LIBRARY OF THE UNIVERSITY OF CALIFORNI Class EXERCISES IN SURVEYING FOR FIELD WORK AND OFFICE WORK WITH QUESTIONS FOR DISCUSSION INTENDED FOR USE IN CONNECTION WITH THE AUTHOR'S BOOK PLANE SURVEYING BY JOHN CLAYTON TRACY, C.E. ASSISTANT PROFESSOR OF STRUCTURAL ENGINEERING SHEFFIELD SCIENTIFIC SCHOOL OF YALE UNIV-EBSITV FIRST EDITION FIRST THOUSAND NEW YORK JOHN WILEY & SONS LONDON: CHAPMAN & HALL, LIMITED 1909 O COPYRIGHT, 1908, 1909, BY JOHN CLAYTON TRACY Stanbopc ipresa F. H. GILSON COMPANY BOSTON. U.S.A. PREFACE. THIS book of exercises is intended for use in connection with the author's book on ''Plane Surveying." Part I consists of exercises in field work grouped according to the nature of the work. Part II consists of exercises in office computations and in plotting. (See page 110.) The whole constitutes a thorough introductory course in fundamental principles and methods. So far as the author is aware, this is the first attempt to offer a collection of problems in both field work and office work, and the first book of exercises to be published for use in connection with a text-book. The author is convinced from his own experience that the most effective course in field work is not that in which the chief aim is to make a complete survey of a more or less extensive territory. On the contrary, much time may be lost in undertaking work for which the student is not prepared, and still more may be wasted in requiring the student to repeat the same kind of work day after day merely to cover a given amount of territory. It is far better to begin with a systematic course of exercises which shall serve not only as a preliminary drill in the use of instruments, but also as a careful study of the various methods which underlie all surveying. Not only will the student be better prepared for subsequent courses in municipal, mining, railway and geodetic surveying, but the effect of such a thorough introductory course will be felt after graduation in whatever kind of surveying he may be engaged. For this reason such a course may well extend over the first two college years. A feature of this book will be found in the questions suggested for discussion at the end of nearly every exercise. Some years ago the author adopted the practice of holding quizzes in the field in connection with each new piece of work. For example, an instructor has charge of a small squad of men to whom problems in chaining have been assigned. As soon as the work is finished, he gathers his men about him, and questions them on various matters pertaining to chaining, such, for example, as the use of the steel tape, the sources of error, different methods of doing the work just completed and the advantages and disadvantages of iii 187737 IV PREFACE. these methods. In some other part of the field another instructor may be holding a similar quiz on leveling or on transit work. These quizzes, if properly conducted; often develop into interest- ing discussions in which many important points are brought out that would entirely escape the student if he were merely required to do the field work assigned. Moreover, the student may be expected to take his text-book into the field with him, and to spend such spare time as he may have in looking up answers to questions which he knows will be asked at the end of the exercise. A. great deal of the recitation work usually held in the class-room will be much more effective if it is transferred to the field in this way. If, however, it is found inexpedient to hold these quizzes and discussions hi the field, the questions may still be used by the earnest student to develop his knowledge of the subject. They should prove particularly valuable to young men in practice who are seeking to perfect themselves in the art of surveying. In most courses it probably will be found necessary to omit some of the exercises given in this book. It is suggested, how- ever, that the student be held responsible for answers to the questions in the omitted exercises, and that, if possible, these questions be made a basis for class-room discussion. Unless otherwise stated, all paragraph and page references are to the author's text-book and not to this book. JOHN C. TRACY. NEW HAVEN, CONNECTICUT. June, 1909. INDEX OF GROUPS. PART I. FIELD WORK. GROUP PAGES E. Introductory Exercises (For the Classroom.) 1-5 C. Exercises in Chaining 6-17 T. Exercises in the Use of the Transit 18-28 Tr. Exercises in Running Transit Lines 29-37 Ts. Transit Surveys 38-41 Tp. Special Problems in Transit Surveying 42-50 L. Exercises in Leveling 51-62 Ls. Special Problems in Leveling 6368 Co. Exercises in the Use of the Compass 69-72 S. Exercises in the Use of the Stadia 73-78 P. Exercises in the Use of the Plane Table 79-83 To. Exercises in Topographic Surveying 84-88 M. Exercises in Determining a True Meridian 89-94 I. A Study of Surveying Instruments 95-101 A. Adjustments 102-109 PART II. OFFICE WORK. G. General Methods of Computation 111-115 B. Calculation of Bearings 116-120 L. Latitudes and Departures 121-124 T. Triangulation 125-127 O. Omitted Measurements 128-136 A. Areas 137-143 E. Earthwork Calculation 144-150 P. Exercises in Plotting 151-160 Q. Questions Pertaining to Mapping 161-168 TABLE OF CONTENTS. PART I. FIELD WORK. GROUP E. Introductory Exercises. (For the Classroom.) EXERCISK PAGE E-l. Definitions, Fundamental Principles and General Methods 1 E 2. General Discussion of Errors 3 E 3. General Discussion of Field Notes 5 E-4. Single Stroke Lettering 5 GROUP C. Exercises in Chaining. (XL Reading a Chain Tape 6 C-2. Chaining Distances Greater than the Length of the Tape 7 C-3. Chaining on a Slope 8 C-4. Standardizing the Tape 8 C-5. Experiments with a Steel Tape 9 C 6. Measuring a Base Line 10 C-7. A Chain Survey of an Irregular Polygon 11 C-8. To Chain from a Given Point over the Brow of a Hill to Another Point 11 C-9. Laying Out a Right Angle with a Tape by the 3-4-5 Method 12 C-10. To Erect a Perpendicular with a Tape at Any Point in a Line 12 C-ll. To Run One Line Parallel to Another Line 13 C-12. To Measure between Two Points When a Corner of a Building Intervenes 13 C-13. To Measure to an Inaccessible Point 13 C-14. To Measure the Angles of a Triangle with a Tape . . 14 C-15. A Chain Survey of a Plot of Ground with Irregular Boundary Lines and Containing Buildings 15 C 16. Discussion of Errors in Linear Measurements 16 vii Vlll TABLE OF CONTENTS. GROUP T. Exercises in the use of the Transit. EXERCISE PAGE T-l. Setting up a Transit 18 T-2. Prolonging a Straight Line , 19 T-3. Balancing in a Transit 20 T-4. Reading the Limb of a Transit 20 T-5. Reading Angles (Limb and Vernier) 22 T-6. Measuring Horizontal Angles with a Transit 23 T-7. Practice in Doubling Horizontal Angles 24 T-8. Measuring Vertical Angles with a Transit 24 T-9. To Find the Intersection of Two Lines of Sight 25 T-10. To Establish a Line of Sight Parallel to a Fence or to a Building 26 T-ll. To Measure an Angle Formed by Two Intersecting Lines Without Setting up over Either Line 26 T-l 2. Discussion of the Errors in Angular Measurement . . 27 GROUP Tr. Exercises in Running Transit Lines. Tr-1. Survey of a Triangle or Polygon by the Direct Angle Method 30 Tr-2. Measuring the Exterior Angles of a Triangle or Polygon 31 Tr-3. Doubling the Interior Angles of a Triangle or Polygon 31 Tr-4. Doubling the Exterior Angles of a Triangle or Polygon 31 Tr 5. Measuring the Interior Angles of a Triangle or Polygon, and Checking by Calculated Bearings . . 32 Tr-6. Measuring the Interior Angles of a Triangle or Polygon to the Left 33 Tr-7. Measuring the Exterior Angles of a Triangle or Polygon to the Left 33 Tr 8. Measuring the Deflection Angles of a Triangle or Polygon 34 Tr-9. Survey of a Triangle or Polygon by the First Azimuth Method 34 Tr 10. Survey of a Triangle or Polygon by the Second Azimuth Method 36 Tr-11. Survey of a Triangle or Polygon by the Method of Bearings 36 Tr-1 2. Questions on the Methods of Running Transit Lines or Traverses . . 37 TABLE OF CONTENTS. GROUP Ts. Transit Surveys. EXERCISE PAGE Ts-1. Discussion of the Methods of Locating Details 38 Ts-2. Discussion of the Methods of Keeping Field Notes. 39 Ts-3. Discussion of Practical Questions of Field Work in Transit Surveying 40 Ts-4. Transit Survey of a Small Area 41 GROUP Tp. Special Problems in Transit Surveying. Tp 1 . Practice in Triangulation 42 Tp-2. 'To Prolong a Straight Line through an Obstacle ... 43 Tp-3. To Run a Line between Two Given Points When an Obstacle Intervenes 44 Tp-4. To Measure between Two Points When One is Inaccessible 44 Tp-5. To Measure between Two Points When Both are Inaccessible 45 Tp-6. To Measure the Height of an Inaccessible Point. ... 45 Tp-7. Perpendiculars and Parallels 46 Tp-8. To Stake Out a Building 46 Tp-9. To Locate Piers for a Bridge 48 Tp-10. To Stake Out a Circle with a Transit and a Tape . . 49 GROUP L. Exercises in Leveling. L-l . Setting up the Level 51 L-2. Reading the Leveling Rod 52 L-3. To Find the Probable Error of Sighting and of Set- ting a Target 53 L 4. Comparison of Readings Taken With and Without the Target 54 L-5. To Test the Sensitiveness of a Level Bubble 54 L-6. Differential Leveling 55 (A Study of the Theory.) L-7. Differential Leveling 56 (A Study of Field Methods.) L-8. Profile Leveling 58 L 9. Discussion of Errors in Leveling 58 L-10. Use of the Hand Level 60 L 11. Trigonometric Leveling 60 L 12. Barometric Leveling 62 TABLE OF CONTENTS. GROUP Lp. Special Problems in Leveling. EXERCISE . PACK Lp-1. Reciprocal Leveling 63 Lp-2. To Set Stakes on a Grade Between Two Fixed Points (Special Method} 63 Lp-3. To Set Grade Stakes between Two Fixed Points When the Ground is Uneven 65 Lp-4. Use of the Gradienter 66 Lp-5. To Estimate Cut and Fill for Grading 66 Lp-6. To Stake Out a Vertical Curve 67 GROUP Co. Exercises in the Use of the Compass. Co 1. The Error of Sighting and Reading a Compass .... 69 Co-2. Reading Bearings 70 Co-3. Compass Survey of a Polygon 71 GROUP S. Exercises in the Use of the Stadia. S-l. To Test the Stadia Interval 73 S-2. To Measure Distances on Level Ground by the Stadia 74 S 3. To Measure Horizontal Distances on Sloping Ground by the Stadia 75 S 4. To Obtain Vertical Distances or Elevations by the Stadia 75 ' S-5. Stadia Survey of a Polygon 77 (Azimuth Method.) GROUP P. Exercises in the Use of the Plane Table. P-l. Plane-table Survey of a Polygon 79 (Method of Radiation.) P-2. Plane-table Survey of a Polygon 80 (Method of Progression, or Traversing.) P-3. Plane-table Survey of a Polygon 80 (Method of Radio-Progression.) P-4. Plane-table Survey of a Polygon 81 (Method of Intersection.) P-5. Plane-table Survey of a Polygon 81 (Method of Resection.) P-6. Discussion of Plane-table Surveying 81 P-7. Plane-table Practice in the Three-point Problem ... 82 TABLE OF CONTENTS. XI GROUP To. Exercises in Topographic Surveying. XERCISE PAGE To-1 . Running in a Contour 84 To-2. Topographic Survey of a Small Area ' 85 (Direct Method of Running in Contours with a Spirit Level.) To-3. Topographic Survey of a Small Area F 6 (Contours Interpolated by Spirit Leveling.) To-4. Topographic Survey of a Small Area 86 (Contours Interpolated by the Vertical-Angle Method.) To-5. Questions Pertaining to Topographic Surveying 87 GROUP M. Exercises in Determining a True Meridian. M 1. Determination of a Meridian by Observations on Polaris at Elongation 89 M-2. Determination of a Meridian by a Single Observation on the Sun 91 M-3. Determination of a Meridian with a Solar Attach- ment 93 GROUP I. A Study of Surveying Instruments. 1-1. The Vernier, the Magnetic Needle and the Level Bubble : 95 1-2. Theory of Lenses 96 1-3. The Telescope 97 1-4. Chains and Tapes 98 1-5. The Transit 98 1-6. The Level 99 1-7. Leveling Rods and Stadia Rods 99 1-8. The Compass, the Plane Table and the Sextant 100 1-9. The Care of Instruments 100 GROUP A. Adjustments. A-l. Preliminary Discussion of Adjustments 102 A-2. Adjustments of the Transit 103 A-3. Adjustments of the Wye-Level 105 A-4. Adjustments of the Dumpy Level 107 A-5. Adjustments of the Compass 108 A-6. Adjustments of the Plane Table 108 A-7. Adjustments of the Sextant 109 TABLE OF CONTENTS. PART II. OFFICE WORK. GROUP G. General Methods of Computation. EXERCISE PAGE G-l . Short Cuts in Arithmetical Work Ill G-2. Consistent Accuracy in Computations 112 G-3. Trigonometric Relations Between the Sides of a Right-Angled Triangle 113 G-4. Use of Logarithms 114 GROUP B. Calculation of Bearings. B-l. Calculation of Bearings from Angles 116 B-2. Changing the Bearings of All Lines of a Traverse by a Given Amount 118 B-3. Calculation of Angles from Bearings 118 GROUP L. Latitudes and Departures. L. Calculation of Latitudes and Departures 121 GROUP T. Triangulation. T 1. Computations for Triangulations 125 T-2. Miscellaneous Problems in Triangulation 127 GROUP O. Omitted Measurements. O-l. To Calculate the Bearing and Length of an Omitted Side of a Polygon 128 O-2. To Calculate the Omitted Bearing of One Side and the Omitted Length of Another Side of a Polygon . 129 O-3. To Calculate the Omitted Lengths of Two Sides of a Polygon 131 O-4. To Calculate the Omitted Bearings of Two Sides of a Polygon 133 O-5. Calculation of Omitted Measurements 134 (Miscellaneous Problems ) TABLE OF CONTENTS. xiii GROUP A. Areas. EXERCISE ' PAGE A-l . The Use of the Planimeter 137 A-2. To Compute Areas Directly from Field Measure- ments 137 A-3. Calculation of Areas from Offsets 138 A 4. Calculation of Areas from Latitudes and Double Longitudes 139 A-5. To Part Off a Required Area by a Line Having a Given Direction 141 A-6. To Part Off a Required Area by a Line Starting from a Given Point 142 GROUP E. Earthwork Calculation. E-l. To Calculate Earthwork by the Method of Unit Areas 144 (All Cut or all MIL) E-2. To Calculate Earthwork by the Method of Unit Areas 146 (Irregular Boundaries.) E-3. To Estimate Cut and Fill by the Method of Unit Areas 147 E-4. To Calculate Cut and Fill by the Method of Unit Areas 148 (Irregular Boundaries.) E-5. Calculation of Earthwork for Ditches and Embank- ments 149 E-6. To Estimate Cut and Fill from a Contour Map .... 150 GROUP P. Exercises in Plotting. P-l. Use of Drawing Instruments 151 P-2. Methods of Plotting Angles 153 P-3. Plotting Traverses with a Protractor 155 P-4. Tangent Method of Plotting Traverses 156 P-5. Chord Method of Plotting Traverses 156 P-6. Plotting Traverses by Bearings 157 (Tangent Method.) P-7. Plotting Traverses by Bearings 158 (Chord Method.) P-8. Plotting Traverses by Azimuths 158 P-9. Plotting Traverses by Latitudes and Departures ... 159 P-10. Plotting the Survey on Page 184 160 TABLE OF CONTENTS. GROUP Q. Questions Pertaining to Mapping. EXERCISE PAGE Q-l. Working up Field Notes Preparatory to Plotting. . . 161 Q-2. Plotting the Map 162 Q-3. Plotting Traverses 163 Q-4. Plotting Details 164 Q-5. Finishing the Map 165 Q-6. Profiles 168 OF THE / UNIVERSITY } OF Exercises in Plane Surveying, PART I. FIELD WORK. GROUP E. INTRODUCTORY EXERCISES. This group of exercises, consisting principally of questions for class-room discussion, are intended to cover those fundamental principles and methods which the student should know before beginning field work. The exercise on single-stroke lettering is inserted for the benefit of students who have not had a systematic course in freehand lettering. Exercise E-l. Definitions, Fundamental Principles and General Methods. Reference: Chapter I, pp. 1-8. Questions: 1. What is meant in geodesy by the earth's sur- face? p. 1. 