Library A 0_IJ R IS E in the Chas. C. Swafford, M. S. // Instructor in Civil Engineering University of California Prepared Especially for the Extension Di% T ision of the University of California Berkeley., California 1920 Engineering Library TY OF CUJPOtNlA. INTENSION Course 1A-1B Elements of Survey ing Stafford Contents Preface, Plan of the Coarse, Introduction. Assignment I. Definitions, Outline of Purpose o;? Surveying. II. Linear Measurements. III. Nature of Errors in Measurements. IV. Adjustment of Errors in Measurements. V. The Compass and its Uses. VI. Compass Surveying. VII. The Level and its Uses. VIII. Problems in Leveling. IX. The Transit and its Uses. X. The Transit, Ranging Lines and Measuring Angles. XI. The Transit, Observing on Polaris for Azimuth c.nd Altitude. XII. The Transit, Solar Apparatus. XIII. Adjustment of Instruments. XIV. Adjustment of Instruments (continued). XV. Land Surveying, Methods of. XVI. Lanu Surveying, Latitudes and Departures, Co-ordinates. XVII. Land Surveying, D. M- D's., Areas. XVIII- Land Surveying, Supplying Omissions, Parting off Land. 793909 Page 2 XIX. Surveying of the public Lands, U. 3. System. XX. Surveying of the Public Lands, Tangent and Secant Methods, Convergence xA \\Z of Meridians. XXI. Stadia Methods of Surveying. XXII. Stadia Methods of Surveying, Stadia Reductions. XXIII. Profile Leveling, Contours, Grede Lines. XXIV. Cross Section Leveling, Volumes, Excavations. XXV- Cuts and Fills, Sectioning and Contours. XXVI. City Surveying, Rectangular Systens. XXVII. City Surveying, Co-ordinate Systems. XXVIII. Simple Curves, Streets and Roads. XXIX. Railroad Surveying, Tangents and Curves. XXX. Railroad Surveying, Stopping, Location, Grading. XXXI. Base Line, tieasurements and Establishing of. XXXII. Triangulation, Method and Uee. XXXIII. Topographical Surveying, Details end Contours. XXXIV. The Planetable and its Uses. XXXV. Planetable , Methods anfi Problems. XXXVI. Mine Surveying. XXXVII. Hydrographic Surveying. XXXVIII. Mapping and Office Work, Computing. XXXIX. Rights, Duties, and Privileges of the Surveyor. XL- Rights, Duties, and Judicial Functions of the Surveyor. Page 3 ?I,iM OF THE COURSE This course in flane Surveying, offered by -the Extension Division of the University of California comprises forty (40) assign- ments, each treating some definite part of the subject, and aiming in their scope to present the work embraced in the regular surveying courses, Civil Engineering 1A, IB, and C. E. 3, College of Civil Engineering, University of California; excepting,, that in the Ex- tension course the problems for i'ield tolution and practice with instruments have bean specially selected and arranged for your work without the immediate supervision of an instructor. The problem work, therefore, calls for special care on your part. You should givo pM-ticular heed to (1st) the purpose; (2nd) the method; (3rd) the care, refinements, and precision of the work; (4th) the record of field notes - sketches, illustrations, diagrams, etc. , and (5th.) the result - the lesson learned or the training gained. i'he plan of this eourse requires that you shall study each assignment in acarei'ul, painstaking way; not merely reading it over, nor even committing it to memory, out maicing a full and critical study of each part, so as to gain a clear understanding of the thought content, to see its bearing on the subject of surveying as a whole, and thus to acquire step by step the science of surveying. References are given in each assignment to standard texts on the subject of surveying, including Tracy, Johnson, Breed & Hosmer, and Raymond. Of course it is not likely that you -.Till have Plan of the Course Page 4 access to all or even several of these works, but it is hoped that you may have one or more of them, and also a manual of surveying instruments, such as published by manufacturers descriptive of their good?. These manuals contain much information, tables, hints, etc. , of value to the engineer. Attention is called here to those issued to the engineering profession by Bausch rjid Lorab, Rochester, New York, C. L. Berger and Sons, Boston, Mass., W. and L. E. Gurley, Troy, New York, and others, which may be procured at small cost by addressing the firms named above. The General Land Office, Washing- ton, D- C., aleo has in preparation, a Manual on the Survey of the Public Lands, which may be had for a small fee from the Bureau of Publication, Washington, D. C. Preferences will frequently be made to the foregoing works, and their careful perusal is earnestly urged. Accompanying each assignment are questions to be answered, problems to be solved, and field exercises to be worked out. Your papers are to be sent to the Extension Division for correction, ctiticism, and grading. Your standing in the course is determined by the paper you submit; your progress will depend largely upon your ability to profit by corrections and criticisms. Each paper submitted by you should be worked out carefully and punctiliously, special care being taken to place everything, from heading to close, in a neat, orderly arrangement, with any field notes, tabulations, or sketches to accompany the same. Field notes and sketches (or illustrations) should preferably be dene in (not write) pencil. Use a 4H pencil and letter/,the notes in approved styles. Plan of tiie csou^pe Page 5 (See illustrative, specimen pages of Field Jotes displayed in the assignments fro-i time to time.) The required computatioas :auet lie _pade in. good arrangement, all operations clearly stated, and conclusions or answers properly indicated. The actual work of multiplication, division, extraction of roots, etc., need not be returned with the solutions, but the processes and their results niust De clearly shown. Logarithmic computations should gi've the logarithms used. Where slide rules or other natherr-atical devices are employed, the fact thould be definitely stated. Where tables are used, the title and source should likewise oe mentioned. This of course does not apply to tables of logarithms, powers and roots, etc. In all your trork bear in mind that the purpose is to acquire a knowledge, not only of the subject of surveying as a theory, but of its methods and expressions as embodied in principles, problems, notes, maps, and instruments employed in fisld and office. Without such knowledge ai\d the training that goes with it, hand in hand, this course v/ill be of small avail in giving you the real training intended. Page 6 INTRODUCTION Surveying, like most other work in applied science, is an art. Ihe strict application of formulas is not alvays possible. In many problems special devices must be used, the correct solution of a problem must be set aside and approximations, more or less crude, must be substituted to meet the rights of legal ownership under the laws of the land. Errors in previous surveys of the same lines or areas must be considered and corrected or the new survey must con- form to the actual conditions met with on the ground in spite of written evidence to the contrary. To accomplish such things, then, the surveyor must be a man at once honest, sincere, dependable, energetic, ingenious, and ob- servant. He must be patient in his work and with other people. Ke must be ready to give an unbiased opinion as to the rights of dispu- tants when called upon to do so, and should always be sure of the correctness of his -.Tork or its limitations before submitting results to his employer or to other persons. Ee cust have a thorough know- ledge of the fundamental principles of surveying such as will enable him to solve all of his problems correctly. In these assignments it is assumed that you have a good under- standing of the common processes of arithmetic, algebra, plane geom- etry, plane trigonometry, and the use of mechanical dravring instru- ments. It should be your endeavor to acquire a knowledge of solid geometry, spherical trigonometry, and physics, as of real value to you in your studies and in later work in the field. Introduction Ps-ge 7 It is further a,isuued that you will ha\e ajcese to the common equipment of the surveyor eo that the exolam-tic-ns of the various instruments and the n.ethods of their p.cijustmtnt and use can be understood. Lacking such equipment, it will be hard for you to grasp the significance of iruch of the course and your studies will result in a training in geometry and not^ in a training in surveying methods. You are cautioned not to juaip to the common and errcnetras conclusion that v/hat is said in this series of assignments is all that can be said on the subject of surveying. It is not iatended to give you s^ill as a surveyor, but to ^ive a good, general train- ing in the fundamental principle^ underlying the methods of the surveyor in the field and office, coupled with a brief discussion of the common problems met in practice. Skill, in any line of endeavor , coiaeg only through long practice. The real student will find suggestions that will lead him far into the fiolci of surveying. /ssr'gnnent 1 Page 8. DEFINITIONS FOREWORD: In this assignment it is intended to give a general outline of the purpose of surveying and some of the common conceptions, as- sumptions, and definitions necessary to a proper understanding of the more detailed problems which are to follow. (1) SURVEYING The process of measuring lengths of lines, magnitudes of angles, and differences of elevation at or near the surface of the earth for the purpose of establishing Boundaries, establishing elevations of points, map-making, and construction, is called surveying. (2) TEE EARTH The earth is approximately a sphere. The diameter from pole to pole oeing 26.37 miles shorter than a diameter in the plane of the equator, the equatorial diameter, as computed oy Clarke in 1866 and adopted by the U. L. Coaet and Geodetic Survey, is 41,852,124 feet, rrhile the corresponding polar diameter is 41,710,242 feet. (3) a; A- LEVEL The fundamental datum for elevation of points on or near the surface of the earth is taken at sea level. The layer of water covering the greater portion of- the earth has nearly the same form as the mean globe. A slight variation is due to the action of the tides and the rotation of the earth on its axis. Since there is a rise end fall of tide twice a day for any one locality, it has been Klem. of Sur- Li. As -signment 1 Page & decided that the MEM UVF.L of the ocean ivill be called sea-level. In all the common determinations of elevations sea- level is said to have a zero elevation. A point well worth remembering is that most of the problems in surveying occur above sea-level. (4) MERIDIANS Imaginary lines passing completely around the earth and forming a closed carte two points of which lie in the north and south poles of the globe are called meridians. To illustrate Imagine that a person starts from a point on the surface of the earth and travels due north until the north pole is reached. Continuing through the pole in the same line, not deviating from it in any way, his course will be due south until he reaches the south pole; thence he travels due north to his original position. The path traversed IE a meridian. The geometric and astronomical definition is as follows: A meridian is the line on the surface of the earth cut by a plane passing through the north and south poles of the earth. (5) PARALLELS OF L^IIUDE Parallels of latitude are imaginary circles on the surface of the earth lying in planes parallel to the plane of the equator and having their centers in the line joining the poles (polar axis) of the earth. (6) LONGITUDE The angular distance from an agreed fundamental meridian to Elea. of Curv. IA Assignment 1 Page 10 the Eeridian through the point in question, measured in the plane of the parallel of latitude through the same point, is the longi- tude. The fundamental (prime) meridian for English-speaking countries is through Greenwich, England. Longitudes are measured East and West from Greenwich, in each direction from to 180. It is to te noted that Icngf.tu^e is the angular Treasure EO that while meridians converge toward the pole and the linear dis- tance between them gets less as the pole is approached, the longiv tude of tne meridian in question does not change so long as neither pole is passed. (7) LATITUDE Latitude is the angular distance from the Equator north or south to the parallel of lax itude through the point in question, measured in the plane of the meridian through the point. Latitudes range fron degrees at the Equator to 90 degrees at the poles. (8) A PLUMb LINE The line determined by the position of a cord at the lower end of T'hich is fastened s. weip,ht sufficient to stretch the cord into a straight line, the cord then being suspended in quiet air and allowed tc come to rest, is called a plumo line. A common definition of a pluino line is about as follows: h. plunb line is a line pointing toward the center of the earth, i. e., a radius of the earth prolonged. This, however, ie not strictly true owing to the rotation of the earth and the irregular distribution of mass within the earth. In the northern hemisphere, generally, the plumb lines point slightly south of the Elem. of Surv. Ih. Assignment 1 Page 11 geometric center or the earth. Ko two plumb lines are parallel. A plumb line points to what is known as the ZENITH or the point in the heavens vertically overhead at any point on the earth's surface, -t-'he NADIR is a corresponding point found by prolonging the plumb line through the earth to a diametrically opposite point on the celestial sphere. (9) A LEVEL SURFACE A surface that is everywhere perpendicular to the plumb line is a level surface. Thus, it is seen that a level surface 1 is not a flat surface or a plane, but a curved surface conforming, approximately, to the general shape of the earth. (10) A LEVEL LIRE A level line, is then, a line in this curved surface and is, itself, curved. What is ordinarily called a level line is a geo- metrically straight line tangent to a level line at which point of tangency the straight line is perpendicular to the plumb line. A level line, on the other hand, is perpendicular to any plumb line that may intersect it. (11) A STRaiGET LINE j- A straight line on the surface of the earth is really a portion of a great circle of the earth, e. g. , a meridian is a straight line. A geometrically straight line is one that would be tangent to a sphere if drawn perpendicular to a plumb line, or would pass through the surface of the sphere in two points or would net touch the sphere at all - all three cases are possible. Elem. of Surv. IA Assignment 1 Page 12 A geometrically straight line would possibly cut through the sur- face of the earth in more than two places owing to inequalities of the surface. (12) GEODETIC SURVEYING The branch of surveying which, owing to the extent of the surveys, t^K&E into consideration the spheroidal fora of the earth is termed Geodetic Surveying. That is, in the survey of a conti- nent or state the lines are so long that they are appreciably curved and the aigles of the various polygons making up the sur- vey are angles of spherical polygons. This type of surveying is very difficult, requiring men of special training and experience in the prosecution of both field and office worK. (13) PLANE SURVEYING The branch of survey ing" which, owing to the limited extent of the surveys, does not consider the spheroidal fora of the earth, is known as plane surveying. In other words the assumptions made are. 1. A level surface is a flat surface. 2. A level line is a straight line. 3. A "straight line " on the earth is geometrically straight. 4. All plumb lines within the limits of the survey are parallel. 5. All angles of a polygon are plane angles. This branch, the more common and elementary of the two types, will constitute the greater part of the work of this course. (14) NATURE CF MEASUREMENTS IK PLh.NE SURVEYING Surveying, as before stated, is for the purpose of determining Elem. of Surv. 1A Assignment 1 -Page 13 the relative positions of points and lines on or near the surface of the earth. A moment's reflection will bring you to the reali- zation that in the use of the common equipment of the surveyor the only things possible to determine are the positions of points, the distances (both horizontal and vertical) between them. sand, the an- gles between lines and planes. In determining the shape of a field, a building, or a cross-section in question the surveyor merely takes a sufficient number of measurements to enable him to establish, geometrically, the outline of the thing measured. Where possible, it . may be best to take direct measure- ments of the line or angle under consideration. Uut direct measure- ments may involve physical difficulty and lack of what is called "reasonable accuracy". For example, it may be possible to tape the length of a line across a swarip or a line across hill and valley for a long distance, but the error in the work and the physical exertion will often be far in excess of vrhat is reasonable. It is usually possible to find some device by which such a determination may be made indirectly so that the result will be more accurate aid the work of measurement far less troublesome and exhausting. (15) ACCURACY VERSUS COST Nearly all work in surveying can be cone very accurately, but it is neither necessary or advisable to attempt to maintain a high degree of accuracy in all problems, nor to make all measure- ments in different parts of the same problems equally accurate. If & survey is being made tc find the acreage of a field planted to a certain crop and the area is to be determined to the nearest one-tenth Elem. of Surv. LA. Assignment 1 jPage 15 the nearest ten or fifteen minutes and the lengths of the courses to the nearest link (7.92 inches). A later survey is made of the property, but the value of the land has increased to $200 per acre, in which case the surveyor should measure the boundaries of this field with the transit end steel tape determining the angles to the nearest minute and the distances to the nearest one-tenth of a foot. It is not to be understood that this accuracy is in strict ratio to the increase in value of the property nor should it be assumed that the new measurements will check the old but roughly. If the original surveyor executed his work properly and, though his meas- urements were more or less in error, set proper monuments which can be found intact by the later surveyor, the new survey will be a survey of the same field. Unfortunately, however, the earlier surveyors did not consider the possibility of a -.later survey or were apt to be careless about such things; consequently much labor and expense is the lot of the surveyor who raakes the more accurate survey. It is seen that a large question of rights and what con- liae stitutes the true position of a boundary>is opened up by such problems. (16) KINDS OF MEASuREioENTS MADE B. PLaNfc SURVEYING TiShen the area of a field is given as 160 acres the area meant is that of the polygon formed by projecting the boundaries of the field onto a horizontal plane. It may be that the boundaries of the field are on a hillside and do not lie in horizontal planes, but experience has shown that it would be difficult and impracticable to consider the lengths of the lines that would be found by following Elera. of Surv. 1A A&.JJ gvime.it 1 -i'ags 16 the slope of the ground. It is seen, then, that what is called the length of a line in surveying is the horizontal projection of the line (Hhis rule holds unless the length is qualified by sone term such as "slope length", etc.). Usually the measurement of a line is made so that the horizontal projection is obtained directly. In a similar manner the angles of a polygon on the ground are given as the angles between the horizontal projections of the lines. In problems where the difference in elevation of points is important the measurements of such quantities are usually made so as to give, directly, the vertical distance. There are numerous cases arising -mere the distance between two points will be measured on a slope. In such cases the vertical angle must be determined or the vertical difference in elevation between the extremities of the line must be found IB sake it pos- sible to compute the horizontal distance. *lhere the slope distance and the vertical angle, or where the slope distance and horizontal distance are given, it will be possible to find the difference in elevation. INhen such a slope measurement is taken, it is subse- quently reduced to a horizontal distance and to a vertical difference in elevation. fo summarize: 1. Distances are usually horizontal. 2. Differences in elevation are usually vertical. 3. Angles are measured in a horizontal plane or a vertical plt-ne. Elem. of Surv. IA A3Rigr.ir?nt 1 Page 17 4. If measurements of any linear or angular quantities are are made in other ways, they are usually reduced to vertical and horizontal quantities. HIOBIEMS 1. Show by a diagram, carefully constructed, that no two plumb-lines are parallel. 2. Show that a level line between two points on the earth s surface is longer than a straight line joining the same points. 3. Prom the values of equatorial and polar diameters given on page 8, Assignment 1, compute the length of one minute of arc of longitude, and the mean length of one minute of arc of latitude. 4. Determine v;hich is greater, the superficial areaof a sloping field or the acre&.ge as measured by the usual methods of plane survey- ing. Give reasons for following the methods of surveying land areas. You will confer a favor upon the instructor and upon other students if you point out anything in the assignments of this course which seems to be obscure. Write questions to the instructor, if you care to do so, when you send in your papers. TOHTEfiSIiT OF CALIFORNIA EXTENSION D IV IE 10?! CORKESi'CNDLNCE-STUDr CCUK23S IN TECHiiZICAL SUBJECTS Course IA. Elements of Surveying Stafford Assignment 2 LINEAR MRjiSUREMFPTS FOREWORD : 1'he following discussion treats of the measurements of linear distances, particularly of those along the surface of the earth, or what are generally designated as horizontal measurements. V?- r ious types of measuring devices will be illustrated and des- cribed snd particular emphasis laid upon the more common methods of measurement. -oOo- (17) A LIK&.aF. DISI&NCE the distance or length in a straight line between two points, is called the linear distance between those points. The linear measure or magnitude of the distance is expressed in terms of some established or arbitrary unit, e. g. , a linear distance of 9 feet may be expressed as 9 feet, 3 yards, 108 inches, or 3 2/5 paces, the unit used depending, largely, upon the ncture of the problem and the custom of the person making the measurement. (18) THE UNITS OF LiEASUHE The units used in the u nited States by the surveyor are either of English or French origin, the English units have the following relations: Linear Measure Square Measure 12 inches = 1 foot 144 sq. inches = 1 sq. foot 3 feet - 1 yard S sq. feet = 1 sq. yard 5 1/2 yards = 1 rod 30 1/4 6}. yards - 1 sq. rod 16 1/2 feet = I rod 160 sq. rods = i &cre 32C rods = 1 mile 43,560 sq, feet = 1 acre 5280 feet - 1 mile 640 acres = 1 sq. mile Volumes 1728 cu. inches - 1 cu. foot 27 cu. feet = 1 cu. yard. Elem. of auv. IA /.ssignn-sn* 2 P-ge 2 An important system of units, of Bnglish origin, used by the surveyor is based on the rod. A measuring device knovn as the Gun- tec's Chain is made just 4 rods or 66 feet long. This length is divided into 100 parts, called links, each link having a length of 7.92 inches, hence a table of measures ma^ be written in terms of links and chains : Linear Measure Square Measure 10 links = 1 rod 100 sq. links = 1 sq. rod 100 links = 1 chain 10000 sq. links = 1 eq. chain 1 chain = 66 feet 1 sq. chain = 16 sq. rods. 80 chains = 1 mile 10 sq. chains = 1 acre 6400 sq. chains = 1 eq. mile French units of measure are cased on the International Metre and are as follows: Linear Measure 10 millimetres - 1 centimetre 10 centimetres = 1 decimetre 10 decimetres = 1 metre = 39.37 inches (English) 10 metres = 1 dekametre 10 dekametres = 1 hektometre 10 hektometres = 1 kilometre = 1000 metres Square Measure Volumes 100 sq. mm. = 1 sq. cm. 1000 cu. mm. = 1 cu. cm. 100 sq. cm. = 1 sq. dm. 1000 cu. cm. = 1 cu. dm. 100 sq. dm. = 1 so. m. 10GO cu. dm. = 1 cu.. m. 100 sq. m. =1 are 100 ares = 1 hectare IOC hectares - 1 sq. kilometre In times past and in the present day the English system of units has been the more common in this country. The Gunter's chain was usad in much of the land surveying done in the early history of nearly ever v , part of the United States. The reason is obvious when one under stands the eimple relation of the chain to the mile, acre, and section. Consequently many deeds and records Llem. of 3urv, iA As *? ig^merrt 2 Page 3 of early surveys appear in chains. For this reason the surveyor should become thoroughly familiar with this system of units. Later surveys, v:here land has become valuable, usually give X the linear dimensions in feet and fractions thereof. In this con- nection it should be pointed out that the surveyor usually takes measurements of linear quantities in feet and decimal fractions of feet. In other words, a foot is divided into tenths, hundredths, etc., and a distance is given as 756.78 feet rather than as feet, inches, and fractions of inches. Again, the surveyor should be familiar v/ith the relation of decimal fractions of a foot to inches so that he can quickly change measurements to ordinary units for mechanics and others net farailiar with such fractions. TAbLS I Decimals of a Foot in Inches. Decinal of a foot Inches 0.01 0.0* : 1/8- 1/2- 0.08 1- 0.17 ^ Z+ 0.25 = 3 exact C. 37. 4- 0.50 - 6 axact G.7o 9 exact for rough approximations the above equivalents will be found convenient, though for precise work it Trill be necessary to carry the relation out to a. riner uegree of accuracy. The French (or Metric) system is shown for the reason that many surveyors are constantly leaving this country for countries where such units are employed and because much of the work of the Elero. of Surv. 1& Assignment 2 ^age 4. U. S, Coast e.nd Geodetic purvey and other depaitaents of tne United States Government is carried out in metric units. (19) MEASURING DEVICES The commonest measuring device is the ordinary foot or two- foot rule. A finely constructed rule or scale of one foot in length is used by the engineer or surveyor. The graduations are given, usually, in deciirai fractions of inches. This rule is used chiefly for office work in making, scale drawings, out it will often have a proper plr.ce in precise measurements in the field. The common type of folding carpenter's rale is well known to almost everyone interested in the mechanical arts. This rale is graduated variously most commonly in inches and sixteenths of an inch. The yard stick and ten-foot pole are used in short, rough measurements. Formerly the ten-foot pole r/as used in certain types of surveying problems. The cloth tap a , jade of a linen strip and graduated to feet and inches, while used in some rough work, has very little place in most surveying vork, as it soon streiches or shrinks so as to make it too inaccurate for reliability. The metallic tape is made by weaving fine bronze wires lengthwise through a heaiy linen strip. This strip, usually fifty feet long, is painted a light color and the foot marks a^id frac- tional division are printed on it. The bronze wires are supposed to prevent undue shrinkage and stretching, but they serve their purpose indifferently. Such tapes, however, are found very useful Elem. of Surv. -".A Assignrient 2 -^age S in taking, short measurements on offsets, and for road cross-sectioning. These tapes may be obtained aither mounted or unmounted, 'fhe reel is usually enclosed in a heavy leather box. If the tape is to be ueed where mud and water are encountered it should be taken off the reel as the box will soon beco.ned choked up with debris of various kinds carried in by the tape. The tape should be carefully dried before it is replaced on the reel. The steel tape is the most useful of all the measuring de- vices to the f. urveyor . There are many varieties of steel tapes ranging from very light "ribbon" tapes to hetorr tapes of the rib- bon or "-wire" type. The light tapes are thin strips of spring steel graduated to feet and inches or to feet and decimals (the common types to hundredths of a foot). Such tapes are never over 100 feet long and seldom over 50 feet. Heavy steel tapes such are are used in most surveying are heavy strips of spring steel graduated to feet with the first and last foot divisions on the tape graduated, usually, to tenths of a foot, though they may be purchr.sed with the end feet graduated to hundredths of a foot. I'he graduations are marked in different ways ranging from feet to 100 feet. In the heaviest patterns the tapes are graduated every five or ten feet with the l?.st five or ten feet graduated to feet and the last foot subdivided to tenths of a foot. These tapes are often from 500 to 10CO feet long. The light tapes are usually obtained on metal reels while Elem. of Surv. - 1 -/. Aseignrent 2 Page 6. the heavier patterns are usually ura^ouirtea. It is possible to purchase reele for the heavy tapes. Such a real should be of rather large diameter and of the "open" pattern. The Gunter's chain ^ though fast becoming obsolete) is a sturdy measuring device 66 feet in length. It is made up of a series of links with metal handles at both ends. . link, sd-called, ic -compose ' of a long, bar with a sraall oval link at each end. The length of the three parts of ths link tsucen together is 7-92 inches, or 1/100 cf the length of the chain. The end links are still more complex, the bar link being divided into two parts. On the outer end of the last portion of this link is fastened the handle. The handle is to fastened to the link that it forms a part of the length of the link, and ie adjustable by means of a nut and screw, raa'cing it possible to correct the total length of the chain. Everv tenth link is marked vrith & metal tag - at the tenth link i& a tag with one point, at the twentieth link a tag -vith two points, etc. The fiftieth link is marked oj a tat; of some peculiar design, usually round, oince the chain is marked in a similar manner from both ends, care must be taken to determine TJhether or not the measurement ie less or more than fifty links for fractions of chain-lengths. The engineer's chain is similar i-i type to the Gunter s chain, the only important dil'ference being that this chain is 100 feet long and the links are 1 foot in length. Both of these chains have been replaced by the more convenient and reliable . ol Surv. L*. >,ssignr;ent Z jps.ge 7 Among the man;- useful raeasa: ing devices usea by the engi- neer or surveyor is the stadia. This is t eoaiu-ination of cross- -.vires in the telescope of tae transit or plane table alidade such that tne intercept on '\ rod, as determined Jay these srosswires, multiplied by e certain constant will be the horizontal distance from the instrument to the point Tnfhere the rod is held^ It is not a perfect method, out under some conditions it will be found to be 'vaitfc- ace irate as compared to measurement ".>'itn tne tspe. For approximate determination of distances, pacing, in con- junction with the pedoaeter is found to be very useful. The pedom- eter is a watchlike instrument which records, in one type, the num- ber of paces taken bj the person carrying it. In another type the record is the distance in feet, yards, aietres or liiles - in this pattern it is possible to regulate the inttruaient to the average length of the carrier's pace. (20) ICASU3IKG OF LIHLS The me- e iv -ement of ! !:>. s with the rulr flnd i^ith similar devices has but little ^lace in the ordinary routine of the sur- veyor "work. On the other hand, the measurement of distances by pacing, or v.'itL the ta^c IZif. j pr ^ if _ . of Surv. Assignment 2 Page 16 by the rear chainraan in trying to hold the plumb-bob over an es- tablished point on the ground. Where uneven ground is encountered it is sometimes necessary for both chairmen to use plumb-bobs to make it possible to keep the tape straight an d horizontal, e. g. -a stump may be on line or a smell hunmock nay lie between the e;rlreuiities of the tape. (24) SLOIE iaEASUKEMLNTS FCR lE'fLK&INiyG HORIZONTAL DIST^CES These measurements nay be made in two ways, both are rather laborious and expensive, involving the use o ths level in one case and of the transit in the other. 1 A Fig- 2. In the first method stakes are set along the line at inter- vals less than a tape length and the difference in elevation in feet is determined betv/een successive stake tops by levelling, the distances on the slope between successive stake tops are then measured. From these measurements the hypotenuse and the vertical leg of a right triangle will be known. By geometry there results - in general - see Fig. 2 h = J s z - v 2 t (1) Elem. of Gurv. 1A. Assignment 2 P^ge 17 ^ h being the horizontal distance, e being the slope measurement and v the difference in elevation of the two points. Solving for each partial horizontal distance and summing the quantities, the total horizontal length will be H = hj + hg + h^ -t- -i- h n (2) In cases where the- differences in elevation between successive points are small, an approximation to the right triangle formula will be found quite accurate enough and simpler to use. This formula is derived as follows: From Eq. (1) raay be written V 2 = 6 2 - h ? the second term may be factored making v 2 - (s + h) (s - h) Since s and h_ are nearly the same length if JP is small., assume that they are equal and. apply this assumption to the first paren- thesis, only, calling them both s. Then v^ = 2s (s - h) v^ Vvhaics s - n = - Zl 2 and h = s - _Z . (3) 2s The accuracy of this formula ms-y be seen for the extreme case where e - 100.000 feet and v = 15.000 fest. Ihe accurate solution g,i-vo h - So. 86^ feet while the approximate formula gives h = 93.876 feet, this difference oeing well withiii the com- mon limit of allowable error, e. g. - 1 part in 10,000. Elen.. of Surv. Assignment 2 18 To find the horizontal method by the second method stakes are set along the line in the same manner and the vertical angles measured with a transit. These vertical angles are measured from the horizontal line through one stake to the sloping line drawn from the same stake t,o the next stake on line, see Fig. 3. These angles are represented as o^, o^, c< 5> o< 4 , etc. fhe slope distances, sj_, sg, 53, etc. , are measured. Fig. 5. the relation of parts in a right triangle it follows that i^ = s^ cos o^j (4) and H = h]_ -t hg - h? + . . . + h n (2) as above. Where snail angles are encountered and the slide rule is to be used it is often convenient to use the following formula: h^ - 5^ -- s^ vers o< i (5) The chief advantage in the use of this formula lies in the smaller number of significant figures necessary in obtaining the same accuracy as with the cosine. "Wherever a table of natural cosines is available it is very simple to obtf.in the versed sine Elem. of Sur?. 1^. Assignment 2 i^-ge 19 by remembering the relation versed sine = 1 - cosine. This method ie also good for solutions involving arithmetical computations with natural functions. INSTRUCTIONS CCNCELKEG SOLUTIONS CF PROBLEMS AND IHBIR At the end of each assignment a group of problems is placed 7,-hich you are to solve. These problems will embody the principles set forth in this and the preceding assignments, liiihen you solve these problems they nraet be returned P.S evidence of your mastery of the matter in hnnd. VJhen the problem group has been examined r<.nd errors indicated, it will be graded and returned with copies of the correct solutions attached for your comparison. The solutions of problems should ue \vor.-:ed out upon the special paper furnished ay the associated Students' Store, Berkeley. All solutions are supposed to be your original work, not neat copies of wor don.: on other p.per. All numerical work end, des- cription Must _be siiCTv^i in i-ik. A probln sheulc 'of solved in tuclv a manner that each ttep will appear ">n the problem sheet (liiinor arithmetic processes and interpolations exc-'ctea). Tabulate 3. series of related partial or similar results. Sketches or diagrams should be mace to illustrate the problem whenever possible. These- sketches inrxy be sn.de in pencil, but all lettering and dimensions rust be shown in ink. In aaking sketches reasonable proportions should bt observed and these should Blem. of Surv. 1A. &esi-.nrnt i; Pa^e 20 approximate the conditions given; it is not necessary, however, to drav,- sketches or figures to sce.le, except -.Then specially directed tc do so. Keatness and .general arrangement of parts of a problem must be considered. You should lay out tine method of procedure before starting to solve a ^r olden and then work out each step in a logical order. Five aim-fees spent in "blocking out' ! the -work will save rruch labor and vrorry. A surveyor should learn to De accurate and as direct as possible, since most of his work is of s nature requiring much time in execution and accuracy in results. % Problems will be graded en neatness, arrangement, complete- ness, and correctness - each part of the above list holds a -weight of one-fourth of the total grade possible. References ; y, pp. 31 - 38. Breed & Hosner, pp. 11 - 15, vol. I. Johnson, pp. 5 - 10. Raymond, p> 13 - 22. Elem. of Eurv. 1A Assignment 2 jfege 21 PROBLEMS 1. . man paces a giver, line in both directions, ihe pedometer carried lay him records 1355 paces in one direction and 1560 in the other. If his pace is 30 inches long, what is the length of the line (a) in feet? (b) ir. rods? (c) in miles? (d) in metres? 2. The horizontal distance between tv,-o points, A and B, is desired. Beginning at A the course was divided into the secTions A-l, 1-2, -3, 3-B and the corresponding quantities E and v measured, in feet. Section s v v^_ h E 2s A-l 98.76 5.63 1-2 99.00 10.70 8-8 97.42 7.11 3-B 67.10 3.24 Compute the several horizontal distances, h, and the total length H, to the neereet one-hundredth of a foot, using approximate formula, ^abulete results. 3. To find the horizontal distance between two points, A and 3, slope distances and corresponding vertical angles we're measured. Section s cos. method Vers- method h H h H A-l 73.65 5 10' 1-2 98.65 1056' 2-? 100.05 264C' 3-B 85-00 3100 ! (a) Solve for ths ssveral quantities, h, by the formula in- volving the cosine. J ee 5-place logarithmic tables and logarith- mic functions. Give results to nearest one-hundredth of a foot. (b) Solve for the serial quantities, h, by the versed sine formula, in eac'u cc.se do not use ;aore than three significant figures iii the ->ersec, sine, e. . - 0. C0373, 0.0985, or 0.173. u se natural functions end solve arithmetically. (c) Tabulate the two sets of results and find H froir. each set. UNIVERSITY OF CALIFORNIA EXTENSION DIVISION Correspondence Courses Surveying-La. Elements of Surveying Swafford Assignment 3 ERRORS IN LINEAR MEASUREMENTS FOREWORD : In this assignment it is intended to give an elementary outline cf the errors occurring in linear measurements, their sources, their correction, and their avoidance. (25) TKL TRUE LENGTH Of A LINE The true length of a line can never be definitely known from measurements made by human agency. Mathematically it is pos- siole to state that a line is a definite length, ag. - a line may be said to be 25 feet long, and from the purely mathematical con- sideration this line may be considered just 25 feet long, no more and no less, out snould a line of unknown length be measured Dy any means available to the human race the length may be found to be, probably, 25 feet plus or minus, a length due to some error which in the last analysis will be indeterminable. The magnitude of the error depends upon several factors, some of which may be dealt with definitely, while others are, in the main, irremediable. (26) ERROBS Errors may arise from two general sources, namely, imper- feet ion of instruments, and imperfection of the human functions. the first head ma,, be listed, the erroneous length of a chain or tape, change of length due to a rise or fall of temper- s/tare, irregular graduation of tape or otner scale, ncn-adjustment . oi' Surv, 1;. ..iSSign^e-rit 3 ^age 2 of instrument, arid such injuries to equipment as the kinking of a tape or chain and the Sending of spindles or compass needles. The effect of human imperfection is shown in several ways. Meet of the errors arising from this condition are termed personal errors This general group may be again divided into what are called mistakes and accidental errors. Mistakes are errors that occur in the mind of the observer, as, the v,rong foot mark -.vill be read from the tape; an angle of 52 degrees r.-ill be read as 57 degrees; in long lines a tape length will be lost or added; in attempting to repeat an angle five times, the instrument man may turn off the angle six times. When sighting a long distance on a sunny day, a person may regard the bright side of a rod as the full width. When this supposed iric'th of the rod is bisected, the line of sight is actually to one side of the true point. Here -y;e have a mistake, net an accidental error. Ajccidentai errors are really small mistakes - mistakes which do not come from a confusion of thought, but from the imperfection of human sight and touch, or from inaccuracy in estimation. These errors are exemplified by the small errors created in setting chain- ing pins or in estimating fractional divisions on a tape or circu- lar scale. (27) CUMULATIVE E3KORS Cumulative errors occur from inperiection of equipment or fron known natural causes. A tape may be too long or too short. In the first case the measured length of the line will be shorter than the true length by the product of the error in one tape length Slera. of Surv. IA Assignrnnt 3 Page 3. multiplied by the number of times the tape length is contained in the total length of the line. Should the tape be too short, the reverse would be true. The cumulative errors common to chaining or linear measure- ments are : 1. Incorrect length of tape or other measuring device. 2. Change of temperature, above or below the standard. 5. Incorrect alignment, i.e., both aids of tape not held in a horizontal line or in the vertical plane including the ends of the line measured. 4. Tape not stretched straight and taut but with the ends etill on the line. 5. Sag of the tape due to wind or lack of proper tension. (28) CCUSTABl JSJD VARIABLE ERRORS Conulative errors may be divided into two sub-classes, constant *nd variable errors. There is but one constant error, that due to the erroneous length of tape. All of the other cumu- lative errors are generally variable in their nature. Temperature is constantly changing; the same errors in alignment are not likely to be repeated; the sag of the tape will vary according to the ten- sion in the tape. It is possible that the sag and the temperature effects will be constant under soae conditions and therefore cause constant errors. Likewise the pull applied at the ends of the tape sine to a misunderstanding of what the proper tension should be, may be in error by a constant amount. Accidental errors are in their nature variable. El em. of Surv. lA Assignment 3 Page 4 (29) CORRECTION OF TIE MEASURED LENGTH OF A LINfc The correction for the several kinds of cumulative errors be determined by considering the nature of each error, its law, and its magnitude. While under certain conditions accidental errors follow a precise law, it is impossible to correct them by mathematical means. The only way to eliminate them is to adopt such methods as will make the possibility of such errors as small as possible. It is evident that they can not be entirely avoided. (30) CORRECTION FOR FALSE LENGTH OF TAPE The false length of a tape may be due to several causes. In the process of manufacture the attempt ie made to make the tape Just IOC' feet long when the temperature is 62 degrees Fahrenheit, the pull 12 pounds, and the tape fully supported on a flat surface. This ideal result is seldom if ever realized; hence the tapes pur- chased from the best raaKers are usually slightly in error. After long, continued use the tape is liaole to be kinked in several places. If the kinks have been straightened the tape will probably be too long because of the permanent stretching, of the material . if the :cinxs Jnj\e not been removed the tape will probably be toe short. A tape way be broken and subsequently patched. TShen broken a portion of the t??e may be lost; or the tape may be lapped in repairing; or, if a splice be made, the t\vo snds of the break may not be brought into proper contact. This source of error in the length of the tape is one that should be carefully considered in each individual case, for the greater portion of the tape may be Elen. of Surv. Lt Assignment 3- Page 5 reasonably correct vrhile the error at the patch will affect only such measurements as shall include this part. It should be noted here that an apparently perfect tape or one with kinks in it may have greater error in one part than in another. An; one or all of these conditions may occur in one tape. In a chain there is the possibility of the long links be- coming kinked and thus making the chain too short. On the other hand, there is a possibility of the many oval limes and the rings at the ends of tl'e long links becoming stretched or of the surfaces of contact between theia beconing so worn that the chain will be elongated. It will be remembered that a chain hag over 600 wearing surfaces. The adjustable handles at the ends of the chain are used to correct the total length of the chain but it evidently does not follow that such a means will correct fractional measurenents. If a tape is shorter than standard the measured length of the line will be too great.. If the tape is longer than standard the meapurec length of the line will be too small. By this is meant that a tape nominally 100. CO feet long and so assumed in raaking t.he measurement will, in the first instance, be contained a greater nuinbtr of times in the total length of the line, and, in the second, a lesser number of timss that would a tape of standard length. (It should be observed, th&t the measured length of a lint ie a ratio;- the length of tape to length of line. ) There are two general cases which arise in this connection, one where the error in the length of the tape io uniformly distributed Blem. of Surv. 1>_ Assignment 3 Page 6 throughout the tape, ,? nd another srhere the error is located at a certain place in the tape. In the first case a simple mathematical relation may be set up between the false and the correct values by considering what he.s been said before. This may be stated as follows : l f ; L f = 1 : L c ....... (6) or, put into words: The false (nominal) length, of tape (If) is tc the measured length of the line (Lf) as the correct length of tape (l c ) is to the c or re c t length of t he 1 ine ( L c ) . L C it the correct or adjusted length of the line, Lr. is the measured length of the line, 1 is the correct or standard length of the tape and If ie the false (nominal) length of the tape. From (6) may be written L , = For example, assume that a line has been measured with a taps nominally 100. jO feet long and that when the tape has been corapared with a standard it is found to be actually 99.95 feet long, the error uniformly distributed. The total measured length of the line is 375.66 feet. Substituting in (7) r = _. = 375.47 feet. 100. 00 In the case where the error ia concentrated at one place in the tape the- proolein is carried out in the following manner : First: Determine the magnitude of the error and the portion Elesi. oi' Surv. !. Assignment 3 Page 7 of the tape in which the error exists, i. e. , in the first ten feet, in the vicinity of the fifty-foot raark, etc. Call the error e. Second: Measure the length of the line in the usual manner, using the nominal length of the tape. Note at the last measurement whether or not that portion of the tape is used that contains the error. Third : Count up the number of full tape lengths, each one of which is known to contain the total error, and call this number N. Fourth : If the last fractional measurement involves the error, add one unit to JS making the number (N + 1). If the last fractional measurement does not include the error add nothing to N ("this also applies if the last measurement is a full tape-length and follows from the third step above). JTiftJri: From the foregoing can be written L c = L f + e(N + 1) (8) in the case where the error occurs in the last measurement. Or L c = L f eN (9) when the lest measurement does not contain the error. lo illustrate (8) : Suppose that the measured length of the line is 375.36 feet and that the length of the tape ie 99.95 feet by a comparison with the standard, though nominally 100.00 feet long. Let it further be assumed that an error of 0.05 foot occurs within the first 75 feet of the tape. Then, L c - 375.66 - 0.05 (3 + 1) = 375.46 To show the effect when the error is not in the last measurement, Elem. of Surv. lA Assignment 3 P^ge 8 let it be assumed that the error of 0.05 foot appears in the last 20 feet, all other conditions being the same as above, fhen. L = 375.66 - 0.05 (5) = 375.51 feet. c , The pluc-or -minus signs in (8) and (9) indicate that the tape may be either too long or too short and the correction is to be applied accordingly. (31) CORRECTION FOR CHANGE IK TEMPERATURE The standard temperature for most steel tapes is eet at 62 degrees Fahrenheit. Sines temperatures vary from day to day and from one hour of the day to another, it is a common phenomenon that a tape changes its length accordingly. Most steel tapes expand or contract at the rate of O.OOG0063 to 0.0000065 of their length for every change of one degree Fahrenheit (this quantity is 4 a ratio ard is called the coefficient of expansion; see any text on Physics). Assuming the temperature of all parts of the tape to be the same at any one time the change of the length of any portion of the tape "7ill be proportional to the eh?nge in temperature and to the length of the tape involved. If it is assumed that T is the standard temperature for the tape and that T is the temperature or the tape at tne tiae of measurement of a distance, also that one nominal tape length is measured, i. e,, 100. OO feet, 1 - If 0.0000065 l f (T - I 8 ) ... (10) The plus-or -minus sign indicates that the correction be added when the temperature is higher than standard and vice versa. Elem. of Surv. 1A Assignment 3 Page 9 It will be noted that if the temperature is higher than standard the tape will be too long, hence it will be contained too few times in the length of a line measured under these con- ditions. In other words the line appears too short, hence the principle set forth in (10) holds for the measured length of the line and it may be stated L c = L f O.OOC0065 L f (T - T g ) ... (11) Ihw same rale regarding signs holds as for (10). In making this correction it must be recognized that the temperature will in all probability vary considerably daring the work of measurement. The usual practice is to take the tempera- ture for each tape length measured, to correct each tape length, and then to find the total distance, though in certain types of v;ork the several temperatures are averaged and the average ie used in the application of the formula to the full length of the Ij ne. As an illustration of the method assume that the measured length of the line is 375.66 feet, and that the average temperature ie 78 degrees Fahrenheit. Substituting in (11) L s - 375.66 + (O.OOOuOSd x 37s. 66 (78 - 62;) = 375.70 feet. It is evident that a minute error -will be introduced by using the false length of line or the nominal length of the tape, but this error is so srnali tiiat it may safely be neglected in most surveying work* , cf bur". 1A assignment 3 Page 10 (32) CORHLCT1CN FOR Eitf.OHS DT A ferncujr* in niiynwij (mould w* *%.**, out wh iupU * thing as intentional deviation f*om the straight line is necessary, use should "oe nade of (3). AS i grime :it II, page 17. The distance, in this case, sboula be measured on the hypotenuse of tht right tri- angle and the perpendicular distance from the line to the end of the tape should lie the short leg of the triangle. This correction is o'uviously i'or only those measurements en each side of the ai&rK vhere the end of the tape is off the line, and for no others. (33) CC3ELCiIOBi FOE T^-ffi NOl S^KitiT AKD twill Errcr in this cise i^ia^ bt prevented jy proper care in seeing that the tape is straight and taut. (34) GORI-SCTIOi- FOk iAG OF XrtPE When tL cord cr strip of material of uniform cross section and of perfect flexibility is supported in a horizontal position by suspension from two points it assumes a curved form in a ver- tical plane. This curve is called a catenary; and it is possible . tc compute its foam mathematically. But since the equation of the catenary is awicvrard to use in general work, an approximation is nade "u;y assuming that the curve is a paraoola. Ihis assouiptxon, not go rauch in error as might oe imagined, gives, on th^e v/hole, a very satisfactory solution of the problem. Professor R. 3. 'woodward derives the final equation for the correction of length of a tr.pe in the u. &. Coast and geodetic our- vey Report of 1S92. ihie equauion must be accepted as correct un- til you have haa a thorough trainirig in Calculus end Engineering Mechanics. Eleau of Surv. 1A Assignment 3 Page 11 This final form for tne difference in span for the tape fully supported and for the tape supported at the ends only, may be written 1 / vl\ s : IT \T) (12) Equation (12) is the correction for & full tape length when supported at the two ends of the tape. c s is the correction for the full span given in the same units as 1; 1 is the total i length of span in feet; w is the weight of the tape in pounds per linear foot; ana p is the pull in pounds applied at the ends of the tape. This equation will also apply for any sp?.>n of the tape less than full length, if the spaa is substituted for 1. If the tape is supported at several points we must consider the sag effect per span. In such case the tape is divided into, say, ri spans of 3c feet per span. If n = 1, 2 (13) Sag tends to shorten the span of the tape and causee the measured length of the line to be too long, i.e., the tape is con- tfined too many times in the line measured. The correct length of the line is expressed by the fomula IfcStf-.Cj (14) Cg in (14) is the summation of the several quantities c g found for the several tape lengths measured. If ii tape lengths, including the fractional tape length at the end of the line, be taken, then C. - c s * c- + c s -f -f c s " (15) s S s s s v i Elesi. of Surv. IA ^eignment 3 Page 12 One -should not fall into the error of assuming that the several quant itiee, c s , can be multiplied by N except under special conditions; for an inspection of the formula will indicate that the correction in each c&se depends upon the cube of the span, the number of epane into which the tape is divided, and the pull on the tape, all of which may vary. The weight is usually constant for the tape used. Let the length of the line measured be 375,66 feet, the pull applied at each tape length measured be ,20 pounds, and the weight of the tape per linear foot be 0.010 pound. The assumption is that the conditions in each of the first three hundred-foot measurements are the sr.aie. Then x . 2 - 0.0104 foct. LOO. 00 0.010 x 100. 00 6 2 For the last 75.66 feet, however, 75. G6 0.010 x 75.66 2 From the above, c s = 0.0104 + 0.0104 + 0.0104 -f 0.004-5 ~ 0-0357 foot. and L c - 376.66 - 0.036 - 3^5. 6 feet (to nearest O.ul foot), Ag^in, the nominal length ia used but the error is so slight as to be negligible. The srror caused oy tie lateral sfc.^ cf the tape .;nder vine prc-s&are is so variable that it is impossible to correct it. The effect of wind is to shorten the tape auct iaake the measured length cf the line too long. The only tning to do when the wind is blowing, sufficiently strong to cause a large error is to stop 'ffork until conditions will allow good work to Oe done. t of isurv. l.-x Assignment, 5 Page 13 QUESTIONS 1. In measuring a line what four sources of error are liible to occur? ' 2. Explain the difference in method of applying error in length of tape : (a) 'When the discrepancy is uniformly distributed? ( o) When discrepancy is due to a defect in one part of the tape only. 3. Miy are measured distances made by leveling the tape? What other means may be used to accomplish the same object? 4. Distinguish between an error and mistake in taking measurements. When may a mistake be" called a blunder? 5. Kow may pacing be used as a check on blunders in measurement of a line? OF CALIFORNIA EXTENSION DIVISION CORRESPONDENCE- STUDY COURSES IN TECHNICAL SIBJECIS Course 1A Elements of Surveying Swafford Assignment 4 ERRORS IN LINEAR MEASUREMENTS (continued) FOREWORD : This assignment is a continuation of the discussion of errors in linear measurements with particular eaphasis on accidental errors and the law governing them. A later portion of the assignment is devoted to a discussion of precision, discrepancies, and allowable errors. (35) ACCIDENTAL ERRORS Accidental errors h&ve a tendency to balance each other; they are often called compensating errors. It is unfortunate that this term has been used, for it tends to mislead a person into believing that by some magic or other the effect of accidental errors is eliminated in all cases. In the greater part of the work done in surveying each quantity is measured but once, and furthermore, moat quantities measured are relatively small. KShere the relatix^ely small quantity is measured many times or where large measurements are made so tnat tnere will be many partial measur events every accidental error will have a tendency to bal ance some other. A mathematical study of the law of errors, particularly accidental errors, and the probability of errors (see Merriman's "Method of Least Squares") shows that such errors will come nearer and nearer to balancing each other as the number of meaeurements Elem. of Surv. IA Assignment 4 Page 2 approaches infinity. This meane not that in the measurement of a single tape length, for instance, the accidental error will be eliminated out that the total length of a line measured by repeated trials approaches more nearly the true, unknowable length, the greater the number of times the line is measured. It must not be supposed tnat, if a ion line is measured by a great number of partial measurements, the total length will be more accurately determined than the length of a shorter line of fewer partial measurements made with the same care and under the same conditions. On the contrary, the actual error will in all probab ility be less in the shorter line^ The common accidental errors occuring in linear measurements are caused by 1. Variation of pull or tension in the tape. 2. Error in setting pins or other markers. Kiet&kes or blunders may occur through 1. Incorrect count of tape or chain lengths. 2. Mistakes in reading the graduations. 3. Disturbing of pins or other markers after chey have been set. (36) CORRECTION OF ACCIDENTAL ERRORS Of all the accidental errors lifter 5 the only one Trhich may be corrected mathematically is that due tc variation of pull or tension. This error lies on the border line between a true cumu- lative error and an accidental error, due to the fact that in moat surveying the pull is estimated and not exactly measured. The case does arise, however, where the pull ie measured with a dynamometer or spring balance. When this is che case the Eleir.