 1 . There would undoubtedly be a presumption in such a 
 case that the corner was correctly fixed by the government 
 surveyor where the field notes indicated it to be. 
 
 2. But this is only a presumption, and may be overcome 
 by any satisfactory evidence showing that in fact it was 
 placed elsewhere. 
 
 3. No statute can confer upon a county surveyor the 
 power to "establish" corners, and thereby bind the parties 
 concerned. Nor is this a question merely of conflict 
 between State and federal law. It is a question of property 
 
SURVEYING LAW AND PRACTICE 297 
 
 right. The original surveys must govern, and the laws 
 under which they were made must govern, because the land 
 was bought in reference to them; and any legislation, 
 whether State or federal, that should have the effect to 
 change these, would be inoperative, because disturbing 
 vested rights. 
 
 4. In any case of disputed lines, unless the parties con- 
 cerned settle the controversy by agreement, the determina- 
 tion 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 the 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 continued, 
 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 were, and whether a claim of 
 title has always accompanied the possession, and give all 
 the facts due force as evidence. Unfortunately, it is known 
 that surveyors sometimes, in supposed obedience to the 
 State statute, disregard all evidences of occupation and 
 claim of title, and plunge whole neighborhoods into quarrels 
 and 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 trust- 
 worthy; 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 himself to annoyance and perhaps discredit, 
 since in a legal controversy the law as well as common sense 
 must declare that a supposed boundary line long acquiesced 
 in is better evidence of where the real line should be than 
 any survey made after the original monuments have dis- 
 
298 PRACTICAL SURVEYING 
 
 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 supposed to be. The 
 streets are likely soon to be marked off by fences, and the 
 lots in a block will be measured off from these without 
 looking farther. Now it may perhaps be known in a par- 
 ticular case that a certain monument still remaining was a 
 starting point in the original survey of the town plat; or a 
 surveyor settling in the town may take some central point 
 as the point of departure in his surveys, and assuming the 
 original plat to be accurate, he will then undertake to find 
 all streets and lots by course and distance according to 
 the plat, measuring and estimating from his point of de- 
 parture. This procedure might unsettle every line and 
 every monument existing by acquiescence in the town; it 
 would be very likely to change the lines of streets, and raise 
 controversies everywhere. Yet this is what is sometimes 
 done, the surveyor himself being the first person to raise 
 the disturbing questions. 
 
 Suppose, for example, a particular village street has been 
 located by acquiescence and used for many years and the 
 proprietors in a certain block have laid off their lots in refer- 
 ence to this practical location. Two lot owners quarrel, 
 and one of them calls in a surveyor, that he 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 generally will allow the new 
 survey to unsettle their possessions, but there is always a 
 
SURVEYING LAW AND PRACTICE 299 
 
 probability of finding some 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 should be re- 
 quired to conclude private owners, where it appears that 
 they have accepted a particular line as their boundary, 
 and all concerned have cultivated and claimed up to it. 
 Public policy requires that such lines be not lightly dis- 
 turbed, or disturbed at all, after the lapse of any consider- 
 able 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 acquiescence has fixed the 
 rights of parties as if it were another. But he would do 
 mischief if he were to attempt to "establish" monuments 
 which he knew would tend to disturb settled rights; the 
 farthest he has a right to go, as an officer of the law, is to 
 express his opinion where the monument should be, at the 
 same time that he imparts the information to those who 
 employ him, and who might otherwise be misled, that the 
 same authority that makes him an officer and entrusts him 
 to make surveys, also allows parties to settle their own 
 boundary lines, and considers acquiescence in a particular 
 line or monument, for any considerable period, as strong if 
 not conclusive evidence of such settlement. The peace of 
 the community 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 which the 
 street has 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 
 
300 PRACTICAL SURVEYING 
 
 by the circumstances, i. He is to search for the original 
 monuments, or for the places where they were originally 
 located, and allow these to control. (If he finds them, un- 
 less 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 expression, to in- 
 form us with more or less accuracy where the monuments 
 may be found. 2. If the original monuments are no longer 
 discoverable, the question of location becomes one of 
 evidence merely. It is merely idle for any state statute 
 to direct a surveyor to locate or "establish" a corner, as 
 the place of the original monument, according to some 
 inflexible rule. The surveyor, on the other hand, must in- 
 quire into all the facts, giving due prominence to the acts 
 of the parties concerned, and always keeping in mind, 
 first, that neither his opinion nor his survey can be conclusive 
 upon parties concerned; and, second, that courts and juries 
 may be required to follow after the surveyor over the same 
 ground, and that it is exceedingly desirable that he govern 
 his actions 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 actu- 
 ally 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 in 
 their application ; and as other interesting topics are some- 
 
SURVEYING LAW AND PRACTICE 301 
 
 what 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 meander lines are bound- 
 ary 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 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 important. In this state the rule 
 that the center line is the boundary line, is applied to all 
 great rivers, including the Detroit, varied somewhat by 
 the circumstance of there being a distinct channel for navi- 
 gation, in some cases, with the stream in the main channel, 
 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 lines 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 because of 
 that fact. In many cases it now happens that the meander 
 line is left some distance from the shore by the gradual 
 change of course of the stream, or diminution of the flow of 
 water. Now the dividing line between two government 
 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 the line crossing it at right 
 
302 PRACTICAL SURVEYING 
 
 angles to its side lines. The effect might be that, of two 
 government subdivisions of equal size and cost, one would 
 be of very great value as water-front property, and the other 
 comparatively valueless. A rule which would produce this 
 result would not be just, and it has not been recognized in 
 the law. 
 
 Nevertheless it is not easy to determine what ought to be 
 the correct rule for every case. If the river has a straight 
 course, or one nearly so, every man's equities will be pre- 
 served by this rule : Extend the line of division 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 notes indicated, except 
 as changes in the water may have affected it, and the only 
 inconvenience will be that the division line between differ- 
 ent subdivisions is likely to be more or less deflected where 
 it strikes the meander line. 
 
 This is the legal rule, and it is not limited to government 
 surveys, but applies as well to water lots which appear as 
 such on town plats. (Bay City Gas Light Co., v. The 
 Industrial Works, 28 Mich. Reports, 182.) It often hap- 
 pens, therefore, that the lines of city lots bounded on navi- 
 gable 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 the foregoing rule is incapable of strict 
 application, it is sometimes very difficult to determine what 
 should be done; and in many cases the surveyor may be 
 under the necessity of working out a rule for himself. Of 
 course his action cannot be conclusive, but if he adopts one 
 that follows, as nearly as the circumstances will admit, the 
 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 
 
SURVEYING LAW AND PRACTICE 303 
 
 have a line on 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 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 purpose 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, any- 
 thing which their purchase would include under it cannot 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 expan- 
 sions near their mouths of the rivers passing through them 
 such as the Muskegon, Pere Marquette and Manistee 
 the same rule of bed ownership has been judicially applied 
 as applied to the rivers themselves; and the division lines 
 are extended under the water in the same way. (Rice v. 
 Ruddiman, 10 Mich., 125.) If such a lake were circular, 
 the lines would converge to the center; if oblong or irregu- 
 lar, there might be a line in the middle on which they 
 would terminate, whose course would bear some relation 
 to that of the shore. But it can seldom be important to 
 follow the division line very far under the water, since all 
 private rights are subject to the public rights of naviga- 
 tion and other use, and any private use of the lands incon- 
 sistent with these would be a nuisance, and punishable as 
 
304 PRACTICAL SURVEYING 
 
 such. It is sometimes important, however, 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. (Lorman v. Benson, 8 
 Mich., 18; Peoples Ice Co. v. 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 surveyed 
 out and sold by the United States, can this be done now? 
 
 Others will be suggested by the answers given to these. 
 
 It seems obvious that the rules of private ownership 
 which are applied to rivers cannot be applied to the great 
 lakes. Perhaps it should be held that the boundary is at 
 low-water mark, but improvements beyond this would only 
 become 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 will not interfere with the rights of shore 
 owners. But so far as these public rights can be the sub- 
 ject of ownership, they belong to the state, not to the 
 United States; and so, it is believed, does the bed of the 
 lake also. (Pollord v. Hagan, 3 Howard's U. S. Reports.) 
 
SURVEYING LAW AND PRACTICE 30$ 
 
 But such rights are not generally considered proper sub- 
 jects of sale, but like the right to make use of the public 
 highways, they are held by the state in trust for all the 
 people. 
 
 What is said of the large lakes may perhaps be said also 
 of many of the interior lakes of the state; such, for example, 
 as Houghton, Higgins, Cheboygan, Hurt's, Mullet, Whit- 
 more, and many others. But there are many little lakes 
 or ponds which are gradually disappearing, and the shore 
 proprietorship advances pari passu as the waters recede. 
 If these are of any considerable size say, even a mile 
 across there may be questions of conflicting rights which 
 no adjudication hitherto made could settle. Let any sur- 
 veyor, for example, take the case of a pond of irregular 
 form, occupying a mile square or more of territory, and 
 undertake to determine the rights of the shore proprietors 
 to its bed when it shall totally disappear, and he will find 
 he is in the midst of problems such as probably he has never 
 grappled with, or reflected upon before. But the general 
 rules for the extension of shore lines, which have already 
 been laid down, should govern such cases, or at least should 
 serve as guides in their settlement. 
 
 Where a pond is so small as to be included within the 
 lines of a private purchase from the government, it is not 
 believed the public have any rights in it whatever. Where 
 it is not so included, it is believed they have rights of fish- 
 ery, rights to take ice and water, and rights of navigation 
 for business or pleasure. This is the common belief, and 
 probably the just one. Shore rights must not be so exer- 
 cised as to disturb these, and the states may pass all proper 
 laws for their protection. It would be easy with suit- 
 able 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 
 
306 PRACTICAL SURVEYING 
 
 the state could not drain, even for this purpose, against 
 the will of the shore owners, unless their rights were appro- 
 priated and paid for. 
 
 Upon many questions that might arise between the state 
 as owner of the bed of the 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 judgment. 
 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 enlightenment, but I 
 trust will not be wholly without value. (Finis.) 
 
 Commenting upon the famous address of Judge Cooley 
 it is right to inform the surveyor that in his dealings with 
 owners and their attorneys he will find there is a thing 
 known as "common law" and another thing known as 
 "statute law." The common law governs in the majority 
 of states until a statute is passed to settle if possible un- 
 certainties which lead to lawsuits. These statutes are 
 passed by state legislatures, in which the majority of 
 members usually are attorneys (by courtesy, lawyers) and 
 whenever the legislature meets, one of the first committees 
 appointed is a committee to pass upon the constitutionality 
 of all proposed legislation. The statute referred to, which 
 purported to enable surveyors to "re-establish" corners, 
 was the product of a legislature in which the usual per- 
 centage of membership was lawyers and having the cus- 
 tomary committee to scan all proposed laws. Such statutes 
 have been passed in a number of states but when inter- 
 ested owners object and take a case to the Supreme Court 
 the statute is held to violate the common law 'doctrine of 
 rights of property and to be unconstitutional. The reasons 
 are well stated by Judge Cooley. Yet many surveyors 
 have been compelled by attorneys to follow the letter of 
 the law, even while these same attorneys knew how the 
 case would go if carried to the proper court. Many sur- 
 
SURVEYING LAW AND PRACTICE 307 
 
 veyors made mistakes through believing every law on the 
 statute books to be good law, but it was not customary 
 fifty years ago for legal points to be touched upon in text- 
 books on surveying. 
 
 In his reference to lot lines, Judge Cooley had particular 
 conditions in mind. On the points he brings up there is 
 considerable room for differences of opinion. Rights 
 never run against the public. That is, when a man en- 
 croaches upon a public highway with a fence or building the 
 public still has every right it was given so possession in 
 such a manner does not give title. However, if the man 
 was permitted to place any part of his building on public 
 property he gains an easement thereby for that structure. 
 He cannot be compelled to remove it, without being com- 
 pensated for whatever damages he may thereby suffer, 
 but the public may declare its right in the ground occupied 
 and forbid him making repairs so the structure may remain 
 indefinitely. The public may also, when the structure is 
 razed, compel the new structure to be placed where it prop- 
 erly belongs, thus terminating the easement. This again 
 is governed by the conditions under which the public rights 
 in the highway were acquired. The street may have been 
 given outright to the public. It may have been dedicated 
 merely for highway purposes and nothing said about rever- 
 sion in case of abandonment. Conditions of reversion may 
 have been stated at the time of dedication. The ownership 
 of adjacent lots may be limited to the exact boundaries of 
 the lots and the edge of the highway; it may extend to the 
 center line of the highway. 
 
 Assume a town site to have been carefully surveyed and 
 well monumented. Stakes in some way became lost and 
 the people who took possession were too penurious to pay 
 five dollars or ten dollars for each lot survey. In course of 
 time every fence line and many building lines are not in 
 the position indicated by the records. Some street im- 
 provements are started and the encroachments discovered. 
 The encroachment may be a fence in such a state of dilap- 
 idation that the necessary street grading causes it to fall, 
 whereupon the city seizes the property thus abandoned. 
 The lot owner finds he has lost a strip of land and pro- 
 ceeds to demand it from his neighbor, the trouble running 
 
308 PRACTICAL SURVEYING 
 
 through the block, until at the far end a surplus is dis- 
 covered in the last lot. This owner proposes to fight for 
 the continued possession of the surplus. The monuments 
 are uncovered and a careful re-survey made with the 
 result that it is found every person in the town may have 
 all the land his, or her, deeds called for and the city may 
 also have all the width in each street. Nobody loses but 
 everyone is put to considerable expense and annoyance. 
 This is typical and all the surveyor can do is to show the 
 facts. There will be plenty of lawyers to take either side 
 in a controversy and depending upon the cleverness of the 
 lawyers and the wealth, or poverty, of the litigants a crop 
 of decisions will be given in the lower courts which will 
 disgust intelligent jurists. If the general rule laid down 
 by Judge Cooley be followed everyone should be per- 
 mitted to stay put where found, but his rule referred to 
 carelessness in the original surveys and the subsequent dis- 
 appearance of stakes and monuments. 
 
 Possession implies two things. Physical possession and 
 payment of taxes. If the deed calls for Lot 5 in Block B, 
 the owner will be taxed for Lot 5 in Block B. If his fence 
 encroaches on Lot 6 he obtains an easement so long as the 
 owner of Lot 6 does not object, but the complaisance of 
 that owner gives the encroacher no title to a part of Lot 6 
 without the payment of taxes. He has a sort of squatter 
 claim as it were, modified by the law of the state. It is 
 under some such rule that the case of the carefully sur- 
 veyed town site with the owners too stingy to make surveys 
 must be considered. It is not believable that all the people 
 can be compelled to get off the land of their neighbors after 
 a long undisturbed possession of their neighbor's land, but 
 if they remove the encroachment they cannot again en- 
 croach. Also if the owner of a lot wishes to fully occupy 
 it he has the right to excavate up to his line and if the en- 
 croachment thereby falls the owner of the encroachment 
 cannot claim damages, but if the structure on the property 
 he actually owns is damaged, then he can stand in court. 
 The foregoing is again to be modified by circumstances, 
 all of which become matters of evidence and thereby give 
 all parties concerned some reason for taking their troubles 
 to court. If it were the custom to employ a board of 
 
SURVEYING LAW AND PRACTICE 309 
 
 three arbitrators consisting of an intelligent surveyor, a 
 genuine lawyer and a real estate agent of good reputation 
 and long experience, such a board probably would settle all 
 disputes in exactly the way they would be settled in the 
 Supreme Court of a state where the Supreme Court is 
 indifferent to the dotting of an i or the crossing of a t, 
 provided the case has been sensibly presented to the court 
 and substantial justice is desired. Common law is the 
 crystallization of the common sense of the people for un- 
 numbered generations. Statute law is not always on a 
 plane with common law, many times representing the opin- 
 ions of men elected to serve the people, but really repre- 
 senting some special interest which may be a corporation; 
 and may be an exasperated community. 
 
 The general rule about length of possession, as well as 
 the rule regarding supposed acquiescence, may be often 
 capable of modification. Both cases fail when the undis- 
 turbed possession or acquiescence may be the result of 
 fraud, or of a mistake made many years before. Fraud 
 vitiates all contracts and acquiescence is the performance 
 of a contract or understanding, a contract being a mutual 
 agreement enforceable at law. The surveyor cannot there- 
 fore be too critical of lawyers whose sole interest is that of 
 their clients. The surveyor's interest is merely to ascer- 
 tain the facts and report them. He may, if a man of con- 
 siderable experience, be of great assistance in smoothing 
 the feelings of property owners and can suggest means for 
 avoiding actions at law. The general criticism against 
 lawyers is that, as a rule, they are averse to seeing that a 
 surveyor is properly paid for his work, while charging good 
 fees themselves, and they too often try to tell a surveyor 
 how his work should be done and object when he attempts 
 to show that to recover a line it may be frequently neces- 
 sary to run many lines and take many days to ascertain the 
 facts. When a man requires the services of a physician or 
 surgeon he makes no attempt to secure bids and award the 
 work to the lowest bidder, but rather he secures as soon as 
 possible a man with a good reputation. He acts similarly 
 when engaging a lawyer, proceeding upon the theory that 
 expense cannot be spared when he is compelled to go into 
 court. When the employment of an engineer or surveyor 
 
310 PRACTICAL SURVEYING 
 
 is necessary attempts are made to employ the man who will 
 work for the least money and too often the selection is left 
 to the lawyer, who is generally inclined to employ the 
 cheapest workman. Young men are preferred as a rule 
 because it is supposed they can get over more land in a given 
 time than older men. Surveyors in the course of years 
 acquire a fund of valuable experience, and many times 
 lawyers are employed on property line disputes only once 
 or twice in a lifetime. It is irksome to competent sur- 
 veyors to be employed under the direction of such men, 
 hence the friction between the two professions. 
 
 Occasionally a surveyor becomes so interested in the 
 legal phases of his work that he studies law and is admitted 
 to practice. John Cassan Wait of New York City is an 
 example of this class, for he made a good reputation as a 
 civil engineer and then became a lawyer whose dictum in 
 such matters is considered conclusive. William E. Kern, 
 Attorney-at-law and Civil Engineer, wrote an article on the 
 legal side of surveys which was printed in Engineering News, 
 Vol. 48, and the author obtained permission from his widow 
 to reprint the article in this book. 
 
 A BRIEF DISCUSSION OF THE LAW OF BOUNDARY 
 SURVEYS 
 
 By WILLIAM E. KERN, C. E. 
 Copyright 1902, by William E. Kern. 
 
 A surveyor may enhance his reputation for piety by 
 being able to cite from the Bible, Deut. XXVII: 17 and 
 XIX: 14; Job XXIV: 2; Prov. XXII: 28 and XXIII: 10, 
 the verses mentioned referring to the sinfulness of removing 
 land marks and the penalty therefor. Josephus has 
 something to say on the same subject, and in his first 
 volume of "Antiquities of the Jews," Chapter II, he inti- 
 mates that the first murderer was also the first surveyor. 
 In the I2th book of the "^neid," Virgil recites an unusual 
 use of a monument. The esteem of the ancients is shown 
 by their provision of a special deity, Terminus, to preside 
 over boundary matters, all land marks of stone being re- 
 garded as monuments to his godship. 
 
SURVEYING LAW AND PRACTICE 311 
 
 So much for ancient history. The modern surveyor is 
 often called upon to decide boundary disputes, where the 
 best results would flow from a union of his efforts with those 
 of a counsellor at law. In view of this condition and the 
 many questions arising in the practice of a surveyor, some 
 systematic knowledge of the subject on his part would seem 
 desirable. 
 
 Several works upon surveying contain a number of syllabi 
 of cases involving boundary disputes. These syllabi are 
 condensed statements of the law applicable to a particular 
 combination of circumstances; and have limited utility 
 for the purpose for which they are inserted in such text- 
 books. All laymen must be cautioned against making a 
 general application of an abstract statement of law, as 
 determined in some special case. To illustrate this, the 
 following is quoted from a textbook on surveying: "Seventy 
 acres lying and being in the southwest corner of a section 
 is a good description, and the land will be in a square. 
 W. v. R. 2 Ham. Ohio 327." This is sufficient reference 
 for a lawyer, as he would look up all the facts; but for a 
 surveyor's enlightenment it should be explained that the 
 description read "Seventy acres lying, and being in the 
 southwest corner of," a certain section, etc., "of the lands 
 sold at Steubenville, " and nothing further. The piece in 
 question was part of a larger lot, and had never been marked ; 
 nor was there anything to indicate the intention as to its 
 shape. The court disagreed with the views on both sides 
 and held it must be surveyed as a square. 
 
 The term "boundary" is used in two senses, and may be 
 denned as: A series of lines forming the perimeter of a 
 specific tract of land; or, an object, parcel of land, or body 
 of water contiguous to a specific tract of land. "Land" 
 in legal parlance usually means a portion of the earth's 
 surface, whether dry or covered with water. Every tract 
 of land with which we may be called upon to deal, can be 
 considered as having at some prior time formed part of a 
 larger tract. Incident to the severance was the creation of 
 a new boundary along the line of division. Every bound- 
 ary in dispute may be conceived as having had such an 
 origin. After its inception, however, neighbors on either 
 side may, by friendly agreements between each other; or 
 
312 PRACTICAL SURVEYING 
 
 by long-continued hostile encroachments by one, with 
 neglect to assert rights by the other; or by other acts; 
 affect the legal status of the line so as to alter its original 
 position. 
 
 The data for determining the legal status of a boundary 
 may be divided into two classes: (i) Circumstances con- 
 nected with the severance from a parent tract at the time 
 the line in question first became a boundary, and (2) con- 
 duct of neighbors on either side of the line, which may 
 affect their relative status thereto. 
 
 EXTENT OF GRANT 
 
 The first class above referred to may be designated as 
 Extent of Grant. In discussing this subdivision, it is 
 almost superfluous to say that the rules stated may be 
 limited in their application in specific cases by the princi- 
 ples falling under the second class. In locating a boundary 
 as determined by extent of grant, we must seek the inten- 
 tion of the parties to the primal conveyance. 
 
 Let us suppose that A, being the owner of a large plot 
 of ground, divides it into two portions by a line XY run- 
 ning north and south, and sells all east of XY to B, and sub- 
 sequently all on the west to M. B then sells to C, the 
 latter to D, and D to E; M conveys to N. At a time when 
 N is owning on one side and E on the other, it becomes 
 necessary to 'define the line XY. This must be based 
 upon the description used in A to B, it being the one which 
 created the line X Y. No conveyance in the chain of A to 
 N can affect the rights acquired by E through the con- 
 veyance of A to B. The intermediate deeds should be 
 examined, however, as they frequently throw light upon 
 the original intention, as for example: If M took in 1821, 
 N in 1860, B in 1820, C in 1840, D in 1860, and E in 1880, 
 the monuments called for in 1820 may be gone in 1900, 
 but a surveyor in 1820 might have found vestiges and 
 restored them, or created new marks in correct positions; 
 and another surveyor in 1860, finding the marks of 1820, 
 may have made an accurate mathematical description and 
 set two or more imperishable monuments. 
 