2. When is it necessary to make this distinction in ordinary surveying? 3. What is the shape of the earth's surface? 4. What is the average radius? 5. What is the difference in lengths between the long and short axes, expressed in miles? 6. What is meant by sea-level, and how can it be determined at any given place? 7. Define horizontal plane; vertical plane. 8. What is the difference between a horizontal plane and a level surface? 9. What is the significance of the two terms "plane" and "geodetic" as applied to surveying? p. 2. 10. What is the approximate curvature of the earth's surface per mile? 11. How large an area may be covered by the methods of plane surveying without involving appreciable errors? 12. Give the approximate difference in length between the arc on the earth's surface and a straight line. 13. What four kinds of measurements are made in plane surveying? (Illustrate by means of a sketch.) 14. If a farm is on a side 1 2 INTRODUCTORY EXERCISES. hill, would a map of that farm show the actual area of the surface? 15. What are the units of linear measurement used in surveying in the United States? p. 3. 16. What are the units of angular measurement? 17. In making certain kinds of surveys, as for example, a survey for an architect, is it ever customary for the surveyor to measure in feet and inches? 18. What is a convenient method of converting hundredths of a foot to fractions of an inch? 19. From the table on page 650, change 6 feet 4| inches to feet and decimals of a foot. 20. What is the standard yard? p. 3. 21. In this country where may a steel tape be sent to be compared with a standard length? p. 561. 21. What is the equivalent of one metre in feet? p. 3. 22. What is the equivalent of one vara in inches? 23. What other units of measurements are sometimes used? 24. Why was 66 feet adopted as the length of Gunter's chain? 25. How would you record eight chains, six links? p. 33, 49. 26. Reduce sixty links to feet? 27. Explain the method of locating a given point with reference to two other points by linear measurements only. p. 4. 28. Explain two methods of locating a point by angle and distance. 29. Explain the method of locating a point by angles only. 30. Explain two other methods of locating a point. 31. Into what three parts may the work of a surveyor be divided? p. 5. 32. Of what does the field work consist? 33. Of what does the office work consist? 34. Name some of the important questions in surveying which arise in connection with field work. p. 6. 35. Explain by different illustrations how the purpose of a survey helps to decide some of these questions. 36. What can you say regarding limits of error and consistent accuracy? p. 7. 37. Upon what does speed depend? p. 8. 38. How does system in surveying diminish the chances of error? INTRODUCTORY EXERCISES. O Exercise E-2. General Discussion of Errors. Reference: Chapter II, pages 9 to 19. Questions: 1. What is the true error of any measurement? p. 9. 2. Why is the true error never known? 3. In general, what are the three sources of error? 4. What are the three classes of error? 5. Explain carefully the difference between constant errors and accidental errors, p. 10. 6. Can the constant error involved in any measurement be reduced by repeating the measurement a number of times and taking the mean of all the measurements? 7. What is the object in repeating the measure- ments and taking the mean? 8. Give a few illustrations of what is meant by mistakes; accidental errors; constant errors? 9. Why may variations in sources of constant error be classed as accidental errors? p. 11. 10. How may constant errors from different sources tend to balance each other? 11. How may constant errors from the same source tend to balance each other? 12. Explain the difference between cumulative and compensating errors. 13. Explain the difference between discrepancy and error. 14. How can the discrepancy between two measurements be small yet the error large? p. 12. 15. What can you say as regards the elimination of constant errors? 16. How may mistakes be eliminated? 17. Illustrate how constant errors may be eliminated. 18. How may accidental errors be reduced? p. 13. 19. What can you say as regards the relative importance of errors from different sources? 20. What is meant by an appreciable error? 21. What is the most probable value of a quantity, when several measurements have been made of that quantity? 22. When the sum of several measurements should equal the exact quantity, how is the true error distributed? p. 14. 23. If the true error of a measurement is never known, how can the error of 1' or 60", in the illustration on page 14, Case 1, be called a true error? 24. Explain by illustration how the discrepancy should be distributed, when the sum of several measurements should equal some other measurement. 25. Why, in Case 2, p. 14, is the term "discrep- ancy" used instead of "error"? Questions on the Method of Least Squares: 26. What use is made of the method of least squares? p. 15. 27. Upon what 4 INTRODUCTORY EXERCISES. assumption is this method based? 28. How do the conditions in practice differ from those of the assumption? 29. What effect has this on a most probable value of a quantity as deter- mined by the method of least squares, and how may the prob- able value be made to approach the true value? 30. Define residual; probable error. 31. In surveying, what use is made of most probable values; probable errors? 32. Illustrate how by comparing the probable errors of the means of two sets of observations, the precision of one mean may be compared with that of the other, p. 16. 33. Explain the use made of the prob- able error of a single observation. 34. How can the weights which should be given to different sets of observations be found from the probable errors? 35. Upon what three assumptions are the formulas for calculating probable errors based? p. 17. 36. What additional point should be kept in mind? 37. In order that probable errors may be of any significance, what precautions must be taken? (See Remark, p. 17.) 38. Two chainmen measure a line six times with the following results in feet: 314.124, 314.130, 314.133, 314.128, 314.136, and 314.131. Two other chainmen measure the same line with the following results in feet: 314.134, 314.124, 314.138, 314.122, 314.131, and 314.122. What is the relative precision of the work of the two pairs of chainmen? (See illustration, p. 19.) 39. An angle is measured with a transit eight different times. The degrees and minutes being the same but the seconds varying as follows: 20", 40", 30", 10", 20", 50", 30", and 40". The same angle was measured with another transit with the following result: 40", 20", 20", 30", 40", 30", 30", and 40". Other things being equal, which is the better instrument? What degree of precision may be expected for each instrument in measuring a given angle under conditions similar to those prevailing when the above angles were measured? INTRODUCTORY EXERCISES. 5 Exercise E-3. General Discussion of Field Notes. Reference: Chapter III, pages 20 to 30. Questions: 1. What are field notes? 2. What can you say as regards methods of keeping notes? 3. Give general sugges- tions for keeping notes, p. 21. 4. Into what three parts may field notes be divided? 5. Give general suggestion for record- ing numerical values, p. 22. 6. Give general suggestions for making sketches, p. 23. 7. Give general suggestions concerning explanatory notes. 8. What style of lettering should be used? p. 24. 9. What is a good height for letters? p. 29. 10. How should the letters be spaced? 11. Give additional suggestions for taking notes, p. 29. 12. Discuss special directions for class work. p. 30. Exercise E-4. Single Stroke Lettering. Reference: Article 38, pages 24 to 30. Directions: In this exercise use a 4H pencil and smooth white paper with hard surface. Sharpen the pencil to a fine point. At first, draw a bottom guide line for letters; later, practice lettering without a guide line. In forming the various letters and figures pay particular attention to the direction and sequence of the strokes in each letter and learn the peculiar characteristics of each. The stem of a letter which extends above or below the body of the letter should not be made too long a common mistake. (a) Print the entire alphabet of lower case letters, grouping them in five groups : adgq, bp, ceos, hmnu, and fijklrtvwxyz. (See p. 25.) Repeat this exercise several times. (6) Print the capital letters of the alphabet, grouping them as follows: EFHIKLMNTZ, AVWXY, BPR, CGOQ, and SJDU. (See p. 28.) Repeat this exercise several times. (c) Practice making the numerals from 1 to 0. (See p. 27.) (d) Rule bottom guide lines about a quarter of an inch apart, and print in single stroke lettering, with the letters spaced as closely together as possible, the introductory note on page XXVII. GROUP C. EXERCISES IN CHAINING. The first six exercises of this group offer the student an opportunity to form correct habits in the use of the steel tape, and, by a study of the sources of error, to learn what precautions must be taken to attain a given degree of precision. The remaining exercises involve special problems in chaining, most of which are based on well known geometric constructions. If it is necessary to omit any of these latter problems in the field, the omitted problems may be given in the class-room as blackboard exercises, and the corresponding questions discussed by the class. Exercise C-l. Reading a Chain Tape. References: Page 32, 48; p. 33, 50 (a) (b) (c) (d). Equipment: Chain tape and chain pins. Directions: 1. On a level piece of ground, stick two chain pins at random about 50 feet apart. 2. Measure the distance between these two pins to the nearest tenth of a foot. 3. Repeat the measurement, the two chainmen interchanging positions. Questions: 1. Why should the chainman at the end of the tape make the final reading? 50 (c). 2. Why should the other chainman read the number on each side of the required reading? p. 33. 3. What kind of tape may be used to render these pre- cautions unnecessary? 50 (d). 4. Does the surveyor use decimal parts of a foot or inches? p. 3. 5. When would it be desirable to use a tape graduated to inches? 6. Are linear measurements in surveying ever made along an inclined surface? p. 2. 7. What is the equivalent of one vara in inches? p. 3. 8. What is the equivalent of one meter in feet? p. 3. 9. What other units of measurements are sometimes used? p. 3. 10. Why was 66 feet adopted as the length of Gunter's chain? p. 3. 11. What is a convenient method of converting hundredths of a foot to fractions of an inch? p. 3. 12. What is the difference be- tween Gunter's chain and an Engineer's chain? p. 559. 13. How would you record 10 chains 7 links? p. 33, 49. 14. Compare the three kinds of tapes, p. 559. 15. What are the relative merits of a chain and a tape? Remark, p. 33. 16. Describe the different kinds of steel tapes, and the standard methods of gradu- ating tapes, pp. 560, 561. 6 EXERCISES IN CHAINING. 7 Exercise C-2. Chaining Distances Greater than the Length of the Tape. References: Page 34, 50 (e) and (f), 51 and 52; p. 37, 56, 1-12. Equipment: Two range poles; steel tape and chain pins. Directions: 1. Ascertain just what points to take for the ends of the tape. 2. Set two chain pins about 300 or 400 feet apart on comparatively level ground. 3. Chain the distance between these two pins. 4. Repeat the measurement, the two chainmen interchanging positions. By means of the tape tables on pages 48 and 49 ascertain whether the discrepancy between the two measurements indicates poor work or good work. Suggestions: Special attention should be paid by the instructor to the method of sticking the pins as this is usually the source of greatest error on level ground. Insist on the use of signals, page 35, and caution men in regard to pulling up pins that mark the ends of a line. Questions: 1. What is the approximate pull used in ordinary chaining? p. 42, 65. 2. Why should the pins be stuck at right angles to the line? 3. How are the pins used to keep count of tape lengths? 4. How can the head chainman keep himself approxi- mately in line? 5. In accurate work what precaution should be taken after each pin is stuck? p. 37, 56 (12). 6. If there is a slight difference between the first and second measurements of the line, what is this difference called? p. 11, 20. 7. If this dis- crepancy is small, is the error of chaining necessarily small? p. 11, 20, 21. 8. If the tape used in measuring is too long or too short, is this a source of constant error? p. 10. 9. Are the errors from sticking the pins accumulative or compensating? p. 11, also p. 42, 66. 10. If in lining in, pins are set as much as 6 inches first one side, then the other of the line, will the resulting error be comparatively large or small? p. 41, 63, p. 51, 76. 11. How may the error in chaining be expressed by a ratio? p. 39, 58. 12. If a line is measured with a tape that is too long will the result be too long or too short, and what is the correspond- ing algebraic sign? p. 39, 59. 13. What are some of the things a chainman should do? 14. What are some of the things a chain- man should not do? p. 38. 15. If in chaining a line AB whose true length is 400 feet, the first pin is stuck 18 inches to the left 8 EXERCISES IN CHAINING. of the line, the second pin 16 inches to the right, the third pin 24 inches to the right, what would be the difference between the true length and the measured length of AB } provided no other errors are involved? 16. If in measuring along a line the end of the tape should fall at a place where a pin cannot be stuck, as, for example, in the middle of a brook, how would you proceed? Exercise C-3. Chaining on a Slope. References: Page 35, 53. Equipment: Steel tape, chain pins, range poles, plumb bob. Directions: 1. Measure the distance between two points on a comparatively steep slope by the second method, page 36, work- ing down hill. 2. Repeat the measurement, chaining again down hill, but the two chainmen interchanging positions. 3. By means of the tables on pages 48 and 49, ascertain whether the discrepancy between the two measurements indicates work that is excellent, good, fair or passable. Suggestions: 1. Pay special attention to throwing the weight of the body against the arm in holding the down-hill end of the tape. p. 36. 2. If the pin is stuck upright it interferes with the plumb bob one reason for stick- ing it in slanting at right angles to the line. 3. A common error is the hold- ing of the down-hill end of the tape too low. The instructor, standing opposite the middle of the tape, should watch this at first. 4. Three men can work to much better advantage than two in chaining on a slope. Questions: 1. Why is it better to chain down hill than up hill? 2. If a line is chained on the slope instead of in horizontal stretches, how may the result be corrected? Page 35, 53 (a). Exercise C-4. Standardizing the Tape. References: Pages 39-44 and p. 561. Equipment: Steel tape to be tested ; scale, spring balance, turn- buckles, thermometer, magnifying glass. Directions: Assuming that a permanent standard has already been established, proceed as directed for "Testing the Tape," page 562. 