- of Surv. IA Aceignment 4 page 3 effect of pull coraes into the group of cumulative and correctable errors. It has been found that if a bar of steel one square inch in cross section r.nd of any length 1, is stressed by applying loads at the ends of the bar, the bar will be either stretched or shoi-tened 1 by "'""'- ',;. of its length for ever./ pound of load applied. Hence 3G,OOC,vA>0 if the area of cross section of the tape and the .nagnitude of the pull applied are ! , -*^ Vp< ' ov Kr A^ ,^^ i,-^j aa^lTKCMa' If -S-1 ^l .:--' i O'a ^'ifeW -L] S\ r^ ^ ^W ^p^ / v A 7^p\^K, 5^/)p J - ( ^^s^gkJ -v-^_' _^ ^z Xv ,_, v -^->^- r "-7-r : \ -.: SS^ o < to c\^ A ^ i * -i B e ||, f II |j' =3-1 Sag |Ja E w o l^x I SI !*l 4. "H ^ -s a HI tsi *>5, 2:3" ^5? 2^3 1=f lil isi ill ^3* c c iig u to S *0 M! .II 1 i l I P,? -2 c ?-5? O B 3 o fS| ri -So^.S 1W 6 o ** all a ll!l w e "? Ill ^ u T-ali |8I] x||S s e ii E'Venu of Surv. I/. Assignment 5 Page 3 applied in discussions involving either true or magnetic north, the limitation being set by the nature of the problem end method used. (42) DECLINATION OF THE NEEDLE Magnetic surveys of the earth's surface show that the pointing OJL the needle uoee not iollow a regular law. The angle by which the needle points away from the geographic or true north is called by surveyors the declination _o the needle, which is the same as the nautical term Variation. If the needle points to the to east of true north, the declination is east and if,, the vest of the true meridian, tne declination is west. The declination of the needle is different for different localities on the surface of the earth. if an observer traversed a line along v;hich the declination of the needle is constant, his path would be an irregular line. This line is called an isogonic line, a., line oi constant variation of declination of the magnetic needle. Figure ^ is a chart shoeing ths positions of such lines for the United States. There is P. line extending from the region of Lake buperior to south Carolina v/hich is aartceti 0. This is the Agonic Line or line cf no declination. If the needle is read at any point along this line the magnetic north will coincide i-rith the true north. To the east of this line the declination is viest and to the west of this, line the declination is east. E n .err-.. c* Suv-t. 1... Assignment 5 Fnge 4 (43) CH7WF- I";.. DnC ...I':.*: ION Th? declination of the needle is not. constant. In fact, the declination of the needle is changing continually. Secular variation is a slrv ?hartc cav.psl, apparently, by the slow shifting of the magnetic poles of the earth.: The law of the change or its caus? is not clearly understood. In the United States the change in declination from this ccuso is about three minutes per year on the average. This change is not the sane for all localities; the greatest rate of change is near the Agonic Line and the rate diminishes as the distance from the Agonic Line increases. Lines indicating the annual change are shown on the chart. Annual variation is a small change of approximately one minute of arc. Diurnal or Daily "^'- r iation is a change in the declination of the needle occurring each day. The anoar.t of the variation depends upon the season, of the year and the tiua of day, r -is 3 ing fron 3 to 12 minutes.;. The needle is in its most e?3~cerly position at about 8 a.m. and in is most westerly position at aoout 1:30 p.m.; the mean positions occur at sbout 10 F...m.. ar.d t:Z-0 n-" 1 ' Irregular yarintions cone from a variety of causes such as electrical or magnetic storms, during which tnc .noveaents of the needle are so erratic that it is impossible to read it accurately and from local attraction, which will .cause a definite deflection of the needle. Elem. of Surv. 1 A Assignment 5 Pag& 5 Of the variations in declination the most important are the Secular and Diurnal Variations. The Annual Variation is so small that it is unimportant. In the case of Irregular Variation due to magnetic storms, etc., all that can be done is to wait until the disturbance subsides. Local attraction is easily cor- rected in most cases. (44) THE SURVEYOR'S COMPASS The surveyor's corr.ppss is a deviee for obtaining the di- rection of lines. A compass box is mounted on a vertical spindle. The spindle turns in a hollow vortical sleeve at the lower end of which is a ball and socket joint. The whole is mounted upon a tripod so that the instrument may be set firmly over a point. The tripod is provided with a loop or hook immediately under the center of the vertical spindle so that a plumb-bob may be suspended from it while setting the instrument up over a point. The bottoir. of the compass box is a dial. In this dial, or art the frame carrying the dial, are two sm.ll spirit levels which are set parallel to tl.e bottom of the compass box and therefore perpendicular to the vertical axis of the instrument; one level These are used in leveling the dial. ie parallel, the other perpendicular, to the line of sight. A The dial is divided into quadrants, one end of the line being marked N or with sorae conventional device to indicate north. Diametrically opposite is placed S for south. The ends of the line at right angles to the first line are marked E and W. On a circle which is raised a short distance abo-ve the dial are found the graduations, usually degrees and half degrees. .. . - - . - . '''..I.-': -. '." . ' ,-.:' ' - ' ' ' '-... " .'. El em. of i>urv. IA A&siganent 5 Page 6 At the point corresponding, to north end to south the graduations are narked 0, while the points E and W are marked 90. The inter- vening divisions are conveniently marked in tens so that readings may "be easily made. On the frame carrying the compass box, or on the sides of the compass box, are mounted two standards or sights. These are vertical bars several inches high in which narrotv slits are cut for sighting. la certain types of compass there are just two nar- row slits, while in others there is a very narrow slit in one stand- ard a$d a wider slit in the other standard in which is mounted a fine wire or thin metal strip. These standards are fastened to the compass box on the line marked K and S, which line is the line of sight of the instrument. in the center of the compass box is a fine, needle-like pivot on which the magnetic needle swings, i'his pivot is very carefully centered and should be handled delicately. On this pivot swinge the needle, supported on a jewelled bearing similar to the bearing on the balance wheel in a watch. The jewel, either of glass or of some serai-precious stone, is mounted at the center of the needle in a circular setting. It has a small conical de- pression cut in it into which the pivot fits. The rhole needle mechanism is very delicate; hence a lever is placed in the lower portion of the compass box by means of which it is possible to raise the needle off the pivot while the instru- ment is being moved from one place to snother. E^em. of Surv. LA. Assignment 5 Page 7 The compass box is covered with a thin sheet of glass to protect the delicate parts within from dust and damage. (4-5) BEARING OF A LINE In surveying with a compass the angular quantities deter- mined by means of the compass itself are called bearings. The bearing of a line may be defined as the angle which the line makes with a line of standard direction called a meridian (i.e. either magnetic meridian or geographical meridian). Bearings, either true or magnetic, are reckoned from the north to the east or west, and from the south to the east or west. As stated above, the graduated circle of the compass is narked from at N and to 90 at E and W, hence bearings range from zero degrees to ninety degrees only. To illustrate: N E (or 70; N 15 W; S 8S C E; S 11 W. Where the bearing is exactly K or S or E or IV, it may be so indi- cated by usihg for brevity the single letter, but .it often prevents confusion to indicate auoh a bearing as in the first exanple above. Bearings are never more than 80 from the N or S, respectively. The bearings are determined by the position of the needle with reepect to the markings on the dial, and by the position of the line sighted. Usually magnetic bearings are obtained. These may be subsequently changed to true bearings if the magnetic dec- lination is known for the time and locality (see Computation of Angles and Bearings, Art. 54, Assignment VI). A mechanical de- vice known as a declination arc is found on some compasses, by Blem. of Surv. IA ^.ssjg.nnent 5 Pag-e 8 means of which iA 5.s possible to set the compass circ.le so as to obtain the _tue bearing directly, (See Leolinaticai AT?.., Art. 48.) TO GST UP A COMPASS To set up a compass over a point , the procedure is as follows 1. The instrument ioan stands the cciapass before him so that the two legs of the tripod are toward him and the third leg away from him. 2. He grasps the two nearer ] sgs in one hand and the. third leg in the other, strings the th5rd leg a-.ay from him till the angle between the legs is about 30, places tha foot of the leg on the ground about three- feet beyond the point over which the instrument is to oe set, and plants it firaly. 3. He then takes ihe other legs, one in each hand, amd swings the whole instrument and tripod toward him, opening the legs of the tripod c.s he does so, until the center of the tripod ie approximately ovtr ths point on the grcuud and the le^s c,bout eynniet-'ically placed. Th^ legs are then set on the ground. 4. The plumb-bob is then suspended from the Ijcp on the tri- pod. In doing thin c-".xe is taken to tie a "running bcw knot" in the string sufficiently far below the loop so tha j :, the liei^ht of the plumb-bob above the ground may be regulatsd. 5. H determines the position of the pj.urab-bob with reepect about to the point on the ground and moves the feet of the tripod/\until the plumb-bob hangs directly over the point. Thiu shifting requires considerably practice before the surveyor becomes adept at it. Elsm. of Surv. IA Assignment 5 Page 9 5. lo level the instrument he grasps the compass box or frams (never the standards) firmXy in both hands and tilts the head of the instrument about the ball and socket joint until both small bubbles in the level tubes remain in the center. As a cautionary measure it will be well to bring one bubble to the cen- ter of ita tune at a time. He swings the head about the vertical axis until the bubble tubes are turned 180 from their former po- sition and notes whether the buobles remain in the center. If they do not he brings each bubole half-way back toward the center of its tube, making the dial of the compass level and the vertical axis truly vertical. The reason for this will be apparent after the Adjustment of the Level Tuoes has been read. 7. He releases the needle by means of the lever, and allows it to swing fresiy. The instrument is then ready for use. 47) TO TAKE THE BEARING OF A. LINE, The instrument Oeing set up over one end of a line, it is possible to obtain the bearing of the line as follows: 1. The needle, for any locality, points in a fixed direction, namely, toward the magnetic pole. (Magnetic north.) 2. The line of sight, as determined oy the slits in the standards, swings about the vertical axis past the needle. 3. The dial with graduated circle, rigidly fastened to the standards, swings about the vertical axis under the needle. It must be Kept clearly in mind that the needle is sta- tionary for any one ''set-up" and that the dial end the line of sight move. E.Um. cC Surv. IA 10 In placing th-3 3 and W on the dial their positions are r*-."srrfia from the ^aot wid west points of the horizon. This fol- lows directly from the three considerations ircaediately above and will be clear after a reference to Figures 5 and 6. True ngrth ^ magnetic north graduated circle Fig. 5 - 6 Figure 5 shov/s the compass box set so tiiat the line of sight points toward the magnetic north, in which case it coincides with direction of the needle. The magnetic declination is taken as E'.em. of Surv, l Assignment 5 Page 11 17 19' East arid is so shown on the figure, i. e^, nagnetic north is east of the true north. To obtain fehe bearing of e. 15 re, sight en a signal at the far end of the line from where the instrument is set up. Sighting consists of turning the compass DOX until the slits in the standards are in the same vertical plane witn the line on the ground, that is, ths signal will appear in the middle of the farthest slit or will be covered by the metal strip in tnat slit when sighting through the nearer slit. Assuming that Figure 6 represents these conditions, it is seen that the needle still points to the magnetic north and that the line of sight pointe in a northwesterly direction. Since the sero has moved with tne line of sight the Worth end of the needle points toward a graduation, eay 30 30' from the, zero but to the .Bright or eastward i'raa 0. The line has a bearing that is really N 30 SO 1 W. An examination of the dial shows that ths 90 point in the right-hand portion of the dial is marked W, as above inai- cated. It is now apparent why the letters t; and W are reversed on the dial. The bearing shown is the magnetic bearing of the line. (48) TtiL Di/CLD'Ai'IOtt ArtC The declination arc is a device which makee it .possible to take the true bearing of a line directly. This arc is festened rigidly to the portion of the compass box which carries the full, graduated circle and extends about 35 degrees in both directions from zero. It is commonly graduated to half degrees. This scale is similar to the main scale of the compass. Elem. of s>'..rv. ]A A83gnnnt 5 Page 12 Fastened tc the portion of the compass box carrying the dial is a small vernier s--:a.l3, or merely an index line* The ver- nier scale is graduated tVcm sere. In the ii> i-?.d ).e , to thirty in both directions. (ics .Theory of the VerriJ.tr.) 3y Eisnns cf the index mark or vernier it is possible to set off an angle on the> declination arc. In Figure 5 the declination arc is shown in the "North" position of the compass box. The declination arc ie so set on the compass box that when the zero on it coincides with the zero of the vernier or the index nark, the zero on the main circle lies in the line of sight. This is true no matter where the declination are may be fastened to the compass box. The line of sight is connected with the portion of the box carrying the dial *nd vernier or index mark. Referring to Figure 7 let it be desired that bearinge taken with the magnetic needle be the true bearings of the lines. The needle will always point toward the magnet io north. From the fig- ure it will be seen t"..?t if the needle is to read zero degrees when the line of sight is pointing torard the true north it will be necessary to swing the line of sight WEST of the zero on the graduated circle. This is accomplished by the use of the declina- tion arc and vernier. Using the saae magnetic declination as be- fore the setting is shown in the figure. The needle points to zero and the line of eight points to the true north. It must be noted that the marks indicating the cardinal points of the compass Elem. of aurv. Assignment 5 Page 13 True North ^Magnetic North 1719 ? Vtrnier roads 17l9 r E True K. North- K \1719' 3030 f Magnetic North Beitring (True) S 3030' E Bearing (Magnetic 1719' = *' ~^^ S 4749' > Fig. 7 Fig. 8 turn around under the graduated circle with the line of sight so that readings should oe referred to the graduations on the circle. The markings on the dial ere used to indicate the general position of the line of sight, only. This setting is made but once for any locality and the parts should be clamped securely so that it will not be changed during the course of the survey in all cases in which the declination arc is set off as explained above. Figure 8 shows the compass set for a bearing of S 30 30' E. The relative positions of the parts of the compass ^should be Elera. of Surv. Lfi. Assignment 5 Page 14 carefully noted and compared with their positions in Figures 5, 6 and 7. (49) READING TEE COMPASS KEEDIE ThiB is a simple operation, but it should be carefully done. Most compasses are now made so that the sighting ie done from the S end of the dial through the N end. If this be the oase the north end of the needle should be read at all times to determine the bear- ings of the lines sighted. That is, always place the eye in sight- ing at the south standard in whatever direction the sighting is made and read the graduated scale at the north end of the needle. If the compass be so constructed that it is possible to sight from either the S or K end the following rules should be ob- > served : 1. If sighting from the S end of the compass through the N end or if the H en'i cf the compass is towerd the point sighted, / read the north end of the needle. . If eight ing fron the N end of the compase through the S end or r..f the b snd of th r v compose is toward the point sighted, read the south erd of the needle. In reading the tearing it is always best to look along the needle from the end opposite from th-3 one being read. This prolongs the line of the needle until it intercepts the graduated circle and eliminatee parallax or apparent sidevise displacement of the needle. References : Tracy, pp. 293 - 300. Breed & Hosner, pp. 16 - 30, vol. I. Johnson, pp. 13 - 57. Elem. c Surv. TA. Assignment 5 Page 15 PROBIEMS 1. The magnetic oearing of a line is S 16 30' W. The declination of the needle is 11 15' E. Draw n figure similar to those shov.ii in the asoi^-uient to show tne position of the coTipass with respect to the line and with re ape it to the true meridian. Make the dial three inches in dianeter, outside dimensions, 2. Draw P. diagram shewing the conpass set to give ihe true bearing of the line shov.n in Problem 1. TOiat ie this bearing? 3. Draw a diagram showing the conpass sighted along a. line, the true bearing of TriiicU is Ii 56 W, so as to give the magnetic bearing. I; the declination is 10 M, what is this bearing? 4. The ceolination of the needle at the present time at r. certain place is 17' nif- The secular variation is 5' eastward per year. What "/ill the declination of the needle be in five years'; OF CALIFORNIA EXTENSION DIVISION Course 1A Elements ol Surveying Swafford Assignment 6 COMPAbS KUhVEYING FOREWORD: This assignment deals with compass surveying methods in- cluding traversing with the compass, computation of angles from bearings, bearings from angles, detection and correction of local . attraction, and adjustments of the compass (50) TEAVEHSIHG WITH THE COMPASS The compass was one of the earliest surveying devices used in modern surveying for determining the directions of lines, and while it i little used today except for reconnaissance and the checking of more accurate work, the method of traversing with this instrument should be thoroughly understood. Traversing is the process of bounding a lot or field or measuring the angles and perimeter of the field. In other words, it is the process of traveling around the perimeter of a polygon in such a way as to determine its dimensions. There are three common cases -which arise in this work:- 1. That in vhich the actual corners of the fields may be oc- cupied by the compass and in which the lines may be measured di- rectly. 2. That in -rhich the corners of the field cannot be occupied by the instruments, but in which the lengths of the lines can be measured directly. 3. That in which the corners of the fields cannot be occupied ' ' ' . to keep the rod plumb. Elem. of Surv. lA /.ss.'.gnrent 6 Page 2 by the instrument and in which the lins cannot be measured flirectly. The first case, which is ths simplest, will be explained in this assignment. The second and third cases, which are not so easy of solution, will be considered under Traversing, Assignment X, Art. 94. Kefer to rigure 9, which represents complete field notes for a compass survey of a farm. Ihe procedure was as follows: Beginning at the point marked A the compass was set up over the stake and properly leveled, The instrument mao then sighted at the point L and noted the bearing, U 72 15' E, this value being recorded in the third column on the left-hand page of th^ noteoooi between D end A- This bearing is called the back or reverse . oearing of the line DA. r_ the bearing of the line AD. This terminology should be very carefully noted eo that the student will not be con- fused in later discussions. The circle in the instrument used was subdivided into quarter degrees of 15' divisions but all readings were taken to the nearest 5 minutes by estimation. After the above reading was recorded, the instrument man sighted on the point 3 and the forward bearing of the line AB or, simply, the bearing of the line AB was recorded in the second col- umn on the line between A and B_. A sight was then taken on the point for the purpose of checking the work. Ihis bearing was noted on the sketc.. on tne right hand page. The rodman held a rod over the points D and A being careful to keep the rod plumb. Blem. of ^urv. 1A Assignment 5 Page 3 The instrument man then moved tr the points b, C, and D respectively, wher^ he repeated the process as at A. At the point C a sight was taken to point A. ihe bearings read in each instance were recorded in their proper places in the f ieldbook. The distances were then chained and recorded in the sixth column on the left hand page. The right hand page shows a sketch of the field which should always be a part of good field notes. Such a sketch should show the shape and dimensions of the survey approximately to scale, the natural position of ireportant^and artificial features, the name of the owner and the naiies of adjacent owners, the meridian (both true and magnetic if possible; , and such explanatory notes as would make the findings clear to any later surveyor who might have oc- casion to use them. It is generally the experience of surveyors to find that most field notes are not full enough. The tabulated notes should tally with the quantities recorded on the similarly designated sketch and should be similarly designated. The notes on the left hand page include a column ior the stations occupied, a column each for the forward and reverse oear- ings of the lines, a column ior local attraction, another for the corrected bearings, and one for the distances. The lower portion of the page shows the interior angles cf tbr field and the 'is>iel check for such a survey. (51) LOCAL ATTRITION Local attraction is the disturbance at any particular point t. Elem. of Surv. IA. Assignment 6 Page 4 or station which caueea the needle to swing out of the magnetic meridian. It is c^-sed by the presence of magnetic metals nearby, or by electric power lines in the vicinity. The surveyor should take reasonable care that he does not carry iron keys or a knife, and that he does not leave any portable objects close to the instrument, such as chains, tapes, iron aligning rods, etc., that would disturb the needle. An annoying source of variable local attraction is found in steel buttons or clasps on clothing worn by the instrument man or his assistants. Local attraction is appreciable over varying areas depending upon the magnitude of the source, but it is usually felt within a very restricted area and the disturbance decreases as the distance increases. The Law of Magnetic Attraction or Repulsion, is that magnetic attraction varies directly as the intensity and inversely as the square of the distance from the source. Where the local attraction cannot oe removed the magnitude of the disturbance of the needle must be determined and corrections made in the notes. (52) DETECT ION AND CORRECTION OF LOCAL ATTRACTION Again referring to Figure 9, columns 2 and 3, it will be seen that the forward and the reveise bearing of the line AB are just 180 degrees apart. Thie condition should hold for the two bearings of any line surveyed and would indiaate that there was no disturbance of the needle at either of the two points. It might be, however, that the disturbance at both of the points was the saae in intensity - t but opposite in direction, although this would be exceptional. Elern. of turv. Page 5 (53) CORRECTION FOK LGC-vL *.!&: 2 103 J-f the forward and th<= bacic bearing check, that is, if they differ by 180, it is reasonable to assune that there ie no local attraction at either end of the line. It cannot be assumed, how- ever, that, if they do not check, the local attraction is at either or both ends; this must be determined by taking bearings on other lines connecting with the two points. For example a closed tra- verse may be taken to illustrate a method of adjusting the local attraction. The following set cf obeerved bearings, given ooth in sketch and tabulated form, will mc.ke this plain:- / v B /ell x \ / N &&* ^ \ v L/n^ if.Jet.'M'r'? if j r: ~IT " 7; n 5. O^dli'"^-? ! 1 (-r>'fC ' -'OM -^ 1 . 4 fcs,^ / s *v ^v< ^\ TT 4 G N ,f>t < "4 o ' ^ j oo H 58*40 r V-~ s^\0, ^-V 3-/i ': i-tv'^oV ^ 50 ' S38*40'W ^c* *\ \ / x 3-CJS5tt:; 4-ao' S3O)0'e d*& T&0\ < X r C-5| 1 4^/i c :5o'w -PC-' N 30" W'^' ^ ^^V^ *" "/ J ^ c-^ 1 : ' ; -/5'.Yi l j -30' 56/ c 45'5V \ /N \ .x A5>> ^ % or if X-I45E 00 W^ C 45" \/^ D ^ "^t^I^J^i : oo 1 /V35V/V A-DI h S ^ C ")'X.! F~ HO -''- *- . W". S35^' i- - - ... . 1. , ' 1 in. Elem. of Surv. 1A /.ssf.gnrrent 6 Page 6 At A and D it may be assumed, Bjjaoe th forward and back bearings check, that there is no local attraction] hence A - B IB probably correct, N 58 40' E; and B - A should be adjusted to read S 58 40' W; likewise D - C is probably correct, N 61 45' E; hence C - D should be adjusted to S 61 45' W. Evidently there is then local attraction from a source af- fecting the needle at both B and C. Here the forward and back Dear- ings (B-C, C-B) differ by 40', and one half of this amount added td 29* 50' and also subtracted from 30 50' will bring the bearings into agreement. Thus for this ease the adjustments are made. The data is tabulated as shown above, The interior angles may now be computed and will be found to check, as they should (nearly) in practice. 54) COMPUTATION OF IMKRIOR MGIEi FRO,.: BEARINGS By referring to the adjoining figure, the bearings being the. adjusted values in the traverse just cited;, it is easily ob- served that the -?.ng.U ,-BC is the sum of the baok bearing B-A and BC D ." ' El em. of Surv. Page 7 the forward bearing B-C. ^ence -d-A&C = &840' * 5010 ! = 8850'. Likewise, angle D is the sum of the back bearing D-C and the for- ward bearing D-A. tience CDA = 6145' + 3500' = 9645'. The angle BAD is found by adding the forward bearing of .-,-B to the back bearing of A-B and subtracting this sum from 180. 180 - (5840 f -* 3500') = 86 20 1 . Also /BCD = 180 - (6145' + 30 C 10 J ) - 88 05'. Many rules are given by the various authors of texts on surveying, but by far the best and the simplest method is as follows: Sketch the meridian (magnetic or true; in accordance with the bearing) and determine the desired angle by geometric addition. A further illustration using general values for angles and bearings here follows : = L MtIV Pi L - o ihe e^L If -f io n , Elem. of Surv. 1A Assijnw^nt 6 Page 8 Check: The sum of the inter j.cr angles of any polygon is equal to twice as many righ + angles as the figure has sidse, J.BSS four right angles. The figure hare has five sides, Twice 5 - 10, from which subtract 4, giving a remainder 6 Therefore, a five sided figure gives 6 right angles or 540. In general S = (2n - 4) 90. This formula is also \7titten S - (n-2)l83 c . (55) TRAVERSING BY DEFLECTION ANGLES Traversing by deflection anglss is somstim^s a convenient method, the angles both exterior and interior being easily calcu- lated. If the deflection angles are measured around a field con- secutively in either direction (either all right of ail left de- flection) these angles constitute the exterior angler of the poly- gon; their sum for any polygon is 360. 'i'he interior angle at any vertex is 190 minus the exterior angle at that vertex} the sum of all the interior angles is 180 times the number of side? minus 360, or in general S = 180n - 360 = (2n-4) 90. 56) RANGING OUT LINbS Ranging out is best done by observing the deflect? oa angle and recording it as right (R) or left (L) at sach charge of bear- ing. This is especially true for a traverse that does not close, as a road, rail-road, boundary line, or ditch, the rangirg of which consists of several parts of straight lines or lines and aurves. To measure a deflection angle, set up the instrument at the forward end of a line and back-sight on the other end. Then Elera. of Surv. Assignment 6 placing the eye at the opj.oe1t.e si^ht, turn the Jine of sight right or left as the case may be inrf rert the angle passed over by the line of sight when bisecting the next point in order. As follows : To range cut the line A B C D E by deflection set the in strument at A and back- sight on 0. ^'hen turn fore -sight on B and . L A ^^-3\ 13 "^'^ * -"'- p read a R. Next occupy B, and back-sight on A; turn lore-sight on C and read b. Then at C read c L; at D, d L. The aigle A is ' 180 - a. 3 is ISO - b, etc. A he direction of lines, or their bearings are often given in azimuth. Ihe azimuth of a line is its bearing measured in de- grees in clockwise direction from either the north or south point of the horizon; either the magnetic or the true north (or south) may be chosen as zero azimuth. This method will be more fully ex- plained later, under transit surveying. REFERENCES : Iracy, pp. 378 - 383 Breed & Hosmer, pp. 29 - 30, Vol. I. Johnson, pp. 34 - 38. Raymond, pp. 83 - 85. . .? .. ,; . ..T .;, :.- .;.-..< .. ; , -' - ' , ',. ! ',,''-. . ' ' " ' ' ,"'<' .-'.- : ' . . . * ; -.'...; , ' ' .' I El em. of ^urv. IA PROBLEMS c5 rnu.ent 6 P&ge 10 1. The courses ana bearings of a certain traverse are at follows: A-B, S 35? 52' 3: 3-C. N ?134 ! E; C-l, K 1826' fc; D-E, due West; E-A, S 4109 ! W. Find the interior angles and the error of closure, toake a sketch of the field. 2. A triangular field was surveyed with compass and the deflection angles v-ere taker, as fellows: at A-B., 9315'; B-C, 15030'; C-A, 11611'. Compute the interior angles, and give the discrepancy. Sketch the field. 3. A certain survey recorded magnetic bearings as follows: 1-2, K 4315' E; 2-3, S 7318' E; 3-4, S 83'23' W; 4-1, N 8017' E. if the magnetic declination at the time was 1215' E, what was the true bearing of each line? OF CALIFORNIA EXTENSION DIVISION Corresoondence Courses Surveying-lA Elements of Purveying Swafford Assignment 7 THE LE\EL AID I'i'S USE FOREWORD : This assignment will deal v;ith the level, one of the most important instruments used by the surveyor. Many of the principles treated in this part of the course are fundamental. You are urged to master the subject of leveling, not omitting the slightest de- tail, because much that follows in the subject of surveying depends upon the material given here. THE LEVEL In its usual form, the level is an instrument for determining the heights (elevations) of points, lines, and surfaces from some known surface. It consists of a spirit level and a telescope, so combined and adjusted as to enable us to determine elevations. The illustration, Plate I, shows a plain type of surveyor's level, which will be fully explained in a subsequent paragraph in this lecture. For the present let as turn attention to a few definitions of fundamental impcrtence. level surface is one that is everywhere perpendicular to :he plumb-line drawn to a point on that surface. Such a surface is not a plane, but is curved in all directions. As it will ap- proach very nearly a spherical surface, for the present purpose will speak of it as such. Elem. cf Surr. IA Assignment 7 Page 2 The surface of a body of water at rest is level; and a large body of water, such as a lake or ocean, is known to be curved. In the more exact work of leveling, the surveyor actually measures and determines the curvature; in the less exact work he determines only a tangent plane or several tangent planes which suffice for '. the purpose in hand. A level line is any element of the level (curved) surface. A plane can have only straight lines lying in its surface, a plane tangent to a le\el surface meets that surface in only one point, the intersection of the plumo-line at that point. Hence a straight liue and a level line coincide only in that point (called point of tangency) where a line drawn directly to the earth's center meets both lines. The points along any vertical line may have surfaces passed through them and thus a vast number of concentric spherical surfaces may be found. These surfaces are designated by the heights of a point in each above (or below) a certain agreed surface called Datum or datum level. Datum ie generally taken to be mean sea level, although any other surface may be assumed for reference. Points, lines, or surfaces referred to the datum surface are said to be so much above (or below) datum, the distance being given in the usual linear unite. Since it is not alwayst practicable to use mean sea Ie ve 1 as datum, other reference planes or surfaces nay be selected. For .Tioet purposes these are just as good; but where the established Eleni. of urv. 1A Assignment 7 Page 3 levels are to be of a permanent and extensive character, see-level is the best reference plane and should be studiously sought. [i) IHT. SPIRIT IVBL The spirit level is a simple instrument consisting of a ightly sealed glass tube, or phial, almost filled with liquid, ter, alcohol, or ether, imprisoning a small quantity of air in the part not filled with liquid. This air constitutes the bubble s it is called, on which observation is made in the use of the pirit level. The tube with its contents of liquid and air is enerally briefly called "oubole tube". The bubble -tube it lightly carved, so that when placed longitudinally in a horizontal oeture, the convex surface is uppermost. 'Ihus the bubble of en- losed air ie forced by the many times heavier liquid to the summit f this curve, and thus caused to occupy a fixed position for a iven horizontal one of the containing tube. Just ae the plumb-bob nd its cord enable one to determine a vertical line, so the bubble- ube may determine for us a horizontal or level line; and by the ombined use of plumb-lrne and Dubble-tube, aiany desiraole quen- ities may be measured, lines located, and relations estaolished. he spirit level enters int,c the construction of many useful in- truiaents and in its use aecones ths surveyor's constant reliance 'or determining horizontality and perpendicularity. i n he compass, escribed in Assiga-nent 5, usually carries two bubble-tubes. The pirit level is an essential part of the engineer's level, the sur- yor's or engineer's transit, and the plane-table alidade. It is Elem. of Surv. lA Assignment 7 page 4 also used for leveling tapes, for plumbing rods, etc. The spirit-level is sometimes made in a circular form that is very convenient for some purposes; for example, for. leveling the ple.ne-table and for plumbing rods. The upper, curving surf ace of the circular level, aads of glass, is a portion of a spherical surface, the sphere being of great radius as compared with the size of the level. The imprisoned air, which is the bubble in this form as in the tube form, rises to the highest part of the glass cover, and so the bubble is made to occupy the center of the circle when the plane of the level is horizontal. This accomplishes all that can ordinarily be accomplished by means of tiso bubble- tubes placed at right angles to each other; a level plane may be determined by the single circular level, which is often of great advantage. IE) PRINCIPLES OF THE ENGINEER'S ILVEL' 1. If a horizontal line be made to rotate about a point in itself, but always maintainxng a horizontal position, it will generate a horizontal plsne. 2. Straight lines lying in this plane will be horizontal lines. 3. All lines passing through the point about which the given line rotates will be perpendicular, to a plumb-line at that point. In the engineer s level the rotating line is the line of :ollimation of a telescope which may be prolonged as the line c si^ht to an indefinite distance. Ihis line of collimation is made perpendicular to a plumb- line, called the vertical axis of the instrument, about which the telescope rotates, therefore, in a horizontal plane. Elsm. of ourv. lA Assignment 7 Pege 5 Conceive then this line determined by the telescope's axis sweeping out a horizontal plane tangent to the true (spherical) level surface at the point of intersection with that surface of a plurao-line, the vertical axis of the instrument, fhe height of this plane aeasured along the plumb-line (vertical axis) from some assumed plane of reference (datura plane) is called the height of instrument (K. I. ) Figure 14 In the illustration (Fig. 14} L. T. is a. level telescope sup- ported upon a tripod that is in adjustment, carefully set, and "leveled up" %-ith the tripod legs firaly planted in well chosen isitione in the midst of e numoer of points at the surface of the ground, a, o, c, c, e, f, and also &n estao lished point B.lt. (bench marie) shovn to the left. The telescope rotated about ite vertical axis will determine the horizontal plane A B C D E F G. If now MS measure ti.e heights oi the horizontal plane above the points on the ground we obtain the quantities aA, bB, cC, dD, eE, JET nd (B.h- ) G. Now the height of B.k. above the plane of refer- ence, called datum level, is known, riold a rod on B.fcL. and direct the line oi sight toward it and read the distance (B.M. )fi. Add UNIVERSITY OF CALIFORNIA EXTENSION DIVISION CORRESPONDENCE COURSES IN ENGINEERING SUBJECTS PLANE SUBVEYING COURSE X-lA PLATE I 16-iNCH DUMPY LEVEL Length of telescope, 16"; diameter of objective, 1%"; magnifying power, 28x; total weight, 25 Ibs. PLATE II PRECISION ENGINEERS ' Y-LEVEL Length of telescope, 18"; diameter of objective, 1%"; magnifying power, 33x; total weight, 27% Ibs. Elem. of Surv. 1A Assignment 7 Page 6 this reading of the rod to the height of B.M. which will give tne height of the level plane above datum. This is called the height cf instrument, H.I. The plane iiJCDLF is at the height H.I. above Datum. To find the height of a, b, c, etc. above datum, hold a rod (level-rod; at a, fa, c, etc. and direct the line of sight to it and read the heights cA, b3, cC, etc. subtract these r eadings from H.I. and the differences will be the elevations of the points on the ground above Datum. Note well that the rod-reading on 3.M. giving the quantity (B.M. )G is added to the height of B.M. to give the H.I., and that the rod-readings aA, b3, cC, etc. are subtracted from H.I. to give the elevation of the points on the ground, hence, since the Sp- reading is added, we regard it always as a -f quantity, called a plus sight, and, fcr e. like reeson, the point whose elevation is to be ceterminea we regard as a - quantity, called a minus sight. The terms B^clc-sight for plus sight and Fore-si^ht for minus sight are also commonly ussd. You should get the foregoing con- ceptions clearly in ;nind, as uuch confusion is often occasioned 'by a vague understanding of these very simple matters. The following is B orief description of the engineer's level. For simplicity the Ijarapy level is chosen. The student should procure a descriptive catalogue or manual cf surveying in- struments such as is published and for sale at small cost by the Elem. of Surv. 1A Assignment 7 Page 7 leading instrument makers - L.C. Berber & Sons, Huff and Buff, W. & L. E. Gurley, riausch & Lomb, Ihe A- Lietz Co., and others. The manuals go much beyond the text-book on the subject, 'and usually give a deal of information, tables, and material of great value to student and engineer alike. The best way to know the in- struments is by intelligent handling of them guided by a skilled instrument man or by following directions given in good text-booke or the manuals najied above. TH1 DUMPY il/vEL Ae shown in the illustration, Plate I., the Dumpy Level con- sists of a horizontal bar fixed to a vertical axis which turns freely in a well made socket, io the bar a telescope and a spirit level are firmly attached. Ihe whole is supported upon a tripod and a leveling hsad. The latter is used to set the bubble in a central position in its tuoe, thus Cringing the line of coliimation of the telescope (which should be parallel to the .axis of the bubble tube) into a horizontal position. ihe line of coliimation can then be made to sweep out a horizontal plane by turning it upon the vertical axis of the instrument, prolonged indefinitely in the line of sight. The telescope contains two optical ports, the ooject-glass at the larger end and the eye piece at the smaller end of the tele- esope tube, i'he object-glass is a douole-convex lens usually of the compound variety to overcome the tendency to color dispersion of light by unequal refraction, called chromatic aberration. This lens is s Iso shaped in & manner to reduce as far as possible a Elem. of Surv. IA Assignment 7 distortion known as spherical aben:ati_on. Both of these faults of simple lenses, would be very trouolesorae, if the skill of the in- strument maker could not combine and construct lenses so as prac- tically to eliminate aberration. For a full explanation of these matters you should consult a reliable work or. optics; in general, a good high-school text on physics will be sufficient. When the light from the various parts of sny object to tvbich the large end of the telescope is directed passes through the double-convex lens, it is made to converge within the tube in such a way as to form a real but diminutive image of the object sighted. This real iiaage is inverted. If now, the real image be viewed by another lens, the eye-piece, it will be magnified so that its details will be more distinct aid it -Brill appear nearer and also larger. In some instruments a compound eye-piece is em- ployed, which re-inverts the image that was inverted by the object- glass; thus the thing viewed through the telescope appears in its normal position or erect. On this account a telescope with the compound eye-piece is called an erecting instrument. The sort having a simple (single) iene for eye-piece is an inverting instrument. The inverting type is preferable for many optical reasons; and when once tne engineer oecoraee accustomed by long use to the inverted itua^es, he may prefer an inverting to an erecting instru- ment. The inverted image is brighter, and more snarply defined, and it gives a flatter field, all of which are very desirable. However, it is not intended to disparage the erecting eye-piece, as Elem. of Surv. 1A Alignment 7 Page 9 all makers construct excellent instruments cf this latter kind, snd many engineers- prefer to use them. Within the limits of ordinary surveying the erecting instrument is quite adequate and very con- venient. ^^ , _ J _ J ___ J _ --..-, - .._ .,_ T ,. -_.-^r. . . ~^ r """" ~' Of- ll 12 V Section of Erecting Telescope Figure 15 The accompanying cut shows a section of the telescope. ' is the objective lens and 1, I-,', 1_, lg ere a comoination of lenses constituting the eye-piece. The lenses are named in order,!, the object lens; 1., the amplifier; lp, the field lens; 1* the eye-lens. Together they form a compound magnifier (microscope) which enlarges end re-inverts the real image formed by the objective-lens, 0. This real image falls at C, at which is placed a ring-like diaphragm carrying two fine threads, either of platinum or spider web, called oroes-hairs or cross-wires. Thus the image viewed through the tele- Esope has the image of the cross-hairs lying distinctly upon it The line joining the intersection of the cross-hairs and the op- tical-center of the objective-lens with the optical-axis of the telescope, vriien in coincidence form the ''Line cf Collimation" to reference has several times been made. (61) THE Another common form of engineer's level is the vllye Level shewn in Plate II, in which the horizontal bar, instead of having El era. of 3ur7. IA Assignment 7 Page 10 the telescope attached to it, carries at each end & clutch for holding the telescope called a %e (or Y) since it resembles the letter Y. The wyes are accurately ground to shape on their supporting surfaces in ivhich rest two collars forming a fixed part of the telescops-tube, The collars also are very accurately ground to cylindrical shape and of equal diameter, so that, when resting in the supporting surfaces of the wyes, the optical axis of the telescope (the line of collimation) is perpendicular to the ver- tical axis in whichever of its two postures it may lie. Thus the telescope may be turned end for end in the wyes, or rotated about its axial line when resting in it. Each wye has fitted to it a cover-clip that fastens down either with a pin or other device. In one collar (usually the eye-end ) is a notch into which a pin on the clip fits, thus holding the tube firmly and preventing its rotating when the clip and pin are down. The bubole-tube in the v;ye-level is fastened by short col- umn-like supports to the telescope-tube so that it may be adjusted to be parallel to and lie in the same plane (i.e. truly parallel) with the line of coliimation of the telescope. See Assignment XIII on Adjustment of Instruments. The wye-level has lor.g been in favor with engineers on ac- count of the readiness with which certain instrumental adjustments oaj be made, and perhaps on account of the belief that it embodies some features thought to be indispensable in an instrument of presicion. Tnese features are, however, rather more fanciful and Elem. of Surv. lA Assignment 7 Page 11 traditional than real or even convenient. The dumpy-level is coming more and more in favor every day; and for precise leveling other and unquestionably better forms have superseded the wye- level. Still the -rye is by no means an obsolete pattern, and is etill preferred by many surveyors. You are, therefore, particularly advised to acquaint yourself with the structure, adjustment, and use of this instrument. (62) THE HAED-iEVLL The hand-level is a simple and very useful leveling tool. It is specially adapted to reconnaissance and preliminary work, or to levels Tfhere a low degree of accuracy is permissible. The Locke hand-level is the simplest and most commonly used type, but others of a more refined and elaborate construction are also made by most manufacturers of instruments. They are der scribed in catalogues and manuals. The Aoney hand-level and clin- ometer, and the Atwood combined hand-level, clinometer, and com- pass (a really universal instrument) are the principal rei'ined forms of this useful little tool. The very elaborate preciss-levels, such as are employed in the Ccast and Geodetic Surveys and other extensive leveling work, are not in general use by engineers ~nd will not be described in the present course. The manuals referred to furnish illustrations anc! descriptions for those who desire to know more about them. The level in any of its forms is used to determine heights . Elen. of Surv. Assignment 7 Page 12 or elevations from an assumed datura, to measure differences of elevations or altitude, to run profiles, lay off cross-sections for fill and excavation, locate contours, etc., etc. the work of such nature" will be explained and amplified in subsequent assignments. Trill ch will deal specifically v/ith each class of problems - 1. What do you understand by H.I-? 2. Explain 3. S. , F-S. , B.fci. 3. Why is a level line not a straight line? 4. A level set up on the shore of a lake had its line of sight in a plane 6.5 feet aoove the water in the lake. The readings taken on a rod held at various points gave at A 4. 3 ft., . at B 8.7 ft., at C 6.5 ft.; how do the heights of these points compare %vith the level of the water in the lake? References : Tracy, Chapters XIX and XX. Raymond, Ch- III. Johnson, pp. 55 - 56, 60 - 64. Breed & Hosraer, pp. 72 - 78, Vol. I. UMVLhbHY Oi' CALIFORNIA EXXLwSIUlM DIVISION Correspondence Courses Surveying-lA Elements of Surveying Stafford Assignment 8 LEVELING PROiiLLwS FOREWORD : fhis aid the following assignment will describe the use of trie engineer's level (A) for determining Difference in Elevation; (B) for Profile Leveling; (C) for Estaolishing 3ench Maries; (D) for Cross-section Leveling; (E) for Leveling for (1) Excavation (tiorrovv- pits) and (2) Fills; and (F) for Finding trades and Contours. The methods of making notes foj the various problems here treated will also be shown. f'63) 3EKLRAL The level should always be set up on firm ground at a con- venient station for observing as many points and as large a range as is consistent with good work within the limits of accuracy sought. It is rarely necessary to set the level over a point, the elevation of which is desired, or immediately upon a line joining two points to be measured. Indeed, such a setting will often frus- trate the purpose in hand cr greatly interfere with accuracy in observation, as will be perceived if you recur to the conception of a horizontal plane at elevation H. I. (64) BENCH MARKS If elevations of points are desired above any agreed plane of reference (or datum level;, a bench mark (B.M. ) of known elevation must begin or dote the work. Eleivi. of Surv. 1A Assignment 3 Page 2 Should the extent of the leveling require that the level be moved from place to place, points of firm and often of durable na- ture must be selected or improvised on which an observation (fore- sight) may be made at one setting end another observation (back - sight) may be made at the following setting. These are called Turning Points, perhaps because the measurements are made from a new horizontal plane svept out by the line of sight at each setting. (65) Turning points are usually selected or established as the work of leveling pro eeds, cfld much depends upon their kind, i'hey should always be sufficiently firm to support the rod held upon them while ooth fore-sight and a back-sight can. be taKen upon them. They should be situated in open spaces where both sights may be clearly taker, free from obstruction, not too near or yet too remote from the instrument. The distances from two successive turning points (T.P's) to tue instrument should be about equal. This last requirement is for the purpose of eliminating any error in the parallelism of the axis cf the bubble-tube with the line of coll inat ion of the telescope. For, if the vertical axis be truly vertical, but the lint of sight for any reason oe not perpendicular to it, then, when the bubble is at mid-point of its tube for each sighting on two points equidistant from the instrument, these two points will be in the sane level plane. The line of sight in such case "sweeps out" the surface of a cone end the vertical axis of the instrument is also the axis of this cone. The position of the cone is either erect, stand in upon its oase, if the line ol sight is inclined downward, or inverted, standing upon its apex, if the . of Surv, 1A Assignment 8 line of sight is inclined upward. Movv, since the two points are at equal distances from the instrument (i.e. from the vertex of the cone) and the axis of the ccne is vertical, a horizontal plane perpendicular to the vertical ajcis may be made to pass through both points, hence they are at the same elevation and a line joining them is a level line. (66) TEE IEVEL ROD For the purpose of making vertical measurement a rod is used, on which is merged off a scale of linear measure, feet and inches, or oetter, feet and decimals of feet, or in other cases, meters and fractions of meters. The common form is in two parts, each about 7 1/2 feet long, 1 1/2 inches wide, 3/4 inches thick, graduated upward upon the front side and continuing downivard on the reverse side of each part. The parts are fitted together and are held in place oy two bronze sleeves, The upper sleeve, turned toward the back, has a vernier acale on its edge, by which the least interval capable of being measured mey be determined. The upper sleeve also carries a clamp sere./, with a large milled head, by means of which the rod, when extended, may be held firmly in position. Thus the rod raay be used either short (up to 7 feet) or lon^ (to about 13 1/2 feet when fully ex- tended). The lower end of the rod is fitted with &. metallic shoe. (e?) A TARGET The target, made of .aetal in circular or other convenient shape, about 5 inches in diameter, has an oblong opening in its . . : . : .: :] . : .. ; . . Elem. of Surv. IA Assignment 8 page 4 middle part. It is suitably fitted to slide up and down upon the front section of the rod. On one edge of the opening a second ver- nier is attached for reading the least measures as with the first vernier. The face of the target is divided into four equal quad- rants inrhich are colored alternately white and red, their horizontal line of division being coincident with the zero of the vernier scale. A clamp-screw similar to the one mentioned above is carried by the target, by means of which the target may tie set at any point along the front section of the red. It may also be fixed at the 7 foot mark of the rod on a segment of the front section that is firmly glued and screwed upon the rear section. The best rods are made of well seasoned, straight-grained maple wood, neatly fashioned ano highly finished, and the divisions are accurately and distinctly narked in black and red upon a white ground. (08) I'flE VERNIERS The above mentioned verniers are constructed ae follows: For a direct-reading vernier, a space equal to 9 of the smallest divisions on the rod is divided into 10 equal parts on the vernier. Let us take the common graduation in feet, tenths of feet, and (the least division on such s rod) hundredths of feet, i'hen in the vernier for this graduation (Direct type) will consist of 9/100 feet divided into 10 equal parts, end each vernier interval will be 9/1000 feet. If now the horizontal line of the target, which coincides with the zero or lowest line of the vernier divisions, be e*i at any point or. the rod, say at 4 ft. , 3 tenths, and 6 hun- dredths and some fraction above the 6 hundredths mark, the vernier '. .' :j 1 .' * v ' '. I Elera. of Surv. IA Assignment S Page 5 will determine for us the number of thousandths oeyond the 6 hun- dredths mark. For example, with the setting of target as given above, 4.36+ ft., count the vernier divisions beyond the 6 marie to that vernier division which coincides vrith a hundreqth division Une of the ro. This will give the number of thousandths of a foot to be appended to the reading aoove. For instance, if the . full numoer be ? vernier divisions, the exact/., rod-reading will be 4.367 any ft. And so f or < other cases. In reading a direct vernier always count along forward in the direction of the main rod sc*le and add this count to the reading upon the rod as shown by the last full division. This will be upward on the front section of the rod, and downward on the re- verse side of the rod when using the rod extended. Generally, and for good reasons, a better form of vernier, called the Retrograde Vernier, is now employed, i'his is constructed by talcing a space equal to eleven least rod divisions and subdividing, it into 10 equal parts. Thus one division of the vernier is (on the retrograde vernier; 11/10 of one hundredth of a foot, to if the zero of the vernier scale fall at a point oeyona a full hundredth division on the rod-scale ana the vernier scale be applied in a reverse (retrograde,; direction to tht scale divisions on the rod, the count on the vernier scale backward to the line of coincidence will measure the number of thousandths of a foot oeyond the last i'ull hundredth rae.rk. ^ence re-^c tue i'ront of the rod upvard es always, but read the retrograde vernier downward) likewise on the Elem. of Surv. IA Assignment 8 reverse scale for long rod, read the rod divisions downward and count on the vernier scale upward. The following illustration ' shows settings for both direct and retrograde verniers for both short-rod and long-rod readings. You are advised strongly to make many similar settings and record the readings for a rod to which you may have access. Check back the readings recorded, Dy resetting the target for each of the whole series of observations. ~ *l -.O i~- - ? J- N-; v - o i 4 LJJLJ_LL r*- r ' 1 ' -I ' ! \ v_>* *o *- < \ / / / (69) USL OF THE LEVLL hOD When taking readings on a target rod, be sure to close the rod down to its shortest, length for short-rod, or set the target exactly on the highest mark for long-rod.. Also be careful that nothing such as mud, gravel, or other foreign matter intrudes itself between the shoe and the point (B.M. , T.P. , or other sur- face) the height of which ic sought. This could cause an error of several thousandths or even hundredtns of feet, thus voiding the accuracy of the measurements. A proper beiich mark is usually- free from foreign natter or can easily be rendered so. fuming points should be selected upon a stone firmly imbedded in the ground, the root of a tree, or & stump or post, a curb-stone, Elem. of Surv. 1A Assignment 3 Page 7 doorsill, etc. ; T.P. 's may also be put down in the fora of wooden stakes or metal pins driven well into the ground. A good form of T.P. is easily made of a triangular piece of light plate iron, which has a round head rivet set in its center and the corners turned dovn sharply at righx angles to the plate, as snown in the figure. When such a turning point is pressed down into the -eatth, especially in light, marshy, or sandy soil, the rod can be held upon the rivet head at the center, thus in- suring an excellent support. If a hole is punched in the plate so that a cord or chain ft 9. 