 It may be urged that the original deed cannot be found, 
 
SURVEYING LAW AND PRACTICE 313 
 
 or is too indefinite to be of any service. The attempt, how- 
 ever, ought always be made; if it fails we must do the 
 next best thing. In a recent case the mesne conveyances 
 of a plot within the limits of a large city differed about 
 100 ft., in frontage on a street. An examination of the 
 original deed disclosed a call for the edge of a fast land along 
 the side of a swamp; a city plan, showing topography, 
 fixed the position of the water line and solved the problem. 
 The intention of the parties is generally to be gathered 
 from the description in the instrument of conveyance. 
 Description we will define as a statement designed to identi- 
 fy a specific tract of land. Every specific tract of land is 
 included between lines of definite length, making equally 
 definite angles with each other and with the meridian, and 
 containing a definite area; and any angle point may be 
 conceived as located at a definite distance and direction 
 from some well-known and fixed point. Some specific 
 tracts are marked at angle points or elsewhere by natural 
 or artificial monuments; some are known by particular or 
 distinguishing names. A complete description would 
 contain all of these elements so far as they exist. 
 
 MONUMENTS 
 
 Monuments are objects established or used to indicate 
 the boundary of a specific tract of land. They may be 
 natural or artificial, the distinction being too obvious to 
 require definition. Under this head are included varieties 
 of land, as swamps, meadows, forests, pasture, etc.; bodies 
 of water and water courses; hillocks, cliffs, trees, rocks, 
 buildings, walls, fences, stones, stakes, pits, etc. 
 
 Ambiguity in a description may arise from conflict 
 between elements of a different sort, or between those of 
 the same sort. Monuments may disagree with the mathe- 
 matical description, or with each other, or be missing; the 
 dimensions may disagree with each other or with the area. 
 When discrepancies arise, the cardinal rule to follow is, 
 effectuate the intention of the original parties. Where there 
 is sufficient evidence to explain the conflict and point out 
 the mistake, the line must be run upon the reformed data 
 whether it cause the rejection of a monument or dimension. 
 
314 PRACTICAL SURVEYING 
 
 It most frequently happens, however, that no such evidence 
 is at hand, in which event the following rules are to be 
 applied : 
 
 Monuments are the best evidence of location of a bound- 
 ary, and natural monuments are to be preferred to all other 
 forms of description. This rule is so important that a list 
 of cases is added containing one from nearly every state 
 in the Union. A perusal of any one case will show the reason 
 for the rule, and probably, in addition, refer to other cases 
 in the same state which may be looked up, as the scope of 
 this article will not permit of stating all the authority for 
 the rules stated.* 
 
 This rule, of course, applied only where the monuments 
 were adopted as such, either by mention in the description 
 or by reference to or use of a survey wherein they are estab- 
 lished or used (Beckman v. Davidson, 162 Mass. 347). 
 In such cases it is not necessary that they be seen by the 
 grantor to bind him. No matter how exactly the survey 
 closes or agrees with the content, if the monuments do not 
 agree therewith, the former falls. If a line is marked on 
 the ground in any manner, though described as straight, it 
 must follow the breaks in the marked line if the latter is 
 not straight. An instance of this kind frequently occurs 
 in running up a creek; several reaches are often combined 
 in one course; in such case the line must follow the water. 
 This occurred in Spring v. Hewston, 52 Cal. 442. 
 
 In Esmond v. Tarbox, 7 Me. 61, a survey was made by 
 
 * Ayers v. Watson, 137 U. S. 584; Wright v. Wright, 34 Ala. 194; 
 Stoll v. Beecher, 94 Cal. i; Nichols v. Turney, 15 Conn. 101; Nivin v. 
 Stevens, 5 Harr. (Del.) 272; Andreu v. Watkins, 26 Flo. 390; Harris v. 
 Hull, 70 Ga. 831; Bolden v. Sherman, no 111. 418; Shepherd v. Nave, 
 125 Ind. 226; Yocum v. Haskins, 81 Iowa 436; Bruce v. Morgan, I B. 
 Mon. (Ky.) 26; Lebeau v. Bergeron, 14 La. Ann. 494; Oxton v. Groves, 
 68 Me. 371 ; Wood v. Ramsey, 71 Md. 9; Woodward v. Nims, 130 Mass. 
 70; Bruckner v. Lawrence, i Doug. (Mich.) 19; Newman v. Foster, 
 3 How. (Miss.) 383; Harding v. Wright, 119 Mo. i; Johnsons. Preston, 
 9 Neb. 474; Cunningham v. Curtis, 57 N. H. 158; Kalbfleisch v. Oil Co., 
 43 N. J. L. 259; Thayer v. Finton, 108 N. Y. 394; Redmond v. Stepp, 
 100 N. C. 212; Hare v. Harris, 14 Ohio 529; Anderson v. McCormick, 
 1 8 Oreg. 301; Morse v. Rollins, 121 Pa. 537; Faulwood v. Graham, 
 I. Rich. L. (S. C.) 491; Lewis v. Oakley, 19 Heis (Tenn.) 483; Wyatt 
 v. Foster, 79 Tex. 413; Grand Trunk Co. v. Dyer, 49 Vt. 74; Coles v. 
 Wooding, 2 P. & H. (Virg.) 189; Teass v. St. Albans, 38 W. Va. i; 
 Miner v. Brader, 65 Wis. 537. 
 
SURVEYING LAW AND PRACTICE 315 
 
 one surveyor and marked with monuments, and a plan 
 made by another; the deed recited the latter only. A dis- 
 agreement between the two having been found, it was 
 held that the monuments governed. In Wilson v. Bass, 
 6 Tenn. no, a course called for a known point, and then 
 crossing a river at 200 poles continued 213^ poles to its 
 terminus. It was found that the true distance to the river 
 was 213 poles. It was held that as the parties had agreed 
 that it was 200 poles to the river, the line must extend 13 
 poles beyond the same. In Frey v. Baker, 7 Ky. L. Rep. 
 663, testator divided a tract into two pieces, and described 
 the dividing line as beginning at a point at a fence, and 
 running thence west along the fence. It was shown that 
 the fence did not run east and west, and that testator was 
 ignorant of that fact. It was held that the fence and not an 
 imaginary line due west was the proper line. 
 
 Wendell v. Jackson, 8 Wend. 183, is an early but inter- 
 esting New York case. A patent had been issued for a 
 tract described as beginning at the easterly 
 corner of township No. 20, etc., and proceeding 
 by various courses and distances, Fig. 211. 
 The third course ran 8*50 153 chains to the 
 side of Schroon Lake, with content as 3500 
 .acres. The actual distance to the lake on the 
 third course FG was less than one-half of the 
 distance stated. A survey for an adjacent 
 tract, made seven days later, began at a clump 
 of rocks on the side of Schroon Lake, at a corner 
 of the preceding patent, thence followed the same around 
 N 50 W 153 chains, S 40 W 105.5 chains S 31 15'. E 
 330 chains to corner of town 20, etc. It was necessary to 
 locate the first patent to determine the title to the land 
 northwest of EF and southeast of the line //. The town- 
 ship corner used as a beginning was well marked by a stake 
 and stones, and was at a point in a swamp difficult of access. 
 It was argued on one side that the call for the beginning at 
 D should be disregarded, that the second survey following 
 the first so closely, and having been made so short a time 
 before, should throw light on the latter, and that the first 
 should start at A on the side of the lake, and, reversing the 
 courses, run backwards until arriving at A. This would 
 
316 PRACTICAL SURVEYING 
 
 locate the land at AJIHK. It was held by a divided court 
 of 14 to 5 that the beginning D being a known point must 
 be used and the line run to the side of the lake at A , thence 
 to beginning, or to B. The point being, the supremacy of 
 the call for a natural monument on the "side of a lake." 
 This case might have been decided otherwise if many 
 members of the court had ever had any practical experience 
 at surveying. It is quite probable that the first survey was 
 begun at a point H, and the line between H and D not run. 
 owing to the difficulty of getting over a bog. The adding of 
 an estimated distance HD was forgotten, and AK and HK 
 calculated to make the figure close. 
 
 On the other hand, in White v. Leming, it was held that 
 where rejecting the call for one monument would reconcile 
 the other parts of the deed and leave enough to identify 
 the land, the rule was not applicable. 
 
 As between natural and artificial monuments the former 
 will have the greatest weight. Stakes, being perishable and 
 so easily moved, are regarded as very inferior marks; in 
 one case, Huffman v. Walker, 83-N. C. 411, the identifica- 
 tion of the location was considered impossible. Where a 
 monument called for is lost, its original position must be 
 ascertained if possible. This may be done by the testi- 
 mony of witnesses as to its former location, by indications 
 on the ground, or, in the absence of these, by such methods 
 as appear to be the nearest to certainty. 
 
 RESTORING LOST SECTION CORNERS 
 
 In restoring lost section or quarter corners, various and 
 conflicting rules have been stated by the land office and 
 courts. In order to understand how to restore corners of 
 the public land surveys, some knowledge of the method of 
 making the original surveys should first be had. The 
 details of the latter may be learned from several textbooks, 
 or better still, from the manual of instructions issued by 
 the government to deputy land surveyors. It may be 
 summarized briefly as follows: In several different parts of 
 the western country, initial points have been established 
 from which base lines have been run towards all four points 
 of the compass. The north and south line, being straight 
 
SURVEYING LAW AND PRACTICE 317 
 
 its entire length, is referred to as the principal meridian 
 for its section of the domain. The east and west line is 
 called the principal base, and is likewise straight its full 
 distance. From points on the meridian, at intervals which 
 are multiples of six miles, lines are extended east and west 
 and are known as standard parallels. On the base and 
 parallels, similar multiples of six miles are laid off, and from 
 their termini guide meridians are extended north to the 
 adjacent parallels. The blocks thus formed are subdivided 
 into six-mile rectangles by lines due east and west and north 
 and south, these blocks being called townships. Sections 
 of one mile square are next surveyed by beginning at a 
 point on the south boundary of a township one mile west 
 of its south east corner, and running slightly west of north, 
 so that each mile point shall be one mile from the east 
 boundary of the township, all error being thrown towards 
 the north and west. Each half-mile point is marked by 
 a "quarter corner." 
 
 In 1885 the General Land Office issued a pamphlet en- 
 titled "Restoration of Lost and Obliterated Corners." 
 This seems to be a very good article except, perhaps, in the 
 method it directs for finding the center of a section. Where 
 a government corner is missing, and after diligent inquiry 
 its former position cannot be ascertained, the surveyor 
 should proceed as follows: If the point was on a base, 
 parallel or meridian, it must be restored to a position on a 
 line between the nearest corners on the same parallel or 
 base, and at a distance which is a proportional part of the 
 whole distance between the known corners. To illustrate, 
 suppose that the intersection of a meridian with a parallel 
 is marked, and the nearest corner south is a section corner 
 two miles away. Three marks are missing. The distance 
 measures 159.2 chains, and the field notes of the original 
 survey show the distances going north to be 40 chains to 
 first-quarter corner, 40 to the section corner, 40 to the next 
 quarter, and 39 chains to close. The excess of 0.2 must be 
 equated all along the line so that the distances become 40.05, 
 40.05, 40.05, 39.05, respectively, and corners are set at these 
 distances on a line between the two known corners. 
 
 In the same manner, points on the boundaries of a* town- 
 ship should be set at equated distances, and on straight 
 
318 PRACTICAL SURVEYING 
 
 lines between the nearest marks on the same boundary, 
 except in the case of the township corners when on a parallel 
 or base. Owing to the system of completing each township 
 by itself, its corners are not necessarily on straight lines. 
 To re-locate a township corner, not on a base or parallel, 
 a trial line should be run between the nearest corners north 
 and south and the distance measured and compared with 
 the field notes. A point is then marked on the trial line 
 at the equated distance. From this point the distances to 
 nearest points east and west are to be measured, added and 
 compared with the field notes, and an equated distance as- 
 certained. The point first fixed is to be moved to the east 
 or west to suit its equated distance in that direction. 
 
 The same course is to be followed in re-locating section 
 corners. Let us suppose (Fig. 212) the section corner A 
 .and quarter corners B and C are lost and we wish to re-lo- 
 ^ cate A. D, E, F and G are found 
 
 marked. A trial line is run from 
 E to G and the distance meas- 
 ured, temporary corners being 
 marked at A and B, B being 
 
 placed at 39 chains from E and A 
 
 f. 
 
 C 
 
 , Measured 
 
 II 1 54 
 
 <L 40 chains from B, these being the 
 
 j distances in the original field notes. 
 
 The whole length having been re- 
 L^is ported as 119 chains is found to 
 
 F measure but 118.70 a deficiency of 
 
 .30 chains. This being equated 
 
 between EB, BA and AG, B must be moved N o.io 
 chains and A placed 0.20 chains north of their first tem- 
 porary positions, D to F is then measured and found 
 111.54 chains, as compared with no chains on the official 
 plat. This is an excess of 1.54 chains, and duly equated 
 would make DC 30.42, CA and AF each 40.56. A is then 
 to be set 40.56 chains from F and 39.90 chains from G. 
 The quarter corners are to be set on a straight line be- 
 tween their section corners, hence B is to be located on 
 a line from A to E 39.90 chains from A, and quarter 
 corner C on a line between D and A and 40.56 from A. 
 
SURVEYING LAW AND PRACTICE 319 
 
 FINDING CENTER OF SECTION 
 
 Two methods have been mentioned for finding the center 
 of a section. The pamphlet above referred to suggests 
 that it be placed at the intersection of lines run from the 
 opposite quarter corners. Another method was to place 
 it at a point equidistant from the quarter corners, except 
 in the north and west tiers of sections, where it would be 
 located as described above for section corners, using the 
 quarters as a basis. From a legal standpoint the following 
 would probably be preferable: 
 
 When completely marked, the exterior of a section has 
 eight monuments. From carelessness, these frequently 
 disagree with the field notes. Quarters are often out of 
 line and nearer one corner than the other. In spite of 
 these defects, when set, the marks must not be altered in 
 position. Again, fractional subdivisions are described as 
 "quarter sections," indicating the idea of quantity; hence 
 a quarter section should be surveyed as one-fourth of the 
 area of the section as physically defined. This should be 
 done by first making a careful survey of the entire section 
 and calculating its actual area. A point should then be 
 fixed so that lines run from it to the quarter corners east, 
 west and south would include exactly one-fourth of the 
 entire calculated area. This would determine a boundary 
 for the S E and S W quarters. A second 
 point on the line between the E and W A 
 quarters at such position, that a line 
 drawn from it to the quarter corner north 
 would divide the north half equally, 
 should then be fixed. 
 
 The section might then have two cen- 
 ters, as it were, but usually one point 
 would answer the requirements. Fig. 213 FlG 
 
 will illustrate a possible condition. Sup- 
 pose the field notes show a section 80 by 80, with quarter 
 corners all 40 chains from the section corners, and a meas- 
 urement shows an area of 640 acres, but the quarter corners 
 at incorrect distances as indicated. A point / is found by 
 calculation, so that by joining H, /, FI and DI the S E and 
 S W quarters are each 1 60 acres. A second point / is 
 
320 PRACTICAL SURVEYING 
 
 found, so that by joining BJ the N E and N W quarters are 
 equal to each other and the other two. The patentee of the 
 N W quarters cannot complain at his irregular lines, as he 
 receives more land than by running straight lines between 
 the quarters corners, as indicated by the land-office circular. 
 
 FAULTY DESCRIPTION 
 
 Where a description by course and distance does not 
 close in itself, courses may be reversed if necessary, and 
 clearly erroneous, as for example: A description read, 
 "Beginning at the center of a railroad at its intersection 
 with the B road, thence S 20 W along said center of said 
 railroad 150 ft., thence N 70 W 50 ft. to a point, thence 
 N 20 E 150 ft. to a point, thence N 70 W 50 ft. to begin- 
 ning. " Here the fourth course was plainly in error, and 
 was read as S 70 E, as there was evidence to show that 
 the fourth and not the second course was erroneous. A 
 similar error occurred in Brown et al v. Hage, 21 How. 320, 
 where a patent read, "Beginning at a sycamore standing 
 on the edge of the Shanadoah River, and extending thence 
 down the said river (N 48 W 200 ch.), etc., etc." The 
 title to a large area depended on the retention or rejection 
 of the words in the parenthesis, and the court said it was 
 clear that to go down the river would not be northwesterly, 
 it being a matter of common knowledge that the river ran 
 in the opposite direction at this point, 
 and the words were rejected. 
 
 Where, however, a natural monument 
 is called for by mistake, such calls will be 
 rejected. Thus in Land Co. v. Thompson, 
 83 Tex. 169, several grants began "on the 
 side of Devil's River," under a mistaken 
 apprehension as to its real position, no 
 FIG. 214 survey having been made on the ground. 
 
 It was held that the surveys must be made 
 by course and distance, recourse having been had to other 
 established marks on the east, and the land between the 
 true and false position of the river was excluded from the 
 grants calling for it. The two positions are shown in the 
 annexed sketch, Fig. 214. 
 
SURVEYING LAW AND PRACTICE 321 
 
 Patents calling for fractional lots along rivers do not 
 take the land between the meander survey line and the 
 river in Nebraska, but this state seems to differ from the 
 others, in decisions on the point. In Jefferis v. Land Co., 
 134 U. S. 178 and Ayers v. Watson, the contrary was held, 
 and United States statutes of 1796, 1800, 1805, 1820, 1832, 
 and Sec. 2395, Rev. Stat. U. S. cited to sustain the doctrine 
 that the land between the meander lines and river does pass 
 with the fractional lots, even though the river be navigable. 
 Quantity is usually subordinated to both course and dis- 
 tance as being the least certain element. Occasionally it 
 may govern, however. In one case a deed called for the 
 side of a certain highway, which at the time of the dis- 
 pute was obliterated. The rear line was defined on the 
 ground. Several positions for the lost highway were found. 
 It was held that the one agreeing most nearly with the area 
 described should be preferred, as none agreed with the 
 course and distance. 
 
 Calls for adjoining surveys or tracts must be observed. 
 This question most frequently arises in patents from state 
 governments. In .the same manner a plan or plat when 
 referred to must be considered a part of the description. 
 But where the plan conflicts with monuments of an actual 
 survey on the ground, the monuments govern; unless the 
 survey was subsequent to the plan, when, in such case, the 
 latter controls. 
 
 The term more or less is construed as importing inexact- 
 ness in all quantities to which it is annexed. It might seem 
 ridiculous to describe a lot in feet, inches and thirty- 
 seconds of an inch, and add to each length "more or less," 
 but if adjacent properties or other monuments are called 
 for, and the distances to such marks disagree with those of 
 the description, the statement of the indefiniteness of the 
 distances in the description becomes of some value, but, at 
 the most, of very slight account, as the monuments would 
 control as well without the words more or less as with 
 them. Cases where they are of importance will seldom 
 arise. 
 
 When course and distance disagree with each other, that 
 will prevail which under all the circumstances and evidence 
 appears to be the most certain. 
 
322 PRACTICAL SURVEYING 
 
 The cases generally arise in this way: The opposite ends 
 of two adjoining courses being agreed upon as definitely 
 determined and located, and one course being extended to 
 meet the other reversed to run from its opposite terminus, 
 the point of intersection is found to disagree with the dis- 
 tances. The general rule is that courses prevail over dis- 
 tances. Thus, in Curtis v. Aaronson, 49 N. J. L. 68, 1886, 
 both parties agreed until the i6th course was run. This 
 was described as S 24 E 29 chains, thence 865 15' W 
 151.5 chains to pine on east side of Shoal Branch. The 
 position of this pine was not disputed, nor was the initial 
 point of the i6th course; but by running the latter S 24 E 
 to intersect the iyth course extended N 65 15' E from the 
 pine, the i6th distance would be lengthened to 96.5 chains 
 in lieu of 29 chains as stated. The surveyors in re-running 
 the entire survey where the lines were not disputed, found 
 the courses all approximately correct, but the distances 
 very erroneous. This fact, among others determined the 
 court to hold that the disputed courses must be inter- 
 sected and the disputed distances ignored. 
 
 On the other hand, in a somewhat similar case in Ken- 
 tucky in 1821, the discrepancy between accuracy of courses 
 as compared with distances did not appear, and the court 
 directed the distances to be maintained, as by inter- 
 secting the courses the distances would be very greatly 
 distorted. 
 
 Areas are regarded as the least certain of the elements of 
 a description. So where courses and distances do not in- 
 clude the same area as that stated, the latter must yield, 
 unless intention is clearly expressed that an area of specific 
 quantity shall be conveyed, in which latter case the area 
 and not the distances should control in construing the 
 description of the land. Sanders v. Golding, 45 Iowa 
 
 463- 
 
 Plats or maps referred to in a deed are to be considered 
 as a part of the description. The following illustrates 
 their status: In Vance v. Fore, 24 Cal. 435, a deed referred 
 to another deed, previously made, for a description of the 
 premises granted, and also to a map not contemporaneous 
 with the last deed. The description in the older deed called 
 for no monument at the initial point, nor could any be 
 
SURVEYING LAW AND PRACTICE 323 
 
 identified with accuracy. The first course terminated at 
 "the base of the mountain, thence running at right angles, 
 following down the base of the mountains." The char- 
 acter of the country was such that witnesses might reason- 
 ably differ in their location of the lines. The map referred 
 to showed all the natural and artificial monuments found on 
 the ground, such as streams, buildings and roads. The 
 descriptions as set forth in the old deed and on the map 
 conflicted. It was held that the description shown on the 
 plat was the most certain under the circumstances, and the 
 least likely to be affected with mistakes, hence to be fol- 
 lowed in making the re-survey. 
 
 There is an apparent conflict in the cases -as to the rela- 
 tive precedence of plats, and the elements of course, dis- 
 tance, etc., but, as explained before, all cases must be 
 considered from the standpoint of principle involved. The 
 principle in this case being that that which is shown to be 
 the most certain will prevail. 
 
 In Heaton v. Hodges, 14 Me. 66, no survey could be ascer- 
 tained to have been made, and the monuments shown on 
 the plan could not be identified; certain distances on the 
 plan disagreed with measurements of established lines. 
 It was held that the disputed lines were to be first scaled 
 from the plat and the lengths so ascertained, altered in 
 such proportion as the length of the accepted lines bore to 
 their dimensions on the plat. 
 