2. Repeat the test two or more times. Suggestion: Sometimes in place of a permanent standard such as that described on page 562, the tape is compared with a standard tape of known length, kept for that purpose. EXERCISES IN CHAINING. FORM OF NOTES. Tape Tested. Observed Length. Temper- ature. Correction for Temp. Corrected Length. Questions: 1. What are the usual standard temperatures and standard pulls for a 100 foot steel tape? Pages 561 and 42. 2. What is the average change in length for each 15 degrees change in temperature? p. 40, 62 (a). 3. What is the average change in length per 15 pounds pull? p. 43, 69 (7). 4. Where can a tape be sent for standardizing? p. 561. 5. What are some of the condi- tions to be fulfilled in any device for standardizing tapes? p. 561. 6. What was the object in using a steel strip for the standard of the Boston Water Works? p. 562. 7. A line measured with a 100 foot tape is found to be 826.34 feet long. What was the true length of the line, (a) if the tape was 100.024 feet long? (b) if the tape was 99.87 feet long? 8. Required to lay off the distance of 784 feet with a tape (a) 100.04 feet long; (b) 99.9 feet long. Exercise C-5. Experiments with a Steel Tape. References: Page 40, 62 (c), p. 41, 64. Equipment: Steel tape, scale, spring balance, turn-buckles, thermometer, magnifying glass, 2 plumb bobs. Directions: 1. Perform the experiment of 62 (c), p. 40. 2. Test the length of a tape first under a 12 pound pull, then under a 20 pound pull. Repeat several times. 3. With the ends of the tape supported a short distance above the floor or ground, the rest of the tape being unsupported, find the tension required to make the tape of standard length. Use plumb bobs to mark the ends of the tape and turn-buckles to hold the tape steady at the standard pull. Questions: 1. A line measured in summer with a 100 foot tape, 70 F. temperature, 12 pound pull is found to be 438.945 feet long; measured again in winter at a temperature of 10 F., with the same tape and same pull, what should be its length (tape supported in both cases on level supports)? p. 40, 62. 2. A tape is 100 feet long, I inch wide, ^ inch thick and weighs 10 EXERCISES IN CHAINING. .0005 pounds per inch; standard at 62 F., 12 pound pull. A line measured when the temperature is 90 F. and the pull 20 pounds with the tape supported at the two ends only is found to be 93.41 feet long. Required the corrected length of the line. 3. Com- pare the pull required to balance the sag as obtained in the third experiment above with that obtained theoretically ^rom the for- mula at the bottom of page 41, using the tape described in the preceding question. Exercise C-6. Measuring a Base Line. References: Page 57, 82. Equipment: Steel tape, scale, spring balance, thermometer, magnifying glass (if desired a turn-buckle device for stretching the tape). Directions: 1. On a level stretch of ground set stakes with their tops on a level, making a line several hundred feet long. Measure the length of this base line a number of times as directed on page 57. 2. Find the probable error as explained on page 58 (read also pp. 15-19). Suggestions: As it will require considerable time to set stakes, this may be done once for all and the same stakes used by different squads. Questions: 1. What should be the maximum discrepancy between two measurements of a line 1000 feet long, that the work may be classed as excellent? pp. 48, 49. 2. Is a large error more likely to occur from a variation in temperature or a variation in pull? p. 43. 3. Give the relative importance of different sources of error, p. 43. 4. State some of the requirements correspond- ing to a ratio of precision of 1/50000. p. 51. 5. When it is impracticable to set stakes with their tops on a level, how would you proceed? p. 57. 6. How can tripods be employed in chain- ing a base line through thick underbrush? p. 58. 7. Sketch a device for supporting and stretching a tape. 8. Give ideal con- ditions for base line measurement. 9. What is the object in having the end of the tape fall on a different stake each time a line is chained? 10. Describe other apparatus for measuring base lines, p. 194. 11. Give some practical suggestions for setting supporting stakes, protecting end hubs, etc. p. 57, EXEKCISES IN CHAINING. . 11 Exercise C-7. A Chain Survey of an Irregular Polygon. Reference: Page 56, 80. Equipment: Chain or tape, chain pins, range poles, plumb bob. Directions: 1. Set 5 stakes at random, forming an irregular polygon each side of which is at least 150 feet long. 2. Survey this polygon with a tape as explained on page 56. 3. Compute the area of the polygon by finding the area of each of the triangles into which it is subdivided from formula 12 on page 408. Suggestions: A survey of some plot of land having straight boundary lines may be substituted if desired. For a survey of a plot with irregular boundary lines see page 15 of this book. Field Notes: 1. A sketch showing all measurements. 2. The computations for areas neatly arranged. Questions: 1. How can the angles of the polygon be found from the original measurements? 2. Give the general method for making any chain survey, p. 56. 3. If in the survey shown in Fig. 80, p. 56, it were impossible to see all four corners from any one station, how would you proceed? Exercise C-8. To Chain from a Given Point over the Brow of a Hill to Another Point. Reference: Page 55, 79. Equipment: Tape, chain pins, sight poles, plumb bobs. Directions: Set two poles, one over the brow of a hill from the other, and chain the distance between them, as explained on page 55. Questions: 1. Give two methods of " ranging in " when a valley intervenes between the two points, p. 56, 79 (c). 2. Give the method of chaining over a high wall. p. 54. 3. Give a method of chaining a line between two points when woods intervene, p. 54. 12 EXERCISES IN CHAINING. Exercise C-9. Laying Out a Right Angle with a Tape by the 3-4-5 Method. Reference: Page 60. Equipment: Tape and chain pins. Directions: 1. Set two pins in the ground 80 feet apart and with this line as one side, stake out an 80 foot square. 2. Check by measuring the two diagonals of the square. Field Notes: Make a sketch showing each distance measured or laid off and the discrepancy between the two diagonals. Questions: 1. Explain the difference between the methods of laying out a right angle with a cloth tape and with a steel tape (a 100 foot tape being used in each case). 2. How would you lay off a right angle with a 50 foot tape? 3. If one diagonal of a square as staked out is longer than the other how would you pro- ceed to correct the mistake? 4. If the two diagonals are exactly equal, does this necessarily prove that the square has been staked out correctly? 5. How closely should the lengths of the two diagonals of an 80 foot square agree to indicate good work? 6. How would you proceed to stake out the house on page 210, using only the tape? Exercise C-10. To Erect a Perpendicular with a Tape at Any Point in a Line. Reference: Pages 61, 62 and 63. Equipment: Tape and chain pins. Directions: 1. Set two pins at random and at one of them erect a perpendicular to the line between them by the method of 85 (b), p. 61. 2. Check by the 3-4-5 method. Field Notes: Make a sketch showing all measurements. Questions: 1 . What is the difference between the methods used for geometrical constructions in drafting and those used for the same constructions in chaining? Illustrate by sketches. 2. What method in chaining corresponds to erecting a perpen- dicular by intersecting arcs and how may the perpendicular erected by this method be checked by the use of the same method? p. 61. 3. Illustrate, by sketches, methods of dropping a perpen- dicular to a given line from any given point, p. 62. EXERCISES IN CHAINING. 13 Exercise C-ll. To Run One Line Parallel to Another Line. Reference: Page 62 and page 63. Equipment: Tape and chain pins. Directions: Assume three points at random and through one of them establish a line parallel to the line between the other two, using two different methods. Field Notes: Make a sketch showing all measurements. Question: Discuss the advantages and disadvantages of the three methods given on page 63. Exercise C-12. To Measure between Two Points when a Corner of a Building Intervenes. Reference: Page 64, 87 (a). Directions: 1. Assume two points at least 40 to 60 feet apart and imagine a corner of a building to intervene. Find the dis- tance between the two points by one of the methods explained on page 64 and check the result by the other method. 2. Check also by a direct measurement. Suggestion: Having completed the exercise with an imaginary obstacle, it is well to repeat the work with the corner of a real building intervening between the two given points. Field Notes: Make a sketch showing all measurements made during the exercise, and the final results. Exercise C-13. To Measure to an Inaccessible Point. Reference: Page 64, 87 (b). Equipment: Chain and chain pins. Directions: 1. Set two stakes or pins about 200 feet apart. 2. Find the distance between the two points by two different methods. 3. Check results by direct measurement. Questions: 1. Could the methods used in transit work (p. 222, 277 (c) ) be used in chaining? 2. Explain at least two other 14 EXERCISES IN CHAINING. methods that might be used with a tape. 87 (c). 3. Describe a rough method of measuring the height of an inaccessible point. 87 (d). 4. Describe at least two methods of prolonging a straight line through an obstacle using only the tape. p. 66. (If time permits this may be done in the field as an additional exer- cise.) 5. Show by a sketch one method of measuring between two inaccessible points. 87 (e). Field Notes: 1. A sketch showing all measurements. 2. Arith- metical work neatly arranged. Exercise C-14. To Measure the Angles of a Triangle with a Tape. References: Page 63, 86, and pages 458-460. Equipment: Tape and chain pins. Directions: Set three stakes or chain pins at random to form a triangle with no side less than 100 feet long. 2. Measure each interior angle of the triangle with the tape using the tangent method. 3. Repeat the problem using the chord method. Suggestions: 1. Take a reasonably long distance for the base in the tangent method and the radii in the chord method. 2. Pay particular attention to lining in the errors are chiefly here. Field Notes: 1. A sketch of the triangle showing all measure- ments for the first method ; a similar sketch for the second method. 2. All arithmetical work neatly arranged. 3. A comparison of results of the first and second methods. 4. Actual error in the sum of the angles. Questions: I. Illustrate how to measure angles of various sizes from to 360 by the tangent method; by the chord method. 2. What is the most convenient base or radius? 3. Upon what does the accuracy of the tangent method depend? 4. When is it advantageous to use the chord method instead of the tangent method? 5. Explain the sine and cosine method, p. 63. EXERCISES IN CHAINING. 15 Exercise C-15. A Chain Survey of a Plot of Ground with Irregular Boundary Lines and Containing Buildings. References: Page 56, 80, also pp. 138 to 140. Equipment: Steel tape, chain pins, stakes, hatchet and range poles. Directions: Make a survey of a small plot of ground in which some of the boundary lines are curved or irregular. There should also be some buildings to be located. Establish a network of triangles for reference lines as explained on page 56, 80 (c), and locate the different details as explained on pages 138-140. Suggestions: If desired this exercise may be conducted as a recitation, the different methods being discussed by the class as a whole, or by each squad, and each method illustrated by sketches on the blackboard or in the note-book. Field Notes: 1. A complete sketch showing all measurements made in the field. 2. Computations for area neatly arranged. Questions: 1. Give the methods of locating a point from two given points by linear measurement only. p. 4. 2. Explain method of locating a building by means of offsets, p. 138, 198 (a). 3. Explain the method of locating a regular curve by offsets. 198 (b). 4. Explain the method of locating an irregular out- line like the bank of a stream, by linear measurements only. 5. What three points must be kept in mind in using the offset method? 198 (d). 6. Explain the "plus system" often used in the offset method. Note, p. 139. 7. Explain how to locate a building by three measurements, one of which is a tie line. Figure 199 (a). 8. Is it necessary that the tie line should have one end at some point on the building? p. 140. 9. Explain how to locate a building by two pairs of intersecting measurements. Figure 199 (b). 10. Explain the use of tie lines in getting the directions of fences. 199 (c). 11. How may the area of the plot surveyed be computed? pp. 410-413. (If time permits, compute the area of the plot surveyed.) 16 EXERCISES IN CHAINING. Exercise C-16. Discussion of the Errors in Linear Measurements. References: Chapter V, pages 39 to 53. Questions: 1. What are the four general sources of error in chaining? p. 39. 2. How is an error expressed by a ratio? 3. If a line is measured with a tape that is too short, will the result be too short or too long, and what is the corresponding algebraic sign? 4. Why is the error in the length of a chain likely to be relatively greater than that in a tape? 5. Show by an illustra- tion how to apply a correction for an error in the length of a tape when chaining between two points; when laying off a given distance, p. 40. 6. How can an error due to the tape not being horizontal be eliminated? 7. When is the error due to temperature plus, and when minus? 8. What is the approxi- mate change in the length of a 100-foot steel tape for a change in temperature of 90 F.? 9. When should temperature be taken into account and when not? 10. What precautions are often taken in base line measurement to eliminate the error due to temperature? 11. What approximate rate of change may be remembered for a 100-foot tape? 12. If the length of a city block is measured in summer and then again in winter with the same tape, what discrepancy due to change in temperature may be expected? 13. Describe an experiment which may be tried to show the effect of a small change in temperature upon the length of a steel tape. 14. What is meant by the error in alignment, and is this source of error relatively important? p. 41. 15. In ordinary chaining, is it necessary for the transit- man to line the chainman in with the instrument? 16. Is an error due to sag plus or minus? 17. What three ways are there of eliminating the error due to sag? 18. How may the error due to uneven pull be eliminated? p. 42. 19. How may the errors in marking tape lengths be reduced? 20. In what kind of work is this source of error the greatest? 21. Name four common mistakes made in reading tapes. 22. In extremely accurate work, what precautions may be taken in reading the tape? 23. Summarize the eleven sources of errors in chaining, p. 43. 24. From what two sources are errors compensating? 25. What errors are always plus? 26. Discuss the relative importance of the sources of errors, pointing out from which sources impor- EXERCISES IN CHAINING. 