1 7 may be' attached, it can easily be lifted up and carried from point to point, be careful also that the T.P. is even and horizontal, for if not, the rod might be too high in one position, and too low in another. (70) THL USE OF THE TARGE I In leveling, the use of the target greatly facilitates the work of accurate sighting and at long ranges, several hundred feet distant, it is nearly if not quite ind ispensaDle to good work. Several forms of targets are in use, but the circular with quad- rantal divisions is perhaps the best. The angular target which is in reality tv/o half ellipses viewed at an angle of 45 degrees and thus appearing as a circular disk, is ouch in favor with many engineers. Juch a target presents a sharply defined vertical edge Elen. of Surv. 1A Assignment 3 (the intersection of the tivo semi-ellipses) , which assists the instrument man in determining whether or not the red is in the X'ertical plane through hie station; and the straightness of two horizontal lines, continuous as one line, enables him to know that the rod is also in the vertical plane at right angles to the first plane, or, in other words, plumb. For, if the rod be in the first vertical plane the vertical cross-hair will coincide with the intersection, and if not in the second vertical plane, the horizontal divisional lines will forr. either a wide vee C^") when the rod leans toward the instrument, or will form a wide inverted vee (^\) when it leans bacicward, as shown Oy the hori- zontal cross-hair- (71) PLUtiB ROD, HUD IEVLLS It is very important' in level work that the rod be held plumb when a reading is taken, as any deviation from perpendicu- larity to the level surface will give a false height, and the true height is the thing sought. Rod Levels are frequently used, especially in nice v;ork. These are either two small spirit-level tubes encased in a bronze block that fits squarely upon a corner of the rod, or a single level set in ?. block that may DC applied by the rod-man when re- quired. The circular or bulleeye level is also used for this jjur- pose. It is supported upon the rod by a bracket that secures the true position. A plumb-boo and line ma- also De used. The rod -man I * Elen. of Surv. 1A Assignment 8 Page 9 can also often secure the rod 1 8 verticality by aligning the same by eye with convenient structures near at hand or in the not too remote distance. Still another way is to place the rod in the vertical plane through the instrument as nearly as possible, which is the less difficult feat of the two; then at a signal from the instrument man to 'Hvave rod", he may slowly tilt the rod forward and backward while the man at instrument determines by trial the highest po- sition of the horizontal line of the target. This last is th least satisfactory method, but in the absence of other, better means, it should not be neglected. (72) CHOOSING SETTINGS AND T. P.'e. In moving from setting to setting of the level, or in se- lecting turning points, it is essential that ooth rod-man and in- strument man use careful judgment in each case. Often, after much trouble in choosing such positions, they are found unsfetis- factory from the fact that the level is either too high or too low. This is especially liable to De the case on very uneven ground or in regions where rocks and trees or brush render sight difficult or impossible. At times the levelers must "hasten slowly", or at lenst exercise much ingenuity and good judgment. Often the longest rod is not long enough or the level plane at H. I. cuts the ground below the foot of the rod. Nothing can then be done but either to set up the instrument anew or to select a more suitable turning point, if it is this that makes the difficulty. , El era. of Surv. IA Assignment 8 Page 10 By roughly sighting along the barrel of the telescope when approxi- mately adjusted to level, an advantageous setting may often be quickly determined. It has been the purpose of the author to give detailed directions, as the student of this course is supposed to do all instrument and field work unassisted by sn instructor to direct his operations. Much must always remain vague until by experience in handling instruments and laying out woric in the field, indis- tinct, misunderstood, or erroneous mental images (such as one may acquire from bocks of description and illustration) are cleared up and corrected by actual practice. Therefore, it is v. r ith earnest sincerity that you are again urged to delve into texts and manuals on surveying in order that you may secure an intimate acquaintance with both instruments and surveying methods. The references at the end of each chapter to standard works will be of much value to those who have access to and will properly use such books. (73) GENERAL PROBLEM. IN LEVELING. A - Difference in Elevation The problem is to determine the difference in elevation between two points A and B. Set up the level at any convenient place about equidistant from the two points to be ooeerved and follow directions, ooserving the cautions given in the preceding part of this lecture. Level the instrument, reversing over each diagonally opposite pair of foot screws, the rod-man must now hold the rod upon the point A in a vertical position with target out of range of line of sight ! I . . ' :, ' v Elem. of Surv. IA Assignment 3 Page 11 so that the instrument :nan may read through the teleecope the approximate height of tne horizontal plane. Then call to or signal this number v r -reading) to the red --man, ?rho will quickly set the target as directed, but hold it if within comfortable reach, and rr.ovc it up or down until signalled to clamp it. At the signal "wave rod", he will tilt it forward and oaclcward. Determine whether the target is r.t the highest point; that is, whether the line of sight ie never below the horizontal line of the middle of the target <\rsd -vhether it cuts this line- but once at each forward or backward rucveirent of the rod when it is "waved". This is the correct target setting, which the rodiaan nov^ reads fcr himself, F.nc, ir. psssing the: instrument tc occupy the point B, the instru- ment man also reode the setting and records it in hie field book: when it ie found to igree -.vith the reading determined Dy the rod- rr.an- fce calls "checs" if correct; otherwise the rod should again be held upon the point and re-read to insure aosolute agreement. At point b the sa.-ae process is repeated and the elevation of B taken - that is, the rod in ooth cases has measured the distances frorc the points A and B to the level-plane svept out by the line of sight, or vhat is the sz.xe thing, the distance of the two points below this letel-plane. ihe difference jetveen the two rod readings is the difference in elevation sought. If the two points A and B are very distant fr?m each othtr, cr if the intervening ground is very sloping, or if the t\/o points can not both be observed from a convenient setting of the instrument, it bsco/aes necessary to select one or more turning points and to observe on these, finally closing or. the second given point. Flem. of Surv. 1A Assignment 8 Page 12 To illustrate this and tc furnish a correct form of notes, tsuce the following: First setting (1) Rod-reading on A 4.376 ft.; (2) roc-reading on T.P.,, 6.843 ft.; reaove level to second setting; (3; rod-reading on T.P.p 3.572 ft.; (4; rod-reading on T.P.p, 5.831 ft.; third getting of instrument; (5) rod-reading on T.P.g, 2.7-*3 ft.; (6) rod-reading on T.P.,, 6.745 ft. ; fourth setting; (7) jcd-reading on i.P. , (on the long-rod, i.e., rod extended with the target O clamped at the topmost graduation; note that this reading is made on the back part of the red and downward) , 9.357 ft.; (8) rod-reading on 8, 3.852 ft., which completes the course from A to B through three T.P's. We now proceed to tabulate these in notebook form as follows : p/ f f e re i > + ic( L&^e Is Z M43 5-35T foicjkT cn jjc.fc. T P, on firm fml<&f* in /^. //> r/*^ /^^3 b is i 42^3f i /: A- The illustration is intended to show t\vo pages of Field IJote-book of approved form. Note:- It is best, when the difference in elevation should be measured with precision that a series of return levels be run Elem. of Surv. 1A Assignment 3 Page 13 beginning upon B and closing upon A. A 'he T.P's. of the return levels should be ehosen at any set of points other than those used in working from A to 3. Thus a new set of determinations is made which should "check" closely, within aoout 0.005 of a foot of the first determination, (74) GEHERAL PROBLEM IN LEVELING B - Profile Leveling This problem will undertake to get the elevation of a series of points, called stations along a line A either straight or crooked; so as to chart the profile. Assume a line staked off by tape or chain measures giving stations along the line at regular intervals of 100 ft., 50 ft., or any other desired number of feet and levels taken at these and intermediate stations. A line, as shown below, is the horizontal 4-tf> 4r56__ U 1 2. . ' " projection along which a profile is desired. he line A B C D is laid off in stations (at A); 0+50; 1; 1+20 (at B); 2 (which is also used as a turning point); 2+50; 3 (at C; ; 3+50; 4,; and 4+75 (at D). L-^ and Lg are positions at which the level is set up, the same being about equidistant from B.M. (a bench mark at known or assumed elevation) and from which rod-readings can conveniently be taken at each full or intermediate station. With level set at L. and carefully leveled up, read the rod at B.M. (+sight; ; then rod-readings at 0, 0+50, 1, 1+20, Z will be - sight readings. ; . / !.'' , ' Elem, of i_urv. 1A Assignment 8 Page 14 The instrument is then set up at L, , carefully leveled as before, and a + sight taken upon 2 as a turning point. After this read the rod held &t 2+50, 3, 3*60, 4, -i+50, and 4+75. hecord in note- book ae follows: -' 8.5 : //I ; Fj>. i 0,. t i Mtt ' '*- j^rS* , I I 1 D- /^- fat .(JQO & &bo\/ OcTilrft). ; ISO. Of 50 EE 119. ... Uu i f A/y ^ . ! 1 I- V, 1 |4.-^-$ 119. 009 h- 50 , j 17?. ~" 2-5-00 8 5S *j 179 ISO TTP^MM^ S^d^ 7-^ ^t5o l i n$. ^s mci-Jfcet 3^0- 2^/70 TFT i 765 3i~o J7.53C IftO. "" r^.r^j^ * 4t**P-bCRLttS, I-nFGLKT-SChEYvo AS stated aoove the nlidade-p.la.te and the graduated-plate, the outer graduated rim of Trhich is called the limb, may oe moved together or separately iu azimuth. In order that this :aay oe ac- complished, each pifcte has a clamp-screw oy which the upper may oe secured to the lower or the lower to the foot-plate. When in this clamped position i*v it. deei;rabie to rotate either plate through a sinall en&le, for careful &ud nice adjustments, each plate is fur- nished with r t?ngent- screw to accomplish this object. They are called tangent -scrfevvs, because they act along a tangent to what may be assumed to be the plate's circumference. The location and the form oi eacb of the p.bo^e e crews are such as to enable- the instrument nian to find a particular one at pleasure, jjn exaini- nation of a transit or oi the detailed illustration, Plete V, will reveal them to you. (79) I.EV AT^tsched to the foot-plate, iu which the spi.idlos are fittea and in which thej- turn, ars four (or in some instruments only three ; lugs extending radially, intc which fit smooch, well-cut screws, having large milled heads, one in each lug. These screws, called foot-screv,s, rest upon the fixed tripod head and permit the spindles to be shifted until they are or ought into a vertical position. Since the two plates are fixed at right angler, each to its own Eien. of Surv. IA Assignment 9 Page 4 spindle, oy adjusting the sp:.ndies until they are vertical, the plates are raaae horizontal at the same time. (so) cu'jMP AKD TANGENT SCREV, OL TELESCOPE Attached to the right hand standard are the clamp to lock the horizontal axis at the trunnion, and the tangent screw to give it gradual motion through a small angle in the vertical plane of the telescope. The three tangent screws aow described bear against opposing springs thet hole the moving part firmly, except vfaen pushed forward or withdrawn backward by the screw. (81) LEVELS UPON iL-IL The alidade-plate has two small spirit-levels attached at right angles to each ether oy adjustable screv/s, which enaole the levels to be orought into parallelism with the plate, i.e. at right angles to the vertical axic through the spindles. In some transits, one of these levels is attached to the left hand st-?ndard th^xt supports the telescope, but its office is the same as if it werelying upon the plate. (82) COMPASS ^HD- JisAGK&TIC NELDLL Commonly a transit has P compass plate ann a magnetic needle fixed to the alidade-plate, by which magnetic bearings may be taken, and magnetic declinations may be determined. The magnetic needle- is of especial value in checking angles measured upon the graduaied or lower plate of the instrument. (33) LEVEL ON TELESCOH, A spirit-level, several inches in length, and usually of ..'.'. 1 ..,.'. . . ... . UNIVERSITY OF CALIFORNIA EXTENSION DIVISION CORRESPONDENCE COURSES IN ENGINEERING SUBJECTS PLANE SURVEYING COURSE X-lA UJftlW SSW*y*ltfer trf* Ml PLATE V Cross-section of a Transit. CORRECTION No. 29 should read "Shade Holder" instead of "Vernier Glass Fr No. 30 should read "Vernier Glass Frame" instead of "Shade Holder." Eleza. of Surv- Lu Assignment 9 Page 5 great sensitiveness, is connected parallel to the telescope by mear.s of short columns. By * use of this level the line of sight may be r.ade horizontal ae in the engineer's leveling instruments already descrioeo. This enables the engineer tc fix the horizontal plane for the usual purposes of leveling, or for securing angles of elevation or depression, etc. (84) VERTIGO, CIRCLE OR VERTICAL /JRC Attached to the horizontal axle of the teleecope (with com- plete transit) is a graduated circle or a graduated arc, with ac- companying verniers, for measuring vertical angles (angles of elevation or of depression). (95) OTHER ACCESSORIES A variation arc is soraeti.nes connected with the compass- plate. J-t is often of great convenience in setting off the dec- lination. A gradienter screw is in some transits substituted for the usual tangent-screw to the horizontal axis. Uy this small angles in elevation (or depression) may be set or determined independent of the vertical arc. It is a screw of very refined make, the thread of which ano the divisions on its head being EO graduated as to secure an elevation of one foot at a distance of 100 feet or 200 feet for one full turn of the screw. (66) IRE GRADUATION OF THE LOWER PLATE The lo^'er plat is graduated in degrees and fractions of degrees, usually to halves or thirds; i.e., the smallest division UNIVERSITY OF CALIFORNIA EXTENSION DIVISION CORRESPONDENCE COURSES IN ENGINEERING SUBJECTS PLANE SURVEYING COURSE X-U METHODS OF GRADUATING SURVEYING INSTRUMENTS AND POPULAR STYLES OF VERNIERS FURNISHED A FIG. 21 Double vernier reading to 30". Circle graduated to 20'. V Fi<;. 22 Double vernier reading to 20". Circle graduated to 15'. 10 FIG. 23 vernier reading to 10". Circle graduated to 10' with one row of figures. Bleu, of ourv. 1A Assignment 9 Page 6 on the graduated Hub IE 50 minutes or 20 minutes of arc. In some instrunents the division is carried to 10 minutes of arc but this is rather exceptional, being too small for ease in reading, and of no especial advantage. X'he divisions are grouped in spaces oi b and 10 degrees marked on the scale by lines somewhat longer tnan the usual division lines and numbered at the 10, 20, SO, etc., manes. The zero of the scale is indicated by a or A rhe numbering is generally in both directions, preferably through 360; one &et of numbers being nearer to the inner p^.rt of the graduated ring and reading clockwise; the outer set reading, therefore, counter-clockwise. For convenience in reading, the characters are slanted in the di- redtion of numbering. Some makers colcr the inner or clockwise set in black, the outer or counter-clockwise in red. These and other devices are intended to assist in reading in the right direction and to prevent contusion of scales. (87) VtHWIERS The vernier as adapted to reading linear scales (see Assign- ment 8) has already been explained. Now we turn to the circular vernier. On the level-rod, verniers of two kinds are used, the direct- reading vernier anu the retrograde vernier. As the retrograde ver- nier is rarely used on transits, the direct type only need be dis- cussed here. One form cf vernier is constructed by taking 29 degrees UNIVERSITY OF CALIFORNIA EXTENSION DIVISION CORRESPONDENCE COURSES IN ENGINEERING SUBJECTS PLANE SURVEYING COURSE X-lA METHODS OF GRADUATING SURVEYING INSTRUMENTS AND POPULAR STYLES OP VERNIERS FURNISHED FIG. 24 Single vernier reading to 20". Circle graduated to 20' with two rows of figures. 10 20 '"I ^rt^'W rH^^r ao ilo FIG. 25 Vernier reading to 2'. Circle graduated to single degree. 3ft A FIG. 26 Double vernier reading to single minutes. Circle graduated to 30'. Elera. of Surv. 1L .ussigmaent 9 Page 7 (when the smallest division on the limb is the whole degree) and subdividing it into 30 equal parts for the vernier scale. The value then of one division on the vernier will differ by 1/30 of 1 degree, or 2', from a single division on the limb; this value ie called the "least count " of the vernier. Another commonly used vernier is formed by subdividing 2S half-degrees (the smallest rivision on the limb being a 30* space) into 30 equal parts on the vernier scale; here we have ** = I 1 as the least count. 29 + 1 A vernier constructed upon a scale of 20 minute divisions in which 39 of these civisions are divided into 40 parts on the vernier would give -?. least count of 1/2 minute or 30"; thus ii 20' is the smallest li.ab division and 39 the vernier number, then 20' 39 * l vill equal - = 1/2' = ?0". The following rule may now oe given: R'JLE. ^Co find the least count of a vernier : Divide the value of the smallest division on the limb by the nuraber of such divisions plus one required in forming the ver- nier scale. In general, To find the least count of a vernier: Let I.e. = least count, d = the value in circular units of the smallest division on th* limb, n the number of these smallest divisions th&t make n-f-1 spaces on the vernier-scale; then d I.e. =. ~ , n + 1 unit In this formula the resulting unit value will be that^in which _d is expressed. This of course may then be reduced to any required unit . Elem. oi' Surv. Fage 8 Verniers are of two types, single and double. The double type comprises tv.o single verniers extending in opposite directions from the initial point, or index, the numbering also extending, in opposite ways. Thus, with this double vernier, vrtien the alidade plo.te has moved over the graduated liiab clockwise, the left-hand vernier is observed; out when in counter-clockwise direction, the right-hand vernier is used, always reading, (for & direct vernier) in the direction of motion. The vernier reading is then appended or added to the reading on the limb. For economy of space another plan of laying out the vernier scale is sometimes used. In this style one half of the vernier is to the right, the other to the left of the index nark. This form is called the "folding vernier" or "split vernier" and is shown in Figure 26A- ]Tb31oTriii what has already been said respecting F o Idir.g or "Split" Vernier 50 / / UlUdl I 10 30 Figure 26a. verniers in general, no extended description of the folding type is needed, 'fhe net hod of reading the folding vernier is as follows Read the limb to the last full division passed over by the vernier's index; then determine the vernier reading by counting El em. of Surv. L^. Assignment 9 Page 9 along the vernier in the direction of motion of the alidade-plate to the line of coincidence as usual. If the coincidence does not occur over the first half of the vernier scale, then continue the count from the opposite end of the vernier toward the index, to the line of coincidence. Add the vernier reading to the reading on the lirr.b. Here the smallest division on the limb is 30'; 29 smallest intervals are divided into 30 parts cm the vernier; hence the least count of the vernier is one minute. In the illustration the movement of the alidade plate taken clockwise has brought the vernier index past 42; reading the vernier forward to the left, we find no coincidence to the left half and, therefore, returning to the extreme right and again counting forward to the left the coincidence is found at 25. ihis appended to the limb reading gives 42 5' as the full, correct reading. Much has been said and several illustrations have been given covering the subject of verniers. This has been done in an endeavor to make clear to the student this important and well-nigh indispensable adjunct of the engineer's transit. The student is urged to acquaint himself fully with verniers of every variety, in order that he may be able to read angles with ease and dispatch. Also in using any transit or other instrument on which a vernier must be read, be careful to determine its "least count" first of all. UNIVERSITY OF CALIFORNIA EXTENSION DIVISION COBEESPONDENCE COURSES IN ENGINEERING SUBJECTS PLANE SURVEYING COURSE X-lA PLATE VI TRANSIT THEODOLITE PLATE VII MOUNTAIN AND MINING TRANSIT FITTED WITH BURT SOLAR ATTACHMENT lien, cf Surv. lA Assignment 9 Page 10 (38) THE VARIOUS UbLS Of 'IHE i'RANI.C A. Ranging-out Lines. 2he fact that the transit telescope can ba rotated in both a vertical and a horizontal plane enables us to make use of it for determining direction and for prolonging lines at any angle desired. For purposes of surveying, lines must be continued or pro- longed in certain desired positions upon the earth s surf ace , as in setting out a line of division between properties, a boundary line or fence line. Such a line must often be carried from point to point either in & straight or raore often in a broken or curved line. Ranging out lines ie also practiced in road, railroad, canal, ditch, and other construction, and in most problems in land surveying. (See especially burvey of the Jrublic Lands, Assignment XIX..) Through homogeneous media light is propagated in straight lines; through any such media, therefore, light emanating from any distant source is brought to the eye in a path that does not deviate fror, a straight line unless its course be interfered with or de- flected by intervening agency causing reflection or refraction. This principle is the fundamental one in all uses of the telescope in surveying and in astrcnor.ical measurement. As mentioned in the description of the telescope on the engineer's level, the line of coll iraat ion may be prolonged indefinitely into "the line of sight". Cbeerve that when the trrnsit telescope is directed toward a dis- tant object, light from thr-t object enters the large lens and is Elera. of burv. 1A Assignment 9 Page 11 fccu?ed a^ tne intersection of the cross-hairs. In fact, a line joining the object viewed and the mid-point of the two cross-hairs is a straight line traversed by a ray of light. TJe are certain then that we can retrace this line of eight as a straight line, unless it is bent from the straight course, by something that de- flects it - a reflector placed in its path, or media of rarer or censer kind which changes its direction, iiut the purposes and the practice of surveying are such as to enable us to eliminate these interferences or to r&ctify them with directness and fidelity to the principle of "sighting". Ihe line of sight in the transit telescope may be directed up or down by its rotstion upon the horizontal axis or t o right or left oy turning the plate and the telescope supports about the vertical axis- Therefore : To prolong a straight, line Set up the transit* at one extremity of the line ; place a marker at the other end; fix the cross-hairs upon the marker (in coiorron parlance "oisect the point"). Now by shifting the telescope upward about the horizontal axis (i.e. by -earning it upon its trunnions) extend the line of eight beyond the farther extremity of the gix'eri line and place markers to fix it upon the ground. Markers, as pins and stakes, or rods are commonly those described under "measuring lines with chain or tape". * To "set up a tr'nsit" means to place it so that the vertical axis points to the zenith through a point of occupency on the earth. In that case the plumb-line attached to the transit axis will be directly in line with the given pcint, vhen the plate-bubbles are at center- ( It is assumed that the instrument is in adjustment as far as the vertical axis and plate levels are concerned.) See Assignment XIV on adjustments. Elsrn. of Surv. lA Assignment 9 Page 12 If nov; it is desired to prolong the line in the opposite direction, clamp the plates, adjust the line of sight to bisect the most distant point of the line, in its original length or pro- longed, and invert the telescope on the trunnions so that the line of s iht may be directed in the opposite direction; place markers (pins or stakes) as before. Instead of inverting the telescope (called plunging) it may be reversed in direction by turning, the alidaderoiate through a horizontal angle of 380?. ^his assumes that there is no eccen- tricity either of the alidade-plate or of the vernier indexes by ?/hich the angle of 130 is detsrmned, in other words that these parts are in adjustment. If the plate adjustnents are not true, the line will not be straight but bent at the point over which the instrument ie set. To eliminate such an error use a principle called double sighting. (8S) DOUBLE SIGHTING consists in sighting Double eight ing^upoa one extremity, B, of a line when the transit occupies the other exiremity A; this brings the line of sight in a vertical plane through the line joining the two points. Turn the alidade-plate through 180*; sight thi telescope upon a required point C, in the reverse direction; now reverse through 180 bisecting the first point 3; plunge the telescope (i.e. invert it) and if the point C is egain bisected the line AB, and its pro- longation AC, form a straight line; if not, choose a point midway Elem. of surv. 1A ..ssignnent 9 Page 13 between C and the point C' next it; this mid-point will lie in the prolongation of A3, as required. In the above method oy douole- eighting it is assumed that the vertical axis is truly vertical and that the horizontal axis is truly at right angles to it. (Sec. Adjustment of the Transit, Assignment XIV.) (90) b_, MEASURING HCHlZOFL-iI The transit is used to measure angles in the horizontal plane. These angles are the bearings of lines, deflection angles for change' of direction, azimuths from any assumed point of di- rection, or mgles included between any two lines, and: hence directly the interior angles of any concavs figure, as those of a triangle or other polygon in a horizontal plane. (a) To measure angles of bearing* Set up the transit over one extremity of the line; set the vernier-index (that of the A vernier) on zero of the graduated plate. With the alidade plate clamped in this position, and the lower motion screw undamped direct the line of sight in line of the assumed meridian (magnetic, true, or any other chosen meridian), Y.OVT clamp the lower motion adjusting by means of the lower tangent screw. Unclamp the alidade-plate and sight on a rod or marker hedd on the line whose bearing is required and rtad the angle moved over by the vernier index. This will be the angle _of bearing. (Compare this method v.-itii th?t explained in Compass Surveying, Assignment \I.) In rending the an^ie always note carefully the direction in which the vernier index raovss snd observe the numbering on the piste Fler. of Surr. 1A Assignment y for this mcvetrent - the inner eet of figures on the limb, if the irovement is clockwise; the outer set of figures, if the movement is cour.ter-clocln.vise. The bearing., would be recorded as N 2312'E, or K23 C 12'W, or as S2512 I E (or W) as the case may be. Bearings must always be so designated. (See Compass Surveying.) (b) To measure a deflection angle; Set up the transit over one extremity of a - line A. With A vernier set at zero of the limb and the alidade-plate clamped, plunge the telescope i^i.e. invert from its normal position, which means vic,h the eye-end near the A vernier). With the telsscope in this inverted position sight upon a marker (a rod, pin, or stake) held at the other extremity cf the line B, the lower motion being used for this purpose. Clamp the lower motion and by means of the tangent -sere* 1 ' of the lower notion bisect the marker. Turn the telescope back to normal, she line of sight is now in the prolonga- tion of the line jlB. Unclamp the alidade-elate, sight in the di- rection of the deflected line AC; clamp the upper plate and bring the line of sight exactly upon the point selectea in AC; i.e. bi- sect it. Kead the angle and record its deflection as right or left in accordance with the movement of the alidade; thus, Deflection 1827 I R (right), er Defi. i827' L (left), ps the case may oe. It is al.rays essential that the designation right (R) or left (L) ge given in registering deflection angles. The further uses of the transit will be continued in the next assignment. Elem. of fcurv, 1^ Assignment 9 Page 15 QUESTIONS 1. l\iame some advantages in the use of the transit as coir.- pp. red to the corapass. 2. A vernier is constructed by dividing 49 thirty minute di- visions into 50 equal parts; what is the least-count of tnis vernier? (urv. 1A Assignment 10 Page 6 lengths with chain or tape (or by stadia) and taking angles by instrument. We have already explained traversing by means of compass and chain in ^Compass Surveying. Traversing by means of transit and tape will be explained here. And while in case of compass work the chain gave sufficient accuracy of line measurement, a more exact or refined set of determinations is secured by use of transit and tape. A steel tape with graduations in feet, tenths, and hundredths of feet is most suitable for use with a transit reading angles to minutes and even seconds. Traverses are of two classes: open, such as lines of wagon road, street, railroad, canal, etc.; ana closed, such as fields of various polygonal forms. In the latter the traverse is said to cloee, if, after having gone completely around, the last point as determined by measured sides and angles, falls exactly at the point of beginning. As this is seldom the case, there is then a greater or less degree of "error of closure' caused either by measurement of lines or measurement of angles, or by both, the presumption being that errors arise from the two sources. The detection and adjustment of errors due to both causes are not so easily made in open traverses as in those which by their very nature c?Qse. fscT^r c Figure 28 Elem. of Surv. IA Assignment 10 page 7 Figure 29 A line of road starting at A and ending at F v.lth deflections at B, C, D, E (Fig. 28) will afford no such check as that of closure to be found in case of a field of the form (or any other form) shown in the accompanying figure (Fig. 29). Here is shown .what may graph- ically represent an error of closure M'M which is a line in magnitude and direction sufficient to close the polygon. Evidently, if the sum of the de- flection (exterior) angles be greater or less than 360, error is manifest, but may be due either to line measure or to angle measure, or to both. Such error is usually, in practice, distributed by proportion to the sides and angles so as to satisfy conditions and cause the figure to close in fact as well as in theory. The manner of dealing with this phase of the problem will be treated in Assignment XVI on Land Surveying. 94) DIFFERENT WAYS OF ioEASUKMG ANGLES If we consider any closed figure bounded by straight lines the angles of which it ie desired to measure with a transit, there are four ways in which this may be done. The method chosen is a matter of convenience, as the same results may be accomplished by any one of the four. Elem. of Sur-r. 1A Assignment 10 Page 8 In traversing a polygonal figure the exterior angles may be measured one by one, which amounts to a measurement of the deflection angles in order around the field. The interior angles at each cor- ner may be found by subtracting the exterior (deflection; angles from 180. In another method the interior angles may be measured directly, thus avoiding a troublesome computation; but this method is not so commonly used as the deflection method, which lends it- self to progressive steps, and allows the records to be easily kept and the errors checked. Another way to accomplish the reading of angles is to set up within the figure and take the angles in series by sighting to each station (vertex; of the polygon in order. This is best done permitting the angular measure as shown on the graduated circle -,o accumulate. The closing reading \vill be the initial reading plus 30 - i.e. a perigon. This affords a ready, although not an ab- solute, check upon the accuracy of the work and in some cases this method is to be preferred to others. The fourth way of making angle measurement is by azimuths referred to sore line assumed as zero, called a reference line. This reference line may be any convenient line, as a bounding line of a figure, magnetic north and south line, starting from either the north or scuth point. Likewise true north and south may be made the line of reference, by convention and universal practice azimuths are always read to the right, i.e. the telescope is turned Elem. of Surv. lA Assignment 10 Page 9 clock-vise; but either the north point or the south point (magnetic or true meridian) may be the starting point on the zero azimuth. As the azimuths are conveniently checked fcy means of the magnetic needle (an adjunct of every complete transit), it is well to taifee azimuths from the magnetic meridian. In making astronomical ooser- vations for aeridicn (longitude), latitude, time, etc., in solar observations or upon some star, it is beet to consider azimuths with reference to the true meridian as this simplifies computations and is the general practice. The ccmpass needle may in this case also be used for checking. It is especially convenient when the compass plate is furnished vdth a variation arc to set off the daclination by that means so that the "check ' angle may oe read directly; otherwise it will be nec- essary to add (or subtract) the declination for each reading -of angle. Every line has two azimuths, the forward azimuth (or 6 imply the azimuth) taker, -.irith transit at the back end of the line, and the back azimuth taken with transit at the forward end of the line, As these two, azimuth and b?ck azimuth, differ by 180, the back azimuth of any line whose azimuth is known may be had by adding (or subtracting) 180 from the azimuth, or vice versa. Orienting the instrument is an operation that is fundamental in taking azimuths. This consists in bringing the line of sight, with plates clamped (the alidade plate and its vernier preferably Elera. of Surr. 1A ^.s^Ignment 10 Page 10 at zero) into parallelism with the line of reference, i.e. to zero azimuth. Thit orientation is, hence, the first step in measuring angleE by azimuths. There are two ways of taking azimuths with the transit, in one the back-sight is taken with telescope plunged, and the other the telescope is always normal. Both methods have their advantages and di sad-vantages. The first method (plunging telescope for back- sights) does not require a setting of the vernier after orientation; with the second niethod (telescope normal), the vernier must be re- set for each angle. This latter operation is troublesome and is likely to introduce & snail error caused in setting the vernier, which may, hovrever, be jaore or less compensated. Xhe t>ac-azizautii may always be obtained by adding 180 to the azimuth - a check that should be automatically applied vhether recorded or not. If the transit is in adjustment as regards its line of colliraftion and if the horizontal axis is truly horizontal and at right angles to the lins of eight, the first method is preferable. (See Assignment XIV on Ad justnents ol_ the_ Iransit. ) 55) AI'iCrLLS BY i-^HLTKIOl* io measure angles when great refinement is required, or when it is desired to "check" angle readings effectively, the method knovm as "Measuring Angles by Kepetition" is resorted to. This consists in turning off tne given angle between any two lines again and again from two to six times or more, and adopting the mean (average) of such readings ae the true or recorded angle. To Flerr.. of Surv. 1A Assignment 10 Page 11 \ / illustrate: Suppose the angle XOY in the adjoining figure is X\ / 'i sought. With transit at and graduated plate clanped, vernier at zero, orient the instrument on line OX by bisecting X. Clamp lower plate, unclamp e.lidade plate avd sight Y, carefully bisedting with tangent-screr; Figure 30. read angle measured. Ihis is (approximately) the angle XOY. Now repeat the angle measurement say, five times, thus: Without disturbing the upper motion, release the lower clamp-screw and turn telescope bock to X, making the necessary fine adjustment in bisecting X by means of the loger t&ngent- screw; unclarap the upper pl^te and again bisect Y. Clamp the upper ple.te when in that, position and ::iake the fine adjustment with the tangent-screw ae before. Ihis is the first measurement. It is irnmaterial whether "ve re-read the angle at each measurement or not ; in practice it is not so read. Now by means of the lower motion sight back on X, then by upper motion sight on Y (second measurement) and so continue until the angle has been repeated five times; i.e. called six repet itione. Evidently an accumulated angular quantity six timee the angle XOY (as observed at the first reading) h?s beer, turned off on th~= graduated plate. This accumulated quantity divided by six gives the angular value of angle XOY> and if the instrument has been manipulnted with care and the points X and Y exactly bisected in each setting, the value for the angle thus obtained is much less liable to error than would be the single first reading or perhaps arr lesser number than six repetitions. Elem. cf Surv. J.A Assignment 10 Page 12 Furthermore the transit ar.d vernier are graduated to read, say to one min- ute of arc, but by this method it is possible to measure the angle to within a few eec ends oi arc, since by the accumulation of small differences, practically impossible of dstection in observing the vernier, the total difference is by division distributed in the average, thus giving a more truly correct determination. In further illustration let us assume that the angle XOY by first measurement was 317', b^ the first repeating 4635 ! . Here evidently the I/ 1 of the first determination was slightly under the mark. While at the tnd of the fifth repeating (for the six repetitions, as ?/e say) the total angular quantity read upon the plate is 139 45'; wh ich now c'ivided Dy 6 gi\es 23n7'30" as the most probable value of an^le XOY. And this seams tc have been revealed oy the second reading, but it is accepted with greater assurance from the evidence cf the six repetitions. The student is urged to acquaint himself vith this method of "reading angles by repetition", both theoretically and prac- tically, as it furnishes the engineer his greatest reliance in angular measurement. It is especially useful where the degree of accuracy sought exceeds the limits of plate md vernier graduation, and is a most reliable 'bheclc" even vrhere the greater degree of refinement is not desired. References : Breed & riosmer pp. 105, 108, 109-111, Vol. I. Tracy pp 152-156 hayraoad pp. 102-105 Johnson pp. 93- 98 Blera. of Surv. 1A Assignment 10 page 13 Exercises in Transit Use 1. Stake out a quadrangular field by pacing, no side less than 150 feet. Assume zero azimuth to be magnet ic-Jiorth. Set up at most westerly corner aid measure the azimuth of each side by either method of taking azimuths. Record the magnetic bearings as a check against large ecrors in angles. 2. Lay out upon the ground a triangle, sides approximately 150 feet; not an equilateral triangle, but no angle less than 30 nor more than 120. Measure each angle by repetition, making six repetitions with telescope nonr.al, beginning with A, vernier set at 0, Then with telescope inverted ootain angle by six repetitions, starting with rernier set at 270. Check each set of readings. Also check by taking the sum of interior angles. UNIVEKLJT.; 01 Cr.IT/CFdJA FXiTN^ T ON DIVISION CGRRLSPuKOLK'Cb CGl'rteEb IK NOIii!r.,U<.I;'G CU3JECTS Course 1A Zl. exeats o:? Survey ing Swafford OBbERVP'iu ?VK j^ijHA]* ^{D LA^/CJl'E FOREWORD : This assijnnent will treat oi the several methods employed by surveyors in locating the North and South line through any po- sition on the earth and also of determining the surveyor's angular distance north or gjuth of the equator. These comprise the sur- veyor's co-ordinates upon the earth's surface, which fix his po- sition; in relation to these the direction and course of all lines in surveying are run. 56) PRELIMINARY CONSIBEkAlIGNS A few definitions and explanations are necessary at this place, in order that certain terrrs and famdarr.ental facts relative to the subject may clearly be understood Leridian is & term applied to the north and south line passing through any gi\en point upon the. earth's surface. ine term is also applied to a great circle of the celestial sphere mafle by the intersection of a pl&ne pa&sing through the observer's north and south points (the uorth and south points o^' the neavens) and including the zenith and nadir, such a plar.e passing through the earth determines a great circle of that uody. The center of the circle is the center of the globe end the circle Is the troce of the plane where it cute the earth's sphere. . of ourv. 1*. Assignment 11 Page 2 Only In changing position in a rorth end soutn line does one me intain .the same meridian, as any slightest movement to east or west causes change of r-ieridinn ; hence, for any position upon the earth's surface there c?n be one and only one meridian. Should a Eur\eyor ^.ovt alon his raeridiJn northward, he would approach the earth's north pole, the point or. the surface pierced by the axis of rotation. tSo also if he should proceed southward lie t. ould eventually reach the south pols. Midway be- tween these two polee liss the equator - ninety degrees from either pole. The Equator is a great circle upon the surface of the earth formed by a plane -which passes through the center of the earth at right angles to the axis of rotation. Later it will oe necessary to distinguish the earth's equator from the celestial equator, but in the present consideration the distinction is not of importance. Since there ie no natural fisced point taken in an east and west direction upon the earth's surface, a certain rceridian is chcsen i'or reference callec 1 a prime meridian, notably that passing through Greenwich, England. This is commcnlj used by astronomers am? navigators for convenience in reckoning distances east and west and for purpose? of t ime or longitude . Longitude and time are expressed either in decrees, minutes and seconds of ?.rc or in hours, minutes, and seconds or time; these units oeing readily convertible, one into the other- Elera. of Surv. 1A Assignment il Page Z prirv Cther^iueridians are also used, as taris by the French; Berlin by the Germans; bt. Petersburg oy the Russians; etc. In the United States we occasionally mate use of the meridian oi Wcshington, D.C. ss a. prLue reference; oat since this causes some confusion, it is better for many reasons to adhere to Greenwich as the one prime meridian. We rill 'always do so in these assignments. Lonitud_e is the di stance- either -east or Tsrest of the prime neridian expressed either in degrees or hours; commonly in degrees by the surveyoc an,: hours by the navigator. Furthermore, the longitude extend e ljo on either side of the piime meridian and is expressed as so many degrees east longitude or west longitude. Latitude is distance north or south of the equator, measured in degrees off-arc on the meridian at any given place. As the equator is at quadrant distance from either pole the range of latitude is from at the equator to 90 at the poia. Correspond- ing to terrestrial latitude is celestial (astronomical^ latitude conceived as measured upon an arc of the celestial sphere. We also use the measure of pqlar distance or cc- latitude, v.'hich is the complement oi the latitude. Since the fixing of the observer's position by means oi these co-ordinates, latitudes --nd longitudes, is al./ays oi j..rune importance, we shall proceed to explain how they are deter:nin>"d. If we could sie,ht upon the north or south pole of the earth and also lay off arc distances upon a meridian upon the earth's surface directly, v/e :.iight have a.;i ideal method oi observation. Elen. of 3urv; lA Assignment 11 Page 4 But this method is impossible. Cth&r i..etLods, however, accomplish our purpose even better than thnt of direct measurement. The sun and the fi::ed stnrs afford us a means of determining meridian and latitude chat is available at all times v;her. the sky is clear enough to observe them. Especially favorable for this work is the sun, which may be viewed by day, and the Pole Star (Polaris), Which is generally visiole in thenorthern hemisphere at night. Other stars may be used, but our 7v r ork will be limited to the methods of observing upon these fvo. On any clear night you may see Polar i, a star of tha fifth magnitude, with vhich all are more or less familiar. It, is readily distinguished by being in line with the tv;o extreme stars of greater magnitude in the bowl of the Dipper (Constellation of Ursa Major). On the opposite side of the Pole Star, aaout equidistant to the Dipper, is another constellation shaped like the letter \'i or 1L (Sigma of the Greek alphabet;; this is the constellation Cassiopeia, and these constellations are of great convenience in locating and determining just the phase in v;hioh Polaris (the Pole Star; may be at any given time. Thest seers cnartec as here shown enable us to locate the celestial north pole, i.e. the point which the &xis of the earth prolonged would pierce if extended in space indefinitely. For Polaris is quite near the pole (107'20") and is no'? approaching nearer and nearer to that position. This star revolves about the pole 'once in twenty -four hours, and trrice in that period of time it is exactly on the- meridian, once Lien, of 3urv. lA Assignment 11 Page 4a. ///"> -A c "7 Cassiopeia 611 }* Polaris Xp-rie <3 ^ i'ca f'ajcr (Big Dipper) Polaris at Upper Culmination Turn Chart to agree with Stars' Positions Figure 31 El em. of Surv. IA. . AtSi&nment II Page 5 above the pole (upper culmination) and cace Delo?; the pole (lowe.r culmination). In that time also it is once at the extreme right (eastern elongation) and oiwe at the extreme left (western elonga- tion). TAihsn the conditions anci ti.aes are favorable it is very convenient to observe upon Polaris \. r hen in one of the four positions or phases given above, i. e. , at either upper or lower culmination or at eastern or western eloagati.cn, as the necessary computations for determining meridian or latitude are somewhat simpler than those required when observation is made at any other time. Mote on the ac9ompauying chart the relative positions of the pole, Polaris, and the stars in the two constellations, the Dipper, and delta ($ ) Cassiopeiae. Hold the chart toward the north before you and turn it counter-clockwise observing the phases of sulmination and elongation as thus illustrated. Now if a line of sight be directed at the Pole Star the instant it is either at upper or lower culmination, this line of sight will evidently lie in the plane of the meridian, since each a plane passes through the earth's center, the zenith, and the north pole, Hence, the meridian or north and south line is de- termined. Either the compass of the transit may be used for directing this line of sight and also for transferring this line to the earth's surface where it becomes cf practical use tc the surveyor. Note that the methods here to be descrioed give one ia reality the zero azimuth of Polaris with respect ^o the observer's north and south line upon the earth's surface. Elem. of 3urv. 1A Assignment 11 Page 6 (97) TC OBSERVE ON POLARIS iVITH COMP-uSS AT CULMINATION Choose an open space unobstructed uy trees or buildings, extending 600 feet or more northward, on a clear night when the northern constellations are plainly visible. Let the time chosen be half an hour before the time cf culmination as obtf.ined from the Nautical Almanac or Ephemeris. (A brief list of computed times is given with this assignment. For oiher times you may .make your o:vn computations. See tables at close of this assignment. ) bet up the compass over a steke having a point marked upon it. Direct the line of sight through the slits in the standards and note that the star will be on the meridian (say in upper cul- mination) a few minutes after tne line joining Sigma (s) Ursae ( Majoris and Delta () Cassiopeiae passing through the star is vertical, i'he compass line of sight is now in a north and south line and should be fixed in this position. A forward point -should be set in the morning light when a etake may be readily placed. The magnetic declination may also be readily determined by reading the angle msde oy the needle at this time. A line joining the stakes is the true north and south line lying in the plane of the meridian. As it is difficult to view the stars (sigma Ursae kajoris, Polaris and delta Cassiopeia*} conveniently through the standards of the compass, a modification of this method is ^aede as follows: Suspend a long plumb line with a heavy bob evincing in a bucket of water to prevent its oscillation. Back of this about Siem. of Surv. 1A Assignment 11 page 7 20 or 30 feet place a board upon two upright stakes four cr five feet high ,and upon this ooerd set the rear standard, rera-jved from the compass for this purpose; bring the slit in ths standard, the plumb line, and the star in line as described -above and complete the operation in daylight as before directed. At upge culmination Foifrie is oest seen, while at lower culmination the details of adjustments of the line of sight ere mere readily Accomplished. Uoth observations are accompanied by difficulties; the observer must judge which, under given conditions, will give the best results. Much will depend upon the character of the view and upon the state of the ataosphere; if cloud or haze interferes with a clear view at lower culmination, choose the time of upper culmination instead. At certain times of ye.r and at given latitudes one or the other of these culminations is invisible on account of daylight conditions, The hour o the night when either culmination occurs may influence one in selecting the best time. >) CAUTIONS IK OBSERVING OK KSLAKI& AT CULMINATION As the star's apparent motion is at right r.ngles to the plane of the meridian, it changes rather rapidly >nd it is, there- fore, necessary to be alert, prepared beforehand, and to clamp the compass plate at the instant the star is on the meridian. Since the movement of the star is westward at upper culmination and eastward at lower culmination, it is advisable to follow the star in these directions, keeping it well in line at every point Eleni. of Surv. 1A Assignment 11 kge 8 of its path. Have everything in readiness, ths time of culmination exactly computed, and your watch set tc standard or mean solar tine in accordance with the sort of time chosen and for which the com- putations have been made. To bungle you;- work means that you must wait 12 or 24 hours for a second observation. (99) TO OBSERVE ON POLARIS A.T ELONOAi'IOW The transit will be used in this explanation of the method of observing for azimuth of Solaris, although the compass also could be employed. The transit is used here so ths/t you may com- pare the two instruments under similar uses. It may be remarked, however, that the transit is the more satisfactory one in any case, on account of the better vision, the meane for finer adjustment, and the ease with which the stat maj be followed and especially located at the critical moment. Note that the star et eastern elongation is moving upward and will so appear in a transit with erecting eyepiece; with an. inverting telescope the reverse movement is seen. Also at western elongation the opposite is true, the star moving downward at the extreme position to the left. The time of eastern elongation occurs 6h 03.4m after iower culmination, while that of western elongation is oh 54.6m later than upper culmination. The : declination (or polar distance) of Polaris is constantly changing, since the star is approaching the pole at a -varying rate, Elem. of Surv. lA Assignment 11 Page 9 which is now about 18. C" per yet.r. In consequence, -che ozimuih of the star at elongation vr.riss. The formula for solution of the spherical trisngle obtained fraa data of the observer's latitude (or as given in this formula the co-latitude) and the polar dis- tance is: , sin Polar dist,. siu star s azimuth at e long =: ^i" ;"la"tid7 ' or ' in logs, log sin azimuth = log sin P. D. + co-log sin eo-lat. Tables giving, the azimuth of the star at elongation (eastern and isrestern) are compiled, a brief table of ti\is nature being appeaded to this assignment, ar.d are & ready means of obtaining these figures without labored computation. 1'he engineer in practice should avail but himself of such help6 >A since the student, must master principles, he should forego the uee of ready-made stuff to the detriment of sound training. (100) DIRECTIONS IN DETAIL FOR USE OF THE I'RAN&IT IN THIS OBSERVATION AT ELONGATION A short time before the st?.r reaches, say eastern elongation, set up the transit in a suitable place with an open view extending 600 feet or more to the northward- Have at hand a ready means of illuminating the cross hairs in the transit, and by shortening the telescope, focus the objective and adjust the eyepiece upon Polaris and see that there ia neither parallax ncr aberration dus to im- proper adjustment of the telescope. Carefuil^ level the transit plates and clamp the alidade plate, setting the A vernier to zero, using the tangent screw. Also have the clamp for the -vertical m. of 3urv. 1A Assignment 11 Page 10 motion lightly set so that the telescope nay be readily transited (plunged). The vertical cross hair is the one used in this ob- servation and should of course be in perfect vertical adjustment. As the change in azimuth is very slow compared with such dhange at culmination, 'there will be ample time to make the two sightings, the first vrith telescope normal, the second with plate reversed and telescope plunged, thus eliminating any slight lack of adjust- ment. tie careful in both positions to see that the star appears to "thread" the vertical cross hair, which will be with an upward movement in the erecting instrument and with a dxwnward movement in the inverting instrument at eastern elongation, the case v;e have chosen for these directions. In reversing for the observation with telescope plunged the graduated plate is undamped and when the star is again bisected, or brought to the vertical cross hair, the final adjustment is made with upper plate tangent screw, if the B vernier now reads exactly zero (that is, A has moved through just 180), the telescope has pointed to the stars at elongation in both positions. If there is a difference in th two readings, A vernier and B vernier, then one half of the difference must De added (or subtracted) from the A zero when the telescope is re- turned to normal position. X'hus the instrumental correction has been set off on the limb and if now the aiimuth, taken from the table or computed from the formula given above (sin asimuth = -SHi 2_ -Jv sin co- latitude 7 be turned off on the limb to the right - the telescope moving to the left - the line of sight is pointing in the plane of the meridian. Elem. of 3urv. 1A Asaigamcnt II Page il To establish tha meridian line on. the earth, .line in a for- ward stake 600 or more feet aray and also set another stake under the plumb bob point of the transit. These two points determine the meridian line. , The cross hairs ma./ be conveniently illuminated by holding an electric torch or a bulls-eye lantern reflecting into the ob- jective tube. To effectively do this, a piece of glazed paper or tracing cloth through which a central opening half an inch in diam- eter may be mounted upon the objective end of the telescope ; set this at an angle of approximately 45 and the light held at the side will be reflected downward through the tube. (101) CAUTIOwS IN OBSERVING In general the cautions given for observing at culmination apply in this case. In addition to these see that the cross hairs are distinct and that the star IE sharply defined in the field of view,; also that the reflector described above does not interfere with a clear view. When once sighted the star should be kept in view and followed as closely as possible by means of the tangent 6crev.-8 both in azimuth and altitude. As before stated the star at about the time of elongation does not change in azimuth perceptioly for some minutes, but when the least change is noticed, cease to follow the star as it is then no longer t.t (or near) elongation. (102) TO FIND IHL LATITUDE OF IhE OBSEKVLK It will be noted that at the equator the pole of the heavens is on the horizon; also as the observer moves northward the altitude Elera. oi Surv. La Assignment 11 Page 12 of the pole increases in exact proportion with the latitude; at the north pole oi the earth the celestial pole is in the zenith; hence the latitude at any point on the earth's surface is equal to the altitude of the celestial pole. Therefore, with the vertical circle and level on telescope, in perfect adjustment (See Assignment XIV, Adjustments of the Transit) bieect the Pole star at culmination , either upper or lower. Re?.d the angle of elevation on Vertical Circle. If at upper culmination, subtract the polar distance of the star and correct for refraction in altitude as given in the table appended to this lecture. If at lower culmination add the polar distance and correct for refraction in altitude as before. The net result in either case is the latitude of the place. (103) CAUTIONS Hi OBSEhVlfoG FOR L/uITUDL Choose the time for beginning of the observation half au hour before culmination and follow the star for a few minutes at least before that time, faote that whereas the star changes rapidly in ajirauth at culmination, on the contrary it moves very slowly in altitude at these times'and for several ninutes before and after culmination has (approximately) the same altitude. Therefore, without too great presumption, the observer r..ay use considerable deliberation in sighting the star for latitude. But atmospheric refraction causes the star to appear higher above the horizon than it is in reality; hence, do not neglect to reduce the observed altitude by the amount of the refraction. Note that the refraction . To Determine the Azimuth of Polaris at Any Given Standard Time To Compute Angles AOC and NOD. J3ef in if i Polaris at lower culrn* Polaris at eastern elc t EW U.W.S.L.E.. U W, L, t. o'L.OK'U.OA Declination of Tola POU-POU.'TOS. Polar distance of ?e.fi SoJ Azimuth of Polaris otW.H 50M Azimuth of Polaris ot E < 5, Polaris as observed ath SOD Azimuth of Polaris as obi m 5) Mean &n at observer Ml mean noon V, Vernal equinox at obs ;cr local mean noon- V Vernal equinox when "Po it S Mean Son at local hou m^i when Polaris i& at ,- AOB "^ '5 Local mean time when lew AOV Z Observer's s'idereal hr* when Polaris is qt , VtOA -^ is Observer's sidereal tie* ascension of oOserve ipt when Polaris is at V.OA * 16 Right ascension of m n at observer's local *IM VtOC * 15 Right ascension of PolM t - AOC V t OA minus V,OC. Houon of Polaris at 5,. Bearing of Polaris a j ( . bNOD Note. Right ascension is measured from the vernalMJ in the direction WAE.K up to 2.4- hours. Hour or en measured from the South, A,in the directic to up to t4- hours. ELxam pie . Find the azimuth of Polaris on Apri)-' at 9:3O P.M., Pacific Standard Tim* r BerKeley, California. Latitude, N. 37* Longitude, IEZ.* 15' -4-Z." W. 6*" "> t? The Celestial Sphere. Known Data. I. From the observers watch '. Standard time of observation of Polaris at 3, i.e. hour angle for Standard meridian cor- responding to AOB for observer's meridian. 2.. From the nautical almanac; Right ascension of the mean sun at the previous Greenwich mean noon. Tnis corresponds at the Greenwich meridian to the s'idereal hour angle V.oA for the observer's meridian. Right ascension of Polaris at S,. Declination of Polaris. 3. From a U.5.G.S. map or b^ observation : Observer's latitude. Right ascension of mean sun t Ci . . Greenwich mean noon on Apf^-i^ife- . r . . Local time of observation is sfand- Greenwich Sidereal time at Greenw.ch mean noon egoals Qrd Vlme ^ 8 h mcr -, dlQn mirio ., 9-3 the right ascension of the mean Sun at Greenwich mean noon. Observer's sidereal time at observer's local mean noon equals right ascension of mean sun at previous Greenwich mean noon plos increase m right ascension during the mea*^ time period between the previous Greenwich mean no* 1 * and the observer's local mean noon. Observer's sidereal time when Polaris is at S, equals th above sidereal time plus the further increase in right ascension during the meantime period between the observer's local mean noon and the ob- server's local mean time of Observation)*)* local hour onqleMB Sidereal time gains on mean time 9.8565 seconds in one mean time hour. Observers local mean time differs from observer's standard time b>j difference in longitude between the standard and the observer's meridian divided by IS; it is earlier than standard time when the observer is west ot the standard meridian. The hour angle of Polaris eguals observer's sidereal time m'mus the right ascension of Polaris. The ai-irnwth of polaris at S. is fownd by solving the spherical triangle PZS, for the angle b at Z-. l"9 Increase >n ngkr ascension in the period between Greenwich mean noon and the local mean time of observation equals * n. **. >S h \, Sidereal time when Polaris is at 5," '0 ': ascension of Polaris " t oo, cos. t J_J! 9.fc,i - 0.4 SO tan b 5 cot - 0.4-4-3 4. o . I Z O 7 toa tint 9 84-01^5 ,. . -. ^ log 4-o.'l8 log tan b a b* o" sr ig When t .* Smaller than I8O* Polaris Is west of the meridian) when areoter than i6O*. Polaris is east of the meridian- wef of nort4i az-imuth of Poland i 179" OO 1 41" .. 8" 1" S* Elem. ofSurv. 