 In Lampe v. Kennedy, 45 Wis. 23, several deeds were 
 made from a plat. The courses and distances did not agree 
 with the plat, but the latter prevailed. In the same man- 
 ner, plats were held to control quantity, in 109 111. 46, 
 and Hathaway v. Power, 5 Hill (N. Y.) 453. 
 
 In Beaty v. Robertson, 130 Ind. 589, it was said that where 
 the plat and field notes of a government survey conflicted, 
 the former showed the lines as fixed by the surveyor-gen- 
 eral, and were those by which the land was, sold, hence to be 
 taken. 
 
 The last method of describing land is by a particular 
 name. In earlier times this was necessarily the most 
 common. In such cases disputed boundaries are estab- 
 lished by the evidence of witnesses familiar with the oldest 
 physical location of the lines, or the admissions of inter- 
 
324 PRACTICAL SURVEYING 
 
 ested parties. This class would include, however, descrip- 
 tion by lot and block number as per a plat. Town-site 
 plats usually show in addition to the dimensions of the lots, 
 stone monuments set at intersection of specified offset 
 range lines at street corners. The monuments are fre- 
 quently found to be inaccurately set, and various expedients 
 have been adopted by later surveyors to eliminate the 
 error. In some cases the total error of each block has been 
 thrown in the adjacent street; in still others, the streets 
 have been made the proper width and the error equated 
 through the lots. Probably the best method by analogy 
 with the decisions on lost government corners, would be 
 to measure the distance between monuments and compare 
 the same with the distance between the same points as 
 shown on the plat. The difference should then be equated 
 for every foot alike, whether streets or lots. 
 
 STANDARD OF MEASUREMENTS 
 
 Congress has never exercised its constitutional right to 
 fix a standard of linear measurement, but an official standard 
 having been adopted by the United States surveyors, a so- 
 called United States standard foot has arisen, and several 
 states have enacted statutes adopting this standard, which, 
 in the absence of Congressional action, is lawful for the 
 respective states so adopting them, and all standards used 
 in contravention of such state standards are illegal in these 
 states. This gives rise to some important questions, which 
 will now be considered. 
 
 Where a piece of ground is conveyed and the descrip- 
 tion used is an illegal standard, that standard must be used 
 to establish its boundaries; but if it cannot be shown that 
 the illegal standard was adopted by the parties, then the 
 construction must be based upon the legal standard of the 
 state. To make the case more concrete: If A owns 500 ft. 
 front of unimproved land and sells 100 ft. from one end, this 
 must be measured in legal standard; but if he owns one 
 lot, described as either 20 or 100 ft., but by an illegal 
 standard, and the further words are added, "being the 
 same premises, " etc., then he would take the same premises, 
 
SURVEYING LAW AND PRACTICE 325 
 
 or all that his grantor took in the recited conveyance, 
 whether more or less. In other words, it is a question of 
 intention; if both parties meant the same thing, then that 
 is what they contracted for, or else nothing. 
 
 PROPERTY ABUTTING ON ROADS AND STREETS 
 
 Where land is conveyed by side of a road, street or alley, 
 the usual practice is that, by implied grant, half the road is 
 conveyed with the abutting lot. In some states the fee 
 of the streets is in the public; in such cases the abutter 
 stops at the side of the road. In a few isolated cases the 
 law has been held to be that a conveyance did not pass 
 title to the center by implication." 
 
 Thus, in Union Cemetery v. Robinson, 5 Wh. (Pa.) 18, 
 the general principle that a conveyance passed a fee to the 
 center of the road was not doubted, but it was said that as 
 in the case at bar the street described was only on paper, 
 and unopened, and further, that the description being in 
 feet, inches and fractions of an inch, much stress should be 
 laid upon the minuteness of the measurements, as showing 
 an intention to limit the conveyance to the side of the road 
 or street. This case has very properly been overruled in 
 the same and other states, both as respects the importance 
 of the refinement of measurements and as to the fact that 
 the street was unopened. The leading case on the subject 
 being Paul v. Carver, 26 Pa. 223. The principle that a 
 conveyance along a road carries the boundary to the cen- 
 ter of that road has been followed in practically every state 
 of the Union, although there have been instances where 
 courts have been swayed by exceptional circumstances to 
 hold that an intention was manifest to exclude the road. 
 The words "by the side of" have been given that effect; 
 also a grant of right of way over the strip included in the 
 street. The following from Paul v. Carver is a forcible 
 statement of the reason of the rule: "The rule had its 
 origin in a regard to the nature of the grant. Where land 
 is laid out in town lots, with streets and alleys, the owner 
 receives full consideration for the streets and alleys in the 
 increased value of the lots. The understanding always is 
 that houses may be erected fronting on the streets with 
 
PRACTICAL SURVEYING 
 
 windows and doors, doorsteps and vaults. If a right of 
 property in the streets might under any circumstances be 
 exercised by the grantor, he might deprive his grantee of 
 the means of entry into or exit from his house, and of all 
 the enjoyments of light and air, and might thereby deprive 
 him of the means of deriving any benefit from his pur- 
 chase." 
 
 Where, however, the original line between two adjacent 
 tracts existed before the road was laid out, and was not 
 coincident with the center line of the road; then it, and not 
 such center line, is the boundary. 
 
 Where the conveyance is along the margin of a river or 
 other water way, the decisions are conflicting, and no pre- 
 cise rule can be stated. In the majority of the states, 
 however, a conveyance by the side of a tidal or navigable 
 river or sea carries the fee to high-water mark, and on un- 
 navigable rivers and ponds to their center. The side lines 
 are projected at right angles with the thread of the stream, 
 unless otherwise provided for in the deed. 
 
 BOUNDARIES FIXED BY AGREEMENT OR A QUIESCENCE 
 
 Boundaries may be fixed by agreement of the owners 
 affected, and in such cases bind those who take from them. 
 But if either party was led to agree by a mistake in a meas- 
 urement made for the purpose, he may repudiate his agree- 
 ment if he does so within a short time after discovery of 
 the mistake. Coon v. Smith, 29 N. Y. 392. An agreement 
 to fix a boundary where it is indefinite need not be in writ- 
 ing. Turner v. Baker, 64 Mo. 218. 
 
 In each state of the Union statutes exist prescribing a 
 length of time during which various suits may be brought, 
 and upon failure to bring such suit the cause is lost. The 
 periods of these several "Statutes of Limitations," vary in 
 the several states. A boundary, though erroneous, if 
 acquiesced in by the parties for a term beyond that of the 
 appropriate statute, becomes fixed, and as suit could not be 
 brought to rectify it, it should be surveyed as maintained. 
 
 Estoppel is a legal principle by which one who has led 
 another to believe that certain conditions arc true, is after- 
 
SURVEYING LAW AND PRACTICE 327 
 
 wards precluded from showing such conditions are false, 
 where the innocent party would thereby be injured. So 
 boundaries may become fixed by estoppel. Thus, in 
 Sheridan v. Barret, one led another to believe that their 
 boundary was in a certain location, and to build a wall in 
 a position based upon that fact. The first was compelled 
 to adopt the boundary which he had falsely represented as 
 being correct. See also New York Co. v. Gardner, 25 S. W. 
 737; Jordan v. Deaton, 23 Ark. 704. 
 
 In re-marking a boundary, we are first to satisfy ourselves 
 that no line has been acquiesced in, agreed upon or main- 
 tained by adverse possession for the statutory period. If 
 not, the deeds are to be examined and the line traced on the 
 ground as originally run ; if no vestiges of the latter remain, 
 then the elements of description are to be observed in the 
 order named, bearing in mind that that which is most 
 certain will prevail over the less certain. (Finis.) 
 
 GOVERNMENT SURVEYS 
 
 When a district was established for the disposal of public 
 lands an Initial Point of which there are 31 in the United 
 States was selected within the district. Through this 
 point was run a north and south line called the Principal 
 Meridian; and an east and west line called the Principal 
 Base. 
 
 At intervals of 24 miles on the Principal Meridian, 
 measured from the initial point, Standard Parallel or Cor- 
 rection Lines run east and west parallel with the Principal 
 Base. Beginning from the Initial Point they are known as 
 1st, 2nd, 3rd, etc., Standard Parallel North (or South). 
 
 From the Principal Base and Standard Parallels, true 
 north lines called guide Meridians run at intervals of 24 
 miles. Thus the land is divided into ."checks" approx- 
 imately 24 miles square. 
 
 The standard parallels were called Correction Lines be- 
 cause the Guide Meridians being true Meridians converged 
 on account of the spherical shape of the earth, so a "check" 
 is narrower on the north than on the south. On each 
 standard parallel two stakes not far apart are set for each 
 guide meridian, which has caused many lawsuits because the 
 
328 PRACTICAL SURVEYING 
 
 first settlers, their attorneys and surveyors knew little or 
 nothing of astronomy and geodesy. 
 
 Each check is divided into townships measuring six 
 miles north and south and six miles wide on the south end. 
 The north and south township boundaries are parallel but 
 the east and west boundaries being true meridians con- 
 verge, hence there are again two stakes marking meridians 
 where parallels intersect them. 
 
 Each township is divided into sections one mile square 
 (approximately) and in surveying the sections (subdivid- 
 ing a township), monuments were supposed to be set every 
 half mile. The corners are called "section" corners and 
 the half-mile corners are called "Quarter Corners, " because 
 they divide the section into quarters. No stakes were 
 set in the interior of a section. 
 
 The original surveys were made by contract, all contracts 
 being awarded to the lowest bidders. Many ignorant men 
 obtained contracts and so much trouble arose that after a 
 time farcical examinations were held to license surveyors. 
 Political influences vitiated this attempt to improve matters. 
 The business of making government surveys assumed large 
 proportions and many frauds were perpetrated by dishonest 
 contractors, some of whom turned in field notes for surveys 
 that were not made. 
 
 The intelligent men in the land department tried to check 
 fraud by having each survey examined before paying the 
 contract price. The work of examining surveys was let 
 also by contract to the lowest bidders, with results any man 
 of common sense could have foretold. At the present time 
 examiners hold office for life with a fair salary and are 
 selected by means of competitive civil service examinations. 
 
 Cheap work was done; dishonest work was done; cor- 
 ners were seldom made of permanent material; early 
 settlers were careless; many surveyors who made the first 
 re-surveys were very ignorant; state legislatures un- 
 fortunately thought corners could be "re-established"; the 
 government altered methods a number of times; on all 
 surveys prior to 1864 an inaccurate method was used to 
 indicate courses on true lines ; these were a few of the many 
 causes for trouble over "government lines," and the Land 
 Department was compelled to issue instructions to guide 
 
SURVEYING LAW AND PRACTICE 329 
 
 surveyors in making re-surveys in states in which the land 
 was ' ' sectionized . ' ' 
 
 The pamphlet is revised from time to time and may be 
 procured free of cost from the General Land Office, Wash- 
 ington, D. C. The title is "Restoration of Lost or Obliter- 
 ated Corners and Subdivisions of Sections." It should 
 be owned and carefully studied by every land surveyor. 
 
 The government " Manual of' Surveying Instructions for 
 the Survey of the Public Lands of the United States and 
 Private Land Claims," is supplied by the Government 
 Printing Office, Washington, D. C., for seventy-five cents, 
 postpaid. Instructions for surveying and marking mining 
 claims may be obtained from the Surveyor-General for the 
 district in which the claims lie. 
 
 The student after reading this chapter is ready to appre- 
 ciate the statement that most of the trouble over bound- 
 ary lines is caused by the ignorance of the men who make 
 the first re-surveys. They are disposed to follow field 
 notes rather than monuments; or possession when monu- 
 ments cannot be found. Field notes and maps are often in 
 error, when compared with modern standards of accuracy, 
 but nevertheless must be used as a guide in re-tracement 
 surveys, and they always stand in court until evidence is 
 introduced to destroy their credibility. The surveyor must 
 refuse to accept work from clients, or their attorneys, who 
 attempt to dictate as to how a survey must be conducted. 
 A single line cannot fix the location of an obliterated monu- 
 ment. 
 
 To subdivide a section all the exterior lines must be re- 
 traced and correct bearings and lengths obtained. Then, 
 and not until then, smaller parcels may be cut out and safely 
 rnonumented. The plat and field notes should show this. 
 
CHAPTER VIII 
 
 ENGINEERING SURVEYING 
 
 Surveyors and civil engineers use the same instruments 
 and in schools where surveying is taught as a branch of 
 mathematics instruction is given in the use of instruments 
 and the computations connected therewith. From this 
 point surveying is divided into : 
 
 1. Land surveying. 
 
 2. Engineering surveying. 
 
 Land surveying, strictly, is concerned only with the divid- 
 ing of land; to determine areas and to re-locate missing 
 monuments and boundaries. Leveling is really engineering 
 work but it is a part of the work performed by all land 
 surveyors because they are surveyors. 
 
 Engineering surveying consists in making surveys for 
 railways, highways, canals, ditches, and the setting out 
 of work, such as bridges, dams, large buildings, tunnels, 
 etc., together with the computation of quantities of mate- 
 rials. Surveys for the purpose of making contour maps, 
 selecting the gradients for roads, surveying for the location 
 of sewers and computing the amount of earthwork falls 
 naturally to local surveyors who specialize in land survey- 
 ing, because such work may be done by anyone familiar 
 with the use of instruments, surveying methods and draft- 
 ing. Mining surveying is engineering work which may be 
 equally well done by all qualified land surveyors. Surveys 
 made for the purpose of preparing good maps belong both 
 to the land surveyor and the civil engineer. In this chapter 
 it is proposed to present some fundamental information 
 relative to good practice in engineering surveying. 
 
 MINING SURVEYS 
 
 Surveys made to determine the boundaries of mining 
 claims on the surface of the ground are called surface sur- 
 veys. No further instructions are required for such work 
 
 33 
 
ENGINEERING SURVEYING 331 
 
 than have been given for land surveys. The work is done 
 with a transit and steel tape with the greatest possible 
 accuracy. When the location of a mining claim is made it 
 must be done in accordance with instructions issued by 
 the Commissioner of the General Land Office, Washington, 
 D. C, and the local rules and regulations of the United 
 States Surveyor General within whose district the claim 
 is located. 
 
 Underground surveys are made to show the workings 
 of mines. The top of the shaft, or the mouth of the tunnel 
 where either appears on the surface, is located and tied to 
 some known corner or permanent object on the surface. 
 A meridian line is chosen and set out on the surface for 
 some distance and the ends marked with good stakes, so a 
 long backsight may be obtained. If a tunnel is to be sur- 
 veyed, a point on this line is set opposite the mouth of 
 the tunnel and the line of the tunnel is run out, and tied 
 to the meridian. The line is carried through the work- 
 ings by backsight and foresight, all angles being double 
 centered and all tack points being set by double centering. 
 When a survey is made on the surface, it is possible to run 
 around the land and make a closed survey, thereby obtain- 
 ing a check. When the survey is run underground this 
 opportunity to make a check does not occur, except when 
 a point may be projected up through a shaft, or be carried 
 through another tunnel to the surface. The greater num- 
 ber of mining surveys, however, take the lines well under- 
 ground and unless much care is used a mistake may be 
 made in setting down R for Z,, or vice versa, and a map 
 made from such incorrect notes will be considerably in 
 error. 
 
 Double centering is always advisable, but if the transit 
 is in perfect adjustment and the surveyor is very careful, 
 it is well to run the line in with azimuths first. Then 
 follow with a carefully run double centered line. On 
 account of the presence of ores or of tracks and metal tools 
 the needle cannot check underground surveys. In coal 
 mines the needle may sometimes be used to advantage in 
 the approximate location of rooms for placing same on 
 working maps, but the cases are few when the needle may 
 be used. 
 
33 2 PRACTICAL SURVEYING 
 
 The preservation of points is a vexing question in mine 
 surveys. When the ground is squeezing and sinking it is 
 impossible to preserve points for any length of time and 
 much of the work of a mining surveyor is concerned with 
 the preservation of lines. Usually it is best to drive nails 
 in the timber over the tunnels, these nails having in them 
 holes through which a plumb-bob string is tied. Flat- 
 headed horseshoe nails with holes drilled in the heads are 
 commonly used. To set up a transit it is necessary to have 
 points underneath on the ground so temporary points are 
 used. These temporary points are often made of pointed 
 nails driven through a piece of lead made in the shape of a 
 shallow cone and weighing several pounds. The point of 
 the nail is at the top. A plumb-bob is suspended from the 
 nail in the roof and the temporary hub placed on the ground 
 so the point of the plumb-bob is directly above the point 
 of the nail. The bob is then hung on one side and the 
 transit placed over the point. Backsights and foresights 
 are taken on the nails in the roof. On account of strong 
 currents of air in mine workings all lines should be as thin 
 as possible and plumb-bobs should weigh not less than two 
 pounds. If they swing too much they should be immersed 
 in buckets of oil with the lower end free of the bottom of the 
 bucket. 
 
 Various forms of plumb-bobs are made for use in mines 
 having lamps in the shank, so sights may be taken on the 
 flame. The writer found in his experience that the best 
 sight is a candle about two inches or less in length. When 
 lighted the candle is to some degree transparent and the 
 wick showing dimly through the luminous wax or paraffine 
 makes a very good point on which to sight. These lights 
 may be set on the ground by using the suspended plumb-bob 
 in the manner mentioned for setting temporary transit 
 points. For reading the verniers it was formerly custom- 
 ary to have tallow candles, but today small pocket electric 
 lamps are used, many instrument makers now supplying 
 reading lamps made especially for use in mines and tunnels. 
 
 The methods described are those used in all ordinary 
 tunnel work, the driving of very costly and important 
 tunnels for railways being work that is entrusted only to 
 men having good experience in this particular specialty. 
 
ENGINEERING SURVEYING 333 
 
 In mining work and in the driving of ordinary tunnels the 
 surveyor is interrupted often by the passing of cars, laborers, 
 etc., and as he is not permitted to interrupt the regular 
 work by his operations, they have the right of way. This 
 calls for the exercise of considerable patience and necessi- 
 tates rapid work. Mistakes, however, are not forgiven, 
 interrupted work being considered no excuse for mistakes. 
 When the surveyor can choose his own time to do the work 
 he works on Sunday, provided the mine shuts down on that 
 day, something seldom done, or he works when the mine is 
 shut down, which is a frequent occurrence in some lines of 
 mining, especially in a dull season. Usually, however, a 
 mine is run night and day and the surveyor is the least 
 considered of all the laborers, but his work nevertheless 
 must be kept up. 
 
 Underground surveys are made for the purpose of keeping 
 up the office maps and also for guiding the miners. When 
 done to keep up the maps a line is run down the tunnels, 
 and stakes set opposite openings in the face, or opposite 
 entrances to rooms, at regular stations, 50 to 100 ft. apart. 
 From these stakes approximate measurements (to the near- 
 est half foot) are made forward and back on line to the 
 edges and a measurement taken normal to the line in to the 
 face at each end and at the station. If closer results are 
 wanted closer measurements are made. The mine maps, 
 however, are on a scale that will hardly show a measure- 
 ment of less than 2 ft. anything less being estimated. 
 When the survey is made for the purpose of guiding the 
 miners, it is necessary to set three points on a line pointing 
 the direction in which the work is to proceed. The fore- 
 man, or boss, then hangs three plumb-bobs to these points 
 and uses them for sighting purposes. He will carry his 
 work in quite a distance, setting points ahead from which 
 to project his line, when he gets too far away to see the 
 three lines plainly that were left by the surveyor. When 
 he feels there is danger that he may be working to one 
 side the surveyor is sent for to set three more points, one 
 close in to the face. 
 
 Careful work is all that is necessary in ordinary under- 
 ground work, much care being necessary to prevent injury 
 to person and instrument. The best mining surveyors 
 
334 PRACTICAL SURVEYING 
 
 are men who have worked as miners before studying sur- 
 veying. 
 
 The greatest work of a mining surveyor is the carrying 
 of a line from the surface to the bottom of a shaft. Per- 
 haps a shaft is being sunk in a certain place and it is in- 
 tended to make an upraise to complete the job, or an up- 
 raise is to be started to reach a certain point on the surface. 
 The traverse cannot be closed until the shaft, or upraise, 
 is finished, and a small error in running the lines may cost 
 thousands of dollars. 
 
 First the point on the surface is located and a line run 
 carefully to the shaft, two points being set at the shaft, 
 one on either side and carefully marked with a tack. All 
 angles are double centered. The sight on the surface may 
 of course be several hundred feet but the line projected 
 down into the tunnel cannot be wider than the shaft, and 
 from this short base in the bottom the underground lines 
 must be run with such a degree of accuracy that the hole 
 at the starting point will be dug in the proper place. Hav- 
 ing marked the base line on the surface set the transit 
 over one of the points at the edge of the shaft and sight 
 down the shaft, placing a hub and tack there. For this 
 purpose it is often necessary to have an extra telescope on 
 the transit, swinging clear of the edge of the plate. When 
 the hub is set below, take the transit to the bottom and from 
 a backsight on the tack above, project the line through the 
 tunnel. 
 
 Work, such as that just described, cannot always be 
 accomplished because of the depth of the shaft. It then 
 becomes necessary to set points in the bottom of the shaft 
 by using plumb-bobs. These bobs should weigh not less 
 than ten pounds and be suspended by fine piano wires 
 instead of cord, provided with reels at the top to keep the 
 points off the ground. A cord is stretched across the top 
 of the shaft from one point to the other. The heavy bobs 
 are then let down on each side of the shaft, as close as 
 possible to the sides without touching same. The bobs 
 must be immersed in buckets of oil to prevent vibration and 
 swinging. When the lines are perfectly still the surveyor 
 goes back into the tunnel and gets himself in line with the 
 plumb lines, which are illuminated by his helpers. His 
 
ENGINEERING SURVEYING 335 
 
 transit is set up and leveled and a sight taken on the 
 wires. If he sees both, he moves the transit to one side and 
 tries again. By trial he finally gets in a position where 
 one line only is seen, the other being hidden behind it. 
 He reverses the telescope and tries again, proceeding in 
 this manner until he is satisfied the center of the transit 
 is exactly in line with the two wires and that the instru- 
 ment is level. He then sets two points, one ahead and one 
 back of the instrument as points on his underground base. 
 
 The student is referred for more information on mining 
 surveys to the best American books on the subject: 
 
 " Underground Surveying," by L. W. Trumbull, E. M. 
 ($3.00) and " Mine Surveying," by Edward B. Durham, 
 E.M. ($3.50.) 
 
 The best English book was written by the late Bennett H. \ 
 B rough. 
 