17 tant errors are most likely to occur, and what sources may be neglected in ordinary work. 27. In judging the relative importance of errors, what points should be kept in mind? p. 44. 28. What largely determines limits of error in a given survey? 29. Should all measurements in the same survey be made with equal care? 30. In general, what distances should be measured carefully and what less carefully? p. 45. 31. What are some of the conditions that affect the accuracy of chaining? 32. If the accidental errors amount to 0.02 foot for 100 feet in using a 100-foot tape, what would be the probable error in measuring a line 900 feet long? 33. Give the general rule for finding the probable error. 34. Why is not the discrepancy between two measurements of the same line affected by constant errors? 35. If the discrepancy between two measurements of the length of a tape is. 0.01, what is it likely to be between two measure- ments of a line sixteen tape lengths long? 36. What can you say regarding the value of the formula at the top of page 46? 37. Discuss the remark at the top of page 46, and explain why the last sentence in that remark is true. 38. Explain the first method of determining a coefficient of precision for chaining. 39. Explain a method which may be used for ordinary chaining. 40. A line is chained twice under favorable conditions, measure- ments being made with an ordinary chain, plumb-bob, chain- pins and average speed. The two results were, respectively, 618.1, 618.3. Would the work be classed excellent, good, or fairly good? (See Table 1, p. 48.) 41. If a steel tape had been used in place of a chain, other conditions being the same, what would be the allowable discrepancy in good work between the two measurements in the preceding question? 42. Apply Table 2, page 49, to questions 40 and 41. 43. Explain how the table on page 51 may be used to indicate the precautions which are necessary to attain certain degrees of precision, p. 52. 44. Discuss the matter of combining errors from different sources, p. 52. GROUP T. EXERCISES IN THE USE OF THE TRANSIT. The exercises in this group offer the student an opportunity to form correct habits in manipulating the transit and in reading angles. As habits formed early in the course are likely to cling to one long after graduation, and as the transit is perhaps the most important instrument used in survey- ing, no suggestion, however trivial it may seem, should be ignored if it is likely to help one to become skilful and accurate in the use of the transit. Exercise T-l. Setting up a Transit. References: Pages 84-89, 109-115. Equipment: A transit for each student, or as many transits as are available. Directions: 1. Set up the transit on comparatively level ground, paying special attention to the suggestions on page 87. 2. Repeat the work, selecting a place on a side hill. Suggestions: At the beginning of the exercise, it is well for each instructor to show his squad how to set up a transit, emphasizing at the same time the importance of the suggestions on page 87, 114. The members of the squad may then be drilled individually under close supervision of the instructor, several transits being used simultaneously, if possible. If this exercise is given before the exercises in the use of the level, it is well for the instructor to explain briefly the principal parts of the telescope (see photograph, page 551), and, if thought best, to add to the questions given below those on pages 96 and 97 of this book. Questions: 1. Give some of the precautions to be observed (a) in the use of clamps and screws; (b) in the treatment of lenses; (c) in standing a transit up on a floor, p. 84. 2. Give pre- cautions taken in carrying a transit, p. 85. 3. Give other precautions, pp. 607-609. 4. Tie a sliding knot as described on p. 85. (If the knot slides too freely, take an extra turn of the string about the hook.) 5. How are the cross-hairs brought in focus? p. 85. 6. Give the rule for focusing the object glass on distant objects, p. 86. (If desired, the theory of lenses (pp. 545-549) and the construction of the telescope (pp. 549-558) may be studied in connection with Questions 5 and 6.) 7. Give suggestions for sighting the telescope, p. 85, 111. 8. Give some 18 EXERCISES IN THE USE OF THE TRANSIT. 19 practical suggestions for manipulating the tripod, p. 86, 113, 114. 9. Give suggestions for leveling up. p. 88. 10. Explain how the telescope level is sometimes used for leveling up. p. 89. 11. If the plumb-bob is as much as i to ^ of an inch off the tack, how much will this affect an angle if sights are 450 feet long? p. 89 and p. 105, 147. 12. Give some points to remember in leveling up. p. 89, 115. Exercise T-2. Prolonging a Straight Line. Reference: Pages 90-93, 116-122. Equipment: Transit, hatchet, and stakes. .Directions: 1. Set two stakes at random, at least 200 feet apart, in such a place that the line between them can be prolonged another 200 feet. 2. Set up over the end of the line to be pro- longed, and set a third stake in line about 200 feet away, using the method of double reverse, p. 92. 3. Set up over the other end of the original line and see if all three stakes are in line. Suggestion: Drive the stakes only part way into the ground so that they may be pulled up easily. Mark the exact points with a small lead-pencil cross. The upper clamp should be set once for all at the beginning of the exercise, and left undisturbed during the remainder of the exercise. The student should then work wholly with ihelower clamp and corresponding slow motion screw, and thus become accustomed to using them for backsighting . Not until Exercise T-4 should he concern himself with the use of the upper clamp and the corresponding slow motion screw. It may be well, at the close of the exercise, for the instructor to explain briefly by means of the figures on pages 588 and 590 what errors of adjust- ment may be eliminated by reversing in azimuth and altitude. Field Notes: A short statement of the work and of the dis- crepancy, if any, found in the final check of the third step above. Questions: Define backsight, foresight, permanent backsight, and permanent foresight, p. 90. 2. Explain the use of the upper and lower clamps and the corresponding tangent screws, p. 91, 119, pp. 563-566. 3. Give signals for lining in. p. 146. 4. Ex- plain the method of reversing in azimuth and altitude ; what are other synonymous terms? p. 92. 5. In setting up to prolong a straight line, what precaution may be taken? p. 89 (14); also note, p. 92. 6. What four methods are there for prolonging a straight line? 20 EXERCISES IN THE USE OF THE TRANSIT. Exercise T-3. Balancing in a Transit. References: Page 93 and page 201. Equipment: Transit, hatchet, stakes, and sight poles. Directions: Set two stakes several hundred feet apart, and balance in the transit about half way between them as explained on page 201. 2. When the transit is apparently on line, test by the method of double reverse. Exercise T-4. Beading the Limb of a Transit. References: Pages 67-75. Equipment: A transit for each student, or as many transits as are available. Directions: 1. Clamp the plates of a transit at random, read the number of degrees (clockwise), estimate the number of minutes, and record the total reading. Pay no attention to the vernier reading. . 2. Record the correct reading as given by the instructor (who will read the vernier). 3. If the transit has two verniers 180 degrees apart, repeat for the reading across the circle. 4. Repeat the whole exercise until from 6 to 10 different readings have been made and recorded by each student. Suggestions: If a number of transits are available for each squad, they may be set up within a few feet of each other, and the plates of each transit set at random by the instructor. Each student may then read in turn the limb of each transit. No vernier readings should be permitted, as the object of the exercise is to make each student thoroughly familiar with the method of reading the limb before he undertakes to read the vernier. After the student has become accustomed to reading the angles clockwise, he may be drilled in reading the angles counter-clockwise. At the beginning of the exercise it is well for the instructor to explain briefly how the upper and lower plates may be turned independently or together, and to illustrate the use of the upper and the lower clamps and the corresponding slow motion screws. The photograph on page 565 should be useful in this connection. If transits with special forms of numbering are available, as, for example, one like that described in Article 95 (d), page 73, a special drill should be given in reading such instruments. EXERCISES IN THE USE OF THE TRANSIT. 21 FORM OF NOTES. Number of Reading. Value. Instructor's Reading. 1 37 15' 37 18' 2 108 40' 108 38' ' Questions: 1. Which plate, the limb or that carrying the vernier, must necessarily turn when the telescope is moved side- wise? pp. 67, 563. 2. Give the three common systems of num- bering the limb gradations, p. 67. 3. Give four common methods of subdividing the degree on the limb. p. 68. 4. Give the three steps in reading an angle, p. 68. 5. In which step are important mistakes liable to occur? Remark, p. 68. 6. Explain what is meant by reading an angle clockwise and counter-clockwise right and left (p. 68), and why are the first two terms preferable to the last two terms? Remark, p. 69. 7. Give general method of reading a limb. 91 (b), p. 68. 8. Give common mistakes in reading a limb. p. 69. 9. If angles are always measured clockwise, what system of numbering is best for the transit (p. 71) and how many rows of numbers would be needed? p. 67. 10. In measuring angles clockwise, which row of numbers on page 70 is ignored? 11. Why is the portion of the limb near the 180-degree mark a place where mistakes are liable to occur? p. 71, 94 (b). 12. Why is it important to check readings by estimating angles with the eye? p. 71, 94 (b). 13. What mistake is illustrated by most of the incorrect readings on page 70? p. 71, 94 (c). 14. What mistake is illustrated by the reading of 171 30' in place of 191 30'? p. 70, 94 (c). 15. If a transit has only one row of numbers (clockwise), and it is desired to read an angle counter- clockwise, how would you proceed? 94 (d), p. 71. 16. If a transit has only one row of numbers, and that is the half-circle system, how would you read an angle clockwise which is greater than 180? p. 73, 95 (c) and (d). 17. When gradations are numbered by a combination of the full-circle and quadrant sys- tems, what is a common mistake? p. 75, 96 (b), and at what two points on the limb are mistakes most likely to occur? (Bottom of p. 75.) Note: As a result of this exercise two things should be impressed upon the mind of the student, first the importance of reading the limb correctly before attempting to read the vernier, and, second, the closeness with which an angle may be read without the use of a vernier. 22 EXERCISES IN THE USE OF THE TRANSIT. Exercise T-5. Beading Angles; Limb and Vernier. References: Pages 76-83. Equipment: A transit for each student, or as many transits as are available; preferably transits should read to minutes. Directions: 1. Clamp the plates of a transit at random, and read both the limb and vernier clockwise, recording the total reading. 2. Record the correct reading as given by the instructor. 3. Repeat for the vernier across the circle. 4. Repeat the whole exercise until from six to ten readings have been made and recorded by each student. Suggestions: Several transits may be used by each squad as in the preced- ing exercise. It is well, however, to confine the work to transits reading to minutes until the method is thoroughly understood, then transits reading 30, 20, or 10 seconds may be used. Finally,. students should be drilled in read- ing angles counter-clockwise. FORM OF NOTES. Same as that for the preceding exercise. Questions: 1. What is the principle of the vernier? pp. 538, 539. 2. What is the difference between direct and retrograde verniers? p. 540. 3. Explain how to determine the smallest reading or least count of a vernier, p. 540; also p. 76, 97 (c). 4. In the vernier reading to minutes, how large an arc is covered by the space that is called a minute? p. 76, 97. 5. Give the general method of reading a vernier. 98. 6. Which side of a double vernier should be read? 99. 7. Why should the vernier reading be estimated from the limb before the vernier is read? 99 (2). 8. How many marks should be examined closely in reading the vernier? 99 (3). 9. When may an angle be read to 30 seconds on a vernier reading to minutes, and what is the greatest error involved in reading such a vernier? 99 (3). 10. What pre- caution should be used in setting a vernier? 99 (4). 11. When there are two verniers 180 apart, which is ordinarily used? 99. 12. What are some of the common mistakes in reading verniers? p. 77. 13. Describe at least three different verniers, each of which reads to minutes? pp. 78, 79. 14. Describe a common EXEECISES IN THE USE OF THE TRANSIT. 23 form of vernier reading to 30 seconds, p. 80. 15. Describe a common form of vernier reading to 20 seconds, p. 81. 16. De- scribe a common form of vernier reading to 10 seconds, p. 82. 17. Describe some special forms of verniers, p. 83. Note: Different forms of verniers may be assigned to students to be drawn on paper outside of the regular exercise. Exercise T-6. Measuring Horizontal Angles with a Transit. References: Pages 93-98, 125-133. Equipment: Transit, stakes, hatchet, and sight poles. Directions: 1. Set four stakes ^4, B, D, and E, at random to form an irregular polygon, no side of which is less than 150 feet in length. 2. Set a stake C near the center of this polygon. 3. Set up the transit over C, and measure in succession the horizontal angles ACB, BCD, DCE, and EC A, all measured clockwise. Make a sketch showing the values of the angles. 4. Check by comparing the sum of the four angles with 360. Questions: 1. Give the general method of measuring horizontal angles, p. 93. 2. Which clamp and tangent screw is used for each operation? Bottom of p. 93, top of p. 94. 3. Explain the difference between taking the angles to the right and to the left, p. 94. 4. Explain how a number of angles may be taken from the same backsight, p. 94. 5. Is the vernier always set at before measuring an angle? p. 94. 6. If there are two verniers, which is used? p. 95. 7. Why form the habit of estimating the angles by eye? p. 95. 8. Precautions to take after backsighting. p. 95. 9. Precautions to take in measuring a large number of angles from the same backsight, p. 95, 129 (6), (7), (8). 10. In taking angles clockwise, is it necessary to turn the telescope clock- wise? 11. Name some of the sources of error in measuring the horizontal angles, p. 96. 12. Give some suggestions for using the vernier in reading the angles, p. 96. 13. Name some common mistakes in reading angles, p. 97. 14. Relative importance of errors, p. 104. 15. Relative importance of error in setting up the transit, p. 104. 16. If a pole is f inch in diameter and is held 100 feet from the transit, what would be the angular error caused by bringing the vertical cross-hair to coincide with the edge 24 EXERCISES IN THE USE OF THE TRANSIT. of the pole instead of with its center? Bottom p. 105. 