1A A- 3ir;r_Q'-nc, II Page 15 V ie greatest near the horizon and d^ro.tsis t-jF-ard tha zenith; the correction for refraction in altitude rili a? ways be less at uoper culmination (when the star's observed altitude is greatest) and greater at lower culmination (when the observed altitude is least). The transit used in this work should be equipped with a full ver- tical circle and the angle read both in normal and inverted .position of telescope to eliminate graduation error. The mean of these twp readings is to be taken as the observed altitude. This with the corrections for polar distance and refraction give the latitude ae the net result. References Breed & Hosmer, pp. 212 - 220, Vol. I. Tracy, pp. 354 - 360 Raymond, pp. 89 93 Johnson, pp. 30-36 As exercises in connection with this assignment it is recom- mended that you make such observationa as the means at your command permit. You may study the heavens to locate Ursa Major and Cassi- opeia and especially by means of these to locate the North Star. An instructive exercise would be the determination of the meridian by the plumb-line method, which is possible witho-ut ex- pensive apparatus. In Addition to this the magnetic declination may also be determined,. of Surv. 1A Adf igrtart- 11 Page 14 Questions. 1. The declination of Polaris on July 1, 1921 is 88 52 '46 1 , 1 what is the azimuth of the star-- at elongation for latitude 3731' North? 2. Find the times of upper and lov:er culmination of Polaris on November 10, 1921. of Surv. 1A n-ssigninerri, li Page 15 1921 July 1 July 15 Aug. 1 Aug. 15 Sept. 1 Sept. 15 Oct. 1 Oct. 15 Nov. 1 Nov. . 15 Dec. 1 Dec . 15 TABLE I. AZH.IUTH OF POLARIS AT ELONGATION, 1921 Latitude AZ imuth Latitude Azimuth 30 117. ' 2 31 1 18. 1 32 1 19. 33 1 19. 8 34 1 20. 7 35 1 21. 7 36 1 22. 8 37 1 23. 8 38 1 24. 9 39 1 26. 1 40 1 27. 3 4tO 127. '3 41 1 28, 5 42 C 1 29. 9 43 1 31. 4 44 1 32. 9 45 1 34. 6 46 1 36. 3 47 1 38. 1 48 1 40. 49 1 42. 50 1 44. 1 TABLE II TIME OF CULMINATIONS OF POLARIS Meridian of Greenwich - Mean lime - Civil Date Upper Culmination h m 8 6 57 33 A. ttg 6 02 46 II 4 56 15 II 4 01 27 II 2 54 51 11 1 59 59 II 57 13 II 02 14 It 10 51 28 P.'id. 9 56 21 H 8 53 19 n 7 58 05 n Lower Culmination h m s 6 55 43 P.M. 6 00 56 n 4 54 25 it 3 59 37 n 2 53 01 ti 1 58 09 ir 55 23 II 00 24 II 10 53 18 A.M. 9 58 11 it 8 55 09 n 7 59 55 M Difference for one day: 3.7 minutes UNIVERSITY OF CALIFORNIA EXTENSION DIVISION CORRESPONDENCE COURSES IN ENGINEERING SUBJECTS Course 1A Eleraents of Surveying Assignment 12 SOLAH OBSERVE IONS - SOLAR INSTRUMENT FOREWORD : This assignment v/ill first deal with the necessary astronom- ical phenomena preliminary to methods of solar observations for meridian, latitude, and time; following these will be taken up the solar attachment and its use in solution of the spherical tri- angle for determining the observer's meridian; and finally, the methods of direct observations on the sun for azimuth, latitude, and time. (104) THE SUN AND ITS MOTIONS IN THE HEAVENS in In its motions the celestial sphere the sun passes through certain unique positions aid appears to an observer upon the earth to traverse a somewhat peculiar path, changing in its equatorial distances, its time of rising;, of culmination (or transit over any meridian), and of setting. The cause of these seemingly irregular movements is due to the fact that the earth's axis is inclined at an angle (2327 ( nearly) to the ecliptic or the plane of the path in whioh the earth re- volves around the sun. While the earth in its diurual rotation and annual revolution is passing through these two cycles of change, the observer upon the earth's surface sees the sun as though it were moving upon the celestial sphere. Hence the sun appears to El en. of Surv. Assignment 12 Page 2 rise at a different point on the eastern horizon, set at a different point on the western horizon, and pass from rising to setting over a slightly different path each day. Xhe explanation is probably better made by saying that the plane of the ecliptic and the plane of the earth's equator are in- clined to each other by the angle of 2327' (nearly). Since these planes are so inclined they intersect each other and the two points upon the equator (celestial equator) cut by the ecliptic are called the equinoxes, because when the earth is at these points, the days snd nights are equal, the sun rising at 6 a..K. and setting at 6 p.m. The vernal equinox is the name given to the one which takes place about the 21st of March; while the autumnal equinox occurs about the 22nd of September. Twice a year the sun seems to reach a most northerly and most southerly point in its course, and these are designated the summer solstice (June 21st) and the winter solstice (December 22nd) respectively. So in March the sun appears on tine equator, and in June about 2327' northward, the change being sloxv, from day to day. The angular distance measured upon the great circle (a meridian circle) passing through the sun and reckoned from the equator, is called the sun's declination. The declination may be either north (+) or south (-), its magnitude varying from to 2327'. From June to September its north declination is decreasing, reaching at September 22nd; then changing to south declination it increases until Dec. 22nd.; and finally decreases in south declination until it again reaches the equator at the time of the vernal equinox. Elem. of Surv. 1A Assignment 12 Page 3 The complement of the declination is the polar distance; 90 - d = P.D. is the simple equation expressing this relation, and, as the declination is positive (-*-) when north and negative (-) when south, the P.D. varies between 90 C - 2327' and 90 -f 2327', i. e. between 6633 ! and 11327'. Ihe sun's position is also determined by its altitude above the observer's horizon, or rather the complement of the altitude (90 - altitude) which is the zenith distance. The azimuth of the sun is the arc on the horizon measured between the meridian and the vertical circle through the sun. Reck- oned from the south point of the horizon, generally, the azimuth is 180 greater than when taken from the north point. The right ascension is the sun's distance in arc from the vernal equinox measured eastward upon the celestial equator. An- other co-ordinate called hour angle is the arc of the equator measured westward from the meridian to the hour circle through the sun. These two arcs are conveniently expressed in time units, hours, minutes and seconds, out may readily be reduced tc degrees, minutes, and seconds (, ', ") as occasion requires. Kight ascen- sion is the hour angle reckoned from the vernal equinox. From the foregoing considerations we have three systems of c o-ordi nates , as follows: (1) The Horizon system employing altitude and azimuth; (2) the Equatorial system, which makes use of the declination and hour UNIVERSITY OF CALIFORNIA EXTENSION DIVISION CORRESPONDENCE COURSES IN ENGINEERING SUBJECTS PLANE SURVEYING COURSE X-lA PLATE IX MOUNTAIN AND MINING TRANSIT FITTED WITH SMITH SOLAR ATTACHMENT PLATE VIII ENGINEERS ' TRANSIT PITTED WITH SAEGMULLER SOLAR ATTACHMENT El em. of Surv. IA Assignment 12 THE; CI.LESIIAL SKffJxE Page 4 0, the earth at center of sphere ES1NN Plane of aorizon SQZPN Meridian of observer EQW Plane of equator MR I Sun's path Angular distance of sun from equator at any time is sun's declination, which is continually changing. Alt. Sun's altitude Sun - Z Zenith distance PN = QZ Latitude ZP = 90 - QZ Go-Latitude Zenith distance is the com- plement of the altitude, i.e. , 90 - altT~ Co-Latitude complement of latitude - i.e., 90 - Lat. Polar distance complement of declination - i.e. 90 - Declination. And, since the declination may be either +, when north, or -, when south, this arc (Polar Dist.) will vary between 6633' and 11327'. Arc NX, the measure of angle, Sun-ZP, is the sun's azimuth. Arc of equator QY, measure of ang.le Sun-PZ , is the sun's hour-angle. Refraction Corrections for Altitude Alt. Ref. Alt. Ref. Alt. Ref. 10 5 '16" 18 2' 56" 30 1'40" 11 4 43 19 2 46 35 1 22 12 4 24 20 2 37 40 1 09 13 4 04 21 2 29 45 58 14 3 47 22 2 22 50 48 15 3 32 23 2 IE GO 33 16 3 18 24 2 09 70 21 17 3 07 25 2 03 80 10 Elera. of Surv. IA Assignment 12 Page 5 angle (or Right Ascension); and (3) the Terrestrial system, using the latitude and the longitude, which define the observer's po- sition upon the earth. This last is the one which must be deter- mined by the surveyor, and hence all observations of an astronomi- cal nature follow this system. By reference to the diagram (Fig. 32) showing the celestial sphere, with the accompanying explanations, the foregoing definitions will be made more intelligible. )5) DETERMINATION OF MERIDIAN The solar attachment must be carefully adjusted (see Assign- ment XIV for detailed directions for these adjustments) . Having determined the declination settings for the times of observation, bring the solar telescope and the transit telescope into the same vertical plane by sighting both upon the same point. (a) Set off the declination for the time of observaxion on the vertical circle with transit telescope pointing southward; if the declination is north, depress the telescope; if south, elevate it. (b) Clamp the transit telescope in this position, and level the solar telescope; the solar is now in the horizon and the angle between the solar and the transit telescopes is that of the sun's declination. Xhe r.ngle formed by the polar axis and the solar telescope will be the sun's polar distance (90 declination). (c) Unclamp the horizontal axis of th* transit telescope Elera. of Surv. IA Assigiiaent 12 Page 6 and set off the co-latitude of the observer's position on the ver- tical circle. The latitude may be taken from a map or may better be determined by a previous observation for latitude. The polar axis now points to the Pole. (d) With the lower motion of the transit and that of the solar undamped, (if in the northern hemisphere, pointing in a southerly direction) the two telescopes may now be turned, each about its vertical axis by means of the tangent screws, until the sun can be followed in its path, its iraag^e falling symmetrically upon the field of the solar, thus: As the sun changes less than one minute of arc per hour, the time is ample for making the setting of the sun's image and may be ]/ sin co-lat. sin Z.D. which lends itself readily to logarithmic computation; thus log cos 1/2 A - -r-(log ain 8 "''log sin (S-P.D. ) * colog sin co-lat. -f colog sin Z.t. ) (103) TO OBSERVE TIME iwirh SOLAR }.iake the observation for meridian ae above directed. Note the time on watch at the instant that the sun is centered in field d" Breed & Hosmer Vol. I, pages 63 - 7Q Elein. of Surv. 1A Assignment 12 Page 9 of sclar and clamp the solar telescope. Set the forward stake with a tack center by pointing the transit telescope without changing the solar. Return the transit telescope to a horizontal by means of its level, and clamp the notion. The polar axis now points to the zenith. By inclining the solar telescope set a stake with a tack center bisected by the vertical cross-hair at the middle of the solar field. This stake should be 200 feet or more distant. Then by a use of the transit limb, measure the angle bet'.veen the meridian previously set and the last mark. This is the hour angle or Right Ascension in , ', ", which must be reduced to time units, hrs. , rains., sees. Compare the time thus obtained with the watch and set the latter fast or slow as required; or, if it is not desirable to change the watch, this difference may be noted as a correction for all time observations. If the watch time is the so-called "standard" time, this must be changed to local mean time before comparison is made. QUESTIONS: 1. (a) Change 14* 37' 42" to time measure. (b) Change 3 h 40 m 15 s to arc measure. 2. An observer in longitude 124 07' 58" notes the time of his watch is 8:43-1/4 A.M., Standard Time. Required: The Mean local time of the observation. REFERENCES ; Tracy Appendix I, page 620 Raymond Pages 116 - 126 Johnson Pages 99 - 102. Breed & Hosmer Vol. I, pages 63 - 70 iY OF CALIFORNIA bX'i'LiiSLJK II VISION .NCF. comsbs iw ENGINEERING SUBJECTS Course 1A Elements of Surveying Swafford Assignment 13 ADJUSTMENT OF INSTRUMENTS FOREWOHD : This assignment will deal with the usual adjustments of surveying instruments, special attention being given to those made in the field. The methods of detecting instrumental errors of the more serious kinds - auch as would require the skill and appliances of the instrument maker - and of correcting them, will also be pointed out. As references will be made to the various sections of this assignment in the matter of instrumental adjustments, a simple system of numbering the several paragraphs has been carried out. The scheme will be quite evident after a few moments of study. 09) (A) SPIRIT LEVEL (1) Adjustment depends upon a principle called reversion. If the bubble-tube is placed longitudinally parallel to a truly horizontal line and the position of the bubble noted, the bubble should have the same position if the bubole-tube is reversed through 180. In Figure 33 the bubble-tube .has the position MN m r~ Figure 33 '? i.par,-, 34 on the horizontal line ^B j the bubble is at 0, the middle of the tube. In Fig. 34 the line AB remains unchanged, but the DubDle-tuoe is reversed through 180 to the position HH; the bubble should now Elem. of Surv. IA Assignment 13 Page 2 be in the same relative position in the tube, i.e. at 0. If the bubble-tube were revolved upon a. horizontal plane through 360, the relative position of the bubble would not change. (2) Experiment may be made by the student with a carpenter's common level, as follows: piece of Joint up the edge of a/^board about 6 inches wide by 3 feet long, so as to have a true, fiducial edge. Clamp lightly in a bench-vise and, by placing the level upon it, bring it to a hori- zontal position in the vise. Reverse the level. Does the bubble stand at the middle of the tube as before? If not, what must be done with the board in the vise to cause the bubble to return to the mid-position? Do this, and again reverse the level to the first position. What do you notice regarding the bubble now? Alter the slope of the board in the vise so as to cause the bubble to approach half-way to mid-position, and reverse the level. You will now note that the bubble will occupy the same relative position in the tube no matter which direction the level has with respect to the straight edge. Hence : (3) To adjust a carpenter's level. Use a straight edge as in the above experiment; level up carefully; reverse the level through 180; bring the bubble half- way to center by tilting the straight-edge in the vise; correct the setting of the bubble-tube in the level-block. In the ad- justable level, a screv: acting at one end of the brass protecting plate enables one to raise or lower that end of the bubble-tube. Elem. of Surv. 1A Assignment 13 Page 3 Some levels require the protecting plate to be removed, revealing the adjusting screw beneath. Still other levels, those of the cheaper kind, have the tube imbedded in cement (plaster of Paris) and are not adjustable. The principle of reversion ijs applied jln making level ad- justments _on all surveying instruments and will frequently be re- ferred to in what follows respecting adjustments. (See A-l, 1st adjustment. ) (110) (B) ADJUSTMENTS OF THE SUKVEYOR'S COMPASS (a) Ihe needle in the compass should be (1) Balanced on the pivot, (2) straight, that is ir. the same vertical plane through the pivot, (3) well-magnetized so that it responds promptly to the earth's action. (1) To balance the needle, level the plate carefully and observe whether the ends of the needle are equally disposed with respect to the graduated circle; if not, move the coil of fine wire on the south end so that the needle balances. To do this it is necessary to remove the glass cover and take the needle from the box. (2) To straighten the needle. By observing both ends of the needle it is possible to determine, first, if the needle is straight; second, if the pivot on which it turns is in the line of sight of the two standards. If the pivot is at the center of the graduated circle, and the needle straight the readings of the opposite ends will -show a constant difference of 180. If the Elea. of Surv. LA. Assignment 13 Page 4 needle is concave to the right, the right hand segment of the circle will be less than 180 and consequently the left hand segment greater than 180 for every position of the needle. In such case remove the needle and support it near its ends, laying it upon the concave side, and press it gently to make it straight, test it frequently by returning it to the pivot and observing if it is straight. If the needle is straight and the pivot is not at the center of the graduated circle, the needle reading of its two ends will differ by exactly 180 in only two positions, which will be themselves 180 apart. All other readings than those just men- tioned will differ from 180, one .segment being less than 180, the other greater than 180. To adjust the pivot in such case, determine the line of no difference, remove the needle and care- fully press the pivot in the direction of this line forward or ' backward so that the needle will give readings that differ by 180, when at right angles to the line of no deviation. The pivot having been adjusted and the needle straightened, adjustment (3) is made. (3) To Magnetize the Needle Remove the needle from the box and support it throughout its length; stroke it with a bar or horse-shoe magnet, applying the north end of the magnet to the south end of the needle, or the south end of the magnet to the north end of the needle . The needle can also be quickly magnetized by bringing it into contact with the poles of a dynamo or motor while current is passing through them. Of course these must be of the direct current (D.C.) type, and the Elem. of Surv. 1&, Assignment 13 Page 5 needle placed in position with south end at * pole and north end at - pole of the motor or generator. If a mistake has been made at first trial, the error may be corrected by reversing the needle in a second trial and allowing it to remain in contact long enough to remedy the fault. The following conditions should now obtain; 1) The standards should oe vertical vidhen the compass plate is horizontal. 2) The line of sight should lie in a plane passing through the slits in the two standards and including the central pivot on which the needle swings and the exact north and south points of the grad- uated plate. These matters require the attention of the maker. 11) (C) DUMPY LEVEL (1) T_o adjust the horizontal cross-hair to true horizontality, sight on some well defined point at a distance of about 50 feet, setting one end of the horizontal cross-hair upon it. Turn the telescope about the vertical axis and if the point remains on the cross-hair, the latter is horizontal; if not, it requires adjusting. To do this, loosen the four capstan-head screws that hold the cross- hair reticule, turn the latter to right or left as the case may be until the distant point is made to "thread" the horizontal cross- heir throughout its length. This adjustment is then complete. (See Print A-i, Preliminary.) (2) To coMirr-.ate the telescope, unscrew the plate which holds the pinion operating the object tube. This will permit the tube containing the optical parts to be rotated, how sight upon a Elem. of Surv. 1A Assignment 13 Page 6 distant point, rotate the tube, and ooserve the position of point of intersection of crose-hair with respect to the chosen point. If they remain in coincidence throughout a complete revolution of the optical tube, the line of collimation is correct. If they do not, loosen the opposite pair of screws holding the cross-hair reticule and tighten first the one that dra?;s the reticule toward a correct position; repeat with the other pair of opposite screws to bring the cross-hairs into position in the second direction. "When finally the point and cross-hairs are coincident tighten the reticule screws, being careful not to overstrain them. The focus of the objective now falls upon the optical axis of the telescope, and at the point of intersection of the cross-haire; i.e., the instrument is collimated. In some levels this adjustment cannot be made. (3) Tc> make the axis of_ ou bole-tube parallel with the line of collimation, set up the level L at a place where two points A and B about 300 or 400 feet apart and at nearly equal distances held from the instrument can conveniently be sighted. Read a rodA.at each point in turn and record Rod Reading at A 4.527 11 " " B 5.213 A - B 1.314 True Diff. in Elevation Instrument at LJ. Remove level to position L.^, ten feet oeyond 3 and approxi- mately in line with A and B. head the rod set upon A md B as ADJUSTMENTS OF THE DUMPY LEVEL PRELIMINARY Focus eye piece on cross-hairy , and make horizontal cross \ per pen a '/cu/ar to ax 15 of rotation of instrument. FIRST ADJUSTMENT > Bubble To make axis of bubble-tube perpendicular to ax/5 of rotation , or vertical axis, of instrument. FI6. 1 Axis of bubble -tube Horizontal line Level instrument carefully Turn instrument end for end on OKI of rotation (this reverses bubble--iie t The apparent error (?e) is twice ht true error (a) Bring bubble halfvyay back to cente: by means of adjusting screws af t1 of bubble- tube II | Horizontal line VT Center bubble by means of leveling scrt 5 FIG IV C.T YVttKC" _^ ADJUSTMENT .-- Line of sight To make l/ne (f sight parallel to axis of bubble-tube , Direct or 'PecfAajustment " Fie.v 6.103 c. 6019 '"Correct position of line of sight (Horizontal line) First position of line of si 3H -Any distort say 30 fe Set instrument up, in line with and half-way between two hubs, B and C, which are from ?00 to 400 feet apart. Pead rod on each hub. As lengths of B.5. and ' f:S are equal , errors of adjustment are eliminated and the true difference in elevations of B and C b obtained by subtracting the rod readings. Now set up again near either hub, say at D, and again take rod readings on B and C. If the instrument is in adjustment the difference in elevation of B and C f as obtained from these last two readings , will agree with the true difference in elevations. If the instrument is not in adjustment determine the amount of error. TO COPPECT EPPOP Dumpy level :- With bubble centered , bring line of sight correct rod reading at B by means of the capstan screws that move the cross -hair ring, Wye level - By means of the I eve I ing -screws bring line of 5/ght to correct reading at B, then bring bubble of tube by turning the capstan nufe at the end of bubble- tube. Form of Note? Instrument at A D Pod at heading d 2.938 ' C 0,972 1,966 True difference in e/evation. C 4,127 d e.ofe 1 .892 Fabe difference in elevation. Thb b less than the true difference in e/evation, therefore the line of sight is inclined downwards. (The hub at being lower than the hub at C) True difference in elevation 1,966 fly/se difference in elevation 1,892 Error in distance L 0,074 feet Total error in line of sight in the distance Dd , &4 +30) x 0,074 = 0.084 feet Target setting at 5 to give horizontal line is 6, 019 +0.084 = 6,103* Note:- To determine whether the correction is to he or subtracted draw a figure in your field book representing the specific case In hand. 5how clearly relative elevation" of hubs. * For exact adjustment this rod reading should be corrected by adding the effect of curvature and refraction which, in this i Is approximately 0,001 feet. The target setting at B would then be 6.104 El em. of Surv. 1A Assignment 13 Page 7 before; we shall now have, suppose Rod Reading at A 7.235 " B 5.685 A - B 1.552 False difference in Elevation Instrument at L,,. Subtract the true difference in elevation from the false difference and take is-, of this difference. 40 False Difference in Elevation 1.552 True " " " 1.314 0.238 which multiplied by ~ = 0.244. This quantity deducted from last rod-reading at A (7. 35 - 0.244 = 6.991) will be the target setting for A, level at Lg, in order that the line of sight (line of collimation prolonged} may be horizontal. With the target, therefore, set at 6.991 and rod held on A, direct the telescope to the target by means of the leveling screws of the instrument, and while it is in this position raise or lower the level-tune by means of the adjusting screws at one end of the tube until the buoble is exactly in the middle of its ran. It should be apparent now, that whenever the bubble is at the middle of its tube, the lime oi sight will be truly horizontal. The foregoing is called the "Peg Method" of adjusting the level and telescope, and should be well mastered by the student, as the same routine is followed in making this adjustment in case of the Wye-level, or the Transit, (See A-3, Second Adjustment.) ( See Fig. A- 1- ) Elem. of Surv. 1A Assignment 1? Page 6 (112) SETTING UP AND PhLLIMINARY ADJUSTMEHTS These relate to instruments in general and are supplemental to the directions furnished in connection with other assignments of this course. For purposes of field adjustment the instrument should be set up in a shaded spot on reasonably level ground, in a locality free from interference, ef.rth shakings and local magnetic attrac-- tions. In compass adjustments avoid proximity to electric lines or pipes buried in the ground. The stretch of open space (different for different cases) should be of sufficient extent to permit all adjustments to be completed with one setting of the instrument, or, in case two or more settings are required, to permit them to be made with ease and facility, so as to secure the advantage of freedom, stability, and range. L13) PRELIMINARY Plant a tripod firmly, spreading the legs at angle of about 60, so as to secure a broad base which adds to stability. Press the shoes well into the earth, which should be solid and free from loose sand or marsh. Bring the head of the instrument as nearly level as possiole in the foregoing processes, so that the final up work of leveling A may be done with ease and dispatch. Examine clamp screws, pivots, etc., to see that all parts are free to turn without requiring undue force. If a telescope is a part of the instrument, uncover the objective and eye-end, and move the objective tube in to the shortest telescope length, direct Elein. of burv. IA Assignment 13 .Page 9 it toward the sk^ , away from the sun, and observe the appearance of the cross-hairs, fhese should stand out clear and sharply defined - even the particles of dust invariably present should be plainly visible. Sight upon sortie well defined distant object and observe for parallax, by shifting the eye first to one side of the to eyepiece, then/vthe other - then up and down in similar way, and note if the cross-hairs appear to move over the field of view. If they do, parallax is present, and must be removed by careful ad- justment of the eyepiece. Many telescopes have a set-screw for clamping the eyepiece when in adjustment. The eye-piece should be clamped if the instrument is to be used by one person, as in that case the eye-piece setting will not require change. Next level up the plate or what is really the case, bring tke axis of the instrument into verticality by means of the level- ing screws on the tripod head. This important step should be done as follows: Set a level in line with two diagonally opposite footscrews, as A and B in the adjoining figure. (Fig. 35) Grasp a screw between thumb and fore- finger, one in each hand and proceed to turn both in or put, uniformly, and observe the direction in trhich the Fig. 35 bubble moves - it will alv.-ays follow the right-hand, finger or the left-hand thumb - and thus bring the bubble to the mid-point Elem. of Surv. 1A Assignment 13 Page 10 of its tube. Next set the level over the other pair of foot-screws and proceed as before, barring possible necessary adjustment, de- scription of which will follow these preliminaries, after a few trials the level (or levels) will be &t mid-point for all positions. The level-plate will then oe horizontal and the vertical axis truly vertical. In manipulating these foot screws it may happen that they "bind", and can be moved only with increasing difficulty; this is due to one screw traveling faster than its opposite. The "binding" will be removed if you continue to turn either screw in a right- hand direction (clockwise when viewing the bottom of the screw); but the quickest method is to turn both screws together to the right (clockwise). Sometimes the binding of one pair of screws is due to the position of the other pair, and the latter should receive attention. In no event permit the screws to bind and of all things, do not continue to turn them when they do bind, except in the manner required to loosen them. All screws used in manip- ulating the instrument - foot-screws, clamp-screws, tangent screws, etc. - must at all ti;aes turn easily and freely. When they do not, they need attention. Do not oil screws or bearing parts. Oil will cause grit and dust to adhere to such surfaces and this in time will cause the parts to wear rapidly. Elem. of Svrv. 1A Assignment 13 Page 11 *<__ == > - is El m Figure 36 3 Suppose a level-tube connected cy supports s s 1 with the line AB and it is desired to make the line truly horizon- tal. Adjust the line so that the bubole is at center 0; re- verse to position 180 as at b, Figure 36, by revolving about C. The buoDle will now be at the high point j>; but by shortening the support s 1 the tube may be brought into parallelism with AB as at , the bubble is then at ^, midway between o and p. If now the line BA is tilted until the oubble again takes the position o, as in d, BA will be parallel to the bubble- tube and in a horizontal position the axis of rotation, C, will also be vertical (a plumb line). REFEEENCES : Tracy Raymond Johnson Breed & Hosiner Pages 581 - 606 63, 79, 108 15, 63, 86, 102 25, 56, 70, 89 Elements of Surveying-lA Assignment 13 Page QUESTIONS: 1. The lines of a tape survey of a field are noted on the adjoining figure. Compute the areas in acres. 2. 3y a single set up at the angles about (were measured and the distances to M, N, P, & R measured; from the data thus obtained compute the area of the field Angles MON 60 NOP 87 FOR 92 ROM 117 15' 30' 30' 45' Distances MO 425.0 feet NO 430.5 feet PO 395.3 feet RO 512.6 feet UNIVEKSLCY OF CALIFQhillA EXTENSION DIVISION CORRESPONDENCE CCUHSES IN ENGINEERING SUBJECTS Course 1A Elements of Surveying Stafford Assignment 14 ADJUSTMENT OF IIvSTRUMLMTo (Cont.) (114) (D) AEJuSTiaENTS UF i'KE K>f_ the wyes perpendicular t_o the vertical axis. Level up the instrument and turn 180 upoa the vertical axis; bring the bubble one half way oacK ta center Dy means of leveling- screws and adjust to center by raising or lowering one of the wyes by means of the capstan screws supporting the wye. Repeat the ad- justment to insure accuracy. (See A-5, Third Adjustment.) L5) (E) ADJUSTMENTS OF THE TRANSIT A coaplete transit, that is, ons having a bubble-tube on the ADJUSTMENTS OF THE WYE LEVEL PZELIMINAPY ADJUSTMENTS :- Same as for the clumpy /eve/. Fii?5T ADJUSTMENT :- Line of sight. To make line of sight coincide with the axis of the collar^ Without leveling set cross- hairs on a distant, point "o" . devolve telescope ha/f-way around in the wyes, (not end for end) intersection now appear , at v . TO COFPECT EPPOP By means of the capstar screws that move the cross -hair ring bring th< intersect ion of the cross -hairs to a point mid- FK5.VI wa y between V 'and "o". When this adjustment is accomplished the intersection c the cross- hairs will remain fixed on one point throughout o complete, rotation of the telescope, . ADJUSTMENT .- Bubble (a) To make the axis of the bubble-tube parallel to // bottom of the wyes, ,5ottom of wyes __ n -^ Horizontal Jine Axjs_ of bubb|e : tube Horizontal line . . . . Level up ana clamp horizontal mot/on. FIG.VE ' - Fio.vnr Unfasten clips, remov^ telescope from the wyes, tun end for end, and replace in v* a * if 1 T ~1_ t Axis J of bubble -tube FIG.K bubble half- way back to center by mean? of adjusting screws at end of bu- tube. Bottom of wyes Axis of bubble-tube Center bubble by me> of leveling screws. FIG.X (b) Id make the axis of the bubb/e-tube lie in the same plane with the axis of the collars- Level up and clamp horizontal motion. Loosen clips, ther, swing bubble- tube through small angle by revolving telesco/st If the bubble moves away from the center it should be adjustea TO COFFECT EWOP Bring the bubble all the way back the center by means of the capstan screws which control the lateral motion of the bubble- tube, (cj Repeat bubble adjustment (a) , Note:- If the collars are the same size these aajustm] will make the axis of the bubble -tube parallel to the line of s, ' This is an indirect method of making the adjustment, for a di.i method u$e the "peg adjustment ", THIPD ADJUSTMENT :- Wyes. To make the axis of the bubble -tube perpendicular the axis of rotation , or vert/'ca/ axis of instrument, FIG. XI FIG. xii FIG.XHI J [ Horizonfeil line FIG, Bottom of wyes _____ - Axis of bubble -tube l"i$~" Center bubble by men, of leveling screws, Horizontal line' 'A ' L. ~ ~ ~~^~ l~ - - f(/ instrument end ax/5 of rotation, error (24>) 15 twice the true w. Horizontal line ' --W- Bring buhble half-way* '~ i r ., to center by mean^ of the law capstan nuts at the end? of /e I eve/ bar, Bottom of wyes Level bar Center bubble by mi of leveling Eiem. of Surv. 1A Acsignment 14 Page 2 telescope and a vertical arc in addition tc the usual plain instru- ment (Plate III, Assignment IX) is preferred for the work described hers, but the surveyor should learn to adjust "any old" instrument that he may be called upon to use, for all adjustments become more or less deranged by the action of the elements, etc., or by rough or careless usage, or through accident. Preliminary : Set up the transit in an open space, about 200 to 400 feet long, on as nearly level ground as possible, preferably in the shade. Place the instrument over a huo or stake set in the ground by means of the plumb-bob; center it over a definite point in the hub. This done, it is unnecessary to select any other initial point throughout many of the adjustments that follow; if the in- strument is removed to any other setting during a given adjustment, this new setting should then be sufficiently identified for occu- pancy at any later time. ( 1 ) Jto make the plate bubble-tubes parpendicul&r t o the vertical axis of i^he instrument : Put one of the level-tubes parallel to a pair of diagonally opposite foot-screws; this orings the other tube parallel to the remaining pair of screws. By manipulating the foot-screws as di- rected for the leveling instrument, Assignment XIII, Art. 113, bring each bubble to the middle of its tubs; turn the plate on the lower spindle through 133 , which rill reverse ooth ouOoles over their respective set of foot-screws, an.?, note the position of each f Elem. of Surv. 1A Assignment 14 Page 3 bubble in turn. If they are at mid-point for each position, no adjustment is necessary; if not, bring either (or both) bubbles one-half way back to center by means of the foot-screws aid then exactly to center by means of the screws that hold the level-tutoe to the plate. Repeat the whole process for test and subsequent adjustment - one repetition is not always enough. Some of the adjustments to follow do not require thie bubblertube-on-plate adjustment, but any other adjustment is made with greater facility and assurance when this is done. This se- cures a normal position of the transit (i.e. vertical axis truly vertical), which is always desirable. In some transits one of the bubble-tubes just spoken of as situated on the horizontal plate, is not so situated, but is attached? to a standard that supports the horizontal axis (the trunnions) of the telescope; the principle is the same, however, and the elevated position of this level-tube in no vay affects the result. (2) To make the upright cross-hai r vertical : Direct the line of sight to some sharply defined point at about 50 ft. from the instrument; focus one end of the upright hair exactly upon it; and then, Dy turning the telescope tube on the trunnions, note whether the point "tracks" upon the cross-hair. If it does, no adjustment is needed; if net, loosen all four screws by that hold the cross-hair reticule, and turn it around gently tapping the screws. Reset the screws, but t?e very careful not to over- strain arty of them in eo doing. In the acsence of a convenient Elem. of Surv. 1A Assignment 14 Page 4 point for sighting upon, a plumb-line may be suspended and used instead. Although this plumb-line requires more time and trouble, it is sometimes preferred. If the horizontal cross-hair is placed (as it should be by the maker) at right angles to the upright one, the horizontal hair is now in adjustment also. However, it is more important .that the vertical cross-hair be truly vertical; hence the reason for testing and correcting this one primarily. ^o make the line f sight lie _in a plane _at right angles to the horizontal axis of the telescope : requires This /> two adjustments - (a) When the line of collimation X of the telescope is horizontal; (b) when the line of collimation is in any other position in altitude. (a) With transit set over a fixed hub P as described in (1) direct the line of sight toward a point A about 200 ft. distant and at a^out the same elevation as that of the telescope (H.I. of transit). Turn the alidade plate 180 about the vertical axis and set another point B at about the same distance as for A and at H. ! Now invert the telescope (plunge it) and sight again upon A. If A is bisected by the line of sight, the adjustment (a) is correct; if not, olamp the instrument and set a marker A' in the line of sight near A. Take 1/2 of the distance AA 1 , and set if off from A toward A'; call this point D. With the plates and the telescope both clamped in position, loosen the right and left screws that hold the cross-hair reticule, aid tighten the one that will bring the vertical cross-hair over the point D. Eleou of Surv. 1A Assignment 14 Page 5 Note: To test this adjustment, three points in line through P, at H.I. , "will be on a straight line. Every position of the telescope when bisecting one point will, by reversing or plunging (inverting) also bisect the other point. (b) Set up the transit in a locality where observation may be made on some high point. A, as the top of a tall building, a spire, or a flag-pole, and carefully level the plate. As this adjustment for is to correct the intersection of two vertical planes the point aloft should be high, but not at a great horizontal distance from the instrument. Sight first upon the high point; clamp the plates while bisecting the point, turn the telescope on ite horizontal axis (in the trunnions), and sight a marker placed at a convenient distance, if possible directly below A and at about the elevation of the center of the instrument (H.I. ). Call this point on the ground B. Reverse on either spindle, plunge the telescope and again bisect A the point aloft. Turn the telescope down on the trunnions and direct toward B; if B is bisected by the line of sight the horizontal axis is in adjustment; if B is not in the line of sight place another marker B in line near B. The ad- justment is then made by raising or lowering the movable end or trunnion of the horizontal axis, by means of the adjusting screws in cap at movable end, until the line of sight bisects the point midway betweefa B and B*. Go through this whole process a second time for test and subsequent adjustment. Elem. of Surv. 1A Assignment 14 Page 6 ( 4 ) 12 ad J ust the Level _on Telescope. This may be done by the Peg Method describee in connection with the Dumpy Level, used likewise with the Wye level. (See Assignment XIII.) A simple method of making this adjustment is the following: Set two stakes A and B_ at the same level. This can be done by choosing a nearly level stretch and carefully setting the stakes by means of the transit telescope, or better, by means of an engineer 'e level. Or if a sheet of water of considerable extent may be near e.t hand this may be used for determining a level line by driving the stakes so that each projects one or several inches above the surface and at the same level. Now with the transit set up and most carefully leveled near A, sight on a target set at H.I. on Bj adjust the level on telescope by means of the screws that hold it to the telescope, bringing the bubble to the center. For test and subsequent adjustment set up at B and observe on target set at H.I. on A. (Here H.I. means height of the line of collimation at intersection of vertical axis, i.e., the center of the instrument.) (5) T adjust the Vernier Index of the Vertical .arc. With the telescope bubble exactly in adjustment and at the center of its run, and with plate levels in perfect adjustment, loosen the screws that hold the vernier of the vertical arc (or circle) and move until the or index (A) coincides exactly with the proper limb division, or 180. If there are t\vo opposite verniers, both should be adjusted to read exactly and 180 when both the telescope and its level are horizontal and in adjustment. Slem. of Surv. 1A At signment 1.4 Page 7 This adjustment is important and most convenient, out by no means essential, as the vernier error nay be observed without moving it into correct adjustment and applying this as correction to readings of vertical angle for every case, adding or subtracting the correction ae circumstances may require. Some transits have a small level attached to the vernier of the vertical arc, which ob- viates the adjustment betv/eea telescope level and arc, as this level and arc adjustment is difficult to make and therefore trouD- lesome. (6) Adjustment or the Gradienter. This is made when the preceding adjustments are made. Loosen the swinging arm (a sort of lever) that is actuated by the gradienter screw, which is likewise the tangent screw to the hori- zontal axis. Move the milled head and knife gauge both to zero and then tighten the arm upon the trunnion. Be sure that the settings of any of the transit parts concerned in this adjustment are not disturbed in making this adjustment. 116) GENERAL OBSERVATION ON ADJUSTING Besides remembering that the adjustment of instruments is / of great importance, you should give attention to other things necessary in sound practice. First. The adjustments just descrioed for the transit snould be gone over in the order they are grven, unless it is obvious thp.t some of them are correct and hence do not require attention. Some adjustments can not be made until certain ones precedent Eleu. 01' Surv. IA Aesigmnent 14 Page 8 thereto have been made; e.g., adjustment 3b must follow 3a to accom- plish the correction, and therefore 3a should at least be tested out. Second,. The order of procedure in any particular adjustment will be (1) the desired relation: (?) the test; (3) the method of adjusting; (4) a final test, usually made with greater care, which will prove of immense satisfaction to the careful engineer. Third. In making nmy alteration by loosening, tightening, or shifting screws or other parts, do all in a workmanlike manner. A slotted screw should be turned with a screw-driver that fits the slot, a capstan screw with a pin that fits the hole in it, milled-head screws with the thumb and finger, not with pliers or a wrench. Never leave screws loosened where they should be firm, nor overstrain them in tightening them. Fourth. Tfnen a general overhauling of an instrument is required, do not attempt to bring any given part into perfect ad- justment oy a "once- over", but proceed by making the series of corrections as accurately as possible in the order given above. Mien the whole series has been followed out, return to number one an3 repeat each one for test. Of course if test does not reveal that the error has been corrected, repeat the process necessary to accom- plish this. (117) (F) ADJUSi&Ei^S OF THE SOUR AXiACEivENT (Saegmuller). (See Plate VIII, Assignment XII.) These adjustments must be made after attaching the solar upon the transit telescope. Elein. of Surv. 1A Assignment 14 Page 9 Preliminary . See that the screw that connects the solar is turned down to a firm position. Bring the levels of the alidade plate to cen- ter and also the level on transit telescope; set the index of the vertical circle. (1) To raake the axis of the bubule-tuibe in the solar telescope parallel t_o t.he axis _or the same. Thie is accomplished exactly as for the transit telescope by the method of reversion. ( 2) To make the polar ay. is _o the solar in line with the vertical axis of the transit. lie sure that the transit's vertical axis is truly vertical, then bring the bubble of the solar telescope to center, and test for all positions through 360. If not in adjustment, employ the method of reversion. (3) To make the lines f collimation _in the two telescopes parallel. With the preceding adjustments fully made, sight ooth tele- scopes upon a vertical line ( a suspended plumb line or corner of a building) at 100 feet or more away. Clamp the telescopes in this position and carefully level the bubble on transit telescope so as to bring the lines of sight of Doth telescopes into the same plane. Now measure the distance between the exes of the telescopes along the polar axis. Draw two distinct parallel lines on a piece of paper, or better, of cardboard, at this same distance apart. Fasten up the card with the lines horizontal at 100 feet or more away, and at the same level, end direct the telescopes toward it, aligning Elein. of Surv. 1A Assignment 14 Page 10 the transit cross-huir upon -che lower line. The solar cross-hair should then be on the upper line when its bubble is at center; if not, alter the position of the cross-hair reticule in the solar telescope until this is accomplished. Test by revolving the whole instrument around upon the spindle cf the lor.-er plate of transit. The bubbles of the alidade-plate, the bubble of the transit tele- scope, End also that of the solar telescope should remain at cen- ter for all positions rhrough 360. The lines of sight of both telescopes should intercept the same interval (that between their axes) at all distances. All seven adjustments, two of the solar and five of the transit, must be made with scrupulous care to se- cure satisfactory results when using the solar. )) (G) ADJUSTMENTS OF THE FLAKE-TABLE Plane-tables embody the same parts as the transit. The lev- eling head corresponds to the similar part in the transit; the board or planchette to that of the lower plate; the ruler, sur- mounted with sights or telescope, to the alidade of the transit. The levels are variously attached; some have only one level upon the ruler, others hrve two levels at right angles so attached, while still others have no level attached to the ruler, bat are furnished with a bar-level that cay be placed at any desired po- sition on the board, for the purpose of adjusting the table to a horizontal position. (1) i'he adjustment of the leveling device in any case is ac- complished by the method of reversion, the levels after oeing Elem. of Burv. 1A Assignment 14 Page 11 turned through 180 are brought half way to center, first by the leveling head, then by the level adjusting screws. In case of the bar- level the adjustment may be made in a manner similar to that described for the carpenter's level in Assignment XIII. ( 2 ) o make _the_ vertical c ircle (arc) read zero when the tele- scope bubole is at center. Carefully level the telescope and set the vernier index at zero of the arc. (3) To make the edge of the ruler a true fiducial line. Upon a smooth surface draw a fine pencil line the full length of the ruler; reverse the ruler end for end setting it against the line and note whether the line and the ruler edge again coincide, -tf not, the edge should be straightened. This requires skill and necessary tools; the instrument maker is the best one to do this. If, as might happen, one half of the ruler's edge curves in one direction aid the other half in the opposite direction (i.e. half is concave, half convex), a reliable test is to move each half along the line to the opposite part; the devia- tion in either portion will then immediately reveal itself. (119) In conclusion on the subject of adjustments a few generali- zations are here presented. Be sure that any adjustment is required oefore making it, and that the method is fully understood before attempting any al- terations, and then use the proper means and proper tools. A Elera. of Surv. 1A Assignment 14 Page 12 novice should always work under the advice and direction of someone having the Knowledge and skill necessary in any case. Needless taking of instruments apart, cutting, filing or scraping parts to fit or attempt to repair and adjust them must never be done except in extreme cases. Generally send the instru- ment to the maker. Good work can often be done with instruments out of adjust- ment, provided the user knows how to determine certain relations. For example, the plate levels ueed not be in adjustment provided one knows what position their bubbles should have when the vertical axis is truly vertical. For the horizontal plate is horizontal if the plate bubbled remain in the same positions in their tubes throughout a complete revolution. Again, the line of sight need not be parallel to the axis of the level on telescope provided the index of the vernier of the vertical arc (circle) is at zero when the line of sight is horizontal; the converse condition is also permissible provided one knows and applies the index error of the vertical circle. Again, in running levels with an engineer's level or transit, it is enough that the level (attached to the telescope- in either instrument) be at the middle of its run, or any other del inite po- sition, when taking sights at equal distances^ from the instrument. But the liability to error is too great, and corrections are too easily neglected in all such cases, and it is, therefore, always Elem. of Surv. 1A Assignment 14 Page 13 best to have instruments fully and nicely adjusted. It is in such conditions that work can be done easily, rspidly, and with the satisfactory assurance that the liability to instrumental error has been mostly eliminated. Among the problems set for student work several will be given in the v,-ork of adjusting. REFERENCES : Tracy pp. 581-606 Raymond pp. 63, 79, 108 Johnson pp. 15, 63, 86, 102 Breed, H. , pp. 25, 56, 70, 89, Vol. I QUESTIONS: 1. The needle of a compact is bent* Hov may the error in bearing from this cause be eliminated? 2. In compass use v;hich of the two plate levels is more important and why? (Give reasons) 3. How may a temporary setting of the plates of a transit in a horizontal position be secured without adjusting the plate levels? 4. If a transit has a full vertical circle, ho^' may a level line be determined without the aic of the level on telescope? 5. How may a level line be determined by use of a transit in the absence of a vertical circle and a level on telescope? UNIVERSITY OF CALIFORNIA EXIENSION DIVISION CORRESPONDENCE COURSES IN ENGINEERING SUBJECTS Course 1A Elements of Surveying Stafford Assignment 15 kEIHODb OF LAND SURVEYING FOREWORD This assignment will set forth the several methods of Land Surveying, explain the purposes of such surveys, and give details of the procedure in each case. (120) PURPOSES OF UND SURVEYING (a) To lay out the bounding lines and locate their intersection; to fix necessary points so that they may be identified and re-lo- cated by reference to certain known or established lines or points; to sake a record in the form of notes of data used in the survey; and to map such a survey in suitable manner. (b) To retrace the lines and the angles of any survey, and to re-establish any or all points in this survey for the purpose of identifying the various features of a tract or parcel of land. (c) To measure established lines and angles of any tract for the purpose of determining the content or area of such tract. (d) To run lines in extent and direction for the purposes of eub-dividing or "parting off" of lands by parcels either in form or content. Land surveys nay be executed either by use of the compass and chain, as in ferin trectr; or by transit and taps in cases re- quiring more refined iueat>urements, as in city surveying. (See Assignment XXVI on City Surveying. ) Elem. of Surv. 1A Assignment 15 Pag 2 (121) Since in most cases the tract of land to be laid out or surveyed forms part of the U._ S. Public Land Survey , it is necessary to follow established lines of such survey. This requires a know- ledge of the methods and practice of the Public Land Surveys, which will be treated in Assignments XIX and XX of this course, The work of the surveyor in such cases is to reproduce the lines and points as established by the Public Land Survey. This becomes a simple matter of identifying certain points previously set. As an example of this, suppose that a certain tract is described as the N. E. Quarter of Section 23, Township 2 North, Range 3 East, Mt. Diablo Meridian and Base. Here is clearly a case of identifying township and section corners. The number of the Township (2 North) and Range (3 East) are the first important designations and these constitute the co-ordinates necessary for locating the region of this particular tract of land which is further described as being the Northeast portion (1/4 ecfuare mile) of Section Number 23- The nearest Township corner is the Southeast corner of T2 N, R3 E of Mt. Diaolo Base Line monument and is north 6 miles and east 18 miles from said B.L. monument. Thence to the south- east corner of Section 23 is two miles north by one mile west where the southeast corner of Section 23 should be found. Then, C!.LXC by measuring 80 chainsAaorth, the northeast corner of section 23 is also identified. Jf the point 40 chains north from the south- east corner was located in the running, of the line northward, it Elem. of Surv. Assignment 15 Page 3 may now be checked by measuring southward from the northeast cor- ner and n proper monument established. Also the northwest corner must be determined by measuring westward from the northeast corner of Section 23 a distance of 40 chains and checking by measuring the full 80 chains (1 mile) to the northwest corner of Section 23- A monument is set at the 40 chain point. Thence measure 40 chains south and check by measuring 40 chains east to the first 1/4 section division established on the east line of Section 23. All this is necessary since the Public Land Survey sets monuments only at sec- tion corners and at quarter-section points on boundaries of sections. interior All/\quarter section corners must be located on the land by the local surveyor. A reference to the shaded quarter section in Section 23 of the adjoining diagram of a township will further elucidate the foregoing description and / Sac. 13 -ic Sec. 15 See. 22 Sec. 27 Sec. 14 / ch 1/4 Sec/" hC v Sec. 24 Sec. 25 o. S.ec. 26 procedure. The function of the U. S. surveyor was completed Figure 37 with the establishment oy ^ o ^r him of the sect; on corner Sec. 25 monument e; the local sur- veyor usj.ng tbvc-s monuments sets the quartei -section corners for his client. A knowledge of the methods er.d practice of Surveys of the Public Land is estential to the surveyor who Elem. of Surv. 1A Assignment 15 Faga 4 would do work worthy of an expert and faithfully serve those em- ploying him. As a large variety of problems of the nature of the foregoing present themselves for solution, you are advised to ac- quaint yourself with the instructions furnished by the General Land Office of the U. S. as no specific instructions are furnished from any other source. Deeds to lands give descriptions from which the surveyor ia required to retrace or lay out the property lines showing the po- sition of corners and giving the angles or bearings of lines. These descriptions usually name the parcel or lot of land and often give "metes and bounds" which are the lengths ard directions of lines bounding the tract. From the data so supplied the sur- veyor goes into the field, sets up the instrument at a corner of the property that is distinctly marked (or perhaps most convenient ) and proceeds to measure angles and courses as set forth in the de- scription. It is his function to reproduce to the best of his ability the points and lines there given and to mark by proper stakes of other means these points upon the land. At least one prominent corner should be tied in. That is, two or more points, the bearing and distance of which are carefully measured, are chosen and described in order that the point so tied may be readily relocated, should the stake or monument marking it be removed. A farm or city lot which is to be fenced or on which build- ings or other improvements are to De erected should be carefully surveyed before beginning such improvements. The location of the Elem. of Surv. 1A Assignment 15 corners, fence lines <*nd sitee of buildings, ditches, drainage lines, etc., can consistently form part of such survey, and should be executed with a precision in accordance with the im- portance of the work. Carelessness in this regard has often been the cause of serious and expensive mistakes and has led to trouble and litigation, which a correct survey would have prevented. This is especially true of divisional lines in both farm and city prop- erty, especially in the latter where buildings are frequently made to follow property lines. In case a survey is to be made to relocate the points and lines of an old survey where the bearings are given with relation to the magnetic meridian, it is essential that the former declina- nation of the magnetic needle be shown. Should the declina- tion at time of the original survey be a part of the record (which should be noted on the original notes, but unfortunately is not always so) and should the present declination also be determined, this becomes a simple matter. But in the absence of such data they must faithfully be recomputed for proper guidance in the survey. Assignment VI on Compass Surveying gives adequate instruction for this purpose, but at least one course should be rerun on the ground for the purpose of fixing such course with respect to the resurvey and of properly orienting the property. Any variation of the dec- lination should be correctly applied to every course in the re- surveys. The surveyor has no warrant for establishing points or run- ning lines otherwise than as given in the original. This my not m. of Surv. 1A Assignment, 15 Page 6 apply in case of a monument or stake that may Oe lost or obliterated, in "which case the surveyor relocates and reports the restoration of such monument to the oest of his ability and in conformity with the remaining features of the survey in which the locations are known and distinct. (See Assignment XL on the Judicial Functions of the Surveyor . ) Each survey for location or resurvey should embody in the report a full and complete set of notes and in most cases a map of the tract, showing tie lines, witness marks, courses in bearing and distance, interior angles, and any explanatory remarks that may render the notes clear and intelligible to others. A clearly executed sketch is usually made a part of the notes and should not be omitted when needed to ciarify otherwise obscure or ambigu- ous data. A little care in this matter often adds to the value of the report and obviates the necessity of much labor otherwise. The survey of a tract for the purpose of finding the content or area constitutes a third purpose and consists generally in re- running the courses, measuring the distances and checking the is points upon the tract of land. If the survey for area only, it is not essential that attention be given to the orientation or even to the declination provided the corners of the tract can otherwise be readily located. In such case the oearings and dis- tances as related to the form of the tract itself are sufficient. However, it is most expedient that the survey for area be conducted Elem. of 3urv. 1A Assignment 15 Page 7 in the usual manner by traversing with compass and chain, or, v;here special accuracy is required, with transit and tape. Bearings may be either magnetic or true bearings. Where the latter are to be taken with the compass, it is convenient to set off the declination with the variation arc and to take such bearings directly in the field. These may be computed, however, from recorded magnetic bearings in lieu of setting off the variation. The surveyor should, if practicable, traverse the entire traet, set up at each corner, take the back and forward bearings, and check by measuring interior angles. He should measure all sides of the field. If it contains any meander lines such as those of large streams, ponds, or lakes, a set of eff-sets from a meas- ured straight line to the meander should be measured and charted. In measuring across streams and bodies of water or across ravines or past any unavoidable obstruction sufficient data should be taken in the field to enable a correct computation to be made subsequently. Hence, check up all such data taken in the field before leaving it and be assured that such are Tall and correct. This will simplify the office work. Upon completion of the survey the error of closure should be noted. This may be determined by measuring the distance from the end of the last course as taken in the field to the point of beginning of the first course end dividing this small distance by the measured perimeter of the fie.'d. If this error of closure exceeds 1 in 500, the measurements, especially of doubtful courses, . . Elem. or ourv. 1A Assignment 15 Page 8 should be repeated for check or correction. In case of transit surveys, where great accuracy is required, this error of c losure ought not to exceed 1 in 5000 or perhaps 1 in 10,000. The error of closure depends upon both the angle measurement and the line measurement, and when ooth are made -.vith equal care, the error should be distributed accordingly. Check lines, such as diagonals, are often of value in locating the presence of ervor and where time or expense are not too great these lines should be measured. Too great pains cannot be taken where precision is required^ (122) Two other methods of survey for area (a) measurement of sides with a sufficient number of diagonals to divide the figure into geometric figures convenient for computing their areas. Most all fields may be divided into triangles by a suitable number of diagonals joining non-consecutive vertices. In such case the trigonometric formula - Area (of each triangle) - i/S(S-a)(S-b)(S-c) where a, b, c, are the lengths of the sides and S = 1/2 (a-fb+c). Frequently the computation is simplified when a rectangular or trapezoidal portion may be measured off. If any portion be bounded oy irregular curved lines as in case of the meander of a stream or body of water, etc., suitable off-eets should be made fron. a straight diagonal line and the area computed and added to the remainder of the field. Assignment XVII will consider in raore detail the mathematical methods of computing areas. Elera. of Surv. 1A Assignment 15 Page 9 (123) The remaining method of surveying for area (b), by a single set-up of the instrument, may now t>e considered. Select any convenient point within the field or at one corner, if suitable triangles can be laid out from such setting: measure the angles (bearings) formed by lines radiating from the point occupied to the corner of the field; also measure the lines including these angles and by trigonometry compute the areae of all the triangles formed ty the sides of the traverse and the radiating lines. The total angle about a central set-up ill of course be 360 and is a convenient check for the angle measurement, but is by no means absolute ^ the discrepancy should be small. If the set-up is made at one corner then the interior angle at this corner should ae measured as a check. The trigonometric formula for computing area in this case ie Area = 1/2 the product of the including sides and the sine of 0/3 the included angle. Example: Given in the triangle^the sides OA, " I 3, and the angle 0; then area - 1/2 OA x 03" x sin 0. B Fig. 38 When employing the method of a single set-up it is most convenient to take stadia measurements oi the radiating lines, which are usually sufficiently accurate for the uses of such surveys. These last two methods (a) with tape only, and (b; by a single set-up are quite suitable for obtaining the areas covered by crops, meadow lands, etc., and are especially applicaole to measurements Elem. 01 Surv. 1A Assignment 15 Page 10 of such lands when devastated by fire, or for estimating the ex- tent of damage for insurance companies, railroads, and others interested in loss adjustment. (124) The subject of subdivision and parting-off of land will be specially treated in a suosequent assignment (XVIII) but a few- general directions are given here. Before proceeding to a subdivision it is essential that a full end correct survey and nap of the entire tract be made, if such is not already available. This survey should show the loca- tion of all corners with a suitable number of tie-lines, courses of boundaries, meanders, roads, buildings, and other improvements. In the office the suDdi visions should be determined by measurement or computation and if expedient should be clearly charted upon the map. ?he actual subdivisional survey should be the locating of these protracted points and lines upon the land, marking the same with convenient monuments or stakes of a more or less permanent character. If portions are to be set aside for public or joint communal use, such as roads, lanes, streets, or parks, the proper designations must be shown upon the map and staked out in the field. It. is sometimes required to subdivide into aliqout parts - halves, thirds, fifths, etc. in the distribution of estates as devised by will or otherwise, or to part-off such proportional parts as may be required. Here the problem becomes one of a math- ematical nature. As these prooiems are purely geometric or sat Elem of Surv. 1A Assignment 15 . Page 11 most trigonometric in their scope, it is unnecessary to deal with them here. When the divisional lins is to be run perpendicular to a given side, as a line of road or street, or parallel to a given side, or otherwise in direction, the "falling" of the line and its direction must be first determined and such lines AE are required to accomplish this ere then accurately staked off in the field. (See Assignment XVIII on "parting off" of land.) References : Breed & Hosmer, Vol. I, Chap V. Tracy, Chap. XII. Raymond, Chap. VII, Johnson, Chap. VII. QUESTIONS: 1. Explain how a line maybe prolonged where an obstruction, such as a building, stands upon a line. 2. How may a line j)ro longed across a river or other body of water? 