 HYDROGRAPHIC SURVEYS 
 
 Hydrographic surveys are surveys of rivers, lakes and 
 harbors. The survey of the shore line is made in the 
 customary manner of traverse surveys, the hydrographic 
 work being the survey of the 
 land under the water. Some- 
 times in making surveys of 
 rivers a method of triangula- 
 tion may be used as shown in 
 Fig. 215. 
 
 Assistants first set Stakes on FlG ' 2I S- Triangulation survey 
 
 , . , c , , , . of river banks. 
 
 each side ot the river at dis- 
 tances approximately equal to the width of the river. The 
 distance AB is carefully measured with a steel tape and 
 the transit is set on each stake on that side of the river. 
 From A, with a foresight on B, angles are taken to stakes I 
 and 2. The transit is then set on B and from a backsight 
 on A, angles are read to stakes I, 2, 3 and C. The transit 
 is then set on C and from a backsight on B angles are read 
 to stakes 2, 3, 4 and D. Proceeding in this manner for a 
 mile or so another distance is carefully measured with a 
 steel tape. By plane trigonometry the lengths of all the 
 lines are computed and the new base line is also computed 
 as a check on the work. The work is carried forward from 
 
336 
 
 PRACTICAL SURVEYING 
 
 the measured length of the new base line, these measured 
 bases being introduced at fairly regular intervals to keep 
 errors within reasonable limits. The limit of error is first 
 fixed for the work and if the error at any base line is too 
 great it may be wise to- have the measured bases at closer 
 intervals. In the figure EF is a measured base. The 
 distance to the shore line is measured on offset lines from 
 the numbered stakes, or the helpers on the banks may have 
 stadia rods and the shore line be located thus from the 
 transit. 
 
 Soundings may be made from boats, men in the boats 
 holding stadia rods which are read from a station on shore. 
 This is only possible in still water. For the proper sound- 
 ing and survey of lakes and harbors along the shores of 
 lakes or the sea, several methods are in vogue. 
 
 In one the boat containing the men making the soundings 
 keeps as closely as possible to a line by means of three stakes 
 set on shore while two observers at stations on shore read 
 angles from a base line to the boat with transits or sextants. 
 Sometimes the boat is rowed at a certain rate on a line 
 determined by sighting on range stakes, and soundings are 
 taken at definite intervals of time. Sometimes an observer 
 in the boat takes angles with a sextant to three points on 
 shore, thus introducing the "Three Point Problem," one 
 of the standard problems inserted in textbooks paying con- 
 siderable attention to the mathematical side of surveying. 
 Problem. Given three points in a triangle and the dis- 
 tances between them AB = 317 it., AC = 308 ft., and 
 BC = 478 ft.; also the angles at a point 
 D which these distances subtend in the 
 same plane with them, i.e., ADB = 24 
 50', and ADC = 27 44'; to find the 
 distance. of the station D from each of 
 them. 
 
 Construct the triangle ABC and on the 
 line BC, set off at C the angle BCd = 
 ADB = 24 50'; and at B set off the 
 FIG. 216. Three- angle CBd = ADC = 2? ^ p om t d 
 
 is located by the intersection of the lines 
 from B and C. Through the three points B, d, C draw 
 a circle. From A draw a line through d and produce it 
 
ENGINEERING SURVEYING 337 
 
 to an intersection with the circumference, thus locating the 
 point D. Draw lines from D to B and C. 
 
 Three sides being known in the triangle ABC, the angle B 
 = 39 25' 14.6" '; then ABd = ABC + dBC = 67 9' 14", when 
 ^4 and d are on different sides of BC, or = 11 41' 14.6", 
 when ^4 and d are on the same side of BC as in the present 
 case. 
 
 In the triangle BCd are given the side BC and the angles 
 B and C. Then the side Bd = 252.7 ft. 
 
 In the triangle ABd are given the sides AB and Bd 
 with the included angle ABd. Then the angle AdB = 
 I3i 53' 53", andBAd = 36 25' 53". 
 
 Then in the triangle ABD are given the angles and the 
 side AB. We find BD = 448.066 ft., and AD= 661.738 
 ft. In the triangle DBC are given the angles and the 
 side BC. We find DC = 591-5^3 ft. 
 
 If the triangle ABC is reversed so the point A is the 
 point nearest D, the angle BAd = 46 47' 32.2"; then BD 
 = 550.153 ft., AD = 282.25 ft. and CD = 528.4 ft. 
 
 1. If D be within the triangle, as at d, make the angles 
 BCD and CBD = supplements of BaA and AdC. 
 
 2. When D is in one of the sides, describe a segment on 
 BC containing the given angle. 
 
 3. If A and B be in a straight line with D, then BC and 
 CA subtend the same angle BDC. Solve the triangle DBC 
 after finding the angle at B. 
 
 4. If the three points A, B, C lie in a straight line, the 
 first operation will not be required. The other operations 
 are unchanged. 
 
 When making soundings of a river an approved method 
 is to stretch a cord across, having knots at definite inter- 
 vals and row a boat along the line, taking soundings at 
 each knot. This is possible only in rivers with sluggish 
 current. Another method is to set stations along the 
 bank at distances about equal to the width of the river and 
 row a boat across at an angle in each of the squares thus 
 formed, taking soundings at as nearly regular intervals as 
 possible. Each square has two diagonals and when the 
 notes are plotted it is assumed that the spaces between 
 soundings are equal to the length of the diagonal divided 
 by the number of soundings. In addition to the diagonal 
 
338 PRACTICAL SURVEYING 
 
 lines, soundings are taken on a straight line normal to the 
 banks of the river from the stake on one bank to the stake 
 on the opposite bank. When studying a navigable river 
 for the purpose of locating a bridge, the government re- 
 quires an accurate map of the banks for half a mile below 
 and one mile above the proposed bridge site. Soundings 
 must be shown at approximately 100 ft. intervals for the 
 entire distance on the map. The right-hand bank of the 
 river is the bank on the right hand of the observer when 
 traveling with the current. 
 
 Current observations must also be given in hydrographic 
 surveys of rivers. These are best obtained by measuring 
 a definite base line along one bank of the river and sight- 
 ing across the river perpendicularly at each end. At the 
 upper end floats are released and timed as they cross the 
 upper line, being timed again as they pass the lower line. 
 Bottles partly filled with water or sand and having in them 
 small rods carrying flags, make good floats for the purpose. 
 One bottle with a white flag should be traced down the 
 middle of the river, one with a red flag close to the right 
 bank and one with a blue flag close to the left bank. To 
 avoid confusion let the middle float go and after the recorder 
 makes his entry of the time, release a second float, and 
 when that record is made, the third. The rate of flow is 
 generally given in feet traveled per second. To determine 
 accurately the amount of water flowing in a stream it will 
 be necessary to use more than three floats to get the stream 
 lines, and in each line of floats there must be three: One 
 to give the velocity near the surface, one the velocity at 
 about half the depth and one to give the velocity close to 
 the bottom, so that for such work it will be necessary to 
 first make soundings. 
 
 To obtain the velocity at different depths the floats will 
 consist of very thin rods wire is best with flags at the 
 upper ends and weights at the bottom, regulated to keep 
 at a definite depth, which is not a difficult matter. Sound- 
 ing lines are prepared so they will not stretch when wet, 
 by wetting and wrapping around a tree or post as tightly as 
 possible and leaving there until dry. This operation is 
 repeated several times and takes out all the stretch, after 
 which they may be measured off in five-foot lengths, with 
 
ENGINEERING SURVEYING 
 
 339 
 
 colored cloths for markers amd small tags to indicate the 
 length. The weight on the sounding lines should be very 
 heavy so the line will be taut and there may be a reasonable 
 certainty that the depth obtained is vertical and not in- 
 clined. Tags at five-foot intervals are all that are neces- 
 sary, for lesser intervals may be measured with a rule by 
 the observer in the sounding boat. 
 
 Soundings in water are wanted for a number of reasons 
 but an engineer or surveyor in private practice usually 
 does such work to ascertain the yardage in dredging oper- 
 ations. The bottom must be first surveyed and after the 
 dredging is completed, or when a progress estimate is 
 wanted, must be again surveyed. A method used for 
 such work is to make soundings from a scow, the positions 
 occupied by the scow being obtained from the shore. 
 
 Fig. 217 is a diagrammatic 
 representation of a sounding 
 device used for such work. 
 Several graduated rods are 
 attached to the scow by pass- 
 ing through holes in boards 
 projecting over the edges, the ^fW^r 
 
 lower ends of the rods being ^ 
 
 \ 
 
 FIG. 217. Sounding from barge. 
 
 fastened to inclined pieces. 
 
 These inclined pieces are 
 
 hinged at the upper end where they are fastened to the scow 
 
 and the lower end of each drags on the bottom. Every 
 
 rise and fall in the bottom is read by an observer on the 
 
 scow as the area is "swept" over. 
 
 For complete information on hydrographic work the stu- 
 dent is referred to " Hydrographic Surveying," by Samuel 
 H. Lea ($2.00). 
 
 ROUTE SURVEYS 
 
 Route surveys are made for the location of railways, 
 highways, canals, pipe lines, etc., where a study must be 
 made of the possibilities for obtaining a proper grade. 
 With railways a one per cent grade (i ft. rise in 100 ft. of 
 distance) is steep. For common highways a 5 per cent 
 grade is about the limit on good roads, although grades 
 of 10 per cent are not uncommon. The grade of an electric 
 
340 PRACTICAL SURVEYING 
 
 road may be considerably higher than that of a steam road, 
 but pulling trains and wagons up a grade is expensive, 
 amounting to a vertical lift in a given distance equal to the 
 difference in elevation between the ends. The grade for 
 drainage ditches should be as great as possible in order to 
 carry the water without washing the banks and cutting 
 gullies in the side of the hill. This is true also of ditches 
 used in mining operation. Open earth ditches have a very 
 steep grade when it exceeds 10 ft. in one mile and not 
 all soil will stand as much grade. For drainage ditches it 
 is frequently difficult to get a grade greater than 2 ft. 
 in one mile. In irrigation work the problem is to take the 
 water to the highest point on the tract to be irrigated and 
 when this tract is reached to keep it high to serve the great- 
 est possible area. If the grade of an irrigation ditch is too 
 light it will silt up rapidly and aquatic plants will grow 
 and interfere with the flow. The problem is then to nicely 
 adjust the grade to keep the ditch, or canal, on high ground 
 and yet have a rapid enough flow to prevent undue silting 
 and growth of grass. Three feet in one mile is a big grade 
 for an irrigation ditch and some have a fall of only 8 ins. 
 to the mile. 
 
 A simple method of survey for roads, ditches and canals 
 consists in setting grade pegs according to methods given 
 in the chapter on leveling. The line follows very closely 
 the contours of the surface and is necessarily crooked. 
 A better way is to set the grade pegs farther apart, 100 to 
 200 ft., and follow this line with a transit line regularly 
 stationed. Elevations of the ground at each station are 
 taken and also side slopes so contours may be placed on 
 the map. A grade line is then selected to give as long 
 straight sections and as easy curves as may be advisable. 
 The grade line is plotted on profile paper and the cuts and 
 fills balanced. The line is then staked out on the ground 
 from the field notes picked out on the map. 
 
 A railway survey proceeds upon practically the same 
 system but the locating engineer goes ahead and picks 
 out the route by eye without running preliminary levels, 
 using a hand level to keep him close to grade. On this 
 line selected by the chief of party the transitman directs 
 the chainmen in setting stakes at stations one hundred 
 
ENGINEERING SURVEYING 
 
 341 
 
 common 
 
 feet long and takes angles at all changes in direction. The 
 leveler follows and gets the elevations at each stake. The 
 topographer slopeman he is called on some roads takes 
 the side slopes and the draftsman plats all the information 
 on a map, the leveler making a profile each evening of the 
 day's work. On the map the chief of party picks out the 
 line, which is then run in and "located" on the ground. 
 
 For full information on field and office methods of mak- 
 ing route surveys the student is referred to "Railroad 
 Location Surveys and Estimates," by F. Lavis, M. Am. 
 Soc. C. E. ($3.00). 
 
 CONTOURS 
 
 Many surveys are made for the purpose of obtaining 
 information about the shape of the surface of the ground. 
 This is called topographical surveying. The 
 method today for showing 
 the surface of the ground is 
 by means of contours and 
 every surveyor should know 
 how to make surveys for the 
 purpose of preparing con- 
 tour maps and should know 
 how to use such maps. A 
 surveyor given a tract of 
 land to subdivide should 
 first prepare a contour map 
 in order that he may lay the 
 roads on proper grades and 
 take care of sewerage and 
 drainage. Properly made 
 contour maps may be used 
 also in earthwork calcula- 
 tions. 
 
 FIG. 218. Contours and profile of hill. 
 
 In the upper part of Fig. 218 is shown a contour map of 
 a hill and the lower part shows a side view of the hill with 
 the contour lines dotted across the face. Contours are 
 level lines defining the shape of the surface of a hill, or 
 hollow. Assume that the hill in Fig. 218 was submerged 
 by water which stood at a depth of 26 ft. long enough 
 to leave a mark on the hillside at that height. The water 
 
342 
 
 PRACTICAL SURVEYING 
 
 then receded to a depth of 25 ft. and left a mark. It 
 receded 5 ft. at a time, standing at each level long 
 enough for debris to collect on the hillside at each step, 
 so that the respective elevations were marked on the sur- 
 face. Water surfaces are always level so each line of debris 
 marked a level line around the hill at the elevations shown. 
 A side view shows the lines as level, while the plan, or 
 map, gives the shape of the hillside. 
 
 In the upper part of Fig. 219 is shown a map of a hill- 
 side, having two projections. The straight lines show lines 
 
 FIG. 219. Selecting grades on contours. 
 
 connecting points A and B and C and D. The contours 
 are marked on the map and it is desired to obtain a uni- 
 form grade connecting C and D. First draw a straight 
 line from C to D and on this line note the elevations of the 
 points / and P. On the profile paper below plot these 
 points, and also the elevations of intermediate points and 
 the wavy line C, /, P, D is obtained. A straight line 
 therefore will not give a uniform grade. Following the 
 dotted line and plotting it on the profile a uniform grade 
 is obtained so that it is represented by the straight line CD 
 on the profile. 
 
 The dotted line is obtained in actual practice by drawing 
 
ENGINEERING SURVEYING 343 
 
 the straight line on the profile and then projecting the 
 points where it intersects the horizontal lines on the profile 
 paper up to an intersection with the contour lines on the 
 map. The intersection being obtained with the contour line 
 in each case, a dotted line is drawn from one contour to 
 the next as shown. 
 
 The line AB is similarly studied. First the straight line 
 was drawn on the map resulting in the very wavy line A, 
 L, N, M, B. Next a crooked line on the map was selected 
 resulting in the wavy profile A, E, F, H, B, which has 
 smaller depressions and less marked elevations than the 
 first line. To obtain a proper grade line on the map a 
 straight line should be drawn on the profile from A to B 
 and this line projected up to the map. 
 
 In making route surveys all data are obtained which are 
 necessary to show the contours for some distance either 
 side of the surveyed line. A grade is picked out on the 
 profile made by the leveler and the point where the grade 
 crosses each contour is marked by stations. These points 
 are laid off on the contour map and a line sketched in to 
 indicate the line of uniform grade. The chief engineer, or 
 chief of party, then lays a line as close to this as possible 
 so cuts and fills will be fairly equalized and the line be as 
 free as possible from great curvature and abrupt changes 
 in direction. 
 
 Fig. 220 is reproduced from an article by Dr. W. G. Ray- 
 mond, in Vol. VI, Transactions of the Technical Society of 
 the Pacific Coast, p. 72 (1889). The description is as fol- 
 lows: "A small reservoir is to be built on a hillside, and will 
 be partly in excavation and partly in embankment. The 
 contours are spaced 5 ft. apart. The top of the wall 
 shown by the full lines making the square is 10 ft. wide 
 and at an elevation of 660 ft. The reservoir is 20 ft. deep, 
 with inside and outside side slopes of 2 to I, making the 
 bottom elevation 640 ft. and 20 ft. square, the top being 
 100 ft. square on the inside. The dotted lines are contours 
 of the finished surfaces inside and outside the reservoir. 
 The areas in fill all fall within the broken line marked 
 abcdefghik, and the cut areas all fall within the broken line 
 marked abcdejgo. These broken lines are grade lines. 
 The areas of fill and cut are readily traced by following 
 
344 
 
 PRACTICAL SURVEYING 
 
 the closed figures formed by contours of equal elevation, 
 thus: 
 
 At 640 ft. level the area in fill is pst. 
 
 At 650 ft. level the area in fill is Imnuvxl. 
 
 At 650 ft. level the area in cut is I, 2, 3 uxl. 
 
 The other areas are as easily traced. The closed figures 
 formed by the intersecting contours of equal elevation are 
 
 640 
 
 640 
 
 (680 
 
 08Q 070 6(50 650. 
 
 FIG. 220. Computing earthwork from contour map. 
 
 horizontal areas of cut or fill separated by the common 
 vertical and perpendicular distances between successive 
 contours. These areas may be measured and the quanti- 
 ties intercepted between computed by the prismoidal, or 
 other formula, used in earthwork computations. Where 
 the grade contours do not intersect natural contours of 
 equal elevation, but themselves form closed areas, those 
 areas are to be measured. 
 
 The author worked for men who actually ran out the 
 contours on the ground for the purposes of map making. 
 A level was used to insure the lines being kept at the proper 
 elevation and stakes were set on these level lines at regular 
 
ENGINEERING SURVEYING 
 
 345 
 
 intervals. A transit party followed the level party taking 
 angles and making measurements to stakes so the lines 
 could be plotted on paper. This is a slow method and 
 seldom used. 
 
 A method often followed is to set stakes in rectangles or 
 squares as already described, taking the elevation of the 
 ground at the stake. This information is platted in the 
 office and contours drawn. On extensive surveys the rec- 
 tangles may have sides several hundred feet in length, 
 while for surveys of small tracts, such as city lots, small 
 reservoirs, etc., the dimensions may be from 10 to 50 ft. 
 It is not necessary, however, to set many stakes in making 
 topographical surveys. 
 
 PLANE TABLE WORK 
 
 One of the oldest map-making methods is that of the 
 plane table. It is in high favor for certain classes of work 
 today and a number of government engineers and sur- 
 veyors and men with government experience display great 
 sensitiveness if plane 
 tabling is criticized. 
 
 The author at one 
 time was a fairly good 
 "plane tabler" but must 
 confess a strong prefer- 
 ence for the transit and 
 stadia for making topo- 
 graphical surveys. The 
 transit is a universal sur- 
 veying instrument, while 
 the plane table costs as 
 much and is good for one 
 
 kind of work only, work FlG . 22I Plane table 
 
 that does not come often 
 
 enough to the average engineer, or surveyor, to justify the 
 expenditure provided he has a transit equipped with stadia 
 wires. The author has never been able to quite see where 
 the plane table possesses any advantage over the transit and 
 stadia, especially since modern plane tables have telescopes 
 equipped with stadia wires, and depend less than formerly 
 upon the principle of intersection. 
 
346 PRACTICAL SURVEYING 
 
 The plane table is illustrated in Fig. 221. A sheet of 
 paper is fastened to the board and the map drawn as 
 rapidly as the sights are taken. Damp weather and very 
 dry, hot, sunshiny weather affects the paper and the working 
 hours during the day are not long. The map is accurate 
 only for the scale to which it is drawn. A reduction of 
 course increases the apparent, but not the relative, accu- 
 racy while all errors are multiplied by an enlargement of the 
 map. 
 
 When a topographical survey is made with transit and 
 stadia, notes are placed in a book with such sketches as are 
 advisable and from these notes maps may be made to any 
 scale. The degree of accuracy is independent of the scale 
 and depends on the care with which the notes are taken and 
 recorded. The work may be done in any weather when 
 the graduations can be read on the rods and for more 
 hours in the day than are possible with a plane table. For 
 most of the work of this nature transit work is far more 
 rapid than plane table work. 
 
 For complete information on plane table work, as well as 
 on all kinds of topographical surveys the student is referred 
 to "Topographic, Trigonometric and Geodetic Surveying," 
 by Herbert M. Wilson, M. Am. Soc. C. E. ($3.50). 
 
 STADIA WORK 
 
 The writer assumes that every surveyor has a transit 
 equipped with stadia wires, so a description of methods is 
 in place in this book. The principles underlying the use 
 of stadia wires were explained in a preceding chapter. 
 
 To make a stadia survey for topographical purposes, a 
 number of points are selected at which the transit is set 
 and from each point side shots are taken to locate the in- 
 equalities of the ground, natural objects, fences, buildings, 
 etc. The first step is the selection of the instrument sta- 
 tions. The maximum length of sight from station to 
 station should not exceed 1000 ft., and the best average 
 length for sights is less than 700 ft. The instrument sta- 
 tions therefore should be selected with these limits in mind 
 and each should be on a point from which as many side 
 shots as possible may be taken. After the instrument sta- 
 
ENGINEERING SURVEYING 347 
 
 tions are selected it is a good plan to run a line of levels 
 with a good level and obtain the ground elevation at each 
 point. This then leaves the angular elevations as merely 
 a check. If time is an object and the survey is not very 
 important the leveling may be omitted and the elevations 
 obtained by means of vertical angles. 
 
 After each point is selected and a stake set, which does 
 not require a tack in the top, the transit is set up on each 
 station precisely as though a closed farm survey was being 
 made. The distance is read forward and back on each 
 course, also the vertical angles. The horizontal angles 
 are usually azimuths and when this survey is completed the 
 survey is closed in the usual way and the errors distributed. 
 Then each point is given its co-ordinate numbers so it may 
 be plotted by latitudes and departures. When the frame- 
 work is platted the side shots from each station are platted. 
 In addition to the readings taken with the stadia rods 
 intersection angles are read from each station to as many 
 other stations as can be seen, simply to check the work. 
 
 From the foregoing paragraph the student must not 
 imagine that the work is gone over twice; once for the 
 framework and once for the side shots. When each sta- 
 tion is occupied and the instrument oriented all the sights 
 possible from that station are read and recorded. When 
 the office work is done the reductions for distance and eleva- 
 tion are made first for the station framework and the results 
 plotted before the work of reducing and plotting the side 
 shots begins. 
 