17. Is the difference between the sum of the four angles and 360 the true error of that sum or simply a discrepancy? p. 4 14, 24 (b). 18. How would you distribute this error? p. 14, 24 (c). 19. If two angles ACB and BCD were measured separately, and then checked by measuring ACD, would the difference between the sum of the first two angles and the measured value of ACD be a true error, or simply a discrepancy, and how would it be dis- tributed? 24 (d). 20. What method corresponding to the work done in this exercise is sometimes used in work of high precision? p. 100, 139. Exercise T-7. Practice in Doubling Horizontal Angles. Reference: Page 99, 138. Equipment: Same as for the preceding exercise. Directions: Repeat the preceding exercise, but double each angle as explained in 138. Questions: 1. Explain the method of measuring angles by repetition, p. 99. 2. What is the chief precaution necessary in manipulating clamps and tangent screws when repeating an angle? p. 100. 3. When is it necessary to add multiples of 360? Illustrate how this is done. 4. Explain how to lay off an angle accurately, p. 101, 142; also p. 218. Exercise T-8. Measuring Vertical Angles with a Transit. Reference: Page 98, 135. Equipment: Transit, stakes, and hatchet. Directions: 1. Drive two stakes A and B about 200 feet apart, B being considerably lower than A. Set up over A and measure the vertical angle which the line of sight from the transit to B makes with the horizontal. 2. Assume some well-defined point at least 50 feet above the ground and 200 feet from the transit; call this point C, and measure the angle which the line of sight EXERCISES IN THE USE OF THE TRANSIT. 25 from the transit to C makes with the horizontal. 3. Measure the vertical angle between the lines of sight to B and to C. Make a sketch showing the values of all angles. 4. Check by compar- ing the last angle measured with the sum of the other two angles. Questions: 1. How should the vertical arc be adjusted, assum- ing that the transit is in adjustment otherwise? p. 98, 135. 2. How would you proceed if neither the vertical arc nor the corresponding vernier were adjustable? 3. What is the "index error," and how are the algebraic signs applied to it? p. 98. 4. In measuring the vertical angles as directed above, where is the vertex of the angle? 5. If it were required to measure a vertical angle, one side of which was determined by two points on the ground, how would you proceed, assuming that all points are accessible? 6. Is it possible to double a vertical angle with a transit as ordinarily made? 7. How can a transit be made so that it will be possible to double vertical angles? Exercise T-9. To Find the Intersection of Two Lines of Sight. Reference: Page 202. Equipment: Transit, stakes, hatchet, wire brads, short pieces of string. Directions: 1. Set four stakes at random to form a polygon, no side of which is less than 150 feet in length. 2. Find the inter- section of the diagonals of this quadrilateral as explained in 257. Questions: 1. Explain the method of referencing a point. 259. 2. Give other methods of referencing a point, p. 54. 3. How would you reference a point for setting a merestone? p. 203, 259. Illustration. 4. Give two methods of referencing a line. 260. 5. Explain the importance of referencing at least one line in a survey. 260. Remark. 26 EXERCISES IN THE USE OF THE TRANSIT. Exercise T-10. To Establish a Line of Sight Parallel to a Fence or to a Building. Reference: Page 201, 256. Equipment: Transit, tape, stakes, hatchet, sight pole. Directions: 1. Assume a station about three feet from a fence or a building. 2. Set the transit over the station and establish a line of sight parallel to the fence or to the building as directed on page 201. Field Notes: Make a sketch showing the fence or building, the transit line and the distance used. Describe the method referring to the sketch in order to make it plain. Questions: 1. What method would you use when the parallel lines are a considerable distance apart? p. 224, 278 (b). Exercise T-ll. To Measure the Angle Formed by Two Intersecting Lines without Setting Up over Either Line. Reference: Page 202, 258. Equipment: Transit, tape, hatchet, stakes, sight pole, pieces of string. Directions: Measure the angle at any place where two definite fence lines meet, choosing preferably a corner where the transit cannot be set up inside of the fence line, using the method of 258. Suggestion: If it is inconvenient to use actual fence lines, drive three stakes to form a triangle no side of which is less than 200 feet in length, and measure the interior angle at one of the vertices without setting up over any vertex or over any point within the triangle. Three parties may work on the same triangle, one at each vertex, and the sum of the three results may then be compared with 180 as a check. Questions: 1. What method can be used when the transit can be set up over a point inside of the corner? 2. How are division fences usually set? 3. How are division walls usually set? 4. How are fences along a street usually set? 5. How are street lines usually marked? 3ISES IN THE USE OF THE TRANSIT. 27 Exercise T-12. Discussion of the Errors hi Angular Measurement. Reference: Chapter X, p. 104. ?: 1. What in general are six sources of error in measuring angles? p. 104. 2. What can you say regarding the relative importance of errors from these different sources? 3. If the plumb-bob is slightly off the tack is the resulting error likely to be large? p. 105. 4. When is the error due to the plates being out of level likely to be large? 5. Why is the error due to either source greater for short sights than for long sights in measuring angles? Why is the reverse true in laying off angles? 6. If the bob is inch off the station, what, approxi- mately, are the resulting errors for sights of 100 and 1000 feet respectively? 7. Give some idea of the errors involved when one edge of the limb is lower than the other. 8. In measuring an angle what values should be kept in mind when judging the importance of errors of sighting? (Note (6), p. 106.) 9. What values in laying off angles? 10. Explain by means of these values the importance of holding the sight pole plumb. 11. Sup- pose that in measuring an angle to the nearest minute the length of sight is 800 feet and only the top of the pole is visible, how much could it be out of plumb without materially affecting the result? 12. What are some of the natural sources of error? 13. Explain how the sun shining on one side of the instrument and not on the other might affect a line of sight. 14. Give some precautions which may be taken to avoid mistakes; to eliminate constant errors; to eliminate accidental errors. 15. In what two ways may the precision of angular measurement be judged? p. 107. 16. Explain each of these two methods. 17. Explain the fundamental difference between checking angles by the true error of their sum and by the discrepancy between their sum and a measured angle. 18. If the sum of the interior angles of a polygon agrees with the check of 175, p. 119, does this mean that the measured value of each angle is exactly right? 19. If the interior angles of a triangle are measured with a coarsely graduated instrument why is their sum more likely to be exactly 180 than if a finely graduated instrument were used? 20. What are some of the conditions which effect the precision of angular measurement? p. 108. 21. Suppose that the ratio 28 EXERCISES IN THE USE OF THE TRANSIT. of precision for chaining has been fixed at Ysl^S) what is the permissible error for an angle? (See table at bottom of p. 108.) 22. If an error of 30" is made in laying off an angle what is the error in feet at a distance of 1500 feet from the transit? 23. What can you say regarding the permissible angular error in ordinary transit surveying? p. 109. 24. For limits j-jj^ or higher what are the permissible angular errors, and how may this degree of precision be attained? 25. What is meant by consistent accuracy as applied to the relation between linear and angular measurement? 26. When is it not desirable to main- tain this consistent accuracy? 27. Suppose that contour points in a survey should be located to the nearest foot and the longest sight does not exceed 600 feet, how accurately must angles be read, and will it be necessary to use the vernier? 28. How accurately must an angle be measured to locate a point 800 feet from the transit within \ inch? 29. A corner of a building is 60 feet from a transit and an angle and a distance are measured to it solely for purposes of plotting to a scale of 1 inch = 50 feet, how accurately should each be measured? (Use 500 feet in the table at the bottom of page 108, changing the position of the decimal point.) 30. If the permissible error in a single angle is 20", what is the permissible error in the sum of the interior angles of a 25-sided polygon? p. 110. 31. What would be the permissible error in the same polygon if the sides averaged 400 feet in length, and the largest discrepancy allowed between two measurements of the same line were 0.1 foot? 32. Is it the general tendency in transit surveying to overestimate the accuracy required in angular measurement and underestimate that required for linear measurements, or vice versa? GROUP TB. EXERCISES IN RUNNING TRANSIT LINES. General Directions for Exercises Tr-1 to Tr-11. The following eleven exercises are intended to fix in mind the methods explained in Chapter XII, p. 116. These exercises are of the greatest importance, since they are preparatory to actual surveys to be taken up later in the course. In all problems drive three or more stakes at random to form a triangle or polygon, no side of which is less than 150 feet. Mark each stake so that different parties will not use each other's stations by mistake. It is not necessary to drive stakes flush with the ground; indeed, it is better to drive them only partway in. In the top of each stake drive a wire brad, leaving enough of the nail projecting above the stake to serve as a sight, thus avoiding the necessity wherever possible of holding a pencil or a pole on the stake. In the direct angle methods measure all angles to the right, unless otherwise stated. Unless it is desirable to gain additional practice in chaining, measurements of the lengths of the sides of triangles or polygons may be omitted. In most of the exercises two men can work to advantage in a party. Each should check the other's work, keep complete notes, and divide the work of running the transit. Thus, for example, if the party consists of two men, one can run the transit at one station, the other at the next station, and so on, alternating. In the azimuth method, however, it is desirable for one man to run the transit entirely around the triangle or polygon, and then the exercise should be repeated, the other man running the transit. It is well to choose a comparatively level piece of ground for all of these exercises, where no difficulty will be encountered in sighting. Equipment for Exercises Tr-l to Tr-11: A transit reading pref- erably to minutes, and for all exercises except deflection angle exercises, graduated according to the full circle system. Also hatchet, stakes, wire brads, crayon, and in some cases sight poles. If the lengths of sides are to be measured, a tape and plumb- bob should be added to the equipment. 29 30 EXERCISES IN RUNNING TRANSIT LINES. Exercise Tr-1. Survey of a Triangle or Polygon by the Direct Angle Method. References: Pages 116-120. 162-178. Equipment: See page 29 of this book. Directions: 1. See general directions, page 29 of this book. 2. Survey the triangle or polygon without doubling angles, measuring the interior angles. Keep notes according to Form 1, p. 170. Follow as far as possible the method of procedure on p. 154, 220 (a). Questions: 1. Define transit station, transit line, and transit angle, p. 116. 2. What are transit lines used for? p. 116, 165. 3. Give the general method of running transit lines, p. 116, 166. 4. Define transit-line angle and explain how to avoid ambiguity in the notes, p. 117, 167. 5. Define deflection angle, a system of transit lines, traverse, triangulation. 6. What is the differ- ence between angles to the right and angles to the left? p. 118. 7. If angles are measured to the right, in what direction must one go around a polygon in order to measure interior angles? p. 118, 173. 8. Give the rule for a closing check on transit-line angles. 9. If the ratio of precision in chaining for a given survey is ^0 Dl D2 D3 D4 D5 - 17.6, 16.1, 18.4, 20.2, 17.4, 16.9, 15.4, 14.5, 13.2, DQ EQ El E2 E3 E4 E5 EG FO = 15.4, 23.8, 21.2, 18.0, 17.8, 18.9, 21.1, 22.3, 24.0, Fl F2 F3 F4: F5 FQ = 25.1, = 26.4, = 26.9, = 25.3, = 23.8, = 27.1. If the ground is graded to a level surface whose elevation is 12 ft., find the amount of earth to be removed. Answer in cubic yards. 144 EARTHWORK CALCULATION. 145 312. Same as Problem 311 except it is required to find the amount of earth which must be filled in to bring the entire lot to a level surface whose elevation is 30 ft. 313. Same as Problem 311 except that the unit area is a 50-ft. square and the elevation of the level surface is 8.4 ft. 314. Same as Problem 312 except that the unit area is a 50-ft. square and the elevation of the level surface is 28.1 ft. 315. Same as Problem 311 except the unit area is a 50-ft. square and the elevation of the level surface is 9.9 ft. 316. Same as Problem 312 except the unit area is a 50-ft. square and the elevation of the level surface is 29.2 ft. When the finished surface slopes in one direction only. 317. Assume the data given in Problem 311. The elevation of the finished surface at AQ and FQ is 12 ft., and this surface slopes downward at the rate of 1 ft. per 100 ft. to the line between AQ and FQ which is horizontal. Find the amount of earth to be removed. 318. Assume the data given in Problem 311. The elevation of the finished surface at AQ and FQ is 30 ft., and this surface slopes upward at the rate of 0.5 ft. per 100 ft. to the line between AQ and FQ which is horizontal. Find the amount of earth to t>e added. 319. Same as Problem 317 except the slope is 1.5 ft. per 100 ft. 320. Same as Problem 318 except the slope is 1.0 ft. per 100 ft. When the finished surface slopes in two directions. 321. Assume the data given in Problem 311. The plane of the finished surface is determined by two lines, one from .40 to -46, the other from AQ to FQ. The elevation of this surface at AQ is 13 ft. ; at 46, 11.5 ft. ; at FQ, 12.5 ft. Find the amount of earth to be removed. 322. Same as Problem 321 except the elevation of the finished surface at AQ is 30 ft. ; at AQ, 31.2 ft. ; at FQ, 29.0 ft. Find the amount of earth to be added. 323. Same as Problem 321 except the unit area is a 50-ft. square and the elevation of the finished surface at AQ is 10 ft.; at AQ, 11. 8 ft.; and at FQ, 12.3 ft. 146 EARTHWORK CALCULATION. 324. Same as Problem 322 except the unit area is a 50-ft. square and the elevation of the finished surface at AO is 28.1 ft.; at A6, 29.9 ft.; and at FQ, 30.9 ft. Questions: 1. What is meant by a 3 per cent grade or gradient? p. 428. 2. How are the elevations at different points corre- sponding to a given grade calculated? p. 428. 