3. Shov; how a parcel of land may be surveyed by traversing when the corners cannot "be occucied owing to fences or buildings. 4. Explain the use of random lines in surveying. What do you understand by the slope or off-set of such a line? OF CALIFORNIA EXIENSIOK DIVISION CORRESPONDENCE COURSES IN ENGINEERING SUBJECTS Course 1A Elements of Surveying Assignment 16 LATITUDES AMD DEPARTURES CO-ORDINATES FOREWORD : The positions of points in a plane are often conveniently expressed by the distances from certain assumed axes, which are the co-ordinates - to the right or left, abcissae; above or below, ordinates. Since the bearings of lines are chosen with respect to a meridian (either true or magnetic) the common designation of rectangular co-ordinates is conveniently made the distance north or south on the meridian, called latitude and the distance east or west from the meridian, called departure, ihe present assign- ment will explain the manner of so representing courses in sur- veying and the application of co-ordinates to Land Surveying. 125) the R A The course AB in Figure 39 is defined oy giving the angle of bearing 9 and the distance AB_. The latitude {distance north on the meridian) Am, is found by ~ ~~ rr ** ' f ' f. rt Bultiplying A3 (course distance) by the cosine of 0. The departure (distance east froa the meridian) An, is found by multiplying AB (course distance) by the . sine of 0. >E Here the Gearing of course AcJ is northeast, or we ssy it lies in the N.E. D* pa st use . Fig. 39 i. of 3urv. 1A Assignment 16 page 2 quadrant; in this case the latitude is a north latitude and the departure is an east departure. Had the bearing been ^QVi, then the latitude would also be a north latitude, but the departure would be a wesfe departure. So also in the S.E. quadrant the latitude is south, the departure east; in the S.W. quadrant the latitude is south, the departure west. For convenience latitudes north are plus, latitudes south, minue ; departures east are plus, departures west, minus. When entering into computations these signs are always prefixed. North latitude and south latitude are also called northings and southings respectively; east and west departures are known as eastings and ge at ings, respectively. The latitude then is the product of the distance into the cosine of the bearing; and if north, it is positive (t-), if south, it is negative (-). Lat. = Dist. x cos Bearing. The departure is likewise the produce, of the distance into the sine of the bearing; and, if east, it is positive (f) ; if west, it is negative (<). Dep. = Dist. x sin Bearing. In Figure 40, the angles by bearings and the latitudes and departures are graphically represented. The course AB has N. Lat. - S, B. Dep. - 6; these may also be taken as the lines Em snd Am respectively. Blem. of ourv. 1A Assignment 16 page 3 D 10 Distances and Bearings of Courses Course Bearing Distance A - B N e E 9 2 + 6 2 B - C S E 13 2 + 4 2 10 C - D S P W 10 2 + 10 2 D - A N 7"W 92 + 5 Calculation of Distances Figure 40 A - B = vTl? - B - C = C - D = D - A = Calculation of Angles of Bearing = 3341.'5 Tan 6 = 6/9 = 0.6667N = 18 00.3 " $ - 13/4 =3. 2500 / P- 45 00.0 * / = 10/10 = L 0000 (" Natural Functions T= 38 00.6 " T = 9/5 = 1.8000J Tabulation of Results 10.82 13.60 14.14 10.30 Course Bearing i _ A - B 3341.' 5 10. 82 ch. 9.00 6. 00 +9.0 +6.0 B - C 72 53. 8 13. 60 " 3.99 13. 00 -4.0 +13. C - D 45 00. 14. 14 " 10.00 10. 00 -10.0 -10. D - A 60 56. 7 10. 30 " 5-00 9. 00 +5.0 -9.0 A - B C-D D - A Lat. 9. 002 3. 989 9. 999 5. 002 Lat. log Lat. 0. 95<137 0. 60079 0. 99994 0. 69916 log Lat. log cos 9. 92014 9. 46725 9. 84949 9. 68632 log cos log Dist. 1. 03423 1. 13354 1. 15045 1. 01284 log Dist log sin 9. 74407 9. 98035 9. 84949 9. 94159 log sin log Dep. 0. 77830 1. 11389 0. 99994 0. 5433 log Dep. Dep. 6. 002 12 .999 9. 999 9. 002 A - B B - C C-D D - A ;I'j:n. of burv. 1A Assignment 16 Page 4 The computation of latitudes and departures from the course (given angle of bearing and distance) is much shortened by logarithmic method. The formulae Lat. - Dist. x cos Bearing, and Dep. = Dist. x sin Bearing, are conveniently put in the logarithmic form as f ollows : log Lat. = log Dist. + log cos Bearing, and log Dep. = log Dist. + log sin Bearing. Example : Given bearing N2345'E, distance 463.7 ft. To compute the latitude and Departure. Arrange in the following convenient form: Lat. = 424.43 (a) Add (c) and (d) which give log Lat., log Lat. = 2.62761 (b) log cos Bear. = 9.96157 (c) then the anti-log is 424.43, (a), log Dist. - 2.66624 (d) log sin Bear. = 9.60503 (e) Again add (d^) and (e); this gives log Dep., log Dep. - 2.27127 (f) Dep. = 186.75 (g) (f), and the anti-log is 186.75, (g). Cases involving the latitudes and departures of a number of courses may be worked out by arrangement in successive columns, the letters designating each course written at the top (and bottom) of its respective column. The arrangement is shown in the example of a closed traverse displaced in Figure 40. The sum of the latitudes of any closed traverse is zero. Likewise the sum of the departures of any closed traverse is zero. Or expressed in mathematical language r Lat. =0; 2. Dep. = 0. Elem. of Surv. 1A Assignment Ifc Page 5 In traversing the field the surveyor will go as far north as he does south, and as far east as ha does west. This may also be put in the familiar terms : The total northing is equal to the total southing; and the total easting is equal to the total westing. If the latitude (or departure) of any course is wanting, it may be found Dy subtracting the sum of the given north latitudes (or east departures) from the sum of south latitudes (or west de- partures), the algebraic difference being the missing co-ordinate. (126) ERRORS Should the latitudes or departures not balance, i.e., should the sura of the north latitudes Qe greater or less than the sum of the south latitudes, the amount of such difference is called the error in latitude; likewise any discrepancy in the departures is the error in departure. The errors in latitude and departure having been determined, it is then necessary to find the lineal amount of error of closure. This last is the hypotenuse of the right triangle of which the sides about the right angle are the error in latitude and the error in departure. Hence the lineel error in closure is the square root of the sum of the squares of theee two co-ordinate quantities. Ihue lineal error of closure = y(error in lat)^ + (error in + d 2 Bleu, of Surv. 1A Assignment 16 -Page 8 The error of closure is usually expressed as a ratio - the lineal error divided by the perirr.eter of the traverse. Error of closure = 1 perimeter Suppose, for example, that the sum of the north latitudes and sum of the south latitudes differ by 15 links, and the sums of the east and lyest departures differ by 20 links, the total perimeter ol the traverse being 99.12 chains; then = y/15 2 + 20 k = 25; and Error of closure = . ? , which is usually expressed in a 9912 fraction with numerator unity. This is found by dividing Doth numerator and denominator by the number expressing the numerator of the above fraction, E].. In this case 25 * 25 = 1. 9912 * 25 ~ 396+, or nearly 400. Hence we say, the error of closure is one in 400; i.e. , E C = 1/400. This error, 1/400, is a very large error. In compass surveys, the error of closure should never exceed 1/500, and is seldom greater than 1/2000. A transit survey error of closure should evidently be much less than this; 1/lQpOO is a maximum, end often, as in city surveys, 1/40,000 is sought. The error of closure may be due to errors in angle measure- ment, which usually occur when the angles measured lie between lines that are short, or when measuring angles rith the compass. (See Assignment VI, Compass Surveying.) The error may result from difficulty in measuring lines, a failure to keep the tape Elem. Of Surv. 1A Assignment 16 Page 7 taut, or failure to level it over uneven ground. Also, mistakes may occur in reading the tape, although these may usually be de- tected from gross discrepancies that should be removed by remeas- uring the courses in which such mistakes seem to occur. (127) ADJUSTING LATITUDES AND DEPARTURES Before proceeding to use latitudes and departures in com- putations, or for mapping, they should be adjusted proportionally, so that the northings and southings, and eastings and westings balance. This is generally called "balancing the survey", and the adjusted co-ordinates are called "balanced latitudes", "bal- anced departures". Balancing is accomplished by distributing the error in latitude and also the error in departure proportionally to their respective co-ordinates. The errors in line measurement usually result in measuring the line top long, as may be seen when we consider that errors here arise from sag, alignment, and lack of tension. Therefore it is better practice to deduct from the greater side than to add to the lesser. Errors in Angle Measurement. Where measurements of angles are made with compass the error may be large, 5' to 15', through- out the course. Such error should be distributed in parts to the respective angles in which error is most likely to occur. Angles measured by the transit are more precisely determined, and differ oy a few seconds or one or two minutes; these errors when small may be applied to the most doubtful angle. In any case where the Elea. oi'Surv. 1A Assignment 16 Page 8 error in angle showF a gross mistake (a blunder in reading), as for example where angle measure does not check with magnetic bearings, a remeasurement of angle should be made. In all these considerations it is necessary that every angle and every line should be measured in the field, as any part omitted while it may _be_ supplied by computation explained further on (Assignment XVIII), no check can be hacl upon the work and hence the attempt to adjust latitudes and departures is futile. In balancing the latitudes and departures of a compass survey the following rule is applied: Having determined the latitude error, subtract such propor- tional part of the error from each computed latitude as the length of its course bears to the perimeter of the field. Also to adjust departures subtract a proportional part of the error in departure from the departure of each course, the corrections in all cases should be applied in such a way as to reduce the difference between the sums of the two columns, so that the resulting difference is zero; i.e., the sums balance. In a transit survey the correction is made by proportional parts of the error in the latitude or departure to each course, as compared with the total latitude or total departure without regard to algebraic signs. This subject may best be explained by taking a concrete example and following the computations, step by step. As such example, we take the data of the following traverse, a chain and Eieau 01 3urv. 1A Assignment 16 Page 9 compass survey of a quadrangular field. The data given are the course, bearing and distance. A - B B - C C - D D - A Balanced Bearing N4530'E : Dist. K . Lat . S.Lat. E.Dep. W.Dep. Lat. Dep. 2.87 2.01 2.05 +2.01 +2.05 S5110'E 5.05 3.17 3.93 -3.19 +2.93 S5310'W 4.63 2.44 3.93 -2.45 -3.94 N2915'W Totals 4.16 3.63 2.03 +3.63 -2.04 16.71 5.64 5.61 5.61 5.98 5.96 5.96 0.00 0.00 Error in Lat . 0.03 0.02 s Error in Dep. Error of closure = V.03 2 + .02 2 = 0.036; i.e. , 1/464 The logarithmic computations are given for the latitudes and departures recorded in the above table in the following scheme: Course A-B B-C C-D D-A Lat. 2.01 3.17 2.44 3.63 log Lat. 0. 30354 0. 50060 0. 38776 0. 55985 log cos 9.84566 9.79731 9.72218 9.94076 log Dist. 0.45788 0.70329 0.66558 0.61909 log sin 9.85324 9.89152 9.92921 9.68897 log Dep. 0.31112 0.59481 0.59479 0. 30806 Dep. 2.05 3.93 3.93 2.03 Course A-B 3 - C C-D D-A N Figure 41 Elem. of Surv. 1A Assignment 16 Page 10 Problea: As an exercise in computing and tabulating the data, in the computation of latitudes and departures, and in balancing them use the data given in Figure 41. REFERENCES : Tracy Pages 384 - 398 Raymond " 141 - 148 Johnson " 185 - 193 Breed & Hosmer " 400 - 408, Vol. I UNIVSRSIiY OF C/.LIFORKIA EXTENSION IdVISIOI; CORRESPONDENCE COURSES IN ENGINEERING SUBJECTS Course 1A Elements of Surveying Swafford Assignment 17 Double Meridian Distances - Areas FOREWORD This assignment will explain the meaning and computation of Double Meridian Distance, Double Parallel Distances, and the Com- putation of Areas. (129) DEFINITIONS The Meridian distance of a line is the horizontal distance of its middle point from the meridian (line) of reference. This meridian of reference may be the magnetic meridian, the true me- ridian, or any assumed line. In Figure 42, the line MN has a meridian distance equal to Ch which joins the mid-point of MN and is perpendicular to the me- ridian of reference XY. Ma is the north latitude, aN the east .X a Figure 42. departure. But Ch = 1/2 aN; i.e., the departure of the course MN is double the merid- ian distance (D-M.D. ) ae de- fined above. Hence to find the double meridian distance of the first course take the departure of that course. D.M.D. of MN = aN. The meridian distance of the course BO is bj, the per- pendicular from the mid-point Elera. of Surv. 1A Assigr.mert 17 Page 2 of NO to the meridian of reference XY. But bj = 1/2 (aN + dC) , since the line joining the mid-points of the non-parallel sides of a trapezoid is parallel to the bases and equal to one-half the sum of the bases. That, is, aN + dls; -r sO = 2 bj (since dO = ds + sO). In other words, the D.M.D. of the second coarse is equal to the departure of the first course plus the D.fc.D. of the first course plus the departure of the second course. D.M.D. of JJO = jaH + aN (= dsj -r sO. The meridian distance of _OP is ek, the perpendicular from the mid-point of OP to XY. ek = 1/2, (gP + dp), which is to say, that k joining the mid-points of the non-parallel sides of the trapezoid dppg is equal to one-half of the parallel sides. Hence, the D.M.D. of OP = 2ek = gP + dO. But gP - dt = do - tO, aid _dp_ = aN + sO. Therefore, by substitution, D. it D. of OP = dO - tO -f aN + sO = aN_ + _eO - tO -f aN + sQ = ^ [ -t- aN + sO -t- sO - to But aN * aN_ + = D. M. D. of the second course, and -to =: departure of the third course, which is a west departure, and therefore negative. Lastly, by construction, fl is the meridian distance of the course MP and is equal to 1/2 gP. Consequently the D. M. D. of the last course is , like the D. M. D. of the first course, equal to the departure of that course. Or thus : D. M. D. f M = 2 1 = gP = dt. Elem. of Surv. 1A Assignment 17 Page 3 Computing from the previous course, D. M.D. of FM = D. M.D. 3rd C * Dep. 3rd C + Dep. 4th C. = 2afl + 2sO -to + (-to) + i'g) but gP = dt = 2dO - 2 to -dt = dt = gP This value, gP, you will note, is equal to the departure of the last course but of contrary sign. This is true of any closed traverse and is a convenient check upon the computations of the D. M.D. 's and should always be observed to that end. Of course it is required that the latitudes and departures are adjusted; i.e., balanced. We will now take up the specific case of a simple traverse, showing the computation of the double meridian distances. Course Bearing Dist. N.Lat. . Lat. E.Dep. W.pep. D.M.E. A - B S3532'E 8.6 ch. 7.0 5.0 5.0 B - C N7134'E 9-5 " 3-0 9-0 19.0 C - D N1826'B 12.7 " 12.0 4.0 32.0 D-E Due West 11.0 " 0.0 11.0 25.0 E - A S41 11'W 10.6 " 3.0 7.0 7.0 -,15.0 15.0 18.0 18.0 =^ I X balanced It will be noted that the latitudes aid departures balance, and we shall now proceed with the computations as follows: Elem, of Surv. l.i. ^Rsisuiusnt 1 Page 4 5.0 D. M.D. of 1st course (= Dep. of that course) 5.0 Dep. of 1st course 9.0 " of 2nd " 19.0 D.M.D. of 2nd course 9.0 Dep. of 2nd course 4.0 " of 3rd course 32.0 D. M. D. of 3rd course 4.0 Dep. 3rd course -11.0 Dep. 4th course 25.0 D-M.D. of 4th course -11.0 Dep. 4th course - 7.0 Dep. 5th course 7.0 D. M.D. 5th course. This is numerically equal to the departure of the 5th (last) course, but of contrary sign. There- fore, the computation checks. (130) RULE FOR COMPUTING D.M.D. 's (a) The D.M.D. of the first course is the departure of that course. (b) The D.M.D. of the second course is the sum of the departure of the first course, the D.M.D. of the first course, and the departure of the second course. (c) The D. M. D. of the nth course is the sum of the D. M. D- of the (n-l)th course, the departure of the (n-l)th course, and the departure of the nth coarse. (d) The D.M.D. of the last course is numerically equal to the departure of the last course, but of contrary sign. (131) AREAS By referring to the traverse MMOPM, Figure 42 of this aseign- raent, it will be seen that the figure there shown is composed of the 3 1 tin. GJ.'' 3urv. i.', ',sff5. ; ,: t r.7 Page 5 trapezoids aNQd and dQPg , which together include the area of the field plus the areas of the triangles aNM and The area of the trapezoid is equal to the product of the meridian distance into the latitude, or twice the area of each trapezoid is the product of the D.M.D. of any course into the latitude of that course. It will be seen that certain of these products are positive and others negative. By choosing the meridian of reference, XY, through the extreme west point, M, the resulting signs of all the D.M.D. 's are positive, ficnce, the products of D.M.D- 's and s outh latitudes will be minus areas, and the products of D.M.D. 's and north lati- tudes will be plus areas. The difference between the minus areas therefore and plus areas will A be the area of the field. This then gives us the Rule for Computing the Area o the Field from Latitudes end D.M.D. 's as follows: (a) Multiply the latitude of each course by its D.M.D. hadng regard to sign. (b) The algebraic sum of the products is the double area; divide by 2 to find the area. If the course distances have been measured in chains, the products will be square chains. This may be reduced to acres by dividing by 10, since 10 sq. ch. = 1A. Simply remove the decimal point one place toward the left. Had the lines of the traverse been measured in feet, the resulting area would have been square feet; and since 43,560 sq. ft. Elein. of Surv. 1A Ajssignraent 1? Page 6 equal 1 A, to reduce to acres, divide by this number (43..560;. The following is a tabulation of the latitudes, departures, D.M.D. 's and products of the five course traverse contained on a previous page : Course A - B B - C C - D D - E E - A Bearings and Distances omitted Lat. Pep. - 7.0 * 5-0 + 3.0 + 9-0 + 12.0 + 4.0 0.0 - 11.0 - 8.0 - 7.0 B. to. D. + Prod. - Prod. 5.0 35.00 19.0 57.00 32.0 384.00 25.0 0.00 7.0 Totals 56.00 441.00 91.00 91.00 2)350.00 175.00 sq. ch. 17.5 acres We now give the following data, showing a complete ^tabulation of bearings, distances , etc. of the traverse, of a quadrangular field: i i ! A-B N8400 'W 9.04 0.945 8.990 k 0.95 ',- 8.98 - 8.98 ! 8.631; 3 -C S2115'W 12.34 11.501 4.472 1-11.50 j- 4.45 -22.41 257. 7150 ! c -r N7215'E 12-92 3.939 12.302 h- 3.95 i+12.32 -14.54 67.433* D- A H 930'E 6.68 6.588 1.105 ;+ 6.60 !+ 1.11 - 1.11 i 1 40.98 11. 472 : 11. 501 ! 13. 407 13.462 bTiancedT 1 257. 7150 i?3. SUBLIVILiIM} LaWD Most of the problems in the parting off and subdividing of land are purely geometric. Many othere may be solved by the simple methods of trigonometry, but a few must De dealt with by resort to co-ordinate meant similar to the foregoing.* Let it be required to cut off a given area from the field ABODE by s. line starting at the corner E. First cut off the triangle EBA and compute the distance E3 and its Figure *7 bearing. Deduct the area of triangle EBA from the required area, the residue will be the area of triangle EbH, and it is therefore required to find H in BC. Kence find HP, the perpendicular upon EB- Area of / EBh - BE * ^ 2 .'.HP - 2 x BE We now have the angle HBP, angle BPH, and angle PHB , also the perpendicular HP; from these the distance BH may be computed. Also compute the angle 3EH, i.e., the bearing of EH. With transit * You will find many examples of this nature in text books on sur- veying: the work of Gillespie snd that of Carhart are specially citeo. Sleia. of Surv. 1A 18 Page 5 or compass run the line EH aad check the falling of the line on C oy measuring the distance KB. Note: If the area of SAB is greater than the required area to be cut off, construct the triangle EH'B and compute BH'. It ie sometimes required to run a line perpendicular to a given sice of a fielc as a road skirting the side of such field and to part off i^ this means a definite area. Suppose that a certain number of acres are to oe cut off from the right hand portion of the field MNOPR oy a line perpendicular to the road extending along MR (Figure 48). '=^_^---^--- Road Figure 48 Erect the perpendicular PS forming the triangle RSP and having computed the area deduct it' from the required area. Assume a point I and draw TX also perpendicular to MR. The area of the trapezoid I SEX, may now be measured and computed. If this area Flen. of S Assignment i differs from the residual area as found above, adjust the line TX Dy successive trials until the small difference becomes less than any appreciable quantity. This may oe done by office computation and the true line run in the field. Had it been required to run the dividing line at a given angle other than perpendicular, a trial line 'might have be en .\rrom a point, say, on MR having the required bearing, the area of the part so cut off computed and deducted from the required area. Then trial lines run at each distance from the first cut-off line, adjusting the same until the area equals the residual area or differs by an inappreoiaole quantity. A field having an irregular bounding line as the meander in Figure 49 is to be divided by a line, as KM, running from the point K and cutting off a given area. In this case the survey of the field should be made to include the line BC and a series of off-sets taken to the meander line. Compute the area of the portion bounded by B and the meander line and deduct this value from the required area to be cut off. From K run KL par- allel to BC, forming the trapezoid KLBC and Figure 4S t }._ Page 7 compute the area of the trapezoid deducting this much also from the required area. The residual area is now that of the triangle K1.M_; KL being computed from the foregoing, MP and ML may be found by using the area of the triangle KLM and the bearing of AB. If the part bounded by the straight line BC and the meander BranopC be considered, it ie readily Been that the area of this section is approximately the sum of the areas of the several trapezoids composing it. iaicing the distances between off-sets equal along BC, then BC (an+bnH-co-fdp) will equal the area of the irregular section. (137) More accurate formulae for computing the areas of irregular boundary sections are Area = 1/2 1 (h^ -f h) for each trapezoid, 1 being a segment along BC, and h^, hg being the off -sets at two successive points. This formula is known as the Trapezoidal Rule. Simpson's _ 1/3 Rule. In this three off-sets at regular intervals are taken and the formula becomes: Area = 1/3 1 (hi -f 4hg + h 3 ). Substituting L for 2 1 in the above formula we have L/6 (h, -e 4hg + hj) which is the well known Prismoidal Formula applied to areas. Still another formula known as Simpson's 3/8 Rule is: Area = 3/8 1 (h^ + 3hg + 3h 3 + h 4 ). Elem. of Surv. 1A Assignment 18 page 8 (138) CHANGING BOUNDARY LINES It is sometimes desirable to change the boundary or divis- ional lines of property, as in Evraight-ening a crooked line be- tween adjoining properties, or placing a line to alter the boundary without changing the content. To illustrate this we give a common case of the adjusting of the divisional line between properties, the area remaining the same. The line oetween the property of Jones and brown ar shown in Figure 50 is to be made straight by a line starting at A and run in such a vray as to make no alteration in the land area of the property of either owner. A The bearing snd the distances of the ad- jacent lines, as kN, OP and also of A3 , bC, are known and it is first required to find the bearing of the auxiliary line AC. Then from B run a line parallel to AC (i.e., having the same bearing). Compute the back bearing of AD and run a line from D to A. The triangles P,C end ADC are equal in area having a common base AC of equal altitudes - the vertices 3 end D falling on a parallel to the base, flence the line AD is the line required. It is also .evident that the triangle AOB is equal in area to COD, and that by the establishment of the new divisional line Jones receives from Brown as much as he yields by this adjustment. Elem. of Surv. 1A, Assignment 18 Page 9 QUESTIONS: 1. Given the following transit traverse; calculate the error of closure and balance the traverse. Assume a reference meridian through point A and calculate the D. M. D. of each course from this meridian. Find the area in acres. Course AB BC CD DE EF FG GA Bearing s s N N N S S 19 44 22 60 8 69 88 33' 12 42 02 55 19 36 E E E E W w W Distance (feet) 267.50 301.89 266.80 323.61 195.65 354.50 321.20 Note: In solving this proolem use the standard forms prepared by th~e"~Department of Civil Engineering. Submit original computations, not a copy. 2. In the following notes of a compass survey, the length and bearing of one of the courses were omitted. Supply the missing data. Course AB BC CD DA REFERENCES: Bearing N 30 30' E S 64 20 E S 13 15 W omitted Pages 421 - 427 163 - 165 213 - 219 Hosmer " 414 - 419, Distance (chains) 28.37 34.21 41.90 omitted Vol. I Tracy Raymond Johnson Breed & - Course 1A UNIVERSITY OF CjuLIFduslA EXTENSION DIVISION CGrtRESPGtiDMCE COURSES IN BJGINEEKING SJdJLCTS Elements of Surveying Assignment 19 SURVEY OF THE PUBLIC LiNDS. (13S) FOBETNOKD In 1785 the Congress of the United States oy enactment decreed that th public domain should be divided by survey into v townships six miles square, containing 36 sections, each one mile square. This seemingly simple plan carried with it so many un- looked-for complications, that an elaborate scheme was required to carry out the ideal conception of covering a spherical surface with rectangular tracts. We need not follow the evolution of this scheme. The purpose of the Assignments 19 and 20 is to show how the present methods are applied in Public Land Survey. In general, then, the scheme provides for the laying out of meridians and parallels of latitude primarily determined by an initial point carefully chosen and established. The rectangular subdivisions are laid out to conform with these primary lines. (140) INITIAL POINTS Initial points from which the lines of the Public Surveys are to be extended are established whenever necessary under di- rection of the Commissioner of the General Land Office. These points are selected with a view to the survey of extensive agri- cultur&l areas within reasonable geographical limitations. The position of an initial point in latitude and longitude is determined by accurate field astronomical methods. Elem. of 3urv. 1A Assignment 19 Page 2 Three such points in California are iit. Diablo, long. 12154'4S", lat. 3751'30"; Humbolot, long. 12407'll", lat. 4925'04"; San dernardine, long. 11656'15", lat. 34 07'10". Suitable monuments marking these points have been erected on lit. Diablo, Bit. Pierce (Huroboldt) , and Mt. San Bernardino as affording at once a permanent and prominent mark from which the primary lines are extended.* (141) PRINCIPAL iaERItlAN This line is extended nortn and south through the initial point and conforms "with, the true meridian. On the principal mer- idian are set quarter section corners and section corners at 40 and 80 chains respectively, and regular township corners at 430 at chains. Meander corners are set ail meaaderable bodies of water, as streams, lakes, etc. Such meridional lines are rigorously de- termined oy astromomical means, the line measurements are made by refined methods, usually two or more times for check. Every reasonable effort is exercised to insure the accuracy of the alignment and the measured lengths. If the tests show a deviation of 3'PO" or a discrepancy of more than 20 links in 80 chains (1 mile;, then the line is remeasured ,to reduce the error and also the line is changed in azimuth to reduce the error in alignrr.ent. The es- tablishment of a Principal Meridian oeing fundamental, the precise determination is essential as it affects all surveys related to it. 2) BASE LINE I From the initial point the Base Line is extended east and * For a complete list of Initial points see Manual, of Surveying Instructions, Chap. Ill, issued by the General Land Office, Wash- ington, D. C. Elern. of Surv. 1A Assignment 19 Page 3 west on a true parallel of latitude. Upon this line, as upon the Principal Meridian, quarter-section and section corners are estab- lished at intervals of 40 chains and 80 chains respectively and standard township corners at 480 chains; also meander corners where the line intersects bodies of water. A straight line as projected through any point in an east and west direction is the arc of a great circle of the sphere (the earth). It will not be coincident with a parallel of latitude, but may be conceived as a tangent at the c oioirou point from which the parallel may be run through points determined by off-sets. This is known as the Tangent Method of running a parallel. It is conveniently used in open country sparsely covered with forest or underbrush, where thicfc brueh end close-set trees do not inter- fere seriously with the measuring of off-sets. If the country is not open, another and more expeditious method, known as the secant method, of running parallels of latitude is employed. Ihe secant, also the arc or a great circle, is run from a point south of the initial point or township corner, which is carefully determined, the distance from the corner being a function of the latitude. The secant is then projected east or west as the case may be. It will cut the parallel at the one mile and five mile points and off- sets will locate the other mile points on the parallel. This and the tangent method will be fully explained in Assignment XX. If meridional lines are accurately run at the mile and quarter-section (1/2 mile) points and east and west lines run through Elem. of Surv. 1A Assignment 19 Page 4 such meridians, a eeriee of such east and west lines will lie approximately in F. curve which follows the parallel of latitude for each point. To lay out a parallel toy such method requires, however, that meridians be established by strict astronomical means and that a small correction to alignment be applied at each mile point oa the east or "west line. In some cases, where from the nr.ture of the territcr^ surveyed, neither the tangent nor the. secant method is conveniently applicable, then the "solar method 11 , as this third method is called, may "os preferable. (145) STANDARD P^ These are parallels of latitude which run in the same manner as the oase-iine at intervals 24 miles apart, north and south of the base line. The 24 mile intervals are chosen as they include 4 intervals of 6 miles each which is a dimension of the township. Care is exercised in laying out standard parallels as they constitute lines of control in the suodivision Oi the 24 mile tracts into townships. t) GUIDE MERIDIANS Guide meridians are extended north end south from the base line e. t intervals of 24 miles east and west of the principal mer- idian. As all meridians are not parallel lines, but converge to- ward the pole, they will not intersect the standard parallel north or south at intervals of 24 miles, but the intersection cf the guide meridian v/ith the standard parallel is marked nd such a point is named a "Closing township corner". Also in running guide Llem. of Surv. 1A Assignment 19 Page 5 3rd Ste indard Par a 11 el North 16 i ~~ ; f Toifnsljiips ~ - Second Standard parallel Nor th T t ^ (a f 2 1 mi . * i^ * -P CS -P CM <0 M to First Standard Parall^ North w W -P to q c to c c 3 H d Point, Ease Line, Principal Meridian, Standard Parallels, and Guide Meridians. These constitute the primary subdivision of the Public Lands. The 24 mile tracts are subdi- t vided into townships 6 wilts square and these in tarn into sections one mile square, 36 sec- tions in each township. The townships are numbered in each range north and south, the tiers of tov.-nships con- .st Standard 1 -T Parallel North 1 1 JT4K fe3E ' L I iaeridian East 09 cT s rt 13 tSK R2W H 'd HI **! fa O '3 i i3ase Line u> *T3~ t H and 32, which will be projected westward parallel to the south boundary of the township setting a temporary quarter-section Elem. of Surv. lA AE6ig,rinint 19 Page 13 corner at 40 chains, to an intersection with the west boundary of the township, where the falling will be measured and the bearing of the true line calculated, whereupon the line between sections 30 and 31 will be permanently marked between the section corners and the quarter-section corner thereon will be established at 40 chains from the east, there oy placing the fractional measurement in the west half mile as required by law"*. The sections are not subdivided in the field by U.S. sur- veyors unless by special provision, but certain subdivisions-of- section lines are always projected upon the maps (official plats) and the local surveyor who may be employed by entrymen to run such lines in the field is compelled to correlate the conditions as found upon the ground with those shown upon the approved plat. The unit of disposal under the general law is the quarter- quarter- sect ion of 40 acres; the square mile (640 acres) is the unit of subdivision; while the unit of survey is the township of 36 sections (each one square mile). (148) The function of the United States Surveyor has been ful- filled when he has properly executed and monuraented his survey and -returned an official record of it in the shape of complete detailed field notes and a plat. The function of the local sur- veyor begins when he is employed as an expert to identify the lands which have passed into private ownership; this may be a *This quotation and others to follow are from the Manual of Sur- veying Instructions, Survey of the Public Lands. Elena, of Surv. lA Assignment 19 Page 14 simple or most complex problem, depending largely upon the condition of the original monuments as affected principally by the lapse of time since the execution of the official survey. Also the local surveyor must when called upon be prepared to make the suodivisions of section, to discover the locations of "lost or obliterated" cor- ners. To do this effectively requires a full knowledge of the methods of the survey, the rules applicable to these matters as laid down for the practice of surveyors, and the rulings of courts in certain cases. The suoject of "lost and obliterated" corners, and also the judicial functions of the surveyor will be treated in subsequent assignments. PROBLEMS: 1. A uarcel of land is dsecrioeo as north half of the north- east quarter of Section 16 in Tovnsttip 4 North, Ran^e 3 East, Mt. Diaolo Base and Meridian. Draw s map shoeing: Ihe initial poi.it, b given in hours, minutes, and seconds, this v>lue should be reduced to degrees, minutes, and seconds; and if the longitude is measured in linear units, this should be re- duced to angular measure by finding the value of one degree using a value for r (the radius of the parallel arc) from the equation r = R cos Lat. While this is not precisely correct, the mean value R is sufficiently near for purposes of land surveying. (150) LINEAR CGi^VEHGLulCE In practice the linear convergence is set off in feet or chains. This may be determined from the formula C/l = sin MSN, in which C is the linear convergence 1 is the distance on the meridian, and MSK ie the kngle of convergence. Ihis equation Elem. of burv. LA Assignment 20 reduced is C = 1 sin MSN. As the meridians approach each other toward the pole, the correction IE subtractive northward, additive southward. A principal meridian is always established in the field by astronomical me ens, i.e., it is a true meridian as nearly as human agencies can determine. Meridians to the east of the prin- cipal meridian converge toward the principal westward; likewise those to the west converge toward the principal eastward, iience the closing corners in the east cjuadrar.gle fall west of fetandard corners in the next quadrangle north, and such closing corners in the west fall east or their respective standard corners. The linear convergence when determined for any township length of 6 miles may be set off 1/6, 1/3, l/, 2/3, 5/6 at each successive section corner^ this simplifies computations. 51) TANGENT METHOD Or RUNNING A PARALLEL CUhVE If it is desired to set out a true latitude curve east (or west) of a given point the Tangent Method may be used as follows: If a line be run due east (or west) from any given point on the earth 'e surface, such a line, which is the arc of a great circle, departs from a true parallel by an amount which is a function of the latitude. At the equator the departure is zero, the departure increasing as the latitude increases. This may be seen by observing that a so-called straight line east or west is the arc of a circle having the earth's center for arc center, while Elem. of Sur-v. 1A Assignment Page 4 parallels of latitude are the circles formed oy planes parallel to the equatorial plane. A line projected east (or west) from any point in a parallel of latitude north of the equator will depart ir.ore and more to the south as it is carried east (or west) of the point. It will be tangent therefore only rt point of beginning. This straight line is, hence, called the tangent am is conveniently mad* use of in laying out the parallel curve. \ Latitude > 4534'51"N Off -sets in Linics 37 Figure 56 By reference to Figure 56 (exaggerated for sake of clearness) the method may be reedily followed, thus : Set un the- transit sna turn cff an angle of 90 and project the tange.it a distance of six miles, the nieagureraents being c^ae i'cr each corner point at 40, 80, 120, etc. chains. Measure proper Elem. of Sarv. 1A Ai.si to njnent 20 Page 5 offsets north from the tangent to the- parallel, and upon the latter establish the corners. Standard field Tables, prepared 'ay the Gen- eral Land Office and contained in the kanual , give the bearing angles and also the off -sets for each mile from 1 to 5. The hall -mile off- sets should be determined by interpolation; the qj arter-sectioa corners should be checked accordingly. The form of record is also given in the Manual by specimen field notes. The tangent method is in favor in open or untimbered country. In a timbered country the blazing on the true line and the measure- ment of large off-sets are made with difficulty; hence the secant method is used. (152) SECANT aJ&ItiOD OF h'JNNIBG A LKTITUDt CUKVL This is a modification of the tangent method. Range out a line east (or west) for a distance of six miles, cutting the true parallel of latitude at the first and fifth mile corners anf iorining a tangent to an imaginary latitude curve at the middle mile point. Reference to Figure 57 will make plain the bearings and relative points of secant line md parallel curve, and the direction anc? order of the offsets required to accomplish the purpose. A point on the meridian south of the township corner from which the parallel is to be run is located o.y computing, the dis- tance being e function of the latitude. Then from the meridirn the proper deflection angle is taken from taole prepares by the General Land Office. Off-sets are ;aeasurec and corners established Elem. of Surv. 1A Assignment 20 Page 6 SEC AM 'I WLIHOD Off -sets in links Secant (5 TT 1 .. i . .-.,.!.- 400 440 480 True Parallel 40 30 160 240 32C Distances in Chains Figure 57 at every 40 and 80 chains; these also are tafcen from a table of off- gets. It will oe noted that the distances south of the parallel on the meridional lines east and west are symmetrical, that the secant cuts the curve at the first and fifth mile points, that between these two points the parallel lies south of the secant and at the and 6 mile points north of the secant. Furthermore, the off- sets are all short compared with most off -sets on a tan- gent an<5 the secant, therefore, follows more nearly the line of the curve; this last fact renders the secant method in some re- spects preferable. 53) RESTORATION OF LOST OK OBLITERATED C OWNERS Perhaps nothing in the work of th3 surveyor gives more trouble and therefore! cells for greater care than the restoration of lost aid obliterated corners. The best practice evolved through many years and regulated by the rules laid down by the General Land Hero, of Surv. 1A As ? igrment 20 Page 7 Office and further defined by many court rulings is embodied in a chapter of the instructions contained in the Manual, to which you are referred. Some general rules and principles may be set forth here. A distinction should be made in the use of the terms "corner" and "monument" as follows: A corner means a point determined oy the surveying process, while the term monument signifies the physical structure erected to mark the corner point upon the earth. Again, distinction is made between lost and obliterated corners. If the physical evidence of location of a corner has been removed, but the marking can be restored upon adequate testi- mony or other available evidence the corner is not lost out is classed as obliterated. Where both the physical evidences and the immediate means of locating a corner are wanting, end lines must be run from other points and measurements by distance and angle taken to locate a missing corner, it is classed as a lost corner. A special procedure must, hence, be followed for the res- toration. Surveys once made and returned to the Commissioner by a Surveyor General and thus made of record in the Public Land Sur- vey, are fixed and immutable for all time. Therefore, the sur- veyor has no authority to fix any point except Dy the means di- rected for that purpose. The surveyor then should note the dif- ference between the regulations for the original survey of the Elem. of Sury. 1A Assignment 20 Page 8 Public Lands and such as relate to identification of official surveys and the replacement of missing monuments on such surveys. In determining whether a corner is lost or merely obliterated, the eurve-yor should examine into each case at every angle. The search for physical evidences having proved fruitless, he may take the corroborating testimony of witnesses, who can point out certain reliable evidences, as the intersect! one of walls, fences, center lines of roads, stumps of bearing tress, etc. Nor should he neg- lect to search for ouried evidences, witness corners, and line tre^s as supplying reliable evidences of a valuable nature. When all eiich evidences fail tlie corner may be classed as a lost corner. L54) jffiTHODS OF RESTORING LOST CORNERS A "single proportionate" measurement is one made in a single direction, east and west, or north and south, between two deter- minate points on the line. It consists in locating the corner in question a proportionate distance whether the true distance or not, by taking such proportionate measurement for tht fractional distance! as the whole measured distance is to the whole record distance. When the lost corner is determined by measurements in two directions, the "double proportionate" method is employed. That is, proportionate measurements are instituted in both directions and should this irethod locate the corner Doth in latitude and longitude the same shall constitute the point required. If the measurements should not result in locating a point in common, then by cardinal off-sets, in latitude and departure, the corner shall be established. Elera. of Surv. 1A Assignment 20 Page 9 If measurements are taken upon a principal meridian or a base line, or upon an established meridional line or parallel of latitude, the single proportionate method applies; out where the corner cannot fulfill these conditions, then measurements are taken in two, three, or four directions and double proportion must be resorted to. Double proportionate measurement is generally applicable to the restoration of lost corners of four townships and lost in- terior corners of four sections. Monuments north and south should control the latitudinal position of a lost corner, and monuments east and west should con- trol the longitudinal position. Each identified original corner should be given a controlling weight inversely proportional to its distance from the lost corner. Lost exterior section and quarter-section corners are re- stored by single proportionate measurements between the nearest identified corners on opposite sides of the missing corner, north and south on a meridional line, or east and west on a latitudinal line, after the township corners have be an identified or relocated. Lost interior quarter-section corners are to be restored by single proportionate measurements between the adjoining section corners, after the section corners have been identified or relo- cated. (Note the emphasis on "after".) Elem. of Surv. 1A Assignment 20 Page 10 Lost meander corners, originally established on a line across a mer,nderaole body of -.rater and marked upon the oppos.vte side of it will oe relocated by a single proporcionate measurement, after the section or quarter-section corners upon opposite sides of the missing meander corner have been identified or relocated. There are otaer provisions for the relocation of corners and retrF.ceinent of lines which are beyond th-= scope oi these assignments, and the student must not presume to follow the meager directions he^e ^ii/en. Here we give only thf; outline of controlling, principles rnd it is not desired to cover these up with a large mass of details. You are. referred to the "Manual of Surveying Instructions" and to a brief treatise on "Restoration of Lost and Obliterated Corners" issued by the Ceneral Land Office, Washington, D. C. PROBLEMS : 1. Whet is the an ; :ular convergence of meridians 6 miles apart in latituoe 4i/'? The length of 1 of longitude in latitude 40 i& 53.U&5 miles. What is the linear convergence in links at 6 miles? 1, 2, 3, 4, 5, miles ? The N.W. quarter of the ji.W. quarter of a certain section measured 20.16 chains by 18.24 chains; what is the content in acres? :. i ..* CAUtOIOJIA EXTENSION DIVISION Correspondence Ccurse Course IB Element e of Surveying Svrafford Assignment 21 STADIA SURVEYING FOREWORD This assignment is designed to explain the principle and method of the stadia, the scope and limitations of its uee, and also the application of stadia measurements to the various branches of surveying - land, railroad, topographical, etc. (166) The Stadia It. is a rod marked with linear units of measure which may be read afar off, and from the measures thus obtained heights and distances may oe computed!. The name "Telemeter" has been proposed and is occasionally used for stadia; this name (tele = afar, meter = measure) like telescope, telegraph, telephone, in- deed fits the thing and describes its use, but usages change slowly and stadia still persists in the language of surveying. 57) Telescope and Stadia - Hairs To aid in reading the distant rod a telescope is used. This gives large range and great definitenese to the distant markings and characters. Also two fine wires or spider-webs are introduced at the principal focus of the telescope and these are apart spsced at such a distance^that they intercept a given interval on the rod at any given distance from the center of the telescope. It is common practice, and for purposes of computation quite con- venient, to space these stadia vires (stadia-hairs) so that an ni. ol Surv. 1H Assignment 21 Page 2 interval of 1 foot is intercepted on the image oi the rod when the latter is 100 feet from the instrument; by the law for similar triangles, at 200 feet the interval would tie 2 feet; at 300 feet, 3 feet; at 1000 feet, 10 feet; etc.; the lav? holding, of course, for all distances snail or great. Thus, by observing through a telescope furnished with these stadia-hairs, directed upon a rod held at some distant point, and by knowing the ratio, of the distance to the rod intercept, the distance of the rod from a point near the observer may be computed. (158) Theory of Lenses Before proceeding further with the subject of this assign- ment, it is best to explain briefly the phenomena of light passing through the lenses of a telescope, as follows: r- f r Figure 58 t rrr r r r r are parallel ra;ys oi light (the rays from the sun, or a star of oi other very distant luminous point are prac- tically parallel). Such ra^s passing through the double convex lens, L, are refracted toward the axis of the lens upon their passage, and are again refracted in passing through the air, and converge at F, the principal fccus. -~~ ----.-".- V- F tlera. of Surv. IB Assignment 21 Page 3 f, and fg are conjugate foci; that is, light rays emanating from f^ are focused at fg, and vice versa. If f^ is the measured distance from L to the focus in the left, fp the distance of its conjugate focus, and F the distance of the principal focus, then by the law of optics, + = f l f 2 F In the following figure, let AB oe the rod intercept as seen through the telescope and included between the stadia-hairs, ab the image (inverted ae shown), i the stadia-hair interval, and f and fg the distance of image and rod respectively; then by geometry: Figure 60 fg : f j :: AB : ab. Putting S for AB (the rod interval), i for ab, the stadia-hair interval and writing in fractional form, f *? we ha.ve 2 _ _ f l i L5.9) Principles of Stadia Method These are two: the first being that of the law of proportion for similar triangles, which is purely geometric, ihe second is the law for conjugate foci of double convex lenses. 1) When light from an infinitely distant source falls upon a convex lens the rays are parallel and are focused on the opposite Elem. of Surv- IB Assignment 21 Page 4 side at a point called the principal focue, F. If the light ema- nates from a near (or finite) source, fg, the rays are divergent and, by the law of refraction, are in such cases brought to a focus, f , that falls beyond the principal focus. Again, the point f^ may be considered as the radiant point and fp the focus. Hence, f and t* are called conjugate foci. The law of optics above alluded to may now be stated as follows: f 2 F (1) Or in words thus: The reciprocal of the principal focal distance (F) is equal to the sum of the reciprocals of the distance of the conjugate foci, all measured from the optical center of the lens. 2) Suppose now that an interval upon the stadia-rod is imaged between the two stadia-hairs situated at the rod's conju- gate focus. Lines drawn through the optical center of the lens, not being refracted, are therefore straight lines, the ray from the upper end of the rod interval A will fall at a on the lower cross-hair (image inverted). AB r rod interval, ab . stadia-hair interval. Now call the rod interval _S, the stadia-hair interval i, and the distances of the conjugate foci, f ^ and f ? as before, and by geometry we have : fo S f I (2) Solve the simultaneous equations (1; and (2) for f ? , i.e. the dis- tance of the rod from the objective end of the telescope, and we Kleia. or Surv. IB Assignment 21 Page 5 find: f 2 = y- S + F (3) Thus it is seen that the distance from the objective to the rod is made up of two parts, F s , which is variable depending upon the rod interval, and the focal distance, F, which is con- stant for any given instrument. Moreover, it is always desirable to measure the distance of the rod not from the objective but from the center of instrument (i^e. from the vertical axis). It is, therefore, necessary to add another quantity which we will call c, the distance of the center of the instrument from the objective. Hence we combine the quantities F and c, giving a value, F + c, which is regarded as a constant for a given instrument, and has a range in value of from 6.75 ft. to 1.50 ft., usually regarded approximately one foot. The distance c in instruments that have the adjustable ob- jective, is variable, but the error introduced thus is a negligible quantity. It may always be regarded as constant, and F+c deter- i> mined for any instrument is, as also , in instruments having fixed stadia -hairs, and, the principal focal length of the ob- jective, being constant, is a constant ratio. i F The ratio L is determined by the maker, and the interval between the stadia-hairs fixed accordingly. In some instruments the interval is made adjustable and may then be set by the user to suit either a given rod or a desired ratio, . i - Eiera. 