 The instrument is first oriented. This means the line 
 through o and 180 lies in the selected meridian, which 
 may or may not be the true meridian. It is convenient, 
 however, to use the true meridian so angles may be checked 
 by the needle and to assist the man who makes the map to 
 keep himself located. People being trained from infancy 
 to look to the north as a starting point, the average man is 
 usually unable to read a map until it is oriented, a habit the 
 surveyor loses with experience. The instrument properly 
 oriented, a sight is taken to the station ahead, the azimuth 
 read, the rod read and the vertical angle read, all being 
 recorded in the field book. Keeping the lower clamp tight 
 the instrument is swung to right and left and the rod read 
 
348 PRACTICAL SURVEYING 
 
 as it is held on different points by the rodman, or men, for 
 on some ground an instrument man can keep two rodmen 
 busy. When he has three or four rodmen busy he must 
 have a recorder at his side to whom he can call off the read- 
 ings on the rod and the angles. When all the sights are 
 taken which are thought necessary from this station the 
 plates are set to the angle first read to the station ahead 
 and the telescope pointed properly. It will be found fre- 
 quently that the instrument has been slightly disturbed by 
 the intermediate sighting so this must be corrected by the 
 lower tangent screw, thus not disturbing the angle. How- 
 ever, if the plate is set to read the correct azimuth it will 
 not be necessary to touch the lower tangent screw, for the 
 instrument is then carried ahead to the next station. 
 
 Set the instrument over the next station. Level care- 
 fully and reversing the telescope sight back on the station 
 just left. Without in any way disturbing the angle on the 
 plate get an exact sight by means of the lower clamp and 
 tangent screws. Having clamped the transit in position, 
 read the angle, and if any change has taken place, bring 
 the plate and vernier to indicate the correct reading and 
 then re-set the line of sight on the last station. When 
 this is done and the telescope transited to read in a for- 
 ward direction it will lie in the line of sight between the 
 two stations with the proper azimuth indicated on the 
 plates. Now unclamp the plate, read the azimuth, vertical 
 angle and rod to the next station, record the readings, and 
 take the side shots as was done from the previous station. 
 Some writers do not advise the reversing of the telescope, 
 thinking it best to set the plate on vernier B to read an 
 angle 180 different from the forward azimuth from the 
 previous station. With this angle set on the plate the back- 
 ward sight is taken on the previous station with the tele- 
 scope erect, after which the plate clamp is loosened and 
 the instrument may be revolved on its vertical axis to take 
 sights in all directions. The author could never see a good 
 reason for this. It increases the number of operations; 
 it renders mistakes possible because the instrument man 
 might get mixed up in adding the 180, and it causes loss 
 of time. To keep the azimuth set on the plate and reverse 
 the telescope introduces an automatic check. 
 
ENGINEERING SURVEYING 349 
 
 All the line readings should be taken forward and back 
 and all the elevations checked by the positive and nega- 
 tive vertical angles. The side shots do not require such 
 repeating and the instrument man must be constantly on 
 guard to avoid setting down wrong readings. The side 
 shots are taken to every object it is desired to locate on the 
 map, fence corners, corners of buildings, bridges, etc., and 
 it is well to make a sketch of these on the opposite page. 
 The principal side shots, however, are taken to inequalities 
 of the surface of the ground to obtain their location and 
 elevation. An experienced instrument man with expe- 
 rienced rodmen will find 300 shots a day about an average 
 day's work, except in such rare cases where more than 50 
 shots can be taken from one station. The average number 
 of sights from one station in average farming country with 
 the average number of natural objects is about 20 to a 
 station, the number from some stations being less than 5. 
 A fault with beginners is to take too many sights to obtain 
 ground elevations. This is better, however, than to take 
 so few that the map has to be pretty thoroughly "guessed 
 out" when the contours are platted. The rule is to have 
 the rod held at each point where there is a decided change 
 in slope. An experienced rodman who has also helped to 
 make maps can cut down the working time of a party 
 one-half. Plentiful use of sketches is advisable. 
 
 The author for many years used loose sheets in his field 
 book having two lines ruled at right angles and crossing 
 in the middle of the sheet. From this as a center ten 
 circles were drawn with intervals between the lines of one- 
 tenth of an inch. Using the vertical line as the meridian 
 when a sight was taken to any fence, building, etc., and 
 the rod read, a ruler was laid from the center of the circle 
 at as nearly as possible the same angle with the meridian 
 and a point made at the distance indicated by the rod 
 reading, the interval between two circles being assumed 
 as 100 ft. After all these sights to objects were marked a 
 sketch was made showing each in its relative position. The 
 same method was pursued when getting the edges of steep 
 banks, large boulders, etc. Sheets, such as the author 
 used, can be drawn on tracing cloth by any surveyor and 
 printed by the direct process, showing dark lines on a 
 
350 PRACTICAL SURVEYING 
 
 white ground, or they may be duplicated by the mimeo- 
 graph, the hektograph or the clay process. The writer 
 had a zinc etching made showing three sets of circles, 
 somewhat smaller than recommended above, on one page 
 with heavy division lines, so sights from one station would 
 not be confused with sights from the other. From this 
 cut he had several thousand sheets printed. The sheets 
 being held in the fold of the book by a rubber band each 
 could be removed as fast as it was filled and the center of 
 the circle used at each station was numbered with the num- 
 
 FIG. 222. Stadia survey note book. 
 
 ber of the station. A copyrighted field book on a similar 
 plan is sold by instrument dealers, but as the author and 
 several of his friends used the method for a number of years 
 before the copyright was granted for the book mentioned, 
 no surveyor need fear to make his own sheets. A variation 
 on this was used by one surveyor, who had a rubber stamp 
 made with the concentric circles and the intersecting nor- 
 mal lines. This stamp was used on each page where it was 
 desired to make sketches, as it is not necessary to make 
 sketches from each station. The stamp was in a thin metal 
 case containing an ink pad and had a folded handle on 
 
ENGINEERING SURVEYING 351 
 
 top, so it could be carried in the vest pocket. A blotter 
 was necessary to prevent smudging of notes. 
 
 For ordinary work a Philadelphia leveling rod with alter- 
 nate hundredths black may be used. For long sights 
 graduations of a more pronounced character should be 
 used. The form of mark on the face of a rod may be 
 pointed or square, opinion today tending towards the dis- 
 carding of angular points and favoring square block-like 
 graduations. The author favored a combination of acute 
 and square marks a few years ago, but now prefers the 
 square marks. 
 
 The graduation shown in Fig. 223 was devised by Prof. 
 W. H. Burger of Northwestern University. The gradua- 
 tions for even feet are placed on the left and those for odd 
 
 FIG. 223. Stadia rod. 
 
 numbered feet on the right. Opposite each half foot is 
 placed a dot. *As with all stadia rods it is optional with 
 the user whether the feet are indicated by figures, it being 
 an advantage to omit them, for it will then make no differ- 
 ence if the rodman accidentally holds the rod upside down. 
 The instrument man can readily direct the middle hair to 
 the height of the telescope and read the number of gradu- 
 ations intercepted between the stadia wires. Some men 
 prefer to have foot marks on the rods and in such event 
 they are one-tenth of a foot high, the bottom touching the 
 division line, the figure being placed in the white space. 
 
 To make a rod use a piece of clear white pine, well 
 seasoned, three inches wide by one or if ins. thick, sur- 
 faced and sandpapered. Give it a thin filler coat and' on 
 top of that three coats of pure white paint. Let each coat 
 dry thoroughly and sandpaper it before applying the next 
 coat. With a steel tape carefully measure the rod and 
 make cuts at the foot divisions. On the ends of the rod 
 place iron or brass strips to prevent undue wear and abra- 
 sion. Make the pattern for one foot in length on tough 
 drawing paper and cut it out with a sharp knife. Lay 
 
35 2 PRACTICAL SURVEYING 
 
 this stencil on the foot division marks and with a fine- 
 pointed pencil trace the outline of the figure on the painted 
 surface. Use good black oil paint and a fine camel's hair 
 brush and fill in the marks. A steady hand is required. 
 On the back of the rod may be hung some device to insure 
 its verticality or the rodman may carry a rod level, this 
 being as good a device as can be used. In stadia work the 
 judgment of the rodman cannot be of service, for the rod 
 must be as nearly vertical as it is possible to get it and it 
 cannot be waved as in leveling operations. The length 
 
 FIG. 224. Protractor for platting stadia survey. 
 
 of the rod depends upon the work to be done, the writer 
 having 8-ft. and 12-ft. rods. Long rods may be made con- 
 venient for carrying by hinging them. Good stadia rods 
 may be purchased. 
 
 The notes are reduced in the office very rapidly, it being 
 best to use two men, one to call off and the other to look 
 up the tables, or use the diagram, and reduce the rod 
 bearing to horizontal distance and obtain the difference in 
 elevation. The difference in elevation is added to, or sub- 
 tracted from (according to the positive or negative sign 
 
ENGINEERING SURVEYING 353 
 
 of the vertical angle), the elevation of the ground at the 
 instrument station. When the instrument is set up the 
 surveyor measures with a tape or rod the height of the axis 
 of the telescope above the ground and directs the middle 
 wire to this height on the rod. The line of sight is then 
 parallel to a straight line drawn from the hub to the foot 
 of the rod. In the office the last two columns in the field 
 book are filled. 
 
 PLATTING STADIA WORK 
 
 To plat the work, the instrument stations are first placed 
 on paper by any approved method, the author doing this 
 work by computing the latitude and departure of each point 
 and platting by co-ordinates. Through each station draw 
 a fine line for the meridian and a perpendicular through it. 
 
 Take a 14-in. paper protractor, untrimmed, and cut out 
 the interior with a radius of 5^ ins., making the diameter 
 of the cut-out portion II ins. The figured graduations go 
 from o to 360, clockwise as on the transit plate. Draw 
 a fine ink line from o to 180 and from 90 to 270, the 
 lines terminating at the edges of the interior cut-out space. 
 When the protractor is placed on the perpendicular lines 
 through a station so the lines coincide with those above 
 mentioned, the station is at the center of the graduated 
 circle. The protractor is to 'be oriented so angles read on 
 it will coincide with the same angles read on the transit. 
 Fasten the protractor in place with two thumb tacks near 
 the upper edge, so it may be flopped over without disturb- 
 ing the adjustment, when plotting long sights. 
 
 Having decided upon the scale, glue to the bottom of 
 the scale at the zero end projecting beyond the edge from 
 | in. to J in. a piece of tough paper J in. wide. Put a fine 
 needle through this projecting strip at the zero graduation 
 and push the point through the station. The scale can 
 now be swung around the circle freely. 
 
 An assistant calls off the azimuth from the field book and 
 the draftsman swings the scale to this reading on the grad- 
 uated circle. The assistant calls off the horizontal dis- 
 tance, the draftsman scaling it, and putting a dot on the 
 paper. The draftsman then writes down the elevation 
 
354 
 
 PRACTICAL SURVEYING 
 
 read to him by the assistant. The shots are plotted in this 
 manner at each station. Sketches are transferred from 
 the field book to the map and the protractor moved to the 
 following station. When the plotting is completed each 
 point occupied by a rod shows on the map with the ground 
 elevation, and all buildings, fences, etc., are shown in light 
 lines. The next step is to put in the contours. 
 
 570 
 
 INTERPOLATING CONTOURS 
 
 In Fig. 225 is shown a method described by the author 
 in Engineering News, May 10, 1900, while Fig. 226 illus- 
 trates a method described 
 by H. F. Bascom, C.E., 
 in Engineering News, June 
 21, 1900. 
 
 In Fig. 225 two points 
 are shown with the eleva- 
 tions. A fine line is drawn 
 connecting them. A piece 
 of graduated paper is 
 placed at an angle with 
 the line with the gradua- 
 tions corresponding to the 
 lower elevation coinciding 
 
 FIG. 225. Interpolating contours, 
 other point and 
 
 touch the 
 
 squared paper correspond- 
 ing to the elevation of the 
 point. A straight-edge is 
 now placed under the tri- 
 angle. The triangle is 
 moved along the straight- 
 edge and as it succes- 
 sively reaches the num- 
 bered graduations a point 
 is marked at the inter- 
 section of the edge of 
 the triangle with the line 
 
 with it. A triangle is then 
 placed so one edge will 
 also the graduation on the 
 
 FIG. 226. Interpolating contours. 
 
 connecting the plotted points. 
 In Fig. 226 the triangle and straight-edge are not used. 
 
ENGINEERING SURVEYING 355 
 
 A strip of ruled paper is placed with the edge touching the 
 two plotted points. A second strip with graduations is 
 placed over it as shown. From each figured graduation 
 the intersecting line on the under piece of paper is followed 
 to the edge and the contour point marked. Contours are 
 seldom drawn at closer intervals than 5 ft. in height, de- 
 pending on the slope. On easy slopes the interval may be 
 less than 5 ft., while on very steep slopes the interval may 
 be as great as 25, or even 50, ft. When all the contour 
 points are marked the draftsman draws in the contours in 
 lead pencil. The instrument man and his assistants, hav- 
 ing a good knowledge of the shape of the surface of the 
 land, look over the map and suggest corrections before 
 the contours are inked in. Contours are usually drawn in 
 light brown or red lines. 
 
 PHOTOGRAPHIC TOPOGRAPHY 
 
 The author, on May 5, 1893, read before the Technical 
 Society of the Pacific Coast, a paper with the above title. 
 The paper is here given with the final paragraph only 
 omitted. In the intervening 22 years the camera has been 
 used on many surveys and its use is increasing. The 
 principles, however, are the same as when the paper was 
 prepared. The best book on the subject is "Phototop- 
 ographic Methods and Instruments," by J. A. Flemer, a 
 recent work ($5.00). 
 
 The object of this paper is to present a method of sur- 
 veying which will be a valuable auxiliary to plane-table 
 and stadia work, and in some cases is extremely useful alone. 
 It is by no means new, but is not very well known, and its 
 advantages are so great that engineers who have much 
 topographic surveying to do should understand it. 
 
 The principle depends upon the art. of projecting per- 
 spective views upon a horizontal plane, and was first used 
 by French naval officers in the beginning of this century in 
 the survey of coast lines. Perspective drawings were 
 made of certain places from two or more positions, and sex- 
 tant angles taken to several objects in the landscape, and 
 the angles recorded on the drawings. These drawings 
 were afterwards used in the mapping of the shore line, and 
 
356 PRACTICAL SURVEYING 
 
 the accuracy of the work depended upon the skill with 
 which the sketches were made. 
 
 The invention of photography made it possible to do 
 better work, and Colonel Laussedat of the French army 
 made experiments covering a period of years to perfect 
 the system. He published a work on the subject in 1865, 
 and no improvements have been made since, unless multi- 
 tudinous patents for cameras can be styled improvements. 
 
 The system is extensively used in Europe, and is very 
 little known in this country, where, of all places, one would 
 think it most valuable from an economical standpoint. 
 
 Instances can be given of the time in which various sur- 
 veys have been made by this method, but such records are 
 of no use unless the ground were known, but the French 
 engineers generally considered that a topographical survey 
 and map, when a camera with horizontal and vertical 
 circles is used, can be made in one- third the time required 
 by other methods. 
 
 Any camera may be used provided it is perfectly level 
 when the view is taken, and the smallest size adapted for 
 the work is one with a 5 by 8-in. plate. Smaller cameras 
 may be used in the same manner as a sketching pad, to 
 carry away unimportant details, but are of little practical 
 use unless most excellently made. All lenses must be good. 
 Although an ordinary camera may be used, still it is better 
 to have one for the purpose, provided with two levels on 
 top at right angles, and four leveling screws beneath. The 
 box should be solid, and focusing done by means of the objec- 
 tive slide. If the camera has a compass, or a horizontal 
 limb, and a vertical limb, so that up or down sights may be 
 taken, it will be complete. 
 
 Glass negatives are the most accurate to use, but films 
 on account of portability are more convenient. The weight 
 of the glass in the field is a drawback. 
 
 There are several adjustments of the camera which must 
 not be neglected. The first is called " the test for register. " 
 The film on the sensitive plate must exactly re-place the 
 surface of the ground glass. To do this set the instrument 
 up and focus for a distant view. Make a scratch to show 
 the relative positions of the plate and tube. Take out the 
 ground glass, and put in one with a transparent film. Focus 
 
ENGINEERING SURVEYING 
 
 357 
 
 on this, and make another mark. In actual work this 
 difference must be allowed for by changing the focus after 
 the removal of the ground glass, so the film on the plate 
 will be in the right position. The instrument maker should 
 see to the register, but it is as well to test his work. 
 
 As everything depends upon the focal distance this must 
 be accurately determined. Lens makers usually state the 
 focal distance, but as it is liable to vary, the operator had 
 better determine it himself. For simple convex lens, 
 double or piano convex lens, measure from optic center to 
 surface of ground glass. For double compound lenses pro- 
 ceed as follows (Fig. 227) : 
 
 6' 
 
 FIG. 227. 
 
 Set up several stakes in the ground distant from about 
 two or three hundred feet as SS'S"S'"S"". With transit 
 at 0, measure angles SOS", etc. Set up the camera at O, 
 level it carefully, make the image 5" coincide with a ver- 
 tical line through the center of the plate, and photograph 
 the stakes. The greater the distance apart on the plate 
 of the stakes the more accurate will be the determination of 
 the focal length. GG' represents the plate, OP the focal 
 length. Measure s"s"" on plate, then 
 
 OP = 
 
 s"s"' 
 tan a 
 
 SOS" = a. 
 
 For a test of distortion of the lens, with OP just found, 
 compute the angles SOS', SOS", etc., and if they agree with 
 angles taken with the transit the lens is free from distortion. 
 
358 PRACTICAL SURVEYING 
 
 Next, the horizon of the view must be found. Find the 
 center of the ground glass and draw a vertical and hori- 
 zontal line through it. Level the instrument carefully, 
 and set beside it an engineer's level, with the telescope at 
 same height as the lens of the camera. With the level find 
 some object in the distance. Turn the camera to this 
 object and move the object slide up and down until the 
 object is exactly at the intersection of the lines on the 
 ground glass; the object slide is then in its normal position, 
 and a scratch on the slide will determine that position for 
 all time. This scratch should be marked zero, and gradu- 
 ations should extend above and below it. The lower grad- 
 uations should have a minus sign before their numbers. 
 The plate holder should have four fine needles, so inserted 
 in the frame that their shadows will be photographed. 
 When the picture is developed, lines scratched on the plate 
 and connecting these points will occupy the same positions 
 as the lines drawn on the ground glass. The maker can 
 fix these needles in place. 
 
 The horizontal line represents the horizon of the picture, 
 and is the trace of a line on a level with the center of the 
 instrument. The object of graduating the vertical move- 
 ment of the object slide is to provide for a changing of the 
 horizon when necessary to limit sky views. By noting the 
 number on this index when the view is taken, the actual 
 horizon of the picture is set off from the horizon of the instru- 
 ment when plotting. 
 
 One more adjustment and we are done. This is to meas- 
 ure the field of view. Half the length of the plate, divided 
 by the focal length gives the tangent of half the horizontal 
 angle. The horizontal angle is the field of view we require, 
 and dividing 360 by this angle, gives us the number of 
 views needed to go around the circle. 
 
 Half the width of the plate, divided by the focal length, 
 gives us the tangent of half the vertical angle. As a gen- 
 eral proposition it may be stated that the greater the focal 
 length, the smaller the field of view and the greater the 
 accuracy in the work. The smaller the focal length, the 
 greater the field of view, greater rapidity (because fewer 
 views) and less accuracy. 
 
 To take a view, set up the camera and level it carefully. 
 
ENGINEERING SURVEYING 359 
 
 Adjust to focal length and set the object slide to the most 
 favorable position and note index number for fixing horizon 
 in views. Then adjust the stop, set in the plate holder and 
 verify the leveling. The levels are apt to get a little out 
 during all the handling. When all is ready, take the picture. 
 
 THE PLATTING 
 
 In the test for distortion, the whole idea of the method for 
 using the proof is given. The proofs are conical projec- 
 tions, and the optic center the point of view. The objects 
 represented are so far distant that their images are formed 
 on the same focal plane, and the point of sight remains 
 constant as in perspective drawing. 
 
 On the plate draw the horizontal line (the horizon) and 
 the vertical line, from the shadows of the points of the 
 needles. If the objective was above or below the horizon, 
 then instead of drawing it, draw a line parallel to it above 
 or below, as indicated by the index number observed. 
 From the vertical line measure to the right or left to the 
 object you wish to locate, and divide this distance by the 
 focal length, this will give the tangent of the horizontal 
 angle from the line of sight. From the horizontal line, 
 which is a trace of the plane of the optic center, measure 
 up or down, as the case may be, to the object, and divide 
 this distance by the focal length to obtain the tangent of 
 the vertical angle. 
 
 Every point located on the map must show in two views 
 at least. These views are taken from points previously 
 fixed by the triangulation or by direct measurement. The 
 points from which the views are taken must be plotted, and 
 from these points lines drawn on the bearings given in the 
 field notes when the view was taken. On these bearings 
 lay off the focal distance and at the end of this line draw 
 one at right angles to represent the plate. On the line 
 representing the plate, lay off on either side the distances 
 from the vertical to the object, and from the point of view 
 draw lines through these points. The lines through two 
 plates produced to an intersection locate the objects. 
 
 Fig. 228 illustrates well the method of plotting. OO f rep- 
 resent the points from which the views were taken. GG f 
 
PRACTICAL SURVEYING 
 
 and G"G'" the plates, OP and O'P' the line of direction of 
 sight. ABC, etc., and A'B'C, etc., represent on the plate 
 the objects to be located and their positions on the maps are 
 shown by the points of intersection. 
 
 FIG. 228. 
 
 To fax the elevation, measure the distance from point of 
 sight to object, and multiply into tangent of vertical angle 
 already found ; add to the elevation of the point from which 
 the sight was taken, the height of instrument, and add 
 or subtract, according to whether the point is above or 
 below the horizon, the height above ascertained. This will 
 give the elevation of the object above datum. 
 
 Spherical aberration does not interfere with the accuracy 
 of the work, provided the focal length is ascertained by 
 means of a point near the extremity of the plate in the 
 horizon. (See Fig. 227.) 
 
 FIELD WORK 
 
 The field work may be performed in one of three ways, 
 or a combination of all. 
 
 i. The ground may be triangulated with the transit 
 and views taken from the triangulation points with the 
 camera, the direction of the views to be ascertained by 
 azimuths from the lines between stations. These azimuths 
 are to be taken by a compass, or by means of a horizontal 
 limb. 
 
ENGINEERING SURVEYING 
 
 2. The camera may be used in connection with a pocket 
 compass, the work starting from a measured base. 
 
 3. The work may be done with a camera alone, fitted 
 with a compass, horizontal limb and vertical circle. In 
 this case the triangulation is carried on with the work. 
 With ordinary care either method is good. The camera 
 must be rigid and the plate truly vertical. 
 
 Below is a form of record. 
 
 
 
 View. 
 
 
 
 Station. 
 
 Number. 
 