3. What is the cut or fill at any point, and the corresponding algebraic signs? p. 429. 4. What is a polyhedron; a truncated prism; a right- truncated prism? p. 429. 5. What is the volume of a truncated prism equal to? 6. Give the general method of calculating earthwork from unit areas, p. 429. 7. Explain the method of procedure when the finished surface slopes in one direction; when it slopes in two directions. 8. Explain the method of procedure when the outline of an area is not rectangular, p. 431. 9. Give suggestions: (a) For a systematic method of com- putation, p. 431 ; (b) as regards the size of a unit square, p. 432. 10. In very uneven ground how may the method of unit rec- tangles or squares be modified to obtain greater accuracy? p. 432. Exercise E-2. To Calculate Earthwork by the Method of Unit Areas. (Irregular Boundaries.) This exercise is the same as the preceding exercise except that the outline of the area to be graded is not rectangular, but irregular. The instructor should make up a problem similar to that indicated in Fig. 467 (g), p. 431, or Fig. 468, p. 432, estab- lishing the elevations for the finished surface in such a way that it will lie entirely below or above the original surface, i.e., so that the grading will be all cut or all fill. EARTHWORK CALCULATION. 147 Exercise E-3. To Estimate Cut and Fill by the Method of Unit Areas. Reference: Page 432, 468. Directions: Before beginning the computation, make a sketch as directed in the method of procedure at the bottom of p. 430. Determine by interpolation the points where there is neither cut nor fill. (See p. 433.) Calculate separately the cut and the fill, each by the method of 467 (e), p. 430. For the methods of interpolation see 469, p. 433; also p. 497. Problem: (To be assigned from the thirteen problems given below.) PROBLEMS. When the finished surface is level. 331. A rectangular plot is divided into 20-ft. squares and the corners are numbered according to the system shown in Fig. 402 (c), p. 341. The elevations at each corner in feet are as follows: AQ = 42.4, 53 = 41.6, C6 = 36.0, E2 = 38.7, F5 = 33.2, Al = 41.3, 4 = 39.9, DO = 41.6, E3 = 36.3, ,P6 = 31.7, A2 = 42.1, B5 = 39.2, Dl = 40.9, E4 = 37.2, GO = 45.8, A3 = 43.2, 56 = 37.1, 1)2 = 41.0, #5 = 36.1, Gl = 43.4, 44 = 39.6, CO = 44.2, D3 = 41.4, EG = 35.0, G2 = 41.2, A5 = 38.7, Cl = 45.6, Z>4 = 40.3, FO = 43.2, G3 = 39.7, AQ = 36.3, C2 = 43.9, D5 = 39.1, Fl = 42.1, G4 = 37.4, BO = 43.1, C3 = 42.1, Z>6 = 36.9, F2 = 37.8, G5 = 31.0, Bl = 44.0, C4 = 41.8, EQ = 42.6, F3 = 35.4, GQ = 30.0. B2 = 42.8, C5 = 39.9, El = 41.4, F4 = 34.6, Calculate the cut and fill required to grade the plot to a level surface whose elevation is 40.0 ft. Answer in cubic yards. 332. Same as Problem 331 except the elevation of the level surface is 39.0 ft. 333. Same as Problem 331 except that the unit area is a 50-ft. square and the elevation of the level surface is 37.0 ft; 148 EARTHWORK CALCULATION. 334. Same as Problem 331 except that the unit area is a 50-ft. square and the elevation of the level surface is 42.0 ft. 335. What should be the elevation of the level surface in Prob- lem 331 in order to make cut and fill equal? When the finished surface slopes in one direction only. 336. Assume the data given in Problem 331. The elevation of the finished surface at AO is 40.0 ft. ; atA6, 38.5 ft. ; at GO, 40.0 ft. ; and at G6, 38.5 ft. Calculate the cut and fill in grading to a plane surface determined by these four elevations. 337. Same as Problem 336 except the elevations are as follows: AO, 41.0 ft.; A6, 38.0 ft.; GO, 41.0 ft.; GQ, 38.0 ft. 338. Same as Problem 336 except the unit area is a 50-ft. square and the elevations are as follows: AQ, 39.0 ft.; AQ, 37.5 ft.; GO, 39.0 ft. ; G6, 37.5 ft. 339. Same as Problem 336 except the unit area is a 50-ft. square and the elevations are as follows: AO, 41.0 ft.; AQ, 39.2 ft.; GO, 41.0ft.;G6, 39.2ft. When the finished surface slopes in two directions. 340. Assume the data given in Problem 331. The plane of the finished surface is determined by two lines, one from AQ to AQ, the other from AQ to G6. The elevation of this surface at AQ is 42.0 ft'. ; at AQ, 40.2 ft. ; at G6, 38.4 ft. Calculate the cut and fill. 341. Same as Problem 340 except the elevation of the finished surface at AQ is 41.8 ft.; at AQ, 38.2 ft.; at G6, 39.4 ft. 342. Same as Problem 340 except the unit area is a 50-ft. square and the elevation of the finished surface at AO is 40.2 ft.; at A6, 39 .Oft.; at G6, 37.2ft. 343. Same as Problem 340 except the unit area is a 50-ft. square and the elevation of the finished surface at AO is 39.0 ft.; at A6, 40.2 ft. ; at G6, 38.4 ft. Exercise E-4. To Calculate Cut and Fill by the Method of Unit Areas. (Irregular Boundaries.) This exercise is the same as Exercise E-2, p. 146,* except that the finished surface is such that the calculations will be for both cut and fill, instead of for all cut, or for all fill. EARTHWORK CALCULATION. 149 Exercise E-5. Calculation of Earthwork for Ditches and Em- bankments. Reference: Page 434, 470. Directions: Find the area of each cross-section in succession, either by the use of the planimeter or by calculation, and apply the "end area" formula to the solids between the respective cross-sections as explained in 470. Note: If desired, the problems may also be solved by the "prismoidal formula." The problems are made simple tmrposely, it being taken for granted that, in most cases, the student will study this part of the subject to better advantage in connection with railway surveying. PROBLEMS. 351. A ditch is 8.0 ft. wide at the bottom, and falls 0.5 ft. in 100 ft.; its sides slope outward 1 horizontal to 1 vertical. The elevation at the bottom of the ditch at + = 20.0 ft. The elevations of the three points on the surface of the ground at each station are as follows: at +0, C (the center height) = 32.1 ft., R (the right-hand edge of the ditch) = 31.1 ft., L (the left-hand edge of the ditch) = 33.6 ft.; at + 50, C = 31.6 ft., R = 30.8 ft., L = 32.4 ft.; at 1+ 00, C = 30.1 ft., R = 28.7 ft., L = 31.0 ft.; at 1 + 60, C = 28.2 ft., R = 26.4 ft., L = 29.1 ft.; at 2+ 00, C = 24.3 ft., R = 23.1 ft., L = 25.3 ft. Calculate the earth to be removed from the portion of the ditch given above. 352. Same as Problem 351 except the bottom of the ditch is 10.0 ft. wide and the sides slope outward l horizontal to 1 vertical. 353. Assume the elevations on the surface of the ground to be the same as in Problem 351. It is desired to build an embank- ment, the top of which is 8.0 ft. wide and at an elevation of 36.0 ft. The sides slope 1$ horizontal to 1 vertical. Calculate the total fill. 354. Same as Problem 353 except the top of the embankment is 10.0 ft. wide and at an elevation of 38.0 ft. The sides slope l horizontal to 1 vertical. 150 EARTHWORK CALCULATION. Questions: 1. Define prismoid. 2. A transverse slope is de- noted by the horizontal distance which corresponds to a vertical distance of one unit. How does this differ from the method ordinarily used in denoting a longitudinal slope (commonly called "grade")? 3. Give the "end area" formula. 4. Give the "prismoidal" formula. 5. Which gives the better results? 6. In the "prismoidal " formula is the area M the mean of the two end areas; how is it obtained? 7. What is meant by " three- level " cross-sections? 8. For "three-level" sections how may the work of computation be shortened? Exercise E-6. To Estimate Cut and Fill from a Contour Map. Reference: Page 435. Directions: The problem in this exercise is to estimate the cut and fill from a contour map, using one or more methods explained in 471, p. 435. A contour map already plotted may be used, or a fictitious map similar to that shown in Fig. 471 (a), p. 435, may be drawn and used as a basis for computation. Questions: 1. Explain in detail the first method, p. 435. 2. Explain the second method, p. 436. 3. Explain the third method, p. 436, illustrating it by means of a sketch. GROUP P. EXERCISES IN PLOTTING. Exercise P-l. Use of Drawing Instruments. References: Pages 438 to 442 and pages 445 to 449. Preparations of instruments: I. First of all, sharpen the pencil properly and then keep it sharp. See remark, p. 438. Every student is required to keep at hand a piece of sandpaper and a cloth which may be suspended from the corner of the drawing table for convenience. 2. Insert a hard lead in the compasses and sharpen it to a cone shape point. See p. 442, 478 (1). 3. Adjust the pivot needle, p. 442, 478 (2). 4. Make a pricking point by inserting a needle in a wooden or a wax handle. See p. 418, 486. 5. Test the edge of the T-square for straight- ness. See footnote, p. 438. 6. Test the right hand edge of the drawing board for straightness by bringing the upper edge of the T-square into contact with it. In a similar manner see if the sur- face of the board is a plane. 7. Test the triangles as directed in the footnote, p. 439. 8. Either during this exercise or previous to the next exercise, the student should prepare a protractor similar to that described on p. 456, 493 (c). Note: While the student is supposed to have had a course in mechanical drawing before beginning this course, nevertheless, it will be found advan- tageous to devote the first exercise in plotting to a drill in those methods of drafting which are of especial importance in mapping. The exercise will be of little value, however, unless the student is careful to observe the precautions outlined above, precautions which, if taken throughout the course, will enable him to secure more accurate results than he otherwise could. Additional suggestions for drafting should be reviewed in a recita- tion on the questions given at the end of this exercise. PROBLEMS. 361. (1) Draw a horizontal line about 10" long. (Read 483, p. 446.) (2) Set the zero mark of the scale at an assumed point A on the horizontal line, and without moving the scale lay off in succession to a scale of 1" = 20' 0" the following distances : 151 152 EXERCISES IN PLOTTING. AB = 30 ft., BC f =42 ft., OD = 65.5 ft., and DE=21.5 ft. Check: AE = 159.0 ft. (Read 475, p. 440, and 486, p. 448.) (3) Draw a second horizontal line about 10" long, and in a similar manner lay off (to a scale of 1" = 40'-0"): + 40, + 60, 1 + 10, 1 + 75, 2 + 38, 3 + 17. (Read 486 (a), p. 449.) Note: In inspecting the above work the instructor will note: (1) If the pencil is sharpened correctly. (2) If fine, almost invisible prick-marks and freehand circles have been used. (3) If lengths are correct. 362. (1) Draw a line about 6" long at 45 to the horizontal, and mark two points on it A and B, 128.2 ft. apart to a scale of 1" = 30' 0". (2) By means of the compasses and scale, estab- lish a point C 98.5 ft. from A and 120.6 ft. from B. Draw only one arc. (Read 478, p, 442.) Note: In inspecting the above work the instructor will note if the com- passes were used correctly, and if the three lengths are correct. If a visible hole was left in the paper by the pivot needle of the compasses, it is evident that either the legs were not bent, or too much pressure was exerted. 363. (1) Draw a line at random about 10" long and inclined about 10 to the horizontal. (2) Mark off on this line 421 ft. to a scale of 1" = 50' 0". (3) At one end of this length erect a perpendicular about 10" long by using two triangles. (Read 474 (1), p. 439.) (4) Lay off 328 ft. on this perpendicular. (5) Com- plete the 421 X 328 ft. rectangle without further measurement by drawing parallel lines with the two triangles. (Read 474 (3), p. 439.) (6) Check the rectangle by measuring the lengths of the two diagonals. Note: In inspecting the above work the instructor will note: (1) If light hair-lines have been used, and if the lines intersect sharply at cor- ners, running beyond instead of stopping short. (2) If right methods of erecting perpendiculars and drawing parallel lines were employed. (3) If the lengths of the sides of a rectangle and of its diagonals are correct. Questions: 1. What is the purpose of sharpening one end of the pencil to a wedge shape edge? p. 438. 2. How close to the ruling edge should pencil lines be drawn? p. 438. 3. Give additional suggestions for penciling, p. 446. 4. Give suggestions for the use of the T-square, p. 438. 5. Why should not the T-square be used against the top or bottom of the board for draw- ing vertical lines? p. 439. 6. What is the incorrect way of erect- ing a perpendicular line by means of a triangle? 7. How should the triangles be used for erecting perpendiculars; for drawing parallel lines? p. 439. 8. Explain the use of the decimal scale, p. 440. 9. Give additional suggestions for laying off measure- ments as regards marking points; use of dividers; arithmetical EXERCISES IN PLOTTING. 153 work; estimating fractions of a foot; setting the compasses to a given radius; detecting large errors; detecting small errors; avoiding the use of the wrong edge of the scale; laying off plus stations, p. 448. 10. What is the proper use of the dividers; the improper use; use of proportional dividers? p. 441. 11. Give suggestions for the use of the compass, p. 442. 12. Give sug- gestions for the use of the curve-ruler, p. 442. 13. Give sugges- tions for fastening the paper to the board, p. 445. 14. Give precautions to insure neatness, p. 445. 15. Give suggestions for erasing as regards erasing pencil lines; keeping the eraser clean; erasing ink lines; restoring the surface of the paper; use of an. erasing shield ; cleaning tracings, p. 447. Exercise P-2. Methods of Plotting Angles. Reference: Chapter XXXVIII, p. 455. Directions: The angles in each problem of this exercise should be plotted with as great accuracy as the method used in that problem will permit. Use hair-lines and prick the points. PROBLEMS. 371 . Draw a horizontal line 396 ft. long to a scale of I" = 50' - 0". By means of the protractor lay off at one end of this line an angle of 38 20' and at the other end an angle of 51 40'. The two lines thus obtained and the original line should form a right- angled triangle ; test the right angle by means of the two instru- mental triangles. (In using the protractor, minutes of arc are estimated.) 372. Same as Problem 371 except use the tangent method for plotting angles, p. 458. 373. Same as Problem 371 except use the cosine and sine method, p. 459. 374. Same as Problem 371 except use the chord method, p. 459. 375. Compare the results by drawing an arc of 6" radius for each of the angles in the above problems, and by means of the dividers see if the corresponding chords are of the same length in all four figures, 154 EXERCISES IN PLOTTING. Questions: 1. Name five methods of plotting angles, p. 455. 2. Give suggestions for centering the protractor, p. 455. 3. Give three methods of laying off an angle greater than 180, p. 456. 4. How closely can angles be plotted with an ordinary pro- tractor? p. 456. 5. What is a vernier protractor? p. 456. 6. What is the advantage of the form of protractor shown on p. 456? 7. Give suggestions for making such a protractor, p. 457. 8. Explain the tangent method of plotting angles, p. 458. 9. Why is it advantageous to use a 10" base? 10. When an angle exceeds 45 how is it plotted? 11. Explain methods of plotting angles a little greater or less than 90; 180; 270. 