01 Surv. IB Assignment 21 Page 6 T? Although the ratio ~- is usually fixed by the maker and the quantity F+c depends upon the construction of the optical parts of the teleecope, it is best, always, to determine these constants for any given instrument, and the student is urged to "try out" the instrument to that end, in the manner shown as follows: (160) To determine the constant F+c With an engineer's scale measure the distance from the middle of the objective lens to the reticule that carries the stadia-hairs, when the instrument is focused on a very distant object, such as a star, the sun, or some other very remote point. By so focusing the instrument, the rays of light entering the objective are made practically parallel and, therefore, the image is at the principal focus. This measured distance is F. Now with the instrument focused upon some object at a mean distance of oo- servation (which is ordinarily about 500 feet from the observer), measure the distance from the middle of the objective to the ver- tical axis of the instrument (i.e. the center of the telescope, which is the horizontal axis of the same). This gives C. The sum of these two distances, F+C , is the constant required. TT> 51) To determine the ratio i Set up the transit on a level stretch of 600 to 1000 feet. By means of markers placed at varying intervals establish ten or more points in a straight and level line. Level the telescope so that the line of sight may be truly horizontal. With the leveling rod and two targets obtain rod intervals for each distance set off Elem. of Surv. IB Assignment 21 PS e 7 by the markers (pins), setting the lower target on the lower cross- hair and the upper target on the upper cross-hair (for each dis- tance in all cases the rod must be plumb). Measure the distance of each pin from the transit point; deduct the constant F-fc, pre- in each case viously determined, as above, for the measured distance^(or better, set off the distance F-fc in front of the transit point and take all measures of distance from this forward point). The rod in- terval (i.e. the distance between the targets) in each case F divided into the measured distance will give the ratio -j. The V average of all these -r's is the mean constant of the instrument. CAUTION: Should any one or more of the determinations differ widely from the general observed value, such may be considered in error and of course rejected in ootaining the nearest mean, . I' For further explanation of these methods see the numerical examples at the end of the assignment. (162) Inc lined Sights Our treatment of the matter of stadia measurement so far has been confined to such as may be made on level ground, in which case the horizontal distance is measured direct. But it is seldom that this condition obtains; sights are taken to points aoove or below the level of the instrument. Such are called inclined sights, and the distances so measured must be reduced to the horizontal for purposes of surveying and mapping. If the rod could be inclined over the point of observation so that the line of sight was per- pendicular to the rod at that point, the distance obtained thus from the roc-reading (stadia interval) would express the slope Lln. or Surv. Id Assignment 21 Page 8 distance, i'ha horizontal distance, in that case, would be the slope- distance multiplied into the cosine of the angle of elevation (or depression). The vertical height of the point (over which the rod isi held) would also be the slope-distance into the sine of the angle of elevation (or depression). In the following figure, the horizontal distance H.D. is required. 01 v 2 i, f s <<; fc (F+c) sin c< ^F+c;cos c HD = AB = sin ex H I. cos c ex (F+c)cos B Figure 61 Here j> is the rod-reading, when the line of sight is directed to the point on the rod held vertically above the point P on the ground, the rod being inclined toward the transit, so that the line of sight (i.e. line of collimation of the telescope) will be perpendicular to the rod ; this point jp is also at a distance above P, equal to the height of the instrument, H. I. The slope-distance measured along the line of sight from the center of the instrument P to p on the rod is 4 s * (F+c) and the horizontal distance, A.B., m. of Surv. IB Assignment 21 Page 9 is evidently the slope-distance times the cosine of the angle of elevation oC Expressed in symbols thus: A B = H.D. = Qa + (F-t-c) J cos o< (1) Likewise the difference in elevation of the point P over transit point A is: B P = V.H. = | s -t- (F+c) 6inc< (2) Having the horizontal distance, we may find the difference in elevation simply by multiplying by the tangent: V.H. = H.D. tan 0^= ( s + (F+c) ) cos o( tan o< (By trigonometry sin f = cos c< tan oc ) Of course in all cases where distances are taken on level F (horizontal) sights the simple formula, _ s * (F+c) is very con- venient and should be used, both because it minimizes liability to error and because it simplifies computation. Also, it has been proposed (and indeed some instruments have been so constructed) that the stadia-hairs be placed vertical to read upon a rod held horizontal instead of being placed horizontal to read on a rod held upright but perpendicular to the line of sight. But both these methods are not always convenient in practice and the usual v.-ay is to hold the rod vertical over the point and apply the necessary reductions for obtaining the horizontal distance, H.D. and vertical heights, V.H. This calls lor a further consid- eration of the method known as: 163 ) Inclined Sights with Rod Vertical In the following figure let AB be the horizontal distance, Elem. of Surv. IB Assignment 21 Page 10 equal the rod-reading (on rod held vertical) AC = BD = height of instrument (H. !.)> which is also equal to the distance of the middle cross-hair above the point P; hence PB = DO = V. H. Like- wise F+c, =3E shown, ma^ be reduced to (F+c) cos o( , or (F+c) sin o( as required for H. D. or V. H- 9 rXf^-" \< (F+c) sin o< M I fe )cos c< D i 1- Figure 62. It can be shown that the horizontal distance AB - H. D. = F 2 __ E . cos o< + (F+c) cos c< (3) and that the difference in elevation TJl : V.H. = . s . cos c< . sin o* + (F+c) sin o< (4) Thus two important formulas, for the determination of the two desired quantities, H.D. and V.H. are here explicitly set forth, and you should become familiar v/ith them, for frequent reference will be made to them in further considerations and in computations. Elem. of Surv. IB Assignment 1 Page 11 Certain changes are made in these formulas, 3 and 4, for the purpose of simplifying computation, and also for the construc- tion of tables suitable for stadia reduction. If we examine the products (Ffc) sin o< and (FTC) cos o* and compare these with the p values (F+C) sinc*cos,o (3 c 24 l ) = 9.99923 0.07841 1.2 = Antilog (f+c) sin g<^ log (f+c) (1.2) = 0.07918 " sin ^ (324')= 8.77310 8.85228 0.07 = Antilog 39.92 = Antilog Elem. of Surv. IB Assignment 21 Page 14 59.92 0.07 39.99 V.D. = 40.0 ft. Thus for small vertical angle the V. D. is the same by the approximate and the exact methods. The H. D. by the two methods differs by l/10th of a foot, which is within the limits of accuracy for mapping. CCMPUTATIOES JOE STJMION E Approx. Method F/i S + (f+c) - 8.67 x 100.2 -4-1.2 V. D. = H.D. x taneo = 868.7 + 1.2 = 799.7 x .2962 = 869. S = 236.87 H. D. V.D. log F/iS + (f+c) = .93947 . log F/iS + (f+c) = 2.93947 " cos<*(l6 30') = 9.98174 " cos (1630') = 9.98174 " = 9.98174 " sin (1630') = 9.45334 2.90295 2.37455 H. D. = 799.7 = Antilog. V. D. = 236.9 = Antilog Exact Method F/i S -Cos*** (f + c) Cos log F/i (100.2) r 2.00087 log (f-fc) (1.2) = 0-07918 " S (3.67) = 0.93802 " cos c*f (I630'p = 9.98174 " cos o<(1630')= 9.98174 6092 " " "^ " = 9 - 98174 2.90237 1.1 = Antilog 798.7 = Antilog 79 J'J 'i * H. D. = 799.8 F/i S-cos o J Page 8 riT* Vve will now plot this section^ and proceed to the computation of the area for D 3 E 8.0 C 8.0 B 1 D 1 ^x3T~^ the purpose of further illustra- tion. Trapezoid LCC'F: 6 x - 5>? 5 ' Q - 20.1 sq. ft. 14.0 Triangle Triangle AEF : Triangle 2 * 3.7 x 5.3 8.0 x 4.0 O - C* - 9.8 = 16.0 " Triangle CC'A 1 : 3 ' -Z2- = 21.0 " Total area of section = 69-1 sq. ft. For section at station 38 * 00 the computation is as # follows (aboreviated form) : 8 x (1.2 + 3.0) 4 2.3 x (9.8 + 12.5) = 42.5 sq. ft. To compute the volume of the embankment between end-sections at station 37 and station 38, we first find the geometric mean of sections. This is : Mean section = /69. 1 x 42. 5 = 54.2 sq. ft. Then the volume L87) on 100 feet of line is: Vol = 1 y/A x A ' = 100 x 54.2 = 54200 cu. ft., or 200.4 cu. yds. Excavations Cross-sectioning is also employed in determining the amount Elem. of Surv. 13 ii.ssign.nent 24 Page of material removed from excavations or the anount of material required to fill depressions over a limited araa, as in the case of a pit (especially a borrow pit, a cellar, eta.) In this work the surface area is divided into squares or iKctangles (sometimes into triangles) of convenient size. Suppose the case of a borrow pit, Fig. 71, divided into squarss 10 ft. x. 10 ft., a few portions A 12 B 12: C 12< D 12^ E 12 F 12i G 12i H 12( T 5.2 12J7.3 12(. .4 125 .7 12= .9 126 .2 126 .5 .3 .2 .2 ..2 i.O .l .5 12? .0 12< .2 12i .4 121 >.0 12( i.O 12< .0 12! .3 12! .8 121 >.9 12( 5.4 12f .3 12( :.4 12( >.0 12( .7 12( .8 12( i.6 121 i.5 12( i.2 12 >.8 12 i.4 126 .7 12 i.6 12( i.5 12f .3 12J ..7 12C .0 12f .4 12( ..4 12< i.5 12( j.7 12J .8 12! J.9 12C .2 12 i.3 I^SV 12 5.1 12t .1 12< .3 126.4^3r26.5 126.2 126.0 126.6 Fig- 71 being shown as triangles for purposes of illus- tration. Xhe dividing lines are numbered to right in the figure, 0, 1, 2, 3, etc,, and downward, K, B, C, D, etc. , so that references nay be readily made to the points of subdivision of the area (ground surface; ; as the upper left hand cor- ner is AO, B4, D2, G6, F3, and so on. The divisions are staked off on the grcuiic 1 and the designation as above marked upon a stake driven at each point as also its elevation above some convenient B-M- to which all roc-readings are referred. Elera. of t>urv. IB Assignment 24 P a ge 10 The rod-readings are taken at each point and recorced in the notes of the survey, and may also oe shown upon a plot of the area and the depth of cut (or fill) marked upon the stake at that point. is always held upon the ground, not on the stake. A level instrument, or a transit with a level on the telescope, is set up at a convenient place where observations may be made upon a rod held upon the B-M. and upon the cross-section points and rod-readings taken as follows: Back-sight onB.tt. , fore-sights on AO, Al, A2, and so on. The elevation of the finished bottom of the pit (or embankment) above datum having been established, and the H.I. of the instrument determined by adding the 8-S. to the 3.M. > the rod- reading, or H.I. for grade is computed. Then the F-S. readings taken at the several points subtracted from the a. I. for grade will give the depth of cut, or height of fill at each point. We new have the elevation of these points above the uottom of the pit, or the height of the top of the finished eabanKn-ent measured from the ground surface. You will observe that we now have the dimen- sions of a number of prisms, each right section of which is the same in the length and breadth. The dividing up of the ground surface in uniform figures both simplifies the measurement, and as you will presently see, lessens the computations as well. The length of each edge of any particular prism is given by the ele- vation of its corners, obtained as explained above. But we require the mean length of each prism which multiplied by the area of its right-section will give the volume of the prism. To obtain the Elera. of Surv. IB Assignment 24 page 11 mean length of the prism add together the elevation of the several points of the ground surface and divide the sum by the number of points, i.e. the nunber of edges - usually four. Referring to Fig. 71, the prism AC, Al, Bl, BO, would be found by taking 1/4 the sum of elevations at AO, Ai, BO, Bl, and multiplying by 100 (= 10' x 10', the dimensions of the right-sec- tion); the prism Al, A2, B.2, Bl likewise has a volume equal to 100 x(Al + A2 -* Bl +B2/ i 1 and so on. You will see that, tne right section is uniform over the whole area cowered by squares. The right-section H2, H4, J2 is reacily computed, being a triangle 1C x 20 , hence 100 sq. ft.; while the triangles G4, G5, K4, and G5, G6, H5 have each a right section of 1/2 (10 x 10) = 50 sq. ft. A further study of Fig. 71 will make plain that if we take the elevations of all cross-section points, we will have used AC once, Al tv.ice, El four times; or, from another point of view, at every exterior corner the elevation is used once; every outer point other than these is used twice; while every interior point is taken as many tirr.es as there are squares having a common corner. The depth of cut or fill for each point may be listed in columns which are headed 1, 2, 3, 4, to shorr the number of times each quantity enters into the computations. We shall list them by the coordinate numeration, but in actual work the depth of cut (or fill) should take the place of this designation. This is given here, not as numerical example, but as a general illus- tration of the method. Elem. of Surv. IB Assignment 24 Page 12 The total in column 1 is taken 1/4 times; in column 2, 2/4 times; in column 3, 3/4 times; in column 4, 4/4 times. Jhe grand total of these is multiplied by tht area of cross-section of one prism (iO 1 x 1C' = 100 sq. ft.); this, divided ay 27, gives the number of cubic yards. To this must be added the irregular portions, consisting of the three triangular sections partially computed. 4 Irregular Sections ' AO Al G4 Bl A6 A2 H2 32 ^ H2, H4, J2 G6 A3 HI B3 E4 A4 B4 10 * 2 - 100 -q ft J2 A5 B5 2 Jl B6 Cl 16.3 HO C6 C2 16.5 D6 C3 16.6 E6 F6 C4 C5 3)49.4 = 16.5 G5 Dl 113 GO D2 D3 16.5 x 100 27 cu. yd FO D4 EO P5 @ DO El ^)H-i, G4, G5 and G5, G6 , H6 CO E2 16.3 BO S3 16.4 E4 16.4 E5 16.1 Fl 16.2 F2 16.5 F3 6)97.9 = 16.6 F4 F5 16.6 x 100 Gl 27 G2 G3 Elera. of 3urv. IB Assigiuaant 2* .. Page 13 PROBLEMS i C 4 \ ** / *4n Vo - 1. From data .. fci-ven in notes, -platv-84% compute the area of sections at stations 40 and 4l and also the volume in cu. yds. of earth to be moved. Station Surface Grane Left Center Right 41 118.0 110.9 C - 7.1 C 22.0 19.3 110.1 C^g 4.6 C^ Width of roadway in cut 20 feet. 2. From data given in the illustration, Fig. 71, compute the amount of material to be exce.vated to bring the pit to a depth having a uniform elevation at the Oottoa of the pit which will be 110 feet above datum. The cross-section checks are 10' x. 10'. Follow the order given in the columns marked 1, 2, 3, 4 on page 12 of this assignment. References : Tracy pp.428 - 434 Raymond pp.275 - 280 Johnson pp. 394 - 399 Breed i Hosruer pp. 2^0 - 2*4 UNIVERSITY OF CALIFORNIA EXTENSION DIVISION Correspondence Course Course IB Elements of Surveying Stafford Assignment 25 VOLUMES of CUTS and FILLS (Continued) Foreword Extended consideration of cuts and fills and additional methods of computing volumes will be treated in this assignment, as -well as the estiaation of quantities in grading from contours. L88) Over extended areas the methods of computing volumes of earthwork to be moved are modified to suit the greater magnitude of the operation. This is done especially where an approximate value suffices in the estimate. But in some instances the more exact method by the prismoidal formula is both expeditious and readily applicable, and gives more accurate results. The end-area method ie sufficient for most estimates and gives results often within the desired limits of accuracy. The volume of a prisiaoid L is v = "g (A.i + 4AJJJ * Ag). If the dimensions for these computations are taken in feet, the volume will be in cubic feet, -which may then be reduced to cubic yards (the common unit for expressing volumes of earthwork,/ by dividing by 27. In the formula L is the length of the prismoid; A^ and Ag are the end sections; A^ is the mid- section. This mid-section is not the mean of the end sections, but is found oy taking the mean values of dimensions of the two end sections or by actually taking the necessary data at mid-posi- tion for determining the section area et that part. i i Ks S^ T* S * f k i ,P : S. : -^ .^s IN & ^ I tt u v? $S t V fes \ S & ^ \ x Q r 3- 5 ?P 55 ff > ^ ^^ .I6 ^^ 4 ^K ik tSa i 5 - L - -4 5 -S - i* : i CCS I ^ x a , , . it Elera. of Surv. IB Assignment 25 Page 2 'I he cross-sections are usually taken at intervals of 100 feet, 50 feet, or 25 feet. Where the prismoids thus determined are fairly uniform, it is practicable to unite two consecutive stations, thus making the cross-sectional area of the intermediate station the mid- sect ion of the combined prismoid thus formed; in this case the value of L will be 200 feet for full stations, or 100 or 50 for 50 's and 25 's. By referring to Fig. 72, it may readily be understood that the corresponding, dimensions of the mid-section may be obtained by taking the mean of the end-sections. Ci + C? di + dp hs -* ho C m = ; I-, d m = : _, h m = = , etc. The width, w, C* C w ' of the r.oadbed is the same throughout, hence no change is made for this dimension t the mid-section. In computing the cut or fill over large areas the work of computing is greatly simplified by checking off the area graded in dimensions that lend themselves tc ready reduction. The general formula for volume in excavations or fill is V = -^- (-~) in which A = ground area of a uniform section, h = the summation of depths of cuts (or heights of fills). Dimensions are taken in feet, and feduced to cubic yards by dividing by 27. Now write the formula in the form El era. of Gurv. IB. Assignment Pago 2b Figure 72 Elem. of iSurv. IB . Assignment 25 Page 3 If the checking of the surface be in rectangles 30' x 36 ' and h be taken to the nearest tenth of a foot, it may be seen by substi tuting above values that 30 x 36 ,__, . V = (2h) cu. yd. 27 x 4 1080 103 cu. yd. Which means that the volume is found simply by summing the h's and dropping the decimal point. The surface might also be checked 18' x 60', or 12' x 90' - Or, by retaining the decimal point, the dimensions of the rectangular checks may be 9' x 12 ' , or 6' x 18 '. By giving each problem a little thought before beginning the collecting of data it is often possible to simplify the work. ^139) Still another method of finding the volume of earthwork in grading is found in the use of contours as laid down upon a topo- graphic msp of the area. This method is especially applicable to Figure 73 Elem. of Surv. IB Assignment 25 Page 4 large areas, or for estimates generally. Take, for instance, the case of a hill that is desired to be reduced to the level of the surrounding terrain. In Fig. 73 is shown a hill depicted by con- tour lines having an interval of 1 foot. It is desired to reduce this to a level of 125 foot elevation. The fcrea of each plane of horizontal section would best be determined by means of a plani- fully meter. This is an area measuring device which will be discussed in a later assignment; (Assignment 30). That part above contour 127 to the crown of the hill is regarded as a cone, the base being the area of the section at contour 127. The volume of this part will be equal to 1/3 h Ai (- ^\Ai). To find the volume of that # 6 ' * part embraced between contours 127 and 125 it is necessary to know the areas A^, Ag and A,. Here it is better to use the pris- moidal formula regarding area at section of oontour 126 as the mid-section. The volume of this portion will be then V = 7(A-i + 4A^, + A,); hence the total volume is expressed as follows: ^ Total Vol. = C5A.-, + 4A 2 + A_). It is sometimes desirable to grade a large surface, changing its slopes and elevations for landscape purposes. For this work the contours are determined and sketched upon a map of the area with a suitable interval. he contours of the finished surface are then drawn upon the map, preferable in a color different from that of the primary contours. The drawing of the two contour systems over the same area shows prismoids whose lengths are the contour interval and whose end areas may readily be measured by Elem. of Surv. IB Assignment 25 Page 5 planimeter. (Assignment 30) If it is desired to apply the pris- noidal formula, two adjacent prismoids may be comoined and the common section will constitute the mid-section. In Fig. 74 the heavy lines depict the contours with a 5 foot interval as drawn Figure 74 upon a map of an area to be reduced to a grade as shown by contours represented by the long-dashed lines. Tr.e sections in each pris- moid are shaded, the section-areas may be found by planiraeter or othenvise. The volumes will oe V (cut) = (A-, -t- 4Am -r V(fill) = 4a a 2 ) m Here the material removed oy cut is conveniently used to make the necessary fill, any excess of course to oe removed Elem. of Surv. IB Assignment 25 Page 6 Other applications of these methods may be employed on roads, dams, embankments, etc. The ingenuity of the engineer may suggest a large variety of uses of such methods, by which much time and money may be saved; especially is this true in computing earthwork in estimates where a high degree of accuracy would prove expensive. References : Raymond, pp. 281-286 Breed Hosmer , pp. 390-395 Johnson, pp. 399-405 Tracy, pp. 435 Elem. of Surv. IB Assignment 25 Page 7 PhOBLLMS Problem 1. Given the following data of cross-sections of a roadway. Compute the area of the raid-section and find volume in cubic yards by the prismoidal forraula. L r 100 ft., w - 20 ft., slope = 1/1, Sta. Surface Grade Left Center Right c f 72 33.9 26.5 71 30.2 25.0 8.5 18.5 6.3 16.3 7.4 5.2 4.8 14.8 2.6 12.6 Problem 2. A lot 150' x 144' is below grade and is to be filled to bring it to a level of 125 ft. above datum. To compute the volume of material required the ground surface was checked in rectangles having dimensions 30' x 36' and rod-readings taken at intersections, as shown on sketch. Find the volume in cu. yards. llfl.fi 18.9 19.2 19.8 19.0 18. 7 118, 117, 116 115 3 18, 1 18 1 18, 18, 5 18 3 1 3 17, 5 17 6 17. 7 18 O 18. 4 16. 9 16 4 16. 2 16 16J 2 15. 8 16 16 ;2 15 8 15 OF CALIFORhLh. EXTENSION DIVISION Correspondence Course Course IB Elements of Surveying Stafford Assignment 26 CTIY SURVEYING Foreword A general consideration of ^ity Surveying and the laying out of streets and city blocks with their grades will be considered in this Assignment. (190) Instruments In general the instruments used in city surveying should be of a higher degree of refinement than those used in country, farm, and highway work. This is partly for the reason that the property - l?nds, buildings, end other municipal improvements - is of greater value, and partly because of the permanent nature of all structures of a public character. On the farm, the value of the land covered by survey is usually a few hundred dollars per acre, while city lands re.ch a value in many cases of hundreds or even thousands of dollars per foot of frontage upon the street. Furthermore, many buildings are ouilt upon the street line and upon the property lines that divide one landowner's holdings from those of his neighbor, desicles, the grades of streets, surface and underground drainage systems, sewers, waterpipes, curbs, side- walks, andthe like, must be determined with great accuracy. The quantities obtained in the survey enter into computation in esti- mates of costs of materials and construction that run into large and often enormous sums. Elein. of Surv. IB Assignment 26 Page 2 i: ence, in country surveys, the compass, and chain usually suffice, and at best a good transit and a tape. In oity work, however, a more refined transit, reading to 30, 20, or 10 seconds, and a tape carefully compared with a standard is necessary. Measurements to a high degree of accuracy should be taken with. both instruments. With the transit angular values to the nearest 5 or 10 seconds should be obtained, 'with the. tape measurements to the nearest thousandth of a foot are necessary. On much city work leveling should be made to the nearest hundredth of a foot, and on bench marks to the nearest thousandth. For the latter a precise level and a straight, carefully graduated rod, having a vernier reading to thousandths, should be used. You should consult the descriptive catalogues of instrument makers showing the high-grade instruments made Dy them. All makers turn out precise instruments possessing features of special excellence, and intelligent selection cannot be made without a knowledge of several different makes. (191) Most city surveying is conducted in cities already built up, where a fev; hundred or a few thousand people have come to live to- gether before the necessity for accurate surveys h.s been realized. in some cases, however, a city may be laid out at the start, and much of the important basis lines, angles, and grades established before any population has come upon the land. In the former con^ dition there are many things that may interfere with a desired or proper basic survey; in the latter, these interferences oeing Banting, the character of the beginnings in a city survey often Elem. of Surv. IB Assignment 26 Page 3 devolves upon the engineer whose duty it is to devise and perfect a suitable plan. (192) In planning a city the size of the blocks is usually the first consideration, as many things connected with convenience in the residence districts, such as lawns, garden, out-buildings, etc., appeal to the early settlers in a new town, while many con- siderations affect the business district. But by far the most important matters for careful consideration are the widths and grades of streets, surface drainage, and sewers and sewage dis- posal. The directions of streets and their relation to the car- dinal points of the compass also deserve attention. Streets in the business district should be generally 80 to 120 feet in width with small blocks of about 225 feet square divided by an alley parallel to the main streets of 15 feet width. In the residence district olocks may be of 300 to 400 feet in length, the alley may be dispensed with, and streets narrowed to 60 feet. The subdivision into lots may be made 25 feet wide in the business district and 40 to 50 feet wide in the residence section. 5) The position of a tov/n site is usually determined by one or more of several considerations. These are: nearness to a body of water or navigaole stream or to railroad station, or, in some cases, to a political division. Or the site may conform to a subdivision of a U. S. L&nd Survey. To such original, determined location there may, of course, be added from time to time other territory that may render the original plan more or less irregular Elem. of Surv. IB Assignment 26 Page 4 and introduce problems that call for the best skill and judgment of the surveyor. Such problems relate to the streets, grades, surface and under evound drainage, sewers and sewage disposal, as was the case -with the original site. The matter of defining these things is largely in the hands of the engineer employed upon the surveys, 'out the final disposal of these matters is through act of the governing body of the town or city and is usually in the form of ai ordinance or ordinances. Lines, grades, and the like are said to be established when the board of trustees, councilmen, or aldermen affirm certain defined lines and grades by enactment. An "established 1 ' line or grade controls all others within the city. When the line of a street and its width and grades have been established, it devolves upon the city engineer to set proper monuments and bench marks so that other engi- neers, surveyors, and land owners may construct streets, curbs, fences (on property lines), buildings, bridges, and other perma- nent features in conformity therewith. To accomplish this, center- lines and intersections must be carefully determined by survey, stone- bound e that are properly witnessed by tie lines must be set, and their position must be indicated correctly upon suitable maps. It is also expedient to chart profiles of principal streets, so that the ground surface and the establishad grade may be fully shown so that future El em. of fiurv. IB. Assignment 26, page 4a tO eo CC 00 f 1 Grade CM tO 10 U _85_ 0' W ". > Grade -gfo {-, o rH 105 in CM 10 rH in rH CM CM rH in CM rH CM T(< in rH CM in rH in CM o 10' Figure 75. co CM Grade f$ < rH 105' in CM rH rH in CM CM rH in CM tO rH CM r-t in CM LO 1 in CM CD rH CM rH in CM 00 rH in 105' "^ 6 t jxJoft.... A ( ...J...5QA.. .0 a Grade o> -c 05 wj o a o in" 10 105' Ci CO <> y j3 85 0' E A~~" in 105' CM (O 10' 105' 105' CO rH - CM rH to rH m u LOMQ. /2/e /4' w as for longer tangents). The full stations are marked upon the line as also the points of de- flection as 1+ 60, 3 + 40, 54-67, 7 + 75, and these are always measured from the initial point. (211) The instrument used in the preliminary survey may be either the compass or the transit. If the compass is used it may be equipped with sight- ing standards or with a telescope pro- vided with cross hairs for giving the line of sight. But if a compass is em- ployed the local magnetic attraction should be reckoned with, and resort to the use of a transit may prove ad- visable. The compass on the pre- liminary survey has usually been con- sidered sufficiently accurate, while the simplicity of reading bearings of the tangents has brought it much into favor . If the transit is used, it is sufficient to run the line by deflec- tion angles as shown above, Fig. 80. Surveying-IB. Assignment 28, paga 8. The first segment of the line is ranged out, either with or without the use of the transit; the transit is then set upoon a forward point of the segment, a back sight with telescope inverted is taken upon the former station, the telescope inverted to normal and the line continued in line -with the rear segment, and this process is continued to the point where the course of the road may turn to the right or left (i.e., deflect). Vifhea the point of deflection is reached the transit is set up over this point, the upper plate is claused with the A verneir of 0, and trith the telescope plunged a backward sight is taken to any convenient point on the last preceding segment; the lower plate is now claraped and the telescope is then turned to normal, which places the line of sight in line with the continuation of the last segment (or tangent of the road). If now the upper plate be undamped and a sight (forward) be taken on a point on the next succeeding tangent, the vernier plate of the transit -will read the deflection angle at the point occupied. (213) Note keeping on the preliminary survey is of especial importance; all data that can in any way contribute to clearness in determining the next important step, towit, the paper location of the road, must be gathered in systematic arrangement and neat form. Specimen notes are shown on the accompanying sh3et. The student is referred to Searles Ives Handbook of Field Practice in Railroad Surveying. S'jrveyir T,-1B< 23, page 9. Questions to Assignment 28 1. State the general outline of work of the chief engineer on a railroad reconnaissance survey. 2. What are the data required in the preliminary survey? Define deflection angle, tangent, and bearing. Notes of R. R. Preliminary Survey Defl. . 61 60 59 58 57 56 857'R 49 13 'P. 1442'L Magnetic Ang. 2520'L 900'R 49 e K>'R B * o * F 3 N48030'E N2310'E W39 30'E N 920'W 1440'L ! II 455'E 'iR. 81 N4830 'E H3950 'E I-J 945'W Bearing - N2642'E N5159'El (Var. 329') N4302 'E (Var. 312') N 6linvi (V^r. 354') Colo Beof>n (N8 5i'Ejl Last i 56.3 Uote : The notes are arranged to read from the bottom of page upward, as shown by numbering of stations; the notes here given are a coni.inua.tion from a preceding page. Sketch on the right gives principal data to assist in mapping the "Preliminary" and also the "Paper Location". Important topographic features are shovm, but minor details are not necessary, and hence are omitted from sketch. UNI\5,KirfY OF Ci-LIFORKIA SJCIENSIOK DIVISION Correspondence Courses Surveying IE Assignment 29 Swafford Railroad Survey ing, - Tangents and Curves Foreword In this assignment -n extended consideration, of the simple curve as used in railrcad surveys will oe made. The student is directed to the latter part of Assignment 26 vhere the use of the simple curve on streets in City Surveying is treated. (P13) Xangents^anC Curves A line cf railroad is essentially a series of straight lines, called te.nge.ntr, and curves - the latter connecting the former in continuous sequence. In the preliminary survey a succession of straight lines of varying length '.vhich intersected with each other from point to point along the route, were staked out. hs roadbed and especially the tracks cov.id not 02 cf use in carrying trains o\er such a course, end it oe comes necessary to connect the straight segments of ro&c &j carves of vaiyirig degree of curvature. These curves, circular in form, are :uacl : tangent to tns tv;o lines of direction, and are c 1 : aruitrery radius (or degree). lor eny giver angle of del lection of the line, the degree of the curve is chosen, usually from several corns iterations. Matters influencing the choice of degree cf curve are :- proximity to other curves, or distance ap*rt of points of intersection (deflection vertices), nature of grades encountered on the line cr in proximity tc the curve, nature of earth materials on vhich Survey ing IB Assignment 29 Page 2 the curve is to be constructed, approaches to bridges and tunnels, even the procurement of right of way, and the speed of trains operating over the curve. A small degree of curve is always to be sought, all things consideree. Beyond such matters the choice of a degree cf curve is rather an arbitrary mattar left to the judgment of the engineer in charge. The simplicity cf the com- putations should not oe vholly overlooked in the selection; generally the. larger the deflection of the taagents (i.e. the larger the intersection angle) the smaller the degree of curve; hence, the more gradual the curve connecting straight lines. For example, if the intersection cf tangents is 40, a 5 curve would give 3 chords (or hundred-foot stations) on the curve, while a 10 curve .would give out four stations. The former -would permit trains to move at a speed of say 40 miles per hour, the latter at, only 25 miles per hour. In addition the work done en the former would oe much less, the wear on ro^doed and rails, as also the v.ear on rolling stock would alsc oe less for the 5 curve than for the 10 curve. Referring tc Fig. S2, AQ and Mb are tangents intersecting at k- QMB is the intersection an 6 le I (= Central angle C). If nov a curve AN3 is laid out tangent to AQ, at ^ ana to Mi at B, All and Mb are called the tangent distances from the points of intersection (p. I.); A is the point of curve (P.O.) and 3 is the point of tangent (F.T.). Radii from P.O. and p. I, meet at C, the center of curvature. C is the central angle. Surveying 29 Page 3 F.I. Fig. 82 The chord of the curve joining ?. C. s.ad P. T. is called the long chord. ihe distance fron the .Tad-point of the long chord to the midpoint of the curve IE called the middle ordinate ; the distance MN from the miipoint of the curve to the point of inter- section of the tangents is the ex-secant (i.e. the external seg- ment of the secant), C.M. , called the external distance. Ihe mathematical relations of these lines and angles are expressed ceo*KtricaI3.y, or trigonometrically . as folloivg; The angle C is equal to angle 1. I is the supplement of angle AMD and angle *AM6 is supplemental to angle C; hence C = I. Surveying IB Assignment 29 Page 4 Augle IL-'JJ (or MBA) is equal to one-half of angle I. Angle I is equal to angle L, plus angle B, and angle A equals angle 3; hence angle A - g-, angle B = -^ . Tangent Distance is equal to radius of the curve multiplied by the tangent of ons-haif angle C (or or.s half angle I). AJs = AC tan C/2. Long chord is equal to twice the product of the radius of the curve into the eine of one half C. L. C. = 2R sin C/2. (Since C - I, I ma be substituted for C in the last tv/o formulae.) The middle t-rdinate is equal to the radius diminished by the radius into the cosine of one half C ( = ). Mid. Ord. = R - R cos , which reduces to Mid. Ord = R (1 - cos ) = R vers . The external distance is equal to the square root of the tangent squared plus the radius squared diminished by the radius. Ex- Dist. r A/Tan 2 t- R 2 / - R The degree of curve is defined as the angle at the center (or its arc) subtended by a chord of lOu ft. In Fig. 83, AB, a chord of 100 ft. subtends the angle at. C, or arc AmB ; AC and BC are radii of this arc; by trigonometry: A3 : AD :: 2 sin D/2 : r, r being the radius of unit circle . AB is 100. Surveying IB A ssignuent 29 Fage 5 Substituting 10Q& = * S * n ; dividing by 2, 50/R = sin D/2; solving for R, R = . 5Q , . sin D/2 There will be as many chords of 100 ft. in the curve, P. C. to P. T. as the degree of curve is contained times in the central angle. This may be expressed by the equation N = ^ (= ^-), where N is the number of 100 foot chords in the curve, I is the inter- section angle, and D is the degree of curve. The number of chords are, therefore, inversely proportional to the degree of curve; therefore the length of the curve from P.O. to P. T. , the tangent distance, and the radius of the curve increase as the degree of curve decreases. It follows then, that as the degree of curve for any given intersection angle diminishes the flatter the curve becomes (or more nearly approaches a straight line). The character of curve may be readily determined, therefore, by the number of stations on the curve; the radius of the curve, the tangent distance, the length of the curve (from P.C. to P.T. ), ant? the long chord increase with the number or stations (i.e. the number of chords) and therefore inversely as the degree of curve. It is the c cam on practice to choose the degree of curve, as that for any given intersection angle determines the other elements of the curve. Soaie engineers choose a radius of curva- ture; and in some special cases it is expedient to assume the tangent distance, as the point of oeginning of the curve or its relation to bridges, tunnels, etc., may make it desiraole that the curve should oegin or end at a definite point. Surveying IB Assignment 2 Puge 6 (214) T!he Deflection Angle This is the angle which the chord naices v;ith the tangent and is equal to one-half the intersection angle divided ty the numoer of chores in the curve. In r'ig. 84 is shown a curve of four sta- tions (each 100 ft.). The angle BAM, which is the angle formed by the long chord and the tangent, is equal to one half I. W.&, MAb, MAC, MAC, are the departures of the lines Aa, Ab, Ac, Ad, from the tangent AM- The deflection angle is here 1/2 * n; since n Figure 84 in this case is *, then tne deflection a^le is 1/4 cf 1/2 (= ]r - I)- J-' he Deflection angle is used in setting off the curve in the field. A transit is set ap at P.C. sighted Off on ?. I. ; the deflection is then turned A on the limb I/n, I/n, 3I/2n, 21/n, the deflection angle aeing added successively to each preceding angle - r!5) We give in the following pages a few examples to illustrate end shov the methods of computation for the several elements of simple curves. Surveying-13 Assignment 29 Page 7 1. Given the intersection angle, 60; it is desired to lo- cate 5 full stations on the curve which begins at the P. C. What is the degree of curve? Solution: N = I/D D = I/N D = 60% = 12 ans. 2. If the intersection angle is 60, the degree of curve 6, what will be the length of the curve? Solution: Length = 100N N = I/D N = 60/6 = 10 . . Length = 100 x 10 = 1000 ft. ans. 3. The intersection angle is 70; find the radius of a 5 curve and the number of chords in the curve. R = 50/sin 2 = 50 * sin 2 30" = 50 * 0.04362 = 1146.28 ft. ans. 4. In the above example what is the tangent distance? Tan. Dist. = R. tan 1/2 = 1146.28 x tan. 35* = 1146.28 x 0.70021 = 902.64 ft. ans. ) 5. For the same intersection angle and degree of curve what is (a) the number of full stations on the curve? (b) the deflec- tion angle? (c) the long chord? (a) N = I/D = 70/5 = 14 ans. (b) d = 1/2 N = I/2N = 70/28 = 2 3C 1 ans. Surveying-IB Assignment 29 Page 8 (c) L.C. = 2R. sin 1/2 = 2 x 1146.28 x sin 35 = 2 x 1146.28 x 0.57358 = 1314.97 ans. The last solution is conveniently made by use of logarithms log 2 = 0.30103 " 1146.28 = 3.05929 " sin 35 = 9. 75859-10 11 Product =13.11891-10 .*. anti-log = 1314,94 ans. 6. The external distance (a) and the middle ordinate (b) may also be computed for the same curve. (a) Ex. Dist. = y/tan 2 4- R 2 - R 1146. 28 2 - 1146.28 = 644230. 9695 + 1314957.8384 - 1146.28 = 1399.71 - 1146.28 = 253.43 ft. ans. (b) Mid. Ord. = R (1 - cos 1/2) = 1146.28 (1 - cos 35) = 1146.28 (1 - 0.81915} = 1146.28 x 0.18085 = 207.30 ft. Ans. (216) Sub -Chorda In general a chord of a curve is understood to be 100 ft. in length, and the foregoing consideration of the simple curve was made on this assumption. When a chord of less than 100 feet is taken as an element of the curve it is called a suo-chord. The sub-chord may be nominally 50 feet, 25 feet, or any fractional part of the 100 ft. chord; but its actual length is slightly more than its nominal length, since in measuring upon the arc of the curve it is evident that two 50-foot chords are not equivalent to the two chords of any arc sub- tended by a 100 foot chord. Shown in Figure 85. Survey ing -IB Assignment 29 Page 9 The deflection for a sab-chord is one half its subtended arc. Therefore, if 1 is the length of a sub- chord and d 1 is the central angle subtended by 1, then 1 = 2(R sin d'/2). Since R = 50 we have by sub- Fig. 85. sin D/2 stituting this value for R in 1 = 1 - 100 sin D/2 100 sin d'/2 the above equation: x sin d '/2; or whence sin D/2 sin d'/2 = 1/100 sin D/2 Since in the case of small arcs the sine is approximately equal to the arc, when the arc D is not greater than 8 or 10, the error in such assumption (sine = arc) is negligible, and we may write this formula, thus: d'/2 = 1/100 x d/2, or more simply d 1 = 1/100 D; whence 1 = 100 d'/D by proportion 1 : 100 :: d 1 : D; or we say the length of the sub- chord is proportional to the arc. Of course for arcs greater than 10 the error of the fore- going assumption must be reckoned with. Surveying-IB Assignment 29 Page 10 In the figure (fig. 85), it is evident that a correction must be added to a sub-chord of nominal length to give the length neces- sary for say two 50 foot sub-chords to reach a point on the curve also reached by a chord of 100 feet from the same initial point. Should it be required to use a 50 foot tape in setting out a curve, then the distance should be determined oy formula given above and this length used in the v.'ork, (217) Example: To lay out a 15 curve with a 50 foot tape, 7.'hat correction must be added to the tape? Here D = 15, D/2 = 730' and o'/ = 1/iCO x D/2 d'/2 = 50/100 x 7 30' = 3 45' 100 x sin 3 45' 100 x 0.0654 0.1305 (The answer is carried to the third decimal place for illustration only.) When a curve begins with a sub-chord, as in the case where the P.O. does not fall at a full station but at some distance be- yond called a plus, for example: P.O. falls at 29 + 35 ft., this nominal length of sub-chord must De increased; thus the nominal length of this sub-chord is then 100 - 35 = 65 ft. If the degree of curve in this Surveying-IB Assignment 29 Page 11 case is 14, then D/2 = 7 00'; d'/2 = 65/100 x 700' = 4955 or 43S'; 100 x sin 4 33' _ 100 x 0.0793 , Ty , therefore, 1 = - = 65.093. (In sin 7 00' 0.1219 setting out this curve the measurement is carried to the hundredth foot only, i.e., 1 = 65.09 ft.) The location of curves, their mapping, etc., will be treated in Assignment 30. Reference is made to Searles & Ives, Field Engineering, pp. 44-60. Problem to Accompany Assignment 29 Problem: At a point 375.8 feet from station 237 on a line of road the deflection is 36 00' R. Find the elements of a six degree curve. (Find number of chords, length of sub-chords both nominal and corrected length, radius of the curve, tangent distances, the plus to P.C., at P.T., long chord, middle ordinate, and the external distance.) UNIVERSITY OF CALIFORNIA EXTENSION DIVISION Correspondence Courses Surveying-IB Assignment 30 Mr. Stafford Mapping, Location, Gracing. The Flanimeter and Its Use. Foreword .Methods of plotting of points, lines, and details, and also the locating of railroad lines both on paper (the paper location) and in the field ere treated in this assignment. A descriptiqp of the planiaieter anc its use in measuring areas of maps, diagrams, etc. will s.lso receive attention. (213) A map is a pictorial representation of a oortion of land or .vater surface. It may range from the simplest outline or sketch to the finished picture., riving details of houses, -wails, fences, and other salient features. Generally, in surveying, it is a represen- tation to a given scsle of the angles, lines, streams or other bodies of v,-ater, ridges, valleys, and slopes vhich have been secured by surveying methods and are emoodied in suitaole records called notes of the survey. In order to make a map it is necessary to have notes or record of a more or less complete character. The completeness of the map will depend upon the amplitude or scope of the data contained in the notes. It has already been made clear that notes should omit no de- tail in the \~&j of data that can materially assist in producing an effective map. There should oe locations and measurements of points, Survey ing -IB Assignment 30 Page 2 lines, and angles an<^ their relative positions together with ample descriptions. These are often best shown oy naming, numbering, and picturing in proper skstches. A line is shown in length (to scale) and angle (bearing, de- flection, etc.). An angle is represented in its true magnitude in units of circular arc (known as "angle"). Elevations and depressions, slopes, valleys, and hills are indicated oy conventions called hatchings or contours. Streams are shown by the lines that picture irregular water courses as they occur at lov;esr valley lines or "thalwegs" (a German word meaning "vs lleyways") . The convention for streams calls for lines that are very fine near the source and that gradually increase in width to- ward the mouth or point of discharge; in larger streams where the scale permits, two lines representing the banks of such water courses are employed in the representation. Lines, when straight, are drawn qy means of a ruler (straight edge, T-square, triangle); when curved, by use of the compass or a curved ruler; when irregular, as meander lines, profiles, etc. Dy short straight free-hand lines from point to point as previously determined. Angles are set off oy one of three principal methods; by tangents, by chords, or by protractor. These three methods are il- lustrated in the following figures and concrete examples, by which angles of bearing, deflection, or included angle are set off or mapped. Surveying-IB As s ignme nt 50 Page 3 (219) Tangent Method To set off an angle of bearing by the tangent method, measure a distance from the vertex of say ten inches along the meridian; from a table of natural tangents take the tangent of the bearing which should be multiplied Dy 10. Upon the perpendicular drawn to the meridian of ten inches, lay off this distance (10 x tan. of bearing); through the points thus determined draw the re- quired line which will make the required angle tne meridian. Example: On a scale of 1 inch equal 200 feet draw a line 750 ft. long, having a bearing N 38 40' E. Here SN represents the meridian and the required line has its origin at A. Lay off 10 in. along SN from A, giving the point m. Erect a perpendicular mn equal to 10 x tan- gent (natural) of 38 40' (10 x 0.8CO = 8.0). Through An draw the line AB making this 750 ft (by scale .".75 inches). For convenience, construction lines have been drawn to a scale of 3/16 = 1' . Survey ing -13 Assignment 20 Page 4 Since the tangents of angles over 45 are greater than one and the tangent increases rapidly, reaching infinity at 90, for angles greater than 50 it is better to use the cotangent instead of the tangent. To illustrate: Lay off a line that deflects 78 41' R by i use of the cotangent as shown in Fig. 89. In this case through the vertex A dra-w AL perpendicular to thfc directed line AM; lay off 10 in. on AL (= Am) ; at m erect the perpendicular an making ran = 10 x cot 78 41' (10 x 0.200 = ?); now draw AB_ from A through n. The angls MAE is the required angle. Had it been attempted ? (jc sea , to set up the tangent of 78 41' at M it would have required a line 10 x -.99695 inches long or nearly 50 inches, much beyond the limits of the ordinary drafting oo?rd. The tangent method is especially applicaolt and convenient in plotting open traverses such as are made in road and railroad surveys. A study of Fig. 90 will sufficiently illustrate this method. Surveying-IB Assignment 30 Page 5 oc In general the angles here are small and of course the tangent is properly used; but at station 17 * 05 the deflection angle being 59 15' R the cotangent is more conveniently applied. 20) The Chord Method The chord of an arc is the line joining the extremities of the arc. Tables are sometimes, out not commonly, included in the tables of engineering handbooks. Referring to Fig. 91 it will be seen that the chord of any arc of unit circle is equal to twice the sine of one -ha If the angle : In the unit circle Radius r 1; sin angle 6 = -=^2.- Ga 1 c.ngle 6ab = 2 angle aQc and ac = cb or ab = 2ac ; i.e., chord ab s 2 sin angle a6c. Now in A6B, the arc A>3 subtends the same angle 6 and the chord Afl ; by similar Surveying-IB Assignment 30 Page 6 triangles da : 8A = ab : AB ; but 6a = Radius = 1, and ab_ = 2 sin 92; .". 1 : 6A = 2 sin angle 6/2 : 3 ; and AB = 6A x 2 sin angle 6/2. If in a special application of this method we make the radius of the construction arc 5 inches long, then the chord is equal to 10 x sin 0/2. That is, for any given angle, take the natural sine of half this angle, move the decimal point one place to the right and lay this off as the chord to a radius of 5 inches. In plotting, describe the arc and from the point where it cuts the line of direction scale off the straight line, 10 x sin 6/2, with the proper scale; draw the required line through the point thus found. This method is applicable to plotting of any sort and when tables of chords are available is very satisfactory and simple. It is especially applicaole to plotting of interior angles. Where the angle is large, exceeding 70, it is better to use complementary construction, using sine of half the complement of the required angle. (221) protractor Method The protractor is a universal instrument for plotting and measuring angles of any sort, such as bearings, deflections, or azimuths. The protractor is made of various kinds of materials, as paper, brass, celluloid, silver, etc., and of sizes and styles suited to the various needs of mapping. Exquisite instruments made of silver with accurate and small subdivisions and supplied with verniers, are to be had from the instrument makers, and the map draftsman will Surveying-IB Assignment 30 Page 7 find use for these where great accuracy and many angles are to be laid out. A good plain protractor is usually all that is required, aad in work of some kinds a large paper protractor is sufficient and convenient. Such paper protractors may oe purchased in several sizes, 8 inches and 14 inches diameter being, common sizes. The larger size is divided into 1/4 degree divisions; to set off angles closer than 15 minutes of arc it is possible to estimate smaller units as 7.5 minutes cr even 5 minutes. This is usually within the required limits of accuracy in construction but where higher ac- curacy is desired, the tangent method should oe used. Generally it is convenient to have the protractor in semi- circular form; it is so used that the center of the circle is placed at the vertex oi the angle to oe plotted with the diameter of the protractor along the meridian or directed line, the angle in degrees and fractions may then oe set off oy the pencil point or pricker and the line drawn through the points thus determined, A protractor of semi-circular form may be conveniently used against a straight edge or T-square placed along the meridian; the center at the angle vertex. For plotting azimuths, especially when it is requirec to locate many points, as on detail or topographic maps, -^here the location is shown in the notes by azimuth and distance, the full circular protractor is most convenient. For such use, cut out the protractor in the form shewn in Fig. 92. This may oe done with a sharp knife along fine pencil lines previously drawn in the following manner: Survey ing- IB Assignment 3 These two arms are hinged at J and are so shaped near the hinge that the/ may be folded to- gether about the wheel W. The axis of W is made parallel to the arm h, and upon the axle is usually a v.onn gear that operates a disk for counting the revolutions of the wheel. The wheel is sub- divided into 10 numbered divisions, each of v;hich is again sub- divided into 10 equal parts. A vernier is attached to secure a reading to one-tenth of these smallest divisions. The counting device consists of the disk that gives the whole number of revo- lutions of the wheel, the numbered divisions on the wheel give the 10th, the smaller subdivisions show the hundredths of revolutions, and the vernier gives the thousandths. Figure 96 shov.s a reading of 4.516 - 4 is taken from the disk C, 5 from the numbering on the Surveying-13 Assignment 30 Page 13 FlQ.96. v.-heel, 1 (the vernier zero has passed this point), which with the vernier reading 6, completes the recorded num- ber of revolutions. The action of the instrument is as follows; The anchor arm is fixed to the diagram and the tracer point carried around the bounding lines of the figure to be measured. The wheel either rolls or slides over the surface; it rolls when the direction of motion is at any angle to the tracer arm greater than zero and slides (without rotation) when the direction of motion of the tracer is in line with the tracer arm. The rolling of the wheel is either direct (i.e. the numbering on the recording device is increasing) or reverse ( i.e. the numbers decrease). The arrangement of the parts of the instrument is such that if the tracer point is carried clockwise around the figure when the anchor point is outside of the figure, the numbering is increasing, or the reading is direct; if the tracer point is moved counter-clockwise, the wheel rolls nega- tively. Therefore, we speak of the action of the wheel as having a positive or negative roll. (Caution; The instrument should be used on a smooth but not a glossy surface. Smooth drawing surface is best.) Without discussing the theory of the planimeter it will be sufficient to state that the relation of the parts is such that the Surveying-IB Assignment 30 Page 14 area of the figure traceu is equal to the product of the tracer arm times the length of the roll of the wheel. Or thus in symbols: A = Inc. in vhich A = the area in square units, 1 = length of the tracer arm from hinge to tracer point, c = the circumference of the wheel, and n = the number of revolutions of the wheel. All of these quantities are aec.su red in thi same units, i.e. in inches, or centimeters or feet, or other desired units. Example: The length of a tracer arm is 5 inches, diameter of v;heel C.75 inches; if the number of revolutions is 4.516, what is the area of the figure traced "when the anchor point is outside of the figure? A = 5 x 0.75 x 3.14 x 4.516 = 10.54 sq.in. (A = Ixd xTTxn) When the anchor point is fixed within the area to be measured, the behavior of the wheel is such that the number of revolutions re- corded ^.vill ,ive, not the area of the figure, but the area of the figure minus a certain area knov,n as the area of the zero circle (or zero circumference). This is better known as the correction area, since this area must be combined with the result obtained when the anchor point is within the oounding line of the figure traced. We shall call this quantity the "correction area". The Constant of the Planimeter For any fixed length of arm and circumference (diameter) of v/heel there is evidently a constant, which is the product of these two factors in the formula A = Inc, n Deing a variaole depending Surveying-IB Assignment 30 Page 15 upon the extent of Lhe ersa craced. 'Without actually measuring the length of arm and the dia- meter of the '"/heel this constant, Ic, raey easily toe determined as follows : 1. Ley out a figure cf known simple dimensions, as a square 5 in. b;, 5 in., or a circle of convenient radius, the area of which is easily computed, or s rectangle of suitaole dimensions; the pur- pose in any case being to have a figure of suitable size whose area may be readily and precisely computed. 2. Fix the anchor point outside of the figure and in such position that the tracing point may be carried around the bounding lines conveniently. After setting the anchor point trace the lines to see if this condition ootains. 3. Note the initial reading, of the counting device -when the tracing point has been set at some point on the bounding line. This is best at a corner, if the figure is made up of straight lines, or at a marked point on the circumference if a circle. 4. Carry the tracer around the figure clockwise until it has passed ever the entire boundary and rerd the counting device for the completed record. The difference between the initial and the final readings is the number of revolutions, n, for this measurement. 5. We nov: ha\e the area of the figure, A, and the number of revolutions, n. Substituting these quantities in the formula A = i Inc, ive may find the constant of the instrument Ic ; for: Ic = A/n. Surveying-IB Assignment 30 Page 16 Likewise we may measure the length of the tracer arm (from the hinge to the tracer point) and find c, or the diameter of the wheel may be measured directly and c computed from = IT d. But it is sufficient to determine the constant Ic, as ex- plained above, and any areas measured are then easily computed by multiplying the number of revolutions (n) in any case by the con- stant to obtain the area. In determining the constant as above it is always best to make several trials and take the mean of these as the most nearly correct value. (227) To find the value of the "correction area" construct a figure of simple dimensions, fix the anchor point within the bounding line and trace the figure noting the roll of the wheel. When the tracer is carried clockwise and the roll is positive we get a positive value for ri (i.e. +n) but if the roll of the wheel is negative then we have a negative value for n. In the same way if the tracer is carried counterclockwise and the roll of the wheel is positive we have -n; if the roll of the wheel is negative we have +n. Care must be taken to observe these facts. The value of the result as thus obtained is Inc = Ak - Ac, in -vhich _l is the constant (determined as above), n is the number of revolutions and may be either positive or negative, Ak is the com- puted area of the known figure, and Ac is the correction area. Hence the correction area, Ac = Ak - (nlc) Surveying-IB Assignment 30 Page 17 It may be observed that if the roll of the wheel is positive the correction area is smaller than the area traced; if the roll of the wheel is negative the correction area is greater than the area traced; also, if the difference between the initial and the final readings is zero, the correction area is equal to the area of the figure traced. Stated thus: Ac = Ak - (lc.+n) = Ak - Icn, Ak greater than Ac Ac = Ak - (Ic.-n) = Ak + Icn, Ak less than Ac Ac = Ak - (Ic.zero) = Ak - 0, Ak = Ac In using the planimeter for finding the area of a portion of a map in square niles, acres, etc., it is necessary to take the scale of the map into consideration. If the scale is given so many acres to the square inch, or so many square miles to the square inch, then it is only necessary to reduce the number of square inches to acres (or square miles) by multiplying by the proper ratio. The same would be true where the area is found by planimeter in other units and the value of one unit known in some other unit. To illustrate: Suppose the map were drawn to scale of 50 miles to the inch, then the square inch would represent 2500 square miles. Again, if the scale were 1 inch = 400 feet, then 1 sq. in. would equal 160,000 square ft., or - acres. Hence, if the area were found by plani- meter to be m sq. in., then the area in acres would be m acres. 