 Index number. 
 
 Bearing. 
 
 Remarks. 
 
 A 
 
 I 
 
 O 
 
 1951 
 
 
 
 2 
 
 O 
 
 227 
 
 
 B 
 
 I 
 
 2 
 
 84 
 
 
 
 
 
 
 
 The index number may be the same or different for all 
 views at the same station. Any time of the year is good for 
 this work, and any hour of the day when the air is clear. 
 Long distances between stations should be chosen as short 
 bases increase error. A few views only are necessary, as 
 sketches may be made of unimportant places, and these 
 views should be well chosen. A little care exercised in 
 selecting positions will save much office work. 
 
 OFFICE WORK 
 
 Upon the scale used depends the accuracy of the plotting. 
 If the scale is large then very long sights should not be 
 attempted, but if the scale is small then of course the range 
 can be longer. The error in height is in proportion to the 
 distance, and Professor Hardy says that with a focal length 
 of 1.64 ft. this error will not exceed I ft. in 550 yds. 
 Colonel Laussedat has shown that with a scale of -3-^-$ with 
 a focal length of 0.5 m. points, 1500 meters distant, may be 
 represented, and with a scale of ^TrW the operations could 
 be conducted at 4000 meters. 
 
 When the plates are prepared for plotting the office notes 
 are placed in a book in seven columns as follows : 
 
362 PRACTICAL SURVEYING 
 
 FORM OF RECORD TO REDUCE HEIGHT TO COMMON DATUM 
 
 View. 
 
 Distance. 
 
 Point. 
 
 Ref. ft. 
 
 Ref. of sta., 
 
 ft. 
 
 True elev., 
 ft. 
 
 Remarks. 
 
 Va. 
 
 
 I 
 
 -(-IO2 
 
 -{-460 
 
 + 562 
 
 
 
 
 2 
 
 + QO 
 
 
 55 
 
 
 
 
 
 2< 
 
 
 4.7IT 
 
 
 
 
 
 
 
 
 
 The first column is for the views. 
 
 The second column contains the distances to the points. 
 
 The third column the names or numbers of the points. 
 
 The fourth column the height above or below station. 
 
 The fifth column elevation of station. 
 
 The sixth column elevation of point. 
 
 The seventh column for remarks. 
 
 For drawing in contours the fixing of natural and artificial 
 objects on the plan, with their heights noted, will give all 
 the data necessary together with a close inspection of the 
 proofs as the work proceeds. In the case of a bare country, 
 with no buildings, fences or trees, a few painted stakes or 
 flags put in at salient points will serve. 
 
 It is best to work directly from the negatives as the paper 
 positives are too much affected by atmospheric changes. 
 Blue prints are as easy to work from as silver prints if posi- 
 tives are used. 
 
 Surveying by camera has equal advantages with survey- 
 ing by plane-table as it is a graphic process, but it is more 
 exact than the plane-table as atmospheric changes have no 
 effect on the records. It is more rapid in the field work, and 
 is accurate for more than one scale in the plotted work. 
 
 As compared with transit and stadia it is more rapid 
 in the field, and a little quicker in the office. Like it, the 
 records may be kept, and reproduced at any time to differ- 
 ent scales, and it has a further advantage in that all errors 
 of observation are entirely eliminated. Various plans for 
 saving labor in the office work have been suggested, but 
 this paper is already long enough. 
 
 Before closing, attention should be called to one point, 
 namely, that the notes may be sent to a draftsman who 
 never was on the work, and a correct map may be drawn 
 by him. No other method approaches it in this particular. 
 
APPENDIX A 
 ESSENTIALS OF ALGEBRA 
 
 The average person whose mathematical training does not extend 
 beyond that given in the grade school arithmetic is afraid of shadows 
 when reading technical books. 
 
 Many men declare an expression like 
 
 V = -Kh (R* - r 2 ) 
 or 
 
 v = ^kA 
 
 2 
 
 to be algebra. Not having studied algebra they slur over their 
 studies until finally the art of using symbols is acquired to some 
 degree and they believe they know enough of algebra for practical 
 purposes. What they have learned is the ability to evaluate simple 
 formulas but of algebra they know nothing. The two expressions 
 above given are not algebra but are merely arithmetical rules written 
 in mathematical shorthand. 
 
 In ordinary language in a grade school arithmetic the rule for 
 finding the volume of a hollow cylinder appears as follows: 
 
 Multiply the radius of the outside of the cylinder by itself (that is, 
 square it] and subtract from the product the product of the radius of the 
 inside of the cylinder by itself. Multiply the difference between these 
 products by the height of the cylinder and this final product is to be multi- 
 plied by 3.1416. 
 
 Written in mathematical shorthand the rule appears: 
 
 V= irk (R* - r 2 ), 
 
 in which TT = 3.1416, the ratio of the circumference to the diameter 
 (that is the circumference is 3.1416 greater than 
 the diameter), 
 
 363 
 
PRACTICAL SURVEYING 
 
 h = height of cylinder, 
 
 R = radius of the outside of the cylinder, 
 
 r = radius of the inside of the cylinder, 
 
 V = volume (cubic contents) of the cylinder. 
 
 When h, R and r are in feet, V = cubic feet. When h, R and 
 r are in inches, V = cubic inches. Similarly for units in the 
 metric or any other system. 
 
 Enclosing some of the factors in a parenthesis indicates that they 
 are first dealt with, the result being a single factor. 
 
 A frustum of a cylinder is shown in Fig. 229. A rule for the 
 volume as written in a school arithmetic is as follows: 
 
 Multiply the area of the base by the average height. 
 
 Assuming that the student does not know how to obtain the 
 area of the base the rule will be written as follows: 
 
 Square the radius of the bottom. Multiply the product by the aver- 
 age height and this result by 3.1416. 
 
 Written in mathematical shorthand it appears : 
 
 in which V = volume of frustum, 
 
 TT = 3.1416, 
 
 r = radius of base, 
 
 h\ = height of one side, 
 
 hz = height of other side. 
 
 (Note. If the diameter is used instead of the radius divide 3.1416 
 by 4 so the rule appears V = 0.7854 d 2 ( -J-j 
 
 EXAMPLES. 
 
 i. What is the volume of a ring (cylinder) 8 ins. long, inside diam- 
 eter 6 ins. and outside diameter 7 ins.? 
 
 Here h = 8; R = J; r = | ; R 2 = 12.25; r 2 = 95 
 V = 3.1416 X 8 X (12.25 - 9) = 3-1416 X 8 
 X 3.25 = 81.68 cu. ins. 
 
ESSENTIALS OF ALGEBRA 365 
 
 2. What is the volume of a frustum of a solid cylinder 10 ins. in 
 diameter, having one side 9 ins. long and the other side 15 ins. long? 
 
 V = 3.1416 X 5 2 X - " = 942.48 cu. ins. 
 
 No matter how accustomed one may become to handling such 
 expressions the use of letters is generally confusing without some 
 knowledge of algebra, so, for the benefit of students who experience 
 difficulty in understanding the mathematical expressions and using 
 the formulas in this book the author will attempt to present some of 
 the essentials of algebra, which is in fact only an extension of 
 arithmetic. 
 
 Arithmetic, as defined in textbooks, is "the science of numbers and 
 the art of computation." Sir Isaac Newton termed Algebra "uni-. 
 versal arithmetic," for it enables us to reason out in a general way 
 solutions for many problems which by ordinary arithmetic would 
 involve much cut-and-try work. It is therefore an abbreviated 
 method for solving questions relating to numbers and quantities. 
 In proceeding to solve a problem algebraically the algebraic work is 
 completed when the formula is derived, the solution of the problem 
 from that point being done arithmetically. Briefly stated the reason- 
 ing is done by algebra and the work by arithmetic. 
 
 All the fundamental operations of algebra depend upon the single 
 principle that quantity bears no relation to order. When a quantity 
 is to be increased, or diminished, by other quantities the result will 
 be the same no matter in what order the work is done, provided none 
 of the quantities be neglected. 
 
 WHY LETTERS ARE USED IN ALGEBRA 
 
 In arithmetic ten characters, o, i, 2, 3, 4, 5, 6, 7, 8, 9, are used 
 to represent numbers, there being no limit to the size of a number 
 which may be expressed by a proper arrangement of the characters. 
 These characters, or figures, are in themselves sufficient for use in 
 solving all problems in common arithmetic except those in which 
 three numbers are given and it is necessary to find a fourth. The 
 method used was once known as the "Rule of Three," the problems 
 being termed problems in proportion. 
 
366 PRACTICAL SURVEYING 
 
 Proportion involves both multiplication and division. During 
 part of the work some character must be used to represent the un- 
 known quantity until it has a numerical value, the letter x being 
 generally used. The work is expressed in the following manner: 
 
 5 : 10 :: 20 : x, 
 
 which is read, As 5 is to IO, so is 20 to x. 
 
 The two end quantities are known as the extremes; the two middle 
 quantities (or terms) as the means. The product of the means = 
 the product of the extremes, therefore, 
 
 10 X 20 c, 
 
 - = 40 .-. * = 40. 
 
 Algebraically it is written, 
 
 5 20 
 
 10 x 
 
 and it is solved as follows: 
 
 10 
 
 5 # = 10 X 20 = 200, 
 therefore 
 
 200 
 x = - - = 40. 
 
 5 
 
 When two expressions are equal they are written with the sign of 
 equality ( = ) between them and thus a new complete expression 
 
 is obtained known as an equation. -^ = is a typical equation; 
 
 10 x 
 
 5/10 is known as the left side and 2O/X as the right side. The 
 resolution of an equation is accomplished by collecting all the known 
 quantities on one side, the unknown quantity then standing alone 
 on the other side. Algebra is used in separating the known from 
 the unknown quantities, after which the work is arithmetical and 
 the unknown quantity receives a numerical value. Any analogy, or 
 proportion, may be changed into an equation by making the product 
 of the extremes equal the product of the means. 
 
ESSENTIALS OF ALGEBRA 367 
 
 A knowledge of the rules and methods of algebra and common 
 arithmetic enables one to solve equations without difficulty. Given 
 a problem the mathematician first attempts to form a clear idea of 
 its nature and then tries to express its terms and the relation of its 
 parts in algebraic language, using the letters x, y, Z, etc., for the 
 unknown quantities. Usually more time is spent in attempting to 
 express a problem than is spent in solving it. 
 
 When a problem is proposed in which there are several unknowns 
 there must be as many independent equations as there are unknowns. 
 If the problem is properly limited the equations may be readily 
 written. 
 
 Suppose we have two quantities x and y. 
 
 Their sum amounts to 18. 
 
 This is written x + y = 18. 
 
 Their difference is 6; x y = 6. 
 
 Their product is 72; x X y or xy = 72. 
 
 x 
 Their quotient is 2; x -i- y or ~ = 2. 
 
 Expressed as a proportion x : y : : 4 : 2 . 
 
 Expressed as an equation 2 x = 4 y. 
 
 The sum of their squares is 180; X 2 + y 2 = 180. 
 
 The difference of their squares is 108; x 2 y 2 = 108. 
 
 The values of X and y are not material, the above equations being 
 illustrative only. The values are found by solving two simultaneous 
 simple equations. This may be done by addition or subtraction. 
 
 By addition: 
 
 x + y = 18 
 x y = 6 
 
 2 x = 24 
 
 24 
 
 X = -* = 12 
 2 
 
 .*. x = 12 and y 6. 
 Here the y and -{-y = o. 
 
368 PRACTICAL SURVEYING 
 
 By subtraction: 
 
 x + y = 18 
 
 x y = 6 
 
 2 y = 12 
 
 12 
 
 y = = 6. 
 
 2 
 
 Here the signs in the subtrahend were changed from + to 
 and from to +. 
 
 When the relation of one unknown quantity to another is simple, 
 a letter may be used for one of them and an expression for the other 
 deduced from the relation between them. This shortens the work 
 and, in the language of mathematicians, "renders it more elegant." 
 For example, if the difference between two quantities be 5, let y be 
 the less, then x = y + 5 ; or, let x be the greater and y = x 5. 
 
 The work will often be shortened if letters are taken, not for the 
 unknown quantities themselves, but for their sum, difference or any 
 other relation from which their values may be readily found. 
 
 A competent mathematician does not consider the solution of a 
 problem to be complete unless it exhibits all the cases which can 
 occur; and the results which flow from contrary suppositions can 
 only be exhibited by expressions such as those to be considered 
 when we demonstrate algebraic multiplication and division. In 
 practical work algebra is applied only when position or other condi- 
 tions must be assigned to quantities which are frequently to be 
 found under such conditions that they are sometimes to be added 
 and sometimes to be subtracted. The use of the positive ( + ) sign 
 to indicate when a quantity is to be added, and the negative ( ) 
 sign when it is to be subtracted does not alter the meaning affixed 
 to these signs (that is to indicate opposite conditions or states of 
 being) in the algebraic sense, for they are prefixed solely for the 
 purpose of subjecting the quantities to algebraic processes. This 
 will be illustrated when we touch on negative exponents. 
 
 USE OF SIGNS 
 
 In algebra multiplication is indicated by (X) as in arithmetic. 
 When quantities are represented by letters instead of figures, a dot 
 
ESSENTIALS OF ALGEBRA 369 
 
 is sometimes used, for the multiplication sign might be mistaken for 
 an x. Thus a X b and a b have the same meaning. 
 
 The dot is not used between figures for it might be mistaken for a 
 decimal point, although in practice when used to indicate multi- 
 plication it is written in a higher position than a period or 
 decimal point. It is customary to write letters with no sign or 
 mark between them when they are to be multiplied together, so 
 a X b, a b and ab have the same meaning. However, if several 
 letters form one quantity when grouped and this is to be multiplied 
 by a letter or another group of letters, the dot is sometimes used to 
 collect the factors. For example abode means aXbXcXdXe, 
 or a b c d e, but abc de means that abc form one factor and 
 de another, so a continued product is not wanted, but the product 
 of two factors. 
 
 Division is represented in the following ways, in the order in 
 
 which they are favored by mathematicians, 7, a/6, a -5- b. 
 
 b 
 
 The parenthesis is very common in algebraic work. Numbers or 
 letters enclosed within parentheses are treated as one group. For 
 example a (b -f- c -f- d) means ab + ac -f- ad, for a is a factor 
 common to the three letters, which may therefore be first added 
 together and the sum multiplied by a. 
 
 The sign for addition (+) is termed the plus sign in arithmetic 
 and the sign for subtraction ( ) is termed the minus sign. These 
 two signs in arithmetic indicate operations to be performed. 
 
 When used in algebra the plus sign is called the positive sign and 
 the minus sign is called the negative sign. These signs do not indi- 
 cate operations but represent existing conditions. Whatever is not 
 positive must be negative, and, conversely, whatever is not negative 
 must be positive. x 
 
 Addition and subtraction in algebra are independent of the signs, 
 for negative quantities may be added to, or subtracted from, posi- 
 tive or negative quantities; also positive quantities may be sub- 
 tracted from, or added to, negative as well as positive quantities. 
 
 The recognition of the fact that negative quantities actually exist 
 and may be operated on as though they are positive is the first 
 striking difference between algebra and common arithmetic. 
 
370 PRACTICAL SURVEYING 
 
 The money a man possesses is positive. The money he owes is 
 negative. These are elementary facts known to every person. When 
 a bookkeeper balances his books he performs an operation in algebra, 
 or, in mathematical language, "obtains the algebraic sum" of the 
 money passing through his hands. 
 
 A surveyor calls the elevation of the first floor of a building zero, 
 or o (plus or minus zero). All heights above are positive (-f) 
 and all depths below are negative ( ). From the first floor to the 
 cellar floor the distance is 9 ft. ( 9) and from the first to the second 
 floor the distance is 16 ft. ( + 16). 
 
 Find the difference in elevation between the basement floor and 
 the second floor. 
 
 Without any knowledge of algebra a person would say 9 + 16 = 
 25 ft., which is correct. 
 
 The elevation of the second floor is +16 and the elevation of the 
 basement floor is 9. How far is it from the second floor to the 
 basement floor? 
 
 Ans. 9 that is 9 
 
 + 16 i -16 
 
 -25 -25 
 
 The negative sign prefixed to the result shows the measurement 
 to be in a negative direction. 
 How far is it from the basement floor to the second floor? 
 
 Ans. +16 that is +16 
 
 - 9 + 9 
 
 + 25 +25 
 
 The positive sign prefixed to the result shows the measurement to 
 be in a positive direction. 
 
 In the first question the second floor elevation was subtracted 
 from that of the basement floor. In the second question the base- 
 ment floor elevation was subtracted from that of the second floor. 
 Thus the answers show direction as well as amount. A study of 
 the work shows that the distance between two points was found 
 and one quantity had not been lessened by taking it from another 
 as in arithmetic. 
 
ESSENTIALS OF ALGEBRA 371 
 
 The algebraic sum is the difference between the sums of the posi- 
 tive and negative quantities or expressions, hence the rule for sub- 
 traction in algebra: Change the sign of the subtrahend from + to , 
 or from to + , and then obtain the algebraic sum. 
 
 A man has $7.00 and on pay day receives $15.00. He owes a 
 friend $1.50, pays $3.50 for a meal ticket, pays $5.00 for a pair of 
 shoes and $6.00 for a hat. What has he left? 
 
 Place the amounts in two columns. 
 
 + 7-00 
 + 15-00 
 
 + 22.00 
 l6.00 
 
 + 6.00 
 
 In the above example the process of subtraction is purely arith- 
 metical and the result obtained is the algebraic sum of the positive 
 and negative sums of money. 
 
 Suppose the man purchased a suit of clothes for $18.00, the result 
 would be +6.00 18.00 = $12.00, showing he would be 
 in debt to the amount of $12.00. 
 
 A man swims up a stream at the rate of 4 miles per hour against a 
 current of 2 miles per hour. How far does he go in 6 hours? 
 
 Let his rate be +4 and the current be 2, for the directions are 
 opposite. Then in 6 hours he will go6(+4 2) = +12 miles. 
 The positive sign indicates that he went up stream 12 miles. Had 
 the current been 4 miles and the swimmer could only proceed at 
 the rate of +2 miles per hour, then in 6 hours he would find him- 
 self 12 miles down stream, that is 6 ( 4 + 2) = 12 miles. 
 
 Letters as well as figures are used freely in algebraic work to 
 represent either quantities or numbers. When a sign of operation 
 is placed between figures or letters they are termed factors. Thus 
 a, b and C are factors in the expression abc, or a X b X c\ and 2, 6 
 and 7 are factors in the expression 2 X 6 X 7. Factors are num- 
 bers from the multiplication of which a product results. 
 
 A figure and letter are often written together, as 5 a, 6 h, 7 &, 
 meaning 5X0, 6 X h, J X k. The figure and letter together form 
 
372 PRACTICAL SURVEYING 
 
 one factor, each being a coefficient of the factor. The figure is the 
 numerical coefficient and the letter is the literal coefficient of the factor. 
 For the sake of clearness the word "coefficient" is used to indicate 
 the numerical coefficient, the word "letter" indicating the literal 
 coefficient. Figures in arithmetic are characters used to represent 
 numbers. The word "figure" does not always mean a number, so 
 it is common to use the word "number" in mathematical work in- 
 stead of "figure," a custom to which the author will adhere. 
 
 ADDITION AND SUBTRACTION 
 
 The rule for obtaining the algebraic sum as a process of addition, 
 the rule for subtraction having been already given, is as follows: 
 
 Case I. When the quantities are alike. If the signs are alike add 
 the coefficients. If the signs are unlike take the difference between the 
 coefficients. To the sum or difference prefix the sign of the greater and 
 annex the common letter , or letters. 
 
 Case II. When the quantities are unlike. Write them one after 
 another with their proper signs and coefficients. 
 
 EXAMPLES. 
 
 Addition. 3 # ~ 5 b + 4 c ~ 3 d 2 e 
 
 6 a + 2 b 70 4d -j- S e 
 
 Subtraction. 8 ab 2 cd + $ ac 7 ad 
 3 ab + 4 cd + 5 ac 2 ad 
 $ ab 6cd* - 5 ad 
 
 When no sign is written in front of a quantity it is understood that 
 the quantity is positive. 
 
 If a b is to be added to 3 a we may first subtract b from a 
 and add the remainder to 3 a; or subtract b from 3 a and add a to 
 the remainder. The result in either case = 4 a b. Since a and b 
 are unlike quantities the only way their sum can be represented is 
 
 * In the example in subtraction the signs in the subtrahend were 
 mentally changed. 
 
ESSENTIALS OF ALGEBRA 373 
 
 to write a + b', conversely their difference can only be represented 
 by writing a b, or b a. ' 
 
 To subtract a c from 3 a we may first subtract c from a and 
 then subtract the remainder from 3 a\ or add C to $ a and then sub- 
 tract a. Thus, 3 a (a c) = 2 a + c. If a c is to be sub- 
 tracted from 3 a -f- 2 c, subtract a from 3 a and add c, the remainder 
 being 2 a -f 3 c. Thus 3 a + 2 c (a c) = 2 a + $ c. In 
 the foregoing illustrations of subtraction the work has all been done 
 by changing the sign of the quantities to be subtracted. 
 
 A little thought will show that we may add or subtract any two 
 terms, without regard to the other terms with which they may be 
 connected. 
 
 DEFINITIONS 
 
 A term is a simple quantity; as a, ab, 4 be, etc. 
 
 Like terms are those of which the literal parts are the same; as 
 4 ab, ab, 9 ab, etc. 
 
 Unlike terms are those which consist of different letters; as 2 ab, 
 3 be, 5 cd, etc. 
 
 Compound quantities consist of several terms connected by the 
 signs + or . A two term quantity is called a binomial; a three 
 term quantity a trinomial; a four term quantity a quadrinomial; 
 a quantity containing more than four terms is a polynomial, or 
 multinomial. It is customary to speak of all quantities higher than 
 binomials as polynomials. 
 
 Compound quantities are sometimes enclosed within parentheses, 
 which indicates, as already explained, that the group thus enclosed 
 must be first operated upon before using the other factors. When 
 two or more negative signs are enclosed within a parenthesis it 
 indicates continued subtraction. 
 
 When the parenthesis is preceded by a positive sign remove the 
 parenthesis and proceed to work as indicated by the signs. 
 
 When the parenthesis is preceded by a negative sign it means 
 the group as a factor is to be subtracted. Therefore change all the 
 signs within the parenthesis, the + to and the to + , after 
 which remove the parenthesis. 
 
 When several parentheses are encountered first remove the inner 
 one and work to the ends. 
 