12. Give practical suggestions as regards erecting perpendicu- lars; laying off hundredths of an inch; using a base larger or smaller than 10 inches ; checking angles roughly ; detecting small mistakes ; laying off angles in the field, p. 459. 13. Explain the cosine and sine method of plotting angles and give the check, p. 459. 14. Explain the chord method of plotting angles, p. 459. 15. Give practical suggestions for the chord method as regards use of beam compasses and scale; a home-made device ; use of 5" radius ; use of 8", 9", or 12" radius. 16. Explain why, if the radius is 10" and fiftieths on the scale are used, it is not necessary to multiply the sine of half the angle by 2. See Fig. 496 (c), p. 460. 17. What is the largest angle that can be laid off conveniently by the chord method without erecting a perpendicular? 18. Compare the different methods of plotting angles, p. 461. 19. Explain the three-point problem in plotting by the graphic method; by the algebraic method. Methods of Plotting Traverses. (Introductory.) Before beginning to plot surveys from notes taken in the field, it is well for the student to have a preliminary drill in the different methods of plotting traverses. For this purpose it is advisable to select a polygon of four or five sides from among the problems given on p. 122,* and to plot this polygon by each of the general methods explained in Chapter XXXIX, p. 463. He will thus obtain a knowledge of the advantages and disadvantages of the different methods when applied to the same polygon. As pointed out on p. 463, for each of the four methods of plotting EXERCISES IN PLOTTING. 155 traverse lines there are four methods of plotting angles, but it is not worth while to plot the polygon sixteen times for the sake of illustrating each of these methods since many of them are so nearly alike. The methods chosen for the seven succeeding exercises have been selected with a view of bringing out the most important points connected with plotting traverses. In each exercise the choice of scale is left to the student or to the instruc- tor. If desired, different scales may be employed for the different problems, care being taken that the scale is not so large that the polygon will run off the paper, or so small that the accuracy of the work is impaired. It is not advisable to ink the drawings, as time can be spent to better advantage in other work. Exercise P-3. Plotting Traverses with a Protractor. Reference: Page 464, 502. Directions: l. r Assume one side of the polygon in any posi- tion on the paper that will permit the rest of the polygon to be plotted without running off the paper. 2. Plot the polygon, using the protractor for laying off angles, and the scale for laying off lengths. 3. Not only should the line from the last station pass through the first station, but the distance from the former to the latter, as scaled on the drawing, should be equal to that given in the notes. The student should make this test when the polygon is completed. Problem: Plot the polygon of Problem , p. (To be assigned by the instructor from the problems on p. 122.*) Questions: 1. Why is it important to plot traverse lines with great accuracy? p. 463. 2. How great an error in feet may be caused by the width of a line? 3. When may a pro- tractor be used consistently for plotting traverses? p. 464. 4. Give precautions for centering protractor, p. 464. 156 EXERCISES IN PLOTTING. Exercise P-4. Tangent Method of Plotting Traverses. Reference: Page 464, 503 (a). Directions: 1. Make a rough freehand sketch in the note- book and enter on this sketch the lengths and the angles to be plotted. (Angles are not necessarily the same as those given in the notes.) Check this data before proceeding. 2. Make out a table corresponding to that at the top of p. 465. 3. Plot each of the transit lines in succession by the tangent method. Check each angle roughly with the protractor as soon as plot- ted. 4. Check the length as well as the direction of the last line. Problem: Same as the problem in Exercise P-3 except that the tangent method is used for plotting angles. Questions: 1. Explain the tangent method of plotting trav- erses, p. 465. 2. If the closing line passes through the first station and is of the right length on the drawing, what addi- tional check may be applied? p. 465. 3. When a traverse has a large number of sides, what check should be applied at every fourth or fifth station? p. 466, 503 (c). Exercise P-5. Chord Method of Plotting Traverses. Reference: Page 465, 503 (b). Directions: Same as for the preceding exercise except that the table made out in the note-book corresponds to that at the bottom of p. 465. It may also be advisable to plot deflection angles at stations where the interior angles are very obtuse. Problem: Same as the problem in Exercise P-3 except that that the chord method is used for plotting angles. Questions: 1. Explain the chord method for plotting trav- erses, p. 465. 2. Give the checks to be employed in this method, p. 466. 3. Give the method of plotting by deflection angles, p. 466. EXERCISES IN PLOTTING. 157 Exercise P-6. Plotting Traverses by Bearings. (Tangent Method.) Reference: Page 467, 505. Directions: I. If the bearings of the lines are not given, cal- culate them from the angles at the stations. 2. Make a sketch in the note-book showing the bearing of each line and its length. 3. Prepare a table in the note-book corresponding to that at the top of p. 469. 4. Find the direction of each line from its bearing by the method illustrated in Fig. 505 (a), p. 468, or Fig. 505 (b), p. 469, preferably the latter method. Whichever method is used do not fail to test the square and to mark each line as soon as obtained with its letters or its numbers as shown in the figures referred to above. 5. Plot the polygon in any position which will bring it wholly within the limits of the paper, by transferring the direction of each line of the polygon to its proper place in the plot. Check each angle of the polygon roughly as soon as it is obtained, using the protractor for this purpose. 6. Apply the usual tests to the closing line. Problem: Same as the problem in Exercise P-3 except that the sides of the polygon are plotted from their bearings by the tangent method. Questions: 1. Explain the general method of plotting trav- erses by bearings, p. 467. 2. What bearings are plotted? See Note, p. 467. 3. Give two general methods of procedure, p. 467. 4. In either method what is the most convenient meridian to assume? p. 467. 5. When is it convenient to use a T-square with an adjustable head? p. 468. 6. Explain the tangent method of plotting bearings, p. 468. 7. Explain the corresponding modified method, p. 469. 8. Explain the chord method of plotting bearings, p. 469. 9. Explain the corre- sponding modified method, p. 470. 10. Give the checks to be employed, p. 471. 11. Give one of the chief objections to plotting traverses by bearings, p. 471. 12. To what kind of work is this method best suited? 158 EXERCISES IN PLOTTING. Exercise P-7. Plotting Traverses by Bearings. (Chord Method.) Reference: Page 469, 505 (c) and 505 (e). Directions: This exercise might be made exactly like the preceding exercise, using the chord method instead of the tangent method in plotting bearings. This would involve little that is new, hence it is better to employ the method of 505 (e), p. 470, drawing a new meridian at each station. Follow the directions of the preceding exercise except where they conflict with the method used in this exercise. Problem: Same as the problem in Exercise P-3 except that the sides of the polygon are plotted from their bearings, a ref- erence meridian being drawn at each station and the correspond- ing bearing plotted by the chord method. Questions: 1. Explain the method of plotting bearings from a reference meridian at each station, p. 470. 2. Compare this method with the method of the preceding exercise. 3. Outline another method sometimes used for large maps. p. 471. 4. What is the advantage of this method? Exercise P-8. Plotting Traverses by Azimuths. Reference: Page 471, 506. Directions: 1. Change the bearings of the sides of the poly- gon in the preceding exercise to azimuths. 2. Make a free- hand sketch in the note-book, giving the azimuth and length of each line, and the angle which will be plotted at each station. 3. Use either the tangent or chord method for plotting the angles, preparing in advance a table of tangents or of chords to correspond. Apply the usual checks throughout the work. Problem: Same as the preceding exercise except azimuths are plotted instead of bearings. As this in itself involves little that is new, it is desirable to use a method of plotting not employed in either of the two preceding problems. For example, the azimuths may be plotted by the tangent method from meridians drawn through the stations. EXERCISES IN PLOTTING. 159 Exercise P-9. Plotting Traverses by Latitudes and Departures. Reference: Pages 471 to 475, 507. Directions: Follow the method of procedure outlined on p. 472, 507 (b), observing also the practical suggestions at the top of p. 475. Problem: Same as the problem in Exercise P-3 except that the polygon is plotted by the method of latitudes and departures. Questions: 1. What bearings are used in computing latitudes and departures, calculated or magnetic? p. 471. 2. Without going into detail, what is the general method of plotting by latitudes and departures? p. 471. 3. Why is the most west- erly station the most convenient reference point? p. 471. 4. Could any other reference point be used? 5. If azimuths are given in field notes instead of bearings, is it necessary to reduce them to bearings before calculating latitudes and depar- tures? p. 472. 6. Explain three different methods of deter- mining which is the most westerly station, p. 472. 7. Outline in detail the general method of plotting by latitudes and depar- tures, p. 472. 8. What is the object in using a reference- rectangle and measuring from the two nearest sides to locate a point instead of locating it by coordinates by means of a T-square and triangles? p. 473, " Caution." 9. Give practical suggestions for plotting by latitudes and departures as regards (1) plotting large maps; (2) assuming the reference-rectangle so that no side is horizontal or vertical; (3) testing the rec- tangle; (4) plotting each station by its latitude and departure from the preceding station, and the advantage of this method; (5) advantage of drawing the traverse lines so that they stop just short of each station, p. 475. 10. What checks may be employed in this method of plotting? p. 475. Methods of Plotting Traverses Compared: 1. Compare the tangent method and the chord method, p. 475. 2. What is the chief advantage in plotting direct angles or deflection angles instead of bearings or azimuths? p. 475. 3. What are some of the disadvantages of the direct angle method? p. 475. 4. When may the method of plotting by bearings be used? p. 476. 5. What are the advantages of plotting by latitudes and departures? p. 476. 160 EXERCISES IN PLOTTING. Exercise P-10. Plotting the Survey on Page 184. Note: It will be found that the plotting of this survey is exceedingly good practice, as it involves nearly all the common methods of locating points which are used in the field. If desired, recitations Q-l to Q-4 may be held in connection with this exercise. Reference: Comments on the survey are given on p. 184; com- ments on the notes on p. 185. The notes themselves, however, are given on the inset sheet opposite to p. 190, Illustration V. Directions: 1. Plot the transit lines first, using one of the trigonometric methods. 2. Plot the details, using the scale and protractor. Problem: Plot the survey shown on the inset sheet opposite p. 190, Illustration 5. Choose a scale as large as practicable and still have the map fall within the limits of the paper. GROUP Q. QUESTIONS PERTAINING TO MAPPING. Plotting Maps from Field Notes. (Introductory). In most courses in surveying students are required to plot maps from their own notes taken in the field. The preliminary drill in plotting having been completed and this work of plotting field notes having been begun, it is well to hold recitations from time to time on tl^at part of the mapping in which the student is engaged. Thus, for example, before beginning to work up the notes preparatory to plotting, it is well to cover the ground out- lined by the questions in Exercise Q-l. Before beginning to plot, the questions in Exercises Q-2 and Q-3 may be discussed. Before beginning to plot details, Exercise Q-4 may be taken up, and when the map is finally completed in pencil and ready to ink, questions on finishing the map may be discussed. Exercise Q-l. Working Up Field Notes Preparatory to Plotting. Questions: 1. What is one of the first things to do in working up notes? p. 477. 2. In addition to saving time what other object is there in getting together all data and in making all the preliminary calculations before beginning to plot? p. 473. 3. When is it most necessary to correct field measurements? p. 478. 4. What are some of the corrections applied to linear measure- ments? to angular measurements? p. 478. 5. For what purpose are measurements adjusted? p. 478. 6. How may linear meas- urements be adjusted? angular measurements? p. 478. 7. It is often best not to follow any rule in adjusting angles. How is this illustrated by the survey on p. 390? p. 479. 8. In work of great precision how are observed values adjusted? p. 479. 9. What can you say regarding the calculation of bearings: for purposes of plot- ting; true bearings; use of adjusted values in angles; bearings in 161 162 QUESTIONS PERTAINING TO MAPPING. compass surveys? p. 479. 10. What methods are used for supplying missing data and what is the disadvantage in the use of these methods? p. 480. 11. Explain the method of reducing stadia notes by the use of reduction tables; by means of a dia- gram.* p. 481. 12. Why should the abridged method of multiplication be used in connection with reduction tables? 13. Summarize the first five or six steps in preparing data for plotting, p. 483. 14. Give the additional steps, p. 484, for (a) protractor method ; (b) tangent method ; (c) chord method ; (d) latitude and departure method; (e) azimuth method. Exercise Q-2. Plotting the Map. (General Methods and Suggestions.) Questions: 1. What is the best kind of paper for map work? p. 485. 2. What is the best paper for maps that are to be tinted ; for maps to be reproduced? p. 485. 3. What are the advantages and disadvantages of the different kinds of blue print paper? p. 485. 4. What are the most essential requirements for draw- ing instruments? p. 486. 5. Give some of the precautions to be taken in accurate plotting, p. 486. 6. What effect has the shrinkage of the paper and how may shrinkage be partly avoided? p. 486. 7. Why should maps be kept flat? p. 487. 8. Give the general methods employed in mapping, p. 487. 9. Give the general method of procedure, p. 487. 10. What are the most common scales for ordinary maps? p. 488. 11. Why are not the scales of 70 and 90 ft. to the inch commonly used? p. 488. 12. What is the so-called natural scale and what are some of the scales used for government work? p. 488. 13. Give some of the primary considerations in choosing a scale and illustrate by examples, p. 489. 