43560 (1 acre = 43,560 sq. ft.) The simpler makes of planimeter have the tracer arm fixed (not adjustable) hence the constant Ic of such having been determined for Surveying-IB Assignment 30 Page 18 any given unit area, the namber of revolutions alone constitutes the only variable to be observed. Other makes have adjustable arms and are provided with a graduated scale or gauge-marks that enable the setting of the instrument to read directly square centimeters, sq. meters, sq. inches, acres, etc. The planimeter is made in many styles and adapted to many purposes such as finding the areas of cross-sections, profile sec- tions, steam engine indicator diagrams, land surfaces, and in fact finding any area that may enter into a general or specific proolem. References Breed & Hosmer, Vol. I, pp. 445-461 Raymond, See index and pp. 172-179 Johnson, pp. 262-270, and pp. 143-161 Tracy, pp. 485 to 500 . Problems to Assignment 30 1. A quadrangular field has the following: Bearing Defl. angle A-3 S 30 00' E 459 B-C 434 C-D 521 117 00' L D-A 397 Int. Angles : ABC = 115 30'; CD* = 109 30' Construct the figure by tangents and chords, scale 1 inch = 5 ft. Accompany drawing with complete data. 2. Find the constant (lc) of a planimeter which in tracing three times the perimeter of a square 4 in. x 4 in. gave: initial reading = 2.734, a = 4.325, b= 5.926, c = 7.527. 3. If the tracer arm is 5 inches, what is the diameter of the wheel of the planimeter used in proolem 2? 4. The same planimeter was used to trace a portion of a map with scale 1 inch = 4 miles, initial reading 3.472, final reading 5.555. UNIVERSITY OF CALIFORNIA EXTLNS ION DIVISION Correspondence Course Elements of Surveying Surveying ID Swaiford Assignment 31 Triangulation and Base-Line Foreword :- This assignment will treat of Trian&ulation systems as used with Base-Lines in surveying, the methods of measuring a base-line and the location of triangulation stations and the adaptation of the method to topographic surveying. (228) Base-Line :- Primarily, any system of triangulation begins with a base-line accurately determined by the most precise methods of measurement and ite location also accurately determined by astronomical observation. The region covered by a triangulation net is first carefully reconnoitred to obtain the most advantageous points for purposes of observation. These are determined by tneir prominence, intervisidility , and accessioility for the purpose used. The situation of triangulation points should be, gener- ally, o prominent one, such as peaces or summits, headlands, steeples, light-houses, etc.; but where the points are to be used for observing stations, it must De possible to occupy them with a transit or theodolite. Therefore some of the above described points would have to be eliminated from the list given. El em. of Surv. IB Assignment 51 Page 2 Another consideration is that of the location of the proposed base-line, which should be chosen in a comparatively low altitude and preferably in a straight and nearly level stretch of sufficient extent to permit its being projected in a straight line or nearly so. If the region covered by the triangulation system emoraces hills, valleys, woods, and plains, it is evident, therefore, that much attention must be given to the work of re- connaissance , so &s to overcome possiole obstructions to the proper carrying out of the work.,. F __ ^ r (229) Suppose the Base- line AB in the adjoining Fig. 97, representing what is called a base- line net, has been established and accurate- Figure 97 ly measured subsequently described in this assignment. The points C, D, E, F, G, etc. are then chosen so that the work of measuring the necessary angles in the configuration shown may proceed unobstruct- ed. It is essential to good work in this case that A, B ; C, and D shall be intervisible each from all the others, and that the triangles ABC, ABD. .aCD, and BCD shall be well proportioned. By this is meant that the sides should be nearly equal. None of the angles should o o be less than 30 or more than 120 and the ^ore nearly they ap- o proach 60 the better. The diagonal CD, being especially for the Elem. of Surv. lb 31 Page 3 purpose of checking, need not make large anglee with the other lines, but it would be best to secure angles greater than 30 if possible. In the triangle ABC the line AB (base-line) is known; hence, to find the sides AC and BC, the angles CAB and CBA are carefully measured with a transit or theodolite, and the law of applied, thus: AC Sin C , and BC * Sm C So also in the triangle ABD, the sides AD r,nd BD may be computed from the known side AB (base-line) and the angles at A and B. The parts AC, BC, AD, and BD having been determined, and the angles included having oeen measured, we aay proceed to compuie the diagonal CD- This completes a quadrilateral known as the base net, and since the subsequent expansion of the triangulation system raust depend for its correctness upon the accuracy with which the parts of this quadrilateral are determined, it is essential that careful h&ed be taken to perfect the measurements and computations upon which the determined parts depend. The angles of the quad- rilateral are aeasured either with a theodolite or a transit; if the transit is used the method of measuring angles by repetition is employed. The Figure 98 Elem. of Surv IB Assignment 31 Page 4. transit is set up at A, the instrument carefully centered over the exact end of the base-line and the plates carefully leveled. Too great care cannot be taken to set the vertical axis vertical, and to have the tripod firmly planted and the instrument in perfect adjustment- The accuracy of the results in angle measure will depend much upon the handling of the instrument; especially ie this true with respect to the clamping of plates and the use of tangent screws; also attention must be given to the manner of turning the transit on its vertical axis in sighting and, again, in the inverting of the telescope in the act of "plunging". Clamp-screws should be brought firmly to seat, but never clamped very tightly. The very act of urging the screw against its bearing may throw the instrument out of level or cause a slight rotation of the whole about the vertical axis. A tangent screw will act better when opposing the spring, than when the spring fol- lows the screw; so it is better always to set the vernier index before clamping so that a clockwise movement of the tangent screw will be required to bisect the signal or point sighted upon. During the series of readings by repetition the plate of the tran- sit should not be disturbed, even if it appear that the vertical axis is not truly vertical; if it is thought best to readjust the verticality of the instrument, this must be done and a nev; set of readings made (230) Repetitions of angle should be made as follows: With Elem. of Surv. lb Assignn-ent 31. Page 6. the telescope normal, set the A vernier at zero and check by ob- serving the B vernier. Record both readings. Unclamp the lower motion; set intersection of cross-hairs on left-hand signal (i.e. when instrument is at A, sight on C); bisect precisely by means of the lower tangent screw. (Caution: The signal, at C for example, may present a phase such that one side of the signal rod, plumb-bob or other device shall be mistaken for the middle of the same; endeavor to eliminate such a condition.) Unclamp upper mo- and sight right-hand signal on B; clamp the upper motion tioryand perfect the bisection of signal with upper tangent-screw. Read and record full angular readings of both A and B verniers. Continue the repetitions by setting on left-hand signal with low- er motion, then upon right-hand signal with upper motion the . requisite number of times to produce the number of repetitions de- sired, but recording only the final readings of A and B vernier. Now (without disturbing either the setting of the instrument or the vernier index) invert the telescope and take an equal number of readings of angle with telescope "plunged". The mean of the two repetitions is the value of the angle measured. Record this as the mean angle. Let it be supposed that we have just measured angle a at A in Fig. 98. Proceed in like manner to measure angle h at A; then the total angle a + h, and the exterior angle CAD, thus closing the horizon about A. Now proceed to B and measure in turn angle d, angle e, d * e, and exterior angle CBD. Next occupy C, measuring angles, b, c, b * c, and exterior angle ACB ; finally, Elem. of Surv. 13. Assignment 31. Page 6. occupy D, determining angles, g, f, g+f, and exterior angle BDA. All angles are actually measured and not computed from partial OD- servec data; for example, at the last point occupied aoove, i.e. at point D, me a sure g and , then measure g -t- f ; do not assume that the whole angle is equal to the sum of its parts; so also the exterior an^le is measured, not corr.puted from 360 - (g + f ) . The test of the accurac;/ of the work of measurement is of course o th.t g + f e int. angle ADB ; and g + f -t- ext. angle BDA * 360 . These are checks used in the field and should be so applied, a serious discrepancy being at once corrected. (231) Having completed the angle measurement at all four stations of the quadrilateral AC3D, before making computations of the sides and the diagonal CD, it is essential that the angles of the Quadrilateral should be carefully adjusted. I he following conditions should obtain: 1. Ih& sum of all the an 6 les aoout any station o should equal 360 (the field test should apply;. 2. The sun of all the angles of the quadrilateral o should equal 560; i.e. a-irb-rc+d + eTf + g + h" 360 . 3. The sum of the angles of any triangle should be 180; e.g. b + a + h-g* 180, and c+d + e + f* 180. 4. Since angle 2 = angle 4 (vertical angles at 0), then c -1- d = h + g; also, since angle 1 - angle 3, a f b s e + f. In making the adjustments it is assumed that the error in any case mny oe distributed by equal parts to each angle, Elem. of Sun/. IB. Assignment 31. Page 7. 1/3 to each angle of a triangle, 1/4 to each angle of a quadrilater- al and 1/2 to each pair of equals. (232) Finally a test of the sides should be made as follows: Since AC= m Sin < b * c) . and CB = AB Si " (b * G) . it follows Sin d Sin a that; Sin (b + c) Sin (b + c) AC : CB = AB sn d : < S in a Cancelling AB sin (b + c) and applying the test of the primitive AC Sin d proportion = after computing AC and CB. CB Sin a A further equation of condition for the quadrilateral n. should also obtain: Sin a Sin c Sin e Sin g Sin b x SirTd" X Sin f x Sin h = * or expressing the same logarithmically; log Sin a -t- log Sin c + log Sin e + log Sin g - (log. Sin b + log Sin d + log Sin f 4- log Sin h) = The further adjustment of the angles may be necessary in order to bring about the above conditions. (233) Measurement of the Base -Line. The site of the Base-line is first determined, the ex- treaities are staked out, and the measurement of its length then made. Various methods have been employed froa time to tine, but that now followed is with a steel tape of known standard and used in the field under suitable conditions of wind and temperature, Elem. of Surv. liJ. Assignment 31. Page 8. generally at a normal tension for compensating sag. In Assignments 3 and 4, the discussion of linear measure- ment was given fully anc it -will be unnecessary here to repeat these instructions; however , it may be advisable to restate the several corrections in tape measurement as applicable to the case in hand. (a) The tape should be corrected for erroneous length. This requires that the tape be compared with a standard and the error carefully determined. Suppose that the tape is 100 feet long (nominally) and the ar.ount that the tape is longer or shorter than this when compared with standard is found. The correction to the measured line is applied for every hundred feet of length in the measured line. Again if a fractional tape length remains - say 37.175 feet or 69.827 feet, then that portion of the tape used in making this fractional measurement should be tested, by standard to determine the correction for this special segment. Usually it is known or assumed that the discrepancy between the used tape "nd the standard is uniformly distributed throughout the whole length of the tape and the correction is so applied. Hence, to correct for standard, multiply the measured length of the line by the error in one tape length, divide by the length of the standard (100 feet in this case), and add the cor- rection if the tape is longer than the standard, or subtract if shorter than the standard. Elera. of Surv. IB. Assignment 31. Page 9. L c = corrected length of line, L = measured length of line, e = difference between the standard 100 foot tape and the nominal one hundred foot tape. (o) Each measured segment of the line should be cor- rected for temperature. The tape is standard et 62 C Fanr. ; therefore measure- ments taken at other temperatures must be corrected as follows :- The temperature T of the tape is observed usually at two points, a fev feet from en.cn end, the menn of the observed values being tr.lcen for the value of I. The coefficient of expansion for the ordinary steel tape is approximately 0.0000065 - a (The cor- rection t a which is very small nay, of course, be ignored). This coefficient is per foot per decree Fahr. ; hence mul- tiply the difference in temperature (temperature of tape minus the temperature of standard; by the length of line times the coeffici- ent vhich gives the correction to be added if temperature is above standard, to be subtracted if temperature is below standard. L c = 1^ t (x x - T ) 0.0000065 In which 1 s correct length of segment, l, a =r measured length of segment, T^ is the temperature (observed mean), T is the standard temperature, <\nd 0.0000065 the coefficient of linear expansion. Elem. of Surv. lb . Assignment 31. Page 10. (c) Each measured segment should oe corrected for slope. It is evident that even upon a comparatively level stretch, unless the end supports of the tape can be brought to the sane elevation, the length of segment, 1, is not the horizontal dis- tance but the slope distance for that segment. Hence, a reduction to the horizontal is required. This may be secured in one of two ways; either the angle of slope must be measured , or the difference of elevation of the two ends of the segment must be found and the appropriate correc- tion computed and applied. The measurement oi' angles of elope, except with special base-line apparatus (notably the Holden Clinometers, and other instruments) is troublesome; and even with such special apparatus gives no more reliaole results than the method here described, as follows: With engineer's level or transit used as such take rod readings at both ends of the segnent; preferably set up instrument equidistant from the two ends thus eliminating error in adjustment of the line of sight perpendicular to the vertical axis. The dif- ference of the two readings so obtained is the required difference in elevation; this difference may be easily found to the nearest 1/100 of a foot which is well within the degree of accuracy desired, as a difference of 1/100 of a foot is about that of 20" of arc, for a distance of 100 feet (one segment of 100 feet). The tape when compared with standard is supported Elem. of Surv. 13. Assignment 31. Page 11. throughout its length, but in the field this is impracticable; therefore, it is supported only at the ends and a correction for sag must be made, or, what is more feasible, the sag may be compensated by applying a tension just sufficient to accomplish this. (d) The correction for sag is found from the formula ; L f i ( ) = C , in which C is subtractive as the length of the 24 t s s segment, is less by the amount of this correction than the actual length. L is the nominal length of the segment (100 ft), 1 is the length of the sane (if supported at the ends, this is also 100ft) , w is the ?reight of the tape per linear foot, and t the ten- sion applied. (e) The correction for tension or pull is expressed by the formula C = - , in which C p is additive since the SE effect of pull is to stretch the tape, and hence the measured I length of the base-line segment is greater than the actual length. Here L is the nominal length of the tape (100 ft), t is the pull or tension, S is the cross-sectional area of the tape in the same unit as L (i.e. the foot) and E ie the modulus of elasticity of the material of the tape. The modulus is taken as 30,000,000 Ibs. , being that of steel used in tapes of this kind. (f) Nonael tension is a tension which applied to the tape compensates for the sa=c; or since the C is a subtractive cor- S rection and C is additive we may put C s = C and find the Elem. of Sur\. IB. Assignment 31. Page 12, resultant value of t in the equation, thus: ( ) = ; and this solved for t gives t 24 t SE ' VsEfr 2 !* 24 This tension applied to all measurements removes the troublesome computations for sag and tension; the results obtained are equally reliable T?ith other more complex method. (234) Broken Base -Lines If feasible a base-line should be measured in a straight line, but as it is sometimes more important that the case should have the desired extent and location of its extremities than that its measured parts should oe in the same straight line, a "broken base" is often chosen intentionally, and the broken base is then reduced to a straight base, as follows: Suppose that the segments AC and CB of a base have been Fig 99. measured and that these deflect at C by the angle 6. Then the base-line ^B is computed from the trigonometric law of cosines, thus AB = AC^ + CB - 2(AC x CBj cos ACB ; or choosing instead the deflection anrlefr" (18C - ACB) and extracting the square-root of both numbers AB =~\/ A 2 + CB 2 + 2(AC x CB) Elera. of Surv. IB. Assignment 31. Page 13. From the aoove an approximate formula is derived which gives satisfactory results when applied in cases where the angle 6 does not exceed from 3 to 5. The formula is: 2 A3 = AC -f CB + 0.00000004231 AC * CB - AC x CB As logarithmic computation is here desirable the log- arithm of 0.00000004231 is 2.626424 - 10. If the angle 6 is larger than 5, then it would be neces- sary to measure the angles at A and B and compute AB from the law of sines- The procedure in the work of measurement is as follows: 1) Stakes are set in line, by transit set up over A, at in- tervals of a tape length (100 feet). These stakes should be of substantial size (about 2" x 4" - 3 feet long) driven firmly into the ground and cut off square on top. On the tops are fastened small strips of zinc (also 2" x 4") on which lines may be inscrib- ed to mark tape lengths. In lieu of these stages low tripods may be used which carry suitable clock heads on which the zinc plates may be fastened; two such tripods are needed, the rear one being advanced to a forward position for each segment of a tape length, while the other is left in position undisturbed. 2) The measurement is to be made between tack centered hubs r.t the base-line extremities and the point on the ground must be transferred to the top of the 2" x 4" staice or to the tripod head. Elem. of Surv. Id. Assignment 31. Page 14. To do this set up a transit at A, carefully leveling same and centering on the point on hub. Turn off an angle of 90 from the base-line and set a tack centered hub 8 to 12 feet distant from A. Invert the telescope and repeat the setting of the off-set point; if the two points coincide, then the off-set point has been properly placed; if the two points do not coincide, take the mean position. Now set up the transit over this off-set point and sight upon the tack-center in hub A and transfer this to the top of the 2" x 4" stake or tripod, marking the point upon the zinc plate. It is necessary to set the point with telescope normal and also plunged, in order to eliminate any lack of adjustment of the line of sight; if the two pointings differ, take the mean position. 3) Having thus narked upon the zinc plate the initial point t of the base-line, stretch the tape over this mark and the zinc plate next i'n order by means of stretcher bars, one at each end, with dynamometers (.spring Balances) attached for determining the pull. If a normal pull is used, and it is better to use a normal pull, see that the dynamometer records this tension at the instant of comparing, or setting off the distance. A line should be drawn on the zinc plate in the direction of the line at each station and a short line it ri^ht-angles to this over the mark, on the tape chosen for the limit of the measurement; it is best to make this limit the 100 foot mark or 99.9, 99.8 or some other definite div- ision. Of course for computation, the 100 foot length is simplest. Elem. of Surv. IB. Assignment 31. Page 15. 4) While the tape is still in the position used in measuring (i.e. while still suspended in air and thus free from the warmer or colder earth, place a thermometer upon the tape first at one end and then at the other, and record the mean of the two readings, as the temperature of the tape. 5) With a transit or engineers level, set up equidistant from the tivo ends of the segment, take rod-readings at both ends, rod held on top of the stake or tripod. The difference of these two rod readings will be the difference in elevation between these points. Proceed in like manner with the second and each succeed- ing segment, being careful to keep the tape off the ground, and flat with draduations uppermost so they may easily be read, and to observe temperature, tension, and elevation in each case. 6) Record all data as observed in a neat tabular form; the following is suggested; Seg. Meas- Length Mean Temp. Elevation Corrections Length Corrected Total Meas. Length .- * a b diff. Temp. Slope 1 2 99.9 100.0 78 83 7.8 6.7 10.7 8.5 2.9 1.8 +0.010 +0.014 -0.042 -0.016 99.868 99.998 * Add the several segments 7) The total measured length of the line should now be cor- rected for standard, the length of tape being 100 * d, and d is the difference oetween the tape used (nominally 100 feet) and 100 feet of standard. Elem. of Surv. 1 Assignment 31. Page 16, The mean elevation of the ends of the base-line above sea-level should now be determined by running a line of differential levels, from some bench-mark. 8) The reduction to sea- level is made by applying the formula; rj u Correction = _ R in which B = the measured length of the base-line, h = the altitude above sea-level and R = the mean radius of the earth. In Fig. 100 let B = the measured base, h - the altitude above sea-level and R r the earth's mean radius; then B : L :: R * H : R .9 B - L B R + h - R R T h B ' L arcs are proportional to their radii. and since h in the R denominator is very small compared with R, it may be ignored. In conclusion, it is best to take three or more measure- ments of the base-line; as from A to B , from B to A, and again from A to B. The several tape measurements of each segment should be carefully checked, and the several points be properly aliened. Two dynamometers are better than one, placing one at each stretcher ; if one is more delicate than the other, there is some advantage in Elera. of Surv. IB Assignment 31. Page 17. putting the less sensitive at the rear end and the more sensitive at the forward end, as it is only necessary to note the approximate pull at the rear end, keeping the zero of the tape exactly on the marks. Six men can conveniently be employed in base-line measure- ments, two to attend the stretchers; one at each end to observe and mark the tape lengths; one to observe temperature; and the sixth man to handle the transit for giving line, and making off-sets, reading elevations, etc. The man who reads temperatures should act as recorder as also the transit man likewise and they should carefully check all data on entering same. Problem to Accompany Assignment 31. The data for segments 1 and 2 are given on page 15 of this assignment; for the subsequent segments 3 to 8 inclusive the data are as follows : 3 99.8 82 &) 6.8 b) 9.3 4 99.. 9 80 7.3 8.4 5 100.0 79 7.6 7.4 6 100.0 78 7.2 6.3 7 99.9 78 C 4.8 2.* 8 87.6 76 5.7 4.3 Compute the measured length of each segment; the total measured length of base-line. Apply the correction for standard: Tape compared with standard was 99.987 ft. long. The line measured was broken base - first part from station A to 4, where it deflect- ed 2 52', the second part station 4 to end of line. Elem. of Surv. IB. Assignment 31. Page 18. The elevation of end h. aoove sea-level was 125.8 feet; find the mean difference in elevation, and reduce to length at sea-level. The mean value of the earth's radius in feet is taken to be 20,890,600. Log. R = 7.31995 QUIVERS HY OF CALIFORNIA EXTENSION DIVISION Correspondence Course Surveying IB. Elements of Surveying Sv/afford Assignment 52. Topographic Surveying. Foreword. /, general consideration of the purposes and methods of topographical surveying is the subject matter of this assignment; details of the subject v.-ill be treated in the next assignment. (235) Topographical Surveying consists in the obtaining of essential data of territorial extent and relief for the purpose of depicting on proper maps the forms and elevations, descriptive details, and pictorial representations of the physical features of land and water areas. lo do this nany conventional devices, signs, and symbols are employed; these conventions have by long use become niore or less fixed in character, but it is often desirable or expedient to reproduce the symbols in the margin of the map when finished!. These constitute what is known as the legend of the AS the methods .employed include octh territorial extent and relief forns it is necessary to have resort to the various raep.ns for securing both horizontal and vertical control of these representations. The horizontal control is secured by triangula- tion or by traverse; the vertical control by adopting a suitable datum plane to which, through proper bench marks, the elevations may be referred. Elem. of Surv. IB. Assignment 32. Page 2. The purpose and extent of the map or its survey must determine the nature of these controls: A triangulation system of control, for example, may be that of some portion of a geodetic or geological system, or it may consist of a special triangulation system built upon a baseline, assumed and properly located, and suitable to the purpose in hand. A traverse may be run enclosing the territory covered in the survey, or if the region is of great extent several connected traverses may be advantageously employed. A closed traverse is generally adopted, as this permits the usual checks in closure both of the courses and the elevations, the latter being carried forward with the angles and distances as the traverse survey proceeds. An open traverse may be used to advant- age in some cases; especially is this desirable in surveys for roads, railroads, or canals, the usual open traverse forming the framework of the topographical features. As over a large extent of territory the principal features would appear as the mountain ranges, the valleye oetween, and the river courses flowing along the valley ways, so in the smaller areas the ridges and thalwegs and the stream lines consti- tute the most salient forms. Neglecting, then, the slighter sinuosities of the streams of any region, the rivers, creeks, and brooks, furnish at once an outline of the reliefs of that locality. The streams flow in the direction of the valleys and at the lowest levels, while the ridges form the heights along water-partings at the crests of watersheds. . Elem, of Surv. Id. Assignment 32. Page 3. Hence, whether the map desired is of large or small extent, it is important first to determine the water-courses of the region; next, principal elevations of prominent peaks and especially, ele- vations along the intervening ridges which call for location and measurement. These may be considered the skeleton of the map up- on which the nr.ny minor features and smallest details may be joined. (236) Instruments use^d in Topographical Surveying. For this crunch of surveying a larger assortment of in- struments is avails ole and useful than for any other branch. An enumeration of these and their application may profitaoly be noted here. In the priaarj triangulation work a transit of refined qualities or a direction theodolite is used for measuring angles. The usual base-line tepes with the necessary accompanying instru- ments, such es stretchers, dynamometers, and thermometers are em- ployed on the linear measurements of base-lines, traverse lines, etc. Level-rods, stadia rods, and tape-rods are required for the various functions of such instruments. Levels of the several vari- eties - hand-level, dumpy, Wye, and the high grade level known as the "precise", rre used as occasion requires; v/hile the transit with level on telescope is often made to perform the functions of the more specialized engineer's level. For measuring differences of elevation and especially for the determination of the elevation of chief or critical points Elenu of Surv. IB. Assignment. 32. Page 4. the baroneter, either the cistern type or the aneroid, is often suit- ably employed. The aneroid especially is readily available on ac- count of its portability and convenience. Ihe transit and stadia and the stadia in connection with the pl.ne-tnble are valuable as affording a ready means of securing heights -nd distances j and especially on work of details and in taking topography of minor areas these useful means are much and successfully used. For reconnaissance topographical surveys the pocket con- pass and the clinometer ere useful instruments, and on account of their lightness and portaoility are especially in favor for such vjork. The Brunton Pocket Transit, used nuch in mine surveying is here favorably mentioned as a comprehensive substitute for compass, level, and clinometer; it may be used as a hand instrument and in the better types is supplied with a "ball and socket" attachment fitting it for use v;ith a Jacob-staff. The odometer or the more modern cyclometer attached to automobiles is in use where distances traveled upon roads and high- ways by vehicle are desired. Also the pedometer or a "pace-tally" is convenient in keeping count in pacing work, which is much used in taking topography. Ihe plane-table calls for special mention, as, in its Various grades from the most complete to the simplest form known as the traverse plane-table, it offers a most ready and convenient means of traversing, intersection, radiation, and resection methods of Elezn. of Surv. IB Assignment 32. Page 5. mapping directly in the field; in solution of the three-point problem (an application of the method of resection; the plane-table has no superior for rapid and efficient work. The description and use of the plane-taole in surveying will be discussed in Assignments 34 and 35. Thus it is seen that use for a large range of instruments is found iri topographic surveying in its various stages; and an ac- quaintance with these instruments and their uses constitutes a large part of the topographers duties. (237) Transit lines when measured by tape are measured either by leveling the tape or, where the slope- is sensibly uniform, the slope angle may be taken and the horizontal distance computed by use of the formula H. D. = L coso( where H D. is the horizontal distance, L = the measured line on slope and o<^ = the angle of elevation (or depression). Or resort may be had to the formula H. D. - L (1 - versoO as the versed sine of small angles is usually a number of few fig- ures and the slide-rule may be used to advantage. For example, a line measured on slope 1S3.70 ft. ; the angle of slope o^ was 3 C 52' (the cosine of which is 0.99772, and hence the versed sine = 1 - cos = 0.00228); by slide-rule 198.7 x 0.00228 = 0.44; then 198.7- 0.44 = 198.26 Elem. of Surv. IB. Assignment 32. Page 6. Angles may be taken by deflection, bearing, or azimuth, the azimuth method being generally preferred. Often an angle may be taken easily by reading an interior angle when the adjacent points on ooth forward and back sights are conveniently marked. If the azimuth record is required the reduction should be made at once and checked in the field. (239) It is usual to mark transit points on map by a small circle about ^. inch radius enclosing the point, a triangle indi- cates a tr iangulation station and a square is used to mark a stadia station; thus transit point, /\ triangulation station, r I stadia station. The transit point may Decome either a triangu- lation station or a stadia station, in which case one sign may be superimposed upon the other as A*Aor IQ1 . Other conventional topographic signs are ; Dwelling . M _, :" A k & . s sl Barn PXJ ; Ruins i ! ; Church LL___J ; House Fence -, Public Road ^^Z/ > Railroad |- | - 1 --| * I -I (for small scale); Railroad i~ ( - I = I =1 (large scale); path or Trail ,'*-. .---"* ; etc. Besides these there are conventions for cover of various sorts, such as meadows or graesy plains, pine forests, deciduous forests, sand dunes, rocky formations, culti- vated fields (shewing crops of corn, small grain, oeans, etc.), swamp lands, and groves and orchards. All these are usually included in a marginal list desig- nated as "legend" and a map where these or other arbitrary conven- Elem, of 3urv. IB. Assignment 32. Page 7. tions are employed is not complete without a legend. (239) Sines a topographic map is the delineation of the natur- al and artificial features of any locality upon a plane surface by means of the foregoing conventional signs , it follows that a cor- rect representation should be true to facts by a corresponding faithful use of proper conventions. Every point of the map corres- ponds to a definite determined geographic position in accordance with some definite method adopted for showing the speroid surface of the earth on i plane; this method is called the "projection". The representation being in miniature (usually a very diminutive scale) , the distance between any two points on the map is a certain proportional fraction of the distance between the relative points in nature. This ratio constitutes the "scale". The points, besides being represented in projection up- on a horizontal plane, have their elevations relative to a level surface indicated in some conventional way; the usual conventions are contour lines, depicting points of equal elevation at regular horizontal elevations, or hachures consisting of hatched lines of varying depth of shade or interval. The level surface to which the elevations are referred is called the datum plane, and this with its system of determined elevations or bench marks constitutes the vertical control of the survey. The representation of the variations in the vertical ele- ment with reference to the datum plane is called the "relief", and should fairly represent the modeling of the country. Elem. of Surv. IB- Assignment 32. Page 8. Since all topographic surveys are based upon a system of triangulation or a carefully prepared traverse, a sufficient number of points, whose geographical positions have been determined by either one or the other, or ooth of the aoove methods, from the frame work for controlling the less accurate location of the many details. These points should oe properly distriouted over the area covered in the survey, and constitute the horizontal control. The determination of these three elements, the scale, the vertical control, and the horizontal control, is fundamental and no survey or its map is complete without it. f.s before stated, the purpose of a topographic survey is primarily the securing of the necessary data for the real pur- pose, the finished map. Hence the observations made in the field should be full and exact enough to secure this final purpose. It is futile to exceed in number or accuracy of data the requirement of the map, but the collection of data must be carried on in an intelligent manner, guided by a more or less complete knowledge of the related sciences and arts employed. (240) First of all the topographer must be a man of observing habits, endowed with an imagination that will enable him to visual- ize the relief features of the country he attempts to picture. He must be skilled in use of the pencil and trained to sketch the var- ious relief forms directly in the field as well as to supply some lines and details from numerical data and descriptive notes, in Elera. of Surv. IB. Assignment 32. Page 9 addition to hia field sketches. Moreover, a topographer should know from a study of physical features the causes that have brought about the present outlines of relief which he may be called upon to delineate. He should have a knowledge of the forces which have caused the vast and the recent geological changes, such as the up- lifts and subsidences, the volcanic and erosional forces, the action of heat and frost, the wearing caused by ice floes and water; ftnd changes again the/effected by erosion or corroding, by the transporting power of streams, and the action of tides and currents on the pro- duction of land forms. Not all persons can make good topographers; indeed but few of the many ever qualify for good work beyond the crudest stages of skill and knowledge - for this work is artistic in its nature and hence exacts from him who would successfully follow it, much patience and intelligent practical experience. Elem. of Surv. Assignment 32. Page 10. QUESTIONS TO ACCOMPANY ASSIGNMENT 32. 1. After reading Assignment 23, state the several characteristics of contours; also get from a dictionary the meaning, spelling and correct pronunciation of the v;ord "contour". 2. Define scale, horizontal control, vertical control, datum, thalweg. 3. What, in your estimation, are the essential qualifications of a good topographer? What are some other desirable qualifica- tions? References : Breed & Hosmer Johnson Raymond Tracy Vol. I. pp. 30G - 320. Chap. VIII. Chap. IX. Chap. XXVII. UNIVERSITY OF CALIFORNIA EXTENSION DIVISION Surveying-IB Mr. Svrafford Corrsspondence Courses Elements of Surveying Assignment 33 Solution of Problems Computations and Results To be returned to student after submitting Assignment 33. The actual computations of H.D. and V.H. for course A - B are here given. In all other courses and for Side Shots the results only are shown. Course H.D. ,y.a. Elevations (Adj) A B 385.3 +47.76 (47.79) B 247.73 B C 395.9 -26.11 (26.07) C 221.72 C--D 487.6 +38.56 (38.69) D 260.41 D~ 2 630.3 -56. 9S (56.95) E 203.46 E--A 598.4 - 3.4-5 (3.46) A 200.00 +86.42 -86.55 Diff - 0.13 Computations log (98.5) = 1.99344 " S (3.96) = 0.59770 " cos OC(74') = 9,99669 n n n (704!) = 9.99539 384.2 = 2.58452 1.1 385.3 = H.D. A B log 1 (98.5) = 1.99344 " S (3.96) = 0.59770 " cos oC(74 1 ) = 9.99669 " sin f 1 minute of accuracy and lines to 1/100 ft. in accuracy and measurement secured in mapping to a scale of 1 inch equal twenty feet. References ; Breed & Hosraer, Vol II. pp. 191-203 Johnson, pp 117- 123 Raymond, pp 268 - 275 Tracy, pp 318 - 325 UNIVERSITY OF CALIFORNIA. EXTENSION DIVISION Correspondence Courses Survey inr IB Assignment 35, Mr. Stafford PLA.NE-TABIE - THREE - POINT PROBLEM, Etc.. Foreword. This Assignment trill deal with the further use of the plane-table, with a special application in the solution of the Three-point Problem and the Two-point Problem.. (250) The Three-point Problem: A clear statement of the problem is a follows; Given:- the location in the field of three points and their plotted positions on the map and the station in the field of the observer, Required;- to plot this position of the observer upon the map. (The three given points may or nay not be inaccessible )- Suppose that we have the three given points in the field, B Fig. 104 Surveying IB. Assignment 35. Page 2, A, a light-house; B, a church spire; C, a head-land; and D.- which may ba assumed as the location of an observer, as a boat at pea, or any location from which all points A, B, and C are visible but inaccessible. In the problen the distances AB< B C, AC are knoiTn; it is required tc find the distances A D., B D, and CD. To do this it ".rill be necessary to have the angles A D B and B D C; these nf.y be had by taking the bearings of the several lines from D, or their aziauths, or by sinply neasur:'.ng the interior angles by one of several methods. Aboard a boat at sea the angles nay be tal-cen by se::ta;it - usually two simultaneous observations being made, if this is possible. Or, on land a transit or even a corapass may be -used and the requisite angles or bearings be taken. Fig* 105 Surveying IB Assignment 35. page 3. If the angle EGA be constructed equal to angle d (= EDA) and angle EAC 'be constructed equal to d ! (=EDC), the lines AE and CE will intersect at some point E which is on a diagonal of an inscribed quadrilateral AECD.. AC being also a diagonal of the sane quadrilateral; for EGA = ADE (both measured by the same arc AE), and ACS = d 1 , (both measured by the arc EC). It is now possible to locate D since it is upon the circle through EC and also on the prolongation of EB, i-e. at D, which is the intersection of the diagonal with the circumference. Hence the problem is graphically solved. The problem nay also be solved trigonometrically, but the plane-table method is essentially a graphic method and the con- struction given above is necessary only for the purpose of ex- plaining the principles of the plane-table method. (251) Methods o Solution o Three-point Problem; There are several methods of plane-table solution of the three-point problem of which we shall explain four: 1. The Protractor method, 2. Bessel's method, 3. Triangle of Error method, 4. Llano's method. 1. Protractor Method : Set up the plane-table at the observer's station, and center a three-arm protractor on the point d (the point upon the table directly over the observer's station, D). Direct the arms of the protractor to A, B, and C respectively, setting off the Surveying 13- Assignment 35. Page 4. angles ABB, EDO- The points a, b, c, are then plotted up- on the map in the relative positions of A, B, C in the field with distances according to the scale 6f the map. Orient the board approximately by estimation. Place the protractor so that the three arms shall pass through the points a., b_, in their re- spective positions, sighting through a on A; check by sighting through "b on 3 and also through c on C. To do this may re- quire a slight adjusting of the board, for the approximate orientation only partially accomplished the desired position; when the three arras pass through the three points a, b, and c and sight A, B, C, the protractor center must be at d, the plotted position of D. This is known as the mechanical method of solving three- point problem. Instead of a three-ana protractor, a piece of tracing paper or tracing linen may be used. Draw lines upon the tracing paper by means of the alidade by first assuming a point on ths tracing paper and then intersecting upon A, B, C; then secure the proper orientation by placing the tracing so that the points aA, bB, cC are in line. A point may now be pricked upon the map through the assumed point on the tracing. For rapid approximate work, this is a useful method, but should not be re- lied upon for great accuracy. 2. Bessel's Method. This method consists of constructing the angle formed by the lines from D (the position of the observer) to A, B , and C, drawing these directly in the required position upon the nap and '" : -'-' -. ' " '-.' .: . . ".,, " ...' ..'... :...- ....... ......- .- -'i t ., . .-. . . Surveying IB, Assignment, 35, page 5. thus determining the fourth vertex of the quadrilateral. A study of the following f5,gures will make this clear; the three figures L, 11, U, represent the plane -tab la in the three steps of the solution. B N A V A B A \ 4 *. H. Fig. 106 Surveying 13 Assignment 35. Page 6. Set up the plane-table at D; place the alidade upon ca and sight A through a "by turning the table; clainp the board and with alidade centered on C, sight B and draw the line from c toward B of indefinite length; this step is shown in Figure 106 at L. Again place the alidade upon ac and sight C by turn- ing the table; clanp the board and with alidade centered on a, sight B and draw the line fron a toward B to intersect the previous line so drawn at e in the figure; this step is shown in Figure 106 at L Now draw a line through b and e, place alidade on this line and sight B; this orients the nap- Re- sect fron A through a, or fron C through c; both resections should intersect in a coupon point on be prolonged: this point is d, the point upon the nap representing D, the position of the observer, A circle through aec vrill also pass through d; i.e. aecd are the vertices of the inscribed quadrilateral discussed above. (252) The Triangle of Error Method. This is also known as Lehnann ! s aethod, as an exhaustive in- vestigation and discussion of this aethod has been prepared by Prof. H. Lehnann of the U.S. G-eological Survey, in which it is extensively used with gratifying results. By its use the observer's position nay be quickly and accurately mapped, and its use is readily acquired by the trained plane-tatle topographer. The solution by this nethod (Lehnann 1 s) is a simple application of resection from, the signals in the field through the plotted positions of the same points upon the nap. As in the Surveying 1 3, Assignment 35 Page 7. other methods, the solution is practically secured T .vhen the nap has "been correctly oriented. You should always bear in mind, therefore , the necessity for correct orientation and the means of securing this. Figure 107 shows the plane -tat le set for determining the pos i- tion of observer and plotting it upon the nap. The resection has resulted in pro- ducing the small shaded triangle at Fig. 107 t; but if the map had been correctly oriented, i.e. had ac been parallel to AC (fron which it would follow that ab was parallel to AB and that be was parallel to BC), then by the laws of geometry the lines through the vertices of the two tri- angles ABC and abc oust intersect in a point; the existence of a triangle instead cf a point is evidence that correct orienta- tion has not been secured. Hence, to secure correct orientation: turn the board in azimuth in that direction which will tend to dimish the triangle t, (called triangle of error) until the resection lines have a Surveying 1 B. Assignment 35. Page 8. common point of intersection. The position of the table in this illustration is such that the board should be turned counter clock-wise in order to accomplish this result; in other words the point d (plotted position of the observer) trill be found to the right of the triangle in sone such point as d in the figure, Yov. can readily verify the fact that if the point d were to fall beyond the limits of the nap that this method or any method fails. Again if a circle is drawn through the signals in the field A, B, and C, called the "great triangle", any sta- tion of the observer upon this circle in the field becomes inde- terminate - i.e. with the observer's station at any point on this circle the resection lines will intersect in a point as the quadrilateral formed with A, B , C, and D as vertices would constitute an inscribed quadrilateral. It may be seen that in Figure 107 , the point D is necessarily outside of the "great circle", as the circle passing through points A, B, and C is called. The solution is said to be strong when the observer is in certain positions; these are well outside of (or within) the great circle, or when the observer is within the great triangle which is the triangle formed by the three signals. Also the solution is strong when the observer's station is nearer to B (the aiddle signal) than to A or C. Figure 107 , therefore, showing a specially strong solution. Surveying 1 B. Assignment 35. page 9, Since two or three attenpts to determine jl may be made "by trial, this method is kno\vn as "the triangle of error method' 1 . The student will find an admirable treatise on the plane-table by D.. B. Wainwright, published by the U-S- Government, Bureau of Publications, Washington, DC Llano ; s Method . This method is essentially geometric and appeals to many plane-table topographers as a direct and simple solution, con- suming little time and giving results adequately exact for most pur pose s, C. Fig. 108 The plotted positions of a, b, c are drawn upon the map. Bisect ab and draw a perpendicular; place alidade on this perpendicular Surveying IB- Assignment 35. Page 10. "bisector; turn the "board and sight B; resect from A through a and where this line cuts the perpendicular mark e; dravT the perpendicular bisector of "be; place alidade on this line and turn the board sighting B; resect from C through c anc! mark f; with and f as centers and ea and fc as radii draw arcs intersecting each other; the intersection of these arcs is d, the plotted position of the observer. Verify this by centering alidade on _d and sighting A, B, C. NOTE:- The position of d may also be found by drawing a line from d perpendicular to ef ; this point is as far froia ef as the point b is. This method depends for accuracy upon the distance bet\?een e and f ; when this is small, the determination is inacu- rate ; if the tv/o points e and f coincide the problem is in- determinate. The choice of methods is governed by the conditions in each case and by the means at hand for constructing the lines in the figures. Practice and the consequent skill acquired by the to- pographer -will generally enable him to choose a method suited to the case in hand. All methods fail vhen the point A falls off the map. the only remedy is for the observer to change his position to a near- er station, or in any case one more favorable to the proper and accurate solution. (253) The Two-point Problem. The observer's position may be determined when only two t : - - ' ' : /:: ' r Surveying 1 B < Assignment 35. Page 11. points in the field instead of three are given. This is known as the two-point problem, and its solution rests upon the assump- tion .of a third point located "by the observer. This usually is accomplished by setting up a line (of the character of a base- line) by locating two points for observation, one of which is the point sought. It is evident that an understanding of this problem is quite essential to the efficient topographer, since cases arise when it is necessary to use it. A statement of the problem is as f ollCWS ; Given - the location in the field of two points and their plotted positions on the map and the station in the field of the observer. Required - to plot the position of the observer upon the map. Referring to Figure 109, in which A and B are the points in the field and T is the plane-table on which the points a and b have been plotted to scale and the Fig. 109 map supposedly oriented. Assume _d, and intersect on A and B Surveying 1 B. Assignment 35. Page 12. through a and b respectively. Then move table to C and intersect on A and B, through a and b. The intersections at a' and b ' with c and d will form a quadrilateral of the proper form "but incorrectly placed. Hence turn the board through the angle dcd ' and resect again for d (or c. if desired). If a point can be occupied in line with the two given points A and B, the table may be set up at this point, which we may call P (on nap p), and the board oriented by placing the alidade on pba and sighting BA in line; then direct the alidade toward some point Q in the field and draw a line pq; finally occupy Q, backsight on P with alidade on : qp , thus orienting the board. The point q is novr located by resecting from A and B through a and b. This is a quick and satisfactory method practiced by topographers under the conditions indicated above. (254) You should understand that in most plane-table problems the whole board is regarded as a point, and, when compared with the large areas over which such problems extend, it is accurate enough to regard it thus. It is well to realize too the similar- ity between the plane-table parts and those of the transit. The alidade corresponds with the alidade plate of the transit; the board, with its movement in azimuth, corresponds with the lower plate of the transit, etc. Follow out this comparison and note the completeness of this agreement or similarity. Realize, too, that while the transit is an instrument of refinement and precision, the plane-table is often the more economical and sufficiently trustworthy as to results obtained by its use. Surveying IB. Assignment 35. Page 13. QUESTIONS ON THE PIAIE-IABIE 1. State the several adjustments of the plane-table and compare them with their corresponding adjustments in the transit. 2. Is it necessary that the line of sight of the alidade telescope alidade an<3 A ruler's edge lie in the same vertical plane? Give reasons for your replies. 3. Compare the four methods of plane-table surveying with equiva- lent transit methods for the same purposes. REFERENCES : Breed and Hosmer Vol. II, pp. 191 - 222. Tracy 318 to 336 Johnson - Solutions of Three-Point Problem p. 278 et seq. Raymond - Solutions of Three-Point Problem p. 291 et seq. . : UNIVERSITY OF CALIFORNIA EXTENSION DIVISION Correspondence Courses Survey ing- IB Elements of Surveying Mr. Swafford Assignment 56 MINE SURVEYING Foreword: A brief treatment of the subject of Mine Surveying, including the instruments and the -methods employed and some of the problems peculiar thereto^ will constitute the scope of this assigrrnent . The terminology and nomenclature used by miners and others connected rith raining need not be set forth at length in this con- nection; but a few terc.s and names are here defined in order to avoid unnecessaty details and circumlocution in the introductory part of the subject* The essential difference between the surveying subject already dealt vrith in this course and mine surveying lies in the fact that in addition to the usual problems we have those relating to under- ground areas, shafts, tunnels, and so forth. Many of the methods and the instruments used in the work peculiar to the usual surface surveys are found unsuited or wholly inadequate to the proper or convenient accomplishment of underground work; while many of the problems of mine surveying differ materially in their nature from those of general surveying as already treated. (255) A Few Terms Defined : A ^haft is an opening from the surface downward into a mine; it may be vertical or more or less inclined, and is not always straight. This is used for entrance of workmen ''' '< ---'- ' .: .*; ' ; .-.- ' '-'- - ' '' ' .. . :. . '. v - ' -": "'. .. ' .:. - .' Surveying- IB. Elements of Surveying. Assignment 36, page 2. and others into the mine and for introducing machinery and for re- moval of ore or ether material. For these purposes a bucket or cage operated by cables and hoisting engine is used. A tunnel is an opening into a mine and usually extends hori- zontally with two openings; i.e., it passes through the hill or mountain into which it penetrates, although the term is loosely used to indicate a more or less commodious ingress into the mine for purposes similar to those of a shaft. An adit is still another form of opening from the surface, usually at the general level of the diggings and hence is often called an ad it -level. Unlike the tunnel the adit only opens into the mine by one entrance and has no other outlet. On hillside diggings the adit or tunnel is the more logical construction of entrance, it is generally more easily and cheaply built and permits of operations that require a minimum of lifting. The terms floor, roof, and wall are about synonymous with these terms when applied to the corresponding parts of houses and nay be understood without special definitions. Collar is the term used to indicate the timbers around the opening of the shaft. At the entrance of a tunnel or adit the name frame and timbering is sufficiently descriptive. Levels are the series of horizontal wor kings at successive elevations or depths. A connection is a short passage connecting two or more parts of the workings on the same or different levels. ...'". i' '.: .! -I'..'.-:"".- ,' ..;'-." ..-'.. .-..-. " "' ' " ' ' ' ' ' ' ' ' ' '' ' ; -;' ' '' " " ' ' . " ' .. ' ' . . : .. . . . . ; . * : ' .- . . - ' . ' ' ' . " . ' ' ' ; ' . ':* '' ' '' "' '' '- . ' - -' ' - "'.'- emT ''-TO' ' .- . . "-'.':'.'' , ' ' : : ; '. ,. ' ' : Surveying-IB. Elements of Surveying. Assignment 36, page 3. A winze is a connection of the nature of an underground shaft that, while it may connect trro or more levels, has no exierior opening. is A vein is a stratum of ore; usually this/dnibedded in rock of a different sort Irom the ore itself, and the workings are usually directed along the course of the vein. Dip and strils are terms used to define the course of the vein; dip means the angle of inclination of the rock plane to the horison; strilae is the bearing (or "cross-country" direction) of this rock plane in a direction e.t right angles to the dip. A horizontal passage following the vein is called a dr if t . (256) Liine surveying is from its nature divided into two branches differing more or less widely, in that one has to deal with the under ground work, while the other deals with surface measures. Traverses are run on the surface and underground to locate and de- fine the property lines above and belo" ground, to give bearing and distances of tunnels, shafts, connections, drifts, etc. Depths, slopes, and azimuths are determined with the greatest accuracy possible, and these must be connected with the surface survey by lines measured in angle and distance with careful precision. To accomplish this a meridian (true or magnetic) or a line of known azimuth is carried fron the surface into the underground works. The Eiethoda for doing this will be given further or. in this assignment. . . . - -..--. . - .. . ... . .:-;;.:. '"''. - ' ' '- - ' ' ' . - ' .'-.;.. : '.... '.;.':' ' '" -' . .....: .....;:.:,-.: - . ; - ' -- : " '- - ' "' : ' " ' " . '- - . : - -: ,.:-/ : .... .-..:.- : ; '. ' ; . .' . ... - .... . ...:./ -s .':-. . ':;'.' ... -...; -,-.. . '. - :; - - ' . :::.. - ' - ' ; - ... . ; . : Surveying- J.3 Elements of Surveying. Assignment 36, page A In the IO*.T underground passages in mines we meet conditions that differ greatly from those found on the surface. First of all the light is much fainter , often amounting to total darkness, ex- cept for the artificial illumination which is mostly poor and al- v;ays insufficient for surveying purposes. Therefore, lights are used for signals and suitable lanterns for reading rods, gradu- ated circles, and for record ng notes and reading maps, records, etc. Since the passages have low roofs, often only a few feet in height, it is necessary that rods, tripods, and other instruments should be short, or, as in case of tripods, 6f the extensible pat- tern. In some cases it is convenient to mount the transit head upon a bracket screwed into the wall or to place it upon a tr ivet , a sort of very low support having three feet, hence the name. The presence of tracks upon rAiich ars are run for trans- porting the ore, or the travel of men and animals over the floors of the passages, renders the setting of points thereon impractica- ble in most cases; hence hubs, points, and even monuments are placed if possible in the roofs or upon the walls of rooms and pas- sages. This necessitates the suspending of plumb-lines, not from the transit, for example, but above the transit which is set up underneath, the point of the plumb bob centered over a point coun- tersunk in the telescope axle. Plummet lamps or candles may be used for signals; or a screen of paper or cardboard makes a good mark when illuminated ' ' ' " ' ' *'- ; ' 3+ -' ...- ::;. V . --.... ""' ; " ' i: - ' ""-: "':.;::/.;, :--.. .. . .- : ' " ' '"' T ''":' " ':' -. -..-.- V - ; .. . > " ' ; '". ' : ' '- '"' '' ; " : - -.-/ i- : :..-. '' ' " '' ' T "^ '" : ' -- ..- -- - $?..; .... . 7 - '' -' " ''-^ - : ;> >-; ;-.-. - - ; --' : - ' ' & '::. .; ; ; .... :-., ., . . : . ; . ''-' "- .- ' ~'- : '"'- ' '"' '" --,.:/. :...:- . --' V-'-' V-.- ;: .J-s_- : * " ' - ' " ' "'.' '-... -..