374 PRACTICAL SURVEYING 
 
 EXAMPLE. Remove the parentheses in the following expression: 
 
 5 a - [2 a + (- 3 a - 4 b) - (a - 8 b) + 4 a] 
 = 5 a (2 a 30, 46 a + 8 + 4 a) 
 = 5 a - (2 a + 4 b) 
 = $ a 2 a 4 
 = 30-46. 
 
 MULTIPLICATION 
 
 In algebraic work the signs of the quantities must always be 
 considered. When multiplying algebraic quantities like signs pro- 
 duce + and unlike signs . That is, +a X a a 2 ; 
 aa = a 2 and a X a = a 2 . 
 
 Multiply the coefficients and to the product annex the letters of both 
 factors. If the multiplicand is compound multiply each term sepa- 
 rately by the multiplier. 
 
 EXAMPLE. 
 
 $ a 4 6 + 3 c 2 d + e i 
 
 5<* 
 
 25 a 2 20 ab + 15 ac 10 ad + 5 ae $ a 
 
 If the multiplier is compound multiply first by one of its terms, then 
 by another, etc., and add the products. 
 EXAMPLE. 
 
 2 x 2 3 xy +6 
 
 3 x 2 + 3 xy 5 
 
 6 x* - 9 tfy + 18 x 2 
 
 + 6 opy 9 x 2 y 2 + 18 xy 
 
 - lox 2 + i$xy - 30 
 
 6 x 4 3 x*y + 8 x 2 - 9 x*y* + 33 xy 30 
 
 It helps if the quantities are arranged in some order. In the ex- 
 ample given above the product comes out according to the descend- 
 ing values of X } both multiplicand and multiplier having been so 
 
ESSENTIALS OF ALGEBRA 375 
 
 arranged. This carries out the principle that quantity is indepen- 
 dent of order and that the positive and negative signs indicate a con- 
 dition or state of being rather than an operation. 
 
 Powers of the same quantity are multiplied by adding their expo- 
 nents, this fact being dealt with in the chapter where the use of loga- 
 rithms is explained. 
 
 Multiplication is a shortened method of addition as division is a 
 shortened method of subtraction. In multiplying a b by C, we 
 may either first subtract and then multiply, or first multiply and 
 then subtract. The latter is the order in algebra; we first multiply 
 a by C = ac, and then b by c = be. The latter is subtracted 
 from the former, the result being ac be, with the signs the same 
 as those of the multiplicand. 
 
 In multiplying a b by c d, first multiply a b by C as 
 before = ac bc\ then multiply a b by d = ad bd, which 
 subtract from the first product. This is accomplished by changing 
 the signs, when it becomes ad + bd, the signs being the opposite 
 of those in the multiplicand. 
 
 The first and last terms show that quantities with like signs mul- 
 tiplied together produce + , and the other two terms show that 
 those which have unlike signs produce . 
 
 DIVISION 
 
 When the signs are alike the sign of the quotient is +, but if the 
 signs be unlike the sign of the quotient is . 
 
 The above statement is self-evident; for the divisor multiplied 
 by the quotient must produce the dividend with its proper sign.' 
 The whole operation depends upon the principle that the value 
 of a quantity is not altered by both multiplying and dividing it by 
 the same quantity. 
 
 Powers of the same quantity are divided by subtracting the ex- 
 ponent of the divisor from that of the dividend; the remainder is 
 the exponent of the quotient. See the chapter dealing with loga- 
 rithms. 
 
 When the divisor is a simple quantity, write it under the dividend in 
 the form of a fraction; cancel like quantities, and divide the coefficients 
 by their greatest common divisor. 
 
376 PRACTICAL SURVEYING 
 
 EXAMPLES. 
 
 // the dividend is compound divide each term separately by the divisor. 
 EXAMPLE. 
 
 6^ a 3 b 2 (? 4.2 a 2 b 3 (? o ac 
 
 - 2 _ i. be 
 
 14 a 2 b 2 c 2 2 
 
 g divisor is compound arrange the terms of the dividend and 
 divisor according to the powers of some letter. Divide the first term oj 
 the dividend by the first term of the divisor to obtain the first term of the 
 quotient, then multiply the whole divisor by this term, and subtract the 
 product from the dividend; bring down as many terms to the remainder as 
 may be requisite for a new partial dividend, with which proceed as before. 
 EXAMPLE. 
 
 a 3 - 
 a 3 - 
 
 3 
 
 a 2 b 
 
 +.3 
 
 + 2 
 
 ab 2 - V 
 ab 2 
 
 a 
 
 - b 
 
 + 
 
 b 2 
 
 a 2 
 
 2 ab 
 
 
 
 2 
 2 
 
 a 2 b 
 a 2 b 
 
 
 
 
 
 + afc 2 - W 
 + a6 2 - ft 8 
 
 Note that while the signs are written in the partial subtrahends 
 as produced by the multiplication, they are changed, mentally, for 
 each subtraction. 
 
 When in dividing the last remainder is a simple quantity, place 
 the divisor below it in the form of a fraction, and annex it with its 
 proper sign to the quotient. 
 
 FRACTIONS 
 
 Fractions are the same in algebra as in common arithmetic save 
 for the fact that letters are used in algebra. They may be added, 
 
ESSENTIALS OF ALGEBRA 377 
 
 subtracted, multiplied and divided by the rules already given for 
 dealing with whole quantities. They are reduced to a common 
 denominator exactly as in arithmetic. 
 
 NEGATIVE QUANTITIES 
 
 A negative quantity standing by itself has, strictly speaking, no 
 meaning. 
 
 For example if C be the difference between a and b the algebraic 
 expression is a b = c, when a is greater than b', or b a = 
 c, when a is smaller than b. The expression c, however, is 
 impossible arithmetically, for a greater quantity cannot be taken 
 from a less. 
 
 Join the negative quantity to another, as m C and the expres- 
 sion may be subjected to all the operations of algebra, therefore it 
 is proper. If there is any absurdity it does not appear until the 
 result is obtained, when it shows that some condition inconsistent 
 with the other conditions has been admitted into the question. A 
 negative result when it agrees with the steps of the processes in 
 solving a problem is a proper algebraic result, for it points out the 
 impossibility of the conditions and thus has a use, in that it places a 
 limit on the terms of the question. 
 
 In all algebraic work it is necessary to pay close attention to the 
 positive and negative expressions and the forms which result from 
 them. There must be no hesitation in the operations for it has been 
 shown how quantities, and single terms may be added, subtracted, 
 multiplied and divided by one another and how the signs of the 
 results are obtained. These signs do not belong to the terms taken 
 as isolated quantities, but express the relation existing between the 
 terms of the result. 
 
 The product of a b by a b = a 2 - - 2 ab -f b 2 and the 
 product of b a by b a = a 2 2 ab + b 2 . There is nothing 
 in the product to indicate whether a is greater or less than 6, or, 
 looking at it in another way, .if a b = c, whether the product 
 has arisen from -\-c or from c, for each of these squared = -\-C 2 . 
 The square root of -\-c 2 may be either -\-c or c and the square 
 root of c 2 is impossible. 
 
378 PRACTICAL SURVEYING 
 
 The formula a 2 b 2 = (a -f b) (a b) is of great service in 
 algebra and should be memorized. Let 
 
 a 2 + b 2 = a 2 - b 2 X -i = (a + b Vi) (a - b VT). 
 
 The use of negative exponents often simplifies work. We can 
 
 a b 
 
 divide a b by a 2 by writing them in the form of a fraction and 
 
 a 2 
 
 a* a 2 
 
 canceling like quantities, -- - = a 3 -!, or merely subtract the 
 a 2 a 2 
 
 exponent of the divisor from that of the dividend a 5 " 2 = a 3 . 
 Now divide a 2 by a 5 . By writing the work in fractional form 
 
 CL T T 
 
 = . By subtraction, a 2 a b = a" 3 ' so that a" 3 = . 
 a 5 a 3 a 3 
 
 In such examples the negative exponent does not represent a nega- 
 tive quantity, but only shows that the quantity placed in the numer- 
 ator should be in the denominator. In either place it can be subjected 
 to all the rules of algebra, but in some problems the use of the nega- 
 tive exponent is a convenient method for getting rid of fractions. 
 The examples given show that any quantity may be removed from 
 the numerator to the denominator, or vice versa, by changing the 
 
 nj 9 
 
 sign of the exponent. Thus - - = a?bc~ 2 and ab~ 2 c 2 = - 
 
 c* cr 
 
 Roots may be expressed by fractional exponents, as V a = a 2 , 
 and fractions by negative exponents as shown, so the following 
 expressions should be carefully studied: 
 
 xa 
 
 In arithmetic it is often found to be convenient to reduce a com- 
 mon fraction to a decimal, and the negative exponent of a fraction 
 is the result of an equivalent operation. 
 
ESSENTIALS OF ALGEBRA 379 
 
 Notice the ease with which the following expressions are handled 
 by the use of fractional and negative exponents. 
 
 Vx X ^x = #M = x*x* = # M = a* 
 ^ _-*-*_ x l - V x 
 
 Exponents, the student must remember, are added and subtracted, 
 being logarithms. 
 
 SECONDARY OPERATIONS 
 
 To facilitate work a number of rules have been evolved from a 
 study of the development of algebraic expressions when raised regu- 
 larly to higher powers. An expert algebraist is thus enabled to save 
 a great deal of time because he remembers how one factor follows 
 another. He not only can write quickly without actual multiplica- 
 tion, the expansion of an expression to any degree, but conversely 
 can "factor" expressions of any degree. By factoring is meant to 
 reduce an expression to its elemental factors. 
 
 EQUATIONS 
 
 A statement in algebraic language that one quantity equals an- 
 other is an equation. 
 
 A simple equation is one of the first degree. That is, the radical 
 sign and exponents are not used in either the statement or the solu- 
 tion. 
 
 A quadratic equation is of the second degree and the square root 
 of a quantity appearing in it must be obtained. 
 
 A cubic equation is of the third degree; a bi-quadratic equation is 
 of the fourth degree, etc. 
 
 The degree of an equation is the degree of its highest term. 
 
 To solve an equation the following rules must be observed in the 
 order given. 
 
380 PRACTICAL SURVEYING 
 
 ist. Clear of fractions. 
 
 If any term be divided by any quantity, multiply every term by this 
 divisor (denominator} . 
 
 2nd. Write all the known quantities on one side of the equation 
 and the unknown quantities on the other side. Then collect by 
 addition. 
 
 Any term may be transposed from one side of an equation to the other 
 by changing its sign from + to , or from to -\-. 
 
 If a term be found on both sides with the same sign it should be can- 
 celed in both. 
 
 yrd. If the unknown quantity be multiplied by any other, divide 
 both sides by the multiplier. 
 
 In this way the value of an unknown quantity is found when there 
 are no surds or powers. 
 
 4th. If the equation contains a surd (an expression of a root of an 
 algebraic quantity which is not a complete power) bring the surd to one 
 side by itself, take away the radical sign and raise the other side to the 
 corresponding power. 
 
 5//f. // one side of the equation be a complete power take the corre- 
 sponding root of both sides. 
 
 At no time during the process of solving an equation can the 
 sides be unequal and the foregoing rules maintain equality of the 
 sides as follows: 
 
 Rule i. Both sides are multiplied by the same quantity. 
 
 Rule 2. In transposition the same quantity is subtracted from 
 both sides. 
 
 Rule 3. Both sides are divided by the same quantity. 
 
 Rule 4. Both sides are raised to the same power. 
 
 Rule 5. The same root is taken of both sides. 
 
 Solve 2 x = 3_* -h 4- 
 
 4 4 
 
 Clear of fractions, 8^19 = 3^ + 16. 
 
 Transpose, 8 X 3 x = 16 + 19. 
 
 Collect, 5 x = 35. 
 
 Divide by 5, x = ^ = 7. 
 
ESSENTIALS OF ALGEBRA 381 
 
 Solve, (3 x -f i)* + 5 = 10. 
 
 Transpose, (3 x -f- i) 2 = 10 5 = 5. 
 
 Square by Rule 4, 3 x + I = 5 2 = 25. 
 
 Transpose, 3^ = 25 i = 24. 
 
 Divide by 3, x = = 8. 
 
 o 
 
 Solve, 9 x 2 + 9 = 3 x 2 + 63. 
 
 Transpose, 9 X 2 - 3 x 2 = 63 - 9 = 54. 
 
 Collect, 6x 2 = 54. 
 
 Divide by 6, x 2 = *r = 9. 
 
 Extract the root, X = Vq = 3. 
 
 It has been stated that any proportion or analogy may be turned 
 into an equation by making the product of the first and fourth terms 
 equal to the product of the second and third terms. This will now be 
 illustrated. 
 
 2 + x : 6 x :: 15 : 9. 
 
 Then 9 (2 + x) = 15 (6 - x). 
 
 or 18 + 9 x = 90 15 x. 
 
 Transpose, 9 X + 15 x = 90 18 = 72. 
 
 Collect, 24 x = 72. 
 
 Divide by 24, x = = 3. 
 
 x 5 : 2 x :: 5 : 20. 
 Then 20 X x 5 = 5X22; 
 
 or 20 x 100 = 10 x. 
 
 Transpose, 20 x 10 X = 100. 
 Collect, IOX = 100. 
 
 IOO 
 
 Divide by 10. X = = 10. 
 
 10 
 
 The age of a man is a and the age of his son is b. In how many 
 years will the father be n times as old as the son? 
 
382 PRACTICAL SURVEYING 
 
 Solution. As time passes there will be added to each age an incre- 
 ment, x. The ratio n between the ages of the father and son de- 
 pends upon this factor, or increment. Then 
 
 _ 
 
 Transpose, a -f x = nb + nx. 
 
 Collect, nx x = a nb. 
 
 x (n i) = a nb. 
 
 a nb 
 Divide, x 
 
 n i 
 
 Assume the father's age to be 26 and that of the son to be 4. When 
 will the father be twice as old as the son? 
 
 26 - (2 X 4) 
 x = s 3* = 18 years. 
 
 2 1 
 
 a + x _ 26 + 18 44 
 b + x " 4 + 18 "~ 22 " 
 
 Problems. Answers. 
 
 I- 5* + 3 = 2X + I 5- x = 4. 
 
 62 vV x- 
 
 x = 4 x = 6. 
 
 o 
 
 ^ x . x _ _3_ 
 
 3 ' 2 3 4 " 9 i3 
 
 x 2 a 
 
 4. ^ a = '. x = 
 
 x - a 2 
 
 s . _^_ + ^ = 6 . ,, = /6 - 2 
 
 T I /y T /y 
 
 T^ A/ 1 wV 
 
 6. x + (" + r)* = 
 
 (a 2 + * 2 )* V^ 
 
 Equations may also be solved by factoring. 
 
ESSENTIALS OF ALGEBRA 383 
 
 SIMULTANEOUS SIMPLE EQUATIONS 
 
 When there are several unknown quantities there must be an in- 
 dependent equation for each quantity; from these an equation must 
 be deduced which contains only one of the unknowns and this equa- 
 tion may be solved by the preceding rules. Three rules are in gen- 
 eral use for solving simultaneous simple equations. It is immaterial 
 which is used. The object is to eliminate one unknown after another 
 until but one is left. 
 
 Rule I. This appears to be the most regular. 
 
 (a) Find a value for one of the unknowns, assuming all the rest to 
 be known. 
 
 (b) Make these values equal to one another and from them find a 
 value for another unknown. 
 
 (c) Repeat (b) successively with the remaining unknowns until the 
 final equation results. 
 
 Rule II. This is a little shorter than the first, but the reductions 
 are more intricate. 
 
 (a) Find a value for one of the unknown quantities in that equation 
 in which it is the least involved. 
 
 (b) Substitute this value and its powers for that unknown quantity 
 and its powers in all the other equations. 
 
 (c) Proceed as in (b) with these equations to get rid of other unknown 
 quantities. 
 
 Rule III. This is the most simple and expeditious. 
 
 (a) Multiply the equations by such quantities as will make the co- 
 efficients of one of the unknown quantities, or of its highest power, the 
 same in all the equations. 
 
 (b) If these equal terms have like signs subtract them, but if signs are 
 unlike add them, thus giving rise to new equations. 
 
 (c) Continue as in (a) and (b) with the new equations. 
 Examples under Rule III. 
 
 i. Let the equations 'be I2 | . Find x and y. 
 
 5* + 3? = 50) 
 First equation multiplied by 5, 5 # + 5 ;y = 60 
 
 5 * + 3 y = SQ 
 
 2 y = 10 
 
3*4 
 
 PRACTICAL SURVEYING 
 
 Divide, 
 
 IO 
 
 y---s. 
 
 x = 12 5 
 
 * + y + ^ = 53 
 2. Let the equations be x + 2y + $z = 105 
 
 # + 3:v + 4z = 134 
 
 Second equation, x + 2 y -}- $ z = 105 
 First equation, x X y + 2 = 53 
 Subtract, 
 
 . Find X, y and 2. 
 
 y + 2Z = 52= Fourth equation. 
 
 Third equation, #. 
 Second equation, ^ 
 Subtract, 
 
 Fourth equation, 
 Fifth equation, 
 
 42; = 134 
 32 = 105 
 
 y -f 
 
 29 = Fifth equation. 
 
 y -f- 2 Z = 52 
 
 y + z = 29 
 
 a = 23 
 
 y = 29 - 2 = 29 - 23 = 6. 
 
 * = 53 ~ (6 + 23) = 24. 
 
 By substituting these values in the original equations they may be 
 solved and a proof obtained of the accuracy of the work. 
 
 I. 
 
 2. - 
 2 
 
 Problems. 
 
 - $y = 90. 
 
 + $y = 160. 
 
 * =z6. 
 
 x y 
 
 Z = 2. 
 
 5 9 
 
 Answers, 
 x = 30. 
 y = 20. 
 
 x = 20. 
 ? = 18. 
 
 a. 
 
 3- x + y 
 
 x 2 - y> - b. 
 
 2 a 
 
 2 a 
 
ESSENTIALS OF ALGEBRA 385 
 
 3# - U = 2X + y + i x = 13. 
 
 3 5 y = 3- 
 
 The student must remember that the solution of equations in- 
 volving fractions is as easy as the solution of equations in which no 
 fractions appear. They merely take a little more time, and care 
 must be exercised that nothing is missed. It has been stated before 
 that fractions in algebra are treated exactly as fractions are treated 
 in common arithmetic. 
 
 I 
 
 5. x + 100 = y -f z, x = 9 
 
 II 
 
 b + a c 
 
 2 
 a + c b 
 
 2 
 c -f b a 
 
 z + 100 = 3 x + 3 y. z = 63 
 
 6. x + y = a. a 
 
 a; + z = b. y 
 
 y + z = c. i 
 
 7. 
 
 I 
 - x 
 
 + % 
 
 . i 
 
 + -z 
 
 = 62. 
 
 
 2 
 
 3 
 
 4 
 
 
 
 I 
 
 i 
 
 i 
 
 
 
 2 
 
 4 
 
 5* 
 
 -47- 
 
 
 I 
 
 4 
 
 i 
 -\- - v 
 
 5 
 
 +; 
 
 = 38. 
 
 # = 24. 
 = 60. 
 
 2= I2O. 
 
 In problems containing fractions the work is simplified by recast- 
 
 T y 
 
 ing some of the expressions. For example - x may be written - 
 
 2 2 
 
 ,2 2 # 
 
 and - x may be written , etc. 
 o X 
 
386 PRACTICAL SURVEYING 
 
 QUADRATIC EQUATIONS 
 
 A quadratic equation often comes to the engineer in the following 
 form 
 
 M = Rbh 2 , 
 
 then h=\/M 
 
 Rb' 
 
 in which M = resisting moment (should be equal to the bending 
 
 moment), 
 
 R = moment factor (unit moment of resistance), 
 b = breadth of beam, 
 h = height (depth) of beam. 
 
 For a rectangular beam the resisting moment = * - in which 
 
 6 
 
 / = maximum fiber stress (skin stress) and 
 
 A pure quadratic equation contains only the second power of the 
 unknown quantity. 
 
 An affected quadratic equation contains the first and second 
 powers of the unknown quantity. 
 
 Affected quadratic equations assume one of the following three 
 forms: 
 
 i. 
 
 x 2 + ax = +b 
 
 . ,. . -a Va 2 + 46 
 
 in which x = 
 
 2 
 
 2. x 2 ax = +b 
 
 +a Va 2 + 4 b 
 in which x = 
 
 2 
 
 3. x 2 ax = b 
 
 i.. i. +a Va 2 - 4 b 
 in which x = ~ 
 
ESSENTIALS OF ALGEBRA 387 
 
 The student should commit the above formulas to memory. 
 
 Since the square root of x 2 2 ax + a 2 is either a x or x a, 
 the root of the known side of the equation must have both the signs 
 + and before it. Sometimes both give the proper solutions, 
 and at other times only one of them is proper. 
 
 If a positive answer is required the sign of the radical in (i) and 
 (2) must be -f-, but in (3) it may be either -f- or . There is how- 
 ever a limitation in this case for 4 b must not be greater than a 2 , 
 otherwise the quantity below the radical sign would be negative 
 and its root impossible. This happens when the absolute term b is 
 
 greater than - a 2 , the square of - the coefficient of x. 
 4 2 
 
 In solving an affected quadratic several methods are at the service 
 of the mathematician. Some problems are more readily solved by 
 one method than by another. The method best retained by the 
 memory is that of completing the square. The methods in the 
 order of general custom are as follows: 
 
 i. By inspection. 
 
 The sum of the roots is equal to the coefficient of the first power 
 of the unknown quantity with the sign changed, and the product of 
 the roots is equal to the right-hand member of the equation with the 
 sign changed. 
 
 x 2 6 x = 72. 
 
 Sum of the roots, = -f- 6. 
 
 Product of the roots, = 72. 
 
 2. By factoring. 
 
 x 2 6 x = 72. 
 
 x 2 6 x 72 = p. 
 
 (x + 6) (x 12) = o. 
 
 x = 12. 
 
 3. By the rule. 
 
 x 2 - 6x = 72. 
 
 _ 6 V6 2 + 4 X 72 
 
 v ~ 
 
388 PRACTICAL SURVEYING 
 
 4. By completing the square. 
 
 The student should work the following by actual multiplication, 
 observe the formation carefully and memorize the results. 
 
 (a + b) (a + b) = (a + b) 2 = a 2 + 2 ab + b\ 
 (a - b) (a - b) = (a - b) 2 = a 2 - 2 ab + 6 2 . 
 (a - b) (a + b) = a 2 - b 2 . 
 
 To solve a quadratic by completing the square add to both sides 
 the square of half the coefficient of the unknown quantity. 
 
 x 2 6 x -f 9 = 72+9 = 81. 
 
 In the above line the square of one-half of 6 was added to both 
 sides of the equation. 
 