14. Give some of the common scales used: for preliminary surveys of railroads; maps of mines and mining claims; maps for architects, p. 489. 15. What are the advan- tages of the scales 1 in. = 20 ft., 1 in. = 40 ft., and 1 in. = 80 ft.? p. 489. 16. What are some of the considerations which some- times determine the size of the sheet? p. 489. 17. How does the arrangement of the map affect the size of the sheet? p. 489. * If desired, the student may be required to construct a diagram. QUESTIONS PERTAINING TO MAPPING. 163 18. How may the shape and extent of a survey be determined roughly? p. 489. 19. Give two general rules for choosing the scale for a map. p. 490. 20. Why is it advantageous to adopt a small scale for topographic maps? p. 490. 21. What is the object in sometimes drawing maps of the same territory to dif- ferent scales? p. 490. 22. What arrangement of the map is desirable as regards points of the compass? p. 490. 23. Why is this arrangement seldom economical? 24. What effect has the natural approach to a piece of property on the arrangement of the map? p. 490. 25. What is usually the first step in begin- ning a map? p. 490. 26. Where do you begin to plot a map and why work half way around a polygon in either direction? p. 491. 27. Give suggestions for assuming the first line. p. 491. 28. When it is desired to economize space and the map is plotted by latitudes and departures how would you proceed in assuming a reference-meridian, or reference-rectangle? p. 491. 29. When is a preliminary plot unnecessary? p. 491. Exercise Q-3. Plotting Traverses. (Review.) Questions: 1. To what do the different methods of plotting traverses correspond? p. 492. 2. In plotting traverses when is the protractor used? 3. What are the other methods used for plotting traverses? 4. Why should traverse lines be plotted accurately? 5. Summarize five methods of plotting transit lines, giving the checks used in each: (1) each line plotted from the preceding line, p. 492; (2) each line plotted by deflection angle, p. 493; (3) each line plotted from its bearing, p. 493; (4) each line plotted from its azimuth, p. 493; (5) transit lines plotted from latitudes and departures, p. 493. 6. Give the methods of checking linear measurements: (1) any straight line; (2) comparing the lengths of lines with corresponding distances in field notes; (3) checking lines of a traverse; (4) checking two points by indirect measurements. 7. Give the checks for angles : (1) when laid off with a protractor; (2) test for two or more adjacent angles; (3) check for angles plotted by trigonometric method; (4) checking angles between traverse lines. 8. Answer 164 QUESTIONS PEKTAINING TO MAPPING. question on p. 495. 9. Give combination checks: (1) for a reference-rectangle; (2) for a closed traverse; (3) for extra distances taken in the field, p. 495. 10. Give methods of running down errors, p. 495: (1) when the error of closure is parallel or perpendicular to some line in the traverse; (2) when a traverse has been plotted by direct angles; (3) when the direction of the closing line is found to be incorrect; (4) when lines have been plotted by bearings or azimuths; (5) when a point is wrong and the mistake not easily found; (6) effect of long lines and lines north and south or east and west. 1 1 . Give the different methods of plotting a triangulation net. p. 491. Exercise Q-4. Plotting Details. Questions: 1. Give the methods of plotting points located by angles and distances, p. 496. 2. What precautions should be taken to avoid confusion? 3. Give the methods of plotting points located by angles. 4. Give the method of plotting points by linear methods only. 5. How may the straightness of a line located by offsets be tested? 6. Give general suggestions for keeping the drawing free from unnecessary lines, p. 496; for testing check distances; for plotting important points at a station; for plotting doubtful points, p. 497. 7. Explain the graphic method of interpolating contours, p. 497. 8. Why, as a rule, will a point interpolated from two given points be dif- ferent if interpolated between two other points. Remark, p. 497. 9. Explain the tracing cloth method of interpolating contours, p. 497. 10. Give suggestions for the use of this method, p. 498. 11. Describe a home-made device for inter- polating contours, p. 498. 12. What determines the accuracy required in plotting? and give illustrations, p. 499. 13. What contributes to speed in plotting? p. 499. 14. Give the different methods of copying and transferring maps. p. 500. 15. Which methods are used for copying to a different scale from that of the original map. p. 501. QUESTIONS PERTAINING TO MAPPING. 165 Exercise Q-5. Finishing the Map. Questions: 1. In finishing a map what are some of the specifi- cations for workmanship? p. 502. 2. What in general should appear on a map? p. 502. 3. When is the scale of the map shown by a drawing instead of being given in figures? p. 502. 4. Is a magnetic meridian always shown? p. 502. 5. What are the most important requirements for an ordinary property map? p. 502. 6. What are some of the additional requirements prescribed by law in some states? Remark, p. 503. 7. Give some of the requirements for topographic maps. p. 503. 8. Give some of the things which should appear on a map made for an architect, p. 211. 9. Give the general method of procedure in finishing a map. p. 503. 10. Give suggestions for arrangement of (a) border lines, p. 504; (b) lettering, p. 505; (c) titles, p. 505; (d) names of streets, streams, property owners, etc., p. 505; (e) give additional suggestions for arrangement, p. 505. Questions on Inking: 1. Give suggestions for the use of a ruling pen as regards (1) producing clear-cut lines, p. 440; (2) method of holding the pen, p. 440; (3) adjusting the nibs of the pen, p. 441; (4) cleaning the nibs; (5) sharpening the pen; (6) method of inking straight lines (Suggestions (5), (6), and (7), p. 446); (7) order of inking straight lines and circles, p. 447; (8) inking lines that meet in a point, p. 447; (9) inking wide lines, p. 447; (10) mixing india ink, p. 447. 2. Give additional suggestions for inking as regards (1) order of inking lines, p. 506; (2) inking edges of streams and similar indefinite lines, p. 506; (3) inking traverse lines, p. 506; (4) inking broken lines. 3. Give the method of outline shading, p. 506. 4. Give suggestions for the use of colored inks. p. 506. Questions on Tracing: 1. Give general suggestions for tracing as regards (1) which side of the cloth to use, p. 451; (2) prep- aration of the tracing cloth, p. 451 ; (3) effect of too fine lines and figures on the blue print, p. 451; (4) precautions to be taken in erasing on tracing cloth, p. 452; (5) inserting new pieces of cloth, p. 452; (6) effect of moisture on tracing cloth, p. 452; (7) use of tracing paper instead of tracing cloth, p. 452; (8) best ink to use, p. 452; (9) method of cleaning a tracing, 166 QUESTIONS PERTAINING TO MAPPING. p. 452. 2. What are the printing qualities of different colored inks? p. 507. 3. How may tracings be colored with pencils? p. 507. 4. How should streams and shore lines be colored? p. 507. 5. Give method of lettering tracings by shifting let- ters underneath, p. 517. Questions on Lettering: 1. Upon what does good lettering mostly depend? p. 507. 2. What defects in lettering are the least excusable? p. 508. 3. Give some suggestions as regards style of letters, p. 508. 4. Give general method of lettering, p. 509. 5. Give suggestions as regards the size of letters, p. 509. 6. Give suggestions as regards the proportions of letters, p. 510. 7. Give suggestions for penciling the letters as regards (1) use of guide lines, p. 510; (2) estimating the unit for proportioning; (3) blocking out letters; (4) two things to observe in drawing letters, p. 511; (5) devices for lettering on tracing cloth, p. 511. 8. Give the most important points to be remembered in the construction of the various letters.* p. 511. 9. Give suggestions for inking letters as regards: pens used, p. 515; inking outlines of letters; filling in; touching up imperfect corners; testing the pen; precautions for keeping the ink. 10. What are the most common defects in lettering? p. 516. 11. Give suggestions for spacing letters on paper; on tracings, p. 516. 12. Give general suggestions for lettering maps as regards (1) general arrangement, p. 517; (2) letter- ing curved streets or winding rivers; (3) arrangement of two lines that are not centered; (4) positions of names and descrip- tions; (5) printing values of small angles and of a decimal of a foot; (6) distinguishing between angular measurements and linear measurements, p. 518; (7) indicating the forward bear- ing for a line. 13. Give six different methods of making letter- ing more prominent, p. 518. Finishing Topographic Maps: 1. Give suggestions for draw- ing contour-lines as regards (1) color of ink, p. 518; (2) width of lines and kind of pen used; (3) which contours are accentu- ated; (4) marking elevations of accentuated contours; (5) breaks in contour-lines, p. 519; (6) representing intermediate contours. 2. What are some of the common mistakes in drawing contours? p. 519, p. 338. 3. Give suggestions for * This study of the Gothic capitals is best made in connection with a regular course in lettering, or at least in separate exercises devoted to practice in proportioning letters. QUESTIONS PERTAINING TO MAPPING. 167 conventional signs.* p. 519. 4. Give method of drawing small water courses, p. 519. 5. Give suggestions for water-lining, p. 522. 6. Give suggestions for section-lining, p. 522. Questions on Tinting. (See p. 449.) 1. Give the method of mixing the tint. 2. What is the effect of pencil lines? p. 450. 3. Should the drawing be tinted first or inked first? 4. Give suggestions for applying the tint. p. 450. 5. What is the secret of success in tinting? p. 450. 6. How may the tint be removed from outside the boundary line? 7. How may the surplus tint be removed? 8. Why is it well to go over the surface first with clean water? p. 451. 9. What is the best method of securing a dark tint? 10. To what is tinting on ordinary maps usually confined? p. 523. 11. Give some of the conventional tints. Questions on Border-Lines and Titles: 1. (p. 523.) Give general suggestions for drawing border-lines as regards: simplicity; size of rectangle; width of heaviest border-line; method of drawing a heavy border-line. 2. Where is the best place for a title? 3. On what does the size of the largest letters depend? 4. What is the first consideration in designing a title? 5. Give suggestions for: arranging subject-matter; making some lines more prominent than others; centering lines; spacing lines; making lines of unequal length; styles of lettering. 6. Should lower-case letters be used? p. 524. 7. Is it well to make part of the letters inclined and part upright? 8. For what is single- stroke lettering used? 9. When should the title be drawn freehand and when with instruments? 10. Should words be abbreviated in a title? 11. How may a title be centered on tracing cloth? 12. What are some of the mechanical devices for printing titles? 13. What should appear in a title? p. 525. 14. Give the general method of procedure in constructing a title, p. 525. Questions on Meridian Needles, Scales, etc.: 1. How is the true north distinguished from the magnetic north? p. 526. 2. How is the direction of the arrows determined? 3. Why should the magnetic declination be given in figures, if given at all? 4. How is the scale usually given? 5. When is it most necessary to represent it by a drawing, and what is the chief advantage of this? 6. What is the purpose of keys or legends? * A plate of conventional signs may well be drawn either in a separate exercise or in connection with a course in lettering. 168 QUESTIONS PERTAINING TO MAPPING. 7. When should explanatory notes be used and where should they be placed? 8. How may the paper be cleaned when maps are finished; how may tracings be cleaned? 9. What are some of the things that should be stated in a surveyor's certificate? p. 527. 10. By looking up some of the references on p. 528, give the most essential elements of a good filing system. Exercise Q-6.* Profiles. Questions: 1. In plotting profiles, is it customary to use the same scale for laying off elevations that is used for horizontal distances? p. 529. 2. When does a profile represent a true vertical section and when does it not? p. 530. 3. Why are profiles apt to be misleading? Remark, p. 530. 4. What are the three standard styles of profile paper? p. 530. 5. Give suggestions for the choice of scale, p. 531. 6. What are some of the most convenient combinations of papers and scales? p. 531. 7. Give the method of working up notes for profiles, including the checks, p. 531. 8. Give suggestions for method of procedure in plotting profiles as regards: assuming the posi- tion and elevation of base-line, p. 532 ; where to begin the pro- file; how to make 100-ft. stations fall on accentuated vertical lines; how to number accentuated lines, vertical and horizontal; how to plot points; how to check points. 9. Give additional suggestions for: two men working to advantage, p. 533; mark- ing points; continuing a profile which runs off the bottom or top of the paper; avoiding the common mistake of plotting turning-points and benches; plotting accurate profiles; making several copies of profiles. 10. Give examples of the method of plotting the profiles of several related lines, p. 533. 11. Give the method of laying out grades and vertical curves on profile. p. 533. 12. Give suggestions for finishing profiles as regards: representing existing surfaces and proposed changes; ruling the profile or drawing it freehand ; smoothing out sharp angles ; inking the base-line ; methods of marking elevations ; the points at which elevations should be marked; marking cut or fill; marking rate of grade; tinting or coloring profiles; method of showing different materials. 13. Give suggestions for lettering profiles, p. 535. * Recitation to be held in connection with an exercise in plotting profiles. SHORT-TITLE CATALOGUE PUBLICATIONS OF JOHN WILEY & SONS NEW YORK LONDON: CHAPMAN & HALL, LIMITED ARRANGED UNDER SUBJECTS Descriptive circulars sent on application. Books marked with an asterisk (*) are sold at net prices only. All books are bound in cloth unless otherwise stated. AGRICULTURE HORTICULTURE FORESTRY. Armsby's Principles of Animal Nutrition 8vo, $4 00 Budd and Hansen's American Horticultural Manuai: Part I. 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SUGGESTIONS C I ) Reserve the first few pages for an index of exercises. (2) At the top of each page print a descriptive title. Number the exercises to correspond to the numbering in the book of exercise ample, L-6 or T-4. Date each exercise, and give the names, of other members of the party. Do not seribbltt in the note-book. Pay particular attention to neatne ^snt. . For additional suggestions qn keeping field notes read Chapter III on /'> a the author's text-book PLANE SURVEYING. { ! M ' 4 4-4-4. i i M 4-444 m- t-H hill -H-4 m ! I I I j II I ! 4-4-U- THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 5O CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. QC I Oct'63JM LD 21-100m-7,'40 (6936s) YB 1102;