- ' '.- "' "" ' i ~ ' '"' r ' "- "" :. ?;.*. : . ... . - - .' '".,'. "..- .;..; . . * - c .1 ' ' '''' ": - : - .' ..-.. .- ' ' - '': -',.'. ' " 7 '" ' '"" : ' : ' : ' : - - .. '' . ' ": ' :- - ^: . ,; .-V -. ._, .. _ . ..../. ' " "' "" ' " ' ' ''"' : ' ':.- ..... ^ . ;... " "" : - --.:=. ... " . : ." - ( ' . , ': ;,;..' .. ... ' " ' ' ' '" " : "' ; - ' ' ..;'.;.;. .---. --.. . - - . . ' '' ' - . : , V>-;U-i-l';i ;.. Surveying-IB. Elements of Surveying. Assignment 36, page 5. by a suitable light. The transit should be fitted with a ting shade, so that light from a torch or lamp may be reflected dovm the tube for illuminating the cress hairs in the tsle scope. (e of a substan- tial but light pattern. It should have a large objective arid a low magnifying power, as sights are usually short and much light rather than magnification is preferable. Either the erecting or the inverting type may be used, the choice depending upon the user, but the inverting type possesses many advantages. A full vertical circle is essential as the instrument is frequently plunged. The vertical circle should be ivell protec- ted by a cover guard, hence double verniers are convenient. As it is often required to sight vertically (or at great inclina- tion) either upward or downward, the transit should have either a top telescope or a side telescope to enable the transitman to look past the plates, vAiich is impossible with the ordinary trans- it in case of greatly inclined sights. The instrument should al- so be fitted with a prismatic eye piece for use with large verti- cal angles. (258) The top telescope above mentioned consists of an auxiliary telescope attached over the top of the main telescope at a short distance from the main telescope equal to the radius of the hori- zontal base plate and a little more to permit of sighting past the plate. The auxiliary is attached best by two short columns connecting the two telescopes, but several forms of attachment are ' '.. . 1 - '''' " . ,: ./... ;- .-.;. - ... ": - * .' ./ v . . -' ' " : - 'wo'S- J- ' - '; -' 7 " .::.v..; ' : ' ; '-" ' - : -----' '- tg . - ., ;>'. -_-; ..:: j- . ; - '" : : " " "' ; - v '"'--- Surveying-IB. Elements of Surveying. Assignment 36, page 6 made* The side telescope is also an auxiliary telescope fastened to the side of the main telescope but usually in the case of this latter auxiliary it is screwed to the axle of the main telescope and rigidly fixed thereto, so that it may always turn with the main telescope, both remaining in the same plane with the horizontal axis It is distant enough to clear the base plate of any project- ing screw, and is properly counter -balanced by a weight attached to the axle at the opposite trunion. (259) Measuring angles when sighting with the top telescope may be understood by the following consideration: The top telescope and the main telescope are fixed in the same plane with the vertical axis of the instrument, the lines of sight of the two telescopes are parallel, and they are at a defi- nite knovm distance apart, called the eccentricity; hence, all hori- zontal angles measured by use of either the auxiliary or main tele- scopes are the true horizontal angles; vertical angles are differ- ent in the two cases, the angle of error in reading with the top telescope being equal to the angle whose sine is the distance of a point from the instrument (horizontal axis) divided into the ec- centricity of the telescope. . ' --..- :/, - *4 '--<': -H-- - :::< ' :. ' ' ... _ ..... -.-!-, .>;;..;-. "5 ".': .-.' <-i'%> J - ---- -------- -'' . ': :':.: l.j-^*I*- ! : -' v " - '' ;/ ' : : fs i>*** s ' V"' ''"" :: - - r,-.:- ., . - . ... . .-_ ... ;.:T u;iWV U : f ^: .:; '' ' " ' ; .-r.u-v: fa W.*/ ' *!>* t r -* - . ' ' Suvveying-lB. Flemeats of Purveying. Assignment 36, page 7. Fig. 110 Fig j.10 shows a transit set upon an eminence from which it is de- sired to measure the a.ng?.e of de- pression by means cf the top tele- scope, and to compute the height N (N) and the horizontal distance OP. Sighting the point P with the top telescope, read the angle O\ on the vertical circle; measure the slope distance HP. But this gives a false angle of depression inas- much as the top telescope must be depressed more than the main tele- scope in order to sight P, and the angle LRP is the true angle of de- pression which is greater than IMP; hence LHP - IMP is a correction angle to be subtracted from DRP. This angle has for sine, E (the eccentricity) divided by MP (the distance from the horizontal axis to point P). Now angle MPO is equal to the true angle of depres- sion and OP = LAP. cos MPO and OM = MPsin MPO. Tc facilitate computations a table of values for sin (p-c<) at varying distances MP may be compiled. The eccentricity E being a constant, the sine of the correction angle varies inversely as the slope distance MP. """ ' " '--'" ; ' '' " '"' ' ' " " ;.. .'.-. -. .: -.-: . . .."-. '"--" -' - ' ' ; .'-... ::- -,, -.... - -.- .. ..:. ;-.; ': .,.-- :: vl . i '' -"-"' -' : - :X ' S - -' - : ? ; . - : - ...; :.:. . .\ : ":-- M: .. .. - .' . : - ;/ '' ' r* i r '- : - : ' : ~''* -'-- " -'-- ;-... ;.; ' - ,/ : ; : : f |^ f-j ." ' | ;;' '. .: --. : : .. : :: . - -.. j, . . . .... . ..,..,.% -, '- ' - Surveying-IB. Elements of Surveying. Assignment 36, page 8. (260) The following considerations will make clear the use of the side telescope in measuring angles. The side telescope and the main telescope are fixed in the saiae plane with the horizontal axis of the instrument, the lines of sight of the two telescopes are parallel in this plane, and are at a definite known distance apart (the eccentricity in this case); hence, all vertical angles measured by either the main telescope or the side telescope are true vertical angles; but horizontal angles read by the side telescope are false angles of azimuth and must be corrected as follows: /V Fig. Ill shorvs how azimuth may be corrected when horizontal readings are made with pointings of the side telescope. Sightings are made on B and F with the auxiliary tele- scope, but the angle required is BCF; as the angle turned in read- ing with the auxiliary is BXM it is plain that this value is in- creased by SBC (=?) and diminished by FXIvi ( = ot ); hence the true angle at vertex A is FXM - p +CC. Here (3 = angle whose tangent is Ho. /// Surveying- IB Elements of Surveying. Assignment 36, page 9. E * CB, andoC = angle whose tangent is E * CF, E being the eccen- tricity. When CE = CF no correction is required. (261) Solar observations for meridian are best made by direct observation upon the sun, although the usual forms of solar at- tachment may be used. But the combined solar and top telescope attachment is not recommended. The compass in mine surveying is of little use as an instru- ment for taking azimuths or bearings. The presence of iron ores, machinery, rails, and lighting and pov;er lines for electric energy rail practically preclude the use of the compass in this sort of -::orlc. (262 ) REIAJIOH OF SURFACE MD_ UNDERGROUND SURVEYS For the purpose of connecting the surface and underground areas, it is necessary that a common meridian be carried from sur- face into the v/or kings. This may be accomplished by one of several means. The meridian, or a line of known azimuth, may be carried directly into the underground area through a tunnel or an adit; to do this, it is only necessary to prolong a line of the surface into the mine through the adit or tunnel, or to carry a line of conveni- ent bearing out of the tunnel and determine its bearing or azimuth with respect to some known line of the surface survey. The method of carry ihg a meridian into a mine through a shaft will depend upon the nature and size of the opening and pos- sibly too upon the depth of the shaft. Plumb lines may be dropped . ._ . . . . . . . : . ' - - - - ' <- - '. '1 3 h ... ..... . - : ... ...,.., .:.;.::... ..... ... : ..... ' . . ~ ' '- : ' -' '' '* ' .". ~\ ..-" .. -V-Vr .-' ~l .'.-' ; v IVV .. : -' ..... T "I.". /...'.." ".". "- ' '.'...': .'....: *',.." ":; ' ... , .. .. .. ...;... . . . . ._-,_. t . ... . . ... ..; C ,. .., . . .... ... . . ... :..,..-... ,.. . ; .-' . : -. . : .-.-: : --: > : . .:".- ~ . ^ ....... , .. .... ... -. . . . . ,. - - . - - - Surveying-IB. Elementary Surveying. Assignment 56, page 10 from two points upon the collar and the "bearing of the line con- necting them determined; this is practicable when the shaft is verticle, so that the plumb lines hang free of the walls, and the opening is wide enough to allow of a line of sufficient length between the two points. In case there are tvro shafts the latter difficulty is removed; points may be established at the bottom of each shaft by plumbing; connecting line on the surface is then the azimuth 01 the the azimuth of ihe^line joining the t-"o points below. (263) A transit vith auxiliary telescope may te used for the purpose of carrying a meridian into the mine through a diaft, and in case the shaft is inclined, this is the most feasible means. Set up the transit at a convenient place at the mouth of the shaft; backsight on a suitable point on the surface "fith telescope plunged; right the telescope and set a point at the botton of the shaft, if possible, as far as practicable within the mine. Read the angle (deflection, as described below) and calculate the azimuth. Occupy the point below and set off the back azimuth sighting the point at the collar, and from this position lay off a line of convenient length in the mine, determining its azimuth. Too great care cannot be given to the matter, as much depends upon the correctness of the line - all lines of the under prouna workings are connected therewith and a small error may be multi- plied or at least repeated again and again. ' i - ; ->: .- : : -- ' "- .* " '=-- Surveying- IB. Elementary Surveying. Assignment 36, page 11. (264) The distance dorm a shaft is measured by tape; if inclined, by supporting the tape at points along the inclined side; or if the shaft is vertical, by suspending the tape and putting the same under tension by hanging a known weight upon it, If the tape hangs free the tension of the tape will be W-frwl/2, where W is the attached weight, w is the weight of a linear foot of tape, 1 is the length of the tape under tension. The tape may norr be compared with a standard by supporting both tapes throughout their length and ap- plying the tension above determined. Very deep shafts may te Trea- sured by means of a piano wire, which is afterwards compared with standard in like manner . (265) Many measurements of distance are necessary or conveniently made from the head of the transit instead of from a point below the plumb bob and care must be taken to secure the correct measure. Again, since many points are chosen upon the roof or walls of rooms and passages, it is frequently necessary in talcing angles of slope to such points to reckon the "height of point" as veil as the height of instrument. The H-I. may be positive or negative depending; uoon its position with respect to the transit; if the point from whi^h the H-I- is taken is below the transit, it is positive (*); if above the transit, i.e. measured downward, it is negative (-) Likewise, the "height of point" (K.Pt.) is positive when measured from above downward and negative when measured upward. Due regard to these quantities and their signs is of special import in reading angles of slopes- . . - . - " ' - - " "... ..." . . " ":.""' " ' ' ' '. '.'.. -' ; "<:""-..' X- . .----. . -.; ;;;;,_ -- " "T '- B -. - -. * ' -'- : , "" - ....-! ' " - "-T - " :V< - t r-- ; Kfe- .,.' ' Surveying-IB- Elementary Surveying. Assignment 36, page 12. (266) Ir. running traverses at surface and underground they should be connected by starting at a common initial point and, if possible, closing upon a point common to the two traverses* The two traverses thus run will be open traverses but the closing line of one will be identical with the closing line of the other; hence, by computing this closing line for each of the two traverses a reliable check may "be secured. In case the two traverse s A run from a common initial point cannot be closed on the same point, then the closing line of each traverse is comupted and from the data thus secured a closing line between the two is then also computed, but it is evi- dent that no check is possible in this case. Fig. 112 illustrates a surface traverse, A12B and an underground traverse A345B, beginning at A and B closing on B. Here it is required to find the dis- s tance AB. This can T>e done T by finding the missing part, AB and its bearing (See Assignment 18, on supplying missing data). The traverse A12B furnishes the necessary data, as also the traverse A345B; the v-.lue of AB by the trro routes should check. To illustrate any case where the common point of closing is wanting, see Fig. 113. Here the line MN is computed for the surface - v.-*/:. -- - ,.". : " I ' V -.- . ': - .:- ,....- 994', :... -.^ . - :--. 3* ::' .. - ;.."."=:. r- -',:.; ~ -- . : . : ': ~- . : - '_ .-' . - Surveying- 13. Elementary Surveying. Assignment 36, page 13. M "> traverse M12W and this line is then taken as a part of the under- ground (incomplete) trav- erse M3450NM - and the line NO computed by the method of supplying omissions. In this case, however, there is no check upon the work and all lines and bearings must therefore be measured with scrupulous exactness. (267) Mining Claims: The surface extent of mining claims is de- fined by the U. S. Government, and these limits must not be ex- ceeded by claimants, although the limits, less than these, are sometimes regulated by local (State) laws. Coal mine and placet claims are 20 acres in extent, by U. S. Government regulation, and mineral (lode) claims are defined as 1500 feet along the strike of the lode, either in a straight or broken line, and extending 300 feet at right angles to this line en both sides. The end boundaries are required to be parallel to each other. The following diagram, Fig. 114, will further show the form of a mineral claim as defined by U. S. Government laws, for regulating same: . . . '-' ' ' .: - ' -. ; r .-; ' f . ;. Surveying-IB. Elementary Surveying. Assignment 36, page \ ^ V ...-"* y A/ C 1 S The place where the prospecting was conducted is shown at M; from here a line is run N60 10 f E to where it is made to deflect 12 05' R (or its bearing from this point is N 72 15 ' E) thence to C 1 ; the line is also run from M, S 60 10' W to C, the total length of the center line COC 1 is 1500 feet; suppose the bearing of the end line Sit to NW is N 20 00* W, then the other end boundary must also bear N 20 OO 1 W, i.e., the end boundaries must be parallel; the side sboundary lines may be 300 feet (perpendicular distance) on each side of the center line COC 1 . These lines and dimensions may be laid out by the prospec- tor or claimant and the approximate location, and necessary witness marks, such as stakes, bearing trees, etc., described by the clai- mant in filing his claim; an official survey is not required for this purpose. Subsequently before a patent can be secured, it is necessary that an accurate survey shall be made by a qualified Mineral Land " : - ':i---. .- : '.'..: -.. .-.':.-: '* - - --- .--'. ' : " -.- .,' .- .; .:..-. :. - ;. : '- - .- -.-.. . --.-.' :. .;..._ q. .-. . ' ..... -. :. .. Survey ing- IB. Elementary Surveying. Assignment 36, page 15. Surveyor, and ii- is upon his report that a patent is issued. The official surveyor is not rigidly bound by the claimant's survey, but he aims to follow consistently the lines and points previously established when these conform to good practice and do not interfere with the rights of other claimants or established surveys. The survey must show the relation to any existing U. S. Land Survey in the region where situated. This is usually done by giving the true bearings of the boundary lines of the claim, and at least one corner of the claim is tied to a section or quarter section corner where possible. The Deputy Mineral Surveyor is a regularly appointed official +0 inswvc. and is under bond with severe penalties or the faithful performance of his duties. REFERENCES: Breed and Hosmer, Vol. 1, pp. 321-371. Johnson -------- Chap. XI Raymond -------- Chap. XII _,.-*-*-*-*-*- ' ' ; ' : ' .- -.. :', ' '' - ' '' '- ' '-' --?-"- ... - . . ; . ..;...'." Surveying-IB. Elementary Surveying. Assignment '66, page 16. PROBLEMS: 1. A transit v.'ith top-telescope was used to set a point down an inclined shaft; the angle read on the verticle circle was 58 18'. If the slope distance from the horizontal axis was 312.5 feet, and the eccentricity of the auxiliary telescope was 0.33 feet, (a) ^//hat was the correction angle, and what the true angle of depression? (b) what was the horizontal distance? (c) what was the vertical distance? 2. It is required to measure the angle ACB with a transit equipped with a side telescope, set up over the point C; sights were taken on A and B with side telescope and the angle read on the vernier plate was 61 38*> if the distance from C to A was 212.3 feet, and from C to B 187.6 feet, and the eccentricity of the auxiliary telescope was 0.31 feet, 7/hat were the correction angles and the true interior angle ACB? 3. A mining claim is 1500 feet along the center line, and in* eludes 300 feei parallel thereto on either side^ and is furthef described as follows: The place of discovery, M, is 450 feet S 80 GO 1 E of the west boundary of said claim which boundary is made at right angles to the center line of the claim; 600 feet east of the west boundary the center line deflects left 23 15*. Lay out the claim on a scale one inch equals 300 feet, drawing the bounding lines, the center line, and showing the bearing and dis- tance of each line. .. - : '' -~ - - - -' ' - . - : *- - t- ''" - ' - !. -;: ; . ---' ..;/ - -- ;-:.- '" " ' t ~ ! - ' '-'"' "' '*"-"; ''."'.'''-. :'- '" ' ' :,-. ". :.' :..' - . ".. UNIVERSITY OF CALIFORNIA EXTENSION DIVISION Correspondence Courses Surveying-13 Elements of Surveying Mr. Swafford Assignment 37 Hydrographio Surveying Forward: The purpose in this assignment is to isrive somo of the methods in that snecial branch of topographic surveying known as "Hydrographio Surveying". We will not presume to enter upon an elaborate treatment of the subject, but to give rather the problems and methods met v;ith in the ordinary work of surveying. Only those problems which employ the usual tools of the surveyor will be dis- cussed, with perhaps the addition of a few of the simpler but in- despensable instruments required for such work. ( 23G ) Scope _pf the Subje ct : Hydrogra.phic Surveying, although properly classed as topographic sur- veying, is more particularly regarded as a highly specialized branch of the sv.bJTct. In all of its problems it deals with water, water surfaces, depths, streams, etc. Hence any survey that has to do with seas, lakes, harbors, rivers, and other aqueous bodies is classed as hydrographic, and the surveyor is here confronted with^problems calling for treatment peculiar to each case, but concerned always with water, either in quiescent form or flowing. The intimate relation to land topography will generally be apparent. (259) Prpb_lens_o_f the_ ^Subject A fev; of the pr&blems net with in hydrographic surveying are: Surveying-IB. Assignment 3V, page 2. (a) The i.ieasuranent and mapping of the coast lines of lakes, ponds, and seas; (b) the Determination of the forns of sub-aqueo i - i .s areas covered by them; (c) the courses and depth of channels, presence and location of submerged rocks, reefs, and shoals; (d) the rise and fall of tides; (e) and the flow of streams, their slope, nature of channel, change of channel, velocity of flow, and the quantity of discharge and its nature. The scope of the sxi.bject its methods and problems are indeed of prodigious importance and often call for highly specialized treat- ment. They must therefore not be considered of trifling nature sug- gested by this passing view of elementary phases of the work; the student should realize that very often vrorks and structures of irmense importance and involving intricate engineering undertakings are dependent upon the survey conducted in many cases. To enumerate again: The many kinds of engineering ventures depending upon a hydrographic survey are: the improvement of harbors, erection of lighthouses, building of sea-walls, dredging of rivers, buildin-; of bridges, dams, tunnels, drainage of areas, and numerous other works. (270) Survey^ of Shors-line: ".. r hen it is desired to survey the shore-line of a lake or bay, for example, and this for the purpose of determining its boundary or for inclusion upon a map, it is the practice to run a traverse at a convenient distance from the shore-line upon the land, and to then establish the requisite number of points for defining the low-water Surveying-IB. Assignment 37, page 3. shore of the body of water. This traverse is generally of the open type, but even in such a case, though the closing course of the traverse cannot be measured directly, it is often desirable and expedient to secure a closing line by stadia or by triangulation, since such line would afford a valuable check on other elements of the survey. If, as often happens, the coast-survey is an integral part of other topographic features, which have been based upon a triangulation systen, then in turn a suitable net is projected to furnish the re- quired basis of the hydrographic portion; or a special base, in some instances, is carefully established and the triangulation then pro- ceeds from this base by the usual methods to include the hydrographic features sought. If a traverse has been run, off-sets are then taken from des- ignated points on the traverse lines to the points on shore at as frequent intervals as the character of the coast may require. These points may be measured by tape where great accuracy is required and the nature of the ground may permit, or by stadia when a lesser degree of accuracy is allowable or the intervening terrain renders direct measurement inconvenient or impossible; this might be neces- sary on account of intervening marshy stretches, or where precipitous rocks stand in the way of chaining. Triangulation must sometimes be resorted to. The illustration, Fig. 115, will show the combined methods along a coast-line, consisting of a portion of a bay. Here the Surveying-IB, Assignment 37, page 4. E Wearing and Distance by Transit and Stadia Fig. 115 transit points A to Z am located at convenient distance from the shore-line (exaggerated in the figure) and perpendicular off-sets are shown by dashed lines; through the .points thus determined the law- water line SSSSS is drawn. The line AE, shown by a b roken line, is conveniently found by stadia, and its bearing and distance also computed by latitudes and departures, as a check upon other elements of the traverse. The points A, B, C would conveniently be located by triangulation from X and Y, two triangulation stations of the land topography. The bearing and distance of each course of the traverse is found by transit and tape, or by transit and stadia. Besides the careful sketch shown here (drawn approximately to scale), a full set of notes in tabular form is kept and the distances along the transit lines to off-set points as also the lengths of off-sets Surveying-IB, i.ssignruent 37, page 5. are also tabulated. From such data a faithful map of the coast is drawn. (271) Soundings. The measurements of the depth of bodies of water are called soundings and are made at specified points over the water surface for the purpose of determining the nature of the sub-aqueous area. The work is v.sually carried on from a boat by means of poles or lead-lines; if the former, a stiff narrow rod of wood, generally weighted at the lower end, and graduated in feet and tenths is used. In shallow water rods up to 15 feet in length graduated as described may be used to advantage, but longer rods are difficult to handle, and even with the shorter rods much decterity is required, especially in vater in motion from currents of tidal flow. For depths greater than two fathoms (l fathom = 6 feet) lead- lines are employed. These consist of hemp cords, 3/8" to 1/2" size, carrying a lead weight at the lower end, weighing 5 to 20 pounds. The cord is suitably marked by tags at each foot and half-foot interval, after being wet and stretched, so that the shrinkage caused by wetting is in a measure eliminated. As this precaution is only temporary in character, s.nd is likely to be affected by long use, the line and its graduations should be tested from time to time by comparing with a standard length, as a tape or the distance be- tween two fixed points on shore. Leads are of several forms depending upon the purpose; one form has a cup-like cavity at the lower end, which is designed to Survey ing- IB. Assignment 37, page 6. lift a. small quantity of the bottom material, from which the character of the sub-area may be judged- this may be mud, gravel, sandy sludge; or, in the absence of "sample", it would indicate rocky formation. When samples are especially desired, the cup-like cavity is smeared with tallow (called greasing the lead) to which the bed materials will more readily cling. A more elaborate device has a cup suspended at the bottom of the lead by a short stiff bar and a leather cap sliding upon this bar, acts as a cover to prevent the sample from washing out as the lead is raised, As the sounding proceeds it is necessary to locate the point of each sounding. This may be done in various ways. The position of the boat nay be measured "off-shore " by observing the angle by means of the sextant and afterwards computing the distance from known signals on shore. For this purpose it would be necessary to have four men, or more depending upon the kind of boat. Supposing that the boat is propelled by a single oarsman then the crew should con- sist of a leads-man, a recorder, an instrument man; the recorder may also serve the tiller, if the boat is thus equipped. The leads-man is stationed in the bow and the man handling the sextant is immediately behind him, the oarsman occupies the middle seat and the recorder sits astern. Another method requires the reading of angles from stations ashore. This is done by two transit-men with instruments set up at known points that are intervis5.ble and that command a clear view of the water area over which the soundings are to be taken. . This is in Surveying-IB. Assignment 37, page 7, M reality the method of tri:mgulatio:i so common in topographic work generally. Fig. 116 shows points M,N,0,P connected bT r triangulation to a station T, the azimuth and dis- tance of which are carefully determined. AS the soundings A B C are in pro- gress the transits a re set up first at 11 and ^, each sighting first on each other; then, at the instant a sounding is taken both sight upon the leads- man's station simultaneously; the moment of taking the sounding can be signaled from the boat by means of a flag. The transit-men then move to N and 0, ""2. \ and again taking stations and P. This is a very accurate and satis- factory method, but of course involves considerable equipment and two transit-men which, of course, means expense. If the soundings can be taken in range, vrhich is expedient v-hen a profile of the bottom of a stream, or bay or harbor is re- quired, then the method requiring no transit is feasible. This is .one as follows: Two oosts are planted in range with the desired course of sov.ndingc; the posts are erected so that they may always be kept in view from the boa.t from which soundings are taken. It is well to have these signals painted white or some other bright color that is in ocntrast with the back-ground against which: they may be- / viewed from the boat. Fig. 116 Surveying-IB. Assignment 37, page 8. The sounding party in the boat keep these signals in range rowing in as nearly a uniform rate as practicable, and taking soundings at regular time intervals; thus the depths at approximately equal intervals are obtained. The profile may now be plotted in this particular range* If other profiles are desired, the range may be altered accordingly. Problems. I, The following data are those shown in Fig. 115, page 4. Course Bearing Distance A-B N 50 00' E 340.0 ft. B-C N 9000' E 320.0 C-D S 5800' E 240.0 D-E S 000' E 280.0 Required, the bearing and distance of E-A. If A and E are inter- visible, how may the bearing and distance be measured directly? II. Referring to Fig. 116, page 7, the following data are given. TN * 730.0 ft., TO = 680.0 ft., Angle NTO = SS'OO 1 , Angle NOB 2 7500', Angle ONB 2 = 4000'. Required the distances NO, OB , NB . *- UNIVERSITY OP CALIFORNIA EXTENSION DIVISION Correspondence Courses Surveying-13 Elements of Surveying I.Ir. Swafford Assignment 38 Mapping and Office V.[ork - Computing. Foreword'; The work in this .assignment will consist of mapping problems: 1st, A Profile of River Crossing; 2d. three traverses, each by a different method, r.ll to be placed upon one sheet; 3d, A_ Topographic Juap_, in which the data given and worked out in detail ir. Assignment 33 will be voade available. You are specially xirged to make this a particularly profitable set of exercises, and to follow the directions given to the least detail. These exercises will serve as an index of your attainment in the vrork of the course, and will further dis- play your aptitude for carrying out projected work and your mastery of the important details of officev/ork in connection with field data. Consult all references available and such other sources of information on drawing, mapping, and lettering as may be likely tc contribute to your successful completion of the projected maps. In order to prevent the maps from becoming folded or danaged in the mails, send them to the Extension Division in a substantial mailing tube. rl&terial; For the three mapping problems the following material is re- quired which may be purchased from the Xeuffel and Esser Co., Surveying-IB. Assf.gnr.ent 38, page 2. 30 Second St., San Francisco, or from the A Lietz Co., 61 Post St., San Francisco. Approximate cost. 1 yard Plate A Profile Paper 20" wide. $0.25 2 ;-ards Detail Paper 36" wide. .40 1 yard Tracing cloth 36" wide. .45 ^ Total $1.10 This does not allow for enough extra to use in case one sheet is damaged, so proper care should be exercised to prevent mistakes in inking, etc. Mapping Problem 1. Profile. For this problem use 2 ft. of 10 in. Plate A profile paper. Plot the profile of the stream crossing given in the notes below. Horizontal scale 1" = 20* , vertical scale 1" * 4 1 . Place station so that the profile is well spaced on the sheet. Y/rite the elevations on the horizontal lines. At each alternate heavy verti- cal line write the station number just below the lowest horizontal line. Profile of Lorenzo River Crossing. Sta. Elev. Remarks Sta. Elev. Remarks 0+00 158.6 2+10 125.5 +10 157.5 +15 126.3 +15 157.2 +25 129.7 +22 156.8 +30 130.5 +30 155.2 +35 130.6 +4-0 154.2 +40 130.3 +48 152.8 + 50 129.0 +59 150.0 high water 1907 +60 127.4 Surveying-IB. Assignment 38, page 3. Sta. Elev. Remarks Sta, Bl e v. Remarks +75 148.3 +64 126.8 +90 147.0 +76 126.8 +95 147.1 +90 129.0 1+00 146.8 3+00 130.3 +05 145.6 +08 131.0 +10 146.3 +20 131.6 +20 146.0 r35 133.0 +24 145.7 +50 134.8 +32 145.2 +60 135.8 +36 141.2 +75 137.8 +48 140.4 ore sent water level +80 140.4 present v/ater level +70 133.0 +90 148.1 +82 IS0.2 4+00 151.8 +86 128.6 -:-10 153.9 +95 124.9 +20 155.5 2+00 124.8 Plot the profile jLifthtly in pencil (profile paper will stand very little erasing). In plotting each point place the pencil point on the plotted position and drav; tack to the preceding point a free hand straight line. Smooth out this broken line when the profile is inked. Before inking you should check the profile carefully and send it to the Extension Division for the approval 01" your instructor. For inking use a contour pen or a ball pointed pen, making the line about the same width as the heaviest lines of the profile paper. Ink the profile in black. Place the following standard form of" title in the upner middle part of the drawing: University of California Extension Division Plane Surveying Assignment 30 Profile of Lorenzo River Crossing ( Name ) Scales: 1 in. - 20 ft. horizontal 1 in. - 4 ft. vertical Survey ing- IBB. Assignment 58, page. 4. Great care and much taste raust be exercised in lettering, especially in titles. Use guide lines and slope lines and a simple letter of uniform style. Mapping Problem 2. Traversejs. General Instructions. Use detail paper 20" x 21'' with standard border as indicated on page 8 of this assignment. Make the finished sheets 18" x 24" with border 3/4 1 ' inside those dimensions. Considering the north-south line to be parallel to the shorter side of the draw- ing, let the origin of plotting coordinates be at the lower left (3W) corner of the border line. Complete the entire drawing in pencil and return it to the Extension Division for the approval of your instructor. Do not do any inking of the drawing until it has been approved in pencil form. Check it over very carefully before submitting it. There are three traverses to be plotted on the one sheet. The scale for each traverse is l" = 200 1 . Shew directions and length of every course in each trc.verse as in Blueprint Problem 3, Assignment 6, (or Figure 199, Breed and Hosmer). Use standard north point illustrated at the end of this assignment and the form of title already indicated. (C.E. Traverses - Chord, Tangent and Protractor Methods). Place the title in the lower right hand corner of the plate. Use soft pencil for lettering so that erased., mistakes will not shov; by reason of indentation of the paper. Surveying-IE, Assignment ?8, page 5. TRAVERSE "A n Traverse "A" will be plotted by deflection angles, using the tangent method. The deflection angles will be computed from the notes given below. Reference'- Assignment 30, p. 3; Breed & Hosmer; Arts 490-491. Show in this traverse plot a typical construction for a deflection angle less than 45?, and one for a deflection angle between 75 and 90 e . Sta. Deflection Bearing Distance (ft.) pr 6 37' W 932.4 . N16 43 E 500.0 N 2 36 1018.0 . K78 53 E 664.9 . . .325 57 E 900.6 . .321 26 W 468.1 S10 22 E 938.7 . S45 23 W 579.4 , N87 57 W - - - 715.3 Note: Co-ordinates Sta. A (270.0 N, 340.0 E) Check deflection angle and bearing closure. TRAVERSE "B" Traverse "B r! vd.ll be plotted by laying out the bearing angle of each course by the chord method, drawing a north and south con- struction line through each transit point. The bearings are to be computed from the following notes. Reference : Assignment 30, p. 5; Surveying-13. Assignment 38, page 6. Arts. 492-494. Show in this traverse a typical construction for a bearing angle less than 45 and one for a bearing angle between 75 and 90. Sta. Deflection Bearing Distance (ft.) N 26 38' W 1171.3 ( ) 722.9 ( ) 1065.0 ( ) 679.0 ( ) 1015.0 ( ) . . 1180.0 ( ) 955.2 Note: Co-ordinates of Sta. B (430.0 N, 1925,0 E). Check deflection angle closure and bearing closure. TRAVERSE n C" Traverse "C" will be plotted with a 14 1 ' paper protractor, using azimuths from the south point which will be computed from the following notes. Reference : Assignment 30, pp. 6-8; Breed & Hosmer; A r ts. 481-483. gjba. Deflection Azimuth Distance (ft.) C, 73 35' R ' 159 10' 635.0 C 2 32 15 R " ( ) 455.1 C 3 9 15 L ( ) 410.3 B! 41 30' R B i 72 15 R B 3 59 25 L B 4 100 00 R 5 66 6.5 R B S 50 28.5 R B 7 27 10 L B 8 116 15 R B, Surveying-IB- Assignment 38, page 7. Stn. Deflection Azimuth Distance (ft.) C4 61 40' R (: ) 468.0 C g 124 29.5 L ( ) 516.9 Cg 126 4.5 R ( ) 460.2 C ? 32 50- R ( ) 870.8 Cg 91 45 R ( ) 918.1 C Q 29 15 L ( ) 596.0 C, 34 40 R ( ) -450.1 C^ 70 10 R ~ _ ( ) .. 1048.0 C ! Note: Co-ordinates of Sta. C^ (72) .0 N, 3300.0 E) Check deflection and azimuth closure. Happing Problem No. 3. Topographic Mapping Problem Traverse. The field notes 'for the traverse to be used in connection with this problem have already been included in Assign- ment 33. In this assignment you were required to compute horizon- tal distances, differences in elevation, and elevations; and to balance the traverse, adjusting the elevations and distributing any closing difference in proportion to the lengths of the respec- tive sides. Map. Using the traverse commutations indicated plot the traverse by the co-ordinate method, checking by scaling the lengths of the sides, -'rite in pencil on the map the length and bearing of each Survey ing- IB. Assignment ?8, page 8. side. The ruap is to ba plotted on a scale of l" = 60*, contour interval 5 ft., on detail paper of good quality, outside dimensions 16" x 24". Use 3/4" margin and standard border. Assume the true north parallel to the shorter edge of the border, and be careful to select the position of the origin of co-ordinates in such a way as to center the drawing upon the sheet. The title is to be placed in the lower right hand corner of the sheet, and is to be of stan- dard form. The map is to be completed _in pencil only and sub- mitted co the instructor for approval. Do not erase correct con- struction work on the pencil drawing, either before or after approval. Use a 411, or harder pencil for plotting traverse and side-shot points and a softer pencil for drawing contours and for lettering. A tracing in ink upon tracing cloth (using the dull side) is then to be made of the approved map. Tracing and original are taen to be submitted to the instructor, together with computations and tabulations. For details consult references listed below and those given in Assignment 33. References: Breed & Hosmer Vol. 1, Chap., XI and XVII; Vol. II, Chap. XI. Tracy Chap. XI. Raymond Chap. XII, also plates 1-3, 7. Hote: Bear in mind that the originals of all three mapping problems are first to be submitted to the Extension Division in pencil only for the approval of your instructor. They will .then be returned for inking-in and should be re-submitted in final form for the proper grade. Surveying-IB. Assignment 38, page 9 Standard Border and North Point - : / s ^111 UNIVERSITY OF CALIFOKWU EXTENSION DIVISION Correspondence Courses Surveying-IB. Elements of Surveying Mr. Stafford Solutions to_ Assignment _38. pjL:tjjLon.s ?or Iviapping 'roslea 2. >ba. Deflection Bearing Tangents Distarce(ft.) Cotar A^ (G1 20' R) G.5606 0.1524 - - N 6 37' T: 0.1160 932.4 A 2 (2520'R) 0.4314 _ ________ jjfi5 43 ___ _ 500.0 A 4 ---- C763.5'r:) 4 .. --- (7510'R) A g ---- (AG^.O'R) (14C5'L) 0.2509 6 ---- (47'25'R) A ? ---- (3150'L) A 8 ---- (5545'R) I ---- (8120 T R) K 2 38 E - - --- ---- 1018.0 4. 0867 0.244 1 ; 7578 53 I .-.. . ------- S54.9 3.7760 0.254( S25 57 E ------- - ---- 900.6 1.0881 S21 28 W ----------- 463.1 0.6208 S10 22 I] ---------- 938.7 1.4687 0.330S 5A 5 23 W ---------- 579.4 1.0599 U87 57 T .T ------ * --- 715.3 6.5606 0.1524 Note: Co-ordinates Stc.. A (270.0 N, 340.0 E) Check deflection angle and bearing closure. Surveying-IB. Solutions to Assignment 38, page 2, Right 8120' 23 20 76 15 75 10 47 25 55 45 46 40 405 55 Check Larft 14'OS 1 31 50 45 55 360 00 ,405 55 Sta. Deflection Bearing; Chords B-, 41 30' R N 26 38' W 0.4606 B 2 72 15 R I TJ i* 1 ^ *^*7 T? 1 fl *7 7 c\ 9 ' " " ^ W *^;0 O I & J W f O tf B 3 59 25 L ___ _(N 13 48 W) B 4 100 00 R B g 66 6.5 R B g 50 28.5 R B ? - - - 27 10 L (S 4 23 E) 0.0766 Bg 116 15 R (N 68 08 W) 1.8532 En II 26 38 W 0.4606 Pi stance (ft.) Sin oC 2 0. .0 722.9 756.4 0. 0. 0. 0.9381 0.2303 Surveying-IB. Solutions to Assignment 38, page 3. Note: Co-ordinates of Sta. B X (430.0 N, 1925.0 E) Check deflection angle closure and bearing closure, Right Left 41-30 1 5925 l 72 15 27 10 100 00 86 35 G6 06g 360 00 50 2&| 7 446 35 116 15 ^ Check -""^ ***^ 448 35 _Sta_. C - - 73? 35 1 R - - 32 15 R C 5 - - 9 15 L - - 61 40 R Azimuth Pi s tan c e (ft.) Bear ings 15910' 635.0 ft 2C' 50 ! V' (191 25) 455.1 N 11 25 E (182 10) 410.3 N 2 10 E (2-13 50) 438.0 I! 63 50 E C 5 - - 124 29.5 L ~- ........ (US 2Q& 3 6 - - 125 4.5 R i ------- - - - (245 25) Gr, - - 32 50 R i ------ .. --- (278 15) 516.9 N 60 430.2 N 65 25 E 870.8 S 81 45 E Surveying-IB . Solutions to Assignment 33, page 4. -".' Def loction Azimuth Distance ( f t . ) Bearings C 8 91 45 1 R ( I000) 918.1 S 1000 ! Vr C Q 29 15 L _. (340 45) 596.0 S 19 15 E C-, c 34 40 R (15 25) 450.1 S 15 25 W C . 70 10 R - (85 35) 1048.0 S 85 35 W C]_ 15910' Note: Co-ordinates of Sta. C^ (720.0 II, 3300.0 E) Check deflection and azimuth closure. Azimuths Deflections Az q 15S 10' Right V 32 15 R ** "*" " 73 35 C 2 191 25 -.. 32 15 C 3 182 10 12S 04-| 61 40 R 32 50 3 4 243 50 91 45 1 9A 9Q-^- T X^T: cy p Lt c. 119 20-;> 126 04-g- R C 245 25 32 50 R C ? 278 15 70 10 91 45 R Check 370 00 560 C 10 00 ' , 29 15 L 522 59^' C 9 240 45 34 40 R 375 25 360 00 Surveying-13. Solutions to Assignment 38, page 5. Azimuth s , con, C lo 15 25' 70 10 R C 11 85 35 73 35 R C 1 159 10* Check liapping Problem No. 3. For computations of Happing Problem No. 3, see those furnished with Assignment 33, Included with Assignment 38 are the following maps: Profile Map, Traverse ilap ana Computations, Topographic Maps (pencil original and tracing) and computations. UNIVERSITY OF CALIFORNIA EXTENSION DIVISION Correspondence Courses Surveying- IB Elements of Surveying Mr. Swafford Assignment 39 Rights, Duties, and Privileges _of the Surveyor , Foreword: The title of this assignment implies the fact that the surveyor in the pursuit of his work is vested with certain rights _. peculiar to his profession and that he is accorded privileges that might be wanting to one not engaged in such work as he may be called upon to do; also that he is obligated by the quasi-official character of his calling to perform certain functions in a v/ay and to an extent perhaps not required of others. There are certain delicate and diffi- cult relations between the surveyor and his client, things he owes to the latter and to himself. There are some obligations to the employer or the public of which he must be conscious while performing his duties, and he must endeavor always to place himself four-square with his work. Let us quote in this connection from the introduction to the course, Assignment 1, Page Q: To properly accomplish the work of his chosen profession, "The surveyor must be a man at once honest, sincere, dependable, energetic, ingenious, and observant. He must be patient in his work and with other people. He must be ready to give an unbiased , opinion as to the rights of disputants when called upon to do so, and should always be sure of the correctness of his work or its Surveying- IB. Assignment. 39 Page 2. limitations before submitting results to his employer or to others concerned. He must have a thorough knowledge of the fundamental principles of surveying such as will enable him to solve all his problems correctly." It will hence be the purpose in this assignment to follow the outline as suggested by the foregoing. The surveyor nust be just in his decisions, and he must observe ?. balancing of claims by rival parties. Fairness should always be the governing motive in all disputes, since the surveyor's work is often for the purpose of settling controverted cle.ims. The nere fact that one has the knowledge or skill that makes him a surveyor does not vest him with authority or judicial functions. His training ought rather to give him a fair understand- ing of the limitations that beset the practice of his profession, and lead him. to a moderate view of his importance. His functions, apart from ths fa.cts that his labors in any case have discovered, are ad- visory; therefore he should be a good councillor ra'ther than a judge. Men vri.ll listen to his counsels, but few would accept his decrees. The surveyor must also distinguish carefully what privileges are his, what rights he ruay claim, and what duties he must perform. He is a servant, an agent, an officer in some cases with duties fully sot forth by statute; he must be punctilious in the performance of these duties, but must at the same time avoid assuming those that are distinctly not his; he must modestly attend to his business his own and not that of others. Surveying-IB. ^ssigmaent 39 Page 5. In other words he must not trespass. This is a strong word, but trespassers are sometimes punished for their offenses when no of- fense was intended, and the surveyor is peculiarly placed in many instances while endeavoring to perform his duty. A surveyor may pass through or over lands of another in making neasurenents, but he must not commit any nuisance or destroy any property, such as crops, trees, or buildings; in case any in - fringnent of property rights is made the surveyor is liable for tres- pass and may be punished therefor. It is often convenient to remove trees, brush, etc, from a line cf road or other survey; but this can be done only by permission or by securing a right of way. Growing or standing crops of any sort come under the same rules. A surveyor should, in the absence of permission to cross any property where damage by himself or assistants might result, devise some method of determining distance or bearing of line or other data by v/hich it is unnecessary to enter the property. Railroads and private roads, canals, etc., usually secure right of way, or at least an option promise carrying with it the prop- - er permit to enter the property for the purposes of surveying. In such cases the duty (morally) of the surveyor is evidently to do as little damage as possible to crops, timber, and other property. In the public land surveys, over public property, the sur- veyor may exercise a wider liberty, but this should j^ever be construed Surveying -IB Assignment 39 Page 4. to mean v/anton destruction of timber or crops. When in a public land survey, or that of a highway, it is necessary to enter or cross private lands, the surveyor is justified only so far as is absolutely essential in removing or destroying property, but the employing prin- cipal, the state or county, or municipality may be required to satis- fy any just claims for damage committed by such work. Here also the surveyor is to avoid wanton waste or damage. The surveyor's duty respecting all surveys conducted by him is to do all vrork vith the full purpose of securing ample data, the making of suitable sketches for his own guidance and that of others who may come to use the material gathered by him. Notes must be full, clear and complete; they must not pre- sent any ambiguity , or otherwise doubtful matter; figures must be clear and plain so as to afford a ready interpretation; the data re- corded in the notes must show no alterations or erasures; If perchance incorrect quantities are entered in the notes and the necessary alter- ations are to be made, it should be done by crossing out the erroneous entry and writing above (or below) the correct quantities. In'; case a sketch is made to accompany tabulated quantities, the lengths (distances), bearings (angles), and other related parts, such as names of o v mers of adjoining property, cardinal directions, etc., must be indicated in proper place and in suitable form on the sketch, thus connecting the sketch with the tabulated data which it is desired to illustrate and clarify. Surveying-13 Assignment 39 Page 5. Throughout this course your attention has repeatedly been called to the forms of notes approved for various cases; other forms are made use of, and in some cases a different form may recommend itself, but a careful consideration of form is a matter of importance in all cases. Remember that notes of surveys are generally made for others than the surveyor himself to use to read and readily and clearly xmderstnad; hence, any departure from approved form, any omission of important data, or the absence of sketch and explanatory notes may lead to much confusion, and the work of the surveyor may be rendered worthless through faulty or insufficient notes. A practice of so;ae surveyors of purposely suppressing important data, or necessary directions or explanations in order that they may monopolize a survey when made, is to be deprecated; this is a species of dishonesty seldom met with, but instances are common enough to v/arn the novice to avoid any practice so flagrantly unfair unprofessional. This brings us to consider an important matter, which is the ovmership of surveys. If the surveyor has acted in the capacity of an employee, servant, or agent, i.e. has done the \vork for hire and has received compensation for his services, evidently the product of his labors connected therewith belong to the employer. Hence the right to monopolize the results the notes, data, maps, or information gathered for the purpose belong to the one who pays for the survey. If such a one is a private individual, to him; if it is for a company, Surveying- IB Assignment 3S Page 6. firm, or corporation the rights of ownership are vested in the company, firm, or corporation. Under this last classification may come a muni- cipality, county , or state, and in such case the survey becomes a public document and should be made a matter of record. In making a survey the following things should be carefully attended to: 1. The bearings of all lines; preferably the true bearings; but if the magnetic bearings are given instead, then the magnetic de- clination should also be correctly shown or stated. 2. The lengths of all boundaries, offsets, and tie lines. 3. The designated name or number of the tract, block, or lot; if the land is part of a subdivision or additon to a town, this fact should be stated and the official record of such subdivision or addition clearly referred to, 4. In cities the names of streets; also bodies of water bounding sane, and names of contiguous owners. 5. The angles of lines of adjacent lands v. r ith names of -ovmers, S. A plain title, a scale, and a meridian arrow; also the date of the survey and the signature of the surveyor with his official title, if any. the 7. If required,~recording of the survey with all parts complete notes, map, etc,, in the office of a county recorder, with a municipal, state, or government official as required. There is nothing wrong in a surveyor 1 s use of information of such character in the further prosecution of the work of his pro- fession, but he has no rights of monopoly of such information. Surveying-IB Assignment 39 Page 7. He has, when vfork is done as above detailed, done only his duty; if he has done this conscientiously and correctly his rewards are large and sure. Problem to Accompany Assignment 39: As a problem to accompany this assignment set up the notes of an ideal land survey giving the data, sketch, and explanatory notes in acceptable form. Follow the specifications given in the assignment from 1 to 6 inclusive. This means a small map on a sheet 8 i " x 11". The map should be carefully drawn in pencil first and finally inked. Do careful lettering using taste in all the work. Submit this as evidence of your having profited by this course in Place Surveying, - UNIVERSITY OF CALIFORNIA EXTENSION DIVISION Correspondence Courses Surveying-IB Elements of Surveying Mr. Swafford Assignment 40 Rights , Duties, and Judicial Functions of Surveyors . Foreword, In this, the concluding assignment of this course in the Elements of Surveying, it has been deemed best to further call your attention to the fact that while as a surveyor you have certain rights due to your quast -official position and irhile certain important and specific duties *wt tLAsV-C&''6~{, must compel you in an wwiefiireble course, you are also hedged about by other things that limit the scope of your authority and especially your judicial functions. (277) There is a tendency with those whose education and experience has been along mathematical and scientific lines to become irnbued with a high regard for their own judgments and conclusions. A training along such lines, however, should render you modest and cautious; it should make plain the fallibility of all human agencies; it should teach the exercise of deference to the opinions of others and of patience and forbearance. Instead of reaching conclusions hurriedly and pronouncing Judgments boldly a scientifically trained mind should take note of all phases, evidences, or testimonies, balance them carefully, and draw deductions only after full deliberation. Again a hasty or faulty conclusion is sure to carry you Surveying- 13 Elements of Surveying. Assignment 40, page 2. into error that must later be rectified by much labor and sacrifice. The consequences of false conclusions are often far-reaching and attended with confusion and loss to all concerned. They often nrovoke disagreement and expensive litigation. (278) In Assignment 20, Art. 153 the subject of the location of lost and obliterated corners of the rjublic land surveys was briefly treated; the surveyor who assumes to attempt such work should enter upon it only after fully understanding the method in each case and by following explicitly and carefully the full instructions afforded by the General Land Office. In the execution of your duties in such matters you must recognize that \-;hat was done and what was intended to be done when the original survey was made may be at variance; and that it was what was done , not what was intended^ that established the corners and their connecting lines that stand as the tangible record; and that this record must not be changed for any reason, whatever the findings in any case. Vlf hen you have to the best of your ability executed the com- mission intrusted to you, your own status has not been ad- vanced beyond that of an expert and you should neither regard yourself nor permit others to believe that you have any judicial prerogative whatever. You may counsel or advise; you nay use the weight of your knowledge and skill, or the Surveying-IB. Elements of Surveying. Assignment 40, page 3. confidence reposed in your ability arising from long experi- ence and conscientious service to advise or concilliate, but you must not act the judge. (279) Thus it v.ill be seen that the relation of the surveyor to his client is a delicate one; that he owes it to himself to be truthful, nainstaking, honest; he must apply the know- ledge and the arts of his craft to the discovery of the truth and he is then and only then justified in urging his con- clusions and findings for acceptance by those he would serve. The fallibility of man and the imperfections in the sys- tems of surveys, even when the greatest precautions have been taken to establish and monument important surveys, have called for the intervention of the courts in myriads of cases and have called for some notable decisions which have a historic as '.veil as a iudicial bearing. The surveyor is not called upon to know the law as ex- pressed in these decisions, but it is expedient that he should know hov these laws may affect the rights of his client, for he may be asked to advise or arbitrate, and then a knowledge of legal decisions and rulings "rould assist his judgment. (280) "In making a resurvey where there is considerable un- certainty, as in the case of disputed boundaries, the surveyor has no official power to decide any disputed point. He can only act as an expert, and give an opinion as to what is the most Surveying- IB. Elements of Surveying. Assignment 40, page 4, probable solution of the difficulties in question. If the interested parties do not agree to accept his decision, the question must be settled by the courts." (Quoted from Gillispie's Surveying.) A code of rules for the guidance of surveyors is found in the above work on pages 346-351. The student of this course is advised to make himself familiar with them. Another compilation of valuable instructions is to be found in a paper prepared for the Michigan Society of Sur- veyors and Engineers by Justice Cooley of the Michigan Supreme Court. A roprint of this paper is found in Raymond, and also in Johnson, to which the student is referred. This and the foragoine; "rules" have become "classics" of the subject, "The Judicial Functions of the Surveyor." These then taken together with the court decisions above referred to consti- tute the "code" of the surveyor and should be relied upon for guidance in his avork. (281) A few quotations follow, taken from several sources, and designed to show the scope and limitations besetting the surveyor; "No statute can confer upon a county surveyor the power to 'establish' corners, and thereby bind the parties con- cerned. Nor is this a question merely of conflict between state and federal lavr, it is a question of property right." "In any case of disputed lines, unless the parties con- Surveying-IB. Elements of Surveying. Assignment 40, pags 5. cerned settle the controversy by agreement, the determination of it is necessarily a judicial act, and it must proceed upon evidence, and give full opportunity for a hearing," "Nothing in -what has been said can require a surveyor to conceal his ov;n 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, ivhen at the some time he is sat- isfied that acquiescence has fixed the rights of parties as if it were at another. But he would do mischief if he were to attempt to "establish" monuments which he knew would tend to disturb settled rights; the farthest he has a right to go, as an officer of the law, is to express his opinion where the 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 ovm boundary lines, and considers acquiescence in a particular line or monument, for any considerable period, as strong, if not conclusive, evidence of such settlement. (282) Reasonable persons are inclined to give weight to the findings of an expert, as the surveyor is supposed to be, and if he has also the authority of a public office to support hife findings, they are likely to be convincing. But he must Surveying-IB. Elements of Surveying. Assignment 40, page 6. not assume that his decisions aro final as the courts, though competent testimony and other convincing evidence may modify or reverse the conclusions of the surveyor. r 'fhen, in case of disputed boundary the surveyor is called uoon to render the service of locating monuments or running lines, after a complete finding as to the true facts, they should be submitted to the parties concerned with no attempt to urge upon either what would not be exacted of both. In other words the disputants should be led to see the facts as revealed in the survey and to regard these as of import to both. To secure a final settlement it is necessary that they .ipree to abide by the survey, and of course to make this bind- ing the agreement should be reduced to writing and properly drawn and recorded. No parole agreement can secure the title as to Property rights. (282) In conclusion let us urge that you exercise both tact and modesty. You must not assume an arbitrary attitude nor permit yourself to mislead employing parties to believe that you ire possessed of authority when you have none, or that your findings t>ro final and inflexible. The courts are the only "last resort" of parties who disagree and disputants who cannot settle their differences otherwise must do so in the courts. Surveying-IB. Elements of 3u/ in ' Assignment 40, page 7. Questions! Problems 1. Show by diagram, fully d( ileci t how to proceed in laying out the south half of t N '^ quarter of section 15 in T6N, R4B, assuming tha ;ne ^^ township corners on the south are located. 2. How are the four quari section corners of section 6 in any township locad? 3. A rectangular city b- k was subdivided into 12 lots of uniform size 50'xl,' making the block dimensions 300' x 350', A refvey to locate the corners of lot #3 revealed that thelock dimensions were actually 304.8ft. x 247 -5 feet, v/ha-should the dimensions of lot #3 be, and by what mea^ements should the corners thereof be determined? 1 2 3 4r 5 6 iwlnj 10 9 8 7 4. For each of the above cases (given in problems 1, 2, 3) state britfly the functions of the surveyor, and indicate how those functions are limited by law and practice. -: ; ; : UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. LN&lNLEKiNG LlbhiM ( .. MAR 1 6 19! LD 21-100m-9,'48iB399sl6)476 YE 03760 793909 TAri Engineering Library UNIVERSITY OF CALIFORNIA LIBRARY