 Extract the square root of both sides. 
 
 * - 3 = 9- 
 Solve for x. 
 
 x = 9 + 3 = I2 - 
 Sometimes the square of the unknown quantity has a coefficient: 
 
 3# 2 - 6x = 72, 
 
 in which case the equation must first be divided by this coefficient; 
 x 2 2 x = 14. 
 
 To avoid fractions multiply the equation by 4 times the coefficient 
 of x 2 and then add the square of the original coefficient of x. 
 
 Let the equation be 7 x 2 20 x 32. 
 
 Multiply by 4 X 7 = 28, 196 x 2 560 x 896. 
 Add 20 2 = 400, 
 
 196 x 2 $60 x -f 400 = 896 + 400 = 1296. 
 Extract the root, 14 x 20 = 36. 
 
 X = +4, or -!- 
 
 If an equation contains two powers of the unknown quantity and 
 the exponent of one is double that of the other it may be solved 
 like a quadratic. 
 
ESSENTIALS OF ALGEBRA 389 
 
 Let the equation be x & 6 x? = 16. 
 
 Completing the square, X 6 6^ + 9 = 16 + 9 = 25. 
 
 Extracting the root, r* 3 = $. 
 
 Transposing, X? = 3 d= 5 = 8. 
 
 Extracting the cube root, X = 2. 
 
 Problems. Answers. 
 
 i. 8 + x 2 - 6 = 80. 
 
 2. - + 42 - = h 20- 
 
 b - a)*. 
 
 (18 + e 6 Ve + 9)"' 
 
 7. A brick pier 30 ins. wide and 42 ins. long carries a load of 
 450,000 pounds. Design a foundation of the same shape resting on 
 earth having a safe bearing value of 3000 pounds per square foot. 
 What should be the length and breadth of the footing? 
 
 4150,000 
 
 Required area = :iii - J =150 sq. ft. 
 
 3000 
 
 Ratio of sides = -- = 1.4. 
 30 
 
 Let X = width, 
 
 then 1.4 x = length. 
 
 x (1.4*) = 150- 
 1.4 x 2 = 150. 
 
 x 
 
 1.4 
 
 1.4 x = 1.4 X 10.35 = r 4-45 
 Make the footing 14' 6" X io r 4". 
 
390 PRACTICAL SURVEYING 
 
 8. Find a number of which the square shall be 4 times the num- 
 ber together with 5 times the same number. Ans. g. 
 
 g. Find two numbers of which the sum is 133 and their difference 
 is 47. Ans. go and 43. 
 
 10. Find two numbers in the ratio of 4 to 3, so that if i be added 
 to each of them, the sums shall be in the ratio of 9 to 7. 
 
 Ans. 8 and 6. 
 
 11. Divide 72 into two parts so that three times the greater shall 
 exceed twice the less by 121. Ans. 53 and 19. 
 
 12. A general sends 1500 of his troops to the West and J of his 
 entire army to the East. He retains i of the army after detaching 
 1200 to the rear guard. How many men are in the army? 
 
 Ans. 16,200. 
 
 13. A and B started out in life with equal amounts of money. 
 A spent annually $600 more than his income, while B saved $800 
 annually. In 12 years B was twice as rich as A. What was their 
 original capital? 
 
 (x - 60 X 12) = x + 80 X 12. 
 
 Ans. $24,000 each. 
 
 14. An aeroplane going at the rate of 90 miles an hour overtakes 
 another having a start of 180 miles, but going at the rate of 70 miles 
 an hour. How many hours did it require? 
 
 Ans. g hours. 
 
 15. A cistern can be filled with water by one faucet in 12 hours 
 and by another in 8 hours. In what time will it be filled if both 
 run together? Ans. 43 hours. 
 
 1 6. Two men can together do a piece of work in 8 days. One can 
 do it alone in 12 days. How long would it take the other? 
 
 Ans. 24 days. 
 
 17. A can do a piece of work in 50 hours; B in 60 hours and C in 
 75 hours. In what time can the three do it when working together? 
 
 Ans. 20 hours. 
 
 18. A and B together can do a piece of work in 12 hours; A and C 
 together in 20 hours; B and C together in 15 hours. In what time 
 
ESSENTIALS OF ALGEBRA 391 
 
 will they do it, all working together, and in what time will each do 
 it separately? 
 
 xxx 
 
 - H 1 = 2. 
 
 12 20 15 
 
 Ans. Together in 10 hours; A alone in 30 hours; B alone in 20 
 hours; C alone in 60 hours. 
 
 19. The sum of two numbers = 100 and their product = 2100. 
 What are the numbers? Ans. 70 and 30. 
 
 20. Find two numbers of which the difference is 8 and the product 
 is 240. Ans. 20 and 12. 
 
 21. Find two numbers of which the product is 108 and the sum of 
 their squares is 225. Ans. 12 and 9. 
 
 22. An oblong pond was surrounded by a road 7 yards wide, the 
 area of the pond being 15,000 square yards and the area of the road 
 being 3696 square yards. Give the length and breadth of the pond. 
 
 xy = 15,000, and 14 x -h 14 y + 196 = 3696. 
 
 Ans. 100 X 150 yards. 
 
 23. Find two numbers of which the sum is 13 and the sum of their 
 cubes is 637. Ans. 8 and 5. 
 
 24. Sold a cow for $24 and gained as much in per cent as the 
 first cost in dollars. What was paid for the cow? 
 
 x 2 
 
 x H = 24. 
 
 100 
 
 Ans. $20. 
 
 25. A and B started at the same time for a place at a distance of 
 150 miles. A travels 3 miles an hour faster than B, and beats him 
 by 85 hours. At what rate did each travel? 
 
 Ans. A 9 miles per hour, B 6 miles per hour. 
 
 26. A and B distribute each $1200 among some poor families. A 
 gave to 40 more families than B, but B gave $5 more to each family. 
 How many families were helped by each? 
 
 Ans. 120 by A, 80 by B. 
 
 27. A traveling to Boston, overtook at the $oth milestone a 
 band of sheep traveling at the rate of 3 miles in 2 hours; and 2 hours 
 later met a wagon moving at the rate of 9 miles in 4 hours. B, 
 traveling at the same rate as A, overtook the sheep at the 45th 
 
39 2 PRACTICAL SURVEYING 
 
 milestone and met the wagon 40 minutes before he came to the 
 3ist milestone. Where would B be when A reached Boston? 
 
 Let x = distance between them, 
 
 y = rate of their traveling per hour. 
 
 27 9 3 
 
 Ans. x = 25. 
 
 y = 9- 
 
 Problems involving ratio, proportion and variation are handled by 
 the rules given for solving equations. 
 
 TABLE OF FORMULAS 
 
 The accompanying table is of great value in many problems. 
 Let x and y be any two quantities. 
 
 S = x + y, their sum, 
 d = x y, their difference, 
 p = xy, their product, 
 
 q = x/y, their quotient, 
 Z = x 2 + y 2 , the sum of their squares, 
 D = x 2 y 2 , the difference of their squares. 
 
 The use of this table will teach how to state a problem and only a 
 few hints will be here given as to its use. The student is advised to 
 consult it when called upon to solve problems. 
 
 EXAMPLES. 
 
 1. The sum of two numbers is 277 and their difference is 115. 
 Find the greater. 
 
 From the table X - f-^) = ^ + "*) = 196. 
 
 2. The difference of two numbers is 10 and the product is no. 
 Find the greater. 
 
ESSENTIALS OF ALGEBRA 
 
 393 
 
 + " 
 
 to 
 
 ? 
 
 I + 
 
 -^,1 W 
 
 - 1 -^,1 
 
 + 1 + 
 
 1 -CXJ 
 
 " 
 
 N 
 
394 PRACTICAL SURVEYING 
 
 From the table 
 
 d + (d 2 + 4 />)* _ io + (IPO + 476)* 
 
 2 2 
 
 IO + ^576 _ IO + "24 
 
 X 
 
 3. The sum of the squares of two numbers is 250 and the differ- 
 ence of the squares is 88. Find the numbers. 
 
 X = 
 
 , = VsT = Q. 
 
 4. Given the sum s of the products of two quantities, by known 
 multipliers m and n, and also the sum of their products c by other 
 known multipliers p and q; to find the quantities. 
 
 Here mx + ny= s, and px -\- qy = c; multiplying the first 
 equation by p, and the second by m, they become pmx-\- pny = ps, 
 and mpx X mqy = me; subtracting we get npy mqy = 
 ps me ; and dividing by np mq, we obtain 
 
 _ ps me 
 np mq 
 
 In like manner we find 
 
 (75 nc 
 mq np 
 
 5. Given the sum 5* of the quotients of two quantities by known 
 divisors m and n, and also the sum c of their quotients by other 
 known divisors p and q', to find the quantities. 
 
 X "V X "V 
 
 Here - + z = s, and - + - = C, 
 
 m n P % 
 
ESSENTIALS OF ALGEBRA 395 
 
 then nx + ny = mns, and qx + py = pqc\ 
 
 which, resolved as in (4), gives 
 
 pm (ns qc) 
 x " ) 
 
 pn qm 
 
 _ nq (ms pc] 
 qm pn 
 
 Equations may be solved graphically and an interesting book on 
 the subject is entitled, "A Graphical Method for Solving Certain 
 Algebraic Equations," by Prof. G. L. Vose, in Van Nostrand's 
 Science Series, 50 cts. 
 
 The foregoing rapid presentation of the essentials of algebra gives 
 a student all that is necessary in order to have a good working tool. 
 If his interest is aroused and he desires to proceed farther in the 
 study of mathematics he should purchase the text on algebra used 
 in the nearest high school and thus when hard places are encoun- 
 tered assistance may readily be obtained. Correspondence courses 
 in mathematics are given by the University of Chicago, Chicago, 111. 
 
INDEX 
 
 Adjustments of instruments: 
 
 compass, 106. 
 
 dumpy level, 80. 
 
 transit, 223. 
 
 wye level, 79. 
 Algebra, essentials of, 363. 
 Algebraic, formulas, table of, 393. 
 
 theorems of sun, 163. 
 Altitudes, equal, for true meridian, 
 
 288. 
 Angle, checking with needle, 234. 
 
 circular measure of, 169. 
 
 denned, 118. 
 
 supplement of, 118. 
 Angles, deflection, 118. 
 
 double centering of, 229. 
 
 exterior, 118. 
 
 from bearings, 118. 
 
 included, 118. 
 
 logarithmic functions of, 190, 
 203. 
 
 natural functions of, 157, 194. 
 
 repeating, 229. 
 
 reading, with transit, 228. 
 
 tables of functions of, 194, 203. 
 Angular leveling, 245. 
 Arc, length of, latitude and longi- 
 tude, 275. 
 Area, by planimeter, 49. 
 
 by weighing, 49. 
 
 table of units of, 72. 
 Areas, computing, 128. 
 
 J 
 
 Areas, correcting, 20. 
 
 of surfaces, 42. 
 Astronomy, 268. 
 Attraction, local, in, 118. 
 Azimuth, denned, 236. ^ 
 
 from bearings, 234. 
 
 table of errors in, 276. 
 
 Backsight, 89. 
 Barlows' tables, 58. 
 Batter boards, 54. 
 Bearings, from angles, 118. 
 
 from azimuth, 234. 
 
 platting, 124. 
 Bench marks, 89. 
 Boundary surveys, law of, 310. 
 Bounds, whipping the, 6. 
 Broken line, defined, 2. 
 Bubbles, sensitive, importance of, 
 
 79- 
 Building, staking out of, 53. 
 
 Care, of compass, 108. 
 
 of level and transit, 226. 
 Carpenters' level, 75. 
 Chain, surveying, 14. 
 
 Gunter's, 14. 
 Chaining methods, 7. 
 Chain men, necessity for good, 6. 
 Chart, Isogonic, 102. 
 Checking, levels, 91. 
 
 rod reading, 86. 
 
 397 
 
INDEX 
 
 Checking, with needle, 234. 
 
 work, 174. 
 
 Circular measure of angle, 169. 
 Compass, surveying, 100. 
 
 adjustments of, 106. 
 
 care of, 108. 
 
 reading the, 104. 
 
 use of, 109. 
 Computation, of areas, 128. 
 
 of traverses, 172. 
 Contact rods, 16. 
 
 Contours, 341. 
 
 - 
 
 Corners, establishment of, 3, 4. 
 
 extinct, 290. 
 Crelle's tables, 61. 
 Cross, surveyors', 36. 
 
 wires, 78. 
 
 Cubes, use of table of, 55. 
 Curvature of earth, 74, 82. 
 Curved line, definition of, I. 
 
 Datum plane, 82. 
 
 Daylight observations on Polaris, 
 
 286. 
 
 Declination, 100, 102, 271, 276. 
 Deflection, angles, 118. 
 
 of tape, 1 8. 
 Deflections, platting angles by, 
 
 124. 
 
 Departure, defined, 128. 
 Differential leveling, 88. 
 Dioptra, 210. 
 
 Distributing errors, 125, 133. 
 Diurnal variation, 103. 
 Dividing tables, 61. 
 Double centering of angles, 37, 
 
 229. 
 
 Double meridian rule, 131. \ 
 Drafting tools, 23. 
 Dumpy level and adjustments, 79, 
 
 80. 
 
 Earthwork computations from 
 
 contour map, 344. 
 Elevations, positive and negative, 
 
 83- _ 
 
 Ephemeris, defined, 268. 
 Equations, solution of, 379. 
 Error, index, 231. 
 
 limit of, 5. 
 
 probable, 230. 
 Errors, in leveling, 91. 
 
 in measuring, 18. 
 
 distributing, 125, 133. 
 Exponents, 56. 
 Exterior angles, 118. 
 Extinct corners, 290. 
 
 Factors, 55. 
 
 Foresight, 89. 
 
 Functions of angles, 151, 157. 
 
 tables of, 194, 203. 
 
 graphical natural, 159. 
 
 logarithmic, 190, 203. 
 Functions, judicial, of surveyors, 
 293- 
 
 Geodetic surveying, 2. 
 Geometry, practical, 23. 
 Government surveys, 327. 
 Grade stakes, to set, 95. 
 Grades, selecting, on contours, 342. 
 Gradienter on transit, 214. 
 Graduations on transit, 218. 
 Gravity and gravitation, defined, I. 
 Greek letters, 289. 
 Gunter's chain, 14. 
 
 Hand level, 94. 
 Horizontal lines, I. 
 Hubs on surveyed lines, 9. 
 Hydrographic surveys, 335. 
 Hypothenuse of triangle, 58. 
 
INDEX 
 
 399 
 
 Included angles, 118. 
 
 Index error, 231. 
 
 Intersection method for surveys, 
 
 241. 
 
 Invar tape, 17. 
 Isogonic lines and chart, 102. 
 
 Jacob staff, 36. 
 
 Judicial, the, functions of sur- 
 veyors, 293. 
 
 Latitude, 128, 273. 
 
 length of arc of, 275. 
 Law of boundary surveys, 310. 
 Legal knowledge of surveyors, 5. 
 Lot surveys, 261. 
 Level, adjustment of, 80. 
 
 bubbles to be sensitive, 79. 
 
 carpenters', 75. 
 
 care of, 226. 
 
 circular, 37. 
 
 dumpy, 79, 80. 
 
 for rod, 85. 
 
 hand, 94. 
 
 line, I, 73. 
 
 Precision, 79. 
 
 rods, 83. 
 
 tape, 10. 
 
 true and apparent, 73. 
 
 water tube, 77. 
 
 wye, 79. 
 Leveling, 73. 
 
 angular, 245. 
 
 checking results of, 91. 
 
 differential, 88. 
 
 errors in, 91. 
 
 profile, or route, 92. 
 
 the tape, 10. 
 
 turning points, 86, 89. 
 
 with rubber hose, 77. 
 
 with transit, 216. 
 
 Length, tables of units of, 72. 
 Letters, why used in algebra, 
 
 365. 
 
 Limit of error in surveys, 5. 
 Lines, definitions of, I, 73. 
 
 offset, 1 1 6. 
 
 random, 22, 122. 
 
 ranging, 21. 
 
 Local attraction, in, 118. 
 Locating objects, 41. 
 Logarithms, 182. 
 
 tables of, 201. 
 Logarithmic functions of angles, 
 
 190, 203. 
 Longitude, 270. 
 
 length of arc of, 275. 
 
 Map making, 42, 122, 261, 266. 
 Marks, bench, for levels, 89. 
 Mean polar distance, 286. 
 Measurement, errors in, 18. 
 Measures, table of units of, 72. 
 Measuring on slopes, 10. 
 Mensuration, plane, 42. 
 Meridian, determination of, 115, 
 
 277, 284, 288. 
 Meridiograph, 281. 
 Miners' triangle, 74. 
 Mining surveys, 330. 
 Missing, lengths and bearings, 136, 
 
 175- 
 
 corners, 4. 
 Multiplying tables, 61. 
 
 Notes of surveys, II, 115. 
 Negative and positive elevations, 
 
 83- 
 Natural graphical functions of 
 
 angles, 159. 
 Needle reading to check angles, 
 
 234- 
 
400 
 
 INDEX 
 
 Objects, locating, on survey, 41. 
 
 Oblique triangles, solution of, 165. 
 
 Obstacles on line, 41. 
 
 Offset lines, 116. 
 
 Omissions in measurements and 
 
 bearings, 136, 175. 
 Original surveys, 4. 
 
 Paper, profile, 94. 
 Peg readings, 91. 
 Percenter for leveling, 76. 
 Personal equation, 91. 
 Phototopographic surveys, 355. 
 Pins, use of, in surveying, 8. 
 Plane, defined, 2. 
 
 surveying, 2. 
 
 mensuration, 43. 
 
 table, 345. 
 Planimeter, 49. 
 Platting, bearings, 124. 
 
 by deflections, 124. 
 
 latitude and departure, 266. 
 
 stadia notes, 353. 
 Plotting, see platting. Also map 
 
 making. 
 Plats, or maps, of chain surveys, 
 
 42. 
 
 Plumb-bobs, best weight for, 19. 
 Plunging a grade, 96. 
 Points, defined, I. 
 
 turning, in leveling, 86, 89. 
 Polar distance, 286. 
 Polaris, observations on, 284, 286. 
 Poles and rods, 21. 
 Positive and negative elevations, 
 
 83. 
 
 Practical geometry, 23. 
 Precision level, 79. 
 Probable error, 230. 
 Profile, leveling, 92. 
 
 paper, 94. 
 Protractors, kinds of, 128. 
 
 Protractors, use of, 123. 
 Pythagorean theorem, 149. 
 
 Quadratic equations, 386. 
 
 Radian measure of angle, 169. 
 Radiation method for surveys, 
 
 117, 241. 
 
 Random lines, 22, 122. 
 Ranging lines, 21. 
 Ratios, trigonometric, 152, 153. 
 Reading, the compass, 104. 
 
 rods, 85. 
 
 Refraction, 73, 82. 
 Repeating angles, 229. 
 Reproducing maps, 288. 
 Resurveys, 4, 291. 
 Road percenter, 76. 
 Rod, level, 85. 
 
 waving the, 85. 
 Rodmen, instructions for, 84. 
 Rods, and poles, 21. 
 
 contact, 1 6. 
 
 level, 83. 
 
 reading of, 85. 
 Roots, square and cube, 55. 
 Route, leveling, 92. 
 
 surveys, 9, 92, 339. 
 Rubber hose for leveling, 77. 
 
 Sag in tape, effect of, 17. 
 Scales for map making, 24. 
 Self-reading rods, 83. 
 Sensitive bubbles, 79. 
 Sewer grade stakes, 96. 
 Signals for surveyors, 97. 
 Signs, trigonometric ratios, 155, 
 
 use of, in algebra, 368. 
 Simpson's rule, 47. 
 Slide rule, 60. 
 
 Slopes, measuring on, 7, 10. 
 Smoley's tables, 150. 
 
INDEX 
 
 401 
 
 Solar compass, 100. 
 
 observations, 277. 
 Squares, tables of, 55. 
 Stadia surveys, 246, 251, 346. 
 Stakes on surveys, 9, 52, 95. 
 Stations on lines, 9, n. 
 Straight line, I. 
 Straight-edge for leveling, 75. 
 Supplement of angle, 118. 
 Surfaces, 2. 
 Survey, notes, n, 115. 
 
 by radiation, 117. 
 
 with transit, 233. 
 Surveying, chain, 14. 
 
 denned, 2. 
 
 transit, 216. 
 Surveyors, chain, 14. 
 
 cross, 36. 
 
 the judicial functions of, 293. 
 
 signals, 97. 
 Surveys, various kinds of, defined, 
 
 3- 
 
 government, 327. 
 hydrographic, 335. 
 mining, 330. 
 phototopographic, 355. 
 route, 339. 
 stadia, 346. 
 
 Tables, Barlow's, 58. 
 
 Boileau's, 176. 
 
 Crelle's, 61. 
 
 Gurden's, 176. 
 
 Lewis & Caunt's, 176. 
 
 Smoley's, 150. 
 Tables of, algebraic functions, 393. 
 
 functions of angles, 194, 203. 
 
 lengths of arcs of latitude, 275. 
 
 logarithms, 201. 
 
 measures length and area, 72. 
 
 squares, cubes, etc., 62. 
 
 stadia reduction, 251. 
 
 Tables of traverse, 142. 
 
 Tape level, 10. 
 
 Tapes, 1 6. 
 
 Targets on rods, 83. 
 
 Temperature, effect of, 17. 
 
 Theodolite, 211. 
 
 Tidal level as datum plane, 82. 
 
 Time, 270. 
 
 Transit, 212. 
 
 surveying, 210, 222, 233. 
 . adjustment of, 223. 
 
 care of, 226. 
 
 reading angles with, 228. 
 Traverse, computation of, 172. 
 
 methods, 241. 
 
 table for compass, 142. 
 
 table for transit, 177. 
 Trapezoidal rule, 47. 
 Triangles, miners', 74. 
 
 solution of, 58, 149, 161, 165. 
 Trigonometry, 148. 
 Trigonometer, 161. 
 Trigonometric, functions, 151. 
 
 laws, 164. 
 
 ratios, 152, 153. 
 Turning points, 86, 89. 
 
 Variation of needle, 100, 102. 
 
 see Declination. 
 Verniers, for angles, 218. 
 
 on rods, 84. 
 Vertical lines, I. 
 
 Water-tube level, 77. 
 Waving the rod, 85. 
 Weighing to obtain area, 49. 
 Whipping the bounds, 6. 
 Wires, in telescope, 78. 
 
 stadia, 246. 
 Witness stakes, 9. 
 Work, staking out, 52; 
